IVIL ENGINEERS' 
POCKET-BOOK 



A REFERENCE-BOOK FOR ENGINEERS 

CONTRACTORS AND STUDENTS 

CONTAINING RULES, DATA 

METHODS, FORMULAS 

AND TABLES 



BY 



ALBERT I.^FRYE 



SECOND EDITION, COJRBECTED 



NEW YORK: 
D. VAN NOSTRAND COMPANY 

LONDON : 

CONSTABLE AND COMPANY, Ltd. 
1918 



.^^ 



^^^'^ 



■.^' 



Copyright, 1913, 1916, 1918, 

BY 

ALBERT I. FRYE 

All rights reserved 



JAN -2 \m 



r 



Western Newspaper Union, Chicago, 
Electrotypers. 

P. H. GiLsoN Company, Boston, 
Printers. 

'"'clash 211 



'^x 






PREFACE. 

The preparation of notes for this volume was begun several 
years ago, while chief engineer of the Hoffman & Bates Bridge 
and Construction Company, and has been continued systematic- ., 
ally up to the present time. / ^ .■ 

Three separate systems of note-keeping have been employed, 
namely: (1) Two blank-books, size eight by twelve and one-half 
inches, with quadrangular ruling and marginal index, were 
started, one for mathematical and structural data and the other 
for general construction notes, each book containing five hun- 
dred pages in flexible leather binding. (2) Note books of the 
same general style, but of pocket size, were classified — ^for 
bridges, buildings, water-works, sewers, surveys, etc. — and kept 
at hand for general reference and memoranda. (3) The loose- 
leaf method was inaugurated and found to be specially useful in 
consulting practice. The last named is explained in an article 
prepared for the * 'Engineering Record'* and published in the 
issue of January 21, 1911. 

In 1904, the present system was outlined — arranging the 
matter logically in numbered Sections for convenient reference. 
From that time up to the present, the work has gradually been 
crystallized and brought up to date. 

Special attention is invited to the vast number of tables, their 
completeness and arrangement. Although most of them have 
never before appeared in print — at least in their present form — 
yet nearly all of them have been subjected to the test of more 
or less constant use for a number of years, in connection with 
practical work. All of them have been thoroughly checked — 
the proofs of all were checked twice before electrotyping, and the 
proofs of the most important ones were checked again after 
electrotyping. 

The logarithmic tables comprise both the common and 
hyperbolic systems, side by side, the latter being useful in cer- 
tain bridge calculations and in steam engineering. Both the 
logarithmic and trigonometric tables are carried out to five 
decimal places, a sufficient refinement for most engineering opera- 
Ill 



IV PREFACE. 

tions, being much more exact than actual measurements in shop 
or field. 

The tables of cubes and squares, in Section 33, will be found 
useful to structural detailers. They were calculated by the 
incremental additive method, which is self -checking. 

The text is so arranged that all general data may be found 
readily from the Table of Contents, which should be consulted 
as frequently as the Index, the latter being necessary for more 
specific reference. The number and title of each Section appear 
as even-page captions throughout the work. The tables and 
illustrations are numbered from one upward for each Section. 
It has been the aim to begin each subject and paragraph with 
the leading or key word, as a supplementary page index. 

At the end of most of the Sections will be found references 
to valuable data in leading technical publications, which should 
be in the library of every engineer. The reader is advised to 
supplement this data with his own lists, perhaps in a separate 
note book, under the respective Section numbers. 

Acknowledgments are due to Mr.' Porter D. Ford, for his 
kindness m Volunteering to read over portions of the manuscript 
while in preparation, and making many valuable suggestions; 
also to Professor C. H. Peabody, for permission to use the steam 
tables and other material in Section 69; and to many others 
whose names appear in the body of the work, where specific 
credit is given. 

The Publishers, Electro typers and Printers are to be con- 
gratulated oh the neat appearance of the book, and the author 
desires to thank Mr. Jacob Wernli for his excellent work in 
preparing the illustrations. 

ALBERT I. FRYE. 

New York, December, 1912, 



REMARKS ON SECOND EDITION 

Several new tables have been added to this edition 
and the first twenty sections revised. All known 
errors have been corrected. A. I. F. 

New York, September, 1915. 



CONTENTS. 

(Page numbers are given. For Alphabetical Index, see page 1539.) 

Introduction XXIX 

Signs and Abbreviations XXXVII 

SEC. I.— ELEMENTARY ARITHMETIC. 

Ntimbers — Roman System, Arabic System 1 

Primes, Multiples and Factors, 2; Table 3 

Greatest Common Factor; Least Common Multiple 6 

Fractions, 7; Reduced to Decimals, Tables 9 

Decimals, 10; Repeating Decimals H 

Short Methods of Multiplication — Fractions and Decimals 11 

Cancellation 13 

SEC. 2.— POWERS, ROOTS AND RECIPROCALS. 
A. — Engineers' Tables. 

Square Root, 14; Square Roots (and Squares) of Numbers, Tables 16 

Cube Root, 20; Cube Roots (and Cubes) of Numbers, Tables 21 

Square Roots of Fifth Powers of Numbers, Table 25 

Fifth Powers (and Fifth Roots) of Numbers, Table 26 

Pv.eciprocals of Ntmibers, 30; Table 28 

B. — Arithmetical or Common Tables. 

Squares, Cubes, Square Roots, Cube Roots of Numbers 1-1600, Table. . 31 

Square Roots and Cube Roots of Numbers 1600-3200, Table 44 

^Reciprocals of Niunbers 1-1000, Table 51 

SEC. 3.— PRACTICAL ARITHMETIC. 

Proportion, 55; Permutation, Combination, 56; Allegation, Progression 57 

Percentage, Interest, Discount, 58; Simple Interest Table 60 

Table for Finding Number of Days Between Any Two Dates 61 

Equ on of Payments, 61; Compound Interest Table 62 

Partial Payments, Annuities, Sinking Fund, 63; Tables 64, 65 

SEC. 4.— MEASURES, WEIGHTS AND MONEY. 
Fundamental Units. 

Meter — ^Length, Area, Volume 66 

Liter — Capacity (Liquid and Dry) 66 

Gram — Mass (Weight) 67 

General Tables. 

Approximate Equivalents — Metric and English. 68 

English Measures, Metric Equivalents — Long, Surveyors', Mariners'. . . 68 

Lengths — ^Inches and Millimeters 69 ,70 

Lengths — Metric Table; Metric and English, Equivalents (1-9) 70 

Lengths^ — Feet and Inches to Meters, 71; Meters to Feet 75 

Areas — Metric Table; Metric and English, Equivalents (1-9) 79, 80 

Areas — English Land Measure, Texas Land Measure 81 

Measures and Weights of the Philippines, English Equivalents 81 

Volumes — Metric Table; Metric and English, Equivalents (1-9) 81, 82 

Volumes — English Cubic Measure, Metric Equivalents 82 

Capacities (Liquid) — Metric Table; Metric and English, Equiv. (1-9).. 82, 83 

Capacities (Liquid) — Liquid and Apothecaries' Measures 83 

Capacities (Dry) — Metric Table; Metric and English, Equivalents (1-9) 84 

Capacities (Dry) — English Dry Measure, Metric Equivalents 84 

Weights — Metric Table; Metric and English, Equivalents (1-9) 85 

Weights — Apothecaries, Troy, Avoirdupois — Metric Equivalents 86 

Weights — Various Tons and Pounds, Equivalents (1-9) 87 

Simple and Compound Units in Common Use, Equivalents. 88 

Electrical, Mechanical and Heat Units — Equivalents 91 

Foreign Weights and Measures, American Equivalents 92 

Nvunbers — ^Abstract, Duodecimo, Paper — ^Tables .' 95 

V 



VI CONTENTS, 

Money — Domestic and Foreign 95 

Money — Value of Foreign Coins, American Equivalents .*..*. *. *. '. *. '. * 95 

Prices — Comparison of German for Metric; British; and American . . . . 98 

Time Measure ; Circular Measure [[[ 99 

SEC. 5.— ALGEBRA. 

Exponents 100 

Binomial Formula, 101; Completing the Square *..'.'.*.!'.*. 102 

Simiiltaneous Equations !.'!!.* 103 

SEC. 6.— LOGARITHMS OF NUMBERS, 

Common System — Described, 104; Operations, 105; Table 108 

Naperian System — Described, 104; Operations, 106; Table 108 

Slide Rules [[[[ 126 

SEC. 7— PLANE GEOMETRY. 

Angles and Lines; Triangles; Quadrilaterals 128 

Polygons (General) ; Regular Polygons; Circle ] * 129 

Problems in Construction of Figures [ ] 130 

SEC. 8— SOLID GEOMETRY. 

Planes, Angles and Lines; Polyhedrons 1 32 

Prisms and Pyramids .' . * * 133 

Cylinders, Cones and Spheres * * 134 

SEC. 9.— PLANE TRIGONOMETRY. 

Trigonometric Functions — Formulas 136 

Values of Trigonometric Functions in the Four Quadrants ., , I37 

Natural Functions of Angles in the Four Quadrants 138 

Functions of Complement, Supplement, etc., of any angle x I39 

Functions of the Sum and Difference of Two Angles, x and y 1 39 

Functions of one-half x, 2x, 3x and 4x ][ 140 

Inverse Trigonometric Functions \[ 140 

Solution of Right Angle Triangles \\ 141 

Solution of any Triangle. — Circular Measure 142 

Cubic Equations I43 

Table of Natural Sines, Tangents, Cotangents, Cosines, etc 144 

Table of Natural Secants, Cosecants, Exsecants and Coexsecants 167 

Table of Logarithmic Sines, Tangents, Cotangents, Cosines, etc 176 

Table for Finding the Logarithmic Sines and Tangents of Small Angles 198 

SEC. 10.— SPHERICAL TRIGONOMETRY. 

Functions; Right Spherical Triangles, Formulas I99 

Oblique Spherical Triangles — Formulas and Rules 200 

Distance Between Two Points on the Earth's Surface 201 

The Celestial Sphere, 201; Astronomical Time, Tables 202 

SEC. II.— MENSURATION. 
A. — Plane Surfaces, Lines, etc. 

Triangle and Quadrilateral 203 

Regular Polygon, and Table; Circle 204 

Tabular Values of Combinations of ;r, with Logs 205 

Arc and Chord of Circle, 207; Lengths of Circular Arcs 208 

Lengths of Circular Arcs for Radius 1, Table 209 

Lengths of Circular Arcs for Chord 1, Table 210 

Flat Circular Arc, Formulas and Tables 211 

Circular Segments — Formulas, 214, 215; Tables 216-218 

Circular Ring and Circular Zone, Properties of 219 

Circular Lune; Circular Sector; Circle and Square, Relations of 220 

Table of Relations of Circle and Square 221 

Decimals of a Foot for Each ^\ of an Inch, Table 223 

Table of Circles — Circumferences for Given Diameters, Decimals 225 

Table of Circles — Circumferences for Diameters in Feet or Inches 226 

Table of Circles — Areas for Given Diameters in Inches and Fractions. . 230 

Table of Circles — Areas for Given Diameters, Decimals 232 

Table of Circles — Areas (Sq. Ft.) for Given Diameters (Ft., Ins.) 234 

Cycloid 236 

Parabola, Parabolic Segment, Parab'c Half-Segment, Parab'c Spandrel, 237 



CONTENTS. VII 

Lengths of Parabolic Arcs for Chord (Base) 1, Table 238 

Ellipse, 238; Formulas for Circumference of Ellipse 239 

Lengths of Semi-Elliptic Arcs, Table 241 

Segment and Chord of Ellipse 242 

B.— Solids. 

Pappus's Theorem. — Prismoidal Formula 243 

The Five Regular Polyhedrons, Table 243 

Prisms and Cylinders; Frustums of Prisms and Cylinders 244 

Circular Cylindric Wedges and Half -Wedges 245 

Properties of Hollow Cyl's (Pipes, Tanks, Wells), One Foot Long, Table 246 

Pyramids and Cones; Frustums of Pyramids and Cones 248 

Conic Wedge and Frustum; Wedge; Sphere 249 

Areas of the Surfaces of Spheres, Table 251 

Volumes of Spheres, Table 252 

Spherical Segment, 252; Spherical Zone 253 

Hollow Sphere; Circular Segmental Ring 253 

Regular (Circular Ring; Circular Spindle 253 

Parabolic Spindle; Cycloidal Spindle; Paraboloid 254 

ElHpsoid 255 

SEC. 12.— ANALYTIC GEOMETRY. 

Straight Line 256 

Circle ; Parabola 257 

Ellipse 258 

Hyperbola, 259; Equilateral Hyperbola 260 

Cycloid; Spiral of Archimedes; Logarithmic Spiral 260 

Hyperbolic Spiral; Lemniscate of Bemouilli 260 

Helex or Screw; Common Spiral 260 

SEC. 13.— DESCRIPTIVE GEOMETRY. 

Perspective; Cabinet-, Isometric-, Orthographic Projection 261 

Revolved Planes; Projection of the Point 262 

Projection of the Right Line; Projection of Two Lines 262 

Projection of the Plane; Problems of Construction. 263 

SEC. 14.— THE CALCULUS. 
A. — Differential Calculus. 

Differentiation and Differential Coefficient, Defined 266 

Tangent and Normal; Rules for Differentiation '. 267 

Maxima and Minima, with Problems 268 ,269 

Differentiation of Logarithmic and Exponential Functions 270 

Differentiation of Trigonometric Functions 270 

Differentiation of Inverse Trigonometric Functions 271 

Expansion of Functions — By Division, Successive Differentiation 271 

Maclauren's Theorem 271 

Taylor's Theorem 272 

B. — Integral Calculus. 

Integration as a Summation, Defined 272 

Definite Integration a Method of Limits 273 

Formulas for Integration 274 

Areas and Lengths of Plane Curves, Problems 275 

Areas of Curved Surfaces, Problem 276 

Volumes, or Planes of Revolution 277 

SEC. 15.— MECHANICS. 

Fundamental Equations of Motion and Force 278 

Motion Formulas. 

Uniform Motion, No Acceleration 279 

Uniformly Accelerated Motion, No Initial Velocity 279 

Uniformly Accelerated Motion with Positive Initial Velocity 280 

Uniformly Accelerated Motion with Negative Initial Velocity 282 

Table of Falling Bodies [See, also. Table on page 1155] 283 

Summary of Preceding Motion Formulas 284 

The Resultant of Two Constant Velocities 284 

Parabolic Motion — Path of Projectile 285 

Circular Motion— Fly-Wheel 286 



VIII CONTENTS. 

Motion on Inclined Plane 286 

Motion on Cycloidal Curve 286 

Simple and Compoiind Circular Pendulums 287 

Simple Cycloidal Penduliim 287 

Dynamic Formulas. 

Force — Fundamental Relations. — Atwood's Machine 288 

Force — General Relations, Distance Included 289 

Work — Hoisting-Rope Problem 290 

Power — Locomotive Problem 291 

Leverage — Simple Lever 291 

Compound Lever; Inclined Plane; Wedge 292 

Screw; Pulley, Simple and Compound 292 

Toggle. — Impulse and Momentum. — Energy 293 

Composition and Resolution of Forces 294 

Principles of Equilibrium 294 

Polygon of Forces; Moments and Reactions 295 

Center of Gravity and Resultant of a System of Parallel Forces 295 

Resultant of a Distributed Force — Problem 296 

Centriftigal Force — Fly-Wheel Problem 297 

Centrifugal Force — Elevation of Outer Rail on Curve 298 

Forces Acting on Plane Surfaces. 

Bending Moment and Resisting Moment 298 

Resisting Moment and Moment of Inertia 299 

Moment of Inertia and Radius of Gyration 299 

Radius of Gyration and Moment of Inertia 300 

Resistance of Rectangular and Circular Beams Compared 301 

Forces Acting on Solids. 

Moment of Inertia of Solid Body, Defined 302 

Moments of Inertia of Regular Solids, Table 302 

Radius of Gyration of Solids 302 

Center of Gravity of Solids 303 

Center of Oscillation= Center of Percussion 303 

Impact or Collision, 303; Formulas 304 

SEC. 16.— THEORY OF STRESSES IN STRUCTURES. 

Outer and Inner Forces — Loads, Reactions, Stresses 305 

Principles of Static Equilibrium 305 

Methods of Calculation — By Moments, Shears, Graphics 306 

Loads and Reactions Vertical. 

Pratt Truss Calculation — Dead Load Stresses 306 

Reaction at Left Support; Lengths of Members 307 

Trigonometric Ratios for Calculating Chord and Web Stresses 307 

Stresses in Chord Members by Method of Moments — Rules 307 

Stresses in Web Mem^bers by Method of Shears — Rules 308 

Static Equilibrium of Inner and Outer Forces at Joints 309 

Bow's Notation in Graphical Statics 309 

Graphical Method . .' 310 

General Rules for Stress Diagrams 310 

Order of Considering Joints 311 

Graphical Solution of Pratt Truss, with — 312 

Loads at Top Joints; Loads at Top and Bottom Joints 313 

Reactions in Any Direction. 

Roof Truss, Both Ends Fixed, Wind on One Side 314 

Roof Truss, One Roller End, Wind on Either Side 314 

Three-Hinged Arch, Vertical Loads 315 

SEC. 17.— NATURAL HISTORY OF MATERIALS. 
A. — Chemical. 

Composition of Matter. — ^The Old Atomic Theory 316 

Recent Discoveries. — The Corpuscular Theory 316 

The Electronic Theory. — The Chemical Elements 317 

Table of the Chemical Elements 318 

Compounds. — Simple Combinations 321 

Acids, Bases and Salts; Oxides and Hydroxides 321 

Acid Combinations , 321 



CONTENTS, IX 

Periodic Law, 322; Natural System of the Elements 323 

Chemical Substances and their Common Names 324 

B. — Mineralogical. 

Minerals — Hardness and Other Physical Characteristics 324 

Classification of Important Mineral Species 325-328 

Native Elements; Sulphides; Chlorides, Bromides, Iodides 325 

Oxygen Compounds — Oxides, Silicates 326 

Oxygen Compounds — Phosphates, Borates, Sulphates, Carbonates. . <, . . 327 

Hydrocarbons — Petroleum, Naphthalin, Asphaltum, Coal 328 

Blowpipe Characteristics of Minerals 328 

Some of the Important Minerals, Table — 328 

Minerals — Gold, Silver, Potassium, Sodium 328 

Minerals — Lithium, Platinum, Iridium 328 

Minerals — Mercury, Copper, Iron, Cadmium, Zinc 329 

Minerals — Lead, Cobalt, Nickel, Manganese, Calcium ., 329 

Minerals — Magnesium, Tin, Titanium, Thorium, Arsenic 330 

Minerals — Antimony, Bismuth, Sulphur, Tellurium, Uranium 330 

Minerals — Molybdenum, Aluminum, Boron, Hydrogen, Carbon 330 

Silica Minerals — Granite, Sandstone, Bluestone, Slate 331 

Silica Minerals — Fibrous Talc, Compact Talc, Mica, Asbestos 331 

Silica Minerals — Serpentine, Feldspar, Quartz, Infusorial Earth 331 

Silica Minerals — Kaolinite, Clay, Fuller's Earth 331 

Lithology — Rocks and Rock-Formations 331 

Principal Rock-Forming Minerals, Table 332 

The Common Rocks, 333; Table of Composition, Texture, etc 334 

The Common Rocks — Notes on Preceding Table 339 

C. — Botanical. 

Acreage of Timber Land in the United States 340 

Classification of Important Trees, Table — 341-346 

Trees — Soft Pines, Pitch Pines, Larches, Spruces 341 

Trees — Spruces, Hemlocks, Firs, Redwoods, Cedars, Cypresses 342 

Trees — Cypresses, Walnuts, Hickories, Poplars 343 

Trees — Poplars, Willows, Birches, Beeches, Chestnuts, White Oaks. . . . 344 

Trees — White Oaks, Black Oaks, Elms, Sweet Gums 345 

Trees — Maples, Ashes 346 

Tree Data — ^Tallest, Best, Ages, Growth, Important Products 346 

D. — Zoological. 

Animal Species and their Uses to Man 347 

Classification of Animals, Table 347-349 

SEC. 18.— EXPLOSIVES. 
(a). — Mechanical Mixtures. 

Nitrate Mixtures, Gunpowder, Blasting Powder 350 

Weight of Powder in a Hole One Foot Deep, Table 350 

Chlorate Explosive Mixtures 351 

(b). — Chemical Compounds. 

Nitro Substitution Explosives; Nitric -Acid Compounds; Guncotton 351 

Detonation; Smokeless Powders; Nitroglycerin; Dynamite 352 

Unmixed Explosives; Percussion Caps .* 353 

The Handling and Use of Dynamite 353 

Some of the Most Common Commercial Dynamites 354 

List of Permissible Explosives for Use in Coal Mines 354 

SEC. 19.— PRESERVATIVES. 
Paints. 

Pigments — Lead, Zinc, Lampblack, Boneblack, Graphite, Iron, etc .... 355 

Vehicles — Linseed Oil 355 , 356 

Driers; Solvents — ^Turpentine 356 

House Paints — Mixtures — Colors, Table 356 

Special Paints — Aluminum 356 

Special Paints — Bronze, Copper 357 

Varnishes, Lacquers, etc. 

Varnishing; Laquering; Japanning 357 

Galvanizing: and Tinning. 

Galvanizing; Tinning — ^Tin Plate, Teme Plate 357 



X CONTENTS, 

Electro-Plating. 

Electro-Chemistry; Electrolysis; Electro-Metallurgy 357 

Electro-Plating— -Gold, Silver, Copper, Nickel, etc 358 

Preservation of Steel and Iron. 

Oiling, Painting, Asphalting; Removing Mill Scale 358 

Preservation of Timber, 

Soiirces of Decay — ^Wet Rot, Fermentation, Dry Rot, Insect Larvae 359 

Piling; Creosoting; Btimettizing; Miscellaneous Notes 360 

Seasoning — Distribution of Water in Timber 361 

Seasoning — Relation of Water to Decay; What Seasoning Is 362 

Seasoning — Preservative Treatment; Advantages; Methods 363 

Seasoning — Conclusions and Recommendations 365 

Creosote in WeIl=Preserved Timbers. 

Manufacture and Composition of Creosote. — Coal Tar 365 

Analyses of Creosote Extracted From Well-Preserved Timber 367 

Results of Analyses of Extracted Oils, Table 368 

Excerpts and References. 

Protection of Ferric Structures from Corrosion 372 

Painting and Sand-Blast Cleaning of Steel Bridges, Costs 373 

Creosoting Wooden Poles for Electric Line Work, Costs 373 

Corrosion of Steel in Reinforced Cinder Concrete ! 374 

Cleaning Steelwork by Sand-Blast, Painting by Compressed Air, Costs. 374 

Comparison of Various Processes of Preserving Timber, Costs 375 

SEC. 20.— LUMBER AND LUMBERING. 

Stumpage in the United States, Tables 376 

Range of Lumber Prices for Twenty Years 377 

Logging — ^Trees, Time for Cutting, Volume of Standing Timber 378 

Logging — ^Transportation of Logs, 378; Scaling Logs 379 

Lumber — Sawing, Sizing, Planing, Seasoning, Board Measiire ..... 379 

Table of Feet Board Measure of Lumber 380-387 

Grading of Lumber 387 

Classification and Inspection of Yellow Pine Lumber 387 

Riiles for Grading Fir, Spruce, Cedar and Hemlock Lumber 388 

Shingles — Grades and Specifications 390 

Graphical Comparison of Various Log Rules 391 

Commercial Shipping Weights of Various Kinds of Lumber 391 

SEC. 21.— METALLURGY. 

Iron Ore; Pig Iron; Cast Iron 392 

Cast Steel; Malleable Castings; Wrought Iron 393 

Steel — Various Processes — Acid-Bessemer 394 

Steel — Basic Bessemer, Acid Open Hearth, Basic Open Hearth 395 

Steel — Cementation Process; Shear Steel; Crucible Cast Steel 395 

Steel — Open Hearth Cast-, Harveyized-, Vanadiimi-, 396 

Steel — Chrome-, Nickel-, Tungsten- 396 

Alloys of Various Metals — 396 

Bronze — Phosphor-.Manganese-, Aluminum-, Silicon- 397 

Brass— Copper-Zinc Alloys, Table 397 

Tin -base Alloys; Lead -base Alloys; Alzene 398 

Malleable Cast Iron; Nickel Steel, Properties 398 

Vanadium Steel — Structural — Alloys 399 

SEC. 22.— BUILDING STONES AND CEMENTS. 
Natural Building Stones. 

Granite, Basalt, Trap, Greenstone, Limestone 400 

Carbonate of Lime, Dolomite, Hydraulic Limestone 401 

Marl, Travertine, Marble (Domestic and Foreign) 401 

Sandstone (Berea, Medina, Potsdam, Conn. Val., N. J.) 401 

Sandstone, Frost Test; Flagstone; Slate 401 

Cements (Miscellaneous). 

Materials with Cementing Properties 402 

Cements — Boiler, Coppersmith's, Fireproof, Flour, Gas Fitters' 402 

Cements — Iron, Glue, Steam-Pipe 402 

Cements — Keene's Marble 40 d 



CONTENTS, XI 

Cements (Builders). 

Calcium, Lime, Common Lime, Lime Mortar, Lime Plaster 403 

Plaster of Paris, Hydraulic Lime 404 

Hydraulic Cement — Natural, Portland, Slag 404 

Bitumen, Asphalt 404 

Manufacture of Portland Cement — Wet and Dry Processes 405 

Calcination — Rotary Kiln; Grinding the Clinker 405 

Requisites of a Cement in Cement Testing 406 

Method of Testing Cement— A. S. C. E. and A. S. T. M.— 407 

Selection of Sample, Specific Gravity, Fineness 407 

Normal Consistency — Vicat Needle Test 408 

Percentage of Water for Standard Sand Mortars 409 

Time of Setting; Standard Sand 409 

Form of Briquette; Molds; Mixing 410 

Molding; Storage of the Test Pieces 410 

Tensile Strength; Constancy of Volume 411 

Specifications for Cement — A. S. T. M.— 411 

Specifications — Natural Cement, Portland Cement 412 

Specifications for Cement — Engrs. U. S. A. — 413 

Specifications — American Portland Cement 413 

Specifications — Natural Cement, Puzzolan Cement 414 

Artificial Building Stones. 

Brick — Common, Face, Glazed, Vitrified, Terra Cotta 415 

Fire Brick, Paving Brick, Sewer Brick 415 

Concrete — Kinds, Mixture, Proportions, Voids, Economy 416 

Concrete — Cement-Sand Mix, Cement-Sand-Stone Mix 417 

Concrete — Size of Broken Stone 417 

Block Stone — Beton-Coignet, Sand Bricks, Portland Stone 417 

Block Stone — McMurtrie Stone, Ransome Stone, Sorel Stone 417 

Miscellaneous Data. 

Lutes and Cements Useful to Engineers — • 418 

Compositions — Water-Proof, Oil-Proof, Acid -Proof, etc 418 

Cost of Portland Cement •. 418 

A Cement which is Proof Against Sea- Water 418 

SEC. 23.— QUARRYING. 

Sand, Gravel, Rip-rap, Stone — Hand Tools, Channeling Machines 419 

Weight and Specifications of Sullivan Channelers, Table 421 

Explosives; Rock Drills — Hammer, Chum, Percussion 422 

Quarrying by Direct Use of Compressed Air 423 

Cost of Quarrying Rubble and Dimension Stone 423 

Dimensions and Weights of Rand Percussion Rock Drills, Table 424 

SEC. 24.— STONE CUTTING. 

Stones Classified According to Finish. — Tools Employed 426 

Unsquared Stones. — Hand Hammer, Plug and Feathers, etc 426 

Squared S.; Quarry-Faced S. — Face Hammer; Cavil 427 

Pitched-Faced S.; Drafted S.— Chisel; Pitching C; Tooth C 427 

Cut Stones. — Mallet, Pick, Point, Crandall 428 

Cut Stones, — Ax, Pean Hammer, Patent Hammer, Tooth Ax 429 

Cut Stones. — Bush Hammer, Machine Tools 430 

SEC. 25.— MASONRY. 

Kinds of Masonry; Classification of Railroad Masonry, Table 431 

I, — Stone Masonry. 

Definitions of Parts of Wall 431 

Definitions of Kinds of Masonry 432 

Specifications for Stone Masonry, General 433 

Specifications for Bridge and Retaining Wall Masonry 434 

Specifications for Arch Masonry, Culvert M., Dry M 435 

Quantities of Masonry in Railroad Abutments, Table 436 

Table for Finding Weights of Quantities in Preceding Table 437 

11. — Brick Masonry. 

Bonds — English, Flemish, etc.; Brickwork 437 

Mortar Used in Brickwork; Table of Quantities 438 

1 11.^ — Concrete Masonry. 

Rock Crushers; Concrete Mixers — Gravity, Mechanical (Batch) 439 



XII CONTENTS. 

Concrete — Proportions of Cement, Sand and Stone — Mixing 440 

Concete — Placing, Spreading and Ramming; Subaqueous C 440 

Concrete, Subaqueous — Depositing by Tubes, Buckets, Bags 441 

Sub-Foundations — Prepared by Dredging, Cement Grout -. . . . 442 

German Specifications for Concrete 442 

IV. — Reinforced Concrete. 

Uses of Reinforced Concrete 443 

The Preservative Qualities of Cement 444 

The Fire-Resisting Qualities of Concrete 444 

The Proportions Used in Mixing Concrete 444 

Calculations of Reinforced Concrete Beams — 444 

Formulas; Values of / for Three Standard Mixes 445 

Properties of Reinforced Concrete Beams 1 Inch Wide, Table 446 

Formulas for Reinforced Concrete Construction — A. S. C. E 446 

V. — Mixed Masonry. 

Description — Weakness of Bond 449 

VI. — Concrete=Block Masonry. 

Solid Concrete Blocks; Hollow Concrete Blocks 450 

Specifications for Hollow Concrete Building Blocks 450 

Miscellaneous Data. 

Permeability of Concrete Under High Water Pressure 453 

Waterproofing Data — Concrete in Government Fortifications 453 

The Efficiency of Concrete-Mixing Machines 453 

Method of Finishing the Concrete Surfaces of Bridges 454 

Concrete Bridge — Materials Required for Different Mixes 454 

Expansion Joints in Concrete Structures — Reservoirs, Drydocks 454 

Oil-Mixed Concrete as a Waterproofing Material 455 

Specifications for Scrubbed Concrete Surface 455 

SEC. 26.— STEREOTOMY. 

Wall of Building.— Stone Arch 457 

SEC. 27.— WEIGHTS AND SPECIFIC GRAVITIES OF MATERIALS. 

Definitions. 

Mass, Unit of Mass, Gravity Acceleration, Weight 459 

Volume, Density, Specific Gravity 460 

Methods for Determining Specific Gravity. 

Solids Heavier than Water; Solids Lighter than Water 460 

Displacement Method; Porous Substances 460 

Granular Substances; Liquids — 461 

Beaume's Hydrometer, Tweddell's H.; Rousseau's Densimeter 461 

Nicholson's and Fahrenheit's Hydrometers; Refinements 462 

Gases — Standard Pressure, Temperature, Substance 462 

Gases. 
Weight of a Cubic Foot of Dry Air at Various Temperatures, Table. . . . 463 

Weights and Specific Gravities of Gases, Table 464 

Liquids. 

Weight of a Cubic Foot of Water at Various Temperatures, Table 465 

Equivalents of Centigrade and Fahrenheit Scales, Table 465 

Weights and Specific Gravities of Liquids, Tables 468 ,469 

Solids and Miscellaneous. 

Weights and Specific Gravities of Woods, Tables 470-473 

Weights and Specific Gravities of Building Stones, Masonry and 

Cements, Table 474-477 

General Table of Weights and Specific Gravities of Materials, Table 478-482 

Weights of Produce (U. S. Law), Table 482 

Reduction Tables. 

Weight Equivalent for Any Specific Gravity 483 

Weight of Sheets, Bars, Wire, for any Specific Gravity 484 

Weight for Cubic Yard for Any Specific Gravity 484 

Comparison of Various Weights, Capacities and Volumes 485 

SEC. 28.— STRENGTH AND RESISTANCE OF MATERIALS. 

I. — General Principles. 

Stress. — Strain. — Modulus of Elasticity 486 



CONTENTS. XIII 

Elastic Limit. — Yield Point. — Ultimate Strength or Stress 487 

Static Stress. — Repeated Stresses. — Alternating Stresses 487 

Working Stress and Factor of Safety 487 

Resilience or Work of Materials Under Stress — Formulas, Table 488 

Effect of Sudden Loading, and Impact. 489 

IL — ^Tables of Strength of Materials. 
A. — Woods. 

Compression (End) Tests of Timber, 12% and 15% Moisture 490 

Strength Factors for Reducing Moisture from 15% to 12% 491 

Compression (End) Tests of Green Timbers — Above 40% Moisture. . . . 491 

Bending Tests of Timber at Rupture 492 

Bending Tests of Timber at Relative Elastic Limit 493 

Compression Tests of Timber Across Grain 494 

Shearing Tests of Timber With Grain 494 

Relation Between Weight and Strength of Timber 494 

Timber in Tension, Compression, Bearing, Bending and Shear 495 

Remarks on Preceding Table. — Formula for Longitudinal Shear 496 

B. — Metals. 

Tension, Compression, Bending, Shearing, etc 496-507 

Notes on Comparison of Aluminum and Copper in Electric Transmission 496 

Notes on the Relative Conductivity of Silicon-Bronze Wire to Copper. . 497 

Standard Specifications for Gray Iron Castings 498 

High Tensile Strength of Steel in Important Structures 499 

Specifications for Structural Steel for Bridges 500 

Specifications for Open-Hearth Boiler Plate and Rivet Steel 501 

Specifications for Steel Rails, and Steel Castings 503 

Specifications for Steel Axles 504 

Specifications for Steel Forgings 505 

C. — Building Stones, Cements, etc. 

Compression, Tension, Bending, etc 507-512 

Tests of Bluestone and Brick; Rattler Test for Bricks 507 

Tests of Cements; Natural and Portland Compared 508 

Tests of Concrete; Formula Deduced from Tests 508 

Modulus of Elasticity and Coefficient of Expansion of Concrete 510 

Cinder Concrete — Watertown Arsenal Tests 510 

Compressive Strength of Granite, Marble and Masonry 511 

Compressive Strength of Sandstone, Slate and Terra Cotta. 512 

D. — Miscellaneous Materials. 

Strength of Canvas, Cotton, Flax, Glass, Ice, etc 512 

III. — Heat Effects on Various Substances. 

Definitions — A Gas, Liquid, Solid, Critical Point, Critical Temperature. 512 

Definitions — Critical Pressure, Boiling Point, Latent Heat of Vaporiz'n. 513 

Definitions — Melting Point, Latent Heat of Fusion 513 

Liquefaction of Gases — Four Methods; Absolute Zero of Temperature.. 513 

Boiling Point, Freezing Point, etc., of Gases and Liquids, Table 514 

Boiling Point of Substances at Atmospheric Pressure, Table 514 

Melting Ponts of Various Substances, Table 515 

Coefficients of Expansion — Formulas and Table 516 

IV. — Frictional Resistance of Materials. 

Coefficient of Friction of Revolving Journals 517 

Friction of Plane Surfaces Which Have Been Some Time in Contact. . . 517 

Friction of Plane Surfaces in Motion Upon Each Other 519 

Friction of Journals in Motion Upon Their Pillows 520 

Coefficients of Friction and Angles of Repose or Friction 521 

Rolling Friction (Dry) — Formula 521 

V. — Miscellaneous Data. 

Crushing Tests of Brick and Terra Cotta Piers, Tables 522 

The Effect of Fire on Building Materials 523 

Gypsum as a Fireproofing Material 523 

Some Tests of Old Timber. Compared With New 523 

SEC. 29.— PROPERTIES AND TABLES OF PLANE SURFACES. 
1. — Geometrical Figures. 

Any Figure : Axis through c. of g.; Axis at Base; Parallel Axis 524 

Triangle : Axis through Center of Gravity 524 

Triangle ; Axis at Base; Axis through Apex 625 



XIV CONTENTS. 

Triangle, Rectangle, Hollow Rectangle, Square, Hollow Square .... 525 

Parallelogram, Rhomboid 525 

Square, Hollow Square, Trapezoid, Regular Hexagon 526 

Hollow Hexagon, Regular Octagon, Hollow Octagon 527 

Circle, Hollow C, Semicircle, Circular Sector, Circular Half-Segment . . 528 

Ellipse, Hollow E., Parabolic Half -Segment, Parabolic Spandrel 529 

2. — Skeleton Figures With Thin Lines. 

Vertical Web Plate; Straight Line about Parallel Axis 529 

Angle, Tee, Cross, H-Section, Rectangular Cell, Channel, I-Beam 530 

I-Beam, Inclined Lines, Angles, Triangular Cell, Circular Cell 531 

Circular and Semi-Circular Arcs — Five Cases 531 ,532 

Corrugated Sheets — Cycloidal and Screw-Thread Shaped 532 

3. — Block Shapes. 

Flanges, Cross, I-Beam, Z-Bar, Tee, Angle, Channel 533 

I-Beam, Channel, Z-Bar, Tee with Tapered Stem 534 

4. — Rolled Shapes. 

Moments of Inertia About Inclined Axis, Formulas 535 

Moment of Inertia About Inclined Axis, Problem 537 

Max. and Min. Values of / About Inclined Axis, Formulas 537 

Properties of T-Beam; Properties of Channel 537 

Properties of Tee, Z-Bar, Angle 538 

Moments of Inertia of Rectangles, Table 539 ,540 

SEC. 30.— PROPERTIES AND TABLES OF STEEL SHAPES. 

List of Tables and Relevant Tables in All Sections 541 

Steel Rods, Square and Round — Weights and Areas, Table 542 

Steel Plates — Weights and Areas, Table 544 

Steel Angles, Unequal Legs — Properties of. Table 548 

Steel Angles, Equal Legs — Properties of. Table 552 

Steel I-Beams — Properties of. Table 554 

Steel Channels — Properties of. Table 556 

Steel Z-Bars— Properties of, Table 557 

Steel T-Shapes— Properties of. Table 558 

Steel Rail Sections — Dimensions and Properties, Table 560 

References: Steel H-Shapes, Corrugated Sheeting, Memoranda 561 

SEC. 31.— PROPERTIES AND TABLES OF BEAMS AND GIRDERS. 

Working Stress, Load, Moment, Slope and Deflection of Beams — Formulas 562 

Practical Examples in Use of Preceding Formulas 564 

Span and Deflection for Plastered Ceiling — Examples 564 

Longitudinal Shear in Beams — Formulas and Problem 565 

Safe Uniform Loads on Rectangular Beams, Table 566 

Examples in Use of Preceding Table 567 

Steel Beam Box Girders — Properties of, Table, Problem 568 ,569 

Steel Plate-Girders, Complete — Properties of. Table 570 ,571 

Flange Angles of Plate-Girders — Properties of. Table 572-574 

Web Plates of Plate-Girders— Properties of, Table 575-579 

Flange Plates of Plate-Girders — Properties of. Table 580-582 

Bethlehem Girder (Single I) Beams, Table 583 

Bethlehem Special I-Beams, Table 584 

Reinforced Concrete Beams — Working Stresses 585 

Computing the Strength of Reinforced Concrete Beams 585 

Various References — Concrete Beams and Slabs, etc 586 

SEC. 32.— PROPERTIES AND TABLES OF COLUMNS. 

General Stresses.— Shearing Effect 587 

Notation for Column Formulas — Various End Conditions 587 

Formulas — Short Strut — Eccentric Loading — Long Columns 588 

Ritter's Formula for Columns 589 

Author's Formula for Columns 590 

Gordon's Formula for Columns 592 

C. Shaler Smith's Formula for Wooden Columns 593 

Straight-Line Column Formulas — Table 593 

Safe Loads for Wooden Columns, Pin Bearing — Table 594 

Steel Columns — Secondary Stresses — Useful Sections 596 

Channel Columns — Table of Standard Dimensions 596 

Ultimate Strength of Steel Columns — General Table 597 

Z-Bar Columns, Without Side Plates— Table 598 ,599 



CONTENTS. XV 

Z-Bar Columns (14-Inch), With Side Plates — ^Table 600 

Channel Columns (6-Inch), With Flat Ends— Table 601 

Channel Columns (8- Inch), With Flat Ends— Table 602 

Channel Columns (10- Inch), With Flat Ends — ^Table 603 

Phoenix Steel Columns — ^Table of Dimensions, Loads, etc 604 

Cast Iron Coltunns, Rectangular and Round — Table of Strengths . . 606 .607 

Rolled Steel H-Columns (8'' to 14'0— Tables 608 

Reinforced Concrete Columns — ^Working Stresses 609 

Carbon-Steel and Nickel-Steel Columns — Tests and Formulas 609 

Tests of Plain and Reinforced Concrete Columns 610 

SEC. 33— STRUCTURAL DETAILS. 

Rivets — Conventional Rivet Signs — Osbom Code 611 

Rivets — Shearing and Bearing Values — ^Table 612 

Problems in Riveted Joints 612 

General Rivet Spacing, Clearances, etc. — ^Table 613 

Rivet Gages for Standard Steel Shapes — ^Table 614 

Standard Connection Angles for I-Beams and Channels 615 

Weights and Dimensions of Rivets — ^Table 616 

Riveted Joints — Kinds, and Table for Finding Net Areas 617 

Standard Bolts for Fastenings; Standard Screw Threads, Table 618 

Dimensions and Weights of Hot Pressed Nuts, Table 619 

Bolts — ^Table of Dimensions for Heads, Nuts, etc 620 

Weight of 100 Bolts with Square Heads and Nuts, Table 621 

Weights of Nuts and Bolt Heads, Table 621 

Lag Screws — ^Table of Weights — Use of 622 

Wood Screws— Table of Sizes 622 

Cast Iron Separators for I-Beams — ^Table and Remarks 623 

Cast Iron Washers and Plate Washers — ^Table and Remarks 624 

Steel Wire Nails, Miscellaneous — Weights and Dimensions, Table 625 

Steel Wire Nails and Spikes, Standard — ^Tables and Remarks 626 

Spikes and Nails; Tacks; Railroad Spikes; — ^Tables 627 

Boat Spikes; Street Railway Spikes; Cut Nails and Spikes; — ^Tables. . . 628 

Standard Pin-Nuts, with Weight of Pins — Table 629 

Details of Bridge Pins, Cotter Pins, Pin-Nuts, Pilot-Nuts, etc 629 

Pins — Bending Moments, and Bearing Value on Plates — ^Table 630 

Eye Bars — Ordinary and Adjustable — Material to Form Head, etc 631 

Clevises — ^Table of Dimensions and Weights 632 

Sleeve Nuts and Tumbuckles — ^Tables of Weights and Dimensions 633 

Upset Screw Ends for Round and Square Bars, Table 634 

Sleeve Nuts; Lateral Rods, Solid or Upset Eyes, LoopWelded Eyes, Det'ls 634 

Segmental Rollers for Bridge Expansion — Formulas. and Table 635 

Howe Truss Brace Problem — Solution 635 

Tables of Cubes and Squares — List of, and Uses 636 

Finding Bending and Resisting Moments from Table of Squares 637 

Finding Moments of Inertia from Table of Cubes 637 

Solving Triangles from Table of Squares 638 

Table of Cubes of Inches, 0" to 9'', Advancing by 64ths and 32nds 639 

Tables of Cubes of Inches, 9" to 29", Advancing by 16ths 640 

Table of Cubes of Inches, 29" to 109", Advancing by 8ths 641 ,642 

Table of Squares, 0' to 1', Advancing by 64ths of an Inch 643 ,644 

Table of Sqtiares, 1' to lO', Advancing by 32nds of an Inch 645-650 

Table of Squares, 10' to 52', Advancing by 16ths of an Inch 651-664 

Cost of Shop Drawings for Structural Iron and Steel 665 

Tests of Nickel-Steel Riveted Joints 665 

SEC. 34.— METAL GAGES. 

Standard Metal Gages— B. W. G., B. & S., etc.— Table 666 

U. S. Standard Gage for Sheet and Plate Iron and Steel, Table 667 

SEC. 35.— CORDAGE, WIRE AND CABLES. 

Technical Cordage Terms — Make-Up (In Manufactiu-e) 668 

Rope — Technical Terms in Manufacture and Use 668 

Rope — Knots, Hitches, Bends, Splices, etc 668 ,669 

Manila Rope — Manufacture, Strength and Weight 669 

Manila Rope — Weight and Strength — Working Load — ^Tables 670 

Comparison of Wire Gages— B. W. G., B. & S., Edison— Tables 671 

Table of Properties of Roebling Steel Wire — English and Metric 672 

Wire Rope — Manufacture, Use, Strength, Lubrication 673 



XVI CONTENTS. 

Roebling Round Wire-Rope — Table or Properties 674 

Wire Rope Fastenings — Details 675 

Telephone Cable in St. Gothard Tunnel — Description , 676 

SEC. 36.— PIPES AND TUBES. 

Wrought Iron Welded Steam Gas and Water Pipe — Tables 677 ,678 

Lead and Tin Lined Lead Pipe and Tubing, Tables 679 

Weight of Lead, Sheet Lead and Cast Tin 679 

Spiral Riveted Steel Pipe and Specials, Table 680 ,681 

Spiral Riveted Steel Pipe Details 682 

SEC. 37.— BRIDGES. 

Economic Lengths of Spans for River Crossing, Formula .... 683 

Economic Depth of Plate Girders and Trusses, Formulas 684 

Estimating Weights of Bridges 685 

Formula for Additional Length of Eye-Bars to Form Heads 686 

Formulas for Weights of Steel Bridges and Trestles 686 

Formula for Finding Bending Stresses in Eye-Bars 686 

SEC. 38.— RAILROAD BRIDGES. 

Moments and Shears — Beams or Girders — Distributed Loads 688 

Easy Method of Drawing a Moment Parabola 688 

Moments and Shears— Concentrated (Axle, Wheels, etc.) Loads 690 

Engine Diagrams, Axle Loads, with Table 690 

Typical Moment Diagram of Special Locomotive 691 

Bending Moments and Shears from Special Locomotive, Table 692 

Moments and Shears — Spans with Floorbeams , 693 

Concentrated Loads — Maximum Floorbeam Reactions 694 

Concentrated Loads — Positions for Maximum Moment 695 

Chord Stresses in Pratt Truss from Concentrated Loads, Problem 695 

Chord Stresses in Warren Truss from Concentrated Loads 696 

Concentrated Loads — Positions for Maximum Shear 696 

Lateral Bracing — Wind and Curve Pressure 697 

Portal and Intermediate Vertical Bracing, Formulas 698 

General Specifications for Steel Railroad Bridges — 699 

General Description — Material, Types of Bridges, Clearance 699 

Trusses, Girders, Floorbeams, Stringers, Wooden Floor, Guards 700 

Loads — Dead Load, Ljve Load 700 

Effect of Impact, Formula. — Engine Diagrams 701 

Wind Pressure; Momentum of Train; Centrifugal Force 702 

Proportion of Parts — Least Thickness of Material 702 

Permissible Tensile and Compressive Stresses 702 ,703 

Alternate Stresses; Combined Stresses 704 

Transverse Loading of Tension or Compression Members 704 

Shearing and Bearing Stresses 704 

Bending Stress on Pins. — Plate Girders 704 

Provision for Future Increase of Live Load 705 

Details of Construction — Camber; Adjustable Members 705 

Truss Bridges; Lateral and Sway Bracing 705 

Diagonal Bracing; Gusset Plates; Temperature 705 

Bolsters and Expansion Rollers; Bed Plates 705 

Rivets; Tie Plates; Lacing; Pin Plates 705 ,706 

Workmanship — Riveted Work; Planing and Reaming 706 

Workmanship — Eye-Bars; Machine Work 706 

Steel — Manuf actiure ; Properties; Pins; Castings 706 

Cooper's Standard Loading — Axle Loads — Table 707 

Moments, Shears and Fl.-Bm. Reac. — Cooper's Loading — Table 708 

Coefiicients of Impact — Formula and Table 709 

Permissible Compressive Stresses — Soft and Med. Steel — ^Table 710 

Approx. Weight of Steel in Railroad Bridges — Formula, Tables 710 

Plans and Details of Howe Truss 711 

Reinforced Concrete Bridges 712 

Safe Unit Stresses in Structural Timber, Table 713 

SEC. 39.— ELECTRIC RAILWAY BRIDGES. 

Typical Loadings — "L" and "X" 716 

Moments, Shears and Fl.-Bm. Reac. — "L 24" Loading — Table 717 

Moments and Fl.-Bm. Reactions — "K 25" Loading — ^Table 718 

Maximum End Shears from "K 25" Loading — ^Table 719 



CONTENTS. XVII 

SEC. 40.— HIGHWAY BRIDGES. 

Unit Stress Sheets for Various Types of Trusses — Tables 720-726 

Calculation of Stresses by Graphics 725 

Case of Cutting Four Active Members \\\\ 726 

Typical Loading for Highway Bridges — Table ' ] 727 

Live-Load Data for Floor Systems and Spans Under 50 Ft ...'...,. 728 

Maximum Moments and End Shears for "L" Loadings — Table ...'.* .' .* \ .' 728 

Uniform Live Loads for Trusses of Spans Over 50 Ft. — ^Table ] ] 729 

Details of Combination Bridge, 230-Ft. Span ,* ' " 729-736 

Nickel-Steel and Carbon-Steel Spans — Costs, Specifications — Table .737 ,738 
Reinforced Concrete Bridges — Cost Data 738 ^739 

SEC. 41.— CANTILEVER BRIDGES. 

Reactions by Formulas and Influence Diagrams 74O 

Deck Cantilever Bridges; Camber ] *. 74^ 

SEC. 42.— MOVABLE BRIDGES. 

Types — Swing, Traversing, Bascule, Lift, Pontoon ' 742 

Swing Bridges — Drawbridges — Rim-Bearing, Center-Bearing . . 742 

Rim-Bearing Draw — Four Supports — Reactions — Formulas 743 

Continuous Girder — Four Supports — Formulas 743 

Rim-Bearing Draw — Four Supports — Reactions — Table . . . 744 

Rim-Bear'g Draw — Reactions and Moments for Balanced L'ds — ^Table. 745 

Calculation of 315 Ft. Drawbridge — Graphical Solution 746 

Center-Bearing Draw — Three Supports — Mom. and Reac. — Formulas. . 746 

Continuous Girder — Three Supports — Formulas 745 

Center-Bearing Draw — Three Supports — Mom. and Reac. — ^Table 747 

Deck Drawbridge — Center-Bearing — Hints for Calculation 748 

Formulas for Weight of Steel in Swing Bridges * ] 743 

Counterweight Jack-Knife Drawbridge \\ 743 

Steel Bascule Bridges \\ 749 

SEC. 43.— SUSPENSION BRIDGES. 

Theoretical — Curve of Main Cables — The Parabolic Cable 750 

The Catenarian Cable — Formulas and Tables 751 ^752 

Linear or Skeleton Arch. — Graphical Solution of Catenary 753 

The Transformed Catenary 753 

Practical — Cables or Chains; Towers and Backstays 754 

Anchorages. — Cable Wrappings 755 

Manhattan Bridge Details — Description — Live Loads 756 

Manhattan Bridge — Specifications — Material, Allowable Stresses 758 

Longitudinal Section of Anchorage 759 

Weight of Materials in Manhattan Bridge, Table 760 

Economic Considerations 760 

SEC. 44.— ARCHES. 

Theoretical Forms; Transformed-Catenary Arch 761 

Masonry Arches; Parts of an Arch; Kinds of Arches 763 

Further Classification of Arches; Brick Arches 764 

Masonry Arch, a Statically Indeterminate Structure 764 

Curve of Intrados, 764; Thickness of Arch Ring, Formulas 765 

Crown Thickness for Masonry Arches, Table 766 

Springing Thickness for Masonry Arches, Table 767 

Forces Acting on a Masonry Arch 767 

Calculation of Arches — Theoretical Hints 768-770 

Centers for Arches; Parts of the Arch Center 770 

Centers for Arches — Loads, 771; Types of Centers 773 

Striking the Center; Camber of Center 773 

Some Notable Arches That Have Been Built, Table 774 

Some Typical Modern Arches, Table 778 

Concrete Culverts — Quantities 782 

Steel Arches — No-Hinged and Two-Hinged — Position Lines 782 

Plan of 4-Track, Railroad, 2-Hinged Steel Arch Bridge 783 

Three-Hinged Arch — Position Line 783 

Cost of Reinforced Concrete Bridges 784 

SEC. 45.— TRESTLES. 

Pile Trestles, Timber Trestles — Plans and Dimensions 787 

Pile and Timber Trestles— A., T. & S. F. R. R. Plan 790 



XVIII CONTENTS. 

Wooden Stringers — Allowable Bending Moments, Table 791 

Steel Trestles. — Elevated Railroad Trestles 791 ,792 

Reinforced Concrete Trestles : 792 

Cost of Railroad Trestles — Timbers, Steel, and Rein. -Cone 793 

Reinforced Concrete Trestles Actually Built, Described 793 

SEC. 46.— ROOFS. 

Wind Pressure — Velocities; Direct Presstire, Formulas 794 

Direct Normal Wind Pressures, Table 795 

Effect of Wind Suction or Tension 795 

Normal and Component Wind Pressures, Formulas 795 

Normal Wind Pressures on Inclined Surfaces, Table 976 

Wind Pressure on Pitched Roofs, Table 797 

Wind Pressures in Open Sheds, and on Cylinders and Spheres 797 

Snow Loads on Roofs — Latitude-Altitude Diagram 797 ,798 

Roof Coverings 798 

Shingle Roofing; Weight of Shingles Laid on Roofs, Table 799 

Slate Roofing; Number of Slates Laid on Roofs, Table 799 

Weight of Slate; Total Weight per Square of Roof; — Tables 799 ,800 

Tile Roofing — Kinds and Specifications 800 

Tin Roofing — Roofing Tin — Plates 800 

Steel Sheet Roofing; Corrugated Steel R., Formula; Tar-Gravel Roof.. 801 

Cement-Gravel R., Asphalt-Gravel R.; Slag R.; Patented R 802 

Weight of Roofing Materials, Table 802 

Common Types of Roof Trusses; Stress Diagrams 803 ,804 

Unit Stresses in Pratt Roof Trusses, Table 804 

Unit Deductions for Half Truss (Lean-to) 805 

Unit Stresses in Fan and Fink Trusses, Table 805 

Unit Deductions for Half Truss (Lean-to) 806 

Design for Combination Roof Trusses — Spacing of Jack-Rafters 806 

Design of Sheathing and Jack-Rafters 807 

Design of Purlins and Trusses 808 

Table of Stresses, and Stress Sheet, of Roof Truss 809 

Details of Roof -Truss Design — Chord Splice, Chord Block, Keys, etc. . . 810 

Weight of Steel Construction in Roofs, Formulas 810 

Steel Roofs — Approximate Weight of Trusses and Parlins — ^Table 811 

SEC. 47.— BUILDINGS. 

Plastering. — Lathing. — Partitions , , 812 

Wooden Partitions; Hollow Tile Partitions; Other Partitions 813 

Expanded Metal Partitions and Lathes 814 

Floors, Ceilings, etc. — Live and Dead Loads 814 

Live Loads — Minimum for Floors and Roofs — ^Ten Cities — ^Table 815 

Maximum Live Load Possible from People 815 

Loads from Safes, with Problem 816 ,817 

Table of Weights and Dimensions of Heavy Safes 816 

Examples of Floor and Ceiling Construction 817 ,818 

New York City Building Code — Digest — 819-823 

Quality of Materials. — Excavations and Foundations 819 

Wooden Beams, Girders and Columns. — Fireproof Buildings 819 

Iron and Steel Construction. — Floor Loads 820 

Calculations. — Strength of Materials 821 

Reinforced Concrete Construction 822, 823 

Allowable Stresses for Steel and Concrete, Table 823 

Chicago Building Ordinance — Digest — 823, 824 

Reinforced Concrete — Ratio of Moduli of Elasticity 823 

Reinforced Concrete — Adhesion — Bond 824 

Philadelphia Building Laws and Ordinances — Digest — 824 ,825 

Live Loads for Floors and Roofs 824 

Calculations. — Ultimate and Working Stresses 824 

Allowable Pressures. — Reinforced Concrete Construction 825 

Boston Building Law-r-Digest — 825-827 

Materials — Allowable Fiber Stresses 825 

Concrete and Reinforced Concrete Construction 826 

Buffalo Building Laws— Digest — 828 ,829 

Concrete and Reinforced Concrete Construction 828 

Hollow Concrete Blocks 829 

Reinforced Concrete Building — Formulas for Unit Stresses 831 

Reinforced Concrete — Standard Building Regulations 831 



CONTENTS. XIX 

Cutting Structural Steel Work with Oxy-Acetylene Flame 833 

Wind Loads on Mill Buildings 833 

SEC. 48 —RETAINING WALLS. 

Theory of Earth Pressure, 835; Rankine's Theory 839 

Cubic Yards of Masonry in Retaining Walls, Table 841 

Comparative Sections of Thirty Retaining Walls, Table 842 

SEC. 49.— DAMS. 

Common Fixed Types, Described 844 

Stability of Gravity Dams. — Hydrostatic Pressure 845 

Center of Pressure — Formulas 846 

Center of Gravity cf Trapezoid — Analytically and Graphically 847 

The Triangular Dam 847 

Pressure on Foundations, Formulas. 848 

Factor of Safety Against Overturning. — Shear 850 

Pressure on Foundations, Table 851 

Masonry Gravity Dam — Calculations and Design 852-854 

Effect of Profile on Gravity Dams 854 

Dimensions and Quantities for Masonry Dams, Tables 855 ,856 

Dimensions and Quantities for Rock-Fill Dams, Table 857 

Dimensions and Quantities for Earth Dams, Table 858 

High Masonry Dams — Table of Dimensions 859 

Comparative Cross-Sections of High Earth Dams 859 

Rubble-Concrete Dam — Cyclopean Masonry 860 

The Eastwood Multiple-Arch Dam, Described 860 

SEC. 50.— FOUNDATIONS. 

Foundation Beds — Rock, Gravel, etc 863 

Actual Bearing Pressures on Bed Rock, Table 864 

Actual Bearing Pressures on Sand, Table 864 

Actual Bearing Pressures on Gravel, Table . . ." 865 

Actual Bearing Pressures on Clay and Sand, Table 865 

Soils — Practical Test, Selection of Site, Examination 865 

Borings in Soil. — Estimating Loads on Foundations 866 

Bearing Power of Soils, from Various City Codes — Table 867 

Foundations for Machines, Dynamos, etc 867 

Types of Foundation Footings 867 

Steel I-Beam Footing for Independent Piers 868 

Coffer-Dams— Types Described 868 

Sheet Piling— Types — Timber and Steel 869 ,870 

Pile Foundations — Supporting Power of Piles 871 

Pile-Driving Formulas for Drop Hammer and Steam Hammer 871 

Safe Bearing Power of Piles, Table 872 

The Drop Hammer and Steam Hammer, Described 872 

Pile Drivers — Derrick, Power and Water- Jet 873 

Pile Shoes; Pile Foundations; Splicing Piles; Cutting Off Piles 874 

Piles — Dead-Men Piles; Iron Piles; Screw Piles; Disk Piles 874 

Piles — Sand Piles; Concrete Piles; Water- Jet Concrete Piles 875 

Piles — Metal Shell Concrete Piles; Reinforced Concrete Piles 875 

Open Caissons 876 

Crib Piers; Pile Piers; Tubular Piers 877 

Gushing Piers 878 

Platform Cylinder Piers. — Pneumatic Cylinder Piers and Process 879 

Pneumatic Foundations — Caisson and Crib 880 

Coffer-Dam. — ^Freezing Process 882 

Masonry Piers, Design 888 

Contents of Piers by Prismoidal Formula , 889 

SEC. 51o— WHARVES, PIERS AND DOCKS. 

Definitions. — Foundations 892 

Pierhead and Bulkhead Lines. — Construction Methods^. 892 

Piers. — Ferry Slips and Bridge Aprons * 893 

Plans of Ferry Crib, Dolphin and Bridge 894-899 

Reinforced Concrete Wharf Construction 900 

SEC. 52 —BREAKWATERS. 

Types of Breakwaters. . 901 

Quantities and Cost per Lin. Ft. of Breakwater, Table C02 



XX CONTENTS, 

Statistics of Notable Breakwaters, Table 903 

Materials for Concrete in Buffalo Breakwater 904 

SEC. 53.— JETTIES. 

Jetty Construction. — Fascines 905 

SEC. 54.— EARTHWORK. 

Uncertain Cost. — Kind and Quality of Material 906 

Approved Methods of Handling. — Clearing and Grubbing 906 

Loosening Earth. — Loading and Conveying 907 

Superintendence. — Labor 908 

Earth Empankment. — Shrinkage of Earth 909 

Experiments on Shrinkage of Earth 913 

Shrinkage in Volume and Vertical Shrinkage 915 

Performance of Work — Methods and Costs 915 

Diamond Drill Borings, Deep Waterways Survey, Table 917 

Railroad Grading with Wheeled Scrapers, Costs 917 

Trenching and Backfilling for Sewer Pipe, Costs 918 

Earthwork Classification — Solid Rock, Loose Rock, etc 919 

References to Earthwork and Especially to Shrinkage 920 

Machine for Excavation in Frozen Ground, Cost Data 921 

SEC. 55.— ROCK EXCAVATION. 

Open Rock Cuts — Drills, Drilling, etc 922 

Trenching in Rock, Methods; Chicago Drainage Canal 923 

Chicago Drainage Canal — Methods and Costs, Tables 924 

Performance of Work — Methods and Costs 925 

Use of Well Driller for Drilling -Blasting Holes, Costs 926 

SEC. 56.— DREDGING. 

Methods of Measuring Material. — Dredges, Types 927 

Elevator (Bucket) Dredge. — Hydraulic Dredge 928 

Blasting Under Water — Dredging Detroit River 929 

Detroit River — Dredging, Drilling, Scows — Cost Tables 930 

Gold Dredging in California 930 ,939 

Performance of Work — Methods and Costs 931 

Large Elevator Dredge for Work in Boston Harbor, Described 932 

SEC. 57.— TUNNELING. 

Definitions. — Kinds of Tunnels 933 

Methods of Tunneling — Open Cut — "Heading"'vs. "Drift" 933 

Drilling and Blasting. — Timbering. — Lining 934 

Alinement and Grade. — Ventilation 935 

Shield Method. — Dredging Method. — Caisson Method 936 

Performance of Work — Methods and Costs 937 

Some Notable Tunnels that Have Been Built, Data 937 

Some Detail Tunnel Costs, Los Angeles Aqueduct 939 

SEC. 58— SURVEYING, MAPPING AND LEVELING. 

Care of Instruments. — To Adjust the Level 941 

To Adjust the Transit, 942, The Solar Attachment 944 

The Solar Instrument — Description and Use 944 

To Adjust the Solar Attachment 947 

Solar Observations with Transit Alone — Calculations 947 

To Determine the Meridian from the North Star 948 

Polar Distance of Polaris for Lat. 0°, and Any Latitude 949 

Azimuth of Polaris at Elongation Jan. 1, Table 950 

Observations of Polaris at Elongation, or at Any Hour Angle 951 

Time— Civil, Astronomical and Railway 952 

Local Mean Time of Upper Culmination of Polaris, Table 953 

Azimuth of Polaris for Use of Land Surveyors, Table 954 

Polaris Tables for the Year 1912 955a-955f 

Tapes. — Temperature Corrections, Table 956 

Sag and Stretch of Tapes — ^To Correct 957 

Table of Equivalents of Feet and Chains 958 

Methods of Plotting Angles. — Table of Chords 959 

Farm Surveying — Equipment and Method — Adjustment of the Traverse 964 

Office Plan and Finished Map. — City Lot Surveying 966 

Government Land Surveying. — Acts of Congress 967 



CONTENTS, XXI 

General Rules from the Foregoing Acts 96g 

To Restore Lost or Obliterated Comers — Rules 969 

Rules for Subdivision of Sections ] 97O 

Useful Tables in Public Land Surveys — Meridians and Base Lines 971 

Azimuths of the Secant, and Offsets to the Parallel — Table 973 

Azimuths of the Tangent to the Parallel, Table 974 

Offsets from Tangent to Parallel, Table 975 

Correction of Randoms, Table 976 

Convergency of Meridians — Table and Examples 977^ 978 

Lengths of a Degree of Latitude to Minutes, Table 979 

Lengths of a Degree of Longitude to Minutes, Table ' 981 

The Stadia. — Stadia Reduction Table 983 ,984 

Leveling Correction for Curvature and Refraction 987 

Correction for Earth's Curvature, and Refraction — Table 988 

Allowable Errors in Leveling, Table and Formula 989 

Description of Four Stadia Surveys and Their Cost 990 

SEC 59.— RAILROADS. 
A. — General Discussion. 

Existing Mileage, etc. — Economic Principles ggi 

Tractive Force of a Locomotive — Grades — Formulas 992 

Tractive Force of Engines on Various Grades, Table 994 

Allowable Expense for Grade Reduction 995 

Cost of Haul on Various Grades, Table 996 

Economic Considerations of Curvature and Distance 997 

B. — Reconnoissance Survey. 

Aneroid Barometer — Formula, etc 998 

Table of Barometric Elevations. — Barometric Correction Table 999 

C. — Preliminary Survey. 

Field Operations — Locating Engineer and Transitman 1000 

Table of Grades— Ft. per 100 Ft. to Ft. per Mile, Equivalents 1001 

Table of Stations Corresponding to Distance in Miles 1001 

Table of Grades — Angles for Rates of Grades, Equivalents 1002 

Table of Grades— Ft. per Mile to Ft. per 100 Ft., Equivalents 1003 

Table of Grades — Angles for Ft. per Mile, Equivalents 1003 

Duties of Levelman and Topographer. — Mapping 1004 

D. — Location Survey, 

Profiles and Grades 1004 

Parabolic Vertical Curves. — Horizontal Circular Curves 1005 

Table of Radii of Curves — English and Metric 1007 

Table of Tangents and Externals to 1° Curve — English and Metric .... 1009 

Minutes and Seconds to Decimals of Degree or Hour, Table 1010 

Various Problems in Simple Curves. — Compound Curves 1011 

Solution of Compound Curve Problems — ^Table of Formulas 1012 

Reversed Curves — Formulas 1012 

Cubic Parabola. — Spiral Curve. — Easement Curves 1013 

E.— Right of Way. 

Filing the Location ^. 1013 

Purchase and Condemnation \ 1014 

Table for Finding Widths of R.-of-W. for Cuts and Fills 1014 

Tables for Finding Acreage of Right of Way 1015 

F. — Construction. 

Methods of Calculation of Earthwork 1016 

List and Description of Earthwork Tables, with Page No 1017 

Multiplication Tables for Earthwork Calculation 1018-1020 

Slope-Staking and Earthwork Computation 1055 

The Prismoidal Formula and Prismoidal Correction Formula 1055 

Tables of Prismoidal Corrections 1056-1058 

Earthwork Correction for Curvature. — Haul. — Roadbed 1059 

Rails and Fastenings — Standards 1060 

Standard Dimensions of Rails and Fastenings, Table 1061 

Weight of Rails Reduced to Tons per Mile of Track, Table 1063 

Middle Ordinates for Curving Rails, Formulas 1063 

Middle Ordinates for Curving Rails, Tables 1064-1067 

Chord Lengths of Curved Rails, Tables 1066 ,1067 

To Find the Degree of Curve of Laid Track, Formulas 1066 

Track Spikes— Table.— Rail Joints 1068 



XXII CONTENTS. 

Formula for Thickness of Shims in Tracklaying 1069 

Cross Ties, — ^Table of Cubic Feet in Wooden Ties ! .' ! ! 1069 

Table of Feet Board Measure in Wooden Ties \\ 1070 

Bills of Switch Ties for Nos. 6 and 8 Frogs, Tables 1070 

Cross Ties — Best Kinds, Life, and Time to Cut 1071 

Tie Plates — Types, and Advantages. — Rail Braces ! . 1071 

Steel Ties. — Concrete Steel Ties .' " 1072 

Ballast — Kinds, and Amount Required 1073 

Track Gage. — Wheel Gage, M. C. B. Standard * , 1073 

Increase of Gage for Various Curves, Table 1073 

Gages and Half-Gages of Track, with Log Values — Table 1074 

Various Track Gages in Use. — Best Standard Gage 1074 

Turnouts and Switches. — Frogs. — Frog Numbers 1075 

Manganese Steel Frogs. — Spring Rail Frogs 1075 

Properties of Frog Angles and Half Frog Angles, Table 1076 

Movable-Point Frogs 1076 ,1077 

Crossing Frogs. — Stub Switches 1078 

Table for Laying Out Switches in the Field 1078 

Table of Theoretical or Stub Switches 1079 

Table of Radii of Theoretical Turnout Curves 1081 

Table of Theoretical Switches for Any Gage 1082 

Turnout Curves for Stub Switches, Formulas 1083 

Formulas for Double Turnout Curves 1084 

Turnout Curves from Curved Main Track, Formulas 1084 

Split Switches 1084 

Turnouts for Split Switches and Spring Frogs, Table 1085 

Three-Throw Turnouts, Split Switches — ^Table 1086 

Turnouts from Straight Track, Split Switches — ^Table 1087 

Formulas for Split Switches 1087 

Wharton Switch 1088 

Ladder Tracks — Spacing of Frogs — ^Table 1089 

Crossovers — Spacing of Frogs— Table 1090 

Standards of Track Construction on Americ^ Railways 1093 

Street Railway Track Construction and Paving 1094 

SEC. 60— HIGHWAYS. 
A. — Traction. 

Power of a Horse. — Effect of Road Surfaces on Traction 1097 

Effect of Grades on Traction — Formvdas, Tests, Problem 1097 

B. — Roads and Streets. 

Definitions, — Dirt Roads. — Corduroy Roads 1098 

Plank Roads. — Gravel Roads and Walks 1098 

Broken-Stone Pavement. — Hydraulic Cement Pavement 1099 

Cement Sidewalks 1099 

Wood-Block Pavements. — Cobblestone Pavement 1099 

Belgian Block Pavement. — Granite Block Pavement 1100 

Brick Pavement. — Asphalt Pavement 1100 

Asphalt Paving Blocks. — Bituminous-Rock Pavement 1100 

• C, — Pavement Specifications. 

Allegheny County, Pa.: — Road Specifications 1101 

Boston:— Granite Block Pavement — Brick Sidewalk 1102 

Wood Block Pavement — Brick Sidewalk 1103 

Asphalt Pavement. — Bitulithic Pavement 1104 ,1105 

Macadam Roadway — Crushed Stone Sidewalk 1105 

The Proper Construction of Brick Pavements. — (W. P. Blair) 1106 

Cincinnati: — Boulder Pavement 1107 

Detroit: — General. — Brick Pavement, Concrete Foundations 1108 ,1109 

Sheet Asphalt Pavement on Concrete Foundations 1110 

Cedar Block Pavement on Concrete Foundations 1110 

Easton, Pa.: — Macadam and Telford Roads 1111 

Concrete Curbs, Gutters and Sidewalks 1111 

Reinforced Concrete Foundations — Reference 1112 

El Paso, Tex.: — Petrolithic Pavement 1112 

Los Angeles:— Gravelled Streets 1114 

Bituminized Brick Gutters 1115 

Maryland State Highway: — Macadam Construction 1116 

Nat'l Association Cement Users: — Portland Cement vSidewalk 1117 



CONTENTS, XXIII 

Manhattan (N. Y. City): — Granite Block Pavement 1119 

Wood Block Pavement 1120 

Richmond (N. Y. City) : — Iron Slag Block Pavement 1121 

Vitrified Brick Pavement 1121 

Asphalt Block Pavement.— Curb on Concrete Foundation 1122 

Richmond, Ind.: — Street Crowning, Table 1123 

Syracuse, N. Y.: — General. — Vitrified Brick Pavement 1123 

Sandstone Block Pavement 1 124 

Asphalt Sheet Pavement 1125 

Creosoted Wood Block Pavement. — Bitulithic Pavement 1126 

Toronto, Ont.:— Grading 1127 

Cedar Block Pavement. — Concrete. — Asphalt Pavement 1128 

Brick Pavement. — Macadam Roadway. — Concrete Walk 1129 

D. — Care of Road Surfaces. 

Dust Preventives: — Classification 1131 

Tars, their Manufacture and Properties — Coal Tars 1131 

Water-Gas Tar. — Composition of Tars 1132 

Application of Tars to Finished Road Stirfaces 1132 

Use of Tar in Road Construction 1132 

Oils, their Classification and Properties 1133 

Application of Heavy Oils to Surfaces, and Roads 1 1 34 

Specifications for Coal Tars 1135 

Experiments with Dust Preventives: — 1136 

Tar Experiments — Miscellaneous and Cost Data, Tables 1137 

Cost of Applying Calcium Chloride 1138 

Rock Asphalt and Oil Experiments — Cost Data 1138-1140 

Asphalt and Bituminous Rock Deposits of the U. S 1141 

E. — Miscellaneous. 

Paving a Country Road with Brick 1141 

Inverted Macadam Road Construction 1142 

Vitrified Clay Curbing for Streets and Roads 1142 

Sidewalk and Paving Practice in Chicago, Cost Data 1143 

Experience in Dust Suppression on N. J. Roads, Cost Data 1143 

Tests of Various Road Siirfacing Materials, Table 1144 

SEC. 61.— HYDROSTATICS. 

Definitions. — Atmospheric Pressure. — Air. — Water 1145 

Hydrostatic Pressure — Units and Formulas 1146 

Hydrostatic Head and Pressure, Equivalents (1-10), Tables 1146 

Head in Feet for Given Pressiires per Square Inch, Table 1147 

Pressure for Square Inch for Given Heads in Feet, Xable 1148 

Pressure for Square Foot for Given Heads in Feet, Tabje 1149 

Center of Pressure on Submerged Surfaces, Formulas 1150 

Center of Pressure on Orifices, Weirs, etc., Table 1151 

Pressure on Pipes, Tanks, etc. — Flotation. — Buoyancy 1152 

Metacenter. — Laws of Eqmlibrium 1153 

SEC. 62.— HYDRAULICS. 

Definitions. — ^Theory of Flow — Formulas 1154 

Velocity and Discharge, Formulas 1155 ,1156 

Theoretic Velocities for Various Heads, Table (283) ,1155 

Areas of Pipes in Sq. Ft. for Diameter in Ft. and Ins., Table 1157 

Velocity in Pipes or Varying Cross-Section, Formulas 1158 

Losses During Flow Through Pipes, Formulas 1158 

Hydraulic Grade Line. — Velocity of Approach 1159 

Total Head. — Loss of Head due to Friction 1160 

Hydraulic Notation and Formulas 1161 

Economic Sections of Conduits — Maximum Velocity 1161 

Chezy's Hydraulic Formula. — Kutter's Formula 1167 

Values of n (and c) in Kutter's Formula 1108 ,1169 

Coefficients c in Kutter's Formula, Table llVO-1172 

Practical Examples in Use of Kutter's Formula 1172 

The Venturi Meter— 1173 

Meter Register; Manometer; Piezometer Tubes 1174 

Orifices, Tubes, Nozzles and Jets — Formulas and Tables 1175 

Weirs — Standard Weir — ^Theoretic Discharge, Formulas 1177 

Francis' Weir Formulas 1178 

Bazin's Weir Formula; and Value of w, Table 1178 .1179 



XXIV CONTENTS, 



n 



Fteley and Steams' Weir Formulas 1180 

Parmley's Weir Formtila; and Values of C and K, Tables 1180 

Triangiilar and Trapezoidal Weirs 1181 

The Submerged Weir:— 1181 

Fteley and Steams' Formula; and Values of m, Table 1181 

Herschel's Formula; and Values of c, Table 1181 ,1182 

Hydraulic Measurements — ^Tank, Venturi, Weir, etc 1182 

Hook Gage. — Pilot Tube Meter. — Floats 1183 

Current Meters — Rating and Use. — Meter Register 1185 ,1186 

Depth of Thread of Mean Velocity in Rivers 1187 

Values of n in Kutter's Formula Determined for Earth Canals 1187 

Durability of Wood Stave Pipe Actually Laid 1187 

Values of c and n in Kutter's Formula — Experiments, Table 1188 

Bazin's Hydraulic Formula 1189 

Friction of Air in Small Pipes, Formula 1189 

SEC. 63.— WATER SUPPLY. 

Source. — Rainfall — Distribution. — ^Artesian Nomenclature 1190 

Average Monthly Precipitation in the U. S., Table 1191 

Percentage of Rainfall to Average Rainfall, Table 1195 

High Intensities of Rainfall, Notation 1195 

Maximum Intensity of Downpour, Formulas. — Rain Gage 1196 

Maximimi Rates of Rainfall by Preceding Formulas, Table 1197 

Rimoff. — Measuring Stream Discharge 1197 

Runoff Formulas.— Effect of Slope 1198 

Evaporation from Ice, Snow, Water Surface, etc 1199 

Monthly Evaporation from Water Surfaces in the U. S., Table 1199 

Seepage and Evaporation in Canals, etc 1200 

SEC. 64.— WATER WORKS. 
A. — Consumption of Water. 

Water Meters, and Waste of Water 1202 

Population in 17 Cities of the U. S., from 1850 to 1906— Table 1202 

Water Consumption in 17 Cities of the U. S., from 1850 to 1906— Tab. 1203 
B. — Purification of Water. 

Screening. — Sedimentation. — Slow Sand Filtration 1204 

Rapid Sand Filtration — Mechanical. — Copper Sulphate 1204 

C. — Reservoirs. 

Storage Reservoirs. — Distributing Reservoirs 1205 

Reservoir Linings. — Stand Pipes — Formxilas for Design 1206 

D, — Conduits. 

Canals. — Flumes. — Bored Wooden Pipe.— Salt Glazed Pipe 1207 

Masonry Aqueducts — Reinforced Concrete 1208 

Bored Wooden Pipe, Banded. — Wood Stave Pipe, Details 1208 

Wood Stave Pipe and Details, Table 1210-1213 

Notes on Preceding Table — Bands and Shoes 1214 

Discharge in Gallons through Wood Stave Pipe, Table 1214 

Cast Iron Pipe — 1214 

Formulas for Designing Cast Iron Pipe 1215 

Kinds of Pipe Joints. — Bell and Spigot Joint Pipe — 1215 

Cast Iron B. and S. Pipe, with Lead and Hemp Data — ^Table 1216 

Friction Heads in Clean Cast Iron Pipe, Table 1217 ,1218 

Examples in Use of Preceding Table 1219 

Notes on Pipe Laying and Taking Up Laid Pipe 1219 

Weights and Dimensions of C. I. Pipe and Specials, Tables 1219-1267 

Straight Cast Iron Pipe 1220 ,1222 ,1223 ,1243-1246 

Bells of Special Castings 1221, 1223 

Cast Iron Pipe Curves * 1224, 1248, 1249 

Cast Iron Pipe Branches— L's, T's, Crosses 1225-1227,1250-1254 

Cast Iron Pipe Branches— Y's 1227 ,1228 ,1255 ,1256 

Cast Iron Pipe Hydrant Branches. 1229 

Cast Iron Pipe Blow-Off Branches 1230 ,1231 ,1257 ,1258 

Cast Iron Pipe Sleeves 1232 ,1265 

Cast Iron Pipe Increasers and Reducers 1233 ,1259-1264 

Cast Iron Pipe Caps 1234 ,1266 

Cast Iron Pipe Lugs 1247 

Cast Iron Pipe Offsets 1235 ,1249 



CONTENTS, XXV 

Cast Iron Pipe Plugs 1234 ,1267 

Cast Iron Manhole Pipe 1232 ,1259 

Standard Specifications for Cast Iron Pipe 1239 

Turned and Bored Joint Pipe .' .' 1235 

Flanged Joint Pipe 1235 

Standard Flange Pipe, Table 1236 

Flexible Joint Pipe 1238 

Wrought Iron Pipe. — Steel Pipe 1268 

Riveted Steel Pipe — Formulas and Design 1268 

Steel Pipe Rivet Table. — Locking Bar Joint Pipe ! .* 1269 

Lap-Welded Pipe. — Spiral Riveted Pipe 1269 

Pressure Pipe Attachments — 1269 

Sluice-Gate, Stand and Wheel. — Air Relief Valves . 1270 

Metropolitan Air Valves, Table. — Standpipes. — Gate Valves !.*.*! 1271 

Ludlow Bronze Mounted Double Gate Valves — Nomenclatiure 1272 

Weights and Dimensions of M. W. W. Gate Valves, Table .' ! ! 1273 

Dimensions of Ludlow Double Gate Valves, Table 1247 ,1275 

Weights of Ludlow Gates and Valves, Table 1276-1279 

Blow-offs. — Special Pipe Connections 1280 

E. — Distributing System. 

Cast Iron Pipe. — Lead-Melting Furnace. — Matheson Pipe 1280 

Matheson Pipe and Specials, Tables ' ] * 128I 

Converse Pipe, Table. — Pipe Dipping Tank \\'\ i282 

Tapping Machines — Mueller '''' 1283 

Black or Galvanized Pipe — Dimensions and Weights — ^Table .' .* 1284 

Discharge in Gallons through Small Pipes, Table .* ! ! ! 1284 

Gate Valves — Chapman and Eddy Types . . . 1285 

Ludlow Double Gate Valves, Dimensions — ^Table . '. 1286,1287 

Gate Boxes. — Weight of Ludlow Gate Boxes, Table .'l288 

Check Valves. — Pressure Relief Valves. — Hydrants ! ! ! ! 1288 

Hydrants — Nomenclature of Parts . . . 1289 

Weights of Ludlow Hydrants, Table ..*.*.'.'.' 1290 

F. — Miscellaneous Data. 

Costs of Slow and Rapid Sand Filtration I291 

Efficiencies of Riveted Pipe Joints 1291 

Water Purification in Reservoirs, Cost Data 1292 

Steel Pipes for Water Works — Economics 1292 

Pneumatic Calking of Mains with Lead Wool 1293 

Waterproofing the New Ulm Reservoir. 1293 

SEC 65.— SANITATION. 

The Disposal of Refuse. — House Drainage 1295 

Cesspools.^-Sewers — Size and Grade . . . 1296 

Values of r, Vr and oVVfor Circular Sections, Table 1297 

Factors for Using Preceding Table for Other Sections, Table ! ! *. 1298 

Examples in Use of Preceding Table . . . . 1298 

Velocities of Circular Brick Sewers, Table 1299 

Properties of Circular Conduits or Sewers, Formulas 1300 

Properties of Catenary Conduits or Sewers, Formulas ! ! ! ! ! 1301 

Properties of Basket-Handle Conduits or Sewers, Formulas . . . . 1302 

Properties of Gothic Conduits or Sewers, Formulas ! ! . . 1303 

Properties of Egg-Shaped Conduits or Sewers, Formulas. \^. ...... ... 1304 

Velocities in Egg-Shaped Sewers, Table . . .1 305 

Examples in Use of Preceding Table 1306 

Thickness of Brick Sewer Walls, Formula. — Sewer Foundations. . .. . . . 1306 

References to Illustrated Sewer Construction 1306 

Sewer Pipe. — Weight and Cost of Sewer and Culvert Pipe, Tables. . ! .' . 1307 

Sewers and their Classification 1 307 

Location of Sewers. — Manholes.— Catch Basins 1308 

Quantities in Mortar Mix for Sewer Joints, Table ..\ 309 

The Wear of Sewer Inverts .'.1310 

Sulphur and Sand for Sewer Pipe Joints, Table 1310 

Cost of 66- Inch Sewer at Gary, Indiana ...*.'.*..*.*.*] 1310 

SEC. 66.— IRRIGATION. 

General Discussion. — Irrigation Units 1313 

Miner's Inch — Equivalents of Discharge — ^Table 1313 

Equivalents of Discharge of One Cu. Ft. per Sec, Table 1314 



XXVI CONTENTS. 

Units of Volume — Acre-Foot and Acre-Inch 1314 

Rates of Discharge for One Acre-Foot per Day, Table 1314 

Rates of Discharge for One Acre-Foot per Month, Table 1315 

Duty of Water in Irrigation 1315 

Duty of Water — Measiirements at Different Points — Tables 1316 

Duty of Water — Losses in Main Canal Included — ^Table 1316 

Duration of Irrigation Period on Some Canals, Table 1316 

Duty of One Cu. Ft. per Sec, and Inches in 10 days — ^Table 1317 

Canals. — Data on Some Perennial Canals, Table 1317 

Conduits and Fliimes 1317 

Field Location of Irrigation Ditches and Canals 1318 

Cost Date on Drainage of Irrigated Lands. 1319 

SEC. 67.— WATERWAYS. 

Suez Canal. — Cronstadt and St. Pertersburg Canal 1320 

Corinth Canal. — Manchester Ship Canal 1320 

Traffic Data on Suez Canal, Table 1321 

Kaiser Wilhelm Canal. — Elbe and Trave Canal 1322 

Welland Canal. — Sault Ste. Marie Canals 1322 

Canadian Canal Systems. — Lake Borgue Canal 1323 

Chicago Drainage Canal. — Proposed Am. Isthmian Canal 1324 

Distance Data via Panama and Nicaragua Routes 1328 

Harlem River Canal. — Cost of Maintenance of Canals 1329 

Principal Commercial Canals of the U. S. — Table 1329 ,1330 

SEC. 68.— WATER POWER. 

Definitions and Formulas. — Economic Design of Penstock 1332 

Horsepower per Cubic Foot of Flow per Second, Table 1333 

Horsepower-Hours from Storage of One Million Cu. Ft., Table 1334 

Horsepower-Hours from Storage of One Acre-Foot, Table 1335 

Water Motors — Wheels — Current, Undershot, Breast, Overshot 1336 

Impulse Water Wheels 1336 

Single Nozzle Pelton Water Wheel Data, Table 1338 

Quintex Nozzle Pelton Water Wheel Data, Table 1342 

Turbine Water Wheels — Nomenclature of Terms 1342 

Lkdsscs of Energy in Turbines; Efficiencies of Turbines 1343 

Theoretic Horsepower of Turbines. — ^Transmission of Power 1344 

Important Designs for Reference I345 

SEC. 69.— STEAM AND GAS POWER. 
A.— Heat. 

Matter and Energy; Kinds, Forms and Transformation of Energy 1346 

Thermal Energy. — First Law of Thermodynamics I347 

Thermal Units — British, French, British-French 1347 

Mechanical Equivalent of Heat, / I347 

Thermal-Units — Equivalents (1-10), Mechanical Work — ^Table 1348 

British Thermal Units, Power and Work — Equivalents (1-9) — ^Table . .1349 

Examples in Use of Preceding Table 1 350 

B.— Fuel. 

Heating Power of Fuel's — Methods of Determination 1350 

Chemical Analysis.— Coals Classified by Carbon and Volatiles, Table. . .1350 

Proximate Analysis and Heating Values of U. S. Coals, Table 1351 

Ultimate Analysis of Fuel, Defined 1351 

Chemical Analysis of Several Kinds of Solid Fuels, Table 1352 

Calculation of Heat of Combustion — Formulas; Calorimeter 1352 

Practical Boiler Tests — Coal as Fuel — Table 1353 

C— Steam. 

General Discussion — Kinds of Steam 1354 

Superheated Steam and Saturated Steam, Formulas 1355 

Saturated Steam Tables, Described; Heat of Vaporization, Formula. . .1356 

Saturated Steam Tables— Old 1357 ,1358 

Saturated Steam Tables— Revised 1359 ,1360 

Flow of Steam Through Pipes — Formula and Table 1361 

Steam Boilers — Efficiency and Commercial Horsepower Rating 1361 

Consumption of Coal per Boiler Horsepower-Hour 1362 

Kinds of Steam Boilers; Boiler Settings 1362 

Steam Engines; Engine Horsepower, Problem 1363 

Coal Consumption per Horsepower per Hour loo2 



CONTENTS, XXVII 

Value of Wood as Fuel; Principle o£ the Steam Engine 1363 

Steam Engine Cylinder and Double Indicator Diagram 1364 

Mean Effective Pressure — ^Table, Formulas and Problem 1365 

Economic Performance of Steam Engines — Non-Condensing Engines. . . 1 365 
Econ. Perf. of Steam Engines — Condensing, and Compound Engines. . .1366 

Effect of Load Upon Economy of Steam Engines 1 366 

Steam Pumps — Duplex, Centrifugal and Rotary 1366, 1367 

Duty of Pumps — Formula 1367 

D. — Heat (Internal=Combustion) Engines. 

Tests of Heat Engines on Alcohol Fuel 1368 

Conclusions Drawn from Above Tests 1 369 

Properties of Liquid Fuels — Gasoline, Kerosene, Alcohol 1370 

Heat of Combustion of Petroleum Oils and Alcohol 1370 

Air Necessary for Combustion of Liquid Fuels 1371 

Vaporization of Liquid Fuels 1371 

Avagadro's Law of Gases 1372 

Vapor Pressure of Saturation for Various Liquids, Table 1373 

Methods of Testing Heat Engines; Brake Horsepower, Formula 1374 

Indicated Horsepower, Formula 1375 

Fuels Used in Testing Heat Engines — Properties 1375 

Fractional Distillation of Gasoline, Table 1376 

E. — Miscellaneous Data. 

Heat Conductance and Resistance of Various Materials, Table 1377 

Solution of Steam Problems by Entropy Diagrams — (Ref .) 1378 

Unique Direct-Acting Explosion Pump — (Ref.) 1378 

SEC. 70.— ELECTRIC POWER AND LIGHTING. 

Electricity as a Form of Energy 1379 

Electric Power Units — Watt; Kilowatt or Electric Horsepower 1379 

Steam-Electric and Hydro-Electric Problems; Dynamos, Defined 1379 

Transformers, Converters and Boosters, Defined 1380 

Principles of Electricity and Magnetism. — What Electricity Is 1380 

Ether ; Ether Waves; Electricity and Magnetism; Magnetic Field 1380 

The Electro-Magnet, — Induced Currents or Induction 1381 

Farraday's Ring; The Horse-Shoe Magnet; Permanent Magnets 1382 

Principle of the Alternate-Current Dynamo 1382 

Classification of Alternate-Current Dynamos 1383 

Principle of the Continuous-Current Dynamo 1 384 

Classification of Continuous-Current Dynamos 1 384 

Electric Transmission of Power. — Steam and Water Power Compared. .1385 

Alternating Current vs. Continuous Current 1 386 

Long-Distance Transmission 1 386 

The Transmission Line — Aluminum vs. Copper 1386 

Transmission Line — Size of Conductors; Problems 1387 

Copper Wire Table for Electrical Calculations 1388-1391 

National Electric Code — General Outline of Plan 1393 

Class A. — Stations and Dynamo Rooms — Generators; Conductors 1394 

Switchboards; Resistance Boxes and Equalizers 1395 

Lightning Arresters; Care and Attendance; Insulation Resistance 1396 

Motors, 1397; Railway Power Plants 1398 

Storage or Primary Batteries; Transformers 1398 

Class B. — Outside Work, All Systems and Voltages — Wires 1399 

Constant-Potential Pole Lines, Over 5,000 Volts 1400 

Transformers; Grounding Low-Potential Circuits 1401 

Class C. — Inside Work, AH Systems and Voltages — Wires 1403 

Underground Conductors; Table of Carrying Capacities of Wires 1404 

Inside Work, Constant-Current Systems — Wires; Series Arc Lamps 1405 

Incandescent Lamps in Series Circuits 1406 

Inside Work, Constant-Potential Systems — ^Automatic Cut-Outs 1406 

Switches; Electric Heaters 1407 

Inside Work, Constant Low-Potential Systems — Wires 1408 

Armored Cables 1411 

Interior Conduits; Metal Moldings 1412 

Fixtures; Sockets; Flexible Cord 1413 

Arc Lamps on Constant-Potential Circuits; Economy Coils 1414 

Decorative Lighting System; Theater Wiring 1414 

Car Wiring and Equipment of Cars 1417 

Car Houses, 1421; Lighting and Power from Railway Wires 1422 



XXVIII CONTENTS, 

Inside Work, Constant High-Potential Systems — ^Wires 1422 

Transformers, 1422; Series Lamps 1423 

Inside Work, Constant Extra-High-Potential Systems — ^Wires 1423 

Class D. — Fittings, Materials and Details of Construction 1423 

Insulated Wires— General Rules 1423 

Rubber-Covered Wires, 1424; Slow-Burning Weatherproof Wire 1425 

Slow-Burning Wire; Weatherproof Wire ,. 1425 

Flexible Cord, 1426; Fixture Wire, Conduit Wire 1427 

Armored Cable, 1427; Interior Conduits 1428 

Switch and Outlet Boxes; Moldings 1429 

Tubes and Bushings; Cleats 1430 

Flexible Tubing; Switches 1431 

Cut-Outs and Circuit Breakers .... 1434 

Fuses, 1436; Standard Cartridge Enclosed Fuses, Table 1438 

Tablet and Panel Boards; Cut-Out Cabinets; Rosettes .1439 

Sockets, 1440; Hanger-Boards for Series Arc Lamps 1442 

Arc Lamps; Spark Arresters; Insulating Joints; Rheostats 1442 

Reactive Coils and Condensers; Transformers; Lightning Arres. 1443 ,1444 

Class E. — Miscellaneous — Signaling Systems 1444 

Electric Gas Lighting; Moving Picture Machines 1446 

'Insulation Resistance; Soldering Fluid - 1447 

Class F. — Marine Work — Generators, Wires . 1447 

Portable Conductors; Bell or Other Wires; Table of Capacity of Wires 1448 

Switchboards; Resistance Boxes; Switches 1449 

Cut -Outs; Fixtures; Sockets; Wooden Moldings. . . . 1449 .1450 

Interior Conduits; Signal Lights; Motors; Insulation Resistance ,. 1450 

Electrical Standardization Rules of A. I. E. E — Definitions ,1451 

Definitions — Currents; Rotating Machines; Stationary Indue. Appara.1451 

General Classification of Apparatus; Motors — Speed Classification 1452 

Definition and Explanation of Electrical Terms — 1453 

Load Factor; Non-inductive Load and Inductive Load 1453 

Power-Factor and Reactive-Factor, Equations 1453 

Saturation-Factor; Variation and Pulsation 1453 

Performance Specifications and Tests — 1454 

Rating; Wave Shape; Efficiency 1454 ,1455 

Regulation — Definitions, and Conditions for Tests 1461 ,1462 

Insulation Resistance; Dielectric Strength 1462 ,1463 

Conductivity. — Rise of Temperature ; 1466 

Overload Capacities 1469 

Voltages and Frequencies 1470 

General Recommendations — Name Plate, Rheostat Data, etc 1470 

Electrical Notation; Railway Motors 1471 

Photometry and Lamps — Candle-Power, Candle-Lumen, etc 1473 

Sparking Distances in Air, Table , 1474 

Temperature Coefficients of Resistivity of Copper — Formulas, Table . . 1475 

Miscellaneous Data — Properties of Various Kinds of Wire 1476 

Reinforced Concrete Telegraph Poles, Described. . . , 1477 

Cost of Constructing Steam-Driven Electric Power Plants 1477 

Cost of Large Steam Plants, Table; also Smaller Plants 1478 

Rates for Electric Current Charged by Pasadena Plant, Table 1478 

Cost of Overhead Trolley Systems, Table 1479 

SEC. 71.— MISCELLANEOUS DATA AND ILLUSTRATIONS. 

References — Derricks and Cranes, Chimneys. « = 1480 

References — Mechanism and Gearing; Marine Engineering. 1480 

References — Cableways and Conveyors; Revetments. 1481 

References — Well Boring; Machines; Bunkers and Bins 1481 

References — Compressed Air; Heating and Ventilation 1482 

References — Telephones; Mining 1482 

New Helical Spring Formulas , 1482 

Tests of Steel Springs, Table 1484 

References — Solar Power; Valuations and Reports; Contracts 1484 

Metal Hoisting Chains — Oval, Open-Link — Formulas 1484 

GLOSSARY ... 1485-1538 

INDEX 1539-1611 



INTRODUCTION. 

General Discussion. — ^The young engineer or student should realize that, 
although engineering is not an exact science itself, it is, in its entire field, 
founded on the exact sciences, supplemented with experimental data. As 
he becomes more proficient in his work he will discern certain broad, under- 
lying principles which he will begin to use instead of mere methods or rules, 
formerly employed. It should be his aim to master these principles thor- 
oughly. 

MATHEMATICS. 

The various branches of mathematics as taught in our common schools 
are based on methods of operation rather than on mathematical principles, 
for when viewed from the latter standpoint they overlap each other to 
such an extent that the dividing lines are not always clearly marked. For 
instance, the science of number, space, quantity, position, motion, mass, 
force and inertia cannot be taught in logical sequence from our text-books 
as now arranged. 

In view of the above, and of the fact that an attempt has been made 
in the body of this work to classify the subject matter of mathematics 
under the usual headings — Arithmetic, Algebra, Geometry, etc., — it is 
deemed pertinent here to introduce a discussion of the more abstract science 
of number, space, quantity, etc., as preliminary to the main text, recognizing 
here none of the divisions, as such, which are contained in the latter. 

Number.* 

The science of Arithmetic involves the principles of Algebra; these 
principles in turn having been deduced from the theory of number, and 
so on. 



Number is independent of the order of counting. ] ? 3 

— In counting any number of things, the counting of 
the last one contains the numeral word which desig- 
nates the total number, no matter in what order # • • 
they are counted. In the illustration, for exam- ^ 5 \ 
pie, there are five dots in each case. *" 



A Sum is independent of the order of adding. — 

In finding the sum of two or more groups of things, 
it matters not in what order they are added; as 
3 added to 5 equals 5 added to 3, equals 8; or, 3+ 5 = 

5+3=8. 3 plus 5 plus 4 = 5+3+4=4+5+3; 

etc., = 12. 

Using shorthand characters or letters to represent 
numbers the same law holds true: as, a + b = h + a\ 
a + 6 + c = 6 + a+c; etc. 



Fig. 1. 



Fig. 2. 



s 



4 



0( ^ 4* 
A Product is independent of the order of multi= 
plying. — 

4 3's = 4 times 3=4X3=12; or, aXb = db=\2. 

3 4's=3 times 4=3X4=12; or, 6Xa = 6a= 12. ro 

Similarly, 4X 3X 5= (4X 3) X 5 = 4 X (3 X5) = 3X " 
(4X5) =60. And using shorthand, in which the let- ^ 
ter a = 4, 6=3, and (7=5: aXbXc= abc = acb = bac 
= bca = cab = cba = 60. 

Note the 6 (1X2X5) different ways of arranging ^. ^ 

the 3 letters. -rig- 3. 



I 


Z 


3 


4 


z 








3 









*See. also, Clifford's "Common Sense of the Exact Sciences." 

XXIX 



XXX 



INTRODUCTION. 



A Product is independent of the parting of 

(7X3) + (7X2) = 21+14=35, or, by 
shorthand, ah = ae+af=a{e-\-f). Again, 
7X5=(6+1) (3+2) = 6 (3+2) + lX 
(3+2) = 30+5=35 OT,ah={c + d) 
X(e+f) =c {e+f) +c? {e+i) =e{c+d) + 
f{c+d). 



A Power is independent of the 
parting of the index. — 

The first power of a number is the num- 



its 



factors. — Thus, 



7X5. 



ber itself; as a\= a. 



a2 = 



The square of 5 = 5^ = 5X5; of a = 
aa\ in which 2 is the index. 

The c^ihe of 7 = 73= 7i X 72= 7^+2= 7 X 72 
= 71+1+1=7X7X7; of 6=63=6^2 
=bhh. 



1 


2 


3 


4 


5 


6 


7 


z 














3 














4 














5 















= 6 



d-l 



Fig, 4. 



The fourth power of = a4 = a3+i=a2+2=a3a=aa3 = a2a2 = a^ = Va8r 
The/^/^/j^ow;^r of 2 = 25= 2X2X2X2X2= 2(2X2X2X2) = (2X2) (2X2X2)' 

or, 25= 21+4= 22+3= 2X24= 22X 23. 

The square root of the fifth power of 4 = Vis = x/ii+i = VI^X 4 = Vl^X 

Vr= 42vT= 16VT= 16X2 = 32; or VJs = 4 ^ = 4^^ = 42 X4*=i 

4Xjii = 4X4X4^ = 32; or, \/45 = V 42X43 = \/42X42X4 = V^X 

V42XVT= 4X4X2 = 32. 



From the above it will be seen that aO= 1, for a =0^ 



Square of (a + 1) is a special case of (a + 6)2. 
(4+ 1)2= (4+1) (4+1) = 4(4+ 1) + 1 (4+1) = 16 + 

4+4 + 1 = 25. 
(a+ 1)2= (a+ 1) (a+ 1) =a (a+ 1) + 1 (a+ 1) =a2 + 

2a +1. 
(a + 6)2= (a + 6) (a + 6) =a (a + 6) +6 (a + 6) =a2 + 

2a6 + 62. 
Again: (a-l)2 = a2-2a+l; and (a-6)2 = a2-2a& 

+ 62. 
Practical application: (21)2= (20+ 1)2 = (20)2+2 

X 20+ 1 = 400+40+1 = 441: 
also, (19)2= (20- 1)2= (20)2- (2X 20) + 1 = 400- 

40+1 = 361. 






o=4 b«r 

^ 

a 





Fig. 5. 
in the case of 



Note. — By making h extremely small compared with a 
(a + 6)2, it will be seen that the third term h^, as b approach zero, can be 
omitted, as it will be the square of an infinitely small decimal or fraction. 
Hence, the limit of {a + hy as h approaches zero, is, a2 + 2a6, or, in other 
words, the actual increase of (a + 6)2 over a2, where 6 is infinitely small, is 
(a2 + 2a6)-a2=2a6. ' 

This is an elementary principle of the Differential Calculus — ^the method 
of limits. 



(a2-l) = (a + l) (a-1) 
(a2-62) = (a + 6) (a-6). 
(42-1) = (4+1) (4-1) = 4 (4 

4X3+1X3=12+3=15. 
a2_ 1 = (a+ 1) (a- 1) = a (a- 1) + 1 

a2-a + a-l = a2-l. 
a2-62= (a + 6) (a-6) = a (a~b) + 6 (a 



a6 + 6a-62 = a2-62, 



Practical application* 41 X 39 = 
= 402-1 = 1600-1 = 1599. 

From a2= (a+ 1) (a— 1) + 1, we have 
38+1 = 1520+1 = 1521. 



IS a special case of 

1) + 1 (4-1) = 

(a-1) = 

6) = 

(40+1) (40-1) 

392 = 40 X 



a-^i 



M 



a«4 
Fig. 6. 



INTRODUCTION, 



XXXI 



-I'he surface 




Space. 

An Object is composed of layers and bounded by a surface 

of an object has a definite area, but as it has no thickness, 
in itself, it occupies no space. On the other hand, the layers 
composing the object, no matter how thin we may assume 
them theoretically, must occupy space and have thickness. 
In theory we assume such shapes as cones, cylinders, spheres, 
etc., to be composed of an infinite number of layers of 
an infinitesimal thickness, t. 

A Surface is an area with or without lineal boundary. — 

The surface of the Earth, of a chair, of a cube, or of any 
whole object can have no lineal boundary; but the surface 
of part of an object, as of a continent, of one face of a cube, ^. 
or of the top of a table is bounded by lines along the edges ^'^^- '• 
of the surface. Neither the surface nor its bounding lines can have thick- 
ness nor occupy space. 

The surface of an object is its shape, 

A Line is a boundary, division, or an intersection, of surfaces. — ^The 
boundary of a surface is a curved or broken line of definite length, and con- 
tinuous', that is, without terminal points. A good illustration of this is a 
traverse survey, as around a farm. If the field notes do not "close" they 
are made to close by "adjustment." 

A plane surface can be divided by a straight line, of definite length only, 
that is, with terminal points at its boundaries; but it can be divided also 
by a curved or broken continuous line, as for instance by a circle, entirely 
within its boundaries. 





Fig. 8. 



Fig. 9. 





Fig. 10. Fig. 11. 

The intersections of the surfaces of the cube furnish the right angle and 
the square', the intersections of the surfaces of the wedge furnish the triangle 
and the rectangle; and, similarly, the pyramid, in its various forms, furnishes 
the numerous polygons. 

By cutting surfaces, the cone furnishes the point, p, the straight line, p s, 
the triangle, p s t, the circle, C, the ellipse, E, the parabola, P, the hyperbola, 
H, etc. Its properties embrace such a wide range in Analytic Geometry 
that the subject is often termed Conic Sections. 

The boundary of a surface is its shape. 



Quantity. 

Quantity is a summation of units, of one 

_ The first involves continuity, length, or 
weights, lengths, money and numerals. 

The second involves length and breadth, or area; as surface measures. 

The third involves length, breadth and depth, or contents; as measures 
of volume. 



two or three dimensions. — 

number; as time, angles, 



XXXII 



INTRODUCTION. 



Area is a summation of one or more products of two factors each.— 

If we divide the surface of the right a <? 

triangle, Fig. 12, into extremely thin 
strips of length y, and thickness t, 
and then move the apex to one side, 
as in Fig. 13, allowing the thin strips 
to slide on one another, their ends will 
still form straight lines from the apex, 
a, to the base of the triangle, and its 
height and area will remain the same. 
The area of the triangle is the sum- 
mation of the areas {yXt) of all the 
thin strips. The above illustrates two 
principles, namely, simple proportion 
line, ae\ thus — 

y b 
Simple proportion: — = -T' 




— b 

Fig. 12. Fig. 13. 

and the equation of the straight 



Equation of the straight line, ae: :V = -r-^.* 

The straight line is sometimes called a curve of the first degree. 

The same law holds true in the case of any 
figure, as Fig. 14 where the boundary line o^ is a 
curve of the second degree; for it is necessary only 
to find the equation of the curve in terms of x and 
y so that for each value of x we may know the 
value of y. In either case we derive the area from 
the summation of all the infinitesimal strips yt be- 
tween the limits x = h and x = 0. This maybe per- 
formed accurately by the method of the Calculus, 
by^ assuming an infinite number of infinitesimal 
strips. 

If we consider Fig. 12 to be an elastic layer of ^.^ - . 

definite thickness, having the surface as shown, and ^^^- \^' 

apply a lateral pressure at the apex, a, to change its shape to Fig. 13, we 
conceive in Mechanics the terms force, elasticity, stress, strain and shear. 




Length is a summation of the square roots of the sums of squares. 

It is an elementary problem to prove that the square 
H (hypothenuse^) is equal to the square B (base^) plus 
the square P (perpendicular^), by the proposition that H 
is composed of two rectangles B' and P' equal in area to 
B and P, respectively. From this principle the length 
of any curve may be obtained by the summation of its 
infinitesimal parts. 

Thus, required (Fig. 16) to find the length S of the 
curve ac between the limits x = c' and x = a\ We have 
here to find the summation of the infinitesimal tangent 
lengths ds in terms of dy and dx for every change of 
position of the tangent as the values of x and y 
change . It is e vident that each infinitesimal length 

ds = \/dx^-hdy^, hence if we find the values of dx^ 
and c/>'2 from the equation of the curve, by the Calcu- 
lus, we can get the value of ds in general terms. It 
only remains to find the summation of all the values 
of ds, that is, S, and this may be obtained by inte- 
gration, subtracting the result obtained by mak- 
ing x = a\ from the result obtained by making 
x = c'. 

The ratio of the circumference of a circle to its 
diameter =3.141592 + , called tc. 




Volume is a summation of layers ; or area times thickness. — If a beam 
of homogeneous material and of rectangular cross-section be loaded so 



The general equation of the straight line is y = mx-hc, in which mis the 
tangent of the angle, and c is a constant. In the above, c=0. 



INTRODUCTION. 



XXXIII 



that it deflects, as in Fig. 17, there will be a plane 5 — 5 through the middle 
of the beam which will not change from its original length ; that is, there 
will be no stretching or shortening of the fibers along this plane, which 
may be called the neutral axis of the beam. If now the beam be turned 





Fig. 17. 
on its side and again deflected, a new plane, s' — s'^ at right angle to the 
first, will become a new neutral axis. In either case the fibres at the top 
of the beam will be compressed and the fibers at the bottom of the beam 
stretched, equally. Also, the stretching or compression of any fibre in 
the beam will be proportional to its distance from the neutral axis. In 
other words, while the beam loses in volume above the neutral axis it 
gains an equal amount below. The intersection of the neutral axes will 
also be found to be at the center of gravity, c. g., of the section. 
The center of gravity may be defined as the point of inter- 
section of all possible neutral axes, for no matter which way the 
beam may rest, on its corner or otherwise, the neutral axis 
will pass through the center of gravity of the section. 

The center of gravity of any regular section, as a square, Fig. 18. 
rectangle, parallelogram, equilateral triangle, circle orreg- -j^- 
ular polygon is in the center of the figure. (Fig. 18.) 

The center of gravity of a triangular section is one-third 
the height from either base. (Fig. 19.) 

These principles furnish the rules for finding the 
contents of the cylinder, circular ring, cone, paraboloid, 
ellipsoid, or any similarly curved bodies.* 

Consider any section lying wholly on one side of the axis 
of revolution, or center of curvature; multiply the area of 
the section by the distance traversed by its center of grav- 
ity. The product will be the volume of revolution of that 
section. 

For example, the volume of a cone is equal to the area 
of the triangle of revolution X| the base X 2k 



= — X "o X 2?: = -r- = Y X ^rea- of base. 

Position. 



(Fig. 20.) 




The Relative Position of an object may be determined by one, two or 
three quantities, according as it is on a known line, surface or in space. The 
third dimension problem can be reduced to the second by passing a known 
surface or plane through the point in space; and this can be reduced to the 
first by passing a known line through the point on the surface or plane. 
A distance along the line, measured to the object in question, will fix its 
exact position. 

The Position of a Point on a Plane. — 

The polar co-ordinates, angle oc and distance D, locate the point P 
from A. Polar co-ordinates may be reduced to rectangular co-ordinates, x 
and y. Fig. 21. 




Fig. 21. 




Fig. 22. 



See also Pappus's Theorems for surfaces and solids, page 243. 



XXXIV 



INTRODUCTION. 



A point P, Fig. 22, may be located from a fixed point A by a distance 
D, and from a fixed line Y—Yhyo. distance x. If D and x are allowed 
to increase and decrease with constant ratio, the point P will trace (1) 
the parabola, if the ratio oi D to x is unity; (2) the hyperbola, if the 
ratio oi D to X is greater than unity; and (3) the ellipse, if the ratio of D 
to X is less than unity. 




zE 



/ 



Fig. 23. 



Fig. 24. 



A point P, Fig. 23, may be located from two fixed points: A, by 
distance D, and B, by distance d. If the ratio oi D to d is allowed to in- 
crease and decrease, but so their sum will remain constant, the point P 
will trace an ellipse. 

The results of experiments are conveniently platted on cross-section 
paper, and a curve drawn as nearly regular as possible through the aver- 
age position of the points. The curve then serves as a formula for future 
use. (Fig. 25.) 




If the curve is of the second degree it can be represented by a straight 
line if logarithmic cross-section paper, Fig. 25. is used. Hydraulic formu- 
las and experiments are often platted on this kind of paper, for sim- 
plicity. 



INTRODUCTION. XXXV 

Motion. 

A man starts to row a boat, at a constant rate of speed, directly across 
the stream from A to B^ not allowing for any <^ 

ctirrent. On reaching the east bank he finds 5. s 

he has landed at D. If the current was uniform .§ v. 

in all parts of the stream his course was the tj ^^ 

straight line AoD, and his motion down stream .^ v 



Q 






B 



uiiiform in passing from the west to the east 
bank. If the current was very strong at the 
west bank and gradually lessened toward the A 
east bank his course was the line AcD and his 
motion down stream a retarded one — uniform- 
\y retarded if the current at any point was 
directly proportional to its distance from the ^ 
east bank. If the current gradually increased g 
toward the east bank his course was the line t§ 
AhD and his motion down stream an accelerated +, 
one — uniformly accelerated if the current at S 
any point was directional proportional to its s 
distance from the west bank. 

The average velocity of the (surface of the) 
stream is the same for all three cases above -pis 26 

cited if the boat lands at the same point, D. 

Uniform velocity or motion may be represented by a straight line, i. e., 
curve of the first degree; uniformly accelerated or retarded velocity or 
motion, by a curve of the second degree. 

The acceleration per second of time due to the gravity of the earth on 
any body falling in vacuo is practically constant. It is designated by the 
letter g^ and its value is about 32.16 ft. per sec. It varies somewhat 
with the elevation above sea level and with the lattitude of the place. 

Mass (Matter), Force and Inertia. 

These are convenient terms used in Applied Mechanics to express certain 
co-relative ideas. 

We may consider mass as matter, or that property which cannot be 
destroyed, because matter is indestructible, although the body of which 
it is composed may change its form or apparently disappear. Mass is 
proportional to weight, and the unit of weight is one pound. The mass 

W 
of a body is its weight, W, divided by gravity acceleration, g, as M= — ; 

g 
or, the mass of a body is measured by a force, F, which will produce an 

p 
acceleration, a, as M= — . The acceleration in either case is the velocity 
a 

attained at the end of the first second of time. 

Force is a shorthand expression of stating the value of one mass acting 
on another, by pressure or impact. Force = mass X acceleration, F = Ma. 
Its unit is one pound. 

Inertia is a negative term implying inherent inactivity in a relative 
manner: the tendency which a body has to continue doing what it is 
already doing — if at rest to remain at rest, and if in motion to continue 
in motion in the same direction and at the same velocity. 



EXPERIMENTATION. 

Several years ago the late Professor Joseph L. LeConte, in a very elabor- 
ate scientific discussion, "proved conclusively" that the heavier-than-air 
fiying-machine was an absolute impossibility; and yet, today, aerial flights 
in such machines are attracting but passing interest. 

Our college professors are daily teaching the maxim: "There is no con- 
flict between theory and practice." This should be restated as follows: 
"There can be no conflict between correct theory and perfect practice, but 
neither can always be attained." The theory of Professor LeConte was in- 
correct — but nevertheless a theory — and the first experiments in flying, by 
Octave Chanute, were simply imperfect ones. 



XXXVl INTRODUCTION. 

Joseph H. Choate, the able jurist, in an address before a society of engi- 
neers, a few years ago, exposed himself to criticism in stating that engineer- 
ing was an exact science — a statement that is vsry far from the truth. 

The great advance in the whole field of engineering — quite phenomenal 
during the past decade, especially — is due almost wholly to experimental 
work, scientifically conducted. 

INVESTIGATIONS AND REPORTS. 

Before making an investigation of any engineering proposition the engineer 
should draw up memoranda of the objects to be attained, and keep them clearly 
In mind throughout the investigation; otherwise he may have to duplicate 
certain work in acqmring " additional " information, at considerable expense. 

The most valuable reports on engineering projects are those which outline 
the " method " proposed to be employed in performing the work, in connection 
with statement of estimated cost. Estimated costs of work are frequently ex- 
ceeded in actual construction, and especially so when the work is in charge of 
inferior men. 

STANDARD SPECIFICATIONS. 

For all important engineering work, the materials and workmanship should 
be subject to approved specifications. 

The matter of drawing up specifications which will at once insure the desired 
results at reasonable cost (not excessive) is a vital problem in the execution of 
any design. 

Standard specifications are being adopted by the various engineering so- 
cieties and associations, as one of their natural functions, and are rapidly sup- 
planting former specifications by individual engineers. It will be the aim of the 
author to present the most essential elements of these specifications in the body 
of this book. 

MANAGEMENT. 

Scientific management is becoming a potent factor in the industrial 
world, and is destined to possess an increasing sphere of influence in the 
future. It simply means intelligent cooperation: distinctly opposed to the 
old system of bossism, which is more or less militant and invites toadyism— 
a qualification distasteful to every honest workman. 



SIGNS AND ABBREVIATIONS, 



Common Abbreviations. — The following abbreviations, and others not 
listed here, are used constantly thoughout this volume in order to conserve 
space. In general, they appear in the text after the full word has been used 
so there can be no doubt as to the meaning: 

= co-versed sine. 

= (hyp-perp)H-hyp. 

= exsecant. 

= (hyp — base) ^base. 

= co-exsecant. 

= (hyp-perp)-T-perp 

= seconds. 

= minutes. 

= hours. 

= days. 

= months. 

= years. 

= for instance. 

= that is. 

= and so forth. 

= et. cetera. 

= center of gravity. 

= inches. 

= feet. 

= yards. 

= pounds. 

= pints. 

= quarts. 

= gallons. 

= bushels. 

= barrels. 

= lineal. 

= square. 

= cubic. 

= board measure. 

= thousand ft. B. M. 

= horse-power. 

= American wire gauge. 

= Birmingham W. G. 

= revolutions per sec. 

= revolutions per min. 



abut 


= abutment. 


covers 


accel 


= acceleration. 




ans 


= answer. 


exsec 


approx 


= approximate. 




cen 


= center. 


coexsec 


circum 


= circumference. 




diag 


= diagonal. 


sec, s. 


diam 


= diameter. 


min, m. 


dist 


= distance. 


hrs, h. 


grav 


= gravity. 


d. 


hd 


= head. 


mos. 


hor 


= horizontal. 


yrs. 


ht 


= height. 


e.g. 


hyp, hypoth 


= hypothenuse. 


i. e. 


hyp log 


= hyperbolic log. 


etc. 


log 


= logarithm. 




perp 


= perpendicular. 


e.g. 


pres 


= pressure. 


ms. 


pt 


= point. 


ft. 


rad 


= radius. 


yds. 
lbs. 


tang 


= tangent. 


vel, veloc 


= velocity. 


pts. 


vert 


= vertical. 


qts. 


vol 


= volume. 


gals. 


wrt 


= wr9ught. 


bu. 


wt 


= weight. 


bbls. 


sin 


= sine = perp h- hyp. 


lin. 


cos 


= cosine = base -^ hyp. 


sq. 


tan 


= tangent = perp -r- base. 


cu. 


cot, CO tan 


= cotangent. 


B. M. 




= base -H perp. 


M., B. M 


sec 


= secant. 


H. P. 




= hyp -J- base. 


A. W. G. 


CSC.COSCC 


= cosecant. 


B. W. G. 




= hyp-^perp. 


r. p. s. 


vers 


= versed sme. 

= (hyp — base) -5- hyp. 


r. p. m. 



XXXVII 



XXXVIII 



SIGNS AND ABBREVIATIONS. 



Greek Alphabet. — Greek letters are used in the sciences to designate 
certain properties, as angles, latitude, temperature, etc. 



English 


Greek Letters. 




Equiva- 








Used to designate. 








lents. 


Name. 


Capital 


Small 




A 


Alpha 


A 


a 


Angles and coefficients. 


B 


Beta 


B 


/? 


Angles and coefficients. 


G 


Gamma 


r 


r 


Angles, coefficients, and specific gravity. 


D 


Delta 


J 


8 


Angles, coefficients, deflection, variation and 










density. 


E 


Epsilon 


E 


e 


Eccentricity; base of natural logarithms'- 
2.7182818; strain. 


Z 


Zeta 


Z 


C 


Coefficients, co-ordinates. 


E 


Eta 


H 


7) 


Coefficients. 


Th 


Theta 


e 


e,d 


Angles and coefficients. 


I 


Iota 


I 


c 




K 


Kappa 


K 


K 


Coefficients. 


L 


Lambda 


A 


X 


Latitude, angles and coefficients. 


M 


Mu 


M 


fl 


Angles and co-efficients. 


N 


Nu 


N 


V 




X 


Xi 


S 


e 


Coefficients, co-ordinates. 


o 


O micron 










p 


Pi 


n 


TZ 


77 signifies continued product, as 77 3 — 

1X2X3. 
7r=3.14159-H= ratio of circum to dia. =« 

180° of arc. 


R 


Rho 


p 


P 


Rad of gyration; rad of curvature; ratio. 


S 


Sigma 


I 


a, S 


I signifies summation; as IW means the 
summation of all the weights W\ W", 
W"\ etc., in any system. In the Calculus 

it is replaced by the symbol 1 . 

a is used as a coefficient; stress. 


T 


Tau 


T 


T 


Coefficients; temperature, time. 


U. Y 


Upsilon 


r 


w 




Ph 


Phi 


(p 


4> 


Angles and coefficients. 


Ch 


Chi 


X 


X 




Ps 


Psi 


¥ 


^ 




O 


Omega 


Q 


0) 


Coefficients; angular velocities. 



MATHEMATICAL SYMBOLS. 

a.bfC, Represent known or constant quantities. 

- - - x,y,z, Represent unknown or variable quantities. 
»= Equals, is equal to. 

-H Plus, as 3+2 = 5; positive, as +i=+.5; extension, as .^ = 
.14285 + . 

— Minus, as 5—3=2; negative, as — i= — .5; contraction, as i = 
.17-. 

± Plus or minus, as \/4 = ± 2. 

T Minus or plus. 

X Times, multiplied by; as 3X2=6; aXh = a.b=^ab. 

a.b =aXb = ab. 

-«- Divided by; as 8-^2=4; 8::2 = 4; 8/2 = 4; | =4. 

r- Divided by. 
o 

alb Divided by. 



COMMON. GREEK. MATHEMATICAL. XXXIX 



4 

:: : Proportion; as 2:3::4:6; means §= ~; reads "as 2 is to 3 so is 
4 to 6." 6 

4.3 =4^ = 4iU, etc. 

> Is greater than, as 4>3; reads " 4 is greater than 3." 

< Is less than, as 3<4; reads " 3 is less than 4." 

V Is equal to or greater than; as, a 7 4. 

^^ Is equal to or less than; as, 4^a. 

.'. Therefore, hence. 

'.* Because. 

CO Infinity; as, - = oo. 

oc Is proportional to, varies directly with; also, angle alpha. 

I Bar 
— Vinculum 
( ) Parenthesis 
Bracket 



Brace 



a + hXc = c {a-'(-b)=ac-\-ah. 
Abbreviation ; 2[a{h -\- c) + x\= 2a{h + c) -{• 2x, 

denote that the ^ ( or^ ck j_ ^^ _i. ^i _i_ ^ I 

quantities co v- « ( 2[a (6 + c) + :^] + cf } 

ered or enclosed = 2a [a {b -\- c) + x\-{- ad 

must be "taken =2a'^{h-¥c) + 2ax-\-ad 

together. " =2a'^b + 2a^c + 2ax + ad. 



— 2 — 1 

\/ Radical sign, square root. Thus \/a = \/a = a\= "• 

a ""a 

V Cube root. 

n 

V nth root. 

a^ The square or second power of a, as aXa. or aa. 

x^ The nth power oi x. x^= "nii*. x^ = x; x^=l. 

a d Continuation. Thus, a,b, c, d. (Either dots. or dashes may be 

used.) 

2', h", x'" Primes, to distinguish letters ; as a — prime, h—2 — prime or ^ — 
second, x—S — prime or ic — third. 
Xi, a2, Ua Subs, to distinguish letters; x sub 1. a sub 2, h sub 4. 

^ Inverted caret indicates repeating decimal; as, 0.016''6=« 

0.0166666 . 

#— Number, as '#2 = number 2. 

— # Pound, or pounds, as 4#= 4 pounds or 4 lbs. in weight. 
6'— 3" Feet and inches (linear measure) ; as 6 feet 3 inches, or 6 ft. 3 ins. 
')°-18'-15" Degrees, minutes and seconds (of arc) ; as deg., 18 min.. 15 sec. 
or 18m. 15s. 

O Round, diameter. 6'° =6 ft. diameter = 6' dia. 3'° == 3 ins. 
diameter = 3" dia. 

D Square. 2'° = 2 ft. sq.; 3"° = 3 ins. sq. 4°' = 4 sq. ft.; 9°''=- 
9 sq. ins. 

m Cube. 2'Ei=2 ft. cubed; 4"ki=4 ins. cubed. 27'a' = 27 cubioft. 

4- Z. Angle; A angles. 

L Right angle. 

± Perpendicular to. 

II Parallel with. 

(k) Circle. 

^ Triangle. 

fc^ Right angle triangle, 

D Square. 

I I Rectangle. 

nU Parallelogram. 



XL 



SIGNS AND ABBREVIATIONS. 

circumference 



^ 



Pi=3.1416- = 

diameter 
1 80° 
— — = 57.2958° nearly. 



of circle; or, =180° of arc. 



e = Base o£ Naperian, hyperbolic, or natural logarithms' 
2.7182818 + . 
sin a Sine of the angle a. 



sm" 



sin a~i 



Inverse on anti-sine of a; the angle whose sine is a. 

= — L- 

sin a 
= (sin a) -1 

bill u 

d (%2) = 2xdx, 



It is not 



see below. 

1 
sin a* 
d Differential (in Calculus) 



I Sigma, summation. 

J Delta, difference; usually considered as a very small quantity. 



J k 



Integral (in Calculus) ; the reverse of differentiation 
Intergral between limits h and k; 



ion; l2xdx = i 



I 



x=^h 

2xdx = h^-kK 

x = k 



MECHANICAL SYMBOLS. 



L or 1 = length. 
M or m = mass. 
T or t =time. 
V = velocity. 



= acceleration, a = 



M 



Wor w = 

P 

F 

ft. 

I 

E 

D 



-lbs. 



= gravity acceleration, 

W 
= -T-v = 32.16 ft. per sec. 

■''^ per sec. 

=work, or weight. 

= power. 

= force. 

= foot-pounds. 

== moment of inertia. 

= modulus of elasticity. 

= diameter. 



r 

H. P. 
B.H.P. 
L H. P. 
M.E.P, 
r. p m. 
C. G. S. 

A.W.G. 
B.W.G. 
B.T. U, 

cal. 

Ib.-cal. 



= radius. 

= horse-power. 

= brake horse-power. 

= indicated horse-power. 

= mean effective pressure. 

= revolutions per minute. 

= centimeter-gram-seconds 

(system). 
= American wire gauge. 
== Birmingham wire gauge, 
= British thermal unit 
= pound-degree-Fahr. 
= calorie (French), 
= kilogram-degree-Cent. 
= po.und-calorie, 
= pound-degree-Cent . 



ELECTRICAL SYMBOLS. 



E..E. 
P. D. 
C 
R 



M.F. = electro-motive force. 

= potential difference. 

= current. 

= resistance. 

= specific resistance. 

=- quantity. 

= capacity. 
L = inductance. 

A. M. = ampere meter. 

V. M. =volt meter. 

F. M. = field magnet. 

+ = positive pole. 

— = negative pole. 




■o_ 



= volt, potential. 
= ampere. 
= megohm. 

= British Association tinits. 
= microfared. 
= Henry. 
= Joule. 
= kilowatt. 
= complete period 
(alt. current). 
= dynamo. 

= battery. 



\J^~ V ■ > J ■ ■ Vi-V' " 



Galvanometer. Ammeter. 




Voltmeter. 



Wattmeter. 



MATHEMATICAL. MECHANICAL. ELECTRICAL. XLI 



< 9 d 






2.2 



Woo 
O (J 

O KXN 



go 

1-1 CM 
,-40 

.tJO 
w C 



I 

o 

a 
<o 

xn 
u 

O 



•4c 










to 




ro 





•-4 


00 


^ » 


00 


?^ 














00 






T-4 






1-4 rH 






















w . S O S O 0) 






<u "^ S o G B^ 



,i3 
o 






\2 



a.2^ 5 fc^ 0.4.-3 






o . 




• 0^ *. I 

op:: 





^ 


• ^ 





\^ 


^ 



P^ o W OM 



I ^ 






O c^ 








to 










1 




G 


1 


e< 


^9 1 




.0 

a 

a 
S 


•^ ^-^ ^ H^ ^ 






jr 


II II 


II il 


II 11 










3 


St" 







4. -r-> 


xl 

a -I- 


3 a 

■1- >^ 


Is 
a.?. 




"S 






y^ — »/ — 






II 


II II 


II II 






bo 












rt 










t/3 


6 




< — ^ 






H 





/^^ 


y 








•4-> 


5 
§■ 


1 

(D 





.2 ^ 





S 


£ 


^> 








1^ 


•+J 





§1 


a 




4-i 


'ip 


-M y 


»H G rt 


< 




^ 




^itj-^ 


1 









(£w)^ 


H 








o>w 


P^ M ^H^l^ 


U 












J 


1 


•H 


7 T ' 


- 




u 


1 


1 ' 


H 




a 


(U 


nt) 


HN H« ne« H^ 




z 


I-? 




h:j 




< 


6 

5 


hM 


Hlr» 







^ 




^ 








II 


II II 


II II 




z 






T^ <U 






g 






'O -Q 






< 








ft ft 


"cS 




^ 






CJ 


^ 


'^ 










r^ ^ 


d^ 










■M 


f „ -^ 


^a 










00 


g g 






i2 






2i Si 


Xbb 










X 


1^ 1^ 
•!• -I- 








y 




V-i 


M <u 













T3 


<D X 






+3 

0) 


> 


ii 










II 


Jl II 


II II 






-3^ 


"^3 

•43 'd 






as 


^ ^ 


X 








M-t <4-< 


M-l 


:i 











ft 


+J TO 










a.2 ^ 








W)4J- 


t (U OT 


y 4-3 








Pi c 




2 








£§ 


s^ ^ 








mO 


_^ ^ 


:^p^ 










g 


> ffi ^SJ 





XLII 



SIGNS AND ABBREVIATIONS. 



MAGNETIC SYMBOLS. 



N = north pole. 

S = south pole. 

nt = strength of pole. 

H = magnetic force (C. G. S.). 

B = magnetic inductance 

, (C.G.S.). 
I = intensity of magnetization. 



fi = magnetic permeability. 
k == magnetic susceptibility. 
H = horizontal intensity of earth's 

magnetism. 
Z = reluctance. 



MEDICAL SIGNS AND ABBREVIATIONS. 

R (Lat. Recipe), take; aa, of each; lb, pound; 5, oiince; 5. drachm: 
9, scruple; HI, minim, or drop; O or o, pint; f 5, fluid ounce; f 3, fluid 
drachm; as, 5 ss, half an ounce; 5i. one ounce; 5iss, one ounce and a half; 
5ij,two ounces; gr., grain; Q. S., as much as sufficient; Ft. Mist., let a 
mixture be made; Ft. Haust., let a draught be made; Ad., add to; Ad. 
lib., at pleasure; Aq., water; M., mix; Mac, macerate; Pulv., powder; Pil., 
pill; Solv., dissolve; St., let it stand; Sum., to be taken; D., dose; Dil., 
dilute; Filt., filter; Lot., a wash; Garg., a gargle; Hor., Decub., at bed 
time: Inject., injection; Gtt., drops; ss, one-half ; Ess., essence. 



SURVEYING SYMBOLS. 

(U. S. Public Lands Surveys.) 

The following contractions are authorized to be used in the preparation 
of field notes, transcripts, inspection reports and similar records, and no 
others should be introduced: 



A. 


for acres. 


mag. for magnetic 


a. m. 


" forenoon. 


M.C. 


" meander corder. 


A. M. C. 


" aux. meander corner. 


men 


" meridian. 


asc. 


•• ascend. 


mkd. 


'• marked. 


astron. 


" astronomical. 


N. 


'* north. 


bdy. 


" boundary. 


NE. 


" northeast. 


bdrs. 


" boundaries. 


NW. 


" northwest. 


bet. 


" between. 


obs. 


'* observe. 


B. O. 


" bearing object. 


obsn. 


'' observation. 


B.T. 


'* bearing tree. 


p. m. 


" afternoon. 


C.C. 


'' closing corner. 


Pol. 


" Polaris. 


chs. 


" chains. 


Pr. Mer. 


principal meridian. 


cor., cors 


. [] comer, comers. 


Pt. of Tr. 


^1 point of triangulation. 


corr. 


correction. 


i sec. 


quarter section. 


decl. 


" declination. 


R., Rs. 


" range, ranges. 


dep. 


" departure. 


red. 


'* reduce, reduction. 


desc. 


" descend. 


S. 


" south. 


dia. 


" diameter. 


B.C. 


" standard comer. 


diff. 


" difference. 


SE. 


" southeast. 


dist. 


" distance 


sec, sees. 


" section, sections. 


D. S. 


*' deputy surveyor. 


S. M. C. 


" special meander comer. 


E. 


" east. 


sq. 


" square. 


elong. 


" elongation. 


St. Par. 


" standard parallel. 


frac. 


" fractional. 


SW. 


southwest. 


ft. 


" foot, feet. ^ 


T.,orTp. 


" township. 


G.M. 


" guide meridian. 


Ts. orTps. 


townships. 


h., hrs. 


" hour, hours. 


temp. 


" temporary. 


ins. 


" inches. 


U.C. 


" upper culmination. 


lat. 


" latitude. 


var. 


" variation. 


L.C. 


" lower culmination. 


W. 


" west. 


Iks. 


" links. 


W. C. 


" witness comer. 


1. m. t. 


" local mean time. 


w. corr. 


" watch correction. 


long. 


" longitude. 


W. P. 


*' witness point. 


m. 


•• minutes. 


w. t. 


'* watch t:me. 



For ordinary surveying abbrevations see general text on Surveying, 
Section 68. 



1.— ELEMENTARY ARITHMETIC. 

NUMBERS. 
Roman System. — This system is used in the arts, for dates, for ntimber- 
ing chapters in Hterature, and for distinctive numbering where the Arabic 
numerals will not suffice. It employs seven letters, corresponding with the 
Arabic numbers, as follows: 



Roman. . . 


. I 


V 


X 


L 


C 


D 


M 


Arabic . . . 


. 1 


5 


10 


50 


100 


500 


1000 



Higher basic denominations are sometimes represented by the letters 
C, D and M, inverted, but they are not in general use. The following table 
will be found useful in expressing any number by a combination of these 
letters. 

1. — Roman Numerals. 



Roman 


Arabic 










Units. 


Thousands. 


Hundreds. 


Tens. 


Units. 




M 


C 


X 


I 


1 




cc 


XX 


II 


2 




ccc 


XXX 


III 


3 




CD 


XL 


IV 


4 




D 


L 


V 


5 




DC 


LX 


VI 


6 




DCC 


LXX 


VII 


7 




DCCC 


LXXX 


VIII 


8 




DCCCC 


XC 


IX 


9 



To write any number in Roman system: Begin with the highest de- 
nomination of the number — tens, hundreds, thousands, etc. — and pick ov^, 
the letter or letters in the Roman column, corresponding with that denomi- 
nation and opposite the proper figure in the Arabic column; then proceed 
in the same manner with each figure of lower denomination. Thus, 3 = 111; 
34 = XXX IV = XXXIV; 648 = DC XL VIII = DCXLVIII; 1799 = 
M DCC XC IX = MDCCXCIX; 1900 = MDCCCC (preferred) = MCM. 

Arabic System. 





:;3 






'a 


n 


i 


vr\ 


.y 


. 


Tl 




Q) 




*2 


<U 


•»H 


.2 




J^ 




U 


a 










G 
d 


(1) 


hi 


w 


H 


cq 


w 


H 






0.000.987 



to 

<u o ^ 

§ « 









W H P 



5 4,32 1 



• 0) - 
w u 



O 



1 2 3 



4 5 6 







This number is read thus: "Nine hundred eighty-seven million, six 
hundred fifty-four thousand , three hundred twenty-one . . . and one hundred 
twenty-three thousand, four hundred fifty-six, millionths." By moving the 
decimal point to the right or left, the number is respectively multiplied or 
divided by ten, times the number of places moved. 



2 1.— ELEMENTARY ARITHMETIC. 

Definitions and Laws. — Numbers may be either abstract (as 16) or concrete (as 

16 boxes). The four fundamental operations of arithmetic are addition, subtraction, 
multiplication, and division; all necessary in performing calculations. A propo- 
sition is a statement set forth either with or without demonstration. It may be 
(1) an axiom, or self-evident truth, without demonstration; (2) a theorem, or 
truth by demonstration; (3) a problem, or question for solution; (4) an hypoth- 
esis, or tentative or preliminary proposition. ^ A corollary is a deduction from 
one or more propositions. A proof of an operation is performed by a second oper- 
ation, and may be either absolute or merely probable. Axioms: (1) Two or more 
quantities each equal to the same quantity are equal to each other; (2) Restilts 
of similar operations performed on equal quantities are equal; (3) The value of a 
whole quantity is not affected by changing the order of its parts. 

Primes, Multiples and Fa 'tors. — A prime number differs from a mul> 
tiple m that it cannot be factored ; that is, it is not exactly divisible by any 
other number, except 1. There is no known positive rule for detecting all 
prime numbers; tentative rules have been framed from time to time only 
to fail high up in the scale. But negative rules, universal in their appli- 
cation, may be stated as follows: 

(a.) No even number (2 excepted) is a prime, because divisible by 2, 
(b). No number (3 excepted), the sum of whose digits is divisible by 3, is 

a prime, because itself divisible by 3. Example: 171 (1+7+1 = 9) 

is not a prime number, because 9 is divisible by 3. 
(c.) No number (5 excepted) » ending in 5 or 0, is a prime, because divisible 

by 5. Examples: 15, 30, 125, are divisible by 5. 
(d.) No number composed of prime factors can have more than one set of 

prime factors. Example: 1001 = 7X11X13; and 7, 11 and 13 is 

the only set of prime factors. 
From the above we rightly conclude that all numbers, excepting 2 and 5, 
which end in 2, 4, 5, 6, 8 and 0, are multiple or composite; that a multiple 
may end in any figure; and that the ending of prime numbers is limited 
strictly to the digits 1, 3, 7 and 9. Furthermore, we may examine any 
number ending in 1, 3, 7 or 9 to see if the sum of the digits is a multiple 
of 3; if so, the number is composite and divisible by 3; but if not, it may 
be either prime or composite. 

Table 2, following, contains a list of numbers up to 9600, which, by the 
preceding rules and analysis, cannot be detected as prime or composite. 
The numbers are composed of hundreds at the left of the lines, and tens 
and units at the top of the colum.ns. At the intersection of the respective 
line and column of a number will be found the smallest prime factor (above 
unity) of that number. If the number is prime, the intersection will be 
represented by two dots. 

The elimination, by Rule (6), of numbers ending in 1, 3, 7 and 9 which 
may be factored by 3, makes it convenient to separate the table into three 
parts by steps of 300, in order to condense it. The arrangement is as fol- 
lows: 

3n + 0. This part (1) contains hundreds beginning with 0, as 000, 300, 600, 

900, etc. Example: 2400= (3 X 8 + 0) hundreds. 
3n-\-l. This part (2). contains hundreds beginning with 100, as 100, 400, 

700, 1000, etc. Example: 5200= (3 X 17 + 1) hundreds. 
Sn + 2. This part (3) contains hundreds beginning with 200, as 200, 500, 
800, 1100, etc. Example: 3500 = (3 X 11 + ^) hundreds. 

Example. — What kind of a number is 27,489? 

Solution. — ^The sum of its digits is 30, hence 3 is a factor. 3 ) 27489 

From part 2 {91 =3n-\-l) of table, the smallest factor of 9163 is 7. 7) 9163 
From part 2 {13 = 3n-\-l) of table, the smallest factor of 1309 is 7. 7 ) 1309 
From part 2 {l=3n + l) of table, the smallest factor of 187 is 11. 11 ) 187 
From part 1 (0 = 3n-\-0) of table, 17 is found to be a prime. 17 

Answer. — 27489 is found to be a multiple or composite number whose 
prime factors are 3, 7, 7, 11 and 17; and from Rule (d) we learn that these 
are its only prime factors. 



NUMBERS. 

-Primes,* Multiples and Factors. 
Part 1. — (3n-\-0) Hundreds. 



8 a 

CO =3 

N 



000 
300 

600 
900 

1200 



1500 
1800 

2100 
2400 

2700 



3000 
3300 

3600 
3900 

4200 



4500 
4800 

5100 
5400 

5700 



6000 
6300 

6600 
6900 

7200 



7500 
7800 

8100 
8400 

8700 



9000 
9300 



Tens and Units. 



01 07 11 13 17 19 23 



17 



.. .. 13 
17 .. .. 

.. 17 7 



19 11 .. 

.. 13 .. 

11 7 .. 

7 29 .. 



11 7 .. 13 
.. .. 23 .. 



17 37 7 . 
7 23 17 . 



.. 29 13 11 
19 .. 41 .. 



37 .. .. 

.. 31 .. 

.. .. 7 

13 .. 23 

47 .. .. 



.. 7 



23 7 
.. 31 



7 .. 



.. 7 .. 11 .... 41 



7 .. 13 
.. 11 17 



.. .. 19 
11 .. 7 



. .. 61 7 

. 7 .. 47 
11 



.. 13 .. 29 



7 59 



17 .. .. 

.. 7 .. 

7 .. 11 

67 .. . 

19 .. .. 



7 11 13 19 
59 .. 71 .. 

17 13 .. 37 
31 .. 11 7 

.. 7 .. 31 



13 .. 7 
29 37 73 

.. 11 .. 
31 7 13 



11 .. 73 .. 

13 .. 7 .. 



7 .. 23 
47 19 .. 



7 .. 31 .. 23 .. 1] 



71 41 



.. 71 29 7 
67 7 .. .. 



29 31 37 41 43 47 49 



. 11 7 



7 .... 23 .. 
.. .. 17 11 29 



11 .. 29 23 .. 7 .. 
31 .. 11 7 19 .. 43 

19 7 

7 11 .. .. 7 .. 31 

.. .. 7 .. 13 41 .. 



13 7 .... 17 11 .. 
.. .. 47 13 .. .. 17 



19 .... 11 .. 7 41 
.. .. 31 7 .. .. 11 



19 .... 31 7 



7 23 13 19 7 .. .. 
11 .. 7 47 29 37 13 

23 7 11 53 37 . . 19 
61 13 .. 

17 11 7 .. 



. 37 
. 13 



7 .... 23 
17 .. 11 7 



7 19 . . 29 7 17 61 
13 29 7 11 53 .. .. 

.. 7 .. 13 .... 11 



.. 17 .. .. 19 .. .. 

.. 41 17 .. 11 7 47 

11 47 79 7 17 .. 29 

.. .. 11 23 .. .. 7 



7 .. 13 



..11 7 .... 83 
19 7 13 



53 59 61 67 71 73 



19 



.. 7 31 
7 .. 13 



23 11 .. 
.. .. 7 

7 31 19 



17 11 



.. 17 .. 
11 .. 23 



31 11 



43 7 .. 
7 .. .. 

13 .. 7 
59 37 17 



13 41 

7 .. 



17 47 



37 7 

7 .. .. 



19 



17 



11 29 



29 47 
23 43 



.. 7 17 
31 .. 11 



.. 7 13 
7 53 43 



11 13 7 73 29 23 

.. 73 11 .. 13 .. 
.. 23 .. 

59 7 .. 



17 



7 53 



7 .. .. 
.. 29 7 



.. .. 19 
13 11 7 



7 67 
.. 17 



31 41 .. 
79 11 .. 



.. .. 11 
. . 43 37 

19 .. 11 7 31 



11 .. 13 

47 7 11 



.. 47 43 
17 .. 7 



77 


79 83 89 91 


97 


7 






7 




13 


. . . 


. . . 


17 






7 . 


. 13 




17 




11 . 


. 23 







19 .. .. 7 37 
.. .. 7 .. 31 7 



7 .. 37 11 7 13 
.. 37 13 19 47 11 



7 11 



17 .. .. 
11 31 17 



1 19 
. 43 



.. 13 29 7 .. .. 
41 23 7 .. 13 7 



7 11 .. 



7 .. 



23 19 .. 13 .. 

.. 7 19 .. 67 



31 .. 71 .. 29 .. 
11 17 23 



53 .. 



11 



37 



11 ..41 

.. 7 .. 29 .. .. 

19 29 . . 37 23 . . 



. 11 



.. ..71 
7 13 53 



13 .. 7 19 .. 7 
7 61 17 13 7 29 



67 .. 



11 59 19 



29 7 31 61 
.. 83 11 U 



*The smallest prime factors of multiple numbers are given. Prime num- 
bers are indicated by dots. 

Example: To find the prime factors of 2413: The smallest prime factor, 
from above table, is 19; then, 2413-^-19=127. Now, from Part 2, on the 
following page, 127 is found to be a prime number. Hence, the prime 
factors of 2413 are 19 and 127. 



1.— ELEMENTARY ARITHMETIC. 



2. — Primes,* Multiples and Factors. — Continued. 
Part 2. — (5M+i) Hundreds. 



n 


Tens and Units. 


N 


01 03 07 09 


13 


19 


21 


27 


31 


33 37 39 43 49 


51 


57 61 63 


67 


69 


73 


79 81 87 


91 


93 97 99 


inn 






7 


11 


7 




7 


.. .. 11 .. 
19 


ii 


.. 7 .. 




13 

7 




.. .. 11 


400 


.. 13 11 .. 


7 


11 


.. 13 .. 




17 7 .. 


700 


..19 7 .. 


23 


•• 


7 




17 




11 .. .. 7 




.. .. 7 


13 






19 11 .. 


7 


13 .. 17 


1000 


7 17 19 .. 
7 


13 


•• 


•• 


13 


ii 3i 


17 .. 7 .. 
7 13 17 19 




7 .. .. 
23 .. 29 




37 


29 


13 23 . . 




.. .. 7 


1300 


7 .. 19 


13 


7 11 .. 


1600 
1900 


.. 7 .. .. 
.. 11 .. 23 


•• 


i9 


17 


41 


7 


23 


.. 11 31 17 
13 7 29 .. 


13 


.. 11 .. 
19 37 13 




ii 


7 


23 41 7 
.. 7 .. 


19 
11 





2200 


31 .... 47 


•• 


7 


•• 


17 


23 


7 


13 




37 7 31 




•• 


•• 


43 .. .. 


29 


.. .. 11 


2500 


41 ..23 13 
.. .. 7 53 

7 29 13 .. 
19 41 .. 7 


13 

29 

11 


11 

13 


'7 

ii 


7 
11 

53 
23 


19 

31 

47 


17 
13 


43 




.. 13 11 




7 31 
19 13 

.. 19 
.. 23 


.. 29 13 
.. 43 .. 

11 .. .. 
7 59 11 




.. 7 23 


2800 


.. 17 .. 7 

. . 43 7 47 
7 19 11 .. 




.. .. 7 


11 . 13 


3100 
3400 


23 


7 29 .. 


31 23 7 
7 13 .. 


3700 


.. 7 11 .. 


47 


•• 


61 


•• 


7 


•• 


37 .. 19 23 


11 


13 .. 53 




•• 


7 


.. 19 7 




.. ..29 


4000 
4300 


19 

11 13 59 31 


19 


*7 


29 




29 37 
61 7 


11 7 13 .. 

.. .. 43 .. 


i9 


.. 31 17 
.. 7 .. 




13 

17 




.. 7 61 
29 13 41 




.. 17 .. 
23 .. 53 


4600 


43 .. 17 11 
13 .. 7 .. 


7 
17 


31 


7 


7 
13 


11 


41 






. 59 .. 


13 


7 




. . 31 43 
13 17 .. 




13 7 37 


4900 


.. 11 .. 7 




.. 11 7 


.. 19 .. 


5200 


7 11 41 .. 13 

7 37 

.. 7 .. 37 .. 

.. 17 31 41 .. 
37 19 43 13 11 


17 23 

a '.'. 

29 .. 

7 .. 


11 


'7 
59 


11 
19 

■7 


..13 7 29 

7 29 23 31 
13 


59 

7 


7 .. 19 

.. 67 .. 
.. .. 11 


23 
19 

29 


11 
31 




.. .. 17 




67 .. 7 


5500 
5800 


'7 


7 .. 37 
.. .. 7 

37 7 23 
11 .. 13 


7 29 11 
71 .. 17 


6100 
6400 


17 7 .. 11 
41 47 17 .. 


•• 


47 61 .. 
11 7 23 


11 .. .. 

43 73 67 


6700 


.. .. 19 .. 


7 


•• 


11 


7 


53 




.. 23 11 17 


43 


29 .. .. 


67 


7 


13 


.. .. 11 




.. 7 13 


7000 
7300 


..47 7 43 
7 67 . . . . 


7i 


13 


7 


i7 


79 


13 31 .. .. 7 
.. 11 41 7 .. 


11 


.. 23 7 
7 17 37 


37 
53 




11 
73 


.. 73 19 
47 11 83 




41 47 31 
.. 13 7 


7600 
7900 


11 .. .. 7 

.. 7 .. 11 


23 
41 


19 


89 


29 


13 
7 


17 


7 

.. 17 13 .. 


7 


13 47 79 
73 19 .. 


31 


i3 


*7 


7 .. .. 
79 23 7 




7 43 .. 
.. 11 19 


8200 


59 13 29 . . 


43 






19 




•• 


.. 7 .. 73 


37 


23 11 .. 




•• 




17 7 .. 




.. .. 43 


8500 
8800 


.. 11 47 67 
13 .. ..23 


*7 


7 


•• 


*7 


19 


11 


83 

.. .. 37 .. 


17 
53 


43 7 . 
17 .. .. 


13 


11 

7 


i9 


23 .. 31 
13 83 . . 




13 .. .. 
.. 7 11 


9100 


19 .. 7 .. 
7 .. 23 97 


13 


11 


7 


ii 


23 




.. 13 41 7 
... 7 11 




.. .. 7 


89 


53 
17 




67 .. .. 
.. 19 53 




29 17 .. 


9400 


13 


7 .. .. 


11 7 













*The smallest prime factors of multiple numbers are given. Prime num- 
bers are indicated by dots. 

Example: To find the prime factors of 1001: The smallest prime factor, 
from above table, is 7; then, 1001-^7=143, the smallest prime factor of 
which is 11; then, 143-^11=13, a prime number. Hence, the prime factors 
of 1001 are 7. 11 and 13. 



» 



2.- 



NUMBERS. 

-Primes,* Multiples and Factors. — Concluded. 
Part Z.—{3n-\-2) Hundreds. 



-f-S 




Tens and Units. 






iV 


03 09 11 17 21 23 


27 29 33 39 41 47 


51 53 57 59 63 69 71 


77 81 83 87 89 93 99 


200 


7 11 .. 7 13 .. 


13 


.. 11 .. 7 


7 17 .. 13 


500 


.. .. 7 11 .. .. 
11 .. .. 19 .. .. 


17 23 13 7 .. .. 
.... 7 .. 29 7 


19 7 .. 13 


.. 7 11 .. 19 .. 


800 


23 11 13 


7 19 29 


1100 
1400 


.. .. 11 .. 19 .. 
23 .. 17 13 7 .. 


7 .. 11 17 7 31 
11 ., 


.... 13 19 .. 7 .. 
.... 31 .. 7 13 .. 

17 .. 7 .. 41 29 7 


11 .. 7 .. 29 .. 11 
7 


1700 


13 .. 29 17 .. .. 


11 7 .. 37 .. .. 


..13 11 7 


2000 
2300 


.. 7 .... 43 7 
7 .. .. 7 11 23 

19 .. 7 .... 43 
.... 41 .. 23 37 


. . . . 19 . . 13 23 
13 17 


7 .. 11 29 .... 19 
.. 13 .. 7 17 23 .. 


31 7 .. 

.. .. .. 7 


2600 
2900 


37 11 .. 7 19 .. 
..29 7 .. 17 7 


11 7 17 .. 

13 .... 11 


.. 7 

13 11 19 29 7 41 .. 


3200 


.. .. 13 .. .. 11 


7 .. 53 41 7 17 


13 7 .. 


29 17 7 19 11 37 .. 


3500 
3800 


31 11 .. .. 7 13 
.. 13 37 11 .. .. 


43 '7 Wii 23 W 


53 11 ... . 7 43 . . 
.... 7 17 .. 53 7 


7 .. .. 17 37 .. 5> 
.... 11 13 .. 17 7 


4100 
4400 


11 7 .. 23 13 7 
7 .. 11 7 .. .. 


41 11 

19 43 11 23 .. .. 


7 23 11 43 

.. 61 .. 7 .. 41 17 


.. 37 47 53 59 7 13 
11 .. .. 7 67 .. 11 


4700 


..17 7 53 .. .. 

29 .. .. 

.... 47 13 17 .. 


29 .. .. 7 11 47 

11 47 7 .. 71 7 
7 73 .. 19 7 .. 


.. 7 67 .. 11 19 13 

.. 31 13 .. 61 37 11 
.. 53 11 23 31 7 41 


17 7 , 


5000 
5300 


.. .. 13 .. 7 11 .. 
19 .. 7 .. 17 .. .. 


5600 


13 71 31 41 7 .. 
.. 19 23 61 31 .. 


17 13 43 


7 .. 53 


7 13 .. 11 .. ..41 


5900 


.. 7 17 .. 13 19 


11 .. 7 59 67 47 7 


43 .. 31 .. 53 13 7 


6200 


.. 7 7 

7 23 17 7 .. 11 
..11 7 17 19 .. 


13 . . 23 17 79 . . 

61 .. 47 13 31 .. 
7 .. 41 


7 13 .. 11 


.. 11 61 .. 19 7 .. 


6500 


79 7 


... 29 7 11 19 .. 


6800 


13 7 .. 19 


13 7 .. 71 83 61 .. 


7100 
7400 


.. .. 13 11 .. 17 
11 31 .. .. 41 13 


.. .. 7 11 37 7 
7 17 .. 43 7 11 


.. 23 17 .. 13 67 71 
.. 29 .... 17 7 31 


.. 43 11 .. 7 .. 23 
.... 7 .... 59 .. 


7700 


.. 13 11 .. 7 .. 


.. 59 11 71 .. 61 


23 7 17 19 


7 31 43 13 .... 11 


8000 
8300 


53 13 71 

19 7 .... 53 7 


23 7 29 .. 11 13 
11 .. 13 31 19 17 


8? .. 7 .. 11 .. 7 
7 .. 61 13 .... 11 


41 .. 59 7 

.. 17 83 .. .. 7 37 


8600 
8900 


7 .. 79 7 37 .. 
29 59 7 37 11 .. 


.... 89 53 ... . 
79 .. . 7 .. 23 


41 17 11 7 .... 13 
.. 7 13 17 


.... 19 7 

47 7 13 11 89 17 .. 


9200 


.... 61 13 .. 23 


..11 7 .. .. 7 


11 19 .. 47 59 13 73 


37 7 .. 17 


9500 


13 37 .. 31 .. 89 


7 13 .. .. 7 .. 


.. 41 19 11 73 7 17 


61 11 7 .. 43 53 29 



*The smallest prime factors of multiple numbers are given. Prime num- 
bers are indicated by dots. 

Example: To find the prime factors of 3211: The smallest prime factor, 
from above table, is 13; then, 3211-^13=247, the smallest prime factor of 
which is 13; then, 247-^-13=19, a prime number. Hence, the prime factors 
of 3211 are 13, 13 and 19. 



I.— ELEMENTARY ARITHMETIC. 



Greatest Common Factor. — ^The G. C. F. of two or more numbers is 
the greatest factor common to all of them. It will be: 
1st. A factor of all the numbers, therefore no greater than the least num- 
ber; 
2nd. A factor of all the differences and successive differences except 0, 

therefore no greater than the least difference; 
And can be obtained (1) by differences, (2) by factoring, and (3) by divi- 
sion. 
Example: Find the (;. C. F. of 84, 126, 210 and 231. 
1st Method. — By differences. 




1st 

Numbers, diff. 
231 

21 

84 



Fig. 1. 



210 
126 

84 



Ans. — ^The least difference, 21, 
is obviously the G. C. F. as 
it is the greatest common 
factor of the least number 
and of all the differences. 



42 



2nd Method. — By factoring. 



(a.) — Finding all the prime 

factors. 
2 )84 2 )126 2) 210 3 )231 

2)42 3)63 3 )105 7)77 

3)21 3 )21 5)35 11 

7 7 7 



(jb.) — Finding all the 

common prime factors. 

3 )84 126 210 23 1 

7 )28 42 70 77 

4 6 10 11 

3 and 7 are common 
to all— 21. 



(c.) — Finding the 

common factors. 

21 )84 126 210 231 

4 6 10 11 



3 and 7 are common to all — 21. to all — 21. By inspection — 21. 

3rd Method. — By division (this is simply another form of differences). 



84 and 126 

84)126(1 
84 

42)84(2 
84 



210 



231 



42)210(5 
210 



42)231(5 
210 



21)42(2 
42 



Rule. — Start with any two of the numbers. Divide the greater by the 
less and if there is a remainder use this remainder as a divisor of the previous 
divisor, and continue until there is no remainder. The last divisor will 
be the greatest common divisor or G. C. F. of the two numbers. Use this 
G. C. F. so obtained with the next number for a new G. C. F., and so on 
until all the numbers are exhausted. The last divisor without a remainder 
will be the G. C. F, of all the numbers. 

Least Common Multiple. — The L. C. M. of two or more numbers is the 
least number that can be divided exactly by each of them. 

Rule for finding the L. C. M. — Divide the given numbers by the greatest 
(or any) factor common to most of them, and if divisible, set down the 
quotients in the line below; but if not divisible, bring down the numbers 
themselves. Divide the new line of numbers in a similar manner, and also 
each successive line, until no two numbers have a common factor except 1. 
The L. C. M. will be the product of all the factors and the last line of num- 
bers. During the process, if any number is a factor of any other number 
in th6 same line it can be cancelled. 



Example 1. 
Find the L. C. M. of 7, 24, li 

4 
6 

7 

Ans.— 4X6X7X11 = 1848. 



and 264. 



7 


24 


168 


264 


7 


e 


42 


66 


7 




7 


11 



Example 2. 
Find the L. C. M. of 13,28,52 and 84. 

4 I 13 t$ $Z U 



n 



13 21 



11 



Note. — 13 is a factor of 13, and 
7 of 21 

Ans.— 4X13X21 = 1092. 



Note that the first line in each example above could have been divided 
by 2, and then again by 2, instead of by 4; and that the second line in 
Example 1 could have been divided by 2 and then by 3, instead of by 6, 



FRACTIONS. 



The L. C. M. of two numbers is obviously a special case of the above 

in which the product of the 



Example 3. 
Find the L C. M. of 126 and 462. 



126 



462 



21 



77 



11 



Ans.— (6X7) X 3 X 11 = 1386. 



factors is the G. C. F., and the 
last line of numbers are quo- 
tients obtained by dividing the 
respective numbers by their 
G. C. F. Thus, in Example 3. 
the L. C. M. of 126 and 462 is 
their G. C. F.( = 42) multiplied 
by the product of their respective 

^. ^ 126 „ ,462 ,, 
quotients, -pr = 3, and -ttt- = 11. 



FRACTIONS. 



Kinds of Fractions. 

f Proper. -i, |. 

I 

1 



Simple 



Improper. -§. |. 
Mixed. -2f, 5|. 
Compound.— I of 4, f of 3^. 



Complex. ^ , ^ 



L 1 2i A 

3' i' 3r 4r 



, . f Pure. -.25, .625. 
^ lMixed.-l.25, 4.875 



Equivalent values. 



3n 
6n 



3X2 
5X2 



.6. 



3+2 
3 



i =—- = §+1 = 11. 



21= l+f = -V-. 



f of 3i=f X3i= 



Both terms 
of a fraction 
may be mul- 
tiplied or di- 
vided by the 
same num- 
ber without 
affecting its 
value. 



2 6+1 
3^^ 2 



^X^ 



=2i 



^_2i-3i _--^--yXy-f. 

1 10 100 400 40 *• 

4.875 = 4im = 4iB8S = 4|. 



To reduce to lowest terms. — Divide both terms of the fraction by their 
greatest common factor; or factor them as in finding their G. C. F. Thus, 



12 12-^-6 



30 30^6 5'°^ 30 



12 6 



15 



2_ 
5* 



To reduce to a common denominator. — Reduce the fractions (generally) 

to their lowest terms and find the least common multiple of their denomi- 
nators for a common denominator. Then expand both terms of each 
fraction proportionately so that their denominators will be equal to the 
common denominator. Thus, |, |, | and 1*2 = 2. s. f and J. The L. C. M. 



of their denominators = 72. Hence, | = 



9X5 _ 45 _ 24X J _ 24 

72 ~ 72' ^"""^ ' ~ 72 ~ 72' 



To Change — 

(o). An improper fraction to a mixed number. 



177 
32 



Ans. 



Rule. — Divide the numerator by the denomi- 
nator. The quotient is the whole number, 
and the remainder the new numerator. 



Process'. 
32)177(5 
160 

17 



8 1.— ELEMENTARY ARITHMETIC, 

131 
(6.) A mixed number to an improper fraction. 14f = ? Ans. ~q— . 

Rule. — Multiply the integral part by the de- Process. 
nominator, add the numerator, and place the 14 

sum over the denominator. 9 

1264-5=131. 
? 90 

(c.) A whole number to an improper fraction. 6 = Ti- Ans. r-- . 

10 15 

Rule. — Multiply the whole number by the Process. 

given denominator for the required numer- 6 X 15 = 90. 

ator. 

(J.) A compound to a simple fraction. ^ f of f = ? Ans. i^. 

Rule. — Proceed as in multiplication. ("Of" Process. 

means "times.") Place the product of the ? v ^ — A 

numerators over the product of the denomi- 3 8 ~ 12* 

nators, and reduce to lowest terms. Cancel, 4 
generally, before multiplying. 

2\ 

(e.) A complex to a simple fraction. "Si == ^ Ans. f . 

Of 

Rule. — Reduce the numerator and denominator Process. 

each to sim.ple fractions ; then multiply the 2^_^_^w_2_ _ « 
numerator by the denominator inverted. 3J~ -5 ~ 3 T ~" 

Cancellation. — In practical operations where the multiplication of vari- 
ous kinds of numbers, including fractions, is expressed, the process may 
often be shortened by cancellation. Thus, let it be required to find the 
product of loo", 2*5 and |. Clearly, we may place the product of the 
numerators over the product of the denominators for a simple fraction ex- 
pressing the result, or we may go to the other extreme and cancel as 
long as factors will cancel each other. There is, however, a medium 
point where cancellation may sometimes cease for simplicity of operation. 
35 

^, 315X4X5 700 frr.. . ' ^ • riAN 

^^^"' 400X25X0 ^ 10000- ^^^^ denommator is a power of 10.) 

Addition of Fractions. — Reduce to the smallest common denominator 

and add the new numerators. Thus, 1+^ = 1 + 1 = 1 = li; li+f = | + | = 
V = U; U+15f = V-f V=tl + ¥f- = "2¥=17ii; or, l|-fl5f = 16+1+3 = 
16 + 14 + ^1= 1611=1711. 

Subtraction of Fractions. — Reduce to the smallest common denominator 
and find the difference of the new numerators. Thus, ii — | = ii — 1% =1%; 
8i- 3i^e = 81% - 3i^5 = 7fi - 31-^5 = 4ii. 

Multiplication of Fractions. — If necessary, change to simple fractions; 
cancel where advisable, and place the product of the numerators over the 
product of the denominators; reduce to required form. Thus, \X%='\z\ 
' 5 

2iXfXUi=-f-x|-x|=|=U. 

Division of Fractions. — If necessary, change to simple fractions; invert 

29 

the divisor and multiply. Thus, 32 f - 6| = ^^ - ^ = ^ X. ^ = ^ = 4t. 

2 3 

To reduce a fraction to a reciprocal. — Divide the denominator by the 
numerator, writing the quotient in decimal form. Thus, the reciprocal of 

ig = V- = 00 ^ ^ • ^- Hence, instead of dividing a number by {^ we may 
multiply it by 3.2. 

To reduce a common fraction to a decimal. — Divide the numerator by 
the denominator, writing the quotient in decimal form. Thus, 

_ 8 )3.000 



I 



FRACTIONS REDUCED TO DECIMALS. 



9 



3. — Fractions (7ths) Reduced to Decimals. 
Note. — These decimals are all repeating decimals, that is, they contain 
figui^es or groups of figures that may be repeated or annexed indefinitely. 
Thus, \ = .142857^142857^142857^1 = .142857^1, the inverted caret D sig- 
nifying that the figure following is the first of the repeating group. Ex- 
amples: ^ = .l^l = .llim = .llllllll. . .; ^ = .3^3 = 33^3 = .33333^3. etc. 



Fractions. 


Equivalent 
Decimals. 


Fractions. 


Equivalent 
Decimals. 


Fractions. 


Equivalent 
Decimals. 




.142857^1 

.285714^2 


1 


.428571^4 
.571428^5 


1 


.714285^7 
.857142^8 





4. — Fractions (9ths) Reduced 


to Decimals. 




Frac- 
tions. 


Equiv. 
Deci- 
mals. 


Frac- 
tions. 


Equiv. 
Deci- 
mals. 


Frac- 
tions. 


Equiv. 
Deci- 
mals. 


Frac- 
tions. 


Equiv. 
Deci- 
mals. 




.1^1 

.2^2 




.3^3 
.4^4 


i 


5^5 

.6^6 




.7^7 
.8-8 



Note.— a = .222222...; |=.5555....; etc. 

5. — Fractions (IIths) Reduced to Decimals. 





Equiv. 
Deci- 
mals. 


- i 


Equiv. 
Deci- 
mals. 




Equiv. 
Deci- 
mals. 




Equiv. 
Deci- 
mals. 




Equiv. 
Deci- 
mals. 




.09-0 

.18-1 


A 
A 


.27-2 
.36-3 


A 

A 


.45-4 
.54-5 


A 


.63-6 

.72-7 


A 


.81-8 
.90-9 



Note.— A= .272727 ; i\- .636363 ; etc. 

6. — Fractions (12ths) Reduced to Decimals. 



Frac- 
tions. 


Equiv. 
Deci- 
mals. 


Frac- 
tions. 


Equiv. 
Deci- 
mals. 


Frac- 
tions. 


Equiv. 
Deci- 
mals. 


Frac- 
tions. 


Equiv. 
Deci- 
mals. 


A 


.083-3 
.166-6 
.25 


A 


.833-3 
.416-6 
.50 


A 


.583-3 
.666-6 
.75 




.833-3 
.916-6 



Note.— A = .083333...; i«2=. 416666 ; etc. 

7. — Fractions (13ths) Reduced to Decimals. 



Fractions. 


Equivalent 
Decimals. 


Fractions. 


Equivalent 
Decimals. 


Fractions . 


Equivalent 
Decimals. 


A 
A 
A 
A 


.076923-0 
.153846-1 
.230769-2 
.307692-3 




.394615-3 
.461538-4 
.538461-5 
.615384-6 


i 


.692307-6 
.769230-7 
.846153-8 
.923076-9 



Note. — These are repeating decimals, as above. 



10 



1.— ELEMENTARY ARITHMETIC, 
8. — Fractions (64ths) Reduced to Decimals. 






w 


u 




^ 


^* 




« 


M 






^ 




Q 


OJ 




Q 


d 




Q 


52 




o 


ClJ 


i 


'•g 


6 




1 


6 






6 






'o 


4^ 


2 


4) 


■M 


s 


<u 




(U 


+J 


S 


(U 


Z 


fe 


Q 


S 


fe 


Q 


^ 


fe 


Q 


s 


fe 


Q 


1 




.015625 


17 




.265625 


33 




.515625 


49 




.765625 


2 


-^o- 


.03125 


18 


9 

32 


.28125 


34 


M 


.53125 


50 


P 


.78125 


3 




.046875 


19 




.296875 


35 




.546875 


51 




.796875 


4 


1-16 


.0625 


20 


,')-16 


.3125 


36 


9-16 


.5625 


52 


13-16 


.8125 


5 




.078125 


21 




.328125 


37 




.578125 


53 




.828125 


6 


^^ 


.09375 


22 


^ 


.34375 


38 


^. 


.59375 


54 


r. 


.84375 


7 




.109375 


23 




.359375 


39 




.609375 


55 




.859375 


8 


1-8 


.125 


24 


3-8 


.375 


40 


6-8 


.625 


56 


7-8 


.875 


9 




.140625 


25 




.390625 


41 




.640625 


57 




.890625 


10 


^^ 


.15625 


26 


\% 


.40625 


42 


l\ 


.65625 


58 


^? 


.90625 


11 




.171875 


27 




.421875 


43 




.671875 


59 




.921875 


12 


3-16 


.1875 


28 


7-16 


.4375 


44 


11-16 


.6875 


60 


15-16 


.9375 


13 




.203125 


29 




.453125 


45 




.703125 


61 




.953125 


14 


if. 


.21875 


30 


P 


.46875 


46 


n 


.71875 


62 


l\ 


.96875 


15 




.234375 


31 




.484375 


47 




.734375 


63 




.984375 


16 


i->^ 


.25 


32 


i-^ 


.5 


48 


3-4 


.75 


64 


1 


1. 



DECIMALS. 

Addition of Decimals. — Have the decimal 

points "in column" before adding. Example; 



Subtraction of Decimals. — Have the decimal 

points "in column" before subtracting. Example: 



2.0625 
315.25 
.0375 

317.3500 

15.125 

3.71825 
11.40675 



Example: 



Multiplication of Decimals. — Multiply as with 
whole numbers; the product to have as 
many decimal places as the factors com- 
bined. Thus, 2. 3X .05X7.22= .83030 = 
.8303. 

Division of Decimals. — Multiply or divide the 
divisor and dividend by some power of 10 
that will make the divisior a whole number 
of significant figures, marking the new 
decimal point in the dividend by a caret C^). 
The quotient will then contain as many 
decimal places as the new dividend. 

Example: Divide 17.34 by 600? 
600 )^17.34 

.0289 Ans. 

Example: Divide 17.34 by .006? 
.006 ) 17.340^ 

2890. Ans. 

To reduce a decimal to a common fraction. 

a.) Exact decimals. — For the numerator of the fraction, use the significant 
figures of the decimal, the denominator being 1 with as many ciphers 
annexed as there are decimal places in the decimal ; reduce to lowest 
terms. 

Thus. .75=^,^ I; .0072 = ^^ = -!-^. 



6.75 
2.4 

2700 
1350 

16.200 

Example: 
Divide 3573.2 by 2.375. 
Ans. 1504.505 + 

1504.5 05 + 
2.375)3573.200^ 
2375 
11982 
11875 

10700 
9500 

12000 
11875 



12500 
11875 



DECIMALS AND FRACTIONS— SHORT METHODS. 



11 



(&.) Repeating decimals. — Treat the non-repeating and the first repeating 
groups as a whole number; subtract from this the non-repeating 
group treated as a whole number; the difference will be the numerator 
of the fraction. The denominator will be composed of as many 9's as 
there are repeating figures in a group, followed by as many O's as there 
are non-repeating figures. Reduce to lowest terms. 

Example: Reduce .3"^ 3 to a fraction. Example: Reduce .467"^ 6 to a fraction. 
- - 4 



numer. 


3 1 A 


num. 463 


denom. 


-g = i. Ans. 


demon. 990 






ADDITION. 




The following examples serve to illustrate approved methods: 






Same by 


Reduced to Reduced to 


Add: 




columns: Add: 


eighths: decimals: 


257.83 




24 




986.97 




33 153f . 


. . 7 . . . .875 


456.73 




29 25f . 


.. 6 ... .75 


523.82 




25 16i . 


. . 4 . . . .5 


357.19 




23 




Ans. 2582 54 




2582.54 Ans. 196i . 


V- 2.125 


Carried ZZ %% 








Fractions: 


f+i=f + l = l=U. 3U4-U=32|. 


Decimals: 


.754-. 5 =1.25. 31.25+1 


.125=32.375. 


Mechanical adding machines are used by accountants. 






SUBTRACTION. 








Reduced to Reduced to 


f- i = f -1= i. 


Subtract: ' 




eighths: decimals: 


1531 . 


, 


8+6 ... 1+ .75 


.75 -.5 =.25. 


251 . 


^ 


7 . . . .875 


Reduce the fraction 


Ans. 1271 . 




i . . . .875 


to a common denomi- 




nator, for addition or 








subtraction. 






MULTIPLICATION. 








Sufficiently i v * 


1X3 


Accurate: 




Accurate: ^ X f — 


2X4 *• 


12.64 




12.64 


15 7 


6.74 




6.74 3i X 2i = 


^ TX3 = ^^- 


5056 




50 .. 




8848 




885. 3.75 X2i 


= 7.5 + 1.25 = 8.75. 


7584 




7584 .2X .3 = 


= .06 .111 


Ans. 85.1936] 




85.19 


.3 



.0333 
A few short methods of multiplication will be found useful: 
(1). By reducing the multiplier'*' to an improper fraction with a simple 
numerator and denominator. Where the numerator becomes 1, 10, 100, 
1000, etc., it is known as the reciprocal method. In the following, let n rep- 
resent the number to be multiplied: 



» X u = 


X If = 


X 2i = 


X 3i = 



X 6i 



10 « 


8 • 


10 w 


6 • 


10« 


4 • 


10 « 


3 • 


100 « 



n X 



8i = 
X 12i = 
X 16f = 
X 25 = 
X 33i = —^ 



X 6f 



X 50 = — ^ 



100 n 

12 • 

100 « 

8 * 


w X 66f - 
X 83i = 


100 n 
6 ' 


X 1661 = 


100 n 
4 * 


X 333J = 


100 » 

3 • 
100 » 

2 • 





200« 

3 • 

1000 n 

12 • 

1000 w 

6 • 

1000 « 



4X4' 
20 n 

3 

Se e also Reciprocals, page 30. 
* If the multiplier is 11, add \. 
" " " 13i, " \ and multiply by 10. 

•• •• •• 133i, " i " " *• 100. 



12 \.— ELEMENTARY ARITHMETIC. 

(2). When the sum of the unit figures equals 10; and balance of numbers 
are identical; 
11 12 15 24 48 96 124 243 

19 _18 _15 _26 _42 _94 126 __2il 

209 216 225 624 2016 9024 15624 60021 

Rule. — Multiply the unit figures together, occupying two places (see 
first example above) ; and prefix the product of the balance of either num- 
ber by (itself +1). 

(a.) Special Case. — When the last figure is 5 the last two figures of the 
product will be 25\ thus, 252 = 625, 952 = 9025, 1452 = 21025, 
2452 = 60025; all performed mentally. 
(6.) Special Case. — When the last figure in each number is a decimal, and 
their sum is unity, the whole numbers being identical; thus, 
1.1 X 1.9 = 2.09, 2.4 X 2.6 = 6.24, 24 . 3 X 24 . 7 = 600 .21 
(c.) Special Case. — When the whole numbers are identical and the n*umbers 
contain fractions (instead of decimals, as above) whose sum is unity; 
thus, 

liV X 1t% = 2r§^, U X U = 2i, 121 X 12| = 156^- 
(3). The product of any two numbers each ending with the fraction \, or 
with the decimal .5: 

9i X 3i =9X3 + ^-j^ + i = 33i. 

7.5X11.5= 77+9 +.25 = 86.25. 
Rule. — *The product of the whole numbers + half their sum + \. 
(a.) Special Case. — When the numbers are whole numbers ending with 5: 
95 X 35 = 100 ( 27 + 6 + .25) = 3325. 
75 X 115 = 100 ( 77 + 9 + .25) = 8625. 
185 X 305 = 100 (540 + 24 + .25) = 56425-. 
Rule. — Apply the general rule above, considering each number divided 
by 10, and miiltiply the result by 100. 

(4.) ^The product of any two numbers is equal to the square of their mean, 
minus the square of half their difference'. 

24 X 26 = 252 - 1 = 624. 136 X 144 = 1402 - 16 = 19584. 
87 X 93 = 902 - 9 = 8091. 244 X 256 = 2502 - 36 = 62464, 
(5.) ^The converse of (4), of course, holds true and may be applied 
readily in finding the squares of numbers — considering them as means'. 
392 = 38 X 40 + 1 = 1521. 682 = gg x 70 + 22 = 4624. 
Note. — Use for a base, any multiple of 10 nearest the number to be 
squared. 

(6.) XThe square of any number is equal to the square of ( itself ±1), 
called the base, + the sum of the base and the number: 

392 = 402 - 79 = 1521. 712 = 702 + 141 == 5041. 

(7.) Miscellaneous Methods.^ 

a.) To multiply any number « by 11. « = 879^^687 
Note. — Imagine the number, and its product by 10, 

arranged thus: 879687 . , ,.^. -7 ^r-r^rrTT;; 

879687 fo^ addition Ans. 9,676,557 



(b.) To multiply a number n by any number from 12 «= 73 64 

to 19, inclusive: »X 7= + 51 548 

Example: Multilpy 7364 by 17. Ans. 125,188. 

Note. — When the unit figure of the multiplier is 1, 7 364 

the same principle of course holds true; as, for instance, +515 48 

7364 X 71 =522,844. 

(c.) To multiply a number n by any number from 92 100« = 736 400 

to 99, inclusive: - 2w 14 728 

Example: Multiply 7364 by 98. Ans. 721,672 

d.) When the multiplier contains simple factors, 987 

it may be quicker to use the factors: nX 8= 7896 

Example: Multiply 987 by 64. nX 8X 8= 63168. Ans. 



* From Algebra: (a + c) (b + c) = ab + c (a + b) + c^. 

t From Algebra: (x + y) (x — y) = x'^ — y^. 

t From Algebra: (x ± 1)^ = x^ ± 2 x -\- 1. 

T These illustrations may be expanded almost indefinitely. 



DECIMALS AND FRACTIONS— SHORT METHODS. 



13 



9.— Multiplication Table of Fractions. 
Note. — The Table is divided into Parts I and II. The greater factor denotes 
which Part to use and will be found as a heading or footing. The products (at in- 
tersection of line and column) are given to six decimal places. (See also Table 8.) 
[Products of Fractions, in Decimals.] 





Part II. Greater Factor. 7-16 to 15-16. 




15-16 


% 


13-16 


H 


11-16 


H 


9-16 


H 


7-16 




1-32 
1-16 
3-32 
Vb 
5-32 
3-16 
7-32 

,.^ 

5-16 
11-32 

Vs 
13-32 
7-16 
15-32 

y2 

17-32 


.029 297 
.058 594 
.087 891 
.117 188 
.146 484 
.175 781 
.205 078 
.234 375 
.263 672 
.292 969 
.322 266 
.351 563 
.380 859 
.410 156 
.439 453 
.468 750 
.498 047 
.527 344 
.556 641 
.585 938 
.615 234 
.644 531 
.673 828 
.703 125 
.732 422 
.761 719 
.791 016 
.820 313 
.849 609 
.878 906 


.027 344 
.054 688 
.082 031 
. 109 375 
.136 719 
.164 063 
.191 406 
.218 750 
.246 094 
.273 438 
.300 781 
.328 125 
.355 469 
.382 813 
.410 156 
.437 500 
.464 844 
.492 188 
.519 531 
.546 875 
.574 219 
.601 563 
.628 906 
.656 250 
.683 594 
.710 938 
.738 281 
.765 625 


.025 391 
.050 781 
.076 172 
.101 563 
.126 953 
.152 344 
.177 734 
.203 125 
.228 516 
.253 906 
.279 297 
.304 688 
.330 078 
.355 469 
.380 869 
.406 250 
.431 641 
.457 031 
.482 422 
.507 813 
.533 203 
.558 594 
.583 984 
.609 375 
.634 766 
.660 156 


.023 438 
.046 875 
.070 313 
.093 750 
.117 188 
.140 625 
.164 063 
.187 500 
.210 938 
.234 375 
.257 813 
.281 250 
.304 68S 
.328 125 
.351 563 
.375 000 
.398 438 
.421 87-5 
.445 313 
.468 750 
.492 188 
.515 625 
.539 063 
.562 500 


.021 484 
.042 969 
.064 453 
.085 938 
.107 422 
. 128 906 
. 150 391 
.171 875 
. 193 359 
.214 844 
.236 328 
.257 813 
.279 297 
.300 781 
.322 266 
.343 750 
.365 234 
.386 719 
.408 203 
.429 688 
.451 172 
.472 656 


.019 531 
.039 063 
.058 594 
.078 125 
.097 656 
.117 188 
.136 719 
. 156 250 
.175 781 
.195 313 
.214 844 
.234 375 
.253 906 
.273 438 
.292 969 
.312 500 
.332 031 
.351 563 
.371 094 
.390 625 


.017 578 
.035 156 
.052 734 
.070 313 
.087 891 
. 105 469 
. 123 047 
. 140 625 
. 158 203 
.175 781 
. 193 359 
.210 938 
.228 516 
.246 094 
.263 672 
.281 250 
.298 828 
.316 406 


.015 625 
.031 250 
.046 875 
.062 500 
.078 125 
.093 750 
.109 375 
.125 000 
.140 625 
.156 250 
.171 875 
.187 500 
.203 125 
.218 750 
.234 375 
.250 000 


.013 672 
.027 344 
.041 016 
.054 688 
.068 359 
.082 031 
.095 703 
.109 375 
.123 047 
.136 719 
.150 391 
.164 063 
.177 734 
.191 406 













9-16 








19-32 




.140 625 
. 128 906 
.117 188 
. 105 469 
.093 750 
.082 031 
.070 313 
.058 594 
.046 875 
.035 156 
.023 438 
.011 719 


Vs 


21-32 






11-32 




.097 656 
.087 891 
.078 125 
.068 359 
.058 594 
.048 828 
.039 063 
.029 297 
.019 531 
.009 766 


5-16 


11-16 






9-32 


23-32 
25-32 




.062 500 
.054 688 
.046 875 
.039 063 
.031 250 
.023 438 
.015 625 
.007 813 


H 






7-32 




035 156 


3-16 


13-18 






.029 297 
.023 438 
.017 578 
.011 719 
.005 859 


5-32 


27-32 




.015 625 


^ 


Vs 
29-32 






.011 719 
.007 813 
.003 906 


3-32 




.003 906 
.001 953 


1-16 


15-16 






1-32 




















1-16 


H 


3-16 


H 


5-16 


H 


o o 




Part I. Greater Factor, 1-16 to %. 


►Sfa 



Examples: 13-16X 23-32 = 0.583984; 5-16 X 5-32 = 0.048828. 



DIVISION. 



Accurate: 



2.64 

674.82) 1783.16^88 
1349 64 

433 528 
404 892 

28 6368 
26 9928 



Sufficiently accurate: 

2.64 

674.82) 1783.17^ 
1349 64 

433 53 
404 88 

28 65 



There are short special methods of division, but they are usually not 
broad enough in their application to record for general use. 
(a.) Factor the divisor: 4 )_972 

Example: Divide 972 by 16. 4 ) 243 

60 i Ans. 
(6.) Multiply by the reciprocal: 43.2 

Example: Divide 43.2 by 2.5 __.4_ 

17.28 Ans. 

(c.) To divide by a fraction, invert it and multiply, but reducing it to a 
simple decimal or reciprocal if possible: 

i -«- I = i X § = i 56 ^ t = 56 X .8 = 44.8. 

(d.) Cancellation: |- ^ t w !^ — I ° 2i Ans. 



Z X 



It X 
3 



2 



2.— POWERS, ROOTS AND RECIPROCALS. 

The processes of multiplication and division, and of finding the powers, 
roots and reciprocals of numbers may be performed by arithmetic (and 
algebra); by the use of tables; by logarithms;* or by the logarithmic 
slide rule.t 

A. ENGINEERS' TABLES. 

Under the present heading will be found Engineers' Tables of powers, 
roots and reciprocals of numbers arranged in form similar to logarithmic 
tables, so that the above properties of any number can be obtained by 
manipulating the decimal point, and by the use of the proportional parts 
(P. P.) columns, as in finding the logarithms of numbers. These tables 
may also be used inversely in finding the squares, cubes, and inverse 
reciprocals, corresponding respectively to the square roots, cube roots and 
reciprocals — the process being similar to finding the numbers corresponding 
to given logarithms. For a more academic arrangement see Arithmetical 
Tables 9, 10. and 11, following. 

Square Root. — ^Tables 1 and 2, following, comprise a 4-page table of the 
square roots of numbers. The first two pages (Table 1) are for numbers with 
an odd number of -figures to the left of the decimal point ; or, if it is a pure decimal 
the first significant figure must be an even number of places to the right of 
the decimal point. The last two pages (Table 2) are for numbers just the 
reverse — even to the left and odd to the right. The range of the tables may be 
extended by remembering that "changing two decimal points in the square 
= one decimal point in the root — in the same direction." The tables are 
logarithmic in form, containing P. P. columns for extension or interpolation. 

Example. — Required the square root of 907.6. 

Solution. — ^Three figures at the left of the decimal point calls for the 
"odd" table (Table 1), from which the sq. rt. of 907= 30.116, and the pro- 
portional part of the difference, 17, between it and the next higher number, 
908, is found to be 10 (10.2) which, added to 30.116=30.126. Ans. 

The following methods are given for extracting the square root when 
tables are not accessible, or when great accuracy is desired, for large num- 
bers: 

By Algebra. By Arithmetic. 

Square root of a^+ 2ax-\-x^ ? Square root of 576 ? 

a'^-{-2ax + x2ia-hx.Ans. 2a = 40 576 (20+4=24. Ans. 

a2 Assume x= _i 400 

2a-{-x) 2ax-{-x^ 2a + ^=44 )176 a = 20, 

2ax-\-x2 . 176 x= 4. 

Example. — Extract the square root of 46,354.87? 

Remarks. — Be- Process: 

ginning with the .... 

unit figure, point 46354.87(215.301+. Ans. 

off two places each a = 20 ) _^ 

way, forming Assume i»;J= l\ .*. 2ai+^i=41)~63 ao= 2, 

^res ?o bring'doi^n -2=210) JL_ a.= 20. ^.=1. 

each time. Proceed Assume X2= 5 j .*. 2a2+ ^2=425) 2254 ^2=210, ^2=5 
as indicated. ^^^^ 

Or, 215X2 = 430; 4303) 12987 

12909 



_ 2153X2=4306; 430601) 780000 

* See table of logarithms, page 108. 

t For description and use of the slide rule, see page 126. 

14 



ENGINEERS' TABLES— SQUARE ROOTS. 15 

Example. — Extract the square root of 4,635.4 ? 
Note that the position of the decimal Process: 

point affects the pointing off and, conse- ... 

quently, the entire character of there- 4635.40 (68.08+ Ans. 

suit. 36 

128)1035 
1024 



13608 ) 114000 
108864 

To Extract the Square Root of a Fraction. — The process of obtaining the 
square root of a fraction may be reduced to that of extracting the square root of 
a whole number. 

Rule. — Multiply the numerator by the denominator, extract the square root 
of this product, and divide the square root thus obtained by the denominator 
of the fraction. 

Example. — Extract the square root of %. 

Solution.— V^= V2X3 -- 3 = ^6 ^ 3 = 0.8165. 

To Extract the Cube Root of a Fraction. — The process of obtaining the cube 
root of a fraction may be reduced to that of extracting the cube root of a whole 
number. 

Rule. — Multiply the numerator by the square of the denominator, extract 
the cube root of this product, and divide the cube root thiis obtained by the 
denominator of the fraction. 

Example. — Extract the cube root of Yd. 

Solution.— V^^ = \^iX^ -^ 5 = •v/lOO -^ 5 = 0.9283. 

To Extract the mth Root of a Fraction. — 

Rule. — Multiply the numerator by the denominator raised to the m — 1 
power, extract the wth root of this product, and divide the root thus obtained 
by the denominator of the fraction. 

Example. — Extract the wth root of a/b. 

Solution.— v"^ = Vab^^ -^ b. 

The Square Root of the Reciprocal of a Number is obtained by dividing 
the square root of the number, by the number itself. This is evident from the 
above rule for extracting the square root of a fraction, the numerator in the 
present case being 1. 

Example. — Extract the square root of the reciprocal of 13. 

Solution.— ^^^ = ^"13 ^ 13 = 0.27735. 

The Cube Root of the Reciprocal of a Number is obtained by dividing the 
cube root of the square of the number, by the number itself. This is evident 
from the above rule for extracting the cube root of a fraction, the numerator in 
the present case being 1. 

Example. — Extract the cube root of the reciprocal of 17. 

Solution.— v^H7 = ^172 -^ 17 = -v^289 ^ 17 = 0.38891. 

The /nth Root of the Reciprocal of a Number is obtained by dividing the 
mth root of the number raised to the m — I power, by the number itself. 
Example. — Extract the wth root of the reciprocal of n. 

Solution.— Vl/w = V w"""^ -^ n. 



16 2.— POWERS, ROOTS AND RECIPROCALS, 

Odd. Even. 

1.— Square Roots (and Squares*) of Numbers— 



1 





The square must contain an 


odd 




P.P. 


Note. — 




50 48 46 44 42 


50301 . 020406 number of figures to the left of the decimal 


1 


5 5 5 4 4 


ODD EVEN point. Ji the square is a, pure decimal— 


-less 


2 


10 10 9 9 8 


than unity 


— the first significant figure must 


3 


15 14 14 13 13 


be an even number of places 


to the right of the decimal point. 


4 


20 19 18 18 17 


See page 14 for explanation of table. 






5 


25 24 23 22 21 












6 


30 29 28 26 25 












7 
8 


35 34 32 31 29 




. . 














40 38 37 35 34 


Sq. 


Vo 


1 


Z 


3 


4 


5 


6 


7 


8 


9 


9 


45 43 41 40 38 


I.O 


1 .0000 


0050 


0100 


0149 


0198 


0247 


0296 


0344 


0392 


0440 


41 40 39 38 37 


.1 


0488 


0536 


0583 


0630 


0677 


0724 


0770 


0817 


0863 


0909 


1 


—wm. -"WX^ %r^ *J\J %M£ 

4 4 4 4 4 


.2 


0954 


1000 


1045 


1091 


1136 


1180 


1225 


1269 


1314 


1358 


2 


8 8 8 8 7 


.3 


1402 


1446 


1489 


1533 


1576 


1619 


1662 


1705 


1747 


1790 


3 


12 12 12 11 11 


■g .4 


1832 


1874 


1916 


1958 


2000 


2042 


2083 


2124 


2166 


2207 


4 


16 16 16 15 15 


° 1.5 


2247 


2288 


2329 


2369 


2410 


2450 


2490 


2530 


2570 


2610 


5 


21 20 20 19 19 


!S .7 


2649 


2689 


2728 


2767 


2806 


2845 


2884 


2923 


2961 


3000 


6 


25 24 23 23 22 


3038 


3077 


3115 


3153 


3191 


3229 


3266 


3304 


3342 


3379 


7 


*Jt/ U~X. iJU U*J uu 

29 28 27 27 26 


a .8 


3416 


3454 


3491 


3528 


3565 


3601 


3638 


3675 


3711 


3748 


8 


33 32 31 30 30 


o 2.0 


3784 


3820 


3856 


3892 


3928 


3964 


4000 


.4036 


4071 


4107 


9 


37 36 35 34 33 


4142 


4177 


4213 


4248 


4283 


4318 


4353 


4387 


4422 


4457 




36 35 34 33 32 


o< .1 


4491 


4526 


4560 


4595 


4629 


4663 


4697 


4731 


4765 


4799 


1 


\M\f \^\J \M~T \^%9 \Mmt 

4 4 3 3 3 


rt -2 


4832 


4866 


4900 


4933 


4967 


5000 


5033 


5067 


5100 


5133 


2 


7 7 7 7 6 


1 -3 


5166 


5199 


5232 


5264 


5297 


5330 


5362 


5395 


5427 


5460 


3 


11 11 10 10 10 


1 ' 


5492 


5524 


5556 


5588 


5620 


5652 


5684 


5716 


5748 


5780 


4 


14 14 14 13 13 


;^2.5 


5811 


5843 


5875 


5906 


5937 


5969 


6000 


6031 


6062 


6093 


5 


18 18 17 17 16 


V -6 

II .7 


6125 


6155 


6186 


6217 


6248 


6279 


6310 


6340 


6371 


6401 


6 


22 21 20 20 19 


6432 


6462 


6492 


6523 


6553 


6583 


6613 


6&43 


6673 


6703 


7 


25 25 24 23 22 


u .8 


6733 


6763 


6793 


6823 


6852 


6882 


6912 


6941 


6971 


7000 


8 


29 28 27 26 26 


"5 3.0 


7029 


7059 


7088 


7117 


7146 


7176 


7205 


7234 


7263 


7292 


9 


32 32 31 30 29 


7321. 


7349 


7378 


7407 


7436 


7464 


7493 


7521 


7550 


7578 




31 30 29 28 27 


^ .2 


7607 


7635 


7664 


7692 


7720 


7748 


7776 


7804 


7833 


7861 


J 


3 3 3 3 3 


7889 


7916 


7944 


7972 


8000 


8028 


8055 


8083 


8111 


8138 


2 


6 6 6 6 5 


d .3 


8166 


8193 


8221 


8248 


8276 


8303 


8330 


8358 


8385 


8412 


3 


9 9 9 8 8 


5 •' 


8439 


8466 


8493 


8520 


8547 


8574 


8601 


8628 


8655 


8682 


4 


12 12 12 11 11 


.a 3.5 


8708 


8735 


8762 


8788 


8815 


8841 


8868 


8894 


8921 


8947 


5 


16 15 15 14 14 


a .6 


8974 


9000 


9026 


9053 


9079 


9105 


9131 


9157 


9183 


9209 


6 


19 18 17 17 16 


^ .7 


9235 


9261 


9287 


9313 


9339 


9365 


9391 


9416 


9442 


9468 


7 


22 21 20 20 19 


g .8 


9494 


9519 


9545 


9570 


9596 


9621 


9647 


9672 


9698 


9723 


8 


25 24 23 22 22 


9748 


9774 


9799 


9824 


9849 


9875 


9900 


9925 


9950 


9975 


9 


28 27 26 25 24 


•d 4.0 


2.0000 


0025 


0050 


0075 


0100 


0125 


0149 


0174 


0199 


0224 




26 25 24 2322 


<M .1 


0248 


0273 


0298 


0322 


0347 


0372 


0396 


0421 


0445 


0469 


I 


3 3 2 2 2 


bO -2 


0494 


0518 


0543 


0567 


0591 


0616 


0640 


0664 


0688 


0712 


2 


5 5 5 5 4 


.S -3 


0736 


0761 


0785 


0809 


0833 


0857 


0881 


0905 


0928 


0952 


3 


8 8 7 7 7 


S) .4 
^ 4.5 


0976 


1000 


1024 


1048 


1071 


1095 


1119 


1142 


1166 


1190 


4 


10 10 10 9 9 


1213 


1237 


1260 


1284 


1307 


1331 


1354 


1378 


1401 


1424 


5 


13 13 12 12 11 


O .6 


1448 


1471 


1494 


1517 


1541 


1564 


1587 


1610 


1633 


1656 


6 


16 15 14 14 13 


.7 


1679 


1703 


1726 


1749 


1772 


1794 


1817 


1840 


1863 


1886 


7 


18 18 17 16 15 


.8 


1909 


1932 


1954 


1977 


2000 


2023 


2045 


2068 


2091 


2113 


8 


21 20 19 18 18 


.9 


2136 1 2159 1 2181 1 2204 


2226 


2249 


2271 I 2293 


2316 


2338 


9 


23 23 22 21 20 



^Square Roots are obtained directly as in the logarithmic tables, using 
the Proportional Parts tables for interpolation. 
Squares may be obtained by inverse interpolation. 
Example.— Square root of 374 . 9 = 19 . 339 

4- 23 

= 19.362 Ans. 



ENGINEERS' TABLES— SQUARE ROOTS. 



17 



Odd. Even. 

-tFROM 1 TO 10; 100 TO 1,000; 10,000 to 100.000; etc. 



Sq. 


V 


1 


2 


3 


4 


5 


6 


7 


8 


9 




P.P. 


5.0 


2.2361 


2383 


2405 


2428 


24§0 


2472 


2494 


2517 


2539 


2561 




23 22 21 20 


.1 


2583 


2605 


2627 


2650 


2672 


2694 


2716 


2738 


2760 


2782 


1 


2 2 2 2 


.2 


2804 


2825 


2847 


2869 


2891 


2913 


2935 


2956 


2978 


3000 


2 


5 4 4 4 


.3 


3022 


3043 


3065 


3087 


3108 


3130 


3152 


3173 


3195 


3216 


3 


7 7 6 6 


.4 


3238 


3259 


3281 


3302 


3324 


3345 


3367 


3388 


3409 


3431 


4 


9 9 8 8 


5.5 


3452 


3473 


3495 


3516 


3537 


3558 


3580 


3601 


3622 


3643 


5 


12 11 11 10 


.6 


3664 


3685 


3707 


3728 


3749 


3770 


3791 


3812 


3833 


3854 


6 


14 13 13 12 


.7 


3875 


3896 


3917 


3937 


3958 


3979 


4000 


4021 


4042 


4062 


7 


16 15 15 14 


.8 


4083 


4104 


4125 


4145 


4166 


4187 


4207 


4228 


4249 


4269 


8 


18 18 17 16 


O -^ 


4290 


4310 


4331 


4352 


4372 


4393 


4413 


4434 


4454 


4474 


9 


21 20 19 18 


2 6.0 


4495 


4515 


4536 


4556 


4576 


4597 


4617 


4637 


4658 


4678 




21 20 19 


^ -1 


4698 


4718 


4739 


4759 


4779 


4799 


4819 


4839 


4860 


4880 


1 


2.1 2.0 1.9 


:S .2 


4900 


4920 


4940 


4960 


4980 


5000 


5020 


5040 


5060 


5080 


2 


4.2 4.0 3.8 


a .3 


5100 


5120 


5140 


5159 


5179 


5199 


5219 


5239 


5259 


5278 


3 


6.3 6.0 5.7 


i •' 


5298 


5318 


5338 


5357 


5377 


5397 


5417 


5436 


5456 


5475 


4 


8.4 8.0 7.6 


•S 6.5 


5495 


5515 


5534 


5554 


5573 


5593 


5612 


5632 


5652 


5671 


5 


10.5 10.0 9.5 


a .6 


5690 


5710 


5729 


5749 


5768 


5788 


5807 


5826 


5846 


5865 


6 


12.6 12.0 11.4 


d -7 
1 •' 


5884 


5904 


5923 


5942 


5962 


5981 


6000 


6019 


6038 


6058 


7 


14.7 14.0 13.3 


6077 


6096 


6115 


6134 


6153 


6173 


6192 


6211 


6230 


6249 


8 


16.8 16.0 15.2 


6268 


6287 


6306 


6325 


6344 


6363 


6382 


6401 


6420 


6439 


9 


18.9 18.0 17.1 




6458 


6476 


6495 


6514 


6533 


6552 


6571 


6589 


6608 


6627 




19 18 17 




6646 


6665 


6683 


6702 


6721 


6739 


6758 


6777 


6796 


6814 


1 


1.9 1.8 1.7 


6833 


6851 


6870 


6889 


6907 


6926 


6944 


6963 


6981 


7000 


2 


3.8 3.6 3.4 


2 .3 
§ -4 


7019 


7037 


7055 


7074 


7092 


7111 


7129 


7148 


7166 


7185 


3 


5.7 5.4 5.1 


7203 


7221 


7240 


7258 


7276 


7295 


7313 


7331 


7350 


7368 


4 


7.6 7.2 6.8 


«« 7.5 


7386 


7404 


7423 


7441 


7459 


7477 


7495 


7514 


7532 


7550 


5 


9.5 9.0 8.5 


^ .7 


7568 


7586 


7604 


7622 


7641 


7659 


7677 


7695 


7713 


7731 


6 


11.4 10.8 10.2 


7749 


7767 


7785 


7803 


7821 


7839 


7857 


7875 


7893 


7911 


7 


13.3 12.6 11.9 


a .8 


7928 


7946 


7964 


7982 


8000 


8018 


8036 


8054 


8071 


8089 


8 


15.2 14.4 13.6 


•^ .9 


8107 


8125 


8142 


8160 


8178 


8196 


8213 


8231 


8249 


8267 


9 


17.1 16.2 15.3 


•S 8.0 

a .1 


8284 


8302 


8320 


8337 


8355 


8373 


8390 


8408 


8425 


8443 




18 17 16 


8460 


8478 


8496 


8513 


8531 


8548 


8566 


8583 


8601 


8618 


1 


1.8 1.7 1.6 


rt -2 


8636 


8653 


8671 


8688 


8705 


8723 


8740 


8758 


8775 


8792 


2 


3.6 3.4 3.2 


'o 


8810 


8827 


8844 


8862 


8879 


8896 


8914 


8931 


8948 


8965 


3 


5.4 5.1 4.8 


8983 


9000 


9017 


9034 


9052 


9069 


9086 


9103 


9120 


9138 


4 


7.2 6.8 6.4 


-§ 8.5 


9155 


9172 


9189 


9206 


9223 


9240 


9257 


9275 


9292 


9309 


5 


9.0 8.5 8.0 


c^ .6 


9326 


9343 


9360 


9377 


9394 


9411 


9428 


9445 


9462 


9479 


6 


10.8 10.2 9.6 


W) .7 


9496 


9513 


9530 


9547 


9563 


9580 


9597 


9614 


9631 


9648 


7 


12.6 11.9 11.2 


•E -8 


9665 


9682 


9698 


9715 


9732 


9749 


9766 


9783 


9799 


9816 


8 


14.4 13.6 12.8 




9833 


9850 


9866 


9883 


9900 


9917 


9933 


9950 


9967 


9983 


9 


16.2 15.3 14.4 


3.0000 


0017 


0033 


0050 


0067 


0083 


0100 


0116 


0133 


0150 




17 16 15 


0166 


0183 


0199 


0216 


0232 


0249 


0265 


0282 


0299 


0315 


1 


1.7 1.6 1.5 


.2 


0332 


0348 


0364 


0381 


0397 


0414 


0430 


0447 


0463 


0480 


2 


3.4 3.2 3.0 


.3 


0496 


0512 


0529 


0545 


0561 


0578 


0594 


0610 


0627 


0643 


3 


5.1 4.8 4.5 


.4 


0659 


0676 


0692 


0708 


0725 


0741 


0757 


0773 


0790 


0806 


4 


6.8 6.4 6.0 


9.5 


0822 


0838 


0854 


0871 


0887 


0903" 


0919 


0935 


0952 


0968 


5 


8.5 8.0 7.5 


.6 


0984 


1000 


1016 


1032 


1048 


1064 


1081 


1097 


1113 


1129 


6 


10.2 9.6 9.0 


.7 


1145 


1161 


1177 


1193 


1209 


1225 


1241 


1257 


1273 


1289 


7 


11.9 11.2 10.5 


.8 


1305 


1321 


1337 


1353 


1369 


1385 


1401 


1417 


1432 


1448 


8 


13.6 12.8 12.0 


.9 


1464 


1480 


1496 


1512 


1528 


1544 


1559 


1575 


1591 


1607 


9 


15.3 14.4 13.5 



tFor squares 10-100, 1,000-10,000, etc., see following table. 
Example. — Square of 24.64 = 607.1 Ans. 



18 



2.— POWERS, ROOTS AND RECIPROCALS. 



Even. Odd . 

2. — Square Roots (and Squares*) of Numbers— 





















P 


. P. 







Note. — The square must contain an 
even number of figures to the left of 










604020.10305 












EVEN ODD 


the decimal point. 


If the square 


is 




155 


150 145 140 135 




a pure decimal- 


-less than unity — the 


1 


16 


15 


15 


14 14 


first significant figure must be an 


odd 


number of 


2 


31 


30 


29 


28 27 


places to the right of the decimal point. 








3 


47 


45 


44 


42 41 


See page 14 for explanation of table. 










4 


62 


60 


58 


56 54 


















5 


78 


75 


73 


70 68 


















6 


93 


90 


87 


84 81 
98 95 
























7 


109 


105 


102 


Sq. 


V-0 


.1 


.2 


.3 


.4 


.5 


.6 


.7 


.8 


.9 


8 


124 


120 


116 


112 108 
























9 


130 135 131 
130 126 122 


126 122 


10. 


3.1623 


1780 


1937 


2094 


2249 


2404 


2558 


2711 


2863 


3015 


118 114 


1. 


3166 


3317 


3466 


3615 


3764 


3912 


4059 


4205 


4351 


4496 


1 


13 


13 


12 


12 11 


2. 


4641 


4785 


4928 


5071 


5214 


5355 


5496 


5637 


5777 


5917 


2 


26 


25 


24 


24 23 


*J 3. 


6056 


6194 


6332 


6469 


6606 


6742 


6878 


7014 


7148 


7283 


3 


39 


38 


37 


35 34 


o 4. 


7417 


7550 


7683 


7815 


7947 


8079 


8210 


8341 


8471 


8601 


4 


52 


50 


49 


47 46 


hi: 


8730 


8859 


8987 


9115 


9243 


9370 


9497 


9623 


9749 


9875 


5 


65 


63 


61 


59 57 


4.0000 


0125 


0249 


0373 


0497 


0620 


0743 


0866 


0988 


1110 


6 


78 


76 


73 


70 68 


^ 7. 


1231 


1352 


1473 


1593 


1713 


1833 


1952 


2071 


2190 


2308 


7 


91 


88 


85 


83 80 


S 8. 


2426 


2544 


2661 


2778 


2895 


3012 


3128 


3243 


3359 


3474 


8 


104 


101 


98 


94 91 


'^ 9. 
o 20. 


3589 


3704 


3818 


3932 


4045 


4159 


4272 


4385 


4497 


4609 


9 


117 113 110 106 103 


4721 


4833 


4944 


6055 


5166 


5277 


5387 


5497 


5607 


5717 




110 106 102 


98 94 


a 1 


5826 


5935 


6043 


6152 


6260 


6368 


6476 


6583 


6690 


6797 


J 


11 


11 


10 


10 9 


d 2! 


6904 


7011 


7117 


7223 


7329 


7434 


7539 


7645 


7749 


7854 


2 


22 


21 


20 


20 19 


S 3. 

.5 4 


7958 


8062 


8166 


8270 


8374 


8477 


8580 


8683 


8785 


8888 


3 


33 


32 


31 


29 28 


8990 


9092 


9193 


9295 


9396 


9497 


9598 


9699 


9800 


9900 


4 


44 


42 


41 


39 38 


'^ 25. 


5.0000 


0100 


0200 


0299 


0398 


0498 


0596 


0695 


0794 


0892 


5 


55 


53 


51 


49 47 


^ 6. 


0990 


1088 


1186 


1284 


1381 


1478 


1575 


1672 


1769 


1865 


6 


66 


64 


61 


59 56 


II 7. 


1962 


2058 


2154 


2249 


2345 


2440 


2536 


2631 


2726 


2820 


7 


77 


74 


71 


69 66 


o 8. 


2915 


3009 


3104 


3198 


3292 


3385 


3479 


3572 


3666 


3759 


8 


88 


85 


82 


78 75 


d 9. 


3852 


3944 


4037 


4129 


4222 


4314 


4406 


4498 


4589 


4681 


9 


99 


95 


92 


88 85 


I30. 


4772 


4863 


4955 


5045 


5136 


5227 


5317 


5408 


5498 


5588 




90 


88 


86 


84 82 


S 2: 


5678 


5767 


5857 


5946 


6036 


6125 


6214 


6303 


6391 


6480 


1 


9 


9 


9 


8 8 


6569 


6657 


6745 


6833 


6921 


7009 


7096 


7184 


7271 


7359 


2 


18 


18 


17 


17 16 


"^ 3. 


7446 


7533 


7619 


7706 


7793 


7879 


7966 


8052 


8138 


8224 


3 


27 


26 


26 


25 25 


C 4. 


8310 


8395 


8481 


8566 


8652 


8737 


8822 


8907 


8992 


9076 


4 


36 


35 


34 


34 33 


^ 35. 


9161 


9245 


9330 


9414 


9498 


9582 


9666 


9749 


9833 


9917 


5 


45 


44 


43 


42 41 


a 6. 


6.0000 


0083 


0166 


0249 


0332 


0415 


0498 


0581 


0663 


0745 


6 


54 


53 


52 


50 49 


2 7. 


0828 


0910 


0992 


1074 


1156 


1237 


1319 


1400 


1482 


1563 


7 


63 


62 


60 


59 57 


a 8. 


1644 


1725 


1806' 


1887 


1968 


2048 


2129 


2209 


2290 


2370 


8 


72 


70 


69 


67 66 


rt 9. 
1 40. 


2450 


2530 


2610 


2690 


2769 


2849 


2929 


3008 


3087 


3166 


9 


81 


79 


77 


76 74 


3246 


3325 


3403 


3482 


3561 


3640 


3718 


3797 


3875 


3953 




80 


78 


76 


74 72 


« 1. 


4031 


4109 


4187 


4265 


4343 


4420 


4498 


4576 


4653 


4730 


1 


8 


8 


8 


7 7 


-d 2. 


4807 


4885 


4962 


5038 


5115 


5192 


5269 


5345 


5422 


5498 


2 


16 


16 


15 


15 14 


<N 3. 


5574 


5651 


5727 


5803 


5879 


5955 


6030 


6106 


6182 


6257 


3 


24 


23 


23 


22 22 


^ *■ 


6332 


6408 


6483 


6558 


6633 


6708 


6783 


6858 


6933 


7007 


4 


32 


31 


30 


30 29 


§ 45. 
§ 6. 


7082 


7157 


7231 


7305 


7380 


7454 


7528 


7602 


7676 


7750 


5 


40 


39 


38 


37 36 


7823 


7897 


7971 


8044 


8118 


8191 


8264 


8337 


8411 


8484 


6 


48 


47 


46 


44 43 


r^ ^• 


8557 


8629 


8702 


8775 


8848 


8920 


8993 


9065 


9138 


9210 


7 


56 


55 


53 


52 50 


C-> 8. 


9282 


9354 


9426 


9498 


9570 


9642 


9714 


9785 


9857 


9929 


8 


64 


62 


61 


59 58 


9. 


7.0000 


0071 


0143 


0214 


0285 


0356 


0427 


0498 


0569 


0640 


9 


72 


70 


68 


67 65 



* Square Roots are obtained directly as in the logarithmic tables, using the 
Proportional Parts column for interpolation. 
Squares may be obtained by inverse interpolation. 



ENGINEERS' TABLES— SQUARE ROOTS. 



19 



Even. Odd. 

— fFROM 10 TO 100; 1,000 to 10,000; etc. 



Sq. 


V.o 


.1 


.2 


.3 


.4 


.5 


.6 


.7 


.8 


.9 




P. 


P. 




50. 


7.0711 


0781 


0852 


0922 


0993 


1063 


1134 


1204 


1274 


1344 




71 


70 


69 


68 67 


1. 


1414 


1484 


1551 


1624 


1694 


1764 


1833 


1903 


1972 


2042 


1 


7 


7 


7 


7 7 


2. 


2111 


2180 


2250 


2319 


2388 


2457 


2526 


2595 


2664 


2732 


2 


14 


14 


14 


14 13 


3. 


2801 


2870 


2938 


3007 


3075 


3144 


3212 


3280 


3348 


3417 


3 


21 


21 


21 


20 20 


i. 


3485 


3553 


3621 


3689 


3756 


3824 


3892 


3959 


4027 


4095 


4 


28 


28 


28 


27 27 


55. 


4162 


4229 


4297 


4364 


4431 


4498 


4565 


4632 


4699 


4766 


5 


36 


35 


35 


34 34 


6. 


4833 


4900 


4967 


5033 


5100 


5166 


5233 


5299 


5366 


5432 


6 


43 


42 


41 


41 40 


7. 


5498 


5565 


5631 


5697 


5763 


5829 


5895 


5961 


6026 


6092 


7 


50 


49 


48 


48 47 


8. 


6158 


6223 


6289 


6354 


6420 


6485 


6551 


6616 


6681 


6746 


8 


57 


56 


55 


54 54 


u 60. 


6811 


6877 


6942 


7006 


7071 


7136 


7201 


7266 


7330 


7395 


9 


64 


63 


62 


61 60 


7460 


7524 


7589 


7653 


7717 


7782 


7846 


7910 


7974 


8038 




66 


65 


64 


63 62 


<u 1. 


8102 


8166 


8230 


8294 


8358 


8422 


8486 


8549 


8613 


8677 


1 


7 


7 


6 


6 6 


5 2. 


8740 


8804 


8867 


8930 


8994 


9057 


9120 


9183 


9246 


9310 


2 


13 


13 


13 


13 12 


4^ 


9373 


9436 


9498 


9561 


9624 


9687 


9750 


9812 


9875 


9937 


3 


20 


20 


19 


19 19 


8.0000 


0062 


0125 


0187 


0250 


0312 


0374 


0436 


0498 


0561 


4 


26 


26 


26 


25 25 


■I'i; 


0623 


0685 


0747 


0808 


0870 


0932 


0994 


1056 


1117 


1179 


5 


33 


33 


32 


32 31 


1240 


1302 


1363 


1425 


1486 


1548 


1609 


1670 


1731 


1792 


6 


40 


39 


38 


38 37 


^ 7. 


1854 


1915 


1976 


2037 


2098 


2158 


2219 


2280 


2341 


2401 


7 


46 


46 


45 


44 43 


c 8. 
1 9. 


2462 


2523 


2583 


2644 


2704 


2765 


2825 


2885 


2946 


3006 


8 


53 


52 


51 


50 50 


3066 


3126 


3187 


3247 


3307 


3367 


3427 


3487 


3546 


3606 


9 


59 


59 


58 


57 56 


-O 70. 


3666 


3726 


3785 


3845 


3905 


3964 


4024 


4083 


4143 


4202 




61 


60 


59 


58 


>-* 1 . 


4261 


4321 


4380 


4439 


4499 


4558 


4617 


4676 


4735 


4794 


1 


6 


6 


6 


6 


II 2. 


4853 


4912 


4971 


5029 


5088 


5147 


5206 


5264 


5323 


5381 


2 


12 


12 


12 


12 


£ 3. 


5440 


5499 


5557 


5615 


5674 


5732 


5790 


5849 


5907 


5965 


3 


18 


18 


18 


17 


§ 4. 


6023 


6081 


6139 


6197 


6255 


6313 


6371 


6429 


6487 


6545 


4 


24 


24 


24 


23 


S^75. 


6603 


6660 


6718 


6776 


6833 


6891 


6948 


7006 


7063 


7121 


5 


31 


30 


30 


29 


-M 7. 


7178 


7235 


7293 


7350 


7407 


7464 


7521 


7579 


7636 


7693 


6 


37 


36 


35 


35 


7750 


7807 


7864 


7920 


7977 


8034 


8091 


8148 


8204 


8261 


7 


43 


42 


41 


41 


a 8- 


8318 


8374 


8431 


8487 


8544 


8600 


8657 


8713 


8769 


8826 


8 


49 


48 


47 


46 


S '• 


8882 


8938 


8994 


9051 


9107 


9163 


9219 


9275 


9331 


9387 


9 


55 


54 


53 


52 


.? 80. 


9443 


9499 


9554 


9610 


9666 


9722 


9778 


9833 


9889 


9944 




57 


56 


55 


54 


°^l 


9.0000 


0056 


0111 


0167 


0222 


0277 


0333 


0388 


0443 


0499 


1 


6 


6 


6 


5 


0554 


0609 


0664 


0719 


0774 


0830 


0885 


0940 


0995 


1049 


2 


11 


11 


11 


11 


c« 3. 


1104 


1159 


1214 


1269 


1324 


1378 


1433 


1488 


1542 


1597 


3 


17 


17 


17 


16 


.§ ^• 


1652 


1706 


1761 


1815 


1869 


1924 


1978 


2033 


2087 


2141 


4 


23 


22 


22 


22 


1 85. 


2195 


2250 


2304 


2358 


2412 


2466 


2520 


2574 


2628 


2682 


5 


29 


28 


28 


27 


S 6- 


2736 


2790 


2844 


2898 


2952 


3005 


3059 


3113 


3167 


3220 


6 


34 


34 


33 


32 


M 7. 


3274 


3327 


3381 


3434 


3488 


3541 


3595 


3648 


3702 


3755 


7 


40 


39 


39 


38 


C 8. 


3808 


3862 


3915 


3968 


4021 


4074 


4128 


4181 


4234 


4287 


8 


46 


45 


44 


43 


•a 9. 


4340 


4393 


4446 


4499 


4552 


4604 


4657 


4710 


4763 


4816 


9 


51 


50 


50 


49 


S 90. 


4868 


4921 


4974 


5026 


5079 


5131 


5184 


5237 


5289 


5341 




53 


52 


51 


50 


O 1. 


5394 


5446 


5499 


5551 


5603 


5656 


5708 


5760 


5812 


5864 


1 


5 


5 


5 


5 


2. 


5917 


5969 


6021 


6073 


6125 


6177 


6229 


6281 


6333 


6385 


2 


11 


10 


10 


10 


3. 


6437 


6488 


6540 


6592 


6644 


6695 


6747 


6799 


6850 


6902 


3 


16 


16 


15 


15 


4. 


6954 


7005 


7057 


7108 


7160 


7211 


7263 


7314 


7365 


7417 


4 


21 


21 


20 


20 


95. 


7468 


7519 


7570 


7622 


7673 


7724 


7775 


7826 


7877 


7929 


5 


27 


26 


26 


25 


6. 


7980 


8031 


8082 


8133 


8184 


8234 


8285 


8336 


8387 


8438 


6 


32 


31 


31 


30 


7. 


8489 


8539 


8590 


8641 


8691 


8742 


8793 


8843 


8894 


8944 


7 


37 


36 


36 


35 


8. 


8995 


9045 


9096 


9146 


9197 


9247 


9298 


9348 


9398 


9448 


8 


42 


42 


41 


40 


9. 


9499 


9549 


9599 


9649 


9700 


9750 


9800 


9850 


9900 


9950 


9 


48 


47 


46 


45 



tFor squares 1 — 10, 100—1,000, etc., see preceding table 



20 



2.— POWERS, ROOTS AND RECIPROCALS. 



Cube Root. — ^The cube roots of numbers may be obtained from Tables 
4 and 5, as follows: 



Table. 
3 

4 

5 



Whole Numbers (cubes), 

j 1 to 50, 

1 1000 to 50000, etc. 

j 1 to 1000, etc. 

\ Any number. 

j 1 to 1.5, 

1 1000 to 1500, etc. 



Decimals (cubes). Remarks. 

.001 to .050, Special 

.000001 to .000050, etc. table. 

.001 to 1. etc. General 

Any decimal. table. 

.001 to .0015. Special 

.000001 to .0000015, etc. table. 



Note that the special tables. 3 and 5. give results more accurately, 
within their limits, than does the general table, 4. The least accuracy 
from these tables is for numbers just above 1.5, or just above 1500; and 
for the cube roots of such numbers Table 3 may be used, or Tables 9 and 
10, remembering that "changing three decimal points in the cube = one 
decimal point in the root — in the same direction." 

Table 3 gives the cube roots of numbers from 1 to 50 : directly, ad- 
vancing by tenths; and by one interpolation of the P. P. table, advancing 
by hundreths. Thus, the cube root of 24.4 is 2.9004; of 24.45 is 2.9004 + 
20 ==2. 9024, the 20 being obtained from the P.P. table opposite 5 and 
under the difference 40 ( = 9044-9004). 

By manipulating the decimal point, the cube root of 0.02445 is 0.29024; 
of 24450 is 29.024, etc. Note that if the decimal point is changed only 
one or two places in the cube the cube root comprises another set of signifi- 
cant figures, as, cube root of 24.4 = 2.9004; of 2.44=1.3463; of 0.244 = 
0.6249. 

Table 5 is especially useful in finding the cube roots of numbers from 
1 to 1.5, or from 1000 to 1500. The cubes of numbers with four significant 
figures may be obtained directly from the table, and of numbers with five 
significant figures may be obtained by one interpolation of the P. P. table. 
Thus, cube root of 1.145=1.04617; of 1.1452=1.04623, the increment 
being Vio of the difference 30 ( = 4647—4617). Likewise, the cube root of 
1145.2=10.4623. 

Table 4 is a general table, with numbers from 1 to 1000. Its special 
range, however, is for numbers from 50 to 1000. 

Cubes of numbers may be obtained by inverse interpolation. 

For numbers beyond the accurate range of the tables the 'following 
methods are given for extracting the cube root: 



By Algebra. 
Cube root of a^+daH-h Zax^ + o^ ? 



By Arithmetic. 
Cube root of 12,167 ? 



a^+ZaH+Zax^+x^{a+x. Ans. 
o3 Assume x=^Z 



Za^+Zax+x^) 



3a%+3a^2_j_^3 
3a2%+3a%24.^3 



3a2 = 1200 
Zax= 180 

a:2= 9 

Za'^+Zax+x'^^Um 



12167(20+3 = 23. Ans. 
8000 a+x 



Example. — Extract the cubelroot of 46,354.87 ? 



Remarks. — 
Beginn i n g 
with the 
unit figure, 
point off 3 
places each 
way, form- 
ing groups 
of 3 fig- 
ures to be 
brought 
down each 
time. Pro- 
ceed as in- 
dicated. 



ai = 30. 
Assume %i = 5: 



3ai2 = 3(302) =2700 
3aiA;,= 3.30.5= 450 

x^-^= 52= 25 

3175 



4167 
4167 

Process: 

46354.870(35.9+ 
27 



Ans. 



) 19354 
15875 



Assume X2 



= 350. 3a22=3(3502) 

= 9: 302^2 = 3.350.9 

iC32= 92 = 



=367500 
= 9450 

81 

377031 



3479870 



3393279 



00 = 3 
ai = 30 

02 = 350 .., _ 

03 = 3590 rc3=? 



Xx=5 
X2==9 



0^ = 3590. 
Assume x^ = ? 



86591000 



3032= ) 

303^3 = > Ready for another operation. 
a^ 2 = . \ 



ENGINEERS' TABLES— CUBE ROOTS, 



21 



1 to 50 
1,000 to 50,000 

3. — Cube Roots (and Cubes*) of Numbers 1 to 50. 



.001 to .050 
.000,001 to .000,050 



Cube. 


V.o 


.1 


.2 


.3 


.4 


.5 


.6 


.7 


.8 


.9 






P. 


P. 




0. 


0.0000 


4642 


5848 


6694 


7368 


7937 


8434 


8879 


9283 


9655 




140 130 120 110 100 


1. 


i.oooo 


0323 


0627 


0914 


1187 


1447 


1696 


1935 


2164 


2386 


1 


14 


13 


12 


11 10 


2. 


2599 


2806 


3006 


3200 


3389 


3572 


3751 


3925 


4095 


4260 


2 


28 


26 


24 


22 20 


3. 


4422 


4581 


4736 


4888 


5037 


5183 


5326 


5467 


5605 


5741 


3 


42 


39 


36 


33 30 


4. 


5874 


6005 


6134 


6261 


6386 


6510 


6631 


6751 


6869 


6985 


4 


56 


52 


48 


44 40 


5. 


7100 


7213 


7325 


7435 


7544 


7652 


7758 


7863 


7967 


8070 


5 


70 


65 


60 


55 50 


^- 6. 


8171 


8272 


8371 


8469 


8566 


8663 


8758 


8852 


8945 


9038 


6 


84 


78 


72 


66 60 


t 7. 


9129 


9220 


9310 


9399 


9487 


9574 


9661 


9747 


9832 


9916 


7 


98 


91 


84 


77 70 


2 8. 


2.0000 


0083 


0165 


0247 


0328 


0408 


0488 


0567 


0646 


0724 


8 


112 


104 


96 


88 80 


0, 9. 


0801 


0878 


0954 


1029 


1105 


1179 


1253 


1327 


1400 


1472 


9 


126 


117 


108 


99 90 


! 10. 


1544 


1616 


1687 


1758 


1828 


1898 


19^7 


2036 


2104 


2172 




95 


90 


85 


80 76 


.S 11. 


2240 


2307 


2374 


2440 


2506 


2572 


2637 


2702 


2766 


2831 


1 


10 


9 


9 


8 8 


•S 12. 


2894 


2958 


3021 


3084 


3146 


3208 


3270 


3331 


3392 


3453 


2 


19 


18 


17 


16 15 


•S 13. 


3513 


3573 


3633 


3693 


3752 


3811 


3870 


3928 


3986 


4044 


3 


29 


27 


26 


24 23 


au. 


4101 


4159 


4216 


4272 


4329 


4385 


4441 


4497 


4552 


4607 


4 


38 


36 


34 


32 30 


2 15. 
.§ 16. 


4662 


4717 


4771 


4825 


4879 


4933 


4987 


5040 


5093 


5146 


5 


48 


45 


43 


40 38 


5198 


5251 


5303 


5355 


5407 


5458 


5510 


5561 


5612 


5662 


6 


57 


54 


51 


48 46 


§ 17. 


5713 


5763 


5813 


5863 


5913 


5963 


6012 


6061 


6110 


6159 


7 


67 


63 


60 


56 53 


^ 18. 


6207 


6256 


6304 


6352 


6400 


6448 


6495 


6543 


6590 


6637 


8 


76 


72 


68 


64 61 


^ 19. 
« 20. 


6684 


6730 


6777 


6824 


6870 


6916 


6962 


7008 


7053 


7099 


9 


86 


81 


77 


72 68 


7144 


7189 


7234 


7279 


7324 


7369 


7413 


7457 


7501 


7545 




72 


68 


64 


60 56 


•§ 21. 

o 22. 


7589 


7633 


7677 


7720 


7763 


7807 


7850 


7893 


7935 


7978 


1 


7 


7 


6 


6 6 


8020 


8063 


8105 


8147 


8189 


8231 


8273 


8314 


8356 


8397 


2 


14 


14 


13 


12 11 


o 23. 


8439 


8480 


8521 


8562 


8602 


8643 


8684 


8724 


8765 


8805 


3 


22 


20 


19 


18 17 


:g 24. 


8845 


8885 


8925 


8965 


9004 


9044 


9083 


9123 


9162 


9201 


4 


29 


27 


26 


24 22 


•^25. 


9240 


9279 


9318 


9357 


9395 


9434 


9472 


9511 


9549 


9587 


5 


36 


34 


32 


30 28 


42 26. 


9625 


9663 


9701 


9738 


9776 


9814 


9851 


9888 


9926 


9963 


6 


43 


41 


38 


36 34 


.S 27. 


3.0000 


0037 


0074 


0111 


0148 


0184 


0221 


0257 


0293 


0330 


7 


50 


48 


45 


42 39 


O 28. 


0366 
072§ 


0402 


0438 


0474 


0510 


0546 


0581 


0617 


0652 


0688 


8 


58 


54 


51 


48 45 


^ 29. 
6 30. 


0759 


0794 


0829 


0864 


0899 


0934 


0968 


1003 


1038 


9 


65 


61 


58 


54 50 


1072 


1107 


1141 


1176 


1210 


1244 


1278 


1312 


1346 


1380 




52 


48 


44 


40 36 


'S 31. 


1414 


1448 


1481 


1515 


1548 


1582 


1615 


1648 


1682 


1715 


1 


5 


5 


4 


4 4 


^ 32. 


1748 


1781 


1814 


1847 


1880 


1913 


1945 


1978 


2010 


2043 


2 


10 


10 


9 


8 7 


CO 33. 


2075 


2108 


2140 


2172 


2204 


2237 


2269 


2301 


2332 


2364 


3 


16 


14 


13 


12 11 


M 34. 
*S) 35. 


2396 


2428 


2460 


2491 


2523 


2554 


2586 


2617 


2648 


2680 


4 


21 


19 


18 


16 14 


2711 


2742 


2773 


2804 


2835 


2866 


2897 


2927 


2958 


2989 


5 


26 


24 


22 


20 18 


S 36. 
^ 37. 


3019 


3050 


3080 


3111 


3141 


3171 


3202 


3232 


3262 


3292 


6 


31 


29 


26 


24 22 


3322 


3352 


3382 


3412 


3442 


3472 


3501 


3531 


3561 


3590 


7 


36 


34 


31 


28 25 


O 38. 


3620 


3649 


3679 


3708 


3737 


3767 


3796 


3825 


3854 


3883 


8 


42 


38 


35 


32 29 


39. 


3912 


3941 


3970 


3999 


4028 


4056 


4085 


4114 


4142 


4171 


9 


47 


43 


40 


36 32 


40. 


4200 


4228 


4256 


4285 


4313 


4341 


4370 


4398 


4426 


4454 




32 


30 


28 


26 24 


41. 


4482 


4510 


4538 


4566 


4594 


4622 


4650 


4677 


4705 


4733 


1 


3 


3 


3 


3 2 


42. 


4760 


4788 


4815 


4843 


4870 


4898 


4925 


4952 


4980 


5007 


2 


6 


6 


6 


5 5 


43. 


5034 


5061 


5088 


5115 


5142 


5169 


5196 


5223 


5250 


5277 


3 


10 


9 


8 


8 7 


44. 


5303 


5330 

■ 


5357 


5384 


5410 


5437 


5463 


5490 


5516 


5543 


4 


13 


12 


11 


10 10 


45. 


5569 


5595 


5622 


5648 


5674 


5700 


5726 


5752 


5778 


5805 


5 


16 


15 


14 


13 12 


46. 


5830 


5856 


5882 


5908 


5934 


5960 


5986 


6011 


6037 


6063 


6 


19 


18 


17 


16 14 


47. 


6088 


6114 


6139 


6165 


6190 


6216 


6241 


6267 


6292 


6317 


7 


22 


21 


20 


18 17 


48. 


6342 


6368 


6393 


6418 


6443 


6468 


6493 


6518 


6543 


6568 


8 


26 


24 


22 


21 19 


49. 


6593 


6618 


6643 


6668 


6692 


6717 


6742 


6767 


6791 


6816 


9 


29 


27 


25 


23 22 


50. 


6840 


6865 


6889 


6914 


6938 


6963 


6987 


7011 


7036 


7060 













*Cube Roots are obtained directly as in the logarithmic tables, using the 
Proportional Parts tables for interpolation. 

Cubes may be obtained by inverse interpolation. 



2.— POWERS, ROOTS AND RECIPROCALS. 



1 to 500 .001 to .5 

1,000 to 500,000 .000,001 to .0005 

4. — Cube Roots (and Cubes*) of Numbers 1 to 1,000 — 



Cube. 


^0. 


1. 


2. 


3. 


4. 


5. 


6. 


7. 


8. 


9. 




P. 


P. 





> 0-. 


zero I. 0000 


2599 


4422 


5874 


7100 


8171 


9129 


0000 


0801 


240 220 200 190 180 


ff «!-• 


2.1544 


2240 


2894 


3513 


4101 


4662 


5198 


5713 


6207 


6684 


1 


24 


22 


20 


19 18 


P. ^2-. 


7144 


7589 


8020 


8439 


8845 


9240 


9625 


0000 


0366 


0723 


2 


48 


44 


40 


38 36 


gas-. 


3.1072 


1414 


1748 


2075 


2396 


2711 


3019 


3322 


3620 


3912 


3 


72 


66 


60 


57 54 


CQ 4-. 
50 
5-. 


4200 


4482 


4760 


5034 


5303 


5569 


5830 


6088 


6342 


6593 


4 


96 


88 


80 


76 72 


6840 


7084 


7325 


7563 


7798 


8030 


8259 


8485 


8709 


8930 


5 


120 110 100 


95 90 


6-. 


9149 


9365 


9579 


9791 


0000 


D207 


0412 


0615 


0817 


1016 


6 


144 


132 


120 


114 108 


4^ 7-. 


4.1213 


1418 


1602 


1793 


1983 


2172 


2358 


2543 


2727 


2908 


7 


168 


154 


140 


133 126 


1 8-- 


3089 


3267 


3445 


3621 


3795 


3968 


4140 


4310 


4480 


4647 


8 


192 


176 


160 


152 144 


2 9-. 
<u 100 
^ 10-. 

a 11-- 

•5 12-. 


4814 


4979 


5144 


5307 


5468 


5629 


5789 


5947 


6104 


6261 


9 


216 


198 


180 


171 162 


6416 


6570 


6723 


6875 


7027 


7177 


7326 


7475 


7622 


7769 




170 160 150 140 130 


7914 


8059 


8203 


8346 


8488 


8629 


8770 


8910 


9049 


9187 


1 


17 


16 


15 


14 13 


9324 


9461 


9597 


9732 


9866 


0000 


0133 


0265 


0397 


0528 


2 


34 


32 


30 


28 26 


a 13-. 

•g 14-. 
D. 150 
^ 15-. 


5.0658 


0788 


0916 


1045 


1172 


1299 


1426 


1551 


1676 


1801 


3 


51 


48 


45 


42 39 


1925 


2048 


2171 


2293 


2415 


2536 


2656 


2776 


2896 


3015 


4 


68 


64 


60 


56 52 


3133 


3251 


3368 


3485 


3601 


3717 


3832 


3947 


4061 


4175 


5 


85 


80 


75 


70 65 


g 16-. 
.S 17-. 


4288 


4401 


4514 


4626 


4737 


4848 


4959 


5069 


5178 


5288 


6 


102 


96 


90 


84 78 


5397 


5505 


5613 


5721 


5828 


5934 


6041 


6147 


6252 


6357 


7 


119 


112 


105 


98 91 


^ 18-. 


6462 


6567 


6671 


6774 


6877 


6980 


7083 


7185 


7287 


7388 


8 


136 


128 120 


112 104 


^ 19-. 
V 200 
II 20-. 


7489 


7590 


7690 


7790 


7890 


7989 


8088 


8186 


8285 


8383 


9 


153 


144 


135 


126 117 


8480 


8578 


8675 


8771 


8868 


8964 


9059 


9155 


9250 


9345 




120 


110 100 


90 80 


^ 21-. 
3 22-. 


9439 


9533 


9627 


9721 


9814 


9907 


0000 


0092 


0185 


0277 


1 


12 


11 


10 


9 8 


6.0368 


0459 


0550 


0641 


0732 


0822 


0912 


1002 


1091 


1180 


2 


24 


22 


20 


18 16 


« 23-. 


1269 


1358 


1446 


1534 


1622 


1710 


1797 


1885 


1972 


2058 


3 


36 


33 


30 


27 24 


2 24-. 
^ 250 
a 25-. 


2145 


2231 


2317 


2403 


2488 


2573 


2658 


2743 


2828 


2912 


4 


48 


44 


40 


36 32 


2996 


3080 


3146 


3247 


3330 


3413 


3496 


3579 


3661 


3743 


5 


60 


55 


50 


45 40 


•t^ 27-. 


3825 


3907 


3988 


4070 


4151 


4232 


4312 


4393 


4473 


4553 


6 


72 


66 


60 


54 48 


4633 


4713 


4792 


4872 


4951 


5U30 


5108 


5187 


5265 


5343 


7 


84 


77 


70 


63 56 


.§ 28-. 


5421 


5499 


5577 


5654 


5731 


5808 


5885 


5962 


6039 


6115 


8 


96 


88 


80 


72 64 


a 29-. 
^ 300 

.§ 31-. 


6191 


6267 


6343 


6419 


6494 


6569 


6644 


6719 


6794 


6869 


9 


108 


99 


90 


81 72 


6943 


7018 


7092 


7166 


7240 


7313 


7387 


7460 


7533 


7606 




70 


66 


64 


62 60 


7679 


7752 


7824 


7897 


7969 


8041 


8113 


8185 


8256 


8328 


1 


7 


7 


6 


6 6 


g 32-. 
'O 33-. 


8399 


8470 


8541 


8612 


8683 


8753 


8824 


8894 


8964 


9034 


2 


14 


13 


13 


12 12 


9104 


9174 


9214 


9313 


9382 


9451 


9521 


9589 


9658 


9727 


3 


21 


20 


19 


19 18 


CO 34-. 
60 350 
.S 35-. 


9795 


9864 


9932 


0000 


0068 


0136 


0203 


0271 


0338 


0406 


4 


28 


26 


26 


25 24 


7.0473 


0540 


0607 


0674 


0740 


0807 


0873 


0940 


1006 


1072 


5 


35 


33 


32 


31 30 


M 36- 


1138 


1204 


1269 


1335 


1400 


1466 


1531 


1596 


1661 


1726 


6 


42 


40 


38 


37 36 


§ 37-. 


1791 


1855 


1920 


1984 


2048 


2112 


2177 


2240 


2304 


2368 


7 


49 


46 


45 


43 42 


;^ 38-. 


2432 


2495 


2558 


2622 


2685 


2748 


2811 


2874 


2936 


2999 


8 


56 


53 


51 


50 48 


^ 39-. 
400 
40-. 


3061 


3124 


3186 


3248 


3310 


3372 


3434 


3496 


3558 


3619 


9 


63 


59 


58 


56 54 


3681 


3742 


3803 


3864 


3925 


3986 


4047 


4108 


4169 


4229 




58 


56 


55 


54 53 


41-. 


4290 


4350 


4410 


4470 


4530 


4590 


4650 


4710 


4770 


4829 


1 


6 


6 


6 


5 5 


42-. 


4889 


4948 


5007 


5067 


5126 


5185 


5244 


5302 


5361 


5420 


2 


12 


11 


11 


11 11 


43-. 


5478 


5537 


5595 


5654 


5712 


5570 


5828 


5886 


5944 


6001 


3 


17 


17 


17 


16 16 


44-. 
450 
45-. 


6059 


6117 


6174 


6232 


6289 


6346 


6403 


6460 


6517 


6574 


4 


23 


22 


22 


22 21 


6631 


6688 


6744 


6801 


6857 


6914 


6970 


7026 


7082 


7138 


5 


29 


28 


28 


27 27 


46-. 


7194 


7250 


73C6 


7362 


7418 


7473 


7529 


7584 


7639 


7695 


6 


35 


34 


33 


32 32 


47-. 


7750 


7805 


7860 


7915 


7970 


8025 


8079 


8134 


8188 


8243 


7 


41 


39 


39 


38 37 


48-. 


8297 


8352 


8406 


8460 


8514 


8568 


8622 


8676 


8730 


8784 


8 


46 


45 


44 


43 42 


49-. 
500 


8837 


8891 


8944 


8998 


9051 


9105 


9158 


9211 


9264 


9317 


9 


52 


50 


50 


49 48 



*Cube Roots are obtained directly as in the logarithmic tables, using the 
Proportional Parts tables for interpolation. 

Cubes may be obtained by inverse interpolation. 

The dash (-) is to be supplied by figs. to 9 at the head of the respective 
column. 



ENGINEERS' TABLES-^CUBE ROOTS. 



23 



500 to 1,000 
500,000 to 1,000,000 

— And Any Other Numbers; 



.5tol 
.0005 to .001 

OR Decimals. 



Cube. 


^0. 


1. 


2. 


3. 


4. 


5. 


6. 


7. 


8. 


9. 


P. P. 


500 
50-. 


7.9370 


9423 


9476 


9528 


9581 


9634 


9686 


9739 


9791 


9843 




53 52 5150 49 


51-. 


9896 


9948 


0000 


D052 


0104 


0156 


0208 


0260 


0311 


0363 


1 


5 5 5 5 5 


52-. 


8.0415 


0466 


0517 


0569 


0620 


0671 


0723 


0774 


0825 


0876 


2 


11 10 10 10 10 


53-. 


0927 


0978 


1028 


1079 


1130 


1180 


1231 


1281 


1332 


1382 


3 


16 16 15 15 15 


54-. 
550 
55-. 


1433 


1483 


1533 


1583 


1633 


1683 


1733 


1783 


1833 


1882 


4 


21 21 20 20 20 


1932 


1982 


2031 


2081 


2130 


2180 


2229 


2278 


2327 


2377 


5 


27 26 26 25 25 


56-. 


2426 


2475 


2524 


2573 


2621 


2670 


2719 


2768 


2816 


2865 


6 


32 31 31 30 29 


^ 57-. 


2913 


2962 


3010 


3059 


3107 


3155 


3203 


3251 


3300 


3348 


7 


37 36 36 35 34 


58-. 
2 59-. 
<D 600 
g 60-. 


3396 


3443 


3491 


3539 


3587 


3634 


3682 


3730 


3777 


3825 


8 


42 42 41 40 39 


3872 


3919 


3967 


4014 


4061 


4108 


4155 


4202 


4249 


4296 


9 


48 47 46 45 44 


4343 


4390 


4437 


4484 


4530 


4577 


4623 


4670 


4716 


4763 




48 47 46 43 44 


a 61-. 

5 62-. 


4809 


4856 


4902 


4948 


4994 


5040 


5086 


5132 


5178 


5224 


1 


5 5 5 5 4 


5270 


5316 


5362 


5408 


5453 


5499 


5544 


5590 


5635 


5681 


2 


10 9 9 9 9 


C 63-. 


5726 


5772 


5817 


5862 


5907 


5952 


5997 


6043 


6088 


6132 


3 


14 14 14 14 13 


S 64-. 
ft 650 

^ 11'- 
S 67-. 


6177 


6222 


6267 


6312 


6357 


6401 


6446 


6490 


6535 


6579 


4 


19 19 18 18 18 


6624 


6668 


6713 


6757 


6801 


6845 


6890 


6934 


6978 


7022 


5 


24 24 23 23 22 


7066 


7110 


7154 


7198 


7241 


7285 


7329 


7373 


7416 


7460 


6 


29 28 28 27 26 


7503 


7547 


7590 


7634 


7677 


7721 


7764 


7807 


7850 


7893 


7 


34 33 32 32 31 


^ 68-. 


7937 


7980 


8023 


8066 


8109 


8152 


8194 


8237 


8280 


8323 


8 


38 38 37 36 35 


'O 69-. 
'-' 700 
II 70-. 


8366 


8408 


4451 


8493 


8536 


8578 


8621 


8663 


8706 


8748 


9 


43 42 41 41 40 


8790 


8833 


8875 


8917 


8959 


9001 


9043 


9085 


9127 


9169 




43 42 41 40 


B 71-. 
'^ 72- 


9211 


9253 


9295 


9337 


9378 


9420 


9462 


9503 


9545 


9587 


1 


4 4 4 4 


9628 


9670 


9711 


9752 


9794 


9835 


9876 


9918 


9959 


0000 


2 


9 8 8 8 


« 73-. 


9.0041 


0082 


0123 


0164 


0205 


0246 


0287 


0328 


0369 


0410 


3 


13 13 12 12 


2 74-. 
S 750 
S 75-. 


0450 


0491 


0532 


0572 


0613 


0654 


0694 


0735 


0775 


0816 


4 


17 17 16 16 


0856 


0896 


0937 


0977 


1017 


1057 


1098 


1138 


1178 


1218 


5 


22 21 21 20 


;^ 76-. 


1258 


1298 


1338 


1378 


4118 


1458 


1498 


1537 


1577 


1617 


6 


26 25 25 24 


1 77-. 
S 78-. 


1657 


1696 


1736 


1775 


1815 


1855 


1894 


1933 


1973 


2012 


7 


30 29 29 28 


2052 


2091 


2130 


2170 


2209 


2248 


2287 


2326 


2365 


2404 


8 


34 34 33 32 


a 79-. 

r^ 800 

g 80-. 

S 81-. 


2443 


2482 


2521 


2560 


2599 


2638 


2677 


2716 


2754 


2793 


9 


39 38 37 36 


2832 


2870 


2909 


2948 


2986 


3025 


3063 


3102 


3140 


3179 




39 38 37 36 


3217 


3255 


3294 


3332 


3370 


3408 


3447 


3485 


3523 


3561 


1 


4 4 4 4 


g 82-. 


3599 


3637 


3675 


3713 


3751 


3789 


3827 


3865 


3902 


3940 


2 


8 8 7 7 


nd 83-. 


3978 


4016 


4053 


4091 


4129 


4166 


4204 


4241 


4279 


4316 


3 


12 11 11 11 


^ 84-. 


4354 


4391 


4429 


4466 


4503 


4541 


4578 


4615 


4652 


4690 


4 


16 15 15 14 


bp 850 
§, 85-. 


























4727 


4764 


4801 


4838 


4875 


4912 


4949 


4986 


5023 


5060 


5 


20 19 19 18 


gJ 86-. 


5097 


5134 


5171 


5207 


5244 


5281 


5317 


5354 


5391 


5427 


6 


23 23 22 22 


oj 87-. 


5464 


5501 


5537 


5574 


5610 


5647 


5683 


5719 


5756 


5792 


7 


27 27 26 25 


g 88-. 
^ 89-. 
900 
90-. 


5828 


5865 


5901 


5937 


5973 


6010 


6046 


6082 


6118 


6154 


8 


31 30 30 29 


6190 


6226 


6262 


6298 


6334 


6370 


6406 


6442 


6477 


6513 


9 


35 34 33 32 


6549 


6585 


6620 


6656 


6692 


6727 


6763 


6799 


6834 


6870 




35 34 33 


91-. 


6905 


6941 


6976 


7012 


7047 


7082 


7118 


7153 


7188 


7224 


1 


4 3 3 


92-. 


7259 


7294 


7329 


7364 


7400 


7435 


7470 


7505 


7540 


7575 


2 


7 7 7 


93-. 


7610 


7645 


7680 


7715 


7750 


7785 


7819 


7854 


7889 


7924 


3 


11 10 10 


94-. 
950 
95-. 


7959 


7993 


8028 


8063 


8097 


8132 


8167 


8201 


8236 


8270 


4 


14 14 13 


8305 


8339 


8374 


8408 


8443 


8477 


8511 


8546 


8580 


8614 


5 


18 17 17 


96-. 


8648 


8683 


8717 


8751 


8785 


8819 


8854 


8888 


8922 


8956 


6 


21 20 20 


97-. 


8990 


9024 


9058 


9092 


9126 


9160 


9194 


9227 


9261 


9295 


7 


25 24 23 


98-. 


9329 


9363 


9396 


9430 


9464 


9497 


9531 


9565 


9598 


9632 


8 


28 27 26 


99-. 


9666 9699 


9733 


9766 


9800 


9833 


9866 


9900 


9933 9967 9 


32 31 30 


1000 




1 


1 


1 







Ex.— Cube root of 887.2? 



Solution— 9.6082 for 887 
7 for ^ 

Ans. 



9.6089 for 887.2 

The dash (-) is to be supplied by figs. to 9 at the head of the respective 
column. 



24 



2— POWERS, ROOTS AND RECIPROCALS, 



1 to 1.5 
1,000 to 1,500 .001 to .0015 

1,000,000 to 1,500,000 .000001 to .0000015 

5. — Cube Roots (and Cubes*) of Numbers 1,000 to 1,500. 



Cube 


V 0. 


1. 


2. 


3. 


4.- 


5. 


6. 


7. 


8. 


9. 


P. P. 


1000 


























1,00-. 


lO.OOOO 


0033 


0067 


0100 


0133 


0166 


0200 


0233 


0266 


0299 




34 33 32 31 


1,01-. 


0332 


0365 


0398 


0431 


0465 


0498 


0531 


0563 


0596 


0629 


1 


3 3 3 3 


i;o2-. 


0662 


0695 


0728 


0761 


0794 


0826 


0859 


0892 


0925 


0957 


2 


7 7 6 6 


1,03-. 


0990 


1023 


1055 


1088 


1121 


1153 


1186 


1218 


1251 


1283 


3 


10 10 10 9 


1,04-. 
1050 
1,05-. 


1316 


1348 


1381 


1413 


1446 


1478 


1510 


1543 


1575 


1607 


4 


14 13 13 12 


1640 


1672 


1704 


1736 


1769 


1801 


1833 


1865 


1897 


1929 


5 


17 17 16 16 


1.06-. 


1961 


1993 


2025 


2057 


2089 


2121 


2153 


2185 


2217 


2249 


6 


20 20 19 19 


1,07-. 


2281 


2313 


2345 


2376 


2408 


2440 


2472 


2503 


2535 


2567 


7 


24 23 22 22 


+5 1,08-. 


2599 


2630 


2662 


2693 


2725 


2757 


2788 


2820 


2851 


2883 


8 


27 26 26 25 


O 1,09-. 
2 1100 
<o 1,10-. 


2914 


2946 


2977 


3009 


3040 


3071 


3103 


3134 


3165 


3197 


9 


31 30 29 28 


3228 


3259 


3291 


3322 


3353 


3384 


3415 


3447 


3478 


3509 




32 3130 29 


6 1.11-. 


3540 


3571 


3602 


3633 


3664 


3695 


3726 


3757 


3788 


3819 


1 


3 3 3 3 


G 1.12-. 
'S 1,13-. 


3850 


3881 


3912 


3943 


3973 


4004 


4035 


4066 


4097 


4127 


2 


6 6 6 6 


4158 


4189 


4219 


4250 


4281 


4311 


4342 


4373 


4404 


4434 


3 


10 9 9 9 


C 1.14-- 
'3 1150 

a 1,15-. 


4464 


4495 


4525 


4556 


4586 


4617 


4647 


4678 


4708 


4739 


4 


13 12 12 12 


4769 


4799 


4830 


4860 


4890 


4921 


4951 


4981 


5011 


5042 


5 


16 16 15 15 


- 1,16-. 
§ 1,17-. 
•S 1,18-. 


5072 


5102 


5132 


5162 


5192 


5223 


5253 


5283 


5313 


5343 


6 


19 19 18 17 


5373 


5403 


5433 


5463 


5493 


5523 


5553 


5583 


5612 


5642 


7 


22 22 21 20 


5672 


5702 


5732 


5762 


5791 


5821 


5851 


5881 


5910 


5940 


8 


26 25 24 23 


^ 1.19- 


5970 


6000 


6029 


6059 


6088 


6118 


6148 


6177 


6207 


6236 


9 


29 28 27 26 


'O 1200 


























^ 1,20-. 


6266 


6295 


6325 


6354 


6384 


6413 


6443 


6472 


6501 


6531 




30 29 28 27 


II 1.21-. 


6560 


6590 


6619 


6648 


6678 


6707 


6736 


6765 


6795 


6824 


1 


3 3 3 3 


Jg 1,22-. 
3 1,23-. 


6853 


6882 


6911 


6940 


6970 


6999 


7028 


7057 


7086 


7115 


2 


6 6 6 5 


7144 


7173 


7202 


7231 


7260 


7289 


7318 


7347 


7376 


7405 


3 


9 9 8 8 


" 1,24-. 


7434 


7463 


7491 


7520 


7549 


7580 


7607 


7635 


7664 


7693 


4 


12 12 11 11 


^ 1250 


























-^^ 1,25-. 


7722 


7750 


7779 


7808 


7837 


7865 


7894 


7922 


7951 


7980 


5 


15 15 14 14 


G 1,26-. 


8008 


8037 


8065 


8094 


8122 


8051 


8179 


8208 


8236 


8265 


6 


18 17 17 16 


w 1.27-. 


8293 


8322 


8350 


8378 


8407 


8435 


8463 


8492 


8520 


8548 


7 


21 20 20 19 


G 1'28-- 
1 1'29-. 
S 1300 


8577 


8605 


8633 


8661 


8690 


8718 


8746 


8774 


8802 


8831 


8 


24 23 22 22 


8859 


8887 


8915 


8943 


8971 


8999 


9027 


9055 


9083 


9111 


9 


27 26 25 24 


























,-H 1,30-. 


9139 


9167 


9195 


9223 


9251 


9279 


9307 


9335 


9363 


9391 




29 28 27 26 


g 1.31-. 
.S 1,32-. 


9418 


9446 


9474 


9502 


9530 


9757 


9585 


9613 


9641 


9668 


1 


3 3 3 3 


9696 


9724 


9752 


9779 


9807 


9834 


9862 


9890 


9917 


9945 


2 


6 6 5 5 


g 1,33-. 


9972 


0000 


0028 


D055 


0083 


0110 


0138 


0165 


0193 


0220 


3 


9 8 8 8 


'O 1,34-. 


11.0247 


0275 


0302 


0330 


0357 


0384 


0412 


0439 


0466 


0494 


4 


12 11 11 10 


CO 1350 


























5P 1,35-. 


0521 


0548 


0575 


0603 


0630 


0657 


0684 


0712 


0739 


0766 


5 


15 14 14 13 


S 1,36-. 


0793 


0820 


0847 


0875 


0902 


0929 


0956 


0983 


1010 


1037 


6 


17 17 16 16 


g> 1,37-. 
rt 1.38-. 


1064 


1091 


1118 


1145 


1172 


1199 


1226 


1253 


1280 


1307 


7 


20 20 19 18 


1334 


13JB1 


1387 


1414 


1441 


1468 


1495 


1522 


1548 


1575 


8 


23 22 22 21 


r^. 1,39-. 
^ 1400 
1,40-. 


1602 


1629 


1655 


1682 


1709 


1736 


1762 


1789 


1816 


1842 


9 


26 25 24 23 


1869 


1896 


1922 


1949 


1975 


2002 


2028 


2055 


2082 


2108 




27 26 25 


1,41-. 


2135 


2161 


2188 


2214 


2241 


2267 


2293 


2320 


2346 


2373 


1 


3 3 3 


1,42-. 


2399 


2425 


2452 


2478 


2505 


2531 


2557 


2583 


2610 


2636 


2 


5 5 5 


1.43-. 


2662 


2689 


2715 


2741 


2767 


2793 


2820 


2846 


2872 


2898 


3 


8 8 8 


1,44-. 
1450 
1,45-. 


2924 


2950 


2977 


3003 


3029 


3055 


3081 


3107 


3133 


3159 


4 


11 10 10 


3185 


3211 


3237 


3263 


3289 


3315 


3341 


3367 


3393 


3419 


5 


14 13 13 


1,46-. 


3445 


3471 


3496 


3522 


3548 


3574 


3600 


3626 


3652 


3677 


6 


16 16 15 


1,47-. 


3703 


3729 


3755 


3780 


3806 


3832 


3858 


3883 


3909 


3935 


7 


19 18 18 


1.48-. 


3960 


3986 


4012 


4037 


4063 


4089 


4114 


4140 


4165 


4191 


8 


22 21 20 


1 49-. 


4206 


4242 


4268 


4293 


4319 


4344 


4370 


4395 


4421 


4446 


9 


24 23 23 


1§00 



























*Ex.— Cubeof 1.07275? 



Solution— 1.234 for 1.07260 

.0005 for 15 

Ans. 1.2345 for 1.07275 



The dash (-) is to be supplied by figs. to 9 at the head of the respective 
column. 



ENGINEERS' TABLES— SQ. RTS. OF 5th POWERS. 



25 



6. — Square Roots of Fifth Powers op Numbers, Advancing by 0.25. 



N 


Vn^m 


.25 


.50 


.75 


N. 


ViVB.oo 


.25 


.50 


.75 


0. 


Zero. 


.03125 


.17678 


.48714 


50. 


17678. 


17899. 


18123. 


18348. 


1. 


1.0000 


1.7469 


2.7557 


4.0513 


51. 


18575. 


18803. 


19033. 


19265. 


2. 


5.6569 


7.5937 


9.8821 


12.541 


52. 


19499. 


19734. 


19971. 


20210. 


3. 


15.588 


19.042 


22.918 


27.232 


53. 


20450. 


20692. 


20936. 


21181. 


4. 


32.000 


37.237 


42.957 


49.174 


54. 


21428. 


21677. 


21928. 


22180. 


5. 


55.902 


63.154 


70.943 


79.281 


55. 


22434. 


22690. 


22947. 


23207. 


6. 


88.182 


97.656 


107.72 


118.38 


56. 


23468. 


23731. 


23995. 


24261. 


7. 


129.64 


141.53 


154.05 


167.21 


57. 


24529. 


24799. 


25071. 


25344. 


8. 


181.02 


195.50 


210.64 


226.48 


58. 


25620. 


25896. 


26175. 


26456. 


9. 


243.00 


260.23 


278.17 


296.83 


59. 


26738. 


27022. 


27308. 


27596. 


10. 


316.23 


336.36 


357.25 


378.90 


60. 


27886. 


28177. 


28470. 


28765. 


11. 


401.31 


424.50 


448.48 


473.25 


61. 


29062. 


29361. 


29661. 


29963. 


12. 


498.83 


525.22 


552.43 


580.46 


62. 


30268. 


30574. 


30882. 


31191. 


13. 


609.34 


639.06 


669.63 


701.06 


63. 


31503. 


31816. 


32132. 


32449. 


14. 


733.36 


766.54 


800.61 


835.56 


64. 


32768. 


33089. 


33412. 


33736. 


15. 


871.42 


908.19 


945.87 


984.47 


65. 


34063. 


34392. 


34722. 


35054. 


" 16. 


1024.0 


1064.5 


1105.9 


1148.2 


66. 


35388. 


35724. 


36062. 


36402. 


17. 


1191.6 


1235.9 


1281.1 


1327.4 


67. 


36744. 


37088. 


37433. 


37781. 


18. 


1374.6 


1422.8 


1472.1 


1522.3 


68. 


38131. 


38482. 


38835. 


39191. 


19. 


1573.6 


1625.8 


1679.1 


1733.5 


69. 


39548. 


39907. 


40268. 


40631. 


20. 


1788.9 


1845.3 


1902.8 


1961.3 


70. 


40996. 


41363. 


41733. 


42103. 


> 21. 


2020.9 


2081.6 


2143.4 


2206.2 


71. 


42476. 


42851. 


43228. 


43607. 


22. 


2270.2 


2335.2 


2401.4 


2468.7 


72. 


43988. 


44371. 


44755. 


45142. 


1.23. 


2537.0 


2606.5 


2677.1 


2748.9 


73. 


45531. 


45922. 


46315. 


46709. 


24. 


2821.8 


2895.9 


2971.1 


3047.5 


74. 


47106. 


47505. 


47906. 


48309. 


25. 


3125.0 


3203.7 


3283.6 


3364.7 


75. 


48714. 


49121. 


49530. 


49941. 


26. 


3446.9 


3530.4 


3615.1 


3700.9 


76. 


50354. 


50769. 


51186. 


51606. 


27. 


3788.0 


3876.3 


3965.8 


4056.6 


77. 


52027. 


52450. 


52875. 


53303. 


, 28. 


4148.5 


4241.8 


4336.2 


4431.9 


78. 


53732. 


54164. 


54598. 


55033. 


29. 


4528.9 


4627.2 


4726.7 


4827.4 


79. 


55471. 


55911. 


56353. 


56797. 


30. 


4929.5 


5032.8 


5137.5 


5243.4 


80. 


57243. 


57692. 


58142. 


58594. 


31. 


5350.6 


5459.2 


5569.0 


5680.1 


81. 


59049. 


59506. 


59964. 


60425. 


32. 


5792.6 


5906.4 


6021.6 


6138.0 


82. 


60888. 


61354. 


61821. 


62290. 


33. 


6255.8 


6375.0 


6495.5 


6617.4 


83. 


62762. 


63235. 


63711. 


64189. 


34. 


6740.6 


6865.2 


6991.1 


7118.5 


84. 


64669 . 


65151. 


65636. 


66122. 


35. 


7247.2 


7377.3 


7508.8 


7641.7 


85. 


66611. 


67102. 


67595. 


68090. 


36. 


7776.0 


7911.7 


8048.8 


8187.3 


86. 


68588. 


69087. 


69589. 


70093. 


37. 


8327.3 


8468.7 


8611.5 


8755.7 


87. 


70599. 


71107. 


71618. 


72130. 


38. 


8901.4 


9048.5 


9197.1 


9347.2 


88. 


72646. 


73162. 


73681. 


74203 


39. 


9498.6 


9651.6 


9806.0 


9961.9 


89. 


74727. 


75252. 


75781. 


76311. 


p40. 


10119. 


10278. 


10438. 


10600. 


90. 


76843. 


77378. 


77915. 


78454. 


' 41. 


10764. 


10929. 


11095. 


11263. 


91. 


78996. 


79539. 


80085. 


80633. 


o42. 


11432. 


11603. 


11775. 


11949. 


92. 


81184. 


81737. 


82291. 


82849. 


43. 


- 12125. 


12302. 


12480. 


12660. 


93. 


83408. 


83970. 


84534. 


85100. 


44. 


12842. 


. 13025. 


13210. 


13396. 


94. 


85668. 


86239. 


86812. 


87387. 


45. 


13584. 


13774 


13965. 


14157. 


95. 


87965. 


88545. 


89127. 


89711. 


46. 


14351. 


14547. 


14745. 


14944. 


96. 


90298. 


90887. 


91478. 


92072. 


47. 


15144. 


15346. 


15550. 


15756. 


97. 


92668. 


93266. 


93867. 


94470. 


48. 


15963. 


16171. 


16382. 


16593. 


98. 


95075. 


95682. 


96292. 


96904. 


49. 


16807. 


17022. 


17239. 


17458. 


99. 


97519. 


98136. 


98755. 


99376. 


SO. 


17678. 


17899. 


18123. 


18348. 


100. 


100000. 


100626. 


101255. 


101886. 



Note that V N^ = N^ = m "^ ^ = N^ X N^ = N^ X ^ N. Hence, the 
above values may be obtained by multiplying together the square and 
the square root of the respective numbers. 



26 



i.—POWBRS, ROOTS AND RECIPROCALS. 



7. — Fifth Powers* (and Fifth Roots) of Numbers 1 to 10 — 



N 


N^ 


2 


4 


5 


6 


8 


1.00 


Unity 


1.01004 


1.02016 


1.02525 


1.03036 


1.04064 


.01 


1.05101 


1.06146 


1.07199 


1.07728 


1.08260 


1.09330 


.02 


1.10408 


1.11495 


1.12590 


1.13141 


1.13694 


1.14806 


.03 


1.15927 


1.17057 


1.18196 


1.18769 


1.19344 


1.20500 


.04 


1.21665 


1.22840 


1.24023 


1.24618 


1.25216 


1.26417 


1.05 


1.27628 


1.28848 


1.30078 


1.30696 


1.31317 


1.32565 


«• .06 


1.33823 


1.35090 


1.36367 


1.37009 


1.37653 


1.38949 


^ .07 


1.40255 


1.41571 


1.42896 


1.43563 


1.44232 


1.45577 


*H .08 


1.46933 


1.48298 


1.49674 


1.50366 


1.51060 


1.52456 


O .09 


1.53862 


1.55279 


1.56706 


1.57424 


1.58144 


1.59592 


$ 1.0 


Unity 


1.10408 


1.21665 


1.27628 


1.33823 


1.46933 


1 -1 


1.61051 


1.76234 


1.92541 


2.00135 


2.11034 


2.28775 


? -2 


2.48832 


2.70271 


2.93163 


3.05176 


3.17580 


3.43597 


S -3 


3.71293 


4.00746 


4.32040 


4.48403 


4.65259 


5.00490 


^ .4 

a 


5.37824 


5.77353 


6.19174 


6.40973 


6.63383 


7.10082 


7.59375 


8.11368 


8.66171 


8.94661 


9.23896 


9.84658 


§ .6 


10.4858 


11.1577 


11.8637 


12.2298 


12.6049 


13.3828 




14.1986 


15.0537 


15.9495 


16.4131 


16.8874 


17.8690 


B 2.0 


18.8957 


19.9690 


21.0906 


21.6700 


22.2620 


23.4849 


24.7610 


26.0919 


27.4795 


28.1951 


28.9255 


30.4317 


32.0000 


33.6323 


35.3306 


36.2051 


37.0968 


38.9329 


*s -1 


40.8410 


42.8232 


44.8817 


45.9401 


47.0185 


49.2360 


^ .2 


51.5363 


53.9219 


56.3949 


57.6650 


58.9579 


61.6133 


S .3 


64.3634 


67.2109 


70.1583 


71.6703 


73.2082 


76.3633 


II -4 


79.6262 


82.9998 


86.4867 


88.2735 


90.0898 


93.8120 


^ 2.5 


97.6563 


101.626 


105.723 


107.820 


109.951 


114.314 


: :? 


118.814 


123.454 


128.239 


130.686 


133.171 


138.253 


143.489 


148.883 


154.438 


157.276 


160.157 


166.044 


O .8 


172.104 


178.339 


184.753 


188.029 


191.351 


198.136 


S -9 


205.111 


212.283 


219.653 


223.414 


227.226 


235.007 


T§ 3.0 


243.000 


251.209 


259.638 


263.936 


268.292 


277.175 


B i 


286.292 


295.647 


305.245 


310.136 


315.091 


325.189 


335.544 


346.162 


357.047 


362.591 


368.204 


379.637 


?, -3 


391.354 


403.358 


415.655 


421.914 


428.249 


441.147 


J .4 


454.354 


467.876 


481.717 


488.760 


495.884 


510.383 


"a 

^ 3.5 


525.219 


540.398 


555.925 


563.822 


571.808 


588.051 


1 :? 


604.662 


621.646 


639.009 


647.835 


656.758 


674.899 


693.440 


712.385 


731.742 


741.577 


751.517 


771.719 


g .8 


792.352 


813.424 


834.941 


845.870 


856.913 


. 879.343 


^ .9 


902.242 


925.615 


949.470 


961.580 


973.814 


998.655 


M 4.0 


1024.00 ' 


1049.86 


1076 23 


1089.62 


1103.14 


1130.58 


.S .1 


1158.56 


1187.10 


1216.19 


1230.95 


1245.85 


1276.09 


§ .2 
rt .3 


1306.91 


1338.33 


1370.34 


1386.58 


1402.97 


1436.21 


1470.08 


1504.59 


1539.74 


1557.57 


1575.55 


1612.02 


6 ' 


1649.16 


1686.99 


1725.50 


1745.02 


1764.71 


1804.64 


4.5 


1845.28 


1886.65 


1928.77 


1950.10 


1971.62 


• 2015.24 


.6 


2059.63 


2104.80 


2150.75 


2174.03 


2197.50 


2245.07 


.7 


2293.45 


2342.66 


2392.72 


2418.07 


2443.63 


2495.40 


.8 


2548.04 


2601.57 


2655.99 


2683.54 


2711.32 


2767.57 


.9 


2824.75 


2882.87 


2941.94 


2971.84 


3001.98 


3063.00 


5.0 


3125.00 


3188.00 


3252.01 


3284.41 


3317.05 


3383.13 



♦Powers (iV°) are obtained directly as in the logarithmic tables; roots (N), 
by inverse interpolation. 

Note that N^ = N^-^^ = N^ X N^. Hence, the above values may be ob- 
tained by multiplying together the square and the cube of the respective 
ntimbers. 



ENGINEERS' TABLES— FIFTH POWERS, 
— ^And Any Whole Number; or Decimal. 



27 



N 


iNr« 


2 


4 


5 


6 


8 


5.0 


3125.00 


3188.00 


3252.01 


3284.41 


3317.05 


3383.13 


.1 


3450.25 


3518.44 


3587.70 


3622.73 


3658.04 


3729.49 


.2 


3802.04 


3875.72 


3950.54 


3988.38 


4026.51 


4103.64 


.3 


4181.95 


4261.46 


4342.17 


4382.97 


4424.09 


4507.25 


.4 


4591.65 


4677.31 


4764.25 


4808.20 


4852.47 


4942.00 


5.5 


5032.84 


5125.02 


5218.54 


5265.81 


5313.42 


5409.67 


.6 


5507.32 


5606.37 


5706.84 


5757.61 


5808.74 


5912.10 


.7 


6016.92 


6123.22 


6231.02 


6285.49 


6340.34 


6451.18 


.8 


6563.57 


6677.52 


6793.04 


6851.40 


6910.16 


7028.89 


.9 


7149.24 


7271.24 


7394.89 


7457.36 


7520.24 


7647.26 


6.0 


7776.00 


7906.47 


8038.68 


8105.45 


8172.65 


8308.41 




8445.96 


8585.33 


8726.54 


8797.83 


8869.59 


9014.52 


k .2 


9161.33 


9310.05 


9460.69 


9536.74 


9613.28 


9767.83 


.3 


9924.37 


10082.9 


10243.5 


10324.5 


10406.0 


10570.7 


.4 


10737.4 


10906.2 


11077.2 


11163.5 


11250.3 


11425.5 


6.5 


11602.9 


11782.5 


11964.3 


12056.1 


12148.4 


12334.7 


.6 


12523.3 


12714.2 


12907.5 


13004.9 


13103.0 


13300.9 


.7 


13501.3 


13704.0 


13909.1 


14012.6 


14116.7 


14326.8 


.8 


14539.3 


14754.4 


14972.0 


15081.8 


15192.2 


15414.9 


.9 


15S40.3 


15868.3 


16098.9 


16215.3 


16332.2 


16568.3 


7.0 


16807.0 


17048.5 


17292.7 


17415.9 


17539.8 


17789.6 


.1 


18042.3 


18297.8 


18556.3 


18686.6 


18817.6 


19081.9 


.2 


19349.2 


19619.4 


19892.7 


20030.4 


20168.9 


20448.3 


.3 


20730.7 


21016.3 


21305.0 


21450.5 


21596.8 


21891.8 


.4 


22190.1 


22491.5 


22796.3 


22949.9 


23104.4 


23415.7 


7.5 


23730.5 


24048.6 


24370.1 


24532.1 


24695.0 


25023.4 


.6 


25355.3 


25690.6 


26029.6 


26200.3 


26372.1 


26718.1 


.7 


27067.8 


27421.2 


27778.3 


27958.2 


28139.0 


28503.5 


.8 


28871.7 


29243.8 


29619.7 


29809.1 


29999.4 


30383.0 


.9 


30770.6 


31162.1 


31557.5 


31756.7 


31957.0 


32360.4 


8.0 


32768.0 


33179.7 


33595.4 


33804.9 


34015.4 


34439.5 


.1 


34867.8 


35300.4 


35737.3 


35957.4 


36178.5 


36624.1 


.2 


37074.0 


37528.3 


37987.1 


38218.1 


38450.3 


38918.1 


.3 


39390.4 


39867.3 


40348.8 


40591.3 


40834.9 


41325.7 


.4 


41821.2 


42321.4 


42826.4 


. 43080.8 


43336.3 


43851.0 


' 8.5 


44370.5 


44895.0 


45424.4 


45691.0 


45958.8 


46498.2 


.6 


47042.7 


47592.3 


48146.9 


48426.2 


48706.8 


49271.8 


.7 


49842.1 


50417.6 


50998.5 


51290.9 


51584.7 


52176.2 


.8 


52773.2 


53375.6 


53983.6 


54289.6 


54597.0 


55216.0 


.9 


55840.6 


56470.9 


57106.8 


57426.9 


57748.4 


58395.8 


9.0 


59049.0 


59708.0 


60372.9 


60707.6 


61043.7 


61720.4 


.1 


62403.2 


63092.0 


63786.8 


64136.5 


64487.8 


65194.9 


.2 


65908.2 


66627.6 


67353.5 


67718.7 


68085.6 


68824.0 


.3 


69568.8 


70320.1 


71077.9 


71459.2 


71842.1 


72612.9 


.4 


73390.4 


74174.5 


74965.3 


75363.1 


75762.7 


76567.0 


9.5 


77378.1 


78196.0 


79020.9 


79435.9 


79852.7 


80691.4 


.6 


81537.3 


82390.2 


83250.1 


83682.9 


84117.3 


84991.8 


.7 


85873.4 


86762.4 


87658.7 


88109.6 


88562.3 


89473.5 


.8 


90392.1 


91318.2 


92251.9 


92721.6 


93193.3 


94142.2 


.9 


95099.0 


96063.5 


97035.8 


97524.9 


98015.9 


99004.0 


10.0 


100000 













Examples:- 



-5th root of 7500 = 5.95+ .007 = 
5throot of 0.75 = .944. 



5.957. 



28 2— POWERS, ROOTS AND RECIPROCALS. 

lto5. 

8. — Reciprocals of Numbers — 








1* 




















(Subtract the dil.) 


-3— - 


]V"0 


1 


2 


3 


4 


5 


6 


7 


8 


9 




P. F 




g 1.00 


Unity 


.9990 


.9980 


.9970 


.9960 


.9950 


.9940 


.9930 


.9921 


.9911 




-85 


-75 


65-55 


^ .01 


O.9901 


9891 


9881 


9872 


9862 


9852 


9843 


9833 


9823 


9814 


1 


9 


8 


7 6 


•S -02 


9804 


9794 


9785 


9775 


9766 


9756 


9747 


9737 


9728 


9718 


2 


17 


15 


13 11 


g .03 


9709 


9699 


9690 


9681 


9671 


9662 


9653 


9643 


9634 


9625 


3 


26 


23 


20 17 


^ .04 


9615 


9606 


9597 


9588 


9579 


9569 


9560 


9551 


9542 


9533 


4 


34 


30 


26 22 


^ 1.05 


0.9524 


.9515 


.9506 


.9497 


.9488 


.9479 


.9470 


.9461 


.9452 


.9443 


5 


43 


38 


33 28 


.2 -06 


9434 


9425 


9416 


9407 


9398 


9390 


9381 


9372 


9363 


9355 


6 


51 


45 


39 33 


a .07 

J .08 


9346 


9337 


9328 


9320 


9311 


9302 


9294 


9285 


9276 


9268 


7 


60 


53 


46 39 


9259 


9251 


9242 


9234 


9225 


9217 


9208 


9200 


9191 


9183 


8 


68 


60 


52 44 


tS .09 


9174 


9166 


9158 


9149 


9141 


9132 


9124 


9116 


9107 


9099 


9 


77 


68 


59 50 


1 1.0 

to .1 


Unity 


.9901 


.9804 


.9709 


.9615 


.9524 


.9434 


.9346 


.9259 


.9174 




-45 


-35 


-25-22 


O.9091 


9009 


8929 


8850 


8772 


8696 


8621 


8547 


8475 


8403 


1 


5 


4 


3 2 


1 -2 


8333 


8264 


8197 


8130 


8065 


8000 


7937 


7874 


7813 


7752 


2 


9 


7 


5 4 


S .3 


7692 


7634 


7576 


7519 


7463 


7407 


7353 


7299 


7246 


7194 


3 


14 


11 


8 7 


§ '' 


7143 


7092 


7042 


6993 


6944 


6897 


6849 


6803 


6757 


6711 


' 


18 


14 


10 9 


« 1.5 


0.6667 


.6623 


.6579 


.6536 


.6494 


.6452 


.6410 


.6369 


.6329 


.6289 


5 


23 


18 


13 11 


^ .6 


6250 


6211 


6173 


6135 


6098 


6061 


6024 


5988 


5952 


5917 


6 


27 


21 


15 13 


■^ -.7 


5882 


5848 


5814 


5780 


5747 


5714 


5682 


5650 


5618 


5587 


7 


32 


25 


18 15 


.S -8 


5556 


5525 


5495 


5464 


5435 


5405 


5376 


5348 


5319 


5291 


8 


36 


28 


20 18 


CO -9 


5263 


5236 


5208 


5181 


5155 


5128 


5102 


5076 


5051 


5025 


9 


41 


32 


23 20 


< .3 


O.5000 


.4975 


.4950 


.4926 


.4902 


.4878 


.4854 


.4831 


.4808 


.4785 




-18 


-16 


-14-12 


4762 


4739 


4717 


4695 


4673 


4651 


4630 


4608 


4587 


4566 


1 


2 


2 


1 1 


4545 


4525 


4505 


4484 


4464 


4444 


4425 


4405 


4386 


4367 


2 


4 


3 


3 2 


4348 


4329 


4310 


4292 


4274 


4255 


4237 


4219 


4202 


4184 


3 


5 


5 


4 4 


11 .4 


4167 


4149 


4132 


4115 


4098 


4082 


4065 


4049 


4032 


4016 


4 


7 


6 


6 5 


^ 2.5 


O.4000 


.3984 


.3968 


.3953 


.3937 


.3922 


.3906 


.3891 


.3876 


.3861 


5 


9 


8 


7 6 


a .6 


3846 


3831 


3817 


3802 


3788 


3774 


3759 


3745 


3731 


3717 


6 


11 


10 


8 7 


2 .7 


3704 


3690 


3676 


3663 


3650 


3636 


3623 


3610 


3597 


3584 


7 


13 


11 


10 8 


« .8 


3571 


3559 


3546 


3534 


3521 


3509 


3497 


3484 


3472 


3460 


8 


14 


13 


11 10 


o .9 


3448 


3436 


3425 


3413 


3401 


3390 


3378 


3367 


3356 


3344 


9 


16 


14 


13 11 




0.3333 


.3322 


.3311 


.3300 


.3289 


.3279 


.3268 


.3257 


.3247 


.3236 




-11 


-9 


-8 -7 


3226 


3215 


3205 


3195 


3185 


3175 


3165 


3155 


3145 


3135 


1 


1 


1 


1 1 


w .2 


3125 


3115 


3106 


3096 


3086 


3077 


3067 


3058 


3049 


3040 


2 


2 


2 


2 1 


8 .3 


3030 


3021 


3012 


3003 


2994 


2985 


2976 


2967 


2959 


2950 


3 


3 


3 


2 2 


^ .4 


2941 


2933 


2924 


2915 


2907 


2899 


2890 


2882 


2874 


2865 


4 


4 


4 


3 3 


►> 3.5 
< .6 


0.2857 


.2849 


.2841 


.2833 


.2825 


.2817 


.2809 


.2801 


.2793 


.2786 


5 


6 


5 


4 4 


2778 


2770 


2762 


2755 


2747 


2740 


2732 


2725 


2717 


2710 


6 


7 


5 


5 4 


■S -7 


2703 


2695 


2688 


2681 


2674 


2667 


2660 


2653 


2646 


2639 


7 


8 


6 


6 5 


.S .8 


2632 


2625 


2618 


2611 


2604 


2597 


2591 


2584 


2577 


2571 


8 


9 


7 


6 6 


O Q 


2564 


2558 


2551 


2545 


2538 


2532 


2525 


2519 


2513 


2506 


9 


10 


8 


7 6 


73 4.0 


O.2500 


.2494 


.2488 


.2481 


.2475 


.2469 


.2463 


.2457 


.2451 


.2445 




-6 


_ 


5 -4 


6 .1 


2439 


2433 


2427 


2421 


2415 


2410 


2404 


2398 


2392 


2387 


1 


0.6 


0.5 0.4 


•G .2 


2381 


2375 


2370 


2364 


2358 


2353 


2347 


2342 


2336 


2331 


2 


1.2 


1. 


0.8 


0) 


2326 


2320 


2315 


2309 


2304 


2299 


2294 


2288 


2283 


2278 


3 


1.8 


1. 


5 1.2 


2273 


2268 


2262 


2257 


2252 


2247 


2242 


2237 


2232 


2227 


4 


2.4 


2.0 1.6 


:S4.5 


0.2222 


.2217 


.2212 


.2208 


.2203 


.2198 


.2193 


.2188 


.2183 


.2179 


5 


3.0 


2.5 2.0 


? :5 


2174 


2169 


2165 


2160 


2155 


2151 


2146 


2141 


2137 


2132 


6 


3.6 


3.0 2.4 


2128 


2123 


2119 


2114 


2110 


2105 


2101 


2096 


2092 


2088 


7 


4.2 


3.5 2.8 


'm .8 


2083 


2079 


2075 


2070 


2066 


2062 


2058 


2053 


2049 


2045 


8 


4.8 


4.0 3.2 


1 "^ 


2041 


2037 


2033 


2028 


2024 


2020 


2016 


2012 


2008 


2004 


9 


5.4 


4 5 3.6 


65.0 


0.2000 


.1996 


.1992 


.1998 


.1984 


.1980 


.1976 


.1972 


.1969 


.1965 











^Reciprocals to four decimal places on this page. 
Ex.— Find reciprocal of 2.743? Solution— 0.3650 

-4 

Reciprocal of 27.43 = 0.03646: 010.2743 = 3.646; etc. 0.3646 Ans. 



ENGINEERS' TABLES— RECIPROCALS OF NUMBERS. 



29 



5 to 10. 



— Whole and Decimal. 





It 




















(Subtract the dif.) 


li ^ 


Wo 


1 


2 


3 


4 


5 


6 


7 


8 


9 




P. P. 


2 5.0 


0.1 


9960 


9920 


9881 


9841 


9802 


9763 


9724 


9685 


9646 




-38-36-34-32 


a .1 


0.1 9608 


9569 


9531 


9493 


9455 


9417 


9380 


9342 


9305 


9268 


1 


4 4 3 3 


o .2 


9231 


9194 


9157 


9120 


9084 


9048 


9011 


8975 


8939 


8904 


2 


8 7 7 6 


u 3 


8868 


8832 


8797 


8762 


8727 


8692 


8657 


8622 


8587 


8553 


3 


11 11 10 10 


s ■* 


8519 


8484 


8450 


8416 


8382 


8349 


8315 


8282 


8248 


8215 


4 


15 14 14 13 


a 5.5 


0.1 8182 


8149 


8116 


8083 


8051 


8018 


7986 


7953 


7921 


7889 


5 


19 18 17 16 


•S .6 


7857 


7825 


7794 


7762 


7731 


7699 


7668 


7637 


7606 


7525 


6 


23 22 20 19 


"1 -8 


7544 


7513 


7483 


7452 


7422 


7391 


7361 


7331 


7301 


7271 


7 


27 25 24 22 


7241 


7212 


7182 


7153 


7123 


7094 


7065 


7036 


7007 


6978 


8 


30 29 27 26 


1 •' 


6949 


6920 


6892 


6863 


6835 


6807 


6779 


6750 


6722 


6694 


9 


34 32 31 29 


^ 6.0 


0.1 6667 


6639 


6611 


6584 


6556 


6529 


6502 


6474 


6447 


6420 




-28-26-24-22 


^ .1 


6393 


6367 


6340 


6313 


6287 


6260 


6234 


6207 


6181 


6155 


1 


3 3 2 2 


'S -2 


6129 


6103 


6077 


6051 


6026 


6000 


5974 


5949 


5924 


5898 


2 


6 5 5 4 


a -3 


5873 


5848 


5823 


5798 


5773 


5748 


5723 


5699 


5674 


5649 


3 


8 8 7 7 


§: .4 


5625 


5601 


5576 


5552 


5528 


5504 


5480 


5456 


5432 


5408 


4 


11 10 10 9 


o 6.5 


0.1 5385 


5361 


5337 


5314 


5291 


5267 


5244 


5221 


5198 


5175 


5 


14 13 12 11 


^ .6 


5152 


5129 


5106 


5083 


5060 


5038 


5015 


4993 


4970 


4948 


6 


17 16 14 13 


.2 i 


4925 


4903 


4881 


4859 


4837 


4815 


4793 


4771 


4749 


4728 


7 


20 18 17 15 


4706 


4684 


4663 


4641 


4620 


4599 


4577 


4556 


4535 


4514 


8 


22 21 19 18 


I •' 


4493 


4472 


4451 


4430 


4409 


4388 


4368 


4347 


4327 


4306 


9 


25 23 22 20 


J 7.0 


0.1 4286 


4265 


4245 


4225 


4205 


4184 


4164 


4144 


4124 


4104 




-21-19-17-16 


"a .1 


4085 


4065 


4045 


4025 


4006 


3986 


3966 


3947 


3928 


3908 


1 


2 2 2 2 


> -2 

7 .3 


3889 


3870 


3850 


3831 


3812 


3793 


3774 


3755 


3736 


3717 


2 


4 4 3 3 


3699 


3680 


3661 


3643 


3624 


3605 


3587 


3569 


3550 


3532 


3 


6 6 5 5 


II -4 


3514 


3495 


3477 


3459 


3441 


3423 


3405 


3387 


3369 


3351 


4 


8 8 7 6 


-2 7.5 


0.1 3333 


3316 


3298 


3280 


3263 


3245 


3228 


3210 


3193 


3175 


5 


11 10 9 8 


S -6 


3158 


3141 


3123 


3106 


3089 


3072 


3055 


3038 


3021 


3004 


6 


13 11 10 10 


•« 


2987 


2970 


2953 


2937 


2920 


2903 


2887 


2870 


2853 


2837 


7 


15 13 12 11 


2821 


2804 


2788 


2771 


2755 


2739 


2723 


2706 


2690 


2674 


8 


17 15 14 13 


1 -^ 


2658 


2642 


2626 


2610 


2594 


2579 


2563 


2547 


2531 


2516 


9 


19 17 15 14 


a 8.0 


0.1 2500 


2484 


2469 


2453 


2438 


2422 


2407 


2392 


2376 


2361 




-15-14-13-12 


"^ .1 


2346 


2330 


2315 


2300 


2285 


2270 


2255 


2240 


2225 


2210 


1 


2 111 


S .2 


2195 


2180 


2165 


2151 


2136 


2121 


2107 


2092 


2077 


2063 


2 


3 3 3 2 


1 -'^ 


2048 


2034 


2019 


2005 


1990 


1976 


1962 


1947 


1933 


1919 


3 


5 4 4 4 


1905 


1891 


1876 


1862 


1848 


1834 


1820 


1806 


1792 


1779 


4 


6 6 5 5 


;$: 8.5 


0.1 1765 


1751 


1737 


1723 


1710 


1696 


1682 


1669 


1655 


1641 


5 


8 7 7 6 


2 -6 


1628 


1614 


1601 


1587 


1574 


1561 


1547 


1534 


1521 


1507 


6 


9 8 8 7 


C .7 


1494 


1481 


1468 


1455 


1442 


1429 


1416 


1403 


1390 


1377 


7 


11 10 9 8 


o .8 


1364 


1351 


1338 


1325 


1312 


1299 


1287 


1274 


1261 


1249 


8 


12 11 11 10 


a .9 


1236 


1223 


1211 


1198 


1186 


1173 


1161 


1148 


1136 


1123 


9 


14 13 12 11 


1 ^'^ 


0.1 1111 


1099 


1086 


1074 


1062 


1050 


1038 


1025 


1013 


1001 




-13-12-11-10 


0989 


0977 


0965 


0953 


0941 


0929 


0917 


0905 


0893 


0881 


1 


1111 


*§ '.2 


0870 


0858 


0846 


0834 


0823 


0811 


0799 


0787 


0776 


0764 


2 


3 2 2 2 


-0 .3 


0753 


0741 


0730 


0718 


0707 


0695 


0684 


0672 


0661 


0650 


3 


4 4 3 3 


a, .4 
^ 9.5 


0638 


0627 


0616 


0604 


0593 


0582 


0571 


0560 


0549 


0537 


4 


5 5 4 4 


0.1 0526 


0515 


0504 


0493 


0482 


0471 


0460 


0449 


0438 


0428 


5 


7 6 6 5 


1 :? 

ti -8 


0417 


0406 


0395 


0384 


0373 


0363 


0352 


0341 


0331 


0320 


6 


8 7 7 6 


0309 


0299 


0288 


0277 


0267 


0256 


0246 


0235 


0225 


0215 


7 


9 8 8 7 


0204 


0194 


0183 


0173 


0163 


0152 


0142 


0132 


Q121 


0111 


8 


11 10 9 8 


c« .9 


0101 


0091 


0081 


0070 


0060 


0050 


0040 


0030 


0020 


0010 


9 


12 11 10 9 


g 10.0 


0.1 0000 

























tReciprocals to five decimal places on this 
Ex. — Find reciprocal ot 1.2368 by 
the inverse method. 



page 



Solution— 8.08 for 0.12376 

.005 ^ 

8.085 for 0.12368 
/. Ans.=0.8085 for 1.2368 



30 



-POWERS, ROOTS AND RECIPROCALS. 



8a.— Square Roots of Reciprocals of Numbers (n). 

(Advancing by Tenths, from 1 to 21.) 

From Formula (page 15); ^1/n = "^n -^ n. 









By Logarithms 


: Log Vl/„ = i, 


:olog n. 








n 


.0 


.1 


.2 


.3 


.4 


.5 


.6 


.7 
.76696 


.8 
.74536 


.9 


1 


1.00000 


.95346 


.91287 


.87706 


.84515 


.81650 


.79057 


.72548 


2 


.70711 


.69007 


.67420 


.65938 


.64550 


.63246 


.62017 


.60858 


.59761 


.58722 


3 


.57735 


.56796 


.55902 


.55048 


.54233 


.53452 


.52705 


.51988 


.51299 


.50637 


4 


.50000 


.49386 


.48795 


.48224 


.47673 


.47140 


.46625 


.46126 


.45644 


.45175 


5 


.44721 


.44281 


.43853 


.43437 


.43033 


.42640 


.42258 


.41885 


.41523 


.41169 


6 


.40825 


.40489 


.40161 


.39841 


.39528 


.39223 


.38925 


.38633 


.38348 


.38069 


7 


.37796 


.37529 


.37268 


.37012 


.36761 


.36515 


.36274 


.36037 


.35806 


.35578 


8 


.35355 


.35136 


.34922 


.34711 


.34503 


.34300 


.34100 


.33903 


.33710 


.33520 


9 


.33333 


.33150 


.32969 


.32791 


.32616 


.32444 


.32275 


.32108 


.31944 


.31782 


10 


.31623 


.31466 


.31311 


.31159 


.31009 


.30861 


.30715 


.30571 


.30429 


.30289 


11 


.30151 


.30015 


.29881 


.29748 


.29617 


.29488 


.29361 


.29235 


.29111 


.28989 


12 


.28868 


.28748 


.28630 


.28513 


.28398 


.28284 


.28172 


.28061 


.27951 


.27842 


13 


.27735 


.27629 


.27524 


.27420 


.27318 


.27217 


.27116 


.27017 


.26919 


.26822 


14 


.26726 


.26631 


.26537 


.26444 


.26352 


.26261 


.26171 


.26082 


.25994 


.25906 


15 


.25820 


.25734 


.25649 


.25565 


.25482 


.25400 


.25318 


.25238 


.25158 


.25078 


16 


.25000 


.24922 


.24845 


.24769 


.24693 


.24618 


.24544 


.24470 


.24398 


.24325 


17 


.24254 


.24182 


.24112 


.24042 


.23973 


.23905 


.23837 


. 237-69 


.23702 


.23636 


18 


.23570 


.23505 


.23440 


.23376 


.23313 


.23250 


.23187 


.23125 


.23063 


.23002 


19 


.22942 


.22881 


.22822 


.22763 


.22704 


.22646 


.22588 


.22530 


.22473 


.22417 


20 


.22361 


.22305 


.22250 


.22195 


.22140 


.22086 


.22033 


.21979 


.21926 


.21874 



Reciprocals. — The reciprocal of a number is 1 divided by that number. 
Thus, the reciprocal of 2 77 = .361, and likewise the reciprocal of .361 is 
2.77; or 2.77=-V.36i and .361 = V2-77- Hence, to divide any quantity by a 
number, as 2.77, is equivalent to multiplying it by its reciprocal, V2.77 or .361. 

As multiplication is usually performed more readily than division, it is 
convenient to multiply by the reciprocal of a number rather than to divide 
by the number itself. 

Table 8* comprises the reciprocals of numbers from 1 to 10, advancing 
by decimals as in logarithmic tables, so as to include a wide range of num- 
bers. Note that changing the decimal point n places in the number equals 
a change of n places in the opposite direction in the reciprocal; also that the 
differences between any reciprocal and the following one is a minus difference, 
hence the proportional part must be subtracted. 

Example. — What is the reciprocal of 79.15? 

Solution. — From the table, the reciprocal of 7.91 = .12642; the propor- 
tional part for 5 (under the difference,— 16) is —8. Hence, the reciprocal 
of 7.915 is .12634, and of 79.15 is 0.012634. Ans. 

B. ARITHMETICAL OR COMMON TABLES. 

The preceding tables are arranged in decimal form, and are called 
Engineers' Tables ; while the arithmetical tables are in the form most com- 
monly used, and are arranged as follows: — 

Table 9 — Squares, cubes, square roots, cube roots, of numbers 1 to 1000, page 31: 
Nos. 0—130, page 31. Nos. 650— 910, page 36. Nos. 1430-1560, page 42. 
•• 130—390. " 32. " 910—1170, " 38. " 1560—1600, " 43. 

" 390—650, " 34. " 1170—1430, " 40. 
Table 9a— Squares of numbers 1600 to 1810, page 43. 
Table 10 — Square roots and cube roots of numbers 1600 to 3200, page 44: 
Nos. 1600—2120, page 44. Nos. 2640—3160, page 48. 

" 2120—2640, '• 46. " 3160—3200, " 50. 

Table 10a— Square roots of numbers 3200 to 3515. page 50. 
Table 11 — Reciprocals of numbers 1 to 1000, page 51: 

Nos. 1—325, page 51. Nos. 650— 975, page 53. 

" 325—650, •' 52. " 975—1000, " 54. 

Table 11a— Reciprocals of numbers 1001 to 1350, page 54. 



* Pages 28 and 29; 



COMMON TABLES— SQUARES, CUBES, ROOTS. 



31 



9. — Squares, Cubes, Square Roots, Cube Roots, of Numbers 

1 to 1600. 



No. 


Square 


Cube. 


Sq. Rt. 


Cu. Rt. 


No. 


Square 


Cube. 


Sq. Rt. 


Cu. Rt. 











O.OOOOOOO 


0.0000000 


65 


42 25 


274 625 


8.0622577 


4.0207256 


1 


1 


1 


1.0000000 


1.0000000 


6 


43 56 


287 496 


.1240384 


.0412401 


2 


4 


8 


.4142136 


.2599210 


7 


44 89 


300 763 


.1853528 


.0615480 


3 


9 


27 


.7320508 


.4422496 


8 


46 24 


314 432 


.2462113 


.0816551 


4 


16 


64 


2.0000000 


5874011 


9 


47 61 


328 509 


.3066239 


.1015661 


T) 


25 


125 


2.2360680 


1.7099759 


70 


49 00 


343 000 


8.3666003 


4.1212853 


6 


36 


216 


.4494897 


.8171206 


1 


50 41 


357 911 


.4261498 


.1408178 


7 


49 


343 


.6457513 


.9129312 


2 


51 84 


373 248 


.4852814 


.1601676 


8 


64 


512 


.8284271 


2.0000000 


3 


53 29 


389 017 


.5440037 


.1793392 


9 


81 


729 


3.0000000 


.0800837 


4 


54 76 


405 224 


.6023253 


.1983364 


10 


1 00 


1 000 


3.1622777 


2.1544347 


75 


56 25 


421 875 


8.6602540 


4.2171633 


11 


1 21 


1 331 


.3166248 


.2239801 


6 


57 76 


438 976 


.7177979 


.2358236 


12 


1 44 


1 728 


,4641016 


.2894286 


7 


59 29 


456 533 


.7749644 


.2543210 


13 


1 69 


2 197 


.6055513 


.3513347 


8 


60 84 


474 552 


.8317609 


.2726586 


14 


1 96 


2 744 


.7416574 


.4101422 


9 


62 41 


493 039 


.8881944 


.2908404 


15 


2 25 


3 375 


3.8729833 


2.4662121 


80 


64 00 


512 000 


8.9442719 


4.3088695 


16 


2 56 


4 096 


4.0000000 


.5198421 


1 


65 61 


531 441 


9.0000000 


.3267487 


17 


2 89 


4 913 


.1231056 


.5712816 


2 


67 24 


551 368 


.0553851 


.3444815 


18 


3 24 


5 832 


.2426407 


.6207414 


• 3 


68 89 


571 787 


.1104336 


.3620707 


19 


3 61 


6 859 


.3588989 


.6684016 


4 


70 56 


592 704 


.1651514 


.3795191 


20 


. 4 00 


8 000 


4.4721360 


2.7144177 


85 


72 25 


614 125 


9.2195445 


4.3968296 


1 


4 41 


9 261 


.5825757 


2.7589243 


6 


73 96 


636 056 


.2736185 


.4140049 


2 


4 84 


10 648 


.6904158 


.8020393 


7 


75 69 


658 503 


.3273791 


.4310476 


3 


5 29 


12 167 


7958315 


.8438670 


8 


77 44 


681 472 


.3808315 


.4479602 


4 


5 76 


13 824 


.8989795 


.8844991 


9 


79 21 


704 969 


.4339811 


.4647451 


25 


6 25 


15 625 


5.0000000 


2.9240177 


90 


81 00 


729 000 


9.4868330 


4.4814047 


6 


6 76 


17 576 


.0990195 


.9624960 


1 


82 81 


753 571 


.5393920 


.4979414 


7 


7 29 


19 683 


.1961524 


3.0000000 


2 


84 64 


778 688 


.5916630 


.5143574 


8 


7 84 


21 952 


.2915026 


.0365889 


3 


86 49 


804 357 


.6436508 


.5306549 


9 


8 41 


24 389 


.3851648 


.0723168 


4 


88 36 


830 584 


.6953597 


.5468359 


30 


9 00 


27 000 


5.4772256 


3.1072325 


95 


90 25 


857 375 


9.7467943 


4.5629026 


1 


9 61 


29 791 


.5677644 


.1413806 


6 


92 16 


884 736 


.7979590 


.5788570 


2 


10 24 


32 768 


.6568542 


.1748021 


7 


94 09 


912 673 


.8488578 


.5947009 


3 


10 89 


35 937 


.7445626 


.2075343 


8 


96 04 


941 192 


.8994949 


.6104363 


4 


11 56 


39 304 


.8309519 


.2396118 


9 


98 01 


970 299 


.9498744 


.6260650 


35 


12 25 


42 875 


5.9160798 


3.2710663 


100 


1 00 00 


1 000 000 


10.0000000 


4.6415888 


6 


12 96 


46 656 


6.0000000 


.3019272 


1 


1 02 01 


1 030 301 


.0498756 


.6570095 


7 


13 69 


50 653 


.0827625 


.3322218 


2 


1 04 04 


1 061 208 


.0995049 


.6723287 


8 


14 44 


54 872 


.1644140 


.3619754 


3 


1 06 09 


1 092 927 


.1488916 


.6875482 


9 


15 21 


59 319 


2449980 


.3912114 


4 


1 08 16 


1 124 864 


.1980390 


.7026694 


40 


16 00 


64 000 


6.3245553 


3.4199519 


105 


1 10 25 


1 157 625 


10.2469508 


4.7176940 


1 


16 81 


68 921 


.4031242 


.4482172 


6 


1 12 36 


1 191 016 


.2956301 


.7326235 


2 


17 64 


74 088 


.4807407 


.4760266 


7 


1 14 49 


1 225 043 


.3440804 


.7474594 


3 


18 49 


79 507 


.5574385 


.5033981 


8 


1 16 64 


1 259 712 


.3923048 


.7622032 


4 


19 36 


85 184 


.6332496 


.5303483 


9 


1 18 81 


1 295 029 


.4403065 


.7768562 


45 


20 25 


91 125 


6.7082039 


3.5568933 


110 


1 21 00 


1 331 000 


10,4880885 


4.7914199 


6 


21 16 


97 336 


.7823300 


.5830479 


11 


1 23 21 


1 367 631 


.5356538 


.8058955 


7 


22 09 


103 823 


.8556546 


.6088261 


12 


1 25 44 


1 404 928 


.5830052 


.8202845 


8 


23 04 


110 592 


.9282032 


.6342411 


13 


1 27 69 


1 442 897 


.6301458 


.8345881 


9 


24 01 


117 649 


7.0000000 


.6593057 


14 


1 29 96 


1 481 544 


.6770783 


.8488076 


50 


25 00 


125 000 


7.0710678 


3.6840314 


115 


1 32 25 


1 520 875 


10.7238053 


4.8629442 


1 


26 01 


132 651 ' 


.1414284 


.7084298 


16 


1 34 56 


1 560 896 


7703296 


.8769990 


2 


27 04 


140 608 


.2111026 


.7325111 


17 


1 36 89 


1 601 613 


8166538 


.8909732 


3 


28 09 


148 877 


.2801099 


.7562858 


18 


1 39 24 


1 643 032 


.8627805 


.9048681 


4 


29 16 


157 464 


.3484692 


.7797631 


19 


1 41 61 


1 685 159 


.9087121 


.9186847 


55 


30 25 


166 375 


7.4161985 


3.8029525 


120 


1 44 00 


I 728 000 


10.9544512 


4.9324242 


6 


31 36 


175 616 


.4833148 


.8258624 


1 


1 46 41 


I 771 561 


11.0000000 


9460874 


7 


32 49 


185 193 


.5498344 


.8485011 


2 


1 48 84 


1 815 848 


.0453610 


.9596757 


8 


33 64 


195 112 


.6157731 


.8708766 


3 


1 51 29 


I 860 867 


.0905365 


.9731898 


9 


34 81 


205 379 


.6811457 


.8929965 


4 


1 53 76 


1 906 624 


.1355287 


.9866310 


60 


36 00 


216 000 


7.7459667 


3.9148676 


125 


1 56 25 


1 953 125 


11.1803399 


5.0000000 


1 


37 21 


226 981 


.8102497 


.9364972 


6 


1 58 76 


2 000 376 


.2249722 


.0132979 


2 


38 44 


238 328 


.8740079 


.9578915 


7 


1 61 29 


2 048 383 


.2694277 


.0265257 


3 


39 69 


250 047 


9372539 


.9790571 


8 


1 63 84 


2 097 152 


3137085 


.0396842 


4 


40 96 


262 144 


8.0000000 


4.0000000 


9 


1 66 41 


2 146 689 


.3578167 


.0527743 


65 


42 25 


274 625 


8.0622577 


4.0207256 


130 


1 69 00 


2 197 000 


11.4017543 


5.0657970 



32 



2.— POWERS, ROOTS AND RECIPROCALS. 



9. — Squares, Cubes, Square Roots, Cube Roots, op Numbers 
1 TO 1600 — Continued. 



No. 


Square 


Cube. 


Sq. Rt. 


Cu. Rt. 


No. 


Square 


Cube. 


Sq. Rt. 


Cu. Rt. 


130 


169 00 


2 197 000 


11.4017543 


5.0657970 


195 


3 80 25 


7 414 87£ 


13.9642400 


5.7988900 


1 


17161 


2 248 091 


.4455231 


.0787531 


6 


3 84 16 


7 529 536 


14.0000000 


.8087857 


2 


174 24 


2 299 968 


.4891253 


.0916434 


7 


3 88 09 


7 645 373 


.0356688 


.8186479 


3 


176 89 


2 352 637 


.5325626 


.1044687 


8 


3 92 04 


7 762 392 


.0712473 


.8284767 


4 


179 56 


2 406 104 


.5758369 


.1172299 


9 


3 96 01 


7 880 599 


.1067360 


.8382725 


135 


182 25 


2 460 375 


11.6189500 


5.1299278 


200 


4 00 00 


8 000 000 


14.1421356 


5.8480355 


6 


184 96 


2 515 456 


.6619038 


.1425632 


1 


4 04 01 


8 120 601 


.1774469 


.8577660 


7 


187 69 


2 571 353 


.7046999 


.1551367 


2 


4 08 04 


8 242 408 


.2126704 


.8674643 


8 


190 44 


2 628 072 


.7473401 


.1676493 


3 


4 12 09 


8 365 427 


.2478068 


.8771307 


9 


1 93 21 


2 685 619 


.7898261 


.1801015 


4 


4 16 16 


8 489 664 


.2828569 


.8867653 


140 


1 96 00 


2 744 000 


11.8321596 


5.1924941 


205 


4 20 25 


8 615 125 


14.3178211 


5.8963685 


1 


198 81 


2 803 221 


.8743422 


.2048279 


6 


4 24 36 


8 741816 


.3527001 


.9059406 


2 


2 0164 


2 863 288 


.9163753 


.2171034 


7 


4 28 49 


8 869 743 


.3874946 


.9154817 


3 


2 04 49 


2 924 207 


.9582607 


.2293215 


8 


4 32 64 


8 998 912 


.4222051 


.9249921 


4 


2 07 36 


2 985 984 


12.0000000 


.2414828 


9 


4 36 81 


9 129 329 


.4568323 


.9344721 


145 


2 10 25 


3 048 625 


12.0415946 


5.2535879 


210 


4 4100 


9 261 000 


14.4913767 


5.9439220 


6 


2 13 16 


3 112 136 


.0830460 


.2656374 


11 


4 45 21 


9 393 931 


.5258390 


.9533418 


7 


2 16 09 


3 176 523 


.1243557 


.2776321 


12 


4 49 44 


9 528 128 


.5602198 


.9627320 


8 


2 19 04 


3 241 792 


.1655251 


.2895725 


13 


4 53 69 


9 663 597 


.5945195 


.9720926 


9 


2 22 01 


3 307 949 


.2065556 


.3014592 


14 


4 57 96 


9 800 344 


.6287388 


.9814240 


150 


2 25 00 


3 375 000 


12.2474487 


5.3132928 


215 


4 62 25 


9 938 375 


14.6628783 


5.9907264 


1 


2 28 01 


3 442 951 


.2882057 


.3250740 


16 


4 66 56 


10 077 696 


.6969385 


6.0000000 


2 


2 3104 


3 511 808 


.3288280 


.3368033 


17 


4 70 89 


10 218 313 


.7309199 


.0092450 


3 


2 34 09 


3 581 577 


.3693169 


.3484812 


18 


4 75 24 


10 360 232 


.7648231 


.0184617 


4 


2 37 16 


3 652 264 


.4096736 


.3601084 


19 


4 79 61 


10 503 459 


.7986486 


.0276502 


155 


2 40 25 


3 723 875 


12.4498996 


5.3716854 


220 


4 84 00 


10 648 000 


14.8323970 


6.0368107 


6 


2 43 36 


3 796 416 


.4899960 


.3832126 


1 


4 88 41 


10 793 861 


.8660687 


.0459435 


7 


2 46 49 


3 869 893 


.5299641) .3946907 


2 


4 92 84 


10 941 048 


.8996644 


.0550489 


8 


2 49 64 


3 944 312 


.5698051 


.4061202 


3 


4 97 29 


11 089 567 


.9331845 


.0641270 


9 


2 52 81 


4 019 679 


.6095202 


.4175015 


4 


5 01 76 


11 239 424 


.9666295 


.0731779 


160 


2 56 00 


4 096 000 


12.6491106 


5.4288352 


225 


5 06 25 


11 390 625 


15.0000000 


6.0822020 


1 


2 59 21 


4 173 281 


.6885775 


.4401218 


6 


5 10 76 


11 543 176 


.0332964 


.0911994 


2 


2 62 44 


4 251 528 


.7279221 


.4513618 


7 


5 15 29 


11 697 083 


.0665192 


.1001702 


3 


2 65 69 


4 330 747 


.7671453 


.4625556 


8 


5 19 84 


11852 352 


.0996689 


.1091147 


4 


2 68 96 


4 410 944 


.8062485 


.4737037 


9 


5 24 41 


12 008 989 


.1327460 


.1180332 


165 


2 72 25 


4 492 125 


12.8452326 


5.4848066 


230 


5 29 00 


12 167 000 


15.1657509 


6.1269257 


6 


2 75 56 


4 574 296 


.8840987 


.4958647 


1 


5 33 61 


12 326 391 


.1986842 


.1357924 


7 


2 78 89 


4 657 463 


.9228480 


.5068784 


2 


5 38 24 


12 487 168 


.2315462 


.1446337 


8 


2 82 24 


4 741632 


.9614814 


.5178484 


3 


5 42 89 


12 649 337 


.2643375 


.1534495 


9 


2 85 61 


4 826 809 


13.0000000 


.5287748 


4 


5 47 56 


12 812 904 


.2970585 


.1622401 


170 


2 89 00 


4 913 000 


13.0384048 


5.5396583 


235 


5 52 25 


12 977 875 


15.3297097 


6.1710058 


1 


2 92 41 


5 000 211 


.0766968 


.5504991 


6 


5 56 96 


13 144 256 


.3622915 


.1797466 


2 


2 95 84 


5 088 448 


.1148770 


.5612978 


7 


5 61 69 


13 312 053 


.3948043 


,1884628 


3 


2 99 29 


5 177 717 


.1529464 


.5720546 


8 


5 66 44 


13 481 272 


.4272486 


.1971544 


4 


3 02 76 


5 268 024 


.1909060 


.5827702 


9 


5 7121 


13 651 919 


.4596248 


.2058218 


175 


3 06 25 


5 359 375 


13.2287566 


5.5934447 


240 


5 76 00 


13 824 000 


15.4919334 


6.2144650 


6 


3 09 76 


5 451 776 


.2664992 


.6040787 


1 


5 80 81 


13 997 521 


.5241747 


.2230843 


7 


3 13 29 


5 545 233 


.3041347 


.6146724 


2 


5 85 64 


14 172 488 


.5563492 


.2316797 


8 


3 16 84 


5 639 752 


.3416641 


.6252263 


3 


5 90 49 


14 348 907 


.5884573 


.2402515 


9 


3 20 41 


5 735 339 


.3790882 


.6357408 


4 


5 95 36 


14 526 784 


.6204994 


.2487998 


180 


3 24 00 


5 832 000 


13.4164079 


5.6462162 


245 


6 00 25 


14 706 125 


15.6524758 


6.2573248 


1 


3 27 61 


5 929 741 


.4636240 


.6566528 


6 


6 05 16 


14 886 936 


.6843871 


.2658266 


2 


3 3124 


6 028 568 


.4907376 


.6670511 


7 


6 10 09 


15 069 223 


.7162336 


.2743054 


3 


3 34 89 


6 128 487 


.5277493 


.6774114 


8 


6 15 04 


15 252 992 


.7480157 


.2827613 


4 


3 38 56 


6 229 504 


.5646600 


.6877340 


9 


6 20 01 


15 438 249 


.7797338 


.2911946 


185 


3 42 25 


6 331625 


13.6014705 


5.6980192 


250 


6 25 00 


. 15 625 000 


15.8113883 


6.2996053 


6 


3 45 96 


6 434 856 


.6381817 


.7082675 


1 


6 30 01 


15 813 251 


.8429795 


.3079935 


7 


3 49 69 


6 539 203 


.6747943 


.7184791 


2 


6 35 04 


16 003 008 


.8745079 


.3163596 


8 


3 53 44 


6 644 672 


.7113092 


.7286543 


3 


6 40 09 


16 194 277 


.9059737 


.3247035 


9 


3 57 21 


6 751269 


.7477271 


.7387936 


4 


6 45 16 


16 387 064 


.9373775 


.3330256 


190 


3 6100 


6 859 000 


13.7840488 


5.7488971 


255 


6 50 25 


16 581 375 


15.9687194 


6.3413257 


1 


3 64 81 


6 967 871 


.8202750 


.7589652 


6 


6 55 36 


16 777 216 


16.0000000 


.3496042 


2 


3 68 64 


7 077 888 


.8564065 


.7689982 


7 


6 60 49 


16 974 593 


.0312195 


.3578611 


3 


3 72 49 


7 189 057 


.8924440 


.7789966 


8 


6 65 64 


17 173 512 


.0623784 


.3660968 


4 


3 76 36 


7 301 384 


.9283883 


.7889604 


9 


6 70 81 


17 373 979 


.0934769 


.3743111 


195 


3 80 25 


7 414 875 


13.9642400 


5.7988900 


260 


6 76 00 


17 576 000 


16.1245155 


6.3825043 



COMMON TABLES— SQUARES, CUBES, ROOTS. 

9. — Squares, Cubes, Square Roots, Cube Roots, of Numbers 
1 TO 1600— Continued. 



No. 


Square 


Cube. 


Sq. Rt. 


Cu. Rt. 


No. Square 


Cube. 


Sq. Rt. 


Cu. Rt. 


260 


6 76 00 


17 576 000 


16.1245155 


6.3825043 


325 


10 56 25 


34 328 125 


18.0277564 


6.8753443 


1 


6 8121 


17 779 581 


.1554944 


.3906765 


6 


10 62 76 


34 645 976 


.0554701 


.8823888 


2 


6 86 44 


17 984 728 


.1864141 


.3988279 


7 


10 69 29 


34 965 783 


.0831413 


.8894188 


3 


6 9169 


18 191 447 


.2172747 


.4069585 


8 


10 75 84 


35 287 552 


.1107703 


.8964345 


4 


6 96 96 


18 399 744 


.2480768 


.4150687 


9 


10 82 41 


35 611 289 


.1383571 


.9034359 


265 


7 02 25 


18 609 625 


16.2788206 


6.4231583 


330 


10 89 00 


85 937 000 


18.1659021 


6.9104232 


6 


7 07 56 


18 821 096 


.3095064 


.4312276 


1 


10 95 61 


36 264 691 


.1934054 


.9173964 


7 


7 12 89 


19 034 163 


.3401346 


.4392767 


2 


11 02 24 


36 594 368 


.2208672 


.9243556 


8 


7 18 24 


19 248 832 


.3707055 


.4473057 


3 


11 08 89 


36 926 037 


.2482876 


.9313008 


9 


7 23 61 


19 465 109 


.4012195 


.4553148 


4 


11 15 56 


37 259 704 


.2756669 


.9382321 


270 


7 29 00 


19 683 000 


16.4316767 


6.4633041 


335 


11 22 25 


37 595 375 


18.3030052 


6.9451496 


1 


7 34 41 


19 902 511 


.4620776 


.4712736 


6 


11 28 96 


37 933 056 


.3303028 


.9520533 


2 


7 39 84 


20 123 648 


.4924225 


.4792236 


7 


1135 69 


38 272 753 


.3575598 


.9589434 


3 


7 45 29 


20 346 417 


.5227116 


.4871541 


8 


11 42 44 


38 614 472 


.3847763 


.9658198 


4 


7 50 76 


20 570 824 


.5529454 


.4950653 


9 


11 49 21 


38 958 219 


.4119526 


.9726826 


275 


7 56 25 


20 796 875 


16.5831240 


6.5029572 


340 


11 56 00 


39 304 000 


18.4390889 


6.9795321 


6 


7 61 76 


21 024 576 


.6132477 


.5108300 


1 


11 62 81 


39 651 821 


.4661853 


.9863681 


7 


7 67 29 


21 253 933 


.6433170 


.5186839 


2 


11 69 64 


40 001 688 


.4932420 


.9931906 


8 


7 72 84 


21 484 952 


.6733320 


.5265189 


3 


1176 49 


40 353 607 


.5202592 


7.0000000 


9 


7 78 41 


21 717 639 


.7032931 


.5343351 


4 


11 83 36 


40 707 584 


.5472370 


.0067962 


280 


7 84 00 


21 952 000 


16.7332005 


6.5421326 


345 


11 90 25 


41 063 625 


18.5741756 


7.0135791 


1 


7 89 61 


22 188 041 


.7630546 


.5499116 


6 


11 97 16 


41 421 736 


.6010752 


.0203490 


2 


7 95 24 


22 425 768 


.7928556 


.5576722 


7 


12 04 09 


41 781 923 


.6279360 


.0271058 


3 


8 00 89 


22 665 187 


.8226038 


.5654144 


8 


12 11 04 


42 144 192 


.6547581 


.0338497 


4 


8 06 56 


22 906 304 


.8522995 


.5731385 


9 


12 18 01 


42 508 549 


.6815417 


.0405806 


285 


8 12 25 


23 149 125 


16.8819430 


6.5808443 


350 


12 25 00 


42 875 000 


18.7082869 


7.0472987 


6 


8 17 96 


23 393 656 


.9115345 


.5885323 


1 


12 32 01 


43 243 551 


.7349940 


.0540041 


7 


8 23 69 


23 639 903 


.9410743 


.5962023 


2 


12 39 04 


43 614 208 


.7616630 


.0606967 


8 


8 29 44 


23 887 872 


.9705627 


.6038545 


3 


12 46 09 


43 986 977 


.7882942 


.0673767 


9 


8 35 21 


24 137 569 


17.0000000 


.6114890 


4 


12 53 16 


44 361 864 


.8148877 


.0740440 


290 


8 41 00 


24 389 000 


17.0293864 


6.6191060 


355 


12 60 25 


44 738 875 


18.8414437 


7.0806988 


1 


8 46 81 


24 642 171 


.0587221 


.6267054 


6 


12 67 36 


45 118 016 


.8679623 


.0873411 


2 


8 52 64 


24 897 088 


.0880075 


.6342874 


7 


12 74 49 


45 499 293 


.8944436 


.0939709 


3 


8 58 49 


25 153 757 


.1172428 


.6418522 


8 


12 8164 


45 882 712 


.9208879 


.1005885 


4 


8 64 36 


25 412 184 


.1464282 


.6493998 


9 


12 88 81 


46 268 279 


.9472953 


.1071937 


295 


8 70 25 


25 672 375 


17.1755640 


6.6569302 


360 


12 96 00 


46 656 000 


18.9736660 


7.1137866 


6 


8 76 16 


25 934 336 


2046505 


.6644437 


1 


13 03 21 


47 045 881 


19.0000000 


.1203674 


7 


8 82 09 


'26 198 073 


.2336879 


.6719403 


2 


13 10 44 


47 437 928 


.0262976 


.12693G0 


8 


8 88 04 


26 463 592 


.2626765 


.6794200 


3 


13 17 69 


47 832 147 


.0525589 


.1334925 


9 


8 94 01 


26 730 899 


.2916165 


.6868831 


4 


13 24 96 


48 228 544 


.0787840 


.1400370 


300 


9 00 00 


27 000 000 


17.3205081 


6.6943295 


365 


13 32 25 


48 627 125 


19.1049732 


7.0465695 


1 


9 06 01 


27 270 901 


.3493516 


.7017593 


6 


13 39 56 


49 027 896 


.1311265 


.1530901 


2 


9 12 04 


27 543 608 


.3781472 


.7091729 


7 


13 46 89 


49 430 863 


.1572441 


.1595988 


3 


9 18 09 


27 818 127 


.4068952 


.7165700 


8 


13 54 24 


49 836 032 


.1833261 


.1660957 


4 


9 24 16 


28 094 464 


.4355958 


.7239508 


9 


13 61 61 


50 243 409 


.2093727 


.1725809 


305 


9 30 25 


28 372 625 


17.4642492 


6.7313155 


370 


13 69 00 


50 653 000 


19.2353841 


7.1790544 


6 


9 36 36 


28 652 616 


.4928557 


.7386641 


1 


13 76 41 


51064 811 


.2613603 


.1855162 


7 


9 42 49 


28 934 443 


.5214155 


.7459967 


2 


13 83 84 


51 478 848 


.2873015 


.1919663 


8 


9 48 64 


29 218 112 


.5499288 


.7533134 


3 


13 91 29 


51 895 117 


.3132079 


.1984050 


9 


9 54 81 


29 503 629 


.5783958 


.7606143 


4 


13 98 76 


52 313 624 


.3390796 


.2048322 


310 


9 6100 


29 791 000 


17.6068169 


6.7678995 


375 


14 06 25 


52 734 375 


19.3649167 


7.2112479 


11 


9 67 21 


30 080 231 


.6351921 


.7751690 


6 


14 13 76 


53 157 376 


.3907194 


.2176522 


12 


9 73 44 


30 371328 


.6635217 


.7824229 


7 


14 21 29 


53 582 633 


.4164878 


.2240450 


13 


9 79 69 


30 664 297 


.6918060 


.7896613 


8 


14 28 84 


54 010 152 


.4422221 


.2304268 


14 


9 85 96 


30 959 144 


.7200451 


.7968844 


9 


14 36 41 


54 439 939 


.4679223 


.2367972 


315 


9 92 25 


31 255 875 


17.7482393 


6.8040921 


380 


14 44 00 


54 872 000 


19.4935887 


7.2431565 


16 


9 98 56 


31 554 496 


.7763888 


.8112847 


1 


14 5161 


55 306 341 


.5192213 


.2495045 


17 


10 04 89 


31 855 013 


.8044938 


.8184620 


2 


14 59 24 


55 742 968 


5448203 


.2558415 


18 


101124 


32 157 432 


.8325545 


.8256242 


3 


14 66 89 


56 181 887 


.5703858 


.2621675 


19 


10 17 61 


32 461 759 


.8605711 


.8327714 


4 


14 74 56 


56 623 104 


.5959179 


.2684824 


320 


10 24 00 


32 768 000 


17.8885438 


6.8399037 


385 


14 82 25 


57 066 625 


19 6214169 


7.2747864 


1 


10 30 41 


33 076 161 


.9184729 


.8470213 


6 


14 89 96 


57 512 456 


.6468827 


.2810794 


2 


10 36 84 


33 386 248 


.9443584 


.8541240 


7 


14 97 69 


57 960 603 


.6723156 


.2873617 


3 


10 43 29 


33 698 267 


.9722008 


.8612120 


8 


15 05 44 


58 411072 


.6977156 


.2936330 


4 


10 49 76 


34 012 224 


18.0000000 


.8682855 


9 


15 13 21 


58 863 869 


.7230829 


.2998936 


325 


10 56 25 


34 328 125 


18.0277564 


6.8753443 


390 


15 2100 


59 319 000 


19.7484177 


7.3061436 



34 



2.— POWERS, ROOTS AND RECIPROCALS. 



9. — Squares, Cubes, Square Roots, Cube Roots, op Numbers 
1 TO 1600 — Continued. 



No. 


Square 


Cube. 


Sq. Rt. 


Cu. Rt. 


No. 


Square 


Cube. 


Sq. Rt . 


Cu. Rt. 


390 


15 21 00 


59 319 000 


19.7484177 


7.3061436 


455 


20 70 25 


94 196 375 


21.3307290 


7.6913717 


1 


15 28 81 


59 776 471 


.7737199 


.3123828 


6 


20 79 36 


94 818 816 


.3541565 


.6970023 


2 


15 36 64 


60 236 288 


.7989899 


.3186114 


7 


20 88 49 


95 443 993 


.3775583 


.7026246 


3 


15 44 49 


60 698 457 


.8242276 


.3248295 


8 


20 97 64 


96 071 912 


.4009346 


.7082388 


4 


15 52 36 


61 162 984 


.8494332 


.3310369 


9 


2106 81 


96 702 579 


.4242853 


.7138448 


395 


15 60 25 


61 629 875 


19.8746069 


7.3372339 


460 


21 16 00 


97 336 000 


21.4476106 


7.7194426 


6 


15 68 16 


82 099 136 


.8997487 


.3434205 


1 


2125 21 


97 972 181 


.4709106 


.7250325 


7 


15 76 09 


62 570 773 


.9248588 


.3495966 


2 


21 34 44 


98 611128 


.4941853 


.7306141 


8 


15 84 04 


63 044 792 


.9499373 


.3557624 


3 


21 43 69 


99 252 847 


.5174348 


.7361877 


9 


15 92 01 


63 521 199 


.9749844 


.3619178 


4 


21 52 96 


99 897 344 


.5406592 


.7417532 


400 


16 00 00 


64 000 000 


20.0000000 


7.3680630 


465 


21 62 25 


100 544 625 


21.5638587 


7.7473109 


1 


16 08 01 


64 481 201 


.0249844 


.3741979 


6 


21 71 56 


101 194 696 


.5870331 


.7528606 


2 


16 16 04 


64 964 808 


.0499377 


.3803227 


7 


21 80 89 


101 847 563 


.6101828 


.7584023 


3 


16 24 09 


65 450 827 


.0748599 


.3864373 


8 


21 90 24 


102 503 232 


.6333077 


.7639361 


4 


16 32 16 


65 939 264 


.0997512 


.3925418 


9 


21 99 61 


103 161 709 


.6564078 


.7694620 


405 


16 40 25 


66 430 125 


20.1246118 


7.3986363 


470 


22 09 00 


103 823 000 


21.6794834 


7.7749801 


6 


16 48 36 


66 923 416 


.1494417 


.4047206 


1 


22 18 41 


104 487 111 


.7025344 


.7804904 


7 


16 56 49 


67 419 143 


.1742410 


.4107950 


2 


22 27 84 


105 154 048 


.7255610 


.7859928 


8 


16 64 64 


67 917 312 


.1990099 


.4168595 


3 


22 37 29 


105 823 817 


.7485632 


.7914875 


9 


16 72 81 


68 417 929 


.2237484 


.4229142 


4 


22 46 76 


106 496 424 


.7715411 


.7969745 


410 


16 81 00 


68 921 000 


20.2484567 


7.4289589 


475 


22 56 25 


107 171 875 


21.7944947 


7.8024538 


11 


16 89 21 


69 426 531 


.2731349 


.4349938 


6 


22 65 76 


107 850 176 


.8174242 


.8079254 


12 


16 97 44 


69 934 528 


.2977831 


.4410189 


7 


22 75 29 


108 531 333 


.8403297 


.8133892 


13 


17 05 69 


70 444 997 


.3224014 


.4470342 


8 


22 84 84 


109 215 352 


.8632111 


.8188456 


14 


17 13 96 


70 957 944 


.3469899 


.4530399 


9 


22 94 41 


109 902 239 


.8860686 


.8242942 


415 


17 22 25 


71 473 375 


20.3715488 


7.4590359 


4SC 


23 04 00 


110 592 000 


21.9089023 


7.8297353 


16 


17 30 56 


71 991 296 


.3960781 


.4650223 


1 


23 13 61 


111284 641 


.9317122 


.8351688 


17 


17 38 89 


92 511 713 


,4205779 


.4709991 


2 


23 23 24 


111 980 168 


.9544984 


.8405949 


18 


17 47 24 


73 034 632 


.4450483 


.4769664 


3 


23 32 89 


112 678 587 


.9772610 


.8460134 


19 


17 55 61 


73 560 059 


.4694895 


.4829242 


4 


23 42 56 


113 379 904 


22.0000000 


.8514244 


420 


17 64 00 


74 088 000 


20.4939015 


7.4888724 


485 


23 52 25 


114 084 125 


22.0227155 


7.8568281 


1 


17 72 41 


74 618 461 


.5182845 


.4948113 


6 


23 61 96 


114 791256 


.0454077 


.8622242 


2 


17 80 84 


75 151 448 


.5426386 


.5007406 


7 


23 71 69 


115 501 303 


.0680765 


.8676130 


3 


17 89 29 


75 686 967 


.5669638 


.5066607 


8 


23 81 44 


116 214 272 


.0907220 


.8729944 


4 


17 97 76 


76 225 024 


.5912603 


.5125715 


9 


23 91 21 


116 930 169 


.1133444 


.8783684 


425 


18 06 25 


76 765 625 


20.6155281 


7.5184730 


490 


24 01 00 


117 649 000 


22.1359436 


7.8837352 


6 


18 14 76 


77 308 776 


.6397674 


.5243652 


1 


24 10 81 


118 370 771 


.1585198 


.8890946 


7 


18 23 29 


77 854 483 


.6639783 


.5302482 


2 


24 20 64 


119 095 488 


.1810730 


.8944468 


8 


18 3184 


78 402 752 


.6881609 


.5361221 


3 


24 30 49 


119 823 157 


.2036033 


.8997917 


9 


18 40 41 


78 953 589 


.7123152 


.5419867 


4 


24 40 36 


120 553 784 


.2261108 


.9051294 


430 


18 49 00 


79 507 000 


20.7364414 


7.5478423 


495 


24 50 25 


121 287 375 


22.2485955 


7.9104599 


1 


18 57 61 


80 062 991 


.7605395 


.5536888 


6 


24 60 16 


122 023 936 


.2710575 


.9157832 


2 


18 66 24 


80 621 568 


.7846097 


.5595263 


7 


24 70 09 


122 763 473 


.2934968 


.9210994 


3 


18 74 89 


81 182 737 


.8086520 


.5653548 


8 


24 80 04 


123 505 992 


.3159136 


.9264085 


4 


18 83 56 


81 746 504 


.8326667 


.5711743 


9 


24 90 01 


124 251 499 


.3383079 


.9317104 


435 


18 92 25 


82 312 875- 


20.8566536 


7.5769849 


500 


25 00 00 


125 000 000 


22.3606798 


7.9370053 


6 


19 00 96 


82 881 856 


.8806130 


.5827865 


1 


25 10 01 


125 751 501 


.3830293 


.9422931 


7 


19 09 69 


83 453 453 


.9045450 


.5885793 


2 


25 20 04 


126 506 008 


.4053565 


.9475739 


8 


19 18 44 


84 027 672 


.9284495 


.5943633 


3 


25 30 09 


127 263 527 


.4276615 


.9528477 


9 


19 27 21 


84 604 519 


.9523268 


.6001385 


4 


25 40 16 


128 024 064 


.4499443 


.9581144 


440 


19 36 00 


85 184 000 


20.9761770 


7.6059049 


505 


25 50 25 


128 787 625 


22.4722051 


7.9633743 


1 


19 44 81 


85 766 121 


21.0000000 


.6116626 


6 


25 60 36 


129 554 216 


.4944438 


.9686271 


2 


19 53 64 


86 350 888 


.0237960 


.6174116 


7 


25 70 49 


130 323 843 


.5166605 


.9738731 


3 


19 62 49 


86 938 307 


.0475652 


.6231519 


8 


25 80 64 


131 096 512 


.5388553 


.9791122 


4 


19 7136 


87 528 384 


.0713075 


.6288837 


9 


25 90 81 


131 872 229 


.5610283 


.9843444 


445 


19 80 25 


88 121 125 


21.0950231 


7.6346067 


510 


26 01 00 


132 651 000 


22.5831796 


7.7895697 


6 


19 89 16 


88 716 536 


.1187121 


.6403213 


11 


26 11 21 


133 432 831 


.6053091 


.9947883 


7 


19 98 09 


89 314 623 


.1423745 


.6460272 


12 


26 21 44 


134 217 728 


.6274170 


8.0000000 


8 


20 07 04 


89 915 392 


.1660105 


.6517247 


13 


26 31 69 


135 005 697 


.6495033 


.0052049 


9 


20 16 01 


90 518 849 


.1896201 


.6574138 


14 


26 41 96 


135 796 744 


.6715681 


.0104032 


450 


20 25 OC 


91 125 000 


21.2132034 


7.6630943 


515 


26 52 25 


136 590 875 


22.6936114 


8.0155946 


1 


20 34 01 


91 733 851 


.2367606 


.6687665 


16 


26 62 56 


137 388 096 


.7156334 


.0207794 


2 


20 43 04 


92 345 408 


.2602916 


.6744303 


17 


26 72 89 


138 188 413 


.7376340 


.0259574 


3 


20 52 09 


92 959 677 


.2837967 


.6800857 


18 


26 83 24 


138 991832 


.7596134 


.0311287 


4 


20 61 16 


93 576 664 


.3072758 


.6857328 


19 


26 93 61 


139 798 359 


.7815715 


.0362935 


455 


20 70 25 


94 196 375 


21.3307290 


.6913717 


520 


27 04 00 


140 608 000 


22.8035085 


8.0414515 



COMMON TABLES— SQUARES, CUBES, ROOTS. 



35 



9. — Squares, Cubes, Square Roots, Cube Roots, of Numbers 
1 to 1600— Continued. 



Square Cube. 



27 04 00 
27 14 41 
27 24 84 
27 35 29 
27 45 76 
27 56 25 
27 66 76 
27 77 29 
27 87 84 

27 98 41 

28 09 00 
28 19 61 
28 30 24 
28 40 89 
28 51 56 
28 62 25 
28 72 96 
28 83 69 

28 94 44 

29 05 21 
29 16 00 
29 26 81 
29 37 64 
29 48 49 
29 59 36 
29 7025 
29 81 16 

29 92 09 

30 03 04 
30 14 01 
30 25 00 
30 36 01 
30 47 04 
30 58 09 
30 69 16 
30 80 25 

30 9136 

31 02 49 
31 13 64 
3124 81 
31 36 00 
31 47 21 
31 58 44 
31 69 69 
31 80 96 

31 92 25 

32 03 56 
32 14 89 
32 26 24 
32 37 61 
32 49 00 
32 60 41 
32 71 84 
32 83 29 

32 94 76 

33 06 25 
33 17 76 
33 29 29 
33 40 84 
33 52 41 
33 64 00 
33 75 61 
33 87 24 

33 98 89 

34 10 56 
34 22 25 



140 608 000 

141 420 761 

142 236 648 

143 055 667 

143 877 824 

144 703 125 

145 531 576 

146 363 183 

147 197 952 

148 035 889 

148 877 000 

149 721 291 

150 568 768 

151 419 437 

152 273 304 

153 130 375 

153 990 656 

154 854 153 

155 720 872 

156 590 819 

157 464 000 

158 340 421 

159 220 088 

160 103 007 

160 989 184 

161 878 625 

162 771 336 

163 667 323 

164 566 592 

165 469 149 

166 375 000 

167 284 151 

168 196 608 

169 112 377 

170 031 464 
170 953 875 
171879 616 

172 808 693 

173 741 112 

174 676 879 

175 616 000 

176 558 481 

177 504 328 

178 453 547 

179 406 144 

180 362 125 

181 321 496 

182 284 263 

183 250 432 

184 220 009 

185 193 000 

186 169 411 

187 149 248 

188 132 517 

189 119 224 

190 109 375 

191 102 976 

192 100.033 

193 100 552 

194 104 539 

195 112 000 

196 122 941 

197 137 368 

198 155 287 

199 176 704 

200 201 625 



S<l. Rt. 



22.8035085 
.8254244 
.8473193 
.8691933 
.8910463 

22.9128785 
.9346899 
.9564806 
.9782506 

23.0000000 

23.0217289 
.0434372 
.0651252 
.0867928 
.1084400 

23.1300670 
.1516738 
.1732605 
.1948270 
.2163735 

23.2379001 
.2594067 
.2808935 
.3023604 
.3238076 

23.3452351 
.3666429 
.3880311 
.4093998 
.4307490 

23.4520788 
.4733892 
.4946802 
.5159520 
.5372046 
3.5584380 
.5796522 
.6008474 
.6220236 
.6431808 

23.6643191 
.6854386 
.7065392 
.7276210 
.7486842 

23.7697286 
.7907545 
.8117618 
.8327506 
.8537209 

23.8746728 
.8956063 
.9165215 
.9374184 
.9582971 

23.9791576 

24.0000000 
.0208243 
.0416306 
.0624188 

24.0831891 
.1039416 
.1246762 
.1453929 
.1660919 

24.1867732 



Cu. Rt. 

8.0414515 
.0466030 
.0517479 
.0568862 
.0620180 

8.0671432 
.0722620 
.0773743 
.0824800 
.0875794 

8.0926723 
0977589 
.1028390 
.1079128 
.1129803 

8 1180414 
.1230962 
.1281447 
.1331870 
.1382230 

8.1432529 
.1482765 
.1532939 
.1583051 
.1633102 

8.1683092 
.1733020 
.1782888 
.1832695 
.1882441 

8.1932127 
.1981753 
.2031319 
.2080825 
.2130271 

8.2179657 
.2228985 
.2278254 
.2327463 
.2376614 

8.2425706 
.2474740 
.2523715 
.2572633 
.2621492 

8.2670294 
.2719039 
.2767726 
.2816355 
.2864928 

8.2913444 
.2961903 
.3010304 
.3058651 
.3106941 

8.3155175 
.3203353 
.3251475 
.3299542 
.3347553 

8.3395509 
.3443410 
.3491256 
.3539047 
.3586784 

8.3634466 



No. 

585 
6 
7 
8 
9 

590 
1 
2 
3 
4 

595 
6 
7 
8 
9 

600 
1 
2 
3 
4 

605 



610 

11 

12 

13 

14 

615 

16 

17 

18 

19 

620 

1 

2 

3 

4 

625 



630 
1 
2 
3 
4 

635 



640 
1 
2 
3 
4 

645 
6 
7 



650 



Square 



34 22 
34 33 
34 45 
34 57 
34 69 
34 81 

34 92 

35 04 
35 16 
35 28 
35 40 
35 52 
35 64 
35 76 

35 88 

36 00 
36 12 
36 24 
36 36 
36 48 
36 60 
36 72 
36 84 

36 96 

37 08 
37 21 
37 33 
37 45 
37 57 
37 69 
37 82 

37 94 

38 06 
38 19 
38 31 
38 44 
38 56 
38 68 
38 81 

38 93 

39 06 
39 18 
39 31 
39 43 
39 56 
39 69 
39 81 

39 94 

40 06 
40 19 
40 32 
40 44 
40 57 
40 70 
40 83 

40 96 
4108 
4121 
4134 

41 47 
4160 
4173 
4186 
4199 

42 12 
42 25 



Cube. 



200 201 625 

201 230 056 

202 262 003 

203 297 472 

204 336 469 

205 379 000 

206 425 071 

207 474 688 

208 527 857 

209 584 584 

210 644 875 

211 708 736 

212 776 173 

213 847 192 

214 921 799 

216 000 000 

217 081 801 

218 167 208 

219 256 227 

220 348 864 

221 445 125 

222 545 016 

223 648 543 

224 755 712 

225 866 529 

226 981 000 

228 099 131 

229 220 928 

230 346 397 

231 475 544 

232 608 375 

233 744 896 

234 885 113 

236 029 032 

237 176 659 

238 328 000 

239 483 061 

240 641 848 

241 804 367 

242 970 624 

244 140 625 

245 314 376 

246 491 883 

247 673 152 

248 858 189 

250 047 000 

251 239 591 

252 435 968 

253 636 137 

254 840 104 

256 047 875 

257 259 456 

258 474 853 

259 694 072 

260 917 119 

262 144 000 

263 374 721 

264 609 288 

265 847 707 

267 089 984 

268 336 125 

269 586 136 

270 840 023 

272 097 792 

273 359 449 

274 625 000 



24. 



24 



Sq. Rt. 



24. 



24 



24, 



24 



24 



25 



25 



25 



25. 



25 



25. 



1867732 
,2074369 
2280829 
,2487113 
.2693222 
.2899156 
.3104916 
.3310501 
.3515913 
.3721152 
.3926218 
.4131112 
.4335834 
.4540385 
.4744765 
4948974 
.5153013 
.5356883 
.5560583 
.5764115 
.5967478 
.6170673 
.6373700 
.6576560 
.6779254 
.6981781 
.7184142 
.7386338 
.7588368 
.7790234 
7991935 
.8193473 
.8394847 
.8596058 
.8797106 
8997992 
.9198716 
.9399278 
.9599679 
.9799920 
.0000000 
.0199920 
.0399681 
.0599282 
.0798724 
.0998008 
.1197134 
.1396102 
.1594913 
.1793566 
1992063 
.2190404 
.2388589 
.2586619 
.2784493 
.2982213 
3179778 
3377189 
3574447 
3771551 
3968502 
4165301 
4361947 
4558441 
4754784 
4950976 



Cu. Rt. 

8.3634466 

.3682095 
.3729668 
.3777188 
.3824653 

8.3872065 
.3919423 
.3966729 
.4013981 
.4061180 

8.4108326 
.4155419 
.4202460 
.4249448 
.4296383 

8.4343267 
.4390098 
.4436877 
.4483605 
.4530281 

8.4576906 
.4623479 
.4670000 
.4716471 
.4762892 

8.4809261 
.4855579 
.4901848 
.4948065 
.4994233 

8.5040350 
.5086417 
.5132435 
.5178403 
.5224321 

8.5270189 
.5316009 
.5361780 
.5407501 
.5453173 

8.5498797 
.5544372 
.5589899 
.5635377 
.5680807 

8.5726189 
.5771523 
.5816809 
.5862047 
.5907238 

8.5952380 
.5997476 
.6042525 
.6087526 
.6132480 

8.6177388 
.6222248 
.6267063 
.6311830 
.6356551 

8.6401226 
.6445855 
.6490437 
.6534974 
.6579466 

8.6623911 



36 



Z.—POWERS, ROOTS AND RECIPROCALS. 



9. — Squares, Cubes, Square Roots, Cube Roots, of Numbers 
1 to 1600— Continued. 



Square 



42 25 00 
42 38 01 
42 51 04 
42 64 09 
42 77 16 

42 90 25 

43 03 36 
43 16 49 
43 29 64 
43 42 81 
43 56 00 
43 69 21 
43 82 44 

43 95 69 

44 08 96 
44 22 25 
44 35 56 
44 48 89 
44 62 24 
44 75 61 

44 89 00 

45 02 41 
45 15 84 
45 29 29 
45 42 76 
45 56 25 
45 69 76 
45 83 29 

45 96 84 

46 10 41 
46 24 00 
46 37 61 
46 51 24 
46 64 89 
46 78 56 

46 92 25 

47 05 96 
47 19 69 
47 33 44 
47 47 21 
47 61 00 
47 74 81 

47 88 64 

48 02 49 
48 16 36 
48 30 25 
48 44 16 
48 58 09 
48 72 04 

48 86 01 

49 00 00 
49 14 01 
49 28 04 
49 42 09 
49 56 16 
49 70 25 
49 84 36 

49 98 49 

50 12 64 
50 26 81 
50 41 00 
50 55 21 
50 69 44 
50 83 69 

50 97 96 

51 12 25 



Cfube. 



274 625 000 

275 894 451 

277 167 808 

278 445 077 

279 726 264 

281 Oil 375 

282 300 416 

283 593 393 

284 890 312 

286 191 179 

287 496 000 

288 804 781 

290 117 528 

291 434 247 

292 754 944 

294 079 625 

295 408 296 

296 740 963 

298 077 632 

299 418 309 

300 763 000 

302 111711 

303 464 448 

304 821 217 

306 182 024 

307 546 875 

308 915 776 

310 288 733 

311 665 752 

313 046 839 

314 432 000 

315 821 241 

317 214 568 

318 611 987 

320 013 504 

321 419 125 

322 828 856 

324 242 703 

325 660 672 

327 082 769 

328 509 000 

329 939 371 

331 373 888 

332 812 557 

334 255 384 

335 702 375 

337 153 536 

338 608 873 

340 068 392 

341 532 099 

343 000 000 

344 472 101 

345 948 408 

347 428 927 

348 913 664 

350 402 625 

351 895 816 

353 393 243 

354 894 912 

356 400 829 

357 911 000 

359 425 431 

360 944 128 

362 467 097 

363 994 344 
365 525 875 



Sq. Rt. 



Cu. Rt. iNo. 



25.4950976 
.5147016 
.5342907 
.5538647 
.5734237 

25.5929678 
.6124969 
.6320112 
.6515107 
.6709953 

25.6904652 
.7099203 
.7293607 
.7487864 
.7681975 

25.7875939 
.8069758 
.8263431 
.8456960 
.8650343 

25.8843582 
.9036677 
.9229628 
.9422435 
.9615100 

25.9807621 

26.0000000 
.0192237 
.0384331 
.0576284 

26,0768096 
.0959767 
.1151297 
.1342687 
.1533937 

26.1725047 
.1916017 
.2106848 
.2297541 
.2488095 

26.2678511 
.2868789 
.3058929 
.3248932 
.3438797 

26.3628527 
.3818119 
.4007576 
.4196896 
.4386081 

26.4575131 
.4764046 
.4952826 
.5141472 
.5329983 

26.5518361 
.5706605 
.5894716 
.6082694 
6270539 

26.6458252 
.6645833 
.6833281 
.7020598 
.7207784 

26.7394839 



.6623911 715 
.666831M 16 
.6712665 17 



.6756974 

.6801237 

i.6845456 

.6889630 

.6933759 

.6977843 

.7021882 

5.7065877 

.7109827 

.7153734 

.7197596 

.7241414 

5.7285187 

.7328918 

.7372604 

.7416246 

.7459846 

5.7503401 

.7546913 

.7590383 

.7633809 

.7677192 

^.7720532 

.7763830 

.7807084 

.7850296 

.7893466 

^.7936593 

.7979679 

.8022721 

.8065722 

.8108681 

3.8151598 

.8194474 

.823730 

.8280099 

.8322850 

B.8365559 

.8408227 

.8450854 

.8493440 

.8535985 

3.8578489 

.8620952 

.8663375 

.8705757 

.8748099 

B.8790400 

.8832661 

.8874882 

.8917063 

.8959204 

3.9001304 

.9043366 

.9085387 

.9127369 

.9169311 

B.9211214 

.9253078 

.9294902 

.9336687 

.9378433 

8.9420140 780 



730 



4 
735 



740 



745 



755 



760 
1 



765 



770 
1 



4 

775 



Square 



Cube. Sq. Rt. 



51 12 25 
51 26 56 
51 40 89 
51 55 24 
51 69 61 
51 84 00 

51 98 41 

52 12 84 
52 27 29 
52 41 76 
52 56 25 
52 70 76 
52 85 29 

52 99 84 

53 14 41 
53 29 00 
53 43 61 
53 58 24 
53 72 89 

53 87 56 

54 02 25 
54 16 96 
54 3169 
54 46 44 
54 61 21 
54 76 00 

54 90 81 

55 05 64 
55 20 49 
55 35 36 
55 50 25 
55 65 16 
55 80 09 

55 95 04 

56 10 01 
56 25 00 
56 40 01 
56 55 04 
56 70 

56 85 16 

57 00 25 
57 15 36 
57 30 49 
57 45 64 
57 60 81 
57 76 00 

57 91 21 

58 06 44 
58 21 69 
58 36 96 
58 52 25 
58 67 56 
58 82 89 

58 98 24 

59 13 61 
59 29 00 
59 44 41 
59 59 84 
59 75 29 

59 90 76 

60 06 25 
60 21 76 
60 37 29 
60 52 84 
60 68 41 
60 84 00 



365 525 875 

367 061 696 

368 601 813 

370 146 232 

371 694 959 

373 248 000 

374 805 361 

376 367 048 

377 933 067 
379 503 424 

381 078 125 

382 657 176 

384 240 583 

385 828 352 
387 420 489 

389 017 000 

390 617 891 

392 223 168 

393 832 837 
395 446 904 

397 065 375 

398 688 256 

400 315 553 

401 947 272 
403 583 419 

405 224 000 

406 869 021 
408 518 488 
410 172 407 
411830 784 
413 493 625 

415 160 936 

416 832 723 
418 508 992 

420 189 749 

421 875 000 
423 564 751 

425 259 008 

426 957 777 
428 661 064 
430 368 875 

432 081 216 

433 798 093 
435 519 512 

437 245 479 

438 976 000 
440 711081 
442 450 728 

444 194 947 

445 943 744 
447 697 125 
449 455 096 

451 217 663 

452 984 832 
454 756 609 
456 533 000 
458 314 011 

460 099 648 

461 889 917 
463 684 824 
465 484 375 
467 288 576 

469 097 433 

470 910 952 
472 729 139 
474 552 000 



26. 



26. 



27 



27 



27 



7394839 
7581763 
7768557 
7955220 
8141754 
8328157 
,8514432 
,8700577 
,8886593 
.9072481 
.9258240 
.9443872 
.9629375 
.9814751 
.0000000 
.0185122 
.0370117 
.0554985 
.0739727 
.0924344 
.1108834 
.1293199 
.1477439 
.1661554 
.1845544 
.2029410 
.2213152 
.2396769 
.2580263 
.2763634 
.2946881 
.3130006 
.3313007 
.3495887 
.3678644 
.3861279 
.4043792 
.4226184 
.4408455 
.4590604 
.4772633 
.4954542 
.5136330 
.5317998 
.5499546 
.5680975 
.5862284 
.6043475 
.6224546 
.6405499 
.6586334 
.6767050 
.6947648 
.7128129 
.7308492 
.7488739 
.7668868 
.7848880 
.8028775 
.8208555 
.8388218 
.8567766 
.8747197 
.8926514 
.9105715 
.9284801 



Cu. Rt. 



8.9420140 
.9461809 
.9503438 
.9545029 
.9586581 

8.9628095 
.9669570 
.9711007 
.9752406 
.9793766 

8.9835089 
.9876373 
.9917620 
.9958829 

9.0000000 

9.0041134 
.0082229 
.0123288 
.0164309 
.0205293 

9.0246239 
.0287149 
.0328021 
.0368857 
.0409655 

9.0450417 
.0491142 
.0531831 
.0572482 
.0613098 

9.0653677 
.0694220 
.0734726 
.0775197 
.0815631 

9.0856030 
.0896392 
.0936719 
.0977010 
.1017265 

9.1057485 
.1097669 
.1137818 
.1177931 
.1218010 

9.1258053 
.1298061 
.1338034 
.1377971 
.1417874 

9.1457742 
.1497576 
.1537375 
.1577139 
.1616869 

9.1656565 
.1696225 
.1735852 
.1775445 
.1815003 

9.1854527 
.1894018 
.1933474 
.1972897 
.2012286 

9.2051641 



COMMON TABLES— SQUARES, CUBES, ROOTS. 



37 



9. — Squares, Cubes, Square Roots, Cube Roots, of Numbers 
1 TO 1600 — Continued. 



Square 



60 84 00 

60 99 61 

61 15 24 
61 30 89 
61 46 56 
61 62 25 
61 77 96 

61 93 69 

62 09 44 
62 25 21 
62 41 00 
62 56 81 
62 72 64 

62 88 49 

63 04 36 
63 20 25 
63 36 16 
63 52 09 
63 68 04 

63 84 01 

64 00 00 
64 16 01 
64 32 04 
64 48 09 
64 64 16 
64 80 25 

64 96 36 

65 12 49 
65 28 64 
65 44 81 
65 61 00 
65 77 21 

65 93 44 

66 09 69 
66 25 96 
66 42 25 
66 58 56 
66 74 89 

66 91 24 

67 07 61 
67 24 00 
67 40 41 
67 56 84 
67 73 29 

67 89 76 

68 06 25 
68 22 76 
68 39 29 
68 55 84 
68 72 41 

68 89 00 

69 05 61 
69 22 24 
69 38 89 
69 55 56 
69 72 25 

69 88 96 

70 05 69 
70 22 44 
70 39 21 
70 56 00 
70 72 81 

70 89 64 

71 06 49 
71 23 36 
71 40 25 



Cube. 



474 552 000 
476 379 541 
478 211 768 

480 048 687 

481 890 304 
483 736 625 
485 587 656 
487 443 403 
489 303 872 
491 169 069 

493 039 000 

494 913 671 
496 793 088 
498 677 257 
500 566 184 
502 459 875 
504 358 336 
506 261 573 
508 169 592 
510 082 399 

512 000 000 

513 922 401 
515 849 608 
517 781 627 
519 718 464 
521 660 125 
523 606 616 
525 557 943 
527 514 112 
529 475 129 
531 441 000 
533 411 731 
535 387 328 
537 367 797 
539 353 144 
541 343 375 
543 338 496 
545 338 513 
547 343 432 
549 353 259 
551 368 000 
553 387 661 
555 412 248 
557 441 767 
559 476 224 
561 515 625 
563 559 976 
565 609 283 
567 663 552 
569 722 789 
571 787 000 
573 856 191 
575 930 368 
578 009 537 
580 093 704 
582 182 875 
584 277 056 
586 376 253 
588 480 472 
590 589 719 
592 704 000 
594 823 321 
596 947 688 
599 077 107 
601 211 584 
603 351 125 



Sq. Rt. 



27 



28. 



28. 



28, 



28, 



28 



28, 



28 



28, 



29. 



9284801 
.9463772 
.9642629 
.9821372 
.0000000 
0178515 
0356915 
0535203 
0713377 
0891438 
1069386 
1247222 
1424946 
1602557 
1780056 
.1957444 
.2134720 
.2311884 
.2488938 
.2665881 
.2842712 
.3019434 
.3196045 
.3372546 
.3548938 
.3725219 
.3901391 
.4077454 
.4253408 
.4429253 
.4604989 
.4780617 
.4956137 
.5131549 
.5306852 
.5482048 
.5657137 
.5832119 
.6006993 
.6181760 
.6356421 
.6530976 
.6705424 
.6879766 
.7054002 
7228132 
.7402157 
.7576077 
.7749891 
.7923601 
.8097206 
.8270706 
.8444102 
.8617394 
.8790582 
.8963666 
.9136646 
.9309523 
.9482297 
.9654967 
.9827535 
.0000000 
.0172363 
,0344623 
.0516781 
0688837 



Cu. Rt. 



9.2051641 
.2000962 
.2130250 
.2169505 
.2208726 

9.2247914 
.2287068 
.2326189 
.2365277 
.2404333 

9.2443355 
,2482344 
.2521300 
.2560224 
.2599114 

9.2637973 
.2676798 
.2715592 
.2754352 
.2793081 

9.2831777 
.2870440 
.2909072 
.2947671 
.2986239 

9.3024775 
.3063278 
.3101750 
.3140190 
.3178599 

9.3216975 
.3255320 
.3293634 
.3331916 
.3370167 

9.3408386 
.3446575 
.3484731 
.3522857 
.3560952 

9.3599016 
.3637049 
.3675051 
,3713022 
.3750963 

9.3788873 
.3826752 
.3864600 
.3902419 
.3940206 

9.3977964 
.4015691 
.4053387 
.4091054 
.4128690 

9 4166297 
.4203873 
.4241420 
.4278936 
.4316423 

9.4353880 
.4391307 
.4428704 
.4466072 
.4503410 

9.4540719 



No. 



845 
6 

7 



850 
1 
2 
3 
4 

855 
6 
7 
8 
9 

860 
1 
2 
3 
4 

865 
6 
7 



870 
1 
2 
3 
4 

875 
6 
7 



4 

885 
6 

7 



6 

4 

895 



900 
1 
2 
3 
4 

905 



9 
910 



Square 



7140 25 
71 57 16 
71 74 09 

71 91 04 

72 08 01 
72 25 00 
72 42 01 
72 59 04 
72 76 09 

72 93 16 

73 10 25 
73 27 36 
73 44 49 
73 61 64 

73 78 81 
11 96 00 

74 13 21 
74 30 44 
74 47 69 
74 64 96 
74 82 25 

74 99 56 

75 16 89 
75 34 24 
75 51 61 
75 69 00 

75 86 41 

76 03 84 
76 21 29 
76 38 76 
76 56 25 
76 73 76 

76 91 29 

77 08 84 
77 26 41 
77 44 00 
77 61 61 
77 79 24 

77 96 89 

78 14 56 
78 32 25 
78 49 96 
78 67 69 

78 85 44 

79 03 21 
79 21 00 
79 38 81 
79 56 64 
79 74 49 

79 92 36 

80 10 25 
80 28 16 
80 46 09 
80 64 04 

80 82 01 

81 00 00 
81 18 01 
81 36 04 
81 54 09 

81 72 16 
8190 25 

82 08 36 
82 26 49 
82 44 64 
82 62 81 
82 81 00 



Cube. 



603 351 125 
605 495 736 
607 645 423 
609 800 192 
611 960 049 
614 125 000 
616 295 051 
618 470 208 
620 650 477 
622 835 864 
625 026 375 
627 222 016 
629 422 793 
631 628 712 
633 839 779 
636 056 000 
638 277 381 
640 503 928 
642 735 647 
644 972 544 
647 214 625 
649 461 896 
651 714 363 
653 972 032 
656 234 909 
658 503 000 
660 776 311 
663 054 848 
665 338 617 
667 627 624 
669 921 875 
672 221 376 
674 526 133 
676 836 152 
679 151 439 
681 472 000 
683 797 841 
686 128 968 
688 465 387 
690 807 104 
693 154 125 
695 506 456 
697 864 103 
700 227 072 
702 595 369 
704 969 000 
707 347 971 
709 732 288 
712 121 957 
714 516 984 
716 917 375 
719 323 136 
721 734 273 
724 150 792 
726 572 699 
729 000 000 
731 432 701 
733 870 808 
736 314 327 
738 763 264 
741 217 625 
743 677 416 
746 142 643 
748 613 312 
751 089 429 
753 571 000 



Sq. Rt. 



29. 



29 



29 



29 



29 



29.' 



30 



30 



30. 



0688837 
.0860791 
.1032644 
.1204396 
.1376046 
.1547595 
.1719043 
.1890390 
.2061637 
.2232784 
.2403830 
.2574777 
.2745623 
.2916370 
.3087018 
.3257566 
.3428015 
.3598365 
.3768616 
.3938769 
.4108823 
.4278779 
.4448637 
.4618397 
.4788059 
.4957624 
.5127091 
.5296461 
.5465734 
.5634910 
.5803989 
.5972972 
.6141858 
.6310648 
.6479342 
.6647939 
.6816442 
.6984848 
.7153159 
.7321375 
.7489496 
.7657521 
.7825452 
.7993289 
.8161030 
.8328678 
.8496231 
.8663690 
.8831056 
.8998328 

9165506 
.9332591 
.9499583 
.9666481 
.9833287 
.0000000 
.0166620 
.0333148 
.0499584 
.0665928 
.0832179 
.0998339 
.1164407 
.1330383 
.1496269 

1662063 



Cu. Rt. 



9.4540719 
.4577999 
.4615249 
.4652470 
.4689661 

9.4726824 
.4763957 
.4801061 
.4838136 
.4875182 

9.4912200 
.4949188 
.4986147 
.5023078 
.5059980 

9.5096854 
.5133699 
.5170515 
.5207303 
.5244063 

9.5280794 
.5317497 
.5354172 
.5390818 
.5427437 

9.5464027 
.5500589 
.5537123 
.5573630 
.5610108 

9.5646559 
.5682982 
.5719377 
.5755745 
.5792085 

9.5828397 
.5864682 
.5900939 
.5937169 
.5973373 

9.6009548 
.6045696 
.6081817 
.6117911 
.6153977 

9.6190017 
.6226030 
.6262016 
.6297975 
.6333907 

9.6369812 
.6405690 
.6441542 
.6477367 
.6513166 

9.6548938 
.6584684 
.6620403 
.6656096 
.6691762 

9.6727403 
.6763017 
.6798604 
.6834166 
.6869701 

9.6905211 



2.— POWERS, ROOTS AND RECIPROCALS, 

9. — Squares, Cubes, Square Roots, Cube Roots, of Numbers 
1 TO 1600 — Continued. 



Square 



Cube. 



Sq. Rt. 



Cu. Rt. 



No. 



Square. 



Cube. 



Sq. Rt. 



Cu. Rt. 



82 81 00 

82 99 21 

83 17 44 
83 35 69 
83 53 96 
83 72 25 

83 90 56 

84 08 89 
84 27 24 
84 45 61 
84 64 00 

84 82 41 

85 00 84 
85 19 29 
85 37 76 
85 56 25 
85 74 76 

85 93 29 

86 11 84 
86 30 41 
86 49 00 
86 67 61 

86 86 24 

87 04 89 
87 23 56 
87 42 25 
87 60 96 
87 79 69 

87 98 44 

88 17 21 
88 36 00 
88 54 81 
88 73 64 

88 92 49 

89 11 36 
89 30 25 
89 49 16 
89 68 09 

89 87 04 

90 06 01 
90 25 00 
90 44 01 
90 63 04 

90 82 09 

91 01 16 
9120 25 
91 39 36 
91 58 49 

91 77 64 
9196 81 

92 16 00 
92 35 21 
92 54 44 
92 73 69 

92 92 96 

93 12 25 
93 31 56 
93 50 89 
93 70 24 

93 89 61 

94 09 00 
94 28 41 
94 47 84 
94 67 29 

94 86 76 

95 06 25 



753 571 
756 058 
758 550 
761 048 
763 551 
766 060 
768 575 
771 095 
773 620 
776 151 
778 688 
781 229 
783 777 
786 330 
788 889 
791 453 
794 022 
796 597 
799 178 
801 765 
804 357 
806 954 
809 557 
812 166 
814 780 
81"7 400 
820 025 
822 656 
825 293 
827 936 
830 584 
833 237 
835 896 
838 561 
841 232 
843 908 
846 590 
849 278 
851 971 
854 670 
857 375 
860 085 
862 801 
865 523 
868 250 
870 983 
873 722 
876 467 
879 217 
881 974 
884 736 
887 503 
890 277 
893 056 
895 841 
898 632 
901 428 
904 231 
907 039 
909 853 
912 673 
915 498 
918 330 
921 167 
924 010 
926 859 



000 
031 
528 
497 
944 
875 
296 
213 
632 
559 
000 
961 
448 
467 
024 
125 
776 
983 
752 
089 
000 
491 
568 
237 
504 
375 
856 
953 
672 
019 
000 
621 
888 
807 
384 
625 
536 
123 
392 
349 
000 
351 
408 
177 
664 
875 
816 
493 
912 
079 
000 
681 
128 
347 
344 
125 
696 
063 
232 
209 
000 
611 
048 
317 
424 
375 



30.1662063 
.1827765 
.1993377 
.2158899 
.2324329 

30.2489669 
.2654919 
.2820079 
.2985148 
.3150128 

30.3315018 
.3479818 
.3644529 
.3809151 
.3973683 

30.4138127 
.4302481 
.4466747 
.4630924 
.4795013 

30.4959014 
.5122926 
.5286750 
.5450487 
.5614136 

30.5777697 
.5941171 
.6104557 
.6267857 
.6431069 

30.6594194 
.6757233 
.6920185 
.7083051 
.7245830 

30.7408523 
.7571130 
.7733651 
.7896086 
.8058436 

30.8220700 
.8382879 
.8544972 
.8706981 
.8868904 

30.9030743 
.9192497 
.9354166 
.9515751 
.9677251 

30.9838668 

31.0000000 
.0161248 
.0322413 
.0483494 

31.0644491 
.0805405 
.0966236 
.1126984 
.1287648 

31.1448230 
.1608729 
.1769145 
.1929479 
.2089731 

31.2249900 



9.6905211 
.6940694 
.6976151 
.7011583 
.7046989 

9.7082369 
.7117723 
.7153051 
.7188354 
.7223631 

9.7258883 
.7294109 
.7329309 
.7364484 
.7399634 

9.7434758 
.7469857 
.7504930 
.7539979 
.7575002 

9.7610001 
.7644974 
.7679922 
.7714845 
.7749743 

9.7784616 
.7819466 
.7854288 
,7889087 
.7923861 

9.7958611 
.7993336 
.8028036 
.8062711 
.8097362 

9.8131989 
.8166591 
.8201169 
.8235723 
.8270252 

9.8304757 
.8339238 
.8373695 
.8408127 
.8442536 



975 
6 
7 
8 



4 

985 
6 
7 
8 
9 

990 



995 
6 

7 



1000 



1005 



1010 
11 
12 
13 
14 

1015 
16 
17 
18 
19 



9.8476920 1020 



.8511280 
.8545617 
.8579929 
.8614218 

9.8648483 
.8682724 
.8716941 
.8751135 
.8785305 

9.8819451 
.8853574 
.8887673 
.8921749 
.8955801 



21 
22 
23 
24 

1025 
26 
27 
28 
29 

1030 
31 
32 
33 
34 



.8989830 1035 



.9023835 
.9057817 
.9091776 
.9125712 



39 



9.9159624 1040 



95 06 25 
95 25 76 
95 45 29 
95 64 84 

95 84 41 

96 04 00 
96 23 61 
96 43 24 
96 62 89 

96 82 56 

97 02 25 
97 21 96 
97 41 69 
97 61 44 

97 81 21 

98 01 00 
98 20 81 
98 40 64 
98 60 49 

98 80 36 

99 00 25 
99 20 16 
99 40 09 
99 60 04 
99 80 01 
00 00 00 
00 20 01 
00 40 04 
00 60 09 

00 80 16 

01 00 25 
01 20 36 
01 40 49 

01 60 64 
0180 81 

02 01 00 
02 21 21 
02 41 44 
02 61 69 

02 81 96 

03 02 25 
03 22 56 
03 42 89 
03 63 24 

03 83 61 

04 04 00 
04 24 41 
04 44 84 
04 65 29 

04 85 76 

05 06 25 
05 26 76 
05 47 29 
05 67 84 

05 88 41 

06 09 00 
06 29 61 
06 50 24 
06 70 89 

06 91 56 

07 12 25 
07 32 96 
07 53 69 
07 74 44 

07 95 21 

08 16 00 



926 859 375 
929 714 176 
932 574 833 
935 441 352 
938 313 739 
941 192 000 
944 076 141 
946 966 168 
949 862 087 
952 763 904 
955 671 625 
958 585 256 
961 504 803 
964 430 272 
967 361 669 
970 299 000 
973 242 271 
976 191 488 
979 146 657 
982 107 784 
985 074 875 
988 047 936 
991 026 973 
994 Oil 992 
997 002 999 
000 000 000 
003 003 001 
006 012 008 
009 027 027 
012 048 064 
015 075 125 
018 108 216 
021 147 343 
024 192 512 
027 243 729 
030 301 000 
033 364 331 
036 433 728 
039 509 197 
042 590 744 
045 678 375 
048 772 096 
051 871 913 
054 977 832 
058 089 859 
061 208 000 
064 332 261 
067 462 648 
070 599 167 
073 741 824 
076 890 625 
080 045 576 
083 206 683 
086 373 952 
089 547 389 
092 727 000 
095 912 791 
099 104 768 
102 302 937 
105 507 304 
108 717 875 
111934 656 
115 157 653 
118 386 872 
121 622 319 
124 864 000 



31 



31 



31 



31 



32, 



32, 



32. 



.2249900 
.2409987 
.2569992 
.2729915 
.2889757 
.3049517 
.3209195 
.3368792 
.3528308 
.3687743 
.384709 
.4006369 
.4165561 
.4324673 
.4483704 
4642654 
.4801525 
.4960315 
.5119025 
.5277655 
.5436206 
.5594677 
.5753068 
.5911380 
.6069613 
.6227766 
.6385840 
.6543836 
.6701752 
.6859590 
.7017349 
.7175030 
.7332633 
.7490157 
.7647603 
.7804972 
.7962262 
.8119474 
.8276609 
.8433666 
.8590646 
.8747549 
.8904374 
.9061123 
.9217794 
9374388 
.9530906 
.9687347 
.9843712 
.0000000 
.0156212 
.0312348 
.0468407 
.0624391 
.0780298 
,0936131 
1091887 
1247568 
1403173 
1558704 
1714159 
1869539 
2024844 
2180074 
2335229 
2490310 



10, 



10, 



10, 



10. 



.9159624 
.9193513 
.9227379 
.9261222 
.9295042 
.9328839 
.9362613 
.9396363 
.9430092 
.9463797 
.9497479 
.9531138 
.9564775 
.9598389 
.9631981 
.9665549 
.9699095 
.9732619 
.9766120 
.9799599 
.9833055 
.9866488 
.9899900 
.9933289 
.9966656 
.0000000 
.0033322 
.0066622 
.0099899 
.0133155 
.0166389 
.0199601 
.0232791 
.0265958 
.0299104 
.0332228 
.0365330 
.0398410 
.0431469 
.0464506 
.0497521 
.0530514 
.0563485 
.0596435 
0629364 
.0662271 
.0695156 
.0728020 
.0760863 
,0793684 
.0826484 
.0859262 
.0892019 
,0924755 
,0957469 
,0990163 
,1022835 
1055487 
1088117 
1120726 
1153314 
1185882 
1218428 
1250953 
1283457 
1315941 



COMMON TABLES— SQUARES, CUBES, ROOTS. 



39 



9. — Squares, Cubes, Square Roots, Cube Roots, of Numbers 
1 to 1600 — Continued. 



Wo. 

1040 
41 
42 
43 
44 

1045 
46 
47 
48 
49 

1050 
51 
52 
53 
54 

1055 
56 
57 
58 
59 

1060 
61 
62 
63 
64 

1065 
66 
67 
68 
69 

1070 
71 
72 
73 
74 

1075 
76 
77 
78 
79 

1080 
81 
82 
83 
84 

1085 
86 
87 
88 
89 

1090 
91 
S2 
93 
94 

1095 
96 
97 



Square 



1100 



3 

4 

1105 



0816 00 
08 36 81 
08 57 64 
08 78 49 

08 99 36 

09 20 25 
09 41 16 
09 62 09 

09 83 04 

10 04 01 
10 25 00 
10 46 01 
10 67 04 

10 88 09 

11 09 16 
1130 25 
11 51 36 
11 72 49 

11 93 64 

12 14 81 
12 36 00 
12 57 21 
12 78 44 

12 99 69 

13 20 96 
13 42 25 
13 63 56 

13 84 89 

14 06 24 
14 27 61 
14 49 00 
14 70 41 

14 9184 

15 13 29 
15 34 76 
15 56 25 
15 77 76 

15 99 29 

16 20 84 
16 42 41 
16 64 00 

16 85 61 

17 07 24 
17 28 89 
17 50 56 
17 72 25 

17 93 96 

18 15 69 
,18 37 44 
18 59 21 

18 81 00 

19 02 81 
19 24 64 
19 46 49 
19 68 36 

19 90 25 

20 12 16 
20 34 09 
20 56 04 

20 78 01 

21 00 00 
21 42 01 
21 44 04 
21 66 09 

21 88 16 

22 10 25 



Cube. 



124 864 
128 111 
131 366 
134 626 
137 893 
141 166 
144 445 
147 730 
151 022 
154 320 
157 625 
160 935 
164 252 
167 575 
170 905 
174 241 
117 583 
180 932 
184 287 
187 648 
191 016 
194 389 
197 770 
201 157 
204 550 
207 949 
211 355 
214 767 
218 186 
221 611 
225 043 
228 480 
231 925 
235 376 
238 833 
242 296 
245 766 
249 243 
252 726 
256 216 
259 712 
263 214 
266 723 
270 238 
273 760 
277 289 
280 824 
284 365 
287 913 
291 467 
295 029 
298 596 
302 170 
305 751 
309 338 
312 932 
316 532 
320 139 
323 753 
327 373 
331 000 
334 633 
338 273 
341 919 
345 572 
349 232 



000 
921 
088 
507 
184 
125 
336 
823 
592 
649 
000 
651 
608 
877 
464 
375 
616 
193 
112 
379 
000 
981 
328 
047 
144 
625 
496 
763 
432 
509 
000 
911 
248 
017 
224 
875 
976 
533 
552 
039 
000 
441 
368 
787 
704 
125 
056 
503 
472 
969 
000 
571 
688 
357 
584 
375 
736 
673 
192 
299 
000 
301 
208 
727 
864 
625 



Sq. Rt. 



32 



32 



32 



32 



32 



32, 



32 



32 



32 



33 



33 



.2490310 
.2645316 
.2800248 
.2955105 
.3109888 
.3264598 
.3419233 
.3573794 
.3728281 
.3882695 
.4037035 
.4191301 
.4345495 
.4499615 
.4653662 
.4807635 
.4961536 
.5115364 
.5269119 
.5422802 
.5576412 
.5729949 
.5883415 
.6036807 
.6190129 
.6343377 
.6496554 
.6649659 
.6802693 
.6955654 
.7108544 
.7261363 
.7414111 
.7566787 
.7719392 
.7871926 
.8024389 
,8176782 
,8329103 
,8481354 
,8633535 
,8785644 
,8937684 
,9089653 
,9241553 
.9393382 
,9545141 
.9696830 
,9848450 
,0000000 
,0151480 
,0302891 
.0454233 
,0605505 
,0756708 
,0907842 
,1058907 
,1209903 
,1360830 
,1511689 
,1662479 
,1813200 
,1963853 
,2114438 
,2264955 
.2415403 



Cu. Rt. 

10.1315941 
.1348403 
.1380845 
.1413266 
.1445667 

10.1478047 
.1510406 
.1542744 
.1575062 
.1607359 

10.1639636 
.1671893 
.1704129 
.1736344 
.1768539 

10. 
.1832868 
.1865002 
.1897116 
.1929209 

10.1961283 
.1993336 
.2025369 
.2057382 
.2089375 

10.2121347 
.2153300 
.2185233 
.2217146 
.2249039 

10.2280912 
.2312766 
.2344599 
.2376413 
.2408207 

10.2439981 
.2471735 
.2503470 
.2535186 
.2566881 

10.2598557 
.2630213 
.2661850 
.2693467 
.2725065 

10.2756644 
.2788203 
.2819743 
.2851264 
.2882765 

10.2914247 
.2945709 
.2977153 
.3008577 
.3039982 

10.3071368 
.3102735 
.3134083 
.3165411 
.3196721 

10.3228012 
.3259284 
.3290537 
.3321770 
.3352985 

10.3384181 



No 
1105 



9 

1110 
11 
12 
13 
14 

1115 
16 
17 
18 
19 
,1800714 1120 
21 
22 
23 
24 

1125 
26 
27 
28 
29 

1130 
31 
32 
33 
34 

1135 
36 
37 



1140 
41 
42 
43 
44 

1145 
46 
47 
48 
49 

1150 
51 
52 
53 
54 

1155 
56 
57 
58 
59 

1160 
61 
62 
63 
64 

1165 
66 
67 



69 
1170 



Square 

22 10 25 
22 32 36 
22 54 49 
22 76 64 

22 98 81 

23 21 00 
23 43 21 
23 65 44 

23 87 69 

24 09 96 
24 32 25 
24 54 56 
24 76 89 

24 99 24 

25 21 61 
25 44 00 
25 66 41 

25 88 84 

26 11 29 
26 33 76 
26 56 25 

26 78 76 

27 01 29 
27 23 84 
27 46 41 
27 69 00 

27 91 61 

28 14 24 
28 36 89 
28 59 56 

28 82 25 

29 04 96 
29 27 69 
29 50 44 
29 73 21 

29 96 00 

30 18 81 
30 41 64 
30 64 49 

30 87 36 

31 10 25 
31 33 16 
31 56 09 

31 79 04 

32 02 01 
32 25 00 
32 48 01 
32 71 04 

32 94 09 

33 17 16 
33 40 25 
33 63 36 

33 86 49 

34 09 64 
34 32 81 
34 56 00 

34 79 21 

35 02 44 
35 25 69 
35 48 96 
35 72 25 

35 95 56 

36 18 89 
36 42 24 
36 65 61 
36 89 00 



Cube. 



1349 
1352 
1356 
1360 
1363 
1367 
1371 
1375 
1 378 
1 382 
1386 
1389 
1393 
1397 
1 401 
1 404 
1408 
1412 
1416 
1420 
1423 
1 427 
1431 
1435 
1439 
1442 
1446 
1450 

454 

458 
1462 
1466 
1469 
1473 
1 477 
1481 
1 485 
1 489 
1 493 
1497 
1 501 
1 505 
1 509 
1 512 

516 
1 520 
1 524 
1 528 
1 532 
1 536 
1 540 
1 544 
1 548 
1 552 
1 556 
1 560 
1 564 
1 568 
1 573 
1 577 
1 581 
1 585 
1 589 
1 593 

597 
1601 



232 625 
899 016 
572 043 
251 712 
938 029 
631 000 
330 631 

036 928 
749 897 
469 544 
195 875 
928 896 
668 613 
415 032 
168 159 
928 000 
694 561 
467 848 
247 867 
034 624 
828 125 
628 376 
435 383 
249 152 
069 689 
897 000 
731 091 
571 968 
419 637 
274 104 
135 375 
003 456 
878 353 
760 072 
648 619 
544 000 
446 221 
355 288 
271 207 
193 984 
123 625 
060 136 
003 523 
953 792 
910 949 
875 000 
845 951 
823 808 
808 577 
800 264 
798 875 
804 416 
816 893 
836 312 
862 679 
896 000 
936 281 
983 528 

037 747 
098 944 
167 125 
242 296 
324 463 
413 632 
509 809 
613 000 



Sq. Rt. 



33 



33 



33 



33, 



33 



33 



33. 



33 



33 



33 



34 



34, 



34, 



,2415403 
2565783 
2716095 
2866339 
3016516 
3166625 
3316666 
3466640 
3616546 
3766385 
3916157 
4065862 
4215499 
4365070 
4514573 
4664011 
4813381 
4962684 
5111921 
5261092 
5410196 
5559234 
5708206 
5857112 
6005952 
6154726 
6303434 
6452077 
6600653 
6749165 
6897610 
7045991 
7194306 
7342556 
7490741 
7638860 
7786915 
7934905 
8082830 
8230691 
8378486 
8526218 
8673884 
8821487 
8969025 
9116499 
9263909 
9411255 
9558537 
9705755 
9852910 
0000000 
0147027 
0293990 
0440890 
0587727 
0734501 
0881211 
1027858 
1174442 
1320963 
1467422 
1613817 
1760150 
1906420 
,2052627 



Cu. Rt. 

10.3384181 
.3415358 
.3446517 
.3477657 
.3508778 

10.3539880 
.3570964 
.3602029 
.3633078 
.3664103 

10.3695113 
.3726103 
.3757076 
.3788030 
.3818965 

10.3849882 
.3880781 
.3911661 
.3942523 
.3973366 

10.4004192 
.4034999 
.4065787 
.4096557 
.4127310 

10.4158044 
.4188760 
.4219458 
.4250138 
.4280800 

10.4311443 
.4342069 
.4372677 
.4403267 
.4433839 

10.4464393 
.4494929 
4525448 
.4555948 
.4586431 

10.4616896 
.4647343 
.4677773 
.4708185 
.4738579 

10.4768955 
.4799314 
.4829656 
.4859980 
.4890286 

10.4920575 
.4950847 
.4981101 
.5011337 
.5041556 

10.5071757 
.5101942 
.5132109 
.5162259 
.5192391 

10.5222506 
.5252604 
.5282685 
.5312749 
.5342795 

10.5372826 



40 



2— POWERS, ROOTS AND RECIPROCALS, 



9. — Squares, Cubes, Square Roots, Cube Roots, op Numbers 
1 TO 1600 — Continued. 



No. 

1170 
71 
72 
73 
74 

1175 
76 
77 
78 
79 

1180 
81 
82 
83 
84 

1185 
86 
87 



1190 
91 
92 
93 
94 

1195 
96 
97 
98 
99 

1200 
1 
2 
3 
4 

1205 
6 
7 
8 
9 

1210 
11 
12 
13 
14 

1215 
16 
17 
18 
19 

1220 
21 
22 
23 
24 

1225 
26 
27 
28 
29 

1230 
31 
32 
33 
34 

1235 



Square 



Cube. Sq. Rt. 



Cu. Rt. 



No. 



Square 



Cube. 



Sq. Rt. 



Cu. Rt. 



36 89 00 

37 12 41 
37 35 84 
37 59 29 

37 82 76 

38 06 25 
38 29 76 
38 53 29 

38 76 84 

39 00 41 
39 24 00 
39 47 61 
39 71 24 

39 94 

40 18 56 
40 42 25 
40 65 96 

40 89 69 

41 13 44 
41 37 21 
41 61 00 

41 84 81 

42 08 64 
42 32 49 
42 56 36 

42 80 25 

43 04 16 
43 28 09 
43 52 04 

43 76 01 

44 00 00 
44 24 01 
44 48 04 
44 72 09 

44 96 16 

45 20 25 
45 44 36 
45 68 49 

45 92 64 

46 16 81 
46 41 00 
46 65 21 

46 89 44 

47 13 
47 37 96 
47 62 25 

47 86 56 

48 10 89 
48 35 24 
48 59 61 

48 84 00 

49 08 41 
49 32 84 
49 57 29 

49 81 76 

50 06 25 
50 30 76 
50 55 29 

50 79 84 

51 04 41 
51 29 00 
51 53 61 

51 78 24 

52 02 89 
52 27 56 
52 52 25 



601 613 
605 723 
609 840 
613 964 
618 096 
622 234 
626 379 
630 532 
634 691 
638 858 
643 032 
647 212 
651 400 
655 595 
659 797 
664 006 
668 222 
672 446 
676 676 
680 914 
685 159 
689 410 
693 669 
697 936 
702 209 
706 489 
710 777 
715 072 
719 374 
723 683 
728 000 
732 323 
736 654 
740 992 
745 337 
749 690 
754 049 
758 416 
762 790 
767 172 
771 561 
775 956 
780 360 
784 770 
789 188 
793 613 
798 045 
802 485 
806 932 
811 386 
815 848 
820 316 
824 793 
829 276 
833 767 
838 265 
842 771 
847 284 
851 804 
856 331 
860 867 
865 409 
869 959 
874 516 
879 080 
883 652 



000 34.2052627 



211 

448 
717 
024 
375 
776 
233 
752 
339 
000 
741 
568 
487 
504 
625 
856 
203 
672 
269 
000 
871 
888 
057 
384 
875 
536 
373 
392 
599 
000 
601 
408 
427 
664 
125 
816 
743 
912 
329 
000 
931 
128 
597 
344 
375 
696 
313 
232 
459 
000 
861 
048 
567 
424 
625 
176 
083 
352 
989 
000 
391 
168 
337 
904 
875 



10.5372825 1235 



.2198773 
.2344855 
.2490875 
.2636834 

34.2782730 
.2928564 
.3074336 
.3220046 
.3365694 

34.3511281 
.3656805 
.3802268 
.3947670 
.4093011 

34.4238289 
.4383507 
.4528663 
.4673759 
.4818793 

34.4963766 
.5108678 
.5253530 
.5398321 
.5543051 

34.5687720 
.5832329 
.5976879 
.6121366 
.6265794 

34.6410162 
.6554469 
.6698716 
-.6842904 
.6987031 

34.7131099 
.7275107 
.7419055 
,7562944 
.7706773 

34.7850543 
.7994253 
.8137904 
.8281495 
.8425028 

34.8568501 
.8711915 
.8855271 
.8998567 
.9141805 

34.9284984 
.9428104 
.9571166 
.9714169 
.9857114 

35.0000000 
.0142828 
.0285598 
.0428309 
.0570963 

35.0713558 
.0856096 
.0998575 
.1140997 
.1283361 

35.1425668 



10, 



,5402837 
,5432832 
,5462810 
,5492771 
,5522715 
5552642 
,5582552 
5612445 
5642322 
5672181 
5702024 
5731849 
5761658 
,5791449 



36 
37 
38 
39 

1240 
41 
42 
43 
44 

1245 
46 
47 
48 
49 



10.5821225 1250 



10, 



10.^ 



10, 



10, 



10, 



5850983 
5880725 
5910450 
5940158 
5969850 
5999525 
6029184 
6058826 
6088451 
611806q 
6147652 
6177228 
6206788 
6236331 
6265857 
6295367 
6324860 
6354338 
6383799 
6413244 
6442672 
6472085 
6501480 
6530860 
6560223 
6589570 
6618902 
6648217 
6677516 
6706799 
6736066 
6765317 
6794552 
6823771 
6852973 
6882160 
6911331 
6940486 
6969625 
6998748 
7027855 
7056947 
7086023 
7115083 
7144127 
7173155 
7202168 
7231165 
7260146 
7289112 



51 
52 
53 
54 

1255 
56 
57 
58 
59 

1260 
61 
62 
63 
64 

1265 
66 
67 
68 
69 

1270 
71 
72 
73 
74 

1275 
76 
77 
78 
79 

1280 
81 
82 
83 
84 

1285 
86 
87 
88 
89 

1290 
91 
92 
93 
94 

1295 
96 
97 
98 
99 

1300 



1 52 52 25 
1 52 76 96 
1 53 01 69 
1 53 26 44 
1 53 51 21 
1 53 76 00 
1 54 00 81 
1 54 25 64 
1 54 50 49 
1 54 75 36 
1 55 00 25 
1 55 25 16 
1 55 50 09 
1 55 75 04 
1 56 00 01 
1 56 25 00 
1 56 50 01 
1 56 75 04 
1 57 00 09 
1 57 25 16 
1 57 50 25 
1 57 75 36 
1 58 00 49 
1 58 25 64 
1 58 50 81 
1 58 76 00 
1 59 01 21 
1 59 26 44 
1 59 51 69 
1 59 76 96 
1 60 02 25 
1 60 27 56 
1 60 52 89 
1 60 78 24 
1 61 03 61 
1 61 29 00 
1 61 54 41 
1 61 79 84 
1 62 05 29 
1 62 30 76 
1 62 56 25 
1 62 81 76 
1 63 07 29 
1 63 32 84 
1 63 58 41 
1 63 84 00 
1 64 09 61 
1 64 35 24 
1 64 60 89 
1 64 86 56 
1 65 12 25 
1 65 37 96 
1 65 63 69 
1 65 89 44 
1 66 15 21 
1 66 41 00 
1 66 66 81 
1 66 92 64 
1 67 18 49 
1 67 44 36 
1 67 70 25 
1 67 96 16 
1 68 22 09 
1 68 48 04 
1 68 74 01 
1 69 00 00 



883 652 
888 232 
892 819 
897 413 
902 014 
906 624 
911 240 
915 864 
920 495 
925 134 
929 781 
934 434 
939 096 
943 764 
948 441 
953 125 
957 816 
962 515 
967 221 
971 935 
976 656 
981 385 
986 121 
990 865 
995 616 
000 376 
005 142 
009 916 
014 698 
019 487 
024 284 
029 089 
033 901 
038 720 
043 548 
048 383 
053 225 
058 075 
062 933 
067 798 
072 671 
077 552 
082 440 
087 336 
092 240 
097 152 
102 071 
106 997 
111 932 
116 874 
121 824 
126 781 
131 746 
136 719 
141 700 
146 689 
151 685 
156 689 
161 700 
166 720 
171 747 
176 782 
181 825 
186 875 
191 933 
197 000 



875 

256 

053 

272 

919 

000 

521 

488 

907 

784 

125 

936 

223 

992 

249 

000 

251 

008 

277 

064 

375 

216 

593 

512 

979 

000 

581 

728 

447 

744 

62535, 

096 

163 

832 

109 

000 

511 

648 

417 

824 

875 

576 

933 

952 

639 

000 

041 

768 

187 

304 

125 

656 

903 

872 

569 

000 

171 

088 

757 

184 

375 

336 

073 

592 

899 

000 



35.142566810. 
.1567917 
.1710108 
1852242 
.1994318 

35.2136337 
2278299 
2420204 
.2562051 
2703842 

35.2845575 
.2987252 
.3128872 
.3270435 
.3411941 

35.3553391 
.3694784 
.3836120 
.3977400 
4118624 

35.4259792 
.4400903 
4541958 
.4682957 
.4823900 

35.4964787 
.5105618 
.5246393 
5387113 
.5527777 
.5668385 
5808937 
.5949434 
.6089876 
6230262 

35.6370593 
.6510869 
.6651090 
6791255 
.6931366 

35.7071421 
7211422 
.7351367 
.7491258 
7631095 

35.7770876 
.7910603 
.8050276 
8189894 
.8329457 

35.8468966 
8608421 
.8747822 
.8887169 
.9026461 

35.9165699 10, 



10 



10, 



10, 



.9304884 

.9444015 

.9583092 

.9722115 

35.9861084 

36.0000000 

.0138862 

.0277671 

.0416426 

36.0555128 10 



7289112 
7318062 
7346997 
7375916 
7404819 
7433707 
7462579 
7491436 
7520277 
7549103 
7577913 
7606708 
7635488 
7664252 
7693001 
7721735 
7750453 
7779156 
7807843 
7836516 
7865173 
7893815 
7922441 
7951053 
7979649 
8008230 
8036797 
8065348 
,8093884 
8122404 
8150909 
8179400 
8207876 
8236336 
8264782 
8293213 
8321629 
8350030 
8378416 
8406788 
8435144 
8463485 
8491812 
8520125 
8548422 
8576704 
8604972 
8633225 
8661464 
81589687 
8717897 
8746091 
8774271 
8802436 
8830587 
8858723 
,8886846 
8914952 
8943044 
8971123 
8999186 
9027235 
9055269 
9083290 
9111296 
9139287 



COMMON TABLES— SQUARES, CUBES, ROOTS. 



41 



9. — Squares, Cubes, Square Roots, Cube Roots, op Numbers 
1 TO 1600 — Continued. 



No. 



Square 



Cube. 



Sq. Rt. 



Cu. Rt. 



No. 



Square 



Cube. 



Sq, Rt. 



Cu. Rt. 



1300 
1 
2 
3 
4 

1305 



1310 
11 
12 
13 
14 

1315 
16 
17 
18 
19 

1320 
21 
22 
23 
24 

1325 
26 
27 
28 
29 

1330 
31 
32 
33 
34 

1335 
36 
37 
38 
39 

1340 
41 
42 
43 
44 

1345 
46 
47 
48 
49 

1350 
51 
52 
53 
54 

1355 
56 
67 
58 
69 

1360 
61 
62 
63 
64 

1365 



1 69 00 00 
69 26 01 
69 52 04 
1 69 78 
1 70 04 16 
1 70 30 25 
1 70 56 36 
1 70 82 49 
1 71 08 64 
I 71 34 81 
1 71 61 00 
I 71 87 21 
1 72 13 44 
1 72 39 69 
1 72 65 96 
1 72 92 25 
1 73 18 56 
173 44 
1 73 71 24 
1 73 97 61 
1 74 24 00 
1 74 50 41 
1 74 76 84 
1 75 03 29 
1 75 29 76 
1 75 56 25 
1 75 82 76 
1 76 09 29 
1 76 35 84 
1 76 62 41 
1 76 89 00 
1 77 15 61 
1 77 42 24 
1 77 68 89 
1 77 95 56 
1 78 22 25 
1 78 48 96 
178 75 
1 79 02 44 
1 79 29 21 
1 79 56 00 
1 79 82 81 
1 80 09 64 
1 80 36 49 
1 80 63 36 
1 80 90 25 
1 81 17 16 
1 8144 
1 81 71 04 
1 81 98 01 
1 82 25 00 
1 82 52 01 
1 82 79 04 
1 83 06 09 
1 83 33 16 
1 83 60 25 
1 83 87 36 
1 84 14 49 
1 84 41 64 
1 84 68 81 
1 84 96 00 
1 85 23 21 
1 85 50 44 
1 85 77 69 
1 86 04 96 
1 86 32 25 



197 000 000 
2 202 073 901 
2 207 155 608 
2 212 245 127 
2 217 342 464 
2 222 447 625 
2 227 560 616 
2 232 681 443 

237 810 112 



2 242 946 629 
248 091 000 
253 243 231 
2 258 403 328 
2 263 571 297 
2 268 747 144 
2 273 930 875 
2 279 122 496 
2 284 322 013 
2 289 529 432 
2 294 744 759 
2 299 968 000 
2 305 199 161 
2 310 438 248 
2 315 685 267 
2 320 940 224 
2 326 203 125 
2 331 473 976 
2 336 752 783 
2 342 039 552 
2 347 334 289 
2 352 637 000 
2 357 947 691 
2 363 266 368 
2 368 593 037 
2 373 927 704 
2 379 270 375 
2 384 621 056 
2 389 979 753 
2 395 346 472 
2 400 721 219 
2 406 104 000 
2 411 494 821 
2 416 893 688 
2 422 300 607 
2 427 715 584 
2 433 138 625 
2 438 569 736 
2 444 008 923 
2 449 456 192 
2 454 911 549 
2 460 375 000 
2 465 846 551 
2 471 326 208 
2 476 813 977 
2 482 309 864 
2 487 813 875 
2 493 326 016 
2 498 846 293 
2 504 374 712 
2 509 911 279 
2 515 456 000 
2 521 008 881 
2 526 569 928 
2 532 139 147 
2 537 716 544 
2 543 302 125 



36.0555128 
.0693776 
.0832371 
.0970913 
.1109402 

36.1247837 
.1386220 
U 524550 
.1662826 
.1801050 

36.1939221 
.2077340 
.2215406 
.2353419 
.2491379 

36.2629287 
.2767143 
.2904946 
.3042697 
.3180396 

36.3318042 
.3455637 
.3593179 
.3730670 
.3868108 

36.4005494 
.4142829 
.4280112 
.4417343 
.4554523 

36.4691650 
.4828727 
.4965752 
.5102725 
.5239647 

36.5376518 
.5513338 
.5650106 
.5786823 
.5923489 

36.6060104 
.6196668 
.6333181 
.6469644 
.6606056 

36.6742416 
.6878726 
.7014986 
.7151195 
.7287353 

36.7423461 
.7559519 
.7695526 
.7831483 
.7967390 

36.8103246 
.8239053 
.8374809 
.8510515 
.8646172 

36.8781778 
.8917335 
.9052842 
.9188299 
.9323706 

36.9459064 



10.9139287 

.9167265 

.9195228 

.9223177 

.9251111 
10.9279031 

.9306937 

.9334829 

9362706 

.9390569 
10.9418418 1375 

.9446253 

.9474074 

.9501880 

.9529673 
10.9557451 

9585215 

.9612965 

9640701 

.9668423 
10.9696131 

.9723825 

.9751505 

.9779171 

9806823 

10.983446211390 

91 

92 



1365 
66 
67 
68 
69 

1370 
71 
72 
73 
74 



76 
77 
78 
79 

1380 
81 
82 
83 
84 

1385 
86 
87 



.9889696 

.9917293 

.9944876 
10.9972445 1395 
11.0000000 

.0027541 

.0055069 

.0082583 
11.0110082 1400 

.0137569 

.0165041 

.0192500 

.0219945 
11.0247377 

.0274795 

.0302199 

.0329590 

.0356967 
11.0384330 1410 

.0411680 

.0439017 

.0466339 

.0493649 
11.0520945 

.0548227 

.0575497 

.0602752 

.0629994 
11.0657222 1420 

.0684437 

.0711639 

.0738828 

.0766003 
11.0793165 

.0820314 

.0847449 

.0874571 

.0901679 
11.0928775 1430 



94 



1405 



11 
12 
13 
14 
1415 
16 
17 
18 
19 



21 
22 
23 
24 
1425 
26 
27 
28 
29 



86 32 25 
86 59 56 

86 86 89 

87 14 24 
87 41 61 
87 69 00 

87 96 41 

88 23 84 
88 51 29 

88 78 76 

89 06 25 
89 33 76 
89 61 29 

89 88 84 

90 16 41 
90 44 00 
90 7161 

90 99 24 

91 26 89 
91 54 56 

91 82 25 

92 09 96 
92 37 69 
92 65 44 

92 93 21 

93 21 00 
93 48 81 

93 76 64 

94 04 49 
94 32 36 
94 60 25 

94 88 16 

95 16 09 
95 44 04 

95 72 01 

96 00 00 
96 28 01 
96 56 04 

96 84 09 

97 12 16 
97 40 25 
97 68 36 

97 96 49 

98 24 64 
98 52 81 

98 81 00 

99 09 21 
99 37 44 
99 65 69 
99 93 96 
00 22 25 
00 50 56 

00 78 89 

01 07 24 
01 35 61 
01 64 00 

01 92 41 

02 20 84 
02 49 29 

02 77 76 

03 06 25 
03 34 76 
03 63 29 

03 91 84 

04 20 41 
04 49 00 



2 543 302 125 
548 895 896 
2 554 497 863 
560 108 032 
2 565 726 409 
571 353 000 
576 987 811 
582 630 848 
2 588 282 117 
593 941 624 
599 609 375 
605 285 376 
2 610 969 633 
2 616 662 152 
2 622 362 939 
2 628 072 000 
2 633 789 341 
639 514 968 
645 248 887 
2 650 991 104 
2 656 741 625 
2 662 500 456 
2 668 267 603 
2 674 043 072 
2 679 826 869 
2 685 619 000 
2 691 419 471 
2 697 228 288 
2 703 045 457 
2 708 870 984 
2 714 704 875 
2 720 547 136 
2 726 397 773 
2 732 256 792 
2 738 124 199 
2 744 000 000 
2 749 884 201 
2 755 776 808 
2 761 677 827 
2 767 587 264 
2 773 505 125 
2 779 431 416 
2 785 366 143 
2 791 309 312 
2 797 260 929 
2 803 221 000 
2 809 189 531 
2 815 166 528 
2 821 151 997 
2 827 145 944 
2 833 148 375 
2 839 159 296 
2 845 178 713 
2 851 206 632 
2 857 243 059 
2 863 288 000 
2 869 341 461 
2 875 403 448 
2 881 473 967 
2 887 553 024 
2 893 640 625 
2 899 736 776 
2 905 841 483 
2 911 954 752 
2 918 076 589 
2 924 207 000 
1 



36.9459064 
.9594372 
.9729631 
.9864840 

37.0000000 

37.0135110 
.0270172 
.0405184 
.0540146 
.0675060 

37.0809924 
.0944740 
.1079506 
.1214224 
.1348893 

37.1483512 
.1618084 
.1752606 
.1887079 
.2021505 

37.2155881 
.2290209 
.2424489 
.2558720 
.2692903 

37.2827037 
.2961124 
.3095162 
.3229152 
.3363094 

37.3496988 
.3630834 
.3764632 



11. 



11. 



11. 



.4032084 
37.4165738 
.4299345 
.4432904 
.4566416 



37.4833296 
.4966665 
.5099987 
.5233261 
.5366487 

37.5499667 
.5632799 
.5765885 



11. 



.6031913 

37.6164857 
.6297754 
.6430604 
.6563407 
.6696164 

37.6828874 
.6961536 
.7094153 
.7226722 
.7359245 

37.7491722 
.7624152 
.7756535 
.7888873 
.8021163 

37.8153408 

I 



0928775 
0955857 
0982926 
1009982 
1037025 
1064054 
1091070 
1118073 
1145064 
1172041 
1199004 
1225955 
1252893 
1279817 
1306729 
1333628 
.1360514 
.1387386 
.1414246 
.1441093 
.1467926 
.1494747 
.1521555 
.1548350 
.1575133 
.1601903 
.1628659 
.1655403 
.1682134 
.1708852 
.1735558 
.1762250 
.1788930 
.1815598 
.1842252 
.1868894 
.1895523 
.1922139 
.1948743 
.1975334 
2001913 
.2028479 
.2055032 
.2081573 
.2108101 
.2134617 
.2161120 
.2187611 
.2214089 
.2240554 
.2267007 
.2293448 
.2319876 
.2346292 
.2372696 
.2399087 
.2425465 
.2451831 
.2478185 
.2504527 
.2530856 
.2557173 
.2583478 
.2609770 
.2636050 
.2662318 



«2 



2.-'P0WERS, ROOTS AND RECIPROCALS, 



9. — Squares, Cubes, Square Roots, Cube Roots, op Numbers 
1 to 1600 — Continued. 



No. 



Square 



Cube. 



Sq. Rt. 



Cu. Rt. 



No, 



Square 



Cube. 



Sq. Rt. 



Cu. Rt. 



1430 
31 
32 
33 
34 

1435 
36 
37 
38 
39 

1440 
41 
42 
43 
44 

1445 
46 
47 
48 
49 

1450 
61 
52 
53 
54 

1455 
56 
57 
58 
59 

1460 
61 
62 
63 
64 

1465 
66 
67 
68 
69 

1470 
71 
72 
73 
74 

1475 
76 
77 
78 
79 

1480 
81 
82 
83 
84 

1485 
86 
87 



1490 
91 
92 
93 
94 

1495 



2 04 49 00 
2 04 77 61 
2 05 06 24 
2 05 34 89 
2 05 63 56 
2 05 92 25 
2 06 20 96 
06 49 69 
2 06 78 44 
2 07 07 21 
2 07 36 00 
2 07 64 81 
2 07 93 64 
2 08 22 49 
2 08 51 36 
2 08 80 25 
2 09 09 16 
2 09 38 09 
2 09 67 04 
2 09 96 01 
2 10 25 00 
2 10 54 01 
2 10 83 04 
2 11 12 
2 11 41 16 
2 11 70 25 
2 11 99 36 
2 12 28 49 
2 12 57 64 
2 12 86 81 
2 13 16 00 
2 13 45 21 
2 13 74 44 
2 14 03 69 
2 14 32 96 
2 14 62 25 

14 91 56 

15 20 89 
15 50 24 

15 79 61 

16 09 00 
16 38 41 

2 16 67 84 
2 16 97 29 
2 17 26 76 
2 17 56 25 
2 17 85 76 
2 18 15 29 
2 18 44 84 
2 18 74 41 
2 19 04 00 
2 19 33 61 
2 19 63 24 
2 19 92 89 
2 20 22 56 
2 20 52 25 
2 20 8196 
2 21 11 
2 21 41 44 
2 21 71 21 
2 22 01 00 
2 22 30 81 
2 22 60 64 
2 22 90 49 
2 23 20 36 
2 23 50 25 



2 924 207 000 
2 930 345 991 
2 936 493 568 
2 942 649 737 
2 948 814 504 
2 954 987 875 
2 961 169 856 
2 967 360 453 
2 973 559 672 
2 979 767 519 
2 985 984 000 
2 992 209 121 

2 998 442 888 

3 004 685 307 
3 010 936 384 
3 017 196 125 
3 023 464 536 

029 741 623 

036 027 392 

3 042 321 849 

3 048 625 000 

3 054 936 851 

3 061 257 408 

3 067 586 677 

3 073 924 664 

3 080 271375 

3 086 626 816 

3 092 990 993 

3 099 363 912 

105 745 579 

3 112 136 000 

3 118 535 181 

3 124 943 128 

3 131 359 847 

137 785 344 

144 219 625 

150 662 696 

157 114 563 

163 575 232 

170 044 709 

176 523 000 

183 010 111 

189 506 048 

196 010 817 

1 3 202 524 424 



37.8153408 
.8285606 
.8417759 
.8549864 
.8681924 

37.8813938 
.8945906 
.9077828 
.9209704 
.9341535 

37.9473319 
.9605058 
.9736751 



11.26623181495 2 23 50 25 



S 215 578 176 
3 222 118 333 
3 228 667 352 
3 235 225 239 
3 241 792 000 
3 248 367 641 
3 254 952 168 
3 261 545 587 
3 268 147 904 
3 274 759 125 
3 281 379 256 
3 288 008 303 
3 294 646 272 
3 301 293 169 
3 307 949 000 
3 314.613 771 
3 321 287 488 
3 327 970 157 
3 334 661 784 
3 341 362 375 



3 

4 

1505 



38.0000000 

38.0131556 
.0263067 
.0394532 
.0525952 
.0657326 

38.0788655 
,0919939 
.1051178 
.1182371 
.1313519 

38.1444622 
.1575681 
.1706693 
.1837662 
.1968585 

38.2099463 
.2230297 
.2361085 
.2491829 
.2622529 

38.2753184 
.2883794 
.3014360 
.3144881 
.3275358 

38.3405790 
.3536178 
.3666522 
.3796821 
.3927076 
3 209 046 87538.4057287 



.2688573 

.2714816 

.2741047 

.2767266 
11.27934721500 

.2819666 

.2845849 

.2872019 

.2898177 
11.2924323 

.2950457 

.2976579 

.3002688 

.3028786 
11.3054871 

.3080945 

.3107006 

.3133056 

.3159094 
11.3185119 

.3211132 

.3237134 

.3263124 

.3289102 
11.3315067 

.3341022 

.3366964 

.3392894 

.3418813 
11.3444719 

.3470614 

.3496497 

.3522368 

.3548227 



2 23 80 16 
2 24 10 09 
2 24 40 04 
2 24 70 01 
2 25 00 00 
2 25 30 01 
2 25 60 04 
2 25 90 09 



.4187454 
.4317577 
.4447656 
.4577691 

38.4707681 
.4837627 
.4967530 
.5097390 
.5227206 

38.5356977 
.5486705 
.5616389 
,5746030 
.5875627 

38.6005181 
.6134691 
.6264158 
.6393582 
.6522962 

38.6652299 



9 

1510 
11 
12 
13 
14 

1515 
16 
17 
18 
19 

1520 
21 
22 
23 
24 

1525 
26 
27 
28 
29 



3 341 362 375 
3 348 071 936 
3 354 790 473 
3 361 517 992 
3 368 254 499 
3 375 000 000 
3 381 754 501 
3 388 518 008 
3 395 290 527 



2 26 20 16 3 402 072 064 



2 26 50 25 
2 26 80 36 



3 408 862 625 
3 415 662 216 



2 27 10 493 422 470 843 



11.35740751530 



.3599911 
.3625735 
.3651547 
.3677347 
11.3703136 
.3728914 
.3754679 
.3780433 
.3806175 



11.3831906 1540 



.3857625 
.388333^ 
.3909028 
.3934712 

11.3960384 
.3986045 
.4011695 
.4037332 
.4062959 

llr4088574 
.4114177 
.4139769 
.4165349 
.4190918 

11.4216476 
.4242022 
.4267556 
.4293079 
.4318591 

11.4344092 



2 27 40 64 
2 27 70 81 
2 28 01 00 
2 28 31 21 
2 28 61 44 
2 28 91 69 
2 29 21 96 
2 29 52 25 
2 29 82 56 
2 30 12 89 
2 30 43 24 
2 30 73 61 
2 31 04 00 
2 31 34 41 
2 31 64 84 
2 31 95 29 
2 32 25 76 
2 32 56 25 
2 32 86 76 
2 33 17 29 
2 33 47 84 
2 33 78 41 



31 
32 
33 
34 
1535 
36 
37 
38 
39 



41 
42 
43 
44 

1545 
46 
47 
48 
49 

1550 
51 
52 
53 
54 



2 34 09 006 581 577 000 



2 34 39 61 
2 34 70 24 
2 35 00 89 
2 35 31 56 
2 35 62 25 
2 35 92 96 
2 36 23 69 
2 36 54 44 
2 36 85 21 
2 37 16 00 
2 37 46 81 
2 37 77 64 
2 38 08 49 
2 38 39 36 
2 38 70 25 
2 39 01 16 
2 39 32 09 
2 39 63 04 
2 39 94 01 
2 40 25 00 
2 40 56 01 
2 40 87 04 
2 41 18 09 
2 41 49 16 



1555 2 41 80 25 



56 
57 
58 
59 
1560 



2 42 11 36 
2 42 42 49 
2 42 73 64 
2 43 04 81 
2 43 36 00 



3 429 288 512 

3 436 115 229 

3 442 951 000 

3 449 795 831 

3 456 649 728 

463 512 697 

470 384 744 

3 477 265 875 

3 484 156 096 

3 491 055 413 

3 497 963 832 

3 504 881 359 

3 511 808 000 

3 518 743 761 

525 688 648 

3 532 642 667 

3 539 605 824 

546 578 125 

553 559 576 

3 560 550 183 

3 567 549 952 

3 574 558 889 



3 588 604 291 
3 595 640 768 
3 602 686 437 
3 609 741 304 
3 616 805 375 
3 623 878 656 
3 630 961 153 
3 638 052 872 
3 645 153 819 
3 652 264 000 
3 659 383 421 
3 666 512 088 
3 673 650 007 
3 680 797 184 
3 687 953 625 
3 695 119 336 
3 702 294 323 
3 709 478 592 
3 716 672 149 
3 723 875 000 
3 731 087 151 
3 738 308 608 
3 745 539 377 
3 752 779 464 
3 760 028 875 
3 767 287 616 
3 774 555 693 
3 781 833 112 
3 789 119 879 
3 796 416 000 



38.6652299 
.6781593 
.6910843 
.7040050 
.7169214 

38.7298335 
.7427412 
.7556447 
.7685439 
.7814389 

38.7943294 
.8072158 
.8200978 
.8329757 
.8458491 

38.8587184 
.8715834 
.8844442 
.8973006 
.9101529 

38.9230009 
.9358447 
.9486841 
.9615194 
.9743505 

38.9871774 

39.0000000 
.0128184 
.0256326 
.0384426 

39.0512483 
.0640499 
.0768473 
.0896406 
.1024296 

39.1152144 
.1279951 
.1407716 
.1535439 
.1663120 

39.1790760 
.1918359 
.2045915 
.2173431 
.2300905 

39.2428337 
.2555728 
.2683078 
.2810387 
.2937654 

39.3064880 
.3192065 
.3319208 
.344631 
.3573373 

39.3700394 
.3827373 
.3954312 
.4081210 
.4208067 

39.4334883 
.4461658 
.4588393 
.4715087 
.4841740 

39.4968353 



11.4344092 
.4369581 
.4395059 
.4420525 
.4445980 

11.4471424 
.4496857 
.4522278 
.4547688 
.4573087 

11.4598474 
.4623850 
.4649215 
.4674568 
.4699911 

11.4725242 
.4750562 
.4775871 
.4801169 
.4826455 

11.4851731 
.4876995 
.4902249 
.4927491 
.4952722 

11.4977942 
.5003151 
.5028348 
.5053535 
.5078711 

11.5103876 
.5129030 
.515417^ 
.5179305 
.5204425 

11.5229535 
.5254634 
.5279722 
.5304799 
.5329865 

11.5354920 
.5379965 
.5404998 
.5430021 
.5455033 

11.5480034 
.5505025 
.5530004 
.5554973 
.5579931 

11.5604878 
.5629815 
.5654740 
.5679655 
.5704559 

11.5729453 
.5754336 
.5779208 
.5804069 
.5828919 

11.5853759 
.5878588 
.5903407 
.5928215 
.5953013 

11.5977799 



COMMON TABLES— SQUARES, CUBES, ROOTS. 



43 



9. — Squares, Cubes, Square Roots, Cube Roots, of Numbers 
1 to 1600— Concluded. 



No. 



Square 



Cube. 



Sq. Rt 



Cu. Rt 



No. 



Square 



Cube. 



Sq. Rt. 



Cu. Rt. 



1560 
61 
62 
63 
64 

1565 
66 
67 
68 
69 

1570 
71 

, 72 
73 
74 

1575 
76 
77 
78 
79 

1580 



43 36 00 
43 67 21 

43 98 44 

44 29 69 
44 60 96 

44 92 25 

45 23 56 
45 54 89 

45 86 24 

46 17 61 
46 49 00 

46 80 41 

47 11 84 
47 43 29 

47 74 76 

48 06 25 
48 37 76 

48 69 29 

49 00 84 
49 32 41 
49 64 00 



796 416 
803 721 
811 036 
818 360 
825 694 
833 037 
840 389 
847 751 
855 122 
862 503 
869 893 
877 292 
884 701 
892 119 
899 547 
906 984 
914 430 
921 887 
929 352 
936 827 
944 312 



39. 



39, 



39, 



39, 



4968353 
5094925 
5221457 
5347948 
5474399 
5600809 
5727179 
5853508 
5979797 
6106046 
6232255 
6358424 
6484552 
6610640 
6736688 
6862696 
6988665 
7114593 
7240481 
7366329 
7492138 



11, 



11.' 



11. 



5977799 

6002576 

6027342 

6052097 

6076841 

6101 

6126299 

6151012 

6175715 

6200407 

6225088 

6249759 

6274420 

6299070 

6323710 

6348339 

6372957 

6397566 

6422164 

6446751 

6471329 



1580 
81 
82 
83 
84 
57511585 
86 
87 
88 
89 

1590 
91 
92 
93 
94 

1595 
96 
97 



1600 



49 64 

49 95 

50 27 
50 58 

50 90 
5122 

51 53 
5185 

52 17 
52 49 

52 81 

53 12 
53 44 

53 76 

54 08 
54 40 

54 72 

55 04 
55 36 

55 68 

56 00 



3 944 
3 951 
3 959 
3 966 
3 974 

253 981 

96 

3 996 

4 004 
4 012 
4 019 
4 027 
4 034 
4 042 
4 050 
4 057 
4 065 
4 073 
4 080 
4 
4 096 



56 



312 000 
805 941 
309 368 
822 287 
344 704 
876 625 
418 056 
969 003 
529 472 
099 469 
679 000 
268 071 
866 688 
474 857 
092 584 
719 87539 
356 736 
003 173 
659 192 
324 799 
000 000 40 



39 



39 



7492138 
.7617907 
.7743636 
.7869325 
.7994975 
.8120585 
.8246155 
.8371686 
.8497177 
.8622628 
.8748040 
.8873413 
.8998747 
.9124041 
.9249295 
.9374511 
.9499687 
.9624824 
.9749922 
.9874980 
.0000000 



1.6471329 
.6495895 
.652Q452 
.6544998 
.6569534 
L .6594059 
.6618574 
.6643079 
.6667574 
.6692058 
1.6716532 
.6740996 
.6765449 
.6789892 
.6814325 
[.6838748 
.6863161 
.6887563 
.6911955 
.6936337 
[.6960709 



9a.— Squares of Numbers 1600 to 1810. 



No. 


Square. 


No. 

1635 


Square. 


No. 


Square. 


No. 


Square. 


No. 


Square. 


No. 
1775 


Square. 


1600 


2560000 


2673225 


1670 


2788900 


1705 


2907025 


1740 


3027600 


3150625 


01 


2563201 


36 


2676496 


71 


2792241 


06 


2910436 


41 


3031081 


76 


3154178 


02 


2566404 


37 


2679769 


72 


2795584 


07 


2913849 


42 


3034564 


77 


3157729 


03 


2569609 


38 


2683044 


73 


2798929 


08 


2917264 


43 


3038049 


78 


3161284 


04 


2572816 


39 


2686321 


74 


2802276 


09 


2920681 


44 


3041536 


79 


3164841 


1605 


2576025 


164^ 


2689600 


1675 


2805625 


1710 


2924100 


1745 


3045025 


1780 


3168400 


06 


2579236 


2692881 


76 


2808976 


11 


2927521 


46 


3048516 


81 


3171961 


07 


2582449 


42 


2696164 


77 


2812329 


12 


2930944 


47 


3052009 


82 


3175524 


08 


2585664 


43 


2699449 


78 


2815684 


13 


2934369 


48 


3055504 


83 


3179089 


09 


2588881 


44 


2702736 


79 


2819041 


14 


2937796 


49 


3059001 


84 


3182656 


1610 


2592100 


1645 


2706025 


1680 


2822400 


1715 


2941225 


1750 


3062500 


1785 


3186225 


11 


2595321 


46 


2709316 


81 


2825761 


16 


2944656 


51 


3066001 


86 


3189796 


12 


2598544 


47 


2712609 


82 


2829124 


17 


2948089 


52 


3069504 


87 


3193369 


13 


2601769 


48 


2715904 


83 


2832489 


18 


2951524 


53 


3073009 


88 


3196944 


14 


2604996 


49 


2719201 


84 


2835856 


19 


2954961 


54 


3076516 


89 


3200521 


1615 


2608285 


1650 


2722500 


1685 


2839225 


1720 


2958400 


1755 


3080025 


1790 


3204100 


16 


2611456 


51 


2725801 


86 


2842596 


21 


2961841 


56 


3083536 


91 


3207681 


17 


2614689 


52 


2729104 


87 


2845969 


22 


2965284 


57 


3087049 


92 


3211264 


18 


2617924 


53 


2732409 


88 


2849344 


23 


2968729 


58 


3090564 


93 


3214849 


19 


2621161 


54 


2735716 


89 


2852721 


24 


2972176 


59 


3094081 


94 


3218436 


1620 


2624400 


1655 


2739025 


1690 


2856100 


1725 


2975625 


1760 


3097600 


1795 


3222025 


21 


2627641 


56 


2742336 


91 


2859481 


26 


2979076 


61 


3101121 


96 


3225616 


22 


2630884 


57 


2745649 


92 


2862864 


27 


2982529 


62 


3104644 


97 


3229209 


23 


2634129 


58 


2748964 


93 


2866249 


28 


2985984 


63 


3108169 


98 


3232804 


24 


2637376 


59 


2752281 


94 


2869636 


29 


2989441 


64 


3111696 


99 


3236401 


1625 


2640625 


1660 


2755600 


1695 


2873025 


1730 


2992900 


1765 


3115225 


1800 


3240000 


26 


2643876 


61 


2758921 


96 


2876416 


31 


2996361 


66 


3118756 


01 


3243601 


27 


2647129 


62 


2762244 


97 


2879809 


32 


2999824 


67 


3122289 


02 


3247204 


28 


2650384 


63 


2766569 


98 


2883204 


33 


3003289 


68 


3125824 


03 


3250809 


29 


2653641 


64 


2768896 


99 


2886601 


34 


3006766 


69 


3129361 


04 


3254416 


1630 


2656900 


1665 


2772225 


1700 


2890000 


1735 


3010225 


1770 


3132900 


1805 


3258025 


31 


2660161 


66 


2775556 


01 


2893401 


36 


3013696 


71 


3136441 


06 


3261636 


32 


2663424 


67 


2778889 


02 


2896804 


37 


3017169 


72 


3139984 


07 


3265249 


33 


2666689 


68 


2782fi24 


03 


2«n0209 


38 


3020644 


73 


3143529 


08 


3268864 


34 


2669956 


69 


2785561 


04 


2903616 


39 


3024121 


74 


3147076 


09 


3272481 


1635 


2673225 


1670 


2788900 


1705 


2907025 


1740 


3027600 


1775 


3150625 


1810 


3276100 



44 



2.— POWERS. ROOTS AND RECIPROCALS. 



10. — Square Roots and Cube Roots op Numbers 
1600 to 3200. 



No. 


Sq. Rt. 


Cu. Rt. 


No. 


Sq. Rt. 


Cu. Rt. 


No. 


Sq. Rt. 


Cu. Rt. 


No. 


3q. Rt. 


Cu. Rt. 


1600 


40.0000 


11.6961 


1665 


40.8044 


11.8524 


1730 


41.5933 


12.0046 


1795 


i2.3674 


12.1531 


1 


.0125 


.6985 


66 


.8167 


.8547 


31 


.6053 


.0069 


96 


.3792 


.1554 


2 


.0250 


.7009 


67 


.8289 


.8571 


32 


.6173 


.0093 


97 


.3910 


.1576 


3 


.0375 


.7034 


68 


.8412 


.8595 


33 


.6293 


.0116 


98 


.4028 


.1599 


4 


.0500 


.7058 


69 


.8534 


.8618 


34 


.6413 


.0139 


99 


.4146 


.1622 


1605 


40.0625 


11.7082 


1670 


40.8656 


11.8642 


1735 


41.6533 


12.0162 


1800 


42.4264 


12.1644 


6 


.0749 


.7107 


71 


.8779 


.8666 


36 


.6653 


.0185 


1 


.4382 


.1667 


7 


.0874 


.7131 


72 


.8901 


.8689 


37 


.6773 


.0208 


2 


.4500 


.1689 


8 


.0999 


.7155 


73 


.9023 


.8713 


38 


.6893 


.0231 


3 


.4617 


.1712 


9 


.1123 


.7180 


74 


.9145 


.8737 


39 


.7013 


.0254 


4 


.4735 


.1734 


1610 


40.1248 


11.7204 


1675 


40.9268 


11.8760 


1740 


41.7133 


12.0277 


1805 


42.4853 


12.1757 


11 


.1373 


.7228 


76 


.9390 


.8784 


41 


.7253 


.0300 


6 


.4971 


.1779 


12 


.1497 


.7252 


77 


.9512 


.8808 


42 


.7373 


.0323 


7 


.5088 


.1802 


13 


.1622 


.7277 


78 


.9634 


.8831 


43 


.7493 


.0346 


8 


.5206 


.1824 


14 


.1746 


.7301 


79 


.9756 


.8855 


44 


.7612 


.0369 


9 


.5323 


.1846 


1615 


40.1871 


11.7325 


1680 


40.9878 


11.8878 


1745 


41.7732 


12.0392 


1810 


42.5441 


12.1869 


16 


.1995 


.7350 


81 


41.0000 


.8902 


46 


.7852 


.0415 


11 


.5558 


.1891 


17 


.2119 


.7373 


82 


.0122 


.8926 


47 


.7971 


.0438 


12 


.5676 


.1914 


18 


.2244 


.7398 


83 


.0244 


.8949 


48 


.8091 


.0461 


13 


.5793 


.1936 


19 


.2368 


.7422 


84 


.0366 


.8973 


49 


.8210 


.0484 


14 


.5911 


.1959 


1620 


40.2492 


11.7446 


1685 


41.0488 


11.8996 


1750 


41.8330 


12.0507 


1815 


42.6028 


12.1981 


21 


.2616 


.7470 


86 


.0609 


.9020 


51 


.8450 


.0530 


16 


.6146 


.2003 


22 


.2741 


.7494 


87 


.0731 


.9043 


52 


.8569 


.0553 


17 


.6263 


.2026 


23 


.2865 


.7518 


88 


.0853 


.9067 


53 


.8688 


.0576 


18 


.6380 


.2048 


24 


.2989 


.7543 


89 


.0974 


.9090 


54 


.8808 


.0599 


19 


.6497 


.2071 


1625 


40.3113 


11.7567 


1690 


41.1096 


11.9114 


1755 


41.8927 


12.0622 


1820 


42.6615 


12.2093 


26 


.3237 


.7591 


91 


.1218 


.9137 


56 


.9047 


.0645 


21 


.6732 


.2115 


27 


.3361 


.7615 


92 


.1339 


.9161 


57 


.9166 


.0668 


22 


.6849 


.2138 


28 


.3485 


.7639 


93 


.1461 


.9184 


58 


.9285 


.0690 


23 


.6966 


.2160 


29 


.3609 


.7663 


94 


.1582 


.9208 


59 


.9404 


.0713 


24 


.7083 


.2182 


1630 


40.3733 


11.7687 


1695 


41.1704 


11.9231 


1760 


41.9524 


12.0736 


1825 


42.7200 


12.2205 


31 


.3856 


.7711 


96 


.1825 


.9255 


61 


.9643 


.0759 


26 


.7317 


.2227 


32 


.3980 


.7735 


97 


.1947 


.9278 


62 


.9762 


.0782 


27 


.7434 


.2249 


33 


.4104 


.7759 


98 


.2068 


.9301 


63 


.9881 


.0805 


28 


.7551 


.2272 


34 


.4228 


.7783 


99 


.2189 


.9325 


64 


42.0000 


.0828 


29 


.7668 


.2294 


1635 


40.4351 


11.7807 


1700 


41.2311 


11.9348 


1765 


42.0119 


12.0850 


1830 


42.7785 


12.2316 


36 


.4475 


.7831 


1 


.2432 


.9372 


66 


.0238 


.0873 


31 


.7902 


.2338 


37 


.4599 


.7855 


2 


.2553 


.9395 


67 


.0357 


.0896 


32 


.8019 


.2361 


38 


.4722 


.7879 


3 


.2674 


9418 


68 


.0476 


.0919 


33 


.8135 


.2383 


39 


.4846 


.7903 


4 


.2795 


.9442 


69 


.0595 


.0942 


34 


.8252 


.2405 


1640 


40.4969 


11.7927 


1705 


41.2916 


11.9465 


1770 


42.0714 


12.0964 


1835 


42.8369 


12.2427 


41 


.5093 


.7951 


6 


.3038 


.9489 


71 


.0833 


.0987 


36 


.8486 


.2450 


42 


.5216 


.7975 


7 


.3159 


.9512 


72 


.0951 


.1010 


37 


.8602 


.2472 


43 


.5339 


.7999 


8 


.3280 


.9535 


73 


.1070 


.1033 


38 


.8719 


.2494 


44 


.5463 


.8023 


9 


.3401 


.9559 


74 


.1189 


.1056 


39 


.8836 


.2516 


1645 


40.5586 


11.8047 


1710 


41.3521 


11.9582 


1775 


42.1307 


12.1078 


1840 


42.8952 


12.2539 


46 


.5709 


.8071 


11 


.3642 


.9605 


76 


.1426 


.1101 


41 


.9069 


.2561 


47 


.5832 


.8095 


12 


.3763 


.9628 


77 


.1545 


.1124 


42 


.9185 


.2583 


48 


.5956 


.8119 


13 


.3884 


.9652 


78 


.1663 


.1146 


43 


.9302 


.2605 


49 


.6079 


.8143 


14 


.4005 


.9675 


79 


.1782 


.1169 


44 


.9418 


.2627 


1650 


40.6202 


11.8167 


1715 


41.4126 


11.9698 


1780 


42.1900 


12.1192 


1845 


42.9535 


12.2649 


51 


.6325 


.8190 


16 


.4246 


.9722 


81 


.2019 


.1215 


46 


.9651 


.2672 


52 


.6448 


.8214 


17 


.4367 


.9745 


82 


.2137 


.1237 


47 


.9767 


.2694 


53 


,6571 


.8238 


18 


.4488 


.9768 


83 


.2256 


.1260 


48 


.9884 


.2716 


54 


.6694 


.8262 


19 


.4608 


.9791 


84 


.2374 


.1283 


49 


43.0000 


.2738 


1655 


40.6817 


11.8286 


1720 


41.4729 


11.9815 


1785 


42.2493 


12.1305 


1850 


43.0116 


12.2760 


56 


.6940 


.8310 


21 


.4849 


.9838 


86 


.2611 


.1328 


51 


.0232 


.2782 


57 


.7063 


.8333 


22 


.4970 


.9861 


87 


.2729 


.1350 


52 


.0349 


.2804 


68 


.7185 


.8357 


23 


.5090 


.9884 


88 


.2847 


.1373 


53 


.0465 


.2826 


59 


.7308 


.8381 


24 


.5211 


.9907 


89 


.2966 


.1396 


54 


.0581 


.2849 


1660 


40.7431 


11.8405 


1725 


41.5331 


11.9931 


1790 


42.3084 


12.1418 


1855 


43.0697 


12.2871 


61 


.7554 


.8429 


26 


.5452 


.9954 


91 


.3202 


.1441 


56 


.0813 


.2893 


62 


.7676 


.8452 


27 


.5572 


.9977 


92 


.3320 


.1464 


57 


.0929 


.2915 


63 


.7799 


.8476 


28 


.5692 


12.0000 


93 


.3438 


.1486 


58 


.1045 


.2937 


64 


.7922 


.8500 


29 


.5812 


.0023 


94 


.3556 


.1509 


59 


.1161 


.2959 


1665 


40.8044 


11.8524 


1730 


41.5933 


12.0046 


1795 


42.3674 


12.1531 


1860 


43.1277 


12.2981 



COMMON TABLES— SQUARE ROOTS. CUBE ROOTS. 



45 



10. — Square Roots and Cube Roots op Numbers 
1600 TO 3200— Continued. 



No. 


Sq. Rt. 


Cu. Rt. 


No. 


Sq. Rt 


Cu. Rt. 


No. 


Sq. Rt. 


Cu. Rt. 


No. 


Sq. Rt. 


Cu Rt. 


1860 


43.1277 


12.2981 


1925 


43.8748 


12.4397 


1990 


44.6094 


12.5782 


2055 


45.3321 


12.7137 


61 


.1393 


.3003 


26 


.8862 


.4419 


91 


.6206 


.5803 


56 


.3431 


.7157 


62 


.1509 


.3025 


27 


.8976 


.4440 


92 


.6318 


.5824 


57 


.3542 


.7178 


63 


.1625 


.3047 


28 


.9090 


.4462 


93 


.6430 


.5845 


58 


.3652 


.7198 


64 


.1741 


.3069 


29 


.9204 


.4483 


94 


.6542 


.5866 


59 


.3762 


.7219 


1865 


43.1856 


12.3091 


1930 


43.9318 


12.4505 


1995 


44.6654 


12.5887 


2060 


45.3872 


12.7240 


66 


.1972 


.3113 


31 


.9431 


.4526 


96 


.6766 


.5908 


61 


.3982 


.7260. 


67 


.2088 


.3135 


32 


.9545 


.4548 


97 


.6878 


.5929 


62 


.4093 


.7281 


68 


.2204 


.3157 


33 


.9659 


.4569 


98 


.6990 


.5950 


63 


.4203 


.7301 


69 


.2319 


.3179 


34 


.9773 


.4591 


99 


.7102 


.5971 


64 


.4313 


.7322 


1870 


43.2435 


12.3201 


1935 


43.9886 


12.4612 


2000 


44.7214 


12.5992 


2065 


45.4423 


12.7342 


71 


.2551 


.3223 


36 


44.0000 


.4634 


1 


.7325 


.6013 


66 


.4533 


.7363 


72 


.2666 


.3245 


37 


.0114 


.4655 


2 


.7437 


.6034 


67 


.4643 


.7384 


73 


.2782 


.3267 


38 


.0227 


.4676 


3 


.7549 


.6055 


68 


.4753 


.7404 


74 


.2897 


.3289 


39 


.0341 


.4698 


4 


.7661 


.6076 


69 


.4863 


.7425 


1875 


43.3013 


12.3311 


1940 


44.0454 


12.4719 


2005 


44.7772 


12.6097 


2070 


45.4973 


12.7445 


76 


.3128 


.3333 


41 


.0568 


.4741 


6 


.7884 


.6118 


71 


.5082 


.7466 


77 


.3244 


.3354 


42 


.0681 


.4762 


7 


.7996 


.6139 


72 


.5192 


.7486 


78 


.3359 


.3376 


43 


.0795 


.4784 


8 


.8107 


.6160 


73 


.5302 


.7507 


79 


.3474 


.3398 


44 


.0908 


.4805 


9 


.8219 


.6181 


74 


.5412 


.7527 


1880 


43.3590 


12:3420 


1945 


44.1022 


12.4826 


2010 


44.8330 


12.6202 


2075 


45.5522 


12.7548 


81 


.3705 


.3442 


46 


.1135 


.4848 


11 


.8442 


.6223 


76 


.5631 


.7568 


82 


.3820 


.3464 


47 


.1248 


.4869 


12 


.8553 


.6244 


77 


.5741 


.7589 


83 


.3935 


.3486 


48 


.1362 


.4891 


13 


.8665 


.6264 


78 


.5851 


.7609 


84 


.4051 


.3508 


49 


.1475 


.4912 


14 


.8776 


.6285 


79 


.6961 


.7630 


1885 


43.4166 


12.3529 


1950 


44.1588 


12.4933 


2015 


44.8888 


12.6306 


2080 


45.6070 


12.7650 


86 


.4281 


.3551 


51 


.1701 


.4955 


16 


.8999 


.6327 


81 


.6180 


.7671 


87 


.4396 


.3573 


52 


.1814 


.4976 


17 


.9110 


.6348 


82 


.6289 


.7691 


88 


.4511 


.3595 


53 


.1928 


.4997 


18 


.9222 


.6369 


83 


.6399 


.7711 


89 


.4626 


.3617 


54 


.2041 


.5019 


19 


.9333 


.6390 


84 


.6508 


.7732 


1890 


43.4741 


12.3639 


1955 


44.2154 


12.5040 


2020 


44.9444 


12.6411 


2085 


45.6618 


12.7752 


91 


.4856 


.3660 


56 


.2267 


.5061 


21 


.9555 


.6432 


86 


.6727 


.7773 


92 


.4971 


.3682 


67 


.2380 


.5083 


22 


.9667 


.6452 


87 


.6837 


.7793 


93 


.5086 


.3704 


58 


.2493 


.5104 


23 


.9778 


.6473 


88 


.6946 


.7814 


94 


.5201 


.3726 


59 


.2606 


.5125 


24 


.9889 


.6494 


89 


.7056 


.7834 


1895 


43.5316 


12.3747 


1960 


44.2719 


12.5146 


2025 


45.0000 


12.6515 


2090 


45.7165 


12.7854 


96 


.5431 


.3769 


61 


.2832 


.5168 


26 


.0111 


.6536 


91 


.7275 


.7875 


97 


.5546 


.3791 


62 


.2945 


.5189 


27 


.0222 


.6557 


92 


.7384 


.7895 


98 


.5660 


.3813 


63 


.3058 


.5210 


28 


.0333 


.6577 


93 


.7493 


.7915 


99 


.5775 


.3834 


64 


.3170 


.5232 


29 


.0444 


.6598 


94 


.7602 


.7936 


1900 


43.5890 


12.3856 


1965 


44.3283 


12.5253 


2030 


45.0555 


12.6619 


2095 


45.7712 


12.7956 


1 


.6005 


.3878 


66 


.3396 


.5274 


31 


.0666 


.6640 


96 


.7821 


.7977 


2 


.6119 


.3900 


67 


.3509 


.5295 


32 


.0777 


.6661 


97 


.7930 


.7997 


3 


.6234 


.3921 


68 


.3621 


.5317 


33 


.0888 


.6681 


98 


.8039 


.8017 


4 


.6348 


.3943 


69 


.3734 


.5338 


34 


.0999 


.6702 


99 


.8148 


.8038 


1905 


43.6463 


12.3965 


1970 


44.3847 


12.5359 


2035 


45.1110 


12.6723 


2100 


45.8258 


12.8058 


6 


.6578 


.3986 


71 


.3959 


.5380 


36 


.1221 


.6744 


1 


.8367 


.8078 


7 


.6692 


.4008 


72 


.4072 


.5401 


37 


.1331 


.6764 


2 


.8476 


.8099 


8 


.6807 


.4030 


73 


.4185 


.5423 


38 


.1442 


.6785 


3 


.8585 


.8119 


9 


.6921 


.4051 


74 


.4297 


.5444 


39 


.1553 


.6806 


4 


.8694 


.8139 


1910 


43.7035 


12.4073 


1975 


44.4410 


12.5465 


2040 


45.1664 


12.6827 


2105 


45.8803 


12.8159 


11 


.7150 


,4095 


76 


.4522 


.5486 


41 


.1774 


.6847 


6 


.8912 


.8180 


12 


.7264 


.4116 


77 


.4635 


.5507 


42 


.1885 


.6868 


7 


.9021 


.8200 


13 


.7379 


.4138 


78 


.4747 


.5528 


43 


.1996 


.6889 


8 


.9130 


.8220 


14 


.7493 


.4160 


79 


.4860 


.5550 


44 


.2106 


.6909 


9 


.9238 


.8241 


1916 


43.7607 


12.4181 


1980 


44.4972 


12.5571 


2045 


45.2217 


12.6930 


2110 


45.9347 


12.8261 


16 


.7721 


.4203 


81 


.5084 


.5592 


46 


.2327 


.6951 


11 


.9456 


.8281 


17 


.7836 


.4225 


82 


.5197 


.5613 


47 


.2438 


.6971 


12 


.9565 


.8301 


18 


.7950 


.4246 


83 


.5309 


.5634 


48 


.2548 


.6992 


13 


.9674 


.8322 


19 


.8064 


.4268 


84 


.5421 


.5655 


49 


.2659 


.7013 


14 


.9783 


.8342 


1920 


43.8178 


12.4289 


1985 


44.5533 


12.5676 


2050 


45.2769 


12.7033 


2115 


45.9891 


12.8362 


21 


.8292 


.4311 


86 


.5646 


.5697 


51 


.2880 


.7054 


16 


46.0000 


.8382 


22 


.8406 


.4332 


87 


.5758 


.5719 


52 


.2990 


.7075 


17 


.0109 


.8403 


23 


.8520 


.4354 


88 


.5870 


.5740 


53 


.3100 


.7095 


18 


.0217 


.8423 


24 


.8634 


.4376 


89 


.5982 


.5761 


54 


.3211 


.7116 


19 


.0326 


.8443 


1925 


43.8748 


12.4397 


1990 


44.6094 


12.5782 


2055 


45.3321 


12.7137 


2120 


46.0435 


12.8463 



48 



2.— POWERS. ROOTS AND RECIPROCALS. 



10. — Squarb Roots and Cube Roots of Numbers 
1600 TO 3200 — Continued. 



No. 


Sq. Rt. 


Cu. Rt. 


No. 


Sq. Rt. 


Cu. Rt. 


No. 


Sq. Rt. 


Cu. Rt. 


No 


Sq. Rt. 


Cu. Rt. 


2120 


46.0435 


12.8463 


2185 


46.7440 


12.9763 


2250 


47 . 4342 


13.1037 


2315 


48.1144 


13.2287 


21 


.0543 


.8483 


86 


.7547 


.9783 


51 


.4447 


.1056 


16 


.1248 


.2306 


22 


.0652 


.8504 


87 


.7654 


.9802 


52 


.4552 


.1076 


17 


.1352 


.2325 


23 


.0760 


.8524 


88 


.7761 


.9822 


53 


.4658 


.1095 


18 


.1456 


.2344 


24 


.0869 


.8544 


89 


.7868 


.9842 


54 


.4763 


.1115 


19 


.1560 


.2363 


2125 


46.0977 


12.8564 


2190 


46.7974 


12.9862 


2255 


47.4868 


13.1134 


2320 


48.1664 


13.2382 


26 


.1086 


.8584 


91 


.8081 


.9882 


56 


.4974 


.1153 


21 


.1768 


.2401 


27 


.1194 


.8604 


92 


.8188 


.9901 


57 


.5079 


.1173 


22 


.1871 


.2420 


28 


.1303 


.8625 


93 


.8295 


.9921 


58 


.5184 


.1192 


23 


.1975 


.2439 


29 


.1411 


.8645 


94 


.8402 


.9941 


59 


.5289 


.1212 


24 


.2079 


.2458 


2130 


46.1519 


12.8665 


2195 


46.8508 


12.9961 


2260 


47.5395 


13.1231 


2325 


48.2183 


13.2477 


31 


.1628 


.8685 


96 


.8615 


.9980 


61 


.5500 


.1250 


26 


.2286 


.2496 


32 


.1736 


.8705 


97 


.8722 


13.0000 


62 


.5605 


.1270 


27 


.2390 


.2515 


33 


.1844 


.8725 


98 


.8828 


.0020 


63 


.5710 


.1288 


28 


.2494 


.2534 


34 


1952 


.8745 


99 


.8935 


.0039 


64 


.5815 


.1308 


29 


.2597 


.2553 


2135 


26.2061 


12.8765 


2200 


46.9042 


13.0059 


2265 


47.5920 


13.1328 


2330 


48.2701 


13.2572 


36 


.2169 


.8786 


1 


.9148 


.0079 


66 


.6025 


.1347 


31 


.2804 


.2591 


37 


.2277 


.8806 


2 


.9255 


.0099 


67 


.6130 


.1366 


32 


.2908 


.2610 


38 


.2385 


.8826 


3 


.9361 


.0118 


68 


.6235 


.1386 


33 


.3011 


.2629 


39 


.2493 


.8846 


4 


.9468 


.0138 


69 


.6340 


.1405 


34 


.3115 


.2648 


2140 


46.2601 


12.8866 


2205 


46.9574 


13.0158 


2270 


47.6445 


13.1424 


2335 


48.3218 


13.2667 


41 


.2709 


.8886 


6 


.9681 


.0177 


71 


.6550 


.1443 


36 


.3322 


.2686 


42 


.2817 


.8906 


7 


.9787 


.0197 


72 


.6655 


.1463 


37 


.3425 


.2705 


43 


.2925 


.8926 


8 


.9894 


.0217 


73 


.6760 


.1482 


38 


.3529 


.2724 


44 


.3033 


.8946 


9 


47.0000 


.0236 


74 


.6865 


.1501 


39 


.3632 


.2743 


2145 


46.3141 


12.8966 


2210 


47.0106 


13.0256 


2275 


47.6970 


13.1521 


2340 


48.3735 


13.2761 


46 


.3249 


.8986 


11 


.0213 


.0276 


76 


.7074 


.1540 


41 


.3839 


.2780 


47 


.3357 


.9006 


12 


.0319 


.0295 


77 


.7179 


.1559 


42 


.3942 


.2799 


48 


.3465 


.9026 


13 


.0425 


.0315 


78 


.7284 


.1578 


43 


.4045 


.2818 


49 


.3573 


.9046 


14 


.0532 


.0334 


79 


.7389 


.1598 


44 


.4149 


.2837 


2150 


46.3681 


12.9066 


2215 


47.06a8 


13.0354 


2280 


4.7.7493 


13.1617 


2345 


48.4252 


13.2856 


51 


,3789 


.9086 


16 


.0744 


.0374 


81 


.7598 


.1636 


46 


.4355 


.2875 


52 


.3897 


.9106 


17 


.0850 


.0393 


82 


.7703 


.1655 


47 


.4458 


.2894 


63 


.4004 


.9126 


18 


.0956 


.0413 


83 


.7807 


.1675 


48 


.4562 


.2913 


54 


.4112 


.9146 


19 


.1063 


.0432 


84 


.7912 


.1694 


49 


.4665 


.2931 


2155 


46.4220 


12.9166 


2220 


47.1169 


13.0452 


2285 


47.8017 


13.1713 


2350 


48.4768 


13.2950 


56 


.4327 


.9186 


21 


.1275 


.0472 


86 


.8121 


.1732 


51 


.4871 


.2969 


57 


.4435 


.9206 


22 


.1381 


.0491 


87 


.8226 


.1751 


52 


.4974 


.2988 


68 


.4543 


.9226 


23 


.1487 


.0511 


88 


.8330 


.1771 


53 


.5077 


.3007 


59 


.4650 


.9246 


24 


.1593 


.0530 


89 


.8435 


.1790 


64 


.5180 


.3026 


2160 


46.4758 


12.9266 


2225 


47.1699 


13.0550 


2290 


47.8539 


13.1809 


2355 


48.5283 


13.3045 


61 


.4866 


.9286 


26 


.1805 


.0569 


91 


.8644 


.1828 


56 


.5386 


.3063 


62 


.4973 


.9306 


27 


.1911 


.0589 


92 


.8748 


.1847 


57 


.5489 


.3082 


63 


.5081 


.9326 


28 


.2017 


.0609 


93 


.8853 


.1867 


58 


.5592 


.3101 


64 


.5188 


.9346 


29 


.2123 


.0628 


94 


.8957 


.1886 


59 


.5695 


.3120 


2165 


46.5296 


12.9366 


2230 


47.2229 


13.0648 


2295 


47.9062 


13.1905 


2360 


48.5798 


13.3139 


66 


.5403 


.9386 


31 


.2335 


.0667 


96 


.9166 


.1924 


61 


.5901 


.3157 


67 


.5510 


.9406 


32 


.2440 


.0687 


97 


.9270 


.1943 


62 


.6004 


.3176 


68 


.5618 


.9425 


33 


.2546 


.0706 


98 


.9375 


.1962 


63 


.6107 


.3196 


69 


.5725 


.9445 


34 


.2652 


.0726 


99 


.9479 


.1981 


64 


.6210 


.3214 


2170 


46.5833 


12.9465 


2235 


47.2758 


13.0745 


2300 


47.9583 


13.2001 


2365 


48.6313 


13.3233 


71 


.5940 


.9485 


36 


.2864 


.0765 


1 


.9687 


.2020 


66 


.6415 


.3251 


72 


.6047 


.9505 


37 


.2969 


.0784 


2 


.9792 


.2039 


67 


.6518 


.3270 


73 


.6154 


.9525 


38 


.3075 


.0804 


3 


.9896 


.2058 


68 


.6621 


.3289 


74 


.6262 


.9545 


39 


.3181 


.0823 


4 


48.0000 


.2077 


69 


.6724 


.3308 


2175 


46.6369 


12.9565 


2240 


47.3286 


13.0843 


2305 


48.0104 


13.2096 


2370 


48.6826 


13.3326 


76 


.6476 


.9584 


41 


.3392 


.0862 


6 


.0208 


.2115 


71 


.6929 


.3346 


77 


.6583 


.9604 


42 


.3498 


.0882 


7 


.0312 


.2134 


72 


.7032 


.3364 


78 


.6690 


.9624 


43 


.3603 


.0901 


8 


.0416 


.2153 


73 


.7134 


.3383 


79 


.6798 


.9644 


44 


.3709 


.0920 


9 


.0521 


.2173 


74 


.7237 


.3401 


2180 


46.6905 


12.9664 


2245 


47.3814 


13.0940 


2310 


48.0625 


13.2192 


2375 


48.7340 


13.3420 


81 


.7012 


.9684 


46 


.3920 


.0959 


11 


.0729 


.2211 


76 


.7442 


.3439 


82 


.7119 


.9703 


47 


.4025 


.0979 


12 


.0833 


.2230 


77 


.7545 


.3458 


83 


.7226 


.9723 


48 


.4131 


.0998 


13 


.0937 


.2249 


78 


.7647 


.3476 


84 


.7333 


.9743 


49 


.4236 


.1018 


14 


.1041 


.2268 


79 


.7750 


.3495 


2185 


46.7440 


12.9763 


2250 


47.4342 


13.1037 


2315 


48.1144 


13.2287 


2380 


48.7852 


13.3614 



COMMON TABLES— SQUARE ROOTS, CUBE ROOTS, 



47 



10. — Square Roots and Cube Roots op Numbers 
1600 TO 3200 — Continued. 



No. 


Sq. Rt. 


Cu. Rt. 


NO. 


Sq. Rt. 


Cu. Rt. 


No. 


Sq. Rt. 


Cu. Rt. 


No. 


Sq. Rt. 


Cu. Rt 


2380 


48.7852 


13.3514 


2445 


49.4469 


13.4718 


2510 


50.0999 


13.5902 


2575 


50.7445 


13.7065 


81 


.7955 


.3532 


46 


.4571 


.4737 


11 


.1099 


.5920 


76 


.7543 


.7082 


82 


.8057 


.3551 


47 


.4672 


.4755 


12 


.1199 


.5938 


77 


.7642 


.7100 


83 


.8160 


.3570 


48 


.4773 


.4773 


13 


.1298 


.5956 


78 


.7740 


.7118 


84 


.8262 


.3588 


49 


.4874 


.4792 


14 


.1398 


.5974 


79 


.7839 


.7136 


2385 


48.8365 


13.3607 


2450 


49.4975 


13.4810 


2515 


50.1498 


13.5992 


2580 


50.7937 


13.7153 


86 


.8467 


.3626 


51 


.5076 


.4828 


16 


.1597 


.6010 


81 


.8035 


.7171 


87 


.8569 


.3644 


52 


.5177 


.4847 


17 


.1697 


.6028 


82 


.8134 


.7189 


88 


.8672 


.3663 


53 


.5278 


.4865 


18 


.1797 


.6046 


83 


.8232 


.7207 


89 


.8774 


.3682 


54 


.5379 


.4883 


19 


.1896 


.6064 


84 


.8331 


,7224 


2390 


48.8876 


13.3700 


2455 


49.5480 


13.4902 


3520 


50.1996 


13.6082 


2585 


50.8429 


13.7242 


91 


.8979 


.3719 


56 


.5580 


.4920 


21 


.2096 


.6100 


86 


.8527 


.7260 


92 


.9081 


.3738 


57 


.5681 


.4938 


22 


.2195 


.6118 


87 


.8626 


.7277 


93 


.9183 


.3756 


58 


.5782 


.4957 


23 


.2295 


.6136 


88 


.8724 


.7295 


94 


.9285 


.3775 


59 


.5883 


.4975 


24 


.2394 


.6154 


89 


.8822 


.7313 


2395 


48.9387 


13.3794 


2460 


49.5984 


13.4993 


2525 


50.2494 


13.6172 


2590 


50.8920 


13.7330 


96 


.9490 


.3812 


61 


.6085 


.5011 


26 


.2593 


.6190 


91 


.9019 


.7348 


97 


.9592 


.3831 


62 


.6185 


.5030 


27 


.2693 


.6208 


92 


.9117 


.7366 


98 


.9694 


.3849 


63 


.6286 


.5048 


28 


.2792 


.6226 


93 


.9215 


.7383 


99 


.9796 


.3868 


64 


.6387 


.5066 


29 


.2892 


.6244 


94 


.9313 


.7401 


2400 


48.9898 


13.3887 


2465 


49.6488 


13.5085 


2530 


50.2991 


13.6262 


2595 


50.9411 


13.7419 


1 


49.0000 


.3905 


66 


.6588 


.5103 


31 


.3090 


.6280 


96 


.9510 


.7436 


2 


.0102 


.3924 


67 


.6689 


.5121 


32 


.3190 


.6298 


97 


.9608 


.7454 


3 


.0204 


.3942 


68 


.6790 


.5139 


33 


.3289 


.6315 


98 


.9706 


.7472 


4 


.0306 


.3961 


69 


.6890 


.5158 


34 


.3389 


.6333 


99 


.9804 


.7489 


2405 


49.0408 


13.3980 


2470 


49.6991 


13.5176 


2535 


50.3488 


13.6351 


2600 


50.9902 


13.7507 


6 


.0510 


.3998 


71 


.7092 


.5194 


36 


.3587 


.6369 


1 


51.0000 


.7525 


7 


.0612 


.4017 


72 


.7192 


.5212 


37 


.3686 


.6387 


2 


.0098 


.7542 


8 


.0714 


.4035 


73 


.7293 


.5231 


38 


.3786 


.6405 


3 


.0196 


.7560 


9 


.0816 


.4054 


74 


.7393 


.5249 


39 


.3885 


.6423 


4 


.0294 


.7577 


2410 


49.0918 


13.4072 


2475 


49.7494 


13.5267 


2540 


50.3984 


13.6441 


2605 


51.0392 


13.7595 


11 


.1019 


.4091 


76 


.7594 


.5285 


41 


.4083 


.6459 


6 


.0490 


.7613 


12 


.1121 


.4109 


77 


.7695 


.5303 


42 


.4183 


.6477 


7 


.0588 


.7630 


13 


.1223 


.4128 


78 


.7795 


.5322 


43 


.4282 


.6495 


8 


.0686 


.7648 


14 


.1325 


.4146 


79 


7896 


.5340 


44 


.4381 


.6512 


9 


.0784 


.7665 


2415 


49.1426 


13.4165 


2480 


49.7996 


13.5358 


2545 


50.4480 


13.6530 


2610 


51.0882 


13.7683 


16 


.1528 


.4183 


81 


.8096 


.5376 


46 


.4579 


.6548 


11 


.0979 


.7701 


17 


.1630 


.4201 


82 


.8197 


.5394 


47 


.4678 


.6566 


12 


.1077 


.7718 


18 


.1732 


.4220 


83 


.8297 


.5413 


48 


.4777 


.6584 


13 


.1175 


.7736 


19 


.1833 


.4239 


84 


.8397 


.5431 


49 


.4876 


.6602 


14 


.1273 


.7753 


2420 


49.1935 


13.4257 


2485 


49.8498 


13.5449 


2550 


50.4975 


13.6620 


2615 


51.1371 


13.7771 


21 


.2037 


.4276 


86 


.8598 


.5467 


51 


.5074 


.6638 


16 


.1468 


.7788 


22 


.2138 


.4294 


87 


.8698 


.5485 


52 


.5173 


.6655 


17 


.1566 


.7806 


23 


.2240 


.4313 


88 


.8799 


.5503 


53 


.5272 


.6673 


18 


.1664 


.7823 


24 


.2341 


.4331 


89 


.8899 


.5522 


54 


.5371 


.6691 


19 


.1762 


.7841 


2425 


49.2443 


13.4350 


2490 


49.8999 


13.5540 


2555 


50.5470 


13.6709 


2620 


51.1859 


13.1859 


26 


.2544 


.4368 


91 


.9099 


.5558 


56 


.5569 


.6727 


21 


.1957 


.7876 


27 


.2646 


.4387 


92 


.9199 


.5576 


57 


.5668 


.6745 


22 


.2055 


.7894 


28 


.2747 


.4405 


93 


.9300 


.5594 


58 


.5767 


.6762 


23 


.2152 


.7911 


29 


.2849 


.4424 


94 


.9400 


.5612 


59 


.6866 


.6780 


24 


.2250 


.7929 


2430 


49.2950 


13.4442 


2495 


49.9500 


13.5630 


2560 


60.5964 


13.6798 


2625 


51.2348 


13.7946 


31 


.3052 


.4461 


96 


.9600 


.5648 


61 


.6063 


.6816 


26 


.2445 


.7964 


32 


.3153 


.4479 


97 


.9700 


.5667 


62 


.6162 


.6834 


27 


.2543 


.7981 


33 


.3254 


.4497 


98 


.9800 


.5685 


63 


.6261 


.6851 


28 


.2640 


.7999 


34 


.3356 


. 4516 


99 


.9900 


.5703 


64 


.6360 


.6869 


29 


.2738 


.8016 


2435 


49.3457 


13.4534 


2500 


50.0000 


13.5721 


2565 


50.6458 


13.6887 


2630 


51.2835 


13.8034 


36 


.3559 


.4553 


1 


.0100 


.5739 


66 


.6557 


.6905 


31 


.2933 


.8051 


37 


.3660 


.4571 


2 


.0200 


.5757 


67 


.6656 


.6923 


32 


.3030 


.8069 


38 


.3761 


.4590 


3 


.0300 


.5775 


68 


.6754 


.6940 


33 


.3128 


.8086 


39 


.3862 


.4608 


4 


.0400 


.5793 


69 


.6853 


. .6958 


34 


.3225 


.8104 


2440 


49.3964 


13.4626 


2505 


50.0500 


13.5811 


2570 


50.6952 


13.6976 


2635 


51.3323 


13.8121 


41 


.4065 


.4645 


6 


.0600 


.5829 


71 


.7050 


.6994 


36 


.3420 


.8139 


42 


.4166 


.4663 


7 


.0700 


.5847 


72 


.7149 


.7011 


37 


.3517 


.8156 


43 


.4267 


.4681 


8 


.0799 


,5865 


73 


.7247 


.7029 


38 


.3615 


.8174 


44 


.4368 


.4700 


9 


.0899 


.5884 


74 


.7346 


.7047 


39 


.3712 


.8191 


2445 


49.4469 


13.4718 


2510 


50.0999 


13.5902 


2575 


50.7445 


13.7065 


2640 


51.3809 


13.8208 



48 



2.— POWERS, ROOTS AND RECIPROCALS. 



10. — Square Roots and Cube Roots of Numbers 
1600 TO 3200 — Continued. 



No. 


Sq. Rt. 


Cu. Rt. 


No. 


Sq. Rt. 


Cu. Rt. 


No. 


Sq. Rt. 


Cu. Rt. 


No. 


Sq. Rt. 


Cu. Rt. 


2640 


51.3809 


13.8208 


2705 


52 . 0096 


13.9334 


2770 


52.6308 


14.0441 


2835 


53.2447 


14.1531 


41 


.3907 


.8226 


6 


.0192 


.9351 


71 


.6403 


.0458 


36 


.2541 


.1547 


42 


.4004 


.8243 


7 


.0288 


.9368 


72 


.6498 


.0475 


37 


.2635 


.1564 


43 


.4101 


.8261 


8 


.0384 


.9385 


73 


.6593 


.0491 


38 


.2729 


.1581 


44 


.4198 


.8278 


9 


.0481 


.9402 


74 


.6688 


.0508 


39 


.2823 


.1597 


2645 


51.4296 


13.8296 


2710 


52.0577 


13.9419 


2775 


52.6783 


14.0525 


2840 


53.2917 


14.1614 


46 


.4393 


.8313 


11 


.0673 


.9437 


76 


.6878 


.0542 


41 


.3010 


.1631 


47 


.4490 


.8331 


12 


.0769 


.9454 


77 


.6972 


.0559 


42 


.3104 


.1647 


48 


.4587 


.8348 


13 


.0865 


.9471 


78 


.7067 


.0576 


43 


.3198 


.1664 


49 


.4684 


.8365 


14 


.0961 


.9488 


79 


.7162 


.0593 


44 


.3292 


.1680 


2650 


51.4782 


13.8383 


2715 


52.1057 


13.9505 


2780 


52.7257 


14.0610 


2845 


53.3385 


14.1697 


51 


.4879 


.8400 


16 


.1153 


.9522 


81 


.7352 


.0626 


46 


.3479 


.1714 


52 


.4976 


.8418 


17 


.1249 


.9539 


82 


.7447 


.0643 


47 


.3573 


.1730 


53 


.5073 


.8435 


18 


.1344 


.9556 


83 


.7541 


.0660 


48 


.3667 


.1747 


54 


.5170 


.8452 


19 


.1440 


.9574 


84 


.7636 


.0677 


49 


.3760 


.1763 


2655 


51.5267 


13.8470 


2720 


52.1536 


13.9591 


2785 


52.7731 


14.0694 


2850 


53.3854 


14.1780 


56 


.5364 


.8487 


21 


.1632 


.9608 


86 


.7826 


.0711 


51 


.3948 


.1797 


57 


.5461 


.8504 


22 


.1728 


.9625 


87 


.7920 


.0728 


52 


.4041 


.1813 


58 


.5558 


.8522 


23 


.1824 


.9642 


88 


.8015 


.0744 


53 


.4135 


.1830 


59 


.5655 


.8539 


24 


.1920 


.9659 


89 


.8110 


.0761 


54 


.4228 


.1846 


2660 


51.5752 


13.8557 


2725 


52.2015 


13.9676 


2790 


52.8205 


14.0778 


2855 


53.4322 


14.1863 


61 


.5849 


.8574 


26 


.2111 


.9693 


91 


.8299 


.0795 


56 


.4416 


.1879 


62 


.5946 


.8591 


27 


.2207 


.9710 


92 


.8394 


.0812 


57 


.4509 


.1896 


63 


.6043 


.8609 


28 


.2303 


.9727 


93 


.8488 


.0828 


58 


.4603 


.1913 


64 


.6140 


.8626 


29 


.2398 


.9744 


94 


.8583 


.0845 


59 


.4696 


.1929 


2665 


51.6236 


13.8643 


2730 


52.2494 


13.9761 


2795 


52.8678 


14.0862 


2860 


53.4790 


14.1946 


66 


.6333 


.8661 


31 


.2590 


.9779 


96 


.8772 


.0879 


61 


.4883 


.1962 


67 


.6430 


.8678 


32 


.2685 


.9796 


97 


.8867 


.0896 


62 


.4977 


.1979 


68 


.•6527 


.8695 


33 


.2781 


.9813 


98 


.8961 


.0912 


63 


.5070 


.1995 


69 


.6624 


.8713 


34 


.2877 


.9830 


99 


.9056 


.0929 


64 


.5164 


.2012 


2670 


51.6720 


13.8730 


2735 


52.2972 


13.9847 


2800 


52.9150 


14.0946 


2865 


53.5257 


14.2028 


71 


.6817 


.8747 


36 


.3068 


.9864 


1 


.9245 


.0963 


66 


.5350 


.2045 


72 


.6914 


.8765 


37 


.3163 


.9881 


2 


.9339 


.0980 


67 


.5444 


.2061 


73 


.7011 


.8782 


38 


.3259 


.9898 


3 


.9434 


.0996 


68 


.5537 


.2078 


74 


.7107 


.8799 


39 


.3355 


.9915 


4 


.9528 


.1013 


69 


.5630 


.2094 


2675 


51.7204 


13.8817 


2740 


52.3450 


13.9932 


2805 


52.9623 


14.1030 


2870 


53.5724 


14.2111 


76 


.7301 


.8834 


41 


.3546 


.9949 


6 


.9717 


.1047 


71 


.5817 


.2127 


77 


.7397 


.8851 


42 


.3641 


.9966 


7 


.9811 


.1063 


72 


.5910 


.2144 


78 


.7494 


.8868 


43 


.3737 


.9983 


8 


.9906 


.1080 


73 


.6004 


.2160 


79 


.7591 


.8886 


44 


.3832 


14.0000 


9 


53.0000 


.1097 


74 


.6097 


.2177 


2680 


51.7687 


13.8903 


2745 


52.3927 


14.0017 


2810 


53.0094 


14.1114 


2875 


53.6190 


14.2193 


81 


.7784 


.8920 


46 


.4023 


.0034 


11 


.0189 


.1130 


76 


.6284 


.2210 


82 


.7880 


.8938 


47 


.4118 


.0051 


12 


.0283 


.1147 


77 


.6377 


.2226 


83 


.7977 


.8955 


48 


.4214 


.0068 


13 


.0377 


.1164 


78 


.6470 


.2243 


84 


.8073 


.8972 


49 


.4309 


.0085 


14 


.0471 


.1180 


79 


.6563 


.2259 


2685 


51.8170 


13.8989 


2750 


52.4404 


14.0102 


2815 


53.0566 


14.1197 


2880 


53.6656 


14.2276 


86 


.8266 


.9007 


51 


.4500 


.0119 


16 


.0660 


.1214 


81 


.6749 


.2292 


87 


.8363 


.9024 


52 


.4595 


.0136 


17 


.0754 


.1231 


82 


.6843 


.2309 


88 


.8459 


.9041 


53 


.4690 


.0153 


18 


.0848 


.1247 


83 


.6936 


.2325 


89 


.8556 


.9058 


54 


.4786 


.0170 


19 


.0943 


.1264 


84 


.7029 


.2342 


2690 


51.8652 


13.9076 


2755 


52.4881 


14.0187 


2820 


53.1037 


14.1281 


2885 


53.7122 


14.2358 


91 


.8748 


.9093 


56 


.4976 


.0204 


21 


.1131 


.1297 


86 


.7215 


.2374 


92 


.8845 


.9110 


57 


.5071 


.0221 


22 


.1225 


.1314 


87 


.7308 


.2391 


93 


.8941 


.9127 


58 


.5167 


.0238 


23 


.1319 


.1331 


88 


.7401 


.2407 


94 


.9038 


.9144 


59 


.5262 


.0255 


24 


.1413 


.1348 


89 


.7494 


.2424 


2695 


51.9134 


13.9162 


2760 


52.5357 


14.0272 


2825 


53.1507 


14.1364 


2890 


53.7587 


14.2440 


96 


.9230 


.9179 


61 


.5452 


.0289 


26 


.1601 


.1381 


91 


.7680 


.2457 


97 


.9326 


.9196 


62 


.5547 


.0305 


27 


.1695 


.1398 


.92 


.7773 


.2473 


98 


.9423 


.9213 


63 


.5642 


.0322 


28 


.1789 


.1414 


93 


.7866 


.2489 


99 


.9519 


.9230 


64 


.5738 


.0339 


29 


.1883 


.1431 


94 


.7959 


.2506 


2700 


51.9615 


13.9248 


2765 


52.5833 


14.0356 


2830 


53.1977 


14.1448 


2895 


53.8052 


14.2522 


1 


.9711 


.9265 


66 


.5928 


.0373 


31 


.2071 


.1464 


96 


.8145 


.2539 


2 


.9808 


.9282 


67 


.6023 


.0390 


32 


.2165 


.1481 


97 


.8238 


.2555 


3 


.9904 


.9299 


68 


.6118 


.0407 


33 


. .2259 


.1498 


98 


.8331 


.2572 


4 


52.0000 


.9316 


69 


.6213 


.0424 


34 


.2353 


.1514 


99 


.8424 


.2588 


2705 


52.0096 


13.9334 


2770 


52.6308 


14.0441 


2835 


53.2447 


14.1531 


2900 


53.8516 


14.2604 



COMMON TABLES— SQUARE ROOTS, CUBE ROOTS, 



49 



10. — Square Roots and Cube Roots of Numbers 
1600 TO 3200 — Continued. 



No. 


Sq. Rt. 


Cu. Rt. 


No. 


Sq. Rt. 


Cu. Rt. 


No. 


Sq. Rt. 


Cu. Rt. 


No. 


Sq. Rt. 


Cu. Rt. 


2900 


53.8516 


14.2604 


2965 


54.4518 


14.3662 


3030 


55.0454 


14.4704 


3095 


55.6327 


14.5732 


1 


.8609 


.2621 


66 


.4610 


.3678 


31 


.0545 


.4720 


96 


.6417 


.5747 


2 


.8702 


.2637 


67 


.4702 


.3694 


32 


.0636 


.4736 


97 


.6507 


.5763 


3 


.8795 


.2653 


68 


.4794 


.3710 


33 


.0727 


.4752 


98 


.6597 


.5779 


4 


.8888 


.2670 


69 


.4885 


.3726 


34 


.0818 


.4768 


99 


.6687 


.5794 


2905 


53.8981 


14.2686 


2970 


54.4977 


14.3743 


3035 


55.0908 


14.4784 


3100 


55.6776 


14.5810 


6 


.9073 


.2703 


71 


.5069 


.3759 


36 


.0999 


.4800 


1 


.6866 


.5826 


7 


.9166 


.2719 


72 


.5161 


.3775 


37 


.1090 


.4815 


2 


.6956 


.5841 


8 


.9259 


.2735 


73 


.5252 


.3791 


38 


.1181 


.4831 


3 


.7046 


.5857 


9 


.9351 


.2752 


74 


.5344 


.3807 


39 


.1271 


.4847 


4 


.7136 


.5873 


2910 


53.9444 


14.2768 


2975 


54.5436 


14.3823 


3040 


55.1362 


14.4863 


3105 


55.7225 


14.5888 


11 


.9537 


.2784 


76 


.5527 


.3839 


41 


.1453 


.4879 


6 


.7315 


5904 


12 


.9630 


.2801 


77 


.5619 


.3855 


42 


.1543 


.4895 


7 


.7405 


.5920 


13 


.9722 


.2817 


78 


.5711 


.3872 


43 


.1634 


.4911 


8 


.7494 


.5935 


14 


.9815 


.2833 


79 


.5802 


.3888 


44 


.1725 


.4927 


9 


.7584 


.5951 


2915 


53.9907 


14.2850 


2980 


54.5894 


14.3904 


3045 


55.1815 


14.4943 


3110 


55.7674 


14.5967 


16 


54.0000 


.2866 


81 


.5985 


.3920 


46 


.1906 


.4958 


11 


.7763 


.5982 


17 


.0093 


.2882 


82 


.6077 


.3936 


47 


.1996 


.4974 


12 


.7853 


.5998 


18 


.0185 


.2899 


83 


.6168 


.3952 


48 


.2087 


.4990. 


13 


.7943 


.6014 


19 


.0278 


.2915 


84 


.6260 


.3968 


49 


.2178 


.5006 


14 


.8032 


.6029 


2920 


54.0370 


14.2931 


2985 


54.6352 


14.3984 


3050 


55.2268 


14.5022 


3115 


55.8122 


14.6045 


21 


.0463 


.2948 


86 


.6443 


.4000 


51 


.2359 


.5038 


16 


.8211 


.6060 


22 


.0555 


.2964 


87 


.6535 


.4016 


52 


.2449 


.5053 


17 


.8301 


.6076 


23 


.0648 


.2980 


88 


.6626 


.4032 


53 


.2540 


.5069 


18 


.8391 


.6092 


24 


.0740 


.2999 


89 


.6717 


.4048 


54 


.2630 


.5085 


19 


.8480 


.6107 


2925 


54.0833 


14.3013 


2990 


54.6809 


14.4a65 


3055 


55.2721 


14.5101 


3120 


55.8570 


14.6123 


.26 


.0925 


.3029 


91 


.6900 


.4081 


56 


.2811 


.5117 


21 


.8659 


.6138 


27 


.1018 


.3046 


92 


.6992 


.4097 


57 


.2901 


.5133 


22 


.8749 


.6154 


28 


.1110 


.3062 


93 


.7083 


.4113 


58 


.2992 


.5148 


23 


.8838 


.6170 


29 


.1202 


.3078 


94 


.7175 


.4129 


59 


.3082 


.5164 


24 


.8928 


.6185 


2930 


54.1295 


14.3094 


2995 


54.7266 


14.4145 


3060 


55.3173 


14.5180 


3125 


55.9017 


14.6201 


31 


.1387 


.3111 


96 


.7357 


.4161 


61 


.3263 


.5196 


26 


.9106 


.6216 


32 


.1479 


.3127 


97 


.7449 


.4177 


62 


.3353 


.5212 


27 


.9196 


.6232 


33 


.1572 


.3143 


98 


.7540 


.4193 


63 


.3444 


.5228 


28 


.9285 


.6248 


34 


.1664 


.3159 


99 


.7631 


.4209 


64 


.3534 


.5243 


29 


.9375 


.6263 


2935 


54.1756 


14.3176 


3000 


54.7723 


14.4225 


3065 


55.3624 


14.5259 


3130 


55.9464 


14.6279 


36 


.1849 


.3192 


1 


.7814 


.4241 


66 


.3715 


.5275 


31 


.9553 


.6294 


37 


.1941 


.3208 


2 


.7905 


.4257 


67 


.3805 


.5291 


32 


.9643 


.6310 


38 


.2033 


.3224 


3 


.7996 


.4273 


68 


.3895 


.5307 


33 


9732 


.6326 


39 


.2125 


.3241 


4 


.8088 


.4289 


69 


.3986 


.5322 


34 


.9821 


.6341 


2940 


54.2218 


14.3257 


3005 


54.8179 


14.4305 


3070 


55.4076 


14.5338 


3135 


55.9911 


14.6357 


41 


.2310 


.3273 


6 


.8270 


.4321 


71 


.4166 


.5354 


36 


56.0000 


.6372 


42 


.2402 


.3289 


7 


.8361 


.4337 


72 


.4256 


.5370 


37 


.0089 


.6388 


43 


.2494 


.3306 


8 


.8452 


.4353 


73 


.4346 


.5385 


38 


.0179 


.6403 


44 


.2586 


.3322 


9 


.8544 


.4369 


74 


.4437 


.5401 


39 


.0268 


.6419 


2945 


54.2679 


14.3338 


3010 


54.8635 


14.4385 


3075 


55.4527 


14.5417 


3140 


56.0357 


14.6434 


46 


.2771 


,3354 


11 


.8726 


.4401 


76 


.4617 


.5433 


41 


.0446 


.6450 


47 


.2863 


.3371 


12 


.8817 


.4417 


77 


.4707 


.5448 


42 


.0535 


.6466 


48 


.2955 


.3387 


13 


.8908 


.4433 


78 


.4797 


.5464 


43 


.0625 


.6481 


49 


.3047 


.3403 


14 


.8999 


.4449 


79 


.4887 


.5480 


44 


.0714 


.6497 


2950 


54.3139 


14.3419 


3015 


54.9090 


14.4465 


3080 


55.4977 


14.5496 


3145 


56.0803 


14.6512 


51 


.3231 


.3435 


16 


.9181 


.4481 


81 


.5068 


.5511 


46 


.0892 


.6528 


52 


.3323 


.3452 


17 


.9272 


.4497 


82 


5158 


.5527 


47 


.0981 


.6543 


53 


.3415 


.3468 


18 


.9363 


.4513 


83 


.5248 


.5543 


48 


.1070 


.6559 


54 


.3507 


.3484 


19 


.9454 


.4529 


84 


.5338 


.5559 


49 


.1160 


.6574 


2955 


54.3599 


14.3500 


3020 


54.9545 


14.4545 


3085 


55.5428 


14.5574 


3150 


56.1249 


14.6590 


56 


.3691 


.3516 


21 


.9636 


.4561 


86 


.5518 


.5590 


51 


.1338 


.6605 


57 


.3783 


.3533 


22 


.9727 


.4577 


87 


.5608 


.5606 


52 


.1427 


.6621 


58 


.3875 


.3549 


23 


.9818 


.4593 


88 


.5698 


.5622 


53 


.1516 


.6636 


59 


.3967 


.3565 


24 


.9909 


.4609 


89 


.5788 


.5637 


54 


.1605 


.6652 


2960 


54.4059 


14.3581 


3025 


55.0000 


14.4624 


3090 


55.5878 


14.5653 


3155 


56.1694 


14.6667 


61 


.4151 


.3597 


26 


.0091 


.4640 


91 


.5968 


.5669 


56 


.1783 


.6683 


62 


.4243 


.3613 


27 


.0182 


.4656 


92 


.6058 


.5684 


57 


.1872 


.6698 


63 


.4334 


.3630 


28 


.0273 


.4672 


93 


.6147 


.5700 


58 


.1961 


.6714 


64 


.4426 


.3646 


29 


.0364 


.4688 


94 


.6237 


.5716 


59 


.2050 


.6729 


2965 


54.4518 


14.3662 


3030 


55.0454 


14.4704 


3095 


55.6327 


14.5732 


3160 


56.2139 


14.6745 



50 



2— POWERS, ROOTS AND RECIPROCALS. 



10. — Square Roots and Cube Roots op XSI umbers 
1600 to 3200— Concluded. 



No. 


Sq. Rt. 


Cu. Rt. 


No. 


Sq. Rt. 


Cu. Rt. 


No. 


Sq. Rt. 


Cu. Rt. 


No. 


Sq. Rt. 


Cu. Rt. 


3160 


56.2139 


14.6745 


3170 


56.3028 


14.6899 


3180 


56.3915 


14.7054 


3190 


56.4801 


14.7208 


61 


.2228 


.6760 


71 


.3116 


.6915 


81 


.4004 


.7069 


91 


.4889 


.7223 


62 


.2317 


.6776 


72 


.3205 


.6930 


82 


.4092 


.7084 


92 


.4978 


.7238 


63 


.2406 


.6791 


73 


.3294 


.6946 


83 


.4181 


.7100 


93 


.5066 


.7254 


64 


.2494 


.6807 


74 


.3383 


.6961 


84 


.4269 


.7115 


94 


.5155 


.7269 


3165 


56.2583 


14.6822 


3175 


56.3471 


14.6977 


3185 


56.4358 


14.7131 


3195 


56.5243 


14.7284 


66 


.2672 


.6837 


76 


.3560 


.6992 


86 


.4447 


.7146 


96 


.5332 


.7300 


67 


.2761 


.6853 


77 


.3649 


.7007 


87 


.4535 


.7161 


97 


.5420 


.7315 


68 


.2850 


.6868 


78 


.3738 


.7023 


88 


.4624 


.7177 


98 


.5509 


.7331 


69 


.2939 


.6884 


79 


.3826 


.7038 


89 


.4712 


.7192 


99 


.5597 


.7346 


3170 


56.3028 


14.6899 


3180 


56.3915 


14.7054 


3190 


56.4801 


14.7208 


3200 


56.5685 


14.7361 



Note. — For square roots and cube roots of numbers above 3200 see 
Engineers' Tables, preceding. 

10a.— Square Roots of Numbers 3200 to 3515. ' 



No. 


Sq.Rt. 


No. 


Sq.Rt. 


No. 
3290 


Sq.Rt. 
57.3585 


No. 
3335 


Sq.Rt. 


No. 


Sq.Rt. 


No. 
3425 


Sq.Rt. 


No. 


Sq.Rt. 


3200 


56.5685 


3245 


56.9649 


57.7495 


3380 


58.1378 


58.5235 


3470 


58.9067 


01 


.5774 


46 


.9737 


91 


.3672 


36 


.7581 


81 


.1464 


26 


.5320 


71 


.9152 


02 


.5862 


47 


.9825 


92 


.3760 


37 


.7668 


82 


.1550 


27 


.5406 


72 


.9237 


03 


.5951 


48 


56.9912 


93 


.3847 


38 


.7754 


83 


.1636 


28 


.5491 


73 


.9322 


04 


.6039 


49 


57.0000 


94 


.3934 


39 


.7841 


84 


.1722 


29 


.5577 


74 


.9406 


3205 


56.6127 


3250 


57.0088 


3295 


57.4021 


3340 


57.7927 


3385 


58.1808 


3430 


58.5662 


3475 


58.9491 


06 


.6216 


51 


.0175 


96 


.4108 


41 


.8014 


86 


.1893 


31 


.5747 


76 


.9576 


07 


.6304 


52 


.0263 


97 


.4195 


42 


.8100 


87 


.1979 


32 


.5833 


77 


.9661 


08 


.6392 


53 


.0351 


98 


.4282 


43 


.8187 


88 


.2065 


33 


.5918 


78 


.9746 


09 


.6480 


54 


.0438 


99 


.4369 


44 


.8273 


89 


.2151 


34 


.6003 


79 


.9830 


3210 


56.6569 


3255 


57.0526 


3300 


57.4456 


3345 


57.8360 


3390 


58.2237 


3435 


58.6089 


3480 


58.9915 


11 


.6657 


56 


.0614 


01 


.4543 


46 


.8446 


91 


.2323 


36 


.6174 


81 


59.0000 


12 


.6745 


57 


.0701 


02 


.4630 


47 


.8533 


92 


.2409 


37 


.6259 


82 


.0085 


13 


.6833 


58 


.0789 


03 


.4717 


48 


.8619 


93 


.2495 


38 


.6345 


83 


.0169 


14 


.6922 


59 


.0877 


04 


4804 


49 


.8705 


94 


.2580 


39 


.6430 


84 


.0254 


3215 


56.7010 


3260 


57.0964 


3305 


57.4891 


3350 


57.8792 


3395 


58.2666 


3440 


58.6515 


3485 


59.0339 


16 


.7098 


61 


.1052 


06 


.4978 


51 


.8878 


96 


.2752 


41 


.6600 


86 


.0424 


17 


.7186 


62 


.1139 


07 


.5065 


52 


.8965 


97 


.2838 


42 


.6686 


87 


.0508 


18 


.7274 


63 


.1227 


08 


.5152 


53 


.9051 


98 


.2924 


43 


.6771 


88 


.0593 


19 


.7362 


64 


.1314 


09 


.5239 


54 


.9137 


99 


.3009 


44 


.6856 


89 


.0678 


3220 


56.7450 


3265 


57.1402 


3310 57.5326 


3355 


57.9224 


3400 


58.3095 


3445 


58.6941 


3490 


59.0762 


21 


.7539 


66 


.1489 


11 


.5413 


56 


.9310 


01 


.3181 


46 


.7026 


91 


.0847 


22 


.7627 


67 


.1577 


12 


.5500 


57 


.9396 


02 


.3267 


47 


.7112 


92 


.0931 


23 


.7715 


68 


.1664 


13 


.5587 


58 


.9483 


03 


.3352 


48 


.7197 


93 


.1016 


24 


.7803 


69 


.1752 


14 


.5674 


59 


.9569 


04 


.3438 


49 


.7282 


94 


.1101 


3225 


56.7891 


3270 


57.1839 


3315 


57.5760 


3360 


57.9655 


3405 


58.3524 


3450 


58.7367 


3495 


59.1185 


26 


.7979 


71 


.1927 


16 


.5847 


61 


.9741 


06 


.3609 


51 


.7452 


96 


.1270 


27 


.8067 


72 


.2014 


17 


.5934 


62 


.9828 


07 


.3695 


62 


.7537 


97 


.1354 


28 


.8155 


73 


.2101 


18 


.6021 


63 


57.9914 


08 


.3781 


53 


.7622 


98 


.1439 


29 


.8243 


74 


.2189 


19 


.610« 


64 


58.0000 


09 


.3866 


54 


.7707 


99 


.1523 


3230 


56.8331 


3275 


57.2276 


3320 


57.6194 


3365 


58.0086 


3410 


58.3952 


3455 


58.7792 


3500 


59.1608 


31 


.8419 


76 


.2364 


21 


.6281 


66 


.0172 


11 


.4038 


56 


.7878 


01 


.1692 


32 


.8507 


77 


.2451 


22 


.6368 


67 


.0259 


12 


.4123 


57 


.7963 


02 


.1777 


33 


.8596 


78 


.2538 


23 


.6455 


68 


.0345 


13 


.4209 


58 


.8048 


03 


.1861 


34 


.8683 


79 


.2626 


24 


.6541 


69 


.0431 


14 


.4294 


59 


.8133 


04 


.1946 


3235 


56.8771 


3280 


57.2713 


3325 


57.6628 


3370 


58.0517 


3415 


58.4380 


3460 


58.8218 


3505 


59.2030 


36 


.8859 


81 


.2800 


26 


.6715 


71 


.0603 


16 


.4466 


61 


.8303 


06 


.2115 


37 


.8946 


82 


.2887 


27 


.6802 


72 


.0689 


17 


.4551 


62 


.8388 


07 


.2199 


38 


.9034 


83 


.2975 


28 


.6888 


73 


.0775 


18 


.4637 


63 


.8473 


08 


.2284 


39 


.9122 


84 


.3062 


29 


.6975 


74 


.0861 


19 


.4722 


64 


.8558 


09 


.2368 


3240 


56.9210 


3285 


57.3149 


3330 


57.7062 


3375 


58.0948 


3420 


58.4808 


3465 


58.8643 


3510 


59.2*53 


41 


.9298 


86 


.3236 


31 


.7148 


76 


.1034 


21 


.4893 


66 


.8727 


11 


.2537 


42 


.9386 


87 


.3324 


32 


.7235 


77 


.1120 


22 


.4979 


67 


.8812 


12 


.2621 


43 


.9473 


88 


.3411 


33 


.7321 


78 


.1206 


23 


.5064 


68 


.8897 


13 


.2706 


44 


.9561 


89 


.3498 


34 


.7408 


79 


.1292 


24 


.5150 


69 


.8982 


14 


.2790 


3245 


56.9649 


3290 


57.3585 


3335 


57.7495 


3380 


58.1378 


3426 58.5235| 


3470 


58.9067 


3515 


59.2874 



COMMON TABLES— RECIPROCALS OR NUMBERS, 
11. — Reciprocals op Numbers 1 to 1000. 



SI 



No. 


Reciprocal 


No. 


Reciprocal 


No. 


Reciprocal 


No. 


Reciprocal 


No. 


Reciprocal 





Infinite. 


65 


.01538 4615 


130 


.00769 2308 


195 


.00512 8205 


260 


. 00384 6154 


1 


Unity 


6 


.01515 1515 


1 


.00763 3588 


6 


.00510 2041 


1 


.00383 1418 


2 


.50000 0000 


7 


.01492 5373 


2 


.00757 5758 


7 


.00507 6142 


2 


.00381 6794 


3 


.33333 3333 


8 


.01470 5882 


3 


.00751 8797 


8 


.00505 0505 


3 


.00380 2281 


4 


.25000 0000 


9 


.01449 2754 


4 


.00746 2687 


9 


.00502 5126 


4 


.00378 7879 


5 


.20000 0000 


70 


.01428 5714 


135 


.00740 7407 


200 


.00500 0000 


265 


.00377 3585 


6 


.16666 6667 


1 


.01408 4507 


6 


.00735 2941 


1 


.00497 5124 


6 


.00375 9398 


7 


.14285 7143 


2 


.01388 8889 


7 


.00729 9270 


2 


.00495 0495 


7 


.00374 5318 


8 


.12500 0000 


3 


.01369 8630 


8 


.00724 6377 


3 


.00492 6108 


8 


.00373 1343 


9 


.11111 nil 


4 


.01351 3514 


9 


.00719 4245 


4 


.00490 1961 


9 


.00371 7472 


10 


.10000 0000 


75 


.01333 3333 


140 


.00714 2857 


205 


.00487 8049 


270 


.00370 3704 


11 


.09090 9091 


6 


.01315 7895 


1 


.00709 2199 


6 


.00485 4369 


1 


.00369 0037 


12 


.08333 3333 


7 


.01298 7013 


2 


.00704 2254 


7 


.00483 0918 


2 


.00367 6471 


13 


.07692 3077 


8 


.01282 0513 


3 


.00699 3007 


8 


.00480 7692 


3 


.00366 3004 


14 


.07142 8571 


9 


.01265 8228 


4 


.00694 4444 


9 


.00478 4689 


4 


.00364 9635 


15 


.06666 6667 


80 


.01250 0000 


145 


.00689 6552 


210 


.00476 1905 


275 


.00363 6364 


16 


.06250 0000 


1 


.01234 5679 


6 


.00684 9315 


11 


.00473 9336 


6 


.00362 3188 


17 


.05882 3529 


2 


.01219 5122 


7 


.00680 2721 


12 


.00471 6981 


7 


.00361 0108 


18 


.05555 5556 


3 


.01204 8193 


8 


.00675 6757 


13 


.00469 4836 


8 


.00359 7122 


19 


.05263 1579 


4 


.01190 4762 


9 


.00671 1409 


14 


.00467 2897 


9 


.00358 4229 


20 


.05000 0000 


85 


.01176 4706 


150 


.00666 6667 


215 


.00465 1163 


280 


.00357 1429 




.04761 9048 


6 


.01162 7907 


1 


.00662 2517 


16 


.00462 9630 


1 


.00355 8719 


2 


.04545 4545 


7 


.01149 4253 


2 


.00657 8947 


17 


.00460 8295 


2 


.00354 6099 


3 


.04347 8261 


8 


.01136 3636 


3 


.00653 5948 


18 


.00458 7156 


3 


.00353 3569 


4 


.04166 6667 


9 


.01123 5955 


4 


.00649 3506 


19 


.00456 6210 


4 


.00352 1127 


25 


.04000 0000 


90 


.01111 nil 


155 


,00645 1613 


220 


.00454 5455 


285 


.00350 8772 


6 


.03846 1538 


1 


.01098 9011 


6 


.00641 0256 


1 


.00452 4887 


6 


.00349 6503 


7 


.03703 7037 


2 


.01086 9565 


7 


.00636 9427 


2 


.00450 4505 


7 


.00348 4321 


8 


.03571 4286 


3 


.01075 2688 


8 


.00632 9114 


3 


.00448 4305 


8 


.00347 2222 


9 


.03448 2759 


4 


.01063 8298 


9 


.00628 9308 


4 


.00446 4286 


9 


.00346 0208 


30 


.03333 3333 


95 


.01052 6316 


160 


.00625 0000 


225 


.00444 4444 


290 


.00344 8276 


1 


.03225 8065 


6 


.01041 6667 


1 


.00621 1180 


6 


.00442 4779 


1 


.00343 6426 


2 


.03125 0000 


7 


.01030 9278 


2 


.00617 2840 


7 


.00440 5286 


2 


.00342 4658 


3 


.03030 3030 


8 


.01020 4082 


3 


.00613 4969 


8 


.00438 5965 


3 


.00341 2969 


4 


.02941 1765 


9 


.01010 1010 


4 


.00609 7561 


9 


.00436 6812 


4 


.00340 1361 


35 


.02857 1429 


100 


.01000 0000 


165 


.00606 0606 


230 


.00434 7826 


295 


.00338 9831 


6 


.02777 7778 


1 


.00990 0990 


6 


.00602 4096 


1 


.00432 9004 


6 


.00337 8378 


7 


.02702 7027 


2 


.00980 3922 


7 


.00598 8024 


2 


.00431 0345 


7 


.00336 7003 


8 


.02631 5789 


3 


.00970 8738 


8 


.00595 2381 


3 


.00429 1845 


8 


.00335 5705 


9 


.02564 1026 


4 


.00961 5385 


9 


.00591 7160 


4 


.00427 3504 


9 


.00334 4482 


40 


.02500 0000 


105 


.00952 3810 


170 


.00588 2353 


235 


.00425 5319 


300 


.00333 3333 


1 


.02439 0244 


6 


.00943 3962 


1 


.00584 7953 


6 


.00423 7288 


1 


.00332 2259 


2 


.02380 9524 


7 


.00934 5794 


2 


.00581 3953 


7 


.00421 9409 


2 


.00331 1258 


3 


.02325 5814 


8 


.00925 9259 


3 


.00578 0347 


8 


.00420 1681 


3 


.00330 0330 


4 


.02272 7273 


9 


.00917 4312 


4 


.00574 7126 


9 


.00418 4100 


4 


,00328 9474 


45 


.02222 2222 


110 


.00909 0909 


175 


.00571 4286 


240 


.00416 6667 


305 


.00327 8689 


6 


.02173 9130 


11 


.00900 9009 


6 


.00568 1818 


1 


.00414 9378 


6 


.00326 7974 


7 


.02127 6600 


12 


.00892 8571 


7 


.00564 9718 


2 


.00413 2231 


7 


.00325 7329 


8 


.02083 3333 


13 


.00884 9558 


8 


.00561 7978 


3 


.00411 5226 


8 


.00324 6753 


9 


.02040 8163 


14 


.00877 1930 


9 


.00558 6592 


4 


.00409 8361 


9 


.00323 6246 


50 


.02000 0000 


115 


.00869 5652 


180 


.00555 5556 


245 


.00408 1633 


310 


.00322 5806 


1 


.01960 7843 


16 


.00862 0690 


1 


.00552 4862 


6 


.00406 5041 


11 


.00321 5434 


2 


.01923 0769 


17 


.00854 7009 


2 


.00549 4505 


7 


.00404 8583 


12 


.00320 5128 


3 


.01886 7925 


18 


.00847 4576 


3 


.00546 4481 


8 


.00403 2258 


13 


.00319 4888 


4 


.01851 8519 


19 


.00840 3361 


4 


.00543 4783 


9 


.00401 6064 


14 


.00318 4713 


55 


.01818 1818 


120 


.00833 3333 


185 


.00540 5405 


250 


.00400 0000 


315 


.00317 4603 


6 


.01785 7143 


1 


.00826 4463 


6 


.00537 6344 


1 


.00398 4064 


16 


.00316 4557 


7 


.01754 3860 


2 


.00819 6721 


7 


.00534 7594 


2 


.00396 8254 


17 


.00315 4574 


8 


.01724 1379. 


3 


.00813 0081 


8 


.00531 9149 


3 


.00395 2569 


18 


.00314 4654 


9 


.01694 9153 


4 


.00806 4516 


9 


.00529 1005 


4 


.00393 7008 


19 


.00313 4796 


60 


.01666 6667 


125 


.00800 0000 


190 


.00526 3158 


255 


.00392 1569 


320 


.00312 5000 


1 


.01639 3443 


6 


.00793 6508 


1 


.00523 5602 


6 


.00390 6250 


1 


.00311 5265 


2 


.01612 9032 


7 


.00787 4016 


2 


.00520 8333 


7 


.00389 1051 


2 


.00310 5590 


3 


.01587 3016 


8 


.00781 2500 


3 


.00518 1347 


8 


.00387 5969 


3 


.00309 5975 


4 


.01562 5000 


9 


.00775 1938 


4 


.00515 4639 


9 


.00386 1004 


4 


.00308 6420 


65 


.01538 4615 


130 


.00769 2308 


195 


.00512 8205 


260 


.00384 6154 


325 


.00307 6923 



52 2.— POWERS, ROOTS AND RECIPROCALS. 

11. — Reciprocals op Numbers 1 to 1000 — Continued. 



No. 


Reciprocal 


No. 


Reciprocal 


No. 


Reciprocal 


No. 


Reciprocal 


No. 


Reciprocal 


325 


.00307 6923 


390 


.00256 4103 


455 


.00219 7802 


520 


.00192 3077 


585 


.00170 9402 


6 


.00306 7485 


1 


.00255 7545 


6 


.00219 2982 


1 


.00191 9386 


6 


.00170 6485 


7 


.00305 8104 


2 


.00255 1020 


7 


.00218 8184 


2 


.00191 5709 


7 


.00170 3578 


8 


.00304 8780 


3 


.00254 4529 


8 


.00218 3406 


3 


.00191 2046 


8 


.00170 0680 


9 


.00303 9514 


4 


.00253 8071 


9 


.00217 8649 


4 


.00190 8397 


9 


.00169 7793 


330 


.00303 0303 


395 


.00253 1646 


460 


.00217 3913 


525 


.00190 4762 


590 


.00169 4915 


1 


.00302 1148 


6 


.00252 5253 


1 


.00216 9197 


6 


.00190 1141 


1 


.00169 2047 


2 


.00301 2048 


7 


.00251 8892 


2 


.00216 4502 


7 


.00189 7533 


2 


.00168 9189 


3 


.00300 3003 


8 


.00251 2563 


3 


.00215 9827 


8 


.00189 3939 


3 


.00168 6341 


4 


.00299 4012 


9 


.00250 6266 


4 


.00215 5172 


9 


.00189 0359 


4 


.00168 3502 


335 


.00298 5075 


400 


.00250 0000 


465 


.00215 0538 


530 


.00188 6792 


595 


.00168 0672 


6 


.00297 6190 


1 


.00249 3766 


6 


.00214 5923 


1 


.00188 3239 


6 


.00167 7852 


7 


.00296 7359 


2 


.00248 7562 


7 


.00214 1328 


2 


.00187 9699 


7 


.00167 5042 


8 


.00295 8580 


3 


.00248 1390 


8 


.00213 6752 


3 


.00187 6173 


8 


.00167 2241 


9 


.00294 9853 


4 


.00247 5248 


9 


.00213 2196 


4 


.00187 2659 


9 


.00166 9449 


340 


.00294 1176 


405 


.00246 9136 


470 


.00212 7660 


535 


.00186 9159 


600 


.00166 6667 


1 


.00293 2551 


6 


.00246 3054 


1 


.00212 3142 


6 


.00186 5672 


1 


.00166 3894 


2 


.00292 3977 


7 


.00245 7002 


2 


.00211 8644 


7 


.00186 2197 


2 


.00166 1130 


3 


.00291 5452 


8 


.00245 0980 


3 


00211 4165 


8 


.00185 8736 


3 


.00165 8375 


4 


.00290 6977 


9 


.00244 4988 


4 


.00210 9705 


9 


.00185 5288 


4 


.00165 5629 


345 


.00289 8551 


410 


.00243 9024 


475 


.00210 5263 


540 


.00185 1852 


605 


.00165 2893 


6 


.00289 0173 


11 


.00243 3090 


6 


.00210 0840 


1 


.00184 8429 


6 


.00165 0165 


7 


.00288 1844 


12 


.00242 7184 


7 


.00209 6436 


2 


.00184 5018 


7 


.00164 7446 


8 


.00287 3563 


13 


.00242 1308 


8 


.00209 2050 


3 


.00184 1621 


8 


.00164 4737 


9 


.00286 5330 


14 


.00241 5459 


9 


.00208 7683 


4 


.00183 8235 


9 


.00164 2036 


350 


.00285 7143 


415 


.00240 9639 


480 


.00208 3333 


545 


.00183 4862 


610 


.00163 9344 


1 


.00284 9003 


16 


.00240 3846 


1 


.00207 9002 


6 


.00183 1502 


11 


.00163 6661 


2 


.00284 0909 


17 


.00239 8082 


2 


.00207 4689 


7 


.00182 8154 


12 


.00163 3987 


3 


.00283 2861 


18 


.00239 2344 


3 


.00207 0393 


8 


.00182 4818 


13 


.00163 1321 


4 


.00282 4859 


19 


.00238 6635 


4 


.00206 6116 


9 


.00182 1494 


14 


.00162 8664 


355 


.00281 6901 


420 


.00238 0952 


485 


.00206 1856 


550 


.00181 8182 


615 


.00162 6016 


6 


.00280 8989 


1 


.00237 5297 


6 


00205 7613 


1 


.00181 4882 


16 


.00162 3377 


7 


.00280 1120 


2 


.00236 9668 


7 


.00205 3388 


2 


.00181 1594 


17 


.00162 0746 


8 


.00279 3296 


3 


.00236 4066 


8 


.00204 9180 


3 


.00180 8318 


18 


.00161 8123 


9 


.00278 5515 


4 


.00235 8491 


9 


.00204 4990 


4 


.00180 5054 


19 


.00161 5509 


360 


.00277 7778 


425 


.00235 2941 


490 


.00204 0816 


555 


.00180 1802 


620 


.00161 2903 


1 


.00277 0083 


6 


.00234 7418 


1 


.00203 6660 


6 


.00179 8561 


1 


.00161 0306 


2 


.00276 2431 


7 


.00234 1920 


2 


.00203 2520 


7 


.00179 5332 


2 


.00160 7717 


3 


.00275 4821 


8 


.00233 6449 


3 


.00202 8398 


8 


.00179 2115 


3 


.00160 5136 


4 


.00274 7253 


9 


.00233 1002 


4 


.00202 4291 


9 


.00178 8909 


4 


.00160 2564 


365 


.00273 9726 


430 


.00232 5581 


495 


.00202 0202 


560 


.00178 5714 


625 


.0U160 0000 


6 


.00273 2240 


1 


.00232 0186 


6 


.00201 6129 


1 


.00178 2531 


6 


.00159 7444 


7 


.00272 4796 


2 


.00231 4815 


7 


.00201 2072 


2 


.00177 9359 


7 


.00159 4896 


8 


.00271 7391 


3 


.00230 9469 


8 


.00200 8032 


3 


.00177 6199 


8 


.00159 2357 


9 


.00271 0027 


4 


.00230 4147 


9 


.00200 4008 


4 


.00177 3050 


9 


.00158 9825 


370 


.00270 2703 


435 


.00229 8851 


500 


.00200 0000 


565 


.00176 9912 


630 


.00158 7302 


1 


.00269 5418 


6 


.00229 3578 


1 


.00199 6008 


6 


.00176 6784 


1 


.00158 4786 


2 


.00268 8172 


7 


.00228 8330 


2 


.00199 2032 


7 


.00176 3668 


2 


.00158 2278 


3 


.00268 0965 


8 


.00228 3105 


3 


.00198 8072 


8 


.00176 0563 


3 


.00157 9779 


4 


.00267 3797 


9 


.00227 7904 


4 


.00198 4127 


9 


.00175 7469 


4 


.00157 7287 


375 


.00266 6667 


440 


.00227 2727 


505 


.00198 0198 


570 


.00175 4386 


635 


.00157 4803 


6 


.00265 9574 


1 


.00226 7574 


6 


.00197 6285 


1 


.00175 1313 


6 


.00157 2327 


7 


.00265 2520 


2 


.00226 2443 


7 


.00197 2387 


2 


.00174 8252 


7 


.00156 9859 


8 


.00264 5503 


3 


.00225 7336 


8 


.00196 8504 


3 


.00174 5201 


8 


.00156 7398 


9 


.00263 8522 


4 


.00225 2252 


9 


.00196 4637 


4 


.00174 2160 


9 


.00156 4945 


380 


.00263 1579 


445 


.00224 7191 


510 


.00196 0784 


575 


.00173 9130 


640 


.00156 2500 


1 


.00262 4672 


6 


.00224 2152 


11 


.00195 6947 


6 


.00173 6111 


1 


00156 0062 


2 


.00261 7801 


7 


.00223 7136 


12 


.00195 3125 


7 


.00173 3102 


2 


.00155 7632 


3 


.00261 0966 


8 


.00223 2143 


13 


.00194 9318 


8 


.00173 0104 


3 


.00155 5210 


4 


.00260 4167 


9 


.00222 7171 


14 


.00194 5525 


9 


.00172 7116 


4 


.00155 2795 


385 


.00259 7403 


450 


.00222 2222 


515 


.00194 1748 


580 


.00172 4138 


645 


.00155 0388 


6 


.00259 0674 


1 


.00221 7295 


16 


.00193 7984 


1 


.00172 1170 


6 


.00154 7988 


7 


.00258 3979 


2 


00221 2389 


17 


.00193 4236 


2 


.00171 8213 


7 


.00154 5595 


8 


.00257 7320 


3 


.00220 7506 


18 


.00193 0502 


3 


.00171 5266 


8 


.00154 3210 


9 


.00257 0694 


4 


.00220 2643 


19 


.00192 6782 


4 


.00171 2329 


9 


.00154 0832 


390 


.00256 4103 


455 


.00219 7802 


520 


.00192 3077 


585 


.00170 9402 


650 


.00153 8462 



COMMON TABLES— RECIPROCALS OF NUMBERS. 53 

11. — Reciprocals of Numbers 1 to lOOO.-^-Continued. 



No. 


Reciprocal 


No. 


Reciprocal 


No. 


Reciprocal 


No. 


Reciprocal 


No. 


Reciprocal 


650 


.00153 8462 


715 


.00139 8601 


780 


.00128 2051 


845 


.00118 3432 


910 


.00109 8901 


1 


.00153 6098 


16 


.00139 6648 


1 


.00128 0410 


6 


.00118 2033 


11 


.00109 7695 


2 


.00153 3742 


17 


.00139 4700 


2 


.00127 8772 


7 


.00118 0638 


12 


.00109 6491 


3 


.00153 1394 


18 


.00139 2758 


3 


.00127 7139 


8 


.00117 9245 


13 


.00109 5290 


4 


.00152 9052 


19 


.00139 0821 


4 


.00127 5510 


9 


.00117 7856 


14 


.00109 4092 


655 


.00152 6718 


720 


.00138 8889 


785 


.00127 3885 


850 


.00117 6471 


915 


.00109 2896 


6 


.00152 4390 


1 


.00138 6963 


6 


.00127 2265 


1 


.00117 5088 


16 


.00109 1703 


7 


.00152 2070 


2 


.00138 5042 


7 


.00127 0648 


2 


.00117 3709 


17 


.00109 0513 


8 


.00151 9757 


3 


.00138 3126 


8 


.00126 9036 


3 


.00117 2333 


18 


.00108 9325 


9 


.00151 7451 


4 


.00138 1215 


9 


.00126 7427 


4 


.00117 0960 


19 


.00108 8139 


660 


.00151 5152 


725 


.00137 9310 


790 


.00126 5823 


855 


.00116 9591 


920 


.00108 6957 


1 


.00151 2859 


6 


.00137 7410 


1 


.00126 4223 


6 


.00116 8224 


1 


.00108 5776 


2 


.00151 0574 


7 


.00137 5516 


2 


.00126 2626 


7 


.00116 6861 


2 


.00108 4599 


3 


.00150 8296 


8 


.00137 3626 


3 


.00126 1034 


8 


.00116 5501 


3 


.00108 3424 


4 


.00150 6024 


9 


.00137 1742 


4 


.00125 9446 


9 


.00116 4144 


4 


.00108 2251 


665 


.00150 3759 


730 


.00136 9863 


795 


.00125 7862 


860 


.00116 2791 


925 


.00108 1081 


6 


.00150 1502 


1 


.00136 7989 


6 


.00125 6281 


1 


.00116 1440 


6 


.00107 9914 


7 


.00149 9250 


2 


.00136 6120 


7 


.00125 4705 


2 


.00116 0093 


7 


.00107 8749 


8 


.00149 7006 


3 


.00136 4256 


8 


.00125 3133 


3 


.00115 8749 


8 


.00107 7586 


9 


.00149 4768 


4 


.00136 2398 


9 


.00125 1564 


4 


.00115 7407 


9 


.00107 6426 


670 


.00149 2537 


735 


.00136 0544 


800 


.00125 0000 


865 


.00115 6069 


930 


.00107 5269 


1 


.00149 0313 


6 


.00135 8696 


1 


.00124 8439 


6 


.00115 4734 


1 


.00107 4114 


2 


.00148 8095 


7 


.00135 6852 


2 


.00124 6883 


7 


.00115 3403 


2 


.00107 2961 


3 


.00148 5884 


8 


.00135 5014 


3 


.00124 5330 


8 


.00115 2074 


3 


.00107 1811 


4 


.00148 3680 


9 


.00135 3180 


4 


.00124 3781 


9 


.00115 0748 


4 


.00107 0664 


675 


.00148 1481 


740 


.00135 1351 


805 


.00124 2236 


870 


.00114 9425 


935 


.00106 9519 


6 


.00147 9290 


1 


.00134 9528 


6 


.00124 0695 


1 


.00114 8106 


6 


.00106 8376 


7 


.00147 7105 


2 


.00134 7709 


7 


.00123 9157 


2 


.00114 6789 


7 


.00106 7236 


8 


.00147 4926 


3 


.00134 5895 


8 


.00123 7624 


3 


.00114 5475 


8 


.00106 6098 


9 


.00147 2754 


4 


.00134 4086 


9 


.00123 6094 


4 


.00114 4165 


9 


.00106 4963 


680 


.00147 0588 


745 


.00134 2282 


810 


.00123 4568 


875 


.00114 2857 


940 


.00106 3830 


1 


.00146 8429 


6 


.00134 0483 


11 


.00123 3046 


6 


.00114 1553 


1 


.00106 2699 


2 


.00146 6276 


7 


.00133 8688 


12 


.00123 1527 


7 


.00114 0251 


2 


.00106 1571 


3 


.00146 4129 


8 


.00133 6898 


13 


.00123 0012 


8 


.00113 8952 


3 


.00106 0445 


4 


.00146 1988 


9 


.00133 5113 


14 


.00122 8501 


9 


.00113 7656 


4 


.00105 9322 


685 


.00145 9854 


750 


.00133 3333 


815 


.00122 6994 


880 


.00113 6364 


945 


.00105 8201 


6 


.00145 7726 


1 


.00133 1558 


16 


.00122 5490 


1 


.00113 5074 


6 


.00105 7082 


7 


.00145 5604 


2 


.00132 9787 


17 


.00122 3990 


2 


.00113 3787 


7 


.00105 5966 


8 


.00145 3488 


3 


.00132 8021 


18 


.00122 2494 


3 


.00113 2503 


8 


.00105 4852 


9 


.00145 1379 


' 4 


.00132 6260 


19 


.00122 1001 


4 


.00113 1222 


9 


.00105 3741 


690 


.00144 9275 


755 


.00132 4503 


820 


.00121 9512 


885 


.00112 9944 


950 


.00105 2632 


1 


.00144 7178 


6 


.00132 2751 


1 


.00121 8027 


6. 


.00112 8668 


1 


.00105 1525 


2 


.00144 5087 


7 


.00132 1004 


2 


.00121 6545 


7 


.00112 7396 


2 


.00105 0420 


3 


.00144 3001 


8 


.00131 9261 


3 


.00121 5067 


8 


.00112 6126 


3 


.00104 9318 


4 


.00144 0922 


9 


.00131 7523 


4 


.00121 3592 


9 


.00112 4859 


4 


.00104 8218 


695 


.00143 8849 


760 


.00131 5789 


825 


.00121 2121 


890 


.00112 3596 


955 


.00104 7120 


6 


.00143 6782 


1 


.00131 4060 


6 


.00121 0654 


1 


.00112 2334 


6 


00104 6025 


7 


.00143 4720 


2 


.00131 2336 


7 


.00120 9190 


2 


.00112 1076 


7 


.00104 4932 


8 


.00143 2665 


3 


.00131 0616 


8 


.00120 7729 


3 


.00111 9821 


8 


.00104 3841 


9 


.00143 0615 


4 


.00130 8901 


9 


.00120 6273 


4 


.00111 8568 


9 


.00104 2753 


700 


.00142 8571 


765 


.00130 7190 


830 


.00120 4819 


895 


.00111 7318 


960 


.00104 1667 


1 


.00142 6534 


6 


.00130 5483 




.00120 3369 


6 


.00111 6071 


1 


.00104 0583 


2 


.00142 4501 


7 


.00130 3781 


2 


.00120 1923 


7 


.00111 4827 


2 


.00103 9501 


3 


.00142 2475 


8 


.00130 2083 


3 


.00120 0480 


8 


.00111 3586 


3 


.00103 8422 


4 


.00142 0455 


9 


.00130 0390 


4 


.00119 9041 


9 


.00111 2347 


4 


.00103 7344 


705 


.00141 8440 


770 


.00129 8701 


835 


,00119 7605 


900 


.00111 1111 


965 


.00103 6269 


6 


.00141 6431 


1 


.00129 7017 


6 


.00119 6172 


1 


.00110 9878 


6 


.00103 5197 


7 


.00141 4427 


2 


.00129 5337 


7 


.00119 4743 


2 


.00110 8647 


7 


.00103 4126 


8 


.00141 2429 


3 


.00129 3661 


8 


.00119 3317 


3 


.00110 7420 


8 


.00103 3058 


9 


.00141 0437 


4 


.00129 1990 


9 


.00119 1895 


4 


.00110 6195 


9 


.00103 1992 


710 


.00140 8451 


775 


.00129 0323 


840 


.00119 0476 


905 


.00110 4972 


970 


.00103 0928 


11 


.00140 6470 


6 


.00128 8660 


1 


.00118 9061 


6 


.00110 3753 


1 


.00102 9866 


12 


.00140 4494 


7 


.00128 7001 


2 


.00118 7648 


7 


.00110 2536 


2 


.00102 8807 


13 


.00140 2525 


8 


.00128 5347 


3 


.00118 6240 


8 


.00110 1322 


3 


.00102 7749 


14 


.00140 0560 


9 


.00128 3697 


4 


.00118 4834 


9 


.00110 0110 


4 


.00102 6694 


715 


.00139 8601 


780 


.00128 2051 


845 


.00118 3432 


910 


.00109 8901 


975 .'.00102 5641 



54 ^.—POWERS, ROOTS AND RECIPROCALS, 

11. — ^Reciprocals op Numbers 1 to 1000. — Concluded. 



No 


Reciprocal 


No. 


Reciprocal 


No. 


Reciprocal 


No. 


Reciprocal 


No 


. Reciprocal 


975 


.00102 5641 


980 


00102 04C8 


985 


00101 5228 


990 


.00101 0101 


995 .00100 5025 


6 


.00102 4590 


1 


00101 9368 


6 


00101 4199 


1 


.00100 9082 


6 .00100 4016 


7 


.00102 3541 


2 


00101 8330 


7 


00101 3171 


2 


.00100 8065 


7 .00100 3009 


8 


.00102 2495 


3 


00101 7294 


8 


00101 2146 


3 


.00100 7049 


8 .00100 2004 


9 


.00102 1450 


4 


00101 6260 


9 


00101 1122 


4 


.00100 6036 


9 .00100 1001 


980 


.00102 0408 


985 


00101 5228 


990 


00101 0101 


995 


.00100 5025 


1000 .00100 0000 


11a.— Reciprocals of Numbers 1001 to 1350. 




Recip- 




Recip 




Recip 




Recip 




Recip 




Recip- 




Recip- 


No. 


rocal, 


No. 


rocal. 


No. 


roca . 


No. 


rocal. 


No 


rocal. 


No 


rocal. 


No 


rocal. 




.000 




.000 




.000 




.000 


1200 


.000 




.000 


1300 


.000 


1000 




1050 


95238 


1100 


90909 


1150 


86957 


83333 


1250 


80000 


76923 


01 


'99900 


51 


95147 


01 


90827 


51 


86881 


01 


83264 


51 


79936 


01 


76864 


02 


99800 


52 


95057 


02 


90744 


52 


86806 


02 


83195 


62 


79872 


02 


76805 


03 


99701 


53 


94967 


03 


90662 


53 


86730 


03 


83126 


53 


79808 


03 


76746 


04 


99602 


54 


94877 


04 


90580 


54 


86655 


04 


83056 


54 


79745 


04 


76687 


1005 


99502 


1055 


94787 


1105 


90498 


1155 


86580 


1205 


82988 


1255 


79681 


1305 


76628 


06 


99404 


56 


94697 


06 


90416 


56 


86505 


06 


82919 


56 


79618 


06 


76570 


07 


99305 


57 


94607 


07 


90334 


57 


86430 


07 


82850 


57 


79554 


07 


76511 


08 


99206 


58 


94518 


08 


90253 


58 


86356 


08 


82781 


58 


79491 


08 


76453 


09 


99108 


59 


94429 


09 


90171 


59 


86281 


09 


82713 


59 


79428 


09 


76394 


1010 


99010 


1060 


94340 


1110 


90090 


1160 


86207 


1210 


82645 


1260 


79365 


1310 


76336 


11 


98912 


61 


94251 


11 


90009 


61 


86133 


11 


82576 


61 


79302 


11 


76278 


12 


98814 


62 


94162 


12 


89928 


62 


86059 


12 


82508 


62 


79239 


12 


76219 


13 


98717 


63 


94073 


13 


89847 


63 


85985 


13 


82440 


63 


79177 


13 


76161 


14 


98619 


64 


93985 


14 


89767 


64 


85911 


14 


82372 


64 


79114 


14 


76104 


1015 


98522 


1065 


93897 


1115 


89686 


1165 


85837 


1215 


82305 


1265 


79051 


1315 


76046 


16 


98428 


66 


93809 


16 


89606 


66 


85763 


16 


82237 


66 


78989 


16 


75988 


17 


98328 


67 


93720 


17 


89526 


67 


85690 


17 


82169 


67 


78927 


17 


75930 


18 


98232 


68 


93633 


18 


89445 


68 


85616 


18 


82102 


68 


78864 


18 


75873 


19 


98135 


69 


93545 


19 


89366 


69 


85543 


19 


82034 


69 


78802 


19 


75815 


1020 


98039 


1070 


93458 


1120 


89286 


1170 


85470 


1220 


81967 


1270 


78740 


1320 


75758 


21 


97943 


71 


93371 


21 


89206 


71 


85397 


21 


81900 


71 


78678 


21 


75700 


22 


97847 


72 


93284 


22 


89127 


72 


85324 


22 


81833 


72 


78616 


22 


75643 


23 


97752 


73 


93197 


23 


89047 


73 


85251 


23 


81766 


73 


78555 


23 


75586 


24 


97656 


74 


93110 


24 


88968 


74 


85179 


24 


81699 


74 


78493 


24 


75529 


1025 


97561 


1075 


93023 


1125 


88889 


1175 


85106 


1225 


81633 


1275 


78431 


1325 


75472 


26 


97466 


76 


92937 


26 


88810 


76 


85034 


26 


81566 


76 


78370 


26 


75415 


27 


97371 


77 


92851 


27 


88731 


77 


84962 


27 


81500 


77 


78309 


27 


75358 


28 


97276 


78 


92764 


28 


88652 


78 


84890 


28 


81433 


78 


78247 


28 


75301 


29 


97182 


79 


92678 


29 


88574 


79 


84818 


29 


81367 


79 


78186 


29 


75245 


1030 


97087 


1080 


92593 


1130 


88496 


1180 


84746 


1230 


81301 


1280 


78125 


1330 


75188 


31 


96993 


81 


92507- 


31 


88417 


81 


84674 


31 


81235 


81 


78064 


31 


75131 


32 


96899 


82 


92421 


32 


88339 


82 


84602 


32 


81169 


82 


78003 


32 


75075 


33 


96805 


83 


92336 


33 


88261 


83 


84531 


33 


81103 


83 


77942 


33 


75019 


34 


96712 


84 


92251 


34 


88183 


84 


84459 


34 


81037 


84 


77882 


34 


74963 


1035 


96618 


1085 


92166 


1135 


88106 


1185 


84388 


1235 


80972 


1285 


77821 


1335 


74906 


36 


96525 


86 


92081 


36 


88028 


86 


84317 


36 


80906 


86 


77760 


36 


74850 


37 


96432 


87 


91996 


37 


87951 


87 


84246 


37 


80841 


87 


77700 


37 


74794 


38 


96339 


88 


91912 


38 


87873 


88 


84175 


38 


80775 


88 


77640 


38 


74738 


39 


96246 


89 


91827 


39 


87796 


89 


84104 


39 


80710 


89 


77580 


39 


74683 


1040 


96154 


1090 


9174S 


1140 


87719 


1190 


84034 


1240 


80645 


1290 


77519 


1340 


74627 


41 


96061 


91 


91659 


41 


87642 


91 


83963 


41 


80580 


91 


77459 


41 


74571 


42 


95969 


92 


91575 


42 


87566 


92 


83893 


42 


80515 


92 


77399 


42 


74516 


43 


95877 


93 


914S1 


43 


87489 


93 


83822 


43 


80451 


93 


77340 


43 


74460 


44 


95785 


94 


91408 


44 


87413 


94 


83752 


44 


80386 


94 


77280 


44 


74405 


1045 


95694 


1095 


91324 


1145 


87336 


1195 


83682 


1245 


80321 


1295 


77220 


1345 


74349 


46 


95602 


96 


91241 


46 


87260 


96 


83612 


46 


80257 


96 


77160 


46 


74294 


47 


95511 


97 


91158 


47 


87184 


97 


83542 


47 


80192 


97 


77101 


47 


74239 


48 


95420 


98 


91075 


48 


87108 


98 


83472 


48 


80128 


98 


77042 


48 


74184 


49 


95329 


99 


90992 


49 


87032 


99 


83403 


49 


80064 


99 


76982 


49 


74129 


1050 


95238 


1100 


90909 


1150 


86957 


1200 


83333 


1250 


80000 


1300 


76923 


1350 


74074 



Ex.— Reciprocal of 1046 is 0.00095602. 



3.— PRACTICAL ARITHMETIC. 

PROPORTION. 

As Algebra is the shorthand of Mathematics so is Proportion the key to 
many of its operations. Indeed, all mathematical problems may be expressed 
in proportion. 

Ratio is an expression erf the relative magnitude of two quantities in 
the form of a fraction, either term of which may be considered the unit 
of measure. Thus, the ratio of the yard to the foot may be expressed as 
Y 
-= or y : F, in which case, however, Y and F must be considered in the 

same unit of measure, yard or foot. Every ratio implies proportion, as 

^=y=Y; F :l = Y :x\ etc. 

Proportion is the equality of two or more 
ratios, and it may be represented graphically 
by similar triangles, as in Fig. 1, from which 
may be obtained a variety of expressions, as 
follows: 

By proportional lengths (also illustrating con- 
tinued proportion): 

*P p Pi p-hpi P+p P+pi ^ . ^ . .. ^ , . ^ ., 



i?= A == 1 = 1±A = 12+9 ^ 12+3^ ^ 

16 12 4 12+4 16+12 16+4 ~ 16+12 + 4 




etc. (or inverted). 



P p-pi P-p P-Pi P-p-pj ^ , . ^ .. 

B = F-Ti -B--b = B:ib = B-b-b, ^ "'"• ^^' ^^^^^ted). 



12 9-3 12-9 12-3 12- 



16 12-4 16-12 16-4 16-12-4* 



etc. (or inverted). 



"Tj^ IT "^ r" ' ^^d 77 "^ T ^ ^^ may likewise be extended (or inverted). 
By proportional squares (also illustrating compound proportion): 

m : h'^ :/ji2 :: P2+52 :^2+62 : p.-^^b^^-^ or ^' : h^'^ = ^'If/ s ^I' + ^i". 

n^ p^ + 0^ 

202 : 152 : 52 :: 122+162 : 92+ 122 : 32+42 ; or ??^ : 52 = ^H | ]Z J 32 + 42. 

16^ y^ + 12^ 

In the above some of the ratios are compound ratios, hence the term 
compound proportion. 

Simple Proportion or single rule of three deals with two simple, equal 
ratios. Thus, 

Extreme mean mean extreme 

P ', B = p : b 

reads, "As P is to 5 so is p to 6." The first and last terms are called the 

extremes, and the middle terms the means. In any simple problem there will 

* From -^ =-r we have Pb = pB, whence — = -r t or — = — or n-= ■^•. etc. 
DO p o p p B tr 

55 





56 S.—PRACTICAL ARITHMETIC, 

be one unknown term which can be solved by applying the rule: The 
product of the means is equal to the product of the extremes. 

Problem. — If a train travels 280 miles in 7 
hours, how far will it travel in 5 hours? 

Solution. — From similar triangles there is ob- 
tained, 7:5 = 280 : x 
from which, applying the above rule, 
7 rx; = 5 X 280 

Fig. 2. or :*: - ^ ^" = 200 miles. Ans. 

Mean Proportional. — In any proportion where the two means are equa^ 
to each other, they are each said to be a me'an proportional between the 
extremes. In the accompanying geometrical figure, b, an ordinate to the 
circumference, is a mean proportional between a and c, com- 
posing the diameter, as, from similar triangles there will 
h c_=g^""v \ obtain the proportion 

Diameter. a '. b = h '. C 

Fig. 3. or 4 : 6 = 6 : 9 

whence 6 = \/4 X 9; or, the mean = Vproduct of extremes. 

Inverse Proportion. — This term is used in such problems as those where 
the rate of speed or velocity are compared, when the total time, distance, 
amount of work, etc., are given; as, the rate of work is inversely proportional 
to the time occupied in doing it; or, the speed of a train is inversely pro- 
portional to the time used in traveling a certain distance. 

Compound Proportion or double rule of three. — Compound proportion 
is merely simple proportion in which two or more of the ratios are com- 
pounded. 

Problem. — If a man saws 18 cords of wood in 12 days of 9 hours each, 
how many cords of wood can he saw in 15 days of 8 hours each? 

2 5 2 

Solution.- j^-^^ = ^^^ ; whence x = ^^ x ^ = ^^ ^^^^,,. 

A little thought coupled with a graphical conception of the problem 
will always point to a correct grouping of the terms. 

PERMUTATION AND COMBINATION. 

Permutation. — The number of different ways in which any number, N, 
of objects may be arranged in line or counted, is equal to the product of all 
the numbers from 1 to N. Thus, 3 objects as a, b and c may be arranged 
1X2X3=6 different ways in line — abc, acb. bac, bca, cab, cba; 7 objects 
may be arranged 1X2X3X4X5X6X7 = 5040 different ways, and 
so on. 

Combination. — The number of different groups each composed of n 
objects (no two groups to be composed of the same objects) which can be 
formed separately from any number N, of objects, is equal to the product 
of all the numbers from (N — « + 1) to A^, divided by the product of all 
the numbers from 1 to n. (It must be remembered that the objects in each 
group are not permuted.) Thus, 6 objects as a, b, c, d, e and / may be 

5X6 
arranged in groups of 2, =15 different ways; thus. 

1 X ^ 

ab, ac, ad, ae, af, 

be, bd, be, bf, 

cd, ce, cf, 

de, df, 

ef. 

Similarly, 8 objects may be arranged in groups of 4, 

5X6X7X8 



1X2X3X4 



= 70 different ways. 



ALLIGATION AND PROGRESSION. 



57 



ALLIGATION. 

The average cost of a mixture containing various quantities of ingre- 
dients at different unit prices may be obtained 
by dividing the total cost of the mixture by the 
total quantity. Let it be required to find the 
average cost of a mixture as follows: 

10 bbls. cement @ $2.00 = $ 20. 

20 '• " 2.25 = 45. 

30 " " 2.50 = 75. 



<- jfgsp 


— - 


i 


. 


< JfZ.C?0 ^> 

1 

5 


1 


s 


r<- -S2.3H --^ 


^ 


9 



.-.60 = $140. Fig. 4. 

Average price per bbl. = ^- = $2. 33 J. Ans. 
This problem is analogous to finding the position of the center of gravity 
and resultant of a system of concentrated loads, in Mechanics. 

PROGRESSION. 
Arithmetical Progression. — An arithmetical series or progression con- 
sists of a series of terms which uniformly increase or decrease by a common 
constant difference, d. Thus, 
Ascending: 
{d = d).- a, a + d, a+2d, a4-3J, a+4cf, ...; 
(fi=3).- 1, 4, 7, 10, 13, 



d = \). 



2.5.-2, -1.5, 



-.5. 



Descending: 
a, a-d, a-2d, a- M, a- Ad, 
7, 5. 3, 1, - 1, 

f. i 0. -h - h 



(d=-d).- 
(d=-2).- 
(J=-i).- 
are arithmetical series. 

As the difference is constant, any arithmetical series may be represented 
by the equation of a straight line — 

3; = w ^ + c, in the language of Analytic Geometry; 
or 5 = c/ (n— 1)+ a, in the language of Arithmetical Progression; 
in which c = a = the value of the first term (placed at axis Y—Y); 

m = d = the common dift'erence, + or — , between adjacent terms; 
X = n— I = the number of any term considered from (not counting) 

the first term; 
y = s = the value of any term x = n — 1; 
n = the number of any term, counting the first term. 
Fig. 5 illustrates an ascending series, and when 
the first term is a positive quantity. The equation, 
however, l:KDlds good for any case by using the quan- 
tities algebraically. 

Problem 1. — Find the number of the term whose 
value is 10, in a series whose common difference is)^-^ 
i and whose first term is 2. 

Solution. — From y = m x -h c there is obtained, 
10 = i:^ + 2. 
.*. a; = 24 = » - 1. 
or « = 25. Ans. 

Problem 2. — Insert 4 terms between the numbers 3 and 28 and find the 
common difference. 

Solution. — From y = m x + c 

28 = w 5 + 3 (If 4 terms are inserted, 28 = 6th term 
.-. X = 5.) 

2Q 3 

whence m = — = 5 = common difference. Ans. 


An arithmetical mean between two quantities = one-half their sum. 

Geometrical Progression. — A geometrical series or progression consists 
of a series of terms which increase or decrease by a common constant 







1 




i 


1 

y 









Fig. 5. 



factor, f. Thus, 

Ascending: 

(f == r).— a, ar, ar^, ar^, ar^, . . . ; 


Decending: 
/. 1\ a a a a 
(/=7)-- "'-'-^r^--^' 


(/ = 2).- 3, 6,12, 24, 48, ...; 


(/ = J). - 4. 2, 1. J. i. 



are geometrical series. 



58 2.— PRACTICAL ARITHMETIC. 



I 




Any geometrical progression may be represented by a curve whose 
equation is — 

y = cf^, in the language of Analytic Geometry; 

or s = ap^~ \ in the language of Geometrical Progression; 
14 which, c = a == the value of the first term (placed at axis Y—Y); 
f = the common factor or ratio between the adjacent terms; 
X = n—1 = number of any term considered from (not including) 

the first term; 
y = s = the value of any term x = n — 1 ; 

n = the number of any term, counting the first term. 

Fig. 6 illustrates an increasing series, and when 
the first term is a positive quantity; but by using 
the quantities algebraically, the equation holds good 
for any case. 

Problem 1. — Find the value of the 6th term of 
an increasing series whose 1st term is 3, and the com- 
►xmon factor 2. 

Solution. — From 
y = c f^, there is obtained, 
Fig. 6. y (= 5) = 3 X 25 = 3 X 32. (x = n - 1 = 5.) 

.'. s = 96. Ans. 

Problem 2. — Insert 3 terms between the numbers 2 and 10 J, and find 
the common factor. 

Solution. — From y == c f^ there is obtained. 

101 = 2p (If 3 terms are inserted, 10| = 4th term 
.*. X = 4.) 

/^ = ^ = 5iV = fi 
.-. / = I = li Ans. 
Note. — ^The latter part of the above solution may also be performed by 
logarithms. Thus, 

fi =1P == 5.0625 log 5.0625 = 0.704365 (Divide this by 4.) 

Ans. / = 1.5 0.176091 (Find number corresponding.) 
A geometrical mean between two numbers = the square root of their 
product (see Fig. 3, page 56) . 

Compound interest is a good illustration of geometrical progression. 

PERCENTAGE, INTEREST AND DISCOUNT. 

Percentage. — Per cent means hundredths, and rate per cent means any 
given number of hundredths. Thus, 5 per cent, or 5%, means .05, or VioOf 
in which 5 is the rate. It may also be expressed in true ratio, 5 : 100, 
meaning 5 parts of the 100, both terms being of the same denomination. 

To reduce any rate, expressed in two denominations, to a rate per cent, 
one term must first be reduced to the denomination of the other so that a 
true ratio may be expressed. Thus, a rate of grade in feet per mile, as 
52 T% feet per mile, may be expressed in ratio 52 fQ : 5280, as there are 
5280 feet in a mile. Clearly, this is equal to a Vioo. or 1% grade. 

Simple Interest is percentage in which the element of time has to be 
considered. The sum placed at interest is called the principal, and the 
principal plus the interest is called the amount. Simple interest differs 
from compound interest in that the principal is not allowed to increase from 
time to time by its own interest. 

The rate of interest is the rate per cent for one year. There are two 
methods used in calculating interest: 
(o.) The common method, in which a year is considered to be divided into 

12 months of 30 days each. 
(Jb.) The exact method, in which the year is divided into 365 (or 366 if leap 
year) days. 

In either case. 

Simple Interest = Principal ($) X Rate % X Time (years). 

The Common Method. — Table 1, page 60, shows the simple interest on 
various principals ($1. — $1,000.) at various rates (3^% — 7%) for time 
in days and months up to one year. Clearly, by combination and 
factoring, the interest on any principal, at any rate and for any time, may 
readily be obtained. Thus, the interest on $1253, at 4^% for 5 months, 
11 days, is calculated as follows: 



PERCENTAGE, INTEREST AND DISCOUNT, 59 

5 mos. lOd. Id. 



on $1000 (1000 X 1) 


= 18.75 


1.25 


t 


200 ( 200 X 1) 
50 ( 50 X 1) 


= 3.75 


.25 


"^^ 


= .938 


.062 


hE'^ 


3 


= .056 


.004 


Ho 

1— 1 



Therefore, total interest =23.494 + 1.566 + .157 = $25.22. Ana. 

This may be calculated in the ordinary manner, as folJbws: 
(Principal) (Rate %) (Time) 

Interest = 1253 X .045 X jf = $25.22. Ans. 

As the interest on any amount is directly proportional to the rate, 
some prefer to use a round rate like 6%, 6 being a factor of 12 (month^) and 
of 30 (days), for a first result, and then factor this result for the required 
rate. Thus, in the above, 

12.53 
Interest on $1253 at 6% for 5 mos. = 1253 X 2|% =-{ 12.53 



'-1 



6.265 
10 days = 1253 X i% =' 2.089 
1 " = 1^0 above = .209 



At 6% 33.62 

Deduct I 8.40 



At 41%. $25.22 

Ans. 

The Exact Method. — ^This is used by various large financial concerns 
in dealing with each other and where large amoimts are involved. The 
time is the ratio of the number of days for which the principal has been 
loaned, to the number of days in the year. Thus, if the loan is for 217 
days the time will be 217/365 of a year; or, if the 29th of February is included, 
it will be 2i7/3gQ of a year. 

Table 2, page 61, gives the number of days from one date to another cover- 
ing a period of two ordinary years of 365 days each. If one of the years is a 
leap year add one day to each number of days after February 28th. Thus, 
if the interest is to be calculated from February 15th of one year to January 

375 4g 329 

10th of the following year, the time would be - — ^r-^-z = -^rrrz of a year. 

000 000 

If the first year is a leap year the time would be ^^^/zee of a year, adding one 
to each term of the ratio. For example, the interest on $450. from Feb. 15, 
1908, (leap year) to Jan. 10, 1909, at 5% = $450. X .05 X ^^^/sm^ 
$20.29. Ans. 

Discount. — ^The fundamental principle of discount is " money off for 
cash," but most mercantile houses will discount their trade catalogue prices 
to their regular customers on short time payments. 

A certain customer may be favored with a discount of 20%, which 
means that the cost to him will be 80% of the catalogue price; another 
customer, more favored or on account of heavier purchases, may receive a 
discount of "20 and 10"%, making the cost .80 X .90 = 72% of the list 
price; while a third may receive a discount of " 20, 10 and 5" which would 
make the cost .80 X .90 X .95 = 68 Vio %• 

Bank discount is the equivalent interest on the face of a note up to the 
time of its maturity. 

True discount is the equivalent interest on the present worth of a note, 
or principal which, at maturity, will amount to the face of the note. Thus, 
Present worth + interest on same ( = true discount) _ $1 4- interest on same ^ 

Amount (= face of note) Amount of $1 and int. ' 

^ . .-, Face of note 

Or, present worth = -r 7— r^Fi T"^— : 7-: : — ^i-- 

Amount of $1 and mterest, to maturity 

Whence the true discount = face of note — present worth. 

Compound Interest. — The simple interest on any orincipal loaned is due 
annually. But by a special agreement it may become due semi-annually, 
quarterly, or for any other period. If not paid, it is added periodically to 



60 



3.— PRACTICAL ARITHMETIC. 



the principal for a new principal drawing interest; hence, compound 
interest. (See Table 3. page 62.) 

The amount (principal and interest) due at the end of any number of 
years, n, interest payable annually, may be expressed as follows: 
Amount = Principal (1 + Rate %)^. 

If the number of years is large the result is easily obtained by the use 
of logarithms. Further, the above forrriula may represent any general case 
by considering n the number of periods (as semi-annual or quarterly 
periods) of compounding — the proportionate rate per cent to be used for 
that period. 

1. — Simple Interest Table.* 



1 


11 


Time. 


1 
Year 


6 Mo. 


5 Mo. 


4 Mo. 


3 Mo. 


2 Mo. 


IMo. 


20 d. 


10 d. 


1 d. 




$1 


.048 


.0200 


.01666^6 


.013^3 


.01000 


.0066''6 


.00333^3 


.0022^2 


.ooiir'i 


.ooouri 




2 


.080 


.0400 


.03333^3 


.026^6 


.02000 


.0133^3 


.00686^6 


.0044^4 


.00222^2 


.000222^2 




3 


.120 


.0600 


.05000 


.040 


.03000 


.0200 


.01000 


.0066^6 


.00333^3 


.000333^3 




4 


.160 


.0800 


.06666^6 


.053''3 


.04000 


.0266^6 


.01333^3 


.0088^8 


.00444^4 


.000444^4 


4% 


5 


.200 


.1000 


.08333''3 


.066^8 


.05000 


.0333^3 


.01866~^6 


.0111^1 


.00555^5 


.000555^5 




6 


.240 


.1200 


.10000 


.080 


.06000 


.0400 


.02000 


.0133^3 


.00666^6 


.000666^6 




7 


.280 


.1400 


.11666^8 


.093^3 


.07000 


.0466^6 


.02333^3 


.0155^5 


.00777^ 


.000777^7 




8 


.320 


.1600 


.13333^3 


.106^6 


.08000 


.0533^3 


.02666^6 


.0177^7 


.00888^8 


.000888^8 




9 


.360 


.1800 


.15000 


.120 


.09000 


.0600 


.03000 


.0200 


.01000 


.001000 




$1 


.045 


.0225 


.01875 


015 


.01125 


.0075 


.00375 


.0025 


.00125 


.000125 




2 


.090 


.0450 


.03750 


.030 


.02250 


.0150 


.00750 


.0050 


.00250 


.000250 




3 


.135 


.0675 


.05625 


.045 


.03375 


.0225 


.01125 


.0075 


.00375 


.000375 




4 


.180 


.0900 


.07500 


.060 


.04500 


.0300 


.01500 


.0100 


.00500 


.000500 


ii% 


5 


.225 


.1125 


.09375 


.075 


.05625 


.0375 


.01875 


.0125 


.00625 


.000625 




6 


.270 


.1350 


.11250 


.090 


.06750 


.0450 


.02250 


.0150 


.00750 


.000750 




7 


.315 


.1575 


.13125 


.105 


.07875 


.0525 


.02625 


.0175" 


.00875 


.000875 




8 


.360 


.1800 


.15000 


.120 


.09000 


.0600 


.03000 


.0200 


.01000 


.001000 




9 


.405 


2025 


.16875 


.135 


.10125 


0675 


.03375 


.0225 


01125 


.001125 




$1 


.050 


.0250 


.02083^3 


.016^6 


.01250 


.0083^3 


.00416^6 


.0027^7 


.00138^8 


.000138^8 




2 


.100 


.0500 


.04166^6 


.033^^3 


.02500 


.0166^6 


.00833^3 


.0055^5 


.00277^ 


.000277^7 




3 


.150 


.0750 


.06250 


.050 


.03750 


.0250 


.01250 


.0083^3 


.00416^6 


.000416''6 




4 


.200 


.1000 


.08333^3 


.066^6 


.05000 


.0333^3 


.01666^6 


.0111^1 


.00555^5 


.000555^5 


5% 


5 


.250 


.1250 


.10416^6 


.083^3 


.06250 


.0416^6 


.02083^3 


.0138^8 


.00694^4 


.000694^4 




6 


.300 


.1500 


.12500 


.100 


.07500 


.0500 


.02500 


.0166^6 


.00833^3 


.000833^3 




7 


.350 


.1750 


.14583^3 


.116^6 


.08750 


.0583^3 


.02916^6 


.0194^4 


.00972^2 


.000972^2 




8 


.400 


.2000 


.16666^6 


.133^3 


.10000 


.0666^6 


.03333^3 


0222^2 


.oiiin 


.001111^1 




9 


.450 


.2250 


.18750 


.150 


.11250 


.0750 


.03750 


.0250 


.01250 


.001250 




$1 


.060 


.0300 


.02500 


.010 


.01500 


.0100 


.00500 


.0033''3 


.00166^6 


.000166^6 




2 


.120 


.0600 


.05000 


.040 


.03000 


.0200 


.01000 


.0066^6 


.00333^3 


.000333^3 




3 


.180 


.0900 


.07500 


.060 


.04500 


.0300 


.01500 


.0100 


.00500 


.000500 




4 


.240 


.1200 


.10000 


.080 


.06000 


.0400 


.02000 


.0133^3 


.00866^6 


.000666^6 


6% 


5 


.300 


.1500 


.12500 


.100 


.07500 


.0500 


.02500 


.0166^6 


.00833''3 


.000833^3 




6 


.360 


.1800 


.15000 


.120 


.09000 


.0600 


.03000 


.0200 


.01000 


.001000 




7 


.420 


.2100 


.17500 


.140 


.10500 


.0700 


.03500 


.0233^3 


.01166^6 


.001166^6 




8 


.480 


.2400 


.20000 


.160 


.12000 


.0800 


.04000 


.0266^6 


.01333^3 


.001333^3 




9 


.540 


.2700 


.22500 


.180 


.13500 


.0900 


.04500 


.0300 


.01500 


.001500 




$1 


.070 


.0350 


.02916^6 


.023^3 


.01750 


.0116^6 


.00583^3 


.0038''8 


.00194^4 


.000194^4 




2 


.140 


.0700 


.05833^3 


.046^6 


.03750 


.0233^3 


.01166^6 


.0077^7 


.00388''8 


.000388''8 




3 


.210 


.1050 


.087 50 


.070 


.05250 


.0350 


.01750 


.0116^6 


.00583^3 


.000583^3 




4 


.280 


.1400 


. 11666^^6 


.093^3 


.07000 


.0466^6 


.02333^3 


.0155^5 


.00777^7 


.000777^7 


7% 


5 


.350 


.1750 


. 14583^3 


.116^6 


.08750 


.0583^3 


.02916^6 


.0194^4 


.00972^2 


.000972^2 




6 


.420 


.2100 


.17500 


.140 


.10500 


.0700 


.03500 


.0233^3 


.01166^8 


.001166^6 




7 


.490 


.2450 


.20416^6 


.163\3 


.12250 


.0816^6 


.04083^3 


.0272^2 


.0138n 


.00136n 




8 


.560 


.2800 


.23333^3 


.186^6 


.14000 


.0933^3 


.04666^6 


.03 in 


.01555^5 


.001555''5 




9 


.630 


.3150 


.26250 


.210 


.15750 


.1050 


.05250 


0350 


.01750 


=001750 



* Note that all repeating decimals may be extended indefinitely. Thus, 
the interest on $1.00 at 4% for 4 months is given as .013^3 or IJ cents, 
because the decimal .013''3= .01333333...; hence the interest on $1,000,- 
000, at the same rate and for the same time, is $1 3, 333. 3 3 i. Decimals which 
are not repeating decimals are exact. 



EQUATION OF PAYMENTS, 



01 



Example. — Find the amount of $600. 
compounded at 2% semi-annually for 6 
years (12 periods). 



12 



log. 
X log. 



Solution. 
600. = 2.7781513 
1.02 = 0.1032024 



Ans. $760.95 . . . 2.8813537 
Clearly, the process of calculating compound interest is simply geomet- 
rical progression, in which 1 + Rate % is a constant factor between suc- 
cessive terms. 

EQUATION OF PAYMENTS. 

Like Alligation, this is a "center of gravity" problem. It consists in 
finding the average time when a single payment can be made to cancel 
several notes, bearing the same rate of interest, which fall due on 
different dates. 

Problem. — A holds B's notes as follows: $600 due in 1 month; $700 
due in 3 months; $400 due in 4 months; and $300 due in 5 months — all 
bearing the same rate of interest. At what time can B make a single pay- 
ment of the whole amount, $2000, to cancel the obligation equitably? 
Solution.— 600 X 1 = 600. 

700 X 3 = 2100. 
400 X 4 = 1600. 
300 X 5 = 1500. 



5800. 



2000 X X 
.*. Average time x = onon ^ ^*^ months == 2 m., 27 d. Ans. 

2, — ^Table For Finding Number of Days Between Any Two Dates 
IN Two Consecutive Years.* 



First Year. 



90 



213 
214 
215 
216 
217 
218 
219 
220 
221 
222 
223 
224 
225 
226 
227 
228 
229 
230 
231 
232 
233 
234 
235 
236 
237 
238 
239 
240 
241 
242 
243 



244 
245 
246 
247 
248 
249 
250 
251 
252 
253 
254 
255 
256 
257 
258 
259 
260 
261 
262 
263 
264 
265 
266 
267 
268 
269 
270 
271 
272 
273 



274 
275 
276 
277 
278 
279 
280 
281 
282 
283 
284 
285 
286 
287 
288 
289 
290 
291 
292 
293 
294 
295 
296 
297 
298 
299 
300 
301 
302 
303 
304 



305 
306 
307 
308 
309 
310 
311 
312 
313 
314 
315 
316 
317 
318 
319 
320 
321 
322 
323 
324 
325 
326 
327 
328 
329 
330 
33-1 
332 
333 
334 



Second Year. 



366 
367 
368 
369 
370 
371 
372 
373 
374 
10375 



376 
377 
378 
379 
380 
381 
382 
383 
384 
385 
386 
387 
388 
389 
390 
391 
392 
393 
394 
395 
396 



578 
579 
580 
581 
582 
583 
584 
585 
586 
587 
588 
589 
590 
591 
592 
593 
594 
595 
596 
597 
598 
599 
600 
601 
602 
603 
604 
605 
606 
607 
608 



m 



609 
610 
611 
612 
613 
614 
615 
616 
617 
618 
619 
620 
621 
622 
623 
624 
625 
626 
627 
628 
629 
630 
631 
632 
633 
634 
635 
636 
637 
638 



639 670 

640 671 



645 676 



646 
647 
648 
649 
650 
651 
652 
653 
654 
655 
656 
657 
658 
659 
660 
661 
662 
663 
664 
665 
666 
667 
668 
669 



700 
701 
702 
703 
704 
705 
706 
707 
708 
709 
710 
711 
712 
713 
714 
715 
716 
717 
718 
719 
720 
721 
722 
723 
724 
725 
726 
727 
728 
729 
730 



* Subtract the number opposite the first date from the number opposite 
thie last. If the 29th of February is included, add one day. 



02 



3,— PRACTICAL ARITHMETIC. 



3. — Compound Interest Table. 
Amoiint of $1. at compound interest for periods 1 to 50 at various 

rates. 



'periodic 











♦Periodic Rates. 


\ 




Periods. 
















n. 


2% 


3% 


H% 


4% 


4i% 


5% ^ 


6% 


7% 


1 


1.02000 


1.03000 


1.03500 


1.04000 


1.04500 


1.05000 


1.06000 


1.07000 


2 


1.04040 


1.06090 


1.07123 


1.08160 


1.09203 


1.10250 


1.12360 


1.14490 


3 


1.06121 


1.09273 


1.10872 


1.12486 


1.14117 


1.15763 


1.19102 


1.22504 


4 


1.08243 


1.12551 


1.14752 


1.16986 


1.19252 


1.21551 


1.26248 


1.31080 


6 


1.10408 


1.15927 


1.18769 


1.21665 


1.24618 


1.27628 


1.33823 


1.40255 


6 


1.12616 


1.19405 


1.22926 


1.26532 


1.30226 


1.34010 


1.41852 


1.50073 


7 


1.14869 


1 :22987 


1.27228 


1.31593 


1.36086 


1.40710 


1.50363 


1.60578 


8 


1.17166 


1.26677 


1.31681 


1.36857 


1.42210 


1.47746 


1.59385 


1.71819 


9 


1.19509 


1.30477 


1.36290 


1.42331 


1.48610 


1.55133 


1.68948 


1.83846 


10 


1.21899 


1.34392 


1.41060 


1.48024 


1.55297 


1.62889 


1.79085 


1.96715 


n 


1.24337 


1.38423 


1.45997 


1.53945 


1.62285 


1.71034 


1.89830 


2.10485 


12 


1.26824 


1.42576 


1.51107 


1.60103 


1.69588 


1.79586 


2.01220 


2.25219 


13 


1.29361 


1.46853 


1.56396 


1.66507 


1.77220 


1.88565 


2.13293 


2.40985 


14 


1.31948 


1.51259 


1.61870 


1.73168 


1.85194 


1.97993 


» 2. 26090 


2.57853 


15 


1.34587 


1.55797 


1.67535 


1.80094 


1.93528 


2.07893- 


2.39656 


2.75903 


16 


1.37279 


1.60471 


1.73399 


1.87298 


2.02237 


2.18287 


2.54035 


2.95216 


17 


1.40024 


1.65285 


1.79468 


1.94790 


2.11338, 


2.29202 


2.69277 


3.15882 


18 


1.42825 


1.70243 


1.85749 


2.02582 


2.20848 


2.40662 


2.85434 


3 37993 


19 


1.45681 


1.75351 


1.92250 


2.10685 


2.30786 


2.526§5 


3.02560 


3.61653 


20 


1.48595 


1.80611 


1.98979 


2.19112 


2.41171 


2.65330 


3.20714 


3.86968 


21 


1.51567 


1.86029 


2.05943 


2.27876 


2.52024 


2.78596* 


3.39957 


4.14057 


22 


1.54598 


1.91610 


2.13151 


2.36991 


2.63365 


2.92523 


3.60354 


4.43041 


23 


1.57690 


1.97358 


2.20611 


2.46471 


2.75217 


3.07152 


3.81976 


4.74054 


24 


1.60844 


2.03279 


2.28332 


2.56330 


2.87602 


3.22510 


4.04894 


5.07237 


25 


1.64061 


2.09378 


2.36324 


2.66583 


3.00544 


3.38635 


4.29188 


5.42744 


26 


1.67342 


2.15659 


2.44595 


2.77246 


3.14068 


3.55567 


4.54939 


5.80736 


27 


1.70689 


2.22129 


2.53156 


2.88336 


3.28201 


3.73346 


4.82224 


6.21388 


28 


1.74103 


2.28792 


2.62016 


2.99870 


3.42970 


3.92013 


5.11170 


6.64885 


29 


1.77585 


2.35656 


2.71187 


3.11864 


3.58406 


4.11614 


5.41840 


7.11427 


30 


1.81134 


2.42726 


2.80672 


3.24339 


3.74532 


4.32194 


5.74351 


7.61227 


31 


1.84759 


2.50008 


2.90501 


3.37312 


3.91386 


4.53804 


6.08812 


8.14513 


32 


1.88454 


2.57508 


3.00670 


3.50805 


4.08998 


4.76494 


6.45340 


8.71529 


33 


1.92224 


2.65233 


3.11193 


3.64837 


4.27403 


5.00319 


6.8^061 


9.32536 


34 


1.96068 


2.73190 


3.22085 


3.79430 


4.46637 


5.25335 


7.25115 


9.97813 


35 


1.99989 


2.81386 


3.33358 


3.94608 


4.66735 


5.51600 


7.68611 


10.6766 


36 


2.03989 


2.89827 


3.45025 


4.10392 


4.87738 


5.79182 


8.14728 


11.4240 


37 


2.08069 


2.98518 


3.57101 


4.26806 


5.09686 


6.08141 


8.63611 


12.2236 


38 


2.12230 


3.07478 


3.69599 


4.43880 


5.32618 


6.38548 


9.15428 


13.0793 


39 


2.16475 


3.16702 


3.82535 


4.61635 


5.56590 


6.70475 


9.70354 


13.9948 


40 


2.20801 


3.26203 


3.95924 


4.80100 


5.81637 


7.03999 


10.2855 


14.9745 


41 


2.25221 


3.35989 


4.09781 


4.99306 


6.07811 


7.39199 


10.9029 


16.0227 


42 


2.29725 


3.46069 


4.24124 


5.19276 


6.35162 


7.76159 


11.5571 


17.1443 


43 


2.34320 


3.56451 


4.38968 


5.40047 


6.63744 


8.14967 


12.2505 


18.3444 


44 


2.39006 


3.67144 


4.54332 


5.61649 


6.93613 


8.55715 


12.9855 


19.6285 


45 


2.43786 


3.78159 


4.70233 


5.84115 


7.24826 


8.98504 


13.7647 


21.0025 


46 


2.48662 


3.89503 


4.86692 


6.07480 


7.57443 


9.43426 


14.5906 


22.4727 


47 


2.53635 


4.01188 


5.03726 


6.31779 


7.91528 


9.90597 


15.4660 


24.0458 


48 


2.58708 


4.13224 


5.21356 


6.57050 


8.27146 


10.4013 


16.3939 


25.7290 


49 


2.63882 


4.25621 


5.39604 


6.83330 


8.64368 


10.9213 


17.3776 


27.5300 


50 


2.69160 


4.38389 


5.58491 


7.10665 


9.03265 


11.4674 


18.4202 


29.4571 



* Periods may be annual, semi-annual or quarterly, etc. Periodic rates 
are proportioned to the length of the period. Thus, 4% annual = 2% semi- 
annual rate. For explanation of table, see top of page 60. 



ANNUITIES— SINKING FUND, 63 

PARTIAL PAYMENTS. 

In canceling notes by partial payments the following is the rule in 
United States law: 

Whenever the payment or payments equal or exceed the interest a new 
principal shall be formed by adding the interest and deducting the payment 
or payments. That is, the principal cannot be reduced without first can- 
celing the interest. 

A common method, however, is to consider the debt as the original 
principal together with its accumulated interest down to the date of settle- 
ment; this to be canceled by the various payments (considered as separate 
principals) together with their accumulated interest, down to the same 
date. 

ANNUITIES. 

Specified payment of money amounting to a fixed sum each year is 
called an ajinnity:— Certain annuity, when payments extend over certain 
definite periods; Contingent annuity, when payments are contingent on 
certain events; Life annuity, when payments are for life of one or more 
persons; Deferred annuity, when payments begin at some future time or 
event. 

The value of an annuity may be reduced to its — 
Final Value, or amount of all payments. at compound interest to the en4 of 

the annuity; 
Initial value, or equivalent principal which, at compound interest for the 

life of the annuity, would amount to its final value; 
Present value, or equivalent principal which, at compound interest for the 

balance of the life of the annuity, would amount to the value of 

future payments at compound interest to the end of the annuity. 

The fundamental equations which enter into the conversion of annuity 
in one form to its equivalent in another form, are those of geometrical 
progression, in general, and compound interest in particular. 

Final Value of Annuity. — ^The final value of an annuity is directly pro- 
portional to the value of the annual payment. Hence, from a table calcu- 
lated on the basis of an annuity of $1, at different rates of interest and for 
any number of years, the final value of any annuity at the same rate and 
for the same time may be obtained by multiplying the value in the table 
by the annuity. 

Table 4, following, gives the final values of an annuity of $1 at com- 
pound interest rates of 2, 3, 3^, 4, 4^, 5, 6 and 7 per cent up to 50 years. It 
is calculated from the formula: 

-r,. 1 1 (1 + Rate %y - 1 

Fmal value == -^^ -p ^ ' ; 

Rate % 

in which x = the number of years = the number of payments, the final 

value being calculated up to the time of each payment, and also includes 

that payment. 

Present or Initial Value of Annuity. — ^The present value of an annuity 
may be considered the initial value of that part of the annuity comprising 
future payments, the date beginning with the first of these payments. 
Hence the term " Present value " will be used in the broadest sense. Further, 
it is the amount of capital or the principal which, if placed at interest, 
would exactly furnish the annuities for the specified time; that is, the 
principal would be depleted at the end of that time. 

For the same time and at the same rate of interest, the present value 
is directly proportional to the annuity (and to the final value). Hence, in 
Table 5, page 65, to find the present value of any annuity, as $1000, 
multiply the tabular number by that annuity (1000) . The table is calculated 
from the formula: 

Present value = ^^ — =r— — 7=-^ — ; 

Rate % 

in which x = the number of years to run = the number of payments to be 
made, payments being made at the end of each year. 

Sinking Fund. — An annuity may be applied as a sinking fund to cancel 
a debt payable at some future time, as in the case of bonds maturing in, 
say, 20, 30 or 60 years. A plant may thus be made to pay for itself by 



64 



3.— PRACTICAL ARITHMETIC. 



4. — ^FiNAL Value of Annuity op $1 in From 1 to 50 Years or Periods, 









Yearly or Periodic Rates. 






Periods. 


















X 


2% 


3% 


3i% 


4% 


4^% 


5% 


6% 


7% 


1 


1. 


1. 


1. 


1. 


1. 


1. 


1. 


1. 


2 


2.02000 


2.03000 


2.03500 


2.04000 


2.04500 


2.05000 


2.06000 


2,07000 


3 


3.06040 


3.09090 


3.10623 


3.12160 


3.13703 


3.15250 


3.18360 


3.21490 


4 


4.12161 


4.18363 


4.21495 


4.24646 


4.27820 


4.31012 


4.37462 


4.43994 


5 


5.20404 


5.30913 


5.36247 


5.41632 


5.47072 


5.52563 


5.63710 


5.75074 


6 


6.30812 


6.46840 


6.55016 


6.63297 


6.71690 


6.80191 


6.97533 


7.15329 


7 


7.43428 


7.66245 


7.77942 


7.89829 


•8.01916 


8.14201 


8.39385 


8.65402 


8 


8.58297 


8.89232 


9.05170 


9.21422 


9.38002 


9.54911 


9.89748 


10.2598 


9 


9.75463 


10.1591 


10.3685 


10.5828 


10.8021 


11.0266 


11.4913 


11.9780 


10 


10.9497 


11.4638 


11.7313 


12.0061 


12.2882 


12.5779 


13.1808 


13.8165 


11 


12.1687 


12.8078 


13.1419 


13.4863 


13.8412 


14.2068 


14.9716 


15.7837 


12 


13.4121 


14.1920 


14.6019 


15.0258 


15.4640 


15.9171 


16.8700 


17.8885 


13 


14.6803 


15.6178 


16.1129 


16.6268 


17.1599 


17.7130 


18.8822 


20.1407 


14 


15.9739 


17.0863 


17.6769 


18.2919 


18.9321 


19.5986 


21.0151 


22.5505 


16 


17.2934 


18.5989 


19.2956 


20.0236 


20.7840 


21.5786 


23.2760 


25.1291 


16 


18.6393 


20.1569 


20.9709 


21.8245 


22.7193 


23.6576 


25.6725 


27.8881 


17 


20.0121 


21.7616 


22.7049 


23.6975 


24.7417 


25.8405 


28.2129 


30.8403 


18 


21.4123 


23.4144 


24.4996 


25.6454 


26.8551 


28.1326 


30.9057 


33.9991 


19 


22.8406 


25.1168 


26.3571 


27.6712 


29.0636 


30.5389 


33.7601 


37.3790 


20 


24.2974 


26.8703 


28.2795 


29.7780 


31.3714 


33.0660 


36.7857 


40.9956 


21 


25.7833 


28.6765 


30.2693 


31.9691 


33.7831 


35.7193 


39.9927 


44.8652 


22 


27.2990 


30.5368 


32.3287 


34.2479 


36.3034 


38.5052 


43.3923 


49.0058 


23 


28.8450 


32.4529 


34.4602 


36.6178 


38.9370 


41.4305 


46.9960 


53.4362 


24 


30.4219 


34.4265 


36.6663 


39.0825 


41.6892 


44.5020 


50.8157 


58.1767 


25 


32.0303 


36.4592 


38.9497 


41.6458 


44.5652 


47.7271 


54.8647 


63.2491 


26 


33.6709 


38.5530 


41.3129 


44.3116 


47.5706 


51.1134 


59.1565 


68.6766 


27 


35.3443 


40.7096 


43.7589 


47.0841 


50.7113 


56.6691 


63.7060 


74.4839 


28 


37.0513 


42.9309 


46.2904 


49.9674 


53.9933 


60.4025 


68.5282 


80.6978 


29 


38.7923 


45.2188 


48.9106 


52.9661 


57.4230 


64.3226 


73.6400 


87.3466 


30 


40.5682 


47.5753 


51.6224 


56.0847 


61.0071 


66.4389 


79.0584 


94.4610 


31 


42.3796 


50.0026 


54.4291 


59.3281 


64.7524 


70.7608 


84.8019 


102.073 


32 


44.2272 


52.5026 


57.3341 


62.7012 


68.6663 


75.2989 


90.8900 


110.218 


33 


46.1117 


55.0777 


60.3408 


66.2093 


72.7563 


80.0638 


97.3434 


118.934 


34 


48.0340 


57.7300 


63.4528 


69.8576 


77.0303 


85.0670 


104.184 


128.259 


35 


49.9947 


60.4619 


66.6736 


73.6519 


•81.4967 


90.3203 


111.435 


138.237 


36 


51.9945 


63.2758 


70.0072 


77.5980 


86.1641 


95.8364 


119.121 


148.914 


37 


54.0344 


66.1740 


73.4574 


81.7019 


91.0414 


101.628 


127.269 


160.338 


38 


56.1151 


69.1592 


77.0284 


85.9700 


96.1383 


107.710 


135.905 


172.561 


39 


58.2374 


72.23'40 


80.7244 


90.4088 


101.464 


114.095 


145.059 


185.641 


40 


60.4023 


75.4010 


84.5498 


95.0251 


107.030 


120.800 


154.762 


199.636 


41 


62.6103 


78.6630 


88.5090 


99.8261 


112.846 


127.840 


165.048 


214.611 


42 


64.8625 


82.0229 


92.6069 


104.819 


118.924 


135.232 


175.950 


230.633 


43 


67.1598 


85.4836 


96.8481 


110.012 


125.276 


142.994 


187.508 


247.778 


44 


69.5030 


89.0481 


101.238 


115.412 


131.914 


151.143 


199.758 


266.122 


45 


71.8930 


92.7195 


105.781 


121.029 


138.850 


159.700 


212.743 


285.750 


46 


74.3309 


96.5012 


110.483 


126.870 


146.098 


168.685 


226.509 


306.753 


47 


76.8176 


100.396 


115.350 


132.945 


153.672 


178.119 


241.099 


329.226 


48 


79.3540 


104.408 


120.387 


139.263 


161.588 


188.025 


256.565 


353.271 


49 


81.9411 


108.540 


125.601 


145.833 


169.859 


198.426 


272.959 


379.000 


50 


84.5800 


112.796 


130.997 


152.666 


178.503 


209.348 


290.337 


406.530 



annually setting aside a certain amount of its earnings to be placed at 
compound interest. 

Table 6, page 65, shows the annual amount to be set aside in order to 
accumulate $1000 at various rates of interest and for various terms of years, 
compounded annually. It is calculated from the formula: 



ANNUITIES— SINKING FUND, 



65 



Annuity = 1000 



(— 



Rate % 



-d- 



Rate %)^ 
in which x — the number of years = the number of annuities, the payments 
being made at the end of each year. 

Problem. — What semi-annual payments shall be made to a sinking fund 
to create $1,000,000 in 20 years, interest at 4%? 

Solution. — In this case there will be 40 periods compounded at 2%. 
From the table, 40 payments of $16,556 will create $1,000. But, in order 

to create $1,000,000 this must be multiplied by 1000 ( = ^' lOQO^^ ) ' 

Hence the semi-annual payments would be $16,556 each. 

5. — Present Worth, Present Value, or Capitalization of Annuity op $1. 





Rate of Interest. 


Years. 




















2% 


3% 


3i% 


4% 


4i% 


5% 


6% 


7% 


5 


$4.7134 


$4.5796 


$4.5150 


$4.4518 


$4.3899 


$4.3294 


$4.2124 


$4.1602 


10 


8.9824 


8.5301 


8.3165 


8.1108 


7.9127 


7.7217 


7.3602 


7.0238 


15 


12.849 


11.938 


11.517 


11.118 


10.739 


10.380 


9.7123 


9.1079 


20 


16.351 


14.877 


14.212 


13.590 


13.008 


12.462 


11.470 


10.594 


25 


19.524 


17.413 


16.482 


15.622 


14.828 


14.094 


12.783 


11.654 


30 


22.396 


19.600 


18.392 


17.292 


16.289 


15.372 


13.765 


12.409 


35 


24.999 


21.487 


20.000 


18.664 


17.461 


16.374 


14.498 


12.948 


40 


27.355 


23.115 


21.355 


19.793 


18.401 


17.159 


15.046 


13.332 


45 


29.490 


24.519 


22.495 


20.720 


19.156 


17.774 


15.456 


13.606 


50 


31.424 


25.730 


23.456 


21.482 


19.762 


18.256 


15.762 


13.801 


60 


34.761 


27.676 


24.988 


22.623 


20.638 


18.929 


16.161 


14.039 


70 


37.499 


29.123 


26.000 


23.395 


21.202 


19.343 


16.385 


14.160 


80 


39.745 


30.201 


26.749 


23.915 


21.565 


19.596 


16.509 


14.222 


90 


41.587 


31.002 


27.279 


24.267 


21.799 


19.752 


16.579 


14.253 


100 


43.098 


31.599 


27.655 


24.505 


21.949 


19.848 


16.613 


14.269 



6. — Sinking Fund Table. 

Annuities (or Annual Saving) Which Will Create $1,000 in Given 

Number of Years, x, at Various Rates of Interest Compounded 

Annually. 



Years to 


Rate of Interest. 


Run. 


















X 


2% 


3% 


H% 


4% 


4i% 


5% 


6% 


7% 


2 


$495.05 


$492.61 


$491.42 


$490.20 


$489.00 


$487.80 


$485.44 


$483.09 


3 


326.72 


323.56 


321.94 


320.36 


318.77 


317.21 


314.10 


311.05 


4 


242.63 


239.02 


237.26 


235.50 


233.74 


232.01 


228.60 


225.26 


5 


192.16 


188.35 


186.49 


184.63 


182.79 


180.98 


177.39 


173.89 


6 


158.53 


154.61 


152.67 


150.79 


148.88 


147.02 


143.36 


139.80 


7 


134.52 


130.51 


128.57 


126.61 


124.67 


122.82 


119.13 


115.55 


8 


116.51 


112.46 , 


110.48 


108.53 


106.60 


104.72 


101.03 


97.468 


9 


102.52 


98.434 


96.446 


94.493 


92.575 


90.690 


87.022 


83.487 


10 


91.327 


87.231 


85.242 


83.291 


81.360 


79.505 


75.868 


72.377 


12 


74.560 


70.462 


68.484 


66.552 


64.666 


62.826 


59.277 


55.902 


15 


57.826 


53.767 


51.825 


49.941 


48.114 


46.342 


42.963 


39.795 


20 


41.157 


37.216 


35.361 


33.582 


31.876 


30.243 


27.184 


24.393 


25 


31.220 


27.428 


25.674 


24.012 


22.439 


20.952 


18.227 


15.811 


30 


24.650 


21.019 


19.371 


17.830 


16.392 


15.051 


12.649 


10.586 


35 


20.002 


16.539 


14.998 


13.577 


12.270 


11.072 


8.9738 


7.2340 


40 


16.556 


13.262 


11.827 


10.524 


9.3432 


8.2781 


6.4615 


4.9976 


45 


13.910 


10.785 


9.4535 


8.2625 


7.2020 


6.2617 


4.7005 


3.4996 


50 


11.823 


8.8656 


7.6338 


6.5502 


5.6021 


4.7767 


3.4443 


2.4598 


60 


8.7679 


6.1330 


5.0887 


4.2019 


3.4543 


2.2207 


1.8757 


1.2292 


70 


6.6676 


4.3366 


3.4610 


2.7451 


2.1651 


1.6992 


1.0331 


.61952 


80 


5.1607 


3.1118 


2.3849 


1.8141 


1.3708 


1.0296 


.57254 


.31357 


90 


4.0460 


2.2556 


1.6578 


1.2078 


.87316 


.62711 


.31836 


.15905 


100 


3.2027 


1.6467 


1.1593 


.80801 


.55839 


.38314 


.17735 


.08076 



4.— MEASURES, WEIGHTS AND MONEY. 

FUNDAMENTAL UNITS. 

The Metric System, on account of its simplicity, is destined in all proba- 
bility to become the international standard of weights and measures. It 
has been legalized by Great Britain, Russia and the United States; and 
has been adopted by all other European nations, by Mexico, and by many 
South American States. 

Meter. — Length, Area, Volume. The international standard Meter, the 
unit of length, is the distance between two lines on a platinum-iridium bar, 
at 0° centigrade, deposited at the International Bureau of Weights and 
Measures, in Paris. The legalized ratio of the standard meter to the standard 

QQ 97 

yard by the United States* is ||-^ =1.09361111 = tl.0936n; hence, 

the following equivalents: 

Length. 

1 meter= 1.09361^1 yards = 3.28083''3 feet= 39.37 inches. 

1 yard = 0.9U 401 828 80 meter. Log =9.9611371. Co-log = 0.0388629 
1 foot =0.304 800 609 60 meter. " =9.4840158. " =0.5159842 
1 inch = 0.025 400 050 80 meter. " =8.4048346. " =1.5951654 

Area. 

1 square meter= 1.195 985 262 35 sq. yd. = 10.763 867 361^^1 sq. ft. = 
1549.9969 sq. ins. 
lsq.yd. = 0.836 130 704 52sq.met. Log= 9.9222742. Co-log = 0.0777258 
Isq. ft. =0.092 903 411 61 sq.met. "' = 8.9680316. " =1.0319684 
Isq.in. =0.000 645 162 58sq.met. " =6.8096692. " =3.1903308 

Volume. 

1 cubic meter =1.307 942 771 63"cu. yd. = 35.314 454 833 91 cu. ft.= 
61023.377953 cu. ins. 
lcu.yd. = 0.764 559 445 33cu.met. Log = 9.8834113. Co-log = 0.1165887 
leu. ft. =0.028 317 016 49 cu.met. " =8.4520475. " =1.5479525 
Icu.in. =0.000 016 387 16cu.met. " =5.2145038. " =4.7854962 

Liter. — Capacity {Liquid and Dry). The liter, the unit of capacity, is 
the volume of one kilogram of pure water at its maximum density; and 
this is equal to a cubic decimeter, or a cube whose edge is one -tenth of a 
meter =3.937 inches. The capacity of one liter is therefore j^V? of the 
volume of a cubic meter. The following are the United States J equivalents: 

*In Great Britain the meter has been legalized at 39.37079 inches, but 
the length of 39.370432 inches, as adopted by France, Germany, Belgium 
and Russia, is frequently used. 

t Logarithm 1.09361^1 = 0.0388629; co-logarithm = 9.9611371. 

±The Imperial gallon, the British unit of liquid capacity, contains 

/ 277 274\ 
277.274 cubic inches, or 1.200320 ( = — 2^j— ) United States gallons. The 

Imperial bushel, the British unit of dry capacity, contains 8 Imperial gal- 

/ 2218 1Q2\ 
Ions or 2218.192 cubic inches= 1.03151570 ( = irr^^TH" ) United States 



(struck) bushels. British measures of the same denomination as those of 
the United States are hence greater than the latter by these ratios. 

66 



METRIC UNITS— ENGLISH EQUIVALENTS. 

Volume. 

1 liter =1 cubic decimeter = 0.001 cubic meter. 

= 0.001 307 942 77 cu. yd. Lpg = 7.1165887 



67 



= 0.035 314 454 83 cu. ft. 

= 61.023 377 953 cu. ins. (exact.) 
1 cu. yd. = 764.559 445 33 liters, 
leu. ft. = 28.317 016 49 liters. 
1 cu. in. = 0.016 387 16 liter. 



= 8.5479525 
= 1.7854962 
= 2.8834113 
= 1.4520475 
= 8.2145038 



Liquid. 

1 Mter =0.264 170 467 33 U. S. gallon. Log = 9.4218842 

= 1.056 681 869 32 quarts. " =0.0239442 

= 2.113 363 738 64 pints. " =0.3249742 

1 gallon =^231 cu.ins = 0.U3 680 5''5cu. ft. " =9.1260682 

-0.004 951 131 69 cu. yd. " =7.6947045 

= 3.785 434 496 56 liters or cubic dm. " = 0.5781158 

) quart « 0.946 358 624 14 liter. " =9.9760558 



Dry. 

1 liter = 0.028 377 422 99 U. S. bushel. 

= 0.113 509 691 97 peck. 

= 0.908 077 535 78 quart. 
1 bushel = 2150.42 cu.ins. = 1.244 456 018 S'^l cu.ft. 

= 0.046 090 963 65 cu. yd. 

= 0.035 239 281 60_cubic meter. 

= 35.239 281 602 15 liters or cubic dm. 
1 peck =8.809 820 400 54 liters. 
1 quart =1.101 227 550 07 liters. 



Log 



= 8.4529730 
= 9.0550330 
= 9.9581229 
= 0.0949796 
= 8.6636158 
= 8.5470270 
= 1.5470270 
= 0.9449670 
= 0.0418771 



Mass. 

1 liter (=1 cubic decimeter) of pure water at maximum density weighs 

1 kilogram (kilo). 

1 millimeter ( = 1 cubic centimeter) of pure water at maximum density weighs 
1 gram. 

Gram. — Mass (Weight). The Gram, the unit of weight, is the weight of 
a cubic centimeter (1 millimeter) of pure water at its maximum density 
= tAtj of the international kilogram. The bureau of Standards, Washington, 
D. C., gives the fundamental equivalents: 



1 avoirdupois pound =453.592 4277 grams, and 

1 troy pound = f^§g avoir, pound = 373.241 769 078 857 142 8^57 

grams, from which are derived the following 

values: 

1 kilogram = 2.204 622 341 406 avoir, pounds. 

= 2.679 228 539 903 troy pounds. 
1 gram = 0.035 273 957 462 5 avoir, ounce. 

= 0.032 150 742 478 8 troy ounce. 

= 15.432 356 389 84 grains (troy). 
1 avoirdupois ounce = 28. 349 526 731 25 grams. 
I troy ounce =31.103 480 756 65 grams. 

I grain (troy) = 0.064 798 918 243 gram. 



Log = 2.6566658 
" =2.5719903 



Log 



= 0.3433342 
= 0.4280097 
= 8.5474542 
= 8.5071910 
= 1.1884323 
= 1.4525458 
= 1.4928090 
-8.8115677 



88 



i.— MEASURES, WEIGHTS AND MONEY. 



GENERAL TABLES. 

L — Approximate Equivalents — Metric and English. 



acre = .40 hectaf 

bushel =35 liters 35 

centimeter = .39 inch 

cubic centimeter = .061 cubic inch 

cubic foot = .028 cubic meter 



cubic inch =16 

cubic meter =35 

cubic meter = 1.3 

cubic yard = .76 

foot =30 

gallon = 3.8 

grain = .065 

gram =15 

hectar = 2.5 

inch =25 



cubic centimet. 16 

cubic feet 35 

cubic yards 1 

cubic meter 

centimeters. . ... 30 



.4047 

.24 

.3937 

.0610 

.0283 

.39 

.31 

.308 

.7645 

.48 



kilo 

kilometer. 

liter , 

liter , 

meter 

mile , 

millimeter 

ounce (avoirdupois). 



2.2 
.62 
.91 

1.1 

3.3 

16 
.039 
28 



ounce (Troy) =31 



peck, 

pint 

pound 

quart (dry) . . . . 
quart (liquid) . , 
sq. centimeter. , 

sq. foot 

sq. inch 

sq. meter 

sq. meter 

sq. yard 

ton (2,000 lbs.) 
ton (2,240 lbs.) 
ton (metric) . . . , 
ton (metric) . . . . 
yard 



= 6 
= 1 
= 11 



8.8 
.47 
.45 
.1 
.95 
15 
093 
5 
,2 



liters 3.785 

gram 0648 

grains 15.43 

acres 2.471 

millimeters 25.40 

pounds 2.205 

mile 6214 

quart (dry) 9081 

quarts (liquid) ... 1.057 

feet 3.281 

kilometers 1 . 609 

inch 0394 

grams 28.35 

grams 31 .10 

liters 8.809 

liter 4732 

kilo 4536 

liters 1.101 

liter .9464 



sq. inch 

sq. meter 

sq. centimeters . . 6 

sq. yards 1 

10 



sq. feet 

. 84 sq. meter 

.91 metric ton. . . . 

1 metric ton .... 

1.1 ton (2,000 lbs.) 

= .98 ton (2,240 lbs.) 

= .91 meter 

2. — Long Measure, English. 
1 inch (in.) 

12 inches =1 foot (ft.) 

3 feet = 1 yard (yd.) 

5i yards ( = 16^ ft.) =lrod (rd.) 

40 rods ( = 220 yds. =.660 ft.) =1 furlong (fur.) 

8 furlongs ( = 320 rods =1760 yds = 

5280 ft.) =1 statute mile • 

3 miles ( = 24 furs. = 960 rds. = 5280 

yds. = 15840 ft.) '....= 1 league 

3. — Surveyors' Measure (lineal). 



.1550 
.0929 
.452 
.196 
.76 
.8361 
.9072 
1.017 
1.102 
.9842 
.9144 



= 0.025400 Meters 
= 0.304801 

= 0.914402 •• 

= 5.029210 " 

= 201.1684 •• 

= 1609.347 •• 

= 4828.042 " 



= .2011684 Meters 
= 20.11684 " 
= 1609.347 " 



7.92 inches =1 link 

100 links ( = 4 rds. = 22 yds. = 66 ft.) = 1 chain 

80 chains ( 320 rds. = 1760 yds = 5280 ft.) =1 statute mile 
4. — ^Mariners' Measure. 

6 feet =1 fathom = 1.828804 Meters 

120 fathoms ( = 720 ft.) = \ cable length =219.4564 " 

[7i cable lengths( = 880 fathoms = 5280 ft.) = 1 5iafw/^ mt/^ =1609.347 " ] 

1.15246 statute miles (=6085 ft.) ^\ nautic al mile =1% 6^.112 " 

60 nau tical miles = 1 degree. 360 degrees = circumference of the earth. 

*The old nautical mile was given as 7^ cable lengths = 5400 feet, but it is 
now obsolete. The present nautical mile is not an exact term, being equal 
to about one minute of longitude at the equator. The British Admiralty- 
knot is 6080 ft. The nautical mile of the U. S. Coast Survey is 6086.07 ft. •= 
1.152664 statute or land miles. Three nautical miles = 1 league. 



LENGTHS— ENGLISH AND METRIC. 



69 



6. — Lengths— Inches and Millimeters.— Equivalents op Decimal and 

Common Fractions of an Inch in Millimeters. 
From 6^ to 1 Inch. 







tn 


03 




m 

^ 


Milli- 


imals 

of 

[nch. 


^ 






02 


oo 

,C1 


-d 


^ 


Milli- 


imals 

of 

nch. 


in 


tn 




+3 

CO 


a 


J 
■^ 


meters. 




C3 


02 


ra 


is 


f£> 


G 

CM 


i? 


meters. 


li 


-" 


rf* 


00 




CO 


to 




1— 1 


H|N 


nil 


00 




CO 


to 














1 


= .397 


.015625 














33 


= 13.097 


.5-15625 










1 


2 
3 


= .794 
= 1.191 


.03125 
.046875 












17 


34 
35 


= 13.494 
= 13.891 


.53125 
.546875 








1 


2 
3 


4 

5 
6 

7 


= 1.588 

= 1.984 
= 2.381 
= 2.778 


.0625 

.078125 

.09375 

.109375 










9 


18 
19 


36 

37 

38 
39 


= 14.288 

= 14.684 
= 15.081 
= 15.478 


.5625 

.578125 

.59375 

.609375 






1 


2 


4 
5 


8 

9 

10 
11 


= 3.175 

= 3.572 
= 3.969 
= 4.366 


.1250 

.140625 

.15625 

.171875 








5 


10 


20 
21 


40 

41 

42 
43 


= 15.875 

= 16.272 
= 16.669 
= 17.066 


.625 

.640625 

.65625 

.671875 








3 


6 

7 


12 

13 

14 
15 


= 4.763 

= 5.159 
= 5.556 
= 5.953 


.1875 

.203125 

.21875 
.234375 










11 


22 
23 


44 

45 
46 
47 


= 17.463 

= 17.859 
= 18.256 
= 18.653 


.6875 

.703125 

.71875 

.734376 




1 


2 


4 


8 
9 


16 

17 
18 
19 


= 6.350 

= 6.747 
= 7.144 
= 7.541 


.2500 

.265625 

.28125 

.296875 






3 


6 


12 


24 

25 


48 

49 
50 

51 


= 19.050 

= 19.447 
= 19.844 
= 20.241 


.75 

.765625 

.78125 

.796875 








5 


10 
11 


20 

21 
22 
23 


= 7.938 

= 8.334 
= 8.731 
= 9.128 


.3125, 

.328125 

.34375 

.359375 










13 


26 
27 


52 

53 
54 
55 


= 20.638 

= 21.034 
= 21.431 
= 21.828 


.8125 

.828125 

.84375 

.859375 






3 


6 


12 
13 


24 

25 

26 
27 


= 9.525 

= 9.922 
= 10.319 
= 10.716 


.3750 

.390625 

.40625 

.421875 








7 


14 


28 
29 


56 

57 
58 
59 


= 22.225 

= 22.622 
= 23.019 
= 23.416 


.875 

.890625 

.90625 

.921875 








7 


14 
15 


28 

29 
30 
31 


= 11.113 

= 11.509 
= 11.906 
= 12.303 


.4375 

.453125 

.46875 

.484375 










15 


30 
31 


60 

61 
62 
63 


= 23.813 

= 24.209 
= 24.606 
= 25.003 


.9375 

.953125 

.96875 

.984375 


1 


2 


4 


8 


16 


32 


= 12.700 


.5 


1^ 


2 


4 


8 


16 


32 


64 


= 25.400 


1.000 



6. — Lengths- 



-Hundredths of an Inch to Millimeters. 
From 1 to 100 Hundredths. 



Is 






















•o ca 






















V 

Wo 





1 


2 


3 


4 


5 


6 


7 


8 


9 








.254 


.508 


.762 


1.016 


1.270 


1.524 


1.778 


2.032 


2.286 


10 


2.540 


2.794 


3.048 


3.302 


3.556 


3.810 


4.064 


4.318 


4.572 


4.826 


20 


5.080 


5.334 


5.588 


5.842 


6.096 


6.350 


6.604 


6.858 


7.112 


7.366 


,^0 


7.620 


7.874 


8.128 


8.382 


8.636 


8.890 


9.144 


9.398 


9.652 


9.906 


40 


10.160 


10.414 


10.668 


10.922 


11.176 


11.430 


11.684 


11.938 


12.192 


12.446 


50 


12.700 


12.954 


13.208 


13.462 


13.716 


13.970 


14.224 


14.478 


14.732 


14.986 


60 


15.240 


1^.494 


15.748 


16.002 


16.256 


16.510 


16.764 


17.018 


17.272 


17.526 


70 


17.780 


18.034 


18.288 


18.542 


18.796 


19.050 


19.304 


19.558 


19.812 


20.066 


80 


20.320 


20.574 


20.828 


21.082 


21.336 


21.590 


21.844 


22.098 


22.352 


22.606 


90 


22.860 


23.114 


23.368 


23.622 


23.876 


24.130 


24.384 


24.638 


24.892 


25.146 



Example. — 21 hundredths of an inch = 6.334 millimeters. 



70 i.— MEASURES, WEIGHTS AND MONEY. 



7. — Lengths — Millimeters to Decimals op an Inch. 
From 1 to 100 Units. 








1 


3 


3 


4 


5 


6 


7 


8 


9 








.03937 


.07874 


.11811 


.15748 


.19685 


.23622 


.27559 


.31496 


.35433 


10 


.39370 


.43307 


.47244 


.51181 


.55118 


.59055 


.62992 


.66929 


.70866 


.74803 


20 


.78740 


.82677 


.86614 


.90551 


.94488 


.98425 


1.02362 


1.06299 


1.10236 


1.14173 


30 


1.18110 


1.22047 


1.25984 


1.29921 


1.33858 


1.37795 


1.41732 


1.45669 


1.49606 


1.53543 


40 


1.57480 


1.61417 


1.65354 


1.69291 


1.73228 


1.7?165 


1.81102 


1.85039 


1.88976 


1.92913 


50 


1.96850 


2.00787 


2.04724 


2.08661 


2.12598 


2.16535 


2.20472 


2.24409 


2.28346 


2.32283 


60 


2.36220 


2.40157 


2.44094 


2.48031 


2.51968 


2.55905 


2.59842 


2.63779 


2.67716 


2.71653 


70 


2.75590 


2.79527 


2.83464 


2.87401 


2.91338 


2.95275 


2.99212 


3.03149 


3.07086 


3.11023 


80 


3.14960 


3.18897 


3.22834 


3.26771 


3.30708 


3.34645 


3.38582 


3.42519 


3.46456 


3.50393 


90 


3.54330 


3.58267 
* 


3.62204 


3.66131 


3.70078 


3.74015 


3.77952 


3.81889 


3.85826 


3.89763 



I 



Example. — ^21 millimeters = 0.82677 inch. 
8. — Lengths. 



Inches 



Feet. 



Yards. 



Miles. 



10 millimeters ( 



10 centimeters 
10 decimeters = 

10 meters = 

1 

10 dekameters ( = 
1 

10 hectometers ( 
1 

10 kilometers ( = 



millimeter (m m) 
= T^Tj meter) = 
centimeter (c m) 
( =■ f^ meter) = 
decimeter (d m) 



1 meter 



dekameter (Dm) 
= 100 meters) = 
hectometer (Hm) 
= 1000 meters) = 
kilometer (Km) 
= 10000 meters) = 
myriameter (Mm) 



.03937 

.3937 

3.937 

39.37 

393.7 

3937 



.0032808 

.0328083 

.328083^3 

3.28083^3 

32.8083^3 

328.083^3 

3280.83^3 

32808.3^3 



.0010936 

.0109361 

.10936n 

1.09361^1 

10.9361^1 

109.361^1 

1093.61^1 

10936.1^1 



0006213 
0062137 
062137 
621370 
). 21370 



9. — Lengths, Equivalents. 1-10. 



Inches. 



Milli- 
meters, 



Cent! 
Inches, meters. 



Feet. Meters. 



U.S. 
Yards. 



Meters. 



U.S. 
Miles. 



Kilo- 
meters. 



0.03937= 
0.07874= 
0.11811= 
0.15748= 

0.19685= 
0.23622= 
0.27559= 
0.31496= 
0.25433= 



: 1 

: 2 

= 3 

: 4 

= 5 

= 6 

= 7 

: 8 
: 9 

: 25.4001 
■■ 50.8001 
: 76.2002 
aOl.6002 

: 127.0003 
152.4003 
177.8004 
203.2004 

= 228.6005 



0.3937= .1 
0.7874= 2 
1 = 2.54001 

1.1811= 3 



1.5748= 
1.9685= 

2 = 
2.3622= 
2.7559= 

3 = 

3.1496= 
3.5433 = 

4 = 

5 = 

6 = 

7 = 

8 = 

9 = 



4 
5 

5.08001 

6 

7 

7.62002 
8 
9 
10.16002 

12.70003 
15.24003 
17.78004 
20.32004 
22.86005 



1 =0.304801 

2 =0.609601 

3 =0.914402 
3.28083=1 

4 =1.219202 

5 =1.524003 

6 =1.828804 
6.56167=2 

7 =2.133604 



2.438405 
2.743205 



9.84250=3 
13.12333 = 4 

16.40417 = 5 
19.68500=6 
22.96583=7 
26.24667 = 8 
29.52750=9 



1 =0.914402 
1.093611=1 

2 =1.828804 
2.187222=2 

3 =2.743205 
3.280833=3 

4 =3.657607 
4.374444 = 4 

5 =4.572009 

5.468056=5 

6 =5.486411 
6.561667 = 6 

7 =6.400813 

7.655278=7 

8 =7.315215 

8.748889 = 8 

9 =8.229616 
9.842500=9 



0.62137= 1 

1 = 1.60935 
1.24274= 2 
1.86411= 3 

2 = 3.21869 
2.48548= 4 

3 = 4.82804 
3.10685= 5 
3.72822= 6 

= 6.43739 
4.34959= 7 
4.97096= 8 

5 = 8.04674 

5.59233= 9 

6 = 9.65608 

7 =11.26543 

8 =12.87478 

9 =14.48412 



LENGTHS—FEET AND INCHES TO METERS. 



71 



1-250. 



10. — Lengths — Feet to Meters. 
From 1 to 1.000 Units. 



Feet. Meters. 


Feet 


. Meters. 


Feet 


. Meters. 


Feet. Meters. 


Feet. Meters. 




1 
2 
3 
4 


.30480 

.60960 

.91440 

1.21920 


50 

1 
2 
3 
4 


15.24003 
15.54483 
15.84963 
16.15443 
16.45923 


100 

1 
2 
3 
4 


30.48006 
30.78486 
31.08966 
31.39446 
31.69926 


150 45.72009 

1 46.02489 

2 46.32969 

3 46.63449 

4 46.93929 


200 60.96012 

1 61.26492 

2 61.56972 

3 61.87452 

4 62.17932 


5 

6 
7 
8 
9 


1.52400 
1.82880 
2.13360 
2.43840 
2.74321 


5 
6 

7 
8 
9 


16.76403 
17.06883 
17.37363 
17.67844 
17.98324 


5 
6 

7 
8 
9 


32.00406 
32.30886 
32.61367 
32.91847 
33.22327 


5 47.24409 

6 47.54890 

7 47.85370 

8 48.15850 

9 48.46330 


5 62.48412 

6 62.78893 

7 63.09373 

8 63.39853 

9 63.70333 


10 

1 
2 
3 

4 


3.04801 
3.35281 
3.65761 
3.96241 
4.26721 


60 

2 
3 
4 


18.28804 
18.59284 
18.89764 
19.20244 
19.50724 


no 

1 

2 
3 
4 


33.52807 
33.83287 
34.13767 
34.44247 
34.74727 


160 48.76810 

1 49.07290 

2 49.37770 

3 49.68250 

4 49.98730 


210 64.00813 

1 64.31293 

2 64.61773 

3 64.92253 

4 65.22733 


5 
6 

7 
8 
9 


4.57201 
4.87681 
5.18161 
5.48641 
5.79121 


5 
6 
7 
8 
9 


19.81204 
20.11684 
20.42164 
20.72644 
21.03124 


5 
6 

7 
8 
9 


35.05207 
35.35687 
35.66167 
35.96647 
36.27127 


5 50.29210 

6 50.59690 

7 50.90170 

8 51.20650 

9 51.51130 


5 65.53213 

6 65.83693 

7 66.14173 

8 66.44653 

9 66.75133 


20 

1 
2 
3 
4 


6.09601 
6.40081 
6.70561 
7.01041 
7.31521 


70 

1 

i 

4 


21.33604 
21.64084 
21.94564 
22.25044 
22.55525 


120 

1 
2 
3 

4 


36.57607 
36.88087 
37.18567 
37.49047 
37.79528 


170 51.81610 

1 52.12090 

2 52.42570 

3 52.73051 

4 53.03531 


220 67.05613 

1 67.36093 

2 67.66574 

3 67.97054 

4 68.27534 


5 
6 
7 
8 
9 


7.62002 
7.92482 
8.22962 
8.53442 
8.83922 


5 
6 

7 
8 
9 


22.86005 
23.16485 
23.46965 
23.77445 
24.07925 


5 
6 

7 
8 
9 


38.10008 
38.40488 
38.70968 
39.01448 
39.31928 


5 53.34011 

6 53.64491 

7 53.94971 

8 54.25451 

9 54.55931 


5 68.58014 

6 68.88494 

7 69.18974 

8 69.49454 

9 69.79934 


30 

1 
2 
3 
4 


9.14402 

9.44882 

9.75362 

10.05842 

10.36322 


80 

1 
2 
3 
4 


24.38405 
24.68885 
24.99365 
25.29845 
25.60325 


130 

1 
2 
3 

4 


39.62408 
39.92888 
40.23368 
40.53848 
40.84328 


180 54.86411 

1 55.16891 

2 55.47371 

3 55.77851 

4 56.08331 


230 70.10414 

1 70.40894 

2 70.71374 

3 71.01854 

4 71.32334 


5 
6 

7 
8 
9 


10.66802 
10.97282 
11.27762 
11.58242 
11.88722 


5 
6 

7 
8 
9 


25.90805 
26.21285 
26.51765 
26.82245 
27.12725 


5 
6 

7 
8 
9 


41.14808 
41.45288 
41.75768 
42.06248 
42.36728 


5 56.38811 

6 56.69291 

7 56.99771 

8 57.30251 

9 57.60732 


6 71.62814 

6 71.93294 

7 72.23774 

8 72.54255 

9 72.84735 


40 

1 
2 
3 
4 


12.19202 
12.49682 
12.80163 
13.10643 
13.41123 


90 

1 
2 
3 
4 


27.43205 
27.73686 
28.04166 
28.34646 
28.65126 


140 

1 
2 
3 
4 


42.67209 
42.97689 
43.28169 
43.58649 
43.89129 


190 57.91212 

1 58.21692 

2 58.52172 

3 58.82652 

4 59.13132 


240 73.15215 

1 73.45695 

2 73.76175 

3 74.06655 

4 74.37135* 


5 
6 
7 
8 
9 


13.71603 
14.02083 
14.32563 
14.63043 
14.93523 


5 
6 

7 
8 
9 


28.95606 
29.26086 
29.56566 
29.87046 
30.17526 


5 
6 

7 
8 
9 


44.19609 
44.50089 
44.80569 
45.11049 
45.41529 


5 59.43612 

6 59.74092 

7 60.04572 

8 60.35052 

9 60.65532 


5 74.67615 

6 74.98095 

7 75.28575 

8 75.59055 

9 75.89535 

250 76.20015 



1 Inch = . 02540 meter. 

2 inches =. 05080 meter. 

3 Inches =. 07620 meter. 

4 Inches =. 10160 meter. 



5 Inches =. 12700 meter. 

6 inches =. 15240 meter. 

7 iAjhes =. 17780 meter. 

8 inches =. 20320 meter. 



9 Inches = .22860 meter. 

10 inches =.25400 meter. 

11 Inches =.27940 meter. 

12 Inches =.30480 meter. 



72 



i.— MEASURES, WEIGHTS AND MONEY, 



250-500, 



10. — Lengths — Feet to Meters (Continued). 



Feet. Meters. 


Feet 


Meters. 


Feet 


Meters. 


Feet 


Meters. 


Feet 


. Meters. 


250 76.20015 

1 76.50495 

2 76.80975 

3 77.11455 

4 -77.41935 


300 

1 
2 
3 
4 


91.44018 
91.74498 
92.04978 
92.35458 
92.65939 


350 

1 
2 
3 
4 


106.68021 
106.98501 
107.28981 
107.59462 
107.89942 


400 

1 
2 
3 
4 


121.92024 
122.22504 
122.53985 
122.83465 
123.13945 


450 

1 
2 
3 
4 


137.16027 
137.46507 
137.76988 
138.07468 
138.37948 


5 77.72416 

6 78.02896 

7 78.33376 

8 78.63856 

9 78.94336 


5 
6 

7 
8 
9 


92.96419 
93.26899 
93.57379 
93.87859 
94.18339 


5 
6 
7 
8 
9 


108.20422 
108.50902 
108.81382 
109.11862 
109.42342 


5 
6 

7 
8 
9 


123.44425 
123.74905 
124.05385 
124.35865 
124.66345 


5 
6 

7 
8 
9 


138.68428 
138.98908 
139.29388 
139.59868 
139.90348 


260 79.24816 

1 79.55296 

2 79.85776 

3 80.16256 

4 80.46736 


310 

1 
2 
3 
4 


94.48819 
94.79299 
95.09779 
95.40259 
95.70739 


360 

1 
2 
3 
4 


109.72822 
110.03302 
110.33782 
110.64262 
110.94742 


410 

1 
2 
3 

4 


124.96825 
125.27305 
125.57785 
125.88265 
126.18745 


460 

1 
2 
3 

4 


140.20828 
140.51308 
140.81788 
141.12268 
141.42748 


5 80.77216 

6 81.07696 

7 81.38176 

8 81.68656 

9 81.99136 


5 
6 

7 
8 
9 


96.01219 
96.31699 
96.62179 
96.92659 
97.23139 


5 
6 
7 
8 
9 


111.25222 
111.55702 
111.86182 
112.16662 
112.47142 


5 
6 
7 
8 
9 


126.49225 
126.79705 
127.10185 
127.40665 
127.71146 


5 
6 

7 
8 
9 


141.73228 
142.03708 
142.34188 
142.64669 
142.95149 


270 82.29616 

1 82.60097 

2 82.90577 

3 83.21057 

4 83.51537 


320 

1 
2 
3 
4 


97.53620 
97.84100 
98.14580 
98.45060 
98.75540 


370 

2 
3 

4 


112.77623 
113.08103 
113.38583 
113.69063 
113.99543 


420 

1 
2 
3 
4 


128.01626 
128.32106 
128.62586 
128.93066 
129.23546 


470 

2 

3 
4 


143.25629 
143.56109 
143.86589 
144.17069 
144.47549 


5 83.82017 

6 84.12497 

7 84.42977 

8 84.73457 

9 85.03937 


5 
6 

7 
8 
9 


99.06020 
99.36500 
99.66980 
99.97460 
100.27940 


5 
6 

7 
8 
9 


114.30023 
114.60503 
114.90983 
115.21463 
115.51943 


5 
6 
7 
8 
9 


129.54026 
129.84506 
130.14986 
130.45466 
130.75946 


5 
6 
7 
8 
9 


144.78029 
145.08509 
145.38989 
145.69469 
145.99949 


280 85.34417 

1 85.64897 

2 85.95377 

3 86.25857 

4 86.56337 


330 

1 
2 
3 
4 


100.58420 
100.88900 
101.19380 
101.49860 
101.80340 


380 

1 
2 
3 
4 


115.82423 
116.12903 
116.43383 
116.73863 
117.04343 


430 

1 
2 
3 
4 


131.06426 
131.36906 
131.67386 
131.97866 
132.28346 


480 

1 
2 
3 
4 


146.30429 
146.60909 
146.91389 
147.21869 
147.52350 


5 86.86817 

6 87.17297 

7 87.47777 

8 87.78258 

9 88.08738 


5 

6 

7 
8 
9 


102.10820 
102.41300 
102.71781 
103.02261 
103.32741 


5 
6 
7 
8 
9 


117.34823 
117.65304 
117.95784 
118.26264 
118.56744 


5 
6 

7 
8 
9 


132.58827 
132.89307 
133.19787 
133.50267 
133.80747 


5 
6 

7 
8 
9 


147.82830 
148.13310 
148.43790 
148.74270 
149.04750 


290 88.39218 

1 88.69698 

2 89.00178 
6 89.30658 
4 89.61138 


340 

1 
2 
3 
4 


103.63221 
103.93701 
104.24181 
104.54661 
104.85141 


390 

2 
3 
4 


118.87224 
119.17704 
119.48184 
119.78664 
120.09144 


440 

1 
2 
3 
4 


134.11227 
134.41707 
134.72187 
135.02667 
135.33147 


490 

1 
2 
3 
4 


149.35230 
149.65710 
149.96190 
150.26670 
150.57150 


5 89.91618 

6 90.22098 

7 90.52578 

8 90.83058 

9 91.13538 


5 
6 
7 
8 
9 


105.15621 
105.46101 
105.76581 
106.07061 
106.37541 


5 
6 
7 
8 
9 


120.39624 
120.70104 
121.00584 
121.31064 
121.61544 


5 
6 

7 
8 
9 


135.63627 
135.94107 
136.24587 
136.55067 
136.85547 


5 
6 
7 
8 
9 

500 


150.87630 
151.18110 
151.48590 
151.79070 
152.09550 

152.40030 



LENGTHS— FEET TO METERS. 



73 



500-750. 


10. — Lengths— 


-Feet to Meters (Continued) 


, 




Feet 


. Meters. 


Feet 


. Meters. 


Feet 


. Meters. 


Feet. Meters. 


Feet. Meters. 


SCO 

1 

2 
3 
4 


152.40030 
152.70511 
153.00991 
153.31471 
153.61951 


5S0 

2 
3 
4 


167.64034 
167.94514 
168.24994 
168.55474 
168.85954 


600 

1 
2 
3 
4 


182.88037 
183.18517 
183.48997 
183.79477 
184.09957 


650 198.12040 

1 198.42520 

2 198.73000 

3 199.03480 

4 199.33960 


700 

1 
2 
3 

4 


213.36043 
213.66523 
213.97003 
214.27483 
214.57963 


5 
6 
7 
8 
9 


153.92431 
154.22911 
154.53391 
154.83871 
155.14351 


5 
6 
7 
8 
9 


169.16434 
169.46914 
169.77394 
170.07874 
170.38354 


5 
6 

7 
8 
9 


184.40437 
184.70917 
185.01397 
185.31877 
185.62357 


5 199.64440 

6 199.94920 

7 200.25400 

8 200.55880 

9 200.86360 


5 
6 

7 
8 
9 


214.88443 
215.18923 
215.49403 
215.79883 
216.10363 


510 

1 
2 
3 
4 


155.44831 
155.75311 
156.05791 
156.36271 
156.66751 


560 

1 
2 
3 
4 


170.68834 
170.99314 
171.29794 
171.60274 
171.90754 


610 

1 
2 
3 
4 


185.92837 
186.23317 
186.53797 
186.84277 
187.14757 


660 201.16840 

1 201.47320 

2 201.77800 

3 202.08280 

4 202.38760 


710 

1 
2 
3 
4 


216.40843 
216.71323 
217.01803 
217.32283 
217.62764 


5 
6 

7 
8 
9 


156.97231 
157.27711 
157.58192 
157.88672 
158.19152 


5 

6 
7 
8 
9 


172.21234 
172.51714 
172.82195 
173.12675 
173.43155 


5 
6 

7 
8 
9 


187.45237 
187.75718 
188.06198 
188.36678 
188.67158 


5 202.69241 

6 202.99721 

7 203.30201 

8 203.60681 

9 203.91161 


5 
6 
7 
8 
9 


217.93244 
218.23724 
218.54204 
218.84684 
219.15164 


520 

1 
2 
3 
4 


158.49632 
158.80112 
159.10592 
159.41072 
159.71552 


570 

1 
2 
3 
4 


173.73635 
174.04115 
174.34595 
174.65075 
174.95555 


620 

1 
2 
3 
4 


188.97638 
189.28118 
189.58598 
189.89078 
190.19558 


670 204.21641 

1 204.52121 

2 204.82601 

3 205.13081 

4 205.43561 


720 

1 
2 
3 
4 


219.45644 
219.76124 
220.06604 
220.37084 
220.67564 


5 
6 

7 
8 
9 


160.02032 
160.32512 
160.62992 
160.93472 
161.23952 


5 
6 

7 
8 
9 


175.26035 
175.56515 
175.86995 
176.17475 
176.47955 


5 
6 

7 
8 
9 


190.50038 
190.80518 
191.10998 
191.41478 
191.71958 


5 205.74041 

6 206.04521 

7 206.35001 

8 206.65481 

9 206.95961 


5 
6 

7 
8 
9 


220. 98044 
221.28524 
221.59004 
221.89484 
222.19964 


530 

1 

2 
3 
4 


161.54432 
161.84912 
162.15392 
162.45872 
162.76353 


580 

1 
2 
3 
4 


176.78435 
177.08915 
177.39395 
177.69876 
178.00356 


630 

1 
2 
3 
4 


192.02438 
192.32918 
192.63399 
192.93879 
193.24359 


680 207.26441 

1 207.56922 

2 207.87402 

3 208.17882 

4 208.48362 


730 

1 
2 
3 
4 


222.50445 
222.80925 
223.11405 
223.41885 
223.72365 


5 
6 

7 
8 
9 


163.06833 
163.37313 
163.67793 
163.98273 
164.28753 


5 
6 
7 
8 
9 


178.30836 
178.61316 
178.91796 
179.22276 
179.52756 


5 
6 
7 
8 
9 


193.54839 
193.85319 
194.15799 
194.46279 
194.76759 


5 208.78842 

6 209.09322 

7 209.39802 

8 209.70282 

9 210.00762 


5 
6 

7 
8 
9 


224.02845 
224.33325 
224.63805 
224.94285 
225.24765 


540 
1 

2 
3 
4 


164.59233 
164.89713 
165.20193 
165.50673 
165.81153 


590 

1 
2 
3 
4 


179.83236 
180.13716 
180.44196 
180.74676 
181.05156 


640 

2 
3 
4 


195.07239 
195.37719 
195.68199 
195.98679 
196.29159 


690 210.31242 

1 210.61722 

2 210.92202 

3 211.22682 

4 211.53162 


740 

2 
3 

4 


225.55245 
225.85725 
226.16205 
226.46685 
226.77165 


5 
6 
7 
8 
9 


166.11633 
166.42113 
166.72593 
137.03073 
167.33553 


5 
6 

7 
8 
9 


181.35636 
181.66116 
181.96596 
182.27076 
182.57557 


5 
6 

7 
8 
9 


196.59639 
196.90119 
197.20599 
197.51080 
197.81560 


5 211.83642 

6 212.14122 

7 212.44602 

8 212.75083 

9 213.05563 


5 
6 

7 
8 
9 

750 


227.07645 
227.38125 
227.68606 
227.99086 
228.29566 

228.60046 



74 



i.— MEASURES, WEIGHTS AND MONEY. 



750-1000. 



10. — Lengths — Feet to Meters (Concluded). 



Feet 


. Meters. 


Feet 


. Meters. 


Feet 


. Meters. 


Feet. Meters. 


Feet. Meters. 


750 

1 

2 
3 
4 


228.60046 
228.90526 
229.21006 
229.51486 
229.81966 


800 

1 
2 
3 
4 


243.84049 
244.14529 
244.45009 
244.75489 
245.05969 


850 

1 
2 
3 
4 


259.08052 
259.38532 
259.69012 
259.99492 
260.29972 


900 274.32055 

1 274.62535 

2 274.93015 

3 275.23495 

4 275.53975 


950 289.56058 

1 289.86538 

2 290.17018 

3 290.47498 

4 290.77978 


5 
6 

7 
8 
9 


230.12446 
230.42926 
230.73406 
231.03886 
231.34366 


5 
6 

7 
8 
9 


245.36449 
245.66929 
245.97409 
246.27889 
246.58369 


5 
6 
7 
8 
9 


260.60452 
260.90932 
261.21412 
261.51892 
261.82372 


5 275.84455 

6 276.14935 

7 276.45415 

8 276.75895 

9 277.06375 


5 291.08458 

6 291.38938 

7 291.69418 

8 291.99898 
■ 9 292.30378 


760 

1 
2 
3 
4 


231.64846 
231.95326 
232.25806 
232.56287 
232.86767 


810 

1 
2 
3 
4 


246.88849 
247.19329 
247.49810 
247.80290 
248.10770 


860 

1 
2 
3 
4 


262.12852 
262.43332 
262.73813 
263.04293 
263.34773 


910 277.36855 

1 277.67336 

2 277.97816 

3 278.28296 

4 278.58776 


960 292.60859 

1 292.91339 

2 293.21819 

3 293.52299 

4 293.82779 


6 
6 

7 
8 
9 


233.17247 
233.47727 
233.78207 
234.08687 
234.39167 


5 
6 

7 
8 
9 


248.41250 
248.71730 
249.02210 
249.32690 
249.63170 


5 
6 
7 
8 
9 


263.65253 
263.95733 
264.26213 
264.56693 
264.87173 


5 278.89256 

6 279.19736 

7 279.50216 

8 279.80696 

9 280.11176 


5 294.13259 

6 294.43739 

7 294.74219 

8 295.04699 

9 295.35179 


770 

1 
2 
3 
4 


234.69647 
235.00127 
235.30607 
235.61087 
235.91567 


820 

1 
2 
3 
4 


249.93650 
250.24130 
250.54610 
250.85090 
251.15570 


870 
1 
2 
3 
4 


265.17653 
265.48133 
265.78613 
266.09093 
266.39573 


920 280.41656 

1 280.72136 

2 281.02616 

3 281.33096 

4 281.63576 


970 295.65659 

1 295.96139 

2 296.26619 

3 296.57099 

4 296.87579 


5 
6 

7 

8 

' 9 


236.22047 
236.52527 
236.83007 
237.13487 
237.43967 


5 
6 

7 
8 
9 


251.46050 
251.76530 
252.07010 
252.37490 
252.67971 


5 
6 

7 
8 
9 


266.70053 
267.00533 
267.31013 
267.61494 
267.91974 


5 281.94056 

6 282.24536 

7 282.55017 

8 282.85497 

9 283.15977 


5 297.18059 

6 297.48539 

7 297.79020 

8 298.09500 

9 298.39980 


780 

1 
2 
3 
4 


237.74448 
238.04928 
238.35408 
238.65888 
238.96368 


830 

1 
2 
3 
4 


252.98451 
253.28931 
253.59411 
253.89891 
254.20371 


880 
1 
2 
3 
4 


268.22454 
268.52934 
268.83414 
269.13894 
269.44374 


930 283.46457 

1 283.76937 

2 284.07417 

3 284.37897 

4 284.68377 


980 298.70460 

1 299.00940 

2 299.31420 

3 299.61900 

4 299.92380 


5 
6 

7 
8 
9 


239.26848 
239.57328 
239.87808 
240.18288 
240.48768 


5 
6 

7 
8 
9 


254.50851 
254.81331 
256.11811 
225.42291 
255.72771 


5 
6 

7 
8 
9 


269.74854 
270.05334 
270.35814 
270.66294 
270.96774 


5 284.98857 

6 285.29337 

7 285.59817 

8 285.90297 

9 286.20777 


5 300.22860 

6 300.53340 

7 300.83820 

8 301.14300 

9 301.44780 


790 

1 
2 
3 

4 


240.79248 
241.09728 
241.40208 
241.70688 
242.01168 


840 

1 
2 
3 
4 


256.03251 
256.33731 
256.64211 
256.94691 
257.25171 


890 

1 
2 
3 
4 


271.27254 
271.57734 
271.88214 
272.18694 
272.49174 


940 286.51257 

1 286.81737 

2 287.12217 

3 287.42697 

4 287.73178 


990 301.75260 

1 302.05740 

2 302.36220 

3 302.66701 

4 302.97181 


5 
6 
7 
8 
9 


242.31648 
242.62129 
242.92609 
243.23089 
243.53569 


5 
6 
7 
8 
9 


257.55652 
257.86132 
258.16612 
258.47092 
258.77572 


5 
6 
7 
8 
9 


272.79655 
273.10135 
273.40615 
273.71095 
274.01575 


5 288.03658 

6 288.34138 

7 288.64618 

8 288.95098 

9 289.25578 


5 303.27661 

6 303.58141 

7 303.88621 

8 304.19101 

9 304.49581 

1000 304.80061 



LENGTHS—METERS TO FEET, 



75 



-250. 



11. — Lengths — Meters to Feet. 
From 1 to 1,000 Units. 



Meters. Feet. 



Meters. Feet. 



Meters. Feet. 



Meters. Feet. 



Meters. Feet. 



3.28083 

6.56167 

9.84250 

13.12333 

16.40417 
19.68500 
22.96583 
26.24667 
29.52750 

32.80833 
36.08917 
39.37000 
42.65083 
45.93167 

49.21250 
52.49333 
55.77417 
59.05500 
62.33583 

65.61667 
68.89750 
72.17833 
75.45917 
78.74000 

82.02083 
85.30167 
88.58250 
91.86333 
95.14417 

98.42500 
101.70583 
104.98667 
108.26750 
111.54833 

114.82917 
118.11000 
121.39083 
124.67167 
127.95250 

131.23333 
134.51417 
137.79500 
141.07583 
144.35667 

147.63750 
150.91833 
154.19917 
157.48000 
160.76083 



50 



164.04167 
167.32250 
170.60333 
173.88417 
177.16500 

180.44583 
183.72667 
187.00750 
190.28833 
193.56917 



60 196.85000 

1 200.13083 

2 203.41167 

3 206.69250 

4 209.97333 

5 213.25417 

6 216.53500 

7 219.81583 

8 223.09667 

9 226.37750 



70 

1 
2 
3 
4 



80 

1 

■2 



90 

1 
2 
3 
4 

5 
6 

7 



229.65833 
232.93917 
236.22000 
239.50083 
242.78167 

246.06250 
249.34333 
252.62417 
255.90500 
259.18583 

262.46667 
265.74750 
269.02833 
272.30917 
275.59000 

278.87083 
282.15167 
285.43250 
288.71333 
291.99417 

295.27500 
298.55583 
301.83667 
305.11750 
308.39833 

311.67917 
314.96000 
318.24083 
321.52167 
324.80250 



100 

1 
2 



120 

1 
2 
3 
4 



130 

2 
3 
4 

5 
6 

7 



140 

1 
2 



328.08333 
331.36417 
334.64500 
337.92583 
341.20667 

344.48750 
347.76833 
351.04917 
354.33000 
357.61083 

360.89167 
364.17250 
367.45333 
370.73417 
374.01500 

377.29583 
380.57667 
383.85750 
387.13833 
390.41917 

393.70000 
396.98083 
400.26167 
403.. 542 50 
406.82333 

410.10417 
413.38500 
416.66583 
419.94667 
423.22750 

426.50833 
429.78917 
433.07000 
436.35083 
439.63167 

442.91250 
446.19333 
449.47417 
452.75500 
456.03583 

459.31667 
462.59750 
465.87833 
469.15917 
472.44000 

475.72083 
479.00167 
482.28250 
485.56333 
488.84417 



150 

1 
2 
3 
4 



160 



170 

1 
2 



180 

1 
2 
3 
4 

5 
6 

7 



190 

1 
2 
3 
4 



492.12500 
495.40583 
498.68667 
501.96750 
505.24833 

508.52917 
511.81000 
515.09083 
518.37167 
521.65250 

524.93333 
528.21417 
531.49500 
534.77583 
538.05667 

541.33750 
544.61833 
547.89917 
551.18000 
554.46083 

557.74167 
561.02250 
564.30333 
567.58417 
570.86500 

574.14583 
577.42667 
580.70750 
583.98833 
587.26917 

590.55000 
593.83083 
597.11167 
600.39250 
603.67333 

606.95417 
610.23500 
613.51583 
616.79667 
620.07750 

623.35833 
626.63917 
629.92000 
633.20083 
636.48167 

639.76250 
643.04333 
646.32417 
649.60500 
652.88583 



200 

1 
2 
3 
4 

5 
6 

7 
8 



210 

1 
2 



220 

1 
2 
3 
4 



230 

1 
2 
3 
4 

5 
6 
7 
8 
9 

240 

2 
3 
4 



656.16667 
659.44750 
662.72833 
666.00917 
669.29000 

672.57083 
675.85167 
679.13250 
682.41333 
685.69417 

688.97500 
692.25583 
695.53667 
698.81750 
702.09833 

705.37917 
708.66000 
711.94083 
715.22167 
718.50250 

721.78333 
725.06417 
728.34500 
731.62583 
734.90667 

738.18750 
741.46833 
744.74917 
748.03000 
751.31083 

754.59167 
757.87250 
761.15333 
764.43417 
767.71500 

770.99583 
774.27667 
777.55750 
780.83833 
784.11917 

787.40000 
790.68083 
793.96167 
797.24250 
800.52333 

803.80417 
807.08500 
810.36583 
813.64667 
816.92750 



250 820.20833 



76 



250-500 



i.— MEASURES, WEIGHTS AND MONEY. 



11. — Lengths — Meters to Feet (Continued). 



I 



Meters. Feet. 

250- 820.20833 

1 823.48917 

2 826.77000 

3 830.05083 

4 833.33167 

5 836.61250 

6 839.89333 

7 843.17417 

8 846.45500 

9 849.73583 

260 853.01667 

1 856.29750 

2 859.57833 

3 862.85917 

4 866.14000 

5 869.42083 
8 872.70167 

7 875.98250 

8 879.26333 

9 882.54417 

270 885.82500 

1 889.10583 

2 892.38667 

3 895.66750 

4 898.94833 

5 902.22917 

6 905.51000 

7 908.79083 

8 912.07167 

9 915.35250 

280 918.66333 

1 921.91417 

2 925.19500 

3 928.47583 

4 931.75667 

5 935.03750 

6 938 31833 

7 941.59917 

8 944.88000 

9 948.16083 

290 951.44167 

1 954.72250 

2 958.00333 

3 961.28417 

4 964.56500 

5 967.84583 

6 971.12667 

7 974.40750 

8 977.68833 

9 980.96917 



Meters. Feet. 



300 984.25000 

1 987.53083 

2 990.81167 

3 994.09250 

4 997.37333 



320 

1 
2 
3 
4 

5 
6 
7 
8 



330 

1 
2 
3 
4 

5 
6 
7 
8 
9 

340 

1 



1,000.65417 
1,003.93500 
1,007.21583 
1,010.49667 
1,013.77750 



310 1,017.05833 

1 1,020,33917 

2 1.023.62000 

3 1,026.90083 

4 1,030.18167 



1,033.46250 
1,036.74333 
1,040.02417 
1,043.30500 
1,046.58583 

1,049.86667 
1,053.14750 
1,056.42833 
1,059.70917 
1,062.99000 

1,066.27083 
1,069.55167 
1,072.83250 
1,076.11333 
1,079.39417 

1,082.67500 
1,085.95583 
1,089.23667 
1,092.51750 
1,095.79833 

1,099.07917 
1,102.36000 
1 ',105.64083 
1,108.92167 
1,112.20250 

1,115.48333 
1,118.76417 
1,122.04500 
1,125.32583 
1,128.60667 

1,131.88750 
1,135.16833 
1,138.44917 
1,141.73000 
1,145.01083 



Meters. Feet. 



350 



1,148.29167 

1 1,151.57250 

2 1,154.85333 

3 1,158.13417 

4 1.161.41500 



370 

1 
2 



5 
6 

7 
8 
9 

380 

1 
2 
3 
4 

5 
6 

7 
8 
9 



1.164.69583 
1,167.97667 
1,171.25750 
1,174.53833 
1,177.81917 



360 1,181.10000 

1 1,184.38083 

2 1,187.66167 

3 1,190.94250 

4 1,194.22333 

5 1,197.50417 

6 1,200.78500 

7 1,204.06583 

8 1.207.34667 

9 1.210.62750 



1,213.90833 
1,217.18917 
1,220.47000 
1,223.75083 
1,227.03167 

1,230.31250 
1,233.59333 
1,236.87417 
1,240.15500 
1,243.43583 

1,246.71667 
1,249.99750 
1,253.27833 
1,256.55917 
1,259.84000 

1,263.12083 
1,266.40167 
1.269.68250 
1,272.96333 
1,276.24417 



Meters. Feet. 



400 1,312.33333 

1 1.315.61417 

2 1.318.89500 

3 1,322.17583 

4 1,325.45667 



390 1,279.52500 

1 1,282.80583 

2 1,286.08667 

3 1,289.36750 

4 1,292.64833 

5 1,295.92917 

6 1,299.21000 

7 1,302.49083 

8 1,305.77167 

9 1,309.05250 



420 

1 
2 
3 
4 

5 
6 

7 
8 
9 

430 

1 
2 



5 
6 
7 
8 
9 

440 

1 
2 
3 
4 

5 
6 
7 
8 
9 



1,328.73750 
1,332.01833 
1,335.29917 
1,338.58000 
1,341.86083 



410 1.345.14167 

1 1,348.42250 

2 1,351.70333 

3 1.354.98417 

4 1.358.26500 



1,361.54583 
1.364.82667 
1.368.10750 
1.371.38833 
1.374.66917 

1.377.95000 
1,381.23083 
1,384.51167 
1,387.79250 
1,391.07333 

1,394.35417 
1,397.63500 
1,400.91583 
1.404.19667 
1,407.47750 

1,410.75833 
1,414.03917 
1,417.32000 
1,420.60083 
1,423.88167 

1,427.16250 
1,430.44333 
1,433.72417 
1,437.00500 
1,440.28583 

1,443.56667 
1,446.84750 
1,450.12833 
1.453.40917 
1,456.69000 

1,459.97083 
1,463.25167 
1,466.53250 
1,469.81333 
1,473.09417 



Meters. Feet. 



450 



1,476.37500 

1 1,479.65583 

2 1,482.93667 

3 1,486.21750 

4 1,489.49833 



460 



5 
6 

7 
8 
9 

470 

1 
2 
3 



480 

1 
2 
3 
4 

5 
6 

7 



490 



1,492.77917 
1,496.06000 
1,499.34083 
1,502.62167 
1,505.90250 



509.18333 

1 1,512.46417 

2 1,515.74500 

3 1,519.02583 

4 1,522.30667 



1.525.58750 
1.528.86833 
1,532.14917 
1.535.43000 
1.538.71083 

1.541.99167 
1.545.27250 
1.548.55333 
1.551.83417 
1.555.11500 

1.558.39583 
1.561. £7667 
1.564.95750 
1.568.23833 
1,571.51917 

1,574.80000 
1.578.08083 
1.581.36167 
1.584.64250 
1,587.92333 

1,591.20417 
1.594.48500 
1,597.76583 
1,601.04667 
1,604.32750 



1,607.60833 

1 1,610.88917 

2 1,614.17000 

3 1,617.45083 

4 1,620.73167 

5 1.624.01250 

6 1.627.29333 

7 1.630.57417 

8 1.633.85500 

9 1,637.13583 

500 U640. 41667 



LENGTHS— METERS TO FEET. 



77 



500-750. 



11. — Lengths — Meters to Feet (Continued). 



Meters. Feet. 



Meters. Feet. 



Meters. Feet. 



Meters. Feet. 



Meters. Feet. 



500 1,640.41667 

1 1,643.69750 

2 1,646.97833 

3 1,650.25917 

4 1,653.54000 

5 1,656.82083 

6 1,660.10167 

7 1,663.38250 

8 1,666.66333 

9 1.669.94417 

510 1,673.22500 

1 1,676.50583 

2 1,679.78667 

3 1,683.06750 

4 1.686.34833 

5 1,689.62917 

6 1,692.91000 

7 1,696.19083 

8 1,699.47167 

9 1,702.75250 



550 

1 
2 
3 
4 



560 

1 
2 



1,804.45833 600 
1,807.73917 1 
1,811.02000 
1,814.30083 
1,817.58167 



1,706.03333 570 



1 1,709.31417 

2 1,712.59500 

3 1,715.87583 

4 1,719.15667 

5 1.722.43750 

6 1,725.71833 

7 1,728.99917 

8 1,732.28000 

9 1,735.56183 

530 1,738.84167 

1 1,742.12250 

2 1,745.40333 

3 1,748.68417 

4 1,751.96500 

5 1,755.24583 

6 1,758.52667 

7 1,761.80750 

8 1,765.08833 

9 1,768.36917 

540 1,771.65000 

1 1,774.93083 

2 1,778.21167 

3 1,781.49250 

4 1,784.77333 

5 1,788.05417 

6 1,791.33500 

7 1.794.61583 

8 1,797.89667 

9 1,801.17750 



1 
2 
3 
4 

5 
6 

7 
8 
9 

580 

1 
2 
3 
4 

5 
6 

7 



1,820.86250 
1.824.14333 
1,827.42417 
1,830.70500 
1,833.98583 

1,837.26667 
1,840.54750 
1,843.82833 
1,847.10917 
1.850.39000 

1,853.67083 
1,856.95167 
1,860.23250 
1,863.51333 
1,866.79417 

1,870.07500 
1,873.35583 
1,876.63667 
1,879.91750 
1,883.19833 

1.886.47917 
1,889.76000 
1,893.04083 
1,896.32167 
1,899.60250 

1,902.88333 
1,906.16417 
1,909.44500 
1,912.72583 
1,916.00667 

1.919.28750 
1,922.56833 
1,925.84917 
1,929.13000 
1,932.41083 



590 1,935.69167 

1 1,938.97250 

2 1,942.25333 

3 1,945.53417 

4 1,948.81500 

5 1,952.09583 

6 1,955.37667 

7 1,958.65750 

8 1,961.93833 

9 1,965.21917 



1,968.50000 
1,971.78083 
1,975.06167 
1,978.34250 
1,981.62333 

1,984.90417 
1,988.18500 
1,991.46583 
1,994.74667 
1.998.02750 



650 2,132.54167 

1 2,135.82250 

2 2,139.10333 

3 2,142.38417 

4 2,145.66500 



2,148.94583 
2,152.22667 
2,155.50750 
2.158.78833 
2,162.06917 



610 2.001.30833 

1 2.004.58917 

2 2,007.87000 

3 2,011.15083 

4 2,014.43167 

5 2,017.71250 

6 2,020.99333 

7 2.024.27417 

8 2.027.55500 

9 2.030.83583 

620 2.034.11667 

1 2.037.39750 

2 2,040.67833 

3 2.043.95917 

4 2.047.24000 



660 2,165.35100 

1 2,168.63083 

2 2,171.91167 

3 2,175.19250 

4 2.178.47333 



2,050.52083 
2.053.80167 
2.057.08250 
2,060.36333 
2,063.64417 



630 2,066.92500 

1 2,070.20583 

2 2,073.48667 

3 2,076.76750 

4 2,080.04833 



2,083.32917 
2,086.61000 
2.089.89083 
2.093.17167 
2.096.45250 



640 2.099.73333 

1 2,103.01417 

2 2.016.29500 

3 2,109.57583 

4 2,112.85667 



2,116.13750 
2,119.41833 
2.122.69917 
15,125.98000 
2.129.26083 



670 

1 
2 
3 
4 



680 

1 
2 
3 
4 



700 2.296.58333 

1 2.299.86417 

2 2,303.14500 

3 2.306.42583 

4 2,309.70667 

5 2,312 98750 

6 2,316 26833 

7 2,319.54917 

8 2,322.83000 

9 2,326.11083 

710 2,329.39167 

1 2,332.67250 

2 2,335.95333 

3 2.339.23417 

4 2,342.51500 



2,181.75417 
2,185.03500 
2,188.31583 
2,191.59667 
2,194.87750 

2,198.15833 
2,201.43917 
2,204.72000 
2,208.00083 
2.211.28167 

2,214.56250 
2,217.84333 
2,221.12417 
2,224.40500 
2.227.68583 

2.230.96667 
2,234.24750 
2.237.52833 
2,240.80917 
2,244.09000 

2,247.37083 
2,250.65167 
2,253.93250 
2,257.21333 
2,260.49417 



690 2,263.77500 

1 2,267.05583 

2 2,270.33667 

3 2,273.61750 

4 2,276.89833 



2,280.17917 
2.283.46000 
2,286.74083 
2.290.02167 
2.293.30250 



720 

1 



9 



2,345.79583 
2,349.07667 
2,352.35750 
2,355.63833 
2,358.91917 

2,362.20000 
2,365.48093 
2,368.76167 
2,372.04250 
2,375.32333 

2.378.60417 
2,381.88500 
2,385.16583 
2,388.44667 
2,391.72750 



730 2,395.00833 

1 2,398.28917 

2 2,401.57000 

3 2,404.85083 

4 2.408.13167 



2.411.41250 
2.414.69333 
2.417.97417 
2,421.25500 
2,424.53583 



740 2,427.81667 

1 2,431.09750 

2 2,434.37833 

3 2,437.65917 

4 2.440.94000 

5 2,444.22083 

6 2,447.50167 

7 2,450.78250 

8 2,454.06333 

9 2,457.34417 

750 2.460.62500 



78 



i.^MBASURES, WEIGHTS AND MONEY. 



750-1000. 



11. — Lengths — Meters to Feet (Concluded). 



Meters. Feet. 



750 2,460.62500 

1 2,463.90583 

2 2.467.18667 

3 2,470.46750 

4 2,473.74833 



2,477.02917 
2,480.31000 
2,483.59083 
2,486.87167 
2,490.15250 



760 2.493.43333 

1 2,496.71417 

2 2.499.99500 

3 2,503.27583 

4 2,506.55667 

5 2,509.83750 

6 2,513.11833 

7 2,516.39917 

8 2,519.68000 

9 2,522.96083 

770 2,526.24167 

1 2,529.52250 

2 2,532.80333 

3 2,536.08417 

4 2,539.36500 



Meters. Feet. 



800 

1 
2 
3 
4 

5 
6 

7 
8 



810 

2 
3 
4 



2,542.64583 
2,545.92667 
2,549.20750 
2,552.48833 
2,555.76917 



780 2,559.05000 

1 2,562.33083 

2 2,565.61167 

3 2,568.89250 

4 2,572.17333 

5 2,575.45417 

6 2,578.73500 

7 2,582.01583 

8 2,585.29667 

9 2,588.57750 

790 2,591.85833 

1 2,595.13917 

2 2,598.42000 

3 2,601.70083 

4 2,604.98167 

5 2,608.26250 

6 2,611.54333 

7 2,614.82417 

8 2,618.10500 

9 2.621.38583 



820 

1 
2 
3 
4 

5 
6 

7 
8 
9 



2.624.66667 
2.627.94750 
2,631.22833 
2,634.50917 
2,637.79000 

2,641.07083 
2,644.35167 
2,647.63250 
2,650.91333 
2.654.19417 

2.657.47500 
2,660.75583 
2.664.03667 
2.667.31750 
2,670.59833 

2,673.87917 
2,677.16000 
2,680.44083 
2,683.72167 
2,687.00250 

2,690.28333 
2.693.56417 
2,696.84500 
2.700.12583 
2,703.40667 

2,706.68750 
2,709.96833 
2.713.24917 
2.716.53000 
2,719.81083 



Meters. Feet. 



830 2.723.09167 

1 2,726.37250 

2 2.729.65333 

3 2,732.93417 

4 2.736.21500 



2,739.49583 
2,742.77667 
2.746.05750 
2,749.33833 
2,752.61917 



840 2,755.90000 

1 2,759.18083 

2 2,762.46167 

3 2,765.74250 

4 2,769.02333 



2,772.30417 
2,775.58500 
2,778.86583 
2,782.14667 
2,785.42750 



850 
1 

2 



860 

1 
2 
3 
4 

5 
6 

7 



870 

1 
2 
3 
4 



880 



2,788.70833 
2,791.98917 
2,795.27000 
2,798.55083 
2,801.83167 

2,805.11250 
2.808.39333 
2.811.67417 
2.814.95500 
2,818.23583 

2,821.51667 
2,824.79750 
2.828.07833 
2,831.35917 
2.834.64000 

2,837.92083 
2.841.20167 
2,844.48250 
2.847.76333 
2.851.04417 

2.854.32500 
2.857.60583 
2,860.88667 
2.864.16750 
2,867.44833 

2,870.72917 
2,874.01000 
2,877.29083 
2,880.57167 
2,883.85250 



Meters. Feet. 



2,887.13333 

1 2,890.41417 

2 2,893.69500 

3 2,896.97583 

4 2,900.25667 



2.903.53750 
2,906.81833 
2,910.09917 
2,913.38000 
2,916.66083 



890 2,919.94167 

1 2,923.22250 

2 2,926.50333 

3 2,929.78417 

4 2,933.06500 



2,936.34583 
2,939.62667 
2.942.90750 
2,946.18833 
2.949.46917 



2,952.75000 
2,956.03083 
2,959.31167 
2,962.59250 
2,965.87333 

2,969.15417 
2,972.43500 
2,975.71583 
2,978.99667 
2,982.27750 

2,985.55833 
2.988.83917 
2,992.12000 
2,995 40083 
2,998.68167 

3,001.96250 
3.005.24333 
3.008.52417 
3.011.80500 
3,015.08583 

3.018.36667 
3,021.64750 
3.024.92833 
3.028.20917 
3,031.49000 

3.034.77083 
3.038.05167 
3.041.33250 
3,044.61333 
3,047.89417 



930 3.051.17500 

1 3.054.45583 

2 3.057.73667 

3 3,061.01750 

4 3,064.29833 

5 3,067.57917 

6 3,070.86000 

7 3,074.14083 

8 3,077.42167 

9 3,080.70250 

940 3,083.98333 

1 3,087.26417 

2 3,090.54500 

3 3.093.82583 

4 3,097.10667 



900 

1 
2 
3 
4 

5 
6 

7 
8 



910 

1 
2 
3 
4 

5 
6 

7 



920 

1 
2 
3 
4 

5 
6 

7 



3,100.38750 
3,103.66833 
3,106.94917 
3,110.23000 
3,113.51083 



Meters. Feet. 



950 

2 
3 
4 

5 
6 
7 



960 

1 
2 
3 
4 



970 

1 
2 
3 
4 

5 
6 

7 



3,116.79167 
3,120.07250 
2,123.35333 
3,126.63417 
3,129.91500 

3,133.19583 
3,136.47667 
3,139.75750 
3,143.03833 
3,146.31917 

3,149.60000 
3,152.88083 
3,156.16167 
3,159.44250 
3,162,72333 

3,166.00417 
3,169.28500 
3,172.56583 
3,175.84667 
3,179.12750 

3,182.40833 
3,185.68917 
3,188.97000 
3,192.25083 
3.195.53167 

3,198.81250 
3,202.09333 
3,205.37417 
3,208.65500 
3,211.93583 



980 3,215.21667 

1 3,218.49750 

2 3,221.77833 

3 3,225.05917 

4 3,228.34000 



3,231.62083 
3,234-90167 
3,238.18250 
3,241.46333 
3,244.74417 



990 3.248.02500 

1 3.251.30583 

2 3.254.58667 

3 3.257.86750 

4 3,261.14833 

6 3,264.42917 

6 3,267.71000 

7 3,270.99083 

8 3,274.27167 

9 3,277.55250 

1000 3.280.83333 



AREAS— METRIC AND ENGLISH. 



79 



11a.— Lengths— Foreign Measures. 
Russian. 
1 archlne = 16 verskops = 0.71120 meter = 28 inches. 
1 verst = 500 sachines (1 sachine = 7 feet) = 1066.8 meters = 3500 feet. 

Spanish. 
1 vara = 0.8350216 meter = 2.739567 feet = 32.8748 inches. 
Japanese. 
1 shaku = 10 suns = 100 bus = 1000 rins = 10,000 mos = 100,000 shis = 0.303^03 

m^ter= 0.994192 feet = 11.93^03 inches. 
1 pi = 36chos = 2160 kens = 12,960 shakus = 3927.2^72 meters = 12884.727 feet. 

Chinese. 
1 ts'un = 10 fens = 100 lis = 1000 haos = 3.2 centimeters = 1.25984 inches. 
1 chang = 10 ch'ihs = 0.32 meter = 1.04987 feet. 
1 pu = 1.6 meters = 5.24933 feet = 62.992 inches. 

lib.— Areas— Foreign Measures. 

Russian. 

1 dessiatine = 2400 square sachines = 117,600 square feet = 10,925.44 square 
meters = 109.2544 are = 2.69972 acres. 

Spanish. 

1 fanegada = 69,134.08 square feet = 6422.792 square meters = 64.22792 are 

= 1.5871 acres. 

Japanese. 
1 bu = 10 gos = 100 shakus = 98.841753 square feet = 9.182736 square meters. 

Chinese. -" ' 

1 mu = 10 fens = 100 lis = 1000 haos = 614.4 square meters. 
1 ch'ing = 100 mtis = 61,440 square meters. 



12.— Areas. 





Sq. Ins. 


Sq. Feet. 


Sq. Yds. 


Acres. 


1 sq, millimeter (sq. mm) = 


.00155000 


.00001076 






100 sq. millimeters = 










1 sq. centimeter (sq. cm) = 


.15499969 


.00107639 


.0001196 




100 sq. centimeters = 










1 sq. decimeter (sq. dm) = 


15.499969 


.10763867 


.0119599 




100 sq. decimeters ( = 1 centare) = 










1 sq. meter. = 


1549.9969 


10.763867 


1.1959853 


.0002471 


100 sq. meters ( = 1 are) = 










1 sq. dekameter (sq. Dm) = 


154999.69 


1076.3867 


119.59853 


.0247104 


100 sq. dekameters (=1 hectare) 










= 1 sq. hectometer. = 


Sq. Miles 


107638.67 


11959.853 


2.4710439 


100 sq. hectometers = 










1 sq. kilometer (sq. Km) = 


.3861 


10763867 


1195985.3 


247.10439 


100 sq. kilometers == 










1 sq. myriameter (sq. M m) = 


38.61 






24710.439 



Logarithms of Numbers in Table 12. 



Log 15.499969 = 1.1903308 
Log 119.59853 = 2.0777259 



Log 10.763867 
Log 2.4710439 



1.0319683 
0.3928804 



80 



4.— MEASURES, WEIGHTS AND MONEY. 



13. — Areas, Equivalents. 1-10. 



Square 
Inches. 


Square 
Millimeters. 


Square 
Inches. 


Square 
Centimeters. 


Square 
Feet. 




Square 
Meters. 


0.00155 
0.00310 
0.00465 
0.00620 


= 


1 

2 
3 
4 


0.1550 
0.3100 
0.4650 
0.6200 


= 


1 
2 
3 
4 


1 
2 
3 
4 


= 


0.09290 
0.18581 
0.27871 
0.37161 


0.00775 
0.00930 
0.01085 
0.01240 
0.01395 


= 


5 
6 

7 
8 
9 


0.7750 

0.9300 

1 

1.0850 

1.2400 


= 


5 
6 

6.452 

7 

8 


5 
6 
7 
8 
9 


= 


0.46452 
0.55742 
0.65032 
0.74323 
0.83613 


1 
2 
3 
4 


^^ 


645.16 
1,290.33 
1,935.49 
2,580.65 


1.3950 
2 
3 
4 


= 


9 

12.903 
19.355 
25.807 


10.764 
21.528 
32.292 
43.055 


= 


1 

2 
3 
4 


5 
6 
7 
8 
9 


= 


3,225.81 
3,870.98 
4,516.14 
5,161.30 
5,806.46 


5 
6 

7 
8 
9 


= 


32.258 
38.710 
45.161 
51.613 
58.065 


53.819 
64.583 
75.347 
86.111 
96.875 


= 


5 
6 
7 
8 
9 


Square 
Yards. 




Square 
Meters. 


Square 
Miles. 


Square 
Kilometers. 


Acres. 




Hectars. 


1.1960 

2 

2.3920 


= 


0.8361 
1 

0.6723 
2 


0.3861 
0.7722 

1.1583 


= 


1 
2 

2.5900 
3 


1 
2 

2.471 
3 


= 


0.4047 
0.8094 
1 
1.2141 


3 

3.5880 
4 

4.7839 
5 


= 


2.5084 

3 

3.344^ 

4 

4.1807' 


1.5444 

1.9305 

2 

2.3166 

2.7027 


= 


4 
5 

5.1800 

6 

7 


4 

4.942 
5 
6 
7 


= 


1.6187 

2 

2.0234 

2.4281 

2.8328 


5.9799 
6 
7 
7.1759 


s= 


5 

5.0168 
6.8529 
6 


3 

3.0888 
3.4749 
4 


^^ 


7.7700 
8 
9 
10.3600 


7.413 

8 

9 

9.884 


= 


3 

3.2375 
3.6422 
4 


8 

8.3719 
9 

9.5679 
10.7639 


t= 


6.6890 

7 

7.5252 

8 

9 


5 
6 
7 
8 
9 


= 


12.9500 
15.5400 
18.1300 
20.7200 
23.3100 


12.355 
14.826 
17.297 
19.768 
22.239 


= 


5 
6 
7 

8 
9 



AREAS, VOLUMES— METRIC AND ENGLISH. 



81 



14. — Land Measure (Square). 

1 square inch = 

144 square inches = 1 square foot ■■ 

9 square feet = 1 square yard 

30i square yards ( = 272i sq. ft.) = 1 square rod ■■ 

40 square rods ( = 1 2 1 sq. yds. == 

10890 sq. ft.) = 1 rood 

4 roods (=160 sq. rods = 4840 sq. 

yds. = 43560 sq. ft.) = I acre 

640 acres ( = 3,097,600 sq. yds.= 

27,878,400 sq. ft.) =1 square mile- 

36 square miles ( = 23040 acres = 

1,003,622,400 sq. ft.) = 1 township 



6.4516 sq. centimeters 
= 0.09290341 sq. meter. 
= 0.836131 sq. meter. 
= 25.292954 sq. meters. 

= 0.1011718 hectar. 

= 0.4046873 hectar. 

= 258.99985 hectars. 

= 9323.9945 hectars. 



15. — ^Texas Land Measure. 
(Also used in Mexico, New Mexico, Arizona and California.) 



26 


.000,000 


sq. 


varas (sq. 


of 5 


.099 




varas) = 1 league and 
























1 labor 


rr= 


4 


,605 


5 


acres. 


1 


,000,000 


sq. 


varas (sq. 


of 1 


,000 




varas) = 1 labor 


= 




177 


136 acres. 


25 


,000,000 


sq. 


varas (sq. 


Of 5 


,000 




varas) = 1 league 


= 


4 


,428. 


4 


acres. 


12 


,500,000 


sq. 


varas (sq. 


of 3 


,535. 


5 


varas) = i league 


=. 


2 


,214 


2 


acres. 


8 


,333,333 


sq. 


varas (sq. 


Of 2 


,886. 


7 


varas) = ^ league 


= 


1 


,476. 


13 


acres. 


6 


,250,000 


sq. 


varas (sq. 


of 2 


,500 




varas) = i league 


= 


1 


.107. 


1 


acres. 


7 


.225,600 


sq. 


varas (sq. 


of 2 


,688 




varas) 


= 


1 


.280 




acres. 


3 


,612,800 


sq. 


varas (sq. 


of 1 


,900. 


8 


varas) = 1 section 


= 




640 




acres. 


1 


,806,400 


sq. 


varas (sq. 


of 1 


,344 




varas) = i section 


= 




320 




acres. 




903.200 


sq. 


varas (sq. 


of 


950. 


44 


varas) = \ section 


== 




160 




acres. 




451,600 


sq. 


varas (sq. 


of 


672 




varas) = \ section 


= 




80 




acres. 




225.800 


sq. 


varas (sq. 


of 


475 




varas) = 1.16 section = 




40 




acres. 




5,645. 


376 sq. 


varas (sq. 


of 


75. 


1 3 7 varas) = 4, 84 sq. yd 


.=s 




1 




acre. 



To find the number of acres in any number of square 
the latter by 177 (or to be more exact, by 177i), and cut 

1 vara = 33| inches 1,900.8 varas = 1 mile 



varas, multiply 
off six decimals. 



16. — Weights and Measures of the Philippines. 



1 pulgada (12 linea) = 
1 pie = 

1 vara = 

1 gantah = 

1 caban = 



,927 inches. 
11.125 inches. 
33.375 inches. 

.8796 gallon. 
21.991 gallons. 



1 libra (16 onzo) = 

1 arroba = 

1 catty (16 tael) = 

Ipecul (100 catty) = 



1.0144 1b. av. 
25.360 lb. av. 

1.394 lb. av. 
;39.482 lb. av. 



17. — Volumes. 





Cu. Ins. 


Cu. Feet. 


Cu. Yds. 


( 1 cubic c 


m) =1 milliliter (ml)... . = 
= 10 cubic cm) = 

1 centiliter (c 1) = 

= i\j cubic d m) = 

1 deciliter d\) = 

1 cubic dm) = 

1 litter . = 


.06102338 

.61023378 

6.1023378 

61.023378 

610.23378 

6102.3378 

61023.378 
610233.78 


.00003531 

.00035314 

.00353145 

.03531445 

.35314455 

3.5314455 

35.314455 
353.14455 




10 milliliters ( = 




10 centiliters ( = 




10 deciliters ( = 


.00130794 


10 liters (=10 cubic d m) 

\ dekaliter (Dl)....= 
10 dekaliters ( = ^^ cubic meter) = 

\ hectoliter (HI)... .= 
10 hectoliters ( = 1 cubic meter) = 

I kiloliter (K\) = 

10 kiloliters = 1 myrialiter (Ml).. . = 


.01307943 

.13079428 

1.3079428 
13.079428 







82 



i.— MEASURES, WEIGHTS AND MONEY. 
18. — Volumes. Equivalents. 1-10. 



Cubic 
Inches. 



Cubic 
Millimeters 



Cubic Cubic 
Inches. Centimet's 



Cubic 
Feet. 



Cubic 
Meters. 



Cubic Cubic 
Yards. Meters. 



0.000061= 1 

0.000122= 2 

0.000183= 3 

0.000244= 4 



0.000305 = 
0.000366 = 
0.000427 = 
0.000488 = 
0.000549 = 



= 16.387.2 

= 32,774.3 

= 49,161.5 

= 65,548.6 

= 81,935.8 
= 98,323.0 
= 114,710.1 
= 131,097.3 
= 147,484.5 



0.0610 = 
0.1220 = 
0.1831 = 
0.2441 = 

0.3051 = 
0.3661 = 
0.4272 = 

0.4882 = 
0.5492 = 



16.3872 
32.7743 
49.1615 
65.5486 

81.9358 

98.3230 

114.7101 

131.0973 

147.4845 



1 
2 
3 
4 

5 

6 

7 
8 
9 

35.314 = 

70.629 = 

105.943 = 

141.258 = 

176.572 = 
211.887 = 
247.201 = 
282.516 = 
317.830 = 



0.02832 
0.05663 
0.08495 
0.11327 

0.14159 
0.16990 
0.19822 
0.22654 
0.25485 

1 

2 
3 
4 

5 

6 

7 
8 
9 



1 =0.7645 
1.3079=1 

2 =1.5291 
2.6159 = 2 

3 =2.2937 
3.9238=3 

4 =3.0582 

5 =3.8228 
5.2318 = 4 

6 =4.5874 
6.5397=5 

7 =5.3519 
7.8477=6 

8 =6.1165 

9 =6.8810 
9.1556=7 

10.4635 = 8 
11.7715=9 



19. — Cubic Measure. 

1 cubic inch= 16 . 3872 cubic centimeters. 

1728 cubic inches . . . = 1 cubic foot = .02832 cubic meter. 

27 cubic feet ( = 46656 cu. ins.) = 1 cubic yard = . 7646 cubic meter. 
16 cubic feet ( = 27648 cu. ins.) = 1 cord foot = . 45307 cubic meter. 
8 cord feet ( = 4|^ cu. yds.= 
128cu. tt. = 221184cu. ins.) =1 cord (wootf) =3.6246 cubic meter. 



20. — Capacities (Liquid). 



(1 cubic cm) ' = 1 milliliter (ml).. 
10 milliliters ( = 10 cubic c m) = 

1 centiliter (c 1) . . . 
10 centiliters ( = iV cubic d m) = 

1 deciliter (d 1) . . . 
10 deciliters (=1 cubic d m) = 

1 liter 

10 liters ( = 10 cubic d m) = 

1 dekaliter (D 1) . . 
10 dekaliters d^ cubic meter) = 

1 hectoliter (HI).. 
10 hectoliters (=1 cubic meter) = 

Ikiloliter (Kl)... 
10 kiloliters = 1 myrialiters (M 1) 



U.S. 

Apoth 
Scruples 



81153168 

8.1153168 
U.S. 

Liquid 
Gallons. 

26417047 

2.6417047 

26.417047 

264.17047 
2641.7047 



U.S. 

Apoth. 
Drams. 



.27051056 
2.7051056 
27.051056 
270.51056 
2705.1056 
27051.056 



U.S. 

Liquid 
Ounces. 



.03381382 

.33813820 

3.3813820 

33.813820 

338.13820 

3381.3820 

33813.820 
338138.20 



U.S. 

Liquid 
Quarts. 



.00105668 

.01056682 

.10566819 

1.0566819 

10.566819 

105.66819 

1056.6819 
10566.819 



CAPACITIES— METRIC AND ENGLISH. 



21. — Capacities, Equivalents. 1-10. 



Millili- 
ters. 
(cc.) 



U.S. 

Liquid 

Oz. 



= 0.03381 
= 0.06763 
= 0.10144 
= 0.13526 

= 0.16907 
= 0.20288 
= 0.23670 
= 0.27051 
= 0.30432 



29.574=1 

59.147=2 

88.721 = 3 

118.295=4 

147.869=5 
177.442=6 
207.016=7 
236.590=8 
266.163=9 



U.S. 
Millili- Apothe- 
ters. caries' 
(cc.) Drams 



1 =0.2705 

2 =0.5410 

3 =0.8115 
3.6967=1 



= 1.0820 
= 1.3525 
= 1.6231 
= 1.8936 



7.3934=2 

8 =2.1641 

9 =2.4346 
11.0901 = 3 
14.7869=4 

18.4836=5 
22.1803 = 6 
25.8770=7 
29.5737=8 
33.2704 = 9 



U.S. 
Apothe- Millili- 

caries' ters. 
Scruples, (cc.) 



0.8115= 
1 

1.6231= 
2 

2.4346= 

3 

3.2461= 

4 

4.0577= 

4.8692= 
5 

5.6807= 
6 



1 

1.2322 

2 

2.4645 

3 

3.6967 
4 

4.9290 
5 

6 

6.1612 

7 

7.3934 



6.4923= 8 

7 = 8.6257 
7.3038= 9 

8 = 9.8579 

9 =11.0901 



U. S. 
Liquid 
Quarts. Liters. 



1 =0.94636 
1.05668=1 

2 =1.89272 
2.11336=2 

3 =2.83908 
3.17005=3 

4 =3.78543 
4.22673 = 4 

5 =4.73179 

5.28341=5 

6 =5.67815 
6.34009 = 6 

7 =6.62451 

7.39677=7 

8 =7.57088 
8.45345=8 

9 = 8.51723 
9.51014=9 



U.S. 
Liquid 
Gallons. Litera. 



0.26417= 1 

0.52834= 2 

0.79251= 3 

1 = 3.78543 



1.05668= 
1.32085= 
1.58502= 
1.84919= 
2 = 



4 
5 
6 

7 
7.57087 



2.11336= 8 
2.37753= 9 

3 =11.35630 

4 =15.14174 



= 18.92717 
= 22.71261 
= 26.49804 
= 30.28348 
= 34.06891 



22. — Liquid Measure. 

1 gill . . 

'. = 1 pint . . . 

= 1 quart. . 

32 gills) = 1 gallon. 



4 gills 

2 pints ( = 8 gills) . . 

4 quarts ( = 8 pints ■■ 

3U gallons (=126 quarts=252 pints) . . 

2 barrels ( = 63 gals. = 252 qts. = 504 pts.) . 

2 hogsheads( = 4 bbls. = 126 gals = 504qts.) 

2 pipes ( = 8bbls. = 252 gals. = 1008 qts.).. 

1 tiera= 42 gals. 1 puncheon == 84 gals. 



= 0.1183 
= 0.47318 
= 0.94636 
= 3.78543 
= 119.2412 
= 238.4824 



liters 



= 1 barrel 

= 1 hogshead. 

= 1 pipe, or butt = 476 . 9647 

= \iun =953.9295 



23. — Apothecaries' Measure (Fluid). 

1 minim {drop) = . 00006161 liters 

60 minims =1 -fluid drachm, = . 0036967 " 

8 fluid drachms ( = 480 minims) = 1 ilu%d ounce. . = . 029574 

16 fluid ounces (=128 drachms=7680 

minims) = lpint =0.473179 

8 pints (=128 fluid ounces =1024 

drachms =61440 minims) = 1 gallon =3.785434 



84 



i— MEASURES, WEIGHTS AND MONEY. 
24.— Capacities (Dry). 





U.S. 

Dry 

Pints. 


U.S. 

Dry 

Quarts. 


U.S. 
Pecks. 


U.S. 
Bushels. 


(1 cubic c 


m) = 1 milliliter (m 1) . . . = 
= 10 cubic cni) = 

1 centiliter (c 1) = 

= T^cubic d m) = 

1 deciliter (dl)....= 
1 cubic dm)^ 

1 liter = 


.00181616 

.01816155 

.18161551 

1.8161551 

18.161551 

181.61551 

1816.1551 
18161.551 


.00090808 

.00908078 

.09080775 

.90807754 

9.0807754 

90.807754 

908.07754 
9080.7754 






10 milliliters ( = 


.00113510 

.01135097 

.11350969 

1.1350969 

11.350969 

113.50969 
1135.0969 




10 centiliters ( = 
10 deciliters ( = 


.00283774 
.02837742 


10 liters (=10 cubic d m) = 

1 dekaliter (D 1) . . . = 
10 dekaliters ( = ^ cubic meter) =- 

\ hectoliter Yi\)....= 
10 hectoliters ( = 1 cubic meter) = 

Ikiloliter CK\)....= 
10 kiloliters = 1 myrialiter (M 1) . . = 


.28377423 

2.8377423 

28.377423 
283.77423 







25. — Capacities, Equivalents. 1-10. 



U.S. 
Dry' 
Quarts. Liters. 



0.9081=1 

1 =1.1012 
1.8162=2 

2 =2.2025 

2.7242 = 3 

3 =3.3037 
3.6323 = 4 

4 =4.4049 
4.5404=5 

5 =5.5061 
5.4485=6 

6 =6.6074 
6,3565=7 

7 =7.7086 
7.2646=8 

8 =8.8098 
8.1727=9 

9 =9.9110 



U. S. 
Pecks. 



Liters. 



0.11351^ 
0.22702= 
0.34053: 
0.45404= 

0.56755= 
0.68106= 
0.79457= 
0.90808= 



1.02157= 

2 

3 

4 

5 
6 
7 
8 
9 



= 8.80982 

9 

17.61964 
26.42946 
35.23928 

44.04910 
52.85892 
61.66874 
70.47856 
79.28838 



Deka- 
liters. 



U. S. 
Pecks. 



0.8810= 

1.7620= 

2 = 

2.6429 = 

3 = 
3.5239 = 

4 = 
4.4049 = 

5 = 

5.2859 = 

6 = 
6.1669= 

7 = 

7.0479= 
7.9288= 

8 = 
9 



1.1351 

2 

2.2702 

3 

3.4053 
4 

4.5404 
5 

5.6755 
6 

6.8106 
7 

7.9457 

8 

9 

9.0808 
10.2159 



U. S. Hecto- 
Bushels. liters. 



1 =0.35239 

2 =0.70479 
2.83774=1 

3 =1.05718 

4 =1.40957 

5 =1.76196 
5.67548=2 

6 =2.11436 

7 =2.46675 

8 =2.81914 
8.51323 = 3 

9 =3.17154 
11.35097=4 

14.18871 = 5 
17.02645=6 
19.86420=7 
22.70194 = 8 
25.53968=9 



U. S. Hecto- 
Bushels liters 
per per 

Acre. Hectar. 



1 =0.87078 
1.14840=1 

2 =1.74156 
2.29680=2 

3 =2.61233 
3.44519 = 3 

4 =3.48311 
4.59359 = 4 

5 =4.35389 

5.74199=5 

6 = 5.22467 
6.89039-6 

7 =6.09545 

8 =6.96622 
8.03879=7 

9 =7.83700 
9.18719 = 8 

10.33558=9 



26. — Dry Measure. 

1 pint = . 55061 liter. 

2 pints = 1 quart = 1.10123 liters. 

8 quarts ( = 16 pints) = 1 peck = 8 . 80982 liters. 

4 pecks (=32 quarts = 64 pints) =1 struck 6«5/j^/= 0. 35239 hectoliter. 

Note. — 1 heaped bushel =li struck bushels The cone of the heaped 
bushel must not be less than 6 inches high. 



WEIGHTS— METRIC AND ENGLISH. 



85 



27. — Masses (Weights). 





Grains. 


Avoir. 
Ounces. 


Troy 
Ounces. 


Troy 
Pounds. 


(*1 cubic mm) 1 milligram (m g) . = 
10 milligrams (*10 cubic m m) 

= 1 centigram (c g) . . = 


.01543236 

.15432356 

1.5432356 

15.432356 

Avoir. 

Pounds. 

.02204622 

.22046223 

2.2046223 

22.046223 

220.46223 

2204.6223 








.00035274 
.00352740 
.03527396 

.35273957 
3.5273957 
35.273957 
352.73957 
3527.3957 
35273.957 


.00032151 
.00321507 
.03215074 

.32150742 
3.2150742 
32.150742 
321.50742 
3215.0742 
32150.742 




10 centigrams (*iV cubic c m) 

= 1 decigram (d g) . . . = 
10 decigrams (*1 cubic c m) 

= 1 gram = 


.00026792 
.00267923 


1 grams ( * 1 cubic c m) 

= 1 dekagram (Dg) . . = 
10 dekagrams (*1 deciliter) 

= 1 hectogram (H g) . = 
10 hectograms (*1 liter) 

= 1 kilogram (Kilo) . = 
10 kilograms (*10 liters) 

= 1 myriagram (Mg) = 
10 myriagram (*1 hectoliter) 

= 1 quintal = 


.02679229 
.26792285 
2.6792285 
26.792285 
267.92285 


1 quintals ( * 1 cubic meter) 

= 1 miller or tonneau = 


2679.2285 



^Equivalent quantity of water at max. density. 



28. — Weights, Equivalents. 1-10. 



Grains. Grams 



Avoirdu- 
pois 
Ounces. Grams. 



Troy 
Ounces. Grams. 



Avoirdu- 
pois Kilo- 
Pounds, grams 



Troy Kilo- 
Pounds, grams. 



= 0.06480 
= 0.12960 
= 0.19440 
= 0.25920 

= 0.32399 
= 0.38879 
= 0.45359 
= 0.51839 
= 0.58319 



0.03527= 1 

0.07055= 2 

0.10582= 3 

0.14110= 4 



0.03215= 1 

0.06430= 2 

0.09645= 3 

0.12860= 4 



1 =0.45359 

2 =0.90718 
2.20462=1 

3 =1.36078 



15.4324=1 
30.8647 = 2 
46.2971 = 3 
61.7294 = 4 

77.1618=5 

92.5941 = 6 

108.0265=7 

123.4589 = 8 

138.8912=9 



0.17637= 
0.21164= 
0.24692= 
0.28219= 
0.31747= 



: 5 

: 6 

: 7 

: 8 

9 

: 28.3495 
: 56.6991 
: 85.0486 
113.3981 

141.7476 
170.0972 
198.4467 
226.7962 
255.1457 



0.16075= 
0.19290= 
0.22506= 
0.25721= 
0.28936= 



= 5 

: 6 

: 7 

■■ 8 
9 

: 31.10348 
-■ 62.20696 
■■ ^3.31044 
124.41392 

155.517401 

186.62088 

217.72437 

248.82785 

279.93133 



4 

4.40924: 

5 

6 

6.61387= 

7 

8 

8.81849= 
9 

1.02311= 
13.22773= 
15.43236= 
17.63698= 
19.84160= 



=1.81437 
= 2 
: 2.26796 
: 2.72155 
3 

^3.17515 
3.62874 
4 
4.08233 

5 
6 
7 
8 
9 



1 =0.37324 

2 =0.74648 
2.67923=1 

3 =1.11973 

4 =1.49297 

5 =1.86621 
5.35846=2 

6 =2.23945 

7 =2.61269 

8 =2.98593 
8.03769 = 3 

9 =3.35918 
10.71691 = 4 

13.39614 = 5 
16.07537 = 6 
18.75460=7 
21.43383=8 
24.11306=9 



86 i.^MEASURES, WEIGHTS AND MONEY. 



29. — *Apothecaries Weight. 

= 1 grain = . 06480 gram. 

20 grains = 1 scruple. . . . = 1 . 29598 grams. 

3 scruples ( = 60 grains) =1 dram = 3 . 88794 grams. 

8 drams ( = 24 scruple = 480 grains) = 1 ounce = 31 . 10348 grams. 

12 ounces ( = 96 drams=228 scruples = 

5760 grains) = 1 pound = . 37324 kilogram. 



30. — *Troy Weight. 

1 grat'w .....= . 06480 gram. 

24 grains = 1 pennyweight = 1 . 5551 7 grams. 

20 pennyweights ( = 480 grains) =1 ounce = 81 . 10348 grams. 

12 ounces ( = 240 pennyweights = 5760 

grains) = 1 pound = . 37324 kilogram. 



31. — Avoirdupois Weight (Short Tons). 

1 grain = . 06480 gram. 

27^ grains ( = 27.34375 grains) .... = 1 dram = 1.771845 grams. 

16 drams ( = 437i grains) =1 ounce =28.3495 grams. 

16 ounces ( = 256 drams =7000 

grains) = 1 pound = . 4535924 kilogram . 

25 pounds ( = 400 ounces) = 1 quarter = 11 . 339811 kilograms. 

4 quarters =1 hundred weight = 4:5 . 35924 kilograms. 

20 hundred weight (2000 lbs.) =lton = 907 . 18486 kilograms. 



32. — Avoirdupois Weight (Long Ton). 

1 grain =0.06480 gram. 

27H grains (= 27.34375 grains).. = 1 draw = 1.771845 grams. 

16 drams = 1 ounce = 28 . 3495 grams. 

16 ounces = 1 pound =0.4535924 kilogram. 

112 pounds =1 hundred weight = 50.80235 kilograms. 

20 himdred weight . . . (2240 lbs.) = 1 ton = 1016 . 047 kilograms. 



* The grain, ounce and potmd Apothecary and Troy weight are respect- 
ively equivalent. 



WEIGHTS— METRIC AND ENGLISH. 



87 



J. — Comparison op the Various Tons and Pounds in usb 

IN THE United States. 

From 1 to 10 Units. 



Long Tons. 


Short Tons. 


Metric Tons. 


Kilograms. 


Avoirdupois 
Pounds. 


Troy Pounds. 


.00036735 
.00044643 
.00073469 
.00089286 
.00098421 


.00041143 
.00050000 
.00082286 
.00100000 
.00110231 


.00037324 
.00045359 
.00074648 
.00090718 
.00100000 


.37324 
.45359 
.74648 
.90718 

1 


.822857 

1 

1.64571 

2 

2.20462 


1.21528 
2 

2.43056 
2.67923 


.00110204 
.00133929 
.00146939 
.00178571 
.00183673 


.00123429 
.00150000 
.00164571 
.00200000 
.00205714 


.00111973 
.00136078 
.00149297 
.00181437 
.00186621 


1.11973 
1.36078 
1.49297 
1.81437 
1.86621 


2.46857 

3 

3.29143 

4 

4.11429 


3 

3.64583 
4 

4.86111 
5 


.00196841 
.00220408 
.00223214 
.00257143 
.00267857 


.00220462 
.00246857 
.00250000 
.00288000 
.00300000 


.00200000 
.00223945 
.00226796 
.00261269 
.00272155 


2 

2.23945 
2.26796 
2.61269 
2.72155 


4.40924 

4.93714 

5 

5.76000 

6 


5.35846 

6 

6.07639 

7 

7.29167 


.00293878 
.00295262 
.00312500 
.00330612 
.00357143 


.00329143 
00330693 
.00350000 
.00370286 
.00400000 


.00298593 
.00300000 
.00317515 
.00335918 
.00362874 


2.98593 

3 

3.17515 

3.35918 

3.62874 


6.58286 

6.61387 

7 

7.40571 

8 


8 

8.03769 
8.50694 
9 
9.72222 


.00393683 
.00401786 
.00492103 
.00590524 
.00688944 


.00440924 
.00450000 
.00551156 
.00661387 
.00771618 


.00400000 
.00408233 
.00500000 
.00600000 
.00780000 


4 

4.08233 

5 

6 


8.81849 

9 

11.0231 
13.2277 
15.4324 


10.71691. 
10.93750 
13.39614 
16.07537 
18.75460 


.00787365 
.00885786 
.89287 
.98421 

1 


.00881849 
.00992080 

1.10231 
1.12000 


.00800000 

.0090000 

.90718 

1 

1.01605 


8 
9 

907.18 
1.000.00 
1,016.05 


17.6370 
19.8416 
2,000.00 
2.204.62 
2,240.00 


21.48383 
24.11306 

2,430.56 

2.679.23 

2.722.22 


1.78571 

1.96841 

2 

2.67857 

2.95262 


2/ 

2 20464 

2.24000 

3 

3.30693 


1.81437 

2 

2.03209 

2.72155 

3 


1,814.37 
2,000.00 
2,032.09 
2,721.55 
3,000.00 


4,000.00 
4.409.24 
4,480.00 
6,000.00 
6,613.87 


4,861.11 
5.358.46 
5.444.44 
7,291.67 
8,037.69 


3 

3.57143 
3.93683 
4 
4.46429 


3.36000 

4 

4.40924 

4.48000 

5 


3.04814 

3.62874 

4 

4.06419 

4.53592^ 


3.048.14 
3,628.74 
4.000.00 
4.064.19 
4,535.92 


6,720.00 
8,000.00 
8.818.49 
8.960.00 
10,000.00 


8,166.67 

9.722.22 

10.716.91 

10,888.89 

12.152.78 


4.92103 

5 

5.35714 

5.90524 

6 


5.51156 

5.60000 

6 

6.61387 

6.72000 


5 

5.08024 
5.44311 
6 
6.09628 


5,000.00 
5.080.24 
5,443.11 
6.000.00 
6.096.28 


11,023.11 
11.200.00 
12.000.00 
13,227.73 
13.440.00 


13,396.14 
13,611.11 
14,583.33 
16.075.37 
16,333.33 


6.25000 

6.88944 

7 

7.14286 

7.87365 


7 

7.71618 
7.84000 
8 
8.81849 


6.35029 

7 

7.11232 

7.25748 

8 


6,350.29 
7,000.00 
7,112.32 
7,257.48 
8,000.00 


14,000.00 
15,432.36 
15,680.00 
16,000.00 
17,636.98 


17,013.89 
18,754.60 
19.055.56 
19,444.44 
21.433.83 


8 

8.03571 
8.85786 
9 


8.96000 
9 

9.92080 
10.08000 


8.12838 
8.16466 
9 
9.14442 


8,128.38 
8,164.66 
9,000.00 
9,144.42 


17.920.00 
18,000.00 
19,841.60 
20,160.00 


21,777.78 
21.875.00 
24.113.06 
24,500.00 



88 



i.— MEASURES, WEIGHTS AND MONEY. 



34. — Simple and Compound Units in Common Use, Equivalents. 
Base: 1 Meter = 39.37 Inches, as per U. S. Law. 



To reduce A to M. 



Mult, by Log. 



To reduce M to A. 



Log. Mult, by 



M 



LENGTH. 



Mils(1000thsofanln.). 

lOOths of an Inch 

64ths of an inch 

Inches 

Feet 

Yards 

Rods 

Chains (66 ft.) 

Stations (100 It.) 

Miles 



.0254 
.254 
.3968758 
2.540005 
.3048006 
.9144018 
5.029210 
20.11684 
30.48006 
1.609347 



8.4048346 

9.4048346 

9.5986546 

0.4048346 

9.4840158 

9.961137 

0.7014998 

1.3035597 

1.4840158 

0.2066497 



5951654 
5951654 
4013454 
5951654 
5159842 
0388629 
2985002 
6964403 
5159842 
7933503 



39.37 

3.937 

2.51968 

.3937 

28083^3 

09361^1 

1988384 

0497096 

0328083 

.62137 



Millimeters 

Millimeters 

Millimeters 

Centimeters 

Meters 

Meters 

Meters 

Meters 

Meters 

Kilometers 



AREA (Square or circular). 



Square mils 

Square inches 

.Square feet 

Square yards 

Square rods 

Acres 

Quarter sections (160 a.) 

Sections (sq. miles) 

Townships (36 sec). . . . 



.0006452 

6.451626 

.0929034 

.8361307 

25.29295 

0.4046873 

64.75 

258.99985 

9323.9945 



8096692 
8096692 
9680316 
9222742 
4029996 
6071196 
8112402 
4132995 
9696020 



1903308 
1903308 
0319684 
0777258 
5970004 
3928804 
1887598 
5867005 
0303980 



1549.997 

.1549997 

10.76387 

1.195985 

.0395367 

2.4710439 

.015444 

.003861 

.00010725 



Sq. millimeters 
Sq. centimeters 
Sq. meters 
Sq. meters 
Sq. meters 
Hectars 
Hectars 
Hectars 
Hectars 



VOLUME (Cubic or globular). 



Cubic inches 


16.38716 
.02831702 


1.2145038 
8.4520475 
9.8834113 
4.6390879 


8.7854962 
1.5479525 
0.1165887 
5.3609121 


.0610234 

35.31445 

1.3079428 

.00002296 


Cubic cent'mtrs 


Cubic feet 


Cubic meters 


Cubic yards 


.7645594 
43560 


Cubic meters 


Acre feet 


Cubic feet 







CAPACITY (Liquid). 



Quarts (U. S.) 

Gallons (U. S.) 

Barrels (31^ galls.)... 



.9463586 
3.785434 
119.2412 



9.9760558 
0.5781158 
2.0764264 



0239442 
4218842 
9235736 



.056682 
.264170 
0083864 



Liters 
Liters 
Liters 



CAPACITY (Dry). 



Quarts (U. S.) 

Bushels (U. S. struck). 



1.10122 
35.23928 



0.0418771 9.9581229 
1.54702708.4529730 



.9080775 
.0283774 



Liters 
Liters 



Examples to illustrate the use of above table: 
45 mils = 45 X .0254 millimeters. 16 centimeters = 16 X .3937 inches. 



UNIT EQUIVALENTS-SIMPLE AND COMPOUND. 89 

34.— Simple and Compound Units in Common Use, Equivalents. (Cont'd.) 



To reduce A to M. 



Mult, by Log. 



To reduce M to A. 



Log. 



Mult, by 



M 



WEIGHT. 



Ounces (Avoir.) 
Pounds (Avoir.) 
Tons (2000 lbs.) 
Tons (2000 lbs.) 
Tons (2240 lbs.) 
Tons (2240 lbs.) 



28.3496 
.453593 
907.186 
.907186 
1016.05 
1.01605 



1.4525461 
9.6566658 
2.9576964 
9.9576964 
3.0069141 
0.0069141 



8.5474539 
0.3433342 
7.0423036 
0.0423036 
6.9930859 
9.9930859 



.035274 

2.20462 
00110231 

1.10231 
00098421 

.984206 



Grams 
Kilograms 
Kilograms 
Metric tons 
Kilograms 
Metric tons 



MOMENTS. 



Inch-Pounds 

Foot-Pounds 


1152.13 
.138255 


3.06150066.9384994 
9.14068180.8593182 


.000868 
7.23300 


Centlmeter-grms 
Meter-kilograms . 



STRESS PER AREA. 



Pounds per sq. In 

Pounds per sq. f t 

Tons (2000 lbs.) p. sq. It. 



0703067 
4.88243 
9.76486 



8469968 
6886361 
9896661 



1 1530032 
9.3113639 
9.01C3339 



14.2234 
.204816 
.102408 



Kllog.p.sq.cent'r. 
KUog. p.sq.meter 
Met.tonsp.sq.mtr 



WEIGHT PER VOLUME. 



Pounds per cu. In 

Pounds per cu. It 

Tons (20001bs.)p.cu.yd 



27.6797 
16.0184 
1.18655 



1.4421621 
1.2046184 
0.0742851 



5578379 
7953816 
9257149 



0361275 
0624283 
8427813 



Grms.p.cu.cent'r. 
Kllog. p. cu.m'tr 
Met.tonsp.cu.mtr 



VELOCITY. 



Feet per second . . 
Feet per second . , 
Feet per second . . 
Feet per minute . 
Miles per minute. 
Miles per minute. 
Miles per hour. . . 



.304801 
01136364 

.681^81 
01136364 
26.82245 
1.609347 
1.609347 



9.4840158 
8.0555173 
9.8336686 
8.0555173 
1.4284984 
0.2066497 
0.2066497 



0.5159842 
1.9444827 
0.1663314 
1.9444827 
8.5715016 
9.7933503 
9.7933503 



3.28083^3 

88. 

1.46^6 

88. 

.0372822 
.62137 
.62137 



Meters p. second 
Miles p. minute 
Miles per hour 
Miles per hour 
Meters p. second 
Kllomtrs. p. min. 
Kllomtrs. p. hour 



ACCELERATION. 



Feet per sec. per sec. 
(g=32.2— . av.)... 



I 



.30480 



9.4840158 



0.5159842 



3.28083^3 



Meters p.sc.p.sc. 
(g=9.81- av.) 



Examples to illustrate the use of above table: 

100 foot-pounds= lOOX 138255 kg.-meters. 10 miles per min.— 
10X88 ft. per sec. 



90 



i.— MEASURES, WEIGHTS AND MONEY. 



34— Simple and Compound Units in Common Use, Equivalents. (Concl'd.) 



To reduce A to M. 



Mult, by 



Log. 



To reduce M to A. 



Log. 



Mult, by 



M 



DISCHARGE (Cu. ft.. Gallons or Liters.) 



Cu. ft. per second 


60. 


1.7781513 


8.2218487 


.016^6 


Cu. ft. p. minute 


Cu. ft. per second 


3600. 


3.5563025 


6.4436975 


.00027^7 


Cu. ft. per hour 


Cu. ft. per second 


86.4 


1.9365137 


8.0634863 


.01157407 


Ths'd.cu.ft.p.day 


Cu. ft. per second 


2.592 


0.4136350 


9.5863650 


.3858025 


Mill.cu.ft.p. mth. 


Cu. ft. per second 


31.536 


1.4988066 


8.5011934 


.0317098 


Mill.cu.ft. p. year 


Cu. ft. per minute 


60. 


1.7781513 


8.2218487 


.016^6 


Cu. ft. per hour 


Cu. ft. per minute 


1440. 


3.1583625 


6.8416375 


.000694^4 


Cu. ft. per day 


Cu. ft. per minute 


43.2 


1.6354837 


8.3645163 


.02314815 


Thsnd.cu.ft.p.mo 


Cu. ft. per minute 


525.6 


2.7206554 


7.2793446 


.00190259 


Thsnd.cu.ft.p.yr. 


Cu. ft. per minute 


7.480521 


0.8739318 


9.1260682 


.1336805^6 


Galls. (U.S.)p.m. 


Cu. ft. per minute 


28.31702 


1.4520475 


8.5479525 


.03531445 


Liters p. minute 


Gallons per minute 


3.785434 


0.5781158 


9.4218842 


.26417047 


Liters p. minute 


Cu. ft. per second 


1.983471 


0.2974258 


9.7025742 


.5041667 


Acre -ft. per day 



WORK AND POWER (Power=Rate of Work.) 



Foot-pounds 


.138255 


9.1406818 


0.8593182 


7.23300 


Meter-Kilogms. 


Foot-pounds 


1.356284 


0.1323508 


9.8676492 


.737308 


Joules 


Meter-Kilograms 


9.81 


0.9916690 


9.0083310 


.101937 


Joules 


Foot-pounds 


.0012853 


7.1090204 


2.8909796 


778. 


Pnd.-deg. (Fahr.) 


Meter-kilograms 


.0092969 


7.9683386 


2.0316614 


107.5626 


Pnd.-deg. (Fahr.) 


Foot-pounds 


.0012853 


7.1090204 


2.8909796 


778. 


Brit, therm, units 


Foot-pounds 


.0003239 


6.5104139 


3.4895861 


3087.35 


Kilog'm-deg. (C.) 


Meter-kilograms 


.0023428 


7.3697321 


2.6302679 


426.843 


KUog'm-deg. (C.) 


Foot-pounds per sec .... 


.OOIS^'IS 


7.2596373 


2.7403627 


550. 


Mechanical H. P. 


Foot-pounds per sec 


.0013563 


7.1323508 


2.8676492 


737.308 


Electric H. P. 


Foot-pounds per sec 


.0018434 


7.2656205 


2.7343795 


542.475 


Metric H. P. 


Foot-pounds per min. . . 


.00003^03 


5.4814861 


4.5185139 


33000. 


Mechanical H. P. 


Foot-pounds per min. . . 


.0000226 


5.3541995 


4.6458005 


44238.51 


Electric H. P. 


Foot-pounds per min. . . 


.00003072 


5.4874692 


4.5125308 


32548.49 


Metric H. P. 


Foot-pounds per hour. . 


.0000005^05 


3.7033348 


6.2966652 


1980000. 


Mechanical H. P. 


Foot-pounds per hour . . 


.0637675* 


3.5760482 


6.4239518 


2654311. 


Electric H. P. 


Foot-pounds per hour . . 


.06 51206* 


3.7093179 


6.2906821 


1952909. 


Metric H. P. 


Meter-kilog. per min. . . . 


.O3219I8* 


6.3408043 


3.6591957 


4562.424 


Mechanical H. P. 


Meter-kilog per min 


.O3 16350* 


6.2135178 


3.7864822 


6116.207 


Electric H. P. 


Meter-kilog. per min 


.0002^2 


6.3467875 


3.6532125 


4500. 


Metric H. P. 


Meter-kilog. per hour. . . 


.0586530* 


4.5626530 


5.4373470 


273745.5 


Mechanical H. P. 


Meter-kilog. per hour. . . 


.0527250* 


4.4353665 


5.5646335 


366972.5 


Electric H. P. 


Meter-kilog. per hour. . . 


.0537^037* 


4.5686362 


5.4313638 


270000. 


Metric H. P. 


Mechanical H. P 


.7459566 


9.8727135 


0.1272865 


1.34056 


Electric H. P. 


Mechanical H. P 


1.01387 


0.0059832 


9.9940168 


.9863177 


Metric H. P. 


Electric H. P 


1.35916 


0.1332697 


9.8667303 


.73575 


Metric H. P. 







Note. — 1 electric horse-power = 1 kilowatt = 1000 watts = 1000 joules per 
second. 
1 metric horse-power = 75 meter-kilograms per second. 

Examples to illustrate the use of above table: 

8 meter-kilograms = 8X 9.81 joules. 10 British thermal units=10X 
778 ft.-lbs. 



*.0637675-.000 000 37675; 0637^037 = 0.000 003 703 7037 ; etc. 



UNIT EQUIVALENTS— SIMPLE AND COMPOUND, 



91 



35. — Electrical, MechanicaI and Heat Units — Equivalent 

Values. 
(See also preceding table.) 



1 Foot-pound 

= 1.356284 joules. 

= 0.138255 kilogram-meters. 

= 0. 0000003767 5 kilowatt hours. 

= 0.0012853 heat units. 

= 0. 0000005^05 horse-power hour. 

1 Kilogram-meter 
= 9.81 joules. 
= 7.23300ft.-lbs. 
== 0.0000036530 horse-po wer hour. 
= 0.0000027250 kilowatt hour. 
= 0.0092969 heat units. 

1 Joule 

= 0. 737308 It.-lbs. 

= 0.101937 kilogram-meters. 

= 0.000947697 heat units. 

= 1 watt second. 

= 0.00000027^7 kilowatt hour. 

= 0. 000000372378 horse-power hour. 

1 Heat Unit (B.T.U.) 
= 1055.18932 joules. 
= 1055.18932 watt-seconds. 
= 778ft.-lbs. 

= 107.5626 kilogram-meters. 
= 0. 00029311 kilowatt hour. 
= 0. 00039293 horse-power hour. 
= 0. 0000688 lb. carbon oxidized.* 
= 0. 001036 lb. water evaporated from 
and at 2120 F.* 

1 Watt 

= 1 joule per second. 

= 0.00134056 horse power. 

= 3.411711 heat units per hour. 

= 0. 737308 ft.-lb. per second. 

= 0.0035 lb. water evaporated per 

hour.* 
= 44. 23851 It.-lbs. per minute. 
= 0. 00135916 metric horse power. 

1 Kilowatt 
= 1000 watts. 
= 1.34056 horse power. 
= 2654311 It.-lbs. per hour. 
= 44.23851 ft.-lbs. per minute. 
= 737. 308 ft.-lbs. per second. 
== 3411.711 heat units per hour. 
= 56. 86185 heat units per minute. 
= 0. 9476975 heatimits per second. 
= 0.2275 lb. carbon oxidized per 

hour.* 
= 3.53 lbs. water evaporated per 

hour from and at 2120 F.* 

1 Watt Per Square Inch 

= 8.19 heat units per sq. ft. per min.* 
= 6371 ft.-lbs. per sq. ft. per min.* 
= 0. 193 horse power per sq. ft.* 



Kilowatt-hour 

= 1000 watt-hours. 

= 1 . 34056 horse-power hours. 

= 2654311 ft.-lbs. 

= 3600000 joules. 

= 3411.711 heat units. 

= 366972. 5 kilogram meters. 

= 0.235 1b. carbon ozidized with per- 
fect efficiency. 

= 3.53 lbs. water evaporated from and 
at 2120 F.* 

= 22.75 lbs. of water raised from 620 
to 2120 F.* 

Horse Power 
= 745.9566 watts. 
= 0.7459566 kilowatts. 
= 550 ft-lbs. per second. 
= 33 000 ft.-lbs. per minute. 
= 2544. 987 heat units per hour. 
= 42. 41645 heat units per minute. 
= 0. 706941 heat units per second. 
= 0.175 lbs. carbon oxidized per hour.* 
= 2.64 lbs. water evaporated per hour 
from and at2120F.* 

1 Horse Power-hour 

= 0.7459566 kilowatt-hours. 

= 1980000 ft.-lbs. 

= 2544.987 heat units. 

= 273745. 5 kilogram-meters. 

= 0.175 lb. carbon oxidized with per- 
fect efficiency.* 

= 2. 64 lbs. water evaporated Irom and 
at 2120 F.* 

= 17.0 lbs. water raised from 620 to 
2120 F.* 

1 Lb. Carbon Oxidized with Perfect Effi- 
ciency* 
= 1 . 1 1 lb. Anthracite coal oxidized. 
= 2.5 lbs. dry wood oxidized. 
= 21 cubic feet illuminating gas. 
= 4.26 kilowatt-hours. 
= 5.71 horse-power hours. 
= 11315000 ft.-lbs. 
= 15 lbs. water evaporated from and at 

2120 F.* 
= 14544 heat units. 

1 Lb. Water Evaporated from and at 2120 p.* 
= 0.283 kilowatt-hour. 
= 0.379 horse-power hour. 
= 965.7 heat units. 
= 103900 kilogram-meters. 
= 1019000 joules. 
= 751300 ft.-lbs. 
= 0. 0664 lb. of carbon oxidized. 

1 Heat Unit per square ft. per min.* 
= 0. 122 watt per square inch. 
= 0.0176 kilowatt per sq. ft. 
= 0. 0236 horse power per sq. ft. 



* Values by H. W. Leonard. — See The Electrical Engineer, Feb. 25. 1895. 



92 



i.— MEASURES. WEIGHTS AND MONEY. 






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FOREIGN WEIGHTS AND MEASURES. 



93 






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94 



i.— MEASURES, WEIGHTS AND MONEY. 



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MONEY—DOMESTIC AND FOREIGN, 95 



Numbers. 

37. — Abstract Numbers. 

10 units =1 ten = 10 

10 tens = 1 hundred = 100 

10 hundreds =1 thousand = 1 000 

10 thousands =1 ten thousand = 10 000 

10 ten thousands =1 hundred thousand . = 100 000 

10 hundred thousand . = 1 milHon = 1 000 000 

A pure decimal, where the first significant figure is far removed from 
the decimal point, may be abbreviated by a subscript to the first cipher to 
indicate the number of ciphers at the right of the decimal point and to the 
left of the first significant figure. Thus, 

2 millionths. or .000002, may be written, . 062. 

65 hundred millionths, or . 00000065, may be written, 
4 billionths, or . 000000004, may be written. 



38. — Duodecimo Numbers. 
12 units = 1 dozen. 

12 dozen = 1 gross = 144. (20 units = 1 score.) 

12 gross == 1 great gross = 1728. 



39.— Paper. 
24 sheets = 1 quire. 2 reams = 1 bundle = 960. 

20 quires = 1 ream = 480. 5 bundles = 1 bale = 4800. 



Money. 

40. — United States Money. 

10 mills (m) = 1 cent (ct.) (Unit is $1.) 

10 cents = 1 dime (d.) 10 dollars = 1 eagle (E.) 

10 dimes = 1 dollar ($.) 2 eagles = 1 double eagle (EE.) 



41. — Foreign Money. 

English Money: 4 farthings (far.) = 1 penny (d.) ; 12 pence = 1 shilling (s.) 
20 shillings=l pound (£) = l sovereign ( = $4.8665, U. S. money). 
1 guinea = 21 shillings; 1 crown = 5 shillings; 1 florin = 2 shillings. 

French Money: 10 centimes =1 decime; 10 decimes=l franc (fr.) 
( = $0,193, U.S. money). 

German Money: 100 pfennig = 1 mark ( = $0,238, U. S. money). 

Italian Money: 100 centesimi= 1 lira ( = $0,193, U. S. money). 

Russian Money: 100 copecks = 1 ruble ( = $0,515, U. S. money). 

Austro-Hungarian Money: 100 kreutzers= 1 florin. 



06 



i.— MEASURES, WEIGHTS AND MONEY. 



§ I 

P. ^ 

ill I 

> ° o . -^ 

-^ ^ S -" s § § 

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Ir.2 y : 1^ i^s IIIPIIS .o ll .111 

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OOOSO -^O OOOOOOOOOOOOOOOOOOOOOOOOOOO 

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VALUE OF FOREIGN COINS. 



97 



42a. — Value op Foreign Coins and Paper Notes in American Money 
Based Upon the Values Expressed in Table 42. 



^ 


2^ 

ii 


1 

d 
B 
O 


k 

d-i 


Hi 

.aw 

;d^ 

o 


d 

1 

rd 






P 


1 


$4.86,61 


$0.23,8 


$0.19.3 


$0.73.6 


$0.40,2 


$0.49.8 


$0.51,5 


$0.20,3 


2 


9.73,3 


0.47.6 


0.38.6 


1.47.2 


0.80.4 


0.99.6 


1.03 


0.40,6 


3 


14.59,91 


0.71.4 


0.57.9 


2.20,8 


1.20,6 


1.49.4 


1.54.5 


0.60,9 


4 


19.46,6 


0.95,2 


0.77.2 


2.94.4 


1.60,8 


1.99.2 


2.06 


0.81,2 


5 


24.33 .2i 


1.19 


0.96,5 


3.68.0 


2.01 


2.49.0 


2.57.5 


1.01,5 


6 


29.19,9 


1.42,8 


1.15.8 


4.41,6 


2.41.2 


2.98,8 


3.09 


1.21.,8 


7 


34.06,51 


1.66,6 


1.35.1 


5.15,2 


2.81,4 


3.48.6 


3.60.5 


1.42 1 


8 


38.93,2 


1.90,4 


1.54,4 


5.88.8 


3.21,6 


3.98.4 


4.12 


1.62.4 


9 


43.79,81 


2.14,2 


1.73.7 


6.62.4 


3.61,8 


4.48,2 


4.63.5 


1.82.7 


10 


48.66,5 


2.38 


1.93 


7.36.0 


4.02 


4.98.0 


5.15 


2.03 


20 


97.33 


4.76 


3.86 


14.72,0 


8.04 


9.96,0 


10.30 


4.06 


30 


145.99,5 


7.14 


5.79 


22.08.0 


12.06 


14.94,0 


15.45 


6.09 


40 


194.66 


9.52 


7.72 


29.44,0 


16.08 


19.92.0 


20.60 


8.12 


50 


243.32.5 


11.90 


9.65 


36,80.0 


20.10 


24.90.0 


25.75 


10.15 


100 


486.65 


23.80 


19.30 


73.60.0 


40.20 


49.80,0 


51.50 


20.30 



42b.— Value of Foreign Coins in United States Money. 
(Proclaimed by the Secretary of the Treasury, Jan. 1, 1911.) 

Values given below are in deviation from those given in Table 42; otherwise, 
the values given in Table 42 remain unchanged. 



Country. 


Monetary 
Unit. 


Value in 
U.S. 
Gold 

Dollar. 


Country. 


Mone- 
tary 
Unit. 


Value in 
U.S. 
Gold 

Dollar. 


Bolivia 

Central America. 


Boliviano 

Peso 


$0.38,9 
0.40.3 
0.60,4 
0.67,3 
0.65,9 
0.19,3 


India (Brit.)... 
Liberia 


Rupee. . 
Dollar. . 
Balboa. 
Libra . . 
Peso.... 
Dollar.. 


$4.32.41 
1 00,0 


China 


f Shanghai 
TaeH Haikwan 

1 Canton. . 
Mark 


Panama 

Peru 


1.00,0 
4 86,65 


Finland 


Philippine I *ds.. 
Straits Settle't.. 


0.50,0 
42,1 







98 



i.— MEASURES, WEIGHTS AND MONEY. 



43. — Comparison of Prices. 

French and German prices for metric units, British prices for Imperial 
units, and United States prices for United States standard weights and 
measures. 

[Based upon the circular of the Secretary of the Treasury dated Octo- 
ber 1, 1902, fixing the legal equivalent of the (German) mark at 23.8 cents, 
of the (French) franc at 19.3 cents, and the British pound sterling at $4.8665.] 



!i 


la 




1 


i 


ft o 


Is 






?,"-3'e3 


I t 


s5^ 

£« 




QQ . 


o a 




SCO g 


Is 




11 
11 






1 = 


.088 


1 = 


.176 


1 


= .731 


1 




.068 


, 


.203 


2 = 


.175 


2 = 


.353 


2 


= .461 


2 


= 


.136 


2 = 


.405 


3 = 


.263 


3 = 


.529 


3 


= 2.192 


3 


= 


.204 


3 = 


.608 


4 = 


.350 


4 = 


.705 


4 


= 2.922 


4 


= 


.272 


4 = 


.810 


5 = 


.438 


5 = 


.882 


5 


= 3.653 


5 


_ 


.340 


S = 


1.013 


6 = 


.525 


6 = 


1.058 


6 


= 4.384 


6 


= 


.408 


6 = 


1.216 


7 = 


.613 


7 = 


1.234 


7 


= 5.114 


7 


= 


.476 


7 = 


1.418 


8 = 


.700 


8 = 


1.411 


8 


= 5.844 


8 


= 


.544 


8 = 


1.621 


9 = 


.788 


9 = 


1.587 


9 


= 6.575 


9 


= 


.612 


9 = 


1.824 


11.423 = 


1 


5.667 = 


1 


1.369 


= 1 


14.703 


— 


I 


4.935 = 


1 


22.846 = 


2 


11.334 = 


2 


2.738 


= 2 


29.407 


= 


2 


9.871 = 


2 


34.269 = 


3 


17.000 = 


3 


4.106 


= 3 


44.110 


= 


3 


14.806 = 


3 


45.691 = 


4 


22.667 = 


4 


5.475 


= 4 


58.813 


= 


4 


19.742 = 


4 


57.115 = 


5 


28.334 = 


5 


6.844 


= 5 


73.517 


_ 


5 


24.677 = 


5 


68.537 = 


6 


34.001 = 


6 


8.213 


= 6 


88.220 


= 


6 


29.612 = 


6 


79.960 = 


7 


39.668 = 


7 


9.581 


= 7 


102.923 


= 


7 


34.548 = 


7 


91.383 = 


8 


45.334 = 


8 


10.950 


= 8 


117.627 


= 


8 


39.483 = 


8 


102.806 = 


9 


51.001 = 


9 


12.319 


= 9 


132.330 


^= 


9 


44.419 = 


9 




fi . 




u 




f^ 'd 


. 




u 


^ 


u 


1^ 


|3 

s o 


I 
IS 


2 . 

^-2 


m . 










5 c3 ^ 




3=3 


o > 


c3 (U 


O Oj 


««.-s 


o ..S^ 


eS « 




O 13 


o . S 


I^W 


Q< 


^^ 


P>H 


Sh^ 


QP.q 


SW 




Q« 


SftPq 


Qt^m 


1 = 


.108 


1 = 


.218 


J 


= .901 


1 




.084 


, 


.236 


2 = 


.216 


2 = 


.435 


2 


= 1.802 


2 


= 


.168 


2 = 


.472 


3 = 


.324 


3 = 


.653 


3 


= 2.703 


3 


= 


.252 


3 = 


.707 


4 = 


.432 


4 = 


.871 


4 


= 3.604 


4 


= 


.335 


4 = 


.943 


5 = 


.540 


5 = 


1 .088 


5 


= 4.505 


5 


_ 


.419 


5 = 


1.179 


6 » 


.648 


6 = 


1.306 


6 


= 5.406 


6 


= 


.503 


6 = 


1.415 


7 = 


.756 


7 = 


1.523 


7 


= 6.307 


7 


= 


.587 


7 = 


1.650 


8 = 


.864 


8 = 


1.741 


8 


= 7.207 


8 


= 


.671 


8 = 


1.886 


9 = 


.972 


9 = 


1.959 


9 


= 8.108 


9 


= 


.755 


9 = 


2.122 


9.263 = 


1 


4.595 = 


1 


1.110 


= I 


11.923 


_ 


I 


4.241 = 


1 


18.526 = 


2 


9.190 = 


2 


2.220 


= 2 


23.847 


= 


2 


8.483 = 


2 


27.789 = 


3 


13.785 = 


3 


3.330 


= 3 


35.770 


= 


3 


12.724 = 


3 


37.052 = 


4 


18.380 = 


4 


4.440 


= 4 


47.693 


= 


4 


16.965 = 


4 


46.316 = 


5 


22.975 = 


5 


5.550 


= 5 


59.616 


= 


5 


21.207 = 


5 


55.579 =-= 


6 


27.570 = 


6 


6.660 


= 6 


71.540 


= 


6 


25.448 = 


6 


64.842 = 


7 


32.165 = 


7 


7.770 


= 7 


83.463 


=. 


7 


29.689 = 


7 


74.105 = 


8 


36.760 = 


8 


8.880 


= 8 


95.386 


== 


8 


33.931 = 


8 


83.368 = 


9 


41.355 = 


9 


9.990 


= 9 


107.310 


= 


9 


38.172 = 


9 



TIME AND CIRCULAR MEASURES. 99 

Miscellaneous. 

44. — Time Measure. 

1 second (s) =15 seconds (0°-O0'- 15") of longitude. 

60 seconds \ minute (m) = 15 minutes (0°— 15') of longitude. 

60 minutes ( = 3600 seconds) = 

1 hour (h) = 15 degrees (15°) of longitude. 
24 hours (=1440 m= 86400 s) = 

1 solar day (d) = 360 degrees (360°) of longitude. 
7 days ( = 168 h = 10080 m = 604800 s) = 

1 IV^^k 

521 weeks (= 365 d = 8760 h = 525600 m) = 

1 common year. 
62| weeks ( = 366 d = 8784 h = 5270'40 m) = 

1 leap year. 
100 years (=75 common + 25 leap) = 

1 century, 

45. — Circular Measure. 

= 1 second (") =^^ (.06^6) second of time. 

60 seconds = 1 minute {') = 4 seconds of time. 

60 minutes (=3600 seconds) = 1 degree (°) = 4 minutes of time. 

{the angle at the base of a 
right triangle whose alti- 
tude is 1. hypothenuse 2, 
and base \/3. 
90 degrees ( = 6400'= 324000^) 

= 1 right angle (L) = 6 hotirs of time. 
180 degrees (= 10800'= 648000^^) 

= 1 semi-circumference = ;r=3. 14159265.... 
360 degrees ( = 21600'= 1296000") 

= 1 circumference = 2k .radius = tt .diameter. 



5.— ALGEBRA. 

An algebraic equation is a shorthand mathematical expression, and 
every such expression may be transformed by observing certain algebraic 
rules. The first letters of the alphabet, a, b, c, ..... represent the known 
quantities of the equation, and the last letters, . . . x,y, z, the unknown. 
Members of an equation are separated by the sign of equality ( = ); terms 
of a member are separated by plus ( + ) and minus ( — ) signs; and factors 
of a term are separated by the signs of multiplication (X) or (.), either 
expressed or understood. 

Any factor or set of factors of a term may be considered the denomination 

of that term. Thus, in a :*; \/y, either a. x, \/y, ax, a\/y, x\/y or axy/y 
may be chosen as the denomination, as may be expedient. 

Addition and Subtraction. — Place like terms (terms having the same 
denomination) in the same column, and reduce fractions to a common 
denominator, before adding or subtracting: 

. . . o . . /-r . 1 ^' . y^ x^ + y^ 



^x-f-ox'-y-r^a-r ^ 

Vx 


a+ b a -\- b o 4- 6 


-2x + Sx^'y -2\'^ + -^ 
Vx 


(a+Z>)2 (a4-6)2 (y-^) (a + b)^ 
X y xy 


Sum = 2 X + S x^ y - Vd -{ ■— 

^x 

Diff. = 6x + 2x^y-SVd ^n 

Vx 


zVZx± \/l2 X = {^ ± 2) VZx 





]^ote. — In subtraction, reverse the + and — signs of the subtrahend 
and proceed as in addition. 

Exponents. — ^The following hints are given without discussion: 

1 6 _ 

« X n X « = nnn = »»2 = ^2^ = „i+2 = ^2+1 = „3 =__ = «2 = ^^g 

1 1. 
n Vlcy = nVxV~y= n x^ y^= ^ J.^ = — ir~~:i =^Vn'^xy= (n^xyy 



X ' y 



V 



-8x^ _ V-Sx^ ^ (Sy (x^)^ ^ - 2 X _ 2x 

~ 1 1 3 3/2 3y2 

27 3'^ i/27y« (27)^ iy^y 



4 _ 4 9 ___ "^ _ ^ X _ JL ^ 1 /^\ O^ 



n 



X'' 



n+x, fjn-x = a^a =— . 



{^"^y = x'"\ (x") '= x^o, (^x)-"^ = m -°- = JL- . ,(x^y^)2 = ^4^. 



*Any number whose exponent is 0, is equal to 1. 

100 



BINOMIAL FORMULA. 103 

Multiplication and Powers. 
Monomials: w^ X nh = n^ ni = n^^ = wf =\^n^ = — ;• 

a3 X a «2 = o%2. 2\/~3 X 3^5"= 6^15. SVoT^ X Va. x = 3a^2. 

Like signs give plus: a X b = ab. —aX( — b)=ab. 

Unlike signs give mitwws: —aXb=—ab. a X { — b) = —ab. 

Polynomials : Multiply each term of the mutliplicand by each term of the 
multiplier, placing like terms of the products in "column," and add. 

a 62 - a2 6 
a 63 _ a2 62 

3 a ^2 + 4 v'5_ _ ^354 + ^4 Z,3 

VS a2 65-a3 64 

Product = 3 a it:2 V5~+ 20 Product = a2 6^ - 2a^ b^+ a^b^ 

= a2 (65 - 2 a64 + a2 63) . 
^ a^b^ (b - a)2. 

/'^4_^^2 = ^2-i- 9 ^ .^4-^,2 / These two formulas are the foun- 
Vl^.,\ f .\ ^2 9 /-dation of many short methods in 
(^ + y)(^-y)=^'-y'- \ arithmetic. (See page 11). 

(x±y)^ = a^±Sx^y+Zxy^±y^, 

(x-hy) {x^-xy + y^)=x^ + y^. 

(x—y) (x^ + xy + y^)=x^ — y^. 

(x + y + z)'^^x'^ + y'^ + z^+2 X y+2 X z+2 y z. 

(x+y-^z)^ = x^-hy^ + z^+Sx^ (y -}- z) ■}- 3 y^ {x-{-z) + Zz^ (x+y) + 6xyz. 

Binomial formula for expanding the sum or difference of two numbers, to 
any power: 

n n n-1 , w(« — 1) ii-2 ^ , n(n~l)(n—2) n-3 „ 

(a ± X) = a ± na. x -^ j w 2 ^ x^ ± — 1X2X3 — ^ x^ +.,, 
Thus: ^ (a+.)a=a. + 5a%+ |21ia3.. + f-fffla^s + f ^|^|^^ a^4. 

5X4X3X2X1 5 
1X2X3X4X5^ 

=o5 + 5a4:x: + 10 a^ x^ + 10 a^ x^ + 5 a x^ + x^. 
(a + .)^=ai+ia-i. + -^|a-^U2+igy^a-^^ + 
^ (-1) (-H) (-21) -5^ 4 

1X2X3X4 "" "" ••••;• 

2Va 2X22Va3 2X3X23VaS 2X3X4X24\/a7 
3 X 5 X 7 a:5 
2X3X4X5X 2^^ Va9 

(a4-.)l=a^ + ia-|. + i^^ ,-l| ,2 + i^JK^ ^"2! ^ 

^ H-i) (-1!) (-21) ^^31^4 

1X2X3X4 

_ ^^ _^ X 2^2 ^ 2X 5a;3 2X5X8^ 

3 ^a2 2 X 32 </^ 2 X 3 X 33 '^^ 2 X 3 X 4 X 3^ V'^ 

2 X 5 X 8 X 11 a;5 

2X3X4X5X35 '^^ 



102 5.— ALGEBRA. 

Division and Roots. 

x^-^x^ = ^ = x^^^ = r»;0 = 1. x^ -^ x^ == x^^ = x\ 
x^ 

a2b^ ^ ah = (a2-i) (63-i) = a&2. £.^-£. = ^X- = — . 

X n X c ex 

Division: vT /s" /l5 Vl5 VlS , , ,- 

.+3) g-f-5. + 6( . + 2 vT ^ Vr^ V2l^ Vll= ±5-= ± ^^15 

Square Root by Binomial formula'. 
Example: Find the square root of 5? 

1 
Solution : In (a + ic) ^ let a = 4 and x=\\ thence by expansion (see page 101) 

(a + ^) ^ = a^ + — : -rrr H -=: ;== + 



2\/a W a^ 16>/a5 128V a^ 256>/a9 



2X2 8X8 16X32 128X128 

= 2 + .25 - .015625 + .001953 - .000305 = 2.236±. 

Cube Root by Binomial formula'. 

Example: Find the cube root of 9? 

Solution: 8 is the cube of 2 and 1 less than 9; hence from expansion 
formula, page 101, there is obtained, 

{a-\-x)^^ (8+ 1)^ = 8^(1+^)^ = 2 (l+i)l But 

{a+xy = a^ H 5— - — J—— + — i^^T ^"ZT + • • • » Therefore 

3Va2 32 V a5 34\/a8 35\/aii 

</T= 2a+*^^= 2 ri+ -i ?— + — ^ ?^ 1 

» ^viTi; ^|^i.-rg^g 32X82 34X83 3^X84 J 

= 2[1 + .0416^6 - .0017361^1 + .00012054 - .00001005 ] 

= 2 X 1.04004 = 2.08008. Ans. 

Completing the Square. — ^This is performed by adding a certain amoimt 
c^ (a third term), to the first member of the (affected quadratic) equation 
to make it a perfect square, and the same amount to the second member 
so as to preserve the equality. 

Example 1: Find the value of x in the equation x^+ix=21} ^ 

Solution: By adding c"^ to both members we have x'^-^i x + c^ — c^+2\, 
in which the first member is a perfect square, the middle term, ^x, being 
equal* to 2V^2 ^c\vfhence c^=2^=i, 
and x^ + ix + 2"^ = 4 4- 21 = 25. 

Extracting the sq. rt., a; + 2 = ± 5 

and X =±5— 2=3 or— 7, Ans. 

Note. — Make the first term a perfect square before completing the 
square of the first member of the equation. This may be done by multi- 
plying or dividing the whole equation by a constant. The first term must 
be positive. 

* The square of a member of two terms = the square of the first + twice 
the product of the two + the square of the second. 



COMPLETING SQUARE. SIMULTANEOUS EQUATIONS. 103 

Example 2: Solve 2ax^ + 5bx = 201 Here, 

Mult, by 2, div. by a and add c"^: ix^ + —x + c^ = c^-h — 

a a 



Extracting the sq. rt., 2x + c = -* /c^ H , in which 



V-"^' 



106* -,„ ^, , 56 . „ , 2562 60 56 



4 



, , 2562^ 60 56 . 



Remarks on apparent fallacy of the above. To prove that 4 = 5? 
Let 16 - 36 = 25 - 45 
81 
and to complete the square, let ^^ = "j 5 then 

16 - 36 + x= 25 - 45 + -Y. 
4 4 

Extracting the sq. rt.. 4 - | =5-^1 But, ± (4-|) = T (5-i) 



. ,1 . ^ / 1 , IX I whence,really,4— 4^= — 5+4^. 

whence, apparently, 4 = o (or — f = + f) . 1 

In a somewhat similar way it may also be "shown" that 2 = 1, 1 =^ 0, etc. ; 

but the discrepancies are always apparent upon inspection. 

Simultaneous Equations. — To solve any 
problem, the number of equations must be equal to 
the number of unknown quantities. This can be 
understood quite readily by considering that each 
equation is really the equation of a curve of some 
kind. Thus, in Fig. 1, let curve A be represented 
by the equation y = m. x + c, and curve B by the 
equation y^ = ^ a x. The equations of these 
curves are simultaneous equations when the 
curves intersect at any point, as p, with common 
coordinates x^y and yo', and, the problem is solved 
by determining the values of these coordinates. Y\z. 1. 

Thus, by substitution, yo^^ {m x + cy = ^ a x', 

"V ~~ C 'V2 

and xq = == T \ from which we have only to solve the equation 

m 4a 

"V ~~ C 1^2 

{ynx 4- £:)2= 4 a ;*:, or the equation = -j-, each one containing only 

one unknown quantity, x or y. 

Simultaneous equations may be solved by three methods of eliminat- 
ing all but one unknown quantity: 

(1) Elimination by substitution, as above. 

(2) Elimination by addition or subtraction, and by substitution. 
„ . 3ic+43/=18. Mult, by 2: 6ic+8:v=36 
Example. 2^+ Zy=\Z. Mult, by 3: 6^+9y-39 

Whence by subtraction, y=Z 

And, by substituting this value in the first equation, 
3^+4X3=18; x^2. 

(3) Elimination by comparison. 

T, 1 . 3^+ 43;= 18- 18-4:v 13-33/ . „ „ 

Example: 2;c+35;=13. ^ = — F" = ~2"~"' * ' ^"^^^ ''^^' 

If more than two equations are given, eliminate one unknown quantity 
by combining two of the equations, and proceed until one equation, with 
one unknown quantity, remains. Then solve for that unknown quantity 
and substitute its value in one of the equations to obtain the value of another 
unknown quantity. Proceed in this manner, substituting the values thus 
obtained in another equation; and so on. 

For Cubic Equations, see Plane Trigonometry. 



6.— LOGARITHMS OF NUMBERS. 

Logarithms are useful in finding the product, quotient, powers and roots 
of numbers. A system of logarithms may be founded on any base, as o. 
If the base a raised to the mth power = M, then m is the logarithm of 
M to the base a; and conversely, M is the anU-logarithm (that is, the 
number corresponding to the logarithm) of m, to the same base. Thus. 



Let M = a number; m its log to base a. 
Let A/ = a number; n its log to base a. 



Then logs M- 
Thenlog^ N-- 



■m; cram = M. 
n: or a° = N. 



The following formulas and methods illustrate the use of logarithms in 
the process above mentioned: — 



To: 

Multiply M by N. 
Divide M by N. 

Raise M to the Nth 

power. 
Extract theiV^?^ root 

of M. 



Formula: 

MN = a^a^ = a™+^ 
M a™ 

— = = /7m— n 

N a- "" 

M = (a"i) = a 

N N m 

V M = Va"^ = aN 



Process: find anti- 
logarithm of 
(Log of M) + (log of AT) 

(Logof M)~(log of A^) 

(LogofM)XN. 

(LogofM)-^A^. 



Two systems of logarithms are in use, namely: 



Common or Briggs System. 

(Founded by Briggs.) 
In general use for all practical pur- 
poses. 



o^ 


Base a 

= iV /. 


= 10. 

log N 


= 


n. 




100 


= 1 .*. 


log 1 


= 


0. 




101 


= 10 .• 


log 10 


= 


1. 




102 = 100 .-. 
102'«= 302 .♦. 


log 100 
log 302 


= 


2. 
2. 


484- 



Hyperbolic, Natural or Naperian 
System. 
(Founded by Napier.) 
Used in pure mathematical discus- 
sion and in steam engineering. 
Base = e = 2.7182818... 
Derivation of e: 

= (H — j Sis X approaches infinity. 



= 1+1 + 



.2 1.2.3 



+ 



1.2.3.4 



= l+l + J + i+^5+ 

= 2.7182818 

The naperian log. of a number = its common log. X 2 . 3025851 .... 
The common log. of a number = its hyperbolic log. X . 4342945 . . . 
Commonlog. of 2.3025851 = 0.3622157; of 0.4342945 = 9.6377843-10. 

Note from the above that — 

The naperian log. of the naperian base (e=2. 7182818 . . .)= unity. 

The common log. of the naperian base, 2.7182818 = 0.4342945. 

The common log. of the common base (a = 10) = unity. 
The naperian log. of the common base, 10, = 2 . 3025851. 

Common or Briggs System. — ^The logarithm of a number is composed 
of the characteristic, or integral portion to the left of the decimal point, 
and the mantissa or decimal fraction. The mantissa is all that appears in 
any table of logarithms and the degree of accuracy is dependent on the num- 
ber of decimal places used in the mantissa. Vega's tables, to seven decimal 
places, are recommended for general office use in city surveying, and. where 
the results, in general, are required accurately to the sixth or seventh 
significant figure. 



104 



COMMON OR BRIGGS SYSTEM. 



105 



Table 1, following, to five decimal places, will be found compact and 
convenient where the result to five significant figures is sufficiently accu- 
rate. Tables to six decimal places, unless arranged on the "Vega system" 
(in which case they would comprise 180 pages — too bulky for any hand 
book) are not recommended. 

In the logarithm of any number the mantissa is independent of the 
position of the decimal point, while on the contrary the characteristic is 
dependent only on the position of the first significant^ figure of the number 
with relation to the decimal point. Thus in the following examples: 



(a) 


log. 


4021.7 


(b) 


log. 


402.17 


(c) 


log. 


40.217 


id) 


log. 


4.0217 



(/) 



log. 
log. 



.40217 
.040217 



3.60441 
2.60441 
1.60441 
0.60441 
1.60441 
2.60441 









^ „ QJ D »-. bO' 

., CO a ^.Sir>>. 

9.60441 - 10 
8.60441 - 10 



it will be seen that the characteristic is equal, algebraically, to the number 
of places minus one, which the first significant figure of the number occupies 
to the left of the decimal point. In (a) the characteristic is 3; in (b), 2; 
in (d), 0; in (e), — 1; and in (/), — 2. Some mathematicians prefer the 
use of the negative characteristic, but most of them employ the " positive," 
by algebraically adding 10 to the integer and placing — 10 to the right of 
the mantissa or omitting the latter (—10) altogether. For example, log 
.040217 = 8.60441, the —10 being understood and the value of the char- 
acteristic being, of course, — 2. In the case of finding the root of (or divid- 
ing) a pure decimal, however, the — 10 must be employed. 



Example: Find the fourth root of .0081? 

Solution: log .0081 

Quotient obtained by dividing by 4 

Antilog or fotirth root = 0.3. Ans. 



7.90849 -10 

1.97712 - 2.5 

2.5 

9.47712 -10 



To find the logarithm of a number. 

Example: Find the log of 678 . 57? 

Solution: The characteristic is 3 — 1 = 2. The mantissa for the first 
four figures, 67.85, is read directly from Table 1, and = .83155. To this, 
however, must be added 7/jq (the next figure of the number is 7) of the 
difference between .83155 and the log of 67.86 = .83161. This difference 
is 6 and in the proportional parts (P. P.) column under 6 and opposite 7 
will be found the value 4.2, call 4, which added to .83155 = 83159. Hence 
the log of 678 . 57 = 2 . 83159. Ans. 



To find the anti^'Iogarithm (number corresponding to a logarithm). 

Example: 1 .92513 is the logarithm of what number? 

Solution: This is the reverse of finding the logarithm of a number. 
Neglecting, for the present, the characteristic, the next lower mantissa to 
.92513 is .92511 and the number corresponding is 8416. The difference 
between .92511 and the next higher mantissa in Table 1, .92516 is 5, and 



the proportional difference 



92513 - 



.92511 2 ,, . 
-^2511- or -^ calls for 



4 to be added 



92516 

to the fourth figure, i. e., 4 to the fifth place of the number, as shown in 
the P. P. column. Therefore the number, disregarding the decimal point, 
is 84164. The characteristic, 1, calls for two places to the left of the decimal 
point, hence the antilog of 1 . 92513 is 84 . 164. Ans. 



106 



^.—LOGARITHMS OF NUMBERS, 



To multiply one number by another. (Add the logs.) 
Example. — Multiply 23.142 by 85.7? 
Solution . — log 2 3 . 1 4 2 



.*. Antilog = product 



log 85.7 
1983.3 

Ans. 



= 1.36436 

4 = P. P. for 2. 
= 1.93298 



3.29738 
32 



P.P. 



3 - 



To divide one number by another. (Subtract the logs.) 
Example. — Divide 846 . 94 by 36.42? 
Solution. — log 846.94 = 2.92785 

log 36.42 = 1.56134 

.-.Antilog = quotient = 23.255 1.36651 

Ans. 42 



9 = P.P. = 5- 

To raise a number to any power. (Multiply the log by the index of thi 
power.) 

Example. — 25. 3^ = ? 

Solution.— log 25.3 = 1.40312 

Multiply by 3 : 3 

.-. Antilog == 3rd power = 16194 _4. 20936 

Ans. 



25 
11 



P. P. = 4 



To extract the root of a number. 

root.) 



(Divide the log by the index of the 



Example 1. V286.12 = ? 

Solution. — log 286.12 = 2.45655 

Divide by 5 : 

.-. Antilog = 5th root = 3.1096 6T49T31 

Ans. 122 

9 P. P. = 6 

Example 2. — Find the square root of the fifth power of 23.2? 

Solution. — The square root of the fifth power oi N = N^.thereiore 
multiply the log of 23.2 by 2| or, better, divide by .4. 

log 23.2 = 1.36549 
Divide by .4 : 



Antilog = v/23.25 = 2592.5 

Ans. 



3.41372 
63 



9 P. P. = 5 + 



Example 3. 
Solution — 



/. Antilog 



V09463 = ? 



log .09463 = 8.97603 - 10. 
Dividing by 2 = 4.48801.5 -5. 

- 5 



square root = . 30762 

Ans. 



9.48801.5 
799 

2.5 



10 



Naperian, Natural or Hyperbolic System. 

A table of naperian logarithms of numbers from 1 to 10, advancing by 
hundredths, is given as a part of Table 1 of the common logarithms of 
numbers. The range of this table may be extended greatly, to include the 
logarithms of other numbers, as follows: 

For logarithms of numbers from 1 to 10 advancing by thousandths and 
ten thousandths; use the "difference" column in the table. Thus, to find 
the log of 4.6953, add ^Aoo of the difference, 213, to the log of 4.69, equals 
1.54543 + 113 = 1.54656. Ans. 



NATURAL OR HYPERBOLIC SYSTEM. 



107 



For logarithms of numbers from 10 to 100: Divide the number by 10, 
find the log of the quotient, from the table, and add 2 . 302585 (the log of 10). 
Thus, to find the log of 46.72. Log 4.672 = 1.54116 + 43, which, added 
to 2.302585 = 3.84418. Ans. 

Rule. — Add 2 . 302585 to the log of Vio of the number. 

For logarithms of numbers from 100 to 1000: Divided the number by 
100, find the log of the quotient, from the table, and add 2 X 2.302585 = 
4.60517 (the log of 100). 

Rule. — Add 4 . 60517 to the log of Vioo oi the number. 

In general, to find the naperian logarithm of any number, add n X 

2. 302585 to the naperian log of — X the number; the factor n to be selected 



Multiple: 
X 8 = 18.4206807 
X 9 = 20.7232658 
X 10 =23.0258509 



and some power of 10, so as to bring the new number within the range of 
the table, i. e.. from 1 to 10. 

Log of 10. Multiple: Multiple: 

2.3025851 X 2 =4.6051702 X 5 = 11.5129255 

X 3 = 6.9077553 X 6 = 13.8155106 

X 4 = 9.2103404 X 7 = 16.1180957 

To find the antilog, proceed in a manner inverse to the above method 
of finding the log. 

Example. — Of what number is 4.85203 the log? 

Solution. — The highest multiple of 2.3025851 below the given log is 
4 . 60517, obtained by the factor 2 cor- Given by log = 4 . 85203 

responding with the factor 100 of the Multiple = 4.60517 

number. From the table of naperian 
logarithms. Table 1, we find fhat the 
number corresponding to the log of 
the difference, . 24686, is 1 . 28. But the given log, 4 . 85203, is of a number 
100 times as large, therefore its antilog is 1 .28 X 100 = 128. Ans. 

To Find the Naperian Logarithm of a Number, from Table of Common 
Logarithms. — From page 104 we find that the naperian log of a number = its 
common log X 2.3025851, and the common log of 2.3025851 = 0.3622157 (call «). 

Problem.— Find the naperian log of 2100.1. (Common log of 2100.1 =■ 
3.3222400. ) Three Solutions :— 



Diff. 



= 0.24686 



iDBy Multiplication: 

2.3025851 
3.32224 

921 

4605 .... 
46052 ... 
460517 .. 
6907755 . 
69077553 

7.6497403 



(2) By Logarithms: 

Com log 2100.1 = 3.3222400 
(call c) 

Com log c = 0.5214310 
« = 0.3622157 

0.8836467 = 
Com log of 7.6497403 

.-. Nap log 2100.1 = 7.6497403 
by the three methods. 



(3) By Multiples (above): 
Com log 2100.1 =3.3222400 

3. ,6.9077553 

3 .6907755 

2 .0460517 

2 .0046052 

2 .0004605 

4 .0000921 

7.6497403 



Given the Naperian Logarithm of a Number: To Find its Common Loga- 
rithm.— From page 104 we find that common log of a number = its naperian log 
X 0.4342945, and the common log of 0.4342945 = 9.6377843 (call n'). 

Problem.- Find the common log of the number whose naperian log is 7 64974. 
(Common log of 7.64974 = 0.8836467.) Three Solutions:— 



(1) By Mult. 






X » 



(2) By Logarithms: 

Com log 7.64974 = 0.8836467 
«' = 9.6377843 



0.5214310 
Antilog = 3.3222400 

.-. Com log = 3.32224 by the 
three methods. 



(3) By Multiples: 
Nap log = 7.64974 



7. 



6 



3.0400614 
.2605767 
.0173718 
.0039087 
.0003040 
.0000174 



3.3222400 



Multiples. 

0.4342945 
0.8685890 
1.3028834 
1.7371779 
2.1714724 
2.6057669 
3.0400614 
3.4743559 
3.9086503 



108 



Q— LOGARITHMS OF NUMBERS. 
I. — Logarithms. 



I 



No. 


Common Logarithms of Numbers. 


Naperian. 




L. O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 


No. 


Log. Dif. 


100 


00 000 


043 


087 


130 


173 


217 


260 


303 


346 


389 




44 43 42 


1.00 


.00000 995 
.00995 If^ 
.01980 l°i 
.02956 III 
.03922 ^^^ 

957 
.04879 q,. 
.05827 5^5 
.06766 If^ 
.07696 If 
.08618 ^^^ 

913 
.09531 9.- 
.10436 ^"^ 
.11333 °^7 
.12222 °°J 
.13103 ^^^ 

873 
.13976 g.-. 
.14842 866 
.15700 If. 
.16551 if. 
.17395 8^^ 

837 
.18232 ... 
.19062*5" 
.19885 Hi 
.20701 l\l 
.21511 8^" 

803 
.22314 „97 
.23111 797 

.23902 ;^i 

.24686 '„°l 
.25464 7^^ 

772 
.26236 7.7 
.27003 ]^. 
.27763 ;^" 
•28518 ]% 
.29267 '^^ 

743 
.30010 „.« 
.30748 ;^° 

.31481 ;;^ 

.32208 ;^i 
.32930 ^^^ 

717 
.33647 „,« 
.34359 ;J; 
.35066 707 
.35767 '^Jj 
.36464 ^^' 

692 
.37156 .nj, 
.37844 ^°° 

.38526 nrjn 

.39204 %l 
.39878 ^74 
669 
.40547 


1 


432 


475 


518 


561 


604 


647 


689 


732 


775 


817 


] 


4 4 4 


1.01 


2 


860 


903 


945 


988 


030 


072 


115 


157 


199 


242 


2 


9 9 8 


1.02 


3 


01 284 


326 


368 


410 


452 


494 


536 


578 


620 


662 


3 


13 13 13 


1.03 


4 


703 


745 


787 


828 


870 


912 


953 


995 


036 


078 


4 


18 17 17 


1.04 


105 


02 119 


160 


202 


243 


284 


325 


366 


407 


449 


490 


5 


22 22 21 


1.05 


6 


531 


572 


612 


653 


694 


735 


776 


816 


857 


898 


6 


26 26 25 


1.06 


7 


938 


979 


U19 


060 


TOO 


T41 


181 


222 


262 


S02 


7 


31 30 29 


1.07 


8 


03 342 


383 


423 


463 


503 


543 


583 


623 


663 


703 


8 


35 34 34 


1.08 


9 


743 


782 


822 


862 


902 


941 


981 


021 


060 


100 


9 


40 39 38 


1.09 


110 


04 139 


179 


218 


258 


297 


336 


376 


415 


454 


493 




41 40 39 


1.10 




532 


571 


610 


650 


689 


727 


766 


805 


844 


883 


1 


4 4 4 


1.11 


2 


922 


961 


999 


038 


077 


115 


154 


192 


231 


269 


2 


8 8 8 


1.12 


3 


05 308 


346 


385 


423 


461 


500 


538 


576 


614 


652 


3 


12 12 12 


1.13 


4 


690 


729 


767 


805* 


843 


881 


918 


956 


994 


032 


4 


16 16 16 


1.14 


115 


06 070 


108 


145 


183 


221 


258 


296 


333 


371 


408 


5 


21 20 20 


1.15 


6 


446 


483 


521 


558 


595 


633 


670 


707 


744 


781 


6 


25 24 23 


1.16 


7 


819 


856 


893 


930 


967 


U04 


041 


078 


115 


151 


7 


29 28 27 


1.17 


8 


07 188 


225 


262 


298 


335 


372 


408 


445 


482 


518 


8 


33 32 31 


1.18 


9 


555 


591 


628 


664 


700 


737 


773 


809 


846 


882 


9 


37 36 35 


1.19 


120 


918 


954 


990 


027 


063 


099 


T35 


171 


207 


243 




38 37 36 


1.20 


1 


08 279 


314 


350 


386 


422 


458 


493 


529 


565 


600 


1 


4 4 4 


1.21 


2 


636 


672 


707 


743 


778 


814 


849 


884 


920 


955 


2 


8 7 7 


1.22 


3 


991 


026 


061 


096 


132 


167 


202 


237 


272 


S07 


3 


11 11 11 


1.23 


4 


09 342 


377 


412 


447 


482 


517 


552 


587 


621 


656 


4 


15 15 14 


1.24 


125 


691 


726 


760 


795 


830 


864 


899 


934 


968 


003 


5 


19 19 18 


1.25 


6 


10 037 


072 


106 


140 


175 


209 


243 


278 


312 


346 


6 


23 22 22 


1.26 


7 


380 


415 


449 


483 


517 


551 


585 


619 


653 


687 


7 


27 26 25 


1.27 


8 


721 


755 


789 


823 


857 


890 


924 


958 


992 


025 


8 


30 30 29 


1.28 


9 


11 059 


093 


126 


160 


193 


227 


261 


294 


327 


361 


9 


34 33 32 


1.29 


130 


394 


428 


461 


494 


528 


561 


594 


628 


661 


694 




35 34 33 


1.30 


1 


727 


760 


793 


826 


860 


893 


926 


959 


992 


024 


1 


4 3 3 


1.31 


2 


12 057 


090 


123 


156 


189 


222 


254 


287 


320 


352 


2 


111 


1.32 


3 


385 


418 


450 


483 


516 


548 


581 


613 


646 


678 


3 


11 10 10 


1.33 


4 


710 


743 


775 


808 


840 


872 


905 


937 


969 


001 


4 


14 14 13 


1.34 


135 


13 033 


066 


098 


130 


162 


194 


226 


258 


290 


322 


5 


18 17 17 


1.35 


6 


354 


386 


418 


450 


481 


513 


545 


577 


609 


640 


6 


21 20 20 


1.36 


7 


672 


704 


735 


767 


799 


830 


862 


893 


925 


956 


7 


25 24 23 


1.37 


8 


988 


019 


051 


082 


114 


145 


176 


208 


Z39 


270 


8 


28 27 26 


1.38 


9 


14 301 


333 


364 


395 


426 


457 


489 


520 


551 


582 


9 


32 31 30 


1.39 


140 


613 


644 


675 


706 


737 


768 


799 


829 


860 


891 




32 31 30 


1.40 


1 


922 


953 


983 


014 


045 


076 


106 


137 


168 


198 


1 


3 3 3 


1.41 


2 


15 229 


259 


290 


320 


351 


381 


412 


442 


473 


503 


2 


6 6 6 


1.42 


3 


534 


564 


594 


625 


655 


685 


715 


746 


776 


806 


3 


10 9 9 


1.43 


4 


836 


866 


897 


927 


957 


987 


017 


047 


077 


107 


4 


13 12 12 


1.44 


145 


16 137 


167 


197 


227 


256 


286 


316 


346 


376 


406 


5 


16 16 15 


1.45 


6 


435 


465 


495 


524 


554 


584 


613 


643 


673 


702 


6 


19 19 18 


1.46 


7 


732 


761 


791 


820 


850 


879 


909 


938 


967 


997 


7 


22 22 21 


1.47 


8 


17 026 


056 


085 


114 


143 


173 


202 


231 


260 


289 


8 


26 25 24 


1.48 


9 


319 


348 


377 


406 


435 


464 


493 


522 


551 


580 


9 


29 28 27 


1.49 


ISO 


609 


638 


667 


696 


725 


754 


782 


811 


840 


869 






1.50 



LOGARITHMS OF NUMBERS. 
1 . — ^Logarithms . — Continued. 



109 



No. 


Common Logarithms of Numbers. 


Naperian, 




L. O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. 


P. 


No. 


Log. Dif. 


ISO 


17 609 


638 


667 


696 


725 


754 


782 


811 


840 


869 




29 


28 


1.50 


.40547 (.(., 
.41211 Zl 
.41871 If^ 
.42527 f^. 
.43178 ^^1 

647 
.43825 ... 
.44469 J** 
.45108 ^g 
.45742 ^^4 
.46373 ^^^ 

627 
.47000 .00 
.47623 1% 
.48243 fll 
.48858 l\l 
.49470 ^^2 

608 
•S 604 
:5?282 600 
.51879 %' 
.52473 ^^^ 

590 
.53063 KRR 
.53649 f^o 
.54232 IJ 
.54812 ^5" 
.55389 ^'' 

573 

.58222 ^^^ 

557 
.58779 KK. 
.59333 f^. 
.59884 1% 
.60432 If^ 
.60977 ^*^ 

542 
.61519 ^oq 
.62058 ^^^ 
.62594 ^o? 
.63127 S? 
.63658 ^'^^ 

527 
.64185 ,25 
.64710 f^ 
.65233 p-<Q 
.65752 ^1; 
.66269 ^^^ 

514 
.66783 .,1 
.67294 ^JJ 
.67803 ^"5 
.68310 ^^' 
.68813 ^"'* 

502 
.69315 


1 


898 


926 


955 


984 


013 


D41 


070 


099 


127 


156 


1 


3 


3 


1.51 


2 


18 184 


213 


241 


270 


298 


327 


355 


384 


412 


441 


2 


6 


6 


1.52 


3 


469 


498 


526 


554 


583 


611 


639 


667 


696 


724 


3 


9 


8 


1.53 


4 


752 


780 


808 


837 


865 


893 


921 


949 


977 


005 


4 


12 


11 


1.54 


155 


19 033 


061 


089 


117 


145 


173 


201 


229 


257 


285 


5 


15 


14 


1.55 


6 


312 


340 


368 


396 


424 


451 


479 


507 


535 


562 


6 


17 


17 


1.56 


7 


590 


618 


645 


673 


700 


728 


756 


783 


811 


838 


7 


20 


20 


1.57 


8 


866 


893 


921 


948 


976 


003 


030 


058 


085 


112 


8 


23 


22 


1.58 


9 


20 140 


167 


194 


222 


249 


276 


303 


330 


358 


385 


9 


26 


25 


1.59 


160 


412 


439 


466 


493 


520 


548 


575 


602 


629 


656 




27 


26 


1.60 


1 


683 


710 


737 


763 


790 


817 


844 


871 


898 


925 


1 


3 


3 


1.61 


2 


952 


978 


005 


032 


059 


085 


112 


139 


165 


192 


2 


5 


5 


1.62 


3 


21 219 


245 


272 


299 


325 


352 


378 


405 


431 


458 


3 


8 


8 


1.63 


4 


484 


511 


537 


564 


590 


617 


643 


669 


696 


722 


4 


11 


10 


1.64 


165 


748 


775 


801 


827 


854 


880 


906 


932 


958 


985 


5 


14 


13 


1.65 


6 


22 Oil 


037 


063 


089 


115 


141 


167 


194 


220 


246 


6 


16 


16 


1.66 


7 


272 


298 


324 


350 


376 


401 


427 


453 


479 


505 


7 


19 


18 


1.67 


8 


531 


557 


583 


608 


634 


660 


686 


712 


737 


763 


8 


22 


21 


1.68 


9 


789 


814 


840 


866 


891 


917 


943 


968 


994 


019 


9 


24 


23 


1.69 


170 


23 045 


070 


096 


121 


147 


172 


198 


223 


249 


274 




25 


24 


1.70 


1 


300 


325 


350 


376 


401 


426 


452 


477 


502 


528 


1 


3 


2 


1.71 


2 


553 


578 


603 


629 


654 


679 


704 


729 


754 


779 


2 


5 


5 


1.72 


3 


805 


830 


855 


880 


905 


930 


955 


980 


005 


030 


3 


8 


7 


1.73 


4 


24 055 


080 


105 


130 


155 


180 


204 


229 


254 


279 


4 


10 


10 


1.74 


175 


304 


329 


353 


378 


403 


428 


452 


477 


502 


527 


5 


13 


12 


1.75 


6 


551 


576 


601 


625 


650 


674 


699 


724 


748 


773 


6 


15 


14 


1.76 


7 


797 


822 


846 


871 


895 


920 


944 


969 


993 


018 


7 


18 


17 


1.77 


8 


25 042 


066 


091 


115 


139 


164 


188 


212 


237 


261 


8 


20 


19 


1.78 


9 


285 


310 


334 


358 


382 


406 


431 


455 


479 


503 


9 


23 


22 


1.79 


180 


527 


551 


575 


600 


624 


648 


672 


696 


720 


744 




24 


23 


1.80 


1 


768 


792 


816 


840 


864 


888 


912 


935 


959 


983 


1 


2 


2 


1.81 


2 


26 007 


031 


055 


079 


102 


126 


150 


174 


198 


221 


2 


5 


5 


1.82 


3 


245 


269 


293 


316 


340 


364 


387 


411 


435 


458 


3 


7 


7 


1.83 


4 


482 


505 


529 


553 


576 


600 


623 


647 


670 


694 


4 


10 


9 


1.84 


185 


717 


741 


764 


788 


811 


834 


858 


881 


905 


928 


5 


12 


12 


1.85 


6 


951 


975 


998 


021 


D45 


068 


091 


114 


138 


161 


6 


14 


14 


1.86 


7 


27 184 


207 


281 


254 


277 


300 


323 


346 


370 


393 


7 


17 


16 


1.87 


8 


416 


439 


462 


485 


508 


531 


554 


577 


600 


623 


8 


19 


18 


1.88 


9 


646 


669 


692 


715 


738 


761 


784 


807 


830 


852 


9 


22 


21 


1.89 


190 


875 


898 


921 


944 


967 


989 


012 


035 


058 


081 




22 


21 


1.90 


1 


28 103 


126 


149 


171 


194 


217 


240 


262 


285 


307 


1 


2 


2 


1.91 


2 


330 


353 


375 


398 


421 


443 


466 


488 


511 


533 


2 


4 


4 


1.92 


3 


556 


578 


601 


623 


646 


668 


691 


713 


735 


758 


3 


7 


6 


1.93 


4 


780 


803 


825 


847 


870 


892 


914 


937 


959 


981 


4 


9 


8 


1.94 


195 


29 003 


026 


048 


070 


092 


115 


137 


159 


181 


203 


5 


11 


11 


1.95 


6 


226 


248 


270 


292 


314 


336 


358 


380 


403 


425 


6 


13 


13 


1.96 


7 


447 


469 


491 


513 


535 


557 


579 


601 


623 


645 


7 


15 


15 


1.97 


8 


667 


688 


710 


732 


754 


776 


798 


820 


842 


863 


8 


18 


17 


1.98 


9 


885 


907 


929 


951 


973 


994 


016 


038 


060 


081 


9 


20 


19 


1.99 


200 


30 103 


125 


146 


168 


190 


211 


233 


255 


276 


298 








2.00 



110 



Q.— LOGARITHMS OF NUMBERS. 
1. — Logarithms. — Continued. 



No. 


Common Logarithms of Numbers.^ 


Naperian. 




L. O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


I 


». P. 


No. 


Log. Dif. 


200 


30 103 


125 


146 


168 


190 


211 


233 


255 


276 


298 




22 


2.00 


.69315 4Q« 
.69813 Y^ 


1 


320 


341 


363 


384 


406 


428 


449 


471 


492 


514 


1 


2 


2.01 


2 


535 


557 


578 


600 


621 


643 


664 


685 


707 


728 


2 


4 


2.02 


.70310 f^. 


3 


750 


771 


792 


814 


835 


856 


878 


899 


920 


942 


3 


7 


2.03 


.70804 yJ: 


4 


963 


984 


006 


D27 


048 


D69 


091 


112 


133 


154 


4 


9 


2.04 


.71295 ^91 
489 
.71784 4„7 
.72271 f^. 
.72755 f^\ 


205 


31 175 


197 


218 


239 


260 


281 


302 


323 


345 


366 


R 


11 


2.05 


6 


387 


408 


429 


450 


471 


492 


513 


534 


555 


576 


6 


13 


2.06 


7 


597 


618 


639 


660 


681 


702 


723 


744 


765 


785 


7 


15 


2.07 


8 


806 


827 


848 


869 


890 


911 


931 


952 


973 


994 


8 


18 


2.08 


.73237 Tr,i 
.73716 ^79 

478 
.74194 .„. 
.74669 S? 
.75142 *;^ 
.75612 f.l 
.76081 ^69 

466 
.76547 ... 
.77011 Z\ 
.77473 f 2 
.77932 ^^^ 
.78390 ^^S 

456 
.78846 ,„ 
.79299 VTo 
.79751 f,l 
.80200 ifr. 
.80648 ^^^ 

445 
.81093 ,.^ 
.81536 \\l, 
.81978 \f. 
.82418 IVi 
.82855 ^"^^ 

436 
.83291 ,^, 
.83725 Jr>; 
.84157 J^J 
.84587 Zil 
.85015 ^^^ 

427 
.85442 42. 
.85866 l^\ 
.86289 \i\ 
.86710 \\\ 
.87129 *^^ 

418 
.87547 .,fi 
.87963 f.\ 

.88789 ]Jf 
.89200 *^^ 

409 
.89609 407 
.90016 !nR 
.90422 ^06 
.90826 ffl 
.91228 ^^2 

401 
.91629 


9 


32 015 


035 


056 


077 


098 


118 


139 


160 


181 


201 


9 


20 


2.09 


210 


222 


243 


263 


284 


305 


325 


346 


366 


387 


408 




21 


2.10 


1 


428 


449 


469 


490 


510 


531 


552 


572 


593 


613 


1 


2 


2.11 


2 


634 


654 


675 


695 


715 


736 


756 


777 


797 


818 


2 


4 


2.12 


3 


838 


858 


879 


899 


919 


940 


960 


980 


001 


021 


3 


6 


2.13 


4 


33 041 


062 


082 


102 


122 


143 


163 


183 


203 


224 


4 


8 


2.14 


215 


244 


264 


284 


304 


325 


345 


365 


385 


405 


425 


5 


11 


2.15 


6 


445 


465 


486 


506 


526 


546 


566 


586 


606 


626 


6 


13 


2.16 


7 


646 


666 


686 


706 


726 


746 


766 


786 


806 


826 


7 


15 


2.17 


8 


846 


866 


885 


905 


925 


945 


965 


985 


005 


025 


8 


17 


2.18 


9 


34 044 


064 


084 


104 


124 


143 


163 


183 


203 


223 


9 


19 


2.19 


220 


242 


262 


282 


301 


321 


341 


361 


380 


400 


420 




20 


2.20 


1 


439 


459 


479 


498 


518 


537 


557 


577 


596 


616 


1 


2 


2.21 


2 


635 


655 


674 


694 


713 


733 


753 


772 


792 


811 


2 


4 


2.22 


3 


830 


850 


869 


889 


908 


928 


947 


967 


986 


005 


3 


6 


2.23 


4 


35 025 


044 


064 


083 


102 


122 


141 


160 


180 


199 


4 


8 


2.24 


225 


218 


238 


257 


276 


295 


315 


334 


353 


372 


392 


K 


10 


2.25 


6 


411 


430 


449 


468 


488 


507 


526 


545 


564 


583 


6 


12 


2.26 


7 


603 


622 


641 


660 


679 


698 


717 


736 


755 


774 


7 


14 


2.27 


8 


793 


813 


832 


851 


870 


889 


908 


927 


946 


965 


8 


16 


2.28 


9 


984 


003 


D21 


040 


059 


078 


097 


116 


135 


154 


9 


18 


2.29 


230 


36 173 


192 


211 


229 


248 


267 


286 


305 


324 


342 




19 


2.30 


1 


361 


380 


399 


418 


436 


455 


474 


493 


511 


530 


1 


2 


2.31 


2 


549 


568 


586 


605 


624 


642 


661 


680 


698 


717 


2 


4 


2.32 


3 


736 


754 


773 


791 


810 


829 


847 


866 


884 


903 


3 


6 


2.33 


4 


922 


940 


959 


977 


996 


014 


033 


051 


070 


088 


4 


8 


2.34 


235 


37 107 


125 


144 


162 


181 


199 


218 


236 


254 


273 


5 


10 


2.35 


6 


291 


310 


328 


346 


365 


383 


401 


420 


438 


457 


6 


11 


2.36 


7 


475 


493 


511 


530 


548 


566 


585 


603 


621 


639 


7 


13 


2.37 


8 


658 


676 


694 


712 


731 


749 


767 


785 


803 


822 


8 


15 


2.38 


9 


840 


858 


876 


894 


912 


931 


949 


967 


985 


003 


9 


17 


2.39 


240 


38 021 


039 


057 


075 


093 


112 


130 


148 


166 


184 




18 


2.40 


1 


202 


220 


238 


256 


274 


292 


310 


328 


346 


364 


1 


2 


2.41 


2 


382 


399 


417 


435 


453 


471 


489 


507 


525 


543 


2 


4 


2.42 


3 


561 


578 


596 


614 


632 


650 


668 


686 


703 


721 


3 


5 


2.43 


4 


739 


757 


775 


792 


810 


828 


846 


863 


881 


899 


4 


7 


2.44 


245 


917 


934 


952 


970 


987 


005 


023 


041 


058 


076 


5 


9 


2.45 


6 


39 094 


111 


129 


146 


164 


182 


199 


217 


235 


252 


6 


11 


2.46 


7 


270 


287 


305 


322 


340 


358 


375 


393 


410 


428 


7 


13 


2.47 


8 


445 


463 


480 


498 


515 


533 


550 


568 


585 


602 


8 


14 


2.48 


9 


620 


637 


655 


672 


690 


707 


724 


742 


759 


777 


9 


16 


2.49 


250 


794 


811 


829 


846 863 


881 


898 


915 


933 


950 






2.50 



LOGARITHMS OF NUMBERS. 
1. — ^Logarithms. — Continued. 



11] 



No. 


Common Logarithms of Numbers. 


Naperian. 




L. O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


I 


'.P. 


No. 


Log. Dif. 


250 

1 
2 
3 
4 


39 794 
967 

40 140 
312 
483 


811 

985 
157 
329 
500 


829 
002 

175 
346 
518 


846 
019 
192 
364 
535 


863 
037 
209 
381 
552 


881 
054 
226 
398 
569 


898 
071 
243 
415 
586 


915 
088 
261 
432 
603 


933 
106 
278 
449 
620 


950 
123 

295 
466 
637 


1 
2 
3 
4 


18 

2 
4 
5 

7 


2.50 
2.51 
2.52 
2.53 
2.54 


.91629 .qq 
.92028 f^^ 
.92426 ^^° 
.92822 fJ*. 
.93216 "^^^ 

393 
.93609 ^p? 
.94001 f.] 
.94391 ^^" 
.94779 f^, 
.95166 ^^^ 

385 
.95551 ofi, 
.95935 ill 
.96317 oof 
.96698 o^i 
.97078 ^^" 

378 
.97456 q„ 
.97833 VJ. 
.98208 %'l 
.98582 %l 
.98954 "^^ 

371 

.99325 orj. 

.99695 S« 

1.00063 ofi7 

1.00430 oflfi 

1.00796 ^^'^ 

364 

1.01160 oao 

1.01523 VA 

1.02245 OKQ 
1.02604 "^^^ 

358 
1.02962 q,g 
1.03318 VZ 
1.03674 iZ 
1.04028 ill 
1.04380 '^^^ 

352 
1.04732 ... 
1 . 05082 ^?Q 
1.05431 {fr. 
1.05779 Vr^ 
1.06126 "^^^ 

345 
1.06471 oAA 

1.06815 olq 

1.07158 olo 
1.07500 q]: 
1.07841 "^^^ 
340 
1.08181 ooo 
1.08519 V° 
1.08856 iiL 

1.09192 iil 

1.09527 ^*^^ 
334 
1.09861 


255 
6 
7 
8 
9 


654 
824 
993 
41 162 
330 


671 
841 
010 

179 
"347 


688 
858 
027 
196 
363 


705 
875 
044 
212 
380 


722 
892 
061 
229 
397 


739 
909 
078 
246 
414 


756 
926 
095 
263 
430 


773 
943 
111 
280 
447 


790 
960 
128 
296 
464 


807 
976 
145 
313 
481 


5 
6 
7 
8 
9 


9 
11 
13 
14 
16 


2.55 
2.56 
2.57 
2.58 
2.59 


260 

1 
2 
3 
4 


497 
664 
830 
996 
42 160 


514 
681 
847 
012 

177 


531 
697 
863 
029 
193 


547 
714 
880 
045 
210 


564 
731 
896 
062 
226 


581 
747 
913 
078 
243 


597 
764 
929 
095 
259 


614 
780 
946 
111 
275 


631 
797 
963 
127 
292 


647 
814 
979 
144 
308 


1 
2 
3 
4 


17 

2 
3 
5 

7 


2.60 
2.61 
2.62 
2.63 
2.64 


265 
6 
7 
8 
9 


325 
488 
651 
813 
975 


341 
504 
667 
830 
991 


357 
521 
684 
846 
008 


374 
537 
700 
862 
^024 


390 
553 
716 
878 
040 


406 
570 
732 
894 
056 


423 
586 
749 
911 
072 


439 
602 
765 
927 
088 


455 
619 
781 
943 
104 


472 
635 
797 
959 
120 


5 
6 
7 
8 
9 


9 

10 
12 
14 
15 


2.65 
2.66 
2.67 
2.68 
2.69 


270 

1 
2 
3 
4 


43 136 
297 
457 
616 

775 


152 
313 
473 
632 
791 


169 
329 
489 
648 
807 


185 
345 
505 
664 
823 


201 
361 
521 
680 
838 


217 
377 
537 
696 
854 


233 
393 
553 
712 
870 


249 
409 
569 

727 
886 


265 
425 
584 
743 
902 


281 
441 
600 
759 
917 


■ 

4 


16 

2 
3 
5 
6 


2.70 
2.71 
2.72 
2.73 
2.74 


275 
6 
7 
8 
9 


933 
44 091 
248 
404 
560 


949 
107 
264 
420 
576 


965 
122 
279 
436 
592 


981 
138 
295 
451 
607 


996 
154 
311 
467 
623 


012 

170 
326 
483 
638 


028 
185 
342 
498 
654 


044 
201 
358 
514 
669 


059 
217 
373 
529 
685 


075 
232 
389 
545 
700 


5 
6 
7 
8 
9 


8 
10 
11 
13 
14 


2.75 
2.76 
2.77 
2.78 
2.79 


280 

1 
2 
3 
4 


716 
871 
45 025 
179 
332 


731 
886 
040 
194 
347 


747 
902 
056 
209 
362 


762 
917 
071 
225 
378 


778 
932 
086 
240 
393 


793 
948 
102 
255 
408 


809 
963 
117 
271 
423 


824 
979 
133 
286 
439 


840 
994 
148 
301 
454 


855 
010 

163 
317 
469 


1 
2 
3 
4 


15 

2 
3 
5 
6 


2.80 
2.81 
2.82 
2.83 
2.84 


285 
6 
7 
8 
9 


484 
637 
788 
939 
46 090 


500 
652 
803 
954 
105 


515 
667 
818 
969 
120 


530 
682 
834 
984 
135 


545 
697 
849 
000 
150 


561 
712 
864 
015 
165 


576 
728 
879 
030 
180 


591 
743 
894 
045 
195 


606 
758 
909 
060 
210 


621 
773 
924 
075 
225 


5 
6 

7 
8 
9 


I 

11 
12 
14 


2.85 
2.86 
2.87 
2.88 
2.89 


290 

1 
2 
3 
4 


240 
389 
538 
687 
835 


255 
404 
553 
702 
850 


270 
419 
568 
716 
864 


285 
434 
583 
731 
879 


300 
449 
598 
746 
894 


315 
464 
613 
761 
909 


330 
479 
627 
776 
923 


345 
494 
642 
790 
938 


359 
509 
657 
805 
953 


374 
523 
672 
820 
967 


1 
2 
3 
4 


14 

1 
3 
4 
6 


2.90 
2.91 
2.92 
2.93 
2.94 


295 
6 
7 
8 
9 


982 
47 129 
276 
422 
567 


997 
144 
290 
436 
582 


012 

159 
305 
451 
596 


026 
173 
319 
465 
611 


041 
188 
334 
480 
625 


056 
202 
349 
494 
640 


070 
217 
363 
509 
654 


085 
232 
378 
524 
669 


100 
246 
392 
538 
683 


114 
261 
407 
553 
698 


5 
6 
7 
8 
9 


7 
8 

10 
11 
13 


2.95 
2.96 
2.97 
2.98 
2.99 


300 


712 


727 


741 


756 


770 


784 


799 


813 


828 


842 






3.00 



112 



%.— LOGARITHMS OF NUMBERS. 
1. — Logarithms. — Continued. 



No. 


Common Logarithms of Numbers. 


Naperlan. 




L. O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 


No. 


Log. Dif. 


300 

1 
2 
3 
4 


47 712 
857 

48 001 
144 
287 


727 
871 
015 
159 
302 


741 
885 
029 
173 
316 


756 
900 
044 
187 
330 


770 
914 
058 
202 
344 


784 
929 
073 
216 
359 


799 
943 
087 
230 
373 


813 
958 
101 
244 
387 


828 
972 
116 
259 
401 


842 
986 
130 
273 
416 


1 
2 
3 
4 


15 

2 
3 
5 
6 


3.00 
3.01 
3.02 
3.03 
3.04 


1.09861 o„ 
1.10194 {%% 
1.10526 %\l 
1.10856 %%l 
1.11186 ^^^ 

328 
1.11514 007 
1.11841 \f„ 
1.12168 %^' 
1.12493 m 
1.12817 ^^* 

323 
1.13140 «„ 
1.13462 ^^f 
1.13783 fl 
1.14103 \\l 
1.14422 "^^^ 

318 
1.14740 017 
1.15057 %]l 
1.15373 %\l 
1.15688 ^;j 
1.16002 ^^* 

313 
1.16315 0,0 
1.16627 i\i 
1.16938 lYn 
1.17248 ^J" 
1.17557 ^"^ 

308 
1.17865 on« 
1.18173 f^f. 
1.18479 {)!l 
1.18784 ^05 
1.19089 3^5 

303 

1.19392 onQ 
1.19695^03 

1.19996 il\ 
1.20297 ^"i 
1.20597 30° 

299 
1.20896 9QO 
1.21194 %^ 
1.21491 1% 
1.21788 f' 
1.22083 ^^^ 

295 
1.22378 „Q. 
1.22671 fj. 
1.22964 f^l 
1.23256 ^5f 
1.23547 ^^^ 

290 
1.23837 200 
1.24127 ;^" 
1.24415 ;°° 
1.24703 ;°° 
1.24990 |«^ 


305 
6 
7 
8 
9 


430 

572 
714 
855 
996 


444 
586 
728 
869 
(510 


458 
601 
742 
883 
024 


473 
615 
756 
897 
038 


487 
629 
770 
911 
052 


501 
643 

785 
926 
066 


515 

657 
799 
940 
080 


530 
671 
813 
954 
094 


544 
686 
827 
968 
108 


558 
700 
841 
982 
122 


5 
6 

7 
8 
9 


8 
9 

11 
12 
14 


3.05 
3.06 
3.07 
3.08 
3.09 


310 

2 
3 
4 


49 136 
276 
415 
554 
693 


150 
290 
429 
568 
707 


164 
304 
443 
582 
721 


178 
318 
457 
596 
734 


192 
332 
471 
610 

748 


206 
346 
485 
624 
762 


220 
360 
499 
638 
776 


234 
374 
513 
651 
790 


248 
388 
527 
665 
803 


262 
402 
541 
679 
817 


1 
2 
3 
4 


14 

1 
3 
4 
6 


3.10 
3.11 
3.12 
3.13 
3.14 


315 
6 

7 
8 
9 


831 
969 
50 106 
243 
379 


845 
982 
120 
256 
393 


859 
996 
133 
270 
406 


872 
010 

147 

284 
420 


886 
024 
161 
297 
433 


900 
037 
174 
311 
447 


914 
D51 
188 
325 
461 


927 
065 
202 
338 
474 


941 
079 
215 
352 
488 


955 
092 
229 
365 
501 


5 
6 

7 
8 
9 


7 
8 

10 
11 
13 


3.15 
3.16 
3.17 
3.18 
3.19 


320 

1 
2 
3 
4 


515 
651 
786 
920 
51 055 


529 
664 
799 
934 
068 


542 
678 
813 
947 
081 


556 
691 
826 
961 
095 


569 
705 
840 
974 
108 


583 
718 
853 
987 
121 


596 
732 
866 
001 
135 


610 
745 
880 
014 
148 


623 
759 
893 
028 
162 


637 
772 
907 
041 
175 


1 
2 
3 
4 


13 

3 
4 
5 


3.20 
3.21 
3.22 
3.23 
3.24 


325 
6 
7 
8 
9 


188 
322 
455 
587 
720 


202 
335 
468 
601 
733 


215 
348 
481 
614 
746 


228 
362 
495 
627 
759 


242 
375 
508 
640 

772 


255 
388 
521 
654 
786 


268 
402 
534 
667 
799 


282 
415 
548 
680 
812 


295 
428 
561 
693 
825 


308 
441 
574 
706 
838 


5 

\ 

9 


7 
8 
9 

10 
12 


3.25 
3.26 
3.27 
3.28 
3.29 


330 

1 
2 
3 
4 


851 
983 
52 114 
244 
375 


865 
996 
127 
257 
388 


878 
009 

140 
270 
401 


891 
022 
153 
284 
41 A. 


904 

035 
166 
297 
427 


917 
048 
179 
310 
440 


930 
061 
192 
323 
453 


943 
075 
205 
336 
466 


957 
088 
218 
349 
479 


970 
101 
231 
362 
492 


1 
2 
3 
4 


13 

1 
3 
4 
5 


3.30 
3.31 
3.32 
3.33 
3.34 


335 
6 
7 
8 
9 


504 
634 
763 
892 
53 020 


517 
647 
776 
905 
033 


530 
660 
789 
917 
046 


543 
673 
802 
930 
058 


556 
686 
815 
943 
071 


569 
699 
827 
956 
084 


582 
711 
840 
969 
097 


595 
724 
853 
982 
110 


608 
737 
866 
994 
122 


621 
750 
879 
007 
135 


5 
6 
7 
8 
9 


7 
8 
9 

10 
12 


3.35 
3.36 
3.37 
3.38 
3.39 


340 

1 
2 
3 
4 


148 
275 
403 
529 
656 


161 
288 
415 
542 
668 


173 
301 
428 
555 
681 


186 
314 
441 
567 
694 


199 
326 
453 
580 
706 


212 
339 
466 
593 
719 


224 
352 
479 
605 
732 


237 
364 
491 
618 
744 


250 

377 
504 
631 
757 


263 
390 
517 
643 
769 


1 
2 
3 
4 


12 

1 
2 
4 
5 


3.40 
3.41 
3.42 
3.43 
3.44 


345 
6 
7 
8 
9 


782 
908 
54 033 
158 
283 


794 
920 
045 
170 
295 


807 
933 
058 
183 
307 


820 
945 
070 
195 
320 


832 
958 
083 
208 
332 


845 
970 
095 
220 
345 


857 
983 
108 
233 
357 


870 
995 
120 
245 
370 


882 
008 

133 
258 
382 


895 
020 
145 
270 
394 


5 
6 
7 
8 
9 


6 
7 
8 

10 
11 


3.45 
3.46 
3.47 
3.48 
3.49 


350 


407 


419 


432 


444 


456 


469 


481 


494 


506 


518 






3.50 


1.25276 



LOGARITHMS OF NUMBERS. 
1. — ^Logarithms. — Continued. 



113 



No. 


Common Logarithms of Numbers. 


Naperian. 




L. O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P 


. P. 


No. 


Log. Dif. 


350 

1 
2 
3 
4 


54 407 
531 
654 

777 
900 


419 
543 
667 
790 
913 


432 
555 
679 
802 
925 


444 
568 
691 
814 
937 


456 
580 
704 
827 
949 


469 

593 
716 
839 
962 


481 
605 
728 
851 
974 


494 
617 
741 

864 
986 


506 
630 

753 
876 
998 


518 
642 
765 
888 
Oil 


1 
2 
3 
4 


13 

1 
3 
4 
5 


3.50 
3.51 
3.52 
3.53 
3.54 


1.25276 „Q- 
1.25562 ^°S 
1.25846 f^l 

1.26130 ;°* 

1.26413 ^^^ 
282 
1.26695 «„. 
1.26976 ;°} 
1.27257 ;°i 
1.27536 279 
1.27815 ^^^ 
278 
1.28093 „„3 
1.28371 ;i^ 
1.28647 %l 
1.28923 f„l 
1.29198 ^^^ 
275 
1.29473 9-0 

1.29746 ;;^ 
1.30019 ;l% 
1.30291 ;;; 

1.30563 ^^^ 
270 
1.30833 «„« 
1.31103 9^" 
1.31372 269 

1.31641 ;^^ 

1.31909 ^^^ 

267 
1.32176 «.- 
1.32442 Zl 
1.32707 ill 
1.32972 ill 
1.33237 ^^^ 

263 
1.33500 «», 
1.33763 ili 
1.34025 ;^f 
1.34286 il\ 
1.34547 261 

260 
1.34807 ^-0 
1.35067 ;?" 
1.35325 if^ 
1.35584 ifr, 
1.35841 ^^^ 

257 
1.36098 ... 
1.36354 256 

1.36609 iil 
1.36864 ill 
1.37118 ^^* 

254 
1.37372 <,„ 
1.37624 Hi 
1.37877 ili 
1.38128 il\ 
1.38379 ^^^ 

250 
1.38629 


355 
6 
7 
8 
9 


55 023 
145 
267 
388 
509 


035 
157 
279 
400 
522 


047 
169 
291 
413 
534 


060 
182 
303 
425 
546 


072 
194 
315 
437 
558 


084 
206 
328 
449 
570 


096 
218 
340 
461 
582 


108 
230 
352 
473 
594 


121 

242 
364 
485 
606 


133 
255 
376 
497 
618 


5 
6 
7 
8 
9 


7 

8 

9 

10 

12 


3.55 
3.56 
3.57 
3.58 
3.59 


360 

1 
2 
3 
4 


630 
751 
871 
991 
56 110 


642 
763 
883 
003 
122 


654 
775 
895 
D15 
134 


666 
787 
907 
027 
146 


678 
799 
919 
038 
158 


691 
811 
931 
050 
170 


703 
823 
943 
062 
182 


715 
835 
955 
074 
194 


727 
847 
967 
D86 
205 


739 
859 
979 
098 
217 


1 
2 
3 
4 




3.60 
3.61 
3.62 
3.63 
3.64 


365 
6 
7 
8 
9 


229 
348 
467 
585 
703 


241 
360 

478 
597 
714 


253 
372 
490 
608 
726 


265 
384 
502 
620 
738 


277 
396 
514 
632 
750 


289 
407 
526 
644 
761 


301 
419 
538 
656 
773 


312 
431 

549 
667 
785 


324 
443 
561 
679 
797 


336 

455 
573 
691 
808 


5 
6 
7 
8 
9 




3.65 
3.66 
3.67 
3.68 
3.69 


370 

1 
2 
3 
4 


820 

937 

57 054 

171 

287 


832 
949 
066 
183 
299 


844 
961 
078 
194 
310 


855 
972 
089 
206 
322 


867 
984 
101 
217 
334 


879 
996 
113 
229 
345 


891 
008 

124 
241 
357 


902 
019 
136 
252 
368 


914 
031 
148 
264 
380 


926 
043 
159 
276 
392 


1 
2 
3 
4 


12 

1 
2 
4 
5 


3.70 
3.71 
3.72 
3.73 
3.74 


375 
6 

7 
8 
9 


403 
519 
634 
749 
864 


415 
530 
646 
761 

875 


426 

542 
657 

772 
887 


438 
553 
669 

784 
898 


449 
565 
680 
795 
910 


461 

576 
692 
807 
921 


473 

588 
703 
818 
933 


484 
600 
715 
830 
944 


496 
611 
726 
841 
955 


507 
623 
738 
852 
967 


5 
6 
7 
8 
9 


6 
7 
8 

10 
11 


3.75 
3.76 
3.77 
3.78 
3.79 


380 

1 
2 
3 
4 


978 
58 092 
206 
320 
433 


990 
104 
218 
331 
444 


001 

115 
229 
343 
456 


D13 
127 
240 
354 
467 


024 
138 
252 
365 
478 


D35 
149 
263 
377 
490 


047 
161 
274 
388 
501 


058 
172 
286 
399 
512 


070 
184 
297 
410 
524 


081 
195 
309 
422 
535 


1 
2 
3 
4 




3.80 
3.81 
3.82 
3.83 
3.84 


385 
6 
7 
8 
9 


546 
659 
771 
883 
995 


557 
670 
782 
894 
006 


569 
681 
794 
906 
017 


580 
692 
805 
917 
D28 


591 
704 
816 
928 
D40 


602 
715 
827 
939 
051 


614 
726 
838 
950 
062 


625 
737 
850 
961 
073 


636 

749 
861 
973 
084 


647 
760 
872 
984 
095 


5 
6 

7 
8 
9 




3.85 
3.86 
3.87 
3.88 
3.89 


390 

1 
2 
3 
4 


59 106 
218 
329 
439 
550 


118 
229 
340 
450 
561 


129 
240 
351 

461 
572 


140 
251 
362 
472 
583 


151 
262 
373 
483 
594 


162 
273 
384 
494 
605 


173 

284 
395 
506 
1616 


184 
295 
406 
517 
627 


195 
306 
417 
528 
638 


207 
318 
428 
539 
649 


1 
2 
3 
4 


11 

1 
2 
3 

4 


3.90 
3.91 
3.92 
3.93 
3.94 


395 
6 

7 
8 
9 


660 
770 
879 
988 
60 097 


671. 
780 
890 
999 
108 


682 
791 
901 
DIO 
119 


693 
802 
912 
021 
130 


704 
813 
923 
032 
141 


715 
824 
934 
043 
152 


726 
835 
945 
054 
163 


737 
846 
956 
065 
173 


748 
857 
966 
076 
184 


759 
868 
977 
086 
195 


5 
6 
7 
8 
9 


6 
7 
8 
9 
10 


3.95 
3.96 
3.97 
3.98 
3.99 


400 


206 


217 


228 


239 


249 


260 


271 


282 


293 


304 






4.00 



114 



Q.— LOGARITHMS OF NUMBERS. 
1. — Logarithms. — Continued. 



No. 


Common Logarithms of Numbers. 


Naperian. 




L. O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


I 


\ P. 


No. 


Log. TAt. 


400 

1 
2 
3 
4 


60 206 
314 
423 
531 
638 


217 
325 
433 
541 
649 


228 
336 
444 
552 
660 


239 
347 
455 
563 
670 


249 
358 
466 
574 
681 


260 
369 
477 
584 
692 


271 
379 
487 
595 
703 


282 
390 
498 
606 
713 


293 
401 
509 
617 
724 


304 
412 
520 
627 
735 


1 
2 
3 
4 


11 

1 
2 
3 
4 


4.00 
4.01 
4.02 
4.03 
4.04 


1.38629 „.. 
1.38879 250 
1.39128 249 
1.39377 if„ 
1.39624 ^47 

248 
1.39872 „.. 
1.40118 ;46 
1.40364 if. 
1.40610 If. 
1.40854 "^44 

245 
1.41099 „,, 
1.41342 if^ 
1.41585 If, 
1.41828 if, 
1.42070 "^42 
, 241 
1.42311 „,, 
1.42552 ;41 
1.42792 ill 
1.43031 239 
1.43270 ^^^ 

238 
1.43508 „,« 
1.43746 f^l 
1.43984 i;Ji 
1.44220 ill 
1.44456 ^^^ 

236 
1.44692 „- 
1.44927 fJl 
1.45161 ill 
1.45395 234 
1.45629 ^^^ 

232 
1.45861 „oo 
1.46094 ili 
1.46326 iii 
1.46557 i%\ 
1.46787 ^^^ 

231 
1.47018 „q 
1.47247 229 
1.47476 ^;5 
1.47705 9^^ 
1.47933 ^^^ 

227 
1.48160 „7 
1.48387 i% 
1.48614 ii,' 
1.48840 iil 
1.49065 ^^^ 

225 
1.49290 „. 
1.49515 iil 
1.49739 „* 
1.49962 iii 
1.50185 ^^^ 

223 
1.50408 


405 
6 
7 
8 
9 


746 
853 
959 
ftl 066 
172 


756 
863 
970 
077 
183 


767 
874 
981 
087 
194 


778 
885 
991 
098 
204 


788 
895 
002 
109 
215 


799 
906 
D13 
119 
225 


810 
917 
023 
130 
236 


821 
927 
034 
140 
247 


831 
938 
045 
151 
257 


842 
949 
055 
162 
268 


5 
6 
7 
8 
9 


6 

7 

8 

9 

10 


4.05 
4.06 
4.07 
4.08 
4.09 


410 

1 
2 
3 

4 

- 


278 
. 384 
490 
595 
700 


289 
395 
500 
606 
711 


300 
405 
511 
616 
721 


310 
416 
521 
627 
731 


321 

426 
532 
637 

742 


331 

437 
542 
648 

752 


342 

448 
553 
658 
763 


352 
458 
563 
669 
773 


363 
469 
574 
679 

784 


374 
479 
584 
690 
794 


1 
2 
3 
4 




4.10 
4.11 
4.12 
4.13 
4.14 


415 
6 
7 
8 
9 


805 
909 
62 014 
118 
221 


815 
920 
024 
128 
232 


826 
930 
034 
138 
242 


836 
941 
045 
149 
252 


847 
951 
055 
159 
263 


857 
962 
066 
170 
273 


868 
972 
076 
180 

284 


878 
982 
086 
190 
294 


888 
993 
097 
201 
304 


899 
003 

107 
211 
315 


5 
6 
7 
8 
9 




4.15 
4.16 
4.17 
4.18 
4.19 


420 

1 
2 
3 
4 


325 
428 
531 
634 
737 


335 
439 
542 
644 

747 


346 
449 
552 
655 
757 


356 
459 
562 
665 
767 


366 
469 
572 
675 
778 


377 
480 
583 
685 
788 


387 
490 
593 
696 
798 


397 
500 
603 
706 
808 


408 
511 
613 
716 
818 


418 
521 
624 
726 
829 


1 
2 
3 
4 


10 

1 
2 
3 
4 


4.20 
4.21 
4.22 
4.23 
4.24 


425 
6 
7 
8 
9 


839 
941 
63 043 
144 
246 


849 
951 
053 
155 
256 


859 
961 
063 
165 
266 


870 
972 
073 
175 
276 


880 
982 
083 
185 
286 


890 
992 
094 
195 
296 


900 
002 

104 
205 
306 


910 
012 
114 
215 
317 


921 
022 
124 
225 
327 


931 
033 
134 
236 
337 


5 
6 
7 
8 
9 


5 
6 
7 
8 
9 


4.25 
4.26 
4.27 
4.28 
4.29 


430 

1 
2 
3 
4 


347 
448 
548 
649 
749 


357 
458 
558 
659 
759 


367 
468 
568 
669 
769 


377 
478 
579 
679 
779 


387 
488 
589 
689 
789 


397 
498 
599 
699 
799 


407 
508 
609 
709 
809 


417 
518 
619 
719 
819 


428 
528 
629 
729 
829 


438 
538 
639 
739 
839 


1 
2 
3 
4 




4.30 
4.31 
4.32 
4.33 
4.34 


435 
6 
7 
8 
9 


849 
949 
64 048 
147 
246 


859 
959 
058 
157 
256 


869 
969 
068 
167 
266 


879 
979 
078 
177 
276 


889 
988 
088 
187 
286 


899 
998 
098 
197 
296 


909 
008 

108 
207 
306 


919 
018 
118 
217 
316 


929 
028 
128 
227 
326 


939 
038 
137 
237 
335 


5 
6 
7 
8 
9 




4.35 
4.36 
4.37 
4.38 
4.39 


440 

1 
2 
3 
4 


345 
444 
542 
640 
738 


355 
454 
552 
650 
748 


365 
464 
562 
660 
758 


375 
473 
572 
670 
768 


385 
483 
582 
680 

777 


395 
493 
591 
689 

787 


404 
503 
601 
699 
797 


414 
513 
611 
709 
807 


424 
523 
621 
719 
816 


434 
532 
631 
729 
826 


1 
2 
3 
4 


9 

1 
2 
3 
4 


4.40 
4.41 
4.42 
4.43 
4.44 


445 
6 
7 
8 
9 


836 
933 
65 031 
128 
225 


846 
943 
040 
137 
234 


856 
953 
050 
147 
244 


865 
963 
060 
157 
254 


875 
972 
070 
167 
263 


885 
982 
079 
176 
273 


895 
992 
089 
186 
283 


904 
002 

099 
196 
292 


914 
Oil 
108 
205 
302 


924 

021 
118 
215 
312 


5 
6 

7 
8 
9 


4.5 

5.4 

6 

7 
8 


4.45 
4.46 
4.47 
4.48 
4.49 


450 


321 


331 


341 


350 


360 


369 : 


379 


389 


398 


408 






4.50 



LORARITHMS OF NUMBERS. 
1 . — Logarithms . — Continued. 



115 



No. 


Common Logarithms of Numbers. 


Naperian. 




L. O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P 

1 
2 
3 
4 


. P. 


No. 


Log. Dif. 


450 

1 
2 
3 
4 


65 321 
418 
514 
610 
706 


331 
427 
523 
619 
715 


341 
437 
533 
629 
725 


350 
447 
543 
639 
734 


360 
456 
552 
648 
744 


369 
466 
562 
658 
753 


379 
475 
571 
667 
763 


389 
485 
581 
677 

772 


398 
495 
591 
686 
782 


408 
504 
600 
696 
792 


10 

2 
3 
4 


4.50 
4.51 
4.52 
4.53 
4.54 


1.50408 „„ 
1.50630 ;;f 
1.50851 ii] 
1.51072 ii] 
1.51293 221 
220 
1.51513 „.. 
1.51732 2 9 

1.51951 ;\l 

1.52170 ^J^ 
1.52388 218 

218 
1.52606 „.„ 
1.52823 i\l 
1.53039 210 
1.53256 217 
1.53471 215 

216 
1.53687 „., 
1.53902 215 
1.54116 2 * 
1.54330 214 
1.54543 21^ 

213 
1.54756 „.„ 
1.54969 iti 
1.55181 21^ 
1.55393 212 
1.55604 211 

210 
1.55814 p., 
1.56025 211 
1.56235 210 
1.56444 209 
1.56653 20y 

209 
1.56862 „0Q 
1.57070 208 
1.57277 207 

1.57485 20« 
1.57691 20t> 

207 
1.57898 „„. 
1.58104 20b 
1.58309 205 
1.58515 X 
1.58719 ^^* 

205 

1.58924 203 
1.59127 III 
1.59331 2"* 
1.59534203 

1.59737 ''""* 
202 
1.59939 „„« 
1.60141 202 
1.60342 ll\ 
1.60543 2"f 
1.60744 201 
206 
1.60944 


455 
6 

7 
8 
9 


801 
896 
992 
66 087 
181 


811 
906 
001 

096 
191 


820 
916 
Dll 
106 
200 


830 
925 
020 
115 
210 


839 
935 
D30 
124 
219 


849 
944 
039 
134 
229 


858 
954 
049 
143 
238 


868 
963 
058 
153 
247 


877 
973 
068 
162 
257 


887 
982 
077 
172 
266 


5 
6 

7 
8 
9 


5 
6 

7 
8 
9 


4.55 
4.56 
4.57 
4.58 
4.59 


460 

1 
2 
3 
4 


276 
370 
464 
558 
652 


285 
380 
474 
567 
661 


295 
389 
483 
577 
671 


304 
398 
492 
586 
680 


314 

408 
502 
596 
689 


323 
417 
511 
605 
699 


332 
427 
521 
614 
708 


342 
436 
530 
624 
717 


351 
445 
539 
633 

727 


361 
455 
549 
642 
736 


1 
2 
3 
4 




4.60 
4.61 
4.62 
4.63 
4.64 


465 
6 

7 
8 
9 


* 745 

839 

932 

67 025 

117 


755 
848 
941 
034 
127 


764 
857 
950 
043 
136 


773 
867 
960 
052 
145 


783 
876 
969 
062 
154 


792 
885 
978 
071 
164 


801 
894 
987 
080 
173 


811 
904 
997 
089 
182 


820 
913 
606 
099 
191 


829 
922 
015 
108 
201 


5 
6 
7 
8 
9 




4.65 
4.66 
4.67 
4.68 
4.69 


470 
1 
2 
3 

4 


210 
302 
394 
486 
578 


219 
311 
403 
495 
587 


228 
321 
413 
504 
596 


237 
330 
422 
514 
605 


247 
339 
431 
523 
614 


256 
348 
440 
532 
624 


265 
357 
449 
541 
633 


274 
367 
459 
550 
642 


284 
376 
468 
560 
651 


293 

385 
477 
569 
660 


1 
2 
3 
4 


9 

1' 
2 
3 
4 


4.70 
4.71 

4.72 
4.73 
4.74 


475 
6 
7 
8 
9 


669 
761 
852 
943 
68 034 


679 
770 
861 
952 
043 


688 
779 
870 
961 
052 


697 
788 
879 
970 
061 


706 
797 
888 
979 
070 


715 
806 
897 
988 
079 


724 
815 
906 
997 
088 


733 
825 
916 
006 

097 


742 
834 
925 
015 
106 


752 
843 
934 
024 
115 


5 
6 
7 
8 
9 


4.5 

5.4 

6 

7 

8 


4.75 
4.76 
4.77 
4.78 
4.79 


480 
1 
2 
3 

4 


124 
215 
305 
395 
485 


133 
224 
314 
404 
494 


142 
233 
323 
413 
502 


151 
242 
332 
422 
511 


160 
251 
341 
431 
520 


169 
260 
350 
440 
529 


178 
269 
359 
449 
538 


187 
278 
368 
458 
547 


196 
287 
377 
467 
556 


205 
296 

386 
476 
565 


1 
2 
3 
4 




4.80 
4.81 
4.82 
4.83 
4.84 


485 
6 
7 
8 
9 


574 
664 
753 
842 
931 


583 
673 
762 
851 
940 


592 
681 
771 
860 
949 


601 
690 
780 
869 
958 


610 
699 
789 
878 
966 


619 
708 
797 
886 
975 


628 
717 
806 
895 
984 


637 
726 
815 
904 
993 


646 
735 
824 
913 
002 


655 
744 
833 
922 
Oil 


5 
6 

7 
8 
9 




4.85 
4.86 
4.87 
4.88 
4.89 


490 

1 
2 
3 
4 


69 020 
108 
197 
285 
373 


028 
117 
205 
294 
381 


037 
126 
214 
302 
390 


046 
135 
223 
311 
399 


055 
144 
232 
320 
408 


064 
152 
241 
329 
417 


073 
161 
249 
338 
425 


082 
170 
258 
346 
434 


090 
179 
267 
355 
443 


099 
188 
276 
364 
452 


1 
2 
3 
4 


8 
1 

1.6 
2.4 
3 


4.90 
4.91 
4.92 
4.93 
4.94 


495 
6 
7 
8 
9 


461 
548 
636 
723 
810 


469 
557 
644 
732 
819 


478 
566 
653 
740 
827 


487 
574 
662 
749 
836 


496 
583 
671 
758 
845 


504 
592 
679 
767 
854 


513 
601 
688 
775 
862 


522 
609 
697 
784 
871 


531 
618 
705 
793 
880 


539 

627 
714 
801 

888 


5 
6 
7 
8 
9 


4 
5 

5.6 
6.4 

7 


4.95 
4.96 
4.97 
4.98 
4.99 


500 


897 


906 


914 


923 


932 


940 


949 


958 


966 


975 






5.00 



116 



^.—LOGARITHMS OF NUMBERS. 
1. — Logarithms. — Continued. 



No. 


Common Logarithms of Numbers. 


Naperian. 




L. O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


I 


». P. 


No. 


Log. Dif. 


500 

1 
2 
3 
4 


69 897 
984 

70 070 
157 
243 


906 
992 
079 
165 
252 


914 
001 

088 
174 
260 


923 
DIO 
096 
183 
269 


932 
018 
105 
191 
278 


940 
027 
114 
200 
286 


949 
036 
122 
209 
295 


958 
044 
131 
217 
303 


966 
053 
140 
226 
312 


975 
062 
148 
234 
321 


1 
2 
3 
4 


9 

1 

2 
3 
4 


5.00 
5.01 
5.02 
5.03 
5.04 


1.60944 „.. 
1.61144 200 
1.61343 Vil 
1.61542 }99 
1.61741 ^^^ 

198 
1.61939 log 
1.62137 |5° 
1.62334 VSj 
1.62531 ;^7 
1.62728 ^^' 

196 
1.62924 iQfi 
1.63120}^^ 
1.63315 \ll 
1.63511 ^6 
1.63705 ^* 

195 
1.639(J0 iQ, 
1.64094 \l\ 
1.64287 ^.\ 
1.64481 \ll 
1.64673 ^^^ 

193 
1.64866 iQ« 
1.65058 IJ^ 
1.65250 }Jf 
1.65441 \l\ 
1.65632 ^^^ 

191 
1.65823 190 
1.66013 }5" 
1.66203 \ll 
1.66393 J^" 
1.66582 ^"^ 

189 
1.66771 .gjj 
1.66959 }°^ 
1.67147 Ifr. 
1.67335 ;°° 
1.67523 ^^^ 

187 
1.67710 io. 
1.67896 °; 
1.68083 \ff, 
1.68269 Jn^ 
1.68455 ^^^ 

185 
1.68640 .o^ 
1.68825 XS, 
1.69010 JoY 
1.69194 }Sj 
1.69378 ^^* 

184 
1.69562 ,„, 
1.69745 Vii 
1.69928 V^i 

1.70111 ;°^ 

1.70293 ^^^ 
182 
1.70475 


505 
6 
7 
8 
9 


329 
415 
501 
586 
672 


338 
424 
509 
595 
680 


346 
432 
518 
603 
689 


355 
441 
526 
612 
697 


364 
449 
535 
621 
706 


372 
458 
544 
629 
714 


381 
467 
552 
638 
723 


389 
475 
561 
646 
731 


398 
484 
569 
655 
740 


406 
492 
578 
663 
749 


5 
6 

7 
8 
9 


4.5 

5.4 

6 

7 

8 


5.05 
5.06 
5.07 
5.08 
5.09 


510 

1 
2 
3 
4 


757 
842 
927 
71 012 
096 


766 
851 
935 
020 
105 


774 
859 
944 
029 
113 


783 
868 
952 
037 
122 


791 
876 
961 
046 
130 


800 
885 
969 
054 
139 


808 
893 
978 
063 
147 


817 
902 
986 
071 
155 


825 
910 
995 
079 
164 


834 
919 
003 

088 
172 


1 
2 
3 
4 




5.10 
5.11 
5.12 
5.13 
5.14 


515 
6 
7 
8 
9 


181 
265 
349 
433 
517 


189 
273 
357 
441 
525 


198 
282 
366 
450 
533 


206 
290 
374 
458 
542 


214 
299 
383 
466 
550 


223 
307 
391 
475 
559 


231 
315 

399 
483 
567 


240 
324 
408 
492 
575 


248 
332 
416 
500 
584 


257 
341 
425 
508 
592 


5 
6 

7 
8 
9 




5.15 
5.16 
5.17 
5.18 
5.19 


520 

2 
3 
4 


600 
684 
767 
850 
933 


609 
692 
775 
858 
941 


617 
700 
784 
867 
950 


625 

709 
792 
875 
958 


634 
717 
800 
883 
966 


642 
725 
809 
892 
975 


650 
734 
817 
900 
983 


659 
742 
825 
908 
991 


667 
750 
834 
917 
999 


675 
759 
842 
925 
008 


1 
2 
3 
4 


8 

1 

1.6 
2.4 
3 


5.20 
5.21 
5.22 
5.23 
5.24 


525 
6 
7 
8 
9 


72 016 
099 
181 
263 
346 


024 
107 
189 
272 
354 


032 
115 
198 
280 
362 


041 
123 
206 
288 
370 


049 
132 
214 
296 
378 


057 
140 
222 
304 
387 


066 
148 
230 
313 
395 


074 
156 
239 
321 
403 


082 
165 
247 
329 
411 


090 
173 
255 
337 
419 


5 
6 
7 
8 
9 


4 
5 

5.6 
6.4 

7 


5.25 
5.26 
5.27 
5.28 
5.29 


530 

1 
2 
3 
4 


428 
509 
591 
673 
754 


436 

518 
599 
681 
762 


444 

526 
607 
689 
770 


452 
534 
616 
697 
779 


460 

542 
624 
705 

787 


469 
550 
632 
713 
795 


477 
558 
640 
722 
803 


485 
567 
648 
730 
811 


493 
575 
656 
738 
819 


501 
583 
665 
746 
827 


1 
2 
3 
4 




5.30 
5.31 
5.32 
5.33 
5.34 


535 
6 
7 
8 
9 


835 
916 
997 
73 078 
159 


843 
925 
006 

086 
167 


852 
933 
014 
094 
175 


860 
941 
022 
102 
183 


868 
949 
030 
111 
191 


876 
957 
038 
119 
199 


884 
965 
046 
127 
207 


892 
973 
054 
135 
215 


900 
981 
062 
143 
223 


908 
989 
070 
151 
231 


5 
6 
7 
8 
9 




5.35 
5.36 
5.37 
5.38 
5.39 


540 

1 
2 
3 
4 


239 
320 
400 
480 
560 


247 
328 
408 
488 
568 


255 
336 
416 
496 
576 


263 
344 
424 
504 
584 


272 
352 
432 
512 
592 


280 
360 
440 
520 
600 


288 
368 
448 
528 
608 


296 
376 
456 
536 
616 


304 
384 
464 
544 
624 


312 
392 
472 
552 
632 


1 
2 
3 
4 


7 

0.7 
1.4 
2 
3 


5.40 
5.41 
5.42 
5.43 
5.44 


545 
6 

7 
8 
9 


640 
719 
799 
878 
957 


648 

727 
807 
886 
965 


656 
735 
815 
894 
973 


664 
743 
823 
902 
981 


672 
751 
830 
910 
989 


679 
759 
838 
918 
997 


687 
767 
846 
926 
DOS 


695 
775 
854 
933 
013 


703 
783 
862 
941 
020 


711 

791 
870 
949 
028 


5 
6 

7 
8 
9 


3.5 

4.2 

5 

5.6 

6.3 


5.45 
5.46 
5.47 
5.48 
5.49 


550 


74 036 


044 


052 


060 


068 


076 


084 


092 


099 


107 






5.50 



LOGARITHMS OF NUMBERS. 
1 . — ^Logarithms . — Continued . 



117 



No. 


Common Logarithms of Numbers. 


Naperian. 




L. O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 


No. 


Log. Dif. 


550 


74 036 


044 


052 


060 


068 


076 


084 


092 


099 


107 






5.50 


1.70475 iR. 
1.70656 ]°l 
1.70838 }°f 
1.71019 III 
1.71199 ^^" 
181 
1.71380 ... 
1.71560 J°" 
1.71740 \^l 

1.71919 I;; 

1.72098 ^^^ 
179 
1.72277 .„« 
1.72455 ]l° 
1.72633 ]ll 

1.72811 |;° 

1.72988 ^^^ 

178 
1.73166 .„. 
1.73342 JI; 
1.73519 \ll 
1.73695 til 
1.73871 ^^^ 

176 
1.74047 175 
1.74222 ;!,l 
1.74397 ]ll 
1.74572 ]ll 
1.74746 ^^* 

174 
1.74920 .„, 
1.75094 };J 
1.75267 };^ 
1.75440 \!,i 
1.75613 ^^^. 

173 
1.75786 i„„ 
1.75958 ili 
1.76130 Y4 
1.76302 {if 
1.76473 ^^^ 

171 
1.76644 i„i 
1.76815 |;i 
1.76985 };" 
1.77156 :j 
1.77326 ^'^ 

169 
1.77495 170 
1.77665 }^" 
1.77834 \ll 
1.78002 ill 
1.78171 ^^^ 

168 
1.78339 ,.o 
1.78507 }l° 
1.78675 }X? 
1.78842 \l' 
1.79009 ^^' 

167 
1.79176 


1 


115 


123 


131 


139 


147 


155 


162 


170 


178 


186 


1 




5.51 


2 


194 


202 


210 


218 


225 


233 


241 


249 


257 


265 


2 




5.52 


3 


273 


280 


288 


296 


304 


312 


320 


327 


335 


343 


3 




5.53 


4 


351 


359 


367 


374 


382 


390 


398 


406 


414 


421 


4 




5.54 


555 


429 


437 


445 


453 


461 


468 


476 


484 


492 


500 


5 




5.55 


6 


507 


515 


523 


531 


539 


547 


554 


562 


570 


578 


6 




5.56 


7 


586 


593 


601 


609 


617 


624 


632 


640 


648 


656 


7 




5.57 


8 


663 


671 


679 


687 


695 


702 


710 


718 


726 


733 


8 




5.58 


9 


741 


749 


757 


764 


772 


780 


788 


796 


803 


811 


9 




5.59 


560 


819 


827 


834 


842 


850 


858 


865 


873 


881 


889 




8 


5.60 


1 


896 


904 


912 


920 


927 


935 


943 


950 


958 


966 


1 


1 


5.61 


2 


974 


981 


989 


997 


SOS 


012 


020 


028 


035 


043 


2 


1.6 


5.62 


3 


76 051 


059 


066 


074 


082 


089 


097 


105 


113 


120 


3 


2.4 


5.63 


4 


128 


136 


143 


151 


159 


166 


174 


182 


189 


197 


4 


3 


5.64 


565 


205 


213 


220 


228 


236 


243 


251 


259 


266 


274 


5 


4 


5.65 


6 


282 


289 


297 


305 


312 


320 


328 


335 


343 


351 


6 


5 


5.66 


7 


358 


366 


374 


381 


389 


397 


404 


412 


420 


427 


7 


5.6 


5.67 


8 


435 


442 


450 


458 


465 


473 


481 


488 


496 


504 


8 


6.4 


5.68 


9 


511 


519 


526 


534 


542 


549 


557 


565 


572 


580 


9 


7 


5.69 


570 


587 


595 


603 


610 


618 


626 


633 


641 


648 


656 






5.70 


1 


664 


671 


679 


686 


694 


702 


709 


717 


724 


732 


1 




5.71 


2 


740 


747 


755 


762 


770 


778 


785 


793 


800 


808 


2 




5.72 


3 


815 


823 


831 


838 


846 


853 


861 


868 


876 


884 


3 




5.73 


4 


891 


899 


906 


914 


921 


929 


937 


944 


952 


959 


4 




5.74 


575 


967 


974 


982 


989 


997 


005 


C12 


D20 


027 


035 


5 




5.75 


6 


76 042 


050 


057 


065 


072 


080 


087 


095 


103 


110 


6 




5.76 


7 


118 


125 


133 


140 


148 


155 


163 


170 


178 


185 


7 




5.77 


8 


193 


200 


208 


215 


223 


230 


238 


245 


253 


260 


8 




5.78 


9 


268 


275 


283 


290 


298 


305 


313 


320 


328 


335 


9 




5.79 


580 


343 


350 


358 


3&5 


373 


380 


388 


395 


403 


410 




7 


5.80 


1 


418 


425 


433 


440 


448 


455 


462 


470 


477 


485 


1 


0.7 


5.81 


2 


492 


500 


507 


515 


522 


530 


537 


545 


552 


559 


2 


1.4 


5.82 


3 


667 


574 


582 


589' 


597 


604 


612 


619 


626 


634 


3 


2 


5.83 


4 


641 


649 


656 


664 


671 


678 


686 


693 


701 


708 


4 


3 


5.84 


585 


716 


723 


730 


738 


745 


753 


760 


768 


775 


782 


5 


3.5 


5.85 


6 


790 


797 


805 


812 


819 


827 


834 


842 


849 


856 


6 


4.2 


5.86 


7 


864 


871 


879 


886 


893 


901 


908 


916 


923 


930 


7 


5 


5.87 


8 


938 


945 


953 


960 


967 


975 


982 


989 


997 


D04 


8 


5.6 


5.88 


9 


77 012 


019 


026 


034 


041 


048 


056 


063 


070 


078 


9 


6.3 


5.89 


590 


085 


093 


100 


107 


115 


122 


129 


137 


144 


151 






5.90 


1 


159 


166 


173 


181 


188 


195 


203 


210 


217 


225 


1 




5.91 


2 


232 


240 


247 


254 


262 


269 


276 


283 


291 


298 


2 




5.92 


3 


305 


313 


320 


327 


335 


342 


349 


357 


364 


371 


3 




5.93 


4 


379 


386 


393 


401 


408 


415 


422 


430 


437 


444 


4 




5.94 


695 


452 


459 


466 


474 


481 


488 


495 


503 


510 


517 


5 




5.95 


6 


525 


532 


539 


546 


554 


561 


568 


576 


583 


590 


6 




5.96 


7 


597 


605 


612 


619 


627 


634 


641 


648 


656 


663 


7 




5.97 


8 


670 


677 


685 


692 


699 


706 


714 


721 


728 


735 


8 




5.98 


9 


743 


750 


757 


764 


772 


779 


786 


793 


801 


808 


9 




5.99 


600 


815 


822 


830 


837 


844 


851 


859 


866 


873 


880 






6.00 



118 



Q.— LOGARITHMS OF NUMBERS. 
1. — Logarithms. — Continued. 



No. 


Common Logarithms of Numbers. 


Naperian. 




L. O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


r 


. P. 


No. 


Log. Dlf. 


600 


77 815 


822 


830 


837 


844 


851 


859 


866 


873 


880 




8 


6.00 


1.79176 ..» 
1.79342 J^; 
1.79509 ]% 
1.79675 }^° 
1.79840 ^^^ 

166 
1.80006 ,.- 
1.80171 \ll 
1.80336 \ll 
1.80500 \ll 
1.80665 ^^5 

164 
1.80829 ... 
1.80993 \l\ 
1.81156 \% 
1.81319 ]f^ 
1.81482 ^^3 

163 
1.81645 .-, 
1.81808 JS? 
1.81970 \li 
1.82132 \f^ 
1.82294 ^^2 

161 
1.82455 ... 
1.82616 ;^} 
1.82777 }°J 
1.82938 }^i 
1.83098 ^^" 

160 
1.83258 ..« 
1.83418 f^ 
1.83578 }5" 
1.83737 \f^ 
1.83896 ^^^ 

1.84372 }^ 
1.84530 \% 
1.84688 ^^ 

157 
1.84845 .^ 
1.85003 }J° 
1.85160 \% 
1.85317 \% 
1.85473 ^^^ 

157 
1.85630 .KR 
1.85786 YZ 
1.85942 }^° 
1.86097 J^^ 
1.86253 ^^ 

155 
1.86408 ... 
1.86563 \f. 
1.86718 \Z. 
1.86872 ;?* 
1.87026 ^^* 

154 
1.87180 


1 


887 


895 


902 


909 


916 


924 


931 


938 


945 


952 


1 


1 


6.01 


2 


960 


967 


974 


981 


988 


996 


003 


DIO 


017 


025 


2 


1.6 


6.02 


3 


78 032 


039 


046 


053 


061 


068 


075 


082 


089 


097 


3 


2.4 


6.03 


4 


104 


111 


118 


125 


132 


140 


147 


154 


161 


168 


4 


3 


6.04 


605 


176 


183 


190 


197 


204 


211 


219 


226 


233 


240 


5 


4 


6.05 


6 


247 


254 


262 


269 


276 


283 


290 


297 


305 


312 


6 


5 


6.06 


7 


319 


326 


333 


340 


347 


355 


362 


369 


376 


383 


7 


5.6 


6.07 


8 


390 


398 


405 


412 


419 


426 


433 


440 


447 


455 


8 


6.4 


6.08 


9 


462 


469 


476 


483 


490 


497 


504 


512 


519 


526 


9 


7 


6.09 


610 


533 


540 


547 


554 


561 


569 


576 


583 


590 


597 






6.10 


1 


604 


611 


618 


625 


633 


640 


647 


654 


661 


668 


1 




6.11 


2 


675 


682 


689 


696 


704 


711 


718 


725 


732 


739 


2 




6.12 


3 


746 


753 


760 


767 


774 


781 


789 


796 


802 


810 


3 




6.13 


4 


817 


824 


831 


838 


845 


852 


859 


866 


873 


880 


' 




6.14 


615 


888 


895 


902 


909 


916 


923 


930 


937 


944 


951 


5 




6.15 


6 


858 


965 


972 


979 


986 


993 


000 


007 


014 


021 


6 




6.16 


7 


79 029 


036 


043 


050 


057 


064 


071 


078 


085 


092 


7 




6.17 


8 


099 


106 


113 


120 


127 


134 


141 


148 


155 


162 


8 




6.18 


9 


169 


176 


183 


190 


197 


204 


211 


218 


225 


232 


9 




6.19 


620 


239 


246 


253 


260 


267 


274 


281 


288 


295 


302 




7 


6.20 


1 


309 


316 


323 


330 


337 


344 


351 


358 


365 


372 


1 


0.7 


6.21 


2 


379 


386 


393 


400 


407 


414 


421 


428 


435 


442 


2 


1.4 


6.22 


3 


449 


456 


463 


470 


477 


484 


491 


498 


505 


511 


3 


2 


6.23 


4 


518 


525 


532 


539 


546 


553 


560 


567 


574 


581 


4 


3 


6.24 


625 


588 


595 


602 


609 


616 


623 


630 


637 


644 


650 


5 


3.5 


6.25 


6 


657 


664 


671 


678 


685 


692 


699 


706 


713 


720 


6 


4.2 


6.26 


7 


727 


734 


741 


748 


754 


761 


768 


775 


782 


789 


7 


5 


6.27 


8 


796 


803 


810 


817 


824 


831 


837 


844 


851 


858 


8 


5.6 


6.28 


9 


865 


872 


879 


886 


893 


900 


906 


913 


920 


927 


9 


6.3 


6.29 


630 


934 


941 


948 


955 


962 


969 


975 


982 


989 


996 






6.30 


1 


80 003 


010 


017 


024 


030 


037 


044 


051 


058 


065 


1 




6.31 


2 


072 


079 


085 


092 


099 


106 


113 


120 


127 


134 


2 




6.32 


3 


140 


147 


154 


161 


168 


175 


182 


188 


195 


202 


3 




6.33 


4 


209 


216 


223 


229 


236 


243 


250 


257 


264 


271 


4 




6.34 


635 


277 


284 


291 


298 


305 


312 


318 


325 


332 


339 


5 




6.35 


6 


346 


353 


359 


366 


373 


380 


387 


393 


400 


407 


6 




6.36 


7 


414 


421 


428 


434 


441 


448 


455 


462 


468 


475 


7 




6.37 


8 


482 


489 


496 


502 


509 


516 


523 


530 


536 


543 


8 




6.38 


9 


500 


557 


564 


570 


577 


584 


591 


598 


604 


611 


9 




6.39 


640 


618 


625 


632 


638 


645 


652 


659 


665 


672 


679 




6 


6.40 


1 


686 


693 


699 


706 


713 


720 


726 


733 


740 


747 


1 


0.6 


6.41 


2 


754 


760 


767 


774 


781 


787 


794 


801 


808 


814 


2 


1.2 


6.42 


3 


821 


828 


835 


841 


848 


855 


862 


868 


875 


882 


3 


1.8 


6.43 


4 


889 


895 


902 


909 


916 


922 


929 


936 


943 


949 


4 


2.4 


6.44 


645 


956 


963 


969 


976 


983 


990 


996 


003 


mo 


017 


5 


3 


6.45 


6 


81 023 


030 


037 


043 


050 


057 


064 


070 


077 


084 


6 


3.6 


6.46 


7 


090 


097 


104 


111 


117 


124 


131 


137 


144 


151 


7 


4.2 


6.47 


8 


158 


164 


171 


178 


184 


191 


198 


024 


211 


218 


8 


4.8 


6.48 


9 


224 


231 


238 


245 


251 


258 


265 


271 


278 


285 


9 


5.4 


6.49 


650 


291 


298 


305 


311 


318 


325 


331 


338 


345 


351 






6.50 



LOGARITHMS OF NUMBERS. 
1. — Logarithms. — Continued. 



lid 



No. 


Common Logarithms of Numbers. 


Naperian. 




L. O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 


No. 


Log. Dif. 


650 


81 291 


298 


305 


311 


318 


325 


331 


338 


345 


351 






6.50 


1.87180 ... 
1.87334 ]li 
1.87487 \Z 
1.87641 }J* 
1.87794 ^^^ 
153 
1.87947 ,„ 
1.88099 ]ii 

1.88251 ;ii 

1.88403 \li 
1.88555 ^^'^ 
152 
1.88707 ... 
1.88858 J^i 

1.89010 ;^^ 

1.89160 ;^? 
1.89311 ^^^ 

151 
1.89462 15Q 
1.89612 }^" 
1.89762 ;^X 
1.89912 JJ" 
1.90061 ^^^ 

150 
1.90211 J 
1.90360 \ll 
1.90509 \\l 
1.90658 ]ll 
1.90806 ^^^ 

148 
1.90954 14^ 
1.91102 JT° 

1.91250 ;it 

1.91398 ;^7 
1.91545 ^^^ 

147 
1.91692 1,7 
1.91839 \jI 
1.91986 tj' 
1.92132 };; 
1.92279 ^*^ 

146 
1.92425 .,(, 
1.92571 \ll 

1.92716 :2i 

1.92862 \jl 
1.93007 ^^^ 

145 
1.93152 j4g 
1.93297 4^ 
1.93442 J^ 
1.93586 \ll 
1.93730 ^** 

144 
1.93874 ... 

1.94018 \:l 

1.94162 J* 
1.94305 \li 
1.94448 ^" 
143 
1.94591 


1 


358 


365 


371 


378 


385 


391 


398 


405 


411 


418 


1 




6.51 


2 


425 


431 


438 


445 


451 


458 


465 


471 


478 


485 


2 




6.52 


3 


491 


498 


505 


511 


518 


525 


531 


538 


544 


551 


3 




6.53 


4 


558 


564 


571 


578 


584 


591 


598 


604 


611 


617 


4 




6.54 


655 


624 


631 


637 


644 


65L 


657 


664 


671 


677 


684 


5 




6.55 


6 


690 


697 


704 


710 


717 


723 


730 


737 


743 


750 


6 




6.56 


7 


757 


763 


770 


776 


783 


790 


796 


803 


809 


816 


7 




6.57 


8 


823 


829 


836 


842 


849 


856 


862 


869 


875 


882 


8 




6.58 


9 


889 


895 


902 


908 


915 


921 


928 


935 


941 


948 


9 




6.59 


660 


954 


961 


968 


974 


981 


987 


994 


000 


007 


014 




7 


6.60 


1 


82 020 


027 


033 


040 


046 


053 


060 


066 


073 


079 


1 


0.7 


6.61 


2 


086 


092 


099 


105 


112 


119 


125 


132 


138 


145 


2 


1.4 


6.62 


3 


151 


158 


164 


171 


178 


184 


191 


197 


204 


210 


3 


2 


6.63 


4 


217 


223 


230 


236 


243 


249 


256 


263 


269 


276 


4 


3 


6.64 


665 


282 


289 


295 


302 


308 


315 


321 


328 


334 


341 


5 


3.5 


6.65 


6 


347 


354 


360 


367 


373 


380 


387 


393 


400 


406 


6 


4.2 


6.66 


7 


413 


419 


426 


432 


439 


445 


452 


458 


465 


471 


7 


5 


6.67 


8 


478 


484 


491 


497 


504 


510 


517 


523 


530 


536 


8 


5.G 


6.68 


9 


543 


549 


556 


562 


569 


575 


582 


588 


595 


601 


9 


6.3 


6.69 


670 


607 


614 


620 


627 


633 


640 


646 


653 


659 


666 






6.70 


1 


672 


679 


685 


692 


698 


705 


711 


718 


724 


730 


1 




6.71 


2 


737 


743 


750 


756 


763 


769 


776 


782 


789 


795 


2 




6.72 


3 


802 


808 


814 


821 


827 


834 


840 


847- 


853 


860 


3 




6.73 


4 


866 


872 


879 


885 


892 


898 


905 


911 


918 


924 


4 




6.74 


675 


930 


937 


943 


950 


956 


963 


969 


975 


982 


988 


5 




6.75 


6 


995 


001 


008 


D14 


D20 


027 


033 


040 


046 


052 


6 




6.76 


7 


83 059 


065 


072 


078 


085 


091 


097 


104 


110 


117 


7 




6.77 


8 


123 


129 


136 


142 


149 


155 


161 


168 


174 


181 


8 




6.78 


9 


187 


193 


200 


206 


213 


219 


225 


232 


238 


245 


9 




6.79 


6^0 


251 


257 


264 


270 


276 


283 


289 


296 


302 


308 




6 


6.80 


1 


315 


321 


327 


334 


340 


347 


353 


359 


366 


372 


1 


0.6 


6.81 


2 


378 


385 


391 


398 


404 


410 


417 


423 


429 


436 


2 


1.2 


6.82 


3 


442 


448 


455 


461 


467 


474 


480 


487 


493 


499 


3 


1.8 


6.83 


4 


506 


512 


518 


525 


531 


537 


544 


550 


556 


563 


4 


2.4 


6.84 


685 


569 


575 


582 


588 


594 


601 


607 


613 


620 


626 


5 


3 


6.85 


6 


632 


639 


645 


651 


658 


664 


670 


677 


683 


689 


6 


3.6 


6.86 


7 


696 


702 


708 


715 


721 


727 


734 


740 


746 


753 


7 


4.2 


6.87 


8 


759 


765 


771 


778 


784 


790 


797 


803 


.809 


816 


8 


4.8 


6.88 


9 


822 


828 


835 


841 


847 


853 


860 


866 


872 


879 


9 


5.4 


6.89 


690 


885 


891 


897 


904 


910 


916 


923 


929 


935 


942 






6.90 


1 


948 


954 


960 


967 


973 


979 


985 


992 


998 


004 


1 




6.91 


2 


84 Oil 


017 


023 


029 


036 


042 


048 


055 


061 


067 


2 




6.92 


3 


073 


080 


086 


092 


098 


105 


111 


117 


123 


130 


3 




6.93 


4 


136 


142 


148 


155 


161 


167 


173 


180 


186 


192 


4 




6.94 


695 


198 


205 


211 


217 


223 


230 


236 


242 


248 


255 


5 




6.95 


6 


261 


267 


273 


280 


286 


292 


298 


305 


311 


317 


6 




6.96 


7 


323 


330 


336 


342 


348 


354 


361 


367 


373 


379 


7 




6.97 


8 


386 


392 


398 


404 


410 


417 


423 


429 


435 


442 


8 




6.98 


9 


448 


454 


460 


466 


473 


479 


485 


491 


497 


504 


9 




6.99 


700 


510 


516 


522 


528 


535 


541 


547 


553 


559 


566 






7.00 



120 



%.— LOGARITHMS OF NUMBERS, 
1. — ^Logarithms. — Continued. 



No. 


ComTnon Logarithms of Numbers. 


Naperian. 




L. O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 


No. 


Log. Dif. 


700 


84 510 


516 


522 


528 


535 


541 


547 


553 


559 


566 




7 


7.00 


1.94591 ..„ 
1.94734 }J5 
1.94876 \\i 
1.95019 \\l 
1.95161 ^^2 

142 
1.95303 ... 
1.95444 Jl 
1.95586 JJf 
1.95727 }JJ 
1.95869 ^^2 

140 
1.96009 ... 
1.96150 \\\ 
1.96291 Jji 
1.96431 \f. 
1.96571 ^^" 

140 
1.96711 1 Q 
1.96851 \Yn 
1.96991 \f. 
1.97130 ]\l 
1.97269 ^^^ 

139 
1.97408 139 
1.97547 J^^ 
1.97685 \\l 
1.97824 \ii 
1.97962 ^"^^ 

138 
1.98100 i^a 
1.98238 ;^° 
1.98376 J^° 
1.98513 1^; 
1.98650 ^-^^ 

137 
1.98787 .„ 
1.98924 }^; 

1.99061 ;^; 

1.99198 J^^ 
1.99334 ^^^ 

136 
1.99470 .ofi 
1.99606 ;^^ 
1.99742 \il 
1.99877 \il 
2.00013 ^'**' 

135 
2.00148 ... 
2.00283 \%l 
2.00418 Wl 
2.00553 Wl 
2.00687 ^"^^ 

134 
2.00821 .or 
2.00956 \\\ 
2.01089 J^^ 
2.01223 \i\ 
2.01357 ^-^^ 

133 
2.01490 

i 


1 


572 


578 


584 


590 


597 


603 


609 


615 


621 


628 


1 


0.7 


7.01 


2 


634 


640 


646 


652 


658 


665 


671 


677 


683 


689 


2 


1.4 


7.02 


; 3 


696 


702 


708 


714 


720 


726 


733 


739 


745 


751 


3 


2 


7.03 


4 


757 


763 


770 


776 


782 


788 


794 


800 


807 


813 


4 


3 


7.04 


705 


819 


825 


831 


837 


844 


850 


856 


862 


868 


874 


5 


3.5 


7.05 


6 


880 


887 


893 


899 


905 


911 


917 


924 


930 


936 


6 


4.2 


7.06 


7 


942 


948 


954 


960 


967 


973 


979 


985 


991 


997 


7 


5 


7.07 


8 


85 003 


009 


016 


022 


028 


034 


040 


046 


052 


058 


8 


5.6 


7.08 


9 


065 


071 


077 


083 


089 


095 


101 


107 


114 


120 


9 


6.3 


7.09 


710 


126 


132 


138 


144 


150 


156 


163 


169 


175 


181 






7.10 


1 


187 


193 


199 


205 


211 


217 


224 


230 


236 


242 


1 




7.11 


2 


248 


254 


260 


266 


272 


278 


285 


291 


297 


303 


2 




7.12 


3 


309 


315 


321 


327 


333 


339 


345 


352 


358 


364 


3 




7.13 


4 


370 


376 


382 


388 


394 


4oa 


406 


412 


418 


425 


4 




7.14 


715 


431 


437 


443 


449 


455 


461 


467 


473 


479 


485 


5 




7.15 


6 


491 


497 


503 


509 


516 


522 


528 


534 


540 


546 


6 




7.16 


7 


552 


558 


564 


570 


576 


582 


588 


594 


600 


606 


7 




7.17 


• 8 


612 


618 


625 


631 


637 


643 


649 


655 


661 


667 


8 




7.18 


9 


673 


679 


685 


691 


697 


703 


709 


715 


721 


727 


9 




7.19 


720 


733 


739 


745 


751 


757 


763 


769 


775 


781 


788 




6 


7.20 


1 


794 


800 


806, 


812 


818 


824 


830 


836 


842 


848 


1 


0.6 


7.21 


2 


854 


860 


866 


872 


878 


884 


890 


896 


902 


908 


2 


1.2 


7.22 


3 


914 


920 


926 


932 


938 


944 


950 


956 


962 


968 


3 


1.8 


7.23 


4 


974 


980 


986 


992 


998 


004 


mo 


016 


022 


028 


4 


2.4 


7.24 


725 


86 034 


040 


046 


052 


058 


064 


070 


076 


082 


088 


5 


3 


7.25 


6 


094 


100 


106 


112 


118 


124 


130 


136 


141 


147 


6 


3.6 


7.26 


. 7 


153 


159 


165 


171 


177 


183 


189 


195 


201 


207 


7 


4.2 


7.27 


■; 8 


213 


219 


225 


231 


237 


243 


249 


255 


261 


267 


8 


4.8 


7.28 


9 


273 


279 


285 


291 


297 


303 


308 


314 


320 


326 


9 


5.4 


7.29 


730 


332 


338 


344 


350 


356 


362 


368 


374 


380 


386 






7.30 


1 


392 


398 


404 


410 


415 


421 


427 


433 


439 


445 


1 




7.31 


2 


451 


457 


463 


469 


475 


481 


487 


493 


499 


504 


2 




7.32 


3 


510 


516 


522 


528 


534 


540 


546 


552 


558 


564 


3 




7.33 


4 


570 


576 


581 


587 


593 


599 


605 


611 


617 


623 


4 




7.34 


735 


629 


635 


641 


646 


652 


658 


664 


670 


676 


682 


5 




7.35 


6 


688 


694 


700 


705 


711 


717 


723 


729 


735 


741 


6 




7.36 


7 


747 


753 


759 


764 


770 


776 


782 


788 


794 


800 


7 




7.37 


8 


806 


812 


817 


823 


829 


835 


841 


847 


853 


859 


8 




7.38 


9 


864 


870 


876 


882 


888 


894 


900 


906 


911 


917 


9 




7.39 


740 


923 


929 


935 


941 


947 


953 


958 


964 


970 


976 




5 


7.40 


1 


982 


988 


994 


999 


DOS 


Oil 


017 


023 


029 


035 


1 


0.5 


7.41 


2 


87 040 


046 


052 


058 


064 


070 


075 


081 


087 


093 


2 




7.42 


: 3 


099 


105 


111 


116 


122 


128 


134 


140 


146 


151 


3 


1.5 


7.43 


■ ■* 


157 


163 


169 


175 


181 


186 


192 


198 


204 


210 


4 


2 


7.44 


745 


216 


221 


227 


233 


239 


245 


251 


*256 


262 


268 


5 


2.5 


7.45 


6 


274 


280 


286 


291 


297 


303 


309 


315 


320 


326 


6 


3 


7.46 


7 


332 


338 


344 


349 


355 


361 


367 


373 


379 


384 


7 


3.5 


7.47 


'. 8 


390 


396 


402 


408 


413 


419 


425 


431 


437 


442 


8 


4 


7.48 


9 


448 


454 


460 


466 


471 


477 


483 


489 


495 


500 


9 


4.5 


7.49 


750 


506 


512 


518 


523 


529 


535 


541 


547 


552 


558 






7.50 



LOGARITHMS OF NUMBERS. 
1. — ^Logarithms. — Continued. 



121 



No. 


Common Logarithms of Numbers. 


Naperlan. 




L. O 


1 


2 


3 


4 


5 


6 


7 


8 
552 


9 


P 


. P. 


No. 


Log. Dlf. 


750 


87 506 


512 


518 


523 


529 


535 


541 


547 


558 






7.50 


2.01490 ,oi 
2.01624 \i\ 
2.01757 \i{ 
2.01890 \i% 
2.02022 ^^^ 

133 
2.02155 ,„ 
2.02287 \ii 
2.02419 J^, 
2.02551 \ii 
2.02683 "^ 

132 
2.02815 .«, 
2.02946 \ii 
2.03078 ;^f 
2.03209 \i\ 
2.03340 ^^^ 

131 
2.03471 ,30 

2.03601 J^y 

2.03732 \i\ 
2.03862 \Z 
2.03992 "" 

130 
2.04122 .00 
2.04252 5; 
2.04381 29 
2.04511 JpX 
2.04640 ^^^ 

129 
2.04769 .,9 
2.04898 \il 
2.05027 \il 
2.05156 ]il 
2.05284 ^^" 

128 
2.05412 ,«« 
2.05540 \il 
2.05668 };° 
2.05796 ;;° 
2.05924 ^^^ 

127 
2.06051 ,00 
2.06179 }97 

2.06306 J;; 

2.06433 \ii 
2.06560 ^^' 

126 
2.06686 .„ 
2.06813 Jofi 
2.06939 1^ 
2.07065 \il 
2.07191 ^^^ 

126 
2.07317 ,». 
2.07443 \il 
2.07568 }«J 
2.07694 \il 
2.07819 ^^^ 

125 
2.07944 




564 


570 


576 


581 


587 


593 


599 


604 


610 


616 


1 




7.51 


2 


622 


628 


633 


639 


645 


651 


656 


662 


668 


674 


2 




7.52 


3 


679 


685 


691 


697 


703 


708 


714 


720 


726 


731 


3 




7.53 


4 


737 


743 


749 


754 


760 


766 


772 


777 


783 


789 


4 




7.54 


755 


795 


800 


806 


812 


818 


823 


829 


835 


841 


846 


5 




7.55 


6 


852 


858 


864 


869 


875 


881 


887 


892 


898 


904 


6 




7.56 


7 


910 


915 


921 


927 


933 


938 


944 


950 


955 


961 


7 




7.57 


8 


967 


973 


978 


984 


990 


996 


001 


007 


013 


018 


8 




7.58 


9 


88 024 


030 


036 


041 


047 


053 


058 


064 


070 


076 


9 




7.59 


760 


081 


087 


093 


098 


104 


110 


116 


121 


127 


133 




6 


7.60 


1 


138 


144 


150 


156 


161 


167 


173 


178 


184 


190 


1 


0.6 


7.61 


2 


195 


201 


207 


213 


218 


224 


230 


235 


241 


247 


2 


1.2 


7.62 


3 


252 


258 


264 


270 


275 


281 


287 


292 


298 


304 


3 


1.8 


7.63 


4 


309 


315 


321 


326 


332 


338 


343 


349 


355 


360 


4 


2.4 


7.64 


765 


366 


372 


377 


383 


389 


395 


400 


406 


412 


417 


5 


3 


7.65 


6 


423 


429 


434 


440 


446 


451 


457 


463 


468 


474 


6 


3.6 


7.66 


7 


480 


485 


491 


497 


502 


508 


513 


519 


525 


530 


7 


4.2 


7.67 


8 


536 


542 


547 


553 


559 


564 


570 


576 


581 


587 


8 


4.8 


7.68 


9 


593 


598 


604 


610 


615 


621 


627 


632 


638 


643 


9 


5.4 


7.69 


770 


649 


655 


660 


666 


672 


677 


683 


689 


694 


700 






7.70 


1 


705 


711 


717 


722 


728 


734 


739 


745 


750 


756 


1 




7.71 


2 


762 


767 


773 


779 


784 


790 


795 


801 


807 


812 


2 




7.72 


3 


818 


824 


829 


835 


840 


846 


852 


857 


863 


868 


3 




7.73 


4 


874 


880 


885 


891 


897 


902 


908 


913 


919 


925 


4 




7.74 


775 


930 


936 


941 


947 


953 


958 


964 


969 


975 


981 


5 




7.75 


6 


986 


992 


997 


003 


009 


D14 


()20 


025 


031 


037 


6 




7.76 


7 


89 042 


048 


053 


059 


064 


070 


076 


081 


087 


092 


7 




7.77 


8 


098 


104 


109 


115 


120 


126 


131 


137 


143 


148 


8 




7.78 


9 


154 


159 


165 


170 


176 


182 


187 


193 


198 


204 


9 




7.79 


780 


209 


215 


221 


226 


232 


237 


243 


248 


254 


260 




5 


7.80 


1 


265 


271 


276 


282 


287 


293 


298 


304 


310 


315 


1 


0.5 


7.81 


2 


321 


326 


332 


337 


343 


348 


354 


360 


365 


371 


2 


1 


7.82 


3 


376 


382 


387 


393 


398 


404 


409 


415 


421 


426 


3 


1.5 


7.83 


4 


432 


437 


443 


448 


454 


459 


465 


470 


476 


481 


4 


2 


7.84 


785 


487 


492 


498 


504 


509 


515 


520 


526 


531 


537 


5 


2.5 


7.85 


6 


542 


548 


553 


559 


564 


570 


575 


581 


586 


592 


6 


3 


7.86 


7 


597 


603 


609 


614 


620 


625 


631 


636 


642 


647 


7 


3.5 


7.87 


8 


653 


658 


664 


669 


675 


680 


686 


691 


697 


702 


8 


4 


7.88 


9 


708 


713 


719 


724 


730 


735 


741 


746 


752 


757 


9 


4.5 


,7.89 


790 


763 


768 


774 


779 


785 


790 


796 


801 


807 


812 






7.90 


1 


818 


823 


829 


834 


840 


845 


851 


856 


862 


867 


1 




7.91 


2 


873 


878 


883 


889 


894 


900 


905 


911 


916 


922 


2 




7.92 


3 


927 


933 


938 


944 


949 


955 


960 


966 


971 


977 


3 




7.93 


4 


982 


988 


993 


998 


004 


009 


D15 


020 


026 


031 


4 




7.94 


795 


90 037 


042 


048 


053 


059 


064 


069 


075 


080 


086 


5 




7.95 


6 


091 


097 


102 


108 


113 


119 


124 


129 


135 


140 


6 




7.96 


7 


146 


151 


157 


162 


168 


173 


179 


184 


189 


195 


7 




7.97 


8 


200 


206 


211 


217 


222 


227 


233 


238 


244 


249 


8 




7.98 


9 


255 


260 


266 


271 


276 


282 


287 


293 


298 


304 


9 




7.99 


800 


309 


314 


320 


325 


331 


336 


342 


347 


352 


358 






8.00 



122 



^—LOGARITHMS OF NUMBERS. 
1. — Logarithms. — Continued. 



No. 


Common Logarithms of Numbers. 


Naperian. 




L. O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P.P. 


No. 
8.00 


Log. Dif. 


800 


90 309 


314 


320 


325 


331 


336 


342 


347 


352 


358 






2.07944 ,,. 
2.08069 til 
2.08194 }^J 
2.08318 \il 


1 


363 


369 


374 


380 


385 


390 


396 


401 


407 


412 


1 




8.01 


2 


417 


423 


428 


434 


439 


445 


450 


455 


461 


466 


2 




8.02 


3 


472 


477 


482 


488 


493 


499 


504 


509 


515 


520 


3 




8.03 


4 


526 


531 


536 


542 


547 


553 


558 


563 


569 


574 


4 




8.04 


2.08443^^^ 
124 
2.08567 ,„. 
2.08691 ]il 
2.08815 J^J 


805 


580 


585 


590 


596 


601 


607 


612 


617 


623 


628 


5 




8.05 


6 


634 


639 


644 


650 


655 


660 


666 


671 


677 


682 


6 




8.06 


7 


687 


693 


698 


703 


709 


714 


720 


725 


730 


736 


7 




8.07 


8 


741 


747 


752 


757 


763 


768 


773 


779 


784 


789 


8 




8.08 


2.08939 \i^ 

2.09063^^* 

123 

2.09186 ,„. 

2.09310 };! 

2.09433 ]ii 
2.09556 J;^ 
2.09679 ^^"^ 

123 
2.09802 i5« 
2.09924 \ii 
2.10047 \ii 
2.10169 ]ii 
2.10291 ^^^ 

122 
2.10413 .„ 
2.10535 \ii 
2.10657 tit 
2.10779 ;;f 
2.10900 ^^^ 

121 
2.11021 ,„, 
2.11142 ]i] 
2.11263 {;} 
2.11384 ]i] 
2.11505 ^^^ 

2.11866 \fi 
2.11986 ;^" 
2.12106 ^'^^ 

120 
2.12226 ,2n 
2.12346 20 
2.12465 III 
2.12585 f" 
2.12704 ^^^ 

119 
2.12823 110 
2.12942 j;^ 
2.13061 5 
2.13180 Wl 
2.13298 ^^^ 

119 
2.13417 .i„ 
2.13535 Jj^ 
2.13653 }!g 
2.13771 ilfl 
2.13889 ^^^ 

118 
2.14007 


9 


795 


800 


806 


811 


816 


822 


827 


832 


838 


843 


9 




8.09 


810 


849 


854 


859 


865 


870 


875 


881 


886 


891 


897 




6 


8.10 


1 


902 


907 


913 


918 


924 


929 


934 


940 


945 


950 


1 


0.6 


8.11 


2 


956 


961 


966 


972 


977 


982 


988 


993 


998 


004 


2 


1.2 


8.12 


3 


91 009 


014 


020 


025 


030 


036 


041 


046 


052 


057 


3 


1.8 


8.13 


4 


062 


068 


073 


078 


084 


089 


094 


100 


105 


110 


' 


2.4 


8.14 


815 


116 


121 


126 


132 


137 


142 


148 


153 


158 


164 


5 


3 


8.15 


6 


169 


174 


180 


185 


190 


196 


201 


206 


212 


217 


6 


3.6 


8.16 


7 


222 


228 


233 


238 


243 


249 


254 


259 


265 


270 


7 


4.2 


8.17 


8 


275 


281 


286 


291 


297 


302 


307 


312 


318 


323 


8 


4.8 


8.18 


9 


328 


334 


339 


344 


350 


355 


360 


365 


371 


376 


9 


5.4 


8.19 


820 


381 


387 


392 


397 


403 


408 


413 


418 


424 


429 






8.20 


1 


434 


440 


445 


450 


455 


461 


466 


471 


477 


482 


1 




8.21 


2 


487 


492 


498 


503 


508 


514 


519 


524 


529 


535 


2 




8.22 


3 


540 


545 


551 


556 


561 


566 


572 


577 


582 


587 


3 




8.23 


4 


593 


598 


603 


609 


614 


619 


624 


630 


635 


640 


4 




8.24 


825 


645 


651 


656 


661 


666 


672 


677 


682 


687 


693 


5 




8.25 


6 


698 


703 


709 


714 


719 


724 


730 


735 


740 


745 


6 




8.26 


7 


751 


756 


761 


766 


772 


777 


782 


787 


793 


798 


7 




8.27 


8 


803 


808 


814 


819 


824 


829 


834 


840 


845 


850 


8 




8.28 


9 


855 


861 


866 


871 


876 


882 


887 


892 


897 


903 


9 




8.29 


830 


908 


913 


918 


924 


929 


934 


939 


944 


950 


955 




5 


8.30 


1 


960 


965 


971 


976 


981 


986 


991 


997 


002 


007 


1 


0.5 


8.31 




92 012 


018 


023 


028 


033 


038 


044 


049 


054 


059 


2 


1 


8.32 


3 


065 


070 


075 


080 


085 


091 


096 


101 


106 


111 


3 


1.5 


8.33 


4 


117 


122 


127 


132 


137 


143 


148 


153 


158 


163 


4 


2 


8.34 


835 


169 


174 


179 


184 


189 


195 


200 


205 


210 


215 


5 


2.5 


8.35 


6 


221 


226 


231 


236 


241 


247 


252 


257 


262 


267 


6 


3 


8.36 


7 


273 


278 


283 


288 


293 


298 


304 


309 


314 


319 


7 


3.5 


8.37 


8 


324 


330 


335 


340 


345 


350 


355 


361 


366 


371 


8 


4 


8.38 


9 


376 


381 


387 


392 


397 


402 


407 


412 


418 


423 


9 


4.5 


8.39 


840 


428 


433 


438 


443 


449 


454 


459 


464 


469 


474 






8.40 




480 


485 


490 


495 


500 


505 


511 


516 


521 


526 


1 




8.41 


2 


531 


536 


542 


547 


552 


557 


562 


567 


572 


578 


2 




8.42 


3 


583 


588 


593 


598 


603 


609 


614 


619 


624- 


629 


3 




8.43 


4 


634 


639 


645 


650 


655 


660 


665 


670 


675 


681 


4 




8.44 


845 


686 


691 


696 


701 


706 


711 


716 


722 


727 


732 


5 




8.45 


6 


737 


742 


747 


752 


758 


763 


768 


773 


778 


783 


6 




8.46 


7 


788 


793 


799 


804 


809 


814 


819 


824 


829 


834 


7 




8.47 


8 


840 


845 


850 


855 


860 


865 


870 


875 


881 


886 


8 




8.48 


9 


891 


896 


901 


906 


911 


916 


921 


927 


932 


937 


9 




8.49 


850 


942 


947 


952 


957 


962 


967 


973 


978 


983 


988 






8.50 



LOGARITHMS OF NUMBERS. 
1. — Logarithms. — Continued. 



123 



NTo. 


Common Logarithms of Numbers. 


Naperian. 




L. O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


I 


\ P. 


No. 


Log. Dlf. 


850 

2 
3 
4 


92 942 
993 

93 044 
095 
146 


947 
998 
049 
100 
151 


952 
003 

054 
105 
156 


957 
008 
059 
110 
161 


962 
013 
064 
115 
166 


967 
018 
069 
120 
171 


973 
024 
075 
125 
176 


978 
029 
080 
131 
181 


983 
034 
085 
136 
186 


988 
039 
090 
141 
192 


1 
2 
3 
4 


6 

0.6 
1.2 

1.8 
2.4 


8.50 
8.51 
8.52 
8.53 
8.54 


2.14007 ..„ 
2.14124 ;jj 
2.14242 \\° 
2.14359 !}; 
2.14476 ^^^ 
117 
2.14593 ..„ 

2.14710 }}; 

2.14827 }J; 
2.14943 ;j2 
2.15060 ^^^ 

116 
2.15176 1,. 
2.15292 ) ? 
2.15409 )J^ 
2.15524 Wl 
2.15640 ^^^ 

116 
2.15756 ,.. 
2.15871 \\l 
2.15987 j;S 
2.16102 \vt 
2.16217 ^^^ 

115 
2.16332 1,^ 
2.16447 \\l 
2.16562 \\l 

2.16677 ;;j 

2.16791 ^^* 

114 
2.16905 UK 
2.17020 \\l 
2.17134 \\\ 
2.17248 \\\ 
2.17361 ^^^ 

114 
2.17475 ... 
2.17589 \\l 
2.17702 Wl 
2.17816 \\l 
2.17929 ^^"^ 

113 
2.18042 no 
2.18155 \\i 
2.18267 } o 
2.18380 \\% 
2.18493 ^ "^ 

112 
2.18605 1,0 
2.18717 JJo 
2.18830 \\i 
2.18942 Yd 
2.19054 ^^^ 

111 
2.19165 n« 
2.19277 ; 
2.19389 f 
2.19500 \\\ 
2.19611 ^^^ 

111 
2.19722 


855 
6 

7 
8 
9 


197 
247 
298 
349 
399 


202 
252 
303 
354 
404 


207 
258 
308 
359 
409 


212 
263 
313 
364 
414 


217 
268 
318 
369 
420 


222 
273 
323 
374 
425 


227 
278 
328 
379 
430 


232 
283 
334 
384 
435 


237 
288 
339 
389 
440 


242 
293 
344 
394 
445 


5 
6 
7 
8 
9 


3 

3.6 
4.2 
4.8 
5.4 


8.55 
8.56 
8.57 
8.58 
8.59 


860 

1 
2 
3 
4 


450 
500 
551 
601 
651 


455 
505 
556 
606 
656 


460 
510 
561 
611 
661 


465 
515 
566 
616 
666 


470 
520 
571 
621 
671 


475 
526 
576 
626 
676 


480 
531 
581 
631 

682 


485 
536 
586 
636 
687 


490 
541 
591 
641 
692 


495 
546 
596 
646 
697 


1 
2 
3 
4 




8.60 
8.61 
8.62 
8.63 
8.64 


865 

? 

8 
9 


702 
752 
802 
852 
902 


707 

757 
807 
857 
907 


712 
762 
812 
862 
912 


717 
767 
817 
867 
917 


722 
772 
822 
872 
922 


727 
777 
827 
877 
927 


732 
782 
832 
882 
932 


737 
787 
837 
887 
937 


742 

792 
842 
892 
942 


747 
797 
847 
897 
947 


5 
6 

7 
8 
9 




8.65 
8.66 

8.67 
8.68 
8.69 


870 

1 
2 
3 
4 


952 
94 002 
052 
101 
151 


957 
007 
057 
106 
156 


962 
012 
062 
111 
161 


967 
017 
067 
116 
166 


972 
022 
072 
121 
171 


977 
027 
077 
126 
176 


982 
032 
082 
131 
181 


987 
037 
086 
136 
186 


992 
042 
091 
141 
191 


997 
047 
096 
146 
196 


1 
2 
3 
4 


5 

0.5 
1 

1.5 
2 


8.70 
8.71 
8.72 
8.73 
8.74 


875 
6 
7 
8 
9 


201 
250 
300 
349 
399 


206 
255 
305 
354 
404 


211 
260 
310 
359 
409 


216 
265 
315 
364 
414 


221 
270 
320 
369 
419 


226 
275 
325 
374 
424 


231 
280 
330 
379 
429 


236 

285 
335 
384 
433 


240 
290 
340 
389 
438 


245 
295 
345 
394 
443 


5 
6 

7 
8 
9 


2.5 

3 

3.5 

4 

4.5 


8.75 
8.76 
8.77 
8.78 
8.79 


880 

1 
2 
3 
4 


448 
498 
547 
596 
645 


453 
503 
552 
601 
650 


458 
507 
557 
606 
655 


463 
512 
562 
611 
660 


468 
517 
567 
616 
665 


473 
522 
571 
621 
670 


478 
527 
576 
626 
675 


483 
532 
581 
630 
680 


488 
537 
586 
635 
685 


493 
542 
591 
640 
689 


1 
2 
3 
4 




8.80 
8.81 
8.82 
8.83 
8.84 


885 
6 
7 
8 
9 


694 
743 
792 
841 
890 


699 
748 
797 
846 
895 


704 
753 
802 
851 
900 


709 
758 
807 
856 
905 


714 
763 
812 
861 
910 


719 
768 
817 
866 
915 


724 
773 
822 
871 
919 


729 
778 
827 
876 
924 


734 

783 
832 
880 
929 


738 
787 
836 
885 
934 


5 
6 

7 
8 
9 




8.85 
8.86 
8.87 
8.88 
8.89 


890 

1 
2 
3 
4 


939 
988 
95 036 
085 
134 


944 
993 
041 
090 
139 


949 
998 
046 
095 
143 


954 
002 

051 
100 
148 


959 
007 
056 
105 
153 


963 
012 
061 
109 
158 


968 
017 
066 
114 
163 


973 
022 
071 
119 
168 


978 
027 
075 
124 
173 


983 
032 
080 
129 
177 


1 
2 

\ 


4 

0.4 
0.8 
1.2 
1.6 


8.90 
8.91 
8.92 
8.93 
8.94 


895 
6 
7 
8 
9 


182 
231 
279 
328 
376 


187 
236 
284 
332 
381 


192 
240 
289 
337 
386 


197 
245 
294 
342 
390 


202 
250 
299 
347 
395 


207 
255 
303 
352 
400 


211 
260 
308 
357 
405 


216 
265 
313 
361 
410 


221 
270 
318 
366 
415 


226 
274 
323 
371 
419 


5 
6 

7 
8 
9 


2 

2.4 
2.8 
3.2 
3.6 


8.95 
8.96 
8.97 
8.98 
8.99 


900 


424 


429 


434 


439 


444 


448 


453 


458 


463 


468 






9.00 



124 



6.— LOGARITHMS OF NUMBERS. 
1. — ^Logarithms. — Continued. 



No. 


Common Logarithms of Numbers. 


Naperlan. 




L. O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 

1 


No. 


Log. Dlf. 


900 


95 424 


429 


434 


439 


444 


448 


453 


458 


463 


468 






9.00 


2.19722 ,.« 
2.19834 ]]l 
2.19944 }}? 
2.20055 }}} 
2.20166 "^ 
110 
2.20276 „. 
2.20387 \\k 

2.20497 ;;x 

2.20607 }}X 
2.20717 ^^" 

110 
2.20827 no 
2.20937 }}X 
2.21047 \\l 
2.21157 JJ" 
2.21266 ^"^ 

109 
2.21375 iin 
2.21485 J° 
2.21594 ;"S 
2.21703 SJ 
2.21812 ^"^ 

108 
2.21920 ,«Q 
2.22029 \nl 
2.22138 J"^ 
2.22246 }"2 
2.22354 ^"^ 

108 
2.22462 ..« 
2.23570 ;0° 
2.22678 JXo 
2.22786 "« 
2.22894 ^"^ 

107 
2.23001 ,.Q 
2.23109 J^5 
2.23216 J07 
2.23324 "? 
2.23431 1"^ 

107 
2.23538 107 
2.23645 frtfi 
2.23751 X? 
2.23858 X 
2.23965 ^"' 

106 
2.24071 .rtfi 
2.24177 2^ 
2.24284 JSfi 
2.24390 Jnfi 
2.24496 1"'' 

105 
2.24601 ,«- 
2.24707 \ll 
2.24813 }"5 
2.24918 JXfi 
2.25024 ^^^ 

105 
2.25129 


1 


472 


477 


482 


487 


492 


497 


501 


506 


511 


516 


1 




9.01 


2 


521 


525 


530 


535 


540 


545 


550 


554 


559 


564 


2 




9.02, 


3 


569 


574 


578 


583 


588 


593 


598 


602 


607 


612 


3 




9.03 


4 


617 


622 


626 


631 


636 


641 


646 


650 


655 


660 


4 




9.04 


905 


665 


670 


674 


679 


684 


689 


694 


698 


703 


708 


5 




9.05 


6 


713 


718 


722 


727 


732 


737 


742 


746 


751 


756 


6 




9.06 


7 


761 


766 


770 


775 


780 


785 


789 


794 


799 


804 


7 




9.07 


8 


809 


813 


818 


823 


828 


832 


837 


842 


847 


852 


8 




9.08 


9 


856 


861 


866 


871 


875 


880 


885 


890 


895 


899 


9 




9.09 


910 


904 


909 


914 


918 


923 


928 


933 


938 


942 


947 




5 


9.10 


1 


952 


957 


961 


966 


971 


976 


980 


985 


990 


995 


1 


0.5 


9.11 


2 


999 


004 


009 


014 


019 


023 


028 


033 


038 


042 


2 


1 


9.12 


3 


96 047 


052 


057 


061 


066 


071 


076 


080 


085 


090 


3 


1.5 


9.13 


4 


095 


099 


104 


109 


114 


118 


123 


128 


133 


137 


4 


2 


9.14 


915 


142 


147 


152 


156 


161 


166 


171 


175 


180 


185 


5 


2.5 


9.15 


6 


190 


194 


199 


204 


209 


213 


218 


223 


227 


232 


6 


3 


9.16 


7 


237 


242 


246 


251 


256 


261 


265 


270 


275 


280 


7 


3.5 


9.17 


8 


284 


289 


294 


298 


303 


308 


313 


317 


322 


327 


8 


4 


9.18 


9 


332 


336 


341 


346 


350 


355 


360 


365 


369 


374 


9 


4.5 


9.19 


920 


379 


384 


388 


393 


398 


402 


407 


412 


417 


421 






9.20 




426 


431 


435 


440 


445 


450 


454 


459 


464 


468 


1 




9.21 


2 


473 


478 


483 


487 


492 


497 


501 


506 


511 


515 


2 




9.22 


3 


520 


525 


530 


534 


539 


544 


548 


553 


558 


562 


3 




9.23 


4 


567 


572 


577 


581 


586 


591 


595 


600 


605 


609 


4 




9.24 


925 


614 


619 


624 


628 


633 


638 


642 


647 


652 


656 


5 




9.25 


6 


661 


666 


670 


675 


680 


685 


689 


694 


699 


703 


6 




9.26 


7 


708 


713 


717 


722 


727 


731 


736 


741 


745 


750 


7 




9.27 


8 


755 


759 


764 


769 


774 


778 


783 


788 


792 


797 


8 




9.28 


9 


802 


806 


811 


816 


820 


825 


830 


834 


839 


844 


9 




9.29 


930 


848 


853 


858 


862 


867 


872 


876 


881 


886 


890 




4 


9.30 


1 


895 


900 


904 


909 


914 


918 


923 


928 


932 


937 


1 


0.4 


9.31 


2 


942 


946 


951 


956 


960 


965 


970 


974 


979 


984 


2 


0.8 


9.32 


3 


988 


993 


997 


002 


007 


Oil 


016 


021 


025 


030 


3 


1.2 


9.33 


4 


97 035 


039 


044 


049 


053 


058 


063 


067 


072 


077 


4 


1.6 


9.34 


935 


081 


086 


090 


095 


100 


104 


109 


114 


118 


123 


5 


2 


9.35 


6 


128 


132 


137 


142 


146 


151 


155 


160 


165 


169 


6 


2.4 


9.36 


7 


174 


179 


183 


188 


192 


197 


202 


206 


211 


216 


7 


2.8 


9.37 


8 


220 


225 


230 


234 


239 


243 


248 


253 


257 


262 


8 


3.2 


9.38 


9 


267 


271 


276 


280 


285 


290 


294 


299 


304 


308 


9 


3.6 


9.39 


940 


313 


317 


322 


327 


331 


336 


340 


345 


350 


354 






9.40 


1 


359 


364 


368 


373 


377 


382 


387 


391 


396 


400 


1 




9.41 


2 


405 


410 


414 


419 


424 


428 


433 


437 


442 


447 


2 




9.42 


3 


451 


456 


460 


465 


470 


474 


479 


483 


488 


493 


3 




9.43 


4 


497 


502 


506 


511 


516 


520 


525 


529 


534 


539 


4 




9.44 


945 


543 


548 


552 


557 


562 


566 


571 


575 


580 


585 


5 




9.45 


6 


589 


594 


598 


603 


607 


612 


617 


621 


626 


630 


6 




9.46 


7 


635 


640 


644 


649 


653 


658 


663 


667 


672 


676 


7 




9.47 


8 


681 


685 


690 


695 


699 


704 


708 


713 


717 


722 


8 




9.48 


9 


727 


731 


736 


740 


745 


749 


754 


759 


763 


768 


9 




9.49 


950 


772 


777 


782 


786 


791 


795 


800 


804 


809 


813 






9.50 



LOGARITHMS OF NUMBERS. 
1. — Logarithms. — Continued. 



125 



No. 


Common Logarithms of Numbers. 


Naperian. 




L. O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P 


. P. 


No. 


Log. Dif. 


950 


97 772 


777 


782 


786 


791 


795 


800 


804 


809 


813 






9.50 


2.25129 105 
2.25234 \f. 
2.25339 {"^ 
2.25444 \Z 
2.25549 ^^5 
105 
2.25654 .«. 
2.25759 \f. 
2.25863 \ll 
2.25968 ]f. 
2.26072 1*'* 

104 
2.26176 .„. 
2.26280 \l\ 
2.26384 JX; 
2.26488 "* 
2.26592 ^"* 

104 
2.26696 .«, 
2.26799 \}r. 
2.26903 IftJ 
2.27006 \l\ 
2.27109 ^"^ 

104 
2.27213 103 
2.27316 VZ 
2.27419 }X; 
2.27521 21 
2.27624 ^"■* 

103 
2.27727 .n, 
2.27829 Jxf 
2.27932 }^^ 
2.28034 21 
2.28136 ^"^ 

102 
2.28238 in« 
2.28340 Vii 
2.28442 1"^ 
2.28544 \li 
2.28646 ^^^ 

101 
2.28747 102 
2.28849 Jf 
2.28950 J"; 
2.29051 \l\ 
2.29152 ^"^ 

101 
2.29253 ,ni 
2.29354 2} 
2.29455 \l\ 
2.29556 \l\ 
2.29657 ^"^ 

100 
2.29757 101 


1 


818 


823 


827 


832 


836 


841 


845 


850 


855 


859 


1 




9.51 


2 


864 


868 


873 


877 


882 


886 


891 


896 


900 


905 


2 




9.52 


3 


909 


914 


918 


923 


928 


932 


937 


941 


946 


950 


3 




9.53 


4 


955 


959 


964 


968 


973 


978 


982 


987 


991 


996 


4 




9.54 


955 


98 000 


005 


009 


014 


019 


023 


028 


032 


037 


041 


5 




9.55 


6 


046 


050 


055 


059 


064 


068 


073 


078 


082 


087 


6 




9.56 


7 


091 


096 


100 


105 


109 


114 


118 


123 


127 


132 


7 




9.57 


8 


137 


141 


146 


150 


155 


159 


164 


168 


173 


177 


8 




9.58 


9 


182 


186 


191 


195 


200 


204 


209 


214 


218 


223 


9 




9.59 


960 


227 


232 


236 


241 


245 


250 


254 


259 


263 


268 




5 


9.60 


1 


272 


277 


281 


286 


290 


295 


299 


304 


308 


313 


1 


0.5 


9.61 


2 


318 


322 


327 


331 


336 


340 


345 


349 


354 


358 


2 


1 


9.62 


3 


363 


367 


372 


376 


.381 


385 


390 


394 


399 


403 


3 


1.5 


9.63 


4 


408 


412 


417 


421 


426 


430 


435 


439 


444 


448 


4 


2 


9.64 


965 


453 


457 


462 


466 


471 


475 


480 


484 


489 


493 


5 


2.5 


9.65 


6 


498 


502 


507 


511 


516 


520 


525 


529 


534' 


538 


6 


3 


9.66 


7 


543 


547 


552 


556 


561 


565 


570 


574 


579 


583 


7 


3.5 


9.67 


8 


588 


592 


597 


601 


605 


610 


614 


619 


623 


628 


8 


4 


9.68 


9 


632 


637 


641 


646 


650 


655 


659 


664 


668 


673 


9 


4.5 


9.69 


970 


677 


682 


686 


691 


695 


700 


704 


709 


713 


717 






9.70 


1 


722 


726 


731 


735 


740 


744 


749 


753 


758 


762 


1 




9.71 


2 


767 


771 


776 


780 


784 


789 


793 


798 


802 


807 


2 




9.72 


3 


811 


816 


820 


825 


829 


834 


838 


843 


847 


851 


3 




9.73 


4 


856 


860 


865 


869 


874 


878 


883 


887 


892 


896 


4 




9.74 


975 


900 


905 


909 


914 


918 


923 


927 


932 


936 


941 


5 




9.75 


6 


945 


949 


954 


958 


963 


967 


972 


976 


981 


985 


6 




9.76 


7 


989 


994 


998 


D03 


007 


012 


016 


021 


025 


029 


7 




9.77 


8 


99 034 


038 


043 


047 


052 


056 


061 


065 


069 


074 


8 




9.78 


9 


078 


083 


087 


092 


096 


100 


105 


109 


114 


118 


9 




9.79 


980 


123 


127 


131 


136 


140 


145 


149 


154 


158 


162 




4 


9.80 


1 


167 


171 


176 


180 


185 


189 


193 


198 


202 


207 


1 


0.4 


9.81 


2 


211 


216 


220 


224 


229 


233 


238 


242 


247 


251 


2 


0.8 


9.82 


3 


255 


260 


264 


269 


273 


277 


282 


286 


291 


295 


3 


1.2 


9.83 


4 


300 


304 


308 


313 


317 


322 


326 


330 


335 


339 


4 


1.6 


9.84 


985 


344 


348 


352 


357 


361 


366 


370 


374 


379 


383 


5 


2 


9.85 


6 


388 


392 


396 


401 


405 


410 


414 


419 


423 


427 


6 


2.4 


9.86 


7 


432 


436 


441 


445 


449 


454 


458 


463 


467 


471 


7 


2.8 


9.87 


8 


476 


480 


484 


489 


493 


498 


502 


506 


511 


515 


8 


3.2 


9.88 


9 


520 


524 


528 


533 


537 


542 


546 


550 


555 


55S 


9 


3.6 


9.89 


990 


564 


568 


572 


577 


581 


585 


590 


594 


599 


603 






9.90 


1 


607 


612 


616 


621 


625 


629 


634 


638 


642 


647 


1 




9.91 


2 


651 


656 


660 


664 


669 


673 


677 


682 


686 


691 


2 




9.92 


3 


695 


699 


704 


708 


712 


717 


721 


726 


730 


734 


3 




9.93 


4 


739 


743 


747 


752 


756 


760 


765 


769 


774 


778 


4 




9.94 


995 


782 


787 


791 


795 


800 


804 


808 


813 


817 


822 


5 




9.95 


6 


826 


830 


835 


839 


843 


848 


852 


856 


861 


865 


6 




9.96 


2.29858 ;J 
2.29958 \2 


7 


870 


874 


878 


883 


887 


891 


896 


900 


904 


909 


7 




9.97 


8 


913 


917 


922 


926 


930 


935 


939 


944 


948 


952 


8 




9.98 


2.30058 JX 


9 


957 


961 


965 


970 


974 


978 


983 


987 


991 


996 


9 




9.99 


2.30158 ^"" 


1000 


000 000 


043 


087 


130 


174 


217 


260 


304 


347 


391 






10.00 


2.302585 



126 



^—LOGARITHMS OF NUMBERS. 



2.— Naperian (Natural or Hyperbolic) Logarithms of Numbers 1 to 
See Table 1 for general use; also explanation preceding Table 1. 
For Hyperbolic Functions, see Section 9 (end). 



II 



No. 


Units of numbers. 


0. 


1. 


2. 


3. 


4. 


5. 


6. 


7. 


8. 


9. 


0-. 


CX) 


0.00000 


0.69315 


1.09861 


1.38629 


1.60944 


1.79176 


1.94591 


2.07944 


2.19722 


1-. 


2.30259 


2.39790 


2.48491 


2.56495 


2.63906 


2.70805 


2.77259 


2.83321 


2.89037 


2.94444 


2-. 


2.99573 


3.04452 


3.09104 


3.13549 


3.17805 


3.21888 


3.25810 


3.29584 


3.33220 


3.36730 


3-. 


3.40120 


3.43399 


3.46574 


3.49651 


3.52636 


3.55535 


3.58352 


3.61092 


3.63759 


3.66356 


4-. 


3.68888 


3.71357 


3.73767 


3.76120 


3.78419 


3.80666 


3.82864 


3.85015 


3.87120 


3.89182 


6-. 


3.91202 


3.93183 


3.95124 


3.97029 


3.98898 


4.00733 


4.02535 


4.04305 


4.06044 


4.07754 


6-. 


4.09434 


4.11087 


4.12713 


4.14313 


4.15888 


4.17439 


4.18965 


4.20469 


4.21951 


4.23411 


7-. 


4.24850 


4.26268 


4.27667 


4.29046 


4.30407 


4.31749 


4.33073 


4.34381 


4.35671 


4.36945 


8-. 


4.38203 


4.39445 


4.40672 


4.41884 


4.43082 


4.44265 


4.45435 


4.46591 


4.47734 


4.48864 


9-. 


4.49981 


4.51086 


4.52179 


4.53260 


4.54329 


4.55388 


4.56435 


4.57471 


4.58497 


4.59512 


10—. 


4.60517 


4.61512 


4.62497 


4.63473 


4.64439 


4.65396 


4.66344 


4.67283 


4.68213 


4.69135 


11-. 


4.70048 


4.70953 


4.71850 


4.72739 


4.73620 


4.74493 


4.75359 


4.76217 


4.67068 


4.77912 


12-. 


4.78749 


4.79579 


4.80402 


4.81218 


4.82028 


4.82831 


4.83628 


4.84419 


4.85203 


4.85981 


13-. 


4.86753 


4.87520 


4.88280 


4.89035 


4.89784 


4.90527 


4.91265 


4.91998 


4.92725 


4.93447 


14-. 


4.94164 


4.94876 


4.95583 


4.96284 


4.96981 


4.97673 


4.98361 


4.99043 


4.99721 


5.00395 


15—. 


5.01064 


5.01728 


5.02388 


5.03044 


5.03695 


5.04343 


5.04986 


5.05625 


5.06260 


5.06890 


16-. 


5.07517 


5.08140 


5.08760 


5.09375 


5.09987 


5.10595 


5.11199 


5.11799 


5.12396 


5.12990 


17-. 


5.13580 


5.14166 


5.14749 


5.15329 


5.15906 


5.16479 


5.17048 


5.17615 


5.18178 


5.18739 


18-. 


5.19296 


5.19850 


5.20401 


5.20949 


5.21494 


5.22036 


5.22575 


5.23111 


5.23644 


5.24175 


19-. 


5.24702 


5.25227 


5.25750 


5.26269 


5.26786 


5.27300 


5.27811 


5.28320 


5.28827 


5.29330 


20-. 


5.29832 


5.30330 


5.30827 


5.31321 


5.31812 


5.32301 


5.32788 


5.33272 


5.33754 


5.34233 


21-. 


5.34711 


5.35186 


5.35659 


5.36129 


5.36598 


5.37064 


5.37528 


5.37990 


5.38450 


5.38907 


22-. 


5.39363 


5.39816 


5.40268 


5.40717 


5.41165 


5.41610 


5.42053 


5.42495 


5.42935 


5.43372 


23-. 


5.43808 


5.44242 


5.44674 


5.45104 


5.45532 


5.45959 


5.46383 


5.46806 


5.47227 


5.47646 


24-. 


5.48064 


5.48480 


5.48894 


5.49306 


5.49717 


5.50126 


5.50533 


5.50939 


5.51343 


5.5J745 


25-. 


5.52146 


5.52545 


5.52943 


5.53339 


5.53733 


5.54126 


5.54518 


5.54908 


5.55296 


5.55683 


26-. 


5.56068 


5.56452 


5.56834 


5.57215 


5.57595 


5.57973 


5.58350 


5.58725 


5.59099 


5.59471 


27-. 


5.59842 


5.60212 


5.60580 


5.60947 


5.61313 


5.61677 


5.62040 


5.62402 


5.62762 


5.63121 


28—. 


5.63479 


5.63835 


5.64191 


5.64545 


5.64897 


5.65249 


5.65599 


5.65948 


5.66296 


5.66643 


29—. 


5.66988 


5.67332 


5.67675 


5.68017 


5.68358 


5.68698 


5.69036 


5.69373 


5.69709 


5.70044 


30-. 


5.70378 


5.70711 


5.71043 


5.71373 


5.71703 


5.72031 


5.72359 


5.72685 


5.73010 


5.73334 



Ex.— Naperian logarithm of 264 is 5.57595. 

Slide Rules -;— Logarithmic slide rules are instruments graduated on a 
logarithmic basis for performing calculations involving multiplication (in- 
cluding powers) and division (including roots and reciprocals) of numbers. 



LogarHhmlc Base of Upper fixed Sca^e. 

cr> o o Oo O O o Q o . o — • • ^ — ^ — :— i— :-^— =_:<\ioJoJ 

r- I 1-^ I- It ■ I I I - - r 






Upper Fixed Scale, A. 

4-5 6 7 8 9\0 20 30 40 5060 80 100 



I 



I ' I ' I ' I I I I n . 



2 3 4 5 6 7 8 9 \0 ZO 30 40 50 60 60 K)0 

Upper Sliding Sccile,a. 



r-T- T- I ' I I I I 



I I I I IT 



I I II I I 1 1 — I 



Logarii-hmic Base of Upper Sliding Scale. 
Fig. 1. 

The principle of the slide rule is very simple, although some of the instru- 
ments themselves are complex and expensive. 



SLIDE RULES. 127 

The plain slide rule, Fig. 1, is usually from ten to eighteen inches long. 
(The longer the scale, the more accurate.) It consists of the upper and lowei 
"fixed" scales A and B, graduated on one piece and grooved to receive the 
sliding piece, on which are graduated the scales a and b. Note that scales 
A and a are similar, as are also B and b; but that the former are in double 
series (from 1 to 10) while the latter are in single series only. The advantage 
of this system will be explained below. 

Fig. 1 shows the upper fixed and sliding scales, A and a, together with 
their logarithmic bases. These two scales (or the two lower ones, either) 
may be used in performing any simple operation in multiplication or divi- 
sion. For instance, as the scales A and a are now set we can find the product 
of any number multiplied by 2; or, inversely, the quotient of any number 
divided by 2. Thus, 2X1 = 2; 2X2 = 4; 2X3=6, etc. Likewise, 
10-i-2=5; 20-^ 2= 10; etc. Note that the logarithm of the product 
= the sum of the logarithms of the factors; thus, log of 40 (= 1 . 6) is equal 
to log of 20 + log of 2 = 1.3+ 0.3. Similarly, the log of the quotient is 
the log dividend minus the log divisior. Of course the logarithms them- 
selves do not appear on the slide rule, but the numbers are arranged so 
their logarithms form series equally spaced, and the principle remains. To 
multiply any number q by n: Set 1 of scale a opposite q of scale A, and 
opposite n of scale a read the product p on scale A. To divide any number 
p by fi: Set n of scale a opposite p of scale A , and opposite 1 of scale a read 
the quotient q on scale A. To find the reciprocal of a number: Divide 1 by 
that number; or, invert the sliding scale, end for end, with ends of both 
scales, A and a, opposite, and the reciprocal of any number on one scale, 
as A, will be found directly opposite on the other scale, as a. 

To facilitate operations, and for accuracy, each slide rule is provided 
with a movable index with a vertical hair line. 

Fig. 2 shows the ordinary slide rule with double scale. With the 
movable index, the squarer root of any number on scale A will be found 



V5 t 3 4-5 6 7 8910 ZQ 30 40 60 60 80 100 

J . ■ 1,11./ 



^t \.5 S 3 4- 5^6 78910 20 30 40 50 60 80 100 

\ 1.5 £ fe.5 3 4. 5 6 7 

^ ^ h H V-^T ■ ■' I \ J I 



lO t 1.5 ^ ,?.5 3 4^ 5^78 9 10 



I I I I .1 
60 80 10 



Fig. 2. 

directly below on scale B. In like manner, the square of any number on 
scale B is found opposite, on scale A. Furthermore, the cube of any number 
on scale B may be found by multiplying its corresponding square on scale 
A by the number itself on scale a, reading the cube on scale A. Thus, 
Fig. 2, the cube of 1.59 (scale B) is found on scale A opposite 1.59 of 
scale a, and is equal to 4. Clearly, then, the cube root of any number n 
(scale A) is found by making the reading on scale a, opposite n, equal to 
the reading on scale B opposite 1 of scale b, and these equal readings are 
the cube root of the number. Thus, the cube root of 4 = 1.59. 

Thatcher's calculating instrument is a cylinder four inches in diameter 
and about eighteen inches long, which acts as a sliding scale inside a frame- 
work of twenty parallel bars forming the fixed scales. This instrument is 
by far the best on the market. It was sold formerly at S25.00, a price barely 
exceeding the cost of manufacture, and is now listed at $35.00; and with 
reading glass, $45.00. Results may be obtained to four or five decimal 
places. It is nearly as accurate as a five-place logarithmic table. 

Slide rules are used in all logarithmic and trigonometric operations, and 
no engineer should be without one. Books giving full directions in the use 
of the slide rule can be obtained for from 25 to 75 cents. They explain the 
method of solving such equations as: 



ax ax ax^ 



y=T- ^ = T«= y^T'-^'^b 



*2 lax /«. . 



« 



7.— PLANE GEOMETRY 

(See also Mensuration.) 
Angles and Lines. — o )c — Jq - — f b 

Straight line: a b. Oblique lines: a6, cd. 3^ ' 

Broken line: a o d. Oblique angles: A^B, .-^ '^'.-^ 

Acute angle: A. Adjacent angles: A,B\ A^C. 
Obtuse angle: B. Reflex angle: Greater than Fig 1. 

180° and less than 360°. 
Complementary angles: A and C (A+C = 90°); each is the complement of 

the other. 
Supplementary angles: A and B (^4+5 = 180°); each is the supplement of 

the other. 
Right angle: ao e ( = 90°) ; ^ o is perpendicular to a b. 
Straight angle: a o b ( = 180°) ; sides form two right angles. 
Conjugate angles: Two angles, about a point, whose sum equals 360°. 
Vertical angles: A and A ; B and B. 

Triangles. — 

Triangle: Plane bounded by three straight sides, Fig. 2. 

Perimeter: Sum of the sides. 

Angles: The interior angles (sum= 180°). 





Base 




Base 



Fig. 2. 



Fig. 3. 

Successive partial exterior angles: A' B' and C (sum = 
Right triangle: One of its angles a right angle, Fig. o. 
Scalene " No two of its sides equal, Fig. 4. 

Obtuse " One of its angles obtuse. Fig. 4. . 



Fig. 4, 



360«). 






Fig. 5, 



Fig. 6. 



Figs. 7. 



Isosceles triangle: Two of its sides equal, Fig. 5. 
Acute " All of its angles acute. Fig. 5, 

Equilateral '* All of its sides equal, Fig. 6. 
Equiangular " All of its angles equal, Fig. 6, 

Area of any triangle = ^ base X altitude. 

Quadrilaterals. — 

Quadrilateral: Plane bounded by four straight sides, Fig. 8. 

Sum of interior angles (A, B, C, D) = 360°; 

Sum of partial successive exterior angles (A', B\ C, D') = 360®; 

Sum of exterior angles (complete) = 1080°. 
Square: All sides equal, and each angle 90°, Fig. 9. 
Rectangle: Opposite sides parallel, and each angle 90°, Fig. 10. 
Parallelogram: Opposite sides parallel, Fig. 11. 



128 



POLYGONS, CIRCLE, 



129 



OVD 




Fig. 8. 



Fig. 9. 



Fig. 10. 



ZZ7 

Fig. 11. 



n u o\ 



Fig 12. 



Fig. 13. 



Fig. 14. 



Fig. 15. 



Rhomboid: Opposite sides parallel, and all angles oblique, Fig. 12 
Rhombus: All sides equal, all angles oblique. Fig. 13. 
Trapezoid: Two sides, only, parallel, Fig. 14. 
Trapegium: No two sides parallel, Fig. 15. 

Polygons (General). — 

Polygon: Plane bounded by straight sides. Fig. 16. 
.3 = triangle, 4=^ quadrilateral, 5 = pentagon, 6 = hexagon. 
7 = heptagon, 8 = octagon. 9 = nonagon, 10 = decagon, 
11 = undecagon, 12 = dodecagon. 
Sum of interior angles (A, B, C, etc.) = 180° X (number 
of sides— 2). 

Sum of partial successive exterior angles (A\ B' , 
C, etc.) = 360°' 

Sum of exterior angles (complete) = 360° + num- 
ber of sides XI 80°. 

Regular Polygons. — 

Regular polygon: All sides equal and all angles equal, 

Fig. 17. 
Circle: A regular polygon with an infinite number of 

sides. 
Radius of polygon: Radius (r) of circumscribed circle. 
Apothem of polygon: Radius (a) of inscribed circle. 
Perimeter: Sum of the sides. 
Area: ^ apothem X perimeter. 

Circle. — 

Tangent: Touches circumference 

at one point. 
Secant: Intersects circumference 

at two points. 
Diameter: Intersects center and 

is limited by circumference. 
Radius: Distance from center to 

circumference. 
Chord: A secant limited by the 

circumference. 
Arc: A portion of the circumfer- 
ence. 
Segment: Area bounded by arc 

and chord. 
Sector : Area bounded by two radii 

and the intercepted arc. 
Quadrant:^ Sector equal to quarter 

of a circle. 
Semicircle: Sector equal to half a circle. 
Circumference == diameter X n. tt = 

Diameter = circumference X — . — 




Fig. 17. 




Fig. 18. 



3.1415927. 
0.3183099. 



Area of circle = diameter^ X 



4' 



0.7853982. 



130 



7.— PLANE GEOMETRY. 



Problems in Construction of Figures. — 

Intercepting circles: Common chord is at right angle to line joining centers 
of circles, Fig. 19. Used in laying off 90°. 

To lay off 90°, 4^°, 60° and 30°: 



I 





Fig. 19. Fig 20. 

Center of circle: Intersection of perpendiculars bisecting any two chords, 
Fig. 21. 

An inscribed angle in a semicircle = 90°, Fig. 22. 

An angle from a point on the circumference is measured by half the inter- 
cepted arc (angle = i arc), Fig. 23. 

An angle included by a tangent and adjacent chord is measiired by half the 
intercepted arc; (angle = i arc), Fig. 24. 




Fig. 21. Fig. 22. Fig. 23. Fig. 24. 

Equal angles from a point, o, on the circumference subtend equal arcs and 

equal chords. Fig. 25. 
Railway curve: The two preceding propositions are fundamental in the 

laying out of railway curves. The chord is usually 100 ft. with a 




Fig. 25. 



Fig. 26. 



Fig. 27, 



central angle D° which equals the 
degree of curvature. The deflection 
angle d° is always one-half the central 
angle Z)°; hence for one chord, or 
station, the deflection angle is one- 
half the degree of curvature. 

Circle inscribed in a triangle: Center of 
circle is intersection of lines bisecting 
the angles, Fig. 27. Radius is shortest 
distance to either side. 

Circle circumscribing a triangle: Center 
of circle is intersection of perpendicu- 
lars bisecting the sides, Fig. 28. 




Fig. 28. 



CONSTRUCTION OF FIGURES. 



131 



ah = cd; — — 
c 



d 



■J = r* ade and cbf 



Proportion by segments of chords: 

are similar triangles, Fig. 29. 

Mean proportional: By similar triangles, — == r. whence c is a mean 

c 
proportional between a and 6 (a & = c^), Fig. 30. 

Inscribed square and octagon; Circumscribed circle, Fig. 31. 

Circumscribed square and octagon; Inscribed circle. Fig. 32. 




Fig. 30 



Fig. 31. 



Fig. 32. 



Pentagon inscribed in a circle: bd is one side of the pentagon (5 sides). 

Process: Bisect radius at a; ac = ab; bd = be. Step off de, ef, etc., = bd, 

and connect, Fig. 33. 
Decagon inscribed in a circle: Bisect the circular arcs of an inscribed 

pentagon, making twice the number of sides. Fig. 34. 
Pentagon of given side ab: Erect the perpendicular be = ^side ab; produce 

ac to d so that cd = cb; then bd = be = ae = radius of pentagon = 

radius of circumscribed circle, with center at e. Step off a f, f g, 

etc., = ab, and connect. Fig. 35. 




i9 a, 




Fig. 33. 



Fig. 34. 



Fig. 35. 



Hexagon inscribed in a circle: Each side is equal to the radius. Fig. 36. 
Equilateral Triangle: Connect alternate points of hexagon, making half 
the number of sides. Fig. 37. 

Dodecagon: Bisect the circular arcs of an inscribed hexagon, making 
twice the number of sides. Fig. 38. 






Fig. 36, 



Fig. 37. 



Fig. 38. 



8.— SOLID GEOMETRY. 

Planes, Angles and Lines. — 

Plane: Determined by (1) two parallel lines; (2) two intersecting lines; 

(3) three points not in the same straight line; (4) a straight line and a 

point outside of it. 
Straight line: Intersection of two planes not parallel, as a 6, Fig. 1. 
Dihedral angle: The angle between two planes, measvired at right angle to 

each plane and to the line of their intersection, or edge, as A, Fig. 1. 
Right dihedral angle: A dihedral angle that is 90°, Fig. 2. 





Fig. 1. Fig. 2. 

Other angles: Acute, obtuse, complementary, supplementary, adjacent, etc., 

as in Plane Geometry. 
Coordinate planes: V and H, Fig. 3, are planes perpendicular to e&ch other; 

hence, any line v in one plane, if perpendicular to a b, is perpendicular 

also to the other plane. 




Fig. 3. 
Polyhedrons. — 

Polyhedron: Solid bounded by planes. 
Tetrahedron: 4 triangular faces; 6 edges. Fig. 
Hexahedron: 6 square faces; 12 edges. Fig. 5. 





Fig. 4. 



Fig. 5. 



Fig. 6. 





Fig. 8, 



Octahedron: 8 triangular faces; 12 edges, Fig. 6. 
Dodecahedron: 12 pentagonal faces; 30 edges, Fig. 7. 
Icosahedron: 20 triangular faces; 30 edges, Fig. 8. 



132 



PRISMS. PYRAMIDS. 



133 



Prisms. — 

Prism: A polyhedron with two opposite faces parallel and equal polygons, 

the other faces parallelograms, Fig. 9. 
Right prism: Lateral edges perpendicular to bases, Fig. 10. 
Regular prism: A right prism whose bases are regular polygons. 
Oblique prism: Lateral edges are oblique to bases, Fig. 11. 
Triangular prism: Bases are triangles. Fig. 11. 






Fig. 9. 



Fig. 10. 



Fig. 11. 



Quadrangular prism: Bases are quadrilaterals, Fig. 12. 
Parallelopiped: Prism whose bases are parallelograms. 

Right parallelopiped: • Lateral edges are perpendicular to the bases, Fig. 12. 
Rectangular parallelopiped: Six faces are rectangles. Fig. 12. 
Cube: Parallelopiped whose six faces are squares, Fig. 13. 
Volume of any prism = area of base X altitude. 

Truncated prism: Portion included between base and plane section oblique 
to base, Fig. 14. 



/; 


/ 


1 
1 
1 
L 




.' 

^ 


/ 



Fig. 13. 




Fig. 14. 



whose sides are tri- 



Fig. 12. 

Pyramids. — 

Pyramid: Polyhedron whose base is a polygon and 

angles joining a common apex or top, Fig. 15. 
Altitude: Perpendicular distance from apex to plane of base. 
Triangular pyramid: Base is a triangle (.*. solid is a tetrahedron). 
Quadrangular pyramid: Base is a quadrilateral. 
Regular pyramid: Base is a polygon, and apex is directly over its center. 

Fig. 16. 




Fig. 15. 





Fig. 16. 



Fig. 17. 




Irregular pyramid: One that is not regular. 
Volume of any pyramid: ^ area of base X altitude. 

Slant height: Distance along any lateral face from its apex to the middle 
of its base. 



134 



8.— SOLID GEOMETRY. 



Truncated pyramid: That portion between the base and a plane section 

cutting all the sides, Fig. 17. 
Frustum of a pyramid: That portion between the base and a plane section 

parallel with the base, Fig. 18. 

Cylinders. — 

Cylinder = circular cylinder: Two bases are circular and parallel; all sec- 
tions parallel with the bases are circular, Fig. 19. 

Elliptic cylinder: Two bases are elliptical and parallel; all sections parallel 
with the bases are elliptical. 




Fig. 19. 

Right cylinder: Elements are perpendicular to the bases. 
Oblique cylinder: Elements are oblique to the bases. 
Volume of any cylinder = area of base X altitude. 

Cones.-^ 

Cone = circular cone: Base and all sections parallel with it are circular; 
elements composing the sides meet at a common apex or top, Fig. 2o! 
Elliptic cone: Base and all sections parallel with it are elliptical.* 
Alt-itude: Perpendicular distance from apex to plane of base. 
Right cone: Cone with axis perpendicular to base. 
Oblique cone: Cone with axis oblique to base. 





Fig. 20. 



Fig. 21. 




Volume of any cone: \ area of base X altitude. 

Truncated cone: That portion between the base and a plane cutting all the 

elements, Fig. 21. 
Frustum of a cone: That portion between the base and a plane section 

parallel with the base, Fig. 22. 

Spheres. — 

Sphere: Solid whose every section is circular, Fig. 23. 
Radius: Distance from center to surface. 
Diameter: Two radii forming one straight line. 
Great circle: The largest plane section (cuts center of sphere). 
Pole of a circle: End of diameter or axis perpendicular to the plane of the 
circle. 





Fig. 23. 



Fig. 24. 



Fig. 25, 



Fig. 26. 




SPHERES. 136 

Arc of a great circle: Shortest surface distance between two points on sur- 
face of sphere. 

Polar distance of a circle: Distance from nearest pole to circumference of 
circle. 

Quadrant: Polar distance of a great circle. 

Surface of sphere = 4 tt radius^. 
4 

Volume of sphere = -5- tt radius^. 

Zone: Portion of surface between two parallel planes, Fig. 24. 

Lune: Portion of surface between two semi-circumferences of great circles, 

Fig. 25. 
Spherical segment: Portion of sphere between two parallel planes. Fig. 26. 
Spherical pyramid: Apex co^Tesponds to center of sphere, sides are radial 

planes, and base is a spherical polygon Fig. 27. 
Spherical cone: Cone with spherical base, Fig. 27. 



9.— PLANE TRIGONOMETRY. 



Plane Trigonometry deals with the functions of plane triangles, and 
shows how to compute the unknown elements when certain of the known 
elements are given. The six elements of a triangle are its three sides and 
three angles; and three of these, one of which must be a side, must be — 
known in order to solve the triangle. ■ 

Trigonometric functions are the ratios of the sides of a right angle triangle. ■ 
There are six primary ratios, namely, sine, cosine, tangent, cotangent, secant 
and cosecant. In addition to these, however, there are two secondary 
functions sometimes employed, namely, versed sine and cover sed sine; and 
two tertiary functions seldom, used, namely, exsecant and coexsecant. 

The following trigonometric functions or ratios refer to Fig. 1, in 
which — 

h — hypothenuse, 

p = perpendicular, ^ 

h = base, 

A = angle at base, ^/k go' 

B = angle opposite base. B^ 

sine A = cosine B, etc. Fig. 1. 




Primary Functions: 

P 

h 



sin A {= cos B) 



cos A (=sm B) = j- 



tanA ( 



cot 5) =f 
o 



cot A ( = tan B) = 



sec A ( = CSC B) =r 
o 



CSC A ( 



sec 5) = - 
P 



Secondary Functions: 
vrs A { = CVS B) = l — cos A = 1 — r 



h-h 



CVS A { = vrs 5) = 1 — sin A 



1_^ Jli:± 

h h ' 



Tertiary Functions: 
xsc A { = cxc B) = sec A — 1 = 
cxc A ( = xsc B) = CSC A — 1 



h .k-b. 
b b 



1= 



h-p 



1. — Equivalent Values op Primary Functions of Any Angle x In Pure 
Terms of Each of the Other Functions. 



stn 



sin X = Vl— cos2iic 



\/l— sin2:<; 
sinic 


vr 


-sin2ic 


\/l- 


- sin23£; 


sin X 

1 




CSC 

1 



CSC X 

\/csc2j;— 1 

CSC X 

1 

v'csc'^— 1 



= Vcscs^— 1 



CSC X 



1- sin2a; 


cos X 


cot X 


1 


1 \/l+tan2% 




= \/l+cot2:;c 


sin X 


Vl— cos2af tah ;c 





sec X 



\/sec2A;— 1 



'\/csc2ic— 1 

= CSC X 



136 



TRIGONOMETRIC FUNCTIONS. 



137 



Other Equivalent Values of Primary Functions op Any Angle x 

AND (Angle x)^. 



sm X = 



tanrx: 



cos X 
cot X 
s in X 
tan X 
sin X 



cos X 

cot ;»; = — . 

sm X 

tan X 



cos ;Jt: tan ^. 



sin X cot rx:. 



sm X sec nt. 



cos iC CSC X. 



sin2 ic = 1 



cos-^ X, 



1 



sm^ :*; 



1 — sin2 X. 



tan2 :*; = sec^ x — 1. 



= 1 4- 



= 1 + 



cos^ 

1 



cot^ 



sec X 



CSC X = 



sm X 
cot X 
cos ^ 



tan ^ CSC X. sec^ x = 1 + tan^ r^. 



tan^ :x; 

1_ 
cot^ X 

1 



CSC2 iC — 1. — r:; — = "^ — 



sin2 


X 


sin2 


X 


cos2 


X 


cos2 


X 


sin2 


X 


sin2 
cos2 


X 

X 


sin2 


X 



cot X sec X. 



CSC* X 



1 + cot' X, "5 — = 



1 

CSC2 



tan2 X 
cos2 X 

C0t2 X 



Values of Trigonometric 
functions in the four quad=> 
rants. — In Fig. 2 the angle 
Xi lies wholly within the first 
quadrant, 0° to 90°; X2 ex- 
tends into the second 
quadrant, comprising angles 
from 90° to 180°; ^3 extends 
into the third quadrant, com- 
prising angles from 180° to 
270°; and x^ extends into 
the fourth quadrant, com- 
prising angles from 270° to 
360°. 

It will be noticed that in 
all cases the sine falls perpen- 
dicularly upon the axis X — X 
and that the cosine falls per- 
pendicularly upon the 
coordinate axis Y—Y. 
Assuming arbitrarily that, 
when the sine is above the 
axis of X — X, it is plus and 
when below, it is minus; also, 
that when the cosine is to 
the right of the axis Y—Y 
it is plus, and when to the 
left it is minus', we obtain 
the following sign values of 
the fundamental sign func- 
tions: 



'E"**'Quaciran+ 



90^-180® 



1^"^- Quadrant 
0°-90° 







Sine. . 
Cosine 



1st quad, 
+ 
4- 



2nd quad. 

4- 



Fig. 2. 



3rd quad. 



4"^^ Quadrant 
t70°-360* 



4th quad. 

+ 



The sign values may further be extended to the other trigonometric 
functions by remembering that 

sin X ^ cos X 1 1 

tan X = , cot X = —. , sec x = — — , esc x = — : , 

cos X sm X cos x sm x 

vrs x= 1 — cos X, CVS i!c= 1 — sin x, xsc :x; = sec x— 1, cxc r^ic^csc x— 1; 

and that in either multiplication or division, like signs give plus and unlike, 

minus. Hence the following table: 

1st quad. 2nd quad. 3rd quad. 4th quad. 



Sine, cosecant, coexsecant.. 
Cosine, secant, exsecant. . . . 

Tangent, cotangent 

Versed sine, co versed sine . 



138 



9— PLANE TRIGONOMETRY. 



2. — Natural Functions of Angles from 0° to 360°. 



Angle 


Sin 


Cos 


Tan 


Cot 


Sec 


Csc 


Vrs 


Cvs 


Xsc 


Cxc 




0° 





1 





oo 


1 


00 





1 





00 


•4-> 

rj 


15° 

30° 


h 


2 


3 


\r 


2vr 

3 


2 


2-V3 


i 


2vr-3 

3 


i 


§ 


2 


'S 


45° 


2 


V2- 
2 


1 


1 


vr 


vr 


2-V2 


2-V2 


vr-1 


VT-i 


o 


2 


2 




60° 


2 


^ 


vr 


vr 

3 


2 


2V3 
3 


i 


2-V3 


1 


2V3-3 




2 


3 


4^ 
Pi 


75° 

90° 

105° 

120° 
135° 


1 

VT 

2 

vr 

2 




\2" 
2 


00 

-vr 
-1 




vr 

3 
-1 


00 

-2 

-vr 


1 

2\^ 
3 

vr 


1 

1 

2+V2 



2-V3 


00 
-3 

-(vT+i) 




2V^- 3 


2 


2 

2-V2 


3 
\^-i 


O 


2 


2 


^3 


150° 


i 


V3 

2 


V3 
3 


-V3 


-2V3 


2 


2+V3 


i 


2V3+3 
3 


1 




3 


2 


-p 


165° 
180° 
195° 

210° 





-1 

V3 

2 



3 


00 

-vr 


-1 

2V3 
3 


00 
-2 


2 

2+V3 


1 
1 


-2 

2vr+3 
3 


00 
-3 


S 


2 




225° 


2 


2 


1 


1 


-vr 


-vr 


2+V2 


24-V2 


-(V2-+1) 


-(V2+1) 


o 


2 


2 


CO 


240° 


vr 

2 


-^ 


vr 


vr 

3 


-2 


-2V3 


i 


2+V3 


-3 


2Vrh3 




3 


2 


3 




255° 
^270° 

285° 

300° 


-1 

V3- 
2 





oo 




vr 

3 


00 
2 


-1 

2V3 

3 


1 
1 


2 

2+\/3 


CO 

1 


-2 
2V/34-3 


+3 


3 


3 


s 

T1 


315° 


V2~ 
2 


vr 

2 


-1 


-1 


vr 


-\r 


2-V2 


2+V2 


vT-i 


-(Nri-i) 


r 


2 


2 




330° 


-i 


2 


vr 

3 


-vr 


2vr 
3 


-2 


2-V3 


1 


2V3^3 
3 


-3 


2 




345° 
360° 





1 





oo 


1 


00 





1 





00 



FUNCTIONS OF ANGLES. 



139 



Functions of complement, supplement, etc., of any angle x. 

The complement of any angle x is 90° — x. Thus, in Fig. 1, page 136, 
B is the complement of A. 

The supplement of any angle x is 180° — it;, etc. 

In the following Table, the first four primary functions of angles extend- 
ing into either of the four quadrants (see Fig. 2, page 137) are reduced to the 
fiinctions of angles not greater than 90°. 





1st Quadrant. 


2nd Quadrant. 


sin X 
cos X 
tan a; 
cot X 


sin :s;i = cos (90° — iti) 
cos ;t;i = sin (90° -^i) 
tan:»;i = cot (90°-:ri) 
cot i»;i = tan(90°-i»;i) 


( sin A;2 = sin (180°-:x:2 ) ) 
( sin :*:2 = cos ( X9—90°) ) 
(cosa;2=-cos (180'°-^2 ) i 
(cos :x;2=-sin ( 3:2- 90°) j 

(tanrx;2=-tan (180°-^2 ) ' 
(taniC2=-cot ( X2-90°) ' 
(cot A;2=-cot (180°-it2 ) 
(cot it:2=-tan( a:2-90°) 




3rd Quadrant. 


4th Quadrant. 


sin X 
cos X 
taxix 
cot X 


( sin ^3= -sin ( ^^3- 180°) ) 
( sin rr3=-cos (270°-^3 ) \ 
( cos ^3= -cos ( ^3-180°) 
\ cos ^3= -sin (270°-^3 ) 
( taniC3= tan ( ^3-180°) ) 
1 tanit3= cot (270°-:;c3 ) ) 
( cot it:3= cot ( ^3-180°) ) 
\ cot xz= tan(270°-A;3 ) j 


(sin ^4= -sin (360° -rt4 ) ) 
(sin rx:4=-cos ( rr4-270°) j 
(cosit'4= cos (360° -iC4 ) ) 
(cos ^4= sin ( x^-210°) 
(tan^4=— tan (360°-r;c4 ) 
(tana;4=-cot ( ^4-270°) 
(cot ^4=— cot (360°-iC4 ) ) 
( cot x^=- tan ( X4. - 270°) } 



Functions of the sum of two angles {x+y), 

sin {x + y) = sin x cos 3; + cos x sin y. 

cos {x + y) = cos X cos 3/ — sin x sin y, 

, , s tan X + tan y • 

ta.n{x + y) = -j — 7 . 

1 — tan X tan y 

^ , , . cot X cot y—\ 

cot {x + y) = 7 ; 7 . 

cot y + cot X 

sin :x:4-sin y = 2 sin 1 {x-\-y) cos \ {x—y), 

cos ic + cos y = 2 cos i {x-\-y) cos \ {x—y), 



Functions of the difference of two angles {x — y), 

sin {x — y) = sin x cos y — cos x sin y. 
cos {x — y) = cos X cos y + sin rx; sin y. 
tan a: — tan y 



tan {x — y) = 
cot {x — y) = 



1 + tan % tan y 
cot y cot y + 1 

cot 3' — cot X 
sin a; — sin 3; = 2 cos ^ (^ + 3') sin ^ {x — y). 
cos ic — cos 3/ = — 2 sin ^ (^ + 3^) sin i {x — y). 



140 9.— PLANE TRIGONOMETRY, 

Functions of half an angle (^ x). 



. sin X /vers x j 1 

sm i x= z 7— = A — n — = A — 

2 COS ^x ^ 2 y 



COS 



_ sin X _ l|l_ 
2 sin i :c ~ \ 



+ cos X 
~2 



, . 1 — cos ^ sm ic , /I — cos X 

tan t ^ = — : = -7-; = cosec x — cot x 

sm X 1 + cos X 



1-, 



cot ^ X 



1 + COS a; sin x 



sm ic 1 — cos X vers iif cosec x — cot ic 

Functions of twice an angle (2 a:). 

. o o • 2 tan X 

sm 2^=2 sm x cos x = ^ . ^ — x— . 

1 + tan^ X 

cos 2 rs; = cos2 ic — sin^ rx; = 1 — 2 sin^ ic = 2 cos^ x — 1 = 7— r— « 

1 + tan2 ac 

^ n 2 tan it: sin 3 x — sin rr 

tan 2 a; = 



cot 2 ic 



1 — tan2 X cos 3 :v + cos x 
cot2:r-l 



2 cot rjc 
Functions of three times an angle (3x), 

sin 3 X = S sin x—i sin^ it;. 
cos 3 it: = 4 cos^ it: — 3 cos it:. 

_ 3 tan X — tan^ it: 

tan 3 it: 



1-3 tan2 it; 
Functions of four times an angle (4x) . 

sin 4 ic = 4 sin it: cos it: — 8 sin^ it: cos it:. 
cos 4 ic = 1 — 8 cos2 iK:+ 8 cos^ x. 

. 4 tan it: — 4 tan^ x 

tan 4 it: 



1-6 tan2 it: + tannic' 



Inverse Trigonometric Functions. — In logarithms we have seen that the 
anti-logarithm A/" is a number whose logarithm is «; that is, it is the number 
corresponding to the logarithm n. Similarly, in Trigonometry, we have — 

The anti-sine of 5 = angle whose sine is s = angle corresponding to sine 5. 
Thus, sin A=s, or A = sirr^s (Reads A = anti-sine or inverse sine of s.) 

The anti-cosine oi c = angle whose cosine is c = angle corresponding to cosine c. 
Thus, cos A^c, or A=cos-i c (Reads A = anti-cosine or inverse cosine 
of c). 

And so on with the remaining functions. 

This must not be confused with the negative exponent,— l,as n~^= — ; 

fi 

itri = — ; (sin x)~^ = -. ; (cos y)-'^= ; etc. But sin A =sin (sin-^5) =s; 

it: sm ic cos y 

cos A=cos (cos-'^)=<:; etc. 

Examples: J=sin 30° .'. the anti-sine of ^ is 30°. 

\/2 = cos 45° .'. the anti-cosine of \/2 is 45°. 
~T_ 2 __ 

V'3 = tan 60° .'. the anti-tangent of \/3 is 60°. 



SOLUTION OF TRIANGLES. 



141 



Natural and logarithmic trigonometric functions. — On pages 144 to 175 
will be found tables (3 and 4) of natural functions, and on pages 176 to 198 
a table (5) of logarithmic functions, of angles up to 90° or in the first 
quadrant. For angles greater than 90° see Table 1 of functions of any 
angle reduced to function of an angle not greater than 90°. In general, the 
following rules are convenient to memorize: 

Sine of an angle = the cosine of its complement. 

** " " = " sine " supplement. 

Tangent" " = " cotangent *' complement. 

" " " = "—tangent " supplement. 

In using the natural functions the processes of multiplication and divi- 
sion have to be performed, while the logarithmic functions are designed to 
reduce these to the simple processes of addition and subtraction. Log- 
arithmic functions are simply logarithms of the natural functions. 
The tables are as follows: 
Table 3, page 144, Natural Sines, Tangents, Cotangents, Cosines, 

(Versed Sines, Co versed Sines). 
Table 4, page 167, Natural Secants, Cosecants, (Exsecants, Coexse- 

cants) . 
Table 5, page 176; Logarithmic Sines, Tangents, Cotangents, Cosines 
(Secants, Cosecants). 

Solution of right angle triangles. — The six primary functions are all 
that need be employed in the solution of any triangle: 

P. 
h 

b_ 
h 



(1) 
(2) 
(3) 
(4) 
(5) 



sin A ( = cos B) =^, Whence, p=h sin A=h cos B. 
cos A ( = sin jB) = ' 



tan 4 ( = cot5)=|-. 



h=h cos A=h sin B. 
p=b tan A=b cot B. 




cot A ( = 

sec A ( = csc B) 



tan B)-- 

P 



h=p cot A=p tanB. 

h = h sec A =6 esc 5; or h= 



Fig. 3. 



cos A 



9 

sin J5 



(6) 

[Also ;j2 



CSC A ( = sec B) = — , 
P 



h = p CSC A=p sec B ; or h= . . 



sin A cosB 



b^ + p^. Whence /t: 
Example 1. 
Given: A = 32°, /^ = 20. 3. 
Required: p. 

Solution: (1) p = h sin A. 

By natural functions. 

sin A = sin 32°= .52992 

^ = .52992X20.3=10.757. Ans. 
By logarithmic functions. 
log 20.3 = 1.30750 
log sin 32° = 9.72421 
Ans. 10.757. 1.03171 



\/b^ + p^;b ^s/h^-p^; p = Vh^-b^.] 

Example 2. . 
Given: ^ = 25,6 = 20. 
Required: Angle B. 

Solution: (4) tan 5=-. 



tanS^ 



By natural functions. 

"?-25~'^- 
AngleB = tan-i.8 = 38°-40'. Ans. 
By logarithmic functions. 
log 20 = 1.30103 
log 25 = 1.39794 
Ans. 38° -40' 9.90309 



142 9,—PLANB TRIGONOMETRY. 

Example 3. 

Given: ^ = 16^-10', :^ = 40. 

Required: h. 

Solution: (6) h^p-^-siriA. 
By natural functions. 
sin A = sin 16°- 10'= 0. 27843. 
;j=40h-0. 27843=143. 66. Ans. 
By logarithmic functions, 

log 40 = 1.60206 
log sin 16°- 10'= 9.44472 

Ans. 143.66 2.15734 

^.\ Solution of any triangle. — ^The following princi- 

ples, easily memorized, lead to the solution of any 
triangle, right or oblique. 

Sides are proportional to sines of opposite angles 
.(1) 

_, --.. .^ sin B sin C 

Thus, = — ' = . 

a c 

Sin ang opp given side : sin ang opp req side :: given side : req side. . . .(2) 

r^. sin B given side h . ^ . ■, a n j t 

ihus, — -r= : — - — — , m which A, B and b are given. 

sm A required side a 

Sum of sides : diff :: tang half sum of other two angles : tang half diff . ... (3) 

^, 6+c tanH^+O . -.•-.„. -, 

Thus, T = 7 TTT^ — FT. Ill which A, and c are given. 

b — c tan i {B —C) 

The square of any side as a = d^ = b^ + c^—2 be cos A (4) 

^2 4. ^2 _ q2 

Thus, cos A = ^7T . in which a, b and c are given. 

2 be 




• lA (s-b) (s-c) . ... a + b + c . 
sm iA =^j T , in which s = r » given. 



tan iA=^ I -. r — , m which s = ^ — , given. 

^ s (s — a) 2 

Rules in conjunction with the above: 

Given: One side and two angles. Solve for "Req side" in (2). 
Given: Two sides and angle opposite one of them. Solve for "Req 
side" and for one of the angles, in (2). 

Given: Two sides and included angle. Solve for "Tang half diff" in (3). 
Given: Three sides. 

(a). Solve for cos A, in (4), unless A is very small. 

(b). Solve for sin ^ J4, in (4), unless A is very large. 

(c). Solve for tan ^A, in (4), in general preferred. 
If all the angles are required we may use the formulas: 



^ A ■ • i • i ,yS — a)(s — b)(s — c) 

tan i A = , m which r - '- ^— ~ -' 

s —a 



V' 



r 7 

tan i B = r ; tan ^ C = . 

5 —b s —c 

Circular Measure. — It is sometimes convenient in mathematical calcu- 
lations to express angles in circular measure. The unit of circular measure 
is an angle subtended by an arc whose length is equal to the radius of the 
circle. The value of such an angle in common measure =57°— 17'— 45" = 
57.2958°, called a radian. The number of radians in 180° is 3.141592. and 
as this value is equal to n we call 180° = ?: in circular measure. Hence, 

2r = 360° 
TT = 180° 

I = 90°. 



CIRCULAR MEASURE. CUBIC EQUATIONS. 143 

Cubic Equations. — The general form of a cubic equation is 

ao^-^bx^ + cx-{-d = (1) 

Dividing by a, we have 

x^-h-x^-^- x-h- =0 (2) 

a a a 

By substitution, this can be reduced to the second general form 

x^-hBx^-hCx-\-D = (3) 

To eliminate x^. let x = y— -^ , then equation (3) reduces to 

^-^MV (¥-¥--)=<•- <*) 

By substitution, this can be reduced to the third general form 

y^-{-pV-\-q=0 (5) 

Now if we let y = u + v, equation (5) reduces to 

u^-hv^-h(u-\-v) (Zuv + p) +(?-0 (6) 

Whence, 3 w7; + p = 0; u^ + v^ + q=0; u^v^= — -^; and u^-^v^=—q 

And we have for the final equations 

— -I^Aif^ ™ 



-^ _ 

Thereiore y==u + v = lj_l +\\ .Ql + ^+lLl _\/i! + £ 
\ 2^^427 \ 2 ^4^2^ 



^'=-|-VM^- <«> 



' (9) 

27 

And x = y— -z (see above) (10) 

Note that the above is a purely algebraic solution, which can obtain 

only when ~+~is equal to or greater than 0. When j-4- o^ <0 we have 

to resort to the trigometric solution, following. 
Trigonometric Solution. — In the equation, (9), 



y: 



\ 2^\427 \ 2 \427 



g2 /j3 
if J- + ^ < 0, then the roots are imaginary and y is the sum of two imaginary 

quantities. To solve the equation for y, under these conditions, proceed as 



follows: Let —^=-n cos 8, and -j--^-^= —n^ sin^ e\ whence, •- - -^i r,7 • 

Z 4 Z7 v J7 



,,K=^- 



and cos B = — 7r~ - From this, the values of the three roots, y, are 
2m 

2^;rcos |-, - 2V;rcos ^60° - j\ , 2^^cos (l20° - |-) . 

A J B 

And X = y — -r-. 

o 

These calculations can be made readily by the use of logarithms. It is 
well to insert the value of x, thus found, in the original equation, (1), and 
solve as a check. Successive trial values of x may be assumed directly. 



144 



9.— PLANE TRIGONOMETRY. 



3. — Natural Sines, Tangents, Cotangents, Cosines. 

(Versed sine =1 — cosine; co versed sine = 1 — sine.) 
1° 



' 1 Sine. 


Tang. ICotang.l Cosine. 


1 II ' 1 Sine. 


1 Tang. 1 Cotang.j Cosine. 







.0000000 


.000000 


Infinite 


1.000000 


60 





.0174524 


.017455 


57.28996 


.9998477 


60 


1 


.0002909 


.000291 


3437.746 


1.000000 


59 


1 


.0177432 


.017746 


56.3505^ 


.9998426 


59 


2 


.0005818 


.000582 


1718.873 


.9999998 


58 


2 


.0180341 


.018037 


55.44151 


.9998374 


58 


3 


.0008727 


.000872 


1145.915 


.9999996 


57 


3 


.0183249 


.018328 


54.56130 


.9998321 


57 


4 


.0011636 


.001163 859.4363 


.9999993 


56 


4 


.0186158 


.018619 


53.70858 


.9998267 


56 


5 


.0014544 


.001454 


687.5488 


.9999989 


55 


5 


.0189066 


.018910 


52.88211 


.9998213 


55 


6 


.0017453 


.001745 


572.9572 


.9999985 


54 


6 


.0191974 


.019201 


52.08067 


.9998157 


54 


7 


.0020362 


.002036 


491.1060 


.9999979 


53 


7 


.0194883 


.019492 


51.30315 


.9998101 


53 


8 


.0023271 


.002327 


429.7175 


.9999973 


52 


8 


.0197791 


.019783 


50.54850 


.9998044 


52 


9 


.0026180 


.002618 


381.9709 


.9999966 


51 


9 


.0200699 


.020074 


49.81572 


.9997986 


51 


10 


.0029089 


.002908 


343.7737 


.9999958 


50 


10 


.0203608 


.020365 


49.10388 


.9997927 


50 


11 


.0031998 


.003199 


312.5213 


.9999949 


49 


11 


.0206516 


.020656 


48.41208 


.9997867 


49 


12 


.0034907 


.003490 


286.4777 


.9999939 


48 


12 


.0209424 


.020947 


47.73950 


.9997807 


48 


13 


.0037815 


.003781 


264.4408 


.9999928 


47 


13 


.0212332 


.021238 


47.08534 


.9997745 


47 


14 


.0040724 


.004072 


245.5519 


.9999917 


46 


14 


.0215241 


.021529 


46.44886 


.9997683 


46 


15 


.0043633 


.004363 


229.1816 


.9999905 


45 


15 


.0218149 


.021820 


45.82935 


.9997620 


45 


16 


.0046542 


.004654 


214.8576 


.9999892 


44 


16 


.0221057 


.022111 


45.22614 


.9997556 


44 


17 


.0049451 


.004945 


202.2187 


.9999878 


43 


17 


.0223965 


.022402 


44.63859 


.9997492 


43 


18 


.0052360 


.005236 


190.9841 


.9999863 


42 


18 


.0226873 


.022693 


44.06611 


.9997426 


42 


19 


.0055268 


.005526 


180.9322 


.9999847 


.41 


19 


.0229781 


.022984 


43.50812 


.9997360 


41 


20 


.0058177 


.005817 


171.8854 


.9999831 


40 


20 


.0232690 


.023275 


42.96407 


.9997292 


40 


21 


.0061086 


.006108 


163.7001 


.9999813 


39 


21 


.0235598 


.023566 


42.43346 


.9997224 


39 


22 


.0063995 


.006399 


156.2590 


.9999795 


38 


22 


.0238506 


.023857 


41.91579 


.9997156 


38 


23 


.0066904 


.006690 


149.4650 


.9999776 


37 


23 


.0241414 


.024148 


41.41058 


.9997086 


37 


24 


.0069813 


.006981 


143.2371 


.9999756 


36 


24 


.0244322 


.024439 


40.91741 


.9997015 


36 


25 


.0072721 


.007272 


137.5075 


.9999736 


35 


25 


.0247230 


.024730 


40.43583 


.9996943 


35 


26 


.0075630 


.007563 


132.2185 


.9999714 


34 


26 


.0250138 


.025021 


39.96546 


.9996871 


34 


27 


.0078539 


.007854 


127.3213 


.9999692 


33 


27 


.0253046 


.025312 


39.50589 


.9996798 


33 


28 


.0081448 


.008145 


122.7739 


.9999668 


32 


28 


.0255954 


.025603 


39.05677 


.9996724 


32 


29 


.0084357 


.008436 


118.5401 


.9999644 


31 


29 


.0258862 


.025894' 


38.61773 


.9996649 


31 


30 


.0087265 


.008726 


114.5886 


.9999619 


30 


30 


.0261769 


.026185 


38.18845 


.9996573 


30 


31 


.0090174 


.009017 


110.8920 


.9999593 


29 


31 


.0264677 


.026477 


37.76861 


.9996497 


29 


32 


.0093083 


.009308 


107.4264 


.9999567 


28 


32 


.0267585 


.026768 


37.35789 


.9996419 


28 


33 


.0095992 


.009599 


104.1709 


.9999539 


27 


33 


.0270493 


.027059 


36.95600 


.9996341 


27 


34 


.0098900 


.009890 


101.1069 


.9999511 


26 


34 


.0273401 


.027350 


36.56265 


.9996262 


26 


35 


.0101809 


.010181 


98.21794 


.9999482 


25 


35 


.0276309 


.027641 


36.17759 


.9996182 


25 


36 


.0104718 


.010472 


95.48947 


.9999452 


24 


36 


.0279216 


.027932 


35.80055 


.9996101 


24 


37 


.0107627 


.010763 


92.90848 


.9999421 


23 


37 


.0282124 


.028223 


35.43128 


.9996020 


23 


38 


.0110535 


.011054 


90.46333 


.9999389 


22 


38 


.0285032 


.028514 


35.06954 


.9995937 


22 


39 


.0113444 


.011345 


88.14357 


.9999357 


21 


39 


.0287940 


.028805 


34.71511 


.9995854 


21 


40 


.0116353 


.011636 


85.93979 


.9999323 


20 


40 


.0290847 


.029097 


34.36777 


.9995770 


20 


41 


.0119261 


.011927 


83.84350 


.9999289 


19 


41 


.0293755 


.029388 


34.02730 


.9995684 


19 


42 


.0122170 


.012217 


81.84704 


.9999254 


18 


42 


.0296662 


.029679 


33.69350 


.9995599 


18 


43 


.0125079 


.012508 


79.94343 


.9999218 


17 


43 


.0299570 


.029970 


33.36619 


.9995512 


17 


44 


.0127987 


.012799 


78.12634 


.9999181 


16 


44 


.0302478 


.030261 


33.04517 


.9995424 


16 


45 


.0130896 


.013090 


76.39000 


.9999143 


15 


45 


.0305385 


.030552 


32.73026 


.9995336 


15 


46 


.0133805 


.013381 


74.72916 


.9999105 


14 


46 


.0308293 


.030843 


32.42129 


.9995247 


14 


47 


.0136713 


.013672 


73.13899 


.9999065 


13 


47 


.0311200 


.031135 


32.11809 


.9995157 


13 


48 


.0139622 


.013963 


71.61507 


.9999025 


12 


48 


.0314108 


.031426 


31.82051 


.9995066 


12 


49 


.0142530 


.014254 


70.15334 


.9998984 


11 


49 


.0317015 


.031717 


31.52839 


.9994974 


11 


50 


.0145439 


.014545 


68.75008 


.9998942 


10 


50 


.0319922 


.032008 


31.24157 


.9994881 


10 


61 


.0148348 


.014836 


67.40185 


.9998900 


9 


51 


.0322830 


.032299 


30.95992 


.9994788 


9 


52 


.0151256 


.015127 


66.10547 


.9998856 


8 


52 


.0325737 


.032591 


30.68330 


.9994693 


8 


53 


.0154165 


.015418 


64.85800 


.9998812 


7 


53 


.0328644 


.032882 


30.41158 


.9994598 


7 


54 


.0157073 


.015709 


63.65674 


.9998766 


6 


54 


.0331552 


.033173 


30.14461 


.9994502 


6 


55 


.0159982 


.016000 


62.49915 


.9998720 


5 


55 


.0334459 


.033464 


29.88229 


.9994405 


5 


56 


.0162890 


.016291 


61.38290 


.9998673 


4 


56 


.0337366 


.033755 


29.62449 


.9994308 


4 


57 


.0165799 


.016582 


60.30582 


.9998625 


3 


57 


.0340274 


.034047 


29.37110 


.9994209 


3 


58 


.0168707 


.016873 


59.26587 


.9998577 


2 


58 


.0343181 


.034338 


29.12200 


.9994110 


2 


59 


.0171616 


.017164 


58.26117 


.9998527 


1 


59 


.0346088 


.034629 


28.87708 


.9994009 


1 


60 


.0174524 


.017455 


57.28996 


.9998477 





60 


.0348995 


.034920 


28.63625 


.9993908 





Cosine. 


CotangI Tang. | Sine. 


1 ' II 1 Cosine. 


CotangI Tang, j Sine. 


_:, 



Note. — Secant = 1 -^ cosine ; cosecant = 1-^sine. 



NATURAL Sm^S, ETC, 



145 



3.— Natural Sines, Tangents, Cotangents, Cosines. — (Continued), 

(Versed sine = 1 — cosine; co versed sine= 1 — sine.) 
3° 



' 


1 Sine. 


1 Tang. ICotang.l Cosine. 


1 


1 ' 


1 Sine. 


1 Tang! ICotang.j Cosine. 







.0348995 


.034920 


28.63625 


.9993908 


60 





.0523360 


.052407 


19.08113 


.9986295 


60 




.0351902 


.035212 


28.39939 


.9993805 


59 


1 


.0526264 


.052699 


18.97552 


.9986143 


59 


2 


.0354809 


.035503 


28.16642 


.9993704 


58 


2 


.0529169 


.052991 


18.87106 


.9985989 


58 


3 


.0357716 


.035794 


27.93723 


.9993600 


57 


3 


.0532074 


.053282 


18.76775 


.9985835 


57 


4 


.0360623 


.036085 


27.71174 


.9993495 


56 


4 


.0534979 


.053574 


18.66556 


.9985680 


56 


5 


.0363530 


.036377 


27.48985 


.9993390 


55 


5 


.0537883 


.053866 


18.56447 


.9985524 


55 


6 


.0366437 


.036668 


27.27148 


.9993284 


54 


6 


.0540788 


.054158 


18.46447 


.9985367 


54 


7 


.0369344 


.036959 


27.05655 


.9993177 


53 


7 


.0543693 


.054449 


18.36553 


.9985209 


53 


8 


.0372251 


.037250 


26.84498 


.9993069 


52 


8 


.0546597 


.054741 


18.26765 


.9985050 


52 


9 


.0375158 


.037542 


26.63669 


.9992960 


51 


9 


.0549502 


.055033 


18.17080 


.9984891 


51 


10 


.0378065 


.037833 


26.43160 


.9992851 


50 


10 


.0552406 


.055325 


18.07497 


.9984731 


50 


11 


.0380971 


.038124 


26.22963 


.9992740 


49 


11 


.0555311 


.055616 


17.98015 


.9984570 


49 


12 


.0383878 


.038416 


26.03073 


.9992629 


48 


12 


.0558215 


.055908 


17.88631 


.9984408 


48 


13 


.0386785 


.038707 


25.83482 


.9992517 


47 


13 


.0561119 


.056200 


17.79344 


.9984245 


47 


14 


.0389692 


.038998 


25.64183 


.9992404 


46 


14 


.0564024 


.056492 


17.70152 


.9984081 


46 


15 


.0392598 


.039290 


25.45170 


.9992290 


45 


15 


.0566928 


.056784 


17.61055 


.9983917 


45 


16 


.0395505 


.039581 


25.26436 


.9992176 


44 


16 


.0569832 


.057075 


17.52051 


.9983751 


44 


17 


.0398411 


.039872 


25.07975 


.9992060 


43 


17 


.0572736 


.057367 


17.43138 


.9983585 


43 


18 


.0401318 


.040164 


24.89782 


.9991944 


42 


18 


.0575640 


.057659 


17.34315 


.9983418 


42 


19 


.0404224 


.040455 


24.71851 


.9991827 


41 


19 


.0578544 


.057951 


17.25580 


.9983250 


41 


20 


.0407131 


.040746 


24.54175 


.9991709 


40 


20 


.0581448 


.058243 


17.16933 


.9983082 


40 


21 


.0410037 


.041038 


24.36750 


.9991590 


39 


21 


.0584352 


.058535 


17.08372 


.9982912 


39 


22 


.0412944 


.041329 


24.19571 


.9991470 


38 


22 


.0587256 


.058827 


16.99895 


.9982742 


38 


23 


.0415850 


.041621 


24.02632 


.9991350 


37 


23 


.0590160 


.059119 


16.91502 


.9982570 


37 


24 


.0418757 


.041912 


23.85927 


.9991228 


36 


24 


.0593064 


.059410 


16.83191 


.9982398 


36 


25 


.0421663 


.042203 


23.69453 


.9991106 


35 


25 


.0595967 


.059702 


16.74961 


.9982225 


35 


26 


.0424569 


.042495 


23.53205 


.9990983 


34 


26 


.0598871 


.059994 


16.66811 


.9982052 


34 


27 


.0427475 


.042786 


23.37177 


.9990859 


33 


27 


.0601775 


.060286 


16.58739 


.9981877 


33 


28 


.0430382 


.043078 


23.21366 


.9990734 


32 


28 


.0604678 


.060578 


16.50745 


.9981701 


32 


29 


.0433288 


.043369 


23.05767 


.9990609 


31 


29 


.0607582 


.060870 


16.42827 


.9981525 


31 


30 


.0436194 


.043660 


22.90376 


.9990482 


30 


30 


.0610485 


.061162 


16.34985 


.9981348 


30 


31 


.0439100 


.043952 


22.75189 


.9990355 


29 


31 


.0613389 


.061454 


16.27217 


.9981170 


29 


32 


.0442006 


.044243 


22.60201 


.9990227 


28 


32 


.0616292 


.061746 


16.19522 


.9980991 


28 


33 


.0444912 


.044535 


22.45409 


.9990098 


27 


33 


.0619196 


.062038 


16.11899 


.9980811 


27 


34 


.0447818 


.044826 


22.30809 


.9989968 


26 


34 


.0622099 


.062330 


16.04348 


.9980631 


26 


35 


.0450724 


.045118 


22.16398 


.9989837 


25 


35 


.0625002 


.062622 


15.96866 


.9980450 


25 


36 


.0453630 


.045409 


22.02171 


.9989706 


24 


36 


.0627905 


.062914 


15.89454 


.9980267 


24 


37 


.0456536 


.045701 


21.88125 


.9989573 


23 


37 


.0630808 


.063206 


15.82110 


.9980084 


23 


38 


.0459442 


.045992 


21.74256 


.9989440 


22 


38 


.0633711 


.063498 


15.74833 


.9979900 


22 


39 


.0462347 


.046284 


21.60563 


.9989306 


21 


39 


.0636614 


.063790 


15.67623 


.9979716 


21 


40 


.0465253 


.046575 


21.47040 


.9989171 


20 


40 


.0639517 


.064082 


15.60478 


.9979530 


20 


41 


.0468159 


.046867 


21.33685 


.9989035 


19 


41 


.0642420 


.064375 


15.53398 


.9979343 


19 


42 


.0471065 


.047158 


21.20494 


.9988899 


18 


42 


.0645323 


.064667 


15.46381 


.9979156 


18 


43 


.0473970 


.047450 


21.07466 


.9988761 


17 


43 


.0648226 


.064959 


15.39427 


.9978968 


17 


44 


.0476876 


.047741 


20.94596 


.9988623 


16 


44 


.0651129 


.065251 


15.32535 


.9978779 


16 


45 


.0479781 


.048033 


20.81882 


.9988484 


15 


45 


.0654031 


.065543 


15.25705 


.9978589 


15 


46 


.0482687 


.048325 


20.69322 


.9988344 


14 


46 


.0656934 


.065835 


15.18934 


.9978399 


14 


47 


.0485592 


.048616 


20.56911 


.9988203 


13 


47 


.0659836 


.066127 


15.12224 


.9978207 


13 


48 


.0488498 


.048908 


20.44648 


.9988061 


12 


48 


.0662739 


.066419 


15.05572 


.9978015 


12 


49 


.0491403 


.049199 


20.32530 


.9987919 


11 


49 


.0665641 


.066712 


14.98978 


.9977821 


11 


50 


.0494308 


.049491 


20.20555 


.9987775 


10 


50 


.0668544 


.067004 


14.92441 


.9977627 


10 


51 


.0497214 


.049782 


20.08719 


.9987631 


9 


51 


.0671446 


.067296 


14.85961 


.9977433 


9 


52 


.0500119 


.050074 


19.97021 


.9987486 


8 


52 


.0674349 


.067588 


14.79537 


.9977237 


8 


53 


.0503024 


.050366 


19.85459 


.9987340 


7 


53 


.0677251 


.06788a 


14.73167 


.9977040 


7 


54 


.0505929 


.050657 


19.74029 


.9987194 


6 


54 


.0680153 


.068173 


14.66852 


.9976843 


6 


55 


.0508835 


.050949 


19.62729 


.9987046 


5 


55 


.0683055 


.068465 


14.60591 


.9976645 


5 


56 


.0511740 


.051241 


19.51558 


.9986898 


4 


56 


.0685957 


.068757 


14.54383 


.9976445 


4 


57 


.0514645 


.051532 


19.40513 


.9986748 


3 


57 


.0688859 


.069049 


14.48227 


.9976245 


3 


58 


.0517550 


.051824 


19.29592 


.9986598 


2 


58 


.0691761 


.069342 


14.42123 


.9976045 


2 


59 


.0520455 


.052116 


19.18793 


.9986447 


1 


59 


.0694663 


.069634 


14.36069 


.9975843 


1 


60 


.0523360 


.052407 


19.08113 


.9986295 





60 


.0697565 


.069926 


14.30066 


.9975641 







Cosine, 


CotangI Tang. | Sine. 


I'll 1 Cosine. 


1 CotangI Tang. 


Sine. 


"^ 



87^ 



Note. — Secant =l-i- cosine. 



Cosecant = 1+sine. 



146 



9— PLANE TRIGONOMETRY. 



3.— Natural Sines, Tangents, Cotangents, Cosines. — (Continued] , 

(Versed sine =1 — cosine; coversed sine = 1 — sine.) 

5° 



' 1 Sine. 


1 Tang. ICotang.l Cosine. 


1 II ' 1 Sine. 


Tang. ICotang.l Cosine. 







.0697565 


.069926 


14.30066 


.9975641 


60 





.0871557 


.087488 


11.43005 


.9961947 


__ 
60 


1 


.0700467 


.070219 


14.24113 


.9975437 


59 


1 


.0874455 


.087781 


11.39188 


.9961693 


59 


2 


.0703368 


.070511 


14.18209 


.9975233 


58 


2 


.0877353 


.088074 


11.35397 


.9961438 


58 


3 


.0706270 


.070803 


14.12353 


.9975028 


57 


3 


.0880251 


.088368 


11.31630 


.9961183 


57 


4 


.0709171 


.071096 


14.06545 


.9974822 


56 


4 


.0883148 


.088661 


11.27888 


.9960926 


56 


5 


.0712073 


.071388 


14.00785 


.9974615 


55 


5 


.0886046 


.088954 


11.24171 


.9960669 


55 


6 


.0714974 


.071680 


13.95071 


.9974408 


54 


6 


.0888943 


.089247 


11.20478 


.9960411 


54 


7 


.0717876 


.071973 


13.89404 


.9974199 


53 


7 


.0891840 


.089540 


11.16808 


.9960152 


53 


8 


.0720777 


.072265 


13.83782 


.9973990 


52 


8 


.0894738 


.089834 


11.13163 


.9959892 


52 


9 


.0723678 


.072558 


13.78206 


.9973780 


51 


9 


.0897635 


.090127 


11.09541 


.9959631 


51 


10 


.0726580 


.072850 


13.72673 


.9973569 


SO 


10 


.0900532 


.090420 


11.05943 


.9959370 


50 


11 


.0729481 


.073143 


13.67185 


.9973357 


49 


11 


.0903429 


.090713 


11.02367 


.9959107 


49 


12 


.0732382 


.073435 


13.61740 


.9973145 


48 


12 


.0906326 


.091007 


10.98815 


.9958844 


48 


13 


.0735283 


.073727 


13.56339 


.9972931 


47 


13 


.0909223 


.091300 


10.95285 


.9958580 


47 


14 


.0738184 


.074020 


13.50979 


.9972717 


46 


14 


.0912119 


.091593 


10.91777 


.9958315 


46 


15 


.0741085 


.074312 


13.45662 


.9972502 


45 


15 


.0915016 


.091887 


10.88292 


.9958049 


45 


16 


.0743986 


.074605 


13.40386 


.9972286 


44 


16 


.0917913 


.092180 


10.84828 


.9957783 


44 


17 


.0746887 


.074897 


13.35151 


.9972069 


43 


17 


.0920809 


.092473 


10.81387 


.9957515 


43 


18 


.0749787 


.075190 


13.29957 


.9971851 


44 


18 


.0923706 


.092767 


10.77967 


.9957247 


42 


19 


.0752688 


.075482 


13.24803 


.9971633 


41 


19 


.0926602 


.093060 


10.74568 


.9956978 


41 


20 


.0755589 


.075775 


13.19688 


.9971413 


40 


20 


.0929499 


.093354 


10.71191 


.9956708 


40 


21 


.0758489 


.076068 


13.14612 


.9971193 


39 


21 


.0932395 


.093647 


10.67834 


.9956437 


39 


22 


.0761390 


.076360 


13.09575 


.9970972 


38 


22 


.0935291 


.093940 


10.64499 


.9956165 


38 


23 


.0764290 


.076653 


13.04576 


.9970750 


37 


23 


.0938187 


.094234 


10.61184 


.9955892 


37 


24 


.0767190 


.076945 


12.99616 


.9970528 


36 


24 


.0941083 


.094527 


10.57889 


.9955620 


36 


25 


0770091 


.077238 


12.94692 


.9970304 


35 


25 


.0943979 


.094821 


10.54615 


.9955345 


35 


26 


.0772991 


.077531 


12.89805 


.9970080 


34 


26 


.0946875 


.095114 


10.51360 


.9955070 


34 


27 


.0775891 


.077823 


12.84955 


.9969854 


33 


27 


.0949771 


.095408 


10.48126 


.9954794 


33 


28 


.0778791 


.078116 


J2. 80141 


.9969628 


32 


28 


.0952666 


.095701 


10.44911 


.9954517 


32 


29 


.0781691 


.078409 


12.75363 


.9969401 


31 


29 


.0955562 


.095995 


10.41715 


.9954240 


31 


30 


.0784591 


.078701 


12.70620 


.9969173 


30 


30 


.0958458 


.096289 


10.38539 


.9953962 


30 


31 


.0787491 


.078994 


12.65912 


.9968945 


29 


31 


.0961353 


.096582 


10.35382 


.9953683 


29 


32 


.0790391 


.079287 


12.61239 


.9968715 


28 


32 


.0964248 


.096876 


10.32244 


.9953403 


28 


33 


.0793290 


.079579 


12.56599 


.9968485 


27 


33 


.0967144 


.097169 


10.29125 


.9953122 


27 


34 


.0796190 


.079872 


12.51994 


.9968254 


26 


34 


.0970039 


.097463 


10.26024 


.9952840 


26 


35 


.0799090 


.080165 


12.47422 


.9968022 


25 


35 


.0972934 


.097757 


10.22942 


.9952557 


25 


36 


.0801989 


.080458 


12.42883 


.9967789 


24 


36 


.0975829 


.098050 


10.19878 


.9952274 


24 


37 


.0804889 


.080750 


12.38376 


.9967555 


23 


37 


.0978724 


.098344 


10.16833 


.9951990 


23 


38 


.0807788 


.081043 


12.33902 


.9967321 


22 


38 


.0981619 


.098638 


10.13805 


.9951705 


22 


39 


.0810687 


.081336 


12.29460 


.9967085 


21 


39 


.0984514 


.098932 


10.10795 


.9951419 


21 


40 


.0813587 


.081629 


12.25050 


.9966849 


20 


40 


.0987408 


.099225 


10.07803 


.9951132 


20 


41 


.0816486 


.081922 


12.20671 


.9966612 


19 


41 


.0990303 


.099519 


10.04828 


.9950844 


19 


42 


.0819385 


.082215 


12.16323 


.9966374 


18 


42 


.0993197 


.099813 


10.01871 


.9950556 


18 


43 


.0822284 


.082507 


12.12006 


.9966135 


17 


43 


.0996092 


.100107 


9.989305 


.9950266 


17 


44 


.0825183 


.082800 


12.07719 


.9965895 


16 


44 


.0998986 


.100400 


9.960072 


.9949976 


16 


45 


.0828082 


.083093 


12.03462 


.9965655 


15 


45 


.1001881 


.100694 


9.931008 


.9949685 


15 


46 


.0830981 


.083386 


11.99234 


.9965414 


14 


46 


.1004775 


.100988 


9.902112 


.9949393 


14 


47 


.0833880 


.083679 


11.95037 


.9965172 


13 


47 


.1007669 


.101282 


9.873382 


.9949101 


13 


48 


.0836778 


.083972 


11.90868 


.9964929 


12 


48 


.1010563 


.101576 


9.844816 


.9948807 


12 


49 


.0839677 


.084265 


11.86728 


.9964685 


11 


49 


.1013457 


.101870 


9.816414 


.9948513 


11 


50 


.0842576 


.084558 


11.82616 


.9964440 


10 


50 


.1016351 


.102164 


9.788173 


.9948217 


10 


51 


.0845474 


.084851 


11.78533 


.9964195 


9 


51 


.1019245 


.102458 


9.760092 


.9947921 


9 


52 


.0848373 


.085144 


11.74477 


.9963948 


8 


52 


.1022138 


.102752 


9.732171 


.9947625 


8 


53 


.0851271 


.085437 


11.70450 


.9963701 


7 


53 


.1025032 


.103046 


9.704407 


.9947327 


7 


54 


.0854169 


.085730 


11.66449 


.9963453 


6 


54 


.1027925 


.103339 


9.676800 


.9947028 


6 


55 


.0857067 


.086023 


11.62476 


.9963204 


5 


55 


.1030819 


.103634 


9.649347 


.9946729 


5 


56 


.0859966 


.086316 


11.58529 


.9962954 


4 


56 


.1033712 


.103928 


9.622048 


.9946428 


4 


57 


.0862864 


.086609 


11.54609 


.9962704 


3 


57 


.1036605 


.104222 


9.594902 


.9946127 


3 


58 


.0865762 


.086902 


11.50715 


.9962452 


2 


58 


.1039499 


.104516 


9.567906 


.9945825 


2 


59 


.0868660 


.087195 


11.46847 


.9962200 


1 


59 


.1042392 


.104810 


9.541061 


.9945523 


1 


60 


.0871557 


.087488 


11.43005 


.9961947 





60 


.1045285 


.105104 


9.514364 


.9945219 





Cosine. | 


Cotang 1 Tang. | Sine. 


1 ' II 1 Cosine. 


Cotang 1 Tang, j Sine. 


EI 



Note. — Secant =l-i- cosine. 



85° 
Cosecant = 1-Hsine. 



84*= 



NATURAL SINES, ETC. 



147 



3. — Natural Sines, Tangents, Cotangents, Cosines. — (Continued). 

(Versed sine =1 — cosine; coversed sine = 1 — sine.) 

7° 



' 


Sine. 


Tang. 


Cotang.l Cosine. 


1 


1 ' 


Sine. 


Tang. 1 Cotang.l Cosine. 


. 





.1045285 


.105104 


9.514364 


.9945219 


60 





.1218693 


.122784 


8.144346 


.9925462 


60 


1 


.1048178 


.105398 


9.487814 


.9944914 


59 


1 


.1221581 


.123079 


8.124807 


.9925107 


59 


2 


.1051070 


.105692 


9.461411 


.9944609 


58 


2 


.1224468 


.123375 


8.105359 


.9924751 


58 


3 


.1053963 


.105986 


9.435153 


.9944303 


57 


3 


.1227355 


.123670 


8.086004 


.9924394 


57 


4 


.1056856 


.106280 


9.409038 


.9943996 


56 


4 


.1230241 


.123965 


8.066739 


.9924037 


56 


5 


.1059748 


.106575 


9.383066 


.9943688 


55 


5 


.1233128 


.124261 


8.047564 


.9923679 


55 


6 


.1062641 


.106869 


9.357235 


.9943379 


54 


6 


.1236015 


.124556 


8.028479 


.9923319 


54 


7 


.1065533 


.107163 


9.331545 


.9943070 


53 


7 


.1238901 


.124852 


8.009483 


.9922959 


53 


8 


.1068425 


.107457 


9.305993 


.9942760 


52 


8 


.1241788 


.125147 


7.990575 


.9922599 


52 


9 


.1071318 


.107751 


9.280580 


.9942448 


51 


9 


.1244674 


.125442 


7.971755 


.9922237 


51 


10 


.1074210 


.108046 


9.255303 


.9942136 


50 


10 


.1247560 


.125738 


7.953022 


.9921874 


50 


11 


.1077102 


.108340 


9.230162 


.9941823 


49 


11 


.1250446 


.126033 


7.934375 


.9921511 


49 


12 


.1079994 


.108634 


9.205156 


.9941510 


48 


12 


.1253332 


.126329 


7.915815 


.9921147 


48 


13 


.1082885 


.108929 


9.180283 


.9941195 


47 


13 


.1256218 


.126624 


7.897339 


.9920782 


47 


14 


.1085777 


.109223 


9.155543 


.9940880 


46 


14 


.1259104 


.126920 


7.878948 


.9920416 


46 


15 


.1088669 


.109517 


9.130934 


.9940563 


45 


15 


.1261990 


.127216 


7.860642 


.9920049 


45 


16 


.1091560 


.109812 


9.106456 


.9940246 


44 


16 


.1264875 


.127511 


7.842419 


.9919682 


44 


17 


.1094452 


.110106 


9.082107 


.9939928 


43 


17 


.1267761 


.127807 


7.824279 


.9919314 


43 


18 


.1097343 


.110401 


9.057886 


.9939610 


42 


18 


.1270646 


.128103 


7.806221 


.9918944 


42 


19 


.1100234 


.110695 


9.033793 


.9939290 


41 


19 


.1273531 


.128398 


7.788245 


.9918574 


41 


20 


.1103126 


.110989 


9.009826 


.9938969 


40 


20 


.1276416 


.128694 


7.770350 


.9918204 


40 


21 


.1106017 


.111284 


8.985984 


.9938648 


39 


21 


.1279302 


.128990 


7.752536 


.9917832 


39 


22 


.1108908 


.111578 


8.962266 


.9938326 


38 


22 


.1282186 


.129285 


7.734802 


.9917459 


38 


23 


.1111799 


.111873 


8.938672 


.9938003 


37 


23 


.1285071 


.129581 


7.717148 


.9917086 


37 


24 


.1114689 


.112168 


8.915200 


.9937679 


36 


24 


.1287956 


.129877 


7.699573 


.9916712 


36 


25 


.1117580 


.112462 


8.891850 


.9937355 


35 


25 


.1290841 


.130173 


7.682076 


.9916337 


35 


26 


.1120471 


.112757 


8.868620 


.9937029 


34 


26 


.1293725 


.130469 


7.664658 


.9915961 


34 


27 


.1123361 


.113051 


8.845510 


.9936703 


33 


27 


.1296609 


.130764 


7.647317 


.9915584 


33 


28 


.1126252 


.113346 


8.822518 


.9936375 


32 


28 


.1299494 


.131060 


7.630053 


.9915206 


52 


29 


.1129142 


.113641 


8.799644 


.9936047 


31 


29 


.1302378 


.131356 


7.612865 


.9914828 


31 


30 


.1132032 


.113935 


8.776887 


.9935719 


30 


30 


.1305262 


.131652 


7.595754 


.9914449 


30 


31 


.1134922 


.114230 


8.754246 


.9935389 


29 


31 


.1308146 


.131948 


7.578717 


.9914069 


29 


32 


.1137812 


.114525 


8.731719 


.9935058 


28 


32 


.1311030 


.132244 


7.561756 


.9913688 


28 


33 


.1140702 


.114819 


8.709307 


.9934727 


27 


33 


.1313913 


.132540 


7.544869 


.9913306 


27 


34 


.1143592 


.115114 


8.687008 


.9934395 


26 


34 


.1316797 


.132836 


7.528057 


.9912923 


26 


35 


.1146482 


.115409 


8.664822 


.9934062 


25 


35 


.1319681 


.133132 


7.511317 


.9912540 


25 


36 


.1149372 


.115703 


8.642747 


.9933728 


24 


36 


.1322564 


.133428 


7.494651 


.9912155 


24 


37 


.1152261 


.115998 


8.620783 


.9933393 


23 


37 


.1325447 


.133724 


7.478057 


.9911770 


23 


38 


.1155151 


.116293 


8.598929 


.9933057 


22 


38 


.1328330 


.134020 


7.461535 


.9911384 


22 


39 


.1158040 


.116588 


8.577183 


.9932721 


21 


39 


.1331213 


.134316 


7.445085 


.9910997 


21 


40 


.1160929 


.116883 


8.555546 


.9932384 


20 


40 


.1334096 


.134612 


7,428706 


.9910610 


20 


41 


.1163818 


.117178 


8.534017 


.9932045 


19 


41 


.1336979 


.134909 


7.412397 


.9910221 


19 


42 


.1166707 


.117473 


8.512594 


.9931706 


18 


42 


.1339862 


.135205 


7.396159 


.9909832 


18 


43 


.1169596 


.117767 


8.491277 


.9931367 


17 


43 


.1342744 


.135501 


7.379990 


.9909442 


17 


44 


.1172485 


.118062 


8.470065 


.9931026 


16 


44 


.1345627 


.135797 


7.363891 


.9909051 


16 


45 


.1175374 


.118357 


8.448957 


.9930685 


15 


45 


.1348509 


.136094 


7.347861 


.9908659 


15 


46 


.1178263 


.118652 


8.427953 


.9930342 


14 


46 


.1351392 


.136390 


7.331898 


.9908266 


14 


47 


.1181151 


.118947 


8.407051 


.9929999 


13 


47 


.1354274 


.136686 


7.316004 


.9907873 


13 


48 


.1184040 


.119242 


8.386251 


.9929655 


12 


48 


.1357156 


.136983 


7.300178 


.9907478 


12 


49 


.1186928 


.119537 


8.365553 


.9929310 


11 


49 


.1360038 


.137279 


7.284418 


.9907083 


11 


50 


.1189816 


.119832 


8.344955 


.9928965 


10 


50 


.1362919 


.137575 


7.268725 


.9906687 


10 


51 


.1192704 


.120127 


8.324457 


.9928618 


9 


51 


.1365801 


.137872 


7.253098 


.9906290 


9 


52 


.1195593 


.120423 


8.304058 


.9928271 


8 


52 


.1368683 


.138168 


7.237537 


.9905893 


8 


53 


.1198481 


.120718 


8.283757 


.9927922 


7 


53 


.1371564 


.138465 


7.222042 


.9905494 


7 


54 


.1201368 


.121013 


8.263554 


.9927573 


6 


54 


.1374445 


.138761 


7.206611 


.9905095 


6 


55 


.1204256 


.121308 


8.243448 


.9927224 


5 


55 


.1377327 


.139058 


7.191245 


.9904694 


5 


56 


.1207144 


.121603 


8.223438 


.9926873 


4 


56 


.1380208 


.139354 


7.175943 


.9904293 


4 


57 


.1210031 


.121898 


8.203523 


.9926521 


3 


57 


.1383089 


.139651 


7.160705 


.9903891 


3 


58 


.1212919 


.122194 


8.183704 


.9926169 


2 


58 


.1385970 


.139947 


7.145530 


.9903489 


2 


59 


.1215806 


.122489 


8.163978 


.9925816 


1 


59 


.1388850 


.140244 


7.130419 


.9903085 


1 


60 


.1218693 


.122784 


8.144346 


. 9925462 





60 


.1391731 


.140540 


7.115369 


.9902681 






I Cosine. |Cotang| Tang. | Sine. 



Note. — Secant =!-«- cosine. 



II I Cosine. ICotangI Tang, j Sine. | 



83° 
Cosecant ■■ 



82° 



l-Hsine. 



148 



9.— PLANE TRIGONOMETRY, 



3,— Natural Sines, Tangents, Cotangents, Cosines. — (Continued.) 

(Versed sine =1 — cosine; co versed sine=l — sine.) 
9^ 



' 1 sine. 


Tang. 1 Cotang.l Cosine. 


1 II ' 1 Sine. 


Tang. ICotang.l Cosine. 







.1391731 


.140540 


7.115369 


.9902681 


60 





.1564345 


.158384 


6.313751 


.9876883 


60 


1 


.1394612 


.140837 


7.100382 


.9902275 


59 


■ 1 


.1567218 


.158682 


6.301886 


.9876428 


59 


2 


.1397492 


.141134 


7.085457 


.9901869 


58 


2 


.1570091 


.158980 


6.290065 


.9875972 


58 


3 


.1400372 


.141430 


7.070593 


.9901462 


57 


3 


.1572963 


.159279 


6.278286 


.9875514 


57 


4 


.1403252 


.141727 


7.055790 


.9901055 


56 


4 


.1575836 


.159577 


6.266551 


.9875057 


56 


5 


.1406132 


.142024 


7.041048 


.9900646 


55 


5 


.1578708 


.159875 


6.254858 


.9874598 


55 


6 


.1409012 


.142321 


7.026366 


.9900237 


54 


6 


.1581581 


.160174 


6.243208 


.9874138 


54 


7 


.1411892 


.142617 


7.011744 


.9899826 


53 


7 


.1584453 


.160472 


6.231600 


.9873678 


53 


8 


.1414772 


.142914 


6.997180 


.9899415 


52 


8 


.1587325 


.160770 


6.220034 


.9873216 


52 


9 


.1417651 


.143211 


6.982678 


.9899003 


51 


9 


.1590197 


.161069 


6.208510 


.9872754 


51 


10 


.1420531 


.143508 


6.968233 


.9898590 


50 


10 


.1593069 


.161367 


6.197027 


.9872291 


50 


11 


.1423410 


.143805 


6.953847 


.9898177 


49 


11 


.1595940 


.161666 


6.185586 


.9871827 


49 


12 


.1426289 


.144102 


6.939519 


.9897762 


48 


12 


.1598812 


.161964 


6.174186 


.9871363 


48 


13 


.1429168 


.144399 


6.925248 


.9897347 


47 


13 


.1601683 


.162263 


6.162827 


.9870897 


47 


14 


.1432047 


.144696 


6.911035 


.9896931 


46 


14 


.1604555 


.162561 


6.151508 


.9870431 


46 


15 


.1434926 


.144993 


6.896879 


.9896514 


45 


15 


.1607426 


.162860 


6.140230 


.9869964 


45 


16 


.1437805 


.145290 


6.882780 


.9896096 


44 


16 


.1610297 


.163159 


6.128992 


.9869496 


44 


17 


.1440684 


.145587 


6.868737 


.9895677 


43 


17 


.1613167 


.163457 


6.117794 


.9869027 


43 


18 


.1443562 


.145884 


6.854750 


.9895258 


42 


18 


.1616038 


.163756 


6.106636 


.9868557 


42 


19 


.1446440 


.146181 


6.840819 


.9894838 


41 


19 


.1618909 


.164055 


6.095517 


.9868087 


41 


20 


.1449319 


.146478 


6.826943 


.9894416 


40 


20 


.1621779 


.164353 


6.084438 


.9867615 


40 


21 


.1452197 


.146775 


6.813122 


.9893994 


39 


21 


.1624650 


.164652 


6.073397 


.9867143 


39 


22 


.1455075 


.147072 


6.799356 


.9893572 


38 


22 


.1627520 


.164951 


6.062396 


.9866670 


38 


23 


.1457953 


.147369 


6.785644 


.9893148 


37 


23 


.1630390 


.165250 


6.051434 


.9866196 


37 


24 


.1460830 


.147667 


6.771986 


.9892723 


36 


24 


.1633260 


.165548 


6.040510 


.9865722 


36 


25 


.1463708 


.147964 


6.758382 


.9892298 


35 


25 


.1636129 


.165847 


6.029624 


.9865246 


35 


26 


.1466585 


.148261 


6.744831 


.9891872 


34 


26 


.1638999 


.166146 


6.018777 


.9864770 


34 


27 


.1469463 


.149559 


6.731334 


.9891445 


33 


27 


.1641868 


.166445 


6.007967 


.9864293 


33 


28 


.1472340 


.148856 


6.717889 


.9891017 


32 


28 


.1644738 


.166744 


5.997195 


.9863815 


32 


29 


.1475217 


.149153 


6.704496 


.9890588 


31 


29 


.1647607 


.167043 


5.986461 


.9863336 


31 


30 


.1478094 


.149451 


6.691156 


.9890159 


30 


30 


.1650476 


.167342 


5.975764 


.9862856 


30 


31 


.1480971 


.149748 


6.677867 


.9889728 


29 


31 


.1653345 


.167641 


5.965104 


.9862375 


29 


32 


.1483848 


.150045 


6.664630 


.9889297 


28 


32 


.1656214 


.167940 


5.954481 


.9861894 


28 


33 


.1486724 


.150343 


6.651444 


.9888865 


27 


33 


.1659082 


.168239 


5.943895 


.9861412 


27 


34 


.1489601 


.150640 


6.638310 


.9888432 


26 


34 


.1661951 


.168539 


5.933345 


.9860929 


26 


35 


.1492477 


.150938 


6.625225 


.9887998 


25 


35 


.1664819 


.168838 


5.922832 


.9860445 


25 


36 


.1495353 


.151235 


6.612191 


.9887564 


24 


36 


.1667687 


.169137 


5.912355 


.9859960 


24 


37 


.1498230 


.151533 


6.599208 


.9887128 


23 


37 


.1670556 


.169436 


5.901913 


.9859475 


23 


38 


.1501106 


.151830 


6.586273 


.9886692 


22 


38 


.1673423 


.169735 


5.891508 


.9858988 


22 


39 


.1503981 


.152128 


6.573389 


.9886255 


21 


39 


.1676291 


.170035 


5.881138 


.9858501 


21 


40 


.1506857 


.152426 


6.560553 


.9885817 


20 


40 


.1679159 


.170334 


5.870804 


.9858013 


20 


41 


.1509733 


.152723 


6.547767 


.9885378 


19 


41 


.1682026 


.170633 


5.860505 


.9857524 


19 


42 


.1512608 


.153021 


6.535029 


.9884939 


18 


42 


.1684894 


.170933 


5.850241 


.9857035 


18 


43 


.1515484 


.153319 


6.522339 


.9884498 


17 


43 


.1687761 


.171232 


5.840011 


.9856544 


17 


44 


.1518359 


.153617 


6.509698 


.9884057 


16 


44 


.0690628 


.171532 


5.829817 


.9856053 


16 


45 


.1521234 


.153914 


6.497104 


.9883615 


15 


45 


.1693495 


.171831 


5.819667 


.9855561 


15 


46 


.1524109 


.154212 


6.484558 


.9883172 


14 


46 


.1696362 


.172130 


5.809531 


.9855068 


14 


47 


.1526984 


.154510 


6.472059 


.9882728 


13 


47 


.1699228 


.172430 


5.799440 


.9854574 


13 


48 


.1529858 


.154808 


6.459607 


.9882284 


12 


48 


.1702095 


.172730 


5.789382 


.9854079 


12 


49 


.1532733 


.155106 


6.447201 


.9881838 


11 


49 


.1704961 


.173029 


5.779358 


.9853583 


11 


50 


.1535607 


.155404 


6.434842 


.9881392 


10 


50 


.1707828 


.173329 


5.769368 


.9853087 


10 


51 


.1538482 


.155701 


6.422530 


.9880945 


9 


51 


.1710694 


.173628 


5.759412 


.9852590 


9 


52 


.1541356 


.155999 


6.410263 


.9880497 


8 


52 


.1713560 


.173928 


5.749488 


.9852092 


8 


53 


.1544230 


.156297 


6.398042 


.9880048 


7 


53 


.1716425 


.174228 


5.739598 


.9851593 


7 


54 


.1547104 


.156595 


6.385866 


.9879599 


6 


54 


.1719291 


.174527 


5.729741 


.9851093 


6 


55 


.1549978 


.156893 


6.373735 


.9879148 


5 


55 


.1722156 


.174827 


5.719917 


.9850593 


5 


56 


.1552851 


.157191 


6.361650 


.9878697 


4 


56 


.1725022 


.175127 


5.710125 


.9850091 


4 


57 


.1555725 


.157490 


6.349609 


.9878245 


3 


57 


.1727887 


.175427 


5.700366 


.9849589 


3 


58 


.1558598 


.157788 


6.337612 


.9877792 


2 


58 


.1730752 


.175727 


5.690639 


.9849086 


2 


59 


.1561472 


.158086 


6.325660 


.9877338 


1 


59 


.1733617 


.176027 


5.680944 


.9848582 


1 


60 


.1564345 


.158384 


6.313751 


.9876883 





60 


.1736482 


.176327 


5.671281 


.9848078 





1 Cosine. 


CotangI Tang. | Sine. 


1 ' II 1 Cosine. 


1 CotangI Tang. | Sine. 


' 




Note.- 


-Secan 


t =1-^C 


:osine. 


81° 
C 


Jose 


cant=lH 


hsine. 






80° 



NATURAL SINES, ETC. 



149 



10^ 



3.— Natural Sines, Tangents, Cotangents, Cosines. — (Continued.) 

(Versed sine = 1 — cosine ; co versed sine = 1 — sine) . 
11° 



' 1 


Sine. 1 Tang. | 


Cotang.l Cosine. 


1 


' 1 


Sine. 1 Tang. | Cotang.l Cosine. 







.1736482 


.176327 5.671281 


.9848078 


60 





.1908090 


.194380 


5.144554 


.9816272 


60 


1 


.1739346 


.176626 5.661650 


.9847572 


59 


1 


.1910945 


.194682 


5.136576 


.9815716 


59 


2 


.1742211 


.176926 5.652051 


.9847066 


58 


2 


.1913801 


.194984 


5.128622 


.9815160 


58 


3 


1745075 


.177226 


5.642483 


.9846558 


57 


3 


.1916656 


.195286 


5.120692 


.9814603 


57 


4 


.1747939 


.177527 


5.632947 


.9846050 


56 


4 


.1919510 


.195588 


5.112785 


.9814045 


56 


5 


.1750803 


.177827 


5.623442 


.9845542 


55 


5 


.1922365 


.195890 


5.104902 


.9813486 


55 


6 


.1753667 


.178127 


5.613968 


.9845032 


54 


6 


.1925220 


.196192 


5.097042 


.9812927 


54 


7 


.1756531 


.178427 


5.604524 


.9844521 


53 


7 


.1928074 


.196494 


5.089206 


.9812366 


53 


8 


.1759395 


.178727 


5.595112 


.9844010 


52 


8 


.1930928 


.196796 


5.081392 


.9811805 


52 


9 


.1762258 


.179027 


5.585730 


.9843498 


51 


9 


.1933782 


.197098 


5.073602 


.9811243 


51 


10 


.1765121 


.179327 


5.576378 


.9842985 


50 


10 


.1936636 


.197400 


5.065835 


.9810680 


50 


11 


.1767984 


.179628 


5.567057 


.9842471 


49 


11 


.1939490 


.197703 


5.058090 


.9810116 


49 


12 


.1770847 


.179928 


5.557766 


.9841956 


48 


12 


.1942344 


.198005 


5.050369 


.9809552 


48 


13 


.1773710 


.180228 


5.548505 


.9841441 


47 


13 


.1945197 


.198307 


5.042670 


.9008986 


47 


14 


.1776573 


.180529 


5.539274 


.9840924 


46 


14 


.1948050 


.198610 


5.034993 


.9808420 


46 


15 


.1779435 


.180829 


5.530072 


.9840407 


45 


15 


.1950903 


.198912 


5.027339 


.9807853 


45 


16 


.1782298 


.181129 


5.520900 


.9839889 


44 


16 


.1953756 


.199214 


5.019707 


.9807285 


44 


17 


.1785160 


.181430 


5.511757 


.9839370 


43 


17 


.1956609 


.199517 


5.012098 


.9806716 


43 


18 


.1788022 


.181730 


5.502644 


.9838850 


42 


18 


.1959461 


.199819 


5.004511 


.9806147 


42 


19 


.1790884 


.182031 


5.493560 


.9838330 


41 


19 


.1962314 


.200122 


4.996945 


.9805576 


41 


20 


.1793746 


.182331 


5.484505 


.9837808 


40 


20 


.1965166 


.200424 


4.989402 


.9805005 


40 


21 


.1796607 


.182632 


5.475478 


.9837286 


39 


21 


.1968018 


.200727 


4.981881 


.9804433 


89 


22 


.1799469 


.182933 


5.466481 


.9836763 


38 


22 


.1970870 


.201030 


4.974381 


.9803860 


38 


23 


.1802330 


.183233 


5.457512 


.9836239 


37 


23 


.1973722 


.201332 


4.966903 


.9803286 


37 


24 


.1805191 


.183534 


5.448571 


.9835715 


36 


24 


.1976573 


.201635 


4.959447 


.9802712 


36 


25 


.1808052 


.183835 


5.439659 


.9835189 


35 


25 


.1979425 


.201938 


4.952012 


.9802136 


35 


26 


.1810913 


.184135 


5.430775 


.9834663 


34 


26 


.1982276 


.202240 


4.944599 


.9801560 


34 


27 


.1813774 


.184436 


5.421918 


.9834136 


33 


27 


.1985127 


.202543 


4.937206 


.9800983 


33 


28 


.1816635 


.184737 


5.413090 


.9833608 


32 


28 


.1987978 


.202846 


4.929835 


.9800405 


32 


29 


.1819495 


.185038 


5.404290 


.9833079 


31 


29 


.1990829 


.203149 


4.922485 


.9799827 


31 


30 


.1822355 


.185339 


5.395517 


.9832549 


30 


30 


.1993679 


.203452 


4.915157 


.9799247 


30 


31 


.1825215 


.185639 


5.386771 


.9832019 


29 


31 


.1996530 


.203755 


4.907849 


.9798667 


29 


32 


.1828075 


.185940 


5.378053 


.9831487 


28 


32 


.1999380 


.204058 


4.900562 


.9798086 


28 


33 


.1830935 


.186241 


5.369363 


.9830955 


27 


33 


.2002230 


.204361 


4.893295 


.9797504 


27 


34 


.1833795 


.186542 


5.360699 


.9830422. 


26 


34 


.2005080 


.204664 


4.886049 


.9796921 


26 


35 


.1836654 


.186843 


5.352062 


.9829888 


25 


35 


.2007930 


.204967 


4.878824 


.9796337 


25 


36 


.1839514 


.187144 


5.343452 


.9829353 


24 


36 


.2010779 


.205270 


4.871620 


.9795752 


24 


37 


.1842373 


.187446 


5.334869 


.9828818 


23 


37 


.2013629 


.205573 


4.864435 


.9795167 


23 


38 


.1845232 


.187747 


5.326313 


.9828282 


22 


38 


.2016478 


.205876 


4.857271 


.9794581 


22 


39 


.1848091 


: 188048 


5.317783 


.9827744 


21 


39 


.2019327 


.206180 


4.850128 


.9793994 


21 


40 


.1850949 


.188349 


5.309279 


.9827206 


20 


40 


.2022176 


.206483 


4.843004 


.9793406 


20 


41 


.1853808 


.188650 


5.300801 


.9826668 


19 


41 


.2025024 


.206786 


4.835901 


.9792818 


19 


42 


.1856666 


.188952 


5.292350 


.9826128 


18 


42 


.2027873 


.207090 


4.828817 


.9792228 


18 


43 


.1859524 


.189253 


5.283925 


.9825587 


17 


43 


.2030721 


.207393 


4.821753 


.9791638 


17 


44 


.1862382 


.189554 


5.275525 


.9825046 


16 


44 


.2033569 


.207696 


4.814709 


.9791047 


16 


45 


.1865240 


.189855 


5.267151 


.9824504 


15 


45 


.2036418 


.208000 


4.807685 


.9790455 


15 


46 


.1868098 


.190157 


5.258803 


.9823961 


14 


46 


.2039265 


.208303 


4.800680 


.9789862 


14 


47 


.1870956 


.190458 


5.250480 


.9823417 


13 


47 


.2042113 


.208607 


4.793695 


.9789268 


13 


48 


.1873813 


.190760 


5.242183 


.9822873 


12 


48 


.2044961 


.208910 


4.786730 


.9788674 


12 


49 


.1876670 


.191061 


5.233911 


.9822327 


11 


49 


.2047808 


.209214 


4.779783 


.9788079 


11 


50 


.1879528 


.191363 


5.225664 


.9821781 


10 


50 


.2050655 


.209518 


4.772856 


.9787483 


10 


51 


.1882385 


.191664 


5.217442 


.9821234 


9 


51 


.2053502 


.209821 


4.765949 


.9786886 


9 


52 


.1885241 


.191966 


5.209245 


.9820686 


8 


52 


.2056349 


.210125 


4.759060 


.9786288 


8 


53 


.1880898 


.192268 


5.201073 


.9820137 


7 


53 


.2059195 


.210429 


4.752190 


.9785689 


7 


54 


.1890954 


.192569 


5.192926 


.9819587 


6 


54 


.2062042 


.210733 


4.745340 


.9785090 


6 


55 


.1893811 


.192871 


5.184803 


.9819037 


5 


55 


.2064888 


.211036 


4.738508 


.9784490 


5 


56 


.1896667 


.193173 


5.176705 


.9818485 


4 


56 


.2067734 


.211340 


4.731695 


.9783889 


4 


57 


.1899523 


.193474 


5.168631 


.9817933 


3 


57 


.2070580 


.211644 


4.724901 


.9783287 


3 


68 


.1902379 


.193776 


5.160581 


.9817380 


2 


58 


.2073426 


.211948 


4.718125 


.9782684 


2 


59 


.1905234 


.194078 


5.152555 


.9816826 


1 


59 


.2076272 


.212252 


4.711368 


.9782080 




60 


.1908090 


.194380 


5.144554 


.9816272 





60 


.2079117 


.212556 


4.704630 


.9781476 






79° 78° 

Note. — Secant = 1 -^ cosine. Cosecant = 1 -»-sine. 



150 



^.— PLANE TRIGONOMETRY. 



3.— Natural Sines, Tangents, Cotangents, Cosines. — (Continued. 



12° 




(Versed sine 


= 1 — cosine ; 


co\ 
13° 


'■ersed sine = 1 — 


sine.) 




1 


' 1 Sine. 1 Tang. |Cotang.| Cosine. | || ' | Sine. | Tang. |Cotang.| Cosine. | 





.2079117 


.212556 


4.704630 


.9781476 


60 





.2249511 


.230868 


4.331475 


.9743701 


60 


1 


.208x962 


.212860 


4.697910 


.9780871 


59 


1 


.2252345 


.231174 


4.325734 


.9743046 


59 


2 


.2084807 


.213164 


4.691208 


.9780265 


58 


2 


.2255179 


.231481 


4.320007 


.9742390 


58 


3 


.2087652 


.213468 


4.684524 


.9779658 


57 


3 


.2258013 


.231787 


4.314295 


.9741734 


57 


4 


.2090497 


.213773 


4.677859 


.9779050 


56 


4 


.2260846 


.232094 


4.308597 


.9741077 


56 


5 


.2093341 


.214077 


4.671212 


.9778441 


55 


5 


.2263680 


.232400 


4.302913 


.9740419 


55 


6 


.2096186 


.214381 


4.664583 


.9777832 


54 


6 


.2266513 


.232707 


4.297244 


.9739760 


54 


7 


.2099030 


.214685 


4.657972 


.9777222 


53 


7 


.2269346 


.233014 


4.291588 


.9739100 


53 


8 


.2101874 


.214990 


4.651378 


.9776611 


52 


8 


.2272179 


.233320 


4.285947 


.9738439 


52 


9 


.2104718 


.215294 


4.644803 


.9775999 


51 


9 


.2275012 


.233627 


4.280319 


.9737778 


51 


10 


.2107561 


.215598 


4.638245 


.9775386 


50 


10 


.2277844 


.233934 


4.274706 


.9737116 


50 


11 


.2110405 


.215903 


4.631705 


.9774773 


49 


11 


.2280677 


.234241 


4.269107 


.9736453 


49 


12 


.2113248 


.216207 


4.625183 


.9774159 


48 


12 


.2283509 


.234547 


4.263521 


.9735789 


48 


13 


.2116091 


.216512 


4.618678 


.9773544 


47 


13 


.2286341 


.234854 


4.257950 


.9735124 


47 


14 


.2118934 


.216816 


4.612190 


.9772928 


46 


14 


.2289172 


.235161 


4.252392 


.9734458 


46 


15 


.2121777 


.217121 


4.605720 


.9772311 


45 


15 


.2292004 


.235468 


4.246848 


.9733792 


45 


16 


.2124619 


.217425 


4.599268 


.9771693 


44 


16 


.2294835 


.235775 


4.241317 


.9733125 


44 


17 


.2127462 


.217730 


4.592832 


.9771075 


43 


17 


.2297666 


.236082 


4.235800 


.9732457 


43 


18 


.2130304 


.218035 


4.586414 


.9770456 


42 


18 


.2300497 


.236390 


4.230297 


.9731789 


42 


19 


.2133146 


.218340 


4.580012 


.9769836 


41 


19 


.2303328 


.236697 


4.224808 


.9731119 


41 


20 


.2135988 


.218644 


4.573628 


.9769215 


40 


20 


.2306159 


.237004 


4.219331 


.9730449 


40 


21 


.2138829 


.218949 


4.567261 


.9768593 


39 


21 


.2308989 


.237311 


4.213869 


.9729777 


39 


22 


.2141671 


.219254 


4.560911 


.9767970 


38 


22 


.2311819 


.237618 


4.208419 


.9729105 


38 


23 


.2144512 


.219559 


4.554577 


.9767347 


37 


23 


.2314649 


.237926 


4.202983 


.9728432 


37 


24 


.2147353 


.219864 


4.548260 


.9766723 


36 


24 


.2317479 


.238233 


4.197560 


.9727759 


36 


25 


.2150194 


.220169 


4.541960 


.9766098 


35 


25 


.2320309 


.238541 


4.192151 


.9727084 


35 


26 


.2153035 


.220474 


4.535677 


.9765472 


34 


26 


.2323138 


.238848 


4.186754 


.9726409 


34 


27 


.2155876 


.220779 


4.529410 


.9764845 


33 


27 


.2325967 


.239156 


4.181371 


.9725733 


33 


28 


.2158716 


.221084 


4.523160 


.9764217 


32 


28 


.2328796 


.239463 


4.176001 


.9725056 


32 


29 


.2161556 


.221389 


4.516926 


.9763589 


31 


29 


.2331625 


.239771 


4.170644 


.9724378 


31 


30 


.2164396 


.221694 


4.510708 


.9762960 


30 


30 


.2334454 


.240078 


4.165299 


.9723699 


30 


31 


.2167236 


.221999 


4.504507 


.9762330 


29 


31 


.2337282 


.240386 


4.159968 


.9723020 


29 


32 


.2170076 


.222305 


4.498322 


.9761699 


28 


32 


.2340110 


.240694 


4.154650 


.9722339 


28 


33 


.2172915 


.222610 


4.492153 


.9761067 


27 


33 


.2342938 


.241001 


4.149344 


.9721658 


27 


34 


.2175754 


.222915 


4.486000 


.9760435 


26 


34 


.2345766 


.241309 


4.144051 


.9720976 


26 


35 


.2178593 


.223221 


4.479863 


.9759802 


25 


35 


.2348594 


.241617 


4.138771 


.9720294 


25 


36 


.2181432 


.223526 


4.473742 


.9759168 


24 


36 


.3351421 


.241925 


4.133504 


.9719610 


24 


37 


.2184271 


.223831 


4.467637 


.9758533 


23 


37 


.2354248 


.242233 


4.128249 


.9718926 


23 


38 


.2187110 


.224137 


4.461548 


.9757897 


22 


38 


.2357075 


.242541 


4.123007 


.9718240 


22 


39 


.2189948 


.224442 


4.455475 


.9757260 


21 


38 


.2359902 


.242849 


4.117778 


.9717554 


21 


40 


.2192786 


.224748 


4.449418 


.9756623 


20 


40 


.2362729 


.243157 


4.112561 


.9716867 


20 


41 


.2195624 


.225054 


4.443376 


.9755985 


19 


41 


.2365555 


.243465 


4.107356 


.9716180 


19 


42 


.2198462 


.225359 


4.437350 


.9755345 


18 


42 


.2368381 


.243773 


4.102164 


.9715491 


18 


43 


.2201300 


.225665 


4.431339 


.9754706 


17 


43 


.2371207 


.244081 


4.096985 


.9714802 


17 


44 


.2204137 


.225971 


4.425343 


.9754065 


16 


44 


.2374033 


.244390 


4.091817 


.9714112 


16 


45 


.2206974 


.226276 


4.419364 


.9753423 


15 


45 


.2376859 


.244698 


4.086662 


.9713421 


15 


46 


.2209811 


.226582 


4.413399 


.9752781 


14 


46 


.2379684 


.245006 


4.081519 


.9712729 


14 


47 


.2212648 


.226888 


4.407450 


.9752138 


13 


47 


.2382510 


.245315 


4.076389 


.9712036 


13 


48 


.2215485 


.227194 


4.401516 


.9751494 


12 


48 


.2385335 


.245623 


4.071270 


.9711343 


12 


49 


.2218321 


.227500 


4.395597 


.9750849 


11 


49 


.2388159 


.245932 


4.066164 


.9710649 


11 


50 


.2221158 


.227806 


4.389694 


.9750203 


10 


50 


.2390984 


.246240 


4.061070 


.9709953 


10 


51 


.2223994 


.228112 


4.383805 


.9749556 


9 


51 


.2393808 


.246549 


4.055987 


.9709258 


9 


52 


.2226830 


.228418 


4.377931 


.9748909 


8 


52 


.2396633 


.246857 


4.050917 


.9708561 


8 


53 


.2229666 


.228724 


4.372073 


.9748261 


7 


53 


.2399457 


.247166 


4.045859 


.9707863 


7 


54 


.2232501 


.229030 


4.366229 


.9747612 


6 


54 


.2402280 


.247475 


4.040812 


.9707165 


6 


55 


.2235337 


.229336 


4.360400 


.9746962 


5 


55 


.2405104 


.247783 


4.035777 


.9706466 


5 


56 


.2238172 


.229642 


4.354586 


.9746311 


4 


56 


.2407927 


.248092 


4.030755 


.9705766 


4 


57 


.2241007 


.229949 


4.348786 


.9745660 


3 


57 


.2410751 


.248401 


4.025744 


.9705065 


3 


58 


.2243842 


.230255 


4.343001 


.9745008 


2 


58 


.2413574 


.248710 


4.020744 


.9704363 


2 


5S 


.2246676 


.230561 


4.337231 


.9744355 


1 


59 


.2416396 


.249019 


4.015757 


.9703660 


1 


6(1 


.2249511 


.230868 


4.331475 


.9743701 





60 


.2419219 


.249328 


4.010780 


.9702957 





1 Cosine. ICotangl Tang. | Sine. | ' |l 1 Cosine. 


ICotangl Tang. | Sine. | ' 



77° 76* 

Note. — Secant — 1 -•- cosine. Cosecant = 1 -«-sine. 



NATURAL SINES, ETC. 



151 



8.— Natural Sines, Tangents, Cotangents, Cosines. — (Continued.) 

(Versed sine =1 — cosine; co versed sine=l — sine.) 
14° 150 



# 


Sine. 


Tang. ICotang.l Cosine. 


1 


1 ' 


1 Sine. 


1 Tang. 1 Cotang.l Cosine. 


1 





.2419219 


.249328 


4.010780 


.9702957 


60 





.2588190 


.267949 


3.732050 


.9659258 


60 


1 


.2422041 


.249637 


4.005816 


.9702253 


59 


1 


.2591000 


.268261 


3.727713 


.9658505 


59 


2 


.2424863 


.249946 


4.000863 


.9701548 


58 


2 


.2593810 


.268572 


3.723384 


.9657751 


58 


3 


.2427685 


.250255 


3.995922 


.9700842 


57 


3 


.2596619 


.268884 


3.719065 


.9656996 


57 


4 


.2430507 


.250564 


3.990992 


.9700135 


56 


4 


.2599428 


.269196 


3.714756 


.9656240 


56 


5 


.2433329 


.250873 


3.986073 


.9699428 


55 


5 


.2602237 


.269508 


3.710455 


.9655484 


55 


6 


.2436150 


.251182 


3.981166 


.9698720 


54 


6 


.2605045 


.269820 


3.706164 


.9654726 


54 


7 


.2438971 


.251491 


3.976271 


.9698011 


53 


7 


.2607853 


.270132 


3.701883 


.9653968 


53 


8 


.2441792 


.251801 


3.971386 


.9697301 


52 


8 


.2610662 


.270444 


3.697610 


.9653209 


52 


9 


.2444613 


.252110 


3.966513 


.9696591 


51 


9 


.2613469 


.270757 


3.693346 


.9652449 


51 


10 


.2447433 


.252420 


3.961651 


.9695879 


50 


10 


.2616277 


.271069 


3.689092 


.9651689 


50 


11 


.2450254 


.252729 


3.956801 


.9695167 


49 


11 


.2619085 


.271381 


3.684847 


.9650927 


49 


12 


.2453074 


.253038 


3.951961 


.9694453 


48 


12 


.2621892 


.271694 


3.680611 


.9650165 


48 


13 


.2455894 


.253348 


3.947133 


.9693740 


47 


13 


.2624699 


.272006 


3.676384 


.9649402 


47 


14 


.2458713 


.253658 


3.942315 


.9693025 


46 


14 


.2627506 


.272318 


3.672166 


.9648638 


46 


15 


.2461533 


.253967 


3.937509 


.9692309 


45 


15 


.2630312 


.272631 


3.667957 


.9647873 


45 


16 


.2464352 


.254277 


3.932714 


.9691593 


44 


16 


.2633118 


.272943 


3.663757 


.9647108 


44 


17 


.2467171 


.254587 


3.927929 


.9690875 


43 


17 


.2635925 


.273256 


3.659566 


.9646341 


43 


18 


.2469990 


.254896 


3.923156 


.9690157 


42 


18 


.2638730 


.273569 


3.655384 


.9645574 


42 


19 


.2472809 


.255206 


3.918393 


.9689438 


41 


19 


.2641536 


.273881 


3.651211 


.9644806 


41 


20 


.2475627 


.255516 


3.913642 


.9688719 


40 


20 


.2644342 


.274194 


3.647046 


.9644037 


40 


21 


.2478445 


.255826 


3.908901 


.9687998 


39 


21 


.2647147 


.274507 


3.642891 


.9643268 


39 


22 


.2481263 


.256136 


3.904171 


.9687277 


38 


22 


.2649952 


.274820 


3.638744 


.9642497 


38 


23 


.2484081 


.256446 


3.899451 


.9686555 


37 


23 


.2652757 


.275133 


3.634606 


.9641726 


37 


24 


.2486899 


.256756 


3.894742 


.9685832 


36 


24 


.2655561 


.275445 


3.630477 


.9640954 


36 


25 


.2489716 


.257066 


3.890044 


.9685108 


35 


25 


.2658366 


.275758 


3.626356 


.9640181 


35 


26 


.2492533 


.257376 


3.885357 


.9684383 


34 


26 


.2661170 


.276071 


3.622244 


.9639407 


34 


27 


.2495350 


.257686 


3.880680 


.9683658 


33 


27 


.2663973 


.276385 


3.618141 


.9638633 


33 


28 


.2498167 


.257997 


3.876014 


.9682931 


32 


28 


.2666777 


.276698 


3.614046 


.9637858 


32 


29 


.2500984 


.258307 


3.871358 


.9682204 


31 


29 


.2669581 


.277011 


3.609960 


.9637081 


31 


30 


.2503800 


.258617 


3.866713 


.9681476 


30 


30 


.2672384 


.277324 


3.605883 


.9636305 


30 


31 


.2506616 


.258928 


3.862078 


.9680748 


29 


31 


.2677187 


.277637 


3.601814 


.9635527 


29 


32 


.2509432 


.259238 


3.857453 


.9680018 


28 


32 


.2677989 


.277951 


3.597754 


.9634748 


28 


33 


.2512248 


.259548 


3.852839 


.9679288 


27 


33 


.2680792 


.278264 


3.593702 


.9633969 


27 


34 


.2515063 


.259859 


3.848235 


.9678557 


26 


34 


.2683594 


.278578 


3.589659 


.9633189 


26 


35 


.2517879 


.260169 


3.843642 


.9677825 


25 


35 


.2686396 


.278891 


3.585624 


.9632408 


25 


36 


.2520694 


.260480 


3.839059 


.9677092 


24 


36 


.2689198 


.279205 


3.581597 


.9631626 


24 


37 


.2523508 


.260791 


3.834486 


.9676358 


23 


37 


.2692000 


.279518 


3.577579 


.9630843 


23 


38 


.2526323 


.261101 


3.829923 


.9675624 


22 


38 


.2694801 


.279832 


3.573569 


.9630060 


22 


39 


.2529137 


.261412 


3.825370 


.9674888 


21 


39 


.2697602 


.280145 


3.569568 


.9629275 


21 


40 


.2531952 


.261723 


3.820828 


.9674152 


20 


40 


.2700403 


.280459 


3.565574 


.9628490 


20 


41 


.2534766 


.262034 


3.816295 


.9673415 


19 


41 


.2703204 


.280773 


3.561590 


.9627704 


19 


42 


.2537579 


.262345 


3.811773 


.9672678 


18 


42 


.2706004 


.281087 


3.557613 


.9626917 


18 


43 


.2540393 


.262656 


3.807260 


.9671939 


17 


43 


.2708805 


.281401 


3.553644 


.9626130 


17 


44 


.2543206 


.262967 


3.802758 


.9671200 


16 


44 


.2711605 


.281715 


3.549684 


.9625342 


16 


45 


.2546019 


.263278 


3.798266 


.9670459 


15 


45 


.2714404 


.282029 


3.545732 


.9624552 


15 


46 


.2548832 


.263589 


3.793783 


.9669718 


14 


46 


.2717204 


.282343 


3.541788 


.9623762 


14 


47 


.2551645 


.263900 


3.789310 


.9668977 


13 


47 


.2720003 


.282657 


3.537852 


.9622972 


13 


48 


.2554458 


.264211 


3.784848 


.9668234 


12 


48 


.2722802 


.282971 


3.533925 


.9622180 


12 


49 


.2557270 


.264522 


3.780395 


.9667490 


11 


49 


.2725601 


.283285 


3.530005 


.9621387 




50 


.2560082 


.264833 


3.775951 


.9666746 


10 


50 


.2728400 


.283599 


3.526093 


.9620594 


10 


51 


.2562894 


.265145 


3.771518 


.9666001 


9 


51 


.2731198 


.283914 


3.522190 


.9619800 


9 


52 


.2565705 


.265456 


3.767094 


.9665255 


8 


52 


.2733997 


.284228 


3.518294 


.9619005 


8 


53 


.2568517 


.265768 


3.762680 


.9664508 


7 


53 


.2736794 


.284543 


3.514407 


.9618210 


7 


54 


.2571328 


.266079 


3.758276 


.9663761 


6 


54 


.2739592 


.284857 


3.510527 


.9617413 


6 


55 


.2574139 


.266390 


3.753881 


.9663012 


5 


55 


.2742390 


.285172 


3.506655 


.9616616 


5 


56 


.2576950 


.266702 


3.749496 


.9662263 


4 


56 


.2745187 


.285486 


3.502791 


.9615818 


4 


57 


.2579760 


.267014 


3.745120 


.9661513 


3 


57 


.2747984 


.285801 


3.498935 


.9615019 


3 


58 


.2582570 


.267325 


3.740754 


.9660762 


2 


58 


.2750781 


.286115 


3.495087 


.9614219 


2 


59 


.2585381 


.267637 


3.736398 


.9660011 


1 


59 


.2753577 


.286430 


3.491247 


.9613418 


1 


60 


.2588190 


.267949 


3.732050 


.9659258 





60 


.2756374 


.286745 


3.487414 


.9612617 






I Cosine. ICotangI Tang. | Sine. \ ' \\ | Cosine. ICotangl Tang. | Sine. 



Note. — Secant = 1 -f- cosine. 



75° 

Cosecant = 1-^sine. 



74= 



162 



^.— PLANE TRIGONOMETRY. 



16° 



3.— Natural Sines, Tangents, Cotangents, Cosines. — (Continued.) 

(Versed sine =1 — cosine; coversed sine=l— sine.) 
17° 



/ 


Sine. 


Tang. 


1 Cotang.l Cosine. 


1 


1 ' 


Sine. 


Tang. 


Cotang.l Cosine. 







.2756374 


.286745 


3.487414 


.9612617 


60 





.2923717 


.305730 


3.270852 


.9563048 


60 


1 


.2759170 


.287060 


3.483589 


.9611815 


59 


1 


.2926499 


.306048 


3.267452 


.9562197 


59 


2 


.2761965 


.287375 


3.479772 


.9611012 


58 


2 


.2929280 


.306367 


3.264059 


.9561345 


58 


3 


.2764761 


.287690 


3.475963 


.9610208 


57 


3 


.2932061 


.306685 


3.260672 


.9560492 


57 


4 


.2767556 


.288005 


3.472161 


.9609403 


56 


4 


.2934842 


.307003 


3.257292 


.9559639 


56 


5 


.2770352 


.288320 


3.468367 


.9608598 


55 


5 


.2937623 


.307321 


3.253918 


.9558785 


55 


6 


.2773147 


.288635 


3.464581 


.9607792 


54 


6 


.2940403 


.307640 


3.250550 


.9557930 


54 


7 


.2775941 


.288950 


3.460802 


.9606984 


53 


7 


.2943183 


.307958 


3.247189 


.9557074 


53 


8 


.2778736 


.289265 


3.457031 


.9606177 


52 


8 


.2945963 


.308277 


3.243834 


.9556218 


52 


9 


.2781530 


.289580 


3.453267 


.9605368 


51 


9 


.2948743 


.308595 


3.240486 


.9555361 


51 


10 


.2784324 


.289896 


3.449512 


.9604558 


50 


10 


.2951522 


.308914 


3.237143 


.9554502 


50 


11 


.2787118 


.290211 


3.445763 


.9603748 


49 


11 


.2954302 


.309233 


3.233807 


.9553643 


49 


12 


.2789911 


.290526 


3.442022 


.9602937 


48 


12 


.2957081 


.309551 


3.230478 


.9552784 


48 


13 


.2792704 


.290842 


3.438289 


.9602125 


47 


13 


.2959859 


.309870 


3.227154 


.9551923 


47 


14 


.2795497 


.291157 


3.434563 


.9601312 


46 


14 


.2962638 


.310189 


3.223837 


.9551062 


46 


15 


.2798290 


.291473 


3.430844 


.9600499 


45 


15 


.2965416 


.310508 


3.220526 


.9550199 


45 


16 


.2801083 


.291789 


3.427133 


.9599684 


44 


16 


.2968194 


.310827 


3.217221 


.9549336 


44 


17 


.2803875 


.292104 


3.423429 


.9598869 


43 


17 


.2970971 


.311146 


3.213922 


.9548473 


43 


18 


.2806667 


.292420 


3.41973S 


.9598053 


42 


18 


.2973749 


.311465 


3.210630 


.9547608 


42 


19 


.2809459 


.292736 


3.416044 


.9597236 


41 


19 


.2976526 


.311784 


3.207344 


.9546743 


41 


20 


.2812251 


.293052 


3.412362 


.9596418 


40 


20 


.2979303 


.312103 


3.204063 


.9545876 


40 


21 


.2815042 


.293368 


3.408688 


.9595600 


39 


21 


.2982079 


.312422 


3.200789 


.9545009 


39 


22 


.2817833 


.293683 


3.405021 


.9594781 


38 


22 


.2984856 


.312742 


3.197521 


.9544141 


38 


23 


.2820624 


.293999 


3.401361 


.9593961 


37 


23 


.2987632 


.313061 


3.194259 


.9543273 


37 


24 


.2823415 


.294316 


3.397708 


.9593140 


36 


24 


.2990408 


.313381 


3.191003 


.9542403 


36 


25 


.2826205 


.294632 


3.394063 


.9592318 


35 


25 


.2993184 


.313700 


3.187754 


.9541533 


35 


26 


.2828995 


.294948 


3.390424 


.9591496 


34 


26 


.2995959 


.314020 


3.184510 


.9540662 


34 


27 


.2831785 


.295264 


3.386793 


.9590672 


33 


27 


.2998734 


.314339 


3.181272 


.9539790 


33 


28 


.2834575 


.295580 


3.383169 


.9589848 


32 


28 


.3001509 


.314659 


3.178040 


.9538917 


32 


29 


.2837364 


.295897 


3.379553 


.9589023 


31 


29 


.3004284 


.314979 


3.174814 


.9538044 


31 


30 


.2840153 


.296213 


3.375943 


.9588197 


30 


30 


.3007058 


.315298 


3.171594 


.9537170 


30 


31 


.2842942 


.296529 


3.372340 


.9587371 


29 


31 


.3009832 


.315618 


3.168380 


.9536294 


29 


32 


.2845731 


.296846 


3.368745 


.9586543 


28 


32 


.3012606 


.315938 


3.165172 


.9535418 


28 


33 


.2848520 


.297163 


3.365156 


.9585715 


27 


33 


.3015380 


.316258 


3.161970 


.9534542 


27 


34 


.2851308 


.297479 


3.361575 


.9584886 


26 


34 


.3018153 


.316578 


3.158774 


.9533664 


26 


35 


.2854096 


.297796 


3.358000 


.9584056 


25 


35 


.3020926 


.316898 


3.155584 


.9532786 


25 


36 


.2856884 


.298112 


3.354433 


.9583226 


24 


36 


.3023699 


.317218 


3.152399 


.9531907 


24 


37 


.2859671 


.298429 


3.350872 


.9582394 


23 


37 


.3026471 


.317538 


3.149220 


.9531027 


23 


38 


.2862458 


.298746 


3.347319 


.9581562 


22 


38 


.3029244 


.317859 


3.146047 


.9530146 


22 


39 


.2865246 


.299063 


3.343772 


.9580729 


21 


39 


.3032016 


.318179 


3.142880 


.9529264 


21 


40 


.2868032 


.299380 


3.340232 


.9579895 


20 


40 


.3034788 


.318499 


3.139719 


.9528382 


20 


41 


.2870819 


.299697 


3.336699 


.9579060 


19 


41 


.3037559 


.318820 


3.136563 


.9527499 


19 


42 


.2873605 


.300014 


3.333173 


.9578225 


18 


42 


.3040331 


.319140 


3.133414 


.9526615 


18 


43 


.2876391 


.300331 


3.329654 


.9577389 


17 


43 


.3043102 


.319461 


3.130270 


.9525730 


17 


44 


.2879177 


.300648 


3.326141 


.9576552 


16 


44 


.3045872 


.319781 


3.127131 


.9524844 


16 


45 


.2881963 


.300965 


3.322636 


.9575714 


15 


45 


.3048643 


.320102 


3.123999 


.9523958 


15 


46 


.2884748 


.301283 


3.319137 


.9574875 


14 


46 


.3051413 


.320423 


3.120872 


.9523071 


14 


47 


.2887533 


.301600 


3.315645 


.9574035 


13 


47 


.3054183 


.320744 


3.117750 


.9522183 


13 


48 


.2890318 


.301917 


3.312159 


.9573195 


12 


48 


.3056953 


.321064 


3.114635 


.9521294 


12 


49 


.2893103 


.302235 


3.308681 


.9572354 


11 


49 


.3059723 


.321385 


3.111525 


.9520404 


11 


50 


.2895887 


.302552 


3.305209 


.9571512 


10 


50 


.3062492 


.321706 


3.108421 


.9519514 


10 


51 


.2898671 


.302870 


3.310743 


.9570669 


9 


51 


.3065261 


.322027 


3.105322 


.9518623 


9 


52 


.2901455 


.303187 


3.298285 


.9569825 


8 


52 


.3068030 


.322348 


3.102229 


.9517731 


8 


53 


.2904239 


.303505 


3.294833 


.9568981 


7 


53 


.3070798 


.322670 


3.099141 


.9516838 


7 


54 


.2907022 


.303823 


3.291387 


.9568136 


6 


54 


.3073566 


.322991 


3.096059 


.9515944 


6 


55 


.2909805 


.304141 


3.287948 


.9567290 


5 


55 


.3076334 


.323312 


3.092983 


.9515050 


5 


56 


.2912588 


.304458 


3.284516 


.9566443 


4 


56 


.3079102 


.323633 


3.089912 


.9514154 


4 


57 


.2915371 


.304776 


3.281090 


.9565595 


3 


57 


.3081869 


.323955 


3.086846 


.9513258 


3 


58 


.2918153 


.305094 


3.277671 


.9564747 


2 


58 


.3084636 


.324276 


3.083786 


.9512361 


2 


59 


.2920935 


.305412 


3.274258 


.9563898 


1 


59 


.3087403 


.324598 


3.080732 


.9511464 


1 


60 


.2923717 


.305730 


3.270852 


.9563048 





60 


.3090170 


.324919 


3.077683 


.9510565 





1 Cosine. 


Cotangl Tang. 


Sine. 


1 ' II 1 Cosine. 


Cotangl Tang. | Sine. 


' 



Note. — Secant =l-»- cosine. 



73° 

Cosecant = iH-sine. 



72° 



NATURAL SINES, ETC, 



153 



3.— Natural Sines, Tangents, Cotangents, Cosines. — (Continued.) 

(Versed sine =1 — cosine; coversed sine = 1 — sine.) 
19° 



' 


1 Sine. 


Tang. ICotang.l Cosine. 


1 


1 ' 


1 Sine. 


1 Tang. 


ICotang.l Cosine. 


1 





.3090170 


.324919 


3.077683 


.9510565 


60 





.3255682 


.344327 


2.904210 


.9455186 


60 


1 


.3092936 


.325241 


3.074640 


.9509666 


59 


1 


.3258432 


.344653 


2.901468 


.9454238 


59 


2 


.3095702 


.325563 


3.071602 


.9508766 


58 


2 


.3261182 


.344978 


2.898731 


.9453290 


58 


3 


.3098468 


.325884 


3.068569 


.9507865 


57 


3 


.3263932 


.345304 


2.895998 


.9452341 


57 


4 


.3101234 


.326206 


3.065542 


.9506963 


56 


4 


.3266681 


.345629 


2.893270 


.9451391 


56 


5 


.3103999 


.326528 


3.062520 


.9506061 


55 


5 


.3269430 


.345955 


2.890546 


.9450441 


55 


6 


.3106764 


.326850 


3.059503 


.9505157 


54 


6 


.3272179 


.346281 


2.887827 


.9449489 


54 


7 


.3109529 


.327172 


3.056492 


.9504253 


53 


7 


.3274928 


.346606 


2.885113 


.9448537 


53 


8 


.3112294 


.327494 


3.053487 


.9503348 


52 


8 


.3277676 


.346932 


2.882403 


.9447584 


52 


9 


.3115058 


.327816 


3.050486 


.9502443 


51 


9 


.3280424 


.347258 


2.879697 


.9446630 


51 


10 


.3117822 


.328138 


3.047491 


.9501536 


50 


10 


.3283172 


.347584 


2.876997 


.9445675 


50 


11 


.3120586 


.328461 


3.044501 


.9500629 


49 


11 


.3285919 


.347910 


2.874300 


.9444720 


49 


12 


.3123349 


.328783 


3.041517 


.9499721 


48 


12 


.3288666 


.348236 


2.871608 


.9443764 


48 


13 


.3126112 


.329105 


3.038538 


.9498812 


47 


13 


.3291413 


.348563 


2.868921 


.9442807 


47 


14 


.3128875 


.329428 


3.035564 


.9497902 


46 


14 


.3294160 


.348889 


2.866238 


.9441849 


46 


15 


.3131638 


.329750 


3.032595 


.9496991 


45 


15 


.3296906 


.349215 


2.863560 


.9440890 


45 


16 


.3134400 


.330073 


3.029632 


.9496080 


44 


16 


.3299653 


.349542 


2.860886 


.9439931 


44 


17 


.3137163 


.330395 


3.026673 


.9495168 


43 


17 


.3302398 


.349868 


2.858216 


.9438971 


43 


18 


.3139925 


.330718 


3.023720 


.9494255 


42 


18 


.3305144 


.350195 


2.855551 


.9438010 


42 


19 


.3142686 


.331041 


3.020772 


.9493341 


41 


19 


.3307889 


.350521 


2.852891 


.9437048 


41 


20 


.3145448 


.331363 


3.017830 


.9492426 


40 


20 


.3310634 


.350848 


2.850234 


.9436085 


40 


21 


.3148209 


.331686 


3.014892 


.9491511 


39 


21 


.3313379 


.351175 


2.847583 


.9435122 


39 


22 


.3150969 


.332009 


3.011960 


.9490595 


38 


22 


.3316123 


.351501 


2.844935 


.9434157 


38 


23 


.3153730 


.332332 


3.009033 


.9489678 


37 


23 


.3318867 


.351828 


2.842292 


.9433192 


37 


24 


.3156490 


.332655 


3.006110 


.9488760 


36 


24 


.3321611 


.352155 


2.839653 


.9432227 


36 


25 


.3159250 


.332978 


3.003193 


.9487842 


35 


25 


.3324355 


.352482 


2.837019 


.9431260 


35 


26 


.3162010 


.333302 


3.000282 


.9486922 


34 


26 


.3327098 


.352809 


2.834389 


.9430293 


34 


27 


.3164770 


.333625 


2.997375 


.9486002 


33 


27 


.3329841 


.353136 


2.831763 


.9429324 


33 


28 


.3167529 


.333948 


2.994473 


.9485081 


32 


28 


.3332584 


.353464 


2.829142 


.9428355 


32 


29 


.3170288 


.334271 


2.991576 


.9484159 


31 


29 


.3335326 


.353791 


2.826525 


.9427386 


31 


30 


.3173047 


.334595 


2.988685 


.9483237 


30 


30 


.3338069 


.354118 


2.823912 


.9426415 


30 


31 


.3175805 


.334918 


2.985798 


.9482313 


29 


31 


.3340810 


.354446 


2.821304 


.9425444 


29 


32 


.3178563 


.335242 


2.982916 


.9481389 


28 


32 


.3343552 


.354773 


2.818700 


.9424471 


28 


33 


.3181321 


.335566 


2.980040 


.9480464 


27 


33 


.3346293 


.355101 


2.816100 


.9423498 


27 


34 


.3184079 


.335889 


2.977168 


.9479538 


26 


34 


.3349034 


.355428 


2.813504 


.9422525 


26 


35 


.3126836 


.336213 


2.974301 


.9478612 


25 


35 


.3351775 


.355756 


2.810913 


.9421550 


25 


36 


.3189593 


.336537 


2.971439 


.9477684 


24 


36 


.3354516 


.356084 


2.808326 


.9420575 


24 


37 


.3192350 


.336861 


2.968583 


.9476756 


23 


37 


.3357256 


.356411 


2.805743 


.9419598 


23 


38 


.3195106 


.337185 


2.965731 


.9475827 


22 


38 


.3359996 


.356739 


2.803164 


.9418621 


22 


39 


.3197863 


.337509 


2.962884 


.9474897 


21 


39 


.3362735 


.357067 


2.800590 


.9417644 


21 


40 


.3200619 


.337833 


2.960042 


.9473966 


20 


40 


.3365475 


.357395 


2.798019 


.9416665 


20 


41 


.3203374 


.338157 


2.957205 


.9473035 


19 


41 


.3368214 


.357723 


2.795453 


.9415686 


19 


42 


.3206130 


.338481 


2.954372 


.9472103 


18 


42 


.3370953 


.358051 


2.792891 


.9414705 


18 


43 .3208885 | 


.338805 


2.951545 


.9471170 


17 


43 


.3373691 


.358380 


2.790333 


.9413724 


17 


44 


.3211640 


.339129 


2.948722 


.9470236 


16 


44 


.3376429 


.358708 


2.787780 


.9412743 


16 


45 


.3214395 


.339454 


2.945905 


.9469301 


15 


45 


.3379167 


.359036 


2.785230 


.9411760 


15 


46 


.3217149 


.339778 


2.943092 


.9468366 


14 


46 


.3381905 


.359365 


2.782685 


.9410777 


14 


47 


.3219903 


.340103 


2.940284 


.9467430 


13 


47 


.3384642 


.359693 


2.780144 


.9409793 


13 


48 


.3222657 


.340427 


2.937480 


.9466493 


12 


48 


.3387379 


.360022 


2.777606 


.9408808 


12 


49 


.3225411 


.340752 


2.934682 


.9465555 


11 


49 


.3390116 


.360350 


2.775073 


.9407822 


11 


50 


.3228164 


.341077 


2.931888 


.9464616 


10 


50 


.3392852 


.360679 


2.772544 


.9406835 


10 


51 


.3230917 


.341401 


2.929099 


.9463677 


9 


51 


.3395589 


.361008 


2.770019 


.9405848 


9 


52 


.3233670 


.341726 


2.926315 


.9462736 


8 


52 


.3398325 


.361337 


2.767499 


.9404860 


8 


53 


.3236422 


.342051 


2.923535 


.9461795 


7 


53 


.3401060 


.361666 


2.764982 


.9403871 


7 


54 


.3239174 


.342376 


2.920761 


.9460854 


6 


54 


.3403796 


.361994 


2.762469 


.9402881 


6 


55 


.3241926 


.342701 


2.917990 


.9459911 


5 


55 


.3406531 


.362324 


2.759960 


.9401891 


5 


56 


.3244678 


.343026 


2.915225 


.9458968 


4 


56 


.3409265 


.362653 


2.757456 


.9400899 


4 


57 


.3247429 


.343351 


2.912464 


.9458023 


3 


57 


.3412000 


.362982 


2.754955 


.9399907 


3 


58 


.3250180 


.343677 


2.909708 


.9457078 


2 


58 


.3414734 


.363311 


2.752458 


.9398914 


2 


59 


.3252931 


.344002 


2.906957 


.9456132 


1 


59 


.3417468 


.363640 


2.749966 


.9397921 


1 


60 


.3255682 


.344327 


2.904210 


.9455186 





60 


.3420201 


.363970 


2.747477 


.9396926 





iZ 


Cosine. 


CotangI Tang. | Sine. 


'II 1 Cosine. 


1 CotangI Tang. 


Sine. 


' 



7V 



70° 



Note. — Secant = 1 -«- cosine. Cosecant = 1 -&-sine. 



154 



9.— PLANE TRIGONOMETRY, 



20^ 



3.— Natural Sines, Tangents, Cotangents, Cosines. — (Continued.) 

(Versed sine =1 — cosine; coversed sine = 1 — sine.) 
21° 



' 1 Sine. 1 Tang. | Cotang.l Cosine. | || ' | Sine. | Tang. | Cotang.| Cosine. | 





.3420201 


.363970 


2.747477 


.9396926 


60 





.3583679 


.383864 2.605089 


.9335804 


60 


1 


.3422935 


.364299 


2.744992 


.9395931 


59 


1 


.3586395 


.384197 


2.602825 


.9334761 


59 


2 


.3425668 


.364629 


2.742512 


.9394935 


58 


2 


.3589110 


.384531 


2.600565 


.9333718 


58 


3 


.3428400 


.364958 


2.740035 


.9393938 


57 


3 


.3591825 


.384865 


2.598309 


.9332673 


57 


4 


.3431133 


.365288 


2.737562 


.9392940 


56 


4 


.3594540 


.385199 


2.596056 


.9331628 


56 


5 


.3433865 


.365618 


2.735093 


.9391942 


55 


5 


.3597254 


.385533 


2.593806 


.9330582 


55 


6 


.3436597 


.365948 


2.732628 


.9390943 


54 


6 


.3599968 


.385867 


2.591560 


.9329535 


54 


7 


.3439329 


.366277 


2.730167 


.9389943 


53 


7 


.3602682 


.386202 


2.589317 


.9328488 


53 


8 


.3442060 


.366607 


2.727710 


.9388942 


52 


8 


.3605395 


.386536 


2.587078 


.9327439 


52 


9 


.3444791 


.366937 


2.725256 


.9387940 


51 


9 


.3608108 


.386870 


2.584842 


.9326390 


51 


10 


.3447521 


.367268 


2.722807 


.9386938 


50 


10 


.3610821 


.387205 


2.582609 


.9325340 


50 


11 


.3450252 


.367598 


2.720362 


.9385934 


49 


11 


.3613534 


.387539 


2.580380 


.9324290 


49 


12 


.3452982 


.367928 


2.717920 


.9384930 


48 


12 


.3616246 


.387874 


2.578153 


.9323238 


48 


13 


.3455712 


.368258 


2.715482 


.9383925 


47 


13 


.3618958 


.388209 


2.575931 


.9322186 


47 


14 


.3458441 


.368589 


2.713048 


.9382920 


46 


14 


.3621669 


.388543 


2.573711 


.9321133 


46 


15 


.3461171 


.368919 


2.710618 


.9381913 


45 


15 


.3624380 


.388878 


2.571495 


.9320079 


45 


16 


.3463900 


.369250 


2.708192 


.9380906 


44 


16 


.3627091 


.389213 


2.569283 


.9319024 


44 


17 


.3466628 


.369580 


2.705769 


.9379898 


43 


17 


.3629802 


.389548 


2.567073 


.9317969 


43 


18 


.3469357 


.369911 


2.703351 


.9378889 


42 


18 


.3632512 


.389883 


2.564867 


.9316912 


42 


19 


.3472085 


.370242 


2.700936 


.9377880 


41 


19 


.3635222 


.390218 


2.562664 


.9315855 


41 


20 


.3474812 


.370572 


2.698525 


.9376869 


40 


20 


.3637932 


.390554 


2.560464 


.9314797 


40 


21 


.3477540 


.370903 


2.696118 


.9375858 


39 


21 


.3640641 


.390889 


2.558268 


.9313739 


39 


22 


.3480267 


.371234 


2.693714 


.9374846 


38 


22 


.3643351 


.391224 


2.556075 


.9312679 


38 


23 


.3482994 


.371565 


2.691314 


.9373833 


37 


23 


.3646059 


.391560- 


2.553885 


.9311619 


37 


24 


.3485720 


.371896 


2.688919 


.9372820 


36 


24 


.3648768 


.391895 


2.551699 


.9310558 


36 


25 


.3488447 


.372227 


2.686526 


.9371806 


35 


25 


.3651476 


.392231 


2.549516 


.9309496 


35 


26 


.3491173 


.372559 


2.684138 


.9370790 


34 


26 


.3654184 


.392567 


2.547335 


.9308434 


34 


27 


.3493898 


.372890 


2.681753 


.9369774 


33 


27 


.3656891 


.392902 


2.545159 


.9307370 


33 


28 


.3496624 


.373221 


2.679372 


.9368758 


32 


28 


.3659599 


.393238 


2.542985 


.9306306 


32 


29 


.3499349 


.373553 


2.676995 


.9367740 


31 


29 


.3662306 


.393574 


2.540815 


.9305241 


31 


30 


.3502074 


.373884 


2.674621 


.9366722 


30 


30 


.3665012 


.393910 


2.538647 


.9304176 


30 


31 


.3504798 


.374216 


2.672251 


.9365703 


29 


31 


.3667719 


.394246 


2.536483 


.9303109 


29 


32 


.3507523 


.374547 


2.669885 


.9364683 


28 


32 


.3670425 


.394582 


2.534323 


.9302042 


28 


33 


.3510246 


.374879 


2.667522 


.9363662 


27 


33 


.3673130 


.394918 


2.532165 


.9300974 


27 


34 


.3512970 


.375211 


2.665163 


.9362641 


26 


34 


.3675836 


.395255 


2.530011 


.9299905 


26 


35 


.3515693 


.375543 


2.662808 


.9361618 


25 


35 


.3678541 


.395591 


2.527859 


.9298835 


25 


36 


.3518416 


.375875 


2.660456 


.9360595 


24 


36 


.3681246 


.'395928 


2.525711 


.9297765 


24 


37 


.3521139 


.376207 


2.658108 


. 9359571 


23 


37 


.3683950 


.396264 


2.523566 


.9296694 


23 


38 


.3523862 


.376539 


2.655764 


.9358547 


22 


38 


.3686654 


.396601 


2.521424 


.9295622 


22 


39 


.3526584 


.376871 


2.653423 


.9357521 


21 


39 


.3689358 


.396937 


2.519286 


.9294549 


21 


40 


.3529306 


.377203 


2.651086 


.9356495 


20 


40 


.3692061 


.397274 


2.517150 


.9293475 


20 


41 


.3532027 


.377536 


2.648753 


.9355468 


19 


41 


.3694765 


.397611 


2.515018 


.9292401 


19 


42 


.3534748 


.377868 


2.646423 


.9354440 


18 


42 


.3697468 


.397948 


2.512889 


.9291326 


18 


43 


.3537469 


.378201 


2.644096 


.9353412 


17 


43 


.3700170 


.398285 


2.510762 


.9290250 


17 


44 


.3540190 


.378533 


2.641774 


.9352382 


16 


44 


.3702872 


.398622 


2.508639 


.9289173 


16 


45 


.3542910 


.378866 


2.639454 


.9351352 


15 


45 


.3705574 


.398959 


2.506519 


.9288096 


15 


46 


.3545630 


.379198 


2:637139 


.9350321 


14 


46 


.3708276 


.399296 


2.504403 


.9287017 


14 


47 


.3548350 


.379531 


2.634827 


.9349289 


13 


47 


.3710977 


.399634 


2.502289 


.9285938 


13 


48 


.3551070 


.379864 


2.632518 


.9348257 


12 


48 


.3713678 


.399971 


2.500178 


.9284858 


12 


49 


.3553789 


.380197 


2.630213 


.9347223 


11 


49 


.3716379 


.400308 


2.498070 


.9283778 


11 


50 


.3556508 


.380530 


2.627912 


.9346189 


10 


50 


.3719079 


.400646 


2.495966 


.9282696 


10 


51 


.3559226 


.380863 


2.625614 


.9345154 


9 


51 


.3721780 


.400984 


2.493864 


.9281614 


9 


52 


.3561944 


.381196 


2.623319 


.9344119 


8 


52 


.3724479 


.401321 


2.491766 


.9280531 


8 


53 


.3564662 


.381529 


2.621028 


.9343082 


7 


53 


.3727179 


.401659 


2.489670 


.9279447 


7 


54 


.3567380 


.381862 


2.618741 


.9342045 


6 


54 


.3729878 


.401997 


2.487578 


.9278363 


6 


55 


.3570097 


.382196 


2.616457 


.9341007 


5 


55 


.3732577 


.402335 


2.485488 


.9277277 


5 


56 


.3572814 


.382529 


2.614176 


.9339968 


4 


56 


.3735275 


.402673 


2.483402 


.'9276191 


4 


57 


.3575531 


.382863 


2.611899 


.9338928 


3 


57 


.3737973 


.403011 


2.481319 


.9275104 


3 


58 


.3578248 


.383196 


2.609625 


.9337888 


2 


58 


.3740671 


.403349 


2.479238 


.9274016 


2 


59 


.3580964 


.383530 


2.607355 


.9336846 


1 


59 


.3743369 


.403687 


2.477161 


.9272928 


1 


60 


.3583679 


.383864 


2.605089 


.9335804 





60 


.3746066 


.404026 


2.475086 


.9271839 





"" 


Cosine. 


ICotangI Tang. | Sine. | ' |l 1 Cosine. 


ICotangI Tang. | Sine. | ^ 



69° 
Note. — Secant =? 1 -^ cosine. Cosecant = 1-^sine. 



NATURAL SINES, ETC. 



155 



3.-— Natural Sines, Tangents, Cotangents, Cosines. — (Continued.) 

(Versed sine =1 — cosine; co versed sine = 1 — sine.) 
23° 



' 1 sine. 


1 Tang. 1 Cotang.l Cosine. 


1 II ' 1 Sine. 


1 Tang. 1 Cotang.l Cosine. 


l__ 





.3746066 


.404026 2.475086 


.9271839 


60 





.3907311 


.424474 2.355852 


.9205049 


60 


1 


.3748763 


.404364 


2.473015 


.9270748 


59 


1 


.3909989 


.424818 


2.353948 


.9203912 


59 


2 


.3751459 


.404703 


2.470947 


.9269658 


58 


2 


.3912666 


.425161 


2.352046 


.9202774 


58 


3 


.3754156 


.405041 


2.468881 


.9268566 


57 


3 


.3915343 


.425505 


2.350148 


.9201635 


57 


4 


.3756852 


.405380 


2.466819 


.9267474 


56 


4 


.3918019 


.425848 


2.348251 


.9200496 


56 


5 


.3759547 


.405719 


2.464759 


.9266380 


55 


5 


.3920695 


.426192 


2.346358 


.9199356 


55 


6 


.3762243 


.406057 


2.462703 


.9265286 


54 


6 


.3923371 


.426536 


2.344467 


.9198215 


54 


7 


.3764938 


.406396 


2.460649 


.9264192 


53 


7 


.3926047 


.426880 


2.342578 


,9197073 


53 


8 


.3767632 


.406735 


2.458598 


.9263096 


52 


8 


.3928722 


.427223 


2.340692 


.9195931 


52 


9 


.3770327 


.407074 


2.456551 


.9262000 


51 


9 


.3931397 


.427568 


2.338809 


.9194788 


51 


10 


.3773021 


.407413 


2.454506 


.9260902 


50 


10 


.3934071 


.427912 


2.336928 


.9193644 


50 


11 


.3775714 


.407753 


2.452464 


.9259805 


49 


11 


.3936745 


.428256 


2.335050 


.9192499 


49 


12 


.3778408 


.408092 


2.450425 


.9258706 


48 


12 


.3939419 


.428600 


2.333174 


.9191353 


48 


13 


.3781101 


.408431 


2.448389 


.9257606 


47 


13 


.3942093 


.428944 


2.331301 


.9190207 


47 


14 


.3783794 


.408771 


2.446355 


.9256506 


46 


14 


.3944766 


.429289 


2.329431 


.9189060 


46 


15 


.3786486 


.409110 


2.444325 


.9255405 


45 


15 


.3947439 


.429633 


2 327563 


.9187912 


45 


16 


.3789178 


.409450 


2.442298 


.9254303 


44 


16 


.3950111 


.429978 


2.325697 


.9186763 


44 


17 


.3791870 


.409790 


2.440273 


.9253201 


43 


17 


.3952783 


.430323 


2.323834 


.9185614 


43 


18 


.3794562 


.410129 


2.438251 


.9252097 


42 


18 


.3955455 


.430668 


2.321974 


.9184464 


42 


19 


.3797253 


.410469 


2.436233 


.9250993 


41 


19 


.3958127 


.431012 


2.320116 


.9183313 


41 


20 


.3799944 


.410809 


2.434217 


.9249888 


40 


20 


.3960798 


.431357 


2.318260 


.9182161 


40 


21 


.3802634 


.411149 


2.432204 


.9248782 


39 


21 


.3963468 


.431703 


2.316407 


.9181009 


39 


22 


.3805324 


.411489 


2.430193 


.9247676 


38 


22 


.3966139 


.432048 


2.314557 


.9179855 


38 


23 


.3808014 


.411830 


2.428186 


.9246568 


37 


23 


.3968809 


.432393 


2.312709 


.9178701 


37 


24 


.3810704 


.412170 


2.426181 


.9245460 


36 


24 


.3971479 


.432738 


2.310863 


.9177546 


36 


25 


.3813393 


.4^2510 


2.424180 


.9244351 


35 


25 


.3974148 


.433084 


2.309020 


.9176391 


35 


26 


.3816082 


.412851 


2.422181 


.9243242 


34 


26 


.3976818 


.433429 


2.307180 


.9175234 


34 


27 


.3818770 


.413191 


2.420185 


.9242131 


33 


27 


.3979486 


.433775 


2.305342 


.9174077 


33 


28 


.3821459 


.413532 


2.418191 


.9241020 


32 


28 


.3982155 


.434120 


2.303506 


.9172919 


32 


29 


.3824147 


.413872 


2.416201 


.9239908 


31 


29 


.3984823 


.434466 


2.301673 


.9171760 


31 


30 


.3826834 


.414213 


2.414213 


.9238795 


30 


30 


.3987491 


.434812 


2.299842 


.9170601 


30 


31 


.3829522 


.414554 


2.412228 


.9237682 


29 


31 


.3990158 


.435158 


2.298014 


.9169440 


29 


32 


.3832209 


.414895 


2.410246 


.9236567 


28 


32 


.3992825 


.435504 


2.296188 


.9168279 


28 


33 


.3834895 


.415236 


2.408267 


.9235452 


27 


33 


.5995492 


.435850 


2.294365 


.9167118 


27 


34 


.3837582 


.415577 


2.406290 


.9234336 


26 


34 


.3998158 


.436196 


2.292544 


.9165955 


26 


35 


.3840268 


.415918 


2.404316 


.9233220 


25 


35 


.4000825 


.436542 


2.290725 


.9164791 


25 


36 


.3842953 


.416259 


2.402345 


.9232102 


24 


36 


.4003490 


.436889 


2.288909 


.9163627 


24 


37 


.3845639 


.416601 


2.400377 


.9230984 


23 


37 


.4006156 


.437235 


2.287095 


.9162462 


23 


38 


.3848324 


.416942 


2.398411 


.9229865 


22 


38 


.4008821 


.437582 


2.285284 


.9161297 


22 


39 


.3851008 


.417284 


2.396449 


.9228745 


21 


39 


.4011486 


.437928 


2.283475 


.9160130 


21 


40 


.3853693 


.417625 


2.394488 


.9227624 


20 


40 


.4014150 


.438275 


2.281669 


.9158963 


20 


41 


.3856377 


.417967 


2.392531 


.9226503 


19 


41 


.4016814 


.438622 


2.279865 


.9157795 


19 


42 


.3859060 


.418309 


2.390576 


.9225381 


18 


42 


.4019478 


.438969 


2.278063 


.9156626 


18 


43 


.3861744 


.418650 


2.388625 


.9224258 


17 


43 


.4022141 


.439316 


2.276264 


.9155456 


17 


44 


.3864427 


.418992 


2.386675 


.9223134 


16 


44 


.4024804 


.439663 


2.274467 


.9154286 


16 


45 


.3867110 


.419334 


2.384729 


.9222010 


15 


45 


.4027467 


.440010 


2.272672 


.9153115 


15 


46 


.3869792 


.419676 


2.382785 


.9220884. 


14 


46 


.4030129 


.440357 


2.270880 


.9151943 


14 


47 


.3872474 


.420019 


2.380844 


.9219758^ 


13 


47 


.4032791 


.440705 


2.269090 


.9150770 


13 


48 


.3875156 


.420361 


2.378906 


.9218632 


12 


48 


.4035453 


.441052 


2.267303 


.9149597 


12 


49 


.3877837 


.420703 


2.376970 


.9217504 


11 


49 


.4038114 


.441400 


2.265518 


.9148422 


11 


50 


.3880518 


.421046 


2.375037 


.9216375 


10 


50 


.4040775 


.441747 


2.263735 


.9147247 


10 


51 


.3883199 


.421388 


2.373106 


.9215246 


9 


51 


.4043436 


.442095 


2.261955 


.9146072 


9 


52 


.3885880 


.421731 


2.371179 


.9214116 


8 


52 


.4046096 


.442443 


2.260177 


.9144895 


3 


53 


.3888560 


.422073 


2.369254 


.9212986 


7 


53 


.4048756 


.442791 


2.258401 


.9143718 


7 


54 


.3891240 


.422416 


2.367331 


.9211854 


6 


54 


.4051416 


.443139 


2.256628 


.9142540 


6 


55 


.3893919 


.422759 


2.365411 


.9210722 


5 


55 


.4054075 


.443487 


2.254857 


.9141361 


5 


56 


.3896598 


.423102 


2.363494 


.9209589 


4 


56 


.4056734 


.443835 


2.253088 


.9140181 


4 


57 


.3899277 


.423445 


2.361580 


.9208455 


3 


57 


.4059393 


.444183 


2.251322 


.9139001 


3 


58 


.3901955 


.423788 


2.359668 


.9207320 


2 


58 


.4062051 


.444531 


2.249558 


.9137819 


2 


59 


.3904633 


.424131 


2.357759 


.9206185 


1 


59 


.4064709 


.444880 


2.247796 


.9136637 


1 


60 


.3907311 


.424474 


2.355852 


.9205049 





60 


.4067366 


.445228 


2.246036 


.9135455 





^ CoslneTl 


Cotangl Tang. | Sine. | 


'II 1 Cosine. | 


Cotangl Tang. I Sine. | 


""'" 



67" 66° 

Note. — Secant =1-^ cosine. Cosecant = In- sine. 



156 



9.— PLANE TRIGONOMETRY. 



3,— Natural Sines, Tangents, Cotangents, Cosines. — (Continued.) 







(Versed sine 


= 1 — cosine; 


CO versed sine = 1 - 


sine.) 






24'' 25° 


' 1 Sine. 1 Tang. | Cotang.l Cosine. | || ' | Sine. | Tang. | Cotang.i Cosine. | 





.4067366 


.445228 


2.246036 


.9135455 


60 





.4226183 


.466307 


2.144506 


.9063078 


60 


1 


.407U024 


.445577 


2.244279 


.9134271 


59 


1 


.4228819 


.466661 


2.142879 


.9061848 


59 


2 


.4072681 


.445926 


2.242524 


.9133087 


58 


2 


.4231455 


.467016 


2.141253 


.9060618 


58 


3 


.4075337 


.446274 


2.240772 


.9131902 


57 


3 


.4234090 


.467370 


2.139630 


.9059386 


57 


4 


.4077993 


.446623 


2.239021 


.9130716 


56 


4 


.4236725 


.467725 


2.138008 


.9058154 


56 


5 


.4080649 


.446972 


2.237273 


.9129529 


55 


5 


.4239360 


.468079 


2.136389 


.9056922 


55 


6 


.4083305 


.447321 


2.235528 


.9128342 


54 


6 


.4241994 


.468434 


2.134771 


.9055688 


54 


7 


.4085960 


.447670 


2.233784 


.9127154 


53 


7 


.4244628 


.468789 


2.133155 


.9054454 


53 


8 


.4088615 


.448020 


2.232043 


.9125965 


52 


8 


.4247262 


.469143 


2.131542 


.9053219 


52 


9 


.4091269 


.448369 


2.230304 


.9124775 


51 


9 


.4249895 


.469498 


2,129930 


.9051983 


51 


10 


.4093923 


.448718 


2.228567 


.9123584 


50 


10 


.4252528 


.469853 


2.128321 


.9050746 


50 


11 


.4096577 


.449068 


2.226833 


.9122393 


49 


11 


.4255161 


.470209 


2.126713 


.9049509 


49 


12 


.4099230 


.449417 


2.225100 


.9121201 


48 


12 


.4257793 


.470564 


2.125108 


.9048271 


48 


13 


.4101883 


.449767 


2.223370 


.9120008 


47 


13 


.4260425 


.470919 


2.123504 


.9047032 


47 


14 


.4104536 


.450117 


2.221643 


.9118815 


46 


14 


.4263056 


.471275 


2.121903 


.9045792 


46 


15 


.4107189 


.450467 


2.219917 


.9117620 


45 


15 


.4265687 


.471630 


2.120303 


.9044551 


45 


16 


.4109841 


.450817 


2.218194 


.9116425 


44 


16 


.4268318 


.471986 


2.118705 


.9043310 


44 


17 


.4112492 


.451167 


2.216473 


.9115229 


43 


17 


.4270949 


.472342 


2.117110 


.9042068 


43 


18 


.4115144 


.451517 


2.214754 


.9114033 


42 


18 


.4273579 


.472697 


2.115516 


.9040825 


42 


19 


.4117795 


.451867 


2.213037 


.9112835 


41 


19 


.4276208 


.473053 


2.113924 


.9039582 


41 


20 


.4120445 


.452217 


2.211323 


.9111637 


40 


20 


.4278838 


.473409 


2.112334 


.9038338 


40 


21 


.4123096 


.452568 


2.209611 


.9110438 


39 


21 


.4281467 


.473765 


2.110747 


.9037093 


39 


22 


.4125745 


.452918 


2.207901 


.9109238 


38 


22 


.4284095 


.474122 


2.109161 


.9035847 


38 


23 


.4128395 


.453269 


2.206193 


.9108038 


37 


23 


.4286723 


.474478 


2.107577 


. 9034600 


37 


24 


.4131044 


.453620 


2.204487 


.9106837 


36 


24 


.4289351 


.474834 


2.105995 


.9033353 


36 


25 


.4133693 


.453970 


2.202784 


.9105635 


35 


25 


.4291979 


.475191 


2.104415 


.9032105 


35 


26 


.4136342 


.454321 


2.201083 


.9104432 


34 


26 


.4294606 


.475548 


2.102836 


.9030856 


34 


27 


.4138990 


.454672 


2.199384 


.9103228 


33 


27 


.4297233 


.475904 


2.101260 


.9029606 


33 


28 


.4141638 


.455023 


2.197687 


.9102024 


32 


28" 


.4299859 


.476261 


2.099686 


.9028356 


32 


29 


.4144285 


.455375 


2.195992 


.9100819 


31 


29 


.4302485 


.476618 


2.098114 


.9027105 


31 


30 


.4146932 


.455726 


2.194299 


.9099613 


30 


30 


.4305111 


.476975 


2.096543 


.9025853 


30 


31 


.4149579 


.456077 


2.192609 


.9098406 


29 


31 


.4307786 


.477332 


2.094975 


.9024600 


29 


32 


.4152226 


.456429 


2.190921 


.9097199 


28 


32 


.4310361 


.477689 


2.093408 


.9023347 


28 


33 


.4154872 


.456780 


2.189234 


.9095990 


27 


33 


.4312986 


.478047 


2.091843 


.9022092 


27 


34 


.4157517 


.457132 


2.187551 


.9094781 


26 


34 


.4315610 


.478404 


2.090280 


.9020838 


26 


35 


.4160163 


.457483 


2.185869 


.9093572 


25 


35 


.4318234 


.478762 


2.088720 


.9019582 


25 


36 


.4162808 


.457835 


2.184189 


.9092361 


24 


36 


.4320857 


.479119 


2.087161 


.9018325 


24 


37 


.4165453 


.458187 


2.182511 


.9091150 


23 


37 


.4323481 


.479477 


2.085603 


.9017068 


23 


38 


.4168097 


.458539 


2.180836 


.9089938 


22 


38 


.4326103 


,479835 


2.084048 


.9015810 


22 


39 


.4170741 


.458891 


2.179163 


.9088725 


21 


39 


.4328726 


.480193 


2.082495 


.9014551 


21 


40 


.4173385 


.459243 


2.177492 


.9087511 


20 


40 


.4331348 


.480551 


2.080943 


.9013292 


20 


41 


.4176028 


.459596 


2.175822 


.9086297 


19 


41 


.4333970 


.480909 


2.079394 


.9012031 


19 


42 


.4178671 


.459948 


2.174155 


.9085082 


18 


42 


.4336591 


.481267 


2.077846 


.9010770 


18 


43 


.4181313 


.460301 


2.172491 


.9083866 


17 


43 


.4339212 


.481625 


2.076300 


.9009508 


17 


44 


.4183956 


.460653 


2.170828 


.9082649 


16 


44 


.4341832 


.481984 


2.074756 


.9008246 


16 


45 


.4186597 


.461006 


2.169167 


.9081432 


15 


45 


.4344453 


.482342 


2.073214 


.9006982 


15 


46 


.4189239 


.461359 


2.167509 


.9080214 


14 


46 


.4347072 


.482701 


2.071674 


.9005718 


14 


47 


.4191880 


.461711 


2.165852 


.9078995 


13 


47 


.4349692 


.483060 


2.070135 


.9004453 


13 


48 


.4194521 


.462064 


2.164198 


.9077775 


12 


48 


.4352311 


.483418 


2.068599 


.9003188 


12 


49 


.4197161 


.462417 


2.162546 


.9076554 


11 


49 


.4354930 


.483777 


2.067064 


.9001921 


11 


50 


.4199801 


.462771 


2.160895 


.9075333 


10 


50 


.4357548 


.484136 


2.065531 


.9000654 


10 


51 


.4202441 


.463124 


2.159247 


.9074111 


9 


51 


.4360166 


.484495 


2.064000 


.8999386 


9 


52 


.4205080 


.463477 


2.157601 


.9072888 


8 


52 


.4362784 


.484855 


2.062471 


.8998117 


8 


53 


.4207719 


.46o831 


2.155957 


.9071665 


7 


53 


.4365401 


.485214 


2.060944 


.8996848 


7 


54 


.4210358 


.464184 


2.1543U 


.9070440 


6 


54 


.4368018 


.485573 


2.059418 


.8995578 


6 


55 


.4212996 


.464538 


2.15267E 


.9069215 


5 


55 


.4370634 


.485933 


2.057895 


. 8994307 


5 


56 


.4215634 


.464891 


2.151037 


.9067989 


4 


56 


.4373251 


.486293 


2.056373 


.8993035 


4 


57 


.4218272 


.465245 


2.149402 


.9066762 


3 


57 


.4375866 


.486652 


2.054853 


.8991763 


3 


5i 


} .4220909 


.465599 


2.14776J 


i .9065535 


2 


5e 


.4378482 


.487012 


2.053334 


.8990489 


2 


5£ 


.4223546 


.465953 


2.14613( 


3 .9064307 


1 


^^ 


.4381097 


.487372 


. 2.0518U 


.8989215 


1 


6C 


> .4226183 


.466307 


2.14450( 


5 .9063078 





6C 


> .4383711 


.487732 


I 2.050303 .8987940 





1 Cosine. |Cotang| Tang. | Sine. | ' |l 1 Cosine. |Cotang| Tang. | Sine. | ' 




Note.- 


— Secan 


t =l-h 


cosine. 


65 
( 




3ose 


xant = 1 


-^sine. 






6i^ 



NATURAL SINES, ETC. 



157 



3.— Natural Sines, Tangents, Cotangents, Cosines. — (Continued.) 

(Versed sine =1 — cosine; coversed sine = 1 — sine.) 
27° 



' 1 sine. 1 Tang. |Cotang.| Cosine. | 1| ' | Sine. | Tang. | Cotang.| Cosine. | 





.4383711 


.487732 


2.050303 


.8987940 


60 





.4539905 


.509525 


1.962610 


.8910065 


60 


1 


.4386326 


.488092 


2.048791 


.8986665 


59 


1 


.4542497 


.509891 


1.961200 


.8908744 


59 


2 


.4388940 


.488453 


2.047280 


.8985389 


58 


2 


.4545088 


.510258 


1.959791 


. 8907423 


58 


3 


.4391553 


.488813 


2.045770 


.8984112 


57 


3 


.4547679 


.510625 


1.958383 


.8906100 


57 


4 


.4394166 


.489173 


2.044263 


.8982834 


56 


4 


.4550269 


.510991 


1.956978 


.8904777 


56 


5 


.4396779 


.489534 


2.042757 


.8981555 


55 


5 


.4552859 


.511358 


1.955573 


.8903453 


55 


6 


.4399392 


.489894 


2.041254 


.8980276 


54 


6 


.4555449 


.511725 


1.954171 


.8902128 


54 


7 


.4402004 


.490255 


2.039751 


.8978996 


53 


7 


.4558038 


.512093 


1.952770 


.8900803 


53 


8 


.4404615 


.490616 


2.038251 


.8977715 


52 


8 


.4560627 


.512460 


1.951371 


.8899476 


52 


9 


.4407227 


.490977 


2.036753 


.8976433 


51 


9 


.4563216 


.512827 


1.949973 


.8898149 


51 


10 


.4409838 


.491338 


2.035256 


.8975151 


50 


10 


.4565804 


.513195 


1.948577 


.8896822 


50 


11 


.4412448 


.491699 


2.033761 


.8973868 


49 


11 


.4568392 


.513562 


1.947112 


.8895493 


49 


12 


.4415059 


.492061 


2.032268 


.8972584 


48 


12 


.4570979 


.513930 


1.945789 


.8894164 


48 


13 


.4417668 


.492422 


2.030776 


.8971299 


47 


13 


.4573566 


.514298 


1.944398 


.8892834 


47 


14 


.4420278 


.492783 


2.029287 


.8970014 


46 


14 


.4576153 


.514665 


1.943008 


.8891503 


46 


15 


.4422887 


.493145 


2.027799 


.8968727 


45 


15 


.4578739 


.515033 


1.941620 


.8890171 


45 


16 


.4425496 


.493507 


2.026313 


.8967440 


44 


16 


.4581325 


.515401 


1.940233 


.8888839 


44 


17 


.4428104 


.493868 


2.024828 


.8966153 


43 


17 


.4583910 


.515770 


1.938848 


.8887506 


43 


18 


.4430712 


.494230 


2.023346 


.8964864 


42 


18 


.4586496 


.516138 


1.937464 


.8886172 


42 


19 


.4433319 


.494592 


2.021865 


.8963575 


41 


19 


.4589080 


.516506 


1.936082 


.8884838 


41 


20 


.4435927 


.494954 


2.020386 


.8962285 


40 


20 


.4591665 


.516875 


1.934702 


.8883503 


40 


21 


.4438534 


.495317 


2.018908 


.8960994 


39 


21 


.4594248 


.517244 


1.933323 


.8882166 


39 


22 


.4441140 


.495679 


2.017433 


.8959703 


38 


22 


.4596832 


.517612 


1.931945 


.8880830 


38 


23 


.4443746 


.496041 


2.015959 


.8958411 


37 


23 


.4599415 


.517981 


1.930569 


.8879492 


37 


24 


.4446352 


.496404 


2.014486 


.8957118 


36 


24 


.4601998 


.518350 


1.929195 


.8878154 


36 


25 


.4448957 


.496766 


2.013016 


.8955824 


35 


25 


.4604580 


.518719 


1.927822 


.8876815 


35 


26 


.4451562 


.497129 


2.011547 


.8954529 


34 


26 


.4607162 


.519089 


1.926451 


.8875475 


34 


27 


.4454167 


.497492 


2.010080 


.8953234 


33 


27 


.4609744 


.519458 


1.925081 


.8874134 


33 


28 


.4456771 


.497855 


2.008615 


.8951938 


32 


28 


.4612325 


.519827 


1.923713 


.8872793 


32 


29 


.4459375 


.498218 


2.007151 


.8950641 


31 


29 


.4614906 


.520197 


1.922347 


.8871451 


31 


30 


.4461978 


.498581 


2.005689 


.8949344 


30 


30 


.4617486 


.520567 


1.920982 


.8870108 


30 


31 


.4464581 


.498944 


2.004229 


.8948045 


29 


31 


.4620066 


.520936 


1.919618 


.8868765 


29 


32 


.4467184 


.499308 


2.002771 


.8946746 


28 


32 


.4622646 


.521306 


1.918256 


.8867420 


28 


33 


.4469786 


.499671 


2.001314 


.8945446 


27 


33 


.4625225 


.521676 


1.916896 


.8866075 


27 


34 


.4472388 


.500035 


1.999859 


.8944146 


26 


34 


.4627804 


.522046 


1.915537 


.8864730 


26 


35 


.4474990 


.500398 


1.998405 


.8942844 


25 


35 


.4630382 


.522417 


1.914179 


.8863383 


25 


36 


.4477591 


.500762 


1.996953 


.8941542 


24 


36 


.4632960 


.522787 


1.912823 


.8862036 


24 


37 


.4480192 


.501126 


1.995503 


.8940240 


23 


37 


.4635538 


.523157 


1.911469 


.8860688 


23 


38 


.4482792 


.501490 


1.994055 


.8938936 


22 


38 


.4638115 


.523528 


1.910116 


.8859339 


22 


39 


.4485392 


.501854 


1.992608 


.8937632 


21 


39 


.4640692 


.523899 


1.908764 


.8857989 


21 


40 


.4487992 


.502218 


1.991163 


.8936326 


20 


40 


.4643269 


.524269 


1.907414 


.8856639 


20 


41 


.4490591 


.502583 


1.989720 


.8935021 


19 


41 


.4645845 


.524640 


1.906066 


.8855288 


19 


42 


.4493190 


.502947 


1.988278 


.8933714 


18 


42 


.4648420 


.525011 


1.904719 


.8853936 


18 


43 


.4495789 


.503312 


1.986838 


.8932406 


17 


43 


.4650996 


.525382 


1.903373 


.8852584 


17 


44 


.4498387 


.503676 


1.985400 


.8931098 


16 


44 


.4653571 


.525754 


1.902029 


.8851230 


16 


45 


.4500984 


.504041 


1.983963 


.8929789 


15 


45 


.4656145 


.526125 


1.900687 


.8849876 


15 


46 


.4503582 


.504406 


1.982528 


.8928480 


14 


46 


.4658719 


.526496 


1.899346 


.8848522 


14 


47 


.4506179 


.504771 


1.981095 


.8927169 


13 


47 


.4661293 


.526868 


1.898006 


.8847166 


13 


48 


.4508775 


.505136 


1.979663 


.8925858 


12 


48 


.4663866 


.527240 


1.896668 


.8845810 


12 


49 


.4511372 


.505501 


1.978233 


.8924546 


11 


49 


.4666439 


.527612 


1.895332 


.8844453 


11 


50 


.4513967 


.505866 


1.976805 


.8923234 


10 


50 


.4669012 


.527983 


1.893997 


.8843096 


10 


51 


.4516563 


.506232 


1.975378 


.8921920 


9 


51 


.4671584 


.528356 


1.892663 


.8841736 


9 


52 


.4519158 


.506597 


1.973953 


.8920606 


8 


52 


.4674156 


.528728 


1.891331 


.8840377 


8 


53 


.4521753 


.506963 


1.972529 


.8919291 


7 


53 


.4676727 


.529100 


1.890000 


.8839017 


7 


54 


.4524347 


.507329 


1.971107 


.8917975 


6 


54 


.4679298 


.529472 


1.888671 


.8837656 


6 


55 


.4526941 


.507694 


1.969687 


.8916659 


5 


55 


.4681869 


.529845 


1.887343 


.8836295 


5 


56 


.4529535 


.508060 


1.968268 


.8915342 


4 


56 


.4684839 


.530217 


1.886017 


.8834933 


4 


67 


.4532128 


.508426 


1.966851 


.8914024 


3 


57 


.4687009 


.530590 


1.884692 


.8833569 


3 


58 


.4534721 


.508792 


1.965436 


.8912705 


2 


58 


.4689578 


.530963 


1.883369 


.8832206 


2 


59 


.4537313 


.509159 


1.964022 


.8911385 


1 


59 


.4692147 


.531336 


1.882047 


.8830841 


1 


60 


.4539905 


.509525 


1.962610 


.8910065 





60 


.4694716 


.531709 


1.880726 


.8829476 





1 Cosine. 


ICotangl Tang. | Sine. | ' |1 1 Cosine. 


ICotangl Tang. | Sine. | ' 




Note.- 


— Secan 


t =1^ 


cosine. 


63 
( 


3 

:ose 


cant= 1- 


^sine. 






62° 



158 



9,— PLANE TRIGONOMETRY. 



3.— Natural Sines, Tangents, Cotangents, Cosines. — (Continued.) 
(Versed sine =1 — cosine; coversed sine = 1 — sine.) 



28** 



' I Sine. I Tang. | Cot^ng.| Cosine. | || ' | Sine. | Tang. | Cotang.l Cosine. | 






.4694716 


1 


.4697284 


2 


.4699852 


3 


.4702419 


4 


.4704986 


5 


.4707553 


6 


.4710119 


7 


.4712685 


8 


.4715250 


9 


.4717815 


10 


.4720380 


11 


.4722944 


12 


.4725508 


13 


.4728071 


14 


.4730634 


15 


.4733197 


16 


.4735759 


17 


.4738321 


18 


.4740882 


19 


.4743443 


20 


.4746004 


21 


.4748564 


22 


.4751124 


23 


.4753683 


24 


.4756242 


25 


.4758801 


26 


.4761359 


27 


.4763917 


28 


.4766474 


29 


.4769031 


30 


.4771588 


31 


.4774144 


32 


.4776700 


33 


.4779255 


34 


.4781810 


35 


.4784364 


36 


.4786919 


37 


.4789472 


38 


.4792026 


39 


.4794579 


40 


.4797131 


41 


.4799683 


42 


.4802235 


43 


.4804786 


44 


.4807337 


45 


.4809888 


46 


.4812438 


47 


.4814987 


48 


.4817537 


49 


.4820086 


50 


.4822634 


61 


.4825182 


52 


.4827730 


63 


.4830277 


54 


.4832824 


55 


.4835370 


56 


.4837916 


67 


.4840462 


58 


.4843007 


59 


.4845552 1 


60 

t 


.4848096 i 



. 531709! 1 
.532082,1 
. 532455; 1 
.53282911 
.533202 1 
. 533576 jl 
.533950 1 
.53432411 
.534698:i, 
. 535072 il 
.53544611 
.535820,1 
.536195 
.53656911 
.53694411. 
.53731911 
.53769411 
.538069 1 
.538444 1 
.538819 1 
.539195'l 
.539570 1 
.539946:1 
.540322 1. 
.540698 1 
.5410741, 
.5414501 
.541826 1 
.54220211 . 
.542579|1 
.542955 I 
.543332 1 
.543709 1 
.544*86 1 
.544463 1 
.544840 1 
.545217 1 1 
.545595 1 
.545972 1 1 
.546350,1 
.546728,1 
.54710611 
.547484 1, 
.547862; 1. 
.548240 1 
.548618 1. 
.548997 1 
.549375 1 
.549754' 1 
.550133 I 
.550512 1, 
.55089111 
.551270 1, 
551650 1. 
552029 1. 
552409 1. 
552789 1. 
553168 1. 
55354S 1. 
55392S 1. 
554309 1 . 



.880726 


.8829476 


60 





.4848096 


.879407 


.8828110 


59 


1 


.4850640 


.878089 


.8826743 


58 


2 


.4853184 


.876773 


.8825376 


57 


3 


.4855727 


.875458 


.8824007 


56 


4 


.4858270 


.8741451.8822638 


55 


5 


.4860812 


.872833 


.8821269 


54 


6 


.4863354 


.871523 


.8S19898 


53 


7 


.4865895 


.870214 


.8818527 


52 


8 


.4868436 


.868906 


.8817155 


51 


9 


.4870977 


.867600'. 8815782 


50 


10 


.4873517 


.866295 .8814409 


49 


11 


.4876057 


. 864992;. 8S13035 


48 


12 


.4878597 


.863690 .8811660 


47 


13 


.4881136 


.8623891. 8S102S4 


46 


14 


.4883674 


.86 1090 .8808907 


45 


15 


.4886212 


.8597 92 .8807530 


44 


16 


.4888750 


.858496 .SS061 52 


43 


17 


.4891288 


.857201 .SS04774 


42 


18 


.4893825 


.855908 .8S03394 


41 


19 


.4896361 


. 854615!. SS02014 


40 


20 


.4898897 


.8533251.8800633 


39 


21 


.4901433 


.852035 .8799251 


38 


22 


.4903968 


.850747 .8797869 


37 


23 


.4906503 


.849461 .8796486 


36 


24 


.4909038 


.848176 .8795102 


35 


25 


.4911572 


.846892 .8793717 


34 


26 


.4914105 


.845609 .8792332 


33 


27 


.4916638 


.84432s'. 87 90946 


32 


28 


.4919171 


.8430491.8789559 


31 


29 


.4921704 


.8417701.8788171 


30 


30 


.4924236 


.840494'. 87 86783 


29 


31 


.4926767 


.8392181.8785394 


28 


32 


.4929298 


.8379441.8784004 


27 


33 


.4931829 


.836671 


.8782613 


26 


34 


.4934359 


.835399 


.8781222 


25 


35 


.4936889 


.834129 


.8779830 


24 


36 


.4939419 


.832861!. 8778437 


23 


37 


.4941948 


.831593 


.8777043 


22 


38 


.4944476 


.830327 


.8775649 


21 


39 


.4947005 


.829062 


.8774254 


20 


40 


.4949532 


.827799 


.8772858 


19 


41 


.4952060 


.826537 


.8771462 


18 


42 


.4954587 


.825276 


.8770064 


17 


43 


.4957113 


.824017 


.8768666 


16 


44 


.4959639 


.8227591.8767268 


15 


45 


.4962165 


.821502 .8765868 


14 


46 


.4964690 


.820247 .8764468 


13 


47 


.4967215 


.81 8993;. 8763067 


12 


48 


.4969740 


.8177401.8761665 


11 


49 


.4972264 


.816489 .8760263 


10 


50 


.4974787 


.81 523 9 '.87 58859 


9 


51 


.4977310 


.813990 '.8757455 


8 


52 


.4979833 


.8127431.8756051 


7 


53 


.4982355 


.811496 .8754645 


6 


54 


.4984877 


.8102521.8753239 


5 


55 


.4987399 


.8090081.8751832 


4 


56 


.4989920 


.807766 .8750425 


3 


57 


.4992441 


.806525 .8749016 


2 ! 58'. 4994961 1 


.805286 .8747607 


1 59 .4997481 1 


.804047 


.8746197 





60 


.5000000 1 



.554309 
.554689 
.555069 
.555450 
.555831 
.556211 
.556592 
.556973 
.557355 
.557736 
.558117 
.558499 
.558881 
.559262 
.559644 
.560026 
.560409 
.560791 
.561173 
.561556 
.561939 
.562321 
.562704 
.563087 
.563471 
.563854 
.564237 
.564621 
.565005 
.56.5388 
.565772 
.566156 
.566541 
.566925 
.567309 
.567694 
.568079 
.568463 
.568848 
.569233 1. 
.569619 
.570004 I. 
.570389 1, 
.570775!l. 
.57116111, 
.57154711, 
.57193311, 
.572319,1, 
.57270511. 
. 573091 jl, 
.573478;!. 
.573864:1. 
.57425111. 
.574638 1. 
. 575025; 1. 
.57541211. 
.575799,1. 
,576187ll. 
,576574 1. 
576962 1.: 
577350 1, 



.804047 


.8746197 


.802810 


.8744786 


.801575 


.8743375 


.800340 


.8741963 


.799107 


.8740550 


.797875 


.8739137 


.796645 


.8737722 


.795416 


.8736307 


.794188 


.8734891 


.792961 


.8733475 


.791736 


.8732058 


.790512 


.8730640 


.789289 


.8729221 


.788067 


.8727801 


.786847 


.8726381 


.785628 


.8724960 


.784410 


.8723538 


.783194 


.8722116 


.781979 


.8720693 


.780765 


.8719269 


.779552 


.8717844 


.778340 


.8716419 


.777130 


.8714993 


.775921 


.8713566 


.774714 


.8712138 


.773507 


.8710710 


.772302 


.8709281 


.771098 


.8707851 


.769895 


.8706420 


.768694 


.8704989 


.767494 


.8703557 


.766295 


.8702124 


.765097 


.8700691 


.763900 


.8699256 


.762705 


.8697821 


.761511 


.8696386 


.760318 


.8694949 


.759126 


.8693512 


.757936 


.8692074 


.756747 


.8690636 


.755559 


.8689196 


.754372 


.8687756 


.753186 


.8686315 


.752002 


.8684874 


.750819 


.8683431 


.749637 


.8681988 


.748456 


.8680544 


.747276 


.8679100 


.746098 


.8677655 


.744921 


.8676209 


.743745 


.8674762 


.742570 


.8673314 


.741396 


.8671866 


.740224 


.8670417 


.739053 


.8668967 


.737883 


.8667517 


.736714 


.8666066 


.735546 


.8664614 


.734380 


.8663161 


.733214 


.8661708 


.732050 


.8660254 



I Cosine. ICotangl Tang. I Sine. \ ' \\ I Cosine. ICotangj Tang. | Sine. I ' 



61^ 



60° 



Note. — Secant = 1 -s- cosine. 



Cosecant = l-s-sine. 



NATURAL SINES. ETC, 



159 



30^ 



3.— Natural Sines, Tangents, Cotangents, Cosines. — (Continued.) 

(Versed sine =1 — cosine; coversed sine = 1 — sine.) 
31° 



' 1 


Sine. 1 Tang. | 


Cotang.l Cosine. 


1 


' 1 


Sine. 1 Tang. | Cotang.l Cosine. 







.5000000 


.577350 


1.732050 


.8660254 


60 





.5150381 


.600860 


1.664279 


.8571673 


60 


1 


.5002519 


.577738 


1.730887 


.8658799 


59 


1 


.5152874 


.601256 


1.663183 


.8570174 


59 


2 


.5005037 


.578126 


1.729726 


.8657344 


58 


2 


.5155367 


.601652 


1.662088 


.8568675 


58 


3 


.5007556 


.578514 


1.728565 


.8655887 


57 


3 


.5157859 


.602049 


1.660994 


.8567175 


57 


4 


.5010073 


.578902 


1.727406 


.8654430 


56 


4 


.5160351 


.602445 


1.659901 


.8565674 


56 


5 


.5012591 


.579291 


1.726247 


.8652973 


55 


5 


.5162842 


.602841 


1.658809 


.8564173 


55 


6 


.5015107 


.579679 


1.725090 


.8651514 


54 


6 


.5165333 


.603238 


1.657718 


.8562671 


54 


7 


.5017624 


.580068 


1.723934 


.8650055 


53 


7 


.5167824 


.603635 


1.656629 


.8561168 


53 


8 


.5020140 


.580457 


1.722779 


.8648595 


52 


8 


.5170314 


.604032 


1.655540 


.8559664 


52 


9 


.5022655 


.580846 


1.721626 


.8647134 


51 


9 


.5172804 


.604429 


1.654452 


.8558160 


51 


10 


.5025170 


.581235 


1.720473 


.8645673 


50 


10 


.5175293 


.604826 


1.653366 


.8556655 


50 


11 


.5027685 


.581624 


1.719322 


.8644211 


49 


11 


.5177782 


.605224 


1.652280 


.8555149 


49 


12 


.5030199 


.582013 


1.718172 


.8642748 


48 


12 


.5180270 


.605621 


1.651196 


.8553643 


48 


13 


.5032713 


.582403 


1.717023 


.8641284 


47 


13 


.5182758 


.606019 


1.650112 


.8552135 


47 


14 


.5035227 


.582793 


1.715875 


.8639820 


46 


14 


.5185246 


.606417 


1.649030 


.8550627 


46 


15 


.5037740 


.583182 


1.714728 


.8638355 


45 


15 


.5187733 


.606814 


1.647949 


.8549119 


45 


16 


.5040252 


.583572 


1.713582 


.8636889 


44 


16 


.5190219 


.607213 


1.646868 


.8547609 


44 


17 


.5042765 


.583962 


1.712438 


.8635423 


43 


17 


.5192705 


.607611 


1.645789 


.8546099 


43 


18 


.5045276 


.584352 


1.711294 


.8633956 


42 


18 


.5195191 


.608009 


1.644711 


.8544588 


42 


19 


.5047788 


.584743 


1.710152 


.8632488 


41 


19 


.5197676 


.608408 


1.643633 


.8543077 


41 


20 


.5050298 


.585133 


1.709011 


.8631019 


40 


20 


.5200161 


.608806 


1.642557 


.8541564 


40 


21 


.5052809 


.585524 


1.707871 


.8629549 


39 


21 


.5202646 


.609205 


1.641482 


.8540051 


39 


22 


.5055319 


.585914 


1.706732 


.8628079 


38 


22 


.5205130 


.609604 


1.640408 


.8538538 


38 


23 


.5057828 


.586305 


1.705595 


.8626608 


37 


23 


.5207613 


.610003 


1.639335 


.8537023 


37 


24 


.5060338 


.586696 


1.704458 


.8625137 


36 


24 


.5210096 


.610402 


1.638263 


.8535508 


36 


25 


.5062846 


.587087 


1.703323 


.8623664 


35 


25 


.5212579 


.610801 


1.637191 


.8533992 


35 


26 


.5065355 


.587478 


1.702189 


.8622191 


34 


26 


.5215061 


.611201 


1.636121 


.8532475 


34 


27 


.5067863 


.587870 


1.701055 


.8620717 


33 


27 


.5217543 


.611601 


1.635052 


.8530958 


33 


28 


.5070370 


.588261 


1.699923 


.8619243 


32 


28 


.5220024 


.612000 


1.633984 


.8529440 


32 


29 


.5072877 


.588653 


1.698792 


.8617768 


31 


29 


.5222505 


.612400 


1.632917 


.8527921 


31 


30 


.5075384 


.589045 


1.697663 


.8616292 


30 


30 


.5224986 


.612800 


1.631851 


.8526402 


30 


31 


.5077890 


.589436 


1.696534 


.8614815 


29 


31 


.5227466 


.613201 


1.630786 


.8524881 


29 


32 


.5080396 


.589828 


1.695406 


.8613337 


28 


32 


.5229945 


.613601 


1.629722 


.8523360 


28 


33 


.5082901 


.590221 


1.694280 


.8611859 


27 


33 


.5232424 


.614001 


1.628659 


.8521839 


27 


34 


.5085406 


.590613 


1.693155 


.8610380 


26 


34 


.5234903 


.614402 


l.«27597 


.8520316 


26 


35 


.5087910 


.591005 


1.692030 


.8608901 


25 


35 


.5237381 


.614803 


1.626536 


.8518793 


25 


36 


.5090414 


.591398 


1.690907 


.8607420 


24 


36 


.5239859 


.615204 


1.625476 


.8517269 


24 


37 


.5092918 


.591791 


1.689785 


.8605939 


23 


37 


.5242336 


.615605 


1.624417 


.8515745 


23 


38 


.5095421 


.592183 


1.688664 


.8604457 


22 


38 


.5244813 


.616006 


1.623359 


.8514219 


22 


39 


.5097924 


.592576 


1.687544 


.8602975 


21 


39 


.5247290 


.616407 


1.622302 


.8512693 


21 


40 


.5100426 


.592969 


1.686426 


.8601491 


20 


40 


.5249766 


.616809 


1.621246 


.8511167 


20 


41 


.5102928 


.593363 


1.685308 


.8600007 


19 


41 


.5252241 


.617210 


1.620192 


.8509639 


19 


42 


.5105429 


.593756 


1.684191 


.8593523 


18 


42 


.5254717 


.617612 


1.619138 


.8508111 


18 


43 


.5107930 


.594150 


1.683076 


.8597037 


17 


43 


.5257191 


.618014 


1.618085 


.8506582 


17 


44 


.5110431 


.594543 


1.681962 


.8595551 


16 


44 


.5259665 


.618416 


1.617033 


.8505053 


16 


45 


.5112931 


.594937 


1.680848 


.8594064 


15 


45 


.5262139 


.618818 


1.615982 


.8503522 


15 


46 


.5115431 


.595331 


1.679736 


.8592576 


14 


46 


.5264613 


.619221 


1.614932 


.8501991 


14 


47 


.5117930 


.595725 


1.678625 


.8591088 


13 


47 


.5267085 


.619623 


1.613882 


.8500459 


13 


48 


.5120429 


.596119 


1.677515 


.8589599 


12 


48 


.5269558 


.620026 


1.612834 


.8498927 


12 


49 


.5122927 


.596514 


1.676406 


.8588109 


11 


49 


.5272030 


.620429 


1.611787 


.8497394 


11 


SO 


.5125425 


.596908 


1.675298 


.8586619 


10 


50 


.5274502 


.620832 


1.610741 


.8495860 


10 


51 


.5127923 


.597303 


1.674192 


.8585127 


9 


51 


.5276973 


.621235 


1.609696 


.8494325 


9 


52 


.5130420 


.597697 


1.673086 


.8583635 


8 


52 


.5279443 


.621638 


1.608652 


.8492790 


8 


53 


.5132916 


.598092 


1.671981 


.8582143 


7 


53 


.5281914 


.622041 


1.607609 


.8491254 


7 


54 


.5135413 


.598487 


1.670878 


.8580649 


6 


54 


.5284383 


.622445 


1.606567 


.8489717 


6 


55 


.5137908 


.598882 


1.669775 


.8579155 


5 


55 


.5286853 


.622848 


1.605526 


.8488179 


5 


56 


.5140404 


.599278 


1.668674 


.8577660 


4 


56 


.5289322 


.623252 


1.604485 


.8486641 


4 


57 


.5142899 


.599673 


1.667574 


.8576164 


3 


57 


.5291790 


.623656 


1.603446 


.8485102 


3 


58 


.5145393 


.600069 


1.666474 


.8574668 


2 


58 


.5294258 


.624060 


1.602408 


.8483562 


2 


59 


.5147887 


.600464 


1.665376 


.8573171 


1 


59 


.5296726 


.624465 


1.601370 


.8482022 


1 


60 


.5150381 


.600860 


1.664279 


.8571673 





60 


.5299193 


.624869 


1.600334 


.8480481 






I Cosine. ICotangI Tang. | Sine. 



Cosine. |Cotang| Tang. | Sine. | 



59^ 



58" 



Note. — Secant =1 ^cosine. 



Cosecant 



160 



9.— PLANE TRIGONOMETRY, 



32= 



3.— Natural Sines, Tangents, Cotangents, Cosines. — (ContinuedO 

(Versed sine =1 — cosine; coversed sine=l — sine.) 
33° 



/ 


1 Sine. 


Tang. 1 Cotang.l Cosine. 


1 


1 ' 


Sine. 


Tang. 


Cotang.l Cosine. 


1 





.5299193 


.624869 


1.600334 


.8480481 


60 





.5446390 


.649407 


1.539865 


.8386706 


60 


1 


.5301659 


.625273 


1.599299 


.8478939 


59 


1 


.5448830 


.649821 


1.538884 


.8385121 


59 


2 


.5304125 


.625678 


1.598264 


.8477397 


58 


2 


.5451269 


.650235 


1.537905 


.8383536 


58 


3 


.5306591 


.626083 


1.597231 


.8475853 


57 


3 


.5453707 


.650649 


1.536927 


.8381950 


57 


4 


.5309057 


.626488 


1.596198 


.8474309 


56 


4 


.5456145 


.651063 


1.535949 


.8380363 


56 


5 


.5311521 


.626893 


1.595167 


.8472765 


55 


5 


.5458583 


.651477 


1.534972 


.8378775 


55 


6 


.5313986 


.627298 


1.594136 


.8471219 


54 


6 


.5461020 


.651891 


1.533996 


.8377187 


54 


7 


.5316450 


.627704 


1.593107 


.8469673 


53 


7 


.5463456 


.652306 


1.533021 


.8375598 


53 


8 


.5318913 


.628109 


1.592078 


.8468126 


52 


8 


.5465892 


.652721 


1.532047 


.8374009 


52 


9 


.5321376 


.628515 


1.591050 


.8466579 


51 


9 


.5468328 


.653136 


1.531074 


.8372418 


51 


10 


.5323839 


.628921 


1.590023 


.8465030 


50 


10 


.5470763 


.653551 


1.530102 


.8370827 


50 


11 


.5326301 


.629327 


1.588997 


.8463481 


49 


11 


.5473198 


.653996 


1.529130 


.8369236 


49 


12 


.5328763 


.629733 


1.587973 


.8461932 


48 


12 


.5475632 


.654381 


1.528160 


.8367643 


48 


13 


.5331224 


.630139 


1.586949 


.8460381 


47 


13 


.5478066 


.654797 


1.527190 


.8366050 


47 


U 


.5333685 


.630546 


1.585926 


.8458830 


46 


14 


.5480499 


.655212 


1.526221 


.8364456 


46 


15 


.5336145 


.630953 


1.584904 


.8457278 


45 


15 


.5482932 


.655628 


1.525253 


.8362862 


45 


16 


.5338605 


.631359 


1.583883 


.8455726 


44 


16 


.■5485365 


.656044 


1.524286 


.8361266 


44 


17 


.5341065 


.631766 


1.582862 


.8454172 


43 


17 


.5487797 


.656460 


1.523320 


.8359670 


43 


18 


.5343523 


.632173 


1.581843 


.8452618 


42 


18 


.5490228 


.656877 


1.522354 


.8358074 


42 


19 


.5345982 


.632581 


1.580825 


.8451064 


41 


19 


.5492659 


.657293 


1.521389 


.8356476 


41 


20 


.5348440 


.632988 


1.579807 


.8449508 


40 


20 


.5495090 


.657710 


1.520426 


.8354878 


40 


21 


.5350898 


.633395 


1.578791 


.8447952 


39 


21 


.5497520 


.658127 


1.519463 


.8353279 


39 


22 


.5353355 


.633803 


1.577776 


.8446395 


38 


22 


.5499950 


.658544 


1.518501 


.8351680 


38 


28 


.5355812 


.634211 


1.576761 


.8444838 


37 


23 


.5502379 


.658961 


1.517540 


.8350080 


37 


24 


.5358268 


.634619 


1.575747 


.8443279 


36 


24 


.5504807 


.659378 


1.516579 


.8348479 


36 


25 


.5360724 


.635027 


1.574735 


.8441720 


35 


25 


.5507236 


.659796 


1.515620 


.8346877 


35 


26 


.5363179 


.635435 


1.573723 


.8440161 


34 


26 


.5509663 


.660213 


1.514661 


.8345275 


34 


27 


.5365634 


.635844 


1.572712 


.8438600 


33 


27 


.5512091 


.660631 


1.513703 


.8343672 


33 


28 


.5368089 


.636252 


1.571702 


.8437039 


32 


28 


.5514518 


.661049 


1.512746 


.8342068 


32 


29 


.5370543 


.636661 


1.570693 


.8435477 


31 


29 


.5516944 


.661467 


1.511790 


.8340463 


31 


30 


.5372996 


.637070 


1.569685 


.8433914 


30 


30 


.5519370 


.661885 


1.510835 


.8338858 


30 


31 


.5375449 


.637479 


1.568678 


.8432351 


29 


31 


.5521795 


.662304 


1.509880 


.8337252 


29 


32 


.5377902 


':637888 


1.567672 


.8430787 


28 


32 


.5524220 


.662722 


1.508927 


.8335646 


28 


33 


.5380354 


.638297 


1.566666 


.8429222 


27 


33 


.5526645 


.663141 


1.507974 


.8334038 


27 


34 


.5382806 


.63870? 


1.565662 


.8427657 


26 


34 


.5529069 


.663560 


1.507022 


.8332430 


26 


35 


.5385257 


.639116 


1.564659 


.8426091 


25 


35 


.5531492 


.663979 


1.506071 


.8330822 


25 


36 


.5387708 


.639526 


1.563656 


.8424524 


24 


36 


.5533915 


.664398 


1.505121 


.8329212 


24 


37 


.5390158 


.639936 


1.562654 


.8422956 


23 


37 


.5536338 


.664817 


1.504171 


.8327602 


23 


38 


.5392608 


.640346 


1.561654 


.8421388 


22 


38 


.5538760 


.665237 


1.503222 


.8325991 


22 


39 


.5395058 


.640756 


1.560654 


.8419819 


21 


39 


.5541182 


.665657 


1.502275 


.8324380 


21 


40 


.5397507 


.641167 


1.559655 


.8418249 


20 


40 


.5543603 


.666076 


1.501328 


.8322768 


20 


41 


.5399955 


.641577 


1.558657 


.8416679 


19 


41 


.5546024 


.666496 


1.500382 


.8321155 


19 


42 


.5402403 


.641988 


1.557660 


.8415108 


18 


42 


.5548444 


.666917 


1.499436 


.8319541 


18 


43 


.5404851 


.642399 


1.556663 


.8413536 


17 


43 


.5550864 


.667337 


1.498492 


.8317927 


17 


44 


.5407298 


.642810 


1.555668 


.8411963 


16 


44 


.5553283 


.667758 


1.497548 


.8316312 


16 


45 


.5409745 


.643221 


1.554674 


.8410390 


15 


45 


.5555702 


.668178 


1.496605 


.8314696 


15 


46 


.5412191 


.643632 


1.553680 


.8408816 


14 


46 


.5558121 


.668599 


1.495663 


.8313080 


14 


47 


.5414637 


.644044 


1.552688 


.8407241 


13 


47 


.5560539 


.669020 


1.494722 


.8311463 


13 


48 


.5417082 


.644456 


1.551696 


.8405666 


12 


48 


.5562956 


.669441 


1.493782 


.8309845 


12 


49 


.5419527 


.644867 


1.550705 


.8404090 


11 


49 


.5565373 


.669863 


1.492842 


.8308226 


11 


50 


.5421971 


.645279 


1.549715 


.8402513 


10 


50 


.5567790 


.670284 


1.491903 


.8306607 


10 


51 


.5424415 


.645691 


1.548726 


.8400936 


9 


51 


.5570206 


.670706 


1.490965 


.8304987 


9 


52 


.5426859 


.646104 


1.547738 


.8399357 


8 


52 


.5572621 


.671128 


1.490028 


.8303366 


8 


53 


.5429302 


.646516 


1.546751 


.8397778 


7 


53 


.5575036 


.671550 


1.489092 


.8301745 


7 


54 


.5431744 


.646929 


1.545764 


.8396199 


6 


54 


.5577451 


.671972 


1.488157 


.8300123 


6 


55 


.5434187 


.647341 


1.544779 


.8394618 


5 


55 


.5579865 


.672394 


1.487222 


.8298500 


5 


56 


.5436628 


.647754 


1.543794 


.8393037 


4 


56 


.5582279 


.672816 


1.486288 


.8296877 


4 


57 


.5439069 


.648167 


1.542810 


.8391455 


3 


57 


.5584692 


.673239 


1.485355 


.8295252 


3 


58 


.5441510 


.648580 


1.541828 


.8389873 


2 


58 


.5587105 


.673662 


1.484423 


.8293628 


2 


59 


.5443951 


.648994 


1.540846 


.8388290 


1 


59 


.5589517 


.674085 


1.483491 


.8292002 


1 


60 


.5446390 


.649407 


1.539865 


.8386706 





60 


.5591929 


.674508 


1.482561 


.8290376 





1 Cosine. 1 


Cotangl Tang. | Sine. 1 


'II 1 Cosine. | 


Cotangl Tang. | Sine. 


3 



57° 
Note. — Secant = 1 -i- cosine. Cosecant = 1 -s-sine. 



66*= 



NATURAL SINES, ETC. 



161 



34= 



3.— Natural Sines, Tangents, Cotangents, Cosines. — (Continued.) 

(Versed sine =1 — cosine; co versed sine=l— sine.) 
35° 



_:_ 


Sine. 1 Tang. | Cotang.| Cosine. | 


1 


' 1 


Sine. 1 Tang. | 


Cotang.l Cosine. 







.5591929 


.674508 


1.482561 


.8290376 


60 





.5735764 


.700207 


1.428148 


.8191520 


60 


1 


.5594340 


.674931 


1.481631 


.8288749 


59 


1 


.5738147 


.700641 


1.427264 


.8189852 


59 


2 


.5596751 


.675355 


1.480702 


.8287121 


58 


2 


.5740529 


.701074 


1.426381 


.8188182 


58 


3 


.5599162 


.675779 


1.479773 


.8285493 


57 


3 


.5742911 


.701508 


1.425498 


.8186512 


57 


4 


.5601572 


.676202 


1.478846 


.8283864 


56 


4 


.5745292 


.701943 


1.424617 


.8184841 


56 


5 


.5603981 


.676626 


1.477919 


.8282234 


55 


5 


.5747672 


.702377 


1.423736 


.8183169 


55 


6 


.5606390 


.677050 


1.476993 


. 8280603 


54 


6 


.5750053 


.702811 


1.422856 


.8181497 


54 


7 


.5608798 


.677475 


1.476068 


.8278972 


53 


7 


.5752432 


.703246 


1.421976 


.8179824 


53 


8 


.5611206 


.677899 


1.475144 


.8277340 


52 


8 


.5754811 


.703681 


1.421097 


.8178151 


52 


9 


.5613614 


.678324 


1.474221 


.8275708 


51 


9 


.5757190 


.704116 


1.420220 


.8176476 


51 


10 


.5616021 


.678749 


1.473298 


.8274074 


50 


10 


.5759568 


.704551 


1.419342 


.8174801 


50 


11 


.5618428 


.679174 


1.472376 


.8272440 


49 


11 


.5761946 


.704986 


1.418466 


.8173125 


49 


12 


.5620834 


.679599 


1.471455 


.8270806 


48 


12 


.5764323 


.705422 


1.417590 


.8171449 


48 


13 


.5623239 


.680024 


1.470535 


.8269170 


47 


13 


.5766700 


.705858 


1.416715 


.8169772 


47 


14 


.5625645 


.680450 


1.469615 


.8267534 


46 


14 


.5769076 


.706294 


1.415840 


.8168094 


46 


15 


.5628049 


.680875 


1.468696 


.8265897 


45 


15 


.5771452 


.706730 


1.414967 


.8166416 


45 


16 


.5630453 


.681301 


1.467778 


.8264260 


44 


16 


.7773827 


.707166 


1.414094 


.8164736 


44 


17 


.5632857 


.681727 


1.466861 


.8262622 


43 


17 


.5776202 


.707602 


1.413222 


.8163056 


43 


18 


.5635260 


.682153 


1.465945 


.8260983 


42 


18 


.5778576 


.708039 


1.412350 


.8161376 


42 


19 


.5637663 


.682580 


1.465029 


.8259343 


41 


19 


.5780950 


.708476 


1.411479 


.8159695 


41 


20 


.5640066 


.683006 


1.464114 


.8257703 


40 


20 


.5783323 


.708913 


1.410609 


.8158013 


40 


21 


.5642467 


.683433 


1.463200 


.8256062 


39 


21 


.5785696 


.709350 


1.409740 


.8156330 


39 


22 


.5644869 


.683860 


1.462287 


.8254420 


38 


22 


.5788069 


.709787 


1.408871 


.8154647 


38 


23 


.5647270 


.684287 


1.461374 


.8252778 


37 


23 


.5790440 


.710225 


1.408003 


.8152963 


37 


24 


.5649670 


.684714 


1.460463 


.8251135 


36 


24 


.5792812 


.710663 


1.407136 


.8151278 


36 


25 


.5652070 


.685141 


1.459552 


.8249491 


35 


25 


.5795183 


.711100 


1.406270 


.8149593 


35 


26 


.5654469 


.685569 


1.458642 


.8247847 


34 


26 


.5797553 


.711539 


1.405404 


.8147906 


34 


27 


.5656868 


.685996 


1.457732 


.8246202 


33 


27 


.5799923 


.711977 


1.404539 


.8146220 


33 


28 


.5659267 


.686424 


1.456824 


.8244556 


32 


28 


.5802292 


.712415 


1.403674 


.8144532 


32 


29 


.5661665 


.686852 


1.455916 


.8242909 


31 


29 


.5804661 


.712854 


1.402811 


.8142844 


31 


30 


.5664062 


.687281 


1.455009 


.8241262 


30 


30 


.5807030 


.713293 


1.401948 


.8141155 


30 


31 


.5666459 


.687709 


1.454102 


.8239614 


29 


31 


.5809397 


.713732 


1.401086 


.8139466 


29 


32 


.5668856 


.688137 


1.453197 


.8237965 


28 


32 


.5811765 


.714171 


1.400224 


.8137775 


28 


33 


.5671252 


.688566 


1.452292 


.8236316 


27 


33 


.5814132 


.714610 


1.399363 


.8136084 


27 


34 


.5673648 


.688995 


1.451388 


.8234666 


26 


34 


.5816498 


.715050 


1.398503 


.8134393 


26 


35 


.5676043 


.689424 


1.450485 


.8233015 


25 


35 


.5818864 


.715490 


1.397644 


.8132701 


25 


36 


.5678437 


.689853 


1.449582 


.8231364 


24 


36 


.5821230 


.715929 


1.396785 


.8131008 


24 


37 


.5680832 


.690283 


1.448680 


.8229712 


23 


37 


5823595 


.716369 


1.395927 


.8129314 


23 


38 


.5683225 


.690712 


1.447779 


.8228059 


22 


38 


.5825959 


.716810 


1.395069 


.8127620 


22 


39 


.5685619 


.691142 


1.446879 


.8226405 


21 


39 


.5828323 


.717250 


1.394213 


.8125925 


21 


40 


.5688011 


.691572 


1.445980 


.8224751 


20 


40 


.5830687 


.717691 


1.393357 


.8124229 


20 


41 


.5690403 


.692002 


1.445081 


.8223096 


19 


41 


.5833050 


.718131 


1.392501 


.8122532 


19 


42 


.5892795 


.692432 


1.444183 


.8221440 


18 


42 


.5835412 


.718572 


1.391647 


.8120835 


18 


43 


.5695187 


.692863 


1.443286 


.8219784 


17 


43 


.5837774 


.719014 


1.390793 


.8119137 


17 


44 


.5697577 


.693293 


1.442389 


.8218127 


16 


44 


.5840136 


.719455 


1.389940 


.8117439 


16 


45 


.5699968 


.693724 


1.441494 


.8216469 


15 


45 


.5842497 


.719897 


1.389087 


.8115740 


15 


46 


.5702357 


.694155 


1.440599 


.8214811 


14 


46 


.5844857 


.720338 


1.388235 


.8114040 


14 


47 


.5704747 


.694586 


1.439704 


.8213152 


13 


47 


.5847217 


.720780 


1.387384 


.8112339 


13 


48 


.5707136 


.695018 


1.438811 


.8211492 


12 


48 


.5849577 


.721222 


1.386534 


.8110638 


12 


49 


.5709524 


.695^49 


1.437918 


.8209832 


11 


49 


.5851936 


.721665 


1.385684 


.8108936 


11 


50 


.5711912 


.695881 


1.437026 


.8208170 


10 


50 


.5854294 


.722107 


1.384835 


.8107234 


10 


51 


.5714299 


.696313 


1.436135 


.8206509 


9 


51 


.5856652 


.722550 


1.383986 


.8105530 


9 


52 


.5716686 


.696745 


1.435245 


.8204846 


8 


52 


.5859010 


.722993 


1.383139 


.8103826 


8 


53 


.5719073 


.697177 


1.434355 


.8203183 


7 


53 


.5861367 


.723436 


1.382292 


.8102122 


7 


54 


.5721459 


.697609 


1.433466 


.8201519 


6 


54 


.5863724 


.723879 


1.381445 


.8100416 


6 


55 


.5723844 


.698042 


1.432578 


.8199854 


5 


55 


.5866080 


.724322 


1.380600 


.8098710 


5 


56 


.5726229 


.698474 


1.431690 


.8198189 


4 


56 


.5868435 


.724766 


1.379755 


.8097004 


4 


57 


.5728614 


.698907 


1.430803 


.8196523 


3 


57 


.5870790 


.725210 


1.378910 


.8095296 


3 


58 


.5730998 


.699340 


1.429917 


.8194856 


2 


58 


.5873145 


.725654 


1.378067 


.8093588 


2 


59 


.5733381 


.699774 


1.429032 


.8193189 


1 


59 


.5875499 


.726098 


1.377224 


.8091879 


1 


60 


.5735764 


.700207 


1.428148 


.8191520 





60 


.5877853 


.726542 


1.376381 


.8090170 





■J^ 


Cosine. 


Cotangl Tang. | Sine. 


' 1 


1 1 Cosine. 1 


Cotang 


Tang. 


Sine. 


~ 



55° 
Note. — Secant = 1 4- cosine. Cosecant — 1 -^sine. 



54* 



162 



g.—PLANE TRIGONOMETRY. 



36° 



3.— Natural Sines, Tangents, Cotangents, Cosines. — (Continued.) 

(Versed sine = 1 — cosine ; co versed sine = 1 — sine.) 
37° 



' 


Sine. 1 Tang. | Cotang.| Cosine. 


11 ' 1 


Sine. 1 Tang. | Cotang. 


Cosine. 







.5877853 


.726542 


1.376381 


.8090170 


60 





.6018150 


.753554 


1.327044 


.7986355 


60 


1 


.5880206 


.726987 


1.375540 


.8088460 


59 


1 


.6020473 


.754010 


1.326242 


.7984604 


59 


2 


.5882558 


.727431 


1.374699 


.8086749 


58 


2 


.6022795 


.754466 


1.325439 


.7982853 


58 


3 


.5884910 


.727876 


1.373859 


.8085037 


57 


•3 


.6025117 


.754923 


1.324638 


.7981100 


57 


4 


.5887262 


.728321 


1.373019 


.8083325 


56 


4 


.6027439 


.755379 


1.323837 


.7979347 


56 


5 


.5889613 


.728767 


1.372180 


.8081612 


55 


5 


.6029760 


.755836 


1.323036 


.7977594 


55 


6 


.5891964 


.729212 


1.371342 


.8079899 


54 


6 


.6032080 


.756294 


1.322237 


.7975839 


54 


7 


.5894314 


.729658 


1.370504 


.8078185 


53 


7 


.6034400 


.756751 


1.321437 


.7974084 


53 


8 


.5896663 


.730104 


1.369667 


.8076470 


52 


8 


.6036719 


.757209 


1.320639 


.7972329 


52 


9 


.5899012 


.730550 


1.368831 


.8074754 


51 


9 


.6039038 


.757666 


1.319841 


.7970572 


51 


10 


.5901361 


.730996 


1.367995 


.8073038 


50 


10 


.6041356 


.758124 


1.319044 


.7968815 


50 


11 


.5903709 


.731442 


1.367161 


.8071321 


49 


11 


.6043674 


.758582 


1.318247 


.7967058 


49 


12 


.5906057 


.731889 


1.366326 


.8069603 


48 


12 


.6045991 


.759041 


1.317451 


.7965299 


48 


13 


.5908404 


.732336 


1.365493 


.8067885 


47 


13 


.6048308 


.759499 


1.316655 


.7963540 


47 


14 


.5910750 


.732783 


1.364660 


.8066166 


46 


14 


.6050624 


.759958 


1.315861 


.7961780 


46 


15 


.5913096 


.733230 


1.363827 


.8064446 


45 


15 


.6052940 


.760417 


1.315066 


.7960020 


45 


16 


.5915442 


.733677 


1.362996 


.8062726 


44 


16 


.6055255 


.760876 


1.314273 


.7958259 


44 


17 


.5917787 


.734125 


1.362165 


.8061005 


43 


17 


.6057570 


.761336 


1.313480 


.7956497 


43 


18 


.5920132 


.734573 


1.361335 


.8059283 


42 


18 


.6059884 


.761795 


1.312687 


.7954735 


42 


19 


.5922476 


.735021 


1.360505 


.8057560 


41 


19 


.6062198 


.762255 


1.311895 


.7952972 


41 


20 


.5924819 


.735469 


1.359676 


.8055837 


40 


20 


.6064511 


.762715 


1.311104 


.7951208 


40 


21 


.5927163 


.735917 


1.358848 


.8054113 


39 


21 


.6066824 


.763175 


1.310314 


.7949444 


39 


22 


.5929505 


.736366 


1.358020 


.8052389 


38 


22 


.6069136 


.763636 


1.309523 


.7947678 


38 


23 


.5931847 


.736814 


1.357193 


.8050664 


37 


23 


.6071447 


.764096 


1.308734 


.7945913 


37 


24 


.5934189 


.737263 


1.356367 


.804893S 


36 


24 


.6073758 


.764557 


1.307945 


.7944146 


36 


25 


.5936530 


.737712 


1.355541 


.8047211 


35 


25 


.6076069 


.765018 


1.307157 


.7942379 


35 


26 


.5938871 


.738162 


1.354716 


.8045484 


34 


26 


.6078379 


.765480 


1.306369 


.7940611 


34 


27 


.5941211 


.738611 


1.353891 


.8043756 


33 


27 


.6080689 


.765941 


1.305582 


.7938843 


33 


28 


.5943550 


.739061 


1.353068 


.8042028 


32 


28 


.6082998 


.766403 


1.304796 


.7937074 


32 


29 


.5945889 


.739511 


1.352244 


.8040299 


31 


29 


.6085306 


.766864 


1.304010 


.7935304 


31 


30 


.5948228 


.739961 


1.351422 


.8038569 


30 


30 


.6087614 


.767327 


1.303225 


.7933533 


30 


31 


.5950566 


.740411 


1.350600 


.8036838 


29 


31 


.6089922 


.767789 


1.302440 


.7931762 


29 


32 


.5952904 


.740861 


1.34^779 


.8035107 


28 


32 


.6092229 


.768251 


1.301656 


.7929990 


28 


33 


.5955241 


.741312 


1.348958 


.8033375 


27 


33 


.6094535 


.768714 


1.300873 


.7928218 


27 


34 


.5957577 


.741763 


1.348139 


.8031642 


26 


34 


.6096841 


.769177 


1.300090 


.7926445 


26 


35 


.5959913 


.742214 


1.347319 


.8029909 


25 


35 


.6099147 


.769640 


1.299308 


.7924671 


25 


36 


.5962249 


.742665 


1.346501 


.8028175 


24 


36 


.6101452 


.770103 


1.298526 


.7922896 


24 


37 


.5964584 


.743117 


1.345683 


.8026440 


23 


37 


.6103756 


.770567 


1.297745 


.7921121 


23 


38 


.5966918 


.743568 


1.344865 


.8024705 


22 


38 


.6106060 


.771030 


1.296964 


.7919345 


22 


39 


.5969252 


.744020 


1.344049 


.8022969 


21 


39 


.6108363 


.771494 


1.296185 


.7917569 


21 


40 


.5971586 


.744472 


1.343233 


.8021232 


20 


40 


.6110666 


.771958 


1.295405 


.7915792 


20 


41 


.5973919 


.744924 


1.342417 


.8019495 


19 


41 


.6112969 


.772423 


1.294627 


.7914014 


19 


42 


.5976251 


.745377 


1.341602 


.8017756 


18 


42 


.6115270 


.772887 


1.293848 


.7912235 


18 


43 


.5978583 


.745829 


1.340788 


.8016018 


17 


43 


.6117572 


.773352 


1.293071 


.7910456 


17 


44 


.5980915 


.746282 


1.339975 


.8014278 


16 


44 


.6119873 


.773817 


1.292294 


.7908676 


16 


45 


.5983246 


.746735 


1.339162 


.8012538 


15 


45 


.6122173 


.774282 


1.291517 


.7906896 


15 


46 


.5985577 


.747188 


1.338350 


.8010797 


14 


46 


.6124473 


.774748 


1.290742 


.7905115 


14 


47 


.5987906 


.747642 


1.337538 


.8009056 


13 


47 


.6126772 


.775213 


1.289966 


.7903333 


13 


48 


.5990236 


.748095 


1.336727 


.8007314 


12 


48 


.6129071 


.775679 


1.289192 


.7901550 


12 


49 


.5992565 


.748549 


1.335917 


.8005571 


11 


49 


.6131369 


.776145 


1.288418 


.7899767 


11 


50 


.5994893 


.749003 


1.335107 


.8003827 


10 


50 


.6133666 


.776611 


1.287644 


.7897983 


10 


51 


.5997221 


.749457 


1.334298 


.8002083 


9 


51 


.6135964 


.777078 


1.286871 


.7896198 


9 


52 


.5999549 


.749911 


1.333490 


.8000338 


8 


52 


.6138260 


.777544 


1.286099 


.7894413 


8 


53 


.6001876 


.750366 


1.332682 


.7998593 


7 


53 


.6140556 


.778011 


1.285327 


.7892627 


7 


54 


.6004202 


.750821 


1.331875 


.7996847 


6 


54 


.6142852 


.778478 


1.284556 


.7890841 


6 


55 


.6006528 


.751276 


1.331068 


.7995100 


5 


55 


.6145147 


.778946 


1.283786 


.7889054 


5 


56 


.6008854 


.751731 


1.330262 


.7993352 


4 


56 


.6147442 


.779413 


1.283016 


.7887266 


4 


67 


.6011179 


.752186 


1.329457 


.7991604 


3 


57 


.6149736 


.779881 


1.282246 


.7885477 


3 


58 


.6013503 


.752642 


1.328652 


.7989855 


2 


58 


.6152029 


.780349 


1.281477 


.7883688 


2 


59 


.6015827 


.753098 


1.327848 


.7988105 


1 


59 


.6154322 


.780817 


1.280709 


.7881898 


1 


60 


.6018150 


.753554 


1.327044 


.7986355 





60 


.6156615 


.781285 


1.279941 


.7880108 






I Cosine. ICotangI Tang. | Sine. \ ' \\ | Cosine. |Cotang| Tang. | Sine. | 



53° 
Note. — Secant = 1 -r- cosine. Cosecant = 1 -^sine. 



52= 



NATURAL SINES, ETC, 



163 



38° 



3.— Natural Sines, Tangents, Cotangents, Cosines. — (Continued.) 
(Versed sine = 1 — cosine; co versed sine= 1 — sine.) 

QQO 



t 


Sine. 


Tang. 


Co tang. 1 Cosine. 


1 II ' 


Sine. 


Tang. 


Cotang.l Cosine. 


1 





.6156615 


.781285 


1.279941 


.7880108 


60 





.6293204 


.809784 


1.234897 


.7771460 


60 


1 


.6158907 


.781754 


1.279174 


.7878316 


59 


1 


.6295464 


.810265 


1.234162 


.7769629 


59 


2 


.6161198 


.782222 


1.278407 


.7876524 


58 


2 


.6297724 


.810747 


1.233429 


.7767797 


58 


3 


.6163489 


.782691 


1.277641 


.7874732 


57 


3 


.6299983 


.811230 


1.232696 


.7765965 


57 


4 


.6165780 


.783161 


1.276876 


.7872939 


56 


4 


.6302242 


.811712 


1.231963 


.7764132 


56 


5 


.6168069 


.783630 


1.276111 


.7871145 


55 


5 


,6304500 


.812195 


1.231231 


.7762298 


55 


6 


.6170359 


.784100 


1.275347 


.7869350 


54 


6 


.6306758 


.812678 


1.230499 


.7760464 


54 


7 


.6172648 


.784570 


1.274583 


.7867555 


53 


7 


.6309015 


.813161 


1:229768 


.7758629 


53 


8 


.6174936 


.785040 


1.273820 


.7865759 


52 


8 


.6311272 


.813644 


1.229038 


.7756794 


52 


9 


.6177224 


.785510 


1.273057 


.7863963 


51 


9 


.6313528 


.814128 


1.228308 


.7754957 


51 


10 


.6179511 


.785980 


1.272295 


.7862165 


50 


10 


.6315784 


.814611 


1.227578 


.7753121 


50 


11 


.6181798 


.786451 


1.271534 


.7860367 


49 


11 


.6318039 


.815095 


1.226849 


.7751283 


49 


12 


.6184084 


.786922 


1.270773 


.7858569 


48 


12 


.6320293 


.815580 


1.226121 


.7749445 


48 


13 


.6186370 


.787393 


1.270013 


.7856770 


47 


13 


.6322547 


.816064 


1.225393 


.7747606 


47 


14 


.6188655 


.787864 


1.269253 


.7854970 


46 


14 


.6324800 


.816549 


1.224665 


.7745767 


46 


15 


.6190939 


.788336 


1.268494 


.7853169 


45 


15 


.6327053 


.817034 


1.223938 


.7743926 


45 


16 


.6193224 


.788808 


1.267735 


.7851368 


44 


16 


.6329306 


.817519 


1.223212 


.7742086 


44 


17 


.6195507 


.789280 


1.266977 


.7849566 


43 


17 


.6331557 


.818004 


1.222486 


.7740244 


43 


18 


.6197790 


.789752 


1.266219 


.7847764 


42 


18 


.6333809 


.818490 


1.221761 


.7738402 


42 


19 


.6200073 


.790224 


1.265462 


.7845961 


41 


19 


.6336059 


.818976 


1.221036 


.7736559 


41 


20 


.6202355 


.790697 


1.264706 


.7844157 


40 


20 


.6338310 


.819462 


1.220312 


.7734716 


40 


21 


.6204636 


.791170 


1.263950 


.7842352 


39 


21 


.6340559 


.819948 


1.219588 


.7732872 


39 


22 


.6206917 


.791643 


1.263195 


.7840547 


38 


22 


.6342808 


.820435 


1.218865 


.7731027 


38 


23 


.6209198 


.792116 


1.262440 


.7838741 


37 


23 


.6345057 


.820922 


1.218142 


.7729182 


37 


24 


.6211478 


.792590 


1.261686 


.7836935 


36 


24 


.6347305 


.821409 


1.217419 


.7727336 


36 


25 


.6213757 


.793064 


1.260932 


.7835127 


35 


25 


.6349553 


.821896 


1.216698 


.7725489 


35 


26 


.6216036 


.793537 


1.260179 


.7833320 


34 


26 


.6351800 


.822384 


1.215976 


.7723642 


34 


27 


.6218314 


.794012 


1.259426 


.7831511 


33 


27 


.6354046 


.822871 


1.215256 


.7721794 


33 


28 


.6220592 


.794486 


1.258674 


.7829702 


32 


28 


.6356292 


.823359 


1.214535 


.7719945 


32 


29 


.6222870 


.794961 


1.257923 


.7827892 


31 


29 


.6358537 


.823847 


1.213816 


.7718096 


31 


30 


.6225146 


.795435 


1.257172 


.7826082 


30 


30 


.6360782 


.824336 


1.213097 


.7716246 


30 


31 


.6227423 


.795911 


1.256421 


.7824270 


29 


31 


.6363026 


.824825 


1.212378 


.7714395 


29 


32 


.6229698 


.796386 


1.255672 


.7822459 


28 


32 


.6365270 


.825314 


1.211660 


.7712544 


28 


33 


.6231974 


.796861 


1.254922 


.7820646 


27 


33 


.6367513 


.825803 


1.210942 


.7710692 


27 


34 


.6234248 


.797337 


1.254174 


.7818833 


26 


34 


.6369756 


.826292 


1.210225 


.7708840 


26 


35 


.6236522 


.797813 


1.253426 


.7817019 


25 


35 


.6371998 


.826782 


1.209508 


.7706986 


25 


36 


.6238796 


.798289 


1.252678 


.7815205 


24 


36 


.6374240 


.827271 


1.208792 


.7705132 


24 


37 


.6241069 


.798765 


1.251931 


.7813390 


23 


37 


.6376481 


.827762 


1.208076 


.7703278 


23 


38 


.6243342 


.799242 


1.251184 


.7811574 


22 


38 


.6378721 


.828252 


1.207361 


.7701423 


22 


39 


.6245614 


.799719 


1.250438 


.7809757 


21 


39 


.6380961 


.828742 


1.206646 


.7699567 


21 


40 


.6247885 


.800196 


1.249693 


.7807940 


20 


40 


.6383201 


.829233 


1.205932 


.7697710 


20 


41 


.6250156 


.800673 


1.248948 


.7806123 


19 


41 


.6385440 


.829724 


1.205219 


.7695853 


19 


42 


.6252427 


.801151 


1.248204 


.7804304 


18 


42 


.6387678 


.830216 


1.204505 


.7693996 


18 


43 


.6254696 


.801628 


1.247460 


.7802485 


17 


43 


.6389916 


.830707 


1.203793 


.7692137 


17 


44 


.6256966 


.802106 


1.246716 


.7800665 


16 


44 


.6392153 


.831199 


1.203081 


.7690278 


16 


45 


.6259235 


.802584 


1.245974 


.7798845 


15 


45 


.6394390 


.831691 


1.202369 


.7688418 


15 


46 


.6261503 


.803063 


1.245232 


.7797024 


14 


46 


.6396626 


.832183 


1.201658 


.7686558 


14 


47 


.6263771 


.808541 


1.244490 


.7795202 


13 


47 


.6398862 


.832675 


1.200947 


.7684697 


13 


48 


.6266038 


.804020 


1.243749 


.7793380 


12 


48 


.6401097 


.833168 


1.200237 


.7682835 


12 


49 


.6268305 


.804499 


1.243008 


.7791557 


11 


49 


.6403332 


.833661 


1.199527 


.7680973 


11 


50 


.6270571 


.804979 


1.242268 


.7789733 


10 


50 


.6405566 


.834154 


1.198818 


.7679110 


10 


51 


.6272837 


.805458 


1.241529 


.7787909 


9 


51 


.6407799 


.834648 


1.198109 


.7677246 


9 


52 


.6275102 


.805938 


1.240790 


.7786084 


8 


52 


.6410032 


.835141 


1.197401 


.7675382 


8 


53 


.6277366 


.806418 


1.240051 


.7784258 


7 


53 


.6412264 


.835635 


1.196693 


.7673517 


7 


54 


.6279631 


.806898 


1.239313 


.7782431 


6 


54 


.6414496 


.836129 


1.195986 


.7671652 


6 


55 


.6281894 


.807378 


1.238576 


.7780604 


5 


55 


.6416728 


.836624 


1.195279 


.7669785 


5 


56 


.6284157 


.807859 


1.237839 


.7778777 


4 


56 


.6418958 


.837118 


1.194573 


.7667918 


4 


57 


.6286420 


.808340 


1.23710a 


.7776949 


3 


57 


.6421189 


.827613 


1.193867 


.7666051 


3 


58 


.6288682 


.808821 


1.236367 


.7775120 


2 


58 


.6423418 


.838108 


1.193162 


.7664183 


2 


59 


.6290943 


.809302 


1.235631 


.7773290 


1 


59 


.6425647 


.838604 


1.192457 


.7662314 


1 


60 


.6293204 


.809784 


1.234897 


.7771460 





60 


.6427876 


.839099 


1.191753 


.7660444 






I Cosine. ICotangl Tang. | Sine. \ ' \\ \ Cosine. ICotangj Tang. | Sine. | 



51" 



50^ 



Note. — Secant = 1 -J- cosine. Cosecant — 1-s-sine. 



164 



9,— PLANE TRIGONOMETRY. 



4Q' 



3.— Natural Sines, Tangents, Cotangents, Cosines.— (Continued.] 

(Versed sine =1 — cosine; coversed sine = 1 — sine.) 
41° 



JD Sine. 


Tang. 1 Cotang.l Cosine. 


1 II ' 1 Sine. 


Tang. 1 Cotang.l Cosine. 


l_ 





.6427876 


.839099 


1.191753 


.7660444 


60 





.6560590 


.869286 


1.150368 


.7547096 


60 


1 


.6430104 


.839595 


1.191049 


.7658574 


59 


1 


.6562785 


.869797 


1.149692 


.7545187 


59 


2 


.6432332 


.840091 


1.190346 


.7656704 


58 


2 


.6564980 


.870308 


1.149017 


.7543278 


58 


3 


.6434559 


.840587 


1.189643 


.7654832 


57 


3 


.6567174 


.870820 


1.148342 


.7541368 


57 


4 


.6436785 


.841084 


1.188941 


.7652960 


56 


4 


.6569367 


.871331 


1.147668 


.7539457 


56 


5 


.6439011 


.841581 


1.188239 


.7651087 


55 


5 


.6571560 


.871843 


1.146994 


.7537546 


55 


6 


.6441236 


.842078 


1.187538 


.7649214 


54 


6 


.6573752 


.872355 


1.146321 


.7535634 


54 


7 


.6443461 


.842575 


1.186837 


.7647340 


53 


7 


.6575944 


.872868 


1.145648 


.7533721 


53 


8 


.6445685 


.843073 


1.186136 


.7645465 


52 


8 


.6578135 


.873380 


1.144976 


.7531808 


52 


9 


.6447909 


.843570 


1.185437 


.7643590 


51 


9 


.6580326 


.873893 


1.144304 


.7529894 


51 


10 


.6450132 


.844068 


1.184737 


.7641714 


50 


10 


.6582516 


.874406 


1.143632 


.7527980 


50 


11 


.6452355 


.844567 


1.184038 


.7639838 


49 


11 


.6584706 


.874920 


1.142961 


.7526065 


49 


12 


.6454577 


.845065 


1.183340 


.7637960 


48 


12 


.6586895 


.875433 


1.142290 


.7524149 


48 


13 


.6456798 


.845564 


1.182642 


.7636082 


47 


13 


.6589083 


.875947 


1.141620 


.7522233 


47 


14 


.6459019 


.846063 


1.181944 


.7634204 


46 


14 


.6591271 


.876462 


1.140950 


.7520316 


46 


IS 


.6461240 


.846562 


1.181247 


.7632325 


45 


15 


.6593458 


.876976 


1.140281 


.7518398 


45 


16 


.6463460 


.847062 


1.180551 


.7630445 


44 


16 


.6595645 


.877491 


1.139612 


.7516480 


44 


17 


.6465679 


.847561 


1.179855 


.7628561 


43 


17 


.6597831 


.878006 


1.138944 


.7514561 


43 


18 


.6467898 


.848061 


1.179159 


.7626683 


42 


18 


.6600017 


.878521 


1.138276 


.7512641 


42 


19 


.6470116 


.848561 


1.178464 


.7624802 


41 


19 


.6602202 


.879037 


1.137608 


.7510721 


41 


20 


.6472334 


.849062 


1.177769 


.7622919 


40 


20 


.6604386 


.879552 


1.136941 


.7508800 


40 


21 


.6474551 


.849563 


1.177075 


.7621036 


39 


21 


.6606570 


.880068 


1.136274 


.7506879 


39 


22 


.6476767 


.850064 


1.176382 


.7619152 


38 


22 


.6608754 


.880585 


1.135608 


.7504957 


38 


23 


.6478984 


.850565 


1.175688 


.7617268 


37 


23 


.6610936 


.881101 


1.134942 


.7503034 


37 


24 


.6481199 


.851066 


1.174996 


.7615383 


36 


24 


.6613119 


.881618 


1.134277 


.7501111 


36 


25 


.6483414 


.851568 


1.174303 


.7613497 


35 


25 


.6615300 


.882135 


1.133612 


.7499187 


35 


26 


.6485628 


.852070 


1.173612 


.7611611 


34 


26 


.6617482 


.882653 


1.132947 


.7497262 


34 


27 


.6487842 


.852572 


1.172920 


.7609724 


33 


27 


.6619662 


.883170 


1.132283 


.7495337 


33 


28 


.6490056 


.853075 


1.172229 


.7607837 


32 


28 


.6621842 


.883688 


1.131620 


.7493411 


32 


29 


.6492268 


.853577 


1.171539 


.7605949 


31 


29 


.6624022 


.884206 


1.130957 


.7491484 


31 


30 


.6494480 


.854080 


1.170849 


.7604060 


30 


30 


.6626200 


.884725 


1.130294 


.7489557 


30 


31 


.6496692 


.854583 


1.170160 


.7602170 


29 


31 


.6628379 


.885244 


1.129632 


.7487629 


29 


32 


.6498903 


.855087 


1.169471 


.7600280 


28 


32 


.6630557 


.885763 


1.128970 


.7485701 


28 


33 


.6501114 


.855591 


1.168782 


.7598389 


27 


33 


.6632734 


.886282 


1.128308 


.7483772 


27 


34 


.6503324 


.856095 


1.168094 


.7596498 


26 


34 


.6634910 


.886801 


1.127647 


.7481842 


26 


35 


.6505533 


.856599 


1.167407 


.7594606 


25 


35 


.6637087 


.887321 


1.126987 


.7479912 


25 


36 


.6507742 


.857103 


1.166720 


.7592713 


24 


36 


.6639262 


.887841 


1.126327 


.7477981 


24 


37 


.6509951 


.857608 


1.166033 


.7590820 


23 


37 


.6641437 


.888361 


1.125667 


.7476049 


23 


38 


.6512158 


.858113 


1.165347 


.7588926 


22 


38 


.6643612 


.888882 


1.125008 


.7474117 


22 


39 


.6514366 


.858618 


1.164661 


.7587031 


21 


39 


.8645785 


.889403 


1.124349 


.7472184 


21 


40 


.6516572 


.859124 


1.163976 


.7585136 


20 


40 


.6647959 


.889924 


1.123690 


.7470251 


20 


41 


.6518778 


.859629 


1.163291 


.7583240 


19 


41 


.6650131 


.890445 


1.123032 


.7468317 


19 


42 


.6520984 


.860135 


1.162607 


.7581343 


18 


42 


.6652304 


.890967 


1.122375 


.7466382 


18 


43 


.6523189 


.860641 


1.161923 


.7579446 


17 


43 


.6654475 


.891489 


1.121718 


.7464446 


17 


44 


.6525394 


.861148 


1.161240 


.7577548 


16 


44 


.6656646 


.892011 


1.121061 


.7462510 


16 


45 


.6527598 


.861655 


1.160557 


.7575650 


15 


45 


.6658817 


.892534 


1.120405 


.7460574 


15 


46 


.6529801 


.862162 


1.159874 


.7573751 


14 


46 


.6660987 


.893056 


1.119749 


.7458636 


14 


47 


.6532004 


.862669 


1.159192 


.7571851 


13 


47 


.6663156 


.893579 


1.119094 


.7456699 


13 


48 


.6534206 


.863176 


1.158511 


.7569951 


12 


48 


.6665325 


.894103 


1.118439 


.7454760 


12 


49 


.6536408 


.863684 


1.157830 


.7568050 


11 


49 


.6667493 


.894626 


1.117784 


.7452821 


11 


50 


.6538609 


.864192 


1.157149 


.7566148 


10 


50 


.6669661 


.895150 


1.117130 


.7450881 


10 


51 


.6540810 


.864700 


1.156469 


.7564246 


9 


51 


.6671828 


.895674 


1.116476 


.7448941 


9 


52 


.6543010 


.865209 


1.155789 


.7562343 


8 


52 


.6673994 


.896199 


1.115823 


.7446999 


8 


53 


.6545209 


.865718 


1.155110 


.7560439 


7 


53 


.6676160 


.896723 


1.115170 


.7445058 


7 


54 


.6547408 


.866227 


1.154431 


.7558535 


6 


54 


.6678326 


.897248 


1.114518 


.7443115 


6 


55 


.6549607 


.866736 


1.153753 


.7556630 


5 


55 


.6680490 


.897773 


1.113866 


.7441173 


5 


56 


.6551804 


.867246 


1.153075 


.7554724 


4 


56 


.6682655 


.898299 


1.113214 


.7439229 


4 


57 


.6554002 


.867755 


1.152397 


.7552818 


3 


57 


.6684818 


.898825 


1.112563 


.7437285 


3 


58 


.6556198 


.868265 


1.151721 


.7550911 


2 


58 


.6686981 * 


.899351 


1.111912 


.7435340 


2 


59 


.6558395 


.868776 


1.151044 


.7549004 


1 


59 


.6889144 


.899877 


1.111262 


.7433394 


1 


60 


.6560590 


.869286 


1.150368 


.7547096 





60 


.6691306 


.900404 


1.110612 


.7431448 






I Cosine. ICotangl Tang. 1 Sine. 



1 Cosine. jCotangj Tang. | Sine^ | 



49^ 



48= 



Note. — Secant = 1 -^ cosine. 



Cosecant = l-r-sine. 



NATURAL SINES, ETC. 



165 



42' 



3.— Natural Sines, Tangents, Cotangents, Cosines. — (Continued.) 

(Versed sine =1 — cosine; coversed sine = 1 — sine.) 
43° 



' 1 Sine. 1 Tang. | Cotang.| Cosine. | || ' | Sine. | Tang. | Cotang.| Cosine. | 





.6691306 


.900404 


1.110612 


.7431448 


60 


.6819984 


.932515 


1.072368 


.7313537 


60 


1 


.6693468 


.900930 


1.109963 


.7429502 


59 


1 


.6822111 


.933059 


1.071743 


.7311553 


59 


2 


.6695628 


.901458 


1.109314 


.7427554 


58 


2 


.6824237 


.933603 


1.071118 


.7309568 


58 


3 


.6697789 


.901985 


1.108665 


.7425606 


57 


3 


.6826363 


.934147 


1.070494 


.7307583 


57 


4 


.6699948 


.902513 


1.108017 


.7423658 


56 


4 


.6828489 


.934692 


1.069870 


.7305597 


56 


5 


.6702108 


.903041 


1.107369 


.7421708 


55 


5 


.6830613 


.935238 


1.069246 


.7303610 


55 


6 


.6704266 


.903569 


1.106721 


.7419758 


54 


6 


.6832738 


.935783 


1.068623 


.7301623 


54 


7 


.6706424 


.904097 


1.106075 


.7417808 


53 


7 


.6834861 


.936329 


1.068000 


.7299635 


53 


8 


.6708582 


.904626 


1.105428 


.7415857 


52 


8 


.6836984 


.936875 


1.067377 


.7297646 


52 


9 


.6710739 


.905155 


1.104782 


.7413905 


51 


9 


.6839107 


.937421 


1.066755 


.7295657 


51 


10 


.6712895 


.905685 


1.104136 


.7411953 


50 


10 


.6841229 


.937968 


1.066134 


.7293668 


50 


11 


.6715051 


.906214 


1.103491 


.7410000 


49 


11 


.6843350 


.938515 


1.065512 


.7291677 


49 


12 


.6717206 


.906744 


1.102846 


.7408046 


48 


12 


.6845471 


.939062 


1.064891 


.7289686 


48 


13 


.6719361 


.907274 


1.102201 


.7406092 


47 


13 


.6847591 


.939610 


1.064271 


.7287695 


47 


14 


.6721515 


.907805 


1.101557 


.7404137 


46 


14 


.6849711 


.940157 


1.063651 


.7285703 


46 


15 


.6723668 


.908336 


1.100914 


.7402181 


45 


15 


.6851830 


.940706 


1.063031 


.7283710 


45 


16 


.6725821 


.908867 


1.100270 


.7400225 


44 


16 


.6853948 


.941254 


1.062411 


.7281716 


44 


17 


.6727973 


.909398 


1.099628 


.7398268 


43 


17 


.6856066 


.941803 


1.061792 


.7279722 


43 


18 


.6730125 


.909930 


1.098985 


.7396311 


42 


18 


.6858184 


.942352 


1.061174 


.7277728 


42 


19 


.6732276 


.910461 


1.098343 


.7394353 


41 


19 


.6860300 


.942901 


1.060556 


.7275732 


41 


20 


.6734427 


.910994 


1.097702 


.7392394 


40 


20 


.6862416 


.943451 


1.059938 


.7273736 


40 


21 


.6736577 


.911526 


1.097060 


.7390435 


39 


21 


.6864532 


.944001 


1.059320 


.7271740 


39 


22 


.6738727 


.912059 


1.096420 


.7388475 


38 


22 


.6866647 


.944551 


1.058703 


.7269743 


38 


23 


.6740876 


.912592 


1.095779 


.7386515 


37 


23 


.6868761 


.945102 


1.058086 


.7267745 


37 


24 


.6743024 


.913125 


1.095139 


.7384553 


36 


24 


.6870875 


.945653 


1.057470 


.7265747 


36 


25 


.6745172 


.913659 


1.094500 


.7382592 


35 


25 


.6872988 


.946204 


1.056854 


.7263748 


35 


26 


.6747319 


.914192 


1.093861 


.7380629 


34 


26 


.6875101 


.946755 


1.056238 


.7261748 


34 


27 


.6749466 


.914727 


1.093222 


.7378666 


33 


27 


.6877213 


.947307 


1.055623 


.7259748 


33 


28 


.6751612 


.915261 


1.092584 


.7376703 


32 


28 


.6879325 


.947859 


1.055008 


.7257747 


32 


29 


.6753757 


.915796 


1.091946 


.7374738 


31 


29 


.6881435 


.948411 


1.054394 


.7255746 


31 


30 


.6755902 


.916331 


1.091308 


.7372773 


30 


30 


.6883546 


.948964 


1.053780 


.7253744 


30 


31 


.6758046 


.916866 


1.090671 


.7370808 


29 


31 


.6885655 


.949517 


1.053166 


.7251741 


29 


32 


.6760190 


.917402 


1.090034 


.7368842 


28 


32 


.6887765 


.950070 


1.052553 


.7249738 


28 


33 


.6762333 


.917937 


1.089398 


.7366875 


27 


33 


.6889873 


.950624 


1.051940 


.7247734 


27 


34 


.6764476 


.918474 


1.088762 


.7364908 


26 


34 


.6891981 


.951178 


1.051327 


.7245729 


26 


35 


.6766618 


.919010 


1.088126 


.7362940 


25 


35 


.6894089 


.951732 


1.050715 


.7243724 


25 


36 


.6768760 


.919547 


1.087491 


.7360971. 


24 


36 


.6896195 


.952287 


1.050103 


.7241719 


24 


37 


.6770901 


.920084 


1.086857 


.7359002 


23 


37 


.6898302 


.952842 


1.049492 


.7239712 


23 


38 


.6773041 


.920621 


1.086222 


.7357032 


22 


38 


.6900407 


.953397 


1.048880 


.7237705 


22 


39 


.6775181 


.921159 


1.085588 


.7355061 


21 


39 


.6902512 


.953952 


1.048270 


.7235698 


21 


40 


.6777320 


.921696 


1.084955 


.7353090 


20 


40 


.6904617 


.954508 


1.047659 


.7233690 


20 


41 


.6779459 


.922235 


1.084322 


.7351118 


19 


41 


.6906721 


.955064 


1.047049 


.7231681 


19 


42 


.6781597 


.922773 


1.083689 


.7349146 


18 


42 


.6908824 


.955620 


1.046440 


.7229671 


18 


43 


.6783734 


.923312 


1.083057 


.7347173 


17 


43 


.6910927 


.956177 


1.045831 


.7227661 


17 


44 


.6785871 


.923851 


1.082425 


.7345199 


16 


44 


.6913029 


.956734 


1.045222 


.7225651 


16 


45 


.6788007 


.924391 


1.081793 


.7343225 


15 


45 


.6915131 


.957291 


1.044613 


.7223640 


15 


46 


.6790143 


.924930 


1.081162 


.7341250 


14 


46 


.6917232 


.957849 


1.044005 


.7221628 


14 


47 


.6792278 


.925470 


1.080532 


.7339275 


13 


47 


.6919332 


.958407 


1.043397 


.7219615 


13 


48 


.6794413 


.926010 


1.079901 


.7337299 


12 


48 


.6921432 


.958965 


1.042790 


.7217602 


12 


49 


.6796547 


.926550 


1.079271 


.7335322 


11 


49 


.6923531 


.959524 


1.042183 


.7215589 


11 


50 


.6798681 


.927091 


1.078642 


.7333345 


10 


50 


.6925630 


.960082 


1.041576 


.7213574 


10 


51 


.6800813 


.927632 


1.078013 


.7331367 


9 


51 


.6927728 


.960642 


1.040970 


.7211559 


9 


52 


.6802946 


.928173 


1.077384 


.7329388 


8 


52 


.6929825 


.961201 


1.040364 


.7209544 


8 


53 


.6805078 


.928715 


1.076756 


.7327409 


7 


53 


.6931922 


.961761 


1.039758 


.7207528 


7 


54 


.6807209 


.929257 


1.076128 


.7325429 


6 


54 


.6934018 


.962321 


1.039153 


.7205511 


6 


55 


.6809339 


.929799 


1.075500 


.7323449 


5 


55 


.6936114 


.962881 


1.038548 


.7203494 


5 


56 


.6811469 


.930342 


1.074873 


.7321467 


4 


56 


.6938209 


.963442 


1.037944 


.7201476 


4 


57 


.6813599 


.930884 


1.074246 


.7319486 


3 


57 


.6940304 


.964003 


1.037340 


.7199457 


3 


58 


.6815728 


.931428 


1.073620 


.7317503 


2 


58 


.6942398 


.964565 


1.036736 


.7197438 


2 


59 


.6817856 


.931971 


1.072994 


.7315521 


1 


59 


.6944491 


.965126 


1.036133 


.7195418 


1 


60 


.6819984 


.932515 


1.072368 


.7313537 





60 


.6946584 


.965688 


1.035530 


.7193398 


1 CoslneT 


ICotangI Tang. | Sine. | ' 11 1 Cosine. 


ICotangI Tang. | Sine. | ' 




Note.- 


-Secan 


t =l-h 


cosine. 


47 
( 


3 

^ose 


cant=l- 


«-sine. 






46° 



166 



9— PLANE TRIGONOMETRY. 



44< 



3. — Natural Sines, Tangents, Cotangents, Cosines. — (Concluded.) 

(Versed sine = 1 — cosine; coversed sine= 1 — sine.) 
44° 



' 1 


Sine. 1 Tang. | Cotang.| Cosine. 


1 


1 ' 1 


Sine. 1 Tang. | Cotang.| Cosine. 


_1 





.6946584 


.965688 


1.035530 


.7193398 


60 


30 


.7009093 


.982697 


1.017607 


.7132504 


30 


1 


.6948676 


.966251 


1.034927 


.7191377 


59 


31 


.7011167 


.983269 


1.017015 


.7130465 


29 


2 


.6950767 


.966813 


1.034325 


.7189355 


58 


32 


.7013241 


.983841 


1.016423 


.7128426 


28 


3 


.6952858 


.967376 


1.033723 


.7187333 


57 


33 


.7015314 


.984414 


1.015832 


.7126385 


27 


4 


.6954949 


.967939 


1.033122 


.7185310 


56 


34 


.7017387 


.984987 


1.015241 


.7124344 


26 


5 


.6957039 


.968503 


1.032520 


.7183287 


55 


35 


.7019459 


.985560 


1.014651 


.7122303 


25 


6 


.6959128 


.969067 


1.031919 


.7181263 


54 


36 


.7021531 


.986133 


1.014061 


.7120260 


24 


7 


.6961217 


.969631 


1.031319 


.7179238 


53 


37 


.7023601 


.986707 


1.013471 


.7118218 


23 


8 


.6963305 


.970196 


1.030719 


.7177213 


52 


38 


.7025672 


.987282 


1.012881 


.7116174 


22. 


9 


.6965392 


.970761 


1.030119 


.7175187 


51 


39 


.7027741 


.987856 


1.012292 


.7114130 


21 


10 


.6967479 


.971326 


1.029520 


.7173161 


50 


40 


.7029811 


.988431 


1.011703 


.7112086 


20 


11 


.6969565 


.971891 


1.028921 


.7171134 


49 


41 


.7031879 


.989006 


1.011115 


.7110041 


19 


12 


.6971651 


.972457 


1.028322 


.7169106 


48 


42 


.7033947 


.989582 


1.010527 


.7107995 


18^ 


13 


.6973736 


.973023 


1.027724 


.7167078 


47 


43 


.7036014 


.990158 


1.009939 


.7105948 


17 


U 


.6975821 


.973590 


1.027126 


.7165049 


46 


44 


.7038081 


.990734 


1.009352 


.7103901 


16 


15 


.6977905 


.974156 


1.026528 


.7163019 


45 


45 


.7040147 


.991311 


1.008764 


.7101854 


15 


16 


.6979988 


.974724 


1.025931 


.7160989 


44 


46 


.7042213 


.991888 


1.008178 


.7099806 


14 


17 


.6982071 


.975291 


1.025334 


.7158959 


43 


47 


.7044278 


.992465 


1.007591 


.7097757 


13 


18 


.6984153 


.975859 


1.024738 


.7156927 


42 


48 


.7046342 


.993042 


1.007005 


.7095707 


12 


19 


.6986234 


.976427 


1.024141 


.7154895 


41 


49 


.7048406 


.993620 


1.006420 


.7093657 


11 


20 


.6988315 


.976995 


1.023546 


.7152863 


40 


50 


.7050469 


.994199 


1.005834 


.7091607 


10 


21 


.6990396 


.977564 


1.022950 


.7150830 


39 


51 


.7052532 


.994777 


1.005249 


.7089556 


9 


22 


.6992476 


.978133 


1.022355 


.7148796 


38 


52 


.7054594 


.995356 


1.004665 


.7087504 


8 


23 


.6994555 


.978702 


1.021760 


.7146762 


37 


53 


.7056655 


.995935 


1.004080 


.7085451 


7 


24 


.6996633 


.979272 


1.021166 


.7144727 


36 


54 


.7058716 


.996515 


1.003496 


.7083398 


6 


25 


.6998711 


.979842 


1.020572 


.7142691 


35 


55 


.7060776 


.997095 


1 .'002913 


.7081345 


5 


26 


.7000789 


.980412 


1.019978 


.7140655 


34 


56 


.7062835 


.997675 


1.002329 


.7079291 


4 


27 


.7002866 


.980983 


1.019385 


.7138618 


33 


57 


.7064894 


.998256 


1.001746 


.7077236 


3 


28 


.7004942 


.981554 


1.018792 


.7136581 


32 


58 


.7066953 


.998837 


1.001164 


.7075180 


2 


29 


.7007018 


.982125 


1.018199 


.7134543 


31 


59 


.7069011 


.999418 


1.000581 


.7073124 


1 


30 


.7009093 


.982697 


1.017607 


.7132504 


30 


60 


.7071068 


1.00000 


1.000000 


.7071068 






I Cosine. ICotangl Tang. | Sine. \ ' \\ \ Cosine. ICotangl Tang. | Sine. | 



45"= 



45^ 



Note. — Secant =1-^ cosine. 



Cosecant = 1-5-sine. 



NATURAL SECANTS, ETC, 



167 



4. — Natural Secants, Cosecants, (Exsecants, Coexsecants).* 

Secants. 



' 1 


0'> 


P 1 


2° 


3° 


40 


5° 


6<> 


70 


8° 


9» 







1.00000 


1.00015 


1.00061 


1.00137 


1.00244 


1.00382 


1.00551 


1.00751 


1.00983 


1.01247 


60 


1 


.00000 


.00016 


.00062 


.00139 


.00246 


.00385 


.00554 


.00755 


.00987 


.01251 


59 


2 


.00000 


.00016 


.00063 


.00140 


.00248 


.00387 


.00557 


.00758 


.00991 


.01256 


58 


3 


.00000 


.00017 


.00064 


.00142 


.00250 


.00390 


.00560 


.00762 


.00995 


.01261 


57 


4 


.00000 


.00017 


.00065 


.00143 


.00252 


.00392 


.00563 


.00765 


.00999 


.01265 


56 


5 


1.00000 


1.00018 


1.00066 


1.00145 


1.00254 


1.00395 


1.00566 


1.00769 


1.01004 


1.01270 


55 


6 


.00000 


.00018 


.00067 


.00147 


.00257 


.00397 


.00569 


.00773 


.01008 


.01275 


54 


7 


.00000 


.00019 


.00068 


.00148 


.00259 


.00400 


.00573 


.00776 


.01012 


.01279 


53 


8 


.00000 


.00020 


.00069 


.00150 


.00261 


.00403 


.00576 


.00780 


.01016 


.01284 


52 


9 


.00000 


.00020 


.00070 


.00151 


.00263 


.00405 


.00579 


.00784 


.01020 


.01289 


51 


10 


1.00000 


1.00021 


1.00072 


1.00153 


1.00265 


1.00408 


1.00582 


1.00787 


1.01024 


1.01294 


50 


11 


.00001 


.00021 


.00073 


.00155 


.00267 


.00411 


.00585 


.00791 


.01029 


.01298 


49 


12 


.00001 


.00022 


.00074 


.00156 


.00269 


.00413 


.00588 


.00795 


.01033 


.01303 


48 


13 


.00001 


.00023 


.00075 


.00158 


.00271 


.00416 


.00592 


.00799 


.01037 


.01308 


47 


14 


.00001 


.00023 


.00076 


.00159 


.00274 


.00419 


.00595 


.00802 


.01041 


.01313 


46 


15 


1.00001 


1.00024 


1.00077 


1.00161 


1.00276 


1.00421 


1.00598 


1.00806 


1.01046 


1.01318 


45 


If 


.00001 


.00024 


.00078 


.00163 


.00278 


.00424 


.00601 


.00810 


.01050 


.01322 


44 


17 


.00001 


.00025 


.00079 


.00164 


.00280 


.00427 


.00604 


.00813 


.01054 


.01327 


43 


18 


.00001 


.00026 


.00081 


.00166 


.00282 


.00429 


.00608 


.00817 


.01059 


.01332 


42 


19 


.00002 


.00026 


.00082 


.00168 


.00284 


.00432 


.00611 


.00821 


.01063 


.01337 


41 


20 


1.00002 


1.00027 


1.00083 


1.00169 


1.00287 


1.00435 


1.00614 


1.00825 


1.01067 


1.01342 


40 


21 


.00002 


.00028 


.00084 


.00171 


.00289 


.00438 


.00617 


.00828 


.01071 


.01346 


39 


22 


.00002 


.00028 


.00085 


.00173 


.00291 


.00440 


.00621 


.00832 


.01076 


.01351 


38 


23 


.00002 


.00029 


.00087 


.00175 


.00293 


.00443 


.00624 


.00836 


.01080 


.01356 


37 


24 


.00002 


.00030 


.00088 


.00176 


.00296 


.00446 


.00627 


.00840 


.01084 


.01361 


36 


25 


1.00003 


1.00031 


1.00089 


1.00178 


1.00298 


1.00449 


1.00630 


1.00844 


1.01089 


1.01366 


35 


26 


.00003 


.00031 


.00090 


.00180 


.00300 


.00451 


.00634 


.00848 


.01093 


.01371 


34 


27 


.00003 


.00032 


.00091 


.00182 


.00302 


.00454 


.00637 


.00851 


.01097 


.01376 


33 


28 


.00003 


.00033 


.00093 


.00183 


.00305 


.00457 


.00640 


.00855 


.01102 


.01381 


32 


29 


.00004 


.00034 


.00094 


.00185 


.00307 


.00460 


.00644 


.00859 


.01106 


.01386 


31 


30 


1.00004 


1.00034 


1.00095 


1.00187 


1.00309 


1.00463 


1.00647 


1.00863 


1.01111 


1.01391 


30 


31 


.00004 


.00035 


.00097 


.00189 


.00312 


.00465 


.00650 


.00867 


.01115 


.01395 


29 


32 


.00004 


.00036 


.00098 


.00190 


.00314 


.00468 


.00654 


.00871 


.01119 


.01400 


28 


33 


.00005 


.00037 


.00099 


.00192 


.00316 


.00471 


.00657 


.00875 


.01124 


.01405 


27 


34 


.00005 


.00037 


.00100 


.00194 


.00318 


.00474 


.00660 


.00878 


.01128 


.01410 


26 


35 


1.00005 


1.00038 


1.00102 


1.00196 


1.00321 


1.00477 


1.00664 


1.00882 


1.01133 


1.01415 


25 


36 


.00005 


.00039 


.00103 


.00198 


.00323 


.00480 


.00667 


.00886 


.01137 


.01420 


24 


37 


.00006 


.00040 


.00104 


.00200 


.00326 


.00482 


.00671 


.00890 


.01142 


.01425 


23 


38 


.00006 


.00041 


.00106 


.00201 


.00328 


.00485 


.00674 


.00894 


.01146 


.01430 


22 


39 


.00006 


.00041 


.00107 


.00203 


.00330 


.00488 


.00677 


.00898 


.01151 


.01435 


21 


40 


1.00007 


1.00042 


1.00108 


1.00205 


1.00333 


1.00491 


1.00681 


1.00902 


1.01155 


1.01440 


20 


41 


.00007 


.00043 


.00110 


.00207 


.00335 


.00494 


.00684 


.00906 


.01160 


.01445 


19 


42 


.00007 


.00044 


.00111 


.00209 


.00337 


.00497 


.00688 


.00910 


.01164 


.01450 


18 


43 


.00008 


.00045 


.00113 


.00211 


.00340 


.00500 


.00691 


.00914 


.01169 


.01455 


17 


44 


.00008 


."00046 


.00114 


.00213 


.00342 


.00503 


.00695 


.00918 


.01173 


.01461 


16 


45 


1.00009 


1.00047 


1.00115 


1.00215 


1.00345 


1.00506 


1.00698 


1.00922 


1.01178 


1.01466 


15 


46 


.00009 


.00048 


.00117 


.00216 


.00347 


.00509 


.00701 


.00926 


.01182 


.01471 


14 


47 


.00009 


.00048 


.00118 


.00218 


.00350 


.00512 


.00705 


.00930 


.01187 


.01476 


13 


48 


.00010 


.00049 


.00120 


.00220 


.00352 


.00515 


.00708 


.00934 


.01191 


.01481 


12 


49 


.00010 


.00050 


.00121 


.00222 


.00354 


.00518 


.00712 


.00938 


.01196 


.01486 


11 


50 


1.00011 


1.00051 


1.00122 


1.00224 


1.00357 


1.00521 


1.00715 


1.00942 


1.01200 


1.01491 


10 


51 


.00011 


.00052 


.00124 


.00226 


.00359 


.00524 


.00719 


.00946 


.01205 


.01496 


9 


52 


.00011 


.00053 


.00125 


.00228 


.00362 


.00527 


.00722 


.00950 


.01209 


.01501 


8 


53 


.00012 


.00054 


.00127 


.00230 


.00364 


.00530 


.00726 


.00954 


.01214 


.01506 


7 


54 


.00012 


.00055 


.00128 


.00232 


.00367 


.00533 


.00730 


.00958 


.01219 


.01512 


6 


55 


1.00013 


1.00056 


1.00130 


1.00234 


1.00369 


1.00536 


1.00733 


1.00962 


1.01223 


1.01517 


5 


56 


.00013 


.00057 


.00131 


.00236 


.00372 


.00539 


.00737 


.00966 


.01228 


.01522 


4 


57 


.00014 


.00058 


.00133 


.00238 


.00374 


.00542 


.00740 


.00970 


.01233 


.01527 


3 


58 


.00014 


.00059 


.00134 


.00240 


.00377 


.00545 


.00744 


.00975 


.01237 


.01532 


2 


59 


.00015 


.00060 


.00136 


.00242 


.00379 


.00548 


.00747 


.00979 


.01242 


.01537 


1 


60 


1.00015 


1.00061 


1.00137 


1.00244 


1.00382 


1.00551 


1.00751 


1.00983 


1.01247 


1.01543 






I 87° I 



I 85° \ 84° I 83° I 82° | 81° | 80° | ^ 



Cosecants. 
* Exsecant = secant — 1 ; coexsecant = cosecant — 1. 



168 



9.— PLANE TRIGONOMETRY, 



4. — Natural Secants, Cosecants (Exsecants, Coexsecants).* — (Cont'd.) 

Secants. 



I 



' 1 


10*> 1 


11° 1 


120 1 


13° 1 14° 1 15° 1 16° 1 


17° 1 


18° 1 


19° 1 





1.01543 


1.01872 


1.02234 


1.02630 


1.03061 


1.03528 1.04030 


1.04569 


1.05146 


1.05762 


60 


1 


.01548 


.01877 


.02240 


.02637 


.03069 


.03536 


.04039 


.04578 


.05156 


.05773 


59 


2 


.01553 


.01883 


. 02247 


.02644 


.03076 


.03544 


. 04047 


. 04588 


.05166 


. 05783 


58 


3 


.01558 


.01889 


.02253 


.02651 


.03084 


.03552 


.04056 


.04597 


.05176 


.05794 


57 


4 


.01564 


.01895 


.02259 


.02658 


.03091 


.03560 


.04065 


.04606 


.05186 


.05805 


56 


5 


1.01569 


1.01901 


1.02266 


1.02665 


1.03099 


1.03568 


1.04073 


1.04616 


1.05196 


1.05815 


55 


6 


.01574 


.01906 


.02272 


.02672 


.03106 


.03576 


.04082 


.04625 


.05206 


.05826 


54 


7 


.01579 


.01912 


.02279 


.02679 


.03114 


.03584 


.04091 


.04635 


.05216 


.05836 


53 


8 


.01585 


.01918 


.02285 


.02686 


.03121 


.03592 


.04100 


.04644 


.05226 


.05847 


52 


9 


.01590 


.01924 


.02291 


.02693 


.03129 


.03601 


.04108 


.04653 


.05236 


.05858 


51 


10 


1.01595 


1.01930 


1.02298 


1.02700 


1.03137 


1.03609 


1.04117 


1.04663 


1.05246 


1.05869 


50 


11 


.01601 


.01936 


.02304 


.02707 


.03144 


.03617 


.04126 


.04672 


.05256 


.05879 


49 


12 


.01606 


.01941 


.02311 


.02714 


.01352 


.03625 


.04135 


.04682 


.05266 


.05890 


48 


13 


.01611 


.01947 


.02317 


.02721 


.03159 


.03633 


.04144 


.04691 


.05276 


.05901 


47 


14 


.01616 


.01953 


.02323 


.02728 


.03167 


.03642 


.04152 


.04700 


.05286 


.05911 


46 


15 


1.01622 


1.01959 


1.02330 


1.02735 


1.03175 


1.03650 


1.04161 


1.04710 


1.05297 


1.05922 


45 


16 


.01627 


.01965 


.02336 


.02742 


.03182 


.03658 


.04170 


.04719 


.05307 


.05933 


44 


17 


.01633 


.01971 


.02343 


.02749 


.03190 


.03666 


.04179 


.04729 


.05317 


.05944 


43 


18 


.01638 


.01977 


.02349 


.02756 


.03198 


.03674 


.04188 


.04738 


.05327 


.05955 


42 


19 


.01643 


.01983 


.02356 


.02763 


.03205 


.03683 


.04197 


.04748 


.05337 


.05965 


41 


20 


1.01649 


1.01989 


1.02362 


1.02770 


1.03213 


1.03691 


1.04206 


1.04757 


1.05347 


1.05976 


40 


21 


.01654 


.01995 


.02369 


.02777 


.03221 


.03699 


.04214 


.04767 


.05357 


.05987 


39 


22 


.01659 


.02001 


.02375 


.02784 


.03228 


.03708 


.04223 


.04776 


.05367 


.05998 


38 


23 


.01665 


.02007 


.02382 


.02791 


.03236 


.03716 


.04232 


.04786 


.05378 


.06009 


37 


24 


.01670 


.02013 


.02388 


.02799 


.03244 


.03724 


.04241 


.04795 


.05388 


.06020 


36 


25 


1.01676 


1.02019 


1.02395 


1.02806 


1.03251 


1.03732 


1.04250 


1.04805 


1.05398 


1.06030 


35 


26 


.01681 


.02025 


.02402 


.02813 


.03259 


.03741 


.04259 


.04815 


.05408 


.06041 


34 


27 


.01687 


.02031 


.02408 


.02820 


.03267 


.03749 


.04268 


.04824 


.05418 


.06052 


33 


28 


.01692 


.02037 


.02415 


.02827 


.03275 


.03758 


.04277 


.04834 


.05429 


.06063 


32 


29 


.01698 


.02043 


.02421 


.02834 


.03282 


.03766 


.04286 


.04843 


.05439 


.06074 


31 


30 


1.01703 


1.02049 


1.02428 


1.02842 


1.03290 


1.03774 


1.04295 


1.04853 


1.05449 


1.06085 


30 


31 


.01709 


.02055 


.02435 


.02849 


.03298 


.03783 


.04304 


.04863 


.05460 


.06096 


29 


32 


.01714 


.02061 


.02441 


.02856 


.03306 


.03791 


.04313 


.04872 


.05470 


.06107 


28 


33 


.01720 


.02067 


.02448 


.02863 


.03313 


.03799 


.04322 


.04882 


.05480 


.06118 


27 


34 


.01725 


.02073 


.02454 


.02870 


.03321 


.03808 


.04331 


.04891 


.05490 


.06129 


26 


35 


1.01731 


1.02079 


1.02461 


1.02878 


1.03329 


1.03816 


1.04340 


1.04901 


1.05501 


1.06140 


25 


36 


.01736 


.02085 


.02468 


.02885 


.03337 


.03825 


.04349 


.04911 


.05511 


.06151 


24 


37 


.01742 


.02091 


.02474 


.02892 


.03345 


.03833 


.04358 


.04920 


.05521 


.06162 


23 


38 


.01747 


.02097 


.02481 


.02899 


.03353 


.03842 


.04367 


.04930 


.05532 


.06173 


22 


39 


.01753 


.02103 


.02488 


.02907 


.03360 


.03850 


.04376 


.04940 


.05542 


.06184 


21 


40 


1.01758 


1.02110 


1.02494 


1.02914 


1.03368 


1.03858 


1.04385 


1.04950 


1.05552 


1.06195 


20 


41 


.01764 


.02116 


.02501 


.02921 


.03376 


.03867 


.04394 


.04959 


.05563 


.06206 


19 


42 


.01769 


.02122 


.02508 


.02928 


.03384 


.03875 


.04403 


.04969 


.05573 


.06217 


18 


43 


.01775 


.02128 


.02515 


.02936 


.03392 


.03884 


.04413 


.04979 


.05584 


.06228 


17 


44 


.01781 


.02134 


.02521 


.02943 


.03400 


.03892 


.04422 


.04989 


.05594 


.06239 


16 


45 


1.01786 


1.02140 


1.02528 


1.02950 


1.03408 


1.03901 


1.04431 


1.04998 


1.05604 


1.06250 


15 


46 


.01792 


.02146 


.02535 


.02958 


.03416 


.03909 


.04440 


.05008 


.05615 


.06261 


14 


47 


.01798 


.02153 


.02542 


.02965 


.03424 


.03918 


.04449 


.05018 


.05625 


.06272 


13 


48 


.01803 


.02159 


.02548 


.02972 


.03432 


.03927 


.04458 


.05028 


.05636 


.06283 


12 


49 


.01809 


.02165 


.02555 


.02980 


.03439 


.03935 


.04468 


.05038 


.05646 


.06295 


11 


50 


1.01815 


1.02171 


1.02562 


1.02987 


1.03447 


1.03944 


1.04477 


1.05047 


1.05657 


1.06306 


10 


51 


.01820 


.02178 


.02569 


.02994 


.03455 


.03952 


.04486 


.05057 


.05667 


.06317 


9 


52 


.01826 


.02184 


.02576 


.03002 


.03463 


.03961 


.04495 


.05067 


.05678 


.06328 


8 


53 


.01832 


.02190 


.02582 


.03009 


.03471 


.03969 


.04504 


.05077 


.05688 


.06339 


7 


54 


.01837 


.02196 


.02589 


.03017 


.03479 


.03978 


.04514 


.05087 


.05699 


.06350 


6 


55 


1.01843 


.02203 


1.02596 


1.03024 


1.03487 


1.03987 


1.04523 


1.05097 


1.05709 


1.06362 


5 


56 


.01849 


.02209 


.02603 


.03032 


.03495 


.03995 


.04532 


.05107 


.05720 


.06373 


4 


57 


.01854 


.02215 


.02610 


.03039 


.03503 


.04004 


.04541 


.05116 


.05730 


.06384 


3 


58 


.01860 


.02221 


.02617 


.03046 


.03512 


.04013 


.04551 


.05126 


.05741 


.06395 


2 


59 


.01866 


.02228 


.02624 


.03054 


.03520 


.04021 


.04560 


.05136 


.05751 


.06407 


1 


60 


1.01872 


1.02234 


1.02630 


1.03061 


1.03528 


1.04030 


1.04569 


1.05146 


1.05762 


1.06418 







1 79° 


1 78" 


1 77° 


1 76° 1 75° 1 74° 1 73° 


1 72° 


1 71° 


70° 1 ' 



Cosecants. 
* Exsecant = secant — 1 ; coexsecant = cosecant — 1 . 



NATURAL SECANTS, ETC. 



169 



4. — Natural Secants, Cosecants (Exsecants, Coexsecants).* — (Cont'd.) 

Secants. 



' 


20° 


21° 


1 22° 


23° 


24° 


25° 


26° 1 


27° 1 


28° 1 


29° 1 







1.06418 


1.07115 


1.07853 


1.08636 


1.09464 


1.10338 


1.11260 


1.12233 


1.132.57 


1.14335 


60 


1 


.06429 


.07126 


.07866 


.08649 


.09478 


.103.53 


.11276 


.12249 


.13275 


.14354 


59 


2 


.06440 


.07138 


.07879 


. 08663 


.09492 


.10368 


.11292 


.12266 


. 13292 


.14372 


58 


3 


.06452 


.07150 


.07892 


.08676 


.09506 


.10383 


.11308 


.12283 


.13310 


.14391 


57 


4 


.06463 


.07162 


.07904 


.08690 


.09520 


.10398 


.11323 


.12299 


.13327 


.14409 


56 


S 


1.06474 


1.07174 


.07917 


1.08703 


1.09535 


1.10413 


1.11339 


1.12316 


1 . 13345 


1.14428 


55 


6 


.03486 


.07186 


.07930 


.08717 


.09549 


.10428 


.11355 


.12333 


.13362 


.14446 


54 


7 


.06497 


.07199 


.07943 


.08730 


.09563 


.10443 


.11371 


.12349 


.13380 


.14465 


53 


8 


.06508 


.07211 


.07955 


.08744 


.09577 


.10458 


.11387 


.12366 


.13398 


.14483 


52 


9 


.06520 


.07223 


.07968 


.08757 


.09592 


.10473 


.11403 


.12383 


.13415 


.14502 


51 


10 


1.06531 


1.07235 


1.07981 


1.08771 


1.09606 


1.10488 


1.11419 


1.12400 


1.13433 


1.14521 


50 


11 


.06542 


.07247 


.07994 


.08784 


.09620 


.10503 


.11435 


.12416 


.13451 


.14539 


49 


12 


.06554 


.07259 


.08006 


.08798 


.09635 


.10518 


.11451 


.12433 


.13468 


.14558 


48 


13 


.06565 


.07271 


.08019 


.08811 


.09649 


.10533 


.11467 


.12450 


.13486 


.14576 


47 


U 


.06577 


.07283 


.08032 


.08825 


.09663 


.10549 


.11483 


.12467 


13504 


.14595 


46 


15 


1.06588 


1.07295 


1.08045 


1.08839 


1.09678 


1.10564 


1.11499 


1.12484 


1.13521 


1.14614 


45 


16 


.06600 


.07307 


.08058 


.08852 


.09692 


.10579 


.11515 


.12501 


.13539 


.14632 


44 


17 


.06611 


.07320 


.08071 


.08866 


.09707 


.10594 


.11531 


.12518 


.13557 


.14651 


43 


18 


.06622 


.07332 


.08084 


.08880 


.09721 


.10609 


.11547 


.12534 


.13575 


.14670 


42 


19 


.06634 


.07344 


.08097 


.08893 


.09735 


.10625 


.11563 


.12.551 


.13593 


.14689 


41 


20 


1.06645 


1.07356 


1.08109 


1.08907 


1.09750 


1.10640 


1.11579 


1.12568 


1.13610 


1.14707 


40 


21 


.06657 


.07368 


.08122 


.08921 


.09764 


.10655 


.11595 


.12585 


.13628 


.14726 


39 


22 


.06668 


.07380 


.08135 


.08934 


.09779 


.10670 


.11611 


.12602 


.13646 


.14745 


38 


23 


.06680 


.07393 


.08148 


.08948 


.09793 


.10686 


.11627 


.12619 


.13664 


.14764 


37 


24 


.06691 


.07405 


.08161 


.08962 


.09808 


.10701 


.11643 


.12636 


.13682 


.14782 


36 


25 


1.06703 


1.07417 


1.08174 


1.08975 


1.09822 


1.10716 


1.11659 


1.12653 


1.13700 


1.14801 


35 


26 


.06715 


.07429 


.08187 


.08989 


.09837 


.10731 


.11675 


.12670 


.13718 


.14820 


34 


27 


.06726 


.07442 


.08200 


.09003 


.09851 


.10747 


.11691 


.12687 


.13735 


.14839 


33 


28 


.06738 


.07454 


.08213 


.09017 


.09866 


.10762 


.11708 


.12704 


.13753 


.14858 


32 


29 


.06749 


.07466 


.08226 


.09030 


.09880 


.10777 


.11724 


.12721 


.13771 


.14877 


31 


30 


1.06761 


1.17479 


1.08239 


1.09044 


1.09895 


1.10793 


1.11740 


1.12738 


1.13789 


1.14896 


30 


31 


.06773 


.07491 


.08252 


.09058 


.09909 


.10808 


.11756 


.12755 


.13807 


n .14914 


29 


32 


.06784 


.07503 


.08265 


.09072 


.09924 


.10824 


.11772 


.12772 


.13825 


.14933 


28 


33 


.06796 


.07516 


.08278 


.09086 


.09939 


.10839 


.11789 


.12789 


.13843 


.14952 


27 


34 


.06807 


.07528 


.08291 


.09099 


.09953 


.10854 


.11805 


.12807 


.13861 


.14971 


26 


35 


1.06819 


1.07540 


1.08305 


1.09113 


1.09968 


1.10870 


1.11821 


1.12824 


1.13879 


1.14990 


25 


36 


.06831 


.07553 


.08318 


.09127 


.09982 


.10885 


.11838 


.12841 


.13897 


.15009 


24 


37 


.06843 


.07565 


.08331 


.09141 


.09997 


.10901 


.11854 


.12858 


.13916 


.15028 


23 


38 


.06854 


.07578 


.08344 


.09155 


.10012 


.10916 


.11870 


.12875 


.13934 


.15047 


22 


39 


.06866 


.07590 


.08357 


.09169 


.10026 


.10932 


.11886 


.12892 


.13952 


.15066 


21 


40 


1.06878 


1.07602 


1.08370 


1.09183 


1.10041 


1.10947 


1.11903 


1.12910 


1.13970 


1.15085 


20 


41 


.06889 


.07615 


.08383 


.01197 


.10055 


.10963 


.11919 


.12927 


.13988 


.15105 


19 


42 


.06901 


.07627 


.08397 


.09211 


.10071 


.10978 


.11936 


.12944 


.14006 


.15124 


18 


43 


.06913 


.07640 


.08410 


.09224 


.10085 


.10994 


.11952 


.12961 


.14024 


.15143 


17 


44 


.06925 


.07652 


.08423 


.09238 


.10100 


.11009 


.11968 


.12979 


.14042 


.15162 


16 


45 


1.06936 


1.07665 


1.08436 


1.09252 


1.10115 


1.11025 


1.11985 


1.12996 


1.14061 


1.15181 


15 


46 


.06948 


.07677 


.08449 


.09266 


.10130 


.11041 


.12001 


.13013 


.14079 


.15200 


14 


47 


.06960 


.07690 


.08463 


.09280 


.10144 


.11056 


.12018 


.13031 


.14097 


.15219 


13 


48 


.06972 


.07702 


.08476 


.09294 


.10159 


.11072 


.12034 


.13048 


.14115 


.15239 


12 


49 


.06984 


.07715 


.08489 


.09308 


.10174 


.11087 


.12051 


.13065 


.14134 


.15258 


11 


50 


1.06995 


1.07727 


1.08503 


1.09323 


1.10189 


1.11103 


1.12067 


1.13083 


1.14152 


1.15277 


10 


51 


.07007 


.07740 


.08516 


.09337 


.10204 


.11119 


.12084 


.13100 


.14170 


.15296 


9 


52 


.07019 


.07752 


.08529 


.09351 


.10218 


.11134 


.12100 


.13117 


.14188 


.15315 


8 


53 


.07031 


.07765 


.08542 


.09365 


.10233 


.11150 


.12117 


.13135 


.14207 


.15335 


7 


54 


.07043 


.07778 


.08556 


.09379 


.10248 


.11166 


.12133 


.13152 


.14225 


.15354 


6 


55 


1.07055 


1.07790 


1.08569 


1.09393 


1.10263 


1.11181 


1.12150 


1.13170 


1.14243 


1.15373 


5 


56 


.07067 


.07803 


.08582 


.09407 


.10278 


.11197 


.12166 


.13187 


.14262 


.15393 


4 


57 


.07079 


.07816 


.08596 


.09421 


.10293 


.11213 


.12183 


.13205 


.14280 


.15412 


3 


58 


.07091 


.07828 


.08609 


.09435 


.10308 


.11229 


.12199 


.13222 


.14299 


.15431 


2 


59 


.07103 


.07841 


.08623 


.09449 


.10323 


.11244 


.12216 


.13240 


.14317 


.15451 


1 


60 


1.07115 


1.07853 


1.08636 


1.09464 


1.00338 


1.11260 


1.12233 


1.13257 


1.14335 


1.15470 






I 69° i 68° I 67° I 66° | 65° | 64° | 63° | 62° | 61° | 60° | ^ 

Cosecants. 
* Exsecant ■= secant — 1 ; coexsecant = cosecant — 1. 



170 



9.— PLANE TRIGONOMETRY. 



4. — Natural Secants, Cosecants (Exsecants.Coexsecants).* — (Cont'd.). 

Secants. ^^H i 



t 


30° 


31° 


32° 


1 33° 


1 34° 


1 35° 


1 36° 


1 37° 


1 38° 


1 39° 


r^ 





1.15470 


1.16663 


1.17918 


1.19236 


1.20622 


1.22077 


1.23607 


1.25214 


1.26902 


1.28676 


60 


1 


.15489 


.16684 


.17939 


.19259 


.20645 


.22102 


.23633 


.25241 


.26931 


.28706 


59 


2 


.15509 


.16704 


.17961 


.19281 


.20669 


.22127 


.23659 


.25269 


.26960 


.28737 


58 


3 


.15528 


.16725 


.17982 


.19304 


.20693 


.22152 


.23685 


.25296 


.26988 


.28767 


57 


4 


.15548 


.16745 


.18004 


.19327 


.20717 


.22177 


.23711 


.25324 


.27017 


.28797 


56 


5 


1.15567 


1.16766 


1.18025 


1.19349 


1.20740 


1.22202 


1.2373811.25351 


1.27046 


1.28828 


55 


6 


.15587 


.16786 


.18047 


.19372 


.20764 


.22227 


.23764 


.25379 


.27075 


.28858 


54 


7 


.15606 


.16806 


.18068 


.19394 


.20788 


.22252 


.23790 


.25406 


.27104 


.28889 


53 


8 


.15626 


.16827 


.18090 


.19417 


.20812 


.22277 


.23816 


.25434 


.27133 


.28919 


52 


9 


.15645 


.16848 


.18111 


.19440 


.20836 


.22302 


.23843 


.25462 


.27162 


.28950 


51 


10 


1.15665 


1.16868 


1.18133 


1.19463 


1.20859 


1.22327 


1.23869 


1.25489 


1.27191 


1.28980 


50 


11 


.15684 


.16889 


.18155 


.19485 


.20883 


.22352 


.23895 


.25517 


.27221 


.29011 


49 


12 


.15704 


.16909 


.18176 


.19508 


.20907 


.22377 


.23922 


.25545 


.27250 


.29042 


48 


13 


.15724 


.16930 


.18198 


.19531 


.22931 


.23402 


.25948 


.25572 


.27279 


.29072 


47 


14 


.15743 


.16950 


.18220 


.19554 


.20955 


.22428 


.23975 


.25600 


.27308 


.29103 


46 


15 


1.15763 


1.16971 


1.18241 


1.19576 


1.20979 


1.22453 


1.24001 


1.25628 


1.27337 


1.29133 


45 


16 


.15782 


-.16992 


.18263 


.19599 


.21003 


.22478 


.24028 


.25656 


.27366 


.29164 


44 


17 


.15802 


.17012 


.18285 


.19622 


.21027 


.22503 


.24054 


.25683 


.27396 


.29195 


43 


18 


.15822 


.17033 


.18307 


.19645 


.21051 


.22528 


.24081 


.25711 


.27425 


.29226 


42 


19 


.15841 


.17054 


.18328 


.19668 


.21075 


.22554 


.24107 


.25739 


.27454 


.29256 


41 


20 


1.15861 


1.17075 


1.18350 


1.19691 


1.21099 


1.25579 


1.24134 


1.25767 


1.27483 


1.29287 


40 


21 


.15881 


.17095 


.18372 


.19713 


.21123 


.22604 


.24160 


.25795 


.27513 


.29318 


39 


22 


.15901 


.17116 


.18394 


.19736 


.21147 


.22629 


.24187 


.25823 


.27542 


.29349 


38 


23 


.15920 


-.17137 


.18416 


.19759 


.21171 


.22655 


.24213 


.25851 


.27572 


.29380 


37 


24 


.15940 


.17158 


.18437 


.19782 


.21195 


.22680 


.24240 


.25879 


.27601 


.29411 


36 


25 


1.15960 


1.17178 


1.18459 


1.19805 


1.21220 


1.22706 


1.24267 


1.25907 


1.27630 


1.29442 


35 


26 


.15980 


.17199 


.18481 


.19828 


.21244 


.22731 


.24293 


.25935 


.27660 


.29473 


34 


27 


.16000 


.17220 


.18503 


.19851 


.21268 


.22756 


.24320 


.25963 


.27689 


.29504 


33 


28 


.16019 


.17241 


.18525 


.19874 


.21292 


.22782 


.24347 


.25991 


.27719 


.29535 


32 


29 


.16039 


.17262 


.18547 


.19897 


.21316 


.22807 


.24373 


.26019 


.27748 


.29566 


31 


30 


1.16059 


1.17283 


1.18569 


1.19920 


1.21341 


1.22833 


1.24400 


1.26047 


1.27778 


1.29597 


30 


31 


.16079 


.17304 


.18591 


.19944 


.21365 


.22858 


.24427 


.26075 


.27807 


.29628 


29 


32 


.16099 


.17325 


.18613 


.19967 


.21389 


.22884 


.24454 


.26104 


.27837 


.29659 


28 


33 


.16119 


.17346 


.18635 


.19990 


.21414 


.22909 


.24481 


.26132 


.27867 


.29690 


27 


34 


.16139 


.17367 


.18657 


.20013 


.21438 


.22935 


.24508 


.26160 


.27896 


.29721 


26 


35 


1.16159 


1.17388 


1.18679 


1.20036 


1.21462 


1.22960 


1.24534 


1.26188 


1.27926 


1.29752 


25 


36 


.16179 


.17409 


.18701 


.20059 


.21487 


.22986 


.24561 


.26216 


.27956 


.29784 


24 


37 


.16199 


.17430 


.18723 


.20083 


.21511 


.23012 


.24588 


.26245 


.27985 


.29815 


23 


38 


.16219 


.17451 


.18745 


.20106 


.21535 


.23037 


.24615 


.26273 


.28015 


.29846 


22 


39 


.16239 


.17472 


.18767 


.20129 


.21560 


.23063 


.24642 


.26301 


.28045 


.29877 


21 


40 


1.16259 


1.17493 


1.18790 


1.20152 


1.21584 


1.23089 


1.24669 


1.26330 


1.28075 


1.29909 


20 


41 


.16279 


.17514 


.18812 


.20176 


.21609 


.23114 


.24696 


.26358 


.28105 


.29940 


19 


42 


.06299 


.17535 


.18834 


.20199 


.21633 


.23140 


.24723 


.26387 


.28134 


.29971 


18 


43 


.16319 


.17556 


.18856 


.20222 


.21658 


.23166 


.24750 


.26415 


.28164 


.30003 


17 


44 


.16339 


.17577 


.18878 


.20246 


.21682 


.23192 


.24777 


.26443 


.28194 


.30034 


16 


45 


1.16359 


1.17598 


1.18901 


1.20269 


1.21707 


1.23217 


1.24804 


1.26472 


1.28224 


1.30066 


15 


46 


.16380 


.17620 


.18923 


.20292 


.21731 


.23243 


.24832 


.26500 


.28254 


.30097 


14 


47 


.16400 


.17641 


.18945 


.20316 


.21756 


.23269 


.24859 


.26529 


.28284 


.30129 


13 


48 


.16420 


.17662 


.18967 


.20339 


.21781 


.23295 


.24886 


.26557 


.28314 


.30160 


12 


49 


.16440 


.17683 


.18990 


.20363 


.21805 


.23321 


.24913 


.26586 


.28344 


.30192 


11 


50 


1.16460 


1.17704 


1.19012 


1.20386 


1.21830 


1.23347 


1.24940 


1.26615 


1.28374 


1.30223 


10 


51 


.16481 


.17726 


.19034 


.20410 


.21855 


.23373 


.24967 


.26643 


.28404 


.30255 


9 


52 


.16501 


.17747 


.19057 


.20433 


.21879 


.23399 


.24995 


.26672 


.28434 


.30287 


8 


53 


.16521 


.17768 


.19079 


.20457 


.21904 


.23424 


.25022 


.26701 


.28464 


.30318 


7 


54 


.16541 


.17790 


.19102 


.20480 


.21929 


.23450 


.25049 


.26729 


.28495 


.30350 


6 


55 


1.16562 


1.17811 


1.19124 


1.20504 


1.21953 


1.23476 


1.25077 


1.26758 


1.28525 


1.30382 


5 


56 


.16582 


.17832 


.19146 


.20527 


.21978 


.23502 


.25104 


.26787 


.28555 


.30413 


4 


57 


.16602 


.17854 


.19169 


.20551 


.22003 


.23529 


.25131 


.26815 


.28585 


.30445 


3 


58 


.16623 


.17875 


.19191 


.20575 


.22028 


.23555 


.25159 


.26844 


.28615 


.30477 


2 


59 


.16643 


.17896 


.19214 


.20598 


.22053 


.23581 


.25186 


.26873 


.28646 


.30509 


1 


60 


1.16663 


1.17918 


1.19236 


1.20622 


1.22077 


1.23607 


1.25214 


1.26902 


1.28676 


1.30541 







59- 


58° 


57° 


56° 55° 1 


54° 1 


53° 


52° 


51° 


50° 1 ' 



I 



Cosecants. 
* Exsecant = secant — 1 ; coexsecant = cosecant — 1. 



NATURAL SECANTS, ETC. 



171 



4.— Natural Secants, Cosecants (Exsecants, Coexsecants).* — (Cont'd.) 

Secants. 



' 


40° 


41° 


42° 


43° 


1 44° 


1 45° 


46° 


1 47° 


48° 


49° 








1.30541 


1.32501 


1.34563 


1.36733 


1.39016 


1.41421 


1.43956 


1.46628 


1.49448 


1.52425 


60 


1 


.30573 


.32535 


.3'4599 


.36770 


.39055 


.41463 


.43999 


.46674 


.49496 


.52476 


59 


2 


.30605 


.32568 


.34634 


.36807 


.39095 


.41504 


.44042 


.46719 


.49544 


.52527 


58 


3 


.30636 


.32602 


.34669 


.36844 


.39134 


.41545 


.44086 


.46765 


.49593 


.52579 


57 


4 


.30668 


.32636 


.34704 


.36881 


.39173 


.41586 


.44129 


.46811 


.49641 


.52630 


66 


5 


1.30700 


1.32669 


1.34740 


1.36919 


1.39212 


1.41627 


1.44173 


1.46857 


1.49690 


1.52681 


55 


6 


.30732 


.32703 


.34775 


.36956 


.39251 


.41669 


.44217 


.46903 


.49738 


.52732 


54 


7 


.30764 


.32737 


.34811 


.36993 


.39291 


.41710 


.44260 


.46949 


.49787 


.52784 


53 


8 


.30796 


.32770 


.34846 


.37030 


.39330 


.41752 


.44304 


.46995 


.49835 


.52835 


52 


9 


.30829 


.32804 


.34882 


.37068 


.39369 


.41793 


.44347 


.47041 


.49884 


.52886 


51 


10 


1.30861 


1.32838 


1.34917 


1.37105 


1.39409 


1.41835 


1.44391 


1.47087 


1.49933 


1.52938 


50 


11 


.30893 


.32872 


.34953 


.37143 


.39448 


.41876 


.44435 


.47134 


.49981 


.52989 


49 


12 


.30925 


.32905 


.34988 


.37180 


.39487 


.41918 


.44479 


.47180 


.50030 


.53041 


48 


13 


.30957 


.32939 


.35024 


.37218 


.39527 


.41959 


.44523 


.47226 


.50079 


.53092 


47 


14 


.30989 


.32973 


.35060 


.37255 


.39566 


.42001 


.44567 


.47272 


.50128 


.53144 


46 


15 


1.31022 


1.33007 


1.35095 


1.37293 


1.39606 


1.42042 


1.44610 


1.47319 


1.50177 


1.53196 


45 


16 


.31054 


.33041 


.35131 


.37330 


.39646 


.42084 


.44654 


.47365 


.50226 


.53247 


44 


17 


.31086 


.33075 


.35167 


.37368 


.39685 


.42126 


.44698 


.47411 


.50275 


.53299 


43 


18 


.31119 


.33109 


.35203 


.37406 


.39725 


.42168 


.44742 


.47458 


.50324 


.53351 


42 


19 


.31151 


.33143 


.35238 


.37443 


.39764 


.42210 


.44787 


.47504 


.50373 


.53403 


41 


20 


1.31183 


1.33177 


1.35274 


1.37481 


1.39804 


1.42251 


1.44831 


1.47551 


1.50422 


1.53455 


40 


21 


.31216 


.33211 


.35310 


.37519 


.39844 


.42293 


.44875 


.47598 


.50471 


.53507 


39 


22 


.31248 


.33245 


.35346 


.37556 


.39884 


.42335 


.44919 


.47644 


.50521 


.53559 


38 


23 


.31281 


.33279 


.35382 


.37594 


.39924 


.42377 


.44963 


.47691 


.50570 


.53611 


37 


24 


.31313 


.33314 


.35418 


.37632 


.39963 


.42419 


.45007 


.47738 


.50619 


.63663 


36 


25 


1.31346 


1.33348 


1.35454 


1.37670 


1.40003 


1.42461 


1.45052 


1.47784 


1.50669 


1.53715 


35 


26 


.31378 


.33382 


.35490 


.37708 


.40043 


.42503 


.45096 


.47831 


.50718 


.53768 


34 


27 


.31411 


.33416 


.35526 


.37746 


.40083 


.42545 


.45141 


.47878 


.50767 


.53820 


33 


28 


.31443 


.33451 


.35562 


.37784 


.40123 


.42587 


.45185 


.47925 


.50817 


.53872 


32 


29 


.31476 


.33485 


.35598 


.37822 


.40163 


.42630 


.45229 


.47972 


.50866 


.53924 


31 


30 


1.31509 


1.33519 


1.35634 


1.37860 


1.40203 


1.42672 


1.45274 


1.48019 


1.50916 


1.53977 


30 


31 


.31541 


.33554 


.35670 


.37898 


.40243 


.42714 


.45319 


.48066 


.50966 


.54029 


29 


32 


.31574 


.33588 


.57307 


.37936 


.40283 


.42756 


.45363 


.48113 


.51015 


.54082 


28 


33 


.31607 


.33622 


.35743 


.37974 


.40324 


.42799 


.45408 


.48160 


.51065 


.54134 


27 


34 


.31640 


.33657 


.35779 


.38012 


.40364 


.42841 


.45452 


.48207 


.51115 


.54187 


26 


35 


1.31672 


1.33691 


1.35815 


1.38051 


1.40404 


1.42883 


1.45497 


1.48254 


1.51165 


1.54240 


25 


36 


.31705 


-.33726 


.35852 


.38089 


.40444 


.42926 


.45542 


.48301 


.51215 


.54292 


24 


37 


.31738 


.33760 


.35888 


.38127 


.40485 


.42968 


.45587 


.48349 


.51265 


.54345 


23 


38 


.31771 


.33795 


.35924 


.38165 


.40525 


.43011 


.45631 


.48396 


.51314 


.54398 


22 


39 


.31804 


.33830 


.35961 


.38204 


.40565 


.43053 


.45676 


.48443 


.51364 


.54451 


21 


40 


1.31837 


1.33864 


1.35997 


1.38242 


1.40606 


1.43096 


1.45721 


1.48491 


1.51415 


1.54504 


20 


41 


.31870 


.33899 


.36034 


.38280 


.40646 


.43139 


.45766 


.48538 


.51465 


.54557 


19 


42 


.31903 


.33934 


.36070 


.38319 


.40687 


.43181 


.45811 


.48586 


.51515 


.54610 


18 


43 


.31936 


.33968 


.36107 


.38357 


.40727 


.43224 


.45856 


.48633 


.51565 


.54663 


17 


44 


.31969 


.34003 


.36143 


.38396 


.40768 


.43267 


.45901 


.48681 


.51615 


.54716 


16 


45 


1.32002 


1.34038 


1.36180 


1.38434 


1.40808 


1.43310 


1.45946 


1.48728 


1.51665 


1.54769 


15 


46 


.32035 


.34073 


.36217 


.38473 


.40849 


.43352 


.45992 


.48776 


.51716 


.54822 


14 


47 


.32068 


.34108 


.36253 


.38512 


.40890 


.43395 


.46037 


.48824 


.51766 


.54876 


13 


48 


.32101 


.34142 


.36290 


.38550 


.40930 


.43438 


.46082 


.48871 


.51817 


.54929 


12 


49 


.32134 


.34177 


.36327 


.38589 


.40971 


.43481 


.46127 


.48919 


.51867 


.54982 




50 


1.32168 


1.34212 


1.36363 


1.38628 


1.41012 


1.43524 


1.46173 


1.48967 


1.51918 


1.55036 


10 


51 


.32201 


.34247 


.36400 


.38666 


.41053 


.43567 


.46218 


.49015 


.51968 


.55089 


9 


52 


.32234 


.34282 


.36437 


.38705 


.41093 


.43610 


.46263 


.49063 


.52019 


.55143 


8 


53 


.32267 


.34317 


.36474 


.38744 


.41134 


.43653 


.46309 


.49111 


.52069 


.55196 


7 


54 


.32301 


.34352 


.36511 


.38783 


.41175 


.43696 


.46354 


.49159 


.52120 


.55250 


6 


55 


1.32334 


1.34387 


1.36548 


1.38822 


1.41216 


1.43739 


1.46400 


1.49207 


1.52171 


1.55303 


5 


56 


.32368 


.34423 


.36585 


.38860 


.41257 


.43783 


.46445 


.49255 


.52222 


.55357 


4 


57 


.32401 


.34458 


.36622 


.38899 


.41298 


.43826 


.46491 


.49303 


.52273 


.55411 


3 


58 


.32434 


.34493 


.36659 


.38938 


.41339 


.43869 


.46537 


.49351 


.52323 


.55465 


2 


59 


.32468 


.34528 


.36696 


.38977 


.41380 


.43912 


.46582 


.49399 


.52374 


.55518 


1 


60 


1.32501 


1.34563 


1.36733 


1.39016 


1.41421 


1.43956 


1.46628 


1.49448 


1.52425 


1.55572 







49° 


48° 


47° 


46° 1 45° 1 44° 1 43° 


42° 


1 41° 


1 40° 


~^ 



* Exsecant = secant 



Cosecants. 
1 ; coexsecant = cosecant — 1 . 



m 



PLANE TRIGONOMETRY. 



4. — Natural Secants, Cosecants (Exsecants, Coexsecants).* — (Cont'd. 

Secants. 



' 


50° 


51° 


52° 


i 53° 


1 54° 


1 55° 


56° 


1 57° 


1 58° 


1 59° 


i__ 





1.55572 


1.58902 


1.62427 


1.66164 


1.70130 


1.74345 


1.78829 


1.83608 


1.88708 


1.94160 


60 


1 


.55626 


.58959 


.62487 


.66228 


.70198 


.74417 


.78906 


.83690 


.88796 


.94254 


59 


2 


.55680 


.59016 


.62548 


.66292 


.70267 


.74490 


.78984 


.83773 


.88884 


.94349 


58 


3 


.55734 


.59073 


.62609 


.66357 


.70335 


.74562 


.79061 


.83855 


.88972 


.94443 


57 


4 


.55789 


.59130 


.62669 


.66421 


.70403 


.74635 


.79138 


.83938 


.89060 


.94537 


56 


5 


1.55843 


1.59188 


1.62730 


1.66486 


1.70472 


1.74708 


1.79216 


1.84020 


1.89148 


1.94632 


55 


6 


.55897 


.59245 


.62791 


.66550 


.70540 


.74781 


.79293 


.84103 


.89237 


.94726 


54 


7 


.55951 


.59302 


.62852 


.66615 


.70609 


.74854 


.79371 


.84186 


.89325 


.94821 


53 


8 


.56005 


.59360 


.62913 


.66679 


.70677 


.74927 


.79449 


.84269 


.89414 


.94916 


52 


9 


.56060 


.59418 


.62974 


.66744 


.70746 


.75000 


.79527 


.84352 


.89503 


.95011 


51 


10 


1.56114 


1.59475 


1.63035 


1.66809 


1.70815 


1.75073 


1.79604 


1.84435 


1.89591 


1.95106 


50 


11 


.56169 


.59533 


.63096 


.66873 


.70884 


.75146 


.79682 


.84518 


.89680 


.95201 


49 


12 


.56223 


.59590 


.63157 


.66938 


.70953 


.75219 


.79761 


.84601 


.89769 


.95296 


48 


13 


.56278 


.59648 


.63218 


.67003 


.71022 


.75293 


.79839 


.84685 


.89858 


.95392 


47 


14 


.56332 


.59706 


.63279 


.67068 


.71091 


.75366 


.79917 


.84768 


.89948 


.95487 


46 


15 


1.56387 


1.59764 


1.63341 


1.67133 


1.71160 


1.75440 


1.79995 


1.84852 


1.90037 


1.95583 


45 


16 


.56442 


.59822 


.63402 


.67199 


.71229 


.75513 


.80074 


.84935 


.90126 


.95678 


44 


17 


.56497 


.59880 


.63464 


.67264 


.71298 


.75587 


.80152 


.85019 


.90216 


.95744 


43 


18 


.56551 


.59938 


.63525 


.67329 


.71368 


.75661 


.80231 


.85103 


.90305 


.95870 


42 


19 


.56606 


.59996 


.63587 


.67394 


.71437 


.75734 


.80309 


.85187 


.90395 


.95966 


41 


20 


1.56661 


1.60054 


1.63648 


1.67460 


1.71506 


1.75808 


1.80388 


1.85271 


1.90485 


1.96062 


40 


21 


.56716 


.60112 


.63710 


.67525 


.71576 


.75882 


.80467 


.85355 


.90575 


.96158 


39 


22 


.56771 


.60171 


.63772 


.67591 


.71646 


.75956 


.80546 


.85439 


.90665 


.96255 


38 


23 


.56826 


.60229 


.63834 


.67656 


.71715 


.76031 


.80625 


.85523 


.90755 


.96351 


37 


24 


.56881 


.60287 


.63895 


.67722 


.71785 


.76105 


.80704 


.85608 


.90845 


.96448 


36 


25 


1.56937 


1.60346 


1.63957 


1.67788 


1.71855 


1.76179 


1.80783 


1.85692 


1.90935 


1.96544 


35 


26 


.56992 


.60404 


.64019 


.67853 


.71925 


.76253 


.80862 


.85777 


.91026 


.96641 


34 


27 


.57047 


.60463 


.64081 


.67919 


.71995 


.76328 


.80942 


.85861 


.91116 


.96738 


33 


28 


.57103 


.60521 


.64144 


.67985 


.72065 


.76402 


.81021 


.85946 


.91207 


.96835 


32 


29 


.57158 


.60580 


.64206 


.68051 


.72135 


.76477 


.81101 


.86031 


.91297 


.96932 


31 


30 


1.57213 


1.60639 


1.64268 


1.68117 


1.72205 


1.76552 


1.81180 


1.86116 


1.91388 


1.97029 


30 


31 


.57269 


.60698 


.64330 


.68183 


.72275 


.76626 


.81260 


.86201 


.91479 


.97127 


29 


32 


.57324 


.60756 


.64393 


.68250 


.72346 


.76701 


.81340 


.86286 


.91570 


.97224 


28 


33 


.57380 


.60815 


.64455 


.68316 


.72416 


.76776 


.81419 


.86371 


.91661 


.97322 


27 


34 


.57436 


.60874 


.64518 


.68382 


.72487 


.76851 


.81499 


.86457 


.91752 


.97420 


26 


35 


1.57491 


1.60933 


1.64580 


1.68449 


1.72557 


1.76926 


1.81579 


1.86542 


1.91844 


1.97517 


25 


36 


.57547 


.60992 


.64643 


.68515 


.72628 


.77001 


.81659 


.86627 


.91935 


.97615 


24 


37 


.57603 


.61051 


.64705 


.68582 


.72698 


.77077 


.81740 


.86713 


.92027 


.97713 


23 


38 


.57659 


.61111 


.64768 


.68648 


.72769 


.77152 


.81820 


.86799 


.92118 


.97811 


22 


39 


.57715 


.61170 


.64831 


.68715 


.72840 


.77227 


.81900 


.86885 


.92210 


.97910 


21 


40 


1.57771 


1.61229 


1.64894 


1.68782 


1.72911 


1.77303 


1.81981 


1.86970 


1.92302 


1.98008 


20 


41 


.57827 


.61288 


.64957 


.68848 


.72982 


.77378 


.82061 


.87056 


.92394 


.98107 


19 


42 


.57883 


.61348 


.65020 


.68915 


.73053 


.77454 


.82142 


.87142 


.92486 


.98205 


18 


43 


.57939 


.61407 


.65083 


.68982 


.73124 


.77530 


.82222 


.87229 


.92578 


.98304 


17 


44 


.57995 


.61467 


.65146 


.69049 


.73195 


.77606 


.82303 


.87315 


.92670 


.98403 


16 


45 


1.58051 


1.61526 


1.65209 


1.69116 


1.73267 


1.77681 


1.82384 


1.87401 


1.92762 


1.98502 


15 


46 


.58108 


.61586 


.65272 


.69183 


.73338 


.77757 


.82465 


.87488 


.92855 


.98601 


14 


47 


.58164 


.61646 


.65336 


.69250 


.73409 


.77833 


.82546 


.87574 


.92947 


.98700 


13 


48 


.58221 


.61705 


.65399 


.69318 


.73481 


.77910 


.82627 


.87661 


.93040 


.98799 


12 


49 


.58277 


.61765 


.65462 


.69385 


.73552 


.77986 


.82709 


.87748 


.93133 


.98899 


11 


50 


1.58333 


1.61825 


1.65526 


1.69452 


1.73624 


1.78062 


1.82790 


1.87834 


1.93226 


1.98998 


10 


51 


.58390 


.61885 


.65589 


.69520 


.73696 


.78138 


.82871 


.87921 


.93319 


.99098 


9 


52 


.58447 


.61945 


.65653 


.69587 


.73768 


.78215 


.82953 


.88008 


.93412 


.99198 


8 


53 


.58503 


.62005 


.65717 


.69655 


.73840 


.78291 


.83034 


.88095 


.93505 


.99298 


7 


54 


.58560 


.62065 


.65780 


.69723 


.73911 


.78368 


.83116 


.88183 


.93598 


.99398 


6 


55 


1.58617 


1.62125 


1.65844 


1.69790 


1.73983 


1.78445 


1.83198 


1.88270 


1.93692 


1.99498 


5 


56 


.58674 


.62185 


.65908 


.69858 


.74056 


.78521 


.83280 


.88357 


.93785 


.99598 


4 


57 


.58731 


.62246 


.65972 


.69926 


.74128 


.78598 


.83362 


.88445 


.93879 


.99698 


3 


58 


.58788 


.62306 


.66036 


.69994 


.74200 


.78675 


.83444 


.88532 


.93973 


.99799 


2 


59 


.58845 


.62366 


.66100 


.70062 


.74272 


.78752 


.83526 


.88620 


.94066 


.99899 


1 


60 


1.58902 


1.62427 


1.66164 


1.70130 


1.74345 


1.78829 


1.83608 


1.88708 


1.94160 


2.00000 





1 39° 


38° 


1 37° 


36° 


35° 34° 


33° 


32° 


31° 


30° 


Z 



Cosecants. 
* Exsecant =» secant — 1 ; coexsecant = cosecant — 1. 



NATURAL SECANTS, ETC. 



173 



4. — Natural Secants, Cosecants (Exsecants, Coexsecants).* — (Cont'd.) 

Secants. 



' \ 


60° 1 


61° 1 


62° 1 


63° 1 


64° 1 


65° 1 


66° 1 


67° 


68° 


69° 







2.00000 


2.06267 


2.13005 


2.20269 


2.28117 


2.36620 2.45859 


2.55930 2.66947 


2.79043 


60 


1 


.00101 


.06375 


.13122 


.20395 


.28253 


.36768 


.46020 


.56106 


.67139 


.79254 


59 


2 


.00202 


.06483 


.13239 


.20521 


.28390 


.36916 


.46181 


.56282 


.67332 


.79466 


58 


3 


.00303 


.06592 


.13356 


.20647 


.28526 


.37064 


.46342 


.56458 


.67525 


.79679 


57 


4 


.00404 


.06701 


.13473 


.20773 


.28663 


.37212 


.46504 


.56634 


.67718 


.79891 


56 


5 


2.00505 


2.06809 


2.13590 


2.20900 


2.28800 


2.37361 


2.46665 


2.56811 


2.67911 


2.80104 


55 


6 


.00607 


.06918 


.13707 


.21026 


.28937 


.37509 


.46827 


.56988 


.68105 


.80318 


54 


7 


.00708 


.07027 


.13825 


.21153 


.29074 


.37658 


.46989 


.57165 


.68299 


.80531 


53 


8 


.00810 


.07137 


.13942 


.21280 


.29211 


.37808 


.47152 


.57342 


.68494 


.80746 


52 


9 


.00912 


.07246 


.14060 


.21407 


.29349 


.37957 


.47314 


.57520 


.68689 


.80960 


51 


10 


2.01014 


2.07356 


2.14178 


2.21535 


2.29487 


2.38107 


2.47477 


2.57698 


2.68884 


2.81175 


50 


11 


.01116 


.07465 


.14296 


.21662 


.29625 


.38256 


.47640 


.57876 


.69079 


.81390 


49 


12 


.01218 


.07575 


.14414 


.21790 


.29763 


.38406 


.47804 


.58054 


.69275 


.81605 


48 


13 


.01320 


.07685 


.14533 


.21918 


.29901 


.38556 


.47967 


.58233 


.69471 


.81821 


47 


14 


.01422 


.07795 


.14651 


.22045 


.30040 


.38707 


.48131 


.58412 


.69667 


.82037 


46 


15 


2.01525 


2.07905 


2.14770 


2.22174 


2.30179 


2.38857 


2.48295 


2.58591 


2.69864 


2.82254 


45 


16 


.01628 


.08015 


.14889 


.22302 


.30318 


.39008 


.48459 


.58771 


.70061 


.82471 


44 


17 


.01730 


.08126 


.15008 


.22430 


.30457 


.39159 


.48624 


.58950 


.70258 


.82688 


43 


18 


.01833 


.08236 


.15127 


.22559 


.30596 


.39311 


.48789 


.59130 


.70455 


.82906 


42 


19 


.01936 


.08347 


.15246 


.22688 


.30735 


.39462 


.48954 


.59311 


.70653 


.83124 


41 


20 


2.02039 


2.08458 


2.15366 


2.22817 


2.30875 


2.39614 


2.49119 


2.59491 


2.70851 


2.83342 


40 


21 


.02143 


.08569 


.15485 


.22946 


.31015 


.39766 


.49284 


.59672 


.71050 


.83561 


39 


22 


.02246 


.08680 


.15605 


.23075 


.31155 


.39918 


.49450 


.59853 


.71249 


.83780 


38 


23 


.02349 


.08791 


.15725 


.23205 


.31295 


.40070 


.49616 


.60035 


.71448 


.83999 


37 


24 


.02453 


.08903 


.15845 


.23334 


.31436 


.40222 


.49782 


.60217 


.71647 


.84219 


36 


25 


2.02557 


2.09014 


2.15965 


2.23464 


2.31576 


2.40375 


2.49948 


2.60399 


2.71847 


2.84439 


35 


26 


.02661 


.09126 


.16085 


.23594 


.31717 


.40528 


.50115 


.60581 


.72047 


.84659 


34 


27 


.02765 


.09238 


.16206 


.23724 


.31858 


.40681 


.50282 


.60763 


.72247 


.84880 


33 


28 


.02869 


.09350 


.16326 


.23855 


.31999 


.40835 


.50449 


.60946 


.72448 


.85102 


32 


29 


.02973 


.09462 


.16447 


.23985 


.32140 


.40988 


.50617 


.61129 


.72649 


.85323 


31 


30 


2.03077 


2.09574 


2.16568 


2.24116 


2.32282 


2.41142 


2.50784 


2.61313 


2.72850 


2.85545 


30 


31 


.03182 


.09686 


.16689 


.24247 


.32424 


.41296 


.50952 


.61496 


.73052 


.85767 


29 


32 


.03286 


.09799 


.16810 


.24378 


.32566 


.41450 


.51120 


.61680 


.73254 


.85990 


28 


33 


.03391 


.09911 


.16932 


.24509 


.32708 


.41605 


.51289 


.61864 


.73456 


.86213 


27 


34 


.03496 


.10024 


.17053 


.24640 


..32850 


.41760 


.51457 


.62049 


.73659 


.86437 


26 


35 


2.03601 


2.10137 


2.17175 


2.24772 


2.32993 


2.41914 


2.51626 


2.62234 


2.73862 


2.86661 


25 


36 


.03706 


.10250 


.17297 


.24903 


.33135 


.42070 


.51795 


.62419 


.74065 


.86885 


24 


37 


.03811 


.10363 


.17419 


.25035 


.33278 


.42225 


.51965 


.62604 


.74269 


.87109 


23 


38 


.03916 


.10477 


.17541 


.25167 


.33422 


.42380 


.52134 


.62790 


.74473 


.87334 


22 


39 


.04022 


.10590 


.17663 


.25300 


.33565 


.42536 


.52304 


.62976 


.74677 


.87560 


21 


40 


2.04128 


2.10704 


2.17786 


2.25432 


2.33708 


2.42692 


2.52474 


2.63162 


2.74881 


2.87785 


20 


41 


.04233 


.10817 


.17909 


.25565 


.33852 


.42848 


.52645 


.63348 


.75086 


.88011 


19 


42 


.04339 


.10931 


.18031 


.25697 


.33996 


.43005 


.52815 


.63535 


.75292 


.88238 


18 


43 


.04445 


.11045 


.18154 


.25830 


.34140 


.43162 


.52986 


.63722 


.75497 


.88465 


17 


44 


.04551 


.11159 


.18277 


.25963 


.34284 


.43318 


.53157 


.63909 


.75703 


.88692 


16 


45 


2.04658 


2.11274 


2.18401 


2.26097 


2.34429 


2.43476 


2.53329 


2.64097 


2.75909 


2.88920 


15 


46 


.04764 


.11388 


.18524 


.26230 


.34573 


.43633 


.53500 


.64285 


.76116 


.89148 


14 


47 


.04870 


.11503 


.18648 


.26364 


•.34718 


.43790 


.53672 


.64473 


.76323 


.89376 


13 


48 


.04977 


.11617 


.18772 


.26498 


.34863 


.43948 


.53845 


.64662 


.76530 


.89605 


12 


49 


.05084 


.11732 


.18895 


.26632 


.35009 


.44106 


.54017 


.64851 


.76737 


.89834 


11 


50 


2.05191 


2.11847 


2.19019 


2.26766 


2.35154 


2.44264 


2.54190 


2.65040 


2.76945 


2.90063 


10 


51 


.05298 


.11963 


.19144 


.26900 


.35300 


.44423 


.54363 


.65229 


.77154 


.90293 


9 


52 


.05405 


.12078 


.19268 


.27035 


.35446 


.44582 


.54536 


.65419 


.77362 


.90524 


8 


53 


.05512 


.12193 


.19393 


.-27169 


.35592 


.44741 


.54709 


.65609 


.77571 


.90754 


7 


64 


.05619 


.12309 


.19517 


.27304 


.35738 


.44900 


.54883 


.65799 


.77780 


.90986 


6 


55 


2.05727 


2.12425 


2.19642 


2.27439 


2.35885 


2.45059 


2.55057 


2.65989 


2.77990 


2.91217 


5 


56 


.05835 


.12540 


.19767 


.27574 


.36031 


.45219 


.55231 


.66180 


.78200 


.91449 


4 


57 


.05942 


.12657 


.19892 


.27710 


.36178 


.45378 


.55405 


.66371 


.78410 


.91681 


3 


58 


.06050 


.12773 


.20018 


.27845 


.36325 


.45539 


.55580 


.66563 


.78621 


.91914 


2 


59 


.06158 


.12889 


.20143 


.27981 


.36473 


.45699 


.55755 


.66755 


.78832 


.92147 


1 


60 


2.06267 


2.13005 


2.20269 


2.28117 


2.36620 


2.45859 


2.55930 


2.66947 


2.79043 


2.92380 







29° 


28° 


1 27°- 


26° 


1 25° 1 24° 1 23° 1 22° | 21° | 20° | ' 



Cosecants. 
* Exsecant = secant — 1 ; coexsecant = cosecant •- 1. 



174 



^.— PLANE TRIGONOMETRY. 



4. — Natural Secants, Cosecants (Exsecants, Coexsecants).* — (Cont'd.) 

Secants. 



' 1 


70° 1 


71° 


72° 


73° 1 74° 1 75° 1 76° | 77° | 78° | 79o | 





2.92380 


3.07155 


3.23607 


3.42030 


3.62796 


3.86370 


4.13357 


4.44541 


4.80973 


5.24084 


60 


1 


.92614 


.07415 


.23897 


.42356 


.63164 


.86790 


.13839 


.45102 


.81633 


.24870 


59 


2 


.92849 


.07675 


.24187 


.42683 


.63533 


.87211 


.14323 


.45664 


.82294 


.25658 


58 


3 


.93083 


.07936 


.24478 


.43010 


.63903 


.87633 


.14809 


.46228 


.82956 


.26448 


57 


4 


.93318 


.08197 


.24770 


.43337 


.64274 


.88056 


.15295 


.46793 


.83621 


.27241 


56 


5 


2.93554 


3.08459 


3.25062 


3.43666 


3.64645 


3.88479 


4.15782 


4.47360 


4.84288 


5.28036 


55 


6 


.93790 


.08721 


.25355 


.43995 


.65018 


.88904 


.16271 


.47928 


.84956 


.28833 


54 


7 


.94026 


.08983 


.25648 


.44324 


.65391 


.89330 


.16761 


.48498 


.85627 


.29634 


53 


8 


.94263 


.09246 


.25942 


.44655 


.65765 


.89756 


.17252 


.49069 


.86299 


.30436 


52 


9 


.94500 


.09510 


.26237 


.44986 


.66140 


.90184 


.17744 


.49642 


.86973 


.31241 


51 


10 


2.94737 


3.09774 


3.26531 


3.45317 


3.66515 


3.90613 


4.18238 


4.50216 


4.87649 


5.32049 


50 


11 


.94975 


.10038 


.26827 


.45650 


.66892 


.91042 


.18733 


.50791 


.88327 


.32859 


49 


12 


.95213 


.10303 


.27123 


.45983 


.67269 


.91473 


.19228 


.51368 


.89007 


.33671 


48 


13 


.95452 


.10568 


.27420 


.46316 


.67647 


.91904 


.19725 


.51947 


.89689 


.34486 


47 


U 


.95691 


.10834 


.27717 


.46651 


.68025 


.92337 


.20224 


.52527 


.90373 


.35304 


46 


15 


2.95931 


3.11101 


3.28015 


3.46986 


3.68405 


3.92770 


4.20723 


4.53109 


4.91058 


5.36124 


45 


16 


.96171 


.11367 


.28313 


.47321 


.68785 


.93204 


.21224 


.53692 


.91746 


.36947 


44 


17 


.96411 


.11635 


.28612 


.47658 


.69167 


.93640 


.21726 


.54277 


.92436 


.37772 


43 


18 


.96652 


.11903 


.28912 


.47995 


.69549 


.94076 


.22229 


.54863 


.93128 


.38600 


42 


19 


.96893 


.12171 


.29212 


.48333 


.69931 


.94514 


.22734 


.55451 


.93821 


.39430 


41 


20 


2.97135 


3.12440 


3.29512 


3.48671 


3.70315 


3.94952 


4.23239 


4.56041 


4.94517 


5.40263 


40 


21 


.97377 


.12709 


.29814 


.49010 


.70700 


.95392 


.23746 


.56632 


.95215 


.41099 


39 


22 


.97619 


.12979 


.30115 


.49350 


.71085 


.95832 


.24255 


.57224 


.95914 


.41937 


38 


23 


.97862 


.13249 


.30418 


.49691 


.71471 


.96274 


.24764 


.57819 


.96616 


.42778 


37 


24 


.98106 


.13520 


.30721 


.50032 


.71858 


.96716 


.25275 


.58414 


.97320 


.43622 


36 


25 


2.98349 


3.13791 


3.31024 


3.50374 


3.72246 


3.97160 


4.25787 


4.59012 


4.98025 


5.44468 


35 


26 


.98594 


.14063 


.31328 


.50716 


.72635 


.97604 


.26300 


.59611 


.98733 


.45317 


34 


27 


.98838 


.14335 


.31633 


.51060 


.73024 


.98050 


.26814 


.60211 


.99443 


.46169 


33 


28 


.99083 


.14608 


.31939 


.51404 


.73414 


.98497 


.27330 


.60813 


5.00155 


.47023 


32 


29 


r99329 


.14881 


.32244 


.51748 


.73806 


.98944 


.27847 


.61417 


.00869 


.47881 


31 


30 


2.99574 


3.15155 


3.32551 


3.52094 


3.74198 


3.99393 


4.28366 


4.62023 


5.01585 


5.48740 


30 


31 


.99821 


.15429 


.32858 


.52440 


.74591 


.99843 


.28885 


.62630 


.02303 


.49603 


29 


32 


3.00067 


.15704 


.33166 


.52787 


.74984 


4.00293 


.29406 


.63238 


.03024 


.50468 


28 


33 


.00315 


.15979 


.33474 


.53134 


.75379 


.00745 


.29929 


.63849 


.03746 


.51337 


27 


34 


.00562 


.16255 


.33783 


.53482 


.75775 


.01198 


.30452 


.64461 


.04471 


.52208 


26 


35 


3.00810 


3.16531 


3.34092 


3.53831 


3.76171 


4.01652 


4.30977 


4.65074 


5.05197 


5.53081 


25 


36 


.01059 


.16808 


.34403 


.54181 


.76568 


.02107 


.31503 


.65690 


.05926 


.53958 


24 


37 


.01308 


.17085 


.34713 


.5453,1 


.76966 


.02563 


.32031 


.66307 


.06657 


.54837 


23 


38 


.01557 


.17363 


.35025 


.54883 


.77365 


.03020 


.32560 


.66925 


.07390 


.55720 


22 


39 


.01807 


.17641 


.35336 


.55235 


.77765 


.03479 


.33090 


.67545 


.08125 


.56605 


21 


40 


3.02057 


3.17920 


3.35649 


3.55587 


3.78166 


4.03938 


4.33622 


4.68167 


5.08863 


5.57493 


20 


41 


.02308 


.18199 


.35962 


.55940 


.78568 


.04398 


.34154 


.68791 


.09602 


.58383 


19 


42 


.02559 


.18479 


.36276 


.56294 


.78970 


.04860 


.34689 


.69417 


.10344 


.59277 


18 


43 


.02810 


.18759 


.36590 


.56649 


.79374 


.05322 


.35224 


.70044 


.11088 


.60174 


17 


44 


.03062 


.19040 


.36905 


.57005 


.79778 


.05786 


.35761 


.70673 


.11835 


.61073 


16 


45 


3.03315 


3.19322 


3.37221 


3.57361 


3.80183 


4.06251 


4.36299 


4.71303 


5.12583 


5.61976 


15 


46 


.03568 


.19604 


.37537 


.57718 


.80589 


.06717 


.36839 


.71935 


.13334 


.62881 


14 


47 


.03821 


.19886 


.37854 


.58076 


.80996 


.07184 


.37380 


.72569 


.14087 


.63790 


13 


48 


.04075 


.20169 


.38171 


.58434 


.81404 


.07652 


.37923 


.73205 


.14842 


.64701 


12 


49 


.04329 


.20453 


.38489 


.58794 


.81813 


.08121 


.38466 


.73843 


.15599 


.65616 


11 


50 


3.04584 


3.20737 


3.38808 


3.59154 


3.82223 


4.08591 


4.39012 


4.74482 


5.16359 


5.66533 


10 


51 


.04839 


.21021 


.39128 


.59514 


.82633 


.09063 


.39558 


.75123 


.17121 


.67454 


9 


52 


.05094 


.21306 


.39448 


.59876 


.83045 


.09535 


.40106 


.75766 


.17886 


.68377 


8 


53 


.05350 


.21592 


.39768 


.60238 


.83457 


.10009 


.40656 


.16411 


.18652 


.69304 


7 


54 


.05607 


.21878 


.40089 


.60601 


.83871 


.10484 


.41206 


.77057 


.19421 


.70234 


6 


55 


3.05864 


3.22165 


3.40411 


3.60965 


3.84285 


4.10960 


4.41759 


4.77705 


5.20193 


5.71166 


5 


56 


.06121 


.22452 


.40734 


.61330 


.84700 


.11437 


.42312 


.78355 


.20966 


.72102 


4 


57 


.06379 


.22740 


.41057 


.61695 


.85116 


.11915 


.42867 


.79007 


.21742 


.73041 


3 


58 


.06637 


.23028 


.41381 


.62061 


.85533 


.12394 


.43424 


.79661 


.22521 


.73983 


2 


59 


.06896 


.23317 


.41705 


.62428 


.85951 


.12875 


.43982 


.80316 


.23301 


.7492S 


1 


60 


3.07155 


3.23607 


3.42030 


3.62796 


3.86370 


4.13357 


4.44541 


4.80973 


5.24084 


5.75877 







1 19° 


1 18° 


1 17° 


1 16° 1 15° 1 14° 1 13° 1 12° 1 11° 1 10° 1 ' 



Cosecants. 
* Exsecant = secant — 1 ; coexsecant = cosecant — 1. 



NATURAL SECANTS, ETC, 



175 



4. — Natural Secants, Cosecants (Exsecants, Coexsecants).* — (Concl'd.) 

Secants. 



' 


1 80° 


1 81<» 


1 820 


1 83° 


1 84° 


1 85° 


1 86° 


1 87° 


1 88° 


1 89° 


L 





5.75877 


6.39245 


7.18530 


8.20551 


9.56677 


11.47371 


14.33559 


19.10732 


28.65371 


57.29869 


60 


1 


.76829 


.40422 


.20020 


.22500 


.59332 


.51199 


.39547 


.21397 


.89440 


58.26976 


59 


2 


.77784 


.41602 


.21517 


.24457 


.62002 


.55052 


.45586 


.32182 


29.13917 


59.27431 


58 


3 


.78742 


.42787 


.23019 


.26425 


.64687 


.58932 


.51676 


.43088 


.38812 


60.31411 


57 


4 


.79703 


.43977 


.24529 


.28402 


.67387 


.62837 


.57817 


.54119 


.64137 


61.39105 


56 


5 


5.80667 


6.45171 


7.26044 


8.30388 


9.70103 


11.66769 


14.64011 


19.65275 


29.89903 


62.50715 


55 


6 


.81635 


.46369 


.27566 


.32384 


.72833 


.70728 


.70258 


.76560 


30.16120 


63.66460 


54 


7 


.82606 


.47572 


.29095 


.34390 


.75579 


.74714 


.76558 


.87976 


.42802 


64.86572 


53 


8 


.83581 


.48779 


.30630 


.36405 


.78341 


.78727 


.82913 


.99524 


.69M0 


66.11304 


52 


9 


.84558 


.49991 


.32171 


.38431 


.81119 


.82768 


.89323 


20.11208 


.97607 


67.40927 


51 


10 


5.85539 


6.51208 


7.33719 


8.40466 


9.83912 


11.86837 


14.95788 


20.23028 


31.25758 


68.75736 


50 


11 


.86524 


.52429 


.35274 


.42511 


.86722 


.90934 


15.02310 


.34989 


.54425 


70.16047 


49 


12 


.87511 


.53655 


.36835 


.44566 


.89547 


.95060 


.08890 


.47093 


.83623 


71.62285 


48 


13 


.88502 


.54886 


.38403 


.46632 


.92389 


.99214 


.15527 


.59341 


32.13366 


73.14583 


47 


14 


.89497 


.56121 


.39978 


.48707 


.95248 


12.03397 


.22223 


.71737 


.43671 


74.73586 


46 


15 


5.90495 


6.57361 


7.41560 


8.50793 


9.98123 


12.07610 


15.28979 


20.84283 


32.74554 


76.39655 


45 


16 


.91496 


.58606 


.43148 


.52889 


10.01015 


.11852 


.35795 


.96982 


33.06030 


78.13274 


44 


17 


.92501 


.59855 


.44743 


.54996 


.03923 


.16125 


.42672 


21.09838 


.38118 


79.94968 


43 


18 


.93509 


.61110 


.46346 


.57113 


.06849 


.20427 


.49611 


.22852 


.70835 


81.85315 


42 


19 


.94521 


.62369 


.47955 


.59241 


.09792 


.24761 


.56614 


.36027 


34.04199 


83.84947 


41 


20 


5.95536 


6.63633 


7.49571 


8.61379 


10.12752 


12.29125 


15.63679 


21.49368 


34.38232 


85.94561 


40 


21 


.96555 


.64902 


.51194 


.63528 


.15730 


.33521 


.70810 


.62876 


.72952 


88.14924 


39 


22 


.97577 


.66176 


.52825 


.65688 


.18725 


.37948 


.78005 


.76555 


35.08380 


90.46886 


38 


23 


.98603 


.67454 


.54462 


.67859 


.21739 


.42408 


.85268 


.90409 


.44539 


92.91387 


37 


24 


.99633 


.98738 


.56107 


.70041 


.24770 


.46900 


.92597 


22.04440 


.81452 


95.49471 


36 


25 


6.00666 


6.70027 


7.57759 


8.72234 


10.27819 


12.51424 


15.99995 


22.18653 


36.19141 


98.22303 


35 


26 


.01703 


.71321 


.59418 


.74438 


.30887 


.55982 


16.07462 


.33050 


.57633 


101.1119 


34 


27 


.02743 


.72620 


.61085 


.76653 


.33973 


.60572 


.14999 


.47635 


.96953 


104.1757 


33 


28 


.03787 


.73924 


.62759 


.78880 


.37077 


.65197 


.22607 


.62413 


37.37127 


107.4311 


32 


29 


.04834 


.75233 


.64441 


.81118 


.40201 


.69856 


.30287 


.77386 


.78185 


110.8966 


31 


30 


6.05886 


6.76547 


7.66130 


8.83367 


10.43343 


12.74550 


16.38041 


22.92559 


38.20155 


114.5930 


30 


31 


.06941 


.77866 


.67826 


.85628 


.46505 


.79278 


.45869 


23.07935 


.63068 


118.5444 


29 


32 


.08000 


.79191 


.69530 


.87901 


.49685 


.84042 


.53772 


.23520 


39.06957 


122.7780 


28 


33 


.09062 


.80521 


.71242 


.90186 


.52886 


.88841 


.61751 


.39316 


.51855 


127.3253 


27 


34 


.10129 


.81856 


.72962 


.92482 


.56106 


.93677 


.69808 


.55329 


.97797 


132.2223 


26 


35 


6.11199 


6.83196 


7.74689 


8.94791 


10.59346 


12.98549 


16.77944 


23.71563 


40.44820 


137.5111 


25 


36 


.12273 


.84542 


.76424 


.97111 


.62605 


13.03458 


.86159 


.88022 


.92963 


143.2406 


24 


37 


.13350 


.85893 


.78167 


.99444 


.65885 


.08040 


.94456 


24.04712 


41.42266 


149.4684 


23 


38 


.14432 


.87250 


.79918 


9.01788 


.69186 


.13388 


17.02835 


.21637 


.92772 


156.2623 


22 


39 


.15517 


.88612 


.81677 


.04146 


.72507 


.18411 


.11297 


.38802 


42.44525 


163.7033 


21 


40 


6.16607 


6.89979 


7.83443 


9.06515 


10.75849 


13.23472 


17.19843 


24.56212 


42.97571 


171.8883 


20 


41 


.17700 


.91352 


.85218 


.08897 


.79212 


.28572 


.28476 


.73873 


43.51961 


180.9350 


19 


42 


.18797 


.92731 


.87001 


.11292 


.82596 


.33712 


.37196 


.91790 


44.07746 


190.9868 


18 


43 


.19898 


.94115 


.88792 


.13699 


.86001 


.38891 


.46005 


25.09969 


.64980 


202.2212 


17 


44 


.21004 


.95505 


.90592 


.16120 


.89428 


.44112 


.54903 


.28414 


45.23720 


214.8600 


16 


45 


6.22113 


6.96900 


7.92400 


9.18553 


10.92877 


13.49373 


17.63893 


25.47134 


45.84026 


229.1839 


15 


46 


.23226 


.98301 


.94216 


.20999 


.96348 


.54676 


.72975 


.66132 


46.45963 


245.5540 


14 


47 


.24343 


.99708 


. .96040 


.23459 


.99841 


.60021 


.82152 


.85417 


47.09596 


264.4427 


13 


48 


.25464 


7.01120 


.97873 


.25931 


11.03356 


.65408 


.91424 


26.04994 


.74997 


286.4795 


12 


49 


.26590 


.02538 


.99714 


.28417 


.06894 


.70838 


18.00794 


.24869 


48.42241 


312.5230 


11 


50 


6.27719 


7.03962 


8.01565 


9.30917 


11.10455 


13.76312 


18.10262 


26.45051 


49.11406 


343.7752 


10 


51 


.28853 


.05392 


.03423 


.33430 


.14039 


.81829 


.19830 


.65546 


.82576 


381.9723 


9 


52 


.29991 


.06828 


.05291 


.35957 


.17646 


.87391 


.29501 


.86360 


50.55840 


429.7187 


8 


53 


.31133 


.08269 


.07167 


.38497 


.21277 


.92999 


.39274 


27.07503 


51.31290 


491.1070 


7 


54 


.32279 


.09717 


.09052 


.41052 


.24932 


.98651 


.49153 


.28981 


52.09027 


572.9581 


6 


55 


6.33429 


7.11171 


8.10946 


9.43620 


11.28610 


14.04350 


18.59139 


27.50804 


52.89156 


687.5496 


5 


56 


.34584 


.12630 


.12849 


.46203 


.32313 


.10096 


.69233 


.72978 


53.71790 


859.4369 


4 


57 


.35743 


.14096 


.14760 


.48800 


.36040 


.15889 


.79438 


.95513 


54.57046 


1145.916 


3 


58 


.36906 


.15568 


.16681 


.51411 


.39792 


.21730 


.89755 


28.18417 


55.45053 


1718.874 


2 


59 


.38073 


.17046 


.18612 


.54037 


.43569 


.27620 


19.00185 


.41700 


56.35946 


3437.747 


1 


60 


6.39245 


7.18530 


8.20551 


9.56677 


11.47371 


14.3355919.10732 


28.65371 


57.29869 


Infinite 





1 9° 


go I 70 1 go 


5° 1 4° 1 3° 1 2° 1 1° 1 0° 1 ' 



Cosecants. 
* Exsecant «= secant — 1 ; coexsecant = cosecant — 1 . 



176 



^.— PLANE TRIGONOMETRY. 



0** 



5. — Logarithmic Sines, Tangents, Cotangents, Cosines. 
(Secants, Cosecants.)* 



' 1 


Sine. 


Tang. 1 Cotang. | 


Cosine. | 


1 


' 1 


Sine. 1 


Tang. 1 


Cotang. 1 


Cosine. | 






Inf. 


Inf. 























Neg. 


Neg. 


Infinite. 


10.00000 


60 





8.24186 


8.24192 


11.75808 


9.99993 


60 


1 


6.46373 


6.46373 


13.53627 


.00000 


59 


1 


.24903 


.24910 


.75090 


.99993 


59 


2 


.76476 


.76476 


.23524 


.00000 


59 


2 


.25609 


.25616 


.74384 


.99993 


58 


3 


6.94085 


6.94085 


13.05915 


.00000 


57 


3 


.26304 


.26312 


.73688 


.99993 


57 


4 


7.06579 


7.06579 


12.93421 


.00000 


56 


4 


.26988 


.26996 


.73004 


.99992 


56 


5 


7.16270 


7.16270 


12.83730 


10.00000 


55 


5 


8.27661 


8.27669 


11.72331 


9.99992 


55 


6 


.24188 


.24188 


.75812 


.00000 


54 


6 


.28324 


.28332 


.71668 


.99992 


54 


7 


.30882 


.30882 


.69118 


.00000 


53 


7 


.28977 


.28986 


.71014 


.99992 


53 


8 


.36682 


.36682 


.63318 


.00000 


52 


8 


.29621 


.29629 


.70371 


.99992 


52 


9 


.41797 


.•41797 


.58203 


.00000 


51 


9 


.30255 


.30263 


.69737 


.99991 


51 


10 


7.46373 


7.46373 


12.53627 


10.00000 


50 


10 


8.30879 


8.30888 


11.69112 


9.99991 


50 


11 


.50512 


.50512 


.49488 


.00000 


49 


11 


.31495 


.31505 


.68495 


.99991 


49 


12 


.54291 


.54291 


.45709 


.00000 


48 


12 


.32103 


.32112 


.67888 


.99990 


48 


13 


.57767 


.57767 


.42233 


.00000 


47 


13 


.32702 


.32711 


.67289 


.99990 


47 


14 


.60985 


.60986 


.39014 


.00000 


46 


14 


.33292 


.33302 


.66698 


.99990 


46 


15 


7.63982 


7.63982 


12.36018 


10.00000 


45 


15 


8.33875 


8.33886 


11.66114 


9.99990 


45 


16 


.66784 


.66785 


.33215 


10.00000 


44 


16 


.34450 


.34461 


.65539 


.99989 


44 


17 


.69417 


.69418 


.30582 


9.99999 


43 


17 


.35018 


.35029 


.64971 


.99989 


43 


18 


.71900 


.71900 


.28100 


.99999 


42 


18 


.35578 


.35590 


.64410 


.99989 


42 


19 


.74248 


.74248 


.25752 


.99999 


41 


19 


.36131 


.36143 


.63857 


.99989 


41 


20 


7.76475 


7.76476 


12.23524 


9.99999 


40 


20 


8.36678 


8.36689 


11.63311 


9.99988 


40 


21 


.78594 


.78595 


.21405 


.99999 


39 


21 


.37217 


.37229 


.62771 


.99988 


39 


22 


.80615 


.80615 


.19385 


.99999 


38 


22 


.37750 


.37762 


.62238 


.99988 


38 


23 


.82545 


.82546 


.17454 


.99999 


37 


23 


.38276 


.38289 


.61711 


.99987 


37 


24 


.84393 


.84394 


.15606 


.99999 


36 


24 


.38796 


.38809 


.61191 


.99987 


36 


25 


7.86166 


7.86167 


12.13833 


9.99999 


35 


25 


8.39310 


8.39323 


11.60677 


9.99987 


35 


26 


.87870 


.87871 


.12129 


.99999 


34 


26 


.39818 


.39832 


.60168 


.99986 


34 


27 


.89509 


.89510 


.10490 


.99999 


33 


27 


.40320 


.40334 


.59666 


.99986 


33 


28 


.91088 


.91089 


.08911 


.99999 


32 


28 


.40816 


.40830 


.59170 


.99986 


32 


29 


.92612 


.92613 


.07387 


.99998 


31 


29 


.41307 


.41321 


.58679 


.99985 


31 


30 


7.94084 


7.94086 


12.05914 


9.99998 


30 


30 


8.41792 


8.41807 


11.58193 


9.99985 


30 


31 


.95508 


.95510 


.04490 


.99998 


29 


31 


.42272 


.42287 


.57713 


.99985 


29 


32 


.96887 


.96889 


.03111 


.99998 


28 


32 


.42746 


.42762 


.57238 


.99984 


28 


33 


.98223 


.98225 


.01775 


.99998 


27 


33 


.43216 


.43232 


.56768 


.99984 


27 


34 


7.99520 


7.99522 


12.00478 


.99998 


26 


34 


.43680 


.43696 


.56304 


.99984 


26 


35 


8.00779 


8.00781 


11.99219 


9.99998 


25 


35 


8.44139 


8.44156 


11.55844 


9.99983 


25 


36 


.02002 


.02004 


.97996 


.99998 


24 


36 


.44594 


.44611 


.55389 


.99983 


24 


37 


.03192 


.03194 


.96806 


.99997 


23 


37 


.45044 


.45061 


.54939 


.99983 


23 


38 


.04350 


.04353 


.95647 


.99997 


22 


38 


.45489 


.45507 


.54493 


.99982 


22 


39 


.05478 


.05481 


.94519 


.99997 


21 


39 


.45930 


.45948 


.54052 


.99982 


21 


40 


8.06578 


8.06581 


11.93419 


9.99997 


20 


40 


8.46366 


8.46385 


11.53615 


9.99982 


20 


41 


.07650 


.07653 


.92347 


.99997 


19 


41 


.46799 


.46817 


.53183 


.99981 


19 


42 


.08696 


.08700 


.91300 


.99997 


18 


42 


.47226 


.47245 


.52755 


.99981 


18 


43 


.09718 


.09722 


.90278 


.99997 


17 


43 


.47650 


.47669 


.52331 


.99981 


17 


44 


.10717 


.10720 


.89280 


.99996 


16 


44 


.48069 


.48089 


.51911 


.99980 


16 


45 


8.11693 


8.11696 


11.88304 


9.99996 


15 


45 


8.48485 


8.48505 


11.51495 


9.99980 


15 


46 


.12647 


.12651 


.87349 


-.99996 


14 


46 


.48896 


.48917 


.51083 


.99979 


14 


47 


.13581 


.13585 


.86415 


.99996 


13 


46 


.49304 


.49325 


.50675 


.99979 


13 


48 


.14495 


.14500 


.85500 


.99996 


12 


48 


.49708 


.49729 


.50271 


.99979 


12 


49 


.15391 


.15395 


.84605 


.99996 


11 


49 


.50108 


.50130 


.49870 


.99978 


11 


50 


8.16268 


8.16273 


11.83727 


9.99995 


10 


50 


8.50504 


8.50527 


11.49473 


9.99978 


10 


51 


.17128 


.17133 


.82867 


.99995 


9 


51 


.50897 


.50920 


.49080 


.99977 


9 


52 


.17971 


.17976 


.82024 


.99995 


8 


52 


.51287 


.51310 


.48690 


.99977 


8 


53 


.18798 


.18804 


.81196 


.99995 


7 


53 


.51673 


.51696 


.48304 


.99977 


7 


54 


.19610 


.19616 


.80384 


.99995 


6 


54 


.52055 


.52079 


.47921 


.99976 


6 


55 


8.20407 


8.20413 


11.79587 


9.99994 


5 


55 


8.52434 


8.52459 


11.47541 


9.99976 


5 


56 


.21189 


.21195 


.78805 


.99994 


4 


56 


.52810 


.52835 


.47165 


.99975 


4 


57 


.21958 


.21964 


.78036 


.99994 


3 


57 


.53183 


.53208 


.46792 


.99975 


3 


58 


.22713 


.22720 


.77280 


.99994 


2 


58 


.53552 


.53578 


.46422 


.99974 


2 


59 


.23456 


.23462 


.76538 


.99994 


1 


59 


.53919 


.53945 


.46055 


.99974 


1 


60 


8.24186 


8.24192 


11.75808 


9.99993 





60 


8.54282 


8.54308 


11.45692 


9.99974 






I Cosine. I Cotang 1 Tang, j Sine. \ ' \\ | Cosine. | Cotang. | Tang. | Sine, j' 

89° 88° 

*Log secant = colog cosine = 1 — log cosine ; log cosecant = colog sine «=» 
1 — log sine. 

£:^.— Log sec 0°- 30' = 10.00002. £:^.— Log cosec 0°- 30' = 12.05916. 



LOGARITHMIC SINES, ETC. 



177 



6. — Logarithmic Sines, Tangents, Cotangents, Cosines. 
(Secants, Cosecants.)* — (Cont'd.) 
3° 



' 1 sine. 


Tang. 


Cotang. 1 Cosine. 1 || ' | Sine. 


Tang. 


Cotang. 1 Cosine. 







8.54282 


8.54308 


11.45692 


9.99974 


60 





8.71880 


8.71940 


11.28060 


9.99940 


60 


1 


.54642 


.54669 


.45331 


.99973 


59 


1 


.72120 


.72181 


.27819 


.99940 


59 


2 


.54999 


.55027 


.44973 


.99973 


58 


2 


.72359 


.72420 


.27580 


.99939 


58 


3 


.55354 


.55382 


.44618 


.99972 


57 


3 


.72597 


.72659 


.27341 


.99938 


57 


4 


.55705 


.55734 


.44266 


.99972 


56 


4 


.72834 


.72896 


.27104 


.99938 


56 


5 


8.56054 


8.56083 


11.43917 


9.99971 


55 


5 


8.73069 


8.73132 


11.26868 


9.99937 


55 


6 


.56400 


.56429 


.43571 


.99971 


54 


6 


.73303 


.73366 


.26634 


.99936 


54 


7 


.56743 


.56773 


.43227 


.99970 


53 


7 


.73535 


.73600 


.26400 


.99936 


53 


8 


.57084 


.57114 


.42886 


.99970 


52 


8 


.73767 


.73832 


.26168 


.99935 


52 


9 


.57421 


.57452 


.42548 


.99969 


51 


9 


.73997 


.74063 


.25937 


.99934 


51 


10 


8.57757 


8.57788 


11.42212 


9.99969 


50 


10 


8.74226 


8.74292 


11.25708 


9.99934 


50 


11 


.58089 


.58121 


.41879 


.99968 


49 


11 


.74454 


.74521 


.25479 


.99933 


49 


12 


.58419 


.58451 


.41549 


.99968 


48 


12 


.74680 


.74748 


.25252 


.99932 


48 


13 


.58747 


.58779 


.41221 


.99967 


47 


13 


.74906 


.74974 


.25026 


.99932 


47 


14 


.59072 


.59105 


.40895 


.99967 


46 


14 


.75130 


.75199 


.24801 


.99931 


46 


15 


8.59395 


8.59428 


11.40572 


9.99967 


45 


15 


8.75353 


8.75423 


11.24577 


9.99930 


45 


16 


.59715 


.59749 


.40251 


.99966 


44 


16 


.75575 


.75645 


.24355 


.99929 


44 


17 


.60033 


.60068 


.39932 


.99966 


43 


17 


.75795 


.75867 


.24133 


.99929 


43 


18 


.60349 


.60384 


.39616 


.99965 


42 


18 


.76015 


.76087 


.23913 


.99928 


42 


19 


.60662 


.60698 


.39302 


.99964 


41 


19 


.76234 


.76306 


.23694 


.99927 


41 


20 


8.60973 


8.61009 


11.38991 


9.99964 


40 


20 


8.76451 


8.76525 


11.23475 


9.99926 


40 


21 


.61282 


.61319 


.38681 


.99963 


39 


21 


.76667 


.76742 


.23258 


.99926 


39 


22 


.61589 


.61626 


.38374 


.99963 


38 


22 


.76883 


.76958 


.23042 


.99925 


38 


23 


.61894 


.61931 


.38069 


.99962 


37 


23 


.77097 


.77173 


.22827 


.99924 


37 


24 


.62196 


.62234 


.37766 


.99962 


36 


24 


.77310 


.77387 


.22613 


.99923 


36 


25 


8.62497 


8.62535 


11.37465 


9.99961 


35 


25 


8.77522 


8.77600 


11.22400 


9.99923 


35 


26 


.62795 


.62834 


.37166 


.99961 


34 


26 


.77733 


.77811 


.22189 


.99922 


34 


27 


.63091 


.63131 


.36869 


.99960 


33 


27 


.77943 


.78022 


.21978 


.99921 


33 


28 


.63385 


.63426 


.36574 


.99960 


32 


28 


.78152 


.78232 


.21768 


.99920 


32 


29 


.63678 


.63718 


.36282 


.99959 


31 


29 


.78360 


.78441 


.21559 


.99920 


31 


30 


8.63968 


8.64009 


11.35991 


9.99959 


30 


30 


8.78568 


8.78649 


11.21351 


9.99919 


30 


31 


.64256 


.64298 


.35702 


.99958 


29 


31 


.78774 


.78855 


.21145 


.99918 


29 


32 


.64543 


.64585 


.35415 


.99958 


28 


32 


.78979 


.79061 


.20939 


.99917 


28 


33 


.64827 


.64870 


.35130 


.99957 


27 


33 


.79183 


.79266 


.20734 


.99917 


27 


34 


.65110 


.65154 


.34846 


.99956 


26 


34 


.79386 


.79470 


.20530 


.99916 


26 


35 


8.65391 


8.65435 


11.34565 


9.99956 


25 


35 


8.79588 


8.79673 


11.20327 


9.99915 


25 


36 


.65670 


.65715 


.34285 


.99955 


24 


36 


.79789 


.79875 


.20125 


.99914 


24 


37 


.65947 


.65993 


.34007 


.99955 


23 


37 


.79990 


.80076 


.19924 


.99913 


23 


38 


.66223 


.66269 


.33731 


.99954 


22 


38 


.80189 


.80277 


.19723 


.99913 


22 


39 


.66497 


.66543 


.33457 


.99954 


21 


39 


.80388 


.80476 


.19524 


.99912 


21 


40 


8.66769 


8.66816 


11.33184 


9.99953 


20 


40 


8.80585 


8.80674 


11.19326 


9.99911 


20 


41 


.67039 


.67087 


.32913 


.99952 


19 


41 


.80782 


.80872 


.19128 


.99910 


19 


42 


.67308 


.67356 


.32644 


.99952 


18 


42 


.80978 


.81068 


.18932 


.99909 


18 


43 


.67575 


.67624 


.32376 


.99951 


17 


43 


.81173 


.81264 


.18736 


.99909 


17 


44 


.67841 


.67890 


.32110 


.99951 


16 


44 


.81367 


.81459 


.18541 


.99908 


16 


45 


8.68104 


8.68154 


11.31846 


9.99950 


15 


45 


8.81560 


8.81653 


11.18347 


9.99907 


15 


46 


.68367 


.68417 


.31583 


.99949 


14 


46 


.81752 


.81846 


.18154 


.99906 


14 


47 


.68627 


.68678 


.31322 


.99949 


13 


47 


.81944 


.82038 


.17962 


.99905 


13 


48 


.68886 


.68938 


.31062 


.99948 


12 


48 


.82134 


.82230 


.17770 


.99904 


12 


49 


.69144 


.69196 


.30804 


.99948 


11 


49 


.82324 


.82420 


.17580 


.99904 


11 


50 


8.69400 


8.69453 


11.30547 


9.99947 


10 


50 


8.82513 


8.82610 


11.17390 


9.99903 


10 


51 


.69654 


.69708 


.30292 


.99946 


9 


51 


.82701 


.82799 


.17201 


.99902 


9 


52 


.69907 


.69962 


.30038 


.99946 


8 


52 


.82888 


.82987 


.17013 


.99901 


8 


53 


.70159 


.70214 


.29786 


.99945 


7 


53 


.83075 


.83175 


.16825 


.99900 


7 


54 


.70409 


.70465 


.29535 


.99944 


6 


54 


.83261 


.83361 


.16639 


.99899 


6 


55 


8.70658 


8.70714 


11.29286 


9.99944 


5 


55 


8.83446 


8.83547 


11.16453 


9.99898 


5 


56 


.70905 


.70962 


.29038 


.99943 


4 


56 


.83630 


.83732 


.16268 


.99898 


4 


57 


.71151 


.71208 


.28792 


.99942 


3 


57 


.83813 


.83916 


.16084 


.99897 


3 


58 


.71395 


.71453 


.28547 


.99942 


2 


58 


.83996 


.84100 


.15900 


.99896 


2 


59 


.71638 


.71697 


.28303 


.99941 


1 


59 


.84177 


.84282 


.15718 


.99895 


1 


60 


8.71880 


8.71940 


11.28060 


9.99940 





60 


8.84358 


8.84464 


11.15536 


9.99894 





1 Coslne7 


Cotang. 


Tang. 


Sine. 


'11 1 Cosine. 


1 Cotang. 


Tang. 


Sine. 


_1 



87° 86^ 

*Log secant == colog cosine = 1 — log cosine ; log cosecant = colog sine = 
1 — log sine. 

Ex.—UiQ sec 2°- 30' = 10.00041. Ex.—'Loq cosec 2°- 30'= 11.36032. 



178 



^.— PLANE TRIGONOMETRY. 



5. — Logarithmic Sines, Tangents, Cotangents, Cosines. — (Cont'd.) 
(Secants, Cosecants.)* 
4° 5° 



' 1 


Sine. 1 Tang. | Cotang. | 


Cosine. 


1 


' 1 


Sine. 1 


Tang. 1 Cotang. | Cosine. 







8.84358 


8. 84464 


11.15536 


9.99894 


60 





8.94030 


8.94195 


11.05805 


9.99834 


60 


1 


.84539 


. 84646 


.15354 


.99893 


59 


1 


.94174 


.94340 


.05660 


.99833 


59 


2 


.84718 


. 84826 


.15174 


.99892 


58 


2 


.94317 


.94485 


.05515 


.99832 


58 


3 


.84897 


. 85006 


.14994 


.99891 


57 


3 


.94461 


.94630 


.05370 


.99831 


57 


4 


.85075 


.85185 


.14815 


.99891 


56 


4 


.94603 


.94773 


.05227 


.99830 


56 


5 


8.85252 


8.85363 


11.14637 


9.99890 


55 


5 


8.94746 


8.94917 


11.05083 


9.99829 


55 


6 


.85429 


. 85540 


.14460 


.99889 


54 


6 


.94887 


.95060 


.04940 


.99828 


54 


7 


. 85605 


.85717 


.14283 


.99888 


53 


7 


.95029 


. 95202 


.04798 


.99827 


53 


8 


.85780 


. 85893 


.14107 


.99887 


52 


8 


.95170 


. 95344 


.04656 


.99825 


52 


9 


.85955 


.86069 


.13931 


.99886 


51 


9 


.95310 


. 95486 


.04514 


.99824 


51 


10 


8.86128 


8. 86243 


11.13757 


9.99885 


50 


10 


8.95450 


8.95627 


11.04373 


9.99823 


50 


11 


.86301 


.86417 


. 13583 


.99884 


49 


11 


.95589 


.95767 


.04233 


. 99822 


49 


12 


.86474 


.86591 


.13409 


. 99883 


48 


12 


.95728 


.95908 


. 04092 


.99821 


48 


13 


.86645 


.86763 


.13237 


.99882 


47 


13 


.95867 


.96047 


.03953 


.99820 


47 


14 


.86816 


.86935 


.13065 


.99881 


46 


14 


.96005 


.96187 


.03813 


.99819 


46 


15 


8.86987 


8.87106 


11.12894 


9.99880 


45 


15 


8.96143 


8.96325 


11.03675 


9.99817 


45 


16 


.87156 


.87277 


.12723 


.99879 


44 


16 


.96280 


.96464 


.03536 


.99816 


44 


17 


.87325 


. 87447 


.12553 


.99879 


43 


17 


.96417 


.96602 


.03398 


.99815 


43 


18 


.87494 


.87616 


.12384 


.99878 


42 


18 


.96553 


.96739 


.03261 


.99814 


42 


19 


.87661 


. 87785 


.12215 


.99877 


41 


19 


.96689 


.96877 


.03123 


.99813 


41 


20 


8.87829 


8.87953 


11.12047 


9.99876 


40 


20 


8.96825 


8.97013 


11.02987 


9.99812 


40 


21 


.87995 


.88120 


.11880 


.99875 


39 


21 


.96960 


.97150 


.02850 


.99810 


39 


22 


.88161 


.88287 


.11713 


.99874 


38 


22 


.97095 


.97286 


.02715 


. 99809 


38 


23 


. 88326 


. 88453 


.11547 


.99873 


37 


23 


.97229 


.97421 


.02579 


.99808 


37 


24 


.88490 


.88618 


.11382 


.99872 


36 


24 


.97363 


.97556 


.02444 


. 99807 


36 


25 


8.88654 


8.88783 


11.11217 


9.99871 


35 


25 


8.97496 


8.97691 


11.02309 


9.99806 


35 


26 


.88817 


. 88948 


.11052 


.99870 


34 


26 


.97629 


.97825 


.02175 


.99804 


34 


27 


.88980 


.89111 


.10889 


.99869 


33 


27 


.97762 


.97959 


.02041 


. 99803 


33 


28 


.89142 


.89274 


.10726 


.99868 


32 


28 


.97894 


.98092 


.01908 


.99802 


32 


29 


. 89304 


.89437 


.10563 


.99867 


31 


29 


.98026 


.98225 


.01775 


.99801 


31 


30 


8.89464 


8.89598 


11.10402 


9.99866 


30 


30 


8.98157 


8.98358 


11.01642 


9.99800 


30 


31 


.89625 


.89760 


.10240 


.99865 


29 


31 


.98288 


.98490 


.01510 


.99798 


29 


32 


.89784 


.89920 


.10080 


. 99864 


28 


32 


.98419 


.98622 


.01378 


.99797 


28 


33 


. 89943 


. 90080 


.09920 


. 99863 


27 


33 


.98549 


.98753 


.01247 


.99796 


27 


34 


.90102 


.90240 


.09760 


. 99862 


26 


34 


.98679 


. 98884 


.01116 


.99795 


26 


35 


8.90260 


8.90399 


11.09601 


9.99861 


25 


35 


8.98808 


8.99015 


11.00985 


9.99793 


25 


36 


.90417 


.90557 


.09443 


. 99860 


24 


36 


. 98937 


.99145 


.00855 


.99792 


24 


37 


.90574 


.90715 


.09285 


.99859 


23 


37 


.99066 


.99275 


.00725 


.99791 


23 


38 


.90730 


.90872 


.09128 


.99858 


22 


38 


.99194 


.99405 


.00595 


.99790 


22 


39 


.90885 


.91029 


.08971 


. 99857 


21 


39 


.99322 


.99534 


.00466 


.99788 


21 


40 


8.91040 


8.91185 


11.08815 


9.99856 


20 


40 


8.99450 


8.99662 


11.00338 


9.99787 


20 


41 


.91195 


.91340 


.08660 


. 99855 


19 


41 


.99577 


.99791 


.00209 


.99786 


19 


42 


.91349 


.91495 


.08505 


. 99854 


18 


42 


.99704 


8.99919 


11.00081 


.99785 


18 


43 


.91502 


.91650 


.08350 


. 99853 


17 


43 


. 99830 


9.00046 


10.99954 


.99783 


17 


44 


.91655 


.91803 


.08197 


.99852 


16 


44 


8.99956 


.00174 


. 99826 


.99782 


16 


45 


8.91807 


8.91957 


11.08043 


9.99851 


15 


45 


9.00082 


9.00301 


10.99699 


9.99781 


15 


46 


.91959 


.92110 


.07890 


.99850 


14 


46 


. 00207 


.00427 


.99573 


.99780 


14 


47 


.92110 


.92262 


.07738 


. 99848 


13 


47 


.00332 


. 00553 


.99447 


.99778 


13 


48 


.92261 


.92414 


.07586 


. 99847 


12 


48 


.00456 


.00679 


.99321 


.99777 


12 


49 


.92411 


.92565 


.07435 


.99846 


11 


49 


.00581 


. 00805 


.99195 


.99776 


11 


50 


8.92561 


8.92716 


11.07284 


9.99845 


10 


50 


9.00704 


9.00930 


10.99070 


9.99775 


10 


51 


.92710 


.92866 


.07134 


. 99844 


9 


51 


.00828 


.01055 


.98945 


. 99773 


9 


52 


.92859 


.93016 


.06984 


. 99843 


8 


52 


.00951 


.01179 


.98821 


.99772 


8 


53 


. 93007 


.93165 


.06835 


. 99842 


7 


53 


.01074 


.01303 


. 98697 


.99771 


7 


54 


.93154 


.93313 


. 06687 


.99841 


6 


54 


.01196 


.01427 


. 98573 


.99769 


6 


55 


8.93301 


8.93462 


11.06538 


9.99840 


5 


55 


9.01318 


9.01550 


10.98450 


9.99768 


5 


56 


.93448 


.93609 


.06391 


.99839 


4 


56 


.01440 


.01673 


.98327 


.99767 


4 


57 


.93594 


.93756 


.06244 


. 99838 


3 


57 


.01561 


.01796 


. 98204 


.99765 


3 


58 


.93740 


.93903 


.06097 


.99837 


2 


58 


.01682 


.01918 


. 98082 


.99764 


2 


59 


.93885 


.94049 


.05951 


.99836 


1 


59 


.01803 


.02040 


.97960 


.99763 


1 


60 


8.94030 


8.94195 


11.05805 


9.99834 





60 


9.01923 


9.02162 


10.97838 


9.99761 






I Cosine. I Cotang. I Tang. | Sine. \ ' \\ | Cosine. | Cotang. | Tang. | Sine. | 

85° 84= 

*Lcrg secant = colog cosine=l — log cosine; log cosecant = colog sine = 
1 — log sine. 

£^.— Log sec 4°- 30' = 10.00134. Ex.— Log cosec 4°- 30' = 11.10536. 



LOGARITHMIC SINES, ETC. 



179 



5. — Logarithmic Sines, Tangents, Cotangents, Cosines. — (Cont'd.) 
(Secants, Cosecants.)* 

go 70 



/ 


1 Sine. 


1 Tang. 


1 Cotang. 1 Cosine. | 


II ' 


1 Sine. 


1 Tang. 


1 Cotang. 


1 Cosine.! 





9.01923 


9.02162 


10.97838 


9.99761 


60 





9.08589 


9.08914 


10.91086 


9.99675 


60 


1 


.02043 


.02283 


.97717 


.99760 


59 


1 


.08692 


.09019 


.90981 


.99674 


59 


2 


.02163 


.02404 


.97596 


.99759 


58 


2 


.08795 


.09123 


.90877 


.99672 


58 


3 


.02283 


.02525 


.97475 


.99757 


57 


3 


.08897 


.09227 


.90773 


.99670 


57 


4 


.02402 


.02645 


.97355 


.99756 


56 


4 


.08999 


.09330 


.90670 


.99669 


56 


5 


9.02520 


9.02766 


10.97234 


9.99755 


55 


5 


9.09101 


9.09434 


10.90566 


9.99667 


55 


6 


.02639 


.02885 


.97115 


.99753 


54 


6 


.09202 


.09537 


. 90463 


.99666 


54 


7 


.02757 


.03005 


.96995 


.99752 


53 


7 


.09304 


.09640 


.90360 


.99664 


53 


8 


.02874 


.03124 


.96876 


.99751 


52 


8 


.09405 


.09742 


.90258 


.99663 


52 


9 


.02992 


.03242 


.96758 


.99749 


51 


9 


.09506 


.09845 


.90155 


.99661 


51 


10 


9.03109 


9.03361 


10.96639 


9.99748 


50 


10 


9.09606 


9.09947 


10.90053 


9.99659 


50 


11 


.03226 


.03479 


.96521 


.99747 


49 


11 


.09707 


.10049 


.89951 


.99658 


49 


12 


.03342 


.03597 


.96403 


.99745 


48 


12 


.09807 


.10150 


. 89850 


.99656 


48 


13 


.03458 


.03714 


.96286 


.99744 


47 


13 


.09907 


.10252 


.89748 


.99655 


47 


14 


.03574 


.03832 


.96168 


.99742 


46 


14 


.10006 


.10353 


. 89647 


.99653 


46 


15 


9.03690 


9. 03948 


10.96052 


9.99741 


45 


15 


9.10106 


9.10454 


10.89546 


9.99651 


45 


16 


.03805 


.04065 


.95935 


.99740 


44 


16 


.10205 


.10555 


.89445 


.99650 


44 


17 


.03920 


.04181 


.95819 


.99738 


43 


17 


.10304 


.10656 


.89344 


.99648 


43 


18 


.04034 


.04297 


. 95703 


.99737 


42 


18 


.10402 


.10756 


. 89244 


.99647 


42 


19 


.04149 


.04413 


.95587 


.99736 


41 


19 


.10501 


.10856 


.89144 


.99845 


41 


20 


9.04262 


9.04528 


10.95472 


9.99734 


40 


20 


9.10599 


9.10956 


10.89044 


9.99643 


40 


21 


.04376 


.04643 


.95357 


.99733 


39 


21 


.10697 


.11056 


. 88944 


.99642 


39 


22 


.04490 


.04758 


.95242 


.99731 


38 


22 


.10795 


.11155 


.88845 


.99640 


38 


23 


.04603 


.04873 


.95127 


.99730 


37 


23 


.10893 


.11254 


.88746 


.99638 


37 


24 


.04715 


.04987 


.95013 


.99728 


36 


24 


.10990 


.11353 


. 88647 


.99637 


36 


25 


9.04828 


9.05101 


10.94899 


9.99727 


35 


25 


9.11087 


9.11452 


10.88548 


9.99635 


35 


26 


.04940 


.05214 


.94786 


.99726 


34 


26 


.11184 


.11551 


.88449 


.99633 


34 


27 


.05052 


.05328 


.94672 


.99724 


33 


27 


.11281 


.11649 


.88351 


.99632 


33 


28 


.05164 


.05441 


.94559 


.99723 


32 


28 


.11377 


.11747 


. 88253 


.99630 


32 


29 


.05275 


.05553 


.94447 


.99721 


31 


29 


.11474 


.11845 


.88155 


.99629 


31 


30 


9.05386 


9.05666 


10.94334 


9.99720 


30 


30 


9.11570 


9.11943 


10.88057 


9.99627 


30 


31 


.05497 


.05778 


.94222 


.99718 


29 


31 


.11666 


.12040 


. 87960 


.99625 


29 


32 


.05607 


.05890 


.94110 


.99717 


28 


32 


.11761 


.12138 


.87862 


.99624 


28 


33 


.05717 


.06002 


.93998 


.99716 


27 


33 


.11857 


.12235 


.87765 


.99622 


27 


34 


.05827 


.06113 


.93887 


.99714 


26 


34 


.11952 


.12332 


.87668 


.99620 


26 


35 


9.05937 


9.06224 


10.93776 


9.99713 


25 


35 


9.12047 


9.12428 


10.87572 


9.99618 


25 


36 


.06046 


.06335 


.93665 


.99711 


24 


36 


.12142 


.12525 


.87475 


.99617 


24 


37 


.06155 


.06445 


.93555 


.99710 


23 


37 


.12236 


.12621 


.87379 


.99615 


23 


38 


.06264 


.06556 


.93444 


.99708 


22 


38 


.12331 


.12717 


.87283 


.99613 


22 


39 


.06372 


.06666 


.93334 


.99707 


21 


39 


.12425 


.12813 


.87187 


.99612 


21 


40 


9.06481 


9.06775 


10.93225 


9.99705 


20 


40 


9.12519 


9.12909 


10.87091 


9.99610 


20 


41 


.06589 


.06885 


.93115 


.99704 


19 


41 


.12612 


.13004 


.86996 


.99608 


19 


42 


.06696 


.06994 


.93006 


.99702 


18 


42 


.12706 


.13099 


.86901 


.99607 


18 


43 


.06804 


.07103 


.92897 


.99701 


17 


43 


.12799 


.13194 


.86806 


.99605 


17 


44 


.06911 


.07211 


.92789 


.99699 


16 


44 


.12892 


.13289 


.86711 


. 99603 


16 


45 


9.07018 


9.07320 


10.92680 


9.99698 


15 


45 


9.12985 


9.13384 


10.86616 


9.99601 


15 


46 


.07124 


.07428 


.92572 


.99696 


14 


46 


.13078 


.13478 


.86522 


.99600 


14 


47 


.07231 


.07536 


.92464 


.99695 


13 


47 


.13171 


.13573 


. 86427 


.99598 


13 


48 


.07337 


.07643 


.92357 


.99693 


12 


48 


.13263 


.13667 


. 86333 


.99596 


12 


49 


.07442 


.07751 


.92249 


.99692 


11 


49 


.13355 


.13761 


.86239 


.99595 


11 


50 


9.07548 


9.07858 


10.92142 


9.99690 


10 


50 


9.13447 


9.13854 


10.86146 


9.99593 


10 


51 


.07653 


.07964 


.92036 


.99689 


9 


51 


.13539 


.13948 


. 86052 


.99591 


9 


52 


.07758 


.08071 


.91929 


.99687 


8 


52 


.13630 


.14041 


.85959 


.99589 


8 


53 


.07863 


.08177 


.91823 


.99686 


7 


53 


.13722 


.14134 


.85866 


.99588 


7 


54 


.07968 


.08283 


.91717 


.99684 


6 


54 


.13813 


.14227 


.85773 


.99586 


6 


55 


9.08072 


9.08389 


10.91611 


9.99683 


5 


55 


9.13904 


9.14320 


10.85680 


9.99584 


5 


56 


.08176 


.08495 


.91505 


.99681 


4 


56 


.13994 


.14412 


.85588 


.99582 


4 


57 


.08280 


.08600 


.91400 


.99680 


3 


57 


.14085 


.14504 


.85496 


.99581 


3 


58 


.08383 


.08705 


.91295 


.99678 


2 


58 


.14175 


.14597 


. 85403 


.99579 


2 


59 


.08486 


.08810 


.91190 


.99677 


1 


59 


.14266 


.14688 


.85312 


.99577 


1 


60 


9.08589 


9.08914 


10.91086 


9.99675 





60 


9.14356 


9.147ap 


10.85220 


9.99575 






Cosine. I Cotang. i Tang, j Sine. \ ' \\ | Cosine. | Cotang. | Tang, j Sine. | ' 

83^ 82° 

*Log secant = colog cosme=l — log cosine; log cosecant = colog sine = 
1 — log sine. 

Ex.— Log sec 6°- 30'= 10.00280. Ex.—Lo^ cosec 6°- 30'= 10.94614. 



180 



^.— PLANE TRIGONOMETRY. 



5. — Logarithmic Sines, Tangents, Cotangents, Cosines. — (Cont'd.) 
(Secants, Cosecants.)* 
8° 9° 



JM 


Sine. 


Tang. 


Cotang. 


Cosine. | 


II ' 


Sine. 


Tang. 


Cotang. 


Cosine. 1 


\ 





9.14356 


9.14780 


10.85220 


9.99575 


60 





9.19433 


9.19971 


10.80029 


9.99462 


60 


1 


.14445 


.14872 


.85128 


.99574 


59 


1 


.19513 


.20053 


.79947 


.99460 


59 


2 


.14535 


.14963 


. 85037 


.99572 


58 


2 


.19592 


.20134 


.79866 


.99458 


58 


3 


.14624 


.15054 


. 84946 


.99570 


57 


3 


.19672 


.20216 


.79784 


.99456 


57 


4 


.14714 


.15145 


. 84855 


.99568 


56 


4 


.19751 


.20297 


.79703 


. 99454 


56 


5 


9.14803 


9.15236 


10.84764 


9.99566 


65 


5 


9.19830 


9.20378 


10.79622 


9.99452 


55 


6 


.14891 


.15327 


.84673 


.99565 


54 


6 


.19909 


.20459 


.79541 


.99450 


54 


7 


.14980 


.15417 


. 84583 


. 99563 


53 


7 


.19988 


.20540 


.79460 


.99448 


53 


8 


.15069 


.15508 


.84492 


.99561 


52 


8 


.20067 


.20621 


.79379 


. 99446 


52 


9 


.15157 


.15598 


. 84402 


.99559 


51 


9 


.20145 


.20701 


.79299 


.99444 


51 


10 


9.15245 


9.15688 


10.84312 


9.99557 


50 


10 


9.20223 


9.20782 


10.79218 


9.99442 


50 


11 


.15333 


.15777 


.84223 


.99556 


49 


11 


.20302 


.20862 


.79138 


.99440 


49 


12 


.15421 


.15867 


.84133 


.99554 


48 


12 


.20380 


.20942 


.79058 


.99438 


48 


13 


.15508 


.15956 


. 84044 


.99552 


47 


13 


.20458 


.21022 


.78978 


. 99436 


47 


14 


.15596 


.16046 


.83954 


.99550 


46 


14 


.20535 


.21102 


.78898 


.99434 


46 


IS 


9.15683 


9.16135 


10.83865 


9.99548 


45 


15 


9.20613 


9.21182 


10.78818 


9.99432 


45 


16 


.15770 


.16224 


.83776 


.99546 


44 


16 


.20691 


.21261 


.78739 


.99429 


44 


17 


.15857 


.16312 


.83688 


.99545 


43 


17 


.20768 


.21341 


.78659 


.99427 


43 


18 


.15944 


.16401 


.83599 


.99543 


42 


18 


.20845 


.21420 


.78580 


.99425 


42 


19 


.16030 


.16489 


.83511 


.99541 


41 


19 


.20922 


.21499 


.78501 


. 99423 


41 


20 


9.16116 


9.16577 


10.83423 


9.99539 


40 


20 


9.20999 


9.21578 


10.78422 


9.99421 


40 


21 


.16203 


.16665 


.83335 


.99537 


39 


21 


.21076 


.21657 


.78343 


.99419 


39 


22 


.16289 


.16753 


. 83247 


.99535 


38 


22 


.21153 


.21736 


.78264 


.99417 


38 


23 


.16374 


.16841 


.83159 


.99533 


37 


23 


.21229 


.21814 


.78186 


.99415 


37 


24 


.16460 


.16928 


.83072 


. 99532 


36 


24 


.21306 


.21893 


.78107 


.99413 


36 


25 


9.16545 


9.17016 


10.82984 


9.99530 


35 


25 


9.21382 


9.21971 


10.78029 


9.99411 


35 


26 


.16631 


.17103 


.82897 


.99528 


34 


26 


.21458 


.22049 


.77951 


.99409 


34 


27 


.16716 


.17190 


.82810 


.99526 


33 


27 


.21534 


.22127 


.77873 


.99407 


33 


28 


.16801 


.17277 


.82723 


.99524 


32 


28 


.21610 


.22205 


.77795 


.99404 


32 


29 


.16886 


.17363 


.82637 


.99522 


31 


29 


.21685 


.22283 


.77717 


.99402 


31 


30 


9.16970 


9.17450 


10.82550 


9.99520 


30 


30 


9.21761 


9.22361 


10.77639 


9.99400 


30 


31 


.17055 


.17536 


.82464 


.99518 


29 


31 


.21836 


.22438 


.77562 


.99398 


29 


32 


.17139 


.17622 


.82378 


.99517 


28 


32 


.21912 


.22516 


.77484 


.99396 


28 


33 


.17223 


.17708 


.82292 


.99515 


27 


33 


.21987 


.22593 


.77407 


.99394 


27 


34 


.17307 


.17794 


.82206 


.99513 


26 


34 


.22062 


.22670 


.77330 


.99392 


26 


35 


9.17391 


9.17880 


10.82120 


9.99511 


25 


35 


9.22137 


9.22747 


10.77253 


9.99390 


25 


36 


.17474 


.17965 


.82035 


.99509 


24 


36 


.22211 


.22824 


.77176 


.99388 


24 


37 


.17558 


.18051 


.81949 


.99507 


23 


37 


.22286 


.22901 


.77099 


.99385 


23 


38 


.17641 


.18136 


.81864 


.99505 


22 


38 


.22361 


.22977 


.77023 


.99383 


22 


39 


.17724 


.18221 


.81779 


. 99503 


21 


39 


.22435 


.23054 


.76946 


.99381 


21 


40 


9.17807 


9.18306 


10.81694 


9.99501 


20 


40 


9.22509 


9.23130 


10.76870 


9.99379 


20 


41 


.17890 


.18391 


.81609 


.99499 


19 


41 


.22583 


.23206 


.76794 


.99377 


19 


42 


.17973 


.18475 


.81525 


.99497 


18 


42 


.22657 


.23283 


.76717 


.99375 


18 


43 


.18055 


.18560 


.81440 


.99495 


17 


43 


.22731 


.23359 


.76641 


.99372 


17 


44 


.18137 


.18644 


.81356 


. 99494 


16 


44 


.22805 


.23435 


.76565 


.99370 


16 


45 


9.18220 


9.18728 


10.81272 


9.99492 


15 


45 


9.22878 


9.23510 


10.76490 


9.99368 


15 


46 


.18302 


.18812 


.81188 


.99490 


14 


46 


.22952 


.23586 


.76414 


.99366 


14 


47 


.18383 


.18896 


.81104 


.99488 


13 


47 


.23025 


.23661 


.76339 


. 99364 


13 


48 


.18465 


.18979 


.81021 


.99486 


12 


48 


.23098 


.23737 


.76263 


.99362 


12 


49 


.18547 


.19063 


.80937 


.99484 


11 


49 


.23171 


.23812 


.76188 


.99359 


11 


50 


9.18628 


9.19146 


10.80854 


9.99482 


10 


50 


9.23244 


9.23887 


10.76113 


9.99357 


10 


51 


.18709 


.19229 


.80771 


.99480 


9 


51 


.23317 


.23962 


.76038 


.99355 


9 


52 


.18790 


.19312 


. 80688 


.99478 


8 


52 


.23390 


.24037 


.75963 


. 99353 


8 


53 


.18871 


.19395 


.80605 


.99476 


7 


53 


.23462 


.24112 


.75888 


.99351 


7 


54 


.18952 


.19478 


.80522 


.99474 


6 


54 


.23535 


.24186 


.75814 


.99348 


6 


55 


9.19033 


9.19561 


10.80439 


9.99472 


5 


55 


9.23607 


9.24261 


10.75739 


9.99346 


5 


56 


.19113 


.19643 


.80357 


.99470 


4 


56 


.23679 


.24335 


.75665 


.99344 


4 


57 


.19193 


.19725 


.80275 


.99468 


3 


57 


.23752 


.24410 


.75590 


.99342 


3 


58 


.19273 


.19807 


.80193 


.99466 


2 


58 


.23823 


.24484 


.75516 


. 99340 


2 


59 


.19353 


.19889 


.80111 


.99464 


1 


59 


.23895 


.24558 


.75442 


.99337 


1 


60 


9.19433 


9.19971 


10.80029 


9.99462 





60 


9.23967 


9.24632 


10.75368 


9.99335 






I Cosine. I Cotang. I Tang. | Sine. | 



I Cosine, i Cotang. | Tang. | Sine. | 



81° 80° 

*Log secant = colog cosine=l — log cosine; log cosecant = colog sine=« 
1 — log sine. 

£^.— Log sec 8°- 30' = 10.00480. Ex.—Loq cosec 8°- 30' = 10.83030. 



LOGARITHMIC SINES, ETC. 



181 



5. — Logarithmic Sines, Tangents, Cotangents, Cosines. — (Cont'd.) 
(Secants, Cosecants.)* 
10° 11° 



' 1 sine. 


Tang. 


Cotang. 1 Cosine. 


1 II ' 1 Sine. 


Tang. 


1 Cotang. 1 Cosine. | 





9.23967 


9.24632 


10.75368 


9.99335 


60 





9.28060 


9.28865 


10.71135 


9.99195 


60 


1 


.24039 


.24706 


.75294 


. 99333 


59 


1 


.28125 


.28933 


.71067 


.99192 


59 


2 


.24110 


.24779 


.75221 


.99331 


58 


2 


.28190 


.29000 


.71000 


.99190 


58 


3 


.24181 


.24853 


.75147 


.99328 


57 


3 


.28254 


.29067 


.70933 


.99187 


57 


4 


.24253 


.24926 


.75074 


.99326 


56 


4 


.28319 


.29134 


.70866 


.99185 


56 


5 


9.24324 


9.25000 


10.75000 


9.99324 


55 


5 


9.28384 


9.29201 


10.70799 


9.99182 


55 


6 


.24395 


.25073 


.74927 


.99322 


54 


6 


.28448 


.29268 


.70732 


.99180 


54 


7 


.24466 


.25146 


.74854 


.99319 


53 


7 


.28512 


.29335 


.70665 


.99177 


53 


8 


.24536 


.25219 


.74781 


.99317 


52 


8 


.28577 


.29402 


.70598 


. 99175 


52 


9 


.24607 


.25292 


.74708 


.99315 


51 


9 


.28641 


.29468 


.70532 


.99172 


51 


10 


9.24677 


9.25365 


10.74635 


9.99313 


50 


10 


9.28705 


9.29535 


10.70465 


9.99170 


50 


11 


.24748 


.25437 


.74563 


.99310 


49 


11 


.28769 


.29601 


.70399 


.99167 


49 


12 


.24818 


.25510 


.74490 


.99308 


48 


12 


.28833 


.29668 


.70332 


.99165 


48 


13 


.24888 


.25582 


.74418 


.99306 


47 


13 


.28896 


.29734 


.70266 


.99162 


47 


14 


.24958 


.25655 


.74345 


.99304 


46 


14 


.28960 


.29800 


.70200 


.99160 


46 


15 


9.25028 


9.25727 


10.74273 


9.99301 


45 


15 


9.29024 


9.29866 


10.70134 


9.99157 


45 


16 


.25098 


.25799 


.74201 


.99299 


44 


16 


.29087 


.29932 


.70068 


.99155 


44 


17 


.25168 


.25871 


.74129 


.99297 


43 


17 


.29150 


.29998 


.70002 


.99152 


43 


18 


.25237 


.25943 


.74057 


.99294 


42 


18 


.29214 


30064 


.69936 


.99150 


42 


19 


.25307 


.26015 


.73985 


.99292 


41 


19 


.29277 


.30130 


.69870 


.99147 


41 


20 


9.25376 


9.26086 


10.73914 


9.99290 


40 


20 


9.29340 


9.30195 


10.69805 


9.99145 


40 


21 


.25445 


.26158 


.73842 


.99288 


39 


21 


.29403 


.30261 


.69739 


.99142 


39 


22 


.25514 


.26229 


.73771 


.99285 


38 


22 


.29466 


.30326 


.69674 


.99140 


38 


23 


.25583 


.26301 


.73699 


.99283 


37 


23 


.29529 


.30391 


.69609 


.99137 


37 


24 


.25652 


.26372 


.73628 


.99281 


36 


24 


.29591 


.30457 


.69543 


.99135 


36 


25 


9.25721 


9.26443 


10.73557 


9.99278 


35 


25 


9.29654 


9.30522 


10.69478 


9.99132 


35 


26 


.25790 


.26514 


.73486 


.99276 


34 


26 


.29716 


.30587 


.69413 


.99130 


34 


27 


.25858 


.26585 


.73415 


.99274 


33 


27 


.29779 


.30652 


.69348 


.99127 


33 


28 


.25927 


.26655 


.73345 


.99271 


32 


28 


.29841 


.30717 


.69283 


.99124 


32 


29 


.25995 


.26726 


.73274 


.99269 


31 


29 


.29903 


.30782 


.69218 


.99122 


31 


30 


9.26063 


9.26797 


10.73203 


9.99267 


30 


30 


9.29966 


9.30846 


10.69154 


9.99119 


30 


31 


.26131 


.26867 


.73133 


.99264 


29 


31 


.30028 


.30911 


.69089 


.99117 


29 


32 


.26199 


.26937 


.73063 


.99262 


28 


32 


.30090 


.30975 


.69025 


.99114 


28 


33 


.26267 


.27008 


.72992 


.99260 


27 


33 


.30151 


.31040 


.68960 


.99112 


27 


34 


.26335 


.27078 


.72922 


.99257 


26 


34 


.30213 


.31104 


.68896 


.99109 


26 


35 


9.26403 


9.27148 


10.72852 


9.99255 


25 


35 


9.30275 


9.31168 


10.68832 


9-99106 


25 


36 


.26470 


.27218 


.72782 


.99252 


24 


36 


.30336 


.31233 


.68767 


.99104 


24 


37 


.26538 


.27288 


.72712 


.99250 


23 


37 


.30398 


.31297 


.68703 


.99101 


23 


38 


.26605 


.27357 


.72643 


.99248 


22 


38 


.30459 


.31361 


.68639 


, 99099 


22 


39 


.26672 


.27427 


.72573 


.99245 


21 


39 


.30521 


.31425 


.68575 


99096 


21 


40 


9.26739 


9.27496 


10.72504 


9.99243 


20 


40 


9.30582 


9.31489 


10.68511 


9.99093 


20 


41 


.26806 


.27566 


.72434 


.99241 


19 


41 


.30643 


.31552 


.68448 


.99091 


19 


42 


.26873 


.27635 


.72365 


.99238 


18 


42 


.30704 


.31616 


.68384 


.99088 


18 


43 


.26940 


.27704 


.72296 


.99236 


17 


43 


.30765 


.31679 


.68321 


.99086 


17 


44 


.27007 


.27773 


.72227 


.99233. 


16 


44 


.30826 


.31743 


.68257 


.99083 


16 


45 


9.27073 


9.27842 


10.72158 


9.99231 


15 


45 


9.30887 


9.31806 


10.68194 


9.99080 


15 


46 


.27140 


.27911 


.72089 


.99229 


14 


46 


.30947 


.31870 


.68130 


.99078 


14 


47 


.27206 


.27980 


.72020 


.99226 


13 


47 


.31008 


.31933 


.68067 


.99075 


13 


48 


.27273 


.28049 


.71951 


.99224 


12 


48 


.31068 


.31996 


.68004 


.99072 


12 


49 


.27339 


.28117 


.71883 


.99221 


11 


49 


.31129 


.32059 


.67941 


.99070 


11 


50 


9.27405 


9.28186 


10.71814 


9.99219 


10 


50 


9.31189 


9.32122 


10.67878 


9.99067 


10 


51 


.27471 


.28254 


.71746 


.99217 


9 


51 


.31250 


.32185 


.67815 


.99064 


9 


52 


.27537 


.28323 


.71677 


.99214 


8 


52 


.31310 


.32248 


.67752 


. 99062 


8 


53 


.27602 


.28391 


.71609 


.99212 


7 


53 


.31370 


.32311 


.67689 


.99059 


7 


54 


.27668 


.28459 


.71541 


.99209 


6 


54 


.31430 


.32373 


.67627 


.99056 


6 


55 


9.27734 


9.28527 


10.71473 


9.99207 


5 


55 


9.31490 


9.32436 


10.67564 


9.99054 


5 


56 


.27799 


.28595 


.71405 


.99204 


4 


56 


.31549 


.32498 


.67502 


.99051 


4 


57 


.27864 


.28662 


.71338 


.99202 


3 


57 


.31609 


.32561 


.67439 


.99048 


3 


58 


.27930 


.28730 


.71270 


.99200 


2 


58 


.31669 


.32623 


.67377 


.99046 


2 


59 


.27995 


.28798 


.71202 


.99197 


1 


59 


.31728 


.32685 


.67315 


.99043 


I 


60 


9.28060 


9.28865 


10.71135 


9.99195 





60 


9.31788 


9.32747 


10.67253 


9.99040 





1 Cosine. 1 


Cotang.l 


Tang. 1 


Sine. 1 


'11 1 Cosine. | 


Cotang. 1 


Tang. 1 


Sine. 1 


~^ 



*Log secant = colog cosine = 1 — log cosine ; log cosecant = colog sine = 
1 — log sine. 

Ex.— Log sec 10°- 30' = 10.00733. Ex.— hog cosec 10°- 30' = 10.73937. 



182 



9.— PLANE TRIGONOMETRY. 



5. — Logarithmic Sines, Tangents, Cotangents, Cosines. — (Cont'd.) 

(Secants, Cosecants.)* 
12° 13° 



' 


Sine. 


1 Tang. 


1 Cotang. 1 Cosine. | 


1 ' 


Sine. 


1 Tang. 


1 Cotang. 


1 Cosine. 1 





9.31788 


9.32747 


10.67253 


9.99040 


60 





9.35209 


9.36336 


10.63664 


9.98872 


60 


1 


.31847 


.32810 


.67190 


.99038 


59 


1 


.35263 


.36394 


.63606 


.98869 


59 


2 


.31907 


.32872 


.67128 


.99035 


58 


2 


.35318 


.36452 


.63548 


.98867 


58 


3 


.31966 


.32933 


.67067 


.99032 


57 


3 


.35373 


.36509 


.63491 


. 98864 


57 


4 


.32025 


.32995 


.67005 


.99030 


56 


4 


.35427 


.36566 


. 63434 


.98861 


56 


5 


9.32084 


9.33057 


10.66943 


9.99027 


55 


5 


9.35481 


9.36624 


10.63376 


9.98858 


55 


6 


.32143 


.33119 


.66881 


.99024 


54 


6 


.35536 


.36681 


.63319 


.98855 


54 


7 


.32202 


.33180 


.66820 


. 99022 


53 


7 


.35590 


.36738 


.63262 


. 98852 


53 


8 


.32261 


.33242 


.66758 


.99019 


52 


8 


.35644 


.36759 


.63205 


.98849 


52 


9 


.32319 


.33303 


.66697 


.99016 


51 


9 


.35698 


.36852 


.63148 


98846 


51 


10 


9.32378 


9.33365 


10.66635 


9.99013 


50 


10 


9.35752 


9.36909 


10.63091 


9.98843 


50 


11 


.32437 


.33426 


.66574 


.99011 


49 


11 


.35806 


.36966 


.63034 


.98840 


49 


12 


.32495 


. 33487 


.66513 


.99008 


48 


12 


.35860 


.37023 


.62977 


. 98837 


48 


13 


.32553 


.33548 


.66452 


.99005 


47 


13 


..35914 


.37080 


.62920 


.98834 


47 


14 


.32612 


.33609 


.66391 


.99002 


46 


14 


.35968 


.37137 


.62863 


.98831 


46 


15 


9.32670 


9.33670 


10.66330 


9.99000 


45 


15 


9.36022 


9.37193 


10.62807 


9.98828 


45 


16 


.32728 


.33731 


.66269 


. 98997 


44 


16 


.36075 


.37250 


.62750 


.98825 


44 


17 


.32786 


.33792 


.66208 


.98994 


43 


17 


.36129 


.37306 


.62694 


.98822 


43 


18 


.32844 


.33853 


.66147 


.98991 


42 


18 


.36182 


.37363 


.62637 


.98819 


42 


19 


.32902 


.33913 


.66087 


.98989 


41 


19 


.36236 


.37419 


.62581 


.98816 


41 


20 


9.32960 


9.33974 


10.66026 


9.98986 


40 


20 


9.36289 


9.37476 


10.62524 


9.98813 


40 


21 


.33018 


.34034 


.65966 


. 98983 


39 


21 


.36342 


.37532 


.62468 


.98810 


39 


22 


.33075 


.34095 


.65905 


.98980 


38 


22 


.36395 


.37588 


.62412 


. 98807 


38 


23 


.33133 


.34155 


.65845 


.98978 


37 


23 


.36449 


.37644 


.62356 


.98804 


37 


24 


.33190 


.34215 


.65785 


.98975 


36 


24 


.36502 


.37700 


.62300 


.98801 


36 


25 


9.33248 


9.34276 


10.65724 


9.98972 


35 


25 


9.36555 


9.37756 


10.62244 


9.98798 


35 


26 


.33305 


.34336 


.65664 


.98969 


34 


26 


.36608 


.37812 


.62188 


.98795 


34 


27 


.33362 


.34396 


.65604 


.98967 


33 


27 


.36660 


.37868 


.62132 


. 98792 


33 


28 


.33420 


.34456 


.65544 


.98964 


32 


28 


.36713 


.37924 


.62076 


.98789 


32 


29 


.33477 


.34516 


.65484 


.98961 


31 


29 


.36766 


.37980 


.62020 


98786 


31 


30 


9.33534 


9.34576 


10.65424 


9.98958 


30 


30 


9.36819 


9.38035 


10.61965 


9. 9b783 


30 


31 


.33591 


.34635 


.65365 


.98955 


29 


31 


.36871 


.38091 


.61909 


.98780 


29 


32 


.33647 


.34695 


.65305 


. 98953 


28 


32 


.36924 


.38147 


.61853 


. 98777 


28 


33 


.33704 


.34755 


.65245 


.98950 


27 


33 


.36976 


.38202 


.61798 


.98774 


27 


34 


.33761 


.34814 


.65186 


.98947 


26 


34 


.37028 


.38257 


.61743 


.98771 


26 


35 


9.33818 


9.34874 


10.65126 


9.98944 


25 


35 


9.37081 


9.38313 


10.61687 


9.98768 


25 


36 


.33874 


.34933 


.65067 


.98941 


24 


36 


.37133 


.38368 


.61632 


.98765 


24 


37 


.33931 


.34992 


.65008 


.98938 


23 


37 


.37185 


.38423 


.61577 


.98762 


23 


38 


.33987 


.35051 


.64949 


.98936 


22 


38 


.37237 


.38479 


.61521 


.98759 


22 


39 


.34043 


.35111 


.64889 


.98933 


21 


39 


.37289 


.38534 


.61466 


. 98756 


21 


40 


9.34100 


9.35170 


10.64830 


9.98930 


20 


40 


9.37341 


9.38589 


10.61411 


9.98753 


20 


41 


.34156 


.35229 


.64771 


. 98927 


19 


41 


.37393 


.38644 


.61356 


.98750 


19 


42 


.34212 


.35288 


.64712 


.98924 


18 


42 


.37445 


.38699 


.61301 


.98746 


18 


43 


.34268 


.35347 


.64653 


.98921 


17 


43 


.37497 


.38754 


.61246 


.98743 


17 


44 


.34324 


.35405 


.64595 


.98919 


16 


44 


.37549 


.38808 


.61192 


.98740 


16 


45 


9.34380 


9.35464 


10.64536 


9.98916 


15 


45 


9.37600 


9.38863 


10.61137 


9.98737 


15 


46 


.34436 


.35523 


.64477 


.98913 


14 


46 


.37652 


.38918 


.61082 


.98734 


14 


47 


.34491 


.35581 


.64419 


.98910 


13 


47 


.37703 


.38972 


.61028 


.98731 


13 


48 


.34547 


.35640 


.64360 


. 98907 


12 


48 


.37755 


.39027 


.60973 


.98728 


12 


49 


.34602 


.35698 


.64302 


. 98904 


11 


49 


.37806 


.39082 


.60918 


.98725 


11 


50 


9.34658 


9.35757 


10.64243 


9.98901 


10 


50 


9.37858 


9.39136 


10.60864 


9. 98722 


10 


51 


.34713 


.35815 


.64185 


. 98898 


9 


51 


.37909 


.39190 


.60810 


.98719 


9 


52 


.34769 


.35873 


.64127 


.98896 


8 


52 


.37960 


.39245 


.60755 


.98517 


8 


53 


.34824 


.35931 


.64069 


. 98893 


7 


53 


.38011 


.39299 


.60701 


.98712 


7 


54 


.34879 


.35989 


.64011 


. 98890 


6 


54 


.38062 


.39353 


.60647 


.98709 


6 


55 


9.34934 


9.36047 


10.63953 


9.98887 


5 


55 


9.38113 


9.39407 


10.60593 


9.98706 


5 


56 


.34989 


.36105 


.63895 


. 98884 


4 


56 


.38164 


.39461 


.60539 


. 98703 


4 


57 


.35044 


.36163 


.63837 


.98881 


3 


57 


.38215 


.39515 


.60485 


.98700 


3 


58 


.35099 


.36221 


.63779 


.98878 


2 


58 


.38266 


.39569 


.60431 


. 98697 


2 


59 


.35154 


.36279 


.63721 


.98875 


1 


59 


.38317 


.39623 


.60377 


. 98694 


1 


60 


9.35209 


9.36336 


10.63664 


9.98872 





60 


9.38368 


9.39677 


10.60323 


9.98690 





1 Cosine. | 


Cotang. 


Tang. 1 


Sine. 


' 1 


1 1 Cosine. 


Cotang. 


Tang. 


Sine. 


z 



77° 76^ 

*Log secant = colog cosine=l — log cosine; log cosecant = colog sine = 
1 — log sine. 

Ex.— hog sec 12°- 30' = 10.01042. Ex.— Log cosec 12°- 30' = 10.66466. 



LOGARITHMIC SINES, ETC, 



183 



5. — Logarithmic Sines, Tangents, Cotangents, Cosines. — (Cont'd.) 
(Secants, Cosecants.)* 
14° 15! 



/ 


1 Sine. 


1 Tang. 


1 Cotang. 


1 Cosine. | 


1 ' 


1 Sine. 


1 Tang. 


1 Cotang. 


1 Cosine. | 





9.38368 


9.39677 


10.60323 


9.98690 


60 





9.41300 


9.42805 


10.57195 


9. 98494 


60 


1 


.38418 


.39731 


.60269 


.98687 


59 


1 


.41347 


.42856 


.57144 


.98491 


59 


2 


.38469 


.39785 


.60215 


.98684 


58 


2 


.41394 


.42906 


.57094 


.98488 


58 


3 


.38519 


.39838 


.60162 


.98681 


57 


3 


.41441 


.42957 


.57043 


. 98484 


57 


4 


.38570 


.39892 


.60108 


.98678 


56 


4 


.41488 


.43007 


. 56993 


.98481 


56 


5 


9.38620 


9.39945 


10.60055 


9.98675 


55 


5 


9.41535 


9.43057 


10.56943 


9.98477 


55 


6 


.38670 


.39999 


.60001 


.98671 


54 


6 


.41582 


.43108 


.56892 


.98474 


54 


7 


.38721 


.40052 


.59948 


.98668 


53 


7 


.41628 


.43158 


.56842 


.98471 


53 


8 


.38771 


.40106 


.59894 


.98665 


52 


8 


.41675 


.43208 


.56792 


.98467 


52 


9 


.38821 


.40159 


.59841 


.98662 


51 


9 


.41722 


.43258 


.56742 


.98464 


51 


10 


9.38871 


9.40212 


10.59788 


9.98659 


50 


10 


9.41768 


9.43308 


10.56692 


9.98460 


50 


11 


.38921 


.40266 


.59734 


.98656 


49 


11 


.41815 


.43358 


.56642 


.98457 


49 


12 


.38971 


.40319 


.59681 


.98652 


48 


12 


.41861 


.43408 


.56592 


. 98453 


48 


13 


.39021 


.40372 


.59628 


.98649 


47 


13 


.41908 


.43458 


.56542 


.98450 


47 


14 


.39071 


.40425 


.59575 


.98646 


46 


14 


.41954 


.43508 


.56492 


.98447 


46 


15 


9.39121 


9.40478 


10.59522 


9.98643 


45 


15 


9.42001 


9.43558 


10.56442 


9.98443 


45 


16 


.39170 


.40531 


.59469 


.98640 


44 


16 


.42047 


.43607 


.56393 


.98440 


44 


17 


.39220 


.40584 


.59416 


.98636 


43 


17 


.42093 


.43657 


.56343 


.98436 


43 


18 


.39270 


.40636 


.59364 


.98633 


42 


18 


.42140 


.43707 


.56293 


.98433 


42 


19 


.39319 


.40689 


.59311 


.98630 


41 


19 


.42186 


.43756 


.56244 


.98429 


41 


20 


9.39369 


9.40742 


10.59258 


9.98627 


40 


20 


9.42232 


9.43806 


10.56194 


9.98426 


40 


21 


.39418 


.40795 


•. 59205 


.98623 


39 


21 


.42278 


.43855 


.56145 


.98422 


39 


22 


.39467 


.40847 


.59153 


.98620 


38 


22 


.42324 


.43905 


.56095 


.98419 


38 


23 


.39517 


.40900 


.59100 


.98617 


37 


23 


.42370 


.43954 


.56046 


.98415 


37 


24 


.39566 


.40952 


.59048 


.98614 


36 


24 


.42416 


.44004 


.55996 


.98412 


36 


25 


9.39615 


9.41005 


10.58995 


9.98610 


35 


25 


9.42461 


9.44053 


10.55947 


9.98409 


35 


26 


.39664 


.41057 


.58943 


.98607 


34 


26 


.42507 


.44102 


.55898 


.98405 


34 


27 


.39713 


.41109 


.58891 


.98604 


33 


27 


.42553 


.44151 


.55849 


. 98402 


33 


28 


.39762 


.41161 


.58839 


.98601 


32 


28 


.42599 


.44201 


.55799 


.98398 


32 


29 


.39811 


.41214 


.58786 


.98597 


31 


29 


.42644 


.44250 


.55750 


.98395 


31 


30 


9.39860 


9.41266 


10.58734 


9.98594 


30 


30 


9.42690 


9.44299 


10.55701 


9.98391 


30 


31 


.39909 


.41318 


.58682 


.98591 


29 


31 


.42735 


.44348 


..55652 


.98388 


29 


32 


.39958 


.41370 


.58630 


.98588 


28 


32 


.42781 


.44397 


. 55603 


.98384 


28 


33 


.40006 


.41422 


.58578 


.98584 


27 


33 


.42826 


.44446 


.55554 


.98381 


27 


34 


.40055 


.41474 


.58526 


.98581 


26 


34 


.42872 


.44495 


.55505 


.98377 


26 


35 


9.40103 


9.41526 


10.58474 


9.98578 


25 


35 


9.42917 


9.44544 


10.55456 


9.98373 


25 


36 


.40152 


.41578 


.58422 


.98574 


24 


36 


.42962 


.44592 


.55408 


.98370 


24 


37 


.40200 


.41629 


.58371 


.98571 


23 


37 


.43008 


.44641 


.55359 


.98366 


23 


38 


.40249 


.41681 


.58319 


.98568 


22 


38 


.43053 


.44690 


.55310 


. 98363 


22 


39 


.40297 


.41733 


.58267 


.98565 


21 


39 


.43098 


.44738 


.55262 


.98359 


21 


40 


9.40346 


9.41784 


10.58216 


9.98561 


20 


40 


9.43143 


9.44787 


10.55213 


9.98356 


20 


41 


.40394 


.41836 


.58164 


.98558 


19 


41 


.43188 


.44836 


.55164 


.98352 


19 


42 


.40442 


.41887 


.58113 


.98555 


18 


42 


.43233 


.44884 


.55116 


.98349 


18 


43 


.40490 


.41939 


.58061 


.98551 


17 


43 


.43278 


.44933 


.55067 


.98345 


17 


44 


.40538 


.41990 


.58010 


. 98548 


16 


44 


.43323 


.44981 


.55019 


. 98342 


16 


45 


9.40586 


9.42041 


10.57959 


9.98545 


15 


45 


9.43367 


9.45029 


10.54971 


9.98338 


15 


46 


.40634 


.42093 


.57907 


.98541 


14 


46 


.43412 


.45078 


.54922 


. 98334 


14 


47 


.40682 


.42144 


.57856 


.98538 


13 


47 


.43457 


.45126 


. 54874 


.98331 


13 


48 


.40730 


.42195 


.57805 


.98535 


12 


48 


.43502 


.45174 


.54826 


.98327 


12 


49 


.40778 


.42246 


.57754 


.98531 


11 


49 


.43546 


.45222 


.54778 


.98324 


11 


50 


9.40825 


9.42297 


10.57703 


9.98528 


10 


50 


9.43591 


9.45271 


10.54729 


9.98320 


10 


51 


.40873 


.42348 


.57652 


.98525 


9 


51 


.43635 


.45319 


.54681 


.98317 


9 


52 


.40921 


.42399 


.57601 


.98521 


8 


52 


.43680 


.45367 


. 54633 


.98313 


8 


53 


.40968 


.42450 


.57550 


.98518 


7 


53 


.43724 


.45415 


.54585 


.98309 


7 


54 


.41016 


.42501 


.57499 


.98515 


6 


54 


.43769 


.45463 


. 54537 


. 98306 


6 


55 


9.41063 


9.42552 


10.57448 


9.98511 


5 


55 


9.43813 


9.45511 


10.54489 


9.98302 


5 


5& 


.41111 


.42603 


.57397 


.98508 


4 


56 


.43857 


.45559 


.54441 


.98299 


4 


57 


.41158 


.42653 


.57347 


.98505 


3 


57 


.43901 


.45606 


.54394 


.98295 


3 


58 


.41205 


.42704 


.57296 


.98501 


2 


58 


.43946 


.45654 


.54346 


.98291 


2 


59 


.41252 


.42755 


.57245 


.98498 


1 


59 


.43990 


.45702 


.54298 


.98288 


1 


60 


9.41300 


9.42805 


10.57195 


9.98494 





60 


9.44034 


9.45750 


10.54250 


9.98284 






ICosine. ICotang. I Tang. | Sine. \ ' \\ | Cosine. 1 Cotang. | Tang. | Sine. | * 

75° 74° 

*Log secant = colog cosine = 1 — log cosine ; log cosecant = colog sine = 
l — log sine. 

Ex.— Log sec 14°- 30'= 10.01406. Ex.— Log cosec 14°- 30'= 10.60140. 



184 



^.—PLANE TRIGONOMETRY, 



5. — Logarithmic Sines, Tangents, Cotangents, Cosines. 
4 (Secants, Cosecants.)* 



(Cont'd.) 



' 1 Sine. 


1 Tang. 


1 Cotang. 


1 Cosine. 1 II ' 1 Sine. 


1 Tang. 


Cotang. 


Cosine, j 





9.44034 


9.45750 


10.54250 


9.98284 


60 


9.46594 


9.48534 


10.51466 


9.98060 


60 


1 


.44078 


.45797 


.54203 


. 98281 


59 


1 


.46635 


.48579 


.51421 


. 98056 


59 


2 


.44122 


.45845 


.54155 


.98277 


58 


2 


.46676 


.48624 


.51376 


. 98052 


58 


3 


.44166 


.45892 


.54108 


.98273 


57 


3 


.46717 


.48669 


,51331 


. 98048 


57 


4 


.44210 


.45940 


54060 


.98270 


56 


4 


.46758 


.48714 


,51286 


. 98044 


56 


5 


9.44253 


9.45987 


10.54013 


9.98266 


55 


5 


9.46800 


9.48759 


10,51241 


9.98040 


55 


6 


.44297 


.46035 


.53965 


.98262 


54 


6 


.46841 


.48804 


.51196 


. 98036 


54 


7 


.44341 


.46082 


.53918 


.98259 


53 


7 


.46882 


.48849 


,51151 


. 98032 


53 


8 


.44385 


.46130 


.53870 


.98255 


52 


8 


.46923 


.48894 


,51106 


.98029 


52 


9 


.44428 


.46177 


.53823 


.98251 


51 


9 


,46964 


.48939 


,51061 


. 98025 


51 


10 


9.44472 


9.46224 


10.53776 


9.98248 


50 


10 


9.47005 


9.48984 


10,51016 


9.98021 


50 


11 


.44516 


.46271 


.53729 


.98244 


49 


11 


.47045 


.49029 


,50971 


■ .98017 


49 


12 


.44559 


.46319 


.53681 


. 98240 


48 


12 


.47086 


.49073 


,50927 


.98013 


48 


13 


.44602 


.46366 


.53634 


.98237 


47 


13 


.47127 


.49118 


. 50882 


. 98009 


47 


14 


.44646 


.46413 


.53587 


. 98233 


46 


14 


.47168 


.49163 


,50837 


. 98005 


46 


15 


9.44689 


9.46460 


10.53540 


9.98229 


45 


15 


9.47209 


9.49207 


10,50793 


9.98001 


45 


16 


.44733 


.46507 


. 53493 


.98226 


44 


16 


.47249 


.49252 


.50748 


.97997 


44 


17 


.44776 


.46554 


.53446 


. 98222 


43 


17 


.47290 


.49296 


.50704 


.97993 


43 


18 


.44819 


.46601 


.53399 


.98218 


42 


18 


.47330 


.49341 


.50659 


.97989 


42 


19 


.44862 


.46648 


.53352 


.98215 


41 


19 


.47371 


.49385 


.50615 


.97986 


41 


20 


9.44905 


9.46694 


10.53306 


9.98211 


40 


20 


9.47411 


9.49430 


10.50570 


9.97982 


40 


21 


.44948 


.46741 


.53259 


: 98207 


39 


21 


.47452 


.49474 


.50526 


.97978 


39 


22 


.44992 


,46788 


.53212 


. 98204 


38 


22 


.47492 


,49519 


.50481 


.97974 


38 


23 


.45035 


.46835 


.53165 


. 98200 


37 


23 


.47533 


.49563 


. 50437 


,97970 


37 


24 


.45077 


.46881 


.53119 


.98196 


36 


24 


.47573 


.49607 


,50393 


.97966 


36 


25 


9.45120 


9.46928 


10.53072 


9.98192 


35 


25 


9.47613 


9.49652 


10,50348 


9.97962 


35 


26 


.45163 


.46975 


.53025 


.98189 


34 


26 


.47654 


.49696 


,50304 


.97958 


34 


27 


.45206 


.47021 


.52979 


.98185 


33 


27 


.47694 


.49740 


.50260 


.97954 


33 


28 


.45249 


.47068 


.52932 


.98181 


32 


28 


.47734 


.49784 


,50216 


.97950 


32 


29 


.45292 


.47114 


.52886 


.98177 


31 


29 


.47774 


.49828 


.50172 


.97946 


31 


30 


9.45334 


9.47160 


10.52840 


9.98174 


30 


30 


9.47814 


9.49872 


10.50128 


9.97942 


30 


31 


.45377 


.47207 


.52793 


.98170 


29 


31 


.47854 


.49916 


. 50084 


.97938 


29 


32 


.45419 


.47253 


.52747 


.98166 


28 


32 


.47894 


.49960 


.50040 


.97934 


28 


33 


.45462 


.47299 


.52701 


.98162 


27 


33 


.47934 


.50004 


.49996 


.97930 


27 


34 


.45504 


.47346 


.52654 


.98159 


26 


34 


.47974 


.50048 


.49952 


.97926 


26 


35 


9.45547 


9.47392 


10.52608 


9.98155 


25 


35 


9.48014 


9.50092 


10,49908 


9.97922 


25 


36 


.45589 


.47438 


.52562 


.98151 


24 


36 


.48054 


.50136 


.49864 


.97918 


24 


37 


.45632 


.47484 


.52516 


.98147 


23 


37 


.48094 


.50180 


.49820 


.97914 


23 


38 


.45674 


.47530 


.52470 


.98144 


22 


38 


.48133 


. 50223 


.49777 


.97910 


22 


39 


.45716 


.47576 


.52424 


.98140 


21 


39 


.48173 


. 50267 


.49733 


.97906 


21 


40 


9.45758 


9.47622 


10.52378 


9.98136 


20 


40 


9.48213 


9.50311 


10,49689 


9.97902 


20 


41 


.45801 


.47668 


.52332 


.98132 


19 


41 


.48252 


.50355 


,49645 


.97898 


19 


42 


.45843 


.47714 


.52286 


.98129 


18 


42 


.48292 


.50398 


,49602 


.97894 


18 


43 


.45885 


.47760 


.52240 


.98125 


17 


43 


.48332 


. 50442 


,49558 


. 97890 


17 


44 


.45927 


.47806 


.52194 


.98121 


16 


44 


.48371 


. 50485 


,49515 


. 97886 


16 


45 


9.45969 


9.47852 


10.52148 


9.98117 


15 


45 


9.48411 


9.50529 


10.49471 


9.97882 


15 


46 


.46011 


.47897 


.52103 


.98113 


14 


46 


.48450 


.50572 


.49428 


.97878 


14 


47 


.46053 


.47943 


. 52057 


.98110 


13 


47 


.48490 


.50616 


.49384 


,97874 


13 


48 


.46095 


.47989 


.52011 


.98106 


12 


48 


.48529 


.50659 


,49341 


.97870 


12 


49 


.46136 


.48035 


.51965 


.98102 


11 


49 


.48568 


. 50703 


,49297 


. 97866 


11 


SO 


9.46178 


9.48080 


10.51920 


9.98098 


10 


50 


9.48607 


9.50746 


10.49254 


9.97861 


10 


51 


.46220 


.48126 


.51874 


.98094 


9 


51 


.48647 


.50789 


.49211 


. 97857 


9 


52 


.46262 


.48171 


.51829 


. 98090 


8 


52 


.48686 


. 50833 


,49167 


. 97853 


8 


53 


.46303 


.48217 


.51783 


. 98087 


7 


53 


.48725 


.50876 


.49124 


.97849 


7 


54 


.46345 


.48262 


.51738 


. 98083 


6 


54 


.48764 


.50919 


.49081 


. 97845 


6 


55 


9.46386 


9.48307 


10.51693 


9.98079 


5 


55 


9.48803 


9.50962 


10.49038 


9.97841 


5 


56 


.46428 


.48353 


.51647 


.98075 


4 


56 


.48842 


.51005 


.48995 


.97837 


4 


67 


.46469 


.48398 


.51602 


.98071 


3 


57 


.48881 


.51048 


.48952 


,97833 


3 


58 


.46511 


.48443 


.51557 


.98067 


2 


58 


.48920 


.51092 


.48908 


.97829 


2 


59 


.46552 


.48489 


.51511 


. 98063 


1 


59 


.48959 


.51135 


.48865 


.97825 




60 


9.46594 


9.48534 


10.51466 


9.98060 





60 


9.48998 


9.51178 


10.48822 


9.97821 





1 Cosine. 1 


Cotang. 


Tang. 1 


Sine. 1 


'1! 1 Cosine.! 


Cotang. 1 


Tang. 


Sine. 1 ' 



73° 72° 

*Log secant = colog cosine = 1 — log cosine ; log cosecant = colog sine = 
1 — log sine. 

Ex.— Log sec 16°- 30' = 10.01826. Ex.— Log cosec 16°- 30' = 10.54666. 



LOGARITHMIC SINES, ETC. 



185 



5^ — Logarithmic Sines, Tangents, Cotangents, Cosines. — (Cont'd.) 
(Secants, Cosecants.)* 

18° 19! 



/ 1 


Sine. 1 


Tang. 1 Cotang. i Cosine. | 


II 


' 1 


Sine. 1 


Tang. 1 Cotang. | Cosine. | 







9.48998 


9.51178 


10.48822 


9.97821 


60 





9.51264 


9.53697 


10.46303 


9.97567 


60 


1 


.49037 


.51221 


.48779 


.97817 


59 


1 


.51301 


.53738 


.46262 


.97563 


59 


2 


.49076 


.51264 


.48736 


.97812 


58 


2 


.51338 


.53779 


.46221 


.97558 


58 


3 


.49115 


.51306 


.48694 


.97808' 


57 


3 


.51374 


.53820 


.46180 


.97554 


57 


4 


.49153 


.51349 


.48651 


.97804 


56 


4 


.51411 


.53861 


.46139 


. 97550 


56 


5 


9.49192 


9.51392 


10.48608 


9.97800 


55 


5 


9.51447 


9.53902 


10.46098 


9.97545 


55 


6 


.49231 


.51435 


.48565 


.97796 


54 


6 


.51484 


.53943 


.46057 


.97541 


54 


7 


.49269 


.51478 


.48522 


.97792 


53 


7 


.51520 


.53984 


.46016 


.97536 


53 


8 


.49308 


.51520 


.48480 


.97788 


52 


8 


.51557 


.54025 


.45975 


.97532 


52 


9 


.49347 


.51563 


.48437 


.97784 


51 


9 


.51593 


. 54065 


.45935 


.97528 


51 


10 


9.49385 


9.51606 


10.48394 


9.97779 


50 


10 


9.51629 


9.54106 


10.45894 


9.97523 


50 


11 


.49424 


.51648 


.48352 


.97775 


49 


11 


.51666 


.54147 


.45853 


.97519 


49 


12 


.49462 


.51691 


.48309 


.97771 


48 


12 


.51702 


.54187 


.45813 


.97515 


48 


13 


.49500 


.51734 


.48266 


.97767 


47 


13 


.51738 


.54228 


.45772 


.97510 


47 


14 


.49539 


.51776 


.48224 


. 97763 


46 


14 


.51774 


.54269 


.45731 


.97506 


46 


15 


9.49577 


9.51819 


10.48181 


9.97759 


49 


15 


9.51811 


9.54309 


10.45691 


9.97501 


45 


16 


.49615 


.51861 


.48139 


.97754 


44 


16 


.51847 


. 54350 


.45650 


.97497 


44 


17 


.49654 


.51903 


.48097 


.97750 


43 


17 


.51883 


.54390 


.45610 


.97492 


43 


18 


.49692 


.51946 


.48054 


.97746 


42 


18 


.51919 


.54431 


.45569 


.97488 


42 


19 


.49730 


.51988 


.48012 


. 97742 


41 


19 


.51955 


.54471 


.45529 


.97484 


41 


20 


9.49768 


9.52031 


10.47969 


9.97738 


40 


20 


9.51991 


9.54512 


10.45488 


9.97479 


40 


21 


.49806 


.52073 


.47927 


.97734 


39 


21 


.52027 


. 54552 


.45448 


.97475 


39 


22 


.49844 


.52115 


.47885 


.97729 


38 


22 


.52063 


. 54593 


.45407 


.97470 


38 


23 


.49882 


.52157 


.47843 


.97725 


37 


23 


.52099 


.54633 


.45367 


.97466 


37 


24 


.49920 


.52200 


.47800 


.97721 


36 


24 


.52135 


.54673 


.45327 


.97461 


36 


25 


9.49958 


9.52242 


10.47758 


9.97717 


35 


25 


9.52171 


9.54714 


10.45286 


9.97457 


35 


26 


.49996 


.52284 


.47716 


.97713 


34 


26 


.52207 


.54754 


.45246 


.97453 


34 


27 


. 50034 


.52326 


.47674 


.97708 


33 


27 


.52242 


.54794 


.45206 


.97448 


33 


28 


. 50072 


.52368 


.47632 


.97704 


32 


28 


.52278 


.54835 


.45165 


.97444 


32 


29 


.50110 


.52410 


.47590 


.97700 


31 


29 


.52314 


.54875 


.45125 


.97439 


31 


30 


9.50148 


9.52452 


10.47548 


9.97696 


30 


30 


9.52350 


9.54915 


10.45085 


9.97435 


30 


31 


.50185 


.52494 


.47506 


.97691 


29 


31 


.52385 


. 54955 


.45045 


.97430 


2S 


32 


.50223 


.52536 


.47464 


.97687 


28 


32 


.52421 


.54995 


.45005 


.97426 


2£ 


33 


.50261 


.52578 


.47422 


.97683 


27 


33 


.52456 


.55035 


.44965 


.97421 


27 


34 


.50298 


.52620 


.47380 


.97679 


26 


34 


.52492 


.55075 


.44925 


.97417 


26 


35 


9.50336 


9.52661 


10.47339 


9.97674 


25 


35 


9.52527 


9.55115 


10.44885 


9.97412 


25 


36 


.50374 


.52703 


.47297 


.97670 


24 


36 


. 52563 


.55155 


.44845 


.97408 


24 


37 


.50411 


.52745 


.47255 


.97666 


23 


37 


. 52598 


.55195 


.44805 


. 97403 


23 


38 


.50449 


.52787 


.47213 


,97662 


22 


38 


.52634 


.55235 


.44765 


.97399 


22 


39 


. 50486 


.52829 


.47171 


.97657 


21 


39 


.52669 


.55275 


.44725 


.97394 


21 


40 


9.50523 


9.52870 


10.47130 


9.97653 


20 


40 


9.52705 


9.55315 


10.44685 


9.97390 


2C 


41 


.50561 


.52912 


.47088 


.97649 


19 


41 


.52740 


.55355 


.44645 


. 97385 


H 


42 


.50598 


.52953 


.47047 


.97645 


18 


42 


.52775 


.55395 


.44605 


.97381 


U 


43 


.50635 


.52995 


.47005 


.97640 


17 


43 


.52811 


.55434 


.44566 


.97376 


17 


44 


.50673 


.53037 


.46963 


.97636 


16 


44 


.52846 


.55474 


.44526 


.97372 


U 


45 


9.50710 


9.53078 


10.46922 


9.97632 


15 


45 


9.52881 


9.55514 


10.44486 


9.97367 


IS 


46 


.50747 


.53120 


.46880 


.97628 


U 


46 


.52916 


. 55554 


.44446 


.97363 


14 


47 


.50784 


.53161 


.46839 


.97623 


13 


47 


.52951 


. 55593 


.44407 


.97358 


13 


48 


.50821 


.53202 


.46798 


.97619 


12 


48 


.52986 


. 55633 


.44367 


.97353 


12 


49 


. 50858 


.53244 


.46756 


.97615 


11 


49 


.53021 


.55673 


.44327 


.97349 


11 


50 


9.50896 


9.53285 


10.46715 


9.97610 


10 


50 


9.53056 


9.55712 


10.44288 


9.97344 


IG 


51 


.50933 


.53327 


.46673 


.97606 


9 


51 


.53092 


.55752 


.44248 


.97340 


i 


52 


.50970 


.53368 


.46632 


.97602 


8 


52 


.53126 


.55791 


.44209 


.97335 


i 


53 


.51007 


. 53409 


.46591 


.97597 


7 


53 


.53161 


.55831 


.44169 


.97331 


1 


54 


.51043 


.53450 


.46550 


.97593 


6 


54 


.53196 


.55870 


.44130 


.97326 


i 


55 


9.51080 


9.53492 


10.46508 


9.97589 


5 


55 


9.53231 


9.55910 


10.44090 


9.97322 


5 


56 


.51117 


.53533 


.46467 


. 97584 


4 


56 


.53266 


.55949 


.44051 


.97317 


/ 


57 


.51154 


.53574 


.46426 


. 97580 


3 


57 


.53301 


.55989 


.44011 


.97312 




58 


.51191 


.53615 


.46385 


.97576 


2 


58 


.53336 


.56028 


.43972 


.97308 


1 


59 


.51227 


.53656 


.46344 


.97571 


1 


59 


.53370 


.56067 


.43933 


.97303 


1 


60 


9.51264 


9.53697 


10.46303 


9.97567 





60 


9.53405 


9.56107 


10.43893 


9.97299 


C 



I Cosine. 1 Cotang. I Tang. 1 Sine. \ ' \\ I Cosine. | Cotang. | Tang. | Sine. | ' 

71° 70° 

*Log secant = colog cosine=l — log cosine; log cosecant = colog sme== 
1 — log sine. 

Ex.— Log sec 18°- 30' = 10.02304. Ex.— Log cosec 18°- 30' = 10.49852. 



18d 



9.— PLANE TRIGONOMETRY. 



5. — Logarithmic Sines, Tangents, Cotangents, Cosines. — (Cont'd.) 
(Secants, Cosecants.)* 

}0° 21^ 

' I Sine. I Tang. 1 Cotang. I Cosine. I I| ' | Sine. | Tang. | Cotang. | Cosine. | 






9.53405 


9.56107 


10.43893 


9.97299 


60 





9.55433 


9.58418 


10.41582 


9.97015 


60 


1 


.53440 


.56146 


.43854 


.97294 


59 


1 


.55466 


.58455 


.41545 


.97010 


59 


2 


.53475 


.56185 


.43815 


.97289 


58 


2 


.55499 


. 58493 


.41507 


.97005 


58 


3 


.53509 


.56224 


.43776 


.97285 


57 


3 


.55532 


.58531 


.41469 


.97001 


57 


4 


.53544 


.56264 


.43736 


.97280 


56 


4 


. 55564 


.58569 


.41431 


.96996 


56 


5 


9.53578 


9.56303 


10.43697 


9.97276 


55 


5 


9.55597 


9.58606 


10.41394 


9.96991 


55 


6 


.53613 


.56342 


.43658 


.97271 


54 


6 


.55630 


.58644 


.41356 


.96986 


54 


7 


.53647 


.56381 


.43619 


.97266 


53 


7 


.55663 


.58681 


.41319 


.96981 


53 


8 


.53682 


.56420 


.43580 


.97262 


52 


8 


.55695 


.58719 


.41281 


.96976 


52 


9 


.53716 


.56459 


.43541 


.97257 


51 


9 


.55728 


.58757 


.41243 


.96971 


51 


10 


9.53751 


9.56498 


10.43502 


9.97252 


50 


10 


9.55761 


9.58794 


10.41206 


9.96966 


50 


11 


.53785 


.56537 


.43463 


.97248 


49 


11 


.55793 


.58832 


.41168 


.96962 


49 


12 


.53819 


.56576 


.43424 


.97243 


48 


12 


.55826 


. 58869 


.41131 


.96957 


48 


13 


.53854 


.56615 


.43385 


.97238 


47 


13 


.55858 


. 58907 


.41093 


.96952 


47 


14 


.53888 


.56654 


.43346 


.97234 


46 


14 


.55891 


.58944 


.41056 


.96947 


46 


15 


9.53922 


9.56693 


10.43307 


9.97229 


45 


15 


9.55923 


9.58981 


10.41019 


9.96942 


45 


16 


.53957 


.56732 


.43268 


.97224 


44 


16 


.55956 


.59019 


.40981 


.96937 


44 


17 


.53991 


.56771 


.43229 


.97220 


43 


17 


.55988 


.59056 


.40944 


.96932 


43 


18 


.54025 


.56810 


.43190 


.97215 


42 


18 


.56021 


.59094 


.40906 


.96927 


42 


19 


.54059 


. 56849 


.43151 


.97210 


41 


19 


. 56053 


.59131 


.40869 


.96922 


41 


20 


9.54093 


9.56887 


10.43113 


9.97206 


40 


20 


9.56085 


9.59168 


10.40832 


9.96917 


40 


21 


.54127 


.56926 


.43074 


.97201 


39 


21 


.56118 


.59205 


.40795 


.96912 


39 


22 


.54161 


. 56965 


-.43035 


.97196 


38 


22 


.56150 


.59243 


.40757 


.96907 


3^ 


23 


.54195 


.57004 


.42996 


.97192 


37 


23 


.56182 


.59280 


.40720 


. 96903 


37 


24 


.54229 


.57042 


.42958 


.97187 


36 


24 


.56215 


.59317 


.40683 


.96898 


36 


25 


9.54263 


9.57081 


10.42919 


9.97182 


35 


25 


9.56247 


9.59354 


10.40646 


9.96893 


35 


26 


.54297 


.57120 


.42880 


.97178 


34 


26 


.56279 


.59391 


.40609 


.96888 


34 


27 


.54331 


.57158 


.42842 


.97173 


33 


27 


.56311 


.59429 


40571 


.96883 


33 


28 


.54365 


.57197 


.42803 


.97168 


32 


28 


.56343 


.59466 


.40534 


.96878 


32 


29 


.54399 


.57235 


.42765 


.97163 


31 


29 


.56375 


. 59503 


.40497 


.96873 


31 


30 


9.54433 


9.57274 


10.42726 


9.97159 


30 


30 


9.56408 


9.59540 


10.40460 


9.96868 


30 


31 


.54466 


.57312 


.42688 


.97154 


29 


31 


. 56440 


. 59577 


.40423 


.96863 


29 


32 


.54500 


.57351 


.42649 


.97149 


28 


32 


.56472 


.59614 


.40386 


.96858 


28 


33 


. 54534 


.57389 


.42611 


.97145 


27 


33 


.56504 


.59651 


.40349 


.96853 


27 


34 


.54567 


.57428 


.42572 


.97140 


26 


34 


.56536 


.59688 


.40312 


.96848 


26 


35 


9.54601 


9.57466 


10.42534 


9.97135 


25 


35 


9.56568 


9.59725 


10.40275 


9.96843 


25 


36 


.54635 


.57504 


.42496 


.97130 


24 


36 


.56599 


. 59762 


.40238 


.96838 


24 


37 


.54668 


.57543 


.42457 


.97126 


23 


37 


.56631 


.59799 


.40201 


.96833 


23 


38 


.54702 


.57581 


.42419 


.97121 


22 


38 


.56663 


. 59835 


.40165 


.96828 


22 


39 


.54735 


.57619 


.42381 


.97116 


21 


39 


.56695 


.59872 


.40128 


.96823 


21 


40 


9.54769 


9.57658 


10.42342 


9.97111 


20 


40 


9.56727 


9.59909 


10.40091 


9.96818 


20 


41 


.54802 


.57696 


.42304 


.97107 


19 


41 


.56759 


.59946 


.40054 


.96813 


19 


42 


. 54836 


.57734 


.42266 


.97102 


18 


42 


.56790 


.59983 


.40017 


.96808 


18 


43 


.54869 


.57772 


.42228 


.97097 


17 


43 


.56822 


.60019 


.39981 


.96803 


17 


44 


. 54903 


.57810 


.42190 


.97092 


16 


44 


.56854 


.60056 


.39944 


.96798 


16 


45 


9.54936 


9.57849 


10.42151 


9.97087 


15 


45 


9.56886 


9.60093 


10.39907 


9.96793 


15 


46 


.54969 


.57887 


.42113 


. 97083 


14 


46 


.56917 


.60130 


.39870 


.96788 


14 


47 


. 55003 


.57925 


.42075 


.97078 


13 


47 


.56949 


.60166 


.39834 


.96783 


13 


48 


. 55036 


.57963 


.42037 


.97073 


12 


48 


.56980 


.60203 


.39797 


.96778 


12 


49 


.55069 


. 58001 


.41999 


.97068 


11 


49 


.57012 


.60240 


.39760 


.96772 


11 


50 


9.55102 


9.58039 


10.41961 


9.97063 


10 


50 


9.57044 


9.60276 


10.39724 


9.96767 


10 


51 


.55136 


.58077 


.41923 


.97059 


9 


51 


.57075 


.60313 


.39687 


.96762 


9 


52 


.55169 


.58115 


.41885 


.97054 


8 


52 


.57107 


.60349 


.39651 


. 96757 


8 


53 


.55202 


.58153 


.41847 


.97049 


7 


53 


.57138 


.60386 


.39614 


.96752 


7 


B4 


.55235 


.58191 


.41809 


.97044 


6 


54 


.57169 


.60422 


.39578 


.96747 


6 


55 


9.55268 


9.58229 


10.41771 


9.97039 


5 


55 


9.57201 


9.60459 


10.39541 


9.96742 


5 


56 


.55301 


.58267 


.41733 


.97035 


4 


56 


.57232 


.60495 


.39505 


.96737 


4 


57 


.55334 


. 58304 


.41696 


.97030 


3 


57 


.57264 


.60532 


.39468 


.96732 


3 


58 


.55367 


. 58342 


.41658 


.97025 


2 


58 


.57295 


.60568 


.39432 


.96727 


2 


59 


. 55400 


.58380 


.41620 


.97020 


1 


59 


.57326 


.60605 


.39395 


.96722 


1 


60 


9.55433 


9.58418 


10.41582 


9.97015 





60 


9.57358 


9.60641 


10.39359 


9.96717 






I Cosine. I Cotang. I Tang. | Sine. \ ' \\ | Cosine. | Cotang. 1 Tang. | Sine. | ' 

69"^ 68° 

*Log secant = colog cosine = 1 — log cosine ; log cosecant = colog sine = 
1 — log sine. 

Ex.— hog sec 20°- 30'= 10.02841. Ex.— Log cosec 20°- 30' = 10.45567. 



LOGARITHMIC SINES, ETC. 



187 



5. — Logarithmic Sines, Tangents, Cotangents, Cosines. — (Cont'd.) 
(Secants, Cosecants.)* 
22° 23° 



J_ 


Sine. 


Tang. 1 Cotang. 


Cosine. 


j 


' 1 


Sine. 


Tang. 1 Cotang. 


Cosine. 







9.57358 


9.60641 


10.39359 


9.96717 


60 





9.59188 


9.62785 


10.37215 


9.96403 


60 


1 


.57389 


.60677 


.39323 


.96711 


59 i 


j 1 


.59218 


.62820 


.37180 


.96397 


59 


2 


.57420 


.60714 


.39286 


.96706 


58 


2 


.59247 


.62855 


.37145 


.96392 


58 


3 


.57451 


.60750 


.39250 


.96701 


57 


3 


.59277 


.62890 


.37110 


.96387 


57 


4 


. 57482 


.60786 


.39214 


.96696 


56 


4 


. 59307 


.62926 


.37074 


.96381 


56 


5 


9.57514 


9.60823 


10.39177 


9.96691 


55 


5 


9.59336 


9.62961 


10.37039 


9.96376 


55 


6 


.57545 


.60859 


.39141 


.96686 


54 


6 


.59366 


.62996 


.37004 


.96370 


54 


7 


.57576 


.60895 


.39105 


.96681 


53 


7 


.59396 


.63031 


.36969 


.96365 


53 


8 


.57607 


.60931 


.39069 


.96676 


52 


8 


.59425 


.63066 


.36934 


.96360 


52 


9 


.57638 


.60967 


.39033 


.96670 


51 


9 


.59455 


.63101 


.36899 


.96354 


51 


10 


9.57669 


9.61004 


10.38996 


9.96665 


50 


10 


9.59484 


9.63135 


10.36865 


9.96349 


50 


11 


.57700 


.61040 


.38960 


.96660 


49 


11 


.59514 


.63170 


.36830 


. 96343 


49 


12 


.57731 


.61076 


.38924 


.96655 


48 


12 


. 59543 


.63205 


.36795 


.96338 


48 


13 


.57762 


.61112 


.38888 


.96650 


47 


13 


.59573 


.63240 


.36760 


.96333 


47 


14 


.57793 


.61148 


.38852 


.96645 


46 


14 


.59602 


.63275 


.36725 


.96327 


46 


15 


9.57824 


9.61184 


10.38816 


9.96640 


45 


15 


9.59632 


9.63310 


10.36690 


9.96322 


45 


16 


.57855 


.61220 


.38780 


.96634 


441 


16 


.59661 


.63345 


.36655 


.96316 


44 


17 


. 57885 


.61256 


.38744 


.96629 


431 


17 


.59690 


.63379 


.36621 


.96311 


43 


18 


.57916 


.61292 


.38708 


96624 


42 


18 


.59720 


.63414 


.36586 


.96305 


42 


19 


.57947 


.61328 


.38672 


.96619 


41 


19 


.59749 


.63449 


.36551 


.96300 


41 


20 


9.57978 


9.61364 


10.38636 


9.96614 


40 


20 


9.59778 


9.63484 


10.36516 


9.96294 


40 


21 


.58008 


.61400 


.38600 


.96608 


39 


21 


.59808 


.63519 


.36481 


.96289 


39 


22 


. 58039 


.61436 


.38564 


.96603 


38 


22 


.59837 


.63553 


.36447 


.96284 


38 


23 


.58070 


.61472 


.38528 


.96598 


37 


23 


. 59866 


.63588 


.36412 


.96278 


37 


24 


.58101 


.61508 


.38492 


.96593 


36 


24 


.59895 


.63623 


.36377 


.96273 


36 


25 


9.58131 


9.61544 


10.38456 


9.96588 


35 


25 


9.59924 


9.63657 


10.36343 


9.96267 


35 


26 


.58162 


.61579 


.38421 


.96582 


34 


26 


.59954 


,63692 


.36308 


.96262 


34 


27 


.58192 


.61615 


.38385 


.96577 


33 


27 


.59983 


.63726 


.36274 


.96256 


33 


28 


.58223 


.61651 


.38349 


.96572 


32 


28 


.60012 


.63761 


.36239 


.96251 


32 


29 


. 58253 


.61687 


.38313 


.96567 


31 


29 


.60041 


.63796 


.36204 


.96245 


31 


30 


9. 58284 


9.61722 


10.38278 


9.96562 


30 


30 


9.60070 


9.63830 


10.36170 


9.96240 


30 


31 


.58314 


.61758 


.38242 


.96556 


29 


31 


.60099 


,63865 


.36135 


.96234 


29 


32 


.58345 


.61794 


.38206 


.96551 


28 


32 


.60128 


.63899 


.36101 


.96229 


28 


33 


.58375 


.61830 


.38170 


.96546 


27 


33 


.60157 


.63934 


.36066 


.96223 


27 


34 


. 58406 


.61865 


.38135 


.96541 


26 


34 


.60186 


.63968 


.36032 


.96218 


26 


35 


9.58436 


9.61901 


10.38099 


9.96535 


25 


35 


9.60215 


9.64003 


10.35997 


9.96212 


25 


36 


. 58467 


.61936 


.38064 


.96530 


24 


36 


.60244 


.64037 


.35963 


. 96207 


24 


37 


. 58497 


.61972 


.38028 


.96525 


23 


37 


.60273 


.64072 


.35928 


.96201 


23 


38 


.58527 


.62008 


.37992 


.96520 


22 


38 


.60302 


.64106 


.35894 


.96196 


22 


39 


. 58557 


.62043 


.37957 


.96514 


21 


39 


.60331 


.64140 


.35860 


.96190 


21 


40 


9.58588 


9.62079 


10.37921 


9.96509 


20 


40 


9.60359 


9 64175 


10.35825 


9.96185 


20 


41 


.58618 


.62114 


.37886 


.96504 


19 


41 


.60388 


,64209 


.35791 


.96179 


19 


42 


.58648 


.62150 


.37850 


.96498 


18 


42 


.60417 


.64243 


.35757 


.96174 


18 


43 


.58678 


.62185 


.37815 


.96493 


17 


43 


.60446 


.64278 


.35722 


.96168 


17 


44 


.58709 


.62221 


.37779 


.96488 


16 


44 


.60474 


.64312 


.35688 


.96162 


16 


45 


9.58739 


9.62256 


10.37744 


9.96483 


15 


45 


9.60503 


9.64346 


10.35654 


9.96157 


15 


46 


.58769 


.62292 


.37708 


.96477 


14 


46 


.60532 


.64381 


.35619 


.96151 


14 


47 


.58799 


.62327 


.37673 


.96472 


13 


47 


.60561 


.64415 


.35585 


.96146 


18 


48 


.58829 


.62362 


.37638 


.96467 


12 


48 


.60589 


.64449 


.35551 


.96140 


12 


49 


.58859 


.62398 


.37602 


.96461 


11 


49 


.60618 


.64483 


.35517 


.96135 


11 


50 


9.58889 


9.62433 


10.37567 


9.96456 


10 


50 


9.60646 


9.64517 


10.35483 


9.96129 


10 


51 


.58919 


.62468 


.37532 


.96451 


9 


51 


.60675 


.64552 


.35448 


.96123 


9 


52 


.58949 


.62504 


.37496 


.96445 


8 


52 


.60704 


.64586 


.35414 


.96118 


8 


53 


.58979 


.62539 


.37461 


.96440 


7 


53 


.60732 


.64620 


.35380 


.96112 


7 


54 


. 59009 


.62574 


.37426 


.96435 


6 


54 


.60761 


.64654 


.35346 


.96107 


6 


55 


9.59039 


9.62609 


10.37391 


9.96429 


5 


55 


9.60789 


9.64688 


10.35312 


9.96101 


5 


56 


.59069 


.62645 


.37355 


. 96424 


4 


56 


.60818 


.64722 


.35278 


.96095 


4 


57 


.59098 


.62680 


.37320 


.96419 


3 


57 


.60846 


.64756 


.35244 


.96090 


3 


58 


.59128 


.62715 


.37285 


.96413 


2 


58 


.60875 


.64790 


.35210 


.96084 


2 


59 


.59158 


.62750 


.37250 


.96408 


1 


59 


.60903 


.64824 


.35176 


.96079 


1 


60 


9.59188 


9.62785 


10.37215 


9.96403 





60 


9.60931 


9.64858 


10.35142 


9.96073 







Cosine. 


Cotang. 1 


Tang. 1 


Sine. 


' 1 ! Cosine. 


Cotang. 1 


Tang. 


Sine. 


->* 



67° 66" 

*Log secant = colog cosine = 1 — log cosine ; log cosecant = colog sine =» 
1 — log sine. 

Ex.— Log sec 22"- 30' = 10.03438. Ex.— Log cosec 22"- 30' = 10.41716. 



188 



9^PLANE TRIGONOMETRY. 



5. — Logarithmic Sines, Tangents, Cotangents, Cosines. — (Cont'd. 
(Secants, Cosecants.)* 
24° 25° 



i 



' 


Sine. 


Tang. 


Cotang. 


Cosine. 


II ' 


Sine. 


Tang. 


Cotang. 


Cosine. 


__ 





9.60931 


9.64858 


10.35142 


9.96073 


60 





9.62595 


9.66867 


10.33133 


9.95728 


60 


1 


.60960 


.64892 


.35108 


. 96067 


59 


1 


.62622 


.66900 


.33100 


.95722 


59 


2 


.60988 


.64926 


.35074 


. 96062 


58 


2 


.62649 


.66933 


.33067 


.95716 


58 


3 


.61016 


.64960 


.35040 


.96056 


57 


3 


.62676 


.66966 


.33034 


.95710 


57 


4 


.61045 


.64994 


.35006 


.96050 


56 


4 


.62703 


.66999 


.33001 


.95704 


56 


5 


9.61073 


9.65028 


10.34972 


9.96045 


55 


5 


9.62730 


9.67032 


10.32968 


9.95698 


55 


6 


.61101 


.65062 


.34938 


.96039 


54 


6 


.62757 


.67065 


.32935 


.95692 


54 


7 


.61129 


.65096 


.34904 


.96034 


53 


7 


.62784 


.67098 


.32902 


.95686 


53 


8 


.61158 


.65130 


.34870 


.96028 


52 


8 


.62811 


.67131 


.32869 


.95680 


52 


9 


.61186 


.65164 


.34836 


.96022 


51 


9 


.62838 


.67163 


.32837 


. 95674 


51 


10 


9.61214 


9.65197 


10.34803 


9.96017 


50 


10 


9.62865 


9.67196 


10.32804 


9.95668 


50 


11 


.61242 


.65231 


.34769 


.96011 


49 


11 


.62892 


.67229 


.32771 


. 95663 


49 


12 


.61270 


.65265 


.34735 


.96005 


48 


12 


.62918 


.67262 


.32738 


.95657 


48 


13 


.61298 


.65299 


.34701 


.96000 


47 


13 


.62945 


.67295 


.32705 


.95651 


47 


14 


.61326 


.65333 


.34667 


.95994 


46| 


14 


.62972 


.67327 


.32673 


.95645 


46 


15 


9.61354 


9.65366 


10.34634 


9.95988 


45 


15 


9.62999 


9.67360 


10.32640 


9.95639 


45 


16 


.61382 


.6D400 


.34600 


.95982 


44 


16 


.63026 


.67393 


.32607 


. 95633 


44 


17 


.61411 


.65434 


.34566 


.95977 


43 


17 


.63052 


.67426 


.32574 


.95627 


43 


18 


.61438 


.65467 


.34533 


.95971 


42 


18 


.63079 


,.67458 


.32542 


.95621 


42 


19 


.61466 


.65501 


.34499 


.95965 


41 


19 


.63106 


.67491 


.32509 


.95615 


41 


20 


9.-61494 


9.65535 


10.34465 


9.95960 


40 


20 


9.63133 


9.67524 


10.32476 


9.95609 


40 


21 


.61522 


.65568 


.34432 


.95954 


39 


21 


.63159 


.67556 


.32444 


.95603 


39 


22 


.61550 


.65602 


.34398 


.95948 


38 


22 


.63186 


.67589 


.32411 


. 95597 


38 


23 


.61578 


.65636 


.34634 


.95942. 


37 


23 


.63213 


.67622 


.32378 


.95591 


37 


24 


.61606 


.65669 


.34331 


.95937 


36 


24 


.63239 


.67654 


.32346 


.95585 


36 


25 


9.61634 


9.65703 


10.34297 


9.95931 


35 


25 


9.63266 


9.67687 


10.32313 


9.95579 


35 


26 


.61662 


.65736 


.34264 


.95925 


34 


26 


.63292 


.67719 


.32281 


.95573 


34 


27 


.61689 


.65770 


.34230 


.95920 


33 


27 


63319 


.67752 


.32248 


.95567 


33 


28 


.61717 


.65803 


.34197 


.95914 


32 


28 


.63345 


.67785 


.32215 


.95561 


32 


29 


.61745 


.65837 


.34163 


.95908 


31 


29 


.63372 


.67817 


.32183 


.95555 


31 


30 


9.61773 


9.65870 


10.34130 


9.95902 


30 


30 


9.63398 


9.67850 


10.32150 


9.95549 


30 


31 


.61800 


.65904 


.34096 


.95897 


29 


31 


.63425 


.67882 


.32118 


.95543 


29 


32 


.61828 


.65937 


.34063 


.95891 


28 


32 


.63451 


.67915 


.32085 


.95537 


28 


33 


.61856 


.65971 


.34029 


.93885 


27 


33 


.63478 


.67947 


.32053 


.95531 


27 


34 


.61883 


.66004 


.33996 


.95879 


26 


34 


.63504 


.67980 


.32020 


.95525 


26 


35 


9.61911 


9.66038 


10.33962 


9.95873 


25 


35 


9.63531 


9.68012 


10.31988 


9.95519 


25 


36 


.61939 


.66071 


.33929 


.95868 


24 


36 


.63557 


.68044 


.31956 


.95513 


24 


37 


.61966 


.66104 


.33896 


.95862 


23 


37 


.63583 


.68077 


.31923 


.95507 


23 


38 


.61994 


.66138 


.33862 


.95856 


22 


38 


.63610 


.68109 


.31891 


.95500 


22 


39 


.62021 


.66171 


.33829 


.95850 


21 


39 


.63636 


.68142 


.31858 


.95494 


21 


40 


9.62049 


9.66204 


10.33796 


9.95844 


20 


40 


9.63662 


9.68174 


10.31826 


9.95488 


20 


41 


.62076 


.66238 


.33762 


.95839 


19 


41 


.63689 


.68206 


.31794 


. 95482 


19 


42 


.62104 


.66271 


.33729 


.95833 


18 


42 


.63715 


.68239 


.31761 


.95476 


18 


43 


.62131 


.66304 


.33696 


.95827 


17 


43 


.63741 


.68271 


.31729 


.95470 


17 


44 


.62159 


.66337 


.33663 


.95821 


16 


44 


.63767 


.68303 


.31697 


.95464 


16 


45 


9.62186 


9.66371 


10.33629 


9.95815 


15 


45 


9.63794 


9.68336 


10.31664 


9.95458 


15 


46 


.62214 


.66404 


.33596 


.95810 


14 


46 


.63820 


.68368 


.31632 


. 95452 


14 


47 


.62241 


.66437 


.33563 


.95804 


13 


47 


.63846 


.68400 


.31600 


.95446 


13 


48 


.62268 


.66470 


.33530 


.95798 


12 


48 


.63872 


.68432 


.31568 


.95440 


12 


49 


.62296 


.66503 


.33497 


.95792 


11 


49 


.63898 


.68465 


.31535 


.95434 


11 


50 


9.62323 


9.66537 


10.33463 


9.95786 


10 


50 


9.63924 


9.68497 


10.31503 


9.95427 


10 


51 


.62350 


.66570 


.33430 


.95780 


9 


51 


.63950 


.68529 


.31471 


.95421 


9 


52 


.62377 


.66603 


.33397 


.95775 


8 


52 


.63976 


.68561 


.31439 


.95415 


8 


53 


.62405 


.66636 


.33364 


.95769 


7 


53 


.64002 


.68593 


.31407 


. 95409 


7 


54 


.62432 


.66669 


.33331 


.95763 


6 


54 


.64028 


.68626 


.31374 


. 95403 


6 


55 


9.62459 


9.66702 


10.33298 


9.95757 


5 


55 


9.64054 


9.68658 


10.31342 


9.95397 


5 


56 


.62486 


.66735 


.33265 


.95751 


4 


56 


.64080 


.68690 


.31310 


.95391 


4 


57 


.62513 


.66768 


.33232 


.95745 


3 


57 


.64106 


.68722 


.31278 


.95384 


3 


58 


.62541 


.66801 


.33199 


.95739 


2 


58 


.64132 


.68754 


.31246 


.95378 


2 


59 


.62568 


.66834 


,33166 


.95733 


1 


59 


.64158 


.68786 


.31214 


.95372 


1 


60 


9.62595 


9.66867 


10.33133 


9.95728 





60 


9.64184 


9.68818 


10.31182 


9.95366 






I Cosine. I Cotang. I Tang, j Sine. \ ' \\ | Cosine. | Cotang. | Tang. | Sine. \ ' 

65° 64° 

*Log secant = colog cosine = 1 — log cosine ; log cosecant = colog sine = 
I — log sine. 

Ex,— Log sec 24°- 30' = 10.04098. Ex.— Log cosec 24°- 30' = 10.38227. 



LOGARITHMIC SINES, ETC. 



189 



5. — Logarithmic Sines, Tangents, Cotangents, Cosines. — (Cont'd.) 
(Secants,[Cosecants.) * 
26° 27° 



' 1 Sine. 


1 Tang. 


Cotang. 1 Cosine. 


1 II ' 1 Sine. 


1 Tang. 1 Cotang. 


1 Cosine. I 





9.64184 


9.68818 


10.31182 


9.95366 


60 





9.65705 


9.70717 


10.29283 


9.94988 


60 


1 


.64210 


.68850 


.31150 


.95360 


59 


1 


.65729 


.70748 


.29252 


. 94982 


59 


2 


.64236 


.68882 


.31118 


.95354 


58 


2 


.65754 


.70779 


.29221 


.94975 


58 


3 


.64262 


.68914 


.31086 


.95348 


57 


3 


.65779 


.70810 


.29190 


.94969 


57 


4 


.64288 


.68946 


.31054 


.95341 


56 


4 


.65804 


.70841 


.29159 


. 94962 


56 


5 


9.64313 


9.68978 


10.31022 


9.95335 


55 


5 


9.65828 


9.70873 


10.29127 


9.94956 


55 


6 


.64339 


.69010 


.30990 


.95329 


54 


6 


.65853 


.70904 


.29096 


. 94949 


54 


7 


.64365 


.69042 


.30958 


.95323 


53 


7 


.65878 


.70935 


.29065 


.94943 


53 


8 


.64391 


.69074 


.30926 


.95317 


52 


8 


.65902 


.70966 


.29034 


.94936 


52 


9 


.64417 


.69106 


.30894 


.95310 


51 


9 


.65927 


.70997 


.29003 


.94930 


51 


10 


9.64442 


9.69138 


10.30862 


9.95304 


50 


10 


9.65952 


9.71028 


10.28972 


9.94923 


50 


11 


.64468 


.69170 


.30830 


.95298 


49 


11 


.65976 


.71059 


.28941 


.94917 


49 


12 


.64494 


.69202 


.30798 


.95292 


48 


12 


.66001 


.71090 


.28910 


.94911 


48 


13 


.64519 


.69234 


.30766 


.95286 


47 


13 


.66025 


.71121 


.28879 


.94904 


47 


14 


.64545 


.69266 


.30734 


.95279 


46 


14 


.66050 


.71153 


.28847 


. 94898 


46 


15 


9.64571 


9.69298 


10.30702 


9.95273 


45 


15 


9.66075 


9.71184 


10.28816 


9.94891 


45 


16 


.64596 


.69329 


.30671 


.95267 


44 


16 


.66099 


.71215 


.28785 


.94885 


44 


17 


.64622 


.69361 


.30639 


.95261 


43 


17 


.66124 


.71246 


.28754 


.94878 


43 


18 


.64647 


.69393 


.30607 


.95254 


42 


18 


.66148 


.71277 


.28723 


.94871 


42 


19 


.64673 


.69425 


.30575 


.95248 


41 


19 


.66173 


.71308 


.28692 


•. 94865 


41 


20 


9.64698 


9.69457 


10.30543 


9.95242 


40 


20 


9.66197 


9.71339 


10.28661 


9.94858 


40 


21 


.64724 


.69488 


.30512 


.95236 


39 


21 


.66221 


.71370 


.28630 


.94852 


39 


22 


.62749 


.69520 


.30480 


.95229 


38 


22 


.66246 


.71401 


.28599 


.94845 


38 


23 


.64775 


.69552 


.30448 


.95223 


37 


23 


.66270 


.71431 


.28569 


.94839 


37 


24 


.64800 


.69584 


.30416 


.95217 


36 


24 


.66295 


.71462 


.28538 


. 94832 


36 


25 


9.64826 


9.69615 


10.30385 


9.95211 


35 


25 


9.66319 


9.71493 


10.28507 


9.94826 


35 


26 


.64851 


.69647 


.30353 


.95204 


34 


26 


.66343 


.71524 


.28476 


.94819 


34 


27 


.64877 


.69679 


.30321 


.95198 


33 


27 


.66368 


.71555 


.28445 


.94813 


33 


28 


.64902 


.69710 


.30290 


.95192 


32 


28 


.66392 


.71586 


.28414 


. 94806 


32 


29 


.64927 


.69742 


.30258 


.95185 


31 


29 


.66416 


.71617 


.28383 


.94799 


31 


30 


9.64953 


9.69774 


10.30226 


9.95179 


30 


30 


9.66441 


9.71648 


10.28352 


9.94793 


30 


31 


.64978 


.69805 


.30195 


.95173 


29 


31 


.66465 


.71679 


.28321 


. 94786 


29 


32 


.65003 


.69837 


.30163 


.95167 


28 


32 


.66489 


.71709 


.28291 


.94780 


28 


33 


.65029 


.69868 


.30132 


.95160 


27 


33 


.66513 


.71740 


.28260 


.94773 


27 


34 


.65054 


.69900 


.30100 


.95154 


26 


34 


.66537 


.71771 


.28229 


.94767 


26 


35 


9.65079 


9.69932 


10.30068 


9.95148 


25 


35 


9.66562 


9.71802 


10.28198 


9.94760 


25 


36 


.65104 


.69963 


.30037 


.95141 


24 


36 


.66586 


.71833 


.28167 


.94753 


24 


37 


.65130 


.69995 


.30005 


.95135 


23 


37 


.66610 


.71863 


.28137 


.94747 


23 


38 


.65155 


.70026 


.29974 


.95129 


22 


38 


.66634 


.71894 


.28106 


.94740 


22 


39 


.65180 


.70058 


.29942 


.95122 


21 


39 


.66658 


.71925 


.28075 


.94734 


21 


40 


9.65205 


9.70089 


10.29911 


9.95116 


20 


40 


9.66682 


9.71955 


10.28045 


9.94727 


20 


41 


.65230 


.70121 


.29879 


.95110 


19 


41 


.66706 


.71986 


.28014 


.94720 


19 


42 


.65255 


.70152 


.29848 


.95103 


18 


42 


.66731 


.72017 


.27983 


.94714 


18 


43 


.65281 


.70184 


.29816 


.95097 


17 


43 


.66755 


.72048 


.27952 


.94707 


17 


44 


.65306 


.70215 


.29785 


.95090 


16 


44 


.66779 


.72078 


.27922 


.94700 


16 


45 


9.65331 


9.70247 


10.29753 


9.95084 


15 


45 


9.66803 


9.72109 


10.27891 


9.94694 


15 


46 


.65356 


.70278 


.29722 


.95078 


14 


46 


.66827 


.72140 


.27860 


.94687 


14 


47 


.65381 


.70309 


.29691 


.95071 


13 


47 


.66851 


.72170 


.27830 


.94680 


13 


48 


.65406 


.70341 


.29659 


.95065 


12 


48 


.66875 


.72201 


.27799 


.94674 


12 


49 


.65431 


.70372 


.29628 


.95059 


11 


49 


.66899 


.72231 


.27769 


.94667 


11 


50 


9.65456 


9.70404 


10.29596 


9.95052 


10 


50 


9.66922 


9.72262 


10.27738 


9.94660 


10 


51 


.65481 


.70435 


.29565 


.95046 


9 


51 


.66946 


.72293 


.27707 


.94654 


9 


62 


.65506 


.70466 


.29534 


.95039 


8 


52 


.66970 


.72323 


.27677 


.94647 


8 


53 


.65531 


.70498 


.29502 


.95033 


7 


53 


.66994 


.72354 


.27646 


.94640 


7 


54 


.65556 


.70529 


.29471 


.95027 


6 


54 


.67018 


.72384 


.27616 


.94634 


6 


55 


9.65580 


9.70560 


10.29440 


9.95020 


5 


55 


9.67042 


9.72415 


10.27585 


9.94627 


5 


56 


.65605 


.70592 


.29408 


.95014 


4 


56 


.67066 


.72445 


.27555 


.94620 


4 


57 


.65630 


.70623 


.29377 


.95007 


3 


57 


.67090 


.72476 


.27524 


.94614 


3 


58 


.65656 


.70654 


.29346 


.95001 


2 


58 


.67113 


.72506 


.27494 


.94607 


2 


59 


.65680 


.70685 


.29315 


.94995 


1 


59 


.67137 


.72537 


.27463 


. 94600 


1 


60 


9.65705 


9.70717 


10.29283 


9.94988 





60 


9.67161 


9.72567 


10.27433 


9.94593 





1 Cosine. 1 


Cotang.l 


Tang. 1 


Sine. 1 


' II 1 Cosine. 1 


Cotang. i 


Tang. 1 


Sine. 1 ' 



63° 62° 

"^Log secant = colog cosine = 1 — log cosine ; log cosecant = colog sine «= 
1 — log sine. 

Ex.—hoQ sec 26°- 30' = 10.04821. Ex,— Log cosec 26°- 30' = 10,35047. 



190 



9.--PLANE TRIGONOMETRY, 



5. — ^Logarithmic Sines, Tangents, Cotangents, Cosines. — (Cont'd.) 
(Secants, Cosecants.)* 
28° 29° 



' 1 Sine. 


1 Tang. 


1 Cotang. 1 Cosine. | |1 ' | Sine. 


1 Tang. 


1 Cotang. 1 Cosine. 


1 





9.67161 


9.72567 


10.27433 


9.94593 


60 





9.68557 


9.74375 


10.25625 


9.94182 


60 


1 


.67185 


.72598 


.27402 


.94587 


59 


1 


.68580 


.74405 


.25595 


.94175 


59 


2 


.67208 


.72628 


.27372 


.94580 


58 


2 


.68603 


.74435 


.25565 


.94168 


58 


3 


.67232 


.72659 


.27341 


.94573 


57 


3 


.68625 


.74465 


.25535 


.94161 


57 


4 


.67256 


.726»9 


.27311 


.94567 


56 


4 


.68648 


.74494 


.25506 


.94154 


56 


5 


9.67280 


9.72720 


10.27280 


9.94560 


55 


5 


9.68671 


9.74524 


10.25476 


9.94147 


55 


6 


.67303 


.72750 


.27250 


. 94553 


54 


6 


.68694 


.74554 


.25446 


.94140 


54 


7 


.67327 


.72780 


.27220 


. 94546 


53 


7 


.68716 


.74583 


.25417 


.94133 


53 


8 


.67350 


.72811 


.27189 


.94540 


52 


8 


.68739 


.74613 


.25387 


.94126 


52 


9 


.67374 


.72841 


.27159 


. 94533 


51 


9 


.68762 


.74643 


.25357 


.94119 


51 


10 


9.67398 


9.72872 


10.27128 


9.94526 


50 


10 


9.68784 


9.74673 


10.25327 


9.94112 


50 


11 


.67421 


.72902 


.27098 


.94519 


49 


11 


.68807 


.74702 


.25298 


.94105 


49 


12 


.67445 


.72932 


.27068 


.94513 


48 


12 


.68829 


.74732 


.25268 


. 94098 


48 


13 


.67468 


.72963 


.27037 


.94506 


47 


13 


.68852 


.74762 


.25238 


. 94090 


47 


14 


.67492 


.72993 


.27007 


.94499 


46 


14 


.68875 


.74791 


.25209 


. 94083 


46 


15 


9.67515 


9.73023 


10.26977 


9.94492 


45 


15 


9.68897 


9.74821 


10.25179 


9.94076 


45 


16 


.67539 


.73054 


.26946 


. 94485 


44 


16 


.68920 


.74851 


.25149 


. 94069 


44 


17 


.67562 


.73084 


.26916 


.94479 


43 


17 


.68942 


.74880 


.25120 


. 94062 


43 


18 


.67586 


.73114 


.26886 


.94472 


42 


18 


.68965 


.74910 


.25090 


.94055 


42 


19 


.67609 


.73144 


.26856 


.94465 


41 


19 


.68987 


.74939 


.25061 


. 94048 


41 


20 


9.67633 


9.73175 


10.26825 


9.94458 


40 


20 


9.69010 


9.74969 


10.25031 


9.94041 


40 


21 


.67656 


.73205 


.26795 


.94451 


39 


21 


.69032 


.74998 


.25002 


. 94034 


39 


22 


.67680 


.73235 


.26765 


.94445 


38 


22 


.69055 


.75028 


.24972 


.94027 


38 


23 


.67703 


.73265 


.26735 


.94438 


37 


23 


.69077 


.75058 


.24942 


. 94020 


37 


24 


.67726 


.73295 


.26705 


.94431 


36 


24 


.69100 


.75087 


.24913 


.94012 


36 


25 


9.67750 


9.73326 


10.26674 


9.94424 


35 


25 


9.69122 


9.75117 


10.24883 


9.94005 


35 


26 


.67773 


.73356 


.26644 


.94417 


34 


26 


.69144 


.75146 


. 24854 


.93998 


34 


27 


.67796 


.73386 


.26614 


.94410 


33 


27 


.69167 


.75176 


.24824 


.93991 


33 


28 


.67820 


.73416 


.26584 


. 94404 


32 


28 


.69189 


.75205 


.24795 


.93984 


32 


29 


.67843 


.73446 


.26554 


.94397 


31 


29 


.69212 


.75235 


.24765 


.93977 


31 


30 


9.67866 


9.73476 


10.26524 


9.94390 


30 


30 


9.69234 


9.75264 


10.24736 


9.93970 


30 


31 


.67890 


.73507 


.26493 


.94383 


29 


31 


.69256 


.75294 


.24706 


.93963 


29 


32 


.67913 


.73537 


.26463 


.94376 


28 


32 


.69279 


.75323 


.24677 


.93955 


28 


33 


.67936 


.73567 


.26433 


.94369 


27 


33 


.69301 


.75353 


.24647 


.93948 


27 


34 


.67959 


.73597 


.26403 


.94362 


26 


34 


.69323 


.75382 


.24618 


.93941 


26 


35 


9.67982 


9.73627 


10.26373 


9.94355 


25 


35 


9.69345 


9.75411 


10.24589 


9.93934 


25 


36 


.68006 


.73657 


.26343 


.94349 


24 


36 


.69368 


.75441 


.24559 


.93927 


24 


37 


.68029 


.73687 


.26313 


.94342 


23 


37 


.69390 


.75470 


.24530 


.93920 


23 


38 


. 68052 


.73717 


.26283 


.94335 


22 


38 


.69412 


.75500 


.24500 


.93912 


22 


39 


.68075 


.73747 


.26253 


.94328 


21 


39 


.69434 


.75529 


.24471 


.93905 


21 


40 


9.68098 


9.73777 


10.26223 


9.94321 


20 


40 


9.69456 


9.75558 


10.24442 


9.93898 


20 


41 


.68121 


.73807 


.26193 


.94314 


19 


41 


.69479 


.75588 


.24412 


.93891 


19 


42 


.68144 


.73837 


.26163 


.94307 


18 


42 


.69501 


.75617 


.24383 


. 93884 


18 


43 


.68167 


.73867 


.26133 


.94300 


17 


43 


.69523 


.75647 


.24353 


. 93876 


17 


44 


.68190 


.73897 


.26103 


.94293 


16 


44 


.69545 


.75676 


.24324 


. 93869 


16 


45 


9.68213 


9.73927 


10.26073 


9.94286 


15 


45 


9.69567 


9.75705 


10.24295 


9.93862 


15 


46 


.68237 


.73957 


.26043 


.94279 


14 


46 


.69589 


.75735 


.24265 


.93855 


14 


47 


.68260 


.73987 


.26013 


.94273 


13 


47 


.69611 


.75764 


.24236 


. 93847 


13 


48 


.68283 


.74017 


.25983 


.94266 


12 


48 


.69633 


.75793 


.24207 


. 93840 


12 


49 


.68305 


.74047 


.25953 


.94259 


11 


49 


.69655 


.75822 


.24178 


.93833 


11 


50 


9.68328 


9.74077 


10.25923 


9.94252 


10 


50 


9.69677 


9.75852 


10.24148 


9.93826 


10 


51 


.68351 


.74107 


.25893 


. 94245 


9 


51 


.69699 


.75881 


.24119 


.93819 


9 


52 


.68374 


.74137 


.25863 


.94238 


8 


52 


.69721 


.75910 


.24090 


.93811 


8 


53 


.68397 


.74166 


.25834 


.94231 


7 


53 


.69743 


.75939 


.24061 


.93804 


7 


54 


.68420 


.74196 


.25804 


.94224 


6 


54 


.69765 


.75969 


.24031 


.93797 


6 


55 


9.68443 


9.74226 


10.25774 


9.94217 


5 


55 


9.69787 


9.75998 


10.24002 


9.93789 


5 


56 


.68466 


.74256 


.25744 


.94210 


4 


56 


.69809 


.76027 


.23973 


. 93782 


4 


57 


.68489 


.74286 


.25714 


.94203 


3 


57 


.69831 


.76056 


.23944 


.93775 


3 


58 


.68512 


.74316 


.25684 


.94196 


2 


58 


.69853 


.76086 


.23914 


.93768 


2 


59 


.68534 


.74345 


.25655 


.94189 


1 


59 


.69875 


.76115 


.23885 


.93760 


1 


60 


9.68557 


9.74375 


10.25625 


9.94182 





60 


9.69897 


9.76144 


10.23856 


9.93753 





1 Cosine. 1 


Cotang. 


Tang. 1 


Sine. 


' 11 1 Cosine. 


Cotang. 1 


Tang. 1 


Sine. 


r 



61' 



60^ 



*Log secant = colog cosine=l — log cosine; log cosecant = colog sine = 
1 — log sine. 

Ex.— Log sec 28°- 30' = 10.05610. Ex.— Log cosec 28°- 30'= 10.32134. 



LOGARITHMIC SINES, ETC. 



191 



5. — Logarithmic Sines, Tangents, Cotangents, Cosines. — (Cont'd.) 
(Secants, Cosecants.)* 
31° 



' 1 Sine. 


1 Tang. 


1 Cotang. 1 Cosine. 


1 II ' 1 Sine. 


1 Tang. 


1 Cotang. 


1 Cosine. | 





9.69897 


9.76144 


10.23856 


9.93753 


60 





9.71184 


9.77877 


10.22123 


9.93307 


60 


1 


.69919 


.76173 


.23827 


.93746 


59 


1 


.71205 


.77906 


.22094 


.93299 


59 


2 


.69941 


.76202 


.23798 


.93738 


58 


2 


.71226 


.77935 


.22065 


.93291 


58 


3 


.69963 


.76231 


.23769 


.93731 


57 


3 


.71247 


.77963 


.22037 


.93284 


57 


4 


.69984 


.76261 


.23739 


.93724 


56 


4 


.71268 


.77992 


.22008 


.93276 


56 


5 


9.70006 


9.76290 


10.23710 


9.93717 


55 


5 


9.71289 


9.78020 


10.21980 


9.93269 


55 


6 


.70028 


.76319 


.23681 


.93709 


54 


6 


.71310 


.78049 


.21951 


.93261 


54 


7 


.70050 


.76348 


.23652 


.93702 


53 


7 


.71331 


.78077 


.21923 


. 93253 


53 


8 


.70072 


.76377 


.23623 


.93695 


52 


8 


.71352 


.78106 


.21894 


.93246 


52 


9 


.70093 


.76406 


.23594 


.93687 


51 


9 


.71373 


.78135 


.21865 


.93238 


51 


10 


9.70115 


9.76435 


10.23565 


9.93680 


50 


10 


9.71393 


9.78163 


10.21837 


9.93230 


50 


11 


.70137 


.76464 


.23536 


.93673 


49 


11 


.71414 


.78192 


.21808 


. 93223 


49 


12 


.70159 


.76493 


.23507 


.93665 


48 


12 


.71435 


.78220 


.21780 


.93215 


48 


13 


.70180 


.76522 


.23478 


.93658 


47 


13 


.71456 


.78249 


.21751 


.93207 


47 


14 


.70202 


.76551 


.23449 


.93650 


46 


14 


.71477 


.78277 


.21723 


.93200 


46 


15 


9.70224 


9.76580 


10.23420 


9.93643 


45 


15 


9.71498 


9.78306 


10.21694 


9.93192 


45 


16 


.70245 


.76609 


.23391 


.93636 


44 


16 


.71519 


.78334 


.21666 


.93184 


44 


17 


.70267 


.76639 


.23361 


.93628 


43 


17 


.71539 


.78363 


.21637 


.93177 


43 


18 


.70288 


.76668 


.23332 


.93621 


42 


18 


.71560 


.78391 


.21609 


.93169 


42 


19 


.70310 


.76697 


.23303 


.93614 


41 


19 


.71581 


.78419 


.21581 


.93161 


41 


20 


9.70332 


9.76725 


10.23275 


9.93606 


40 


20 


9.71602 


9.78448 


10.21552 


9.93154 


40 


21 


.70353 


.76754 


.23246 


.93599 


39 


21 


.71622 


.78476 


.21524 


.93146 


39 


22 


.70375 


.76783 


.23217 


.93591 


38 


22 


.71643 


.78505 


.21495 


.93138 


38 


23 


.70396 


.76812 


.23188 


.93584 


37 


23 


.71664 


.78533 


.21467 


.93131 


37 


24 


,70418 


.76841 


.23159 


.93577 


36 


24 


.71685 


.78562 


.21438 


.93123 


36 


25 


9.70439 


9.76870 


10.23130 


9.93569 


35 


25 


9.71705 


9.78590 


10.21410 


9.93115 


35 


26 


.70461 


.76899 


.23101 


.93562 


34 


26 


.71726 


.78618 


.21382 


.93107 


34. 


27 


.70482 


.76928 


.23072 


.93554 


33 


27 


.71747 


.78647 


.21353 


.93100 


3? 


28 


.70504 


.76957 


.23043 


.93547 


32 


28 


.71767 


.78675 


.21325 


.93092 


32 


29 


.70525 


.76986 


.23014 


.93539 


31 


29 


.71788 


.78704 


.21296 


.9^084 


31 


30 


9.70547 


9.77015 


10.22985 


9.93532 


30 


30 


9.71809 


9.78732 


10.21268 


9.93077 


30 


31 


.70568 


.77044 


.22956 


.93525 


29 


31 


.71829 


.78760 


.21240 


.93069 


29 


32 


.70590 


.77073 


.22927 


.93517 


28 


32 


.71850 


.78789 


.21211 


.93061 


28 


33 


.70611 


.77101 


.22899 


.93510 


27 


33 


.71870 


.78817 


.21183 


.93053 


27 


34 


.70633 


.77130 


.22870 


.93502 


26 


34 


.71891 


.78845 


.21155 


.93046 


26 


35 


9.70654 


9.77159 


10.22841 


9.93495 


25 


35 


9.71911 


9.78874 


10.21126 


9.93038 


25 


36 


.70675 


.77188 


.22812 


.93487 


24 


36 


.71932 


.78902 


.21098 


.93030 


24 


37 


.70697 


.77217 


.22783 


. 93480 


23 


37 


.71952 


.78930 


.21070 


.93022 


23 


38 


.70718 


.77246 


.22754 


.93472 


22 


38 


.71973 


.78959 


.21041 


.93014 


22 


39 


.70739 


.77274 


.22726 


.93465 


21 


39 


.71994 


.78987 


.21013 


.93007 


21 


40 


9.70761 


9.77303 


10.22697 


9.93457 


20 


40 


9.72014 


9.79015 


10.20985 


9.92999 


20 


41 


.70782 


.77332 


.22668 


.93450 


19 


41 


.72034 


.79043 


.20957 


.92991 


19 


42 


.70803 


.77361 


.22639 


.93442 


18 


42 


.72055 


.79072 


.20928 


.92983 


18 


43 


.70824 


.77390 


.22610 


.93435 


17 


43 


.72075 


.79100 


.20900 


. 92976 


17 


44 


.70846 


.77418 


.22582 


.93427 


16 


44 


.72096 


.79128 


.20872 


.92968 


16 


45 


9.70867 


9.77447 


10.22553 


9.93420 


15 


45 


9.72116 


9.79156 


10.20844 


9.92960 


15 


46 


.70888 


.77476 


.22524 


.93412 


14 


46 


.72137 


.79185 


.20815 


.92952 


14 


47 


.70909 


.77505 


.22495 


.93405 


13 


47 


.72157 


.79213 


.20787 


.92944 


13 


48 


.70931 


.77533 


.22467 


.93397 


12 


48 


.72177 


.79241 


.20759 


.92936 


12 


49 


.70952 


.77562 


.22438 


.93390 


11 


49 


.72198 


.79269 


.20731 


.92929 


11 


50 


9.70973 


9.77591 


10.22409 


9.93382 


10 


50 


9.72218 


9.79297 


10.20703 


9.92921 


10 


51 


.70994 


.77619 


.22381 


.93375 


9 


51 


.72238 


.79326 


.20674 


.92913 


9 


52 


.71015 


.77648 


.22352 


.93367 


8 


52 


.72259 


.79354 


.20646 


.92905 


8 


53 


.71036 


.77677 


.22323 


.93360 


7 


53 


.72279 


.79382 


.20618 


.92897 


7 


54 


.71058 


.77706 


.22294 


.93352 


6 


54 


.72299 


.79410 


.20590 


.92889 


6 


55 


9.71079 


9.77734 


10.22266 


9.93344 


5 


55 


9.72320 


9.79438 


10.20562 


9.92881 


5 


56 


.71100 


.77763 


.22237 


.93337 


4 


56 


.72340 


.79466 


.20534 


.92874 


4 


57 


.71121 


.77791 


.22209 


.93329 


3 


57 


.72360 


.79495 


.20505 


.92866 


3 


58 


.71142 


.77820 


.22180 


.93322 


2 


58 


.72381 


.79523 


.20477 


.92858 


2 


59 


.71163 


.77849 


.22151 


.93314 


1 


59 


.72401 


.79551 


.20449 


.92850 


1 


60 


9.71184 


9.77877 


10.22123 


9.93307 





60 


9.72421 


9.79579 


10.20421 


9.92842 





1 Cosine. 1 


Cotang. 


Tang. 


Sine. 1 


' 11 1 Cosine. 1 


Cotang. 


Tang. 1 


Sine. 1 ' 



59° 58^ 

*Log secant = colog cosine = 1 — log cosine ; log cosecant = colog sine = 
1 — log sine. 

£^.— Log sec 30°- 30' = 10.06468. Ex,—Loq cosec 30°- 30'= 10.29463. 



9.— PLANE TRIGONOMETRY. 



6. — Logarithmic Sines, Tangents, Cotangents, Cosines. — (Cont'd.) 
(Secants, Cosecants.)* 
32° 33° 



' 


1 Sine. 


1 Tang. 


1 Cotang. 1 Cosine. | 


1 ' 


1 Sine. 


1 Tang. 


Cotang. 


1 Cosine. 1 





9.72421 


9.79579 


10.20421 


». 92842 


60 





9.73611 


9.81252 


10.18748 


9.92359 


60 


1 


.72441 


.79607 


.20393 


.92834 


59 


1 


.73630 


.81279 


.18721 


.92351 


5S 


2 


.72461 


.79635 


.20365 


.92826 


58 


2 


.73650 


.81307 


.18693 


. 92343 


5i 


3 


.72482 


.79663 


.20337 


.92818 


57 


3 


.73669 


.81335 


.18665 


. 92335 


57 


4 


.72502 


.79691 


.20309 


.92810 


56 


4 


.73689 


.81362 


.18638 


. 92326 


56 


5 


9.72522 


9.79719 


10.20281 


9.92803 


55 


5 


9.73708 


9.81390 


10.18610 


9.92318 


55 


6 


.7.2542 


.79747 


.20253 


.92795 


54 


6 


.73727 


.81418 


.18582 


.92310 


54 


7 


.72562 


.79776 


.20224 


.92787 


53 


7 


.73747 


.81445 


.18555 


. 92302 


53 


8 


.72582 


.79804 


.20196 


.92779 


52 


8 


.73766 


.81473 


.18527 


. 92293 


52 


9 


.72602 


.79832 


.20168 


.92771 


51 


9 


.73785 


.81500 


.18500 


. 92285 


51 


10 


9.72622 


9.79860 


10.20140 


9.92763 


50 


10 


9.73805 


9.81528 


10.18472 


9.92277 


50 


11 


.72643 


.79888 


.20112 


.92755 


49 


11 


.73824 


.81556 


.18444 


.92269 


49 


12 


.72663 


.79916 


.20084 


. 92747 


48 


12 


.73843 


.81583 


.18417 


. 92260 


48 


13 


.72683 


.79944 


.20056 


.92739 


47 


13 


.73863 


.81611 


.18389 


. 92252 


47 


14 


.72703 


.79972 


.20028 


.92731 


46 


14 


.73882 


.81638 


.18362 


. 92244 


46 


15 


9.72723 


9.80000 


10.20000 


9.92723 


45 


15 


9.73901 


9.81666 


10.18334 


9.92235 


45 


16 


.72743 


.80028 


.19972 


.92715 


44 


16 


.73921 


.81693 


.18307 


. 92227 


44 


17 


.72763 


. 80056 


.19944 


.92707 


43 


17 


.73940 


.81721 


.18279 


.92219 


43 


18 


.72783 


. 80084 


.19916 


.92699 


42 


18 


.73959 


.81748 


.18252 


.92211 


42 


19 


.72803 


.80112 


.19888 


.92691 


41 


19 


.73978 


.81776 


.18224 


. 92202 


41 


20 


9.72823 


9.80140 


10.19860 


9.92683 


40 


20 


9.73907 


9.81803 


10.18197 


9.92194 


40 


21 


.72843 


.80168 


.19832 


.92675 


39 


21 


.74017 


.81831 


.18169 


.92186 


39 


22 


.72863 


.80195 


.19805 


. 92667 


38 


22 


.74036 


.81858 


.18142 


.92177 


38 


23 


.72883 


. 80223 


.19777 


.92659 


37 


23 


.74055 


.81886 


.18114 


.92169 


37 


24 


.72902 


.80251 


.19749 


.92651 


36 


24 


.74074 


.81913 


. 18087 


.92161 


36 


25 


9.72922 


9.80279 


10.19721 


9.92643 


35 


25 


9.74093 


9.81941 


10.18059 


9.92152 


35 


26 


.72942 


.80307 


.19693 


.92635 


34 


26 


.74113 


.81968 


.18032 


.92144 


34 


27 


.72962 


.80335 


.19665 


.92627 


33 


27 


.74132 


.81996 


.18004 


.92136 


33 


28 


.72982 


.80363 


.19637 


.92619 


32 


28 


.74151 


. 82023 


.17977 


.92127 


32 


29 


.73002 


.80391 


.19609 


.92611 


31 


29 


.74170 


.82051 


.17949 


.92119 


31 


30 


9.73022 


9.80419 


10.19581 


9.92603 


30 


30 


9.74189 


9.82078 


10.17922 


9.92111 


30 


31 


.73041 


. 80447 


.19553 


.92595 


29 


31 


.74208 


.82106 


.17894 


.92102 


29 


32 


.73061 


.80474 


.19526 


. 92587 


28 


32 


.74227 


.82133 


.17867 


.92094 


28 


33 


.73081 


. 80502 


.19498 


.92579 


27 


33 


.74246 


.82161 


.17839 


.92086 


27 


34 


.73101 


. 80530 


.19470 


.92571 


26 


34 


.74265 


.82188 


.17812 


.92077 


26 


35 


9.73121 


9.80558 


10.19442 


9.92563 


25 


35 


9.74284 


9.82215 


10.17785 


9.92069 


25 


36 


.73140 


. 80586 


.19414 


.92555 


24 


36 


.74303 


. 82243 


.17757 


. 92060 


24 


37 


.73160 


.80614 


.19386 


.92546 


23 


37 


.74322 


.82270 


.17730 


. 92052 


23 


38 


.73180 


.80642 


.19358 


.92538 


22 


38 


.74341 


.82298 


.17702 


. 92044 


22 


39 


.73200 


. 80669 


.19331 


.92530 


21 


39 


.74360 


.92325 


.17675 


. 92035 


21 


40 


9.73219 


9.80697 


10.19303 


9.92522 


20 


40 


9.74379 


9.82352 


10.17648 


9.92027 


20 


41 


.73239 


.80725 


.19275 


.92514 


19 


41 


.74398 


.82380 


.17620 


.92018 


19 


42 


.73259 


.80753 


.19247 


.92506 


18 


42 


.74417 


. 82407 


.17593 


.92010 


18 


43 


.73278 


.80781 


.19219 


.92498 


17 


43 


.74436 


. 82435 


.17565 


. 92002 


17 


44 


.73298 


. 80808 


.19192 


.92490 


16 


44 


.74455 


.82462 


.17538 


.91993 


16 


45 


9.73318 


9.80836 


10.19164 


9.92482 


15 


45 


9.74474 


9.82489 


10.17511 


9.91985 


15 


46 


.73337 


. 80864 


.19136 


.92473 


14 


46 


.74493 


.82517 


.17483 


.91976 


14 


47 


.73357 


.80892 


.19108 


.92465 


13 


47 


.74512 


.82544 


.17456 


.91968 


13 


48 


.73377 


.80919 


.19081 


.92457 


12 


48 


.74531 


.82571 


..17429 


.91959 


12 


49 


.73396 


.80947 


.19053 


.92449 


11 


49 


.74549 


.82599 


.17401 


.91951 


11 


50 


9.73416 


9.80975 


10.19025 


9.92441 


10 


50 


9.74568 


9.82626 


10.17374 


9.91942 


10 


51 


.73435 


.81003 


.18997 


.92433 


9 


51 


.74587 


. 82653 


.17347 


.91934 


9 


52 


.73455 


.81030 


.18970 


.92425 


8 


52 


.74606 


.82681 


.17319 


.91925 


8 


53 


.73474 


.81058 


.18942 


.92416 


7 


53 


.74625 


.82708 


.17292 


.91917 


7 


54 


.73494 


.81086 


.18914 


.92408 


6 


54 


.74644 


.82735 


.17265 


.91908 


6 


55 


9.73513 


9.81113 


10.18887 


9.92400 


5 


55 


9.74662 


9.82762 


10.17238 


9.91900 


5 


56 


.73533 


.81141 


.18859 


.92392 


4 


56 


.74681 


.82790 


.17210 


.91891 


4 


57 


.73552 


.81169 


.18831 


.92384 


3 


57 


.74700 


.82817 


.17183 


.91883 


3 


58 


.73572 


.81196 


.18804 


.92376 


2 


58 


.74719 


.82844 


.17156 


.91874 


2 


59 


.73591 


.81224 


.18776 


.92367 


1 


59 


.74737 


.82871 


.17129 


.91866 


1 


60 


9.73611 


9.81252 


10.18748 


9.92359 





60 


9.74756 


9.82899 


10.17101 


9.91857 





1 Cosine. 


Cotang. 


Tang. 


Sine. 1 ' II 1 Cosine. 


Cotang. 


Tang. 


Sine. 1 ' 



57° 56^ 

*Log secant = colog cosine = 1 — log cosine ; log cosecant = colog sine = 
1 — log sine. 

Ex.— Log sec 32°- SO" = 10.07397. ^:x:.— Log cosec 32°- 30' = 10.26978. 



LOGARITHMIC SINES, ETC. 193 

p 5. — Logarithmic Sines, Tangents, Cotangents, Cosines. — (Cont'd.) 

(Secants, Cosecants.)* 
34*' 35° 



/ 


Sine. 


1 Tang. 


1 Cotang. 1 Cosine. | 


1 ' 


Sine. 


1 Tang. 


Cotang. 1 Cosine. 







9.74756 


9.82899 


10.17101 


9.91857 


60 





9.75859 


9.84523 


10.15477 


9.91336 


60 


1 


.74775 


.82926 


.17074 


.91849 


59 


1 


.75877 


. 84550 


.15450 


.91328 


59 


2 


.74794 


.82953 


.17047 


.91840 


58 


2 


.75895 


.84576 


.15424 


.91319 


58 


3 


.74812 


.82980 


.17020 


.91832 


57 


3 


.75913 


.84603 


.15397 


.91310 


57 


4 


.74831 


.83008 


.16992 


.91823 


56 


4 


.75931 


.84630 


.15370 


.91301 


56 


6 


9.74850 


9.83035 


10.16965 


9.91815 


55 


5 


9.75949 


9.84657 


10.15343 


9.91292 


55 


6 


.74868 


.83062 


.16938 


.91806 


54 


6 


.75967 


.84684 


.15316 


.91283 


54 


7 


.74887 


.83089 


.16911 


.91798 


53 


7 


.75985 


.84711 


.15289 


.91274 


53 


8 


.74906 


.83117 


.16883 


.91789 


52 


8 


.76003 


.84738 


.15262 


.91266 


52 


9 


.74924 


.83144 


.16856 


.91781 


51 


9 


.76021 


.84764 


.15236 


.91257 


51 


10 


9.74943 


9.83171 


10.16829 


9.91772 


50 


10 


9.76039 


9.84791 


10.15209 


9.91248 


50 


11 


.74961 


.83198 


.16802 


.91763 


49 


11 


.76057 


.84818 


.15182 


.91239 


49 


12 


.74980 


.83225 


.16775 


.91755 


48 


12 


.76075 


.84845 


.15155 


.91230 


48 


13 


.74999 


.83252 


.16748 


.91746 


47 


13 


.76093 


. 84872 


.15128 


.91221 


47 


14 


.75017 


.83280 


.16720 


.91738 


46 


14 


.76111 


. 84899 


.15101 


.91212 


46 


15 


9.75036 


9.83307 


10.16693 


9.91729 


45 


15 


9.76129 


9.84925 


10.15075 


9.91203 


45 


16 


.75054 


.83334 


.16666 


. 91720 


44 


16 


.76146 


. 84952 


.15048 


.91194 


44 


17 


.75073 


.83361 


.16639 


.91712 


43 


17 


.76164 


.84979 


.15021 


.91185 


43 


18 


.75091 


.83388 


.16612 


.91703 


42 


18 


.76182 


. 85006 


.14994 


.91176 


42 


19 


.75110 


.83415 


.16585 


.91695 


41 


19 


.76200 


. 85033 


.14967 


.91167 


41 


20 


9.75128 


9.83442 


10.16558 


9.91686 


40 


20 


9.76218 


9.85059 


10.14941 


9.91158 


40 


21 


.75147 


.83470 


.16530 


.91677 


39 


21 


.76236 


. 85086 


.14914 


.91149 


39 


22 


.75165 


.83497 


.16503 


.91669 


38 


22 


.76253 


.85113 


.14887 


.91141 


38 


23 


.75184 


.83524 


.16476 


.91660 


37 


23 


.76271 


.85140 


.14860 


.91132 


37 


24 


.75202 


.83551 


.16449 


.91651 


36 


24 


.76289 


.85166 


.14834 


.91123 


36 


25 


9.75221 


9.83578 


10.16422 


9.91643 


35 


25 


9.76307 


9.85193 


10.14807 


9.91114 


35 


26 


.75239 


.83605 


.16395 


.91634 


34 


26 


.76324 


.85220 


.14780 


.91105 


34 


27 


.75258 


.83632 


.16368 


.91625 


33 


27 


.76342 


.85247 


.14753 


.91096 


33 


28 


.75276 


.83659 


.16341 


.91617 


32 


28 


.76360 


. 85273 


.14727 


.91087 


32 


29 


.75294 


.83686 


.16314 


.91608 


31 


29 


.76378 


.85300 


.14700 


.91078 


31 


ao 


9.75313 


9.83713 


10.16287 


9.91599 


30 


30 


9.76395 


9.85327 


10.14673 


9.91069 


30 


31 


.75331 


.83740 


.16260 


.91591 


29 


31 


.76413 


.85354 


.14646 


.91060 


29 


32 


.75350 


.83768 


.16232 


.91582 


28 


32 


.76431 


.85380 


.14620 


.91051 


28 


33 


.75368 


.83795 


.16205 


.91573 


27 


33 


.76448 


. 85407 


.14593 


.91042 


27 


34 


.75386 


.83822 


.16178 


.91565 


26 


34 


.76466 


.85434 


.14566 


.91033 


26 


35 


9.75405 


9.83849 


10.16151 


9.91556 


25 


35 


9.76484 


9.85460 


10.14540 


9.91023 


25 


36 


.75423 


.83876 


.16124 


.91547 


24 


36 


.76501 


. 85487 


.14513 


.91014 


24 


37 


.75441 


.83903 


.16097 


.91538 


23 


37 


.76519 


.85514 


.14486 


.91005 


23 


38 


.75459 


.83930 


.16070 


.91530 


22 


38 


.76537 


.85540 


.14460 


.90996 


22 


39 


.75478 


.83957 


.16043 


.91521 


21 


39 


.76554 


.85567 


.14433 


. 90987 


21 


40 


9.75496 


9.83984 


10.16016 


9.91512 


20 


40 


9.76572 


9.85594 


10.14406 


9.90978 


20 


41 


.75514 


.84011 


.15989 


.91504 


19 


41 


.76590 


. 85620 


.14380 


.90969 


19 


42 


.75533 


.84038 


.15962 


.91495 


18 


42 


.76607 


.85647 


.14353 


.90960 


18 


43 


.75551 


.84065 


.15935 


.91486 


17 


43 


.76625' 


.85674 


.14326 


.90951 


17 


44 


.75569 


. 84092 


.15908 


.91477 


16 


44 


.76642 


. 85700 


.14300 


.90942 


16 


45 


9.75587 


9.84119 


10.15881 


9.91469 


15 


45 


9.76660 


9.85727 


10.14273 


9.90933 


15 


46 


.75605 


.84146 


.15854 


.91460 


14 


46 


.76677 


.85754 


.14246 


.90924 


14 


47 


.75624 


.84173 


.15827 


.91451 


13 


47 


.76695 


.85780 


.14220 


.90915 


13 


48 


.75642 


.84200 


.15800 


.91442 


12 


48 


.76712 


. 85807 


.14193 


.90906 


12 


49 


.75660 


. 84227 


.15773 


.91433 


11 


49 


.76730 


.85834 


.14166 


. 90896 


11 


50 


9.75678 


9.84254 


10.15746 


9.91425 


10 


50 


9.76747 


9.85860 


10.14140 


9.90887 


10 


51 


.75696 


.84280 


.15720 


.91416 


9 


51 


.76765 


. 85887 


.14113 


.90878 


9 


52 


.75714 


. 84307 


.15693 


.91407 


8 


52 


.76782 


.85913 


.14087 


.90869 


8 


53 


.75733 


.84334 


.15666 


.91398 


7 


53 


.76800 


.85940 


.14060 


.90860 


7 


54 


.75751 


.84361 


.15639 


.91389 


6 


54 


.76817 


.85967 


.14033 


.90851 


6 


55 


9.75769 


9.84388 


10.15612 


9.91381 


5 


55 


9.76835 


9. 85993 


10.14007 


9.90842 


5 


56 


.75787 


.84415 


.15585 


.91372 


4 


56 


.76852 


.86020 


.13980 


. 90832 


4 


57 


.75805 


.84442 


.15558 


.91363 


3 


57 


.76870 


.86046 


.13954 


. 90823 


3 


58 


.75823 


.84469 


.15531 


.91354 


2 


58 


.76887 


.86073 


.13927 


.90814 


2 


59 


.75841 


.84496 


.15504 


.91345 


1 


59 


.76904 


.86100 


.13900 


. 90805 


1 


60 


9.75859 


9. 84523 


10.15477 


9.91336 





60 


9.76922 


9.86126 


10.13874 


9.90796 





1 Cosine. 


iCotang.l Tang. 


1 Sine: 


1 ' II 1 Cosine. 


1 Cotang. 


1 Tang. 


Sine. 


' 



55° 54° 

*Log secant = colog cosine = 1 — log cosine ; log cosecant = colog sine = 
1 — log sine. 

Ex.— Log sec 34°- 30' = 10.08401. Ex.— Log cosec 34°- 30' = 10.24687. 



194 



9.— PLANE TRIGONOMETRY. 



5. — Logarithmic Sines, Tangents, Cotangents, Cosines — (Cont*d.) 
(Secants, Cosecants.)* 

^_ 37^ __^ 

I Sine. I Tang. | Cotang. | Cosine. j \\ ' \ Sine. | Tang. | Cotang. | Cosine. | 






9.76922 


9.86126 


10.13874 


9.90796 


60 





9.77946 


9.87711 10 


12289 


9.90235 


60 


1 


.76939 


.86153 


.13847 


.90787 


59 


1 


.77963 


.87738 


12262 


.90225 


59 


2 


.76957 


.86179 


.13821 


.90777 


58 


2 


.77980 


.87764 


12236 


.90216 


58 


3 


.76974 


. 86206 


.13794 


.90768 


57 


3 


.77997 


.87790 


12210 


.90206 


57 


4 


.76991 


.86232 


.13768 


.90759 


56 


4 


.78013 


.87817 


12183 


.90197 


56 


5 


9.77009 


9.86259 


10.13741 


9.90750 


55 


5 


9.78030 


9.87843 10 


12157 


9.90187 


55 


6 


.77026 


.86285 


.13715 


.90741 


54 


6 


.78047 


.87869 


12131 


.90178 


54 


7 


.77043 


.86312 


.13688 


.90731 


53 


7 


.78063 


. 87895 


12105 


.90168 


53 


8 


.77061 


.86338 


.13662 


.90722 


52 


8 


.78080 


.87922 


12078 


.90159 


52 


9 


.77078 


.86365 


.13635 


.90713 


51 


9 


.78097 


.87948 


12052 


.90149 


51 


10 


9.77095 


9.86392 


10.13608 


9.90704 


50 


10 


9.78113 


9.87974 10 


12026 


9.90139 


50 


11 


.77112 


.86418 


.13582 


. 90694 


49 


11 


.78130 


. 88000 


12000 


.90130 


49 


12 


.77130 


. 86445 


.13555 


.90685 


48 


12 


.78147 


. 88027 


11973 


.90120 


48 


13 


.77147 


.86471 


.13529 


.90676 


47 


13 


.78163 


. 88053 


11947 


.90111 


47 


14 


.77164 


.86498 


.13502 


.90667 


46 


14 


.78180 


.88079 


11921 


.90101 


46 


15 


9.77181 


9.86524 


10.13476 


9.90657 


45 


15 


9.78197 


9.88105 10 


11895 


9.90091 


45 


16 


.77199 


.86551 


.13449 


.90648 


44 


16 


.78213 


.88131 


11869 


. 90082 


44 


17 


.77216 


.86577 


.13423 


.90639 


43 


17 


.78230 


.88158 


11842 


. 90072 


43 


18 


.77233 


. 86603 


.13397 


.90630 


42 


18 


.78246 


.88184 


11816 


. 90063 


42 


19 


.77250 


.86630 


.13370 


.90620 


41 


19 


.78263 


.88210 


11790 


. 90053 


41 


20 


9.77268 


9.86656 


10.13344 


9.90611 


40 


20 


9.78280 


9.88236 10 


11764 


9. 90043 


40 


21 


.77285 


. 86683 


.13317 


.90602 


39 


21 


.78296 


.88262 


11738 


. 90034 


39 


22 


.77302 


.86709 


.13291 


.90592 


38 


22 


.78313 


.88289 


11711 


.90024 


38 


23 


.77319 


.86736 


.13264 


. 90583 


37 


23 


.78329 


.88315 


11685 


.90014 


37 


24 


.77336 


.86762 


.13238 


.90574 


36 


24 


.78346 


.88341 


11659 


.90005 


36 


25 


9.77353 


9.86789 


10.13211 


9.90565 


35 


25 


9.78362 


9.88367 10 


11633 


9.89995 


35 


26 


.77370 


.86815 


.13185 


.90555 


34 


26 


.78379 


.88393 


11607 


.89985 


34 


27 


.77387 


. 86842 


.13158 


.90546 


33 


27 


.78395 


. 88420 


11580 


.89976 


33 


28 


.77405 


. 86868 


.13132 


.90537 


32 


28 


.78412 


. 88446 


11554 


.89966 


32 


29 


.77422 


.86894 


.13106 


.90527 


31 


29 


.78428 


.88472 


11528 


.89956 


31 


30 


9.77439 


9.86921 


10.13079 


9.90518 


30 


30 


9.78445 


9.88498 10 


11502 


9.89947 


30 


31 


.77456 


.86947 


.13053 


.90509 


29 


31 


.78461 


.88524 


11476 


.89937 


29 


32 


.77473 


.86974 


.13026 


.90499 


28 


32 


.78478 


.88550 


11450 


.89927 


28 


33 


.77490 


. 87000 


.13000 


.90490 


27 


33 


.78494 


.88577 


11423 


.89918 


27 


34 


.77507 


.87027 


.12973 


.90480 


26 


34 


.78510 


. 88603 


11397 


.89908 


26 


35 


9.77524 


9.87053 


10.12947 


9.90471 


25 


35 


9.78527 


9.88629 10 


11371 


9.89898 


25 


36 


.77541 


.87079 


.12921 


.90462 


24 


36 


.78543 


.88655 


11345 


.89888 


24 


37 


.77558 


.87106 


.12894 


.90452 


23 


37 


.78560 


. 88681 


11319 


.89879 


23 


38 


.77575 


.87132 


.12868 


.90443 


22 


38 


.78576 


.88707 


11293 


.89869 


22 


39 


.77592 


.87158 


.12842 


.90434 


21 


39 


.78592 


. 88733 


11267 


.89859 


21 


40 


9.77609 


9.87185 


10.12815 


9.90424 


20 


40 


9.78609 


9.88759 10 


11241 


9.89849 


20 


41 


.77626 


.87211 


.12789 


.90415 


19 


41 


.78625 


.88786 


11214 


.89840 


19 


42 


.77643 


.87238 


.12762 


.90405 


18 


42 


.78642 


.88812 


.11188 


. 89830 


18 


43 


.77660 


. 87264 


.12736 


.90396 


17 


43 


.78658 


.88838 


.11162 


. 89820 


17 


44 


.77677 


.87290 


.12710 


.90386 


16 


44 


.78674 


. 88864 


.11136 


.89810 


16 


45 


9.77694 


9.87317 


10.12683 


9.90377 


15 


45 


9.78691 


9.88890 10 


11110 


9.89801 


15 


46 


.77711 


.87343 


.12657 


.90368 


14 


46 


.78707 


.88916 


.11084 


.89791 


14 


47 


.77728 


.87369 


.12631 


.90358 


13 


47 


.78723 


.88942 


11058 


.89781 


13 


48 


.77744 


.87396 


.12604 


.90349 


12 


48 


.78739 


.88968 


11032 


.89771 


12 


49 


.77761 


.87422 


.12578 


.90339 


11 


49 


.78756 


. 88994 


11006 


.89761 


11 


50 


9.77778 


9.87448 


10.12552 


9.90330 


10 


50 


9.78772 


9.89020 10 


10980 


9.89752 


10 


51 


.77795 


.87475 


.12525 


.90320 


9 


51 


.78788 


.89046 


.10954 


.89742 


9 


52 


.77812 


.87501 


.12499 


.90311 


8 


52 


.78805 


.89073 


.10927 


.89732 


8 


53 


.77829 


.87527 


.12473 


.90301 


7 


53 


.78821 


.89099 


.10901 


. 89722 


7 


54 


.77846 


.87554 


.12446 


.90292 


6 


54 


.78837 


.89125 


.10875 


.89712 


6 


55 


9.77862 


9.87580 


10.12420 


9.90282 


5 


55 


9.78853 


9.89151 10 


10849 


9.89702 


5 


56 


.77879 


.87606 


.12394 


.90273 


4 


56 


.78869 


.89177 


10823 


. 89693 


4 


57 


.77896 


.87633 


.12367 


.90263 


3 


57 


.78886 


. 89203 


.10797 


. 89683 


3 


58 


.77913 


.87659 


.12341 


.90254 


2 


58 


.78902 


.89229 


.10771 


. 89673 


2 


59 


.77930 


.87685 


.12315 


.90244 


1 


59 


.78918 


.89255 


.10745 


.89663 


1 


60 


9.77946 


9.87711 


10.12289 


9.90235 





60 


9.78934 


9.89281 10 


.10719 


9. 89653 






I Cosine. I Cotang. i Tang. | Sine. \ ' \\ 1 Cosine. | Cotang. | Tang. | Sine. | 

*Log secant = colog cosine = 1 — log cosine ; log cosecant = colog sine = 
1 — log sine. 

Ex.— Log sec 36°- 30' = 10.09482. Ex.— Log cosec 36°- 30' = 10.22561- 



LOGARITHMIC SINES, ETC. 



195 



5. — Logarithmic Sines, Tangents, Cotangents, Cosines. — (Cont'd.) 
(Secants, Cosecants.)* 

38° _39f 

' I Sine. I Tang. | Cotang. | Cosine. | || ' | Sine. | Tang. | Cotang. | Cosine. T 






9.78934 


9.89281 


10.10719 


9.89653 


60 





9.79887 


9.90837 


10.09163 


9.89050 


60 


1 


.78950 


.89307 


.10693 


. 89643 


59 


1 


.79903 


.90863 


.09137 


.89040 


59 


2 


.78967 


.89333 


.10667 


.89633 


58 


2 


.79918 


. 90889 


.09111 


.89030 


58 


3 


.78983 


.89359 


.10641 


. 89624 


57 


3 


.79934 


.90914 


.09086 


.89020 


57 


4 


.78999 


.89385 


.10615 


.89614 


56 


4 


.79950 


. 90940 


.09060 


.89009 


56 


5 


9.79015 


9.89411 


10.10589 


9.89604 


55 


5 


9.79965 


9.90966 


10.09034 


9.88999 


55 


6 


.79031 


.89437 


.10563 


.89594 


54 


6 


.79981 


.90992 


.09008 


.88989 


54 


7 


.79047 


.89463 


.10537 


. 89584 


53 


7 


.79996 


.91018 


.08982 


.88978 


53 


8 


.79063 


.89489 


.10511 


.89574 


52 


8 


.80012 


.91043 


.08957 


.88968 


52 


9 


.79079 


.89515 


.10485 


.89564 


51 


9 


. 80027 


.91069 


.08931 


.88958 


51 


10 


9.79095 


9.89541 


10.10459 


9.89554 


50 


10 


9.80043 


9.91095 


10.08905 


9.88948 


50 


11 


.79111 


.89567 


.10433 


.89544 


49 


11 


. 80058 


.91121 


.08879 


.88937 


49 


12 


.79128 


. 89593 


.10407 


.89534 


48 


12 


. 80074 


.91147 


.08853 


.88927 


48 


13 


.79144 


.89619 


.10381 


. 89524 


47 


13 


.80089 


.91172 


.08828 


.88917 


47 


14 


.79160 


.89645 


.10355 


.89514 


46 


14 


.80105 


.91198 


.08802 


. 88906 


46 


15 


9.79176 


9.89671 


10.10329 


9.89504 


45 


15 


9.80120 


9.91224 


10.08776 


9.88896 


45 


16 


.79192 


.89697 


.10303 


.89495 


44 


16 


.80136 


.91250 


.08750 


. 88886 


44 


17 


.79208 


.89723 


.10277 


.89485 


43 


17 


.80151 


.91276 


.08724 


.88875 


43 


18 


.79224 


.89749 


.10251 


.89475 


42 


18 


.80166 


.91301 


.08699 


.88865 


42 


19 


.79240 


.89775 


.10225 


.89465 


41 


19 


.80182 


.91327 


.08673 


.88855 


41 


20 


9.79256 


9.89801 


10.10199 


9.89455 


40 


20 


9.80197 


9.91353 


10.08647 


9.88844 


40 


21 


.79272 


. 89827 


.10173 


. 89445 


39 


21 


.80213 


.91379 


.08621 


.88834 


39 


22 


.79288 


. 89853 


.10147 


.89435 


38 


22 


.80228 


.91404 


.08596 


. 88824 


38 


23 


.79304 


.89879 


.10121 


.89425 


37 


23 


. 80244 


.91430 


. 08570 


. 88813 


37 


24 


.79319 


. 89905 


.10095 


.89415 


36 


24 


.80259 


.91456 


.08544 


. 88803 


36 


25 


9.79335 


9.89931 


10.10069 


9.89405 


35 


25 


9.80274 


9.91482 


10.08518 


9.88793 


35 


26 


.79351 


.89957 


.10043 


.89395 


34 


26 


.80290 


.91507 


.08493 


.88782 


34 


27 


.79367 


.89983 


.10017 


.89385 


33 


27 


.80305 


.91533 


.08467 


. 88772 


33 


28 


.79383 


.90009 


.09991 


.89375 


32 


28 


.80320 


.91559 


.08441 


.88761 


32 


29 


.79399 


.90035 


.09965 


.89364 


31 


29 


.80336 


.91585 


.08415 


.88751 


31 


30 


9.79415 


9.90061 


10.09939 


9.89354 


30 


30 


9.80351 


9.91610 


10.08390 


9.88741 


30 


31 


.79431 


.90086 


.09914 


.89344 


29 


31 


.80366 


.91636 


.08364 


.88730 


29 


32 


.79447 


.90112 


.09888 


.89334 


28 


32 


.80382 


.91662 


.08338 


.88720 


28 


33 


.79463 


.90138 


.09862 


. 89324 


27 


33 


.80397 


.91688 


.08312 


.88709 


27 


34 


.79478 


.90164 


.09836 


.89314 


26 


34 


.80412 


.91713 


.08287 


.88699 


26 


35 


9.79494 


9.90190 


10.09810 


9. 89304 


25 


35 


9.80428 


9.91739 


10.08261 


9.88688 


25 


36 


.79510 


.90216 


.09784 


. 89294 


24 


36 


.80443 


.917«5 


. 08235 


.88678 


24 


37 


.79526 


.90242 


.09758 


.89284 


23 


37 


. 80458 


,91791 


.08209 


.88668 


23 


38 


.79542 


.90268 


.09732 


. 89274 


22 


38 


. 80473 


.91816 


.08184 


. 88657 


22 


39 


.79558 


.90294 


.09706 


.89264 


21 


39 


.80489 


.91842 


.08158 


.88647 


•21 


40 


9.79573 


9.90320 


10.09680 


9.89254 


20 


40 


9.80504 


9.91868 


10.08132 


9.88636 


20 


41 


.79589 


.90346 


.09654 


.89244 


19 


41 


.80519 


.91893 


.08107 


.88626 


19 


42 


.79605 


.90371 


.09629 


.89233 


18 


42 


. 80534 


.91919 


.08081 


.88615 


18 


43 


.79621 


.90397 


.09603 


.89223 


17 


43 


.80550 


.91945 


.08055 


.88605 


17 


44 


.79636 


.90423 


.09577 


.89213 


16 


44 


.80565 


.91971 


.08029 


. 88594 


16 


45 


9.79652 


9.90449 


10.09551 


9.89203 


15 


45 


9.80580 


9.91996 


10.08004 


9.88584 


IS 


46 


.79668 


.90475 


.09525 


.89193 


14 


46 


.80595 


.92022 


.07978 


. 88573 


14 


47 


.79684 


.90501 


.09499 


.89183 


13 


47 


.80610 


.92048 


.07952 


.88563 


13 


48 


.79699 


.90527 


.09473 


.89173 


12 


48 


.80625 


.92073 


.07927 


. 88552 


12 


49 


.79715 


.90553 


.09447 


.89162 


11 


49 


.80641 


.92099 


.07901 


.88542 


11 


50 


9.79731 


9.90578 


10.09422 


9.89152 


10 


50 


9.80656 


9.92125 


10.07875 


9.88531 


10 


51 


.79746 


.90604 


.09396 


.89142 


9 


51 


.80671 


.92150 


.07850 


.88521 


9 


52 


.79762 


.90630 


.09370 


.89132 


8 


52 


.80686 


.92176 


.07824 


.88510 


8 


53 


.79778 


.90656 


.09344 


.89122 


7 


53 


.80701 


.92202 


.07798 


.88499 


7 


54 


.79793 


. 90682 


.09318 


.89112 


6 


54 


.80716 


.92227 


.07773 


.88489 


6 


55 


9.79809 


9.90708 


10.09292 


9.89101 


5 


55 


9.80731 


9.92253 


10.07747 


9.88478 


5 


56 


.79825 


.90734 


.09266 


.89091 


4 


56 


.80746 


.92279 


.07721 


.88468 


4 


57 


.79840 


.90759 


.09241 


.89081 


3 


57 


.80762 


. 92304 


.07696 


.88457 


3 


58 


.79856 


.90785 


.09215 


.89071 


2 


58 


.80777 


.92330 


.07670 


.88447 


2 


59 


.79872 


.90811 


.09189 


.89060 


1 


59 


.80792 


.92356 


.07644 


.88436 


1 


60 


9.79887 


9.90837 


10.09163 


9. 89050 





60 


9.80807 


9.92381 


10.07619 


9.88425 





1 Cosine. | 


Cotang. 


Tang. 


Sine. 


'1 


1 Cosine. 


Cotang. 1 


Tang. 


Sine. 



51° 50° 

*Log secant = colog cosine = 1 — log cosine ; log cosecant == colog sine = 
1 — log sine. 

Ex.—LoK sec 38°- 30' = 10 10646. Ex.— Lor cosec 38°- 30' = 10.20585. 



196 



9.— PLANE TRIGONOMETRY. 



5.— Logarithmic Sines, Tangents, Cotangents, Cosines. — (Cont'd.) 
(Secants, Cosecants.)* 
40° 41° 



' 1 sine. 


Tang. 


Cotang. 1 Cosine. 


II ' 1 Sine. 


Tang. 


Cotang. 1 Cosine. 







9.80807 


9.92381 


10.07619 


9.88425 


60 





9.81694 


9.93916 


10.06084 


9.87778 


60 


1 


. 80822 


.92407 


.07593 


.88415 


59 


1 


.81709 


.93942 


.06058 


.87767 


59 


2 


.80837 


. 92433 


.07567 


. 88404 


58 


2 


.81723 


.93967 


.06033 


. 87756 


58 


3 


. 80852 


. 92458 


.07542 


.88394 


57 


3 


.81738 


.93993 


.06007 


.87745 


57 


4 


.80867 


. 92484 


.07516 


.88383 


56 


4 


.81752 


.94018 


.05982 


. 87734 


56 


5 


9.80882 


9.92510 


10.07490 


9.88372 


55 


5 


9.81767 


9.94044 


10.05956 


9.87723 


55 


6 


. 80897 


.92535 


.07465 


.88362 


54 


6 


.81781 


.94069 


.05931 


.87712 


54 


7 


.80912 


.92561 


.07439 


.88351 


53 


7 


.81796 


.94095 


.05905 


.87701 


53 


8 


. 80927 


. 92587 


.07413 


.88340 


52 


8 


.81810 


.94120 


.05880 


.87690 


52 


9 


.80942 


.92612 


.07388 


.88330 


51 


9 


.81825 


.94146 


.05854 


.87679 


51 


10 


9.80957 


9.92638 


10.07362 


9.88319 


50 


10 


9.81839 


9.94171 


10.05829 


9.87668 


50 


11 


.80972 


.92663 


.07337 


.88308 


49 


11 


.81854 


.94197 


.05803 


.87657 


49 


12 


. 80987 


.92689 


.07311 


.88298 


48 


12 


.81868 


. 94222 


.05778 


.87646 


48 


13 


.81002 


.92715 


.07285 


.88287 


47 


13 


.81882 


.94248 


.05752 


.87635 


47 


14 


.81017 


.92740 


.07260 


.88276 


46 


14 


.81897 


.94273 


. 05727 


.87624 


46 


15 


9.81032 


9.92766 


10.07234 


9.88266 


45 


IS 


9.81911 


9.94299 


10.05701 


9.87613 


45 


16 


.81047 


.92792 


.07208 


.88255 


44 


16 


.81926 


.94324 


.05676 


.87601 


44 


17 


.81061 


.92817 


.07183 


. 88244 


43 


17 


.81940 


.94350 


.05650 


.87590 


43 


18 


.81076 


.92843 


.07157 


.88234 


42 


18 


.81955 


.94375 


.05625 


.87579 


42 


19 


.81091 


.92868 


.07132 


. 88223 


41 


19 


.81969 


. 94401 


.05599 


.87568 


41 


20 


9.81106 


9.92894 


10.07106 


9.88212 


40 


20 


9.81983 


9.94426 


10.05574 


9.87557 


40 


21 


.81121- 


.92920 


.07080 


.88201 


39 


21 


.81998 


. 94452 


.05548 


.87546 


39 


22 


.81136 


.92945 


.07055 


.88191 


38 


22 


.82012 


.94477 


.05523 


.87535 


38 


23 


.81151 


.92971 


.07029 


.88180 


37 


23 


.82026 


. 94503 


.05497 


.87524 


37 


24 


.81166 


.92996 


.07004 


.88169 


36 


24 


.82041 


.94528 


.05472 


.87513 


36 


25 


9.81180 


9.93022 


10.06978 


9.88158 


35 


25 


9.82055 


9.94554 


10.05446 


9.87501 


35 


26 


.81195 


.93048 


.06952 


.88148 


34 


26 


.82069 


.94579 


.05421 


.87490 


34 


27 


.81210 


.93073 


.06927 


.88137 


33 


27 


. 82084 


. 94604 


.05396 


.87479 


33 


28 


.81225 


.93099 


.06901 


.88126 


32 


28 


.82098 


.94630 


.05370 


.87468 


32 


29 


.81240 


.93124 


.06876 


.88115 


31 


29 


.82112 


.94655 


.05345 


.87457 


31 


30 


9.81254 


9.93150 


10.06850 


9.88105 


30 


30 


9.82126 


9.94681 


10.05319 


9.87446 


30 


31 


.81269 


.93175 


.06825 


.88094 


29 


31 


.82141 


.94706 


.05294 


.87434 


29 


32 


.81284 


.93201 


.06799 


. 88083 


28 


32 


.8215'5 


.94732 


.05268 


.87423 


28 


33 


.81299 


.93227 


.06773 


.88072 


27 


33 


.82169 


.94757 


.05243 


.87412 


27 


34 


.81314 


.93252 


.06748 


.88061 


26 


34 


.82184 


.94783 


.05217 


.87401 


26 


35 


9.81328 


9.93278 


10.06722 


9.88051 


25 


35 


9.82198 


9.94808 


10.05192 


9.87390 


25 


36 


.81343 


.93303 


.06697 


.88040 


24 


36 


.82212 


. 94834 


.05166 


.87378 


24 


37 


.81358 


.93329 


.06671 


.88029 


23 


37 


.82226 


.94859 


.05141 


.87367 


23 


38 


.81372 


.93354 


.06646 


.88018 


22 


38 


.82240 


.94884 


.05116 


.87356 


22 


39 


.81387 


.93380 


.06620 


. 88007 


21 


39 


.82255 


.94910 


.05090 


.87345 


21 


40 


9.81402 


9.93406 


10.06594 


9.87996 


20 


40 


9.82269 


9.94935 


10.05065 


9.87334 


20 


41 


.81417 


.93431 


.06569 


.87985 


19 


41 


.82283 


.94961 


.05039 


. 87322 


19 


42 


.81431 


.93457 


.06543 


.87975 


18 


42 


.82297 


.94986 


.05014 


.87311 


18 


43 


.81446 


.93482 


.06518 


.87964 


17 


43 


.82311 


.95012 


.04988 


.87300 


17 


44 


.81461 


. 93508 


.06492 


.87953 


16 


44 


.82326 


. 95037 


.04963 


. 87288 


16 


45 


9.81475 


9.93533 


10.06467 


9.87942 


15 


45 


9.82340 


9.95062 


10.04938 


9.87277 


15 


46 


.81490 


.93559 


.06441 


.87931 


14 


46 


.82354 


.95088 


.04912 


.87266 


14 


47 


.81505 


. 93584 


.06416 


.87920 


13 


47 


.82368 


.95113 


. 04887 


.87255 


13 


48 


.81519 


.93610 


.06390 


.87909 


12 


48 


.82382 


.95139 


.04861 


.87243 


12 


49 


.81534 


.93636 


.06364 


.87898 


11 


49 


.82396 


.95164 


.04836 


.87232 


U 


50 


9.81549 


9.93661 


10.06339 


9.87887 


10 


50 


9.82410 


9.95190 


10.04810 


9.87221 


10 


51 


.81563 


. 93687 


.06313 


.87877 


9 


51 


. 82424 


.95215 


.04785 


.87209 


9 


52 


.81578 


.93712 


.06288 


.87866 


8 


52 


.82439 


. 95240 


.04760 


.87198 


8 


53 


.81592 


.93738 


. 06262 


.87855 


7 


53 


.82453 


. 95266 


.04734 


.87187 


7 


54 


.81607 


.93763 


.06237 


.87844 


6 


54 


.82467 


.95291 


.04709 


.87175 


6 


55 


9.81622 


9.93789 


10.06211 


9.87833 


5 


55 


9.82481 


9.95317 


10.04683 


9.87164 


5 


56 


.81636 


.93814 


.06186 


.87822 


4 


56 


.82495 


. 95342 


.04658 


.87153 


4 


57 


.81651 


.93840 


.06160 


.87811 


3 


57 


.82509 


.95368 


.04632 


.87141 


3 


58 


.81665 


.93865 


.06135 


.87800 


2 


58 


.82523 


.95393 


.04607 


.87130 


2 


59 


.81680 


.93891 


.06109 


.87789 


1 


59 


.82537 


.95418 


.04582 


.87119 


1 


60 


9.81694 


9.93916 


10.06084 


9.87778 





60 


9.82551 


9.95444 


10.04556 


9.87107 





1 Cosine. 


ICotang. 


1 Tang. 


1 Sine. 


'II 1 Cosine. 


1 Cotang. 


1 Tang. 


Sine. 


' 



49° 48° 

*Log secant = colog cosine = 1 — log cosine ; log cosecant = colog sine = 
1 — log sine. 

^Ex.—Log sec 40°- 30^= 10.11895. Ex.— Log cosec 40°- 30'= 10.18746. 



LOGARITHMIC SINES, ETC, 



197 



5.— Logarithmic Sines, Tangents, Cotangents, Cosines. — (Cont'd.) 
(Secants, Cosecants.)* 
42° 43° 



' 1 Sine. 


1 Tang. 


1 Cotang. 1 Cosine. | 1 1 ' | Sine. | Tang. 


1 Cotang. 1 Cosine. 1 


c 


9.82551 


9.95444 


10.04556 


9.87107 


60 





9.83378 


9.96966 


10.03034 


9.86413 


60 




.82565 


.95469 


.04531 


. 87096 


59 


1 


.83392 


.96991 


.03009 


.86401 


59 


2 


.82579 


.95495 


.04505 


.87085 


58 


2 


.83405 


.97016 


.02984 


. 86389 


58 


3 


.82593 


.95520 


.04480 


. 87073 


57 


3 


.83419 


. 97042 


.02958 


.86377 


57 


A 


.82607 


. 95545 


.04455 


. 87062 


56 


4 


. 83432 


.97067 


.02933 


.86366 


56 


5 


9.82621 


9.95571 


10.04429 


9.87050 


55 


5 


9.83446 


9.97092 


10.02908 


9.86354 


55 


6 


.82635 


.95596 


.04404 


.87039 


54 


6 


.83459 


.97118 


.02882 


.86342 


54 


7 


.82649 


.95622 


.04378 


.87028 


53 


7 


.83473 


.97143 


.02857 


.86330 


53 


8 


.82663 


.95647 


.04353 


.87016 


52 


8 


. 83486 


.97168 


.02832 


.86318 


52 


9 


.82677 


. 95672 


.04328 


.87005 


51 


9 


. 83500 


.97193 


. 02807 


.86306 


51 


10 


9.82691 


9.95698 


10.04302 


9.86993 


50 


10 


9.83513 


9.97219 


10.02781 


9.86295 


50 


11 


.82705 


.95723 


.04277 


. 86982 


49 


11 


.83527 


.97244 


.02756 


. 86283 


49 


12 


.82719 


.95748 


.04252 


.86970 


48 


12 


.83540 


.97269 


.02731 


.86271 


48 


13 


. 82733 


.95774 


.04226 


.86959 


47 


13 


. 83554 


.97295 


.02705 


.86259 


47 


14 


.82747 


.95799 


.04201 


.86947 


46 


14 


. 83567 


.97320 


.02680 


.86247 


46 


15 


9.82761 


9.95825 


10.04175 


9.86936 


45 


15 


9.83581 


9.97345 


10.02655 


9.86235 


45 


16 


.82775 


.95850 


.04150 


. 86924 


44 


16 


.83594 


.97371 


.02629 


. 86223 


44 


17 


.82788 


.95875 


.04125 


.86913 


43 


17 


.83608 


.97396 


.02604 


.86211 


43 


18 


.82802 


.95901 


.04099 


.86902 


42 


18 


.83621 


.97421 


.02579 


. 86200 


42 


19 


.82816 


.95926 


.04074 


.86890 


41 


19 


.83634 


.97447 


.02553 


.86188 


41 


20 


9.82830 


9.95952 


10.04048 


9.86879 


40 


20 


9.83648 


9.97472 


10.02528 


9.86176 


40 


21 


. 82844 


.95977 


.04023 


.86867 


39 


21 


.83661 


.97497 


.02503 


,86164 


39 


22 


.82858 


.96002 


.03998 


.86855 


38 


22 


.83674 


.97523 


.02477 


.86152 


38 


23 


.82872 


.96028 


.03972 


. 86844 


37 


23 


.83688 


.97548 


.02452 


.86140 


37 


24 


.82885 


.96053 


.03947 


.86832 


36 


24 


.83701 


.97573 


.02427 


,86128 


36 


25 


9.82899 


9.96078 


10.03922 


9.86821 


35 


25 


9.83715 


9.97598 


10.02402 


9.86116 


35 


26 


.82913 


.96104 


.03896 


.86809 


34 


26 


.83728 


.97624 


.02376 


,86104 


34 


27 


.82927 


.96129 


.03871 


.86798 


33 


27 


.83741 


.97649 


.02351 


, 86092 


33 


28 


.82941 


.96155 


.03845 


.86786 


32 


28 


.83755 


.97674 


.02326 


, 86080 


32 


29 


.82955 


.96180 


.03820 


.86775 


31 


29 


.83768 


.97700 


. 02300 


.86068 


31 


30 


9.82968 


9.96205 


10.03795 


9.86763 


30 


30 


9.83781 


9.97725 


10.02275 


9.86056 


30 


31 


.82982 


.96231 


.03769 


.86752 


29 


31 


.83795 


.97750 


.02250 


. 86044 


29 


32 


.82996 


.96256 


.03744 


.86740 


28 


32 


.83808 


.97776 


.02224 


. 86032 


28 


33 


.83010 


.96281 


.03719 


.86728 


27 


33 


.83821 


.97801 


.02199 


. 86020 


27 


34 


.83023 


.96307 


.03693 


.86717 


26 


34 


.83834 


.97826 


.02174 


. 86008 


26 


35 


9.83037 


9.96332 


10.03668 


9.86705 


25 


35 


9.83848 


9.97851 


10.02149 


9.85996 


25 


36 


.83051 


.96357 


.03643 


. 86694 


24 


36 


,83861 


.97877 


.02123 


. 85984 


24 


37 


.83065 


.96383 


.03617 


.86682 


23 


37 


.83874 


.97902 


.02098 


.85972 


23 


38 


.83078 


.96408 


.03592 


.86670 


22 


38 


.83887 


.97927 


.02073 


. 85960 


22 


39 


.83092 


.96433 


.03567 


.86659 


21 


39 


.83901 


.97953 


.02047 


.85948 


21 


40 


9.83106 


9.96459 
.96484 


10.03541 


9.86647 


20 


40 


9.83914 


9.97978 


10.02022 


9.85936 


20 


41 


.83120 


.03516 


.86635 


19 


41 


.83927 


. 98003 


.01997 


. 85924 


19 


42 


.83133 


.96510 


.03490 


.86624 


18 


42 


.83940 


. 98029 


.01971 


.85912 


18 


43 


.83147 


.96535 


.03465 


.86612 


17 


43 


.83954 


. 98054 


.01946 


. 85900 


17 


44 


.83161 


.96560 


.03440 


.86600 


16 


44 


.83967 


.98079 


.01921 


. 85888 


16 


45 


9.83174 


9.96586 


10.03414 


9.86589 


15 


45 


9.83980 


9.98104 


10.01896 


9.85876 


15 


46 


.83188 


.96611 


.03389 


.86577 


14 


46 


.83993 


.98130 


.01870 


. 85864 


14 


47 


. 83202 


.96636 


.03364 


.86565 


13 


47 


. 84006 


.98155 


.01845 


.85851 


13 


48 


.83215 


.96662 


.03338 


.86554 


12 


48 


.84020 


.98180 


.01820 


. 85839 


12 


49 


.83229 


.96687 


.03313 


.86542 


11 


49 


. 84033 


.98206 


.01794 


, 85827 


11 


50 


9.83242 


9.96712 


10.03288 


9.86530 


10 


50 


9.84046 


9.98231 


10.01769 


9.85815 


10 


51 


.83256 


.96738 


.03262 


.86518 


9 


51 


.84059 


.98256 


.01744 


. 85803 


9 


52 


.83270 


.96763 


.03237 


. 86507 


8 


52 


. 84072 


.98281 


.01719 


.85791 


8 


53 


. 83283 


.96788 


.03212 


.86495 


7 


53 


.84085 


.98307 


.01693 


.85779 


7 


54 


.83297 


.96814 


.03186 


. 86483 


6 


54 


.84098 


.98332 


.01668 


.85766 


6 


55 


9.83310 


9.96839 


10.03161 


9.86472 


5 


55 


9.84112 


9.98357 


10.01643 


9.85754 


5 


56 


.83324 


.96864 


.03136 


.86460 


4 


56 


.84125 


.98383 


.01617 


. 85742 


4 


57 


.83338 


.96890 


.03110 


.86448 


3 


57 


.84138 


. 98408 


.01592 


.85730 


3 


58 


.83351 


.96915 


.03085 


.86436 


2 


58 


.84151 


. 98433 


.01567 


.85718 


2 


59 


.83365 


.96940 


.03060 


.86425 


1 


59 


.84164 


.98458 


.01542 


.85706 


1 


60 


9.83378 


9.96966 


10.03034 


9. 86413 





60 


9.84177 


9.98484 


10.01516 


9.85693 





1 Cosine. 1 


Cotang. 


Tang. 1 


Sine. 1 ' II 1 Cosine. 1 Cotang. 


j Tang. 


Sine. ' 



47° "46° 

*Log secant = colog cosine = 1 — log cosine ; log cosecant = colog sine = 
1 — log sine. 

-E^.— Log sec 42°- 30' = 10.13237. Srv.— Log cosec 42°- 30' =» 10.17032. 



198 



^.-— PLANE TRIGONOMETRY. 



5. — Logarithmic Sines, Tangents, Cotangents, Cosines. — (Concl'd.) 

(Secants, Cosecants.) 
440 440 



' 


Sine. 


Tang. 


Cotang. 1 Cosine. 


1 


1 ' 1 


Sine. 


Tang. 1 Cotang. | Cosine. | 







9.84177 


9.98484 


10.01516 


9.85693 


60 


30 


9.84566 


9.99242 


10.00758 


9.85324 


30 


1 


.84190 


. 98509 


.01491 


. 85681 


59 


31 


.84579 


. 99267 


.00733 


.85312 


29 


2 


.84203 


.98534 


.01466 


.85669 


58 


32 


. 84592 


. 99293 


.00707 


. 85299 


28 


3 


.84216 


.98560 


.01440 


. 85657 


57 


33 


.84605 


.99318 


. 00682 


. 85287 


27 


4 


.84229 


. 98585 


.01415 


.85645 


56 


34 


.84618 


. 99343 


. 00657 


. 85274 


26 


5 


9. 84242 


9.98610 


10.01390 


9.85632 


55 


35 


9.84630 


9.99368 


10.00632 


9.85262 


25 


6 


.84255 


.98635 


.01365 


.85620 


54 


36 


.84643 


.99394 


.00606 


. 85250 


24 


7 


. 84269 


.98661 


.01339 


. 85608 


53 


37 


.84656 


.99419 


.00581 


. 85237 


23 


8 


. 84282 


.98686 


.01314 


. 85596 


52 


38 


.84669 


.99444 


.00556 


. 85225 


22 


9 


.84295 


.98711 


.01289 


. 85583 


51 


39 


.84682 


.99469 


.00531 


.85212 


21 


10 


9.84308 


9.98737 


10.01263 


9.85571 


50 


40 


9.84694 


9.99495 


10.00505 


9.85200 


20 


11 


.84321 


.98762 


.01238 


.85559 


49 


41 


. 84707 


.99520 


.00480 


.85187 


19 


12 


.84334 


.98787 


.01213 


. 85547 


48 


42 


.84720 


.99545 


.00455 


.85175 


18 


13 


.84347 


.98812 


.01188 


.85534 


47 


43 


.84733 


.99570 


.00430 


.85162 


17 


14 


. 84360 


.98838 


.01162 


. 85522 


46 


44 


. 84745 


.99596 


.00404 


.85150 


16 


15 


9.84373 


9. 98863 


10.01137 


9.85510 


45 


45 


9.84758 


9.99621 


10.00379 


9.85137 


IS 


16 


.84385 


.98888 


.01112 


.85497 


44 


46 


.84771 


.99646 


.00354 


.85125 


14 


17 


.84398 


.98913 


.01087 


.85485 


43 


47 


.84784 


.99672 


.00328 


.85112 


13 


18 


.84411 


.98939 


.01061 


.85473 


42 


48 


.84796 


.99697 


.00303 


.85100 


12 


19 


.84424 


.98964 


.01036 


.85460 


41 


49 


. 84809 


.99722 


.00278 


.85087 


11 


20 


9.84437 


9.98989 


10.01011 


9.85448 


40 


50 


9.84822 


9.99747 


10.00253 


9.85074 


10 


21 


.84450 


.99015 


.00985 


.85436 


39 


51 


.84835 


.99773 


.00227 


. 85062 


9 


22 


. 84463 


.99040 


.00960 


.85423 


38 


52 


.84847 


.99798 


.00202 


.85049 


8 


23 


.84476 


.99065 


.00935 


.85411 


37 


53 


.84860 


.99823 


.00177 


. 85037 


7 


24 


.84489 


.99090 


.00910 


.85399 


36 


54 


. 84873 


.99848 


.00152 


.85024 


6 


25 


9.84502 


9.99116 


10. 00884 


9.85386 


35 


55 


9.84885 


9.99874 


10.00126 


9.85012 


5 


26 


.84515 


.99141 


.00859 


.85374 


34 


56 


.84898 


.99899 


.00101 


.84999 


4 


27 


.84528 


.99166 


.00834 


.85361 


33 


57 


.84911 


.99924 


.00076 


.84986 


3 


28 


.84540 


.99191 


.00809 


.85349 


32 


58 


.84923 


.99949 


.00051 


.84974 


2 


29 


. 84553 


.99217 


.00783 


. 85337 


31 


59 


.84936 


.99975 


.00025 


.84961 


1 


30 


9.84566 


9.99242 


10.00758 


9.85324 


30 


60 


9.84949 


10.00000 


10.00000 


9.84949 






I Cosine. | Cotang. | Tang. | Sine. 



Cosine. | Cotang. | Tang. | Sine. | ' 



45= 



45= 



5a. — Table for Finding the Logarithmic Sines and Tangents of 

Small Angles. 
[Values of 5 and T in Formulas Below.*] 



A. 


A(sec.). 


S. 


A. 


A(sec.). 


T. 


A. 


A(sec.). 


T. 


0°00'00" 


0000" 


4.68557 


0°00'00" 


000" 


4.68557 


I°2y40" 


5140" 


4.68566 


0°40'10" 


2410" 


57 


0°03'00" 


180" 


57 


1°25'50" 


5150" 


67 


0°40'20" 


2420" 


56 


0°03'20" 


200" 


58 


1°30'20" 


5420" 


67 


0°5r00" 


3420" 


56 


0°28'40" 


1720" 


58 


1°30'30" 


5430" 


68 


o°5rio" 


3430" 


55 


0°28'50" 


1730" 


59 


1°34'40" 


5680" 


68 


1°09'50" 


4190" 


55 


0°40'20" 


2420" 


59 


1°34'50" 


5690" 


69 


1°10'00" 


4200" 


54 


0°40'30" 


2430" 


60 


1^39' 00" 


5940" 


69 


1°20'40" 


4840" 


54 


0°49'30" 


2970" 


60 


1°39'10" 


5950" 


70 


1°20'50" 


4850" 


53 


0°49'40" 


2980" 


61 


1°43'00" 


6180" 


70 


1°30'10" 


5410" 


53 


0°57'10" 


S430" 


61 


1°43'10" 


6190" 


71 


1°30'20" 


5420" 


52 


0°5r20" 


3440" 


62 


1°46'50" 


6410" 


71 


1°38'50" 


5930" 


52 


1°03'50" 


3830" 


62 


1°47'00" 


6420" 


72 


1°39'00" 


5940" 


51 


1°04'00" 


3840" 


63 


1°50'40" 


6640" 


72 


1°46'50" 


6410" 


51 


1°10'00" 


4200" 


63 


1°50'50" 


6650" 


73 


1°47'00" 


6420" 


50 


1°10'10" 


4210" 


64 


1°54'10" 


6850" 


73 


1°54'10" 


6850" 


50 


1°15^30" 


4530" 


64 


1°54'20" 


6860" 


74 


1°54'20" 


6860" 


49 


1°15'40" 


4540" 


65 


l°5r40" 


7060" 


74 


2°oroo" 


7260" 


49 


1°20'50" 


4850" 


65 


1°57'50" 


7070" 


75 


2°orio" 


7270" 


48 


i°2roo" 


4860" 


66 


2°orio" 


7270" 


75 



* Log sin A=\oQ A (seconds) +S. Log tan A=\og A (seconds) + r. 



LOGARITHMIC SINES. HYPERBOLIC FUNCTIONS. 198a 

Hyperbolic Functions.— As the trigonometric or circular functions are re- 
lated to the circle, so are the hyperbolic functions related to the equilateral 
or rectangular hyperbola. The equation of the circle is x^ ~\- y^ = a^ = r^ = 
(radius)2. The equation of the equilateral hyperbola is «2 — y2 = ^2. The 
radius r (= OP, Fig. 5) of the equilateral hyperbola increases with * and y, a re- 
maining constant. Length of arc HP = s. 




Fig. 5. 



Compared with Trigonometric Functions, Fig. 1, page 136, let OH = hypoth- 
enuse (h) = a, OB = base (b) = x, and BP = perpendicular (p) = y; then, by 
analogy: hyperbolic sin of « = sinh u (= p/h) = y/a, hyper cos of u = cosh u 
(= b/h) = x'/a, hyper tan of m = tanh u = sinh u -r- cosh u (= p/b) = y/x, hyper cot 
of M = coth M = 1 -f- tanh u (= b/p) = x/y, hyper sec of w = sech m = 1 -r- cosh u 
(= h/b) = a/x, hyper esc of m = csch m = 1 -t- sinh u (= h/p) = a/y. 

If we let a = unity, Fig. 5, then u = the area of the sector OPHPiO = s/r = 
loge {x -}- y) = loggg"; sinh u = BP, cosh u = OB, tanh u = HT. 

Exponential Values (Naperian Base e) of Hyperbolic Functions: sinh u = 
\ (e^—er^), cosh u=^ (e» + r^), tanh u = sinh u -r- cosh u = (e^ — er^) -r- (e** + er"^), 
coth w=cosh M4-sinh m= (e^ -\-e-^)-^ (e« — e~^), sech m=1 -^ cosh m = 2 ^ («« + «~**), 
csch M = 1 -i- sinh u = 2 -^ {e^ — e~^). 

Relations Between Hyperbolic Functions: cosh^ u — sinh^w = 1, tanh^ u + 
sech2 M = 1, coth2 u — csch^ u = 1, sinh 2 u = 2 sinh m cosh ;/, cosh 2 u = cosh^w 
-f- sinh2 u, sinh (m dz v) — sinh u cosh v ± cosh m sin r, cosh (uziiv) = cosh m cosh v ± 
sinh M sin v, tanh (m ± ?') = (tanh u ± tanh z;) -t- (1 ± tanh u tanh y). 

Relations Between Hyperbolic and Circular Functions: sin u = —i sinh iu = 
J i (e*^ — e-*"); cos u = cosh fw = ^ (ew + «~»«); tan u= — f tanh iu = — i(g^" — g -t") 
-^ (^«* + ^»); cot u = 7 coth m = i{ei» + r^«) -f- (giw— g-tt*). [i = V— i.] 

Inverse Hyperbolic Functions are written similarly to inverse trigonometric 
functions. If m = sinh v, then sinh~i u = v; and similarly with the cosine, 

tangent, etc. Values of inverse hyperbolic functions: sinh-i u = log^ (« -{-^u^ + 1); 

cosh-i « = loge (M + ^»^ - 1); tanh-i u = h loge }^ ; coth-i u = loge ^^^ • 

Tables of Hyperbolic Functions and Logarithms of same may be found on 
the five following pages. 



198b 



9.— PLANE TRIGONOMETRY, 



6.— Natural Hyperbolic Sines u = Sinh ii = | (c~ — e^), 
(For common logarithms of these values, see opposite page.) 



u 


.00 


.01 


.02 


.03 


.04 


.05 


.06 


.07 


.08 


.09 


Dif 


0.0 


0.0000 


0.0100 


0.0200 


0.0300 


0.0400 


0.0500 


0.0600 


0.0701 


0.0801 


0.0901 


100 


0.1 


0.1002 


0.1102 


0.1203 


0.1304 


0.1405 


0.1506 


0.1607 


0.1708 


0.1810 


0.1911 


101 


0.2 


0.2013 


0.2115 


0.2218 


0.2320 


0.2423 


0.2526 


0.2629 


0.2733 


0.2837 


0.2941 


103 


0.3 


0.3045 


0.3150 


0.3255 


0.3360 


0.3466 


0.3572 


0.3678 


0.3785 


0.3892 


0.4000 


106 


0.4 


0.4108 


0.4216 


0.4325 


0.4434 


0.4543 


0.4653 


0.4764 


0.4875 


0.4986 


0.5098 


110 


0.5 


0.5211 


0.5324 


0.5438 


0.5552 


0.5666 


0.5782 


0.5897 


0.6014 


0.6131 


0.6248 


116 


0.6 


0.6367 


0.6485 


0.6605 


0.6725 


0.6846 


0.6967 


0.7090 


0.7213 


0.7336 


0.7461 


121 


0.7 


0.7586 


0.7712 


0.7838 


0.7966 


0.8094 


0.8233 


0.8353 


0.8484 


0.8615 


0.8748 


129 


0.8 


0.8881 


0.9015 


0.9150 


0.9286 


0.9423 


0.9561 


0.9700 


0.9840 


0.9981 


1.0122 


138 


0.9 


1.0265 


1.0409 


1.0554 


1.0700 


1.0847 


1.0995 


1.1144 


1.1294 


1.1446 


1.1598 


148 


1.0 


1.1752 


1.1907 


1.2063 


1.2220 


1.2379 


1.2539 


1.2700 


1.2862 


1.3025 


1.3190 


160 


1.1 


1.3356 


1.3524 


1.3693 


1.3863 


1.4035 


1.4208 


1.4382 


1.4558 


1.4735 


1.4914 


173 


1.2 


1.5095 


1.6276 


1.6460 


1.5645 


1.6831 


1.6019 


1.6209 


1.6400 


1.6593 


1.6788 


188 


1.3 


1.6984 


1.7182 


1.7381 


1.7583 


1.7786 


1.7991 


1.8198 


1.8406 


1.8617 


1.8829 


205 


1.4 


1.9043 


1.9259 


1.9477 


1.9697 


1.9919 


2.0143 


2.0369 


2.0597 


2.0827 


2.1059 


224 


1.5 


2.1293 


2.1529 


2.1768 


2.2008 


2.2251 


2.2496 


2.2743 


2.2993 


2.3245 


2.3499 


245 


1.6 


2.3756 


2.4015 


2.4276 


2.4540 


2.4806 


2.5075 


2.5346 


2.5620 


2.5896 


2.6175 


269 


1.7 


2.6456 


2.6740 


2.7027 


2.7317 


2.7609 


2.7904 


2.8202 


2.8503 


2.8806 


2.9112 


295 


1.8 


2.9422 


2.9734 


3.0049 


3.0367 


3.0689 


3.1013 


3.1340 


3.1671 


3.2005 


3.2341 


324 


1.9 


3.2682 


3.3025 


3.3372 


3.3722 


3.4075 


3.4432 


3.4792 


3.5156 


3.5523 


3.5894 


357 


2.0 


3.6269 


3.6647 


3.7028 


3.7414 


3.7803 


3.8196 


3.8593 


3.8993 


3.9398 


3.9806 


393 


2.1 


4.0219 


4.0635 


4.1056 


4.1480 


4.1929 


4.2342 


4.2779 


4.3221 


4.3666 


4.4117 


433 


2.2 


4.4571 


4.5030 


4.5494 


4.5962 


4.6434 


4.6912 


4.7394 


4.7880 


4.8372 


4.8868 


478 


2.3 


4.9370 


4.9876 


5.0387 


5.0903 


5.1425 


5.1951 


5.2483 


5.3020 


5.3562 


5.4109 


526 


2.4 


6.4662 


5.5221 


5.5785 


5.6354 


5.6929 


5.7510 


5.8097 


5.8689 


5.9288 


5.9892 


581 


2.5 


6.0502 


6.1118 


6.1741 


6.2369 


6.3004 


6.3645 


6.4293 


6.4946 


6.5607 


6.6274 


641 


2.6 


6.6947 


6.7628 


6.8315 


6.9009 


6.9709 


7.0417 


7.1132 


7.1854 


7.2583 


7.3319 


708 


2.7 


7.4063 


7.4814 


7.5572 


7.6338 


7.7112 


7.7894 


7.8683 


7.9480 


8.0285 


8.1098 


782 


2.8 


8.1919 


8.2749 


8.3586 


8.4432 


8.5287 


8.6150 


8.7021 


8.7902 


8.8791 


8.9689 


863 


2.9 


9.0596 


9.1512 


9.2437 


9.3371 


9.4315 


9.5268 


9.6231 


9.7203 


9.8185 


9.9177 


953 


3.0 


10.018 


10.119 


10.221 


10.324 


10.429 


10.534 


10.640 


10.748 


10.856 


10.966 


105 


3.1 


11.076 


11.188 


11.301 


11.415 


11.530 


11.647 


11.764 


11.883 


12.003 


12.124 


117 


3.2 


12.246 


12.369 


12.494 


12.620 


12.747 


12.876 


13.006 


13.137 


13.269 


13.403 


129 


3.3 


13.538 


13.674 


13.812 


13.951 


14.092 


14.234 


14.377 


14.522 


14.668 


14.816 


142 


3.4 


14.965 


15.116 


15.268 


15.422 


15.577 


15.734 


15.893 


16.053 


16.214 


16.378 


157 


3.5 


16.543 


16.709 


16.877 


17.047 


17.219 


17.392 


17.567 


17.744 


17.923 


18.103 


173 


3.6 


18.285 


1^.470 


18.655 


18.843 


19.033 


19.224 


19.418 


19.613 


19.811 


20.010 


191 


3.7 


20.211 


20.415 


20.620 


20.828 


21.037 


21.249 


21.463 


21.679 


21.897 


22.117 


212 


3.8 


22.339 


22.564 


22.791 


23.020 


23.252 


23.486 


23.722 


23.961 


24.202 


24.445 


234 


3.9 


24.691 


24.939 


25.190 


25.444 


25.700 


25.958 


26.219 


26.483 


26.749 


27.018 


258 


4.0 


27.290 


27.564 


27.842 


28.122 


28.404 


28.690 


28.979 


29.270 


29.564 


29.862 


286 


4.1 


30.162 


30.465 


30.772 


31.081 


31.393 


31.709 


32.028 


32.350 


32.675 


33.004 


316 


4.2 


33.336 


33.671 


34.009 


34.351 


34.697 


35.046 


35.398 


35.754 


36.113 


36.476 


349 


4.3 


36.843 


37.214 


37.588 


37.966 


38.347 


38.733 


39.122 


39.515 


39.913 


40.314 


386 


4.4 


40.719 


41.129 


41.542 


41.960 


42.382 


42.808 


43.238 


43.673 


44.112 


44.555 


420 


4.5 


45.003 


45.455 


45.912 


46.374 


46.840 


47.311 


47.787 


48.267 


48.752 


49.242 


471 


4.6 


49.737 


50.237 


50.742 


51.252 


51.767 


52.288 


52.813 


53.344 


53.880 


54.422 


521 


4.7 


54.969 


55.522 


56.080 


56.643 


57.213 


57.788 


58.369 


58.955 


59.548 


60.147 


575 


4.8 


60.751 


61.362 


61.979 


62.601 


63.231 


63.866 


64.508 


65.157 


65.812 


66.473 


635 


4.9 


67.141 


67.816 


68.498 


69.186 


69.882 


70.584 


71.293 


72.010 


72.734 


73.465 


702 


5.0 


74.203 


74.949 


75.702 


76.463 


77.232 


78.008 


78.792 


79.684 


80.384 


81.192 


770 



Ex.— For u = 2.73, sinh u = 7.6338. 



HYPERBOLIC SINES, WITH LOGARITHMS. 



198c 



7.— Common Logarithms of Hyperbolic Sines u. 

(For natural functions, see opposite page.) 



u 


.00 


.01 


.02 


.03 


.04 


.05 


.06 


.07 


.08 


.09 


Dif. 


0.0 


— oo 


8.00001 


30106 


47719 


60218 


69915 


77841 


84545 


90356 


96483 


9697 


0.1 


1^.00072 


04227 


08022 


11517 


14765 


17772 


20597 


23264 


26762 


28136 


3017 


0.2 


30392 


32541 


34592 


36555 


38437 


40245 


41986 


43663 


45282 


46847 


1808 


0.3 


48362 


49830 


51254 


52637 


53981 


55290 


56564 


57807 


69019 


60202 


1309 


0.4 


61358 


62488 


63594 


64677 


65738 


66777 


67797 


68797 


67779 


70744 


1039 


0.5 


71692 


72624 


73540 


74442 


75330 


76204 


77065 


77914 


78751 


79576 


874 


0.6 


80390 


81194 


81987 


82770 


83543 


84308 


86063 


85809 


86548 


87278 


765 


0.7 


88000 


88715 


89423 


90123 


90817 


91504 


92186 


92869 


93527 


94190 


687 


0.8 


94846 


95498 


96144 


96784 


97420 


98051 


98677 


99299 


99916 


00528 


631 


0.9 


0.01137 


01741 


02341 


02937 


03530 


04119 


04704 


06286 


05864 


06439 


589 


1.0 


07011 


07580 


08146 


08708 


09268 


09826 


10379 


10930 


11479 


12025 


657 


1.1 


12569 


13111 


13649 


14186 


14720 


16253 


16783 


16311 


16836 


17360 


533 


1.2 


17882 


18402 


18920 


19437 


19951 


20464 


20976 


21486 


21993 


22499 


513 


1.3 


23004 


23507 


24009 


24509 


25008 


25505 


26002 


26496 


26990 


27482 


497 


1.4 


27974 


28464 


28952 


29440 


29926 


30412 


30896 


31379 


31862 


32343 


486 


1.6 


32823 


33303 


33781 


34258 


34735 


35211 


36686 


36160 


36633 


37105 


476 


1.6 


37577 


38048 


38518 


38987 


39456 


39923 


40391 


40867 


41323 


41788 


467 


1.7 


42253 


42717 


43180 


43643 


44105 


44567 


45028 


45488 


45948 


46408 


462 


1.8 


46867 


47325 


47783 


. 48241 


48698 


49164 


49610 


50066 


50621 


60976 


466 


1.9 


51430 


61884 


52338 


52791 


53244 


53696 


64148 


54600 


66051 


65502 


452 


2.0 


55953 


56403 


56853 


57303 


67753 


58202 


58660 


59099 


59547 


59996 


449 


2.1 


60443 


60890 


61337 


61784 


62231 


62677 


63123 


63569 


64016 


64460 


446 


2.2 


64905 


65350 


65795 


66240 


66684 


67128 


67672 


68016 


68459 


68903 


444 


2.3 


69346 


69789 


70232 


70675 


71117 


71569 


72002 


72444 


72885 


73327 


442 


2.4 


73769 


74210 


74652 


75093 


75534 


75975 


76416 


76856 


77296 


77737 


441 


2.5 


78177 


78617 


79057 


79497 


79937 


80377 


80816 


81266 


81695 


82134 


440 


2.6 


82573 


83012 


83451 


83890 


84329 


84768 


85206 


86645 


86083 


86622 


439 


2.7 


86960 


87398 


87836 


88274 


88712 


89160 


89688 


90026 


90463 


90901 


438 


2.8 


91339 


91776 


92213 


92651 


93088 


93626 


93963 


94400 


94837 


96274 


437 


2.9 


95711 


96148 


96584 


97021 


97458 


97896 


98331 


98768 


99205 


99641 


437 


3.0 


1.00078 


00514 


00950 


01387 


01823 


02259 


02696 


03132 


03568 


04004 


436 


3.1 


04440 


04876 


05312 


05748 


06184 


06620 


07066 


07492 


07927 


08363 


436 


3.2 


08799 


09235 


09670 


10106 


10542 


10977 


11413 


11849 


12284 


12720 


435 


3.3 


13155 


13591 


14026 


14461 


14897 


15332 


16768 


16203 


16638 


17073 


43^ 


3.4 


17509 


17944 


18379 


18814 


19250 


19685 


20120 


20555 


20990 


21425 


435 


8.5 


21860 


22296 


22731 


23166 


23601 


24036 


24471 


24906 


25341 


25776 


435 


3.6 


26211 


26646 


27080 


27515 


27950 


28386 


28820 


29255 


29690 


30125 


435 


3.7 


30559 


30994 


31429 


31864 


32299 


32733 


33168 


33603 


34038 


34472 


434 


3.8 


34907 


35342 


35777 


36211 


36646 


37081 


37515 


37950 


38385 


38819 


435 


3.9 


39254 


39689 


40123 


40558 


40993 


41427 


41862 


42296 


42731 


43166 


434 


4.0 


43600 


44035 


44469 


44904 


45339 


46773 


46208 


46642 


47077 


47511 


434 


4.1 


47948 


48380 


48815 


49249 


49684 


50118 


50663 


60987 


51422 


51856 


434 


4.2 


52291 


52725 


53160 


53594 


64029 


54463 


54898 


65332 


65767 


56201 


434 


4.3 


66636 


57070 


57505 


57939 


68373 


58808 


59242 


69677 


60111 


60646 


435 


4.4 


60980 


61414 


61849 


62283 


62718 


63152 


63687 


64021 


64466 


64890 


434 


4.5 


65324 


65759 


66193 


66627 


67062 


67496 


67931 


68365 


68799 


69234 


434 


4.6 


69668 


70102 


70537 


70971 


71406 


71840 


72274 


72709 


73143 


73577 


434 


4.7 


74012 


74446 


74881 


75315 


75749 


76184 


76618 


77052 


77487 


77921 


435 


4.8 


78355 


78790 


79224 


79658 


80093 


80527 


80962 


81396 


81830 


82265 


434 


4.9 


82699 


83133 


83568 


84002 


84436 


84871 


85305 


85739 


86174 


86608 


439 


5.0 


87042 


87477 


87911 


88345 


88780 


89214 


89648 


90083 


90517 


90951 


434 



Ex.— For u = 2.73, com log of sinh u = 0.88274. 



198d 9.— PLANE TRIGONOMETRY, 

8.— -Natural Hyperbolic Cosines u = Cosh ii = H«" + ^**). 
(For common logarithms of these values, see opposite page.) 



I 



u 


.00 


.01 


.02 


.03 


.04 


.05 


.06 


.07 


.08 


.09 


Dif. 


0.0 


1.0000 


1.0001 


1.0002 


1.0005 


1.0008 


1.0013 


1.0018 


1.0025 


1.0032 


1.0041 


5 


0.1 


1.0050 


1.0061 


1.0072 


1.0085 


1.0098 


1.0113 


1.0128 


1.0145 


1.0162 


1.0181 


15 


0.2 


1.0201 


1.0221 


1.0243 


1.0266 


1.0289 


1.0314 


1 . 0340 


1.0367 


1.0395 


1.0423 


25 


0.3 


1.0453 


1.0484 


1.0516 


1.0549 


1.0584 


1.0619 


1.0655 


1.0692 


1.0731 


1.0770 


35 


0.4 


1.0811 


1.0852 


1.0895 


1.0939 


1.0984 


1.1030 


1.1077 


1.1125 


1.1174 


1.1225 


46 


0.5 


1.1276 


1.1329 


1.1383 


1.1438 


1.1494 


1.1551 


1.1609 


1. 1669 


1.1730 


1.1792 


57 


0.6 


1.1855 


1.1919 


1.1984 


1.2051 


1.2119 


1.2188 


1.2258 


1.2330 


1.2402 


1.2476 


69 


0.7 


1.2552 


1.2628 


1.2706 


1.2785 


1.2865 


1.2947 


1.303U 


1.3114 


1.3199 


1.3286 


82 


0.8 


1.3374 


1.3464 


1.3555 


1.3647 


1.3740 


1.3835 


1.3932 


1.4029 


1.4128 


1.4229 


95 


0.9 


1.4331 


1.4434 


1.4539 


1.4645 


1.4753 


1.4862 


1.4973 


1.5085 


1.5199 


1.5314 


109 


1.0 


1.5431 


1.5549 


1.5669 


1.5790 


1.5913 


1.6038 


1.6164 


1.6292 


1.6421 


1.6552 


125 


1.1 


1.6685 


1 . 6820 


1.6956 


1.7093 


1.7233 


1.7374 


1.7517 


1.7662 


1.7808 


1.7956 


141 


1.2 


1.8107 


1.8258 


1.8412 


1.8568 


1.8725 


1.8884 


1.9045 


1.9208 


1.9373 


1.9540 


159 


1.3 


1.9709 


1.9880 


2.0053 


2.0228 


2.0404 


2.0583 


2.0764 


2.0947 


2.1132 


2.1320 


179 


1.4 


2.1509 


2.1700 


2.1894 


2.2090 


2.2288 


2.2488 


2.2691 


2.2896 


2.3103 


2.3312 


200 


1.5 


2.3524 


2.3738 


2.3955 


2.4174 


2.4395 


2.4619 


2.4845 


2.5073 


2.5305 


2.5538 


224 


1.6 


2.5775 


2.6013 


2.6255 


2.6499 


2.6746 


2.6995 


2.7247 


2.7502 


2.7760 


2.8020 


249 


1.7 


2.8283 


2.8549 


2.8818 


2.9090 


2.9364 


2.9642 


2.9922 


3.0206 


3.0492 


3.0782 


278 


1.8 


3.1075 


3.1371 


3.1669 


3.1972 


3.2277 


3.2585 


3.2897 


3.3212 


3.3530 


3.3852 


308 


1.9 


3.4177 


3.4506 


3.4838 


3.5173 


3.5512 


3.5855 


3 . 6201 


3.6551 


3.6904 


3.7261 


343 


2.0 


3.7622 


3.7987 


3.8355 


3.8727 


3.9103 


3.9483 


3.9867 


4.0255 


4.0647 


4.1043 


380 


2.1 


4.1443 


4.1847 


4.2256 


4.2668 


4.3085 


4.3507 


4.3932 


4.4362 


4.4797 


4.5236 


422 


2.2 


4.5679 


4.6127 


4.6580 


4.7037 


4.7499 


4.7966 


4.8437 


4.8914 


4.9395 


4.9881 


467 


2.3 


5.0372 


5.0868 


5.1370 


5.1876 


5.2388 


5.2905 


5.3427 


5.3954 


5.4487 


5.5026 


517 


2.4 


5.5569 


5.6119 


5.6674 


5.7235 


5.7801 


5.8373 


5.8951 


5.9535 


6.0125 


6.0721 


572 


2.5 


6.1323 


6.1931 


6.2545 


6.3166 


6.3793 


6.4426 


6.5066 


6.5712 


6.6365 


6.7024 


633 


2.6 


6.7690 


6.8363 


6.9043 


6.9729 


7.0423 


7.1123 


7.1831 


7.2546 


7.3268 


7.3998 


700 


2.7 


7.4735 


7.5479 


7.6231 


7.6990 


7.7758 


7.8533 


7.9136 


8.0106 


8.0905 


8.1712 


775 


2.8 


8.2527 


8.3351 


8.4182 


8.5022 


8.5871 


8.6728 


8.7594 


8.8469 


8.9352 


9.0244 


857 


2.9 


9.1146 


9.2056 


9.2976 


9.3905 


9.4844 


9.5791 


9.6749 


9.7716 


9.8693 


9.9680 


947 


3.0 


10.068 


10.168 


10.270 


10.373 


10.477 


10.581 


10.687 


10.794 


10.902 


11.011 


104 


3.1 


11.122 


11.233 


11.345 


11.459 


11.574 


11.689 


11.807 


11.925 


12.044 


12.165 


115 


3.2 


12.287 


12.410 


12.534 


12.660 


12.786 


12.915 


13.044 


13.175 


13.307 


13.440 


129 


3.3 


13.575 


13.711 


13.848 


13.987 


14.127 


14.269 


14.412 


14.556 


14.702 


14.850 


142 


3.4 


14.999 


15.149 


15.301 


15.455 


15.610 


15.766 


15.924 


16.084 


16.245 


16.408 


156 


3.5 


16.573 


16.739 


16.907 


17.077 


17.248 


17.421 


17.596 


17.772 


17.951 


18.131 


173 


3.6 


18.313 


18.497 


18.682 


18.870 


19.059 


19.250 


19.444 


19.639 


19.836 


20.035 


191 


3.7 


20.236 


20.439 


20.644 


20.852 


21.061 


21.272 


21.486 


21.702 


21.919 


22.139 


211 


3.8 


22.362 


22.588 


22.813 


23.042 


23.273 


23.507 


23.743 


23.982 


24.222 


24.466 


234 


3.9 


24.711 


24.959 


25.210 


25.463 


25.719 


25.977 


26.238 


26.502 


26.768 


27.037 


258 


4.0 


27.308 


27.582 


27.860 


28.139 


28.422 


28.707 


28.996 


29.287 


29.581 


29.878 


285 


4.1 


30.178 


30.482 


30.788 


31.097 


31.409 


31.725 


32.044 


32.365 


32.691 


33.019 


316 


4.2 


33.351 


33.686 


34.024 


34.366 


34.711 


35.060 


35.412 


35.768 


36.127 


36.490 


349 


4.3 


36.857 


37.227 


37.601 


37.979 


38.360 


38.746 


39.135 


39.528 


39.925 


40.326 


386 


4.4 


40.732 


41.141 


41.554 


41.972 


42.393 


42.819 


43.250 


43.684 


44.123 


44.566 


426 


4.6 


45.014 


45.466 


45.923 


46.385 


46.851 


47.. 321 


47.797 


48.277 


48.762 


49.252 


470 


4.6 


49.747 


50.247 


50.752 


51.262 


51.777 


52.297 


52.823 


53.354 


53.890 


54.431 


520 


4.7 


54.978 


55.531 


56.089 


56.652 


57.221 


57.796 


58.377 


58.964 


59.556 


60.155 


575 


4.8 


60.759 


61.370 


61.987 


62.609 


63.239 


63.874 


64.516 


65.164 


65.819 


66.481 


635 


4.9 


67.149 


67.823 


68.505 


69.193 


69.889 


70.591 


71.300 


72.017 


72.741 


73.472 


702 


5.0 


74.210 


74.956 


75.709 


76.470 


77.238 


78.014 


78.798 


79.590 


80.390 


81.198 


776 



Ex.— For u = 2.73, cosh « = 7. 



HYPERBOLIC COSINES, WITH LOGARITHMS. 



198e 



9.— Common Logarithms of Hyperbolic Cosines u. 

(For natural functions, see opposite page.) 



u 


.00 


.01 


.02 


.03 


.04 


.05 


.06 


.07 


.08 


.09 


Dif. 


0.0 


0.00000 


00002 


00009 


00020 


00035 


00054 


00078 


00106 


00139 


00176 


19 


0.1 


00217 


00262 


00312 


00366 


00424 


00487 


00554 


00625 


00700 


00779 


63 


0.2 


00863 


00951 


01043 


01139 


01239 


01343 


01452 


01564 


01681 


01801 


104 


0.3 


01926 


02054 


02187 


02323 


02463 


02607 


02755 


02907 


03063 


03222 


144 


0.4 


03385 


03552 


03723 


03897 


04075 


04256 


04441 


04630 


04822 


05018 


181 


0.5 


05217 


05419 


05625 


05834 


06046 


06262 


06481 


06703 


06929 


07157 


216 


0.6 


07389 


07624 


07861 


08102 


08346 


08593 


08843 


09095 


09351 


09609 


247 


0.7 


09870 


10134 


10401 


10670 


10942 


11216 


11493 


11773 


12055 


12340 


274 


0.8 


12627 


12917 


13209 


13503 


13800 


14099 


14400 


14704 


15009 


15317 


299 


0.9 


15627 


15939 


16254 


16570 


16888 


17208 


17531 


17855 


18181 


18509 


320 


1.0 


18839 


19171 


19504 


19839 


20176 


20515 


20855 


21197 


21541 


21886 


339 


1.1 


22233 


22582 


22931 


23283 


23636 


23990 


24346 


24703 


25062 


25422 


354 


1.2 


25784 


26146 


26510 


26876 


27242 


27610 


27979 


28349 


28721 


29093 


368 


1.3 


29467 


29842 


30217 


30594 


30972 


31352 


31732 


32113 


32495 


32878 


380 


1.4 


33262 


33647 


34033 


34420 


34807 


35196 


35585 


35976 


36367 


36759 


389 


1.5 


37151 


37545 


37939 


38334 


38730 


39126 


39524 


39921 


40320 


40719 


396 


1.6 


41119 


41520 


41921 


42323 


42725 


43129 


43532 


43937 


44341 


44747 


404 


1.7 


45153 


45559 


45966 


46374 


46782 


47191 


47600 


48009 


48419 


48830 


409 


1.8 


49241 


49652 


50064 


50476 


50889 


51302 


51716 


52130 


52544 


52959 


413 


1.9 


53374 


53789 


54205 


54621 


55038 


55455 


55872 


56290 


56707 


57126 


417 


2.0 


57544 


57963 


58382 


58802 


59221 


59641 


60061 


60482 


60903 


61324 


420 


2.1 


61745 


62167 


62589 


63011 


63433 


63856 


64278 


64701 


65125 


65548 


423 


2.2 


65972 


66396 


66820 


67244 


67668 


68093 


68518 


68943 


69368 


69794 


425 


2.3 


70219 


70645 


71071 


71497 


71923 


72349 


72776 


73203 


73630 


74056 


426 


2.4 


74484 


74911 


75338 


75766 


76194 


76621 


77049 


77477 


77906 


78334 


427 


2.5 


78762 


79191 


79619 


80048 


80477 


80906 


81335 


81764 


82194 


82623 


429 


2.6 


83052 


83482 


83912 


84341 


84771 


85201 


85631 


86061 


86492 


86922 


430 


2.7 


87352 


87783 


88213 


88644 


89074 


89505 


89936 


90367 


90798 


91229 


431 


2.8 


91660 


92091 


92522 


92953 


93385 


93816 


94247 


94679 


95110 


95542 


431 


2.9 


95974 


96405 


96837 


97269 


97701 


98133 


98565 


98997 


99429 


99861 


432 


3.0 


1.00293 


00725 


01157 


01589 


02022 


02454 


02886 


03319 


03751 


04184 


432 


3.1 


04616 


05049 


05481 


05914 


06347 


06779 


07212 


07645 


08078 


08510 


432 


3.2 


08943 


09376 


09809 


10242 


10675 


11108 


11541 


11974 


12407 


12840 


433 


3.3 


13273 


13706 


14139 


14573 


15006 


15439 


15872 


16306 


16739 


17172 


433 


3.4 


17605 


18039 


18472 


18906 


19339 


19772 


20206 


20639 


21073 


21506 


433 


3.5 


21940 


22373 


22807 


23240 


23674 


24107 


24541 


24975 


25408 


25842 


433 


3.6 


26275 


26709 


27143 


27576 


28010 


28444 


28878 


29311 


29745 


30179 


434 


3.7 


30612 


31046 


31480 


31914 


32348 


32781 


33215 


33649 


34083 


34517 


433 


3.8 


34951 


35384 


35818 


36252 


36686 


37120 


37554 


37988 


38422 


38856 


434 


3.9 


39290 


39724 


40158 


40591 


41025 


41459 


41893 


42327 


42761 


43195 


434 


4.0 


43629 


44063 


44497 


44931 


45365 


45799 


46233 


46668 


47102 


47536 


434 


4.1 


47970 


48404 


48838 


49272 


49706 


50140 


50574 


51008 


51442 


51876 


434 


4.2 


52310 


52745 


53179 


53613 


54047 


54481 


54915 


55349 


55783 


56217 


434 


4.3 


56652 


57086 


57520 


57954 


58388 


58822 


59256 


59691 


60125 


60559 


434 


4.4 


60993 


61427 


61861 


62296 


62730 


63164 


63598 


64032 


64467 


64901 


434 


4.5 


65335 


65769 


66203» 


66637 


67072 


67506 


67940 


68374 


68808 692431 


434 


4.6 


69677 


70111 


70545 


70979 


71414 


71848 


72282 


72716 


73151 735851 


434 


4.7 


74019 


74453 


74887 


75322 


75756 


76190 


76624 


77059 


77493 


77927 


434 


4.8 


78361 


78796 


79230 


79664 


80098 


80532 


80967 


81401 


81835 


82269 


434 


4.9 


82704 


83138 


83572 


84006 


84441 


84875 


85309 


85743 


86178 


86612 


434 


6.0 


87046 


87480 


87915 


88349 


88783 


89217 


89652 


90086 


90520 


90955 434 , 



Ex.— For u = 2.73, com log of cosh u = 0.88674. 



198f 



^.—PLANE TRIGONOMETRY. 



10.— Natural Hyperbolic Tangents a = Tanh u 



«"- 



(For common logarithms of these values, see Table 11, below.) 



i 



u 


.00 


.01 


.02 


.03 


.04 


.05 


.06 


.07 


.08 


.09 


Dif. 





0.0000 


0.0100 


0.0200 


0.0300 


0.0400 


0.0500 


0.0599 


0.0699 


0.0798 


0.0898 


100 


0,1 


.0997 


.1096 


.1194 


.1293 


.1391 


.1489 


.1587 


.1684 


.1781 


.1878 


98 


2 


.1974 


.2070 


.2165 


.2260 


.2355 


.2449 


.2543 


.2636 


.2729 


.2821 


94 


0.3 


.2913 


.3004 


.3095 


.3185 


.3275 


.3364 


.3452 


.3540 


.3627 


.3714 


89 


0.4 


.3800 


.3885 


.3969 


.4053 


.4137 


.4219 


.4301 


.4382 


.4462 


.4642 


82 


5 


0.4621 


0.4700 


0.4777 


0.4854 


0.4930 


0.5005 


0.5080 


0.5154 


0.5227 


0.5299 


75 


6 


.5370 


.5441 


.5511 


.5581 


.5649 


.5717 


.5784 


.5850 


.5915 


.5980 


68 


7 


.6044 


.6107 


.6169 


.6231 


.6291 


.6352 


.6411 


.6469 


.6527 


6584 


61 


8 


.6640 


.6696 


.6751 


.6805 


.6858 


.6911 


.6963 


.7014 


.7064 


.7114 


53 


0.9 


.7163 


.7211 


.7259 


.7306 


.7352 


.7398 


.7443 


.7487 


.7531 


.7574 


46 


1,0 


0.7616 


0.7658 


0.7699 


0.7739 


0.7779 


0.7818 


0.7857 


0.7895 


0.7932 


0.7969 


39 


1.1 


.8005 


.8041 


.8076 


.8110 


.8144 


.8178 


.8210 


.8243 


.8275 


.8306 


34 


12 


.8337 


.8367 


.8397 


.8426 


.8455 


.8483 


.8511 


.8538 


.8565 


.8591 


28 


1.3 


.8617 


.8643 


.8668 


.8693 


.8717 


.8741 


.8764 


.8787 


.8810 


.8832 


24 


1.4 


.8854 


.8875 


.8896 


.8917 


.8937 


.8957 


.8977 


.8996 


.9015 


.9033 


20 


15 


0.9052 


0.9069 


0.9087 


0.9104 


0.9121 


0.9138 


0.9154 


0.9170 


0.9180 


0.9202 


17 


1.0 


.9217 


.9232 


.9246 


.9261 


.9275 


.9289 


.9302 


.9316 


.9329 


.9342 


14 


1.7 


.9354 


.9367 


.9379 


.9391 


.9402 


.9414 


.9425 


.9436 


.9447 


.9458 


12 


1 8 


.9468 


.9478 


.9488 


.9498 


.9508 


.9518 


.9527 


.9536 


.9545 


.9554 


10 


1.9 


.9562 


.9571 


.9579 


.9587 


.9695 


.9603 


.9611 


.9619 


.9626 


.9633 


8 


2 


0.9640 


0.9647 


0.9654 


0.9661 


0.9668 


0.9674 


0.9680 


0.9687 


0.9093 


0.9699 


6 


2 1 


.9705 


.9710 


.9716 


.9722 


.9727 


.9732 


.9738 


.9743 


.9748 


.9753 


5 


2.2 


.9757 


.9762 


.9767 


.9771 


.9776 


.9780 


.9785 


.9789 


.9793 


.9797 


4 


2.3 


.9801 


.9805 


.9809 


.9812 


.9816 


.9820 


.9823 


.9827 


.9830 


.9834 


4 



11.— Common Logarithms of Hyperbolic Tangents u. 
(For natural functions, see Table 10, above.) 



u 


.00 


.01 


.02 


.03 


.04 


.05 


.06 


.07 


.08 


.09 


Dif. 


0.0 


— 00 


7.99999 


.30097 


.47699 


.60183 


.69861 


.77763 


.84439 


.90216 


.95307 


9678 


0.1 


8.99856 


03965 


07710 


T1151 


14330 


17285 


20044 


22629 


25062 


27357 


2955 


0.2 


9.29529 


31590 


33549 


35416 


37198 


38902 


40534 


42099 


43601 


45046 


1704 


0.3 


46436 


47775 


49067 


50314 


51518 


526S2 


53809 


54899 


55956 


56980 


1164 


0.4 


67973 


58936 


59871 


60780 


61663 


62521 


63355 


64167 


64957 


65726 


858 


5 


.66475 


. 67205 


.67916 


68608 


.69284 


.69942 


.70584 


.71211 


.71822 


.72419 


658 


6 


73001 


73570 


74125 


74667 


75197 


75715 


76220 


76714 


77197 


77669 


518 


7 


78130 


78581 


79022 


79453 


79875 


80288 


80691 


81086 


81472 


81850 


413 


8 


82219 


82581 


82935 


83281 


83620 


83952 


84277 


84595 


84906 


85211 


332 


0.9 


85509 


85801 


86088 


86368 


86642 


86910 


87173 


87431 


87683 


87930 


268 


1 


.88172 


.88409 


.88642 


88869 


.89092 


.89310 


.89524 


.89733 


.89938 


.90139 


218 


1 1 


90336 


90529 


90718 


90903 


91085 


91262 


91436 


91607 


91774 


91938 


177 


1 ?. 


9209S 


92256 


92410 


92501 


92709 


92854 


92996 


93135 


93272 


93406 


145 


1 3 


93537 


93665 


93791 


93914 


94035 


94154 


94270 


94384 


94495 


94604 


119 


1.4 


94712 


94817 


94919 


95020 


95119 


95216 


95311 


95404 


95495 


95584 


97 


1 5 


.95672 


.95758 


. 95842 


95924 


. 96005 


. 96084 


.96162 


.96238 


.96313 


.96386 


79 


1 6 


96467 


96528 


96597 


96664 


96730 


96795 


96858 


96921 


96982 


97042 


65 


1 7 


97100 


97158 


97214 


97269 


97323 


97376 


97428 


97479 


97529 


97578 


53 


1 8 


97626 


97073 


97719 


97764 


97809 


97S52 


97895 


97936 


97977 


98017 


43 


1.9 


98057 


98095 


98133 


98170 


98206 


98242 


98276 


98311 


98344 


98377 


36 


2 


.98409 


.98440 


.98471 


.98502 


.98531 


.98560 


.98589 


.98617 


.98644 


.98671 


29 


2 1 


98697 


98723 


98748 


98773 


98798 


98821 


98845 


98868 


98890 


98912 


23 


2 2 


98934 


98955 


98975 


98996 


99016 


99035 


99054 


99073 


99091 


99109 


19 


2 3 


99127 


99444 


99161 


99178 


99194 


99210 


99226 


99241 


99256 


99271 


10 


2.4 


99285 


99299 


99313 


99327 


99340 


99353 


99366 


99379 


99391 


99403 


13 



10.— SPHERICAL TRIGONOMETRY. 



General Discusssion — Spherical Trigonometry, in its broadest sense, 
treats of the solution of spherical pyramids; and, more directly, of the 
solution of triangular spherical pyramids. 

A triangular spherical pyramid (Fig. 1) is a triangular pyramid cut 
from a sphere; the center of the sphere being the apex of the pyramid, 
with the spherical surface forming the base. The base is therefore bounded 
by the arcs of great circles. 

A spherical triangle is a term applied to the outline of the base of a 
triangular spherical pyramid ; thus. Fig. i, A B C A is a spherical triangle. 
Problems in Spherical Trigonometry are confined usually to the solution of 
spherical triangles. The practical application leads us into the fields of 
Navigation, Geodesy and Astronomy. 

There are ten functions or quantities in any 
spherical triangle, as follows (Fig. 1): Radius, r; 

Tangential angles A B C ; 

Central angles a b c ; 

Lengths of arcs Oi 6i Ci . 

Considering the radius as unity, and remember- 
ing that the arcs of circles are proportional to the 
radii and to the central angles, it is seen that the 
above may be reduced to six primary functions, 
namely: the tangential angles A, B. C, and the 
central angels a, b, c. Three of these must be 
known in order to solve the other three. The 
central angles, a, b, and c, are always shown on the 
sides of the triangles, as in Figs. 2 and 3. 

Right Spherical Triangles.— The spherical 
triangle shown in Fig. 2 is a right spherical triangle, 
C being the right angle and o the center of the 
sphere. The following formulas are given for 
solving right spherical triangles. 



'1 


\ 


y 


'^^\j>'^ 






y ^\ ) 


^ t5 / O 


04-V--* \ (- 


"^^■Q 7 \u 


^^../>^ 



Fig. 1. 



o< 




c-gor 



Fig. 2. 



Formulas: 



sin A 

cos A 
cos c 



cos 



sm c 
tan 6 



cos b ' 
tan A 



„ sin b 
sm B — --. 



tanc 

cos a cos b. 



tana 
sin b 



sm c 
tan a 



cos_A 
cos a 



„ .«,xx "- , D *3,n b 

cos B = . tanS = - — 

tan c sm o 

cos c = cot A cot B. 



Note. — ^The above formulas are all that are necessary, as the value of 
any required function may be determined by transposition. Thus, from 

„ tan b , . tan b , 

fcan B = -; , we have, sm a = 5 , and so on. 

sm a tan B 

Example.— What is the side b if the angle A = 18°- 20' and the hypoth- 
enuse c = 48°- 30'? 

Solution.-^From cos A = , we have, tan & = tan c cos A ; 

tan c 



whence, 



log tan 48°- 30' = 0.05319 
log cos 18°- 20' = 9.97738 

01', as . 03057 = log tan b. 



.*. central angle b = iT^ 
The length of the side (arc) b may be obtained from the table of Circular 
Arcs (see Mensuration, following). 



199 



200 



10.— SPHERICAL TRIGONOMETRY, 



Oblique Spherical Triangles.- 




sm 



sm a 



Formulas. 

sin B 



B 



sin b 

cos a = cos b 
cos b = cos c 
cos c == cos a 
cos A = — cos B 
cos B = — cos C 



sin C 



cos c 

cos a 

cos b 
cos (7 
cos A 



cos 



— cos A cos B 



sin & 
sin cr 

+ sin b 

+ sin c 

+ sin a 

+ sin B 

4- sin C 

+ sin A 



sm 



sm o 

sin c: 
sin a 
sin b 
sin C7 
sin A 
sin 5 



sm g 
sin c 
cos A, 
cos B. 
cos C 
cos a. 
cos b. 
cos c. 



1 



sin i A 
sin i B 

sin i C = 

cos i A = 

cos ^ B = 

cos i C = 



-V— 



sin ( 5 — 6) sin (5 — c) 
sin b sin c 



5 — c) sin (s — a) 



sm c sm a 



/ sin (■? — g) sin (5 — 6) 
y sin a sin 6 



tan i A 
tani5 



4 

/ sin ( 
'^ -V si 

V sin ( 
~li 



sin 5 sin (s — a) 
sin b sin c 

sin 5 sin (s — b) 
sin c sin a 

sin 5 sin (5 — c) 
sin a sin 6 



5 — 6) sin (s — c) 
sin 5 sin (s — a) 



s — c) sin (5 — g) 
sin 5 sin (s — b) 



tan J C = H (^-°). f ( 
\' sm 5 sm (5 — 



5_-6) 



in ia ='y/— 



sin ^ b 



cos 5 cos (5 — A) 
sin B sin C 

cos 5 cos (S — B) 
sin C sin A 



sin i c =-%/ — 



cos i 
cos ^ 6 



a =^ 

-V 



cos 5 cos (S — C) 
sin A sin B 



cos (S—B) cos (S—C) 



sin 5 sin C 



cos (S—C) cos (5— A) 
sin C sin A 



cos I c =-%/ — 



cos (5- A) cos (S-B) 
sin A sin 5 



tan i a = 
tan ^ 6 = 
tan i c = 






COS 5 cos (S — A) 
'cos (S-B) cos(S-C) 

cos 5 cos (S — B) 
cos(S—C ) cos(S— A) 



/ cos 5 cos (S — C) 
\ cos(S-A)cos(S-B) 



Note. — In the above formulas, s=i (a + b + c); S=h (A+B + C). 
sin i (A+B) tan ^ c cos ^ (A+J5) tan i c 



sin h (A-B) 
sin \ (g + b) 
sin i (g — 6 ) 



tani(g — 6) cos i (A— B) 

cot ^ C cos \ (g + b) 

tan i (A-B) cos i (o - b) 

General Rules. 



tan i (g+6) 

cot ^ C 
tani (A +5) 



1. Sines of the sides are proportional to the sines of their opposite angles. 

Example. — In Fig. 2, sin A : sin a : : sin C : sin c. 

2. Cosine of any side equals the product of the cosines of the other two 

sides, plus the product of their sines and the cosine of their included 
angle. Example. — In Fig. 2, cos g = cos b cos c + sin b sin c cos A. 

Special Cases. 

Case I. Given one side (c) and two adjacent angles (A and B). 
1st. Solve for b and c in the following: 

tan h (b — a) = sin h (B — A) cosec ^ (B-\rA) tan \ c. 

tan i (b-\-a) = cos i (B~A) sec J (5 + A) tan } c. 
2nd. Solve for C in the following: 

cot \ C = sin i (6 + g) cosec ^ (6 — g) tan i (B — A). 



SPHERICAL TRIANGLES. CELESTIAL SPHERE, 201 

Case II. Given two sides (b and c) and their included angle (A). 
1st. Solve for B and C in the following: 

tan ^ (B — C) = sin ^ (b — c) cosec i (b + c) cot i A ; 
tan i (5 + Q = cos | (6-c) sec 1 (6 + c:) cot i A. 
2nd. Solve for a in the following: 

tan i a = sin i (B + C) cosec i (B — C) tan ^ (&--c). 

Case III. Given the three sides (a, b and c). Solve for A, S and C, 
tan i A = -: — -, r ; tan \ B = — — rr ; tan^ C- 



sin (s — o) ' sin {s — b) * sin (5 — c) 



-4- 



... , sm (5 — 0) sm (s — b) sm (s — c) 

m which k =^\ : 

sin s 



and 5 = ^ (a + & + c). 

Case IV. Given the three angles {A , B and C) . Solve for a, 6 and c. 

tan \ a = K cos (S — A); tan i b = K cos (5 — 5); tan § c = 
K cos (S — Q ; 



in which K 



V- 



cos (S-A) cos (S-B) cos (S-C) ' 

and 5 = i (A +5 + 0. 

Case V. Given two sides (a and b) and the angle (B) opposite one of them. 
1st. Solve for A in the following: 

sin A = sin a cosec b sin B. 
2nd. Solve for C and c in the following: 

cot ^ C == sin i (b + a) cosec i (b — a) tan i- (5 — ^4); 
tan \ c = sin | (5 + 4) cosec ^ (5 — A) tan \ (b — a). 
Case VI. Given two angles (A and B) and the side (b) opposite one of them. 
1st. Solve for a in the following: 

sin a = sin A cosec B sin b. ^ A=^ /e 

2nd. Solve for c and C in the following: ^"^ ^nT^'^ 

ta.nh c = sin i (B + A) cosec i (S-A) tan ^ (6-a); X \\ X 

cot i C = sin ^(b-\-a) cosec i (6 — a) tan -3 (5 — A). / J V A 

Distance between two points on the Earth's sur= / 1— VC \ 

face. — Given the latitudes and longitudes of two i^"" Lm ""^ n 

points 5 and C, Fig. 4, let it be required to find thejv^^^^ ' 9\?'Y J) 

shortest distance, a, between them. Select a third \ ^qufi-or ^ ' — ""^^Z 

point, A, at the nearest pole; and connect the points \ ^J* I 

A5Cwitharcsof great circles. Produce the meridian V-"" / 

lines A B and A C to the equator. Then will /' be \^ / 

the latitude of C\ l'\ the latitude of B\ and L, the ^""^--____— -^^ 

difference of longitude ; and, in the spherical triangle t^- ^ * 

A 5 C, 6= 90°-/', c= 90°-r, and the angle A = L. ^^^^ 4. 

Solve for Case II, preceding. 

The Celestial Sphere. — In solving astronomical problems, the center of 
the earth is assumed to be the center of the celestial sphere; and its axis 
produced, the axis of said sphere. The extremeties of the axis produced 
are the poles", and the great circle whose plane is perpendicular to the axis 
(at the center), is the equator. A plumb line at any given point is called 
the vertical line, and if produced it intersects the celestial sphere in the 
zenith (above) and in the nadir (below) . The zenith and nadir are the poles 
of any vertical line ; and a great circle whose plane is perpendicular to said 
vertical line, forms the horizon. The meridian is the great circle whose 
plane passes through the zenith and the poles, and the points where it 
intersects the horizon are called the north and south points. The prime 
vertical is the great circle whose plane passes through the zenith perpendicular 
to the meridian, and the points where it intersects the horizon are called 
the east and west points. 

A vertical circle is a great circle whose plane passes through the poles of 
a vertical line, i. e., through the zenith and nadir. It is therefore per- 
pendicular to the horizon. The azimuth of a star is the arc measured on the 
horizon between the north point (or the south point) and the vertical circle 
passing through the star; hence it is the tangential angle at the zenith, 
between the meridian and said vertical circle. The altitude t h oi a, star is 



202 



10.— SPHERICAL TRIGONOMETRY, 



its distance from the horizon measured on a vertical circle. The zenith 
distance = z==90° — h. 

An hour circle is any great circle whose plane passes through the poles 
(and perpendicular to the equator). The hour angle, P, of any star is the 
tangential angle which its hour circle measures {westward from and) v/ith 
the meridian, at the pole; the arc of the angle is measured at the equator. 
The right ascension, R. A., oi 3, star is the distance on the equator from the 
vernal equinox (the point where the sun crosses the equator from south to 
north*), eastward to the hour circle of the star. The declination, d, of a star 
is its distance from the equator, measured on an hour circle. The polar 
distance, p, = 90°—^. 

Astronomical Time. — A solar day is our common day of twenty-four 
hoiurs. Any particular solar day is an apparent solar day, and is the interval 
between two successive passages of the sun across the same meridian. A 
mean solar day is the average length of the solar days in a tropical year. 
A tropical year contains, according to Hansen and Olufsen, 365.2422008 
mean solar days; according to Bessel, 365.24222 mean solar days. On 
accoiint of the earth's revolution around the sun, there is one more sidereal 
day in a year than solar days. Hence, according to Bessel there are 366.- 
24222 sidereal days in a year, and we have the ratios — 

1 solar day = 1.0027379 sidereal day == 1 sid. day+ 3m 56.5555 sid. time. 
1 sid. day = 0.9972696 solar day = 1 solar day- 3m 55.9095 solar time. 

The equation of time is the quantity (time) to be " added," algebraically, 
to the apparent solar time to give the m^an solar time. Its value for any 
day of the year may be found in the solar ephemeris tables of the Nautical 
Almanac. 

The following are conversion tables for m£an solar time and sidereal 
time: 

1. — Mean Solar Time Reduced to Sidereal Time. 



Mean 


Sidereal Time 


Mean 


Sidereal Time 


Mean 


Sidereal 


Hours 






Minutes 






Seconds 


Time 


1 


Ih Om 


9.856s 


1 


Im 


0.164s 


1 


1.003s 


2 


2 


19.713 


2 


2 


0.329 


2 


2.005 


3 


3 


29.569 


3 


3 


0.493 


3 


3.008 


4 


4 


39.426 


4 


4 


0.657 


4 


4.011 


5 


5 


49.282 


5 


5 


0.821 


5 


5.014 


6 


6 


59.139 


6 


6 


0.986 


6 


6.016 


7 


7 1 


8.995 


7 


7 


1.150 


7 


7.019 


8 


8 1 


18.852 


8 


8 


1.314 


8 


8.022 


9 


9 1 


28.708 


9 


9 


1.478 


9 


9.025 


10 • 


10 1 


38.565 


10 


10 


1.643 


10 


10.027 


24 • 


24 3 


56.555 


30 


30 


4.928 


30 


30.082 



2. — Sidereal Time Reduced to Mean Solar Time. 



Sidereal 


Mean Solar Time. 


Sidereal 


Mean Solar 


Sidereal 


Mean Solar 


Hours 




Minutes 




Time 


Seconds 


Time 


1 


Oh 59m 50. 170s 


1 


Om 


59.836s 


1 


0.997s 


2 


1 59 40.341 


2 


1 


59.672 


2 


1.995 


3 


2 59 30.511 


3 


2 


59.509 


3 


2.992 


4 


3 59 20.682 


4 


3 


59.345 


4 


3.989 


5 


4 59 10.852 


5 


4 


59.181 


5 


4.986 


6 


5 59 1.023 


6 


5 


59.017 


6 


5.984 


7 


6 58 51.193 


7 


6 


58.853 


7 


6.981 


8 


7 58 41.364 


8 


7 


58.689 


8 


7.978 


9 


8 58 31.534 


9 


8 


58.526 


9 


8.975 


10 


9 58 21.704 


10 


9 


58.362 


10 


9.973 


24 


23 56 4.091 


30 


29 


55.085 


30 


29.918 



* The autumnal equinox is the point where the sun crosses the equator 
from north to south. 



t 



11.— MENSURATION, 



A —PLANE SURFACES, LINES, ETC. 
Triangle. — ^Three sides. Area = y ; = | a6 sin C; 

a + b-^c 



= s/s (s — a) (5 — b) (s — c), in which s ■■ 



h 



Length of 
above b. 



c=y/a^-{-b^-i-2 ab cos C. Center of gravity is 

Angles A +5 + C= 180°. 

Right-angled triangle. — Angle C= 90°. Area = ^ . Hypoth 



ab 




enuse c= Va'^ + b^; hence a^^/c^-b"^ = \/ {c + b){c-b)\ b = 
Iia = b, then c=lAU2a = 1.4142 6, or a = 6 = 0.7071 c. 

Acute-angled triangle. — Each angle less than 90°. 

Obtuse-angled triangle. — One angle greater than 90° (as Fig. 1). 

Isosceles triangle. — Two sides (therefore two angles) equal. 

Equilateral triangle. — ^Three sides (therefore three angles) equal. 
1, Regular Polygons. 



Vc2 - 



See Table 



/'■ 


\ h 
\ 1 


V-- 


\h 
\ 1 


/ t> 




\ b 


Ai 



Fig. 3. 



Fig. 4. 



Fig. 2. 

Quadrilateral. — Four Sides. 
Trapezium. — An irregular quadrilateral; no two sides parallel. Area may 
be obtained by cutting it into two triangles, and solving. 

Trapezoid. — ^Two sides only, parallel. Fig. 2. Area-=- (b + bj). See 

Section 29 for center of gravity. 
Parallelogram. — Includes the rhomboid, rhombus, rectangle and square. 

Area = length of one side multiplied by perpendicular distance to 

parallel side opposite. 
Rhomboid. — A "skewed" rectangle, Fig. 3; opposite sides (therefore op- 
posite angles) equal. Area = bh = ba sin A=ab sin A=ahi (Fig. 3). 

Cen. of grav is at intersection of two diagonals. 
Rectangle. — Opposite sides equal and parallel, as with the rhomboid; but 

angles 90°, Fig. 4. Side b>a. Area = ab. Diagonal = Va'^ + b"^. Cen. 

of grav. equi-distant from parallel sides, and at intersection of diagonals. 



Fig. 5. Fig. 6. 

Rhombus. — Same as rhomboid (Fig. 3) but with all sides equal. Fig. 5. 

Area = bh = b'^ sin A. 
Square. — All sides equal; angles 90°, Fig. 6. Area = b^. Diagonal =^2 b^=^ 

1.4142 6. Side 6 = diagonal X 0.7071. 



203 



204 



11 .—MEN SURA TION. 



Regular Polygon. — Any number of sides, from the 
triangle, with three sides, to the circle, with an oo number 
of sides. In Table 1, following, 

s = length of each side = 2 R sin ^ oc = 2 r tan | oc; 
n = number of sides in polygon (n 5 = perimeter) ; 
R = radius of circumscribing circle = h s cosec J oc = 

r sec i oc; 
r — apothem = radius of inscribed circle = R cos ^ oc = 

\ s cot ^ OC; 
ex = angle subtended from center by each side = 3 60-5- »; 
/? = internal angle = 1 80° - oc ; 

s r n perimeter X apothem , _, 
Area = — ^r- = ■ z ■ = i n s R cos f oc 




Fig. 7. 
n r^ tan i oc. 



1. — Table of Regular Polygons (Fig. 7). 















Outer 


Inner 




Name 


Central 


Internal 


blQt; cf 


radius R 


radius r 


ta 


03 


m 


02 




of 


Angle 


Angle 


1.1 


73 ro 


1 s 








n 


Polygon 


OC 





H 53 


H ^ 


H ^ 


H ^ 


w *^ 


m s 


3 


Triangle 


IBO^-OO'-OO" 


60°-00'-00" 


1.73205 


3.46410 


.577350 


2.00000 


.28868 


. 50000 


4 


Square 


90 -00 -00 


90 -00 -00 


1.41421 


2.00000 


.707107 


1.41421 


. 50000 


.7.0711 


5 


Pentagon 


72 -00 -00 


108-00-00 


1.17557 


1.45308 


.850651 


1.23607 


.68819 


. 80902 


6 


Hexagon 


60 -00 -00 


120 -00-00 


1.00000 


1.15470 


1.000000 


1.15470 


.86603 


. 86603 


7 


Heptagon 


51 -25-43 


128 -34-17 


.86777 


.96315 


1.152382 


1.10992 


1.03827 


. 90097 


8 


Octagon 


45 -00-00 


135 -00 -00 


.76537 


.82843 


1.306563 


1.08239 


1.20711 


.92388 


9 


Nonagon 


40 -00 -00 


140 -00 -00 


.68404 


.72794 


1.461902 


1.06418 


1.37374 


.93969 


10 


Decagon 


36 -00 -00 


144 -00-00 


.61803 


.64984 


1.618034 


1.05146 


1.53884 


.95106 


11 


Undecagon 


32 -43 -38 


147 -16-22 


.56346 


.58725 


1.774733 


1.04222 


1.70287 


.95943 


12 


Dodecagon 


30 -00 -00 


150 -00-00 


.51764 


.53590 


1.931854 


Ll. 03528 


1.86603 


.96593 


oc 


Circle 





180° 








CO 


Unity 


oo 


Unity 



II 


Name 
of 




Perimeter 


P 


Area A 


Equals 


Equals 


Equals 


Equals 


Equals 


Equals 


Equals 




Polygon 


s 


R 


r 


5 2 


i22 


r2 


P 


n 




times 


times 


times 


times 


times 


times 


times 


3 


Triangle 


3 


5.19615 


10.39230 


.4330127 


1.29904 


5.19615 


r-i-2 


4 


Square 


4 


5.65685 


8.00000 


1.0000000 


2.00000 


4.00000 






5 


Pentagon 


5 


5.87785 


7.26542 


1.7204774 


2.37764 


3.63271 






6 


Hexagon 


6 


6.00000 


6.92820 


2.5980762 


2.59808 


3.46410 






7 


Heptagon 


7 


6.07437 


6.74204 


3.6339124 


2.73641 


3.36502 






8 


Octagon 


8 


6.12293 


6.62741 


4.8284271 


2.82843 


3.31370 






9 


Nonagon 


9 


6.15636 


6.55146 


6.1818242 


2.89254 


3.27573 






10 


Decagon 


10 


6.18034 


6.49838 


7.6942088 


2.93893 


3.24920 






11 


Undecagon 


11 


6 19812 


6.45977 


9.3656399 


2.97352 


3.22989 






12 


Dodecagon 


12 


6.21166 


6.43078 


11.1961524 


3.00000 


3.21539 






00 


Circle 


00 


6.28319 


6.28319 


oo 


3.14159 


3.14159 







Circle. — Infinite number of sides. 
Circumference, p =7td = 3.1416 cJ, = 2 rr = 6.2832 r; 

Area, a = -j- = 0.7854^^2, = ;r r2 = 3.1416 r2; 
4 



Diameter, d =^= 0.318310 p, 

7Z 



Radius, r =~- = 0.159155 />, 

J 7Z 



2J^ = 1.12838^0; 
J~= 0.56419\/ar 




Fig. 8. 



3.141592 6536 Log = 0.497 149 8727 l] — =0.318 309 8862 Log = 9.502 8.50 1273 



POL YGONS— CIRCLE. 



206 



^Tabular Values of Combinations of n, with Logs. 



Number. Logarithm. 


Number. Logarithm. 


Number. 


Logarithm. 


2;r= 6.283185 0.7981799 


-= 0.636620 

It 


9.8038801 


|-= 1.570796 


0.1961199 


37r= 9.424778 0.9742712 


— = 0.954930 


9.9799714 


\= 1.047198 
o 


0.0200286 


4;r= 12. 566371 1.0992099 


i= 1.273240 

Tt 


0.1049101 


^= 0.785398 


9.8950899 


57r=15, 707963 1.196li99 


-= 1.591549 


0.2018201 


-|= 0.628319 


9.7981799 


6;r=18. 849556 1.2753012 


-= 1.909859 

It 


0.2810014 


1"= 0.523599 


9.7189986 


77r=21. 991149 1.3422479 


~= 2.228169 

Tt 


0.3479481 


y= 0.448799 


9.6520519 


87r=25. 132741 1.4002399 


-= 2.546479 

Tt 


0.4059401 


|-= 0.392699 


9.5940599 


97r=28. 274334 1.4513924 


-= 2.864789 

Tt 


0.4570926 


|-= 0.349066 


9.5429074 


Note. — In circular 

measure, 7r=180° y 

=10800 min. I 

=648000 sec. J 
7r2= 9.86960 0.9942997 


180° 
^^=57.295780 

It 

^=114.59156 
10800' _ . 


1.7581226 

2.0591526 

3.5362739 

5.3144251 
1.4914496 


^=.01745329 8.2418774 
^=.00087266 7.9408474 


648000" 206264 81 


g~Q7,=.0O0O04848 4.6855749 
;r4=97.4091 1.9885995 


Tt 

7r3=31. 00628 


-i-2= 0.10132 9.0057003 


^= 0.03225 


8.5085504 


i= 0.01027 

71^ 


8.0114005 


Vir= 1.77245 0.2485749 


^^=1.46459 


0.1657166 


^5^=1. 33134 


0.1242875 


V2^=2. 50663 0.3990899 

/ — 


-^2^=1.84527 


0.2660600 


4 

V27r=l. 58323 

1— - 


0.1995450 


Ji =0.56419 9.7514251 


Wi=0. 68278 


9.8342834 


i/— =0.75113 


9.8757125 


J^ =0.79788 9.9019401 


^^=0.86025 


9.9346267 


4/^=0.89324 


9.9509700 


J^ =1.25331 0.0980599 


y|-=1.16245 


0.0653733 


y|-=l. 11952 


0.0490300 



\ 



For multiples 1 to 9, see next page. 



206 



1 1 —MEN SURA TION . 



Tabular Values of Combinations op tt, with Logs. — Concluded. 
(Multiples 1 to 9.) 



IZ* 


Log. 


7C 


Log. 


Vi 


Log. 


1 

2 
3 

4 
5 
6 
7 
8 
9 


3.1415927 
6.2831853 
9.4247780 
12.5663706 
15,7079633 
18.8495559 
21.9911486 
25.1327412 
28.2743339 


0.4971499 
0.7981799 
0.9742712 
1.0992099 
1.1961199 
1.2753012 
1.3422479 
1.4002399 
1.4513924 


2 
2 
3 
4 
5 
6 
7 
8 
9 


0.3183099 
0.6366198 
0.9549297 
1.2732395 
1.5915494 
1.9098593 
2.2281692 
2.5464791 
2.8647890 


9.5028501 
9.8038801 
9.9799714 
0.1049101 
0.2018201 
0.2810014 
0.3479481 
0.4059401 
0.4570926 


1 
2 
3 
4 
5 
6 
7 
8 
9 


0.5641896 
1.1283792 
1.6925688 
2.2567583 
2.8209479 
3.3851375 
3.9493271 
4.5135167 
5.0777063 


9.7514251 
0.0524551 
0.2285464 
0.3534851 
0.4503951 
0.5295764 
0.5965231 
0.6545151 
7056676 


7r2 


Log. 


1 

7r2 


Log. 


V' 


Log. 


1 
2 
3 
4 
5 
6 
7 
8 
9 


9.8696044 
19.7392088 
29.6088132 
39.4784176 
49.3480220 
59.2176264 
69.0872308 
79.9568352 
88.8264396 


0.9942997 
1.2953297 
1.4714210 
1.5963597 
1.6932697 
1.7724510 
1.8393977 
1.8973897 
1.9485422 


1 
2 
3 
4 
5 
6 
7 
8 
9 


0.1013210 
0.2026420 
0,3039631 
0.4052841 
0.5066051 
0.6079261 
0.7092471 
0.8105682 
0.9118892 


9.0057003 
9.3067303 
9.4828216 
9.6077603 
9.7046703 
9.7838516 
9.8507983 
9.9087903 
9.9599428 


1 

2 
3 
4 
5 
6 
7 
8 
9 


1.7724539 

3.5449077 

5.3173616 

7.0898154 

8.8622693 

10.6347231 

12.4071770 

14.1796308 

15.9520847 


0.2485749 
0.5496049 
0.7256962 
0.8506349 
0.9475449 
1.0267262 
1.0936729 
1.1516649 
1.2028174 


^ 


Log. 


1 

7t^ 


Log. 


'4- 


Log. 


1 
2 
3 
4 
5 
6 
7 
8 
9 


31.0062767 
62.0125534 
93.0188300 
124.0251067 
155.0313834 
186.0376601 
217.0439368 
248.0502134 
279.0564901 


1.4914496 
1.7924796 
1.9685709 
2.0935096 
2.1904196 
2.2696009 
2.3365476 
2.3945396 
2.4456921 


1 

2 
3 

4 
5 
6 
7 
8 
9 


0.0322515 
0.0645030 
0.0967545 
0.1290060 
0.1612575 
0.1935090 
0.2257605 
0.2580120 
0.2902635 


8.5085504 
8.8095804 
8.9856717 
9.1106104 
9.2075204 
9.2867017 
9.3536484 
9.4116404 
9.4627929 


1 
2 
3 
4 
5 
6 
7 
8 
9 


1.4645919 

2.9291838 

4.3937756 

5.8583675 

7.3229594 

8.7875513 

10.2521432 

11.7167351 

13.1813269 


0.1657166 
0.4667466 
0.6428379 
0.7677766 
0.8646866 
0.9438679 
1.0108146 
1.0688066 
1.1199591 


7r4 


Log. 


1 

7r4 


Log. 


Vi 


Log. 


1 
2 
3 
4 
5 
6 
7 
8 
9 


97.4090909 
194.8181818 
292.2272727 
389.6363636 
487.0454545 
584.4545453 
681.8636362 
779.2727271 
876.6818180 


1.9885995 
2.2896295 
2.4657208 
2.5906595 
2.6875695 
2.7667508 
2.8336975 
2.8916895 
2.9428420 


1 
2 
3 
4 
5 
6 
7 
8 
9 


0.0102660 
0.0205320 
0.0307979 
0.0410639 
0.0513299 
0615959 
0.0718619 
0.0821278 
0.0923938 


8.0114005 
8.3124305 
8.4885218 
8.6134605 
8.7103705 
8.7895518 
8.8564985 
8.9144905 
8.9656430 


1 
2 
3 
4 
5 
6 
7 
8 
9 


0.6827841 
1.3655681 
2.0483522 
2.7311363 
3.4139203 
4.0967044 
4.7794885 
5.4622725 
6.1450566 


9.8342834 
0.1353134 
0.3114047 
0.4363434 
0.5332534 
0.6124347 
0.6793814 
0.7373734 
0.7885259 



Logarithms of Numbers 1 to 9, for Reference, 

Log 1 = 0.0000000 Log 4 = 0.6020600 Log 7=0.8450980 

" 2 = 0.3010300 " 5 = 0.6989700 *' 8=0.9030900 

" 3=0.4771213 " 6=0.7781513 " 9=0.9542425 

*See Table 11, pages 224 and 225, for values of n when multiplied by any 
whole number or decimal. (See Foot-note to Table 11.) Table 12 can be 
used in a similar manner. 



CIRCULAR ARC; CHORD. 



207 



Circular Arc ; and Chord. — 

As trigonometricinnctions (a+ ^9=360°), 
oc = central angle in degrees, < 180°. 
fi = central angle in degrees, >180°. 

As circular functions, 

oCi = length of arc a to radius 1. 
/?i = length of arc b to radius 1. 

Also, yo = distance to cen. of grav. g of arc a. 
Yq = distance to cen. of grav. G of arc h. 
Formulas', 



f Y -Jt 

Sinia =sin i /5 = — . Cosia:= — ; cos | ^3 = ^ 







xl— ->: 



Tan i oc = 
Tan i a = 



2 (r 
2^ 
c ' 



h) 



tan i /? = 



2{H — r) ' 



Vers ^oc^ 



vers t i9 = — . 

7" 



tani /? = . 



OCi 



TT CC 

180° 



= 0.01745329 oc; 



tt/? 
180" 



= 0.01745329 /? 

Arcfc = r/3i = 0.01745 r^. 

.01745 c ^ .0174:5 H ^ 



2 sin i 



vers I /i^ 



Area = r0Ci= 0.01745 roc; 

.01745 coc ^ . 01745 h a 

*" 2 sin i ex vers ^ OC 

Chord c= 2 r sin i oc = 2 r sin M = 2V/j {2r-h)= 2VH {2r-H) 

= 2 (r-/i) tan i oc = 2 (r-h) tan ijS = 2 (Z:^-r) tan i a = 2 (H-r) 

tan I ^ 
= 2;tcot ioc = 2/jcot i /3=2 {2r-H) cot i oc = 2 (2r-H) cot i i9. 



Note that ^ chord 



(f) 



is a mean proportional between /i and H; thus 



/j:^::-^:i^.or(;=2x/^^ 



Yo 



'2 
180 g 

ttOC 

180 <: 



Note.- 



2 r sin ^ oc length of chord c re 
length of arc a a 
_ length of chord c _rc 
length of arc h b 



CC, 

2 r sin i /? 



oc oc, 



^1 



180 



= 57.29578; log = 1.7581226 (see under Circle). 



Surface of revolution 5 generated by arc a revolving about axis X — X 
is equal to a multiplied by length of path traveled by point g; 

or, s = -T^' ci yo (in which is the angle of revolution, in degrees) 



;r ^ r2 sin ^ oc 



the general equation. Thus, to get the surface of a 



90 

sphere make oc = 180°, 0= 360°; then 5=4 tt r2= 12.566371 72. 
h = middle ordinate of arc a H = middle ordinate of arc A 

= r (1 — cos ^ ex) = f vers I cx =r(l + cosij9) 

='■- V^'- (i) '= f *"" * ""■ ^ '^+^l''- (t) '=!*"" ^ "• 

Note. — Ordi nate at any point distant x f rom center = h — d, in which 
d = r—y/r^~x^', hence, ordinate = /j— r +\/r'^ — x'^. 

h ^ c ^ h , 

OCi ttOC oc 1 — cos^a"" 2 sin i OC~vers i oc ' 

b ^180b_ 57.29578b ^ ^ -^ ^ ^ ^ 
3x n^ (i l+cosi;9" 2 sin i fi vers ^0' 



^ ,. a 180a 57.29578a 
Radius r = — = = 



208 



n.— MENSURATION, 



Values of cCi and 0i (Fig. 9) 
2. — Lengths of Circular Arcs for Radius 1. 
(Note that the arc is directly proportional to central angle and to radius.) 



1 



Deg. 


Length. 


Deg. 


Length. 


Min. 


Length. 


Sec. 


Length. 


15° 


0.261799 3878 


1° 


.017453 2925 


r 


.000290 8882 


1" 


.000004 8481 


30° 


0.523598 7756 


2° 


.034906 5850 


2' 


.000581 7764 


2" 


.000009 6963 


45° 


0.785398 1634 


3° 


.052359 8776 


3' 


.000872 6646 


3" 


.000014 5444 


60° 


1.047197 5512 


4° 


.069813 1701 


4' 


.001163 5528 


4" 


.000019 3925 


75° 


1.308996 9390 


5° 


.087266 4626 


5' 


.001454 4410 


5" 


.000024 2407 


90° 


1.570796 3268 


6° 


.104719 7551 


6' 


.001745 3293 


6* 


.000029 0888 


135° 


2.356194 4902 


7° 


.122173 0476 


7' 


.002036 2175 


7' 


.000033 9370 


180° 


3.141592 6536 


8° 


.139626 3402 


8' 


.002327 1057 


8" 


.000038 7851 


270° 


4.712388 9804 


9° 


.157079 6327 


9' 


.002617 9939 


r 


.000043 6332 


360° 


6.283185 3072 


10° 


.174532 9252 


10' 


.002908 8821 


10' 


.000048 4814 



Solution. — From Table 2. 



200° 


= 


3.490659 




30° 


= 


.523599 




7° 


= 


.122173 




40' 


= 


.011636 




6' 


== 


.001745 




50" 


= 


.000242 




8" 


— 


.000039 




.3" 


= 


.000001 




log 




4.150094 = 


0.6180580 


log 




6387.42 = 


3.8053255 


Ans. 


26508.4 


4.4233835 



Problem. — Find the length 
of circular arc corresponding to 
a central angle of 2 37° - 46' - 58 . 3" 
and to a radius of 6387.42. 



Note. — Table 3, following page, 
has been prepared from Table 2. 



Raho of Chord Lengths of Circles.— From the preceding page we have 
the equation: Chord c = 2 r sin ^ oc , r being the radius of the circle and « the 
central angle. 

For comparison of lengths of chords of two circles, r and oc varying, we have 
chord c' = 2 r' sin I oc ' 
and chord c" = 2 r" sin ^ oc " 
. £l^ r'sin^oc^ 
•• c" r"sinic:" ^^^ 

By logarithms: log c' = log c" + log r' + log sin I oc ' — log r" — log sin | oc " . . ( la) 
For comparison of lengths of chords of the same circle, r' = r" in equation (1), 
and we have 

.^^sinioc^ 

c" sm 5 cc " 

By logarithms: log c' = log c" +'log'sin | oc ' — log sin | oc " (2a) 

Equation (2a) is useful in finding the lengths of long chords in laying out 
railway curves. 

Problem 1. — A chord c" = 100 ft., of a circular curve, subtends a central 
angle oc " = 4° 00'. Find the length of chord c' which will subtend a central 
angle oc ' = 12° 00', of the same curve. 

Solution. — Using formula (2a), we have, 

log 100= 2.00000 
log sin 6" = 9.01923 



ioc 



Ans.- 



299.51. 



11.01923 
log sin 2° = 8.54282 

log 295.51 = 



ioc" 



2.47641 c' 

Problem 2. — A chord c" = 100 ft., of a circular curve, subtends a central 
angle oc " = 4° 00'. Find the central angle oc ' which will be subtended by a chord 
c' = 300 ft., to the same curve. 

Solution.— log 300 = 2.47712 c' 

log sin 2° = 8.54282 i oc " 

11.01994 
log 100 = 2.00000 c" 

Ans.— oc' = 12°01'. logsine^OO'SO" = 9.01994 §oc 



CIRCULAR ARC; CHORD. 



209 



Values op qc^ and ^i (Fig. 9) 
3.- — ^Lengths op Circular Arcs por Radius 1. 
(Deg., Min., Sec, are central angles; Length is length of subtended arc.) 





Deg. 


Length. 


Deg 


Length. 


Deg. 


Length. 


Min. 


Length. 


Sec. 


Length. 


1 


.0174533 


61 


1.0646508 


121 


2.1118484 


1 


.0002909 


1 


.0000048 


2 


.0349066 


62 


1.0821041 


122 


2.1293017 


2 


.0005818 


2 


.0000097 


3 


.0523599 


63 


1.0995574 


123 


2.1467550 


3 


.0008727 


3 


.0000145 


4 


.0698132 


64 


1.1170107 


124 


2.1642083 


4 


.0011636 


4 


.0000194 


5 


.0872665 


65 


1.1344640 


125 


2.1816616 


5 


.0014544 


5 


.0000242 


6 


.1047198 


66 


1.1519173 


126 


2.1991149 


6 


.0017453 


6 


.0000291 


7 


.1221730 


67 


1.1693706 


127 


2.2165682 


7 


.0020362 


7 


. 0000339 


8 


.1396263 


68 


1.1868239 


128 


2.2340214 


8 


.0023271 


8 


.0000388 


9 


.1570796 


69 


1.2042772 


129 


2.2514747 


9 


.0026180 


9 


.0000436 


10 


.1745329 


70 


1.2217305 


130 


2.2689280 


10 


.0029089 


10 


-.0000485 


11 


.1919862 


71 


1.2391838 


131 


2.2863813 


11 


.0031998 


11 


. 0000533 


12 


.2094395 


72 


1.2566371 


132 


2.3038346 


12 


.0034907 


12 


. 0000582 


13 


.2268928 


73 


1.2740904 


133 


2.3212879 


13 


.0037815 


13 


.0000630 


14 


.2443461 


74 


1.2915436 


134 


2.3387412 


14 


.0040724 


14 


.0000679 


15 


.2617994 


75 


1.3089969 


135 


2.3561945 


15 


. 0043633 


15 


.0000727 


16 


.2792527 


76 


1.3264502 


136 


2.3736478 


16 


.0046542 


16 


.0000776 


17 


.2967060 


77 


1.3439035 


137 


2.3911011 


17 


.0049451 


17 


. 0000824 


18 


.3141593 


7» 


1.3613568 


138 


2.4085544 


18 


.0052360 


18 


.0000873 


19 


.3316126 


79 


1.3788101 


139 


2.4260077 


19 


.0055269 


19 


.0000921 


20 


.3490659 


80 


1.3962634 


140 


2.4434610 


20 


.0058178 


20 


.0000970 


21 


.3665191 


81 


1.4137167 


141 


2.4609142 


21 


.0061087 


21 


.0001018 


22 


.3839724 


82 


1.4311700 


142 


2.4783675 


22 


.0063995 


22 


.0001067 


23 


.4014257 


83 


1.4486233 


143 


2.4958208 


23 


.0066904 


23 


.0001115 


24 


.4188790 


84 


1.4660766 


144 


2.5132741 


24 


.0069813 


24 


.0001164 


25 


.4363323 


85 


1.4835299 


145 


2.5307274 


25 


.0072722 


25 


.0001212 


26 


.4537856 


86 


1.5009832 


146 


2.5481807 


26 


.0075631 


26 


.0001261 


27 


.4712389 


87 


1.5184364 


147 


2.5656340 


27 


.0078540 


27 


0001309 


28 


.4886922 


88 


1.5358897 


148 


2.5830873 


28 


.0081449 


28 


.0001357 


29 


.5061455 


89 


1.5533430 


149 


2.6005406 


29 


.0084358 


29 


.0001406 


30 


.5235988 


90 


1.5707963 


150 


2.6179939 


30 


.0087266 


30 


.0001454 


31 


.5410521 


91 


1.5882496 


151 


2.6354472 


31 


.0090175 


31 


.0001503 


32 


.5585054 


92 


1.6057029 


152 


2.6529005 


32 


.0093084 


32 


0001551 


33 


.5759587 


93 


1.6231562 


153 


2.6703538 


33 


.0095993 


33 


.0001600 


34 


.5934119 


94 


1.6406095 


154 


2.6878070 


34 


.0098902 


34 


.0001648 


35 


.6108652 


95 


1.6580628 


155 


2.7052603 


35 


.0101811 


35 


.0001697 


36 


.6283185 


96 


1.6755161 


156 


2.7227136 


36 


.0104720 


36 


.0001745 


37 


.6457718 


97 


1.6929694 


157 


2.7401669 


37 


0107629 


37 


.0001794 


38 


.6632251 


98 


1.7104227 


158 


2.7576202 


38 


.0110538 


38 


.0001842 


39 


.6806784 


99 


1.7278760 


159 


2.7750735 


39 


.0113446 


39 


.0001891 


40 


.6981317 


100 


1.7453293 


160 


2.7925268 


40 


.0116355 


40 


.0001939 


41 


.7155850 


101 


1.7627825 


161 


2.8099801 


41 


.0119264 


41 


.0001988 


42 


.7330383 


102 


1.7802358 


162 


2.8274334 


42 


.0122173 


42 


.0002036 


43 


.7504916 


103 


1.7976891 


163 


2.8448867 


43 


.0125082 


43 


.0002085 


44 


.7679449 


104 


1.8151424 


164 


2.8623400 


44 


.0127991 


44 


.0002133 


45 


.7853982 


105 


1.8325957 


165 


2.8797933 


45 


.0130900 


45 


.0002182 


46 


.8028515 


106 


1.8500490 


166 


2.8972466 


46 


0133809 


46 


.0002230 


47 


.8203047 


107 


1.8675023 


167 


2.9146999 


47 


.0136717 


47 


.0002279 


48 


.8377580 


108 


1.8849556 


168 


2.9321531 


48 


.0139626 


48 


.0002327 


49 


.8552113 


109 


1.9024089 


169 


2.9496064 


49 


.0142535 


49 


.0002376 


50 


.8726646 


110 


1.9198622 


170 


2.9670597 


50 


.0145444 


50 


.0002424 


51 


.8901179 


111 


1.9373155 


171 


2.9845130 


51 


.0148353 


51 


.0002473 


52 


.9075712 


112 


1.9547688 


172 


3.0019663 


52 


.0151262 


52 


.0002521 


53 


.9250245 


113 


1.9722221 


173 


3.0194196 


53 


.0154171 


53 


.0002570 


54 


.9424778 


114 


1.9896753 


174 


3.0368729 


54 


.0157080 


54 


.0002618 


55 


.9599311 


115 


2.0071286 


175 


3.0543262 


55 


.0159989 


55 


.0002666 


56 


.9773844 


116 


2.0245819 


176 


3.0717795 


56 


.0162897 


56 


.0002715 


57 


.9948377 


117 


2.0420352 


177 


3.0892328 


57 


.0165806 


57 


.0002763 


58 


1.0122910 


118 


2.0594885 


178 


3.1066861 


58 


.0168715 


58 


.0002812 


59 


1.0297443 


119 


2.0769418 


179 


3.1241394 


59 


.0171624 


59 


.0002860 


60 


1.0471976 


120 


2.0943951 


180 


3.1415927 


60 


.0174533 


60 


.0002909 


N( 
radius 


3te. — Len 


gth o 


£ arc is dii 


•ectly 


proportio 


nal to 


central 


angle 


and to 



210 



11.— MENSURATION. 



4. — Lengths of Circular Arcs for Chord 1. 
(Note. — Multiply the tabular number by the length of chord.) 



Thousandths 



.000 .001 



.002 



.003 



.004 



.005 



.006 



.007 



.008 



1.00000 

.00027 

00107 

.00240 

.00426 

1.00665 
.00957 
.01302 
.01698 
.02146 

1.02646 
.03196 
.03797 
.04447 
.05147 

1.05896 
.06693 
,07537 
.08428 
.09365 

.20 1.10347 
,21 .11374 
22 ' .12444 
,23 .13557 
.24 .14714 

1.15912 
.17150 
.18429 
.19746 
.21102 



1.00002 
.00032 
.00117 
.00256 
.00447 

1.00692 
.00989 
.01338 
.01741 
.02192 

1.02698 
.03254 
.03860 
.04515 
.05220 

1.05973 
.06775 
.07624 
.08519 
.09461 

1.10447 
.11479 
.12554 
.13671 



.14832 .14951 



1.16034 
.17276 
.18559 
.19880 
.21239 

1.22636 
.24070 
.25540 
.27044 
.28583 

1.30156 
.31761 
.33399 
.35068 
.36767 

1.38496 
.40254 
.42041 
.43673 .43856 
.45512 .45697 



22495 

23926 

.25391 

.26892 

.28428 

1.29997 
.31599 
.33234 
.34899 
c 36596 

1.38322 
.40077 
.41861 



.47377 
.49269 
,51185 
,53126 
, 55091 

50 1.57080 



1.47565 
.49460 
.51378 
. 53322 
. 55289 



1.00002 
.00038 
.00128 
.00272 
.00469 

1.00720 
.01021 
.01376 
.01784 
.02240 

1.02752 
.03312 
.03923 
.04584 
.05293 

1.06051 
.06858 
.07711 
.08611 
.09557 

1.10548 
.11584 
.12664 
.13785 



1.16156 

.17403 

18689 

.20014 

.21377 

1.22778 
.24216 
.25689 
.27196 
.28739 

1.30315 
.31923 
.33564 
.35237 
.36939 

1.38671 
.40432 
.42221 
.44039 
.45883 

1.47753 
.49651 
.51571 
.53518 
.55487 



1.00003 
.00045 
.00140 
.00289 
.00492 

1.00748 
.01054 
.01414 
.01828 
.02289 

1.02806 
.03371 
.03987 
.04652 
.05367 



1.00004 
.00053 
.00153 
.00307 
.00515 

1.00776 
.01088 
.01453 
.01872 
.02339 

1.02860 
.03430 
.04051 
.04722 
.05441 



1.06130 1.06209 
.06941 .07025 
.07799 .07888 
.08704 
.09654 



1.10650 
.11690 
.12774 
.13900 
.15070 

.16279 
.17530 
.18820 
.20149 
.21515 

1.22920 
.24361 
.25838 
.27349 
.28895 

1.30474 
.32086 
.33730 
.35406 
.37111 

1.38846 
.40610 
.42402 
.44222 
.46069 

1.47942 
.49842 
.51764 
.53714 
.55685 



1.00007 
.00061 
.00167 
.00327 
.00539 

1.00805 
.01123 
.01493 
.01916 
.02389 

1.02914 
.03490 
.04116 
.04792 
.05516 

1.06288 

.07109 

.07977 

08797 .08890 

09752 .09850 



1.10752 
.11796 
.12885 
.14015 
.15189 

1.16402 
.17657 
.18951 
.20284 
.21654 

1.23063 
.24507 
.25988 
.27502 
.29052 

1.30634 
.32249 
.33896 
.35575 
.37283 

1.39021 
.40788 
.42583 
.44405 
.46255 

1.48131 
. 50033 
.51958 
.53910 
.55884 



.10855 
.11904 
.12997 
.14131 
.15308 

1.16526 

.17784 
.19082 
.20419 
.21794 

1.23206 
.24654 
.26138 
.27656 
.29209 

1.30794 
.32413 
.34063 
.35744 
.37455 

1.39196 
.40966 
.42764 
.44589 
.46441 

1.48320 
. 50224 
.52152 
.54106 
.56083 



1.00010 
.00069 
.00182 
.00345 
.00563 

1.00834 
.01158 
.01533 
.01961 
.02440 

1.02970 
.03551 
.04181 
.04862 
.05591 

1.06368 
.07194 
.08066 
.08984 
.09949 

1.10958 
.12011 
.13108 
.14247 
.15428 

1.16650 
.17912 
.19214 
.20555 
.21933 

1.23349 

.24801 
.26288 
.27810 
.29366 

1.30954 
.32577 
.34229 
.35914 
.37628 

1.39372 
.41145 
.42945 
.44773 
.46628 

1.48509 
.50416 
.52346 
.54302 
.56282 



1.00013 
.00078 
.00196 
.00364 
.00587 

1.00864 
.01193 
.01573 
.02006 
.02491 

1.03026 
.03611 
.04247 
.04932 
.05667 

1.06449 
.07279 
.08156 
.09079 
.10048 

1.11062 
.12118 
.13219 
.14363 
.15549 

1.16774 
.18040 
.19346 
.20691 
.22073 

1.23492 
.24948 
.26437 
.27964 
.29523 

1.31115 
.32741 
.34396 
.36084 
.37801 

1.39548 
.41324 
.43127 
.44957 
.46815 

.48699 
.50608 
.52541 
.54499 
.56481 



1.00017 
.00087 
.00210 
.00384 
.00612 

1.00895 
.01228 
.01614 
.02052 
.02542 

1.03082 
.03672 
.04313 
.05003 
.05743 

1.06530 
.07365 
.08246 
.09174 
.10147 

1.11165 
.12225 
.13331 
.14480 
.15670 

1.16899 
.18169 
.19479 
.20827 
.22213 

1.23636 
.25095 
.26588 
.28118 
.29681 

1.31276 
.32905 
.34563 
.36254 
.37974 

1.39724 
.41503 
.43309 
.45142 
.47002 

1.48889 
.50800 
.52736 
.54696 
.56681 



Ex. — For a chord of 215 ft. and rise of 18 ft.: 
0.083721; and the corresponding arc = 215 X (1.01828 + 
1.0186=219.00 ft. Ans. 



The rise -r- chord = 
721 X .00044) = 215 X 



FLAT CIRCULAR ARC. 



211 



4a. — Lengths of Chords for Circular Arc 1. 
(Note. — Multiply the tabular number by the length of arc.) 



Centr 
Angl 


al 
e. 






Degree Units. 










Deg 


0. 


1. 


2. 


3. 


4. 


5. 


6. 


7. 


8. 


9. 


1— 
2- 
3- 

4- 


. 1.00000 
.99873 
.99493 

.98862 
.97982 


0.99999 
.99847 
.99441 
.98785 
.97880 


0.99995 
.99817 
.99387 
.98705 
.97776 


0.99989 
99786 
.99330 
.98624 
.97670 


0.99980 
.99751 
.99271 
.98539 
.97561 


0.99968. 
.99715 
.99209 
.98452 
.97450 


0.99954 
.99675 
.99144 
.98363 
.97336 


0.99938 
.99634 
.99077 
.98271 
.97220 


0.99919 
.99589 
.99008 
.98177 
.97101 


0.99897 
.99542 
.98936 
.98081 
.96980 


6- 
6- 
7- 
8- 
9- 


0.96857 
.95493 
.93896 
.92073 
.90032 


0.96731 
.95344 
.93723 
.91878 
.89816 


0.96603 
.95192 
.93549 
.91681 
.89598 


0.96473 
.95038 
.93372 
.91483 
.89378 


0.96340 
.94882 
.93193 
.91282 
.89156 


0.96205 
.94723 
.93012 
.91079 
.88932 


0.96067 
.94562 
.92828 
.90874 
.88706 


0.95927 
.94399 
.92643 
.90666 
.88478 


0.95785 
.94234 
.92455 
.90457 
.88248 


0.95640 
.94066 
.92265 
.90245 
.88016 


10- 
11— 
12— 
13- 
14- 


0.87782 
.85334 
.82699 
.79889 
.76915 


0.87546 
.85079 
.82426 
.79598 
.76609 


0.87308 
.84822 
.82151 
.79307 
.76302 


0.87068 
.84563 
.81874 
.79195 
.75993 


0.86826 
.84302 
.81595 
.78718 
.75683 


0.86582 
.84040 
.81315 
.78421 
.75371 


0.86337 
.83775 
.81033 
.78123 
.75058 


0.86089 
.83509 
.80750 
.77823 
.74743 


0.85839 
.83241 
.80464 
.77522 
.74427 


0.85588 
.82971 
.80177 
.77219 
.74110 


15- 
16- 
17- 
18— 


0.73791 
.70532 
.67150 
.63662 


0.73471 
.70199 
.66806 


0.73150 
.69865 
.66461 


0.72827 
.69529 
.66114 


0.72503 
.69193 
.65767 


0.72178 
.68855 
.65419 


0.71851 
.68517 
.65069 


0.71532 
.68177 
.64719 




0.71194 
. 67836 
.64367 


0.70863 
.67493 
.64015 



Ex.— For a central angle of 31K deg: the chord is 0.98765 ( =0.98785 minus H difT. 
or 20) times the arc. 




Flat Circular Arc. — 

Exact formulas: 

C2 = chord of half arc =\lh^ + (y) = ysec I oc ; 
e = external secant = r (sec ^ oc — 1) = r (exsec ^ a) 
/ = tangent distance = r tan § oc = ^ sec ^ OC . 
Approximate formulas: 
8c2 — c 



h^ 



X (c—x) 
2r 



ihx 



(c-x); 



8 C2"" 3 a; 



hi ~ quarter ordinate = ih I when ^ =7") » 



c =" chord 
e - ;;; 



(g = cen. of grav. of arc.) 



The coefficients , in Table 5 (a and b), following, give an idea of the ap- 
proximation of the above formulas; and they will also be found convenient 
to use in some cases where fairly correct values, simply, are sought. 



212 



11.— MENSURATION, 



5. — Coefficients to be used with Approximate Formulas (Page, pre- 
ceding) TO give Exact Values. (Fig. lOj 

(a) — Based on Central Angle oc = 10°, 20°, 30°, etc. 

(Note. — Multiply result from approximate formula, by Coefficient.) 



Central ^ 
Angle ^ " 


10° 


20° 


30° 


40° 


50° 


60° 


Values of - = 
c 


.02183 


.04374 


.06583 


.08816 


.11085 


.13397 


Values of - = 


1.00127 


1.00510 


1.01152 


1.02060 


1.03245 


1.04720 


Values of - = 
a 


.99873 


.99493 


.98862 


.97982 


.96857 


.95493 


Values of - = 
r 


.17431 


.34730 


.51764 


.68404 


.84524 


1.00000 



Formulas. 








Coefficients.* 






Sco- 
^- 3 


'- X 


1.00000 


1.00000 


.99999 


.99997 


.99992 


.99984 


c = (8^2- 


3a) X 


1.00000 


.99999 


.99997 


.99991 


.99977 


.99951 


Ife^h 


X 


1.00382 


1.01543 


1.03528 


1.06418 


1.10338 


1.15470 


^-t 


X 


.99810 


.99240 


.98296 


.96985 


.95315 


.93301 


1fht=ih 


X 


1.00017 


1.00064 


1.00143 


1.00256 


1.00401 


1.00581 


^h2=\h 


X 


1.00050 


1.00191 


1.00429 


1.00765 


1.01199 


1.01733 


^yi-lh 


X 


1.00014 


1.00052 


1.00116 


1.00204 


1.00321 


1.00464 



+ r>rc-^r „8smioc— sm^oc. 8sm^oc— IJ arc 

* Coefficient f or a = f • — ; f or c = -. — -. ' 

arc sm t OC 

for e = sec \ cc ; for h = cos^ \ oc ; for hi = f 



r 1 4(l-cosia:) . » 

1 — cos ^ oc c 



cos i oc — cos ^ ex * 
1 — cos ^ a 



-cos I oc 



^1 



t Increase in coefficient is nearly parabolic; that is, nearly proportional 

as)^ 

(20)5 



/■I ON 2 

toa2; thus, the coefficient of hi for oc = 18°, is 1+ .00064 X 757^2= 1.00052, 



using the coefficient of the nearest angle. 

t Decrease in coefficient is nearly parabolic. 



il 



FLAT CIRCULAR ARC. 213 



5. — CoEPPiciENTS TO BE USED WITH APPROXIMATE FORMULAS. — Concluded. 

h 
(b) Based on Rise to Chord, - , of Arc. 

(Note. — Multiply result from approximate formula, by Coefficient.) 



Values of — 
c 


= 


.02 


.04 


.06 


.08 


.10 


.12 


Values of oc 


= 


9° 09' 45'' 


18° 17' 44'' 


27° 22' 16" 


36° 21' 40" 


45° 14' 23" 


53° 58' 59" 


Values of — 
c 


= 


6.3 


3.1 


2.1 


1.6 


1.3 


1.1 


Values of — 
r 


= 


.16 


.32 


.47 


.62 


.77 


.91 


Formulas. 


Coefficients. 


^e =- h 


X 


1.003 


1.013 


1.029 


1.052 


1.083 


1.122 


^-1 


X 


.998 


.994 


.986 


.975 


.962 


.946 


ih=ih 


X 


1.000 


1.001 


1.001 


1.002 


1.003 


1.005 


ih=ih 


X 


1.0004 


1.0016 


1.0036 


1.0063 


1.0098 


1.0140 



t Increase in coefficient is nearly parabolic. 
t Decrease in coefficient is nearly parabolic. 

5a.— Middle Ordinates (or Rise h. Fig. 10) for Arc 1. 
(Note. — Multiply the tabular number by the length of arc.) 



Central 


Minutes. 




Angle. 






Deg. 


00' 


10' 


20' 


30' 


40' 


50' 


P.P. 


0'' 


0.000 000 


0.000 364 


0.000 727 


0.001 091 


0.001 454 


0.001 818 




364 363 362 


1° 


.002 182 


.002 545 


.002 909 


.003 273 


.003 636 


.004 000 


1 


36 36 36 


2° 


.004 363 


.004 727 


.005 090 


.005 454 


.005 818 


.006 181 


2 


73 73 72 


^o" 


.006 545 


.006 908 


.007 272 


.007 635 


.007 999 


.008 362 


3 


109 109 109 


4° 


.008 726 


.009 089 


.009 453 


.009 816 


.010 180 


.010 543 


4 


146 14S 14fl 


5' 


0.010 907 


0.011 270 


0.011 633 


0.011 997 


0.012 360 


0.012 724 


6 


182 181 181 


6' 


.013 087 


.013 450 


.013 814 


.014 177 


.014 540 


.014 904 


6 


218 218 217 


7" 


.015 267 


.015 630 


.015 993 


.016 357 


.016 720 


.017 083 


7 


255 254 253 


8" 


.017 446 


.017 809 


.018 173 


.018 536 


.018 899 


.019 262 


8 


291 290 290 


9° 


.019 625 


.019 988 


.020 351 


.020 714 


.021 077 


.021 440 


9 


328 327 328 


K 


0.021 803 


0.022 166 


0.022 529 


0.022 891 


0.023 254 


0.023 617 




362 36 1 360 


11" 


.023 980 


.024 343 


.024 705 


.025 068 


.025 431 


.025 793 


1 


36 36 36 


12" 


.026 156 


.026 519 


.026 881 


.027 244 


.027 606 


.027 969 


2 


72 72 72 


is; 


.028 331 


.028 694 


.029 056 


.029 418 


.029 781 


.030 143 


3 


109 108 108 


14' 


.030 505 


.030 867 


.031 230 


.031 592 


.031 954 


.032 316 


4 


145 144 144 


15* 


0.032 678 


0.033 040 


0.033 402 


0.033 764 


0.034 126 


0.034 488 


5 


181 180 180 


K 


.034 850 


.035 212 


.035 574 


.035 935 


.036 297 


.036 659 


6 


217 217 216 


17" 


.037 020 


.037 382 


.037 743 


.038 105 


.038 466 


.038 828 


7 


253 253 202 


18* 


.039 189 


.039 550 


.039 912 


.040 273 


.040 634 


.040 996 


8 


290 289 288 


19" 


.041 357 


.041 718 


.042 079 


.042 440 


.042 801 


.043 162 


9 


326 325 324 


20" 


0.043 523 


0.043 883 


0.044 244 


0.044 605 


0.044 966 


0.045 326 








Ex.— For a central angle of 12° 23' the middle ordinate to a circular curve Is 
0.026990 (= 0.026881 + 109) times the length of arc. 



214 



11.— MENSURA TION, 




Circular Segment; and Half Segment. — 

Diameter d = 2r. 
Point g = ) position f segment A ^ with 

gi = ! of center I i (segment A) I coordi- 
G = j of grav- ] segment B j nates 
Gi = J ity of Li (segment B) j 

Formulas for Center of Gravity (Fig. 1 1) : 

4sin2 ioc— sin2 ioc cosi-OC 



For smaller segment: 
OC = 0° to 180° 



For larger segment 
^ = 180° to 360° 

Or, yQ = 



xo=\r 
yo=lr 

Xo=\r 
Yo=lr 



iocj— sin^oc cos^oc 

sin^^g 

^oci— sin^oc cos^oc 

4sinH/? + sinH/? cos ^/? 
§/?! + sin^^ cos§i8 



x=Q\ y=yo. 

x = xo] y=yo. 
x = 0;y=Yo. 
x = Xo\ y=YQ. 

(Geometrical) 

i ^ A 7 

area A of segment 
^^ area A of segment 



^r 



Cb2 + ^ (if-r) 

area B of segment 
c3 



; 1^0 



area B of segment 
when a + ;9 = 360°. 



i/?, + sin|/3 cosiiS 
(^ yo^B_ 
12A' r" 125' Yo A 
Note that sin | a: = sin i 0; cos ^ oc = cos i /?; but sin I OC = cos i ^, etc. 
Table 6, below, gives tabular values. 



6. — Values (Coefficients) of xq, yo, Xq and Yq for Various Values of 

OC AND /?. (Fig. 11.) 

(Multiply the tabular Coefficient by the radius.) 





xo= 


yo= 




Xo= 


Yo= 




Xo = 


yo= 




Xo= 


Ko= 


OC 


r 


r 


a 


r 


r 


OC 


r 


r 


a 


r 


r 




times 


times 




times 


times 




times 


times 




times 


times 


0° 


.00000 


Unity 


360° 


.42441 


.00000 














80° 


.09728 


.97957 


330° 


.42565 


.00369 


120° 


.33920 


.70502 


240° 


.44162 


.17133 


60° 


.18930 


.91994 


300° 


.42211 


.03301 


150° 


.39058 


.56734 


210° 


.44162 


.28849 


90° 


.27124 


.82587 


270° 


.43972 


.08252 


180° 


.42441 


.42441 


180° 


.42441 


.42441 



CIRCULAR SEGMENTS, 



216 



Formulas for Area of Circular Segment (Fig 11; Tables 7, 8). — 
= area of segment not greater than a semi-circle, i. e., oc not > 180°. 
= area of segment not less than a semi-circle, i. e., p not < 180°. 

= ^ arc X radius— — (r — h)=r I— — r sin | OC cos ^ oc j =-k'(ci — r sin oc). (1) 



B 



i arc X radius+ — (H- 



)=r (y+r sin i^ cos \ ^ -jib + r sin /?). (2) 
[(1) and (2), above, are General Formulas.] 
Under "Circular Arc," a and b are given in terms of r, c and h; whence — 



A=r2 



5 = r2 



B 



A=h^ 



A=hc 



.0174533 a (deg.)-sin oc" 



.0174533 /?(deg.)-}-sin 



.0174533 oc(deg.) -sin 



8 sin2 I oc 
.0174533 /?(deg.) 



~i 



area of sector — area of triangle be- 
tween chord and radii. (3) 

area of sector + area of triangle be- 
tween chord and radii. (4) 
c 



in which sin i CX 



2^,tanl cx = -^. (5) 



8 sin2 i /? 
.0174533 oc (deg.) 



sm oc 



2 vers2 | oc 
.0174533 oc (deg.) -sin oc 



c c 

in which sin ^ /? = -jr- ; tan i ^ = ;rr. (6) 
Zr Zn 

h 
in which vers i OC = — = 2 sin2 J oc. (7) 
r 



2h 
in which tan i OC = — 
c 



(8) 



4 sin i oc vers ^oc 

Note. — When radius and central angle are given use equation (1), (2) 
(3) or (4); chord and central angle, use (5) or (6); rise and chord, use (5) 
(6), (7) or (8); radius and rise, use (7). See also foot-note to Table 8. 



6a.— Areas of Circular Segments for Chord Square (c2) = 1. 
(Note. — Multiply the tabular number by the square of the chord.) 



at 

o 

0-. 

1-. 

2-. 
3-. 

4-. 

6-. 
6-. 
7-. 
8-. 
9-. 

10-. 
11-. 
12-. 
13-. 
14-. 

15-. 
16-. 
17-. 
18-. 



Degree Units. 



0. 



.0000000 
.0145601 
.0292078 
.0440360 
,0591394 

0746196 
0905861 
1071606 
1244795 
1426990 

1619994 
1825916 
2047282 
2287093 
2549020 

2837574 
3158368 
.3518490 
.3926989 



. 0014642 
.0160182 
.0306814 
0455323 
.0606689 

0761941 
0922141 
1088565 
1262583 
1445772 

1639967 
.1847318 
2070378 
2312223 
2576594 

2868098 
3192480 
3556993 



2. 



0029548 
0174809 
0321556 
0470316 
0622020 

0777702 
0938482 
1105601 
1280460 
1464664 

1660072 
1868871 
2093661 
2337578 
2604438 

2898950 
3226990 
3595990 



0043562 
.0189407 
0336335 
,0485336 
0637392 

0793532 
0954887 
1122713 
1298431 
1483668 

1680310 
1890582 
2117135 
2367162 
2632559 

2930138 
3261910 
3635486 



0058293 
0204028 
0351121 
0500391 
0649804 

0809413 
0971356 
1139905 
1316500 
1502784 

1700684 
1912457 
2140804 
2388978 
2660960 

2961664 
3297241 
3675496 



0072793 
0218670 
0365937 
0515472 
0668258 

0825347 
0987883 
1157177 
1334661 
1522017 

1721196 
1934497 
2164670 
2415033 
2689647 

2993536 
3333002 
3716031 



6. 



0087348 
0233309 
0380773 
0530589 
0683755 

0841336 
1004493 
1174531 
1352923 
1541368 

1741849 
1956705 
2188736 
2441327 
2718626 

3025768 
3369194 
3757104 



7. 



.0101893 
.0247990 
.0395629 
.0545737 
.0699298 

. 0857380 
.1021166 
.1191968 
. 1371284 
. 1560839 

. 1762645 
.1979185 
.2213008 
2467870 
.2747902 

.3058357 
.3405830 
.3798722 



8. 



0116472 
0262660 
0410512 
0560920 
0714883 

0873483 
1037906 
1209490 
1389748 
1580432 

1783590 
2001639 
2237490 
2494664 

2777482 

3091315 
3442921 
3840814 



.0131014 
0277361 

. 0425422 
0576139 

.0730516 

.0889642 
1054720 
1227097 
1408316 
1600146 

1804680 
.2024370 
2262183 
2521711 
2807370 

3124651 
3480468 
3883651 



Ex. — For a central angle of 42 deg. and a chord length of 20 ft., the area of the 
segment = 0.062202 X 400 = 24.8808 sq. ft. 



216 



11.— MENSURA TION. 



'^,. 



/ 



7. — ^Table of Circular Segments — ^Areas, Etc. 

To find the area: Multiply the tabular coefficient in column 4 by r^\ or the 
coefficient in column 5 by hXc. 



1 


2 


3 


4 


5 


1 


2 


3 


4 


5 


73 (u ti) 






Arej 


I == 


"3 a> tab 






Area 


_ 


2,2 0) 


Rise h 
Radr 


Rise 71 
Chord c 








Rise h 


Rise h 






6<^ 


(Rad)2 


he 


(Rad)2 




Radr 


Chord c 


ho 






times 


times 






times 


times 


1° 


.00004 


.00218 


.0000004 


.6667 


51° 


.0974 


.1131 


.0564860 


.67344 


2 


.00015 


.00436 


.0000035 


.6667 


52 


.1012 


.1154 


.0597802 


.67373 


3 


.00034 


.00655 


,0000119 


.6667 


53 


.1051 


.1177 


.0631945 


.67400 


4 


.00061 


.00873 


.0000284 


.6667 


54 


.1090 


.1200 


.0667304 


.67429 


5 


.00095 


.01091 


.0000554 


.6667 


55 


.1130 


.1223 


.0703895 


.67458 


6 


.00137 


.01309 


.0000956 


.6667 


56 


.1171 


.1247 


.0741734 


.67488 


7 


.00187 


.01528 


.0001518 


.6667 


57 


.1212 


.1270 


.0780835 


.67519 


8 


.00244 


.01746 


.0002266 


.6668 


58 


.1254 


.1293 


.0821214 


.67550 


9 


.00308 


.01965 


.0003226 


.6669 


59 


.1296 


.1316 


.0862885 


.67582 


10 


.00381 


.02183 


.0004424 


.66690 


60 


.1340 


.1340 


.0905861 


.67614 


11 


.00460 


.02402 


.0005886 


.66697 


61 


.1384 


.1363 


.0950156 


.67647 


12 


.00548 


.02620 


.0007639 


.66702 


62 


.1428 


.1387 


.0995783 


.67681 


13 


.00643 


.02839 


.0009709 


.66708 


63 


.1474 


.1410 


.1042755 


.67715 


14 


.00745 


.03058 


.0012121 


.66716 


64 


.1520 


.1434 


.1091084 


.67750 


15 


.00856 


.03277 


.0014902 


.66725 


65 


.1566 


.1457 


.1140781 


.67786 


16 


.00973 


.03496 


.0018076 


.66731 


66 


.1613 


.1481 


.1191859 


.67822 


17 


.01098 


.03716 


.0021671 


.66740 


67 


.1661 


,1505 


.1244328 


.67859 


18 


.01231 


.03935 


.0025711 


.66749 


68 


.1710 


.1529 


.1298200 


.67897 


19 


.01371 


.04155 


.0030222 


.66758 


69 


.1759 


.1553 


.1353484 


.67935 


20 


.01519 


.04374 


.0035229 


.66769 


70 


.1808 


.1576 


.1410189 


.67974 


21 


.01675 


.04594 


.0040756 


.66779 


71 


.1859 


.1601 


.1468326 


.68014 


22 


.01837 


.04814 


.0046829 


.66790 


72 


.1910 


.1625 


.1527903 


.68054 


23 


.02008 


.05035 


.0053473 


.66802 


73 


.1961 


.1649 


.1588928 


.68095 


24 


.02185 


.05255 


.0060712 


.66814 


74 


.2014 


.1673 


.1651410 


.68136 


25 


.02370 


.05476 


.0068570 


.66826 


75 


.2066 


.1697 


.1715356 


.68179 


26 


.02563 


.05697 


.0077073 


.66840 


76 


.2120 


.1722 


1780773 


.68222 


27 


.02763 


.05918 


.0086242 


.66853 


77 


.2174 


.1746 


.1847667 


.68265 


28 


.02970 


.06139 


.0096103 


,66868 


78 


.2229 


.1771 


.1916046 


.68310 


29 


.03185 


.06361 


.0106679 


.66882 


79 


.2284 


.1795 


.1985915 


.68355 


30 


.03407 


.06583 


.0117994 


.66897 


80 


.2340 


.1820 


.2057278 


.68401 


31 


.03637 


.06805 


.0130070 


.66913 


81 


.2396 


.1845 


.2130142 


.68448 


32 


.03874 


.07027 


.0142930 


.66929 


82 


.2453 


.1869 


.2204509 


.68495 


33 


.04118 


.07250 


.0156598 


.66946 


83 


.2510 


.1894 


.2280385 


.68543 


34 


.04370 


.07473 


.0171095 


.66964 


84 


.2569 


.1919 


.2357773 


.68592 


35 


.04628 


.07696 


.0186444 


.66982 


85 


.2627 


.1944 


.2436676 


.68641 


36 


.04894 


.07919 


.0202666 


.67000 


86 


.2686 


.1970 


.2517095 


.68692 


37 


.05168 


.08143 


.0219784 


.67019 


87 


.2746 


.1995 


.2599035 


.68743 


38 


.05448 


.08367 


.0237818 


.67039 


88 


.2807 


.2020 


.2682495 


.68795 


39 


.05736 


.08592 


.0256790 


.67059 


89 


.2868 


.2046 


.2767477 


.68848 


40 


.06031 


.08816 


.0276721 


.67079 


90 


.2929 


.2071 


.2853982 


.68902 


41 


.06333 


.09041 


.0297630 


.67101 


91 


.2991 


.2097 


.2942010 


.68956 


42 


.06642 


.09267 


.0319538 


.67122 


92 


.3053 


.2122 


.3031561 


.69011 


43 


.06958 


.09493 


.0342466 


.67145 


93 


.3116 


.2148 


.3122634 


.69067 


44 


.07282 


.09719 


.0366432 


.67168 


94 


.3180 


.2174 


.3215227 


.69123 


45 


.07612 


.09946 


.0391457 


.67193 


95 


.3244 


.2200 


.3309340 


.69181 


46 


.07950 


.10173 


.0417558 


.67215 


96 


.3309 


.2226 


.3404971 


.69239 


47 


.08294 


.10400 


.0444755 


.67240 


97 


.3374 


.2252 


.3502116 


.69299 


48 


.08645 


.10628 


.0473066 


.67265 


98 


.3439 


.2279 


.3600773 


.69359 


49 


.09004 


.10856 


.0502504 


.67292 


99 


.3506 


.2305 


.3700938 


.69420 


50 


.09369 


.11085 


.0533101 


.67318 


100 


.3572 


.2332 


.3802607 


.69482 



CIRCULAR SEGMENTS. 



217 




7. — Table op Circular Segments — Areas, Etc.— Concluded. 

To find the area: Multiply the tabular coefficient in column 4 by r^; or the 
coefficient in column 5 by h X c. 



1 


2 


3 


4 


5 


1 


2 


3 


4 


5 


^ GJ bib 






Area 


t = 


"3 c) M, 






Area = 


— 


h-r.oi 


niseh 
Rad r 


Rise h 
Chord c 






6<^ 


Rise h 


Rise h 






Cent 
Ang 
OCD 


(Rad)2 


he 


(Rad)2 




Radr 


Chord c 


he 






times 


times 






times 


times 


101« 


.3639 


.2358 


.3905777 


.69545 


141° 


.6662 


.3534 


.9157969 


.72916 


102 


.3707 


.2385 


.4010441 


.69608 


142 


.6744 


.3566 


.9313530 


.73026 


103 


.3775 


.2412 


.4116595 


.69673 


143 


.6827 


.3599 


.9470029 


.73137 


104 


.3843 


.2439 


.4224234 


.69738 


144 


.6910 


.3633 


.9627444 


.73250 


105 


.3912 


.2466 


.4333350 


.69805 


145 


.6993 


.3666 


.9785755 


.73364 


106 


.3982 


.2493 


.4443937 


.69872 


146 


.7076 


.3700 


.9944939 


.73480 


107 


.4052 


.2520 


.4555988 


.69941 


147 


.7160 


.3734 


1.0104975 


.73598 


108 


.4122 


.2548 


.4669495 


.70010 


148 


.7244 


.3768 


1.0265840 


.73717 


109 


.4193 


.2575 


.4784451 


.70080 


149 


.7328 


.3802 


1.0427512 


.73838 


110 


.4264 


.2603 


.4900848 


.70151 


150 


.7412 


.3837 


1.0589969 


.73960 


111 


.4336 


.2631 


.5018675 


.70223 


151 


.7496 


.3871 


1.0753188 


.74084 


112 


.4408 


.2659 


.5137924 


.70295 


152 


.7581 


.3906 


1.0917144 


.74210 


113 


.4481 


.2687 


.5258586 


.70371 


153 


.7666 


.3942 


1.1081816 


.74337 


114 


.4554 


.2715 


.5380649 


.70446 


154 


.7751 


.3977 


1.1247180 


.74466 


115 


.4627 


.2743 


.5504104, 


.70522 


155 


.7836 


.4013 


1.1413210 


.74597 


116 


.4701 


.2772 


.5628940 


.70600 


156 


.7921 


.4049 


1.1579885 


.74730 


117 


.4775 


.2800 


.5755144 


.70678 


157 


.8006 


.4085 


1.1747179 


.74865 


118 


.4850 


.2829 


.5882705 


.70758 


158 


.8092 


.4122 


1.1915068 


.75001 


119 


.4925 


.2858 


.6011611 


.70838 


159 


.8178 


.4158 


1.2083528 


.75140 


120 


.5000 


.2887 


.6141849 


.70920 


160 


.8264 


.4195 


1.2252534 


.75280 


121 


.5076 


.2916 


.6273405 


.71003 


161 


.8350 


.4233 


1.2422059 


.75422 


122 


.5152 


.2945 


.,6406268 


.71087 


162 


.8436 


.4270 


1.2592082 


.75566 


123 


.5228 


.2975 


.6540422 


.71172 


163 


.8522 


.4308 


1.2762575 


.75713 


124 


.5305 


.3004 


.6675853 


.71258 


164 


.8608 


.4346 


1.2933513 


.75861 


125 


.5383 


.3034 


.6812548 


.71345 


165 


.8695 


.4385 


1.3104871 


.76011 


126 


.5460 


.3064 


.6950489 


.71434 


166 


.8781 


.4424 


1.3276623 


.76164 


127 


.5538 


.3094 


.7089613 


.71523 


167 


.8868 


.4463 


1.3448744 


.76318 


128 


.5616 


.3124 


.7230053 


.71615 


168 


.8955 


.4502 


1.3621207 


.76475 


129 


.5695 


.3155 


.7371644 


.71707 


169 


.9042 


.4542 


1.3793987 


.76634 


130 


.5774 


.3185 


.7514418 


.71800 


170 


.9128 


.4582 


1.3967058 


.76795 


131 


.5853 


.3216 


.7658359 


.71895 


171 


.9215 


.4622 


1.4140393 


.76959 


132 


.5933 


.3247 


.7803449 


.71991 


172 


.9302 


.4663 


1.4313966 


.77124 


133 


.6013 


.3278 


.7949671 


.72088 


173 


.9390 


.4704 


1.4487752 


.77293 


134 


.6093 


.3309 


.8097007 


.72187 


174 


.9477 


.4745 


1.4661722 


.77463 


135 


.6173 


.3341 


.8245438 


.72287 


175 


.9564 


.4786 


1.4835852 


.77636 


136 


.6254 


.3373 


.8394947 


.72388 


176 


.9651 


.4828 


1.5010115 


.77812 


137 


.6335 


.3404 


.8545517 


.72491 


177 


.9738 


.4871 


1.5184484 


.77990 


138 


.6416 


.3436 


.8697119 


.72595 


178 


.9826 


.4913 


1.5358933 


.78171 


139 


.6498 


.3469 


.8849743 


.72701 


179 


.9913 


.4957 


1.5533435 


.78354 


140 


.6580 


.3501 


.9003667 


.72808 


180 


unity 


.5000 


1.5707963 


.78540 






/r 


2 \ 


r 














/-2 





ment is greater than a semi-circle and we denote the rise by H ( = 2r — h), 

thenr=i (^+^) iH^y+Jr^-— \c = 2\/H {2r-H). The area of a seg- 

ment greater than a semi-circle, i. e., with rise H, = area of circle (3.1416 r^) 
— area of segment of rise h ( =- 2 r — H) . 



218 U.—MENSURA TION. 



8. — *Areas op Segments op Circles for Diameter 1. 

To find the area of any circular segment: Multiply the tabular area 
below, corresponding to the particular value of rise -&- diam., by the square 
of the diam. 



[^ 



Note that the diam. = rise + (hali chord) ^-j- rise; hence, {diam,.)^ 
' • and, rise -^ diam.= rise^ -i- [rise^ + (half chord)2]. 



rise 



']■ 



m 
I 



li 


Thousandths. 




.000 


.001 


.002 


.003 


.004 


.005 


.006 


.007 


.008 


.009 


.00 


.000000 


.000042 


.000119 


.000219 


.000337 


.000471 


.0UU619 


.000779 


.000952 


.001135 


.01 


.001329 


.001533 


.001746 


.001969 


.002199 


.002438 


.002685 


.002940 


.003202 


.003472 


.02 


.003749 


.004032 


.004322 


.004619 


.004922 


.005231 


.005546 


.005867 


.006194 


.006527 


-03 


.006866 


.007209 


.007559 


.007913 


.008273 


.008638 


.009008 


.009383 


.009763 


.010148 


.04 


.010538 


.010932 


.011331 


.011734 


.012142 


.012555 


.012971 


.013393 


.013818 


.014248 


.05 


.014681 


.015119 


.015561 


.016008 


.016458 


.016912 


.017369 


.017831 


.018297 


.018766 


.06 


.019239 


.019716 


.020197 


.020681 


.021168 


.021660 


.022155 


.022653 


.023155 


.023660 


.07 


.024168 


.024680 


.025196 


.025714 


.026236 


.026761 


.027290 


.027821 


.028356 


.028894 


.08 


.029435 


.029979 


.030526 


.031077 


.031630 


.032186 


.032746 


.033308 


.033873 


.034441 


.09 


.035012 


.035586 


.036162 


.036742 


.037324 


.037909 


.038497 


.039087 


.039681 


.040277 


.10 


.040875 


.041477 


.042081 


.042687 


.043296 


.043908 


.044523 


.045140 


.045759 


.046381 


.11 


.047006 


.047633 


.048262 


.048894 


.049529 


.050165 


.050805 


.051446 


.052090 


.052737 


.12 


.053385 


.054037 


.054690 


.055346 


.056004 


.056664 


.057327 


.057991 


.058658 


.059328 


.13 


.059999 


.060673 


.061349 


.062027 


.062707 


.063389 


.064074 


.064761 


.065449 


.066140 


.14 


.066833 


.067528 


.068225 


.068924 


.069626 


.070329 


.071034 


.071741 


.072450 


.073162 


.15 


.073875 


.074590 


.075307 


.076026 


.076747 


.077470 


.078194 


.078921 


.079650 


.080380 


.16 


.081112 


.081847 


.082582 


.083320 


.084060 


.084801 


.085545 


.086290 


.087037 


.087785 


.17 


.088536 


.089288 


.090042 


.090797 


.091555 


.092314 


.093074 


.093837 


.094601 


.095367 


.18 


.096135 


.096904 


.097675 


.098447 


.099221 


.099997 


.100774 


.101553 


.102334 


.103116 


.19 


.103900 


.104686 


.105472 


.106261 


.1U7051 


.107843 


.108636 


.109431 


.110227 


.111025 


.20 


.111824 


.112625 


.113427 


.114231 


.115036 


.115842 


.116651 


.117460 


.118271 


.119084 


.21 


.119898 


.120713 


.121530 


.122348 


.123167 


.123988 


.124811 


.125634 


.126459 


.127286 


.22 


.128114 


.128943 


.129773 


.130605 


.131438 


.132273 


.133109 


.133946 


.134784 


.135624 


.23 


.136465 


.137307 


.138151 


.138996 


.139842 


.140689 


.141538 


.142388 


.143239 


.144091 


.24 


.144945 


.145800 


.146656 


.147513 


.148371 


.149231 


.150091 


.150953 


.151816 


.152681 


.25 


.153546 


.154413 


.155281 


.156149 


.157019 


.157891 


.158763 


.159636 


.160511 


.161386 


.26 


.162263 


.163141 


.164020 


.164900 


.165781 


.166663 


.167546 


.168431 


.169316 


.170202 


.27 


.171090 


.171978 


.172868 


.173758 


.174650 


.175542 


.176436 


.177330 


.178226 


.179122 


.28 


.180020 


.180918 


.181818 


.182718 


.183619 


.184522 


.185425 


.186329 


.187235 


.188141 


.29 


.189048 


.189956 


.190865 


.191774 


.192685 


.193597 


.194509 


.195423 


.196337 


.197252 


.30 


.198168 


.199085 


.200003 


.200922 


.201841 


.202762 


.203683 


.204605 


.205528 


.206452 


.31 


.207376 


.208302 


.209228 


.210155 


.211083 


.212011 


.212941 


.213871 


.214802 


.215734 


.32 


.216666 


.217600 


.218534 


.219469 


.220404 


.221341 


.222278 


.223216 


.224154 


.225094 


.33 


.226034 


.226974 


.227916 


.228858 


.229801 


.230745 


.231689 


.232634 


.233580 


.234526 


.34 


.235473 


.236421 


.237369 


.238319 


.239268 


.240219 


.241170 


.242122 


.243074 


.244027 


.35 


.244980 


.245935 


.246890 


.247845 


.248801 


.249758 


.250715 


.251673 


.252632 


.253591 


.36 


.254551 


.255511 


.256472 


.257433 


.258395 


.259358 


.260321 


.261285 


.262249 


.263214 


.37 


.264179 


.265145 


.266111 


.267078 


.268046 


.269014 


.269982 


.270951 


.271921 


272891 


.38 


.273861 


.274832 


.275804 


.276776 


.277748 


.278721 


.279695 


.280669 


.281643 


.282618 


.39 


.283593 


.284569 


.285545 


.286521 


.287499 


.288476 


.289454 


.290432 


.291411 


.292390 


.40 


.293370 


.294350 


.295330 


.296311 


.297292 


.298274 


.299256 


.300238 


.301221 


.302204 


.41 


.303187 


.304171 


.305156 


.306140 


.307125 


.308110 


.309096 


.310082 


.311068 


.312055 


.42 


.313042 


.314029 


.315017 


.316005 


.316993 


.317981 


.318970 


.319959 


.320949 


.321938 


.43 


.322928 


.323919 


.324909 


.325900 


.326891 


.327883 


.328874 


.329866 


.330858 


.331851 


.44 


.332843 


.333836 


.334829 


.335823 


.336816 


.337810 


.338804 


.339799 


.340793 


.341788 


.45 


.342783 


.343778 


.344773 


.345768 


.346764 


.347760 


.348756 


.349752 


.350749 


.351745 


.46 


.352742 


.353739 


.354736 


.355733 


.356730 


.357728 


.358725 


359723 


.360721 


.361719 


.47 


.362717 


.363715 


.364714 


.365712 


.366711 


.367710 


.368708 


369707 


.370706 


.371705 


.48 


.372704 


.373704 


.374703 


.375702 


.376702 


.377701 


.378701 


379701 


.380700 


.381700 


.49 


.382700 


.383700 


.384699 


.385699 


.386699 


.387699 


.388699 


389699 


.390699 


.391699 


.50 


.392699 





















^Calculated from formula: Area A =0.3926990817- \x{r^-x'^)^-^rH\rr^ 
in which r = radius of circle, and rx: = radius — rise = 0.5 — rise. 



n- 



Note that sin~i 



the angle (in circular measure) whose natural sine is 



For angles reduced to circular measure, see Tables 2 and 3. 



CIRCULAR RING. 



219 



Let 



Circular Ring ; and Half Circular Ring. — 

R — radius of outer circle = -^ ; 

r = radius of inner circle = -»- ; 

Re— radius of circle with area equivalent to that 

circular ring; 
A = area of circular ring; ??= 3.1416. 




Fig. 12. 



Then R,= \^R'^-r^=V{R + r) {R-r) = \V{D + d){D 



^>=v?= 



A = 7ri?e2 = ;r(i?2_r2) = ;r(i? + r)(i?-r) = -|-(J92-d2) = ^(r> + ci)(D-J); 



R = /r2+ ^ = V'f2 + i?,2; L>= ^2 + 



4A 



V^^-T= 



\/i?2_i?^2; 



V^^-T- 



0.21221 (D^-d^) 



distance 



Formula for Center of Gravity (Fig. 12) : 

X, ^ ,c ' 1 • 0.42441 (i?3-r3) 
For half circular ring, xq=- (/?'> ~^ 

to cen. of grav. g. 

The above value of xo may be obtained from Pappus's Theorem, page 243, 
by using it inversely. Thus, we know that the volume generated by the 
revolution of a plane area (lying wholly on one side of an axis) about its 
axis = the area X the path described by its center of gravity. Hence, 
volume V=2 7c xq A; or 

_ V volume of spherical shell _ J_ J 

° 2;rA 2;: . Area of half circular ring 2n x 



2tz . Area of half circular ring 
0.42441. 



(i?2 _ r2) 



Pappus's Theorem is useful in finding the centers of gravity of any 
figures (lines or areas) of revolution. 



Area 



Zone, and Half Zone, of Circle. — 

Area of zone = Z = area of circle (3.1416 r2) 
— (A-\-B); in which r = radius of circle, A = 
area of upper segment, and B = area of lower 
segment. (See preceding tables and formulas 
for areas of segments.) 

Formulas for Center of Gravity (Fig. 1 3) : 
Let g = cen. of grav. of zone, with coordinates 
x = 0,y=^yo; 

gi = cen. of grav. of half zone, with coor- 
dinates x = xo, y=yo', 

A = area of upper segment, with cen. of 
grav. dist. :Vafrom axis Xi — Xi\ 

Z = area of zone, with cen. of grav. dis- 
tant y^ from Xi — Xt 

B = area of lower segment, with cen. of grav. dist. yh from Xi — Xi; 

C = area of circle (=3. 1416r2), with cen. of grav. distantrfromXi—Xi; 




Then , xq = 



yo 



0.42441 rC-Ait:» -Bx^ 



Cr - Ay^ - By^ 



Ordinates x^, x^,, y^ and yy, 
may be solved by use of formulas 
in connection with Fig. 11. 



220 



11.— MENSURA TION, 








Fig. 14. 

Circular Lune (Fig. 14). — 

Area A = area of segment with rise h minus area of segment with rise h'. 
(Common chord c.) 
See preceding Tables, 7 and 8, of Circular Segments. 

Circular Sector; and Half Sector (Fig. 15). — 
Point g = 1 position f sector A 1 with lx = 0',y = yo 
gi = [of center I ^ (sector A) 1 coordi- I rr = iCo ; 3^ = yo 
G = f of grav- 1 sector B [nates. \ x = 0] y=Yo 
6^i=Jityof lHsector5)J [x = Xo;y==Yo 

Formulas for Center of Gravity and Area (Fig. 15): 
, sin i OC , sin I oc , „sin i oc , ,sin ^ a 
t oCj oCi a A 



yo 

Xq 
Yq 



yo tan i oc = | r 
Fotan i p ==tr 



vers Joe _ ^ ^2 vers Joe _ ^ ^3 vers |QC 



0C1 



3^^ 



a 

sin J /? 



|r3 



vers J 



^1 



=l^2Y££ili = |H 



6 
.sin J OC 



= lr3 



^sin_Jj9_ 

B 
vers J /? 

5 ' 
vers J ex 



Ar^a J5 = Jr2)9i =J6r 



Ar^a A = ir^(Xt = iar 

yo Xo 

3 sin J/? ^ vers J/? . 

In which a = length of arc of sector of area A and central angle OC, 

b = length of arc of sector of area B and central angle IS, 

OCi = .0174533 OC (degrees), (See Tables 2 and 3 of length of 

/?! = .0174533 13 (degrees). circular arcs to radius 1.) 

Relations of Circle and Square. — 

Let D = diam. of circle; C = circum. of circle; A = area of circle. 

d = diag. of square; s = side of square; p = perimeter of square; 
a = area of square. 

Then, D = —= -^\ C= ttD = -yr- A = -7-Z;2 = = _ 
;r 6 1/ 4 4 4;: 



d = 5\/2 =-|V2 



V2a; 5= — ^=y-=\/a ;p=2dV2 =is=-Wa 
V2 4 



2 16 

(For convenience of calculation: ;:= 3.141592+ , log = 0.4971499; 

log = 9.5028501; 



■^ = 0.785398 + , log = 9.8950899; — = 0.318310 
4 7: 



log 4 = 0.6020600; log i = 9.3979400; log 2 = 0.3010300; 

V2"=1.414214-, log = 0.1505150; —?= =0.707107, log = 9.8494850.) 

V2 



CIRCLE AND SQUARE, RELATIONS. 



221 



y 




^ / 


Ci 


/- 


Sm 




u <u 




S u 




fl.<1 


11 11 


:^ 


II II 


cr 


•<;\« 


C/3 






o s 



S «*3 

2 II 



^ o 



> 

00 
CO 

oo 



> 



> 



1-^ 
> 






*u_l"fe'"l> 



> 



> 



H>1 




o 

a 

II 1! 



'> 



II 2 

'? II 

< I Hi-* 

H II 

Q 



> 



> 



o 



•^ <o 



's' 

oo 



> 






til- 



K loo ^ IS \i\-^ 



•X^i^-Biaba; 



*5- = 'a 



1 7 


Vh 


j^ 


00 


r- 


0) 


cj 


oo 


0) 

a^ 


O 


II 






(U 




C| 


Q 


rt 


II II 


• 


1 


^ 


1 




l(M 


cr 

CO 


^« 




<M 


> 



C5 O 



-,.>l 








o^ 






^ r:! 



■«^ CO 

1-H O 

II II 

to ^ 



""^ -^ 






i<N I'M 

>> 



■^ I'M 



l(M 

> 



O 



I Q 
> 

00 
00 
(M 



lo 

> 



^ 



> 

II 



o 
oo t>. 

O 1-i 






® to 






•X:^n'Bnba; 






222 



11.— MENSURA TION, 









1-^ 


K 






















1 






> 

<M 

1^ 
> 




o 


s i 

T-; o 

II 11 


1 ^ 

CO 05 

d 

Q II 


d 

II 


d 
II 




/■ 




& 




g II II 


1 


> 


Q 


l^^l II 


i^ V 


1 1 


1 <M 


II _ 


S ^« 


> 


rt* 


».u 


> 


->h 


^ -b 


1^ 


c^ 


(M 


> 




II 


II II 


II 


II II 


II II 


II 


II 


II 




•^ 


v> ^ 


Q 


t3 «o 


^ « 


-« 


<o 


^ 








Iq 




lo 














> 




^ > 








. 


N 


«, M 




t3 ^ 


o. S 


-« 


«0 


^ 


8 


'd 


% ^ 


c5 


i S 


s ^ 


s 


1 


c5 


feren 


1 


1 1 




d ^ 


OO «5 
*^ CO 

=» II 

II « 


<N 


CO 


d 


6 ^^ 


o 


T-H O 


> 


'ti «o 


-Q 


to 


II 


o II 11 






"!'> 


l<^l<=5' 11 ti 


^ 






> 


>h 




II 


II n 


n 


11 11 


II 11 


II 


11 


II 




^ 




Q 




o 








A^ii'BXih'gi 








(•S^3JB 
D = 




















N 


K 


K 






^ 






Q 


> 


> 


> 






G 


Q 


i ^ 


Q 1 




CO 


■^ 


. 


O 

II 


i i 

<M o 

O |] 

II « 


<0 CO 

s II 

d K 
II > 

w * 


§ 


d 

> 




y: 


<u 

See 


05 
CO 


3 i 

d 


> 


o 
d 


t 




§ II II 


1 1 


^ V 


q 


Ic 


Q 


? 1- 


1 *=i 


1 1 


-^t: 


S '■' 


■^ 


Tt( 


;. ^ 


N 


> 




^IS ^ 


> 


CS| 


1 K 
> 


. 


J^ 




II 


II II 


11 


II II 


II II 


II 


II 


II 




^ 


to e 


j;^ 


-Q to 


Q <^ 


-^ 


«0 


Q 




'^ 


Iq 




ti 


> 








(3 


(U 


CO 


> 

CD -^ 


7:i 


O to 

d g? 




^ 


<o 




VJ 


L^s . 


II 


rtJ " 


CO 

d 


II ^. 


II d 


d 


T-1 


1— t 


<"* S ^ 


-« 


1 


II 


■? II 


l« II 


11 


II 


II 


1 o< 


c 


II 1 « 

I > 


^ 


i-'^L ■" 


> -^ 


^ 


% 


e 


O II 11 


> 




1 


> 1 


1 1 N 


1 


1 


1 


^^ 


<N 


Tt* M* 


rnl ti 


<mI -«ii| ti 


-^l ^ '-1^ 


(nI fe 


^1 ti 


^\ fe 




II 


II II 


II 


II II 


II II 


II 


II 


II 




^ 




Q 




'^^ 








M 


ipr 


ibg 










( 


sa9:).9uiij3d 


IBnbg) 















INCHES TO DECIMALS OF A FOOT, 



223 



10. — Decimals of a Foot for Each Ve* op an Inch. 



Inch 


0" 


V 


2" 


3" 


4" 


5" 


6" 


1" 


8" 


9" 


10» 


11' 


Inch 








.0833 


.1667 


.2500 


.3333 


.4167 


.5000 


.5833 


.6667 


.7500 


.8333 


.9167 





1-64 


.0013 


.0846 


.1680 


.2513 


.3346 


.4180 


.5013 


.5846 


.6680 


.7513 


.8346 


.9180 


1-64 


1-32 


.0026 


.0859 


.1693 


.2526 


.3359 


.4193 


.5026 


.5859 


.6693 


.7526 


.8359 


.9193 


1-32 


3-64 


.0039 


.0872 


.1706 


.2539 


.3372 


.4206 


.5039 


.5872 


.6706 


.7539 


.8372 


.9206 


3-64 


1-16 


.0052 


.0885 


.1719 


.2552 


.3385 


.4219 


.5052 


.5885 


.6719 


.7552 


.8385 


.9219 


1-16 


5-64 


.0065 


.0898 


.1732 


.2565 


.3398 


.4232 


.5065 


.5898 


.6732 


.7565 


.8398 


.9232 


5-64 


3-32 


.0078 


.0911 


.1745 


.2578 


.3411 


.4245 


.5078 


.5911 


.6745 


.7578 


.8411 


.9245 


3-32 


7-64 


.0091 


.0924 


.1758 


.2591 


.3424 


.4258 


.5091 


.5924 


.6758 


.7591 


.8424 


.9258 


7-64 


1-8 


.0104 


.0937 


.1771 


.2604 


.3437 


.4271 


.5104 


.5937 


.6771 


.7604 


.8437 


.9271 


1-8 


9-64 


.0117 


.0951 


.1784 


.2617 


.3451 


.4284 


.5117 


.5951 


.6784 


.7617 


.8451 


.9284 


9-64 


5-32 


.0130 


.0964 


.1797 


.2630 


.3464 


.4297 


.5130 


.5964 


.6797 


.7630 


.8464 


.9297 


5-32 


11-64 


.0143 


.0977 


.1810 


.2643 


.3477 


.4310 


.5143 


.5977 


.6810 


.7643 


.8477 


.9310 


11-64 


3-16 


.0156 


.0990 


.1823 


.2656 


.3490 


.4323 


.5156 


.5990 


.6823 


.7656 


.8490 


.9323 


3-16 


13-64 


.0169 


.1003 


.1836 


.2669 


.3503 


.4336 


.5169 


.6003 


.6836 


.7669 


.8503 


.9336 


13-64 


7-32 


.0182 


.1016 


.1849 


.2682 


.3516 


.4349 


.5182 


.6016 


.6849 


.7682 


.8516 


.9349 


7-32 


15-64 


.0195 


.1029 


.1862 


.2695 


.3529 


.4362 


.5195 


.6029 


.6862 


.7695 


.8529 


.9362 


15-64 


1-4 


.0208 


.1042 


.1875 


.2708 


.3542 


.4375 


.5208 


.6042 


.6875 


.7708 


.8542 


.9375 


1-4 


17-64 


.0221 


.1055 


.1888 


.2721 


.3555 


.4388 


.5221 


.6055 


.6888 


.7721 


.8555 


.9388 


17-64 


9-32 


.0234 


.1068 


.1901 


.2734 


.3568 


.4401 


.5234 


.6068 


.6901 


.7734 


.8568 


.9401 


9-32 


19-64 


.0247 


.1081 


.1914 


.2747 


.3581 


.4414 


.5247 


.6081 


.6914 


.7747 


.8581 


.9414 


19-64 


5-16 


.0260 


.1094 


.1927 


.2760 


.3594 


.4427 


.5260 


.6094 


.6927 


.7760 


.8594 


.9427 


5-16 


21-64 


.0273 


.1107 


.1940 


.2773 


.3607 


.4440 


.5273 


.6107 


.6940 


.7773 


.8607 


.9440 


21-64 


11-32 


.0286 


.1120 


.1953 


.2786 


.3620 


.4453 


.5286 


.6120 


.6953 


.7786 


.8620 


.9453 


11-32 


23-64 


.0299 


.1133 


.1966 


.2799 


.3633 


.4466 


.5299 


.6133 


.6966 


.7799 


.8633 


.9466 


23-64 


f 


.0312 


.1146 


.1979 


.2812 


.3646 


.4479 


.5312 


.6146 


.6979 


.7812 


.8646 


.9479 


t 


25-64 


.0326 


.1159 


.1992 


.2826 


.3659 


.4492 


.5326 


.6159 


.6992 


.7826 


.8559 


.9492 


25-64 


13-32 


.0339 


.1172 


.2005 


.2839 


.3672 


.4505 


.5339 


.6172 


.7005 


.7839 


.8672 


.9505 


13-32 


27-64 


.0352 


.1185 


.2018 


.2852 


.3685 


.4518 


.5352 


.6185 


.7018 


.7852 


.8685 


.9518 


27-64 


7-16 


.0365 


.1198 


.2031 


.2865 


.3698 


.4531 


.5365 


.6198 


.7031 


.7865 


.8698 


.9531 


7-16 


29-64 


.0378 


.1211 


.2044 


.2878 


.3711 


.4544 


.5378 


.6211 


.7044 


.7878 


.8711 


.9544 


29-64 


15-32 


.0391 


.1224 


.2057 


.2891 


.3724 


.4557 


.5391 


.6224 


.7057 


.7891 


.8724 


.9557 


15-32 


31-64 


.0404 


.1237 


.2070 


.2904 


.3737 


.4570 


.5404 


.6237 


.7070 


.7904 


.8737 


.9570 


31-64 


i 


.0417 


.1250 


.2083 


.2917 


.3750 


.4583 


.5417 


.6250 


.7083 


.7917 


.8750 


.9583 


i 


33-64 


.0430 


.1263 


.2096 


.2930 


.3763 


.4596 


.5430 


.6263 


. 7096 


.7930 


.8763 


.9596 


33-64 


17-32 


.0443 


.1276 


.2109 


.2943 


.3776 


.4609 


.5443 


.6276 


.7109 


.7943 


.8776 


.9609 


17-32 


35-64 


.0456 


.1289 


.2122 


.2956 


.3789 


.4622 


.5456 


.6289 


.7122 


.7956 


.8789 


.9622 


35-64 


9-16 


.0469 


.1302 


.2135 


.2969 


.3802 


.4635 


.5469 


.6302 


.7135 


.7969 


.8802 


.9635 


9-16 


37-64 


.0482 


.1315 


.2148 


.2982 


.3815 


.4648 


.5482 


.6315 


.7148 


.7982 


.8815 


.9648 


37-64 


19-32 


.0495 


.1328 


.2161 


.2995 


.3828 


.4661 


.5495 


.6328 


.7161 


.7995 


.8828 


.9661 


19-32 


39-64 


.0508 


.1341 


.2174 


.3008 


.3841 


.4674 


.5508 


.6341 


.7174 


.8008 


.8841 


.9674 


39-64 


f 


.0521 


.1354 


.2188 


.3021 


.3854 


.4688 


.5521 


.6354 


.7188 


.8021 


.8854 


.9688 


i 


41-64 


.0534 


.1367 


.2201 


.3034 


.3867 


.4701 


.5534 


.6367 


.7201 


.8034 


.8867 


.9701 


41-64 


21-32 


.0547 


.1380 


.2214 


.3047 


.3880 


.4714 


.5547 


.6380 


.7214 


.8047 


.8880 


.9714 


21-32 


43-64 


.0560 


.1393 


.2227 


.3060 


.3893 


.4727 


.5560 


.6393 


.7227 


.8060 


.8893 


.9727 


43-64 


11-16 


.0573 


.1406 


.2240 


.3073 


.3906 


.4740 


.5573 


.6406 


.7240 


.8073 


.8906 


.9740 


11-16 


45-64 


.0586 


.1419 


.2253 


.3086 


.3919 


.4753 


.5586 


.6419 


.7253 


.8086 


.8919 


.9753 


45-64 


23-32 


.0599 


.1432 


.2266 


.3099 


.3932 


.4766 


.5599 


.6432 


.7266 


.8099 


.8932 


.9766 


23-32 


47-64 


.0612 


.1445 


.2279 


.3112 


.3945 


.4779 


.5612 


.6445 


.7279 


.8112 


.8945 


.9779 


47-64 


f 


.0625 


.1458 


.2292 


.3125 


.3958 


.4792 


.5625 


.6458 


.7292 


.8125 


.8958 


.9792 


f 


49-64 


.0638 


.1471 


.2305 


.3138 


.3971 


.4805 


.5638 


.6471 


.7305 


.8138 


.8971 


.9805 


49-64 


25-32 


.0651 


.1484 


.2318 


.3151 


.3984 


.4818 


.5651 


.6484 


.7318 


.8151 


.8984 


.9818 


25-32 


51-64 


.0664 


.1497 


.2331 


.3164 


.3997 


.4831 


.5664 


.6497 


.7331 


.8164 


.8997 


.9831 


51-64 


13-16 


.0677 


.1510 


.2344 


.3177 


.4010 


.4844 


.5677 


.6510 


.7344 


.8177 


.9010 


.9844 


13-16 


53-64 


.0690 


.1523 


.2357 


.3190 


.4023 


.4857 


.5690 


.6523 


.7357 


.8190 


.9023 


.9857 


53-64 


27-32 


.0703 


.1536 


.2370 


.3203 


.4036 


.4870 


.5703 


.6536 


.7370 


.8203 


.9036 


.9870 


27-32 


5^64 


.0716 


.1549 


.2383 


.3216 


.4049 


.4883 


.5716 


.6549 


.7383 


.8216 


.9049 


.9883 


55-64 


i 


.0729 


.1562 


.2396 


.3229 


.4062 


.4896 


.5729 


.6562 


.7396 


.8229 


.9062 


.9896 


i 


57-64 


.0742 


.1576 


.2409 


.3242 


.4076 


.4909 


.5742 


.6576 


.7409 


.8242 


.9076 


.9909 


57-64 


29-32 


.0755 


.1589 


.2422 


.3255 


.4089 


.4922 


.5755 


.6589 


.7422 


.8255 


.9089 


.9922 


29-32 


59-64 


.0768 


.1602 


.2435 


.3268 


.4102 


.4935 


.5768 


.6602 


.7435 


.8268 


.9102 


.9935 


59-64 


15-16 


.0781 


.1615 


.2448 


.3281 


.4115 


.4948 


.5781 


.6615 


.7448 


.8281 


.9115 


.9948 


15-16 


61-64 


.0794 


.1628 


.2461 


.3294 


.4128 


.4961 


.5794 


.6628 


.7461 


.8294 


.9128 


.9961 


61-64 


31-32 


.0807 


.1641 


.2474 


.3307 


.4141 


.4974 


.5807 


.6641 


.7474 


.8307 


.9141 


.9974 


31-32 


63-64 


.0820 


.1654 


.2487 


.3320 


.4154 


.4987 


.5820 


.6654 


.7487 


.8320 


.9154 


.9987 


63-64 



Note. — For fractions of an inch reduced to decimals, see Table 8, Ele- 
mentary Arithmetic, page 10. 



224' 11.— MENSURATION. 

11. — Circumferences C of Circles for given — 
Circumferences are directly proportional to the diameters. 



I 



D. 


CO 


1 


2 


3 


4 


5 


6 


7 


8 


9 


0.0 


.000000 


.031416 


.062832 


.094248 


.125664 


.157080 


. 188496 


.219911 


251327 


.282743 


.1 


.314159 


.345575 


.376991 


.408407 


.439823 


.471239 


.502655 


.534071 


. 565487 


. 596903 


.2 


.628319 


.659734 


.691150 


.722566 


.753982 


.785398 


.816814 


.848230 


.879646 


.911062 


.3 


.942478 


.973894 


1.00531 


1.03673 


1.06814 


1.09956 


1.13097 


1.16239 


1.19381 


1.22522 


A 


1.25664 


1.28805 


1.31947 


1.35088 


1.38230 


1.41372 


1.44513 


1.47655 


1.50796 


1.53938 


0.5 


1.57080 


1.60221 


1.63363 


1.66504 


1.69646 


1.72788 


1.75929 


1.79071 


1.82212 


1.85354 


.6 


1.88496 


1.91637 


1.94779 


1.97920 


2.01062 


2.04204 


2.07345 


2.10487 


2.13628 


2.16770 


.7 


2.19911 


2.23053 


2.26195 


2.29336 


2.32478 


2.35619 


2.38761 


2.41903 


2.45044 


2.48186 


.8 


2.51327 


2.54469 


2.57611 


2.60752 


2.63894 


2.67035 


2.70177 


2.73319 


2.76460 


2.79602 


.9 


2.82743 


2.85885 


2.89027 


2.92168 


2.95310 


2.98451 


3.01593 


3.04734 


3.07876 


3.11018 


1.0 


3.14159 


3.17301 


3.20442 


3.23584 


3.26726 


3.29867 


3.33009 


3.36150 


3.39292 


3.42434 


.1 


3.45575 


3.48717 


3.51858 


3.55^00 


3.58142 


3.61283 


3.64425 


3.67566 


3.70708 


3.73850 


.2 


3.76991 


3.80133 


3.83274 


3.86416 


3.89557 


3.92699 


3.95841 


3.98982 


4.02124 


4.05265 


.3 


4.08407 


4.11549 


4.14690 


4.17832 


4.20973 


4.24115 


4.27257 


4.30398 


4.33540 


4.36681 


.4 


4.39823 


4.42965 


4.46106 


4.49248 


4.52389 


4.55531 


4.58673 


4.61814 


4.64956 


4.68097 


1.5 


4.71239 


4.74381 


4.77522 


4.80664 


4.83805 


4.86947 


4.90088 


4.93230 


4.93672 


4.99513 


.6 


5.02655 


5.05796 


5.08938 


5.12080 


5.15221 


5.18363 


5.21504 


5.24646 


5.27788 


5.30929 


.7 


5.34071 


5.37212 


5.40354 


5.43496 


5.46637 


5.49779 


5.52920 


5.56062 


5.59203 


5.62345 


.8 


5.65487 


5.68628 


5.71770 


5.74911 


5.78053 


5.81195 


5.84336 


5.87478 


5.90619 


5.93761 


.9 


5.96903 


6.00044 


6.03186 


6.06327 


6.09469 


6.12611 


6.15752 


6.18894 


6.22035 


6.25177 


2.0 


6.28319 


6.31460 


6.34602 


6.37743 


6.40885 


6.44026 


6.47168 


6.50310 


6.53451 


6.56593 


.1 


6.59734 


6.62876 


6.66018 


6.69159 


6.72301 


6.75442 


6.78584 


6.81726 


6.84867 


6.88009 


.2 


6.91150 


6.94292 


6.97434 


7.00575 


7.03717 


7.06858 


7.10000 


7.13142 


7.16283 


7.19425 


.3 


7.22566 


7.25708 


7.28849 


7.31991 


7.35133 


7.38274 


7.41416 


7.44557 


7.47699 


7.50841 


A 


7.53982 


7.57124 


7.60265 


7.63407 


7.66549 


7.69690 


7.72832 


7.75973 


7.79115 


7.82257 


2.5 


7.85398 


7.88540 


7.91681 


7.94823 


7.97965 


8.01106 


8.04248 


8.07389 


8.10531 


8.13672 


.6 


8.16814 


8.19956 


8.23097 


8.26239 


8.29380 


8.32522 


8.35664 


8.38805 


8.41947 


8.45088 


.7 


8.48230 


8.51372 


8.54513 


8.57655 


8.60796 


8.63938 


8.67080 


8.70221 


8.73363 


8.76504 


.8 


8.79646 


8.82788 


8.85929 


8.89071 


8.92212 


8.95354 


8.98495 


9.01637 


9.04779 


9.07920 


.9 


9.11062 


9.14203 


9.17345 


9.20487 


9.23628 


9.26770 


9.29911 


9.33053 


9.36195 


9.39336 


3.0 


9.42478 


9.45619 


9.48761 


9.51903 


9.55044 


9.58186 


9.61327 


9.64469 


9.67611 


9.70752 


.1 


9.73894 


9.77035 


9.80177 


9.83319 


9.86460 


9.89602 


9.92743 


9.95885 


9.99026 


10.0217 


.2 


10.0531 


10.0845 


10.1159 


10.1473 


10.1788 


10.2102 


10.2416 


10.2730 


10.3044 


10.3358 


.3 


10.3673 


10.3987 


10.4301 


10.4615 


10.4929 


10.5243 


10.5558 


10.5872 


10.6186 


10.6500 


.4 


10.6814 


10.7128 


10.7442 


10.7757 


10.8071 


10.8385 


10.8699 


10.9013 


10.9327 


10.9642 


3.5 


10.9956 


11.0270 


11.0584 


11.0898 


11.1212 


11.1527 


11.1841 


11.2155 


11.2469 


11.2783 


.6 


11.3097 


11.3411 


11.3726 


11.4040 


11.4354 


11.4668 


11.4982 


11.5296 


11.5611 


11.5925 


.7 


11.6239 


11.6553 


11.6867 


11.7181 


11.7496 


11.7810 


11.8124 


11.8438 


11.8752 


11.-9066 


.8 


11.9381 


11.9695 


12.0009 


12.0323 


12.0637 


12.0951 


12.1265 


12.1580 


12.1894 


12.2208 


.9 


12.2522 


12.2836 


12.3150 


12.3465 


12.3779 


12.4093 


12.4407 


12.4721 


12.5035 


12.5350 


4.0 


12.5664 


12.5978 


12.6292 


12.6606 


12.6920 


12.7235 


12.7549 


12.7863 


12.8177 


12.8491 


.1 


12.8805 


12.9119 


12.9434 


12.9748 


13.0062 


13.0376 


13.0690 


13.1004 


13.1319 


13.1633 


.2 


13.1947 


13.2261 


13.2575 


13.2889 


13.3204 


13.3518 


13.3832 


13.4146 


13.4460 


13.4774 


.3 


13.5088 


13.5403 


13.5717 


13.6031 


13.6345 


13.6659 


13.6973 


13.7288 


13.7602 


13.7916 


.4 


13.8230 


13.8544 


13.8858 


13.9173 


13.9487 


13.9801 


14.0115 


14.0429 


14.0743 


14.1058 


4.5 


14.1372 


14.1686 


14.2000 


14.2314 


14.2628 


14.2942 


14.3257 


14.3571 


14.3885 


14.4199 


.6 


14.4513 


14.4827 


14.5142 


14.5456 


14.5770 


14.6084 


14.6398 


14.6712 


14.7027 


14.7341 


.7 


14.7655 


14.7969 


14.8283 


14.8597 


14.8911 


14.9226 


14.9540 


14.9854 


15.0168 


15.0482 


.8 


15.0796 


15.1111 


15.1425 


15.1739 


15.2053 


15.2367 


15.2681 


15.2996 


15.3310 


15.3624 


.9 


15.3938 


15.4252 


15.4566 


15.4881 


15.5195 


15.5509 


15.5823 


15.6137 


15.6451 


15.6765 


5.0 


15.7080 


15.7394 


15.7708 


15.8022 


15.8336 


15.8650 


15.8965 


15.9279 


15.9593 


15.9907 



Diameter may be obtained from circumferences by inverse interpolation, 
j^ote. — Area of surface of Sphere = diameter X circumference = D XC. 
Values in this table are also multiples of n. 



CIRCLES, DIAM, TO CIRCUM., DECIMALS. 



225 



— Diameters D, in Decimals. 
This table may be used like logarithmic tables. 



C. 



15.7080 
16.0221 
16.3363 
16.6504 
16.9646 

17.2788 
17.5929 
17.9071 
18.2212 
18.5354 

18.8496 
19.1637 
19.4779 
19.7920 
20.1062 

20.4204 
20.7345 
21.0487 
21.3628 
21.6770 

21.9911 
22.3053 
22.6195 
22.9336 
23.2478 

23.5619 
23.8761 
24.1903 
24.5044 
24.8186 

25.1327 
25.4469 
25.7611 
26.0752 
26.3894 

26.7035 
27.0177 
27.3319 
27.6460 
27.9602 

28.2743 
28.5885 
28.9027 
29.2168 
29.5310 

29.8451 
30.1593 
30.4724 
30.7876 
31.1018 

31.4159 



15.7394 
16.0535 
16.3677 
16.6819 
16.9960 

17.3102 
17.6243 
17.9385 
18.2527 
18.5668 

18.8810 
19.1951 
19.5093 
19.8235 
20.1376 

20.4518 
20.7659 
21.0801 
21.3942 
21.7084 

22.0226 
22.3367 
22.6509 
22.9650 
23.2792 

23.5934 
23.9075 
24.2217 
24.5358 
24.8500 

25.1642 
25.4783 
25.7925 
26.1066 
26.4208 

26.7350 
27.0491 
27.3633 
27.6774 
27.9916 

28.3058 
28.6199 
28.9341 
29.2482 
29.5624 

29.8765 
30.1907 
30.5049 
30.8190 
31.1332 

31.4473 



15.7708 
16.0850 
16.3991 
16.7133 
17.0274 

17.3416 
17.6558 
17.9699 
18.2841 
18.5982 

18.9124 
19.2265 
19.5407 
19.8549 
20.1690 

20.4832 
20.7973 
21.1115 
21.4257 
21.7398 

22.0540 
22.3681 
22.6823 
22.9965 
23.3106 

23.6248 
23.9389 
24.2531 
24.5673 
24.8814 

25.1956 
25.5097 
25.8239 
26.1381 
26.4522 

26.7664 
27.0805 
27.3947 
27.7088 
28.0230 

28.3372 
28.6513 
28.9655 
29.2796 
29.5938 

29.9080 
30.2221 
30.5363 
30.8504 
31.1646 

31.4788 



15.8022 
16.1164 
16.4305 
16.7447 
17.0588 

17.3730 
17.6872 
18.0013 
18.3155 
18.6296 

18.9438 
19.2580 
19.5721 
19.8863 
20.2004 

20.5146 
20.8288 
21.1429 
21.4571 
21.7712 

22.0854 
22.3996 
22.7137 
23.0279 
23.3420 

23.6562 
23.9704 
24.2845 
24.5987 
24.9128 

25.2270 
25.5411 
25.8553 
26.1695 
24.4836 

26.7978 
27.1119 
27.4261 
27.7403 
28.0544 

28.3686 
28.6827 
28.9969 
29.3111 
29.6252 

29.9394 
30.2535 
30.5677 
30.8819 
31.1960 

31.5102 



15.8336 
16.1478 
16.4619 
16.7761 
17.0903 

17.4044 
17.7186 
18.0327 
18.3469 
18.6611 

18.9752 
19.2894 
19.6035 
19.9177 
20.2319 

20.5460 
20.8602 
21.1743 
21.4885 
21.8027 

22.1168 
22.4310 
22.7451 
23.0593 
23.3734 

23.6876 
24.0018 
24.3159 
24.6301 
24.9442 

25.2584 
25.5726 
25.8867 
26.2009 
26.5150 

26.8292 
27.1434 
27.4575 
27.7717 
28.0858 

28.4000 
28.7142 
29.0283 
29.3425 
29.6566 

29.9708 
30.2850 
30.5991 
30.9133 
31.2274 

31.5416 



15.8650 
16.1792 
16.4934 
16.8075 
17.1217 

17.4358 
17.7500 
18.0642 
18.3783 
18.6925 

19.0064 
19.3208 
19.6350 
19.9491 
20.2633 

20.5774 
20.8916 
21.2058 
21.5199 
21.8341 

22.1482 
22.4624 
22.7765 
23.0907 
23.4049 

23.7190 
24.0332 
24.3473 
24.6615 
24.9757 

25.2898 
25.6040 
25.9181 
26.2323 
26.5465 

26.8606 
27.1748 
27.488.9 
27.8031 
28.1173 

28.4314 
28.7456 
29.0597 
29.3739 
29.6881 

30.0022 
30.3164 
30.6305 
30.9447 
31.2588 

31.5730 



15.8965 
16.2106 
16.5248 
16.8389 
17.1531 

17.4673 
17.7814 
18.0956 
18.4097 
18.7239 

19.0381 
19.3522 
19.6664 
19.9805 
20.2947 

20.6088 
20.9230 
21.2372 
21.5513 
21.8655 

22.1796 
22.4938 
22.8080 
23.1221 
23.4363 

23.7504 
24.0646 
24.3788 
24.6929 
25.0071 



25.3212 
25.6354 
25.9496 
26.2637 
26.5779 

26.8920 
27.2062 
27.5204 
27.8345 
28.1487 

28.4628 
28.7770 
29.0911 
29.4053 
29.7195 

30.0336 
30.3478 
30.6619 
30.9761 
31.2903 

31.6044 



15.9279 
16.2420 
16.5562 
16.8704 
17.1845 

17.4987 
17.8128 
18.1270 
18.4411 
18.7553 

19.0695 
19.3836 
19.6978 
20.0119 
20.3261 

20.6403 
20.9544 
21.2686 
21.5827 
21.8969 

22.2111 

22.5252 
22.8394 
23.1535 
23.4677 

23.7819 
24.0960 
24.4102 
24.7243 
25.0385 



25.3527 
25.6668 
25.9810 
26.2951 
26.6093 

26.9234 
27.2376 
27.5518 
27.8659 
28.1801 

28.4942 
28.8084 
29.1226 
29.4367 
29.7509 

30.0650 
30.3792 
30.6934 
31.0075 
31.3217 

31.6358 



15.9593 
16.2734 
16.5876 
16.9018 
17.2159 

17.5301 
17.8442 
18.1584 
18.4726 
18.7867 

19.1009 
19.4150 
19.7292 
20.0434 
20.3575 



15.9907 
16.3049 
16.6190 
16.9332 
17.2473 

17.5615 
17.8757 
18.1898 
18.5040 
18.8181 

19.1323 
19.4465 
19.7606 
20.0748 
20.3889 



20.6717 20.7031 



20.9858 
21.3000 
21.6142 
21.9283 

22.2425 
22.5566 
22.8708 
23.1850 
23.4991 

23.8133 
24.1274 
24.4416 
24.7558 
25.0699 

25.3841 
25.6982 
26.0124 
26.3265 
26.6407 

26.9549 
27.2690 
27.5832 
27.8973 
28.2115 

28.5257 
28.8398 
29.1540 
29.4681 
29.7823 

30.0965 
30.4106 
30.7248 
31.0389 
31.3531 



21.0173 
21.3314 
21.6456 
21.9597 

22.2739 
22.5881 
22.9022 
23.2164 
23.5305 

23.8447 
24.1588 
24.4730 
24.7872 
25.1013 

25.4155 
25.7296 
26.0438 
26.3580 
26.6721 

26.9863 
27.3004 
27.6146 
27.9288 
28.2429 

28.5571 
28.8712 
29.1854 
29.4996 
29.8137 

30.1279 
30.4420 
30.7562 
31.0704 
31.3845 



31.6673 31.6987 



Diff. between any two successive circumferences = .03 14 4- ; hence, if 
the diameter is extended to the third decimal place (thousandths), add 
(.00314+ multiplied by the thousandth figure), Ex. — Diam. = 7. 523; then 
circum. = 23.G248 + .0094 = 23.6342. 



226 



11.— MENSURATION. 



12. — Circumferences of Circles for Given — 
Circumferences are directly proportional to the diameters. 



Decimal 


.0000 


.083^3 


.1250 


. 166^6 


.2500 


.333^3 


.3750 


.416^6 






r 


'Factions— 


as 12ths of a Foot (Inches), 


and 8ths of an Inch. 


Ft 


12ths 





1 


1.5 


2 


3 


4 


4.5 


5 






12 


12 


12 


12 


12 


12 


12 


12 


Ins. 


8ths 



8 




1 
T 




2 

T 




3 
■8 









.000000 


.261799 


.392699 


.523599 


.785398 


1.04720 


1.17810 


1.30900 




1 


3.14159 


3.40339 


3.53429 


3.66519 


3.92699 


4.18879 


4.31969 


4.45059 




2 


6.28319 


6.54498 


6.67588 


6.80678 


7.06858 


7.33038 


7.46128 


7.59218 


00 


3 


9.42478 


9.68658 


9.81748 


9.94838 


10.2102 


10.4720 


10.6029 


10.7338 


+3 

oo 


4 


12.5664 


12.8282 


12.9591 


13.0900 


13.3518 


13.6136 


13.7445 


13.8754 


a 


5 


15.7080 


15.9698 


16.1007 


16.2316 


16.4934 


16.7552 


16.8861 


17.0170 


6 


18.8496 


19.1114 


19.2423 


19.3732 


19.6350 


19.8968 


20.0277 


20.1586 


s 


7 


21.9911 


22.2529 


22.3838 


22.5147 


22.7765 


23.0383 


23.1692 


23.3001 


•g 


8 


25.1327 


25.3945 


25.5254 


25.6563 


25.9181 


26.1799 


26.3108 


26.4417 


a 


9 


28.2743 


28.5361 


28.6670 


28.7979 


29.0597 


29.3215 


29.4524 


29.5833 


M 


10 


31.4159 


31.6773 


31.8086 


31.9395 


32.2013 


32.4631 


32.5940 


32.7249 


OS 


11 


34.5575 


34.8193 


34.9502 


35.0811 


35.3429 


35.6047 


35.7356 


35.8665 


5 


12 


37.6991 


37.9609 


38.0918 


38.2227 


38.4845 


38.7463 


38.8772 


39.0081 


ca 


13 


40.8407 


41.1025 


41.2334 


41.3643 


41.6261 


41.8879 


42.0188 


42.1497 


o 


14 


43.9823 


44.2441 


44.3750 


44.5059 


44.7677 


45.0295 


45.1604 


45.2913 


15 


47.1239 


47.3857 


47.5166 


47.6475 


47.9093 


48.1711 


48.3020 


48.4329 


^ 


16 


50.2655 


50.5273 


50.6582 


50.7891 


51.0509 


51.3127 


51.4436 


51.5745 


1X4 


17 


53.4071 


53.6689 


53.7998 


53.9307 


54.1925 


54.4543 


54.5852 


54.7161 


18 


56.5487 


56.8105 


56.9414 


57.0723 


57.3341 


57.5959 


57.7268 


57.8577 


_2 


19 


59.6903 


59.9521 


60.0830 


60.2139 


60.4757 


60.7375 


60.8684 


60.9993 


"3 


20 


62.8319 


63.0937 


63.2246 


63.3555 


63.6173 


63.8791 


64.0100 


64.1409 


21 


65.9734 


66.2352 


66.3661 


66.4970 


66.7588 


67.0206 


67.1515 


67.2824 


o 


22 


69.1150 


69.3768 


69.5077 


69.6386 


69.9004 


70.1622 


70.2931 


70.4240 


p 


23 


72.2566 


72.5184 


72.6493 


72.7802 


73.0420 


73.3038 


73.4347 


73.5656 


fc 


24 


75.3982 


75.6600 


75.7909 


75.9218 


76.1836 


76.4454 


76.5763 


76.7072 


£ 


25 


78.5398 


78.8016 


78.9325 


79.0634 


79.3252 


79.5870 


79.7179 


79.8488 


S 


26 


81.6814 


81.9432 


82.0741 


82.2050 


82.4668 


82.728r 


82.8595 


82.9904 


•g 


27 


84.8230 


85.0848 


85.2157 


85.3466 


85.6084 


85.8702 


86.0011 


86.1320 


C3 


28 


87.9646 


88.2264 


88.3570 


88.4882 


88.7500 


89.0118 


89.1427 


89.2736 


HH 


29 


91.1062 


91.3680 


91.4989 


91.6298 


91.8916 


92.1534 


92.2843 


92.4152 


o 


30 


94.2478 


94.5096 


94.6405 


94.7714 


94.0332 


95.2950 


95.4259 


95.5568 


4d 


31 


97.3894 


97.6512 


97.7821 


97.9130 


98.1748 


98.4366 


98.5675 


98.6984 




32 


100.531 


100.793 


100.924 


101.055 


101.316 


101.578 


101.709 


101.840 


33 


103.673 


103.934 


104.065 


104.196 


104.458 


104.720 


104.851 


104.982 


5 


34 


106.814 


107.076 


107.207 


107.338 


107.600 


107.861 


107.992 


108.123 




35 


109.956 


110.218 


110.348 


110.479 


110.741 


111.003 


111.134 


111.265 


a; 


36 


113.097 


113.359 


113.490 


113.621 


113.883 


114.145 


114.145 


114.406 


s 


37 


116.239 


116.501 


116.632 


116.763 


117.024 


117.286 


117.417 


117.548 


.s 


38 


119.381 


119.642 


119.773 


119.904 


120.166 


120.428 


120.559 


120.690 





39 


122.522 


122.784 


122.915 


123.046 


123.308 


123.569 


123.700 


123.831 




40 


125.664 


125.926 


126.056 


126.187 


126.449 


126.711 


126.842 


126.973 


a 


41 


128.805 


129.067 


129.198 


129.329 


129.591 


129.852 


129.983 


130.114 


1 


42 


131.947 


132.209 


132.340 


132.470 


132.732 


132.994 


133.125 


133.256 


43 


135.088 


135.350 


135.481 


135.612 


135.874 


136.136 


136.267 


136.397 


o 


44 


138.230 


138.492 


138.623 


138.754 


139.015 


139.277 


139.408 


139.539 


■S 


45 


141.372 


141.633 


141.764 


141.895 


142.157 


142.419 


142.550 


142.681 


£ 


46 


144.513 


144.775 


144.906 


145.037 


145.299 


145.560 


145.691 


145.822 


P^ 


47 


147.655 


147.917 


148.048 


148.178 


148.440 


148.702 


148. 833 


148.964 




48 


150.796 


151.058 


151.189 


151.320 


151.582 


151.844 


151.975 


152.105 




49 


153.938 


154.200 


154.331 


154.462 


154.723 


154.985 


155.116 


155.247 




50 


157.080 


157.341 


157.472 


157.603 


157.865 


158.127 


158.258 


158.389 



Diameters may be obtained from circumferences by inverse interpolation. 

Note. — Area of surface of 5M^^'^ = diameter X circumference. 

The Circumferences are in the same Denomination as that for which the 
First Column is used. Ex: — Dia. = 14.125 ft. = 14 ft. U ins.: then, 
Circumference = 44.375 ft. 



CIRCLES, DIA. TO CIR., FRAC. AND DEC. 



227 



-Diameters, in Feet and Inches; and in Inches. 
This table may be used like logarithmic tables. 



Decimal 


.5000 


.583^3 


.6250 


.666^6 


.7500 


. 833^3 


.8750 


. 916^6 






Fractions— as 12ths of a Foot (Inches), and Bths of an Inch. 






6 


7 


7.5 


8 


9 


10 


10.5 


11 


Ft. 


12ths 


12 


12 


12 


12 


12 


T2 


12 


12 


Ins. 


8ths 


4 
8 




5 
8 




6 
8 




7 
8 









1.57080 


1.83260 


1.96350 


2.09440 


2.35619 


2.61799 


2.74889 


2.87979 




1 


4.71239 


4.97419 


5.10509 


5.23599 


5.49779 


5.75959 


5.89049 


6.02139 




2 


7.85398 


8.11578 


8.24668 


8.37758 


8.63938 


8.90118 


9.03208 


9.16298 


S 


3 


10.9956 


11.2574 


11.3883 


11.5192 


11.7810 


12.0428 


12.1737 


12.3046 




4 


14.1372 


14.3990 


14.5299 


14.6608 


14.9226 


15.1844 


15.3153 


15.4462 


CX3 


5 


17.2788 


17.5406 


17.6715 


17.8024 


18.0642 


18.3260 


18.4569 


18.5878 


o 


6 


20.4204 


20.6822 


20.8131 


20.9440 


20.2058 


21.4675 


21.5984 


21.7293 


^ 


7 


23.5619 


23.8237 


23.9546 


24.0855 


24.3473 


24.6091 


24.7400 


24.8709 




8 


26.7035 


26.9653 


27.0962 


27.2271 


27.4889 


27.7507 


27.8816 


28.0125 




9 


29.8451 


30.1069 


30.2378 


30.3687 


30.6305 


30.8923 


31.0232 


31.1541 


10 


32.9867 


33.2485 


33.3794 


33.5103 


33.7721 


34.0339 


34.1648 


34.2957 


*M 


11 


36.1283 


36.3901 


36.5210 


36.6519 


36.9137 


37.1755 


37.3064 


37.4373 


,E3 


12 


39.2699 


39.5317 


39.6626 


39.7935 


40.0553 


40.3171 


40.4480 


40.5789 




13 


42.4115 


42.6733 


42.8042 


42.9351 


43.1969 


43.4587 


43.5896 


43.7205 


Ih 


14 


45.5531 


45.8149 


45.9458 


46.0767 


46.3385 


46.6003 


46.7312 


46.8621 


£ 


15 


48.6947 


48.9565 


49.0874 


49.2183 


49.4801 


49.7419 


49.8728 


50.0037 


4^ 


16 


51.8363 


52.0981 


52.2290 


52.3599 


52.6217 


72.8835 


53.0144 


53.1453 




17 


54.9779 


55.2397 


55.3706 


55.5015 


55.7633 


56.0251 


56.1560 


56.2869 


Ph 


18 


58.1195 


58.3813 


58.5122 


58.6431 


58.9049 


59.1667 


59.2976 


59.4285 


m 


19 


61.2611 


61.5229 


61.6538 


61.7847 


62.0465 


62.3083 


62.4392 


62.5701 


"S 


20 


64.4026 


64.6644 


64.7953 


64.9262 


65.1880 


65.4498 


65.5807 


65.7116 


^ 


21 


67.5442 


67.8060 


67.9369 


68.0678 


68.3296 


68.5914 


68.7223 


68.8532 




22 


70.6858 


70.9476 


71.0785 


71.2094 


71.4712 


71.7330 


71.8639 


71.9948 


tH 


23 


73.8274 


74.0892 


74.2201 


74.3510 


74.6128 


74.8746 


75.0055 


75.1364 


24 


76.9690 


77.2308 


77.3617 


77.4926 


77.7544 


78.0162 


78.1471 


78.2780 


o 


25 


80.1106 


80.3724 


80.5033 


80.6342 


80.8960 


81.1578 


81.2887 


81.4196 


s^ 


26 


83.2522 


83.5140 


83.6449 


83.7758 


83.0376 


84.2994 


84.4303 


84.5612 


.£3 


27 


86.3938 


86.6556 


86.7865 


86.9174 


87.1792 


87.4410 


87.5719 


87.7028 


C3 


28 


89.5354 


89.7972 


89.9281 


90.0590 


90.3208 


90.5826 


90.7135 


90.8444 


HH 


29 


92.6770 


92.9388 


93.0697 


93.2006 


93.4624 


93.7242 


93.8551 


93.9860 


g 


30 


95.8186 


96.0804 


96.2113 


96.3422 


96.6040 


96.8658 


96.9967 


97.1276 


4^ 


31 


98.9602 


99.2220 


99.3529 


99.4838 


99.7456 


100.007 


100.138 


100.269 


% 


32 


102.102 


102.364 


102.494 


102.625 


102.887 


103.149 


103.280 


103.411 


fe 


33 


105.243 


105.505 


105.636 


105.767 


106.029 


106.291 


106.421 


106.552 


•S 


34 


108.385 


108.647 


108.778 


108.909 


109.170 


109.432 


109.563 


109.694 


5i 


35 


111.527 


111.788 


111.919 


112.050 


112.312 


112.574 


112.705 


112.836 


% 


36 


114.668 


114.930 


115.061 


115.192 


115.454 


115.715 


115.846 


115.977 


1 


37 


117.810 


118.072 


118.202 


118.333 


118.596 


118.857 


118.988 


119.119 


38 


120.951 


121.213 


121.344 


121.475 


121.737 


121.999 


122.129 


122.260 


p 


39 


124.093 


124.355 


124.486 


124.617 


124.878 


125.140 


125.271 


125.402 




40 


127.235 


127.496 


127.627 


127.758 


128.020 


128.282 


128.413 


128.543 


a 


41 


130.376 


130.638 


130.769 


130.900 


131.161 


131.423 


131.554 


131.685 


a 


42 


133.518 


133.779 


133.910 


134.041 


134.303 


134.565 


134.696 


134.827 


3 


43 


136.659 


136.921 


137.052 


137.183 


137.445 


137.706 


137.837 


137.968 


o 
O 

4i9 


44 


139.801 


140.063 


140.194 


140.324 


140.586 


140.848 


140.979 


141.110 


45 


142.942 


143.204 


143.335 


143.466 


143.728 


143.990 


144.121 


144.251 


£ 


46 


146.084 


146.346 


146.477 


146.608 


146.869 


147.131 


147.262 


147.393 


Pn 


47 


149.226 


149.487 


149.618 


149.749 


150.011 


150.273 


150.404 


150.535 




48 


152.367 


152.629 


152.760 


152.891 


153.153 


153.414 


153.545 


153.676 




49 


155.509 


155.771 


155.902 


156.032 


156.294 


156.556 


156.687 


156.818 


. 


50 


158.650 


158.912 


159.043 


159.174 ' 159.436 1 159.698 


159.829 


159.959 



Diff. between any two successive circumferences in 12ths = .2618; hence 
if the diameter is extended beyond 12ths, this diff. must be used proportion- 
ately. Diff. between any two successive circumferences in 8ths = . 3927; hence, 
if the diameter is extended beyond 8ths, this diff. must be used proportion- 
ately. 

The Circumferences are in the same Denomination as that for which the 
First Column is used. 



228 



11.— MENSURA TION. 



12. — Circumferences of Circles for Given — 
Circumferences are directly proportional to the diameters. 



Decimal 


.0000 


.083^3 


.1250 


. 166^6 


.2500 


.333-^3 


.3750 


.416^6 






Fractions— as 12ths of a Foot (Inches), and 8ths of an Inch. 









1 


1.5 


2 


3 


4 


4.5 


5 


Ft. 


12ths 


12 


12 


I2 


I2 


12 


12 


12 


Vl 


Ins. 


8ths 




8 




1 
8 




2 

8 




3 
8 






50 


157.080 


157.341 


157.472 


157.603 


157.865 


158.127 


158.258 


158.389 




51 


160.221 


160.483 


160.614 


160.745 


161.007 


161.268 


161.399 


161.530 




52 


163.363 


163.625 


163.756 


163.886 


164.148 


164.410 


164.541 


164.672 


o3 


53 


166.504 


166.766 


166.897 


167.028 


167.290 


167.552 


167.683 


167.813 


4J 

oo 


54 


169.646 


169.908 


170.039 


170.170 


170.431 


170.693 


170.824 


170.955 


u 


55 


172.788 


173.049 


173.180 


173.311 


173.573 


173.835 


173.966 


174.097 


a 


56 


175.929 


176.191 


176.322 


176.453 


1^6.715 


176.976 


177.107 


177.^38 


2 


57 


179.071 


179.333 


179.463 


179.594 


179.856 


180.118 


180.249 


180.380 


^ 


58 


182.212 


182.474 


182.605 


182.736 


182.998 


183.260 


183.390 


183.521 




59 


185.354 


185.616 


185.747 


185.878 


186.139 


186.401 


186.532 


186.663 


M 


60 


188.496 


188.757 


188.888 


189.019 


189.281 


1«9.543 


189.674 


189.805 


03 


61 


191.637 


191.899 


192.030 


192.161 


192.423 


192.684 


192.815 


192.946 


5 


62 


194.779 


195.041 


195.171 


195.302 


195.564 


195.826 


195.957 


196.088 


1—1 


63 


197.920 


198.182 


198.313 


198.444 


198.706 


198.968 


199.098 


199.229 


U> 


64 


201.062 


201.324 


201.455 


201.586 


201.847 


202.109 


202.240 


202.371 


a 


65 


204.204 


204.465 


204.596 


204.727 


204.989 


205.251 


205.382 


205.513 


■V3 


66 


207.345 


207.607 


207.738 


207.869 


208.131 


208.392 


208.523 


2U8.654 


1 


67 


210.487 


210.749 


210.879 


211.010 


211.272 


211.534 


211.665 


211:796 


68 


213.628 


213.890 


214.021 


214.152 


214.414 


214.675 


214.806 


214.937 


DQ 


69 


216.770 


217.032 


217.163 


217.293 


217.555 


217.817 


217.948 


218.079 


"3 


70 


219.911 


220.173 


220.304 


220.435 


220.697 


220.959 


221.090 


221.220 


S 


71 


223.053 


222.315 


223.446 


223.577 


223.838 


224.100 


224.231 


224.362 


g 


72 


226.195 


226.456 


226.587 


226.718 


226.980 


227.242 


227.373 


227.504 


p 


73 


229.336 


229.598 


229.729 


229.860 


230.122 


230.383 


230.514 


230.645 


^1 


74 


232.478 


232.740 


232.871 


233.001 


233.263 


233.525 


233.656 


233.787 


o 


75 


235.619 


235.881 


236.012 


236.143 


236.405 


236.667 


236.798 


236.928 


S 


76 


238.761 


239.023 


239.154 


239.285 


239.546 


239.808 


239.939 


240.070 


^ 


77 


241.903 


242.164 


242.295 


242.426 


242.688 


242.950 


243.081 


243.212 




78 


245.044 


245.306 


245.437 


245.568 


245.830 


246.091 


246.222 


246.353 


1—1 


79 


248.186 


248.448 


248.579 


248.709 


248.971 


249.233 


249.364 


249.495 


o 


80 


251.327 


251.589 


251.720 


251.851 


252.113 


252.375 


252.506 


252.636 


4^ 


81 


254.469 


254.731 


254.862 


254.993 


255.254 


255.516 


255.647 


255.778 




82 


257.611 


257.872 


258.003 


258.134 


258.396 


258.658 


258.789 


258.920 


p^ 


83 


260.752 


261.014 


261.145 


261.276 


261.538 


261.799 


261.930 


262.061 


c 


84 


263.894 


264.156 


264.286 


264.417 


264.679 


264.941 


265.072 


265.203 


u 


85 


267.035 


267.297 


267.428 


267.559 


267.821 


268.083 


268. 2i3 


263.344 




86 


270.177 


270.439 


270.570 


270.701 


270.962 


271.224 


271.355 


271.486 


87 


273.319 


273.580 


273.711 


273.842 


274.104 


274.366 


274.497 


274.628 


§ 


88 


276.460 


276.722 . 


276.853 


276.984 


277.246 


277.507 


277.638 


277.769 


Q 


89 


279.602 


279.864 


279.994 


280.125 


280.387 


280.649 


280.780 


280.911 




90 


282.743 


283.005 


283.136 


283.267 


283.529 


283.791 


283.921 


284.052 


c 


91 


285.885 


286.147 


286.278 


286.409 


286.670 


286.932 


287.063 


287.194 


S 


92 


289.027 


289.288 


289.419 


289.550 


289.812 


290.074 


290.205 


290.336 


s 


93 


292.168 


292.430 


292.561 


292.692 


292.954 


293.215 


293.346 


293.477 




94 


295.310 


295. 572 


295.702 


295.833 


296.095 


296.357 


296.488 


296.619 


95 


298.451 


298.713 


298.844 


298.975 


299.237 


299.498 


299.629 


299.760 


£ 


96 


301.593 


301.855 


301.986 


302.116 


302.378 


302.640 


302.771 


302.902 


S 


97 


304.734 


304.996 


305.127 


305.258 


305.520 


305.782 


305.913 


306.043 




■ 98 


307.876 


308.138 


308.269 


308.400 


308.661 


308.923 


309.054 


309.185 




99 


311.018 


311.279 


311.410 


311.541 


311.803 


312.065 


312.196 


312.327 




100 


314.159 


314.421 


314.552 


314.683 


314.945 


315.206 


315.337 


315.468 



Diameters may be obtained from circumferences by inverse interpolation. 

Note. — Area of surface of 5p/t^r^ = diameter X circumference. 

The Circumferences are in the same Denomination as that for which the 
First Column is used. Ex. — Dia. = 62.25 ins. = 62| = 62i ins.: then, 
Circiimference = 195.564 ins. 



CIRCLES, VIA. TO CIR., FRAC, AND DEC, 



229 



-Diameters in Feet and Inches; and in Inches, — Concluded. 
This table may be used like logarithmic tables. 



Decimal 


.5000 


. 583^3 


.6250 


.666^6 


.7500 


.833^3 


.8750 


.916^6 






F 


Factions— as 12ths of a Foot 


(Inches), and 8ths of an Inch. 






6 


7 


7.5 


8 


9 


10 


10.5 


11 


Ft. 


12ths 


12 


12 


12 


12 


T2 


12 


12" 


12 


Ins. 


8ths 


4 
8 




5 
8 




6 
8 




7 
8 






50 


158.650 


158.912 


159.043 


159.174 


159.436 


159.698 


159.829 


159.959 




51 


161.792 


162.054 


162.185 


162.316 


162.577 


162.839 


162.970 


163.101 


m 


52 


164.934 


165.195 


165.326 


165.457 


165.719 


165.981 


166.112 


166.243 


s 


53 


168.075 


168.337 


168.468 


168.599 


168.861 


169.122 


169.253 


169.384 


s 


54 


171.217 


171.479 


171.609 


171.740 


172.002 


172.264, 


172.395 


172.526 


§ 


55 


174.358 


174.620 


174.751 


174.882 


175.144 


175.406 


175.536 


175.667 


<« 


56 


177.500 


177.762 


177.893 


178.024 


178.285 


178.547 


178.678 


178.809 


s 


57 


180.642 


180.903 


181.034 


181.165 


181.427 


181.689 


181.820 


181.951 


•g 


58 


183.783 


184.045 


184.176 


184.307 


184.569 


184.830 


184.961 


185.092 


a 
1— 1 


59 


186.925 


187.187 


187.317 


187.448 


187.710 


187.972 


188.103 


188.234 


60 


190.066 


190.328 


190.459 


190.590 


190.852 


191.114 


191.244 


191.375 


CO 

c-l 


61 


193.208 


193.470 


193.601 


191.732 


193.993 


194.255 


194.386 


194.517 


•s 


62 


196.350 


196.611 


196.742 


196.873 


197.135 


197.397 


197.528 


197.659 


^ 


63 


199.491 


199.753 


199.884 


200.015 


200.277 


200.538 


200.669 


200.800 


a 


64 


202.633 


202.895 


203.025 


203.156 


203.418 


203. 680 


203.811 


203.942 




65 


205.774 


206.036 


206.167 


206.298 


206.560 


206.822 


206.952 


207.083 


•4J 


66 


208.916 


209.178 


209.309 


209.440 


209.701 


209.963 


210.094 


210.225 


r 


67 


212.058 


212.319 


212.450 


212.581 


212.843 


213.105 


213.236 


213.367 




68 


215.199 


215.461 


215.592 


215.723 


215.984 


216.246 


216.377 


216.508 


m 


69 


218.341 


218.602 


218.733 


218.864 


219.126 


219.388 


219.519 


219.650 


o3 


70 


221.482 


221.744 


221.875 


222.006 


222.268 


222.529 


222.660 


222.791 


71 


224.624 


224.886 


225.017 


225.147 


225.409 


225.671 


225.802 


225.933 


o 


72 


227.765 


228.027 


228.158 


228.289 


228.551 


228.813 


228.944 


229.074 


Q 


73 


230.907 


231.169 


231.300 


231.431 


231.692 


231.954 


232.085 


232.216 


a 


74 


234.049 


234.310 


234.441 


234.572 


234.834 


235.096 


235.227 


235.358 




75 


237.190 


237.452 


237.583 


237.714 


237.976 


238.237 


238.368 


238.499 


§ 


76 


240.332 


240.594 


240.725 


240.855 


241.117 


241.379 


241.510 


241.641 


■s 


77 


243.473 


243.735 


243.866 


243.997 


244.259 


244.521 


244.652 


244.782 




78 


246.615 


246.877 


247.008 


247.139 


247.400 


247.662 


247.793 


247.924 


ui 


79 


249.757 


250.018 


250.149 


250.280 


250.542 


250.804 


250.935 


251.066 


o 


80 


252.898 


253.160 


253.291 


253.422 


253.684 


253.945 


254.076 


254.207 




81 


256.040 


256.302 


256.433 


256.563 


256.825 


257.087 


257.218 


257.349 




82 


259.181 


259.443 


259.574 


259.705 


259.967 


260.229 


260.359 


260.490 


C3 


83 


262.323 


262.585 


262.716 


262.847 


263.108 


263.370 


263.501 


263.632 




84 


265.465 


265.726 


265.857 


265.988 


266.250 


266.512 


266.643 


266.774 


<u 


85 


268.606 


268.868 


268 999 


269.130 


269.392 


269.653 


269.784 


269.915 


o 


86 


271.748 


272.010 


272.140 


272.271 


272.533 


272.795 


272.926 


273.057 


s 


87 


274.889 


275.151 


275.282 


275.413 


275.675 


275.937 


276.067 


276.198 


.S3 


88 


278.031 


278.293 


278.424 


278.555 


278.816 


279.078 


279.209 


279.340 


89 


281.173 


281.434 


281.565 


281.696 


281.958 


282.220 


282.351 


282.482 


II 


90 


284.314 


284.576 


284.707 


284.838 


285.100 


285.361 


285.492 


285.623 


B 
:3 


91 


287.456 


287.718 


287.848 


287.979 


288.241 


288.503 


288.634 


288.765 


92 


290.597 


290.859 


290.990 


291.121 


291.383 


291.645 


291.775 


291.906 


o 


93 


293.739 


294.001 


294.132 


294.263 


294.524 


294 786 


294.917 


295.048 


O 


94 


296.881 


297.142 


297.273 


297.404 


297.666 


297.928 


298.059 


298.190 


^ 


95 


300.022 


300.284 


300.415 


300.546 


300.807 


301.069 


301.200 


301.331 


b; 


96 


303.164 


303.425 


303.556 


303.687 


303.949 


304.211 


304.342 


304.473 


S 


97 


306.305 


306.567 


306.698 


306.829 


307.091 


307.352 


307.483 


307.614 




98 


309.447 


309.709 


309.840 


309.970 


310.232 


310.494 


310.625 


310.756 




99 


312.588 


312.850 


312.981 


313.112 


313.374 


313.636 


313.767 


313.897 




100 


315.730 


315.992 


316.123 


316.254 


316.515 


316.777 


316.908 


317.039 



Diff. between any two successive circumferences m 12ths = .2618; hence, 
if the diameter is extended beyond 12ths, this diff. must be used proportion- 
ately. Diff. between any two successive circumferences in 8ths = .3927; hence, 
if the diameter is extended beyond 8ths, this diff. must be used proportion- 
ately. 

The Circumferences are in the same Denomination as that for which the 
First Column is used. 



230 11.— MENSURATION, 

13. — Areas of Circles in Square Inches for Given — 



1 












3 




8 


16 


24 


32 


40 , 






1-64 


.00019 




7.0686 




50.265 


201.06 


452.39 


804.25 


1256.64 






1-32 


.00077 


1-16 


7.3662 


Vs 


51.849 


204.22 


457.11 


810.54 


1264.50 


M 




1-16 


.00307 


Vs 


7.6699 


M 


53.456 


207.39 


461.86 


816.86 


1272.39 


}4, 


^-v 


3-32 


.00690 


3-16 


7.9798 


Vs 


55.088 


210.60 


466.64 


823.21 


1280.31 


^ 


w '^ 


Vs 


.01227 


M 


8.2958 


V2 


56.745 


213.82 


471.44 


829.58 


1288.25 


Vi 


fcl 


5-32 


.01917 


5-16 


8.6179 




58.426 


217.08 


476.26 


835.97 


1296.21 


^/s 


o"o 


3-16 


.02761 


Vs 


8.9462 


M 


60.132 


220.35 


481.11 


842.39 


1304.20 


M 


S'S 


7-32 


.03758 


7-16 


9.2806 


Vs 


61.862 


223.65 


485.98 


848.83 


1312.21 


1/s 


1° 












9 


17 


25 


ZZ 


41 




^g 


H 


.04909 


H 


9.6211 




63.617 


226.98 


490.87 


855.30 


1320.25 




U "3 


9-32 


.06213 


9-16 


9.9678 


Vs 


65.397 


230.33 


495.79 


861.79 


1328.32 


Y^ 


J! 


5-16 


.07670 


Vs 


10.321 


V. 


67.201 


233.71 


500.74 


868.31 


1336.40 


M 


11-32 


.09281 


11-16 


10.680 




69.029 


237.10 


505.71 


874.85 


1344.51 


H 


he areai 
uares of 
sphere 


Vs 


.11045 


H 


11.045 


Vi, 


70.882 


240.53 


510.71 


881.41 


1352.65 


3^ 


13-32 


. 12962 


13-16 


11.416 


hyC 


72.760 


243.98 


515.72 


888.00 


1360.81 


X^ 


7-16 


.15033 


Vs 


11.793 


Va. 


74.662 


247.45 


520.77 


894.62 


1369.00 


x^ 


15-32 


. 17257 


15-16 


12.177 


Vs 


76.589 


250.95 


525.84 


901.26 


1377.21 


Vs 


-*-> cat'*-" 








4 




10 


13 


26 


34 


42 




K 


.19635 




12.566 




78.540 


254.47 


530.93 


907.92 


1385.44 




" opposi 
al to th 
ng diam 


17-32 


.22166 


'i-ie 


12.962 


Vs 


80.516 


258.02 


536.05 


914.61 


1393.70 


Vs 


9-16 


.24850 


Vs 


13.364 


M 


82.516 


261.59 


541.19 


921.32 


1401.98 


M 


19-32 


.27688 


3-16 


13.772 


Vs 


84.541 


265.18 


546.35 


928.06 


1410.29 


Vs 


Vs 


.30680 


H 


14.186 


V2 


86.590 


268.80 


551.55 


934.82 


1418.62 


3^ 


olumn ' 
iportion 
Assumi 


21-32 


.33824 


5-16 


14.607 


Vs 


88.664 


272.45 


556.76 


941.61 


1426.98 


/^ 


11-16 


.37122 


Vs 


15.033 


M 


90.763 


276.12 


562.00 


948.42 


1435.36 


¥. 


23-32 


.40574 


7-16 


15.466 


Vs 


92.886 


279.81 


567.27 


955.25 


1443.77 


Vs 












11 


19 


27 


35 


43 




^S^ 


H 


.44179 


K 


15.904 




95.033 


283.53 


572.56 


962.11 


1452.20 




a o< 


25-32 


.47937 


9-16 


16.349 


Vs 


97.205 


287.27 


577.87 


969.00 


1460.65 


^ 


^"^^ 


13-16 


.51849 


Vs 


16.800 


H 


99.402 


291.04 


583.21 


975.91 


1469.13 


H 


-^^^ 


27-32 


.55914 


11-16 


17.257 


Vs 


101.62 


294.83 


588.57 


982.84 


•1477.63 


/^ 


«^2 


Vs 


.60132 


H 


17.721 


V2 


103.87 


298.65 


593.96 


989.80 


1486.17 


Vl 


29-32 


.64504 


13-16 


18.190 


Vs 


106.14 


302.49 


599.37 


996.78 


1494.72 


/^ 


Ian 
.fcii 
give 


15-16 


.69029 


Vs 


18.665 


H 


108.43 


306.35 


604.81 


1003.78 


1503.30 


M 


31-32 


.73708 


15-16 


19.147 


Vs 


110.75 


310.24 


610.27 


1010.82 


1511.90 


^8 


six 




1 




5 




12 


20 


28 


36 


44 






.78540 




19.635 




113.10 


314.16 


615.75 


1017.87 


1520.53 




O ^ si 


'i-16 


.88664 


i-ie 


20.129 


1^ 


115.47 


318.10 


621.26 


1024.95 


1529.18 


yi 


ract 
at a 
dial 


Vs 


.99402 


Vs 


20.629 


V. 


117.86 


322.06 


626.80 


1032.06 


1537.86 


M 


3-16 


1.1075 


3-16 


21.135 


% 


120.28 


326.05 


632.36 


1039.19 


1546.55 


H 




M 


1.2272 


M 


21.6*8 


14 


122.72 


330.06 


637.94 


1046.35 


1555.28 


Vi 


5-16 


1.3530 


5-16 


22.166 


Vs 


125.19 


334.10 


643.55 


1053.52 


1564.03 


Vs 


Vs 


1.4849 


Vs 


22.691 




127.68 


338.16 


649.18 


1060.73 


1572.81 


M 


heavy type, ^ 
.arged by not: 
jas. Volume 


7-16 


1.6230 


7-16 


23.221 


yi 


130.19 


342.25 


654.84 


1067.95 


1581.61 


Vk 












13 


21 


29 


37 


45 




3^ 


1.7671 


V2 


23.758 




132.73 


346.36 


660.52 


1075.21 


1590.43 




9-16 


1.9175 


9-16 


24.301 


Vs 


135.30 


3D0.50 


666.23 


1082.48 


1599.28 


yk 


H 


2.0739 


Vs 


24.850 


M 


137.89 


354.66 


671.96 


1089.79 


1608.15 


yi 


11-16 


2.2365 


11-16 


25.406 


Vs 


140.50 


3D8.84 


677.71 


1097.11 


1617.04 


Vs 


H 


2.4053 


M 


25.967 




143.14 


363.05 


683.49 


1104.46 


1625.97 


}/2 


.5 eg 


13-16 


2.5802 


13-16 


26.535 


5^ 


145.80 


367.28 


689.30 


1111.84 


1634.92 


^s 


^Zc 


K 


2.7612 


Vs 


27.109 


% 


148.49 


371.54 


695.13 


1119.24 


1643.89 


M 


rt^ <D 


15-16 


2.9483 


15-16 


27.688 


Vs 


151.20 


375.83 


700.98 


1126.66 


1652.88 


ji 


at "^.^ 




2 




6 




14 


22 


30 


38 


46 








3.1416 




28.274 




153.94 


380.13 


706.86 


1134.11 


1661.90 




i-ie 


3.3410 


"H 


29.465 


Vs 


156.70 


384.46 


712.76 


1141.59 


1670.95 


Vs 


•^-^^ 


Vs 


3.5466 


M 


30.680 


M 


159.48 


388.82 


718.69 


1149.08 


1680.01 


H 




3-16 


3.7583 


H 


31.919 


% 


162.30 


393.20 


724.64 


1156.61 


1689.10 


Vs 


M 


3.9761 




33.183 


V2 


165.13 


397.61 


730.62 


1164.15 


1698.23 


H 


S-^s 


5-16 


4.2000 


% 


34.472 


Vs 


167.99 


402.04 


736 . 62 


1171.73 


1707.37 


^8 


|:g^ 


Vs 


4.4301 


H 


35.785 


K 


170.87 


406.49 


742.64 


1179.32 


1716.54 


M 





7-16 


4.6664 


Vs 


37.122 


Vs 


173.78 


410.97 


748.69 


1186.94 


1725.73 


Vs 








7 




15 


23 


31 


39 


47 




Q w). 


H 


4.9087 


^ 


38.485 




176.71 


415.48 


754.77 


1194.59 


1734.94 




1 Sh </: 


9-16 


5.1572 


Vs 


39.871 


Vs 


179.67 


420.00 


760.87 


1202.26 


1744.18 


Vs 


032^ 


^ 


5.4119 


H 


41.282 




182.65 


424.56 


766.99 


1209.95 


1753.45 






11-16 


5.6727 


Vs 


42.718 


y4 


185.66 


429.13 


773.14 


1217.67 


1762.73 


H 


H 


5.9396 


V2 


44.179 


V2 


188.69 


433.74 


779.31 


1225.42 


1772.05 


/^ 




13-16 


6.2126 


Vs 


45.664 


Vs 


191.75 


438.36 


785.51 


1233.18 


1781.39 


H 




3^ 


6.4918 




47.173 


V4. 


194.83 


443.01 


791.73 


1240.98 


1790.76 


^ 


115-16 


6.7771 


48. 707 


Vs 


197.93 


447.69 


797.98 


1248.79 1800.14 1 


j| 



CIRCLES, DIAM. TO AREA, IN INCHES. 



231 



— Diameters in Inches and Fractions. 





48 


56 


64 


72 




80 


88 


96 


104 




j 




1809.6 


2463.0 


3217.0 


4071.5 




5026.5 


6082.1 


7238.2 


8494.9 




^ 


M 


1819.0 


2474.0 


3229.6 


4085.7 


'a 


5042.3 


6099.4 


7257.1 


8515.3 


Y 


a 


M 


1828.5 


2485.0 


3242.2 


4099.8 




5058.0 


6116.7 


7276.0 


8535.8 


Y 




5^ 


1837.9 


2496.1 


3254.8 


4114.0 


3^ 


5073.8 


6134.1 


7294.9 


8556.2 


Y 


i 


3^ 


1847 5 


2507.2 


3267.5 


4128.2 


}/2 


5089.6 


6151.4 


7313.8 


8576.7 


Y 




5^ 


1857.0 


2518.3 


3280.1 


4142.5 


Y% 


5105.4 


6168.8 


7332.8 


8597.3 


Y 




f4 


1866.5 


2529.4 


3292.8 


4156.8 


% 


5121.2 


6186.2 


7351.8 


8617.8 


Y 




% 


1876.1 


2540.6 


3305.6 


4171.1 


K 


5137.1 


6203.7 


7370.8 


8638.4 


Y 


C^ 




49 


57 


65 


73 




81 


89 


97 


105 




^ 




1885.7 


2551.8 


3318.3 


4185.4 




5153.0 


6221.1 


7389.8 


8659.0 




1 


y» 


1895.4 


2563.0 


3331.1 


4199.7 


H 


5168.9 


6238.6 


7408.9 


8679.6 


Y 


M 


1905.0 


2574.2 


3343 . 9 


4214.1 


M 


5184.9 


6256.1 


7428.0 


8700.3 


Y 


u 




1914.7 


2585.4 


3356.7 


4228.5 


Vi 


5200.8 


6273.7 


7447.1 


8721.0 


Y 


& 


V 


1924.4 


2596.7 


3369.6 


4242.9 


3J 


5216.8 


6291.2 


7466.2 


8741.7 


Y 


5^ 


1934.2 


2608.0 


3382.4 


4257.4 


^ 


5232.8 


6308.8 


7485.3 


8762.4 


Y 


. 


M 


1943.9 


2619.4 


3395.3 


4271.8 


M 


5248.9 


6326.4 


7504.5 


8783.2 


Y 


1 


K 


1953.7 


2630.7 


3408.2 


4286.3 


% 


5264.0 


6344.1 


7523.7 


8803.9 


Y 






50 


58 


66 


74 




82 


90 


98 


106 




e!§s 




1963.5 


2642.1 


3421.2 


4300.8 




5281.0 


6361.7 


7543.0 


8824.7 




• § 


M 


1973.3 


2653.5 


3434.2 


4315.4 


'y% 


5297.1 


6379.4 


7562.2 


8845.6 


Y 


^^ 


M 


1983.2 


2664.9 


3447.2 


4329.9 


Ya 


5313.3 


6397.1 


7581.5 


8866.4 


Y 


":s 


^ 


1993.1 


2676.4 


3460.2 


4344.5 


Yk 


5329.4 


6414.9 


7600.8 


8887.3 


Y 


i^:^ 


H 


2003.0 


2687.8 


3473.2 


4359.2 


3^ 


5345.6 


6432.6 


7620.1 


8908.2 


Y 




% 


2012.9 


2699.3 


3486.3 


4373.8 


6^ 


5361.8 


6450.4 


7639.5 


8929.1 


Y 


M 


2022.8 


2710.9 


3499.4 


4388.5 


M 


5378.1 


6468.2 


7658.9 


8950.1 


Y 


J^ 


2032.8 


2722.4 


3512.5 


4403.1 


K 


5394.3 


6486.0 


7678.3 


8971.0 


Y 


M 




51 


59 


67 


75 




83 


91 


99 


107 




-S^^ 




2042.8 


2734.0 


3525.7 


4417.9 




5410.6 


6503.9 


7697.7 


8992.0 




S W 


H 


2052.8 


2745.6 


3538.8 


4432.6 


K 


5426.9 


6521.8 


7717.1 


9013.0 


Y 


CTgJ 


M 


2062.9 


2757.2 


3552.0 


4447.4 


M 


5443.3 


6539.7 


7736.6 


9034.1 


Y 


(U Oi 


^ 


2073.0 


2768.8 


3565.2 


4462.2 


^ 


5459.6 


6557.6 


7756.1 


9055.2 


Y 


H 


2083.1 


2780.5 


3578.5 


4477.0 




5476.0 


6575.5 


7775.6 


9076.3 


Y 


a fj 


t^ 


2093 . 2 


2792.2 


3591.7 


4491.8 


Y, 


5492.4 


6593.5 


7795.2 


9097.4 


Y 




15 


2103.3 


2803.9 


3605.0 


4506.7 


% 


5508.8 


6611.5 


7814.8 


9119.4 


Y 




K 


2113.5 


2815.7 


3618.3 


4521.5 


% 


5525.3 


6629.6 


7834.4 


9139.7 


Y 




52 


60 


68 


76 




84 


92 


100 


108 




ccNiO 




2123.7 


2827.4 


3631.7 


4536.5 




5541.8 


6647.6 


7854.0 


9160.9 




g-o 


yk 


2133.9 


2839.2 


3645.0 


4551.4 


a 


5558.3 


6665.7 


7873.6 


9182.1 


'y 


"o tt 


^ 


2144.2 


2851.0 


3658.4 


4566.4 




5574.8 


6683.8 


7893.3 


9203.3 


Y 


.•2 


2154.5 


2862.9 


3671.8 


4581.3 


Y 


5591.4 


6701.9 


7913.0 


9224.6 


^ 




3^ 


2164.8 


2874.8 


3685.3 


4596.3 


Y 


5607.9 


6720.1 


7932.7 


9245.9 


3^ 





i 


2175.1 


2886.6 


3698.7 


4611.4 


Y 


5624.5 


6738.2 


7952.5 


9267.2 




2185.4 


2898.6 


3712.2 


4626.4 


H 


5641.2 


6756.4 


7972.2 


9288.6 


Y 


2195.8 


2910.5 


3725.7 


4641.5 


% 


5657.8 


6774.7 


7992.0 


9309.9 


Y 


CO-- 




5Z 


61 


69 


77 




85 


93 


101 


109 




u 




2206.2 


2922.5 


3739.3 


4656.6 




5674.5 


6792.9 


8011.8 


9331.3 




i 


2216.6 


2934.5 


3752.8 


467.1.8 


Y 


5691.2 


6811.2 


8031.7 


9552.7 


Y 


^s 


2227.0 


2943.5 


3766.4 


4686.9 


Y 


5707.9 


6829.5 


8051.6 


9374.2 


Y 


oS 


2237.5 


2958.5 


3780.0 


4702.1 


Ys 


5724.7 


6847.8 


8071.4 


9395.6 


Y 


i>^ d 


2248.0 


2970.6 


3793.7 


4717.3 


Y2 


5741.5 


6866.1 


8091.4 


9417.1 




^-B 


2258.5 


2982.7 


3807.3 


4732.5 




5758.3 


6884.5 


8111.3 


9438.6 


Y 


<^^ 


2269.1 


2994.8 


3821.0 


4747.8 


z/ 


5775.1 


6902.9 


8131.3 


9460.2 


Y 


-^ ^ 


2279.6 


3006.9 


3834.7 


4763.1 


Y 


5791.9 


6921.3 


8151.3 


9481.7 


Y 


§x 




54 


62 


70 


78 




86 


94 


102 


no 




Cot 


1 


2290.2 


3019.1 


3848.5 


4778.4 




5808.8 


6939.8 


8171.3 


9503.3 




2300.8 


3031.3 


3862.2 


4793.7 


Y 


5825.7 


6958.2 


8191.3 


9524.9 


Y 


OJ ^ 


2311.5 


3043.5 


3876.0 


4809.0 


Y 


5842.6 


6976.7 


8211.4 


9546.6 


Y 


c3 ^ • 


2322.1 


3055.7 


3889.8 


4824.4 




5859.6 


6995.3 


8231.5 


9568.2 


Y 


is.S 


2332.8 


3068.0 


3903.6 


4839.8 


Y 


5876.5 


7013.8 


8251.6 


9589.9 


Y 




2343.5 


3080.3 


3917.5 


•4855.2 


Y 


5893.5 


7032.4 


8271.7 


9611.6 


Y 


2354.3 


3092.6 


3931.4 


4870.7 


H 


5910.6 


7051.0 


8291.9 


9633.3 


Y 


tOM'^O 


2365.0 


3104.9 


3945.3 


4886.2 


Y 


5927.6 


7069.6 


8312.1 


9655.1 


Y 


"^"^t; 




55 


63 


71 


79 




87 


95 


103 


111 




.S-S-S 


i 


2375.8 


3117.2 


3959.2 


4901.7 




5944.7 


7088.2 


8332.3 


9676.9 




-5 -3 4^ 


2386.6 


3129.6 


3973.1 


4917.2 


Y 


5961.8 


7106.9 


8352.5 


9698.7 


Y 


feoJ 


2397.5 


3142.0 


3987.1 


4932.7 


Y 


5978.9 


7125.6 


8372.8 


9720.5 


Y 


£fg^ 


2408.3 


3154.5 


4001.1 


4948.3 


Y 


5996.0 


7144.3 


8393.1 


9742.4 


Y 


1 1 ^ 


2419.2 


3166.9 


4015.2 


4963.8 


Y 


6013.2 


7163.0 


8413.4 


9764.3 


Y 


« « g 


2430.1 


3179.4 


4029.2 


4979.5 


Y 


6030.4 


7181.8 


8433.7 


9786.2 


Ys 


te^b^-a 


2441.1 


3191.9 


4034.3 


4995.2 


Y 


6047.6 


7200.6 


8454.1 


9808.1 


Y 


Ttt 


2452.0 


3204.4 


4057.4 


6010.9 


Y 


6064.9 


7219.4 


8474.5 


9830.1 


Y 


.S3 



232 



n.—MENSURA TION, 



14. — Areas op Circles in Square Feet* for Given — 

Note. — Diameters in feet are in heavy type, with decimals of 
in column" opposite the areas. 



foot 





,000000 
.007854 
.031416 
.070686 
.125664 
.196350 
.282743 
.384845 
.502655 
.636173 

1 

.785398 
.950332 
1.13097 
1.32732 
1.53938 
1.76715 
2.01062 
2.26980 
2.54469 
2.83529 

2 
3.14159 
3.46361 
3.80133 
4.15476 
4.52389 
4.90874 
5.30929 
5.72555 
6.15752 
6.60520 

3 

7.06858 

7.54768 

8.04248 

8.55299 

9.07920 

9.62113 

10.17876 

10.75210 

11.34115 

11.94591 

4 
12.56637 
13.20254 
13.85442 
14.52201 
15.20531 
15.90431 
16.61903 
17.34945 
18.09557 
18.85741 

5 
19.63495 
20.42821 
21.23717 
22.06183 
22.90221 
23.75829 
24.63009 
25.51759 
26.42079 
27.33971 



6 

28.27433 
29.22467 
30.19071 
31.17245 
32.16991 
33.18307 
34.21194 
35.25652 
36.31681 
37.39281 

7 

38.48451 
39.59192 
40.71504 
41.85387 
43.00840 
44.17865 
45.36460 
46.56626 
47.78362 
49.01670 

8 

50.26548 
51.52997 
52.81017 
54.10608 
55.41769 
56.74502 
58.08805 
59.44679 
60.82123 
62.21139 

9 

63.61725 
65.03882 
66.47610 
67.92909 
69.39778 
70.88218 
72.38229 
73.89811 
75.42964 
76.97687 

10 
78.53982 
80.11847 
81.71282 
83.32289 
84.94867 
86.59015 
88.24734 
89.92024 
91.60884 
93.31316 

1 

95.03318 
96.76891 
98.5203; 
100.2875 
102.0703 
103.8689 
I05.ft832 
107.5132 
109.3588 
111.2202 



12 I 

113.0973 
114.9901 
116.8987 
118.8229 
120.7628 
122.7185 
124.6898 
126.6769 
128.6796 
130.6981 

13 
132.7323 
134.7822 
136.8478 
138.9291 
141.0261 
143.1388 
145.2672 
147.4114 
149.5712 
151.7468 

14 
153.9380 
156.1450 
158.3677 
160.6061 
162.8602 
165.1300 
167.4155 
169.7167 
172.0336 
174.3662 

15 
176.7146 
179.0786 
181.4584 
183.8539 
186.2650 
188.6919 
191.1345 
193.5928 
196.0668 
198.5565 

16 
201.0619 
203.5831 
206.1199 
208.6724 
211.2407 
213.8246 
216.4243 
219.0397 
221.6708 
224.3176 

17 
226.9801 
229.6583 
232.3522 
235.0618 
237.7871 
240.5282 
243.2849 
246.0574 
248.8456 
251.6494 



18 I 

254.4690 
257.30431 
260.1553 
263.0220 
265.9044 
268.8025 
271.7163 
274.6459 
277.5911 
280.5521 

19 
283.5287 
286.5211 
289.5292 
292.5530 
295.5925 
298.6477 
301.7186 
304. 8052 
307.9075 
311.0255 

20 
314.1593 
317.3087 
320.4739 
323.6547 
326.8513 
330.0636 
333.2916 
336.5353 
339.7947 
343.0698 

21 
346.3606 
349.6671 
352.9894 
356.3273 
359.6809 
363.0503 
366.4354 
369.8361 
373.2526 
376.6848 

22 
380.1327 
383.5963 
387.0756 
390.5707 
394.0814 
397.6078 
401.1500 
104.7078 
408.2814 
411.8707 

23 
415.4756 
419.0963 
422.7327 
426.3848 
430.0526 
433.7361 
437.4354 
441.1503 
444.88U9 
448.6273 



24 

452.3893 
456.1671 
459.9606 
463.7698 
467.5947 
471.4352 
475.2916 
479.1636 
483.0513 
486.9547 

25 
490.8739 
494.8087 
498.7592 
502.7255 
506.7075 
510.7052 
514.7185 
518.7476 
522.7924 
526.8529 

26 
530.9292 
535.0211 
539.1287 
543.2521 
547.3911 
551.5459 
555.7163 
559.9025 
564.1044 
568.3220 

27 
572.5553 
576.8043 
581.0690 
585.3494 
589.6455 
593.9574 
598.2849 
602.6282 
606.9871 
611.3618 

28 
615.7522 
620.1582 
624.5800 
629.0175 
633.4707 
637.9397 
642.4243 
646.9246 
651.4407 
655.9724 

29 
660.5199 
665.0830 
669.6619 
674.2565 
678.81 
683.4928 
688.1345 
692.7919 
697.4650 
702.15381 



30 

706.8583 
711.5786 
716.3145 
721.0662 
725.8336 
730.6166 
735.4154 
740.2299 
745.0601 
749.9060 

31 
754.7676 
759.6450 
764.5380 
769.4467 
774.3712 
779.3113 
784.2672 
789.2388 
794.2260 
799.2290 

32 
804.2477 
809.2821 
814.3322 
819.3980 
824.4796 
829.5768 
834.6898 
839.8184 
844.9628 
850.1228 

855.2986 
860.4901 
865.6973 
870.9202 
876.1588 
881.4131 
886.6831 
891.9688 
897.2703 
902.5874 

34 
907.9203 
913.2688 
918.6331 
924.0131 
929.4088 
934.8202 
940.2473 
945.6901 
951.1486 
956.6228 

35 
962.1128 
967.6184 
973.1397 
978.6768 
984.2296 
989.7980 
995.3822 
1000.9821 
1006.5977 
1012.2290 



36 

1017.8760 
1023.5387 
1029.2172 
1034.9113 
1040.6212 
1046.3467 
1052.0880 
1057.8449 
1063.6176 
1069.4060 

37 
1075.2101 
1081.0299 
1086.8654 
1092.7166 
1098.5835 
1104.4662 
1110.3645 
1116.2786 
1122.2083 
1128.1538 

38 
1134.1149 
1140.0918 
1146.0844 
1152.0927 
1158.1167 
1164.1564 
1170.2118 
1176.2830 
1182.3698 
1188.4724 

39 
1194.5906 
1200.7246 
1206.8742 
1213.0396 
1219.2207 
1225.4175 
1231.6300 
1237.8582 
1244.1021 
1250.3617 

40 
1256.6371 
1262.9281 
1269.2348 
1275.5573 
1281.8955 
1288.2493 
1294.6189 
1301.0042 
1307.4052 
1313.8219 

41 
1320.2543 
1326.7024 
1333.1663 
1339.6458 
1346.1410 
1352.6520 
1359.1786 
1365.7210 
1372.2791 
1378.8529 



42 

1385.4424 
1392.0476 
1398.6685 
1405.3051 
1411.9574 
1418.6254 
1425.3092 
1432.0086 
1438.7238 
1445.4546 

43 
1452.2012 
1458.9635 
1465.7415 
1472.5352 
1479.3446 
1486.1697 
1493.0105 
1499.8670 
1506.7393 
1513.6272 

44 
1520.5308 
1527.4502 
1534.3853 
1541.3360 
1548.3025 
1555.2847 
1562.2826 
1569.2962 
1576.3255 
1583.3706 

45 
1590.4313 
1597.5077 
1604.5999 
1611.7077 
1618.8313 
1625.9705 
1633.1255 
1640.2962 
1647.4826 
1654.6847 

46 
1661.9025 
1669.1360 
1676.3853 
1683.6502 
1690.9308 
1698.2272 
1705.5392 
1712.8670 
1720.2105 
1727.5697 

47 
1734.9445 
1742.3351 
1749.7414 
1757.1635 
1764.6012 
1772.0546 
1779.5237 
1787.0086 
1794.5091 
1802.0254 



48 

1809.5574 
1817.1050 
1824.6684 
1832.2475 
1839.8423 
1847.4528 
1855.0790 
1862.7210 
1870.3786 
1878.0519 

49 
1885.7410 
1893.4457 
1901.1662 
1908.9024 
1916.6543 
1924.4218 
1932.2051 
1940.0041 
1947.8189 
1955.6493 

SO 
19&3.4954 
1971.3572 
1979.2348 
1987.1280 
1995.0370 
2002.9617 
2010.9020 
2018.8581 
2026.8299 
2034.8174 

51 

2042.8206 
2050.8395 
2058.8742 
2066.9245 
2074.9905 
2083.0723 
2091.1697 
2099.2829 
2107.4118 
2115.5563 

52. 
2123.7166 
2131.8926 
2140.0843 
2148.2917 
2156.5149 
2164.7537 
2173.0082 
2181.2785 
2189.5644 
2197.8661 

53 
2206.1834 
2214.5165 
2222.8653 
2231.2298 
2239.6100 
2248.0059 
2256.4175 
2264.8448 
2273.2879 
2281.7466 



* Or any other denomination. Note that areas of circles are proportional 
to the squares of their diameters, and that the range of the table may there- 
fore be extended greatly. 

Spheres: Surface = 4 X above areas. Volume = I diam. X above areas. 
(Assuming diam. of sphere = diam. of circle.) 



CIRCLES, DIAM. TO AREA, IN DECIMALS. 



233 



— Diameters in Feet* and Tenths. 

Note that changing 1 decimal point in the diameter = 2 decimal points 
in the area. Ex. — For dia. of 5.41, area equals 22.987. 



.0 



54 

2290.2210 
2298.7112 
2307.2171 
2315.7386 
2324.2759 
2332.8289 
2341.3976 
2349.9820 
2358.5821 
2367.1979 

55 
2375.8294 
2384.4767 
2393.1396 
2401.8183 
2410.5126 
2419.2227 
2427.9485 
2436.6899 
2445.4471 
2454.2200 

56 
2463.0086 
2471.8130 
2480.6330 
2489.4687 
2498.3201 
2507.1873 
2516.0701 
2524.9687 
2533.8830 
2542.8129 

57 
2551.7586 
2560.7200 
2569.6971 
2578.6899 
2587.6985 
2596.7227 
2605.7626 
2614.8183 
2623.8896 
2632.9767 

58 
2642.0794 
2651.1979 
2660.3321 
2669.4820 
2678.6476 
2687.8289 
2697.0259 
2706.2386 
2715.4670 
2724.7112 

59 
2733.9710 
2743.2466 
2752.5378 
2761.8448 
2771.1675 
2780.5058 
2789.8599 
2799.2297 
2808.6152 
2818.0165 



60 

2827.4334 
2836.8660 
2846.3144 
2855.7784 
2865.2582 
2874.7536 
2884.2648 
2893.7917 
2903.3343 
2912.8926 

61 
2922.4668 
2932.0563 
2941.6617 
2951.2828 
2960.9197 
2970.5722 
2980.2405 
2989 9244 
2999.6241 
3009.3395 

62 
3019.0705 
3028.8173 
3038.5798 
3048.3580 
3058.1520 
3067.9616 
3077.7869 
3087.6279 
3097.4847 
3107.3571 

63 
3117.2453 
3127.1492 
3137.0688 
3147.0040 
3156.9550 
3166.9217 
3176.9042 
3186.9023 
3196.9161 
3206.9456 

64 
3216.9909 
3227.0518 
3237.1285 
3247.2209 
3257.3289 
3267.4527 
3277.5922 
3287.7474 
3297.9183 
3308.1049 

65 
3318.3072 
3328.5253 
3338.7590 
3349.0085 
3359.2736 
3369.5545 
3379.8510 
3390.1633 
3400.4913 
3410.8350 



66 

3421.1944 
3431.5695 
3441.9603 
S452.3669 
3462.7891 
3473.2270 
3483.6807 
3494.1500 
3504.6351 
3515.1359 

67 
3525.6524 
3536.1845 
3546.7324 
3557.2960 
3567.8754 
3578.4704 
3589.0811 
3599.7075 
3610.3497 
3621.0075 

68 
3631.6811 
3642.3704 
3653.0754 
366^3.7960 
3674.5324 
3685.2845 
3696.0523 
3706.8359 
3717.6351 
3728.4500 

69 
3739.2807 
3750.1270 
3760.9891 
3771.8668 
3782.7603 
3793.6695 
3804.5944 
3815.5350 
3826.4913 
3837.4633 

70 
3848.4510 
3859.4544 
3870.4736 
3881.5084 
3892.5590 
3903.6252 
3914.7072 
3925.8049 
3936.9182 
3948.0473 

71 
3959.1921 
3970.3526 
3981.5289 
3992.7208 
4003.9284 
4015.1518 
4026.3908 
4037.6456 
4048.9160 
4060.2022 



72 

4071.5041 
4082.8217 
4094.1550 
4105.5040 
4116.8687 
4128.2491 
4139.6452 
4151.0571 
4162.4846 
4173.9279 

73 
4185.3868 
4196.8615 
4208.3519 
4219.8579 
4231.3797 
4242.9172 
4254.4704 
4266.0394 
4277.6240 
4289.2243 

74 
4300.8403 
4312.4721 
4324.1195 
4335,7827 
4347.4616 
4359.1562 
4370.8664 
4382.5924 
4394.3341 
4406.0916 

75 
4417.8647 
4429.6535 
4441.4580 
4453.2783 
4465.1142 
4476.9659 
4488.8332 
4500.7163 
4512.6151 
4524.5296 

76 
4536.4598 
4548.4057 
4560.3673 
4572.3446 
4584.3377 
4596.3464 
4608.3708 
4620.4110 
4632.4669 
4644.5384 

77 
4656.6257 
4668.7287 
4680.8474 
4692.9818 
4705.1319 
4717.2977 
4729.4792 
4741.6765 
4753.8894 
4766.1181 



78 

4778.3624 
4790.6225 
4802.8983 
4815.1897 
4827.4969 
4839.8198 
4852.1584 
4864.5128 
4876.8828 
4889.2685 

79 
4901.6699 
4914.0871 
4926.5199 
4938.9685 
4951.4328 
4963.9127 
4976.4084 
4988.9198 
5001.4469 
5013.9897 

80 
5026.5482 
5039.1225 
5051.7124 
5064.3180 
5076.9394 
5089.5764 
5102.2292 
5114.8977 
5127.5819 
5140.2818 

81 
5152.9974 
5165.7287 
5178.4757 
5191.2384 
5204.0168 
5216.8110 
5229.6208 
5242.4463 
5255.2876 
5268.1446 

82 
5281.0173 
5293.9056 
5306.8097 
5319.7295 
5332.6650 
5345.6162 
5358.5832 
5371.5658 
5384.5641 
5397.5782 

83 
5410.6079 
5423.6534 
5436.7146 
5449.7915 
5462.8840 
5475.9923 
5489.1163 
5502.2561 
5515.4115 
5528.5826 



84 

5541.7694 
5554.9720 
5568.1902 
5581.4242 
5594.6739 
5607.9392 
5621.2203 
5634.5171 
5647.8296 
5661.1578 

85 
5674.5017 
5687.8614 
5701.2367 
5714.6277 
5728.0345 
5741.4569 
5754.8951 
5768.3490 
5781.8185 
5795.3038 

86 
5808. 8048 
5822.3215 
5835.8539 
5849.4020 
5862.9659 
5876.5454 
5890.1407 
5903.7516 
5917.3783 
5931.0206 

87 
5944.6787 
5958.3525 
5972.0420 
5985.7472 
5999.4681 
6013.2047 
6026.9570 
6040.7250 
6054.5088 
6068.3082 

88 
6082.1234 
6095.9542 
6109.8008 
6123.6631 
6137.5411 
6151.4348 
6165.3442 
6179.2693 
6193.2101 
6207.1666 

89 
6221.1388 
6235.1268 
6249.1304 
6263.1498 
6277.1848 
6291.2356 
6305.3021 
6319.3843 
6333.4822 
6347.5958 



90 

6361.7251 
6375.8701 
6390.0308 
6404.2073 
6418.3994 
6432.6073 
6446.8308 
6461.0701 
6475.3251 
6489.5958 

91 
6503.8822 
6518.1843 
6532.5021 
6546.8356 
6561.1848 
6575.5497 
6589.9304 
6604.3267 
6618.7388 
6633.1666 

92 
6647.6100 
6662.0692 
6676.5441 
6691.0347 
6705.5410 
6720.0630 
6734.6007 
6749.1542 
6763.7233 
6778.3081 

93 
6792.9087 
6807.5250 
6822.1569 
6836.8046 
6851.4680 
6866.1471 
6880.8419 
6985.5524 
6910.2786 
6925.0205 

94 
6939.7782 
6954.5515 
6969.3406 
6984.1453 
6998.9658 
7013.8019 
7028.6538 
7043.5214 
7058.4047 
7073.3037 

95 
7088.2184 
7103.1488 
7118.0950 
7133.0568 
7148.0343 
7163.0276 
7178.0366 
7193.0612 
7208.1016 
7223.1577 



96 

7238.2295 
7253.3170 
7268.4202 
7283.5391 
7298.6737 
7313.8240 
7328.9901 
7344.1718 
7359.3693 
7374.5824 

97 
7389.8113 
7405.0559 
7420.3162 
7435.5922 
7450.8839 
7466.1913 
7481.5144 
7496.8532 
7512.2078 
7527.5780 

98 
7542.9640 
7558.3656 
7573.7830 
7589.2161 
7604.6648 
7620.1293 
7635.6095 
7651.1054 
7666.6170 
7682.1444 

99 
7697.6874 
7713.2461 
7728.8206 
7744.4107 
7760.0166 
7775.6382 
7791.2754 
7806.9284 
7822.5971 
7838.2815 

100 
7853.9816 
7869.6975 
7885.4290 
7901.1762 
7916.9392 
7932.7178 
7948.5122 
7964.3222 
7980.1480 
7995.9895 

101 
8011.8467 
8027.7196 
8043.6082 
8059.5125 
8075.4325 
8091.3682 
8107.3197 
8123.2868 
8139.2697 
8155.2682 



* Or any other denomination. Diameters and areas must be in the same 
denomination, i. e., meters, feet or inches, etc. 

Note that areas of circles are proportional to the squares of their dia- 
meters, and that the range of the table may therefore be extended greatly. 



234 



11 .—MENS URA TION. 



15. — Areas of Circles in Square Feet for Given — 
Note. — Diameters in feet are in heavy type, with inches "in column" 
opposite the areas. 





.000000 
.005454 
.021817 
.049087 
.087266 
.136354 
.196350 
.267254 
.349066 
.441786 
.545415 
.659953 

.785398 

.921752 

.069014 

.22718 

.39626 

.57625 

.76715 

.96895 

.18166 

,40528 

,63981 

, 88525 

2 
,14159 
,40885 
,68701 
,97608 
,27606 
,58694 
.90874 
,24144 
,58505 
93957 
30500 
,68134 
3 

06858 
46674 
,87580 
29577 
72665 
16843 
62113 
08473 
5592 
0447 
5410 
0482 
4 

5664 
0954 
6354 
1863 
7480 
3207 
9043 
4988 
1042 
7205 
3478 
9859 



5 

19.6350 
20.2949 
20.9658 
21.6475 
22.3402 
23.0438 
23.7583 
24.4837 
25.2200 
25.9672 
26.7254 
27.4944 

6 
28.2743 
29.0652 
29.8669 
30.6796 
31.5032 
32.3377 
33.1831 
34.0394 
34.9066 
35.7847 
36.6737 
37.5737 

7 
38.4845 
39.4063 
40.3389 
41.2825 
42.2370 
43.2024 
44.1786 
45.1658 
46.1640 
47.1730 
48.1929 
49.2237 

8 
50.2655 
51.3181 
52.3817 
53.4562 
54.5415 
55.6378 
56.7450 
57.8631 
58.9921 
60.1320 
61.2829 
62.4446 

9 

63.6173 
64.8008 
65.9953 
67.2006 
68.4169 
69.6441 
70.8822 
72.1312 
73.3911 
74.6619 
75.9436 
77.2363 



10 

78.5398 
79.8543 
81.1796 
82.5159 
83.8631 
85.2212 
86.5901 
87.9700 
89.3609 
90.7626 
92.1752 
93.5987 
11 

95.0332 
96.4785 
97.9348 



15 

176.715 
178.684 
180.663 
182.654 
184.656 
186.668 
188.692 
190.726 
192.772 
194.828 
196.895 
198.973 

16 
201.062 
203.162 
205.273 



99.4020 207.394 
100.8800 209.527 
211.670 
213.825 
215.990 
218.166 
220.353 
222.551 
224.760 

17 
226.980 
229.211 
231.452 
233.705 
235.969 
238.243 
240.528 
242.824 
245.131 
247.450 
249.778 
252.118 

18 
254.469 
256.831 
259.203 
261.587 
263.981 
266.386 
268.803 
271.230 
273.668 
276.117 
278.576 
281.047 

19 
283.529 
286.021 
288.525 
291.039 
293.564 
296.101 
298.648 
301.206 
303.775 
306 354 
308.945 
311.547 



102.369 
103.869 
105.380 
106.901 
108.434 
109.978 
111.532 

12 
113.097 
114.674 
116.261 
117.859 
119.468 
121.088 
122.718 
124.360 
126.013 
127.676 
129.351 
131.036 

13 
132.732 
134.439 
136.157 
137.886 
139.626 
141.377 
143.139 
144.911 
146.695 
148.489 
150.295 
152.111 

14 
153.938 
155.776 
157.625 
159.485 
161.356 
163.237 
165.130 
167.033 
167.948 
170.873 
172.809 
174.757 



20 

314.159 
316.783 
319.417 
322.062 
324.719 
327.386 
330.064 
332.752 
335.452 
338.163 
340.885 
343.617 

21 
346.361 
349.115 
351. f 
354.656 
357.443 
360.241 
363.050 
365.870 
368.701 
371.542 
374.395 
377.258 

22 
380.133 
383.01 
385.914 
388.821 
391.739 
394.668 
397.608 
400.559 
403.520 
406.493 
409.476 
412.470 

23 
415.476 
418.492 
421.519 
424.557 
427.606 
430.665 
433.736 
436.818 
439.910 
443.014 
446.128 
449.253 

24 
452.389 
455.536 
458.694 
461.863 
465.043 
468.234 
471.435 
474.648 
477.871 
481.105 
484.351 
487.607 



25 

490.874 
494.152 
497.441 
500.740 
504.051 
507.373 
510.705 
514.049 
517.403 
520.768 
524.144 
527.531 

26 
530.929 
534.338 
537.758 
541.188 
544.630 
548.082 
551.546 
555.020 
558.505 
562.001 
565.508 
569.026 

27 
572.555 
576.095 
579.646 
583.207 
586.780 
590.363 
593.597 
597.563 
601.179 
604.806 
608.444 
612.092 

28 
615.752 
619.423 
623.104 
626.797 
630.500 
634.214 
637.940 
641.676 
645.423 
649.181 
652.949 
656.729 

29 
660.520 
664.321 
668.134 
671.957 
675.791 
679.637 
683.495 
687.360 
691.238 
695.126 
699.026 
702.937 



30 

706.858 
710.791 
714.734 
718.688 
722.654 
726.630 
730.617 
734.615 
738.623 
742.643 
746.674 
750.715 

31 
754.768 
758.831 
762.905 
766.990 
771.086 
775.193 
779.311 
783.440 
787.580 
791.730 
795.892 
800.064 

32 
804.248 
808.442 
812.647 
816.863 
821.0901 
825.328 
829.577 
833.837 
838.107 
842.389 
846.681 
850.984 

33 
855.299 
859.624 
863.960 
868.307 
872.665 
877.033 
881.413 
885. 804 
890.205 
894.618 
899.041 
903.475 

34 
907.920 
912.376 
916.843 
921.321 
925.810 
930.310 
934.820 
939.342 
943.874 
948.417 
952.972 
957.537 



35 

962.113 

966.700 

971.298 

975.906 

980.526 

985.157 

989.798 

994.450 

999.114 

1003.788 

1008.47 

1013.17 

36 
1017.88 
1022.59 
1027.32 
1032.06 
1036.81 
1041.57 
1046.35 
1051.13 
1055.92 
1060.73 
1065.55 
1070.37 

37 
1075.21 
1080.06 
1084.92 
1089.79 
1094.67 
1099.56 
1104.47 
1109.38 
1114.31 
1119.24 
1124.19 
1129.15 

38 
1134.11 
1139.09 
1144.09 
1149.09 
1154.10 
1159.12 
1164.16 
1169.20 
1174.26 
1179.32 
1184.40 
1189.49 

39 
1194.59 
1199.70 
1204.82 
1209.95 
1215.10 
1220.25 
1225.42 
1230.59 
1235.78 
1240.98 
1246.19 
1251.41 



40 

1256.64 
1261.88 
1267.13 
1272.39 
1277.67 
1282.95 
1288.25 
1293.56 
1298.87 
1304.20 
1309.54 
1314.89 

41 
1320.25 
1325.63 
1331.01 
1336.40 
1341.81 
1347.23 
1352.65 
1358.09 
1363.54 
1369.00 
1374.47 
1379.95 

42 
1385.44 
1390.95 
1396.46 
1401. 
1407.52 
1413.07 
1418.63 
1424.19 
1429.77 
1435.36 
1440.97 
1446.58 

43 
1452.20 
1457.84 
1463.48 
1469.14 
1474.80 
1480.48 
1486.17 
1491.87 
1497.58 
1503.30 
1509.03 
1514.78 

44 
1520.53 
1526.30 
1532.07 
1537.86 
1543.66 
1549.47 
1555.28 
1561.12 
1566.96 
1572.81 
1578.67 
1584.55 



Areas of circles are proportional to the squares of their diameters. 
Spheres: Surface = 4 X above areas. Volume = f diam. X above areas. 
(Assuming diam. of sphere = diam. of circle.) 



CIRCLES, VIA. IN FT, AND INS., TO AREA. 



235 



— Diameters in Feet and Inches. 

Ex. — For dia. of 65 ft., area equals 3318.31 sq. ft. 

For dia. of 90'- 6'', area equals 6432.61 sq. ft. 



50 

1963.50 
1970.05 
1976.61 
1983.18 
1919.76 
1996.36 
2002.96 
2009.58 
2016.20 
2022.84 
2029.49 
2036.15 

51 
2042.82 
2049.50 
2056.19 
2062.90 
2069.61 
2076.34 
2083.07 
2089.82 
2096.58 
2103.35 
2110.12 
2116.92 

52 
2123.72 
2130.53 
2137.35 
2144.19 
2151.03 
2157.89 
2164.75 
2171.63 
2178.52 
2185.42 
2192.33 
2199.25 

53 
2206.18 
2213.13 
2220.09 
2227.05 
2234.02 
2241.01 
2248.01 
2255.01 
2262.03 
2269.06 
2276.11 
2283.16 

54 
2290.22 
2297.30 
2304.38 
2311.48 
2318.58 
2325.70 
2332.83 
2339.97 
2347.12 
2354.28 
2361.45 
2368.64 



55 

2375.83 
2383.03 
2390.25 
2397.48 
2404.71 
2411.96 
2419.22 
2426.49 
2433.77 
2441.07 
2448.37 
2455.68 

56 
2463.01 
2470.34 
2477.69 
2485.05 
2492.42 
2499.80 
2507.19 
2514.59 
2522.00 
2529.42 
2536.86 
2544.30 

57 
2551.76 
2559.23 
2566.70 
2574.19 
2581.69 
2589.20 
2596.72 
2604.25 
2611.80 
2619.35 
2626.92 
2634.49 

58 
2642.08 
2649.68 
2657.29 
2664.91 
2672.54 
2680.18 
2687.83 
2695.49 
2703.17 
2710.85 
2718.55 
2726.25 

59 
2733.97 
2741.70 
2749.44 
2757.19 
2764.95 
2772.72 
2780.51 
2788.30 
2796.10 
2803.92 
2811.75 
2819.58 



60 

2827.43 
2835.29 
2843.16 
2851.04 
2858.94 
2866.84 
2874.75 
2882.68 
2890.61 
2898.56 
2906.52 
2914.49 

61 
2922.47 
2930.46 
2938.46 
2946.47 
2954.49 
2962.53 
2970.57 
2978.63 
2986.69 
2994.77 
3002.86 
3010.96 

62 
3019.07 
3027.19 
3035.32 
3043.47 
3051.62 
3059.79 
3067.96 
3076.15 
3084.35 
3092.55 
3100.77 
3109.00 

63 
3117.25 
3125.50 
3133.76 
3142.03 
3150.32 
3158.62 
3166.92 
3175.24 
3183.57 
3191.91 
3200.26 
3208.62 

64 
3216.99 
3225.37 
3233.77 
3242.17 
3250.59 
3259.02 
3267.45 
3275.90 
3284.36 
3292.83 
3301.31 
3309.80 



65 

3318.31 
3326.82 
3335.35 
3343.88 
3352.43 
33G0.99 
3369.55 
3378.13 
3386.72 
3395.33 
3403.94 
3412.56 

66 
3421.19 
3429.84 
3438.49 
3447.16 
3455.84 
3464.53 
3473.23 
3481.94 
3490.66 
3499.39 
3508.13 
3516.89 

67 
2525.65 
3534.43 
3543.21 
3552.01 
3560.82 
3569.64 
3578.47 
3587.31 
3596.16 
3605.03 
3613.90 
3622.79 

68 
3631.68 
3640.59 
3649.51 
3658.43 
3667.37 
3676.32 
3685.29 
3694.26 
3703.24 
3712.23 
3721.24 
3730.25 

69 
3739.28 
3748.32 
3757.37 
3766.43 
3775.50 
3784.58 
3793.67 
3802.77 
3811.89 
3821.01 
3830.15 
3839.29 



70 

45 
3857.62 
3866.80 
3875.99 
3885.19 
3894.40 
3903.63 
3912.86 
3922.10 
3931.36 
3940.63 
3949.90 

71 
3959.19 
3968.49 
3977.80 
3987.12 
3996.45 
4005.80 
4015.15 
4024.52 
4033.89 
4043.28 
4052.68 
4062.08 

72 
4071.50 
4080.93 
4090.38 
4099.83 
4109.29 
4118.76 
4128.25 
4137.74 
4147.25 
4156.77 
4166.30 
4175.84 

73 
4185.39 
4194.95 
4204.52 
4214.10 
4223.70 
4233.30 
4292.92 
4252.54 
4262.18 
4271.83 
4281.49 
4291.16 

74 
4300.84 
4310.53 
4320.24 
4329.95 
4339.67 
4349.41 
4359.16 
4368.91 
4378.68 
4388.46 
4398.25 
4408.05 



75 

4417.86 
4427.69 
4437.52 
4447.37 
4457.22 
4467.09 
4476.97 
4486.85 
4496.75 
4506.66 
4516.58 
4526.52 

76 
4536.46 
4546.41 
4556.38 
4566.35 
4576.34 
4586.34 
4596.35 
4606.37 
4616.40 
4626.44 
4636.49 
4646.55 

77 
4656.63 
4666.71 
4676.81 
4686.91 
4697.03 
4707.16 
4717.30 
4727.45 
4737.61 
4747.78 
4757.96 
4768.16 

78 
4778.36 
4788.58 
4798.80 
4809.04 
4819.29 
4829.55 
4839.82 
4850.10 
4860.39 
4870.70 
4881.01 
4891.33 

79 
4901.67 
4912.02 
4922.37 
4932.74 
4943.12 
4953.51 
4963.91 
4974.32 
4984.75 
4995.18 
5005.63 
5016.08 



80 

5026.55 
5037.03 
5047.51 
5058.01 
5068.52 
5079.04 
5089.58 
5100.12 
5110.67 
5121.24 
5131.81 
5142.40 

81 
5153.00 
5163.61 
5174.22 
5184.86 
5195.50 
5206.15 
5216.81 
5227.48 
5238.17 
5248.87 
5259.57 
5270.29 

82 
5281.02 
5291.76 
5302.51 
5313.27 
5324.04 
5334.82 
5345.62 
5356.42 
5367.24 
5378.06 
5388.90 
5399.75 

83 
5410.61 
5421.48 
5432.36 
5443.25 
5454.15 
5465.07 
5475.99 
5486.93 
5497.87 
5508.83 
5519.80 
5530.7 

84 
5541.77 
5552.77 
5563.78 
5574.81 
5585.84 
5596. 
5607.94 
5619.01 
5630.08 
5641.17 
5652.27 
5663,38 



85 

5674.50 
5685.63 
5696.78 
5707.93 
5719.09 
5730.27 
5741.46 
5752.65 
5763.86 
5775.08 
5786.31 
5797.55 

86 
5808.80 
5820.07 
5831.34 
5842.63 
5853.92 
5865.23 
5876.55 
5887.87 
5899.21 
5910.56 
5921.92 
5933.30 

87 
5944.68 
5956.07 
5967.48 
5978.89 
5990.32 
6001.76 
6013.20 
6024.66 
6036.13 
6047.61 
6059.11 
6070.61 

88 
6082.12 
6093.65 
6105.18 
6016.73 
6128.29 
6139.86 
6151.43 
6163 ..02 
6174.63 
6186.24 
6197.86 
6209.49 

89 
6221.14 
6232.79 
6244.46 
6256.14 
6267.83 
6279.53 
6291.24 
6302.96 
6314.69 
6326.43 
6338.18 
6349.95 



90 

6361.73 
6373.51 
6385.31 
6397.12 
6408.94 
6420.77 
6432.61 
6444.46 
6456.32 
6468.20 
6480.08 
6491.98 

91 
6503.88 
6515.80 
6527.73 
6539.67 
6551.62 
6563.58 
6575.55 
6587.53 
6599.53 
6611.53 
6623.55 
6635.57 

92 
6647.61 
6659.66 
6671.72 
6683.79 
6695.87 
6707.96 
6720.06 
6732.18 
6744.30 
6756.44 
6768.58 
6780.74 

93 
6792.91 
6805.09 
6817.28 
6829.48 
6841.69 
6853.91 
6866.15 
6878.39 
6890.65 
6902.91 
6915.19 
6927.48 

94 
6939.78 
6952.09 
6964.41 
6976.74 
6989.08 
7001.44 
7013.80 
7026.18 
7038.56 
7050.96 
7063.37 
7075.79 



95 

7088.22 
7100.66 
7113.11 
7125.57 
7138.05 
7150.53 
7163.03 
7175.53 
7188.05 
7200.58 
7213.12 
7225.67 

96 
7238.23 
7250.80 
7263.38 
7275.98 
7288.58 
7301.20 
7313.82 
7326.46 
7339.11 
7351.77 
7364.44 
7377.12 

97 
7389.81 
7402.51 
7415.23 
7427.95 
7440.69 
7453.43 
7466.19 
7478.96 
7491.74 
7504.53 
7517.33 
7530.14 

98 
7542.96 
7555.80 
7568.64 
7581.50 
7594.36 
7607.24 
7620.13 
7633.03 
7645.94 
7658.86 
7671.79 
7684.73 

99 
7697.69 
7710.65 
7723.63 
7736.61 
7749.61 
7762.62 
7775.64 
7788.68 
7801.71 
7814.76 
7827.82 
7840. 9() 



For a circle 1 ft. in dia., area = 0.7853981634 sq. ft. 
For a circle 1 in. in dia., area = 0.0054541539 sq. ft. 



236 



U.^MEN5URATTON. 




Fig. 16. 

Cycloid. — If a " generating " circle C of diameter d is rolled along a 
straight base or chord c, any point as p'\ starting from a point A, will have 
traced a cycloidal arc a, from A to J5, when the generating circle has 
performed a complete revolution. Moreover, the evolute of the cycloid is 
composed of two half-arcs shown below the base and meeting at the 
point O. Thus, A = \ arc a. 

Properties of the Cycloid (Fig. 1 6) : 



Length of arc a = ^d 



4 c 

4 times diam. of generating circle = — = 1.27324 c. 



Length of chord c=»7r(i=3.1416(i= 



= 0.7854 a. 



Area of cycloidal segment {bet. a and c) = 3 X area of generating circle = Ind"^ = 

idc. 
Distance yo from base to cen of grav g of cycloidal arc (line) = 1 d. 
Distance Yq from base to cen of grav G of cycloidal segment (surface above 

, c) = ^\d. 
Tangent t' at point p' is parallel with t', f at point p" is par with t^. 
Normal n' at point p' is parallel with n\ c' d' at point p" is par with «i. 

Note that a' = a!' = length along base from A to intersection of d with c. 
The extremities of c' d' touch the cycloidal arc above, and the evolute below. 
Area contained between the base and the evolute (below c) = \ n d^=^\ d c. 
If the half-arc A O is inverted it forms a curve along which a body will 
descend, by gravity, from O to A in the least space of time. 
For Motion of Falling Bodies on the Cycloidel Curve, see page 286. 
For Equation of Cycloid, see page 260. 

The Trochoid is a curve described by a point fixed to the rolling circle C, 
but lying either outside or inside its circumference, while tracing the cycloid. 

Epicycloid and Hypocycloid.— If the base AB, Fig. 16, is an arc of a fixed 
circle instead of a straight line, then a point p on the generating circle C will 
trace: (1) an epicycloid if rolled upon the outside of the fixed circle, and (2) a 
hypocycloi d if rolled upon the inside of the fixed circle. 

The Epitrochoid is a curve described by a point fixed either outside or in- 
side the rolling circle C while tracing the Epicycloid. 

The Hypotrochoid is a curve described by a point fixed either outside or 
inside the rolling circle C while tracing the hypocycloid. 



PARABOLA, SEGMENT AND SPANDRIL. 



237 




Parabola ; Parabolic Segment ; Parabolic HaIf=Segment ; and Parabolic 
Spandril. — (For Equation of Parabola, see page 257.) 

Properties of the Parabola (Fig. 17): 
Let /i = height, and c = chord; - may have any value desired. 

Length of area (=A od) = 2/i^ /_£?_+ 1+^log*— (l+^/_fi- + l ) . (Exact.) 

= 2^—+th\ (Approximate t.) 
Area of segment of height h = l base X altitude = ^ c h. 
Length of chord c' of segment of height 



Area of segment of height h' = \ c'W = I ch 
Area of zone of height h 






Area of parabolic spandril, A O B,= ^h X -^ = i c h. 

Dist. Xq to cen of grav d of half segment of height /t = f X ^ = ?6^. 
Dist Yq to cen of grav G and G^ of segment and half segment Qi) = %h, 

Dist Xq to cen of grav g of spandril = f X -^ =1 c. 

it 
Dist 3^0 to cen of grav g of spandril = /jj h. 
T is tangent to parabola at d\ Ti is tangent at A\ T2 at p2' 

i 2 -t 1 -1 C 

* Use hyperbolic (natural) log = common (Briggs) log multhy 2.3025851, 
the common log of which is 0.3622157. . 

t For more exact values, mult result from approx formula by 0.99975 

when — = 0.1; mult by 0.9972 when — =0.2; by 0.99 when — = 0.3; by 
c c c 

0.968 when — = 0.5; by 0.923 when h = c, 
c 



238 



ll.^MENSURA TION, 



Parabola may be drawn (1) by drawing T2 in various positions between 
T and Ti, varying the distance A pi= C p and using the preceding equa- 
tions for position of p2, although the latter is not absolutely necessary. 

Parabola may be drawn (2) by the method adopted for the right-hand 
half of arc a (Fig. 17): dividing the half-chord and the height into equal 
spaces (any number) and joining points of intersection of verticals with 
corresponding inclined lines, as 1 — 5 with — a, 2—6 with — b, etc. 

Parabola may be diawn (3) by laying off ordinates from the base. 
Thus, if base is divided into 8 parts the middle ordinate = (4)2 k = h, in 
which k = a. constant = re /j. Then the ordinates at and 8 are OXSife; 
at 1 and 7 are 1X7^; at 2 and 6 are 12 k; at 3 and 5 are 15 k; at 4 is 16 k. 
It matters not how many divisions of the base are used, whether odd or 
even. If odd, say 11, the middle ordinate = (5.5)2^. Also, the ordinate 
midway between 1 and 2 (Fig. 17) = 1.5X6.5^. 

16. — Lengths of Parabolic Arcs for Chord (Base) 1. 
(The final figure may not be exact in some cases.) 



Height 


Length of Arc 


Height 


Length of Arc 


Height 


Length of Arc 


div. by 


= Chord mult. 


div. by 


= Chord mult. 


div. by 


= Chord mult. 


Chord. 


by 


Chord. 


by 


Chord. 


by 


.01 


1.000 267 


.11 


1.031 389 


.21 


1.107 516 


.02 


1.001 066 


.12 


1.037 171 


.22 . 


1.117 128 


.03 


1.002 396 


.13 


1.043 395 


.23 


1.127 053 


.04 


1.004 251 


.14 


1.050 048 


.24 


1.137 278 


.05 


1.006 627 


.15 


1.057 116 


.25 


1.147 794 


.06 


1.009 519 


.16 


1.064 587 


.26 


1.158 588 


.07 


1.012 918 


.17 


1.072 447 


.27 


1.169 651 


.08 


1.016 814 


.18 


1.080 684 


.28 


1.180 971 


.09 


1.021 199 


.19 


1.089 281 


.29 


1.192 540 


.10 


1.026 061 


.20 


1.098 230 


.30 


1.204 347 



Calculated from the exact equation, preceding. 




w S=2ci- 

Fig. 18. 

Ellipse. — ^The Ellipse is a flattened circle. (See Analytic Geometry, 
Fig. 9, page 258.) 

Notation and Methods of Drawing the Ellipse (Fig. 18) : 

d /rr— r„ X bx 



a = semi-major axis= 



6 = semi-minor axis=c/. 



Vb^+d^=^s^Ku+v)- 



V 



y2 Vb^-y2 



.l = \/a'^-d^=\/(a-{-d){a-d)=- 



ay 



i-t 



2 \/a^— y2 



,..,,,,. d . ^ \/a2-fe2 / 62 
^=» eccentricity or ellipse = — = sin /? = "^ A/ 2* 



PARABOLIC ARCS. ELLIPSE, 239 



4== i (focal distance) = \/o2-62=v' (a + b) (a-b). 

s ( = span) = major axis = 2 a = u-i-v = — = 2<^^2_|_j2_ 



variable abscissa = a ^jl —■±-. 

d2 






Sometimes used in platting (p") 
or laying out the ellipse. 



2 

y = variable ordinate 

- -■ 2 

u -h V = a constant = 2 a (see point p)\ common method of drawing the 

ellipse. 
a — b = constant distance bet axes on line of "elliptic compass" (see pt. p'). 

Other Properties af the Ellipse: 
Dist xq to can of grav gi of quadrant or g of half -ellipse = 0.42441 a. 
Dist yo to cen of grav gi of quadrant or G of half -ellipse == 0.42441 b. 

Dist ro to cen of grav gi of quadrant = \^xo'^ + yo^ = 0.42441 \/a^-hb^. 

Line n is normal to the ellipse at point p, and bisects angle OC. 
Line t is tangent to the ellipse at point p, and is at L with n. 

Area oi ellipse = nab = - — = ^sb =■— b (u + v). (;r= 3.1416.) 
= 0.7854 X major axis X minor axis. 
Formulas for Circumference or Perimeter of Ellipse: 

^2 

Method 1 . — Let / = perimeter; s = major axis = 2a ; e^= (eccentricity) ^ = — ^ == 



(a + b) (a-b). 
a2 



a2 
Terms 12 3 4 6 



Then /= n 



3 , 32.5 , 32.52.7 



[•(-i 



Then/=7r s {l--^e^- 



22. 42 22. 42. 62 22. 42. 62. 82 

6 7 etc, 

3^-5^-7^-9 ,10^ 32.52.72.92.ll \1 

22.42.62.82.102.122 ....yju; 

12 3 4 5 

175 





22.42.62.82.102 


3 




4 


3 


.^4. 


5 


64 


256 ^ 
6 
441 



16384 

7 etc. 

,10 4851 \1 

65536 ^ 1048576 •• • ; J W 

Formulas (1) and (2) may be expressed: /= 7:[sik)] (3) 

in which the continued series ^ is a coefficient of 5, making sk the diameter 
of a circle whose circumference = perimeter of the ellipse. 

To facilitate the use of equation (2): Log 7r = 0.4971499; log i= 9.3979400; 
log 6^4 = 8.6709413; log .h = 8.2907300; log jUh = 8.0286181; log ^Uh = 
7.8279587; log yxfA^TS = 7.6652314. Note that logarithms of e^, e\ e^, e^^, e^^ 
are 2, 3, 4, 5, 6 times log e^. 

Problem 1. — ^The major and minor axes of an ellipse are 36 and 24 ft., 
respectively. Find the perimeter? 

1 Sfl 

Solution.— Major axis s= 36] a=18; 6=12; e^=^ ; log^2= 9.7447275. 

lo^ 

Using 5-place logarithms, we have for the value of ^, by terms: 
1.2 34 5 6 7 

Log 1=9.39794 63_=8.67094 8.29073 8.02862 7.82796 7.66523 

Log g2= 9. 74473 e^ =9.48946 g6 =9.23418 ^8=8^7891 gio =8.72364 gi2 =8.46837 

Sum =9.14267 8.16040 7.52491 7.00753 6.55160 6.13360 

and numbers corresponding to above logarithms are below: 
.-.^=1.00000 - 0.13889 - 0.01447 - 0.00335 - 0.00102 - 0.00036 - 0.00014 
minus the sum of the value of terms abovethe 7th, which, by inspec- 
tion, we will assume to equal 0.00007. Hence, k =0.8417; and the 
perimeter / = 7rs/j= 3.1416 X 36 X 0.8417 = 95.194 ft. Ans. 



240 



n.— MENSURATION, 



Method 2. — Let /=» perimeter; a = semi-major axis; b = semi-minor axis; 

a + b 

Terms: 1 2 

Then/=7r (a+6)[l+^ 



5 

32.52 



,Ei 



£2 4. -^^~R4 J £6 + 

22.42 22.42 62 22.42 62 82 

6 7 

32.52.72 32.52.72.92 



Terms: 1 
Then /=;: (a+b) | 



1 + 



2 

£2 



3 

64 

+ 



22.42.62.82.102' 
4 

£6 



22.42.62.82.102.122 

5 

25 £8 



£12+ (4) 



256 
6 
49 £10 



16384 

7 
441 £12 



](5) 

(6) 



65536 1048576 
Formulas (4) and (5) may be expressed: l=7:(a + b)K 

in which (a + b)K is the diameter of a circle whose circumference = perimeter 

of the elHpse. 

Tofacilitatetheuseof equation (5): Log ;:= 0.4971499; log 1 = 9.397400; 

log g\=8.1938200; log ^^6 = 7.5917600; log i6¥5i = 7.1835201; log 55^38 = 

6.8737162; log T^|||ye = 6.6238387. Note that logarithms of £^, E\ E^ £8, 

£10, £12 are 2, 4, 6, 8, 10, 12 times log £. 

Problem 2. — Solve problem 1 by formula (5) ? 

Solution.— a ^ IS; 6=12; a + 6=30; a-6 = 6; £ = 0.2; log £=9.3010300. 
Using 5-place logarithms, we have for the value of K, by terms: 

1 2 3 4 5 6 7 

Log i=9.39794 b\=8.19382 7.59176 7.18352 6.87372 6.62384 

Log £2 =8.60206 £^ =7.20412 £6= 5.80618 £8 =4.40824 £io =3.01030 £12 =1:61236 

Sum=2.00000 5.39794 .7.39794 9.59176 11.88402 12'.23620 

and numbers corresponding to above logarithms are below: 

.•.K=l +0.010000+0.000025 + 0.00000025H + + 

= 1.0100253; and the perimeter /=7r(a + 6)i^=95.193. Ans. 

Comparison of Methods 1 and 2. — A mere glance at the solution of 
Problems 1 and 2, illustrating the two preceding methods of calculating the 
perimeter of the ellipse, clearly shows the superiority of Method 2: The 5th, 
6th, 7th, etc., terms giving values so small as to be negligible in the present 
instance. Moreover, equation (5), with the accompanying logarithmic 
values given just below it, will be found quite as rapid to use, in many cases, 
as many of oiu* so-called approximate formulas, with, in addition, the 
advantage of accuracy. 



16a.— Values of 2C in Formula (6) for Successive Values of ^ . , . 

a-\-b 

Note. — To obtain diameter of circle whose circumference equals perimeter 
of ellipse: Multiply tabular values (K) by (a + 6). 



E,OT 

a-b 


Hundredths. 


a+y 


.00 


.01 


.02 


.03 


.04 


.05 


.06 


.07 


.08 


.09 


0.0 


1.0000 


1.0000 


1.0001 


1.0002 


1.0004 


1.0006 


1.0009 


1.0012 


1.0016 


1.0020 


0.1 


.0025 


.0029 


.0034 


.0040 


.0047 


.0054 


.0062 


.0070 


.0080 


.0090 


0.2 


1.0100 


1.0110 


1.0122 


1.0133 


1.0145 


1.0158 


1.0173 


1.0186 


1.0200 


1.0215 


0.3 


.0226 


.0245 


.0261 


.0276 


.0291 


.0311 


.0331 


.0349 


.0369 


.0389 


0.4 


1.0404 


1.0431 


1.0450 


1.0472 


1.0494 


1.0516 


1.0538 


1.0561 


1.0885 


1.0608 


0.5 


.0635 


.0661 


.0686 


.0713 


.0740 


.0768 


.0798 


.0827 


.0557 


.0889 


0.6 


1.0922 


1.0954 


1.0986 


1.1016 


1.1048 


1.1083 


1.1115 


1.1157 


1.1193 


1.1229 


0.7 


.1267 


.1306 


.1345 


.1383 


.1423 


.1466 


.1509 


.1550 


.1593 


.1637 


0.8 


1.1677 


1.1721 


1.1768 


1.1813 


1.1859 


1.1903 


1.1950 


1.2000 


1.2049 


1.2100 


0.9 


.2154 


.2207 


.2263 


.2315 


.2374 


.2430 


.2486 


.2546 


.2605 


.2665 


1.0 


1.2732 





















Ex. — For the ellipse a = 3, 6 = 2, the dia. of corresponding circle is 5.05. 



LENGTHS OF SEMI-ELLIPTIC ARCS, 



241 



Table 17, following, will be found useful in calculating the lengths of semi- 
elliptic arcs (A or B) when the semi-diameters a and b are given. 



17.— Lengths op Semi-Elliptic Arcs, A oh B 
For a = Unity, and for Successive Values of - 



Note. — ^To find A or B: Multiply values of co- 
efficient C, in the table, by length of semi-major 
liL axis, or a. Thus, A=B = Ca, 

'^ ArcA=Arcd 

«' >b. [Calculated from Formula (4-5).*] 




2.00000 
2.00061 
2.00193 
2.00394 
2.00657 
2.00971 
2.01334 
2.01740 
2.02188 
2.02675 
2.03198 
2.03757 
2.04349 
2.04971 
2.05624 



06305 

07014 

07749 

08509 

09293 

10100 

10931 

11782 

12655 

13548 

14461 

2.15392 

2.16342 

2.17309 

2.18294 

2.19296 

2.20313 

2.21347 

2.22395 



Dlff. 



.00061 
.00132 
.00201 
.00263 
.00314 
.00363 
.00406 
.00448 
.00487 
.00523 
.00559 
.00592 
.00622 
.00653 
.00681 
.00709 
.00735 
.00760 
.00784 
.00807 
.00831 
.00851 
. 00873 
.00893 
.00913 
.00931 
.00950 
.00967 
.00985 
.01002 
.01017 
.01034 
.01048 



,22395 
.23459 
.24537 
.25629 
.26735 
,27854 
.28986 
30131 
31288 
32457 
33638 
34831 
,36035 
37249 
38475 
39710 
40956 
42211 
43477 
44752 
46036 
47329 
48632 
49943 
51262 
52590 
53926 
55270 
56622 
57982 
59349 
60723 
62105 
63494 



DIfE. 



.01064 
.01078 
.01092 
.01106 
.01119 
,01132 
,01145 
,01157 
,01169 
,01181 
,01193 
,01204 
,01214 
,01226 
,01235 
,01246 
,01255 
,01266 
,01275 
,01284 
,01293 
,01303 
,01311 
,01319 
,01328 
,01336 
,01344 
,01352 
,01360 
,01367 
,01374 
,01382 
,01389 



.90 
.91 
.92 
.93 
.94 
.95 
.96 
.97 
.98 
.99 
1.00 



2.63494 

2.64890 

2.66293 

2.67702 

2.69118 

2.70541 

2.71970 

2.73405 

74846 

76293 

77747 

,79206 

,80671 

82141 

,83617 

85098 

, 86584 

88076 

89573 

91075 

92582 

94094 

95611 

2.97132 

2.98658 

3.00189 

3.01724 

3.03263 

3.04807 

3.06356 

3.07908 

3.09465 

3.11026 

3.12590 

3.14159 



DIfl. 



.01396 
.01403 
.01409 
.01416 
.01423 
.01429 
.01435 
.01441 
.01447 
.01454 
.01459 
.01465 
.01470 
.01476 
.01481 
.01486 
.01492 
.01497 
.01502 
.01507 
.01512 
.01517 
.01521 
.01526 
.01531 
.01535 
.01539 
.01544 
.01549 
.01552 
.01557 
.01561 
.01564 
.01569 



Problem. — A concrete semi-elliptic arch has a span 2 a (see Fig. above) of 
120 ft., and a rise 6 = 13 ft. What is the length of soffit of arch, or arc A'i 

Solution.— From above formula and table: A = Ca\ C, for hla — 13/60 or 
0.21%, is 2.10931 + diff 0.00851 X % = 2.11498; hence, length of arc A = 2.11498 X 
60 = 126.90 ft. 



* Number of terms used in Formula (4) in calculation of this table: 

SOterms for — =0.01; 35, for — =0.06; 20, for — =0.15; 13, for -^ = 0.25; 

o a a a 

7,for— =0.50; 5, for— =0.76; 4. for— =0.90; 3. for— = 0.98; and 2 
a a a a 

terms for— = 0.99. 



242 



n.—MENSURA TION. 



Elliptic Segment ; and Chord. — 

Let A = areas of elliptic segment with chord 

B = area of elliptic segment with chord 
a; 

a = semi-major axis = rad of large circle; 

h = semi-minor axis = rad of small circle; 
b — fc ' = rise of segment A ; 
a — a'= rise of segment B. 






Then, length of chord Ca = 2a 







-,J '• 



length of chord Cb = 26- 

' Fig. 19. 

Area segment A : area whole ellipse : : area seg small circle : area small circle. 



.'. A = (area seg small circle with same chord Ca) X -t" 

b 



(1) 



Area segment B : area whole ellipse :: area seg large circle : area large circle, 



.'. B = (area seg large circle with same chord Cb) X — 

a 



(2) 



(See Tables 7 and 8 of Circular Segments, preceding.) 

Problem 1. — Find the area of segment A of the ellipse a =10, 6 = 8, 
whose chord is distant 6' = 5 from and parallel with the major axis? 

Solution. — Diam of small circle =16, and middle rise h ( = b — b') of arc 
from chord = 8 — 5 = 3. Now from Table 8, of Circular Segments, the area 

corresponding to ^?^^^, or .1875, = . 101943 diam^ ==. lOlUZ X^b^; and 
diam lo 

multiplying this value by -r (see Equation 1) we have. 

Area A = . 101943 X 4a6 =.101943 X 320 = 32.622. Ans. 

b' a' 
It is to be noted that area A = area B when -r = — . 

o a 

Problem 2. — What is the length of chord Ca of the ellipse given in 
Problem 1? 



Solution. — From the above formula, Ca=2a^/l— yr-j '^^\^~Kl 

15.612. Ans. 

17a.— Areas of Elliptic Segments, A or B, for a = Unity. 

b a' b' 

For successive Elliptic Values - , and Segmental Values - or r • 

a a b 

(Note. — Multiply tabular values by the square of the semi-major axis, or a^.) 



h 


Values of a' /a or b'/b. 


a 


1.0 


0.9 


0.8 


0.7 


0.6 


0.5 


0.4 


0.3 


0.2 


0.1 


0.0 


.0 
.1 
.2 
.3 
.4 
.6 
.6 
.7 
.8 
.9 
1.0 


.000000 
.000000 
.000000 
.000000 
.000000 
.000000 
.000000 
.000000 
.000000 
.000000 
.000000 


. 000000 
.005872 
.011745 
.017617 
.023490 
.029362 
.035234 
.041107 
.046679 
.052852 
.058724 


.000000 
.016350 
.032700 
.049050 
.065400 
.081750 
.098100 
.114450 
. 130800 
. 147150 
. 163500 


.000000 
.029550 
.059100 
.088650 
.118200 
. 147750 
. 177300 
.206850 
.236400 
.265950 
.295500 


.000000 
.044730 
.089459 
.134189 
.178918 
.223648 
.268378 
.313107 
.357837 
.402566 
.447296 


.000000 
.061418 
. 122837 
. 184255 
.245674 
.307092 
.368510 
.429929 
.491347 
.552766 
.614184 


.000000 
.079267 
. 158534 
.237802 
.317069 
.396336 
.475603 
554870 
.634138 
.713405 
.792672 


.000000 
.097992 
. 195984 
.293976 
.391968 
.489960 
.587952 
.685944 
.783936 
.881928 
.979920 


000000 
117348 
.234696 
.352044 
.469392 
. 586740 
. 704088 
.821436 
.938784 
1.05613 
1.17348 


. 000000 
.137113 
.274226 
.411340 
.548453 
.685566 
. 822679 
.959792 
1.09691 
1.37113 
1.23402 


.000000 
. 157080 
.314159 
.471239 
.628318 
.785398 
.942478 
1.09956 
1.25664 
1.41372 
1.57080 



ELLIPTIC SEGMENT. PRISMOIDAL FORMULA, 



243 



B.— SOLIDS. 

Pappus's Theorem. — If a plane curve / or area a lies wholly on one side 
of a straight line as axis in its own plane, the surface S or volume V gene- 
rated by its whole or partial revolution about that axis is: 
5 = / X length of path p traversed by cen of grav g of line; or 5 = lp\ 
V = a X length of path P traversed by cen of grav G of area; or V = aP, 

" S„= disttScI froS aS to & } ^^en for one complete revolution. 

^ = *2i:xo\ .'. 5 == 2t:Ixo, and xq = S -^ 2nl (1) 

P =*2;rXo; .-. V = 27raXo, and Xo = y -!- 2;ra (2) 

Thus, equations (1) and (2) are used for finding the surfaces and vol- 
umes of the sphere, cone, cylinder, torus (cylindrical ring), paraboloid, 
ellipsoid, etc.; also of their sectors, segments, zones and frustums. 

It is to be noted also that these equations enable us to find the centers 
of gravity of their lines and areas when their lines, surfaces and volumes 
are known. 

Prismoidal Formula. — ^The volume V of a prismoid is equal to the 
length / multiplied by the mean area A ; and A is equal to i (sum of end 
areas, Oi and 02, + 4 times the middle area am) ; thus 

V = /A= j(ai + ia^ + a2) (3) 

A prismoid is a solid having parallel end faces or areas, joined tbgether 
by regular surfaces or sides, as the sides of prisms, cylinders, cones, pyra- 
mids, wedges, or their frustums, or any lateral combination of same. The 
prismoidal formula will apply also to the sphere, hemisphere and spherical 
segment; to warped-surface solids where the warp is continuous between 
ends of solid; to railroad cuttings that can be decomposed into prisms, 
wedges, etc.; to two equal cones arranged like an hour glass with bases as 
end areas; to the conical wedge bounded on one side by a plane radiating 
from the apex of cone; to the frustums of same; and to many other solids. 



18. — ^Thb Five Regular Polyhedrons. 

(All dihedral or solid angles are equal, and all faces regular polygons. Five 

only.) 











Apothem a. 


Radius r , 






Total 


Total 


or radius of 


or radius of 


Name. 


Bounded 


Surface 5 


Volume V 


inscribed 


circum- 




by 


= (ledge)2 


= (ledge)3 


sphere, 


scribed 






times 


times 


= 1 edge 
times 


sphere, 

= 1 edge 

times 


Tetrahedron 


4^'s 


1.7320508 


0.1178513 


0.2041 


0.6124 


Cube (hexahedron) 


6D's 


6.0000000 


1.0000000 


0.5000 


0.8660 


Octahedron 


8^'s 


3.4641016 


0.4714045 


0.4082 


0.7071 


Dodecahedron. . . . 


12 0's 


20.6457788 


7.6631189 


1.1136 


1.4013 


Icosahedron 


20 ^'s 


8.6602540 


2.1816950 


0.7558 


0.9511 



The volume V of any regular polyhedron is equal to its surface S times 
one-third its apothem a; or, y= iSa; .*. a = SF-r-S. 



* 2;r= 6. 283185. 



244 



11 .—MEN SURA TION 




Fig. 20. 

Prisms and Cylinders. — A prism is a solid with parallel ends and parallel 
side edges. Hence the ends will be equal and similar polygons (regular or 
irregular) , and the sides will be parallelograms. A cylinder is a prism with an 
infinite number of sides. The ends of the cylinder may be circular, elliptic, 
or of any curvature. 

Area. — The surface of any prism or cylinder, whether right or oblique, 
is equal to the two end areas + the perimeter p of any right section s mul- 
tiplied by the length / of any lateral element: or S = 2a-{-pl (Fig. 20). 

Volume. — ^The volume of any prism or cylinder, whether right or oblique, 
is equal to the area of any right section s multiplied by the length / of any . 
lateral element; or V = s I (Fig. 20). 

Also, volume equals area of either end multiplied by the vertical distance 
between the end faces; or, V = ah (Fig. 20). 



Frustum of Prism or Cylinder. — Prism 
or cylinder with end faces not parallel 
(Fig. 21). 

Volume. — Let gx — cen of grav of end 
area ai; g2 of any sectional area 02; gs of 
end area 03. Then 

V = axh-i ; {hx = vert dist from gs to plane Oi) . 

V = a^z \ (^3 = '^^^^ dist from gx to plane 03) . 
F = 02/^2; fe is vert to plane 02, het gx and 

g3-) 

In general, F = area a of any plane 
section multiplied by the perpendicular 
distance h between planes passing through -^ 
centers of gravity of end areas and par- 
allel with the said plane section. If a 
is a right section ao, V = aol. These for- 
mulas also enable us to find the relation 
between certain elements, as aol = axhx = 
02/^2 = ^3^3. Fig. 21. 

Note that Fig. 21' becomes a circular cylindric ungula when the right 
section Oq is a circle, and hence 1=^ (longest side -F shortest side). 




Circular Cylindric Frustum. — This is 
special case of the preceding in which ao is 
right circular section whose perimeter is 
Then, 

Volume y = Oq/ = i oo (/i -I- /2) ; 

V = axhx\ (hx is perp to plane ax.) 

V = a3kz; (hzis perp to plane 03.) 
Area A=ax-\-a3 + pl=ax-ha2 + p ik + k)' 




Fig. 22. 



PRISMS. CYLINDERS; CYLINDRICAL WEDGES. 



245 





Fig. 23 



Fig. 23b. Fig. 23c. 




-d-- 
Fig. 23d. 



Circular Cyllndric Half=Wedges. — The following formulas give the 
Volumes and Areas of half -wedges cut from circular cylinders ;_/j being the 
height of the wedge, measured along the element of the cylinder at ai. 
There are four cases — (a), (6), (c) and (d) — as follows: — 



(a). — Base bi less than radius r; lower edge Ci. 



* Volume 



bill2 



(area of base at bi) (r — bi) 



(Fig. 23a) 



^Area of curved surface, S=-r-[ Cir- (length of arc Oi) (r~bt) ]. (Fig. 23a) 

(b). — Base r = radius of cylinder; lower edge = d. 
^Volume F = f . r%. (Fig. 23b) 

1[Area of curved surface, S=2rh = dh. (Fig. 23b) 

(c). — Base b2 > r and < diameter d; lower edge = C2. 

] . (Fig. 23c) 

(Fig. 23c) 



* Volume V=u-\% +(area of base at 62) (&2-^) 

^Area of curved surface, S= r- [ C2r-1- (27rr = arc 02) (62 — ]• 

02 



(d). — Base d = diameter of cylinder; lower edge at 02. 



(Fig. 23d) 

(Fig. 23d) 
(Fig. 23d) 



h h 

* Volume V= -r-. nr^= -^ (area of circular base). 

\Area of curved surface, S = nrh. 

Center of gravity is at g whether figure is right or oblique. 

Properties of Hollow Cylinders.— For a cylinder one meter in diam. {d =» 1) 
and one meter long (/ = 1), the following properties obtain: 

Circumference = 3. 14159 m. (log = 0.4971499) = 10.30704 ft. 

(log = 1.0131341) 
Surface = 3.14159 sq. m. (log = 0.4971499) = 33.81569 sq. ft. 

(log = 1.5291182) 
Volume = 0.78540 cu. m. (log = 9.8950899) = 27.73591 cu. ft. 

(log = 1.4430424) 
Capacity = 785.398 liters (log = 2.8950899) = 207.4790 U. S. gal. 

(log = 2.3169741) 
Weight (water) = 785.398 kilog. (log = 2.8950899) = 1731.506 lbs. 

(log = 3.2384241) 
For a cylinder one meter in diam. (</ = 1) and one foot long (/' = 1), the fol- 
lowing properties obtain: 

Volume = 0.23939 cu. m. (log = 9.3791057) = 8.45392 cu. ft. 

(log = 0.9270582) 
Capacity = 239.3898 liters (log = 2,3791057) = 63.2397 U. S. gal. 

(log = 1.8009899) 
Weight (water) = 239.3898 kilog. (log = 2.3791057) = 527.7641 lbs. 

(log = 2.7224399) 
Circumference is proportional to d; surface is proportional to dl\ volumCt 
capacity and weight, to dH. 

Table 19, following, is wholly in U. S. measure. 



* Volume is for either right or oblique figure, h being perp height. 
t Area of curved surface, for right figure only. For total surface, add 
inclined face (elliptic) and base (circular). 



246 



1 1 —MENS URA TION. 



19.— Properties of Hollow Cylinders (Pipes, Tanks or Wells), One Foot 

Length. 

Note that Areas, Volumes, Capacities and Weights are proportional to the 
squares of the diameters. 1728 cu. ins. = 7.4805 gallons = 1 cu. ft. = 62.5 lbs 
(nearly) of water; 231 cu. ins. = 1 gallon; 201.974 gallons = 1 cu. yd. 



Diam. 


Hydrau- 
lic Mean 
Radius 

4* 


Cir- 
cum. 
Ins. 


Cir- 
cum. 
= Sur- 
face. 
Ft. 


Vol- 
ume. 
Cu. 
Ins. 


Area = 

Volume. 

Ft. 


Volume. 
Cu. 

Yds. 


Capac- 
ity. 
Gallons. 


Weight 
Walter. 


Ins. 


Ft. 


Lbs. 


Vs 
3-16 

H 
% 


.0104 
.0156 
.0208 
.0312 


.0026 
.0039 
.0052 
.0078 


.392699 
. 589049 
. 785398 
1.17810 


.032725 
.049087 
.065450 
.091875 


. 147262 
.331340 
.589049 
1.32536 


.000085 
.000192 
.000341 
.000767 


.000003 
.000007 
.000013 
.000028 


.00064 
.00143 
.00255 
.00574 


.00533 
.01198 
.02131 
.04794 


1 


.0417 
.0521 
.0625 
.0729 


.0104 
.0130 
.0156 
.0182 


1.57080 
1.96350 
2.35619 

2.74889 


. 130900 
. 163625 
. 196350 
.229074 


2.35619 
3.68155 
5.30144 
7.21585 


.001364 
.002131 
.003068 
.004176 


.000051 
.000079 
.000114 
.000155 


.01020 
.01594 
.02295 
.03124 


.08522 
.13316 
.19175 
.26099 


1 


.0833 


.0208 


3.14159 


.261799 


9.42478 


.005454 


.000202 


.04080 


.34088 


IM 
1^ 
IM 

2 


.1042 
.1250 
.1458 
.1667 


.0261 
.0312 
.0365 
.0417 


3.92699 
4.71239 
5.49779 
6.28319 


.327249 
.392699 
.458149 
.523599 


14.7262 
21.2058 
28.8634 
37.6991 


.008522 
.012272 
.016703 
.021817 


.000316 
.000455 
.000619 
.000808 


.06375 
.09180 
. 12495 
. 16320 


.53263 
.76699 
1.04396 
1.3635 




.1875 
.2083 
.2292 
.2500 


.0469 
.0521 
.0573 
.0625 


7.06858 
7 . 85398 
8.63938 
9.42478 


.589049 
.654498 
.719948 
.785398 


47.7129 
58.9049 
71.2749 
84.8230 


.027612 
.034088 
.041247 
.049087 


.001023 
.001263 
.001528 
.001818 


.20655 
.25500 
.30855 
.36720 


1.7257 
2.1305 
2.5779 
3.0680 


3K 

4>^ 
5 


.2917 
.3333 
.3750 
.4167 


.0729 
.0833 
.0937 
.1042 


10.9956 
12.5664 
14.1372 
15.7080 


.916298 
1.04720 
1.17810 
1.30900 


115.454 
150.796 
190.852 
235.619 


.066813 
.087266 
.110447 
. 136354 


.002475 
.003232 
.004091 
.005050 


.49980 

.65280 

.82620 

1.02000 


4.1759 
5.4541 
6.9029 
8.5221 


5M 

6 

6H 


.4583 
.5 

.5417 
.5833 


.1146 
.1250 
.1354 
.1458 


17.2788 
18.8496 
20.4204 
21.9911 


1.43990 
1.57080 
1.70170 
1.83260 


285.100 
339.292 
398.197 
461.814 


. 16^988 
. 196350 
.230438 
.267254 


.006111 
.007272 
.008535 
.009898 


1.2342 
1.4688 
1.7238 
1.9992 


10.312 
12.272 
14.402 
16.703 


9 


.6250 
.6667 
.7083 
.7500 


.1562 
.1667 
.1771 
.1875 


23.5619 
25.1327 
26.7035 
28.2743 


1 . 96350 
2.09440 
2.22529 
2.35619 


530.144 
603.186 
680.940 
763.407 


.306796 
.349066 
.394063 
.441786 


.011363 
.012928 
.014595 
.016362 


2.2950 
2.6112 
2.9478 
3.3048 


19.176 
21.817 
24.629 
27.612 


9H 
10 
lOH 

11 


.7917 
.8333 
.8750 
.9167 


.1979 
.2083 
.2187 
.2292 


29.8451 
31.4159 
32.9867 
34.5575 


2.48709 
2.61799 
2.74889 
2.87979 


850.586 
942.478 
1039.08 
1140.40 


.492237 
.545415 
.601320 
. 659953 


.018231 
.020201 
.022271 
.024443 


3.6822 
4.0800 
4.4982 
4.9368 


30.765 
34.088 
37.582 
41.247 


IIH 

12 
13 
14 


.9583 
1. 

1.0833 
1.1667 


.2396 
.25 
.2708 
.2917 


36.1283 
37.6991 
40.8407 
43.9823 


3.01069 
3.14159 
3.40339 
3.66519 


1246.43 
1357.17 
1592.79 
1847.26 


.721312 
.785398 
.921752 
1.06901 


.026715 
.029089 
.034139 
.03959 


5.3068 
5.8752 
6.8952 
7.9968 


45.082 
49.087 
57.609 
66.813 


15 
16 

17 
18 


1.2500 
1.3333 
1.4167 
1.5 


.3125 
.3333 
.3542 
.375 


47.1239 
50.2655 
53.4071 
56.5487 


3.92699 
4.18879 
4.45059 
4.71239 


2120.58 
2412.74 
2723.76 
3053.63 


1.22718 
1 . 29626 
1.57625 
1.76715 


.04545 
.05171 
.05838 
.06545 


9.1800 
10.445 
11.791 
13.219 


76.699 
87.266 
98.516 
110.46 


19 
20 
22 
24 


1.5833 
1.6667 
1.8333 
2. 


.3958 
.4167 
.4583 
.6 


59.6903 
62.8319 
69.1150 
75.3982 


4.97419 
5.23599 
5.75959 
6.28319 


3402.34 
3769.91 
4561 . 59 
5428.67 


1.96895 
2.18166 
2.63981 
3.14159 


.07292 
.08080 
.09777 
.11636 


14.729 
16.320 
19.747 
23.501 


123.06 
136.35 
164.99 
196.35 



The Circumference is proportional to the Diameter. 



PROPERTIES OF HOLLOW CYLINDERS. 



247 



19.— Properties of Hollow Cylinders.— Concluded. 



Diam. 


Hydrau- 
lic Mean 
Radius 
d 

4' 


Cir- 
cum. 
Ins. 


Cir. 

cum. 

= Sur- 
face. 
Ft. 


Vol- 
ume. 
Cu. 

Ins. 


Area = 

Volume. 

Ft. 


Volume. 
Cu. 
Yds. 


Capac- 
ity. 
Gallons. 


Weight 

of 
Water. 


Ins. 


Ft. 


Lbs. 


26 
28 
30 
32 


2.1667 
2.3333 
2 5 
2.6667 


.5417 
.5833 
.625 
.6667 


81.6814 
87.9646 
94.2478 
100.531 


6.80678 
7.33038 
7.85398 
8.37758 


6371.15 
7389.03 
8482.30 
9650.97 


3.68701 
4.27606 
4.90874 
5.58505 


. 13656 
.15837 
.18181 
.20685 


27.581 
31.987 
36.720 
41.779 


230.44 
267.25 
306.80 
349.07 


34 
36 

38 
40 


2.8333 
3. 

3.1667 
3.1333 


.7083 
.75 
.7917 
.8333 


106.814 
113.097 
119.381 
125.664 


8.90118 
9.42478 
9.94838 
10.4720 


10895.0 
12214.5 
14609.4 
15079.6 


6.30500 
7.06858 
7.87580 
8.72665 


.23352 
.26180 
.29170 
.32321 


47.165 
52.877 
63.244 
65.280 


394.06 
441.79 
492.24 
545.42 


42 
44 
46 

48 


3.5 

3.6667 
3.8333 
4. 


.875 
.9167 
.9583 
1. 


131.947 
138.230 
144.513 
150.796 


10.9956 
11.5192 
12.0428 
12.5664 


16625.3 
18246.4 
19942.8 
21714.7 


9.62113 
10.5592 
11.5410 
12.5664 


.35634 
.39108 
.42744 
.46542 


71.971 
78.989 
86.333 
94.003 


601.32 
659.95 
721.31 
785.40 


60 
62 
64 
66 


4.1667 
4.3333 
4.5 
4.6667 


1.0417 
1.0833 
1.125 
1.1667 


157.080 
163.363 
169.646 
175.929 


13.0900 
13.6136 
14.1372 
14.6608 


23561.9 

25484.6 
27482.7 
29556.1 


13.6354 
14.7480 
15.9043 
17.1042 


.50501 
.54622 
..58905 
.63349 


102.000 
110.32 
118.97 
127.95 


852.21 
921.75 
994.02 
1069.0 


60 
66 

72 
78 


5. 
5.5 
6. 
6.5 


1.25 
1.375 
1.5 
1.625 


188.496 
207.345 
226.195 
245.044 


15.7080 
17.2788 
18.8496 
20.4204 


33929.2 
41054.3 
48858.0 
57340.3 


19.6350 
23.7583 
28.2743 
33.1831 


.72722 
.87994 
1.0472 
1.2290 


146.88 
177.72 
211.51 
248.23 


1227.2 
1484.9 
1767.1 
2073.9 


84 

90 

96 

100 


7. 
7.5 
8. 
8.3333 


1.75 

1.875 

2. 

2.0833 


263 . 894 
282.743 
301.593 
314.159 


21.9911 
23.5619 
25.1327 
26.1799 


66501.2 
76340.7 
86858.8 
94247.8 


38.4845 
44.1786 
50.2655 
54.5415 


1.4254 
1.6362 
1.8617 
2.0201 


287.88 
330.48 
376.01 
408.00 


2405.3 
2761.2 
3141.6 
3408.8 


108 
120 
132 
144 


9. 
10. 
11. 
12. 


2.25 
2.5 
2.75 
3. 


339.292 
376.991 
414.690 
452.389 


28.2743 
31.4159 
34.5575 
37.6991 


109931. 
135717. 
164217. 
195432. 


63.6173 
78.5398 
95.0332 
113.097 


2.3562 
2.9089 
3.5197 
4.1888 


475.89 
587.52 
710.90 
846.03 


3976.1 
4908.7 
5939.6 
7068.6 


156 
168 
180 
192 


13. 
14. 
15. 
16. 


3.25 
3.5 
3.75 
4. 


490.088 
527.788 
565.487 
603.186 


40.8407 
43.9823 
47.1239 
50.2655 


229361. 
266005. 
305363. 
357435. 


132.732 
153.938 
176.715 
201.062 


4.9160 
5.7014 
6.5450 
7.4467 


992.91 
1151.5 
1321.9 
1547.3 


8295.8 
9621.1 
11045. 
12566. 


204 
216 
228 
240 


17. 
18. 
19. 
20. 


4.25 
4.5 
4.75 
5. 


640.885 
678.584 
716.283 
753.982 


53.4071 
56.5487 
59.6903 
62.8319 


392222. 
439722. 
525938. 
542867. 


226.980 
254.469 
283.529 
314.159 


8.4067 
9.4248 
10.501 
11.636 


1697.9 
1903.6 
2276.8 
2350.1 


14186. 
15904. 
17721. 
19636. 


264 
288 
300 
312 


22. 
24. 
25. 
26. 


5.5 
6. 

6.25 
6.5 


829.380 
904.779 
942.478 
980.177 


69.1150 
75.3982 
78.5398 
81.6814 


666869. 
781729. 
848230. 
917446. 


380.133 
452.389 
490.874 
630.929 


14.079 
16.755 
18.181 
19.664 


2843.6 
3384.1 
3672.0 
3971.6 


23758. 
28274. 
30680. 
33i83. 


336 


28. 
30. 
32. 
34. 

36. 

38. 
40. 
42. 

44. 
46. 
48. 
60. 


7. 
7.5 
8. 
8.5 

9. 

9.6 
10. 
10.5 

11. 
11.6 
12. 
12.5 


1055.58 
1130.97 
1206.37 
1281.77 

1357.17 
1432.57 
1507.96 
1583.36 

1658.76 
1734.16 
1809.56 
1884.96 


87.9646 
94.2478 
100.531 
106.814 

113.097 
119.381 
125.664 
131.947 

138.230 
144.513 
150.796 
157.080 




615.752 
706.858 
804.248 
907.920 

1017.88 
1134.11 
1356.64 
1385.44 

1520.53 
1661.90 
1809.56 
1963.50 


22.806 
26.180 
29.787 
33.627 

37.699 
42.004 
46.542 
61.313 

66.316 
61.552 
67.021 

72.722 


4606.1 
5287.7 
6116.2 
6791.7 

7614.3 
8483.7 
9400.3 
10364. 

11374. 
12432. 
13536. 
14688. 


38484. 


360 




44179. 


384 




50265. 


408 




66746. 


43?, 




63617. 


AH 




70882. 


4P0 




78540. 


604 




86590. 


6?8 




95033. 


65? 




103869. 


676 




113097. 


Aon 




122719. 









See, also, tables on pages 678 and 1157. 



248 



n.— MENSURATION. 



Pyramid and Pyramidic Frustum. — A " regu- 
lar" pyramid is one in which the base is a regu- 
lar polygon; if not, it is " irregular." If the axis, 
from the apex to the cen of grav g of the base, 
is perp to the base it is a "right" pyramid; if 
not, it is "oblique." Fig. 24 shows an oblique 
pyramid and frustum together forming a right 
pyramid. 

Volume Vr of right pyramid = \ (area of base X 

perp height) = \ axh^. 
Volume Vo of oblique pyramid = i (area of base 

X perp height) = \ 02/^2- 

Volume V{ of pyramidic frustum = Vr — Vo = 

i (ai/^i — 02^2)' 

Fig. 24. 

If 02 is parallel with Oi, applying the prismoidal formula, volume 
frustum = — ^ 




Vfof 



lai + a2 + V' O1O2 j , whether pyramid is right or oblique, 

regular or irregular. 

The area of the sides of a right regular pyramid = ^ (perimeter of base X 
least slant height) ; of a right regular frustum with parallel faces = ^ (peri- 
meter of top + perimeter of base) X least slant height. 

For area of an oblique or irregular pyramid or frustum, the sides must be 
calculated — as triangles, or as trapezoids or trapeziums, respectively. No 
simple general formula will apply. 

Center of gravity of pyramid, whether right of oblique, lies in the axis, 
and one-fourth its length from the base. 



Cone and Conic Frustum. — A cone may 
be considered as a pyramid with an infinite 
number of sides. If the axis from the apex 
to the cen of grav g of the base, is perp to 
the base it is a "right" cone; if not, it is 
"oblique." Generally speaking, a right cone 
is understood to have a circular base; and 
an oblique cone, to have an elliptic base. 
Such cones, however, are sometimes termed 
right- and oblique circular cones, to distin- 
guish them from right- and oblique elliptic 
cones whose base of right cone is an ellipse. 
Note that an oblique circular cone may be 
cut from a right elliptic cone; or that an 
oblique elliptic cone may be cut from either 
a circular or an elliptic cone. Fig. 25 shows 
an oblique cone and frustum together forming 
a right cone. 




Fig. 25. 



Volume Vr of right cone = J (area of base X perp height) = J aihi. 
Volume Vo of oblique cone = i (area of base X perp height) = \ 02/^2. 
Volume Vf of conic frustum. =Vt—Vo=\ (ciihi — 02/^2) • 

If a2 is parallel with Oi, applying the prismoidal formula, volume Ffof 

frustiim = ^ o ^ fai+a2 4- Vaia2J .* 

The area of the curved surface (side) of a right cone = \ (perim of base X 
slant height) ; of a right frustum with parallel faces = i(perim of top + perim 
of base) X slant height. 



* If the bottom and top faces of the frustum are circles with radii rx 
and r2, respectively, then ai= nri"^ and a2=7zr2^. If they are elHptical, use 
the formula: area of ellipse = 7ra6, in which a = semi-major axis and 6=^ 
semi-minor axis. ;c= 3.1416. 



PYRAMID. CONE. WEDGE. SPHERE. 



249 



The area of the curved surface .(side) of an oblique elliptic cone of height 

02/^2 
-y-, in 



^2 (Fig. 25), and which has been cut from a right circular cone. 



which /j2 = perp height, 02 = area of elliptic base, andr' = /t2 — =perpdist 

from side of right circular cone to point where axis of same pierces base 02 



of oblique cone. Hence, area= 



02^2 



a2 cos oc . . 
— : — ^~ . Also, area = 
sin /? 



— (volume of 



oblique cone) = 



3Fo 
r' 



Center of gravity of cone, whether right or oblique, lies in the axis and 
one-fourth its length from the base. 



Conic Wedge and Frustum from right cone. — If 
the wedge is cut from the cone by a plane pass-, 
ing through the apex (Fig. 26), with 02 parallel to 
ai, we have, 

Vi = \ aihi ; Ai = ^ hipi. (pi = perim of Oi.) 
V2=i 02/^2; A2=l h2p2' (^2 = perim of 02.) 
Volume of frustum = Fi — 1^2 = i i<^ihi — 02^2) • 
Area of frustum = Ai — A2= 2 {hipt — ^2^2) I 
(ends not included). 

Volume of frustum may also be derived from the 
prismoidal formula; 

thus, Vt = ^n ^ (oi'+a2+4om) in which Om, the area of the middle section, 
= i\/aia2 + i(aj +02); or, V{= ^"q-^ (ai+a2+Voia2J • 

Note. — The above formulas for volume, Vi, V2 and Vt, will apply also if 
the cone is oblique. For an oblique circular cone the above formulas for 
area will not apply; but see formulas for area of oblique elliptic cone above. 




Fig. 26. 



Wedge. — In Fig. 27 let w= width, and /= length 
of base, which must be a parallelogram but not neces- 
sarily a rectangle; let h be the perp height of cutting 
edge above the base; and e the length of cutting 
edge, which must be parallel with /. Then volume 

V=^ -T-(2l-\-e). The two ends and two faces of 

D 

height h may slope in any direction; the two ends 
need not be parallel. 




Sphere. — A sphere is generated by the complete revolution of a semi= 
circle (Fig. 28) about its diameter. The 
volume of the sphere is equal to the area of 
the semicircle X the path described by its 
center of gravity G\ the area of the surface 
of the sphere is equal to the length of 
semi-circular arc X the path described by 
its center of gravity g. Any section of a 
sphere cut by a plane is- a circle; and if 
the plane cuts the center, it is a great circle. 
The axis of a sphere always lies in the 
plane of a great circle. The ratio of vol- Fig. 28. 

lime to surface of a sphere is greater than that of any other solid. 



X-i- 




250 n.—MENSURA TION. 

Distance Yo to G of semi-circle (area) =-*.-= 0.42441 31816 r. 

TT 

Distance yo to g of semi-circular arc = — = 0.63661 97724 r. 

It 

Area of semicircle = ^ = 1.57079 63268 r^. 

Length of semi-circular arc = itr =3.14159 26536 r. 

The following are some of the relations of Volume, Surface, Area of 
Great Cicrle, Circumference, Diameter and Radius, of the sphere: 

Volume of Sphere = %7: radius3 = 4.18879 radius^ = - diam^ 



= 0.52360 diam3= -i- circum3 = 0.016887 circumS 

= t radius X area great circle = f diam X area great circle 
= 1 radius X surface of sphere = i diam X surface of sphere 

= — r— ( = 2.161128) volume of inscribed cube* 
= jT ( = 0.52360) volume of circumscribing cube 

= 's/3 (= 1.732051) volume of maximum inscribed cylinderf 
= § volume of circumscribing cylinder. 
Volumes of spheres are as the cubes of their radii or their diameters. 

Surface of Sphere = ^t: radius2= 12.56637 radius2 = ;r diam^ 

= 3.14159diam2= - circum2= 0.31831 circum2 = diam X circum 

TT 

= 4 X area great circle = area of circle whose radius is equal to diam 
of sphere = 3 X volume -^ radius of sphere. 

= 2 (^ 1.57080) area of surface of inscribed cube 

= r ( = 0.52360) area of surface of circumscribing cube 

= 4 X area of convex surface of inscribed cylinder of m^x convex sur- 
face J 

= convex surface of circumscribing cylinder 

= 3\/3 ( = 5.19615) convex surface of inscribed cone of max convex 

surface^!. 

Surfaces of spheres are as the squares of their radii or their diameters. 
Area of Great Circle of Sphere = n radius2 = 3. 141593 radius2 = - diam2=» 

0. 785398 diam2 = - circum of sphere = \ area of surface of sphere = i volume 
of sphere -r- diam. 

* Edge of inscribed cube= — =( = 0.57735) diam of sphere. 

V3 
t Altitude of inscribed cylinder=fV3 ( = 1.15470) rad of sphere; 

Diam of base of inscribed cylinder = 2\/| (= 1.63299) rad of sphere. 
% Altitude of inscribed cylinder = diam of base = \/2 radius of sphere. 
If Height of inscribed cone = f diam of sphere; diam of base of inscribed 
cone = J V 2 ( = 0.4714) diam of sphere. 



PROPERTIES OF SPHERES. 



251 



Circumference of Sphere = 2?: radius = 6. 28 3 185 radius = ;r diam= 3.141593 
diam = (area of surface of sphere) h- diam = V;r surface of sphere = 



1.77245 ^surface of sphere = VQn^ (= 3.89778) Vvolxime of sphere = 
</69.21763 volume of sphere. 

Diameter of Sphere = 2 radii = - circum = 0.318310 circum 

n 

o , 

= — =r( = 1.12838) Varea great circle = Vl. 273240 area great circle 

=^/— ( = 0.56419) Vsurface of sphere = V 0.318310 surface of sphere 

=^/— (= 1.240701) ^volume of sphere = Vl. 909859 volume of sphere. 

Radius of Sphere = \ diam = -^ — circum = 0.159155 circum 

Ztz 
^ \—{ = 0.56419) Varea great circle = V 0.318310 area great circle 

=^/— (= 0.282095) Vsurface of sphere = VO. 079577 surf ace of sphere 

= 3 /L (= 0.620350) Vvolume of sphere = ^^0.238732 volume of sphere. 

20. — Areas op the Surfaces of Spheres. 
(Surfaces of spheres are proportional to the squares of their diameters.) 



Diam. 


Area Surface 


Logarithm 


Diam. 


Area Surface 


Logarithm 


A 


0.000 766 99 


6.884 7899 


^\-h 


0.000 005 326 322 


4.726 4274 


^' 


0.003 067 96 


7.486 8499 


^\-i\ 


0.000 021 305 29 


5.328 4874 


1*5 


0.012 271 85 


8.088 9099 


^•A' 


0.000 085 221 16 


5.930 5474 


0.049 0874 


8.690 9699 


• ^ 


0.000 340 8846 


6.532 6074 


\ 


0.196 350 


9.293 0299 




0.001 363 538 


7.134 6674 


0.785 398 


9.895 0899 


0.005 454 154 


9.736 7274 


\ 


3.141 593 


0.497 1499 


1*2 


0.021 816 616 


8.338 7874 


0.1 


0.031 415 93 


8.497 1499 


^ 


0.087 266 46 


8.940 8474 


0.2 


0.125 6637 


9.099 2099 


^ 


0.196 349 55 


9.293 0299 


0.3 


0.282 7433 


9.451 3924 


A 


0.349 0659 


9.542 9074 


0.4 


0.502 6548 


9.701 2699 


1% 


0.545 4154 


9.736 7274 


0.5 


0.785 398 


9.895 0899 


\% 


0.785 398 


9.895 0899 


0.6 


1.130 973 


0.053 4524 


/a 


1.069 014 


0.028 9835 


0.7 


1.539 380 


0.187 3460 


1% 


0.396 264 


0.144 9674 


0.8 


2.010 619 


0.303 3299 


1^2 


1.767 146 


0.247 2724 


0.9 


2.544 690 


0.405 6349 


n 


2.181 662 


0.338 7874 


1.0 


3.141 593 


0.497 1499 


u 


2.639 811 


0.421 5728 



Ex. — Surface of sphere i in. in 
diam = 0.033408846 sq. ft. ; therefore, 
surface of sphere V- ins. in diam = 
0.033408846X502= 0.8522115 sq. ft. 
Ex. — Surface of sphere 11 ins. in 
diameter =2.639811 sq. ft. 
Rem,arks. — For Surfaces of Spheres of various diameters, see — 
Foot note to Table 11, of Circles, with diameters in decimals. 
Foot-note to Table 12, of Circles, with diameters in 8ths and 12ths. 
Foot-note to Table 13, of Circles, with diameters in inches and fractions. 
Foot-note to Table 14, of Circles, with diameters in decimals. 
Foot-note to Table 1 5, of Circles, with diameters in feet and inches. 



Ex. — Surface of sphere -^i in. in 
diam = 0.0376699 sq. in.; therefore, 
surface of sphere a\ in. in diam = 
0.0376699 X 52 = 0.019175 sq. in. 

Ex. — Surface of sphere 7 units in 
diameter = 1.539380X 102=153.9380. 



(a) 
(b) 
(c) 
(d) 
(e) 



252 



11.— MENSURATION. 



21. — Volumes of Spheres. 
(Volumes of spheres are proportional to the cubes of their diameters.) 



Diam. 


Volume 


Logarithm 


Diam. 


Volume 


Logarithm 


ii 


0.000 001 997 37 


4.300 4586 


e\-j\ 


0.000 000 001 155 885 


1.062 914§ 




0.000 015 978 96 


5.203 5486 


^\.:l\ 


0.000 000 009 247 085 


1.966 0049 


IL 


0.000 127 8317 


6.106 6386 


r\'j2 


0.000 000 073 9767 


2.869 0949 




0.001 022 654 


7.009 7286 


i.j\ 


0.000 000 591 8136 


3.772 1849 




0.008 181 23 


7.912 8186 


l-i\ 


0.000 004 734 509 


4.675 2749 




0.065 449 84 


8.815 9086 


It\ 


0.000 037 876 07 


5.578 3649 




0.523 5988 


9.718 9986 




0.000 303 008 54 


6.481 4549 


0.1 


0.000 523 5988 


6.718 9986 


a 


0.002 424 069 


7.384 5449 


0.2 


0.004 188 790 


7.622 0886 


i2 


0.008 181 232 


7.912 8187 


0.3 


0.014 137 17 


8.150 3624 


T2 


0.019 392 55 


8.287 6349 


0.4 


0.033 510 32 


8.525 1786 


T2 


0.037 876 07 


8.578 3649 


0.5 


0.065 449 84 


8.815 9086 


6 


0.065 449 84 


8.815 9086 


0.6 


0.113 0973 


9.053 4524 


r5 


0.103 931 94 


9.016 7490 


0.7 


0.179 5944 


9.254 2927 


8 


0.155 1404 


9.190 7249 


0.8 


0.268 0826 


9.428 2686 


9 


0.220 8932 


9.344 1824 


0.9 


0.381 7017 


9.581 7241 


10 


0.303 0085 


9.481 4549 


1.0 


0.523 5988 


9.718 9986 


u 


0.403 304i 


9.605 6330 



Ex. — Volume of sphere b4 in. in 
diam = 0.06199737 cu. in.; therefore, 
volume of sphere 6*^4 in. in diam = 
0.05199737X53 = 0.00024967 cu. in. 

Ex. — Volume of sphere 7 units in 
diameter = 0.1795944X103= 179.5944. 



Ex. — ^Volume of sphere i in. in 
diam = 0.065918136 cu. ft.; therefore, 
volume of sphere ^^- ins. in diam = 
0.065918136X503 = 0.0739767 cu. ft. 

Ex. — Volume of sphere 11 ins. in 
diameter = 0.4033044 cu. ft. 



Remarks. — For Volumes of spheres of various diameters, see — 

(a) Foot note to Table 1 3, of Circles, with diameters in inches and fractions. 

(b) Foot-note to Table 14, of Circles, with diameters in decimals.^ 

(c) Foot-note to Table 1 5. of Circles, with diameters in feet and inches. 




I 
-I ^ ^ 

Fig. 29. 
Spherical Segment. — 

Let a = area of base of segment = 7rr2; 
A = area of great circle = nR'^; 
/i = height of segment. 
Then r = R sin 6; h = R vers ^ = r tan h B. 

Volume of segment = | /a 4- y ) = ^ {^r^ + m = 0.52360 h {Zr'^ + h?) 
= nh^{R-^\ =3.14159/^2 /i^_^^ . 

07, r. 

Convex Surface of segment = 2r.Rh= r^^r volume of sphere = jr^ surface 

of sphere = ^ area great circle = -^— = h X circumference of great circle. 

It is thus seen that area of convex surface is directly proportional to height h 
of segment. Hence, from Table of Spheres find the total surface of the 

sphere for diam=2R, and multiply this result by 77^; or, better, from 

Table 11 or 12, of Circles, multiply the circumference of its great circle by h. 
Area of Base of segment = rrr^. 



SPHERE. RING. SPINDLE. 



253 



Spherical Zone. — Volume of zone = volume of 
sphere minus volume of segments (see Spherical 

Segment, page 252) = |- // ( ^ ■*" ^»^ "^ ^^^ ) = 



1.570796 



Convex Surface of zone = surface of sphere 
minus convex surface of segments (see Spherical 
Segment, page 252) = 27:RH = H X circum of 
great circle. 

Areas of Bases = 7:ri^ and 7:r2^. 




Fig. 30. 



Hollow Sphere. — ^Let i?= radius to outside surface, and r = radius to 

inside surface. Then, 

Volume = 4 TT (i?3 _ ^) = 4. 18879 (R^ - r^) . May also be taken from Table 
of Spheres, by deducting volume of the lesser from volume of greater sphere. 

Surface = convex surface + concave surface = 4 tt (R^ + r2) = 12.56637 times 
(R^-i-r"^). May also be taken from Table of Spheres, by adding together the 
surfaces of the larger and the smaller spheres. 



Circular Segmental Ring. — Let g be the cen 

of grav of area A of segmental section, with o' 
as a center, forming a ring about the axis X — X. 
Also let yo-i-d = radial distance from said axis 
to center of grav g of segment. Then, 

Volume of ring = 27rA (yo + d). For values of 
yq and A see Fig. 11, also Tables 7 and 8 of 
Circular Segments. 

Note that when d = 0, Fig. 31 will represent 
a sphere with a hollow cylindrical core, points 
o' and o'' being identical with o. The above for- 
mula holds true for any section A , provided that Q" 
yo + d = dist to cen of grav g. Fig. 31. 

Convex Surface of ring = 2;ra (Yo + d), in which a == length of circular arc, 
and Yo-{-d = dist to cen of grav of arc. (See Fig. 9; also Tables 2, 3 and 4, of 
Circular Arcs.) 

Concave Surface of ring = 27rr^, in which c = chord of arc = width of ring. 
(See Fig. 9; also Tables 2 and 3 of Circular Arcs.) 




Regular Circular Ring. — For any circular ring 
where section 5 — 5 is a regular polygon, circle, ellipse, 
etc., so that center of grav of area A and of perimeter 
P of section lies midway between outer and inner 
circumferences Fig. 32, we have, 

Volume of ring = - {D-hd)X area A of section s — s. 
Surface of ring = - {D-{-d)X perimeter P of section 



Circular Spindle — Generated by revolving circu- 
lar arc a or segment A about an axis, as X~X. 

Volume =3.14159 ff--^"(^-^) 1 

= 3.14159 (j- ~2Ad) . 
Surface = 3.14159 \h (o + c)- ^ (a-c) ] 




At 
Z- 



->l'S>l 



x 



! \ ctrccr \ 1 



= 6.283185 (cr- 



4/j ' 
ad) = 2n{cr — ad). 




Fig. 33. 



254 11,— MENSURATION, 

Middle Zone (and Frustum) of circular spindle. — Let z (Fig. 33) be height 
of middle zone; then, if A2 = area of zone of segment, 



Volume ==Z.U159 [-|(<^'^-f-) -2A,(il . 



Convex Surface — 6.283185 X length of "zone-arc" Oz X dist from axis 
X — X to cen of grav of a^. 

Segment of circular spindle. — Let s (Fig. 33) be height of segment; then, 
Volum,e = h (vol of spindle minus volume of middle frustum) . 
Convex Surface = J (surf of spindle minus convex surface of middle zone). 

Parabolic Spindle. — Generated by revolving parabolic arc a (assume same 
Fig. 33) or segment A, about axis X — X, 

Volume == ^^nch^ = 1.Q7 551Q ch^ = ^z volume of circumscribing cylinder. 
Convex Surface = 6.283185 X length of arc a X dist from axis X—X to 
cen of grav of arc. 

Middle Zone (and Frustum) of parabolic spindle. — Let z (Fig. 33) be 

height of middle zone, assuming arc a to be parabolic. Then, as — = 0.209440, 

10 

V(?/wm^ = 0. 209440 2 [8/^2+ 3^2+ 4/^^]. Note that ^ = /t ( 1 - |V 

(The above formula may be used in determining the contents of casks 
whose staves are parabolic.) 

Convex 5wr/ac^= 6.283185 X length of "zone-arc" a, X dist from axis 
X — X to cen of grav of a^. 

S^gwenf of parabolic spindle.— Let 5 (Fig. 33) be height of segment ; then, 
Volume = i (volume of spindle minus volume of middle frustum) . 

Convex Surface =i (surface of spindle minus convex surface of middle zone). 



Cycloidal Spindle. — Generated by revolving cycloidal arc a (assume same 
Fig. 33) or segment A about axis X — X. 

Volume = 5A7t%^= 6.16850/13=^^= 0.198944^3= f^a^ = 0.09638o3 = 

167r 512 

c 

i 7c ch'^= 1.9%S50 ch^ = -^ volnnie of circumscribing cylinder. Note that 

, c a 
h= -= 7. 

7: 4 

Convex Surface = i|^ /t2= 16. 7551 6;t2=^ ^2=1. 69765^2 = ^a^=1.0i720a'i 
^jch. 

Paraboloid or Parabolic Conoid. — Gen- 
erated by revolving one-half parabola of arc a, 
height h, base c, and area A (Fig. 34), about 
its vertical axis Y—Y. 

Volume = I c%= 0. 39270 c% = area of base 

X half the height. 

Volume of parabolic segment of height 

h'=^c'^h\ 

Volume of parabolic frustum of height 



—'aJf- 




^(c% - c'%') . Note that c'^^^,, . 

^ \^ Fig. 34. 



SPINDLE. PARABOLOID. ELLIPSOID, 255 



Con.e.Surface^\.S/l[{i,^^^i-ig] 



Ellipsoid. — Generated by — 

(1) Revolving ellipse around its short diam d as the fixed axis; whence, volume = 

fc?D2 = 0.52360 JL>2. 



(2) Revolving ellipse around its long diam D as the fixed axis; whence, volume = 

5 Dd^ = 0.52360 DJ2; 



or, volume = % vol of circumscribing cylinder, (1) of length d and diam D, 

(2) of length D and diam d. 
An Ellipsoid generated by revolving an ellipse about its short diameter or 
axis is called an Oblate Spheroid; and about its long diameter or axis, a Prolate 
Spheroid. 

Convex Surface of Prolate Spheroid = 2irb^ -\ ^^ sin-i e, the notation being 

e 
given under Fig. 18, page 238. Or, eliminating the eccentr icity e, surface 5 == 

2 Trb[b-\ _ COS-i - ) =irb lb -] z:z=z:=zz SlIT^ I. 

V Va2 - ^2 a/ \ Va2 — 52 a / 

Problems and Answers. — 

Maximum Cylinder, inscribed in Oblate Spheroid. — Radius of base of 
cylinder = a ^%; altitude of cylinder = 2 b/^Z; where a and b are the semi- 
axes of spheroid (ellipsoid). 

Maximum Cylinder, inscribed in circular right Cone. — Fig. 25. Height of 
cylinder = ]'i height of cone. Radius of cylinder = Ys radius of base of cone. 
Volume of cylinder = % volume of cone. 

Maximum Cylinder, inscribed in Sphere. — Altitude of cylinder = % r "V^, 
where r = radius of sphere. 

Cylindrical vessel without top, to contain greatest amount for the given sur- 



= y/a-. 



face, 5. — Altitude = radius of base 

T d TT 

Cylinder of maximum convex surface, inscribed in right circular Cone. — 
Surface of cylinder = H Trab, where a = altitude and b = diameter of base of 
cone. 

Cylinder of maximum whole surface, inscribed in Sphere. — ^Altitude of 

cylinder = r'^2 (1 — l/v^5); where r = radius of sphere. 

Maximum Cone, inscribed in Sphere. — Altitude of cone = % diameter of 
sphere. Volume of cone = y27 volume of sphere. 

Maximum Cone, inscribed in Paraboloid. — Fig. 34. Vertex of cone at middle 
point of base: Altitude of cone = J'i h. 

Maximum Cone whose convex surface is constant. — Altitude = V2 times 
radius of base. 

Maximum Rectangle, inscribed in Ellipse. — Fig. 18. Sides of rectangle are 

<* ^2 and b ^2. Area of rectangle = 2ab. 

Maximum Rectangle, inscribed in Parabola. — Fig. 17. Height of rectangle 
= % height of parabola. 

Maximum Parabola, cut from circular right Cone. — Fig. 11, Introduction, 
page xxxi. Area of parabola (= % its own base, times its altitude) is equal 
to \i ab ^^3, where a = slant height of cone and b = diameter of base of cone. 

Properties of Hypocycloid. — Equation of hypocycloid is ^3 -\- y^ = a^. En- 
tire length of arc = 6 a. Entire area within hypocycloid = — -— . 

o 
Properties of Epicycloid. — Equation (polar) of the Epicycloid, when a = 
a 4 (r2 _ fl2)8 

radius of fixed circle and - = radius of rolling circle, is sin^d = — — • 

J Zi Q*r^ 

Length of one loop = 6 a. 




- 'Y Y 



12.— ANALYTIC GEOMETRY. 

(See also Mensuration.) 

Analytic Geometry treats of the algebraic analysis of geometrical 
figures. 

Plane Analytic Geometry, including what is commonly termed Conic 
Sections or sections cut from cones, deals with the analysis of plane ciirves, 
referred to two coordinate axes, X and Y. The coordinate distances to any 
point p (Fig. 1) of the figure are x and y; x being measured from Y 
parallel to X, and y being measured from X parallel to Y. One, usually x, 
is called the abscissa, and the other, usually y, is called the ordinate, to the 
point p. They are dependent variables for any particular figure. The 
coordinate axes, X and Y, lie in the plane of the figure, and their point of 
intersection is called the origin. If the axes are at right angle with each 
other they are called "rectangular coordinate axes," and the variables ic 
and y are called 'rectangular coordinates." This is the usual method. 

Solid Analytic Geometry deals with the analysis of solid figiires and is 
therefore sometimes termed Analytic Geometry of Space. Three coordinate 
axes are used, namely, X Y and Z, intersecting at the origin; and any 
point in space may be located by the three coordinates x. y and z, measured 
parallel to the respective axes. 

Straight Line. — Equation: y=mx+'b, 
X and Y are coordinate axes. 

X = abscissa, measured from axis Y to any point p. 
y = ordinate t measured from axis X to any point p, 

»*= tangent of angle of inclination = . 

X 

6 =a constant = distance on axis Y from the origin O 

to the straight line. ' Fig. 1. 

Note.— For any point p in 1st qudrant, x and y are plus; 2nd quadrant, 
a; is —, y is + ; 3rd quadrant, x and y are minus; 4th quadrant, ic is +, 
y is —. For angle upward to the right, m is -\- ; downward to the right, 
m is — . Straight line cuts Y, above origin O when 6 is 4- ; below, when 
b is —. 

Problem. 1 — A straight line cuts the axis of F 10 feet above the origin, 
and makes an angle of ( + ) 6°— 10' with the axis of X. Solve? 

Solution. — Natural tang of 6°— 10' = .108; ^7 = 10; hence the equation 

3/ = . 108^+10 
from which the value of y may be obtained for any value of x, or vice versa. 

Thus, if i»; = 0, y=10; it;=l, :j;= 10.108; y = 0, x= -4^=^92.6; etc. 

. lOo 

Problem 2. — ^Find the point of intersection of the two following lines, 
and plat them: y= — ^ x-i-2, and y = x—8. 

Equations: Platting: 

Solution. — (1)3^=- ^x+ 2 Wieny = 0,x=i; x=0,y = 2. 

(2) y= X- 8 y = 0,x=S;x = 0,y 8. 

diff. 0=-liA;+10 
.'.%x =10 

Substituting value of x=Ql in either of the above y, 
equations, we obtain y = — If. 

. *. the coordinates of the point p at intersection of 
the two lines are 

x= + 6i, and y= — IJ. 

This is also a graphical illustration of Simul- 
taneous Equations, page 103. 

256 




STRAIGHT LINE, CIRCLE, PARABOLA, 



257 



Circle. — Origin at center of circle. Equation of 
circle: r^^^x^+y^. 




whence 7-= ±VA;2+y2; ^=_j_ Vr2— :v2; y^^^^z^^i^ 

Problem. — A circle of 10 ft. radius is cut by a 
vertical line 8 ft. to the right of its center at the points 
p and pi. What is the lengt h of th e ch ord ppi ? X 

Solution.— ^= ± Vr2-ic2= ± VlOO- 64= ± V36 
= ±6. Hence, for ^, 3;= + 6, and for^i,:y=— 6. 
.*. the length of chord = 12 ft. 

Intersection of circle with an oblique straight line. 
Problem. — At what point in the above circle 
will the straight line :j/ = 2it: — 1 intersect ? 

Solution. — Equating the value of y^ for the circle and for the straight line, 
ar2 = y2__^2=: (2^— 10)2, -yve have, eliminating y and making r2= 100, 
100-%2 = 4;t2_4o^+100 
whence 5x^ = 40rc 

.-. x=8, or 0; 
and substituting these values of x in the above equation of the straightline, 
we have, for rx:= 8, 3/= 6; for x = 0, y= — 10. Hence the line intersects the 
circle at a point x=8,y=Q; which is the same point as p in the above figure. 
It also intercepts the axis of X at a point x=5', and the axis of y at a 
point y = — 10, at which point it also cuts the lowest point of the circle. 
Equation of Tangent to the circle: XiX + yty = r'^. 



-The 
Find 



equation of a circle (Fig. 4) is 
the equation of the tangent to 



Problem. 
x^+y^=^100. 

the circle at point p, whose abscissa a; is 8? 

Solution. — Substituting the valueof :«:( = 8) in the 
equation of the circle, w e have 

y=-{-\/lOO—64i= + Q, plus, since p is above 
the axis of X; w 

and substituting these values of rx: == 8 and 3/ = 6 in the 
above equation of the tangent, and remembering that 
they correspond with Xi and yi of that equation. 
We have, for equation of tangent, S x+Qy=100. 
This tangent cuts the axis of X at x = ^^^, and the 
axis of Y at y = ^^"^ (by making 3^ = in the first case Y 

and a; = m the second). Fig. 4. 

Equation of Normal to the circle: yiX — Xiy = 0. Let it be required to 
find the normal to the circle (Fig. 4) whose equation is x^-{-y^=100. 
Proceeding as in the above solution for the tangent, we obtain for the 
normal, Qx— 8y = 0; or, y=lx. Clearly, this passes through the origin, at O, 
for when x = 0, y = 0. 

Origin not at center of circle. Equation : r^ = 
(it;-o)2+(y-fe)2, whence r= ±y(x-a)^-h(y-by 
= b±Vr'- 




x = a±\/r^-(y-b)^; y = b±Vr'^-ix-a)^. 

Equation of above circle may be reduced to 
x^-\-y2- 2ax - 2by + a2 + 62 - r2 = 0; or letting ~2a = H, 
-26 = 7, and a^ + b^-r^^K, we have, x^+y^-hHx + 
Jy-\-K = 0. 

Equation of tangent to circle: 

XiX+y,y+j(x + Xi) + 'L(y^y^) + K = 0. 

Parabola. 

useful curve. 



Equation: y 
whence y= ± \/nx 



-Next to the circle this is the most 
(See, also. Mensuration, page 237.) 



nx in may be any quantity), 



— :. .. y 



n 

2y 



n= — ; x = 

X 

The latus rectum (L L) 
2x. To find this point — • 

y = 2x = \/nx 
, ', 4%2 = nx 
n 



at a point where 



and n = ix = 2y= 
t 

length of latus rectum. 




258 



12.— ANALYTIC GEOMETRY. 



The focus (F) is at the intersection of the latus rectum with the axis X, 

a distance oi x = y from the origin, O. 
4 

The directrix is a line parallel with the axis of Y and distant j- (the focal 

distance) to the left oi it, so that the axis of Y lies midway between the 
directrix and the latus rectum. The horizontal distance H from the direc- 
trix to any point, as P, on the parabola is equal to the direct distance D, 
from the focus to that point (Fig. 6). Y 



Problem 1. — At what points does the circle, whose 
radius is 3, intersect the parabola whose value of 



Solution. — circle: 

parabola: 

(diff.) 



A;2+8it: = r2 = 



1; y- 



2V2 



Problem 2. — What is the equation of a 
parabola (that is, the value of n) whose base 
is 16, and altit.ude 12? 

Solution. — When x= the altitude = 12; 

3/2 64 16 
y= half the base =8. Then n= — ^fi^^T' 

X \Z 6 

and the equation of the above parabola is 

y'^= —X, so y can be determined for any 

o 
value of X (measured from the top down- 
ward). _Thus, for x=3_y=VTQ = A; ^=6, 
y = 4V2; x=9, 3^ = 4^3; a;= 12, 3; = 4v^4 = 8. 




Fig. 8. 



Equation of Tangent to the parabola: y{y= -^(x-hxt). 

Equation of Normal to the parabola: y^x-^ ^3' = ^i:Vi+ 7ry\' 

Radius of Curvature r of the parabola y'^ = nx is, 
4 / «2\ 1 
At any point, r = -^ (3'^+ -j ) * 

At the vertex, O, r= -. 

Note. — ^The equation of the parabola is variously given as y^ = nXt 
-- 2px, y^=Aax, etc. It is plain that in these equations n = 2p = 4a, etc. 



Ellipse. — ^The ellipse is a flattened circle. 
Equation : ^ -}- ^ = 1 , whence b^x^ + a'^y^ = a^b^ ; 

X =± ^Vb^^; y ■ ^ 
bx 



±V62- 
ay 




semi -major axis; 



= semi-minor axis 



Fig. 9. 
Foci (singular, focus). Fi and F2 are foci, so situated that the length of 
any broken line Fi Pi F2, joining any point Pi of the curve, is a constant. 
Therefore, the distance from either focus to either extremity of the minor 
axis is equal to one-half the major axis, or a. 



PARABOLA. HYPERBOLA, 



259 



Problem. — ^The major axis of an ellipse is 20 (a =10), and the minor 
axis 10 (6 = 5). For a point P whose abscissa x=Q, what is the value of the 
ordinate y? (See Fig. 9.) 



Solution. 



V'c 



-i*;2=AVl00- 36=^^64 = 4. 



Ans. 



Tangent to the ellipse* Let T (Fig. 10) be the tan- 
gent point of the tangent PT to the auxiliary circle \ 
then will the point t, lying vertically under T, be the 
tangent point of the tangent Pt to the ellipse, drawn 
from the same point P, lying on the major axis. 

Equation of Tangent to the ellipse: b'^XiX-\-a^yxy 
= a^b^. ^ 

Equation of Normal to the ellipse: a'^xX — bHiy 

= (a2-62)^jyj. 

x^ y^ 
Radius of Curvatxire r of the ellipse — +t5= 1 is 



At any point, r 



a2 • 62 

(0^2+^,4^2)1 

a^b^ 




At extremity of major ax^s, r = 
At extremity of minor axis, r = 



62 

a ' 
of 

b' 



Fig. 10. 



The false ellipse or oval is simply an approximation to the true ellipse. 
For instance, the true ellipse consists of an infinite number of infinitesimal 

q2 

arcs with radii gradually decreasing in length from r= -j- at extremity of 

minor axis, to r= — at extremity of major axis; whereas the false ellipse or 

oval consists of a definite number of arcs with radii decreasing as above. 
The semi-false ellipse is sometimes called a many-centered arch; the most 
common forms for bridges are the 5-centered and 3-centered. These terms 
may also be applied to the false elliptic arc even if it comprises only part of 
the semi-false ellipse. 

Y 



Hyperbola (Fig. 11). 
whence. 




or62A;2-o2>'2 = a«52 (i) 



±V62+:v2' ±V'.:^2_o2 

Problem. — In a hyperbola the coordinates of a point P are, x=Q, 
y= 4's/3; and o= 3. Find the value of 6_so that other points can be plotted? 
ay 3X4\/3 12\/3 



Solution. 



:=±4. 



± -s/x"^ - o2__ ±y 36 - 9 ±3 V3 
Then for any point, y= ±Wx^- 9; and x= ±W\^ + y'^' 

Equation of Tangent to the hyperbola: bH^x — a'^iy^^^^^'^- 
Equation of Normal to the hyperbola: a'^yix-hb'^Xiy = (a^-^b^)xiyi. 



260 



12.— ANALYTIC GEOMETRY, 



Equilateral Hyperbola. — The hyperbola becomes equilateral when the 
asymptotes are perpendicular to each other: hence the two axes, 2a and 2b 
(Fig. 11) are equal: hence a = b in Equation (1), page 259. 

The equilateral hyperbola is a very useful curve. It enters into the 
solution of the Howe-truss brace problem (see Section 33) . 




Cycloid (Fig. 12).- 



Fig. 12. 
-Equation: x = rveTS- 



— —\/2ry—y^, in which vers~^— 
r r 



=the angle, in circular measure, whose vers= — . 

r 

Hence, x^r<Xi-ND = OD-ND = ON (Fig. 12). Also, 
x = r (oCi — sin oc); (a:i = 0.0174533 ocin degrees.) 
:j; = r (1 — cos ex) =r vers oc; 
r = it:^(0Ci— sin OC)=y- T-ve rs OC. 

Radius of Curvature r = 2 \/2ry = twice the length of the normal. (The 
normal to the curve at ^ is a straight line pD.) 

For other Properties of the Cycloid, see page 236. 

Catenary and Modified Catenary. — (See Suspension Bridges, and Arches.) 

Spiral of Archimedes. — Equation: r = a6. 

Logarithmic Spiral. — Equation: r=a^ . 

Hyperbolic Spiral. — Equation: r6 = a. 

Lemniscate of Bernouilli. — Equation: r^ = a^ cos 2 d. 

Helix or Screw. — A line traced from a fixed point bearing on a cylinder 
which is made to revolve, and at the same time to advance, uniformly. 
The pitch = the spacing of the lines or threads of the screw. Note that if 
the cylinder is developed, i. e., represented on a fiat surface, all the lines or 
threads will be straight. 

Common Spiral. — A line traced from a point bearing on a revolving 
plate and advancing uniformly outward from its center. 



(2a- 

8fl3 



X) 



Cissoid.— Equation: y^ = 

Witch.-Equation: y = (^2 ^4 ^2) 
Cubic Parabola. — Equation: a^y = x^. 
Semi<=Cubic Parabola. — Equation: ay^ = x^. 
Circle. — Equation: r = a sin 6. 
Logarithmic Spiral.— Equation: r = e . 

a 

Parabola. — Equation: r = asec^ - • 
Cardioid. — Equation: r = a{\ — cos0). 



13.— DESCRIPTIVE GEOMETRY. 



Descriptive Geometry deals with graphical methods of representing mag- 
nitudes and of solving problems in Geometry. 

Perspective pictures an 
object as if seen from a 



object as if seen from a ^ior. u t 

finite distance. Fig. 1 shows V.R_ H.L^ |K5. a;iii_____ 

the perspective of a rect- ^'^.:; — — I ___— — — Z^"^^^^ 

angular box. Points VP \ ""^-^-':: :;;:*.=-- T''"*^^~~i::===7'C-'^-^^^ ^"^ 



V.R 



are vanishing points on 
the horizon line HL, which 
is supposed to be on a level 
with the eye. PS is the 
point of sight, directly 
opposite the eye and at 
the intersection of HL 
with the vertical line VL. 
All lines which are par- 



6.L. 




Fig. 1. 
allel in the object must meet at a common vanishing point. If the lines 
are horizontal in the object the VP lies on HL. If the lines are vertical 
in the object the VP lies on the line VL, at an infinite distance from PS\ 
therefore, vertical lines in the object are shown vertical in the perspective. 
Note that the ground line, GL, is horizontal. The perspective of the 
object may occupy any position with reference to PS, HL, and FL; 
i.e., HL may lie above or below the object, or cut it, and VL may lie to 
the right or left of the object, or cut it. 



Cabinet projection pictures an object as if seen 
from an infinite distance. The ground line GL is at 
an angle of 45° with the horizontal, and lines parallel 
with its direction are drawn to ^ scale. (Fig. 2.; 



Isometric projection pictures an object as if seen 
from an infinite distance, with the ground line GL 
30° with the horizontal; and all lines drawn to full 
scale. (Fig. 3.) 



ORTHOGRAPHIC PROJECTION 



Orthographic projection, or Descrip- 
tive Geometry proper, deals graphically 
with the problems of position and dimien- 
sion of the (point.) line, surface and 
solid in space. The ground line GL 
is the intersection of two coordinate 
planes, *V (vertical) and H (horizontal), 
forming four dihedral angles — 1st, 2nd, 
3rd and 4th (Fig. 4). 




261 



262 



13.— DESCRIPTIVE GEOMETRY, 




Revolved planes. — In the isometric figure, Fig. 4, a is a point in 
space, ah is its vertical projection on the horizontal plane H, and a' is its 
horizontal projection on the vertical plane V; further, gq is the horizontal 
projection of ah and. the vertical projection of a', on the ground line. If 
now the vertical plane V is revolved on the ground line GL, so as to coincide 
with the horizontal plane H, the projection a' 
of a will fall on av, so that the distance av ao = 
a' ao = a ah, and ah ao = a a'. 

By revolving the ground line to a horizontal _ 
position as in Fig. 5, with the vertical plane ^~' 
above and the horizontal plane below, it will be 
seen that the point a in space may be repre- 
sented by its projections, av and ah. 

Projection of the point. — A point may 
be situated in the 1st, 2nd, 3rd or 4th 
dihedral angle, or it may be situated in * 
one of the composite planes. Fig. 6 illus- 
trates the system of lettering adopted 
to show the position of any point. A ©• 
point in space is designated by a small ^ 
letter, and its projection by the same let- 
ter with V or h written above, as an 
index. 
Dihedral angle in which point is situated: 

Plane " *' " " " : 

Note. — Incases (2) and (4), /vhand wvh 
represent points / and n respectively equi- 
distant from the coordinate planes. 

Projection of the right line. — The position of a right line in space may 
be determined by the projection of two of its points. Fig. 7 illustrates 
the system of lettering adopted to show the position of any line. A line in 
space is designated by a capital' letter, and its projection by the same letter 
with V or h written above, as an index. 

(1) (E) (3) (4) (5) (6) (7) (8) (9) (10) (II) 

; 4 1 r<' ... r ■ 





Fig. 


5. 






(1) (2) r3) (4) (5) (6) (7) (8) (9) 


K 








. 1 


k 








'gJR-L 


1° 


Fig. 6. 
2° 3° 4° 






TJ 




H V 


V 


H 






front 
upper 


1 


1 


O 



! o 



Dihed ang: 1° 
Plane: 



1° 



^ 






) ! ! \-f^ 






^ 



I I 

'Fig. 7. 
2° 3° 4^ 3° 1° 3° 1** 

H V 

front upper 

Remarks. — B and K pierce H; C, G and Ht pierce the ground line; / 
pierces V, and is parallel to H\ B and G are perpendicular to H\ and J is 
parallel to H and V. 

Projection of two lines. 

Remarks. — 

(1). M and L intersect 
at a, therefore the projec- 
tions of their intersec- 
tion are av and ah. ^_ 

(2). A^ and O are ^ 
not in the same plane, as 
the intersections of their 
projections are not the pro- 
jections of a common point. 

(3). P and Q are par- 
allel, as their common projections are parallel. 

Note. — Two lines which are parallel or intersect, represent a plane, and 
the horizontal and vertical projections of their point of intersection must 
lie in the same perpendicular line. 




{ 



PROBLEMS OF CONSTRUCTION. 



263 



Projection of the plane. — A plane is determined by three points, or a 
point and a right line, or two right lines parallel or intersecting. The lines 
of intersection of a plane in space with the coordinate planes are called 
traces. Fig. 9 shows the vertical and horizontal traces of various planes 
in the 1st dihedral angle. 




Fig. 9. 



Plane .4, 


Plane B, 


Plane C, 


Traces of Traces Traces of 


Plane G 


perpendic- 


perpendic- 


perpendic- 


D make 


oiE. 


F make 


parallel 


ular to ver- 


ular to 


ular to 


acute 




supplemen- 


to 


tical plane. 


horizontal 


both co- 


angles 




tary angles 


ground 




plane. 


ordinate 
planes. 


with 

ground 

line. 




with 

ground 

line. 


me. 



Problems of Construction. — In problems of construction the following 
conventional lines are employed: 

Principal lines (data and results) , full when visible and dots when invisible. 

Auxiliary lines (minor lines), small dashes and dots. 

Construction lines, (joining projections of the same points in space), fine 

dashes. 

Problem 1. — -To find the 
projections of a line in space 
and also its length. (Fig. 10.) 

A right line A in the 1st 
dihedral angle pierces the 
H plane at a point n, 3 ft. 
from the ground line, and 
the V plane at a point m, 
5 ft. from the ground line; 
the distance between _ the 
projections of the points, 
measured parallel with the Fig. 10. 

ground line, being 15 ft. What is the length of the line A in space? 

Solution. — Let the points n and m be represented respectively by their 
vertical and horizontal projections wv whand mv, m^\ then will Av joining 
«v and mv, and Ah, joining wh and mh, be the respective vertical and hori- 
zontal projections of the line A\ and the length of the line will equal A' or 
A", A' being the hypothenuse of a right triangle whose base is Avand altitude 
3 ft.; and A" the hypothenuse of a right triangle with base A.h and altitude 
5 ft^ 
V259^ 




Hence, by scale, 
= 16.09-hft.) Ans. 



A = 16.1 ft. (Analytically, A = \/l52+ 52+ 32= 



Problem 2. — To find where a given right line A, shown in the 1st dihedral 
angle, pierces the coordinate planes V and H \ (Fig. 11.) 

Solution. — Let Av and Ah be the 
vertical and horizontal projections 
of the line A ; then will m^, the point 
where the horizontal projection of A 
intercepts the ground line, be the 
horizontal projection of the point 
where the line pierces the V plane 
(at mv); and mv, the point where theG 
vertical projection of A intercepts the 
ground line, will be the vertical pro- 
jection of the point where the line 
pierces the H plane (at wh) 



Z 

Fig. 11. 
Therefore, the line A pierces the (upper) 




264 



IZ.— DESCRIPTIVE GEOMETRY. 



V plane at mv, by scale 9 ft. from the ground line; and it pierces the (back) 
H plane at n^, by scale 12 ft. from the ground line; the points being hori- 
zontally (parallel with the ground line) 17^ ft. apart. It will be observed 
that the portion of the line between wv and n^ is in the 2nd dihedral angle. 
Problem 3. — To pass a plane P through three given points not in the same 
straight line. (Fig. 12.) 




Fig. 12 

Solution. — Let m, n and o be three points in space, in the 1st dihedral 
angle, so that the horizontal distance (parallel with ground line) between 
the projections of m and w is 5 ft., and between those of n and o, is 2^ ft. 
Also let the perpendicular distances from the points in space, to the coordi- 
nate planes be as follows: m to H, 6ft.; m to V, 2 ft.; n to H, 4: ft.; n to V, 
6 ft. ; to H, 8 ft., and oto V, Z ft. That is, m is 6 ft. from the H plane and 
2 ft. from the V plane, etc. ; 

Let A^ he the horizontal projection of a line A passing through the 
points m and n, then A^wiW pass through mh and «h; also where Ah inter- 
cepts the ground line GL (Problem 2) , p^ is the horizontal projection of the 
point where the line A pierces the V plane, at py. 
In a similar way, A pierces the H plane at <?h. By- 
like analysis, Sv and B^ are the vertical and hori- 
zontal projections of the line B, passing through 
the points n and o in space; and sv and rh are the 
points where the line pierces the V and 
H planes, respectively. It is evident ^9^ 

that py and sv are points on the 
vertical trace VP of the plane P, 
and that rh and gh He in its hori- q- 
zontal trace, HP. 

By scale, the tangent of the 
angle which the trace of each plane makes 
with the ground line is .400, and the point 
of intersection, at G, of the two traces at 
the ground line, is distant 20 ft. from m° 
of point m. 

This problem is very useful iti structural shop 
details, such as finding the end bevel of hip and 
valley rafters, in framing, etc. 

Problem 4. — To find the point b where a per' 
pendicular line L from a given point a pierces a 
given plane P\ also the length of L. (Fig. 13.) 

Data. — Let av and ah be the vertical and 
horizontal projections of the point a, and let VP 
and HP be the traces of the plane P. 




Fig. 



TO FIND LENGTHS AND ANGLES. 



265 




Fig. 14. 



Solution. — If a line in space is perpendicular to a plane, the projec- 
tions of the line are respectively perpendicular to the traces of the plane; 
and conversely. Therefore, draw Lv from av perpendicular to FP, and Lh 
from ah perpendicular to HP, as the vertical and horizontal projections of the 
line L perpendicular to the plane P. Find 6v and 6h, the vertical and 
horizontal projections of h. From the extremities of Lh lay off, at right 
angles, ah a' = ao av, and b^ b' = bo b^. Then L' is the length of the line L 
in space. 

Problem 5. — To find the angle between two given planes, P and Q, 

Suggestions. — Let the two planes 
P and Q, Fig. 14, intersect at the 
line L. Then pass a plane R through 
any point «, on the line L, perpen- 
dicular to P and Q (it will also be 
perpendicular to their common line 
of intersection L) cutting these planes 
by the lines Ri and R2, and cutting 
the H plane, by its horizontal trace 
HR. Then will the lines Ri and i?2, 
intersecting at n, measure the required 
angle between the planes. 

Data. — Let the planes P and Q 
intersect the ground line at points 3 
ft. apart. Let the vertical trace VP 
of the plane P make an angle of 65° 
with the ground line; HP, an angle of 
60°; VQ, 45°; and HQ, 60°. The re- 
quired angle between the planes is OC, 
Fig. 15, deduced as follows: 

Solution. — In Fig. 15, lay 
off, from points 3 ft. apart 
by^ scale, the angles of the 
traces of the planes Pand Q, 
as per the data. VP and 
HP will be the vertical and 
horizontal traces of the 
plane P; VQ and HQ, the 
vertical and horizontal traces 
of the plane Q; and Lh will 
be the horizontal projection 
of their line of intersection 
L. Pass a plane R perpen- 
dicular to the line L, at n, 
cutting the planes P and Q; 
then will the horizontal trace 
HR of the plane R be per- 
pendicular to the horizontal 
projection Lh of the line L, 
at o; and the plane R will be 
perpendicular to both planes 
P and Q. Revolve the tri- 
angle of which the line L is 
the hypothenuse, around its 
perpendicular axis avah so 
that 6h touches the ground 
line at b'\ then will L' rep- 
resent the length of the line 
L, and o'n', perpendicular to 

L\ will be the altitude of the triangular portion of the plane i? between the 
coordinate plane H and the planes P and Q. Revolve this triangle on 
Its base HR; then will «, the apex of the triangle, fall at n'\ and on'' will 
equal o'n\ The required angle between the planes is OC. 




14— THE CALCULUS. 



J 



The Calculus furnishes us with direct and exact methods for solvi 
many ploblems which could be solved otherwise only by indirect approxi 
mations; and the application of its principles provides us with useful 
working formulas in Mensuration, Geometry, Mechanics and other subjects. 

The two processes in calculus are differentiation and integration, each 
being the inverse of the other. The former comprises the subject of Dif- 
ferential Calculus, and is analytic in its nature; the latter comprises Integral 
Calculus, and is synthetic. 

Problems to be solved by the Differential Calculus may be reduced to 
an equation which will be the equation of some curve referred to coordinate 
axes, as in Analytic Geometry. If there are two variables in the equation, 
as X and y, there will be two coordinate axes, X and Y. The process of 
differentiation enables us to find the slope of the curve at any point p whose 
coordinates are x and y; and from this angle of slope, say with the axis of X, 
we may determine (1) the actual change in y for a corresponding change in x, 
and (2) the rate of change in y, at any point on the curve. 

Problems to be solved by the Integral Calculus may be reduced to an 
equation which will be the equation of the slope of some curve reierred. to 
coordinate axes, as in Analytic Geometry. If there are two variables in the 
equation, as x and y, there will be two coordinate axes, X and Y. The 
process of integration enables us to find the equation of the curve, i. e., the 
value of the ordinate y for any value of the abscissa x. 

A. DIFFERENTIAL CALCULUS. 

Differentiation has for its main object the determination, from the equa- 
tion of the problem, of — 

1st, the differential dy, = t\ie actual change in a function, due to a corres- 
ponding change dx of the variable x upon which it depends. 



dy 



== the rate of 



2nd, the differential coefficient , 

a X 

change of a fvmction. 
To illustrate: In the equation y = x'^ let :v = 
the area of a square of which x is the side. Now 
increase x by the distance dx (reads ''differential 
of ic," and does not mean dXx)y an amount 
smaller than any numerical value\ then will the 
new Qxe2iyx = Xi^={x-\-dx)'^ = x'^+2xdx-{-dx'^\ and 
the increment or actual change in y will be 
^y = y^-y = x'^+ 2xdx + dx^ - x^ = 2xdx + dx^. But 
as any infinitesimal quantity, as dx or dx^, which 
is purely additive may be neglected, we have, 
differential dy = 2xdx, 

and the differential coefficient -i— =2x=-—. 

ax 1 

Differential Coefficient. — The differen- 
tial coefficient is simply the natural tangent 
of the angle which a tangent to a curve 
makes with the axis of X. 

The preceding equation, y = x'^, is the 
equation of the parabola, y = nx'^ (see 
Analytic Geometry), in which n= 1. Let 
Fig. 2 represent such a curve and let x 
and y be the coordinates of any point 
p. Also let p' be another point so that 

x-^-dx^x* and y + cfy=y; then will ^ = 

* Note that fiy = increment oiy=y'—y, 
and dx = increment oi x = x' — x. 





X, 






X 


dx 


K 






% 





Fig. L 



(1) 
(2) 




Fig. 2. 



26a 



RULES FOR DIFFERENTIATION, 267 

the natural tangent of the angle ^ of a straight line passing through the 
points pp' . But the points pp' are practically identical, as dx is smaller 
than any numerical value; hence the line tt, passing through the point 
pp', is tangent to the curve at the point p whose coordinates are x and y. 

Problem, — What is the length of the side of a square whose area would 
increase 8 times as fast as the side, for a small increment? 

dy 
Solution. — Make the differential coefficient -j— equal to 8 in equation (2), 

dx 

^=2x; or, 8 = 2a;, .*. x=i. Ans. 
dx 

Tangent and Normal. — We have seen, Fig. 2, that T~ ( = t") repre- 
sents the slope of a tangent to the parabola passing through the point p 
whose coordinates are x and y. If, now, we assign a definite value to either 
X or y the value of the other may be determined from the equation y = x'^, 

d y 
and the point p plotted. Thus, let x=i\ then y=i, and -j- = 2x=2 X^=l, 

d X 

Therefore, the slope of the tangent to the curve at the point p whose co- 

dy 
ordinates are x = \, y=\, is -7- = !; and hence the angle which it makes 

ax 

with the axis of ic is 45°. This angle, 45°, means that for any increment in 
the side of the square whose length is \, the area will receive an equal incre- 
ment. For another example, let x=\\\ then, from y = x^,y=2\\ and from 

d v 

-7^ =2:*;= 3, we have the angle which the tangent to the curve at the point 

dx 

%= li. y = 2i, makes with the axis of X, =the anti-tangent of 3=71°— 34'. 

d "V 
As -5— ( = 2x) = the differential coefficient of y with respect to x, so is 

dx 

-rr- I = =) the differential coefficient of x with respect to y\ that is, 

^y\ 2\/y I 

-7- = TT- = =: \ represents the slope of a normal to the parabola 

dx I 

through the point p whose coordinates are x and y. 
d'V d X 

Therefore, -j- and 3— are, respectively, the slopes of the tangent and 
ax ay 

normal (to the curve) referred to the axis of X. 

Rules for Differentiation. — The differential coefficient of — 

(a) a variable, with respect to itself is 1. 

dy . 

(b) a constant, is 0. 

, = ,^^^=0. . = 8. 11=0. 

(c) the algebraic sum of any number of quantities is the algebraic sum of their 
differential coefficients. 

3, = „ + „; ^='^+^. j, = 4^-6^»; ^ = 12^^-10*. 

dx dx dx dx 

(d) the product of two variables, is the sum of the products of each variable by 

the differential coefficient of the other. 

y=uv\ p^^vp+u^. y==SxHix+l)', ^ = (4ic+l) 6x-\-(3x^)i. 
ax d X dx dx 

(e) the product of a constant and a variable, is the product of the constant 

and the differential coefficient of the variable. 
dy du _ , dy _._ „ 



268 



U.—THE CALCULUS. 



(f) a fraction, is the differential coefficient of the numerator multiplied by 
the denominator, minus the differential coefficient of the denominator 
multiplied by the numerator, this difference being divided by the square 
of the denominator. 

du dv 



y= 



y= 



v' dx 
3%2+2 



dx 



dx 



^2 

2x{Qx+0) 



dy 
dx 



(3:j;2+2)2 3a;2-2 




2x ' dx ix^ 2%2 • 

(g) any power of a variable, is the product of the exponent, the power with^ 
exponent diminished by 1, and the differential coefficient of the variable. 
^ dy „ , diu 

dx dx 



y= 2(^2+ 3)3; ^ = (3x 2)(^2+ 3)2x 2x= 12^(^2+ 3)2. 
ax 



\^'^'^va± 




Maxima and Minima.— One 

of the principal uses of the differ- 
ential calculus is to find the 
maximum or minimum value of 
a function. In any curve, as 
for instance the ellipse, a^^-\-b^x^ 
= a^b^ (see Analytic Geometry), 
let it be required to find thev 
maximum value of the ordinate 
y. From the above equation, 
placing y in the first member, 
we have 023/2 = a^b^ - bH^. Differ- 
entiating, 2a2 y dy = — 2b^ xdx; 
whence the differential coefficient 
dy 2bH b'^x ^ ^ , 

~r' = ~K~^ = — T — tangent of 
dx 2a2y a^y 

angle of inclination of the tangent «, with the axis of X. That is,— ^V" is the 

a^y 

value of m in the equation of the straight line tt.^ (See Analytic Geometry.) 
For any tangent to the ellipse, if w is a minus quantity, the point of 
contact p is in the 1st or 3rd quadrants, and if it is a plus quantity the point 
of contact is in the 2nd or 4th quadrants. This is made evident by noting 
whether the tangent line it slopes upward to the right or downward to the 
right. If the point of contact p is in the first quadrant, x and y are plus 
dy b^ /x \ b^x 

and -7— = ( — ) = — ; if in the second quadrant, x is minus and y is plus, 

dx a2 \y ) a^y 

d v 
3rd quadrant, -7- = 

b^x . ., . ^, ' ^, , dy b^ / X \ b^x "^ 

— r- ; and if in the 4th quadrant, 3— = { J = ^r • 

023; ^ ' dx a^\—yj a^ 

to study the slope of a curve at any point. 

By inverse analysis we may find the values of x and y (including their ^ 
maximum and minimum values) by assigning definite values to the slope ] 
dy 

3— . Thus, when 3; is a maximum we know that the tangent line tt must 
ax 

move so as to be parallel with the axis of X; hence, it will coincide with the 
tangent line TT, touching the point P at the upper extremity of the minor 
axis; or with a parallel tangent through the point P' at its lower extremity. 
And, when 3; is a minimum, tt will be perpendicular to the axis of X, touch- 
ing points on the curve at either extremity of the major axis. In the former 

dy / b'^x\ 

case, 3~ I = ± —r- 1=0; whence x = Q. which value substituted in the 

dx\ a^yj 

general equation makes 3/= ±6, a maximum. And in the latter case. 



ax a"- \y / a'^y 
, dy b^ /-x\ bH ., . ^, 

hence -7-= 51 I = -t" J if m the 

dx a^ \ y / a^y 



a^\-y/ 
Thus, we are able 



dy ( h'^x\ . n • • 

3— I = ±-7-1 =00; whence 3' = 0, a minimum. 

dx \ a^yj 

Maxima and Minima obtain at points where ^ changes from, 

a X 
or from — to +. 



dy 



■h to - 



MAXIMA AND MINIMA. 



269 



Problem 1. — It is desired to suspend a a 
weight W vertically beneath a point P situated '^ 
midway between two level supports A and 5. 
For this purpose two diagonal rods, A 1^ and 
BW, of equal length, are used. At what angle 
a shall the rods be inclined so that a mini- 
mum amount of metal will be required? 

Solution. — ^The stress in each rod ^-^ ^y^ 

secant a, and the length of each rod = d Fig. 4. 

cosecant a = h secant a; hence, the amount of metal required in the 

rods would be, making n a constant, and h variable. 




/W 



sec a 



nhW 



) {h sec a) 

sec2a = ^^(i + tan^a) 

. nhW 



whence -jt^ 
dh 



2 

nhW 

nhW nd^W 
2 ^ 2h 

nW 2nd^W 
2 ih^ 



d^ 



0, for minimum; 



$8000p&rmo/ x=26.63m. } 
O 50 mf"" 




OJ 



.*. inWh'^^inWd^ 

.'. h = d, or a = 45°. 

This proves that the most economical angle for a truss diagonal is 45°. 

Problem 2. — It is proposed to build a rail- 
road from A, a point opposite B and distant 30 
miles, to D a point 50 miles below B. The cost 
of grading on the line BC is $8,000 per mile, 
while the cost of grading from ^ A diagonally 
to any point C is $12,000 per mile. Find the 
position of the point C so that the entire cost 
of grading will be a minimum. 

Solution. — Let y = the cost of grading the 
line, and from the data, 

:v = $12,000 {'AB^-{-BC^)^ + $8,000 (BD 
Let BC = x, AB = 30, and BD = 50; then 

y= 12000 (900 + a;2)^ + 8000 (50-a;). 
^= 6000 (900 + a;2)-^ (2^) -8000 

; 12000^^8000 
V900+:x;2 

/. ft; =2 6. 8 3 miles. Ans. 
Note. — The entire cost of construction and the cost of operation would 
^of course enter as factors to reduce the distance between C and D, for 
economy. 

Problem 3. — What is the maximum cylinder, in 
volume, that can be inscribed in a sphere? 

Solution. — Let i? = radius of sphere; r = radius, 
and /j = altitude, of cylinder. Let V' = volume of 
cylinder; then 



Fig. 5. 
-BC). 



for minimum; 



V=7:r%=7:h(R'^-j\ . 



in which V = a, function of h, both being variable 

with R constant. Differentiating and making :77- = 0, 

an 

we have, jr =^^^- 1^^^ = 0; 

an 

.'.h=--=R = iVJ R, Ans. 
V3 




Fig. 6. 



270 U.—THE CALCULUS. 

Other Forms of Differentiation. — ^There are two forms in which the 
result of differentiation may be expressed, namely, differential coefficient or 
simply differential: 

dy 
y = x^\, -j- = 2x = differential coefficient. 
dx 

y — x'^\ dy = 2xdx = differential. 

d "V 
Hence the differential coefficient -r- may be considered as a fraction, 

dx ' 

the numerator dy being the perpendicular, and the denominator dx the base, 

of an infinitesi mal triang le of which the hypothenuse 

is equal to y/dx^-hdy"^. Multiplying the differential 

coefficient by dx we obtain the differential dy = 2x dx. 

It is well to keep this graphical analysis in mind in 

dealing with problems involving both Differential 

and Integral Calculus, as it then loses much of its 

haze and mystery. 

" Fig. 7. 

Besides the Algebraic forms (a) to (g) , which are universal in their applica' 
tion, the following formulas for differentiation of special functions will be 
found useful for reference. (For Algebraic Functions, see pages 267, 268.) 

Logarithmic and Exponential Functions: 
loga ^ ^^ 




y=\ogg,x; dy = 

X 

y= loge x\ dy= — . 

X 



du 



y 


= loga 


u 


dy 
• dx 


= loga ^ 


dx 

u • 


<iy==loga e 


du 
u 


y 


= loge 


u 


dy 
dx' 


du 
_dx^ 
u * 


dy 


_du 

u ' 




; = 


= au; 




dy 
dx 


loge o- 


„ du 
au -— ; 
dx 


dy = \oQQ 


a-a^du 


/ = 


=eu; 




dy 
dx 


dx 


dy = 


= e^du. 





(h) 



(i) 



(j) y 
(k) y 

(1) y = «v; -r-^vuy-'^-7--\-\ogQ u-uv-r-\ dy — vu'^~'^du + \ogQU'U^dv. 

Trigonometric Functions; 

dy = cos u du. 
dy = —sm udu. 
dy = sec^udu. 
ciy=: — cosec^w du. 
dy = sec u tan u du. 

3— ; dy= — cosec u cot u du. 
dx 

dy = sin u du. 



(m) 


j' = sin u; 


dy du 
-J— = cos u -J- ; 
dx dx 


(n) 


>' = cos u; 


dy . du 


(0) 


y = tan u; 


dx dx 

dy „ du 
—-= — cosec^w^- ; 
dx dx 


(P) 


y = cot u; 


(q) 


y = sec u\ 


dy ^ du 
3— = sec « tan u -j- \ 
dx dx 


(r) 


.'V = cosec u\ 


dy 

-^ = —cosec u cot u 

dx 


(s) 


y = vers u\ 


dy . du 
-7^ = sm w -7— ; 



DIFFERENTIATION FORMULAS. THEOREMS, 271 

Inverse Trigonometric Functions: 

du 

. , dy dx , du 

dx Vl-ii2 Vl-u^ 

du 

. dy dx J du 
(u) :v = cos-i w; __= ; dy^ . 

du 

X 1 dy dx J du 

(V) 3,=tan-'«; ^'J^r^,: '^y=H:¥2- 

/ N ^1 dy dx , du 

(w) j,=cot-'«: ^=-i^r^,: 'i^=-T+ir«- 

(x) y = sec-» w; -r-= ; dy = ^ 

, Jy drc' J du 
(y) y = cosec-iw; j— = ; oy= 7= 

, . i dy dx , du 

EXPANSION OF FUNCTIONS. 

By Division. — y = 7—— = l — x-\-x^ — x^-h x* . 

Successive Differentiation is employed in the expansion of functions and 
will be explained briefly. Let y = uhe any equation in which « is a function 

of X. Then, the 1st differential coefficient \=-j—) is the diff coef of y\ 

the 2nd diff coef ( = -t^) is the diff coef of the first diff coef; the 3rd diff 

coef (-r^) is the diff coef of the 2nd diff coef, etc. For example, let y=Zx^\ 

then P^ = \2x\ ^ = 36^2, etc. 
dx dx^ 

Maclauren's Theorem enables us to expand into a series any function u 
of a single variable x, by the use of successive differential coefficients. Let 
u be the quantity to be expanded, and assume 

u=A+Bx + Cx^ + Dx? + Ex^ , using Indeterminate Coefficients. ( 1) 

Differentiating this value of u successively and making ic=0 in all 
functions after differentiation, we have, 

Making x=^(^: 
y = u (See equation 1.) y =At or A = y, 

^ =B-{-2Cx-{-ZDx^-hiEc^ . f| = -^» or 5= ^. 

|^2C + 2.3Z>.+ 3.4S.--. g = 2C. or C= I@). 

g = 2.3P.2.3.4^«^--. g = 2.3Z.. orI, = ^3 (g) . 

g=,.3.4^.--. g = 2.3.4^. orE = ^L,g). 
etc. 



27? U.—THE CALCULUS. 

Substituting the above values of the indeterminate coefficients in equa- 
tion (1) we have the following working equation: 

, dy , d'^y x'^.d^y x^ .d^y x^ , ,„. 

dx dx^ 2 dx^ 2-3 dx'^ 2- 3- 4 
remembering that 3/ = ^ when, in all the successive differential coefficients, 
x = 0. 

Examples in the use of equation (2). — It is to be noted that Maclauren's 
Theorem may be considered a special case of Taylor's Theorem, following. 
Also, the Binomial Formula (see Algebra) is a special case of Maclauren's 
Theorem. 

Expansion of a few common functions: 

^2 /^3 />ri ^5 

loge ('i- + x)=x — -^ + -^ — -T-+-= . Naperian system. 

(^2 ^z ^4 ^5 \ 

a;— Y + — — -j+-r ) Common system. 

. ,, Common log (1+ a;) 1 AnAnnAAOtn ii- J 

,\M=:rz : , \ ., . \ = ^ ^- , ^_ ^-- = .4342944819 = the mod- 

Napenan log (l + x) 2.30258509 

ulus of the common system = common log e = loga e. 
e =2+J + J-3+-L_ + -L--g+ =2.718281828 + . 

^ ~ 1 1-2 1-2-3 1-2-3-4" • 

sma-«.-j^2— 3+ 1.2-3-4-5~ 1-2- 3-4-5-6-7+ ' 



thus, „at sine of 30o= ^-^ (^) %_L^ (z) '__L^ (|) ' 



+ . 



thus, nat cos of 20°= 1-^ (|) V ^^ (|) '- ^^ (f ) ^ + ' 

Y = l-^+i-^ + i- . .•.7r= 3.1415926536. 

Taylor's Theorem enables us to expand a function of the sum of two 

variables arranged according to the ascending powers of one of the variables. 
Let u, = a, function of (x-hh), be the function to be developed. Then, for 
a working formula, 

^dy . d^ /j2 d^y i^z 

^=^+^•^■^572 •y+^- 172-3+ -(3) 

remembering that y = u when, in all the successive differential coefficients 
of u, h=0. (See Maclauren's Theorem, preceding, for illustration.) 
Expansion of a few common functions: 

log (x + h) =log x+ --2^2+ 3^-4^4 + • 

h^ 

tan (x + h) =tsinx + h secH + h^ sec H tan x+ — sec2^(H- 3 tanZ^;) + 

o 

log il + smx) = x-Y-^-Q — ^+ . 

B. INTEGRAL CALCULUS. 

Integral Calculus is the process of summation or the adding up of an 
infinite number of infinitesimal quantities. Its operation is the inverse of 
differentiation. In the equation y = x^, by differentiation we have dy = 

2xdx, hence the summation or integration of 2xdx= I 2xdx = x'*- I =2— j . 

Suppose, for example, we wish to find the area of a triangle whose altitude 



INTEGRAL CALCULUS. METHOD OF LIMITS, 



273 




is h and base 6, Fig. 8. If we let dA represent, m 
general, the area of any vertical strip of length y 
and width dx, distant x from the axis of Y, then 
the total area of the triangle will equal the summa- 
tion of all these strips between the limits x = h and 

/• x = h /• x=l) 
x=Q\ or, area=i4=| dA =1 ydx. 

The main difficulty in Integral Calculus is in 
forming the equations to be integrated and recog- 
nizing them as related to certain fundamental forms ^_ 
whose integrals are well known, many of which are'^" 
given in tabluar form in the following pages. In the 
above case, x and y are dependent variables, x in- ^ Fig. 8. 

creasing as y decreases, and vice versa. When x = b, y=0; when x=0, 
y = h', and the point P, whose coordinates are x and y, moves along the 
hypothenuse in a straight line between the axes of X and Y, as x and y 
vary. The equation of this straight line, of which y^^mx+b is the general 
equation, is 

y= —-j-x+h 

because the tangent of inclination m of the hypothenuse with the axis of X 

h 
is ~-r- (minus, because downward to the right), and the value of & where 

the hypothenuse intercepts the axis of Y is h. But dA'= ydx = the area of 
an infinitesimal strip or length y, and from the above equation, ydx=' 

— -r-xdx + hdx; hence, 

x<^b 



Area ==A 



/• x<Mb 

■f-i' 



dx-\-hdx 



which, by certain fundamental forms given in the table, reduces to 



^~ b 



+ hx. 



Substituting the value of x=b in the equation, we have 

h b^ 
Area A = — -r- • -^ ■\-hb=^\bh. Ans. 



Definite Integration a Method of Limits. — In the preceding illustration 
of the area of the triangle we referred to the upper limit x^^b, and the lower 
limit x=Q. We will now find the area of a part of the triangle or that etched 
portion, Fig. 8, between the upper limit x= lb, and the lower limit x=\b. 



Thus, Ax 



After integrating, Ax 



Substituting \b and\ 



r x^i 

- -\' 

[x^\b 



dx^rhdx 



+hx 



upper 



lower 



muting 16 and! . / ^ ^b^^^,,\ / h b^ hh\ 

.*. partial area Ax=-r' Ans. 

4 

Note. — Subtract the value obtained by using the lower limit, from the 
value obtained by using the upper limit. 



274 U.—THE CALCULUS. 
Formulas for Integration. — 

(B) I ~ =log «. I ^ ""^ ^^^ ^* 

(C) I a^dic = ^ . I a^ log a dx^O*, 

(D) I ^^(ii»; = e^. I a^^^dr*; = , f 7' • 

J J l + loga 

(E) 



I cos X dx = sm X. j cos 2x dx = sm (x-ha) cos («—«). 

f ^ Cjdx , ^ 

I sm X dx= —cos x. I -^ — j^ =log tan x, 

J J sm 2x 

I sec ^ic da; = tan x, 

I cosec ^xdx^— cot ^. I cosec xdx = log^ / 1 — cos x ^ 
J J \ l+cos* 

I cot X 

I sec a; dx = log (sec r>; + tan re) =log tan ( y"*" t) • 

I cosec X dx=\oe (cosec ic — cot x) =log tan -^ = log^ JlZS^U^, 

J -« \ 1+COSiC 

/*dit; 1 . .X 1^,«? 

I -5- — -„ = — tan-i— = cot-i— . 

Jx^-ha^ a a a a 

/ dx ^ _1_ - x — a ^ _1_ - a-- X 
x^-a^~ 2a ^^ x + a~ 2a °^ a + x' 

rr\\ r dx . 1 ^ t X 

(Q) I — =sin-* — = — COS-*— . 

(R) I Jl =log (*+ \/^^2±^2) . 



(I) I sec X tan xdx= sec it. 

(J) I cosec X cot a; ci;;i; = -- cosec x, 

(K) I tan xdx = \og sec «. 

(L) I cot X dx=log sin ic. 

(M) 

(N) 

(O) 

(P) 



/•ON ■ ^^ 1 ,X 1 - fl? 

(S) I == = — sec-i— = cosec-* — • 

aVx^-a^ a a a a 



(T) I— J?==vers-i-f 
\/2ax-x^ « 



PLANE CURVES— AREAS AND LENGTHS. 



275 



(U) 
(V) 



x^dx=-j\/a^- 



x^+ -^ sm- 



-1 — 

a' 






Areas of Plane Curves. — 

Formula (1). A=\ydx (Fig. 9.) 



= I yc 



Y 3^ 




I 
1 

b 








^ 




r 


\A 





Let Figs. 9 and 10 each represent one- 
half a parabola whose base is 2h and altitude a. 
Required the area of the figure? From the 
general equation of the parabola, y'^=nx, n= 

y2 y^ 62 

— . When x = a, y = b; therefore w =? -^ = — , and 
the equation of the parabola in the figure is 3/2 = y^, 

62 x^ 

— X, From this equation, y = h -^ and x 



ay- 
62- 



^ 


Fig. 9. 

r 

^^ — '^^i\ 
/\ a-x ^ 


1 
b 


V 






Y 





Method (1), vertical strips. 
A = I ydx = I ■— x^ dx. 



f!^ A. 

[ai'i 



Fig. 10. 
Method (2), horizontal strips. 

a y3 

Ln 



= J06. Ans. 



Equivalent 
values. = ah ■ 



\ah = lah. Ans. 



It is thus seen that the same result, fa6, is obtained whether we assume 
the strips to be vertical and summate horizontally or whether we assume 
the strips to be horizontal and summate vertically. 

Again, required the area of the shaded portion of the parabola in Fig. 9, 
Formula (1), between the limits 02 = 4 and ai = 2: 






3 

x-i 



^ 2 


^ 2 




_16 
3 * 


h 4V"2 
V"o 3 


h 

\/~~a 


Lengths of Plane Curves- 


Formula: 


Length L = 


\dL = 



I6-4V 2 ^ _h_ 
3 V"a 



VdA;2 + (iy2. 



2 
Ans. 



(See Fig. 7.) 



\ 
dx, when x is the independent variable. 



Use ^=1 1+(t~) I ^y. when y is the independent variable. 



276 



14.— TH^ CALCULUS. 



Problem. —The catenary, y=Y(^''+^ °j.is the curve which a 

cable assumes when suspended at both ends, as at A and S, Fig. 11. Re- 
quired the length of the catenary from the vertex V, on the axis of Y, 
to any point p whose coordinates are x and y ? 




Solution. — 



hence. £=i (^"^-^"~j 
and dL= i ( e~+e""» j dx (='a/i+ (^) ''^*) 
with limits it = a?, and re =0, L = J j I e " +e " \ dx 
P = L =-|-( ^^ -^"" j. Ans. 



length from V to 



Areas of Curved Surfaces, or surfaces of revolution, — 



Formula (1). S 
Formula (2). S 



= 27:lyds=27:ly 
= 27: 1 xds= 27: 1 Ci 



1+ 
1+ 






dx. (About axis of X.) 



dy (About axis of Y,) 




Y 

Fig. 12. 

Problem. — ^Let it be required to find the area of the shaded zone of a 
sphere of radius r, Fig. 12. 

Solution. — Use Formula (1): 



J 



AREAS OF CURVED SURFACES. VOLUMES, 



277 



Equation of circle is x^ + y^ = r^ 

whence, y2 = f2_ ^2 or y^'s/r^ — x^. 



dy 
dx 



and 

y 



(dy\ 2_£f. 
\dx) y^ 

Substituting value of ( -r- j in Formula (1) there is obtained, 

/•x = b rx^b 

= 2;: I (y^-{-x'^)i dx=27c \rx ==2itr (b-a). Ans. 
J x = a ix=a 

Volumes, or planes of revolution. — 

/*x=a 
Formula (1). F= ;r I y^dx (Plane revolved about axis of X.) 
J x—b 
r*x=c 
Formula (2). V = n\ xMy (Plane revolved about axis of K.) 
•/ a;=rf 

Y 




Fig. 13. 

Problem. — Find the volume generated by the shaded portion of the para- 
bola y^= 4x, Fig. 13, about the axis of A'. 

Solution. — From the equation of the parabola and Formula (1), we have, 



/•o /•a 

V = n\ y^dx =1 ixdx 
Jb J b 

ra 

= 7r 2i»r2=2ff(a2-62). Ans. 



15.— MECHANICS. 



General Discussion. 



The subject of Mechanics deals fundamentally with matter as masSt 
Sit rest and in motion; and with its accompanying phenomena. 

Unit of weight I 
W=l pound. I 

Unit of mass I 



All matter has weight (W) and its mass (M), for any 
particular locality, is directly proportional to its weight. 
The weight of a body is the attracitve force (F) of gravity 



M=? 

Unit of force 
F = 1 pound. 



existing between that body and the earth. 

Motion is a purely relative term: used to express the relation existing 
between two or more bodies when one or more of them is in a state of 
"unrest." 

Unit of velocity! 
u= 1 ft. per sec. I 



If a body starting from rest is allowed to fall unre- 
strained toward the earth, its rate of motion, or velocity 
(v), through space (h or s) will increase constantly with 
the tim^ (t) occupied in its descent, and this increase in 
velocity per second of time is called acceleration (g) or (a). 
Such a motion as described is called uniformly accelerated 
motion, the reverse of uniformly retarded motion; each of 



Unit of space 
h or s=l foot. 
Unit of time 
t=l second. 

Unit of accel. 
g=32.18±ft. 

Unit of accel. 
a = any dist. 
when due to 
other forces 
than gravity. 



which is a form of uniformly varying motion. 

Moreover, if, during its descent and after the body has acquired a 
certain velocity, the action of gravity should suddenly cease, then accelera- 
tion would also cease and the body would continue with a constant or 
uniform motion. 

Fundamental Equations of Motion and Force. — In their relations to 
one another, any force F acting on a mass M so as to produce acceleration 
a, may be considered analogous to the force of gravity W acting on the 
mass M and producing gravity acceleration g. Thus, 

Weight = mass X g acceleration. Or, Force = mass X a acceleration. Or, 
Fg W Wa ,, W ^. -^ Wa F Fg ^^ F 



W = Mg = 



''^M 



M = 



g 



(Bcfuaf-i'on of. 




F=Ma-- 



g 



= hT',M- 



M W 




(Equation of 
Strvrigfhf ^y ^ 

JL 

Fig. 2. 

Ex. 1. — A car weighing 60 tons is impelled on a level track by a hori- 
zontal force of 1 ton. Assuming the rolling friction at 8 lbs. per ton, find 
the acceleration? 

Fg (2000-480) 32.2 ^ .^q .^ 
Ans.- a=^= ' ^Q^^QQQ =0.408 ft. per sec. 



Note. — The acceleration X the length of time of the acting force' 
velocity in ft. per second (see Ex. 3). 



^the 



278 



FUNDAMENTAL EQUATIONS OF MOTION. 



279 



MOTION. 

I. Uniform motion : vq constant, no acceleration. — With uniform motion 
a body moves at a constant velocity vq, imparted to it by a force which 
has been removed; hence there is no acceleration. Note that vq becomes 
the initial velocity in uniformly varying (accelerated or retarded) motions, 
following. 

h hi h — hi^j^ J. ^_ ^ .. _ ^_ ^1 _^~""^i. __ f. u. s 



^«=T=77 



t-u 







S Sx S—Si 



Vq 



(EcfucTfion of 
Straigfhf ^xl.t, 
Line) 



— ^'— s 




(EquaHon of ^ . 
Straight ^^ "' 



Fig. 3. Fig. 4. 

Ex. 2. — What velocity will be required to travel 16 miles in three- 
quarters of an hour? 

. 5 16X5280 _, «_ ,^ 

Ans. — ^0 = 7= "TsVeo" '^ ^^^ ^^^* 

. Relative uniform motion^ — A man walking forward with a velocity vq', 
in a train moving with a velocity vq\ acquires an actual velocity of Fo = 
Vo' + V- Similarly, if walking backward in the train he would acquire a 
velocity of 7^0 = ^0' — ^o". in the direction in which he is walking; or, vq — 
Vq —vq\ in the direction of the moving train. 

II. Uniformly accelerated motion; no initial velocity, vq. — Problems 
that come under this head are those in which the body starts from rest and 
is acted upon by a constant force in the direction of its motion. A train 
starting from a station, or a body falling from a height, are examples. In 
the following, friction is neglected: 

(a). — Acceleration, Time and Velocity. 



v=gt\ t = 



_V__ Vx_ V — Vt 

^~ t ~~h~ t-h' 



(Ecfucrffon of 
Strcrigfhf ^^ \^ "^ 



v = at\t = 



V\ _ v — Vx 



(Ecfuocfion of 
SfraiGfhf 
Line) 




t 
Fig. 5. 

Ex. 3. — What velocity will a body acquire at the end of 10 seconds, 
falling in a vacuum? 

Ans.— z; = gi= 32.2X 10= 322 ft. per sec. 



;t= 



2 ' 7; ' ^ / t-h 



(&). — Time, Velocity and Distance. 

,_i!?.i-if. -..g^, 2 (5-50 




(Equofflon of 
Sfraighf ^y^_\:i. 
Line) ^y^ 



(Equation of 
.Sfrcriqfhf'^J^ 



Fig. 8. 



280 



15.— MECHANICS. 



Ex. 4. — A body falling in a vacuum attains a velocity of 322 ft. per 
sec. at the end of 10 seconds. Through what height has it fallen? 



Ans.- 



^=f = i (322X10) 



1610 ft. 



Note that Y is the average velocity for the time t. 



(c). — Velocity, Distance and Acceleration. 



^^k'^^li'"^'''^^^^-^^^^* 



/ 



(Ecfuafion of y^ 
Parabola)// % 



^^^/^<> ;v-r2 



Fig. 9. 



^=27'" 



2a 



; V- 



\/2os. 



(EcjuaHon of 
ParcrboJa) y/ , ,^ 



-v-riA^ ;v4-2 



Fig. 10. 



Ex. 5. — ^What final velocity will a body acquire in falling 1000 ft. to 
the earth? 

Ans. — Final velocity = z; = 8.02 \/h =8.02X 31.62 = 253.6 ft. per sec. 



{ji). — Distance, Acceleration and Time. 



-Vf- 






249\/;j;g=^';;i=^=16.1/2. 



(Equafion of 
^P^rabola)/^^ 



Fig. 11. 



V2s 25 



2* 



/Equation of ., 
Parabola) // vj 



Fig. 12. 



Ex. 6. — Starting from rest, what acceleration per sec. will cause a 
body to travel 2000 ft. in 10 seconds? 



Ans. — a- 



2s 



YoV = 40 ft. every second. 



Note that the term "acceleration " means " rate or acceleration per 
second " = increase in velocity per second. 



III. Uniformly accelerated motion with positive initial velocity vp. — 

This is a case of uniformly varying motion in which the constant initial 
velocity t;o is in the same direction as the accelerated velocity v produced by 
some constant, acting force. Let V = v-\-VQ~t\iQ resultant velocity at any 
time / measured from the instant that v begins to act; h or 5 = the distance 
traveled in that time; g or a = the rate of acceleration per second. Then 
the following relations exist, neglecting friction: 



FUNDAMENTAL EQUATIONS OF MOTION, 



281 



{A). — Acceleration, Time and Velocity. 



V=v-{-VQ = gt+vo\ v=V—vo\ vo=V—gt. 
V — Vq V V — vo V 



V='V-\-vo=at+vo\ v=V—vo\ vo=V—aU 

V — vo _v _ V — vq ^ V 
a a' t t ' 



Ex. 7. — A stone is thrown from a balloon vertically downward toward 
the earth with a velocity of 100 ft. per sec, striking the earth with a velocity 
of 1000 ft. per sec. "What is the time of its descent? 

, V-Vo 1000-100 ^^^ . 

Ans. — /= - = ^TT-s- — = 28.0 seconds. 

g oZ.l 



(B). — Time, Velocity and Distance. 



h = -~(iV-hvo)=t 



(f ^ = 



i 2/1 

J = — — — = — 

V + Vq V 



+Vq 



V=- -vo; v=2 {^--vo) ; i;o= J - y. 






V + Vq V 



+ Vq 



T7 25 ^ ( S \ S V 

V =—-Vo\v=2 ( j-^0 I ; vq=-- — -y. 



Ex. 8. — At what height above the ground is the balloon in Ex. 7, pre- 
ceding ? 

Ans.— ;t = -|-(F + vo) = 14X1100= 15400 ft. 



(Q. — Velocity, Distance and Acceleration. 

_ V'^-vo^juiv-\-2vo)ju{2V~v) 
^~ 2s 2s 2s 

V^-vg^ ^ v{v+ 2t;o) ju{2V-v) 
2a 2a - • 



^ V'^-V(?ju{v+2vo) ^ v{2V-v) 
^ 2h 2h 2h ' 

, ^ V'^-vo'^ ^ v{v-{-2vo) ^ v{2V-v) 
2g ~ 2g 2g ' 

V=^2gh+v^\ vo = VV^-2gh; u = 



5 = 



2a 



V=\/2as+Vo'^; vq =VV^-2as; v = 



Vvo^+2as — Vo 



\^Vo^+2gh — VQ. ) 'vvo^-i-'zas — VQ. 

Ex. 9. — At a point one mile above the earth's surface a rifle ball is fired 
downward to the earth with an initial velocity of 2200 ft. per sec. With 
what velocity does it strike the earth? 

Ans.— F = V2g;t4-t;o2=\/2X32.2X5280+(2200)2=2276 ft. per sec. 



(D). — Distance, Acceleration and Time. 



^vo\ \^_'!lo 









Ex. 10. — In what length of time will a body travel 2000 ft., if the initial 
velocity is 20 ft. per sec, and the acceleration 10 ft. per sec? 



--'=Vc^)^^"-NO 



2^4000 20 ,., , 

+ -r^r r^ =18.1 SftCOndS, 



282 



Ib.^MECHANiCS. 



IV. Uniformly accelerated motion with negative initial velocity vp. — 

This is a case of uniformly varying motion in which the constant, initial 
velocity vq is opposite in direction to the accelerated velocity v pro- 
duced by some constant, acting force. Let v' = v — vo = t'he resultant 
velocity, in the direction of v at any time t measured from the instant v 
begins to act; h or 5 = the distance (algebraic) traveled in that time; g or a — 
the rate of acceleration per second. Then the following relations exist, 
neglecting friction: 



{A'). — Acceleration, Time and Velocity. 



i/z==i) — i)Q=gt — -UQ'^ v = gt; vo = gt — v'. 



t = 



v'-hvo V 



g = 



v'-^Vq 



v' = v — vo = at — vo; v = at\ VQ — at — v'. 
a a' t t * 



g g ~ . t 

Ex. 11. — From a point distant h (unknown) above the earth, a rock 
is thrown vertically upward with a velocity of 200 ft. per sec, and in falling 
strikes the earth with a velocity of 400 ft. per sec. What length of time 
is the stone in the air ? 

Vq 400+200 



Ans.- 



t = 



g 



32.2 



= 18.63 seconds. 






{B'). — ^TiME, Velocity and Distance. 
t_ 
2 






^ r t ^ . 25 



V' — Vq 



/ 2s^ ,25 o/5_i_ \ 

v'= — + vq\ vo = v' — -; v=2 I y+^o I 



Ex. 12. — In example 11, preceding, find the distance h of the starting 
point above the ground ? 

Ans.— k=Y(v'-vo) = ^^~^ (400- 200) = 1863 ft. 

Note. — From formulas (c) we find that the rock ascended h= -^ = — x 

2g 2g 

= 621 ft.; and then descended ;t = ^ =^-^^ = 2484 ft. See Ex. 13. illus- 



trating this. 



2g 2g 



{€'). — Velocity, Distance and Acceleration. 



g = 



vci' 



2h 



"~2j~ 



v' = V2gh-\-vo'^] vo=Vv'^-2gh. 



vo' 



Vq' 



2s 



2a 



V2as + Vq^ ; vq = \^v'^-2as. 



Ex. 13. — In Example 11, find the distance h of the starting point above 
the ground, using the acceleration instead of the time (Ex. 12)? 



Ans.- 



22 64.4 



(D'). — Distance. Acceleration and Time. 



=V(7)^-T-?- 



14.— In Exar 
und, using th 

;.— h = t (y-^o) =1863 ft. 



2 25 

a 



"=7 (7+"") • 



a ' 



- (!-) 



Ex. 14. — In Example 11, find the distance h pi the starting point above 
the ground, using the time, acceleration and initial velocity? 



Ans. 



UNIFORMLY ACCELERATED MOTION. 



283 



1. — Falling Bodies.* 
(/>== height of fall; / = time in seconds; ?; = final veloc. in ft. pel" sec.) 

.015547 ?;2 



v= gt =32.16/, 
= \/2g^= S.02\/h'. 



V 



t= — =.031095 V, 
S 

g 



^4 



^-Tg 



2 



16.08 f. 





Time 


Height 


i. 


Time 


Height 


6 
2,^ 


Time 


Height 


i. 


Time 


Height 


t 


h 


o '^ 


t 


h 


0) '^ 


t 


h 


^ 


t 


Ji 


> 






> 






> 






> 






.1 


.00311 


.00016 


42. 


1.3060 


27.425 


490. 


15.237 


3732.8 


1040. 


32.339 


16816. 


.2 


.00622 


.00062 


44. 


1.3682 


30.099 


500. 


15.547 


3886.7 


1060. 


32.961 


17469. 


.3 


.00933 


.00140 


46. 


1.4304 


32.897 


510. 


15.858 


4043.8 


1080. 


33.583 


18134, 


.4 


.01244 


.00249 


48. 


1.4926 


35.820 


520. 


16.169 


4203.9 


1100. 


34.204 


18812. 


.5 


.01555 


.00389 


50. 


1.5547 


38.867 


530. 


16.480 


4367.2 


1120. 


34.826 


19502. 


.6 


.01866 


.00560 


55. 


1.7102 


47.030 


540. 


16.791 


4533.5 


1140. 


35.448 


20205. 


.7 


.02177 


.00762 


60. 


1.8657 


55.969 


550. 


17.102 


4703.0 


1160. 


36.070 


20920, 


.8 


.02488 


.00995 


65. 


2.0212 


65.686 


560. 


17.413 


4875.5 


1180. 


36.692 


21648. 


.9 


.02799 


.01259 


70. 


2.1766 


76.180 


570. 


17.724 


5051.2 


1200. 


37.314 


22388. 


1.0 


.03109 


.01555 


75. 


2.3321 


87.452 


580. 


18.035 


5230.1 


1250. 


38.869 


' 24292. 


1.2 


.03731 


.02239 


80. 


2.4876 


99.501 


590. 


18.346 


5411.9 


1300. 


40.423 


26274. 


1.4 


.04353 


. 03047 


85. 


2.6431 


112.33 


600. 


18.657 


5596.9 


1350. 


41.978 


28334. 


1.6 


.04975 


.03980 


90. 


2.7985 


125.93 


610. 


18.968 


5785.0 


1400. 


43.533 


30472. 


1.8 


.05597 


.05037 


95. 


2.9540 


140.31 


620. 


19.279 


5976.3 


1450. 


45.088 


32688. 


2.0 


.06219 


.06219 


100. 


3.1095 


155.47 


630. 


19.590 


6170.6 


1500. 


46.642 


34981. 


2.25 


.06996 


.07871 


110. 


3.4204 


188.12 


640. 


19.901 


6368.0 


1550. 


48.197 


37352. 


2.50 


.07774 


.09717 


120. 


3.7314 


223.88 


650. 


20.212 


6568.6 


1600. 


49.752 


39800. 


2.75 


.08551 


.11757 


130. 


4.0423 


262.74 


660. 


20.523 


6772.3 


1650 


51.307 


42327. 


3.0 


.09328 


.13992 


140. 


4.3533 


304.72 


670. 


20.834 


6979.0 


1700. 


52.865 


44931. 


3.5 


.10883 


.19045 


150. 


4.6642 


349.81 


680. 


21.145 


7188.9 


1750. 


54.416 


47613. 


4.0 


.12438 


.24875 


160. 


4.9752 


398.00 


690. 


21.455 


7401.9 


1800. 


55.971 


50372. 


4.5 


.13993 


.31483 


170. 


5.2865 


449.31 


700. 


21.766 


7618.0 


1850. 


57.526 


53210. 


5.0 


.15547 


.38867 


180. 


5.5971 


503.72 


710. 


22.077 


7837.2 


1900. 


59.081 


56125. 


5.5 


.17102 


.47030 


190. 


5.9081 


561.25 


720. 


22.388 


8059.6 


1950. 


60.635 


59117. 


6.0 


.18657 


.55969 


200. 


6.2190 


621.88 


730. 


22.699 


8285.0 


2000. 


62.190 


62188. 


6.5 


.20212 


.65686 


210. 


6.5299 


685.62 


740. 


23.010 


8513.5 


2100. 


65.299 


68562. 


7.0 


.21766 


.76180 


220. 


6.8409 


752.47 


750. 


23.321 


8745.2 


2200. 


68.409 


75247. 


7.5 


.23321 


.87452 


230". 


7.1518 


822.44 


760. 


23.632 


8979.9 


2300. 


71.518 


82245. 


8.0 


.24876 .99501 


240. 


7.4628 


895.51 


770. 


23.943 


9217.8 


2400. 


74.628 


89551. 


8.5 


.26431 


1.1233 


250. 


7.7737 


971.69 


780. 


24.254 


9458.8 


2500. 


77.737 


97169. 


9.0 


.27985 


1.2593 


260. 


8.0847 


1051.0 


790. 


24.565 


9702.9 


2600. 


80.847 


105100. 


9.5 


.29540 


1.4031 


270. 


8.3956 


1133.4 


800. 


24.876 


9950.1 


2700. 


83.956 


113340. 


10.0 


.31095 


1.5547 


280. 


8.7066 


1218.9 


810. 


25.187 


10200. 


2800. 


87.066 


121890. 


11. 


.34204 


1.8812 


290. 


9.0175 


1307.5 


820. 


25.498 


10454. 


2900. 


90.175 


130750. 


12. 


.37314 


2.2388 


300. 


9.3285 


1399.2 


830. 


25.809 


10710. 


3000. 


93.285 


139920. 


13. 


.40423 


2.6274 


310. 


9.6394 


1494.1 


840. 


26.120 


10970. 


3200. 


99.504 


159200. 


14. 


.43533 


3.0472 


320. 


9.9504 


1592.0 


850. 


26.431 


11233. 


3400. 


105.72 


179720. 


15. 


.46642 


3.4981 


330. 


10.2613 


1693.1 


860. 


26.742 


11499. 


3600. 


111.94 


201490. 


16. 


.49752 


3.9800 


340. 


10.572 


1797.2 


870. 


27.053 


11768. 


3800. 


118.16 


224500. 


17. 


.52865 


4.4931 


350. 


10.883 


1904.5 


880. 


27.364 


12040. 


4000. 


124.38 


248750. 


18. 


.55971 


5.0372 


360. 


11.194 


2014.9 


890. 


27.675 


12315. 


4200. 


130.60 


274250. 


19. 


.59081 


5.6125 


370. 


11.505 


2128.4 


900. 


27.985 


12593. 


4400. 


136.82 


300990. 


20. 


.62190 


6.2188 


380. 


11.816 


2245.0 


910. 


28.296 


12874. 


4600. 


143.04 


328970. 


22. 


.68409 


7.5247 


390. 


12.127 


2364.7 


920. 


28.607 


13159. 


4800. 


149.26 


358200. 


24. 


.74628 


8.9551 


400. 


12.438 


2487.5 


930. 


28.918 


13447. 


5000. 


155.47 


388670. 


26. 


. 80847 


10.510 


410. 


12.749 


2613.5 


940. 


29.229 


13737. 


5500. 


171.02 


470300. 


28. 


. 87066 


12.189 


420. 


13.060 


2742.5 


950. 


29.540 


14031. 


6000. 


186.57 


559690. 


30. 


.93285 


13.992 


430. 


13.371 


2874.6 


960. 


29.851 


14328. 


6500. 


202.12 


656860. 


32. 


. 99504 


15.920 


440. 


13.682 


3009.9 


970. 


30.162 


14628. 


7000. 


217.66 


761800. 


34. 


1.0572 


17.972 


450. 


13.993 


3148.3 


980. 


30.473 


14931. 


7500. 


233.21 


874520. 


36. 


1.1194 


20.149 


460. 


14.304 


3289.7 


990. 


30.784 


15238. 


8000. 


248.76 


995010. 


38. 


1.1816 


22.450 


470. 


14.615 


3434.3 


1000. 


31.095 


15547. 


9000. 


279.85 


1259300. 


40. 


1.2438 


24.875 


480. 


14.926 


3582.0 


1020. 


31.717 


16175. 


10000. 


310.95 


1554700. 



* Ex. — A body falling from a height h of 398 ft. reaches the earth in 
6 seconds, attaining a -final velocity v oi 160 ft. per sec. If the body were 
shot vertically upward with an initial velocity of 160 ft. per sec. it would 
reach the starting point, 398 ft. above the earth, in 5 sec. Each motion 
would be just the inverse of the other. See, also, table on page 1155. 



284 IB.— MECHANICS. 

Summary op Preceding Motion Formulas. 
Notation: 
a =rate of acceleration in feet per second. 
g = gravity acceleration in feet per second. 

V == velocity at time t, in ft. per sec, due to acceleration only. 
Vo = constant or uniform or initial velocity in ft. per sec. 

V = u -f t^o = resultant velocity in ft. per sec. with initial velocity positive, 
t/ =v — Vo = Tes\i\tant velocity in ft. per sec. with initial velocity negative, 
s = direct distance in ft. from point of starting. (Used with a.) 

h = direct distance in ft. from point of starting. (Used with g.) 
t =time in seconds after starting. 

Formulas: 

I. Uniform motion; no acceleration. 

h s , ^ ^ u ^ s 

Vo==--=—. h=vot; s = vot. i= — = — . 

t t Vo Vo 

II. Uniformly accelerated motion; no initial velocity. 
, ^ 2h 2s ^/-^r-r ^/TT— , vt v^ gt^ vt v^ at^ 

g a V V ^^ g yj a' ^ t 2h /2 ' ^ t 2s t^' 

III. Uniformly accelerated motion; initial velocity positive. 

V = gt + Vo = at + vo = — vo= '^0 = \^2gh-\-VQ^= \/2as+ VqK 

t t 

Vo=V-gt=V-at = --^ = ^-^=VV^-2gh=VV^-2as. 
t ^ t ^ 

v=-gt = at = 2 I vo\ =2 (4— ^o) = Vvo^ + 2gh - u. = Vvq^ + 2as - vq. 

h=.-(V + vo) = ^^—=t (j+vo) ; s= j(V-hvo)==—^^=t [-2+^0) . 
V V-vo V^-vo^ 2 (h \ V V-vo V- -vo^ 2 1s \ 

IV. Uniformly accelerated motion; initial velocity negative. 



2}t , 2s 



v' = gt — Vo'=a't~vo — —r-\-Vo'= — -^VQ^ \/2gh + vo^= \/2as-i-Vo^. 
t t 

vo = gt-v'='at-v'=-v'-^=v'-^=Vv'^-2gh=Vv'^-2as. 

x; = g/ = o/=2 (y+^o) =2 /j+uo) = ^2gh + vq^ + vo= V2as + vq^ + vq. 

The resultant of two constant velocities in different directions is a 
motion in a straight line, and may be termed the parallelogram of motions 
or the triangle of motions. Let Vq and vq" represent two velocities of 
magnitude and direction shown in Figs. 13 and 14, and making any 
angle 6 with each other. Then will Vq be the resultant velocity both in 
magnitude and direction: 




Fig. 13. Fig. 14. 

From Trigonometry, W = vq ^ + vq"^ + 2z;o^t;o'' cos ; 
. •. Vo= \/vo"^ + vo*"^ + 2 W cos d. 
Note. — ^The polygon of motions is analogous to the polygon of forces 
(see Figs. 31 to 35). 



RESULTANT VELOCITIES, PARABOLIC MOTION, 



286 



^ Z Z 




Fig. 15. 

Parabolic Motion. — ^The path of a projectile, or of a jet from a nozzle, 
illustrates the resultant of a constant velocity vq in one direction, and a 
variable velocity v in another direction. Let x and y (Fig. 15) be the co- 
ordinates of any point p in the path of the resultant curve; then 
X =i'o^ cos ^ = horizontal distance traveled; 



y = t (t;osin e-^\ 



Vq Sin 6- 
X tan 6- 



vertical distance traveled 



gx' 



2vq^ cos 2 ^ ' 



dy 
dx 



tan 



gx 



Vo^ COS2 d 



nat tan of angle which the tangent to the curve 

d'V 
makes with the axis of X. Making ■;t^=0. we obtain the coordinates 

x' and y of the point p' at the vertex of the curve; thus, 
, vo^ sin d cos . , v^ sin 2 Q 

x = x'-= ; and :v = 3;' = -^^-7r . 

g ' -^ . 2g 

When p falls to the point p" on the axis of X, y = y'' = 0\ and x = x^ = 
2z;o2 sin d cos « « , , , . ... j. . >/ • 
. .*. xr = 2x^, or the horizontal distance from o to p is 

double that from o to the vertex p'. 

The general equation of the time t occupied in moving from o to any 
point p is 



/ ( vq si n ^\ 2 2 y 7^0 sin ^ 



T • r X xt. . / / 1 . t'o'^sinS^X 
In moving from o to the vertex ^ , (making y = — ^ I 



t = 



Vq sin ^ 
g 



In moving from o to p" on the axis of X, (making y = 0), 

2z;o sin 6 



The above discussion is a general case of which the following are special 
cases: 

Case 1. Initial velocity horizontal {6 = 0).— 

x = Vot. y=--Y' 

Case 2. Initial velocity vertical (6== 90°). — 

x=0. y = t (^o-y). 

(The last is a case of Uniformly Retarded Motion, being the reverse of 
"Uniformly Accelerated Motion with negative initial velocity," preceding. 
Note that y corresponds with h of the preceding Motion formulas; and 
that when y becomes a minus quantity, p is below the starting point.) 

When the value of :v is a minus quantity, in any of the above cases, it 
shows that the projectile p has fallen below the axis of X. 



386 



15.— MECHANICS. 



Circular Motion. — Let p he a. point on the rim 
of a fly-wheel of radius r, and revolying at n revo- 
lutions per unit of time. Then the velocity v of the 
point p is 

v = 27zrn. 

(Note that v takes the compound denomination 
of r and n\ i.e., if r= rad in ft., and w = rev per sec, 
then i; = veloc in ft. per sec.) 

Ex. 15. — A driving rope traveling at the rate of 
300 ft. per min runs over a pulley 4 ft. in diam. 
How many revolutions does the pulley make per min? 

Ans. — n = -x — = -7— =23.87 rev per min. Ans. 

2;rr in 



Motion on Inclined Plane. — Neglecting 
friction, bodies falling from A would 
reach a (on inclined plane Aa), b (on 
inclined plane Ab), and H (distant 2r ver- 
tically below) in the same length of time t. 
Thus. 

'2h 




Fig. 16. 



from general formula, t= ^ ~ for AH ; 



as h = —. > 

sm a 



ash = -. — r, 



siniS' 



-V 



2/ 



g sm a 



21' 



for Aa\ 



for Ab. 





^"^^^ 




SK^~~«fe^Q 









\ /^ 




\ / 


J. 


Yx^- 


(U 


/\ 


It 

X 


J.l. 




4 J;/ 







~Y-^b 



H 



^^^^^ Fig. 17. 

Moreover, the velocities at the same elevation would be equal; thus, the 
velocity of the body at a, descending on the inclined plane Aa, would be 
equal to the velocity of the body at a' , falling freely through space. Simi- 
larly, the velocities at b and b' would be equal. 

The following formulas relate to the inclined plane Aa, making an 
angle a with the horizontal: 

v = gism a =v2g/ sm a. 1= r = -r ; . 

2 2g sm a 



g sin a Y 



2/ 



g sm a 



The time occupied in the descent of bodies down inclined planes of the 
same height varies as the length of the planes. Thus, let Aia = 2 X Aa = 
2/; then the time occupied in falling from Ax to a would be double that 
occupied in falling from A to a. 



Motion on Cycloidal Curve. — The 

cycloid (Fig. 18) is often called the 
curve of quickest descent. Let ACB 
be the cycloidal curve (of length id) 
generated by a circle of diam d rolled 
along the plane AB. Then will a body 
starting at A fall to C and to B quicker 
than by any other curve. (For Proper- 
ties of the Cycloid, see page 236.) 




Fig. 18. 



Time occupied in reaching C'. 
Time occupied in reaching B: 



-'4t- i-f- 



By cy- 
cloidal 
curve. 



MOTION. CYCLOIDAL CURVE. PENDULUM. 287 

Time occupied in descent from A to C on the inclined plane AC, is 

t =J — ^ — = 1.862^ /— . as against t = 1.5708^ /— for the cycloid. (See 
^g sm a \ g \ g 

also Simple Cycloidal Pendulum, below.) 

Simple Circular Pendulum. — This consists of a mass suspended from 
a fixed point O by a thread, of length /, considered O 

as haying no weight. The length of time t for a sin- 
gle vibration from A to 5 is 

/ = ;r^/— (1+5-7) (Nearly exact.) / 



-vK-a 



t = Tz^\ — (Approximate but most frequently used; 

^g =— 

practically exact when a does not exceed 2° 30'). Fig. 19. 

From the second equation we get, / = -5 : g=~-7,. The length of a 

a 

pendulum which will give a single vibration in 1 second of time is /=-^ ft. == 

0.101321 g ft. The value of g for any locality may be obtained from the 
formula* — 



g= 32.16954 (1-0.00284 cos 2A) (l -^) ft. per 



sec. 



in which A = latitude of the place, 

h = its elevation in feet above sea level, 
R = radius in feet of the earth at the latitude A, 
= 20 887 510 (1 + 0.00164 cos 2A), 
= 20 900 000 ft., approximately. 
The value of g in England is 32.2 ft.; in New York City^32.17. The 
value of g usually assumed in the U. S. is 32.16; and of \/2g, is 8.02, in 
hydraulics. The length of a pendulum which will make a single vibration 
in one second is therefore /= 0. 101321 g = 0. 101321 X 32 . 16= 3.258 ft. = 
39.1 ins., or a little less than 1 meter (=3.28 ft. = 39.37 ins.). 

The compound circular pendulum consists of a pendulum in which 
the mass is more or less distributed, instead of being concentrated at a single 
point as above (Fig. 19). It may be reduced to the simple circular pen- 
dulum by finding the distance / from O to the center of oscillation, otherwise 
termed the center of percussion. Assuming the whole pendulum to be 
rigid, the center of percussion is such a point that when struck sharply at 
right angle to the direction of the pendulum the latter will begin to oscillate 
of vibrate as a simple pendulum without producing any shock at its upper 
end or axis O. If the pendulum is a rod of uniform cross-section, and 
homogeneous, the center of percussionf will be a point distant /=f of the 
length of the rod from O. 

Simple Cycloidal Pendulum.^ Let Op ( = 2d). Fig. 18, be the length 
of the pendulum vibrating between the cycloidal arcs OA and OB. Then 
will Op be the evolute of the cycloid ACB, and the mass p will trace the 
cycloidal curve. Hence the motion of the mass p will be the same whether 
it is swinging from O or simply rolling freely on ACB, friction neglected. 
Moreover, for any given cycloid the time of descent from any point on the 

curve will be ^ = 7r^ /tt— ; and the time of one oscillation, t =x^ / — . In order 

\2g y g 

a 

that the time of one oscillation may be one second, make 2ci= — r = . 101321 

g\ .'.the length of the cycloidal pendulum, = 2fi, will be the same length 
as the Simple Pendulum, preceding. The cycloidal pendulum is exact for 
any angle of vibration, while the simple pendulum is practically correct 
only for small arcs if the formula is adhered to strictly. 

* See also formula for g under Gravity Acceleration, Weights and Specific 
Gravities of Materials (Section 27) . 

t See formula for Center of Percussion, page 303. 



288 



15.— MECHANICS, 

DYNAMICS. 
Work, Power, Energy, Etc. 



s 



Force. — In the preceding equations of motion, the mass of the moving 
body is not considered — simply abstract motions. When, however, the 
moving body is considered as having mass, it will take on the conception 
of force ( = mass X acceleration) , work (== force X distance) , power ( = rate 
of work = work -4- time), and energy ( = capacity for work); also " impulse " 
( = force X time) , and " momentum " ( = mass X velocity) , — remembering that 
the momentum imparted to a body is equal to the impulse which produces 
it. ("Impact" or "collision" is a blow, or pressure of such short duration 
that it cannot be measured, between two bodies. See page 303.) 

Considering " Force " i^ to be a constant, unbalanced, resultant force 
acting on a total mass M of weight W during the time t, the above relations 
may be summarized as follows: 

(a). Fundamental Relations; Resultant Force F Constant. 

(Distance not included.) 

Impulse = force X time = mass X velocity = momentum 

weight X velocity „ 



Therefore, 
Mass 



M- 



(Force-) Accel 

Weight W = Mg 
(Grarity-) Accel 

Force 



in ft. per sec, 



Time 



Velocity 



F = 



t = 



El = H 

V a 

_F Fg 

M ^ W ' 

Fg ^ 

a 

W _Wa 

M F 

Mv _ Wa 

t " g 

Mv _ Wv 

F ~ Fg 

Ft Fgt ^ . . 

-T7 = -TtT = at m ft. per sec. 



W 
g 

V 

T 

Fgt 

V 

Wv 

Ft 

Wv 

— — in pounds (7) 

gt 



in pounds. . . 
in ft. per sec. 



(1) 
(2) 

(3) 
(4) 
(5) 
(6) 



in seconds. 



(8) 



(9) 



Remarks. — ^The above formulas are for problems in which the force, 
and therefore acceleration, is constant. Equated values may be substi- 
tuted from one formula in another. 

Problem 1. — A railroad train weighing 600 tons, 5 minutes after starting 
attains a velocity of 30 miles per hour. Find the tractive force of the 
engine drawing the train, the resistance being 8 lbs. per ton.? 

Solution. — Tractive force T= unbalanced force F + resistance R. 
n^ X 1 /'TN TT ^^ 600X2000X30X5280 __„ „ , _, 

From formula (7). F^ — == 3216X5X60X3600 ^^^^^ ^^'■' ^^^ ^= 



600X8 = 4800 lbs.; .-. r=F+i? = 5473+ 4800= 10273 lbs. 

Atwood's Machine. 

Problem 2. — Atwood's machine consists of a flexible 
cord passing over a (frictionless) pulley and supporting 
equal weights at each end, with provision for an "un- 
balanced" weight at either end. In Fig. 20, the bal- 
anced weights are 5 lbs., and the unbalanced weight 
is 2 lbs. Find the acceleration of the weights, and the 
tension on the cord ? 

Fg_ 
W 



Ans. 



Solution. — From formula (4), acceleration a 
2X 32.2 



a F 2 
^- =5.37 ft. per sec; and — = 7^ = 77: = i. In con- 
Iz g w iz 

sidering the tension T on the cord, we have to consider 
the weight W' whose mass is M' and acceleration a 




F=t* Fig. 20. 



DYNAMICS— FORCE^GENERAL EQUATIONS. 



289 



as above determined; thus, T -= F' 4- H^' == (formula 7) -^+^^' = 1+5= 
5f lbs. Ans. 

(Note that W in Problem 2 corresponds with the resistance R in Prob- 
lem 1, preceding.) 

Problem 3. — Find the acceleration, and tension 
in the rope. Fig. 21, assuming no friction? (Note 
the similarity to Atwood's Machine, Fig. 20.) 



So lution . — Acceleration a 
2X32.2 



(formula 7) 



Fg 
W 



12 



5.37 ft. 



and — 
g 



i. As there is no 




friction there is no resistance, hence the tension Fig. 21. 

in the rope = r = F' 4- i?(i? = 0)=F' = (formula 7) ^-j 



|?ib&1 
If lbs. Ans. 



(Note that of the total force F (==2 lbs.), — or i of it is employed in 

moving its own mass, and the balance or § (== If lbs.) is employed in moving 
the larger mass. Hence the tension in the rope, as above.) 

In both of the above examples (2 and 3) the velocity at the end of say 
10 seconds (formula 9) =T; = a^=5.37X 10=53.7 ft. per sec. 

(b). General Relations; Resultant Force F Constant. 

(Distance included.) 



(10) 
(11) 
(12) 
(13) 
(14) 
(15) 
(16) 
(17) 
(18) 
(19) 

(20) 



Impulse 


Z 


= Ft = 


=Mat = = 

g 


_2E_^2Pt_2K 
at V at ^ ^ 


' =the re- " 
suit if to- 
tal force 


Momentum 


N- 


= Mv = 


Wv Fv 
g a 


2E 2P 2K ' 
' V a V ^ 


were ex- 
pended 
in one 
second. 


Mass 


M 


W 
g 


F K 
a as 


Pt E Z N 
as as at v 


Accel (ft. per sec 


) a 


F 
M 


Fg V 

' W ~ t ~ 


2s v^ Kg 2P 2P 2P 

t^ 2s Ws Ft Z N " 


Weight (lbs.) 


W 


=Mg = 


Fg Kg 
a as 


Pgt Zg Ng 
as at V 


Accel (ft. per sec 
Force (lbs.) 


) F ■■ 


W 


Wa 




F 


M 
= Ma = 


F 

Wa Wv 
g gt 


2Ws ,K Pt E Z Na 

gt"^ S S S t V " 


Time (seconds) 


t 


V 

a 


2s \2s 


Wvs 2K K Fs 2P Z 
Kg Fv P P Fa F'" 


Velocity (ft . p. sec) v 


= at = 


= 'I -V2a. 


- 2K 2P 2E Na 2K 

Ft F Z F N "" 


Distance (ft.) 


s 


vt 
2 


v^ at^ 
2a 2 


K Pt E 

F F F 


Work (ft.-lbs) 


K 


= Fs = 


Was Wvs 
g gt 


2Ws^ Fvt Fv^ Fat^ 
gf^ 2 2a 2 






Zat 
2 


=^^( = F) 







290 



15.— MECHANICS, 



K 



Power g.i^=;)p-^ 



Energy (ft.-lbs.) E 



' t 

Za^Nv 
2 ~ 2t " 

2 2g 



Was 
gt 



Wvs 
gt' 



2Wf 
gfi 



Fv 
" 2 " 



2at ' 



Fat 
2 



(21) 



Fv^ 
' 2a ^ 



■- same values as f or i^ , above (=K)(22) 



(Remarks. — ^The above formulas are for problems in which the force, 
and therefore acceleration, is constant. Equated values may be substi- 
tuted from one formula in another.) 



Problem 4. — How much energy is expended in raising a weight of 800 
lbs.. 10 ft.? 



Solution. — From formula (20), 
of energy or work. Ans. 



£ = 2^=775=800X10=8000 ft.-lbs. 



Problem 5. — A weight of 4000 lbs. is raised 100 ft. 
in 5 seconds. Find the tension in the hoisting rope, 
the acceleration being uniform? (See Fig. 22.) 

Solution. — ^Tension T = unbalanced force F + the re- 

7P / T;r/^ rf i ^a^^^^^J;fr 4000X200 

sistance R ( = W) = (formula 1 6) — — -hW = 

+ 4000=994+4000=4994 lbs. 



gt' 
Ans. 



32.2 X 25 




Fig. 22. 



4000* 



Work. — ^Work is the transformation of energy, and is simply force 
(F)X distance (s). The foot-pound (one pound raised one foot high, or 
its equivalent) is generally considered the unit of work unless other units 
are specified. The element of time is not a factor. 

The Fundamental Formulas for Work are — 



When the force F is constant, Work (K) =Fs. . . . 

When the force F is uniformly variable, Work (K) = I Fds 

J So 
In which Si and Sq are the limiting values of 5, in feet. 



(23) 
(24) 



Problem 6. — A rope weighing 6 lbs. per lin. ft. and 
400-ft. long, is suspended at one end from a drum. 
Find the least number of ft.-lbs. of work required to 
wind up 100-ft. of it? 

Solution. — Let s (Fig. 23) be the variable length of 
the hanging rope; then 65 will be its weight, and Qs ds 
the. work done in raising it through the distance ds. 
Using formula (24), the total work done in raising 
the rope through 100-ft. of distance is 



/•5j /•400 
K= \Fds'= I Qsds= 
J So J 300 



p400 
^= 3(4002-3002) 



2 
300 
= 210 000 ft.-lbs. 



Ans. 



This is represented graphically in Fig. 24, 
the vertical ordinate F being the force applied 
for any length s of the hanging rope, and the 
shaded area representing the total least work 
performed. Thus, to wind up the whole rope the 
least amount of work required would be (mak- 
ing 50 = 0) 4- • 5i2 = 3 X 4002 = 480 000 ft.-lbs. = area 
whole triangle. 




_J 



Fig. 23. 



dSn^ 




Fig. 24. 



POWER. LEVERAGE. 291 

Power. — Power is the rate of work, that is, it is the amount of work 
done in a unit of time. It bears the same relation to work that velocity- 
bears to motion: 

_, work . ^ - .^ space traveled 

Power = —: , 3ust as velocity = —. — . 

time time 

The unit of power may be: 

(a). The foot-pound-second unit =1 lb. raised 1 ft. in 1 sec. 

(b). The foot-pound-minute unit = 1 lb. raised 1 ft. in 1 min. 

/'^^ n^u^ v.^^c^ r^^-r.r««- ,-.*^u I = ^50 foot-pounds per sec. 

(c) . The horse-power unit | _ 33000 foot-pounds per min. 

or any other measure of work per unit of time. 

The formula for power ^ with the force F constant, is — 

_ „ force X space Fs .._. 

P°^«'-^= itei^ = T ' (25) 

Problem 7. — A locomotive draws a load of 600 tons at the rate of 20 
miles per hour, the resistance being 8 lbs. per ton. Find the horse-power? 

o 1 .• r, Fs 8X600X20X5280 . ., 

Solution. — P=—'= — ft. -lbs. per mm. 

t bO 

8X600X20X5280^ 
= 60X33000 horsepower. 

= 256 h. p. Ans. 

Leverage. — As power (-— j is the rate of work Fs, it is possible through 

mechanism and appliances to accomplish a great amount of work with 
comparatively little power, by increasing the time t of performing the 
work; and vice versa. That the ancients understood this is evidenced by 
their use of the lever, inclined plane, wedge, screw and pulley, in raising 
immense weights W, greater than any available direct force F. All prob- 
lems of this character really come under the principle of leverage, simple 
or compound. 

Notation. 

Let Fi = initial force in lbs. applied to first lever, 

5i = distance in ft. that Fi, or the initial lever end, moves in time t, 

t =time in seconds, 

F\ F^, F'", etc. = intermediate forces in lbs. at successive joints of com- 
pound lever, 

s', s", s'" , etc. = distances in ft. the intermediate joints move in time t, 

F = resultant force in lbs. at end of last lever = weight of mass raised. 

5 = distance in ft. that F, or the final lever end, moves in time t. 

General Formulas. 

Ft Si Fs 

Simple lever formulas: Power P = -~ = -— ; Work K = F^si = Fs (26) 

t t 

jFiSi F's' F'^s" F"'s"' Fs 

Compound lever formulas : Power P = — - — = — — = — — - = — - — = etc . == (27) 

J, I i t, t 

Work X =Fi5i=FV=FV=F'V"=etc.=F5 (28) 

The above formulas neglect friction or loss of energy in the mechanism; 
that is, they assume that the total power and work applied to the mechan- 
ism are transmitted into useful work. 

Simple Lever. — The force Ft re- IF, --""T 

quired to raise the weight W (Fig. 1 \, ^ ^^""'^ 1 ^ 

25) i^s, from formula (26). F, = \ ...-'' Fulcrum ^ If-'W 



Fs Ws Wl / . I s\ 
— = = -7-. (since -7-=— I 



Fig. 25. 



292 



15.— MECHANICS, 



Compound Lever. — Let Fi be the 
force required to raise W. Fi and 
W may be equated with F' acting 
at the joint of the two levers. Thus, 



Fi5i = FV = Fs; 



. . , Silt si 

hut s' = -rr=^: 




li + li IS a ben+ Lever 
V+l is a straight Lever 



1? i 



V 



>♦ & 



*• 51 //i" --^^ ^5i ^ Ik'' 



^F-W 



Fig. 26. 



Note that the acting force multiplied by the continued product of 
the left-hand lever arms = the weight raised multiplied by the continued 
product of the right-hand lever arms. This principle holds true in 
belting and gearing also. 




Fig. 27. 

Inclined Plane. — Let'PT be a weight to be moved up an inclined plane 
of length Sx and height 5, by the force Fi. Then FiSt = Ws, or Fi = 

W — , in which — = sin a. 

Sx Si 

Wedge. — ^The wedge is a double inclined plane. 



Screw. — ^The screw is an inclined plane wound around a cylinder. The 

77" c TTc 

formula —j^ = — applies; in which Fi is the acting force at the end of a 

lever attached to the screw; 5i= distance traversed by the lever end in the 
time /; F = the force acted against by the screw; and s the progressive dis- 
tance traversed by the screw in the time t. Or, we may use the formulas 

FiJ(27r/)24- {^y=FP (Exact) (29) 

Fi(2;r/) =Fp (Approx.) (30) 

in which / = length of lever from cen of screw to applied force Fi, 
and ^ = pitch of screw, of dtam d. 



Pulley. — From formula (26), FiSi = Ws, in which W 
moves the distance 5 when Fi moves the distance Sx. 
In Fig. 28 it is evident that 5i = 2s; hence, 

Another method of determining the forces in pulley 
ropes is by " cutting sections." Thus, imagine all the 
forces to be in equilibrium, and draw a cutting plane 
ah. Then it follows that each of the two ropes cut, sup- 
ports half the weight W, and the cutting plane ch shows 

W 
that Fx has the same stress — . 

Simple and compound pulleys are thus often more 
easily analyzed by the " method of sections," although 
the principle of " work " applies universally. 




IMPULSE AND MOMENTUM. ENERGY. 



293 




Toggle. — If the weight W is raised by the force Fi, use 

formula (26); thus. FiSt = Ws, or Fi=WJ-=Wj. (It is to 

be noted that the required force Fi gradually decreases as 
b decreases, and that an almost infinite force W may be 
raised by a finite force Ft when b approaches zero.) 

Proof: Let Fi force the toggle joint to the right the 
infinitesimal distance Si*, then W will be raised the infini- 

tesimal distance s, that is, -»- for each lever arm /. Now 
as / remains constant, we have, before the movement, 
P = b^ + h^; and after the movement, P = (b-Si)^ -h (h-h-^j . Equa- 
ting. b^-\-h^==b^-2bsi-hSi^-\-h'^+hs + j-, Remembering that the infini- 

52 s 2b 

tesimal quantities ^i^ and-T-=0, this reduces to hs = 2bsi, or— = -r-. The 

same result may be obtained by the parallelogram of forces indicated in 
Fig. 29. 



Fig. 29. 



Impulse and Momentum. — If a constant force F acts for a length of 
time ^ on a Mass M, the latter will acquire a velocity v at the end of that 
time. Furthermore, the product Ft, called impulse, is equal to the product 
Mv, called momentum; or 

(Impulse, Z = ) Ft'=Mv (== Momentum, iV). 

The acting force F (lbs.) may be any amount acting through any length 
of time / (sec), or it may be a force i-times as large as F acting for one 
second on the mass; in either case the velocity v of the mass at the end of 
that time (t seconds in the first case and one second in the last case) will 
be the same, and therefore the momentum will be the same. Hence the 



momentum of a mass M 



f) 



force which in one second will give it that velocity; or, 
amount of force which in one second will bring it to rest. 

If the acting force is uniformly variable instead of constant 
the equation 



moving with a velocity v is equal to that 

inversely, the 
we have 



fMdv= fl 



Fdt or M (t^i— 1^0) 



■/> 



in which vo =the velocity when t equals to, 
aiid_z;i =the velocity when t equals ti. 



Energy. — Energy is capacity for work. The energy of a body is meas- 
ured by the amount of work which it is capable of performing, called poten- 
tial energy; or by the amount of work it does perform, called 
kinetic energy, -o 

For illustration, a cannon ball weighing 60 lbs. is 
raised from A to B, a. distance of 20 feet. The work per- JfcIs 



formed in raising it is, therefore 



/f..= 



F5= 60X20= 1200 



to pos- /• , 
a potential 7^'^^'' 



ft. -lbs. The ball at B may now be considered 

sess, by virtue of its position with reference to A 

(or statical) energy equivalent to 1200 ft. -lbs.; and which, 

if the ball is allowed to fall back to A, will be entirely 

expended in the form of kinetic (or actual) energy when it 

reaches A and its velocity is destroyed. 



Fig. 30. 



294 



15.— MECHANICS. 



From the preceding equation, lFdt= IMdv, multiplying both sides 
of the equation by v, and remembering that vdt = ds, there is obtained, 



Work in as- 1 f = Kinetic en- 

cent stored /• /• i; ^ at A ergy in des- 

up at 5 as \ I Fds= I Mvdv ■ cent expen- 

potential en- J Jo&tB ded at A 

ergy= J 

Or, the general formula, lFds=h M W-vq^) (31) 

If the initial velocity vq =0, then Vi=v and we have, 

(32) 



Energy £; = iMz;2=-!y^ 



Or, as ;i=|^ (lie). 



£=l^;t = Fs = workK. 



(33) 



That is, the energy expended = the work performed. 



Problem 8. — How much energy is expended in shooting a cannon ball 
weighing 50 lbs. with an initial velocity of 2300 ft. per sec? 

o w ^ ^ 1 /ooN • ,. ^^. ^^' 50X5290 000 

Solution. — From formula (32) energy m ft.-lbs. = -j^ — = — o\/ qo i a ^ 

4,112,250 ft.-lbs. Ans. 

If fired vertically it would ascend, from formula (19), 

j^___E E 4 112 250 

"' F W 50 



82 245 ft. 



RESULTANT FORCES. 

Composition and Resolution of Forces. — On page 284 are explained the 
parallelogram and triangle of uniform motions composed of two constant 
initial velocities Vq and vo\ producing the resultant initial velocity Vq. 
From Mechanics we learn that: Forces are proportional to the velocities which 
they will impart to a given body in a unit of time. Hence, if F^ imparts a 
velocity t;o' . and F2 a velocity vq", then will R, the resultant of F^ and F2 
(Figs. 31 and 32) of the parallelogram or triangle of forces, correspond 
with Vo.the resultant of vq and vq" (Figs. 13 and 14) of the parallelogram 
or triangle of motions. Thus, Fig. 31, 
i^ = Fi2 + F22 4- 2 F1F2 cos d, 

otR= \/Fi2 + F22+2 F1F2 cos .d. 
When d =90 °. cos ^ = 0, and 

i? = VFi2+F22. 






Fig. 32. 



Fig. 33. 



Equilibrium. — If, in Fig 32, the resultant R, of the two forces Ft 
and F2, is replaced by a third force F3 equal in magnitude and opposite in 
direction to R, then will the triangle represent a closed triangle of 
forces, or forces which if allowed to act at any common point P (Fig. 33) 
will be in equilibrium. Clearly, any one of the forces of a closed tri- 
angle of forces is equal and opposite to the resultant of the other two. 



FORCE POLYGON. MOMENTS AND REACTIONS. 



295 





Fig. 34. 



Fig. 35. 



Polygon of Forces. — It is evident, from the preceding demonstration, 
that any number of forces as Fi, F2, Fs, F4, and F5, acting in equilib- 
rium at a common point P, Fig. 34, will, if drawn consecutively in ro- 
tation in the direction of the forces, form a closed polygon of forces, Fig. 
35. For any polygon may be cut into triangles by the diagonal lines Ri, 
R2, etc., each diagonal being the resultant of two other forces, and consid- 
ered as replacing them. Thus Ri replaces Fi and F2 so that we may con- 
sider only four forces acting, namely, Ri, F3, F4 and F5; again, /?2 replaces 
Ri i==Fi and F2) and F3 so only three forces remain, namely, R2, F^ 
and F5 — a triangle of forces. Hence, as R2, F4 and F5 are in equilib- 
rium, so are all the forces composing the sides of the polygon in equilib- 
rium. The principle of the force polygon, representing static equilibrium 
of forces, is fundamental to Graphical Statics , in the determination of 
stresses in structures. 



33V 



50' 



W=60' 









■OB 



Moments and Reactions. — Let AB 

be a horizontal lever 50-ft. long, fixed 
at, and free to rotate around, its left 

hand end A (Fig. 36). If now a^o- j-— 

weight P7=60# is applied vertically 50' 

downward at a point d3^-it. distant 

from A, tending to cause the rod to i'lg* 3o. 

rotate around A in a right hand mo- 
tion, it is evident that some force R2 

applied at the other end B of the Ar-r 

lever will preserve equilibrium or pre- fp _-,^"^ 

vent rotation. The value of R2 is ob- ^rT^ 

tained by taking moments M about A, 

as origin; thus, Fig. 37. 

IM = 0: mW = 50R2; .'. i?2 = f ^ = 40#. 

In a similar manner, if the lever is considered as fixed at, and free to 
rotate around, B (Fig. 37), we have IM = 0: UW=50Ri; .•.i?i = iM^=20#. 

Fig. 38 is the combined result of 
the two operations above, considering W=i60 

the lever AB as a beam 50-ft. long sup- 33Vj I |^V 

porting the weight W= 60#, produc- ^Jf. jr et^t — ^ 

ing a reaction of 20# at A and 40# I Rj=ZO^ 

at B. The weight of the beam itself 

is not considered. pig, 33, 

Center of Gravity and Resultant of a System of Parallel Forces. — Ihe 

line of the resultant of a system of parallel forces passes through the cente? 

of gravity oi the system. In Fig. 39, , 

let Pi ( = 10#), P2 ( = 20#) and P3 \ 

( = 30#) be a system of parallel forces I 

or loads acting on a rigid body AB, at p~?h^ 

points 25 ft. apart. Then will R ^"^ 

( = 60#) acting at the point O, be the I 

resultant of the above forces and J^ 

capable of replachig them in certain ^ Xo-33J^- 

problems where the reactions are 

to be determined. In Alligation, page 1 

57, we have a similar problem, that >b 

is, finding the average cost of a Fig. 39. 



50' 



1^- 



B 



E»40 



•25 



E5' 



Px='50'' 



f 



R=60' 



296 



15.— MECHANICS. 



mixture where oarreis of cement replace pounds, and cents replace feet, 

of the present problem; and in which the origin of moments is 200 to the 

left of J4. As a matter of fact, the origin of moments may be at any point, 

but for convenience it will be assumed at A in the present case. Thus, 

taking moments about A, of the loads Pi, P2 and P3, we have — 

I Px 
Distance to center or gravity of resultant = ico = „ ; 

I Px — sum of moments of all the forces about center of moments A ; 
I P =sum of all the forces acting = i?; whence, 
IPx IM ^^^ ^ ^ IM 



IP R 
Taking moments about A as origin: 

I M 



xq 



?: 

fz 



P1 + P2 + P3 
R 



10 X C 

20 X 5 

30 X I 
60 

IPXXQ 





500 

1500 

2000 

IM 

i-p"" 60 -^^*"- 



resultant of a 



distributed 
C 



P=100' 



Therefore, i?=60 1bs.; axid xq = 

Note the similarity of analysis of this problem of concentrated forces and 
the following problem of a distributed force. 

Resultant of a Distributed Force. — ^The 
force is equal to the total force, acting in 
a line passing through its center of grav- 
ity. We have seen (Fig. 39), that the 
horizontal distance Xq to the line of the 
resultant of any system of forces, from 
any point taken as the origin of moments, 
is equal to the sum of the moments of 
the forces about that point, divided by 
the sum of the forces. The same prin- 
ciple _ holds true with a distributed force 
as with a system of forces or concen- 
trated loads. Fig. 40. 

Problem 9. — "L&tAB, Fig. 40, be a girder of 60 ft. span, 5, loaded with a 
distributed force whose intensity, at any point distant x from the left abut- 
•ment, is the ordinate y from the line AB to the line AC', that is, the intensity 
of the force increases uniformly from lbs. at A, to 100 lbs. per lin. ft. at 

j5, so that y = lbs. Ist, find the general equation of the resultant i?, 

s 

and its distance ocq from the left abutment A\ 2nd, find R and xq for the 
span completely loaded from A to B\ Srd, find R and xq for the span loaded 
from 6 = 20' to a =50'. 

o , .. ^ . I Px summation of :t .T(i% forces 2M 

Solution. — 1st. xo= -=r-pr = — -. 7 — j— 7 =~^r~» 

I P summation of ydx forces K 




X-(^)" »/;* ^['s 



Xo 



s: 



100 



In which 



s 



X 



dx 
100 



lOoT^ 100 [^x^ 



a3 


-63 


3 


a2 


-fc2 



= !x 



a2-62* 



a^-bK. 100 



fX 



a2-62 



= sum of the moments of all the infinitesimal forces 

xydx = lM', 
= sum of all the infinitesimal forces ydx'^R; 

= distance from A to center of gravity of the infini- 
tesimal f orces = ico ; 
a = upper limit of any assigned value to x; 
b = lower limit of any assigned value to x, j 



DISTRIBUTED AND CENTRlPUGAL FORCES. 



297 



2nd. For the span completely loaded from A to B, the upper limit of 
x = a=W, and the lower limit of x = b^Q. Substituting these values 
in the general equation, we have, 



^2-62 100 



^^^^= 3000 lbs. 



Xq==1 



a3-63 



fa=40ft. 



a2-62 

3rd. For a span loaded between x = a = 50\ and x = b = 20't we have, 

63 , 117000 



a2-fe2 100 2100^100 ,-__., , o3 



= 37?^ ft. 



62 -» • 2100 

In the same manner we may find the resultant and center of gravity 
of any distributed force, as parabolic, circular, elliptical, etc., by substi- 
tuting the value of y in the equation of the curve before summing up all 
the moments of the ydx forces for lPx = lM, and all the ydx forces for 
IP = R. Note that the resultant R = the area of the curve, and the distance 
xo = the distance to its center of gravity. 



Centrifugal Force. — Centrifugal force is that component of a force 
acting radially outward when a body moves along a curve with a certain 
velocity. In the general equatibn (16) of force, F = Ma, the acceleration 

a in the present instance is — in which v = velocity of the moving body in 

feet per second, and r = radius of curve in feet. Hence, 



Centrifugal force 



r 



g r ' 



(34) 



Problem 10. — Find the tension in a string 10 ft. long, fixed at one end, 
with a weight of 40 lbs. fastened at the other and revolving around a circle, 
with the string as a radius, at a velocity of 30 revolutions per minute ? 

Solution. — In formula (34) Fc is the tension in lbs. in the string; and 

30 
V, in feet per second, is ^ . 2;rr, or v^==7:^ r^. 



Therefore, tension, Fc- 



40 



32.16 
40X9.87X10 



= 122.8 lbs. 



32.16 

Note that in the above case, Fc is a concentrated resultant force acting 
normal to the curve at one point, and r ( = radius of curve) is the distance 
from center of curve to cen of grav of moving body. Compare above prob- 
lem with the two following. 

Problem 11. — Find the radial pressure which a train weighing 1000 
tons and moving 20 miles per hour, will produce on a track laid on a 7 
degree curve. 

Solution. — ^The radius of a 7° curve is 819 ft. Use formula (34): 



Fc = — . — 

g T 



lOOOX 



60X60 / 



tons 



32. 2X 819 

= 32.63 tons =65300 lbs. Ans. 
Note that in the above case the force Fc is 
a distributed force acting normal to the curve, 
throughout the length of the train. 

Problem 12. — Given a ring (Fig. 41) 4 ft. in 
diameter, weighing 100 lbs., and revolving 
about its axis at a speed of 1000 revolutions 
per minute. Find the tension stress in the ring ? 

Solution. — [Note. — Distance of center of 
gravity of the semi-circular ring from center of 

• 1 2r Diam 4 , ^^., , 

circle = ro = —= =— (see page 207).] 




'298 



15.— MECHANICS, 



Consider half the weight of the ring acting at each center of gravity 
of the semi-circle on the axis of X, causing tension at points a and b on the 
axis of Y. Then the total stress at a and b, from formula (34) is 

p^^iu, . :2L. in which { vq =2;rro» = *R^ 



g ro 
^_50_ 400X400X7r 
32.2* 3X3X4 
.'. h Fc= 10845 lbs. =tension at a or b, or at any point. 



(m = revolutions per second 
21690 lbs. 



Ans. 



Problem 13. — Prove that in order for a train to press normally upon 
the track in going around a curve of radius r (feet), the outer rail must be 

G iP- G ifi 

elevated an amount e = = „, ^ (35) 

gr 32.2r ^ ' 

in which ^ = elevation of outer rail in feet, 

v = velocity of train in feet per second, 
G^ = gauge of track in feet, 
r = radius of curve in feet. 

Solution. — ^The train is exerting two compo- 
nent forces, one horizontally due to centrifugal 
force and the other vertically due to gravita- 
tion, which can be reduced to one resultant 
force iacting normal to the calculated inclined 
surface of elevated track. In Fig. 42 let e-i rep- 
resent intensity of centrifugal force, and Gx = 
W = weight of train; then a-Jox is the resultant 
pressure which, in order to be normal to the 
track, must be at right angle Xo ah, G being the 
gauge (practically) and e the required elevation. 

e G 
Through similar triangles 




ei Gx 



Fig. 42. 
but from formula (34), ^i 



Fc— and 

g T 



Gx^W, 



Get 

Gv^ 
gr 

Whence, elevation outer rail== 



^ W v^ 

tr 

g r 

W 
Gv^ 



Required proof. 



32.2r * 

gauge of track X (velocity of train) ^ 
32 . 2 X radius of curve in feet 



MOMENTS OF INERTIA, ETC., OF PLANE SURFACES. 

Bending Moment = Resisting Moment. — We will now consider the 
relation between the moments of the outer forces of a girder and the moments 
of the inner forces. For example, let Fig. 43 represent a wooden beam 
4" wide, 6'' deep and 10' long 



X{ 



between supports, and loaded uni- 
formly with 120 lbs per lin. ft. 
Consider the bending moment 
(of outer forces) and resisting 
moment (of inner forces) about 
the center of moments o, situated 
at the center of the span on the 
neutral axis X — X of the beam. 
Then from the natureof the loading, 

i?j = Pi = P2 = 7?2 = l total load=^^-^^^^ 



;FJ=600 



!P^=GO0 



c-9tt 



^^ 



s 



>.c.g. 



600' 



f 



Tr?^600* 



Fig. 43. 
= 600 lbs.; Pi and P2 each being 



the resultant of half the total load; Rx and R2 the reactions at the points 
of support. Taking moments of the outer forces to the left of the section 
at o, we have — 
Bending moment = Mb =RxX5-Px X 2^= 1500 ft.-lbs. = 18000 inch-lbs. 



MOMENTS: BENDING, RESISTING, INERTIA. 



299 



This moment of 18000 inch-lbs. must of course be resisted by the moment 
of the inner forces acting on the right of the vertical section 5 — s' passing 
through o. In other words, the resist- 
ing moment must equal the bending mo- 
ment. Let Fig. 44 represent an en- 
larged section of the beam at 5 — 5'; F, 
the resultant of the compressive forces , 
acting to the right of the section above "^^ 
the neutral axis o — o\ and Ft the re-Q-ja}-' 
sultant of the tensile forces acting to the "^ 
right of the section below 'the neutral 
axis 0—0. Then, as the neutral axis is 
in the center of the section, will Fc = 
Ft = F and, taking moments about o, we 
have — 




ft»F 



Fig. 44. 



9000 



Resisting mofnent=M& =2 Fyo=MB =18000 inch- lbs., whence F= lbs. 

yo 
But each force F is the resultant of a distributed force varying in intensity 
from lbs. at the center of the section to /« at the top and ft at the bottom 
fiber of the beam, so that the stress in any fiber is directly proportional to 
its distance from the neutral axis; hence, /c = /t = / = the intensity of outer 
fiber stress per square inch ; and the resultant forces F F act in a line pas- 
sing through the center of gravity of each distributed force (see Fig. 40), 
two-thirds the distance from the neutral axis of the beam to the outer fiber; 

.'.yo = 2 inches. As the beam is 4 inches wide, we have F=-^ • -r- = 4X3 



X 



f _. 9000 



(see above), or /=750 lbs per sq. in. That is, the maxi- 



mum or outer fiber stress on the beam is 750 lbs. per sq. in. (See also 
formulas (36) and (36a) for the ordinary methods of finding the maximum 
fiber stress.) 



Resisting Moment = Moment of InertiaX— =— . — ^The general formula 

for the moment of resistance of a beam of any section, is 
// My 
Mr =— ; f = -j- (36) 

In which / = the outer fiber stress per square inch, 

y = distance in inches from neutral axis to outer fiber, 
/ = moment of inertia of section, about the neutral axis. 

bd^ d 

For a rectangular beam of breadth b and depth d, I = -tj: and y — -^', hence, 

lib Z 



equation (36) reduces to MR = i fbd^; f- 



6 Ml 



6 Mb 



d=^l^-^^ (36a) 



bd^ ' fd^ 

which will be found to agree with the preceding analysis, Fig. 4 4. 



"yj fb 



Moment of Inertia* = Area X (radius of gyrrf- 

iion)^ = Ar^. — The moment of inertia, /, about an 
axis X — X, of any plane surface as bh, resting on 
that axis, is the sum of the p;-oducts of the areas of 
the infinitesimal strips x .dy by the squares of their 
distances y from the axis. Thus, for the rectangle 
bh, resting on the axis of X, where x = b, and y varies 
between y = h and 3/ = 0, we have, for beam of height h. 



Moment of inertia =/h 



-/;-'-[:?-?■ 



(37) 



h 

I 



i 






^1 



* For moments of inertia of plane figures. 



of Plane Surfaces, Section 29; also, 
Section 30. 



Y Fig. 45. 
see Properties and Tables, 



see Properties and Tables of Steel Shapes, 



300 



15.— MECHANICS 



If now another similar and equal rectangle is added below the axis of 
X, making the total height 2h = d, it is evident that the above moment of 
inertia will be doubled ; thus (Fig. 45) , 

for a beam of depth d, /d=27h=| bh^ =— (38) 

the general formula for 7 of a rectangular beam, of depth d, about its neutral 
axis. 

Moment of Inertia about a Parallel Axis. — Let I be the moment of inertia 
about an axis X passing through the cen oi grav of the plane figure: /', 
the moment of inertia about a parallel axis X\ distant a from X; and A, 
the area of the plane figure; then will 

/' = / + Aa2. 

Moment of Inertia about an Inclined Axis. — See Properties and Tables 
of Plane Surfaces, Section 29. 



n .. - ^ .. /Moment of inertia // -,. ,. ^ 

Radius of Gyration = ^/ -r ^\~A' — radius of gyra- 



tion r of a plane surface as bh, whose moment 
of inertia is h, and area Ah, is the distance r = 
yi, from the axis of X to a point (or line) at f 
which if the whole area were concentrated the ji 
moment of inertia would remain the same. 7 

From the above and equation (37) we have, %jc. 

since /h = -r- and Ah =bh, 
o 



b --^ 



y,=r«.577h 



Fig. 46. 



/ Ik m ~ h h V"3 

r a beam of 



(39) 



And for a beam of depth ci= 2 h, with axis through cen of grav, equation (38), 

d 

.577 h. 



If,... 



dVs 
2X3 



In other words, the radius of gyration r is the same for a rectangle bh resting 
on the axis of X, as it is for a rectangle bd (in which d = 2h) about its neutral 
axis passing through the center of the section. 

Caution. — ^The radius of gyration r ( = yi), Fig. 46, must not be con- 
fused with the distance yo, Fig, 44, for — 

r = distance from axis of X to cen of grav of moment-ar^aj, while 
yo = " " •" " " " " " " moment -/orc^5. 
In the rectangle bd which, for simplicity, has been referred to throughout 
the discussion, 

tl 2M3 

M y _ 21 ^ ^^ _ ^ 

2F~ bdf bdy bd^ 3 " ^ ' 



yo 



"ylA~yjbd~yji2bd Vi2 



\/~3h 



2vT 
yo=—o—r-- 



1.155r(40) 



.•.r=^^o=.866:Vo(41) 



4 J2 4 
whence, yo^ = y r2 = _ = _h2 ; 



4 ^° 12 



h^ 



and r2 = ^- 3;o2 = ^ = '-^ (42) 

For other sections than rectangular, these ratios will not hold, necessarily. 

Y 



Problem 14. — Find the moment of inertia / and 
radius of gyration r of a circular section (complete 
circle) of diameter D, about its diameter. 

Solution. — The moment of inertia of a complete 
circle is 4 times that of the quadrant. Fig. 47; and 



/ of the quadrant equals 7q 



= fy2 

Jo 



X dy, because x dy 




is the area of each infinitesimal strip, and y^ is the 



Fig. 47. 



RAD OF GYR. CIR. BEAM. CEN. OF GRAY, 301 

square of its distance f rom the axis of X. From the equation of the 

circle, x^ + y'^==R^, x=\/R^— y^. Substituting this value of x in the above 
equation, and integrating by formula V of Integral Calculus, page 275, 
we have, 

/, = f^ yWW=^^ dy = [^ |-(23'' - ^) y/W^y^ + ^ sin-| 

= -^ sin-^ 1; but anti-sine of 1 = 90° = y» 

therefore I^ = -V^ I 
16 



and 7=4 I^ = ^^-j-; 



=» moment of inertia of the circle .... (43) 



Radius of gyration r= yj ^a/"^ "^ \ ^ ^T^'2 ^^^^ 

Resistance of Rectangular and Circular Beams Compared. — The general 

f I MI* 
formula for resistance of a beam is M = — ; or -7-= — • Now if two beams 

y f y 

of equal length and equal loading are to have the same safe resistance it 
is evident that their outer fiber stresses /, as well as their resisting moments 
M, must be respectively equal. It remains then simply to equate the section 

moduli — of the two beams. Thus— 

y 

For a rectangular beam of breadth h and depth d, — = -^, 

For a circular beam of diameter D, — = -— — i- — = -— — , 

Equating the above values of — , we have -i7^=—^* 

y OiS D 



Therefore, for the diameter of the circular beam, D = 2^ / — - — 

07C 



"-';!■ 



,» t: D^ 
and, for the depth of the rectangular beam, d = i. 



^v^ 



b • 

Q 7~)3 

and, for the width of the rectangular beam, b= ^„ ,„ . 

10 a^ 

Thus, a circular beam of 6 J ins. diam will give the same safe resistance as 

a rectangular beam 4" wide and 6" deep, Fig. 44. 

Center of Gravity. — ^The distance to the center of gravity of any plane 
figure, from a fixed axis, is equal to the sum of the moments of each infini- 
tesimal area of the figure about that axis, divided by the total area of the 
figure.t Thus, 

J'xy dy 

About the axis Xi, distance yo = (45) 

Jxdy 
substituting for x its value from the equation of the figure; 

fy X dx 

About the axis Fi, distance xq = (46) 

Jy dx 
substituting for y its value from the equation of the figure. 

* / . 
— is termed the section modulus, and is equal to the moment of inertia 

divided by the distance of the most extreme fiber from the neutral axis, 
t See also Figs. 39 and 40. 



302 



15.— MECHANICS. 



It is not necessary, however, to find the moments of infinitesimal areas; 
in fact, the figure may be divided into any number of areas of such shape 
that their centers of gravity are easily determined. Then the distance 
from the fixed axis to the center of gravity of the plane figure, is equal to 
the sum of the products of each area into the distance of its center of gravity 
from that axis, and this divided by the total area of the figure. Further- 
more, the areas whose moments are thus obtained may form one plane 
figure or any number of plane figures. In the latter case, the center of 
gravity obtained would be the center of gravity of all the figures. The 
exact point of the center of gravity is determined from two coordinate 
axes. 

The center of gravity of a plane figure or system of plane figures or 
areas, connected or isolated, is the position of the resultant of a uniformly 
distributed force over the figure or areas; hence a thin sheet, of any outline, 
may be balanced on a point of support applied at its center of gravity. 

MOMENTS OF INERTIA, ETC., OF SOLIDS. 

Moment of Inertia. — The moment of inertia of a solid body, about a 
fixed axis, is the sum of the products of the weight of each infinitesimal 
particle of matter (composing the body) into the square of its distance 
from said axis. Hence, the moment of' inertia is obtained by the use of 
the Integral Calculus. An approximation to exact values may be obtained 
by assuming a definite number of very small particles, multiplying the 
weight of each by the square of the distance from its center of gravity to 
said axis, and finding the sum of these products. 

For an axis X passing through the cen of grav of the body, the moment 
of inertia Is = I wr^; where w; = the weight cf each element, and r = its dist 
from the axis X. 

For an axis X' parallel with and distant d from X, the moment of inertia 
I'b =/9 -{-Wd^; where W = th.e total weight of the body. Hence, knowing 
the value of L about an axis passing through the cen of gravity, the value 
of 7% about any axis parallel with it can easily be obtained. 

2. — Moment op Inertia op Regular Solids, 



Description. 



About axis X 

through 

c. of g. 



About 11 axisX' 
dist (i from A". 



Sphere of total weight W, and radius r. . . 
Circular plate of rad r; axis perp to plate 



Circular cylinder of 
length 21, and ra- 
dius r 



Axis longitudinal . . . 

Axis perp to axis of 
cylind 



Circular ring, outer radr, inner rad ri ; axis perp 

Rod or bar of uniform cross-section and 
length 21; axis perp to length of rod 



~Wr^ 



2 
2 



W 



w 






2 
3 



W 



W 



w 



w 



w 



w 



■r^+d^ 






* Radius of Gyration. — The radius of gyration of a solid about a fixed 
axis, is the square root of the quotient obtained by dividing the moment 
of inertia about that axis, by the total weight of the body. Thus, from 
Table 2, above, the radius of gyration of a circular plate about a ^^rp axis 

V2~ 



VWr"^ r r' 
'~W = — -=z = — 
2 V2 



The radius of gyration of a body is the distance from the axis to the 
center of gyration or point at which if the whole mass were concentrated 
the moment of inertia would remain the same. 



MOMENTS OF INERTIA, ETC., OF SOLIDS. 



303 



Center of Gravity. — The distance to the center of gravity of any body 
from a fixed plane, is equal to the sum of the moments of the weight of each 
infinitesimal particle of matter of the body into its perp distance from that 
plane, divided by the weight of the body. 

It is not necessary, however, to find the moments of infinitesimal par- 
ticles; in fact, the body may be divided into any number of parts of such 
shape that their centers of gravity are easily determined. Then the distance 
from the fixed plane to the center of gravity of the body, is equal to the 
sum of the products of the weight of each mass into the distance of its 
center of gravity from that plane, and this divided by the total weight of 
the body. Furthermore, the masses whose moments are thus obtained 
may form one body or any number of bodies. In the latter case, the center 
of gravity obtained would be the center of gravity of all the bodies. The 
exact point of the center of gravity, is determined from three coordinate 
planes. 

If the center of gravity of a solid is supported on a point, the whole 
body will be in unstable equilibrium. 

Center of Oscillation = Center of Percussion, about 
Axis O. — The center of oscillation P of a mass 
swinging or oscillating about a fixed axis passing 
through O, isa point at which if the whole mass were 
concentrated the oscillation would remain the same. 
The center of percussion P of the mass suspended 
from the axis at 0,is that point at which if a hori- 
zontal force be applied there will be no shock at O. 
The two points P are identical. Let any mass M, 
Fig. 48, be suspended from an axis at O. Let G 
be the cen of grav of the mass, C the center of gyra- 
tion, and P the center of oscillation or percussion; 
and let yo, r and / be the respective distances to 
these points from the plane X\ r being the radius 
of gyration of the mass about O, and / the radius of 
oscillation. Then will r be a mean proportional be- 
tween yo and /, so that 

,. - .,1 ^- 1 f^ (radius of gyration) 2 

radius of oscillation / = — = ,. ^^ 7 . 

yo dist to cen of grav 




Fig. 



To illustrate: For a rod of length L and of uniform section, 
freely from its upper end: 



(47) 

suspended 



3'o=-2-,r2=— . and /=fL. 



Note that O and P are interchangeable; that is, if P becomes the axis, 
then O will be the center of oscillation or percussion. 

The theory of the compound pendulum is based on the above principles. 



Impact or Collision. — The "line of impact" of two bodies in collision 
is a line normal to the surfaces at the " point of contact," irrespective of the 
relative motion of the two bodies. Impact is — 
Central, when the line of impact coincides with the line joining the centers 

of gravity of the two bodies; eccentric, when it does not. 

Direct, when the line of impact coincides with the line of relative motion 
of the two bodies; oblique, when it does not. 

Central impact may be either direct or oblique. 

Direct impact may be either central or eccentric. 

In all cases of impact, action = reaction. 

Notation. 
= the respective masses, 
==the respective velocities before impact, 

= the respective velocities at any given instant during impact, 
= the respective velocities after impact, 
= common velocity at time of greatest compression, 
= coefficient of restitution in imperfectly elastic contact. 



fHi 




IW2 


Cl, 




C2, 


v'. 




v\ 


Vl, 


V 


V2, 



304 15.--MECHANICS. 

Central Impact Formulas. 

Velocity at time of greatest compression, v= ^ ^ — ^ (48) 

„ „ . ^ J. ^.^ ^. veloc regained after compression v — Vi 

Coefficient of restitution, e = — :; — r- — ; — -—z — ; : — == = 

velocity lost during compression Ci — v 

^ (49) 

For perfectly elastic bodies, e would = 1; but as all bodies are imperfectly 
elastic,^ is always less than unity, showing that there is a loss of energy 
during impact. 

Loss of energy due to inelastic impact = „ , ^ ," r- (Ci — -^2)^ (50) 

Ji {nil + W2) 

Loss of energy due to imperfectly elastic impact = il—e^) >^i»^2 — f^^__^^^2 

I (tWi + W2; 
(51) 



16.— THEORY OP STRESSES IN STRUCTURES. 

OUTER AND INNER FORCES. 

A structure is designed to resist safely the loads which may come 
upon it. These loads (which include the weight of the structure itself) 
together with their attendant reactions, are called the external or outer 
forces', while the stresses which these outer forces produce in the members 
of the structure themselves, are called the internal or inner forces'^ thus— 

Outer forces= { ^^] Retctiim'"^' ^^^^' ^"^'^' ^'^°^' ^^'''^' 
Inner forces = (c.) Stresses in the members. 

Assuming that the loads acting on the structure are known, the first 
step is to find one or more of the reactions', and from these outer forces the 
stresses in the members may be determined by the use of the analytical or 
the graphical methods, or by the two methods combined. In the operation 
of finding the reactions and stresses it is well to remember the following 
principles: 

PRINCIPLES OF STATIC EQUILIBRIUM. 

First. — The algebraic sum of the moments* of the outer forces about 
any point is equal to zero; from which, considering right-hand (clockwise) 
moments plus and left-hand (un-clockwise) moments minus, or vice versa, 
we have — 

Summation of moments=0: i'M = (1) 

Second. — ^The algebraic sum of the components, in any given direction, 
of the outer forces is equal to zero; from which, considering those downward 
or to the right, plus, and those upward or to the left, minus, or vice versa, 
we have — 

Summation of vertical components = 0: IV =0 (2) 

Summation of horizontal components =0: IH = (3) 

Third. — For clearness and simpUcity of calculation, in the application 
of equations (1), (2) and (3), the stresses in certain members of the truss or 
frame-work may temporarily be considered as outer forces, by cutting the 
structure in two by certain planes (straight or curved) properly intersecting 
the members in question — considering that portion of the structure on one 
side of the cutting plane (usually the left side) as a complete structure; 
and the stresses in the members sp cut, as outer forces acting on the section 
to produce equilibrium. That is, these imaginary outer forces are the 
stresses to be calculated. (See Figs. 1 and 2.) Generally, only threef active 
members may be cut ; thus — 

Cut three active members; intersection of either two is) 
origin of moments for calculating the stress in the third [•.... (4) 
member. ) 




Fig. 1. 



* The moment of a force about a given point, as origin, is the product of 
the force and the shortest distance from the origin to the line of the force. 
t See "Notes," page 726, for cutting of four active members. 



305 



306 



IQ.— THEORY OF STRESSES IN STRUCTURES. 



PRINCIPAL METHODS OF CALCULATION. 

Method of Moments. — This method of calculation, using equation (1), 
is usually employed in determining the reactions at the points of support; 
the stresses in the chord members of bridges and roofs and in the web mem- 
bers where the chord members are not parallel; the wind stresses in the 
main vertical members of towers, buildings and similar structtires ; the com- 
pressive stresses in masonry dams; etc. 

Method of Shears. — This method, using equations (2) and (3), is usually 
employed in calculating the stresses in the web members of any frame-work 
connecting parallel chords, as the lateral systems of bridges; the web mem- 
bers of simple bridge and roof trusses; the bracing of towers; the shear 
in dams; etc. 

The stress in any vertical web member which "takes up" all the shear, 
is equal to the algebraic sum of the vertical components of the outer forces 
to the left of the section cutting the member; and the stress in any diagonal 
member is equal to the vertical shear multiplied by the secant of angle of 
inclination of that member with the vertical. 

The term "vertical" may be used in any uniformly specific direction — 
usually at right angle with the axis of the structure and parallel with the 
direction of the prevailing outer forces. To epitomize: 

Stress in any member carrying all the shear = vertical ) fr\ 

shear X secant of angle of inclination with the vertical.) • • • • ^^-^ 

The above five laws are fundamental in the determination of the stresses 
in any structure which is statically determinate. 

Graphical Methods. — The most common methods of determining 
stresses is by graphics, which may be performed by "moment diagrams" 
or by diagrams involving the principle of the "triangle of forces" (see 
Mechanics, Figs. 31 to 35). The latter is called Maxwell's method and 
is the most used. (See Figs. 4, etc., following.) 

PRACTICAL APPLICATION OF PRECEDING PRINCIPLES. 

CASE A. LOADS AND REACTIONS VERTICAL. 

Problem. — Find the dead-load stresses in a 6-panel Pratt truss of 
120-ft. span, 23 ft. in height center to center of chords; assuming the 
dead load per truss at 500 lbs. per lin. ft. and acting at the lower panel 
points, Fig. 2. 



This Half: 

tbmihers of Members', 
' Joints-, 



This Haff : 
Stress Sheet of each kSparr 
Stresses, in lOOOIbunitSf 



Cutting Planes. , ^ ^ * ., Ten5ion,t}Compr^C- 




R.eso P5=lOi) P4=10.0 P3^ 



0.0 ?^m Pr'10.0 



R^ISjO 



Fig. 2. 



Calculation. — Stresses are usually in thousand -pound units and calcu- 
lated to the nearest hundred pounds; thus, 64.7=64700 lbs.; 64.0=64000 lbs.; 
64=64 lbs, , etc. That is, where the stress is abbreviated to the thousand-pound 
unit there should be one figure to the right of the decimal point, whether 
it is a significant figure or merely zero, to show that the stress is in the 
abbreviated form. The sign ( — ) before, or (c) after, the stress indicates 
that the member is in compression, as —64.7= 64.7c; while the signs (-f) 
and (0 indicate tension, as +64.7 = 64.7 ^ 

Load per panel per /m5s= 500 X 20= 10000 lbs. = 10.0. 



the oppo- !^-nih-">k-— 'X,l — >U J<-ni-->{ 

Let any i: *; f^ -* ^ , 

ict to the R.r -ai ->« bl -> o 

of panel *!*• --— Nl ->^'^* 



PRATT TRUSS WITH PARALLEL CHORDS. 307 

Reaction at Left Support. — Assuming the panel lengths as unity and 
taking moments about the right hand abutment B, using equation (1), 
IM=0, we have, 

i?iX6-10.0 (5+4+3+2+l) = 0. 

Hence, reaction Ri = — p— = 25.0, acting upward. 

Lengths of Members. — All horizontal or chord members are 20-ft.; all 
verticals (posts and end suspenders) are 23-ft.; hence, all diagonals (bars 
and end posts) are 

V202+232=30.48-ft. Thus, hor = 20; ver = 23; diag = 30.48. 

Trigonometric Ratios. — The angle oc, Fig. 2, is the angle of inclination 
of the diagonal members with the vertical; and its two functions, tan oc 
and sec oc, are respectively employed in calculating the stresses in the chords 
and diagonal members of bridges with parallel chords. 

Bending Moments and Chord Stresses: 

Let O, Fig. 3, be the origin of moments at any panel point of eitker 
chord, obtained by cutting three active 

members, see equation (4), and let it be . iM , . fi 

required to find the stress in the oppo- 
site chord member so cut. 
number of panel loads Pi act to the 
left of O, and any number of panel 
loads P2 act to the right of O, on -pitr <l 

the span Nl\ in which ^^^' ^• 

/== length in feet of one panel, 
iV=number of panels in the span, 
ni, a=number of panels from Ri to Pi and O, respectively, 
«2. 6=number of panels from R2 to P2 and O, respectively. 

Then, Bendmg moment at = Mo= ^-^ ^r^^ —- (6) 

And if d represents the effective depth of truss, we have, 

_ . , , , Mo Motancc i" Pi «i fc + i" P2 «2 a^ ^ ,^v 

Stress m chord member = —7 = -, = — ^^ — =— tan oc. . (7) 

a I iV 

which are the general equations for finding the bending moments and chord 
stresses without first finding the reactions. 

Equations (6) and (7) are, however, very little used, and it is probably 
best for the beginner to employ equations (8) and (9), following, in which 
the reaction Rt must first be determined. 
Then, Bending moment at = Mo = i?i. al— I Pi. xj (8) 

And, Stress in chord member = -— = (Ria — I Pi Xt) tan OC (9) 

in which i<;i = number of panels from Pi to O. 

Shears and Web Stresses: 

Let O, Fig. 3, be the line of shear cutting usually two or three active 
members, including one web member which takes up all the shear. Then — 

Vertical shear at O = So = i?i - i" Pi (10) 

Stress in web member = (Ri — I Pi) sec OC (11) 

in which oc = angle of inclination of web member with the vertical, 
and I Pi = sum of all loads to left of cutting line. 

Stresses in Chord Members by Method of Moments: IM^O. — In addi- 
tion to determining the amount of stress in the various members of a 
structure, it is of course necessary that the kind of stress, whether 
tensile or compressive, be known in each case. Certain rules may be followed 
in determining the amount and kind of stresses in statically determined struc- 
tures, as included in the following steps: 
First. — Draw a plane through the structure cutting three active members, 

including the one to be calculated. The point of intersection of any 

two of the members will be the center of moments for determining the 

stress in the third. 



308 



U.— THEORY OF STRESSES IN STRUCTURES. 



Second. — For the member in question, find the intersection of the other two 
members or "center of moments" about 'which the moments of the 
outer forces, acting on the section to the left of the cutting plane, are 
to be taken. These will include the reaction at the left abutment; and 
the moment equation, IM = 0, will include also the required stress 
assumed temporarily as an outer force and acting at the right of the 
cutting plane. 

Third. — Take moments about the center of moments, assuming that the 
above last mentioned unknown force (the required stress) is plus ( + ) 
and acts away from the section. All other forces tending to produce 
moments in the same direction as the above may be considered as ( + ) 
forces and their moments as (4-) moments; while all other outer forces, 
that is, forces tending to produce moments in the opposite direction 
may be considered as (— ) forces and their moments as ( — ) moments. 
Fourth. — If, in the equation IM = 0, the result of the required stress is a 
( + ) quantity, the member is in tension; while if it is a ( — ) quantity 
the member is in compression. Thus, 

Assume required stress as 4-5 acting away from the section (12) 

If the result of S is + , the member is in tension (13) 

" " '* " 5 is -- , " " " " compression (14) 

Table 1 is based on the above methods and rules, which will be found 
useful to the young engineer. The experienced engineer can usually tell 
the kind of stress by inspection, but there are many cases in complex struc- 
tures where the rules have to be followed. 

1. — Showing Calculation of Chord Stresses, Fig. 2. 





Cut- 
ting 
Plane 


Method 




Equation. 


Calculation. 


i 


'^ 






6^ 






C/5 


2 


C-C 


IM=0 


u. 


S2 = i?i tan a 


25.0X. 87-4- 21.7 


21.7/ 


1 


u,-u 


IM = Q 


f/l 


Si = Ri tan a 




21.7/ 


3 

4 


d-d 

d-d 


JM=0 
JM=0 


^2 
L3 


S3=(i?iX2-P5Xl) 

tan a 
S4 = -(3i?-3P)tana 


40.0X.87=-l-34.8 
-45.0X.87=-39.1 


34.8/ 
39 Ic 


5 


C-C 


JM=0 


U 


Sr^=-{2R-P) tana 


-40.0X.87=-34.8 


U.^c 



Remarks. 



/i?i = 25.0; 
itan a = .87 

(Also, from 
St=S2 

jZP=PsX 

I24-P4XI. 

2i?=i?iX2 



Stresses in Web Members by Method of Shears: IV =0. — The shear 
method is usually adopted where a single web member in a panel takes 
up all the shear, as in Fig. 2, in which top and bottom chords are parallel 
and at right angle with the prevailing outer forces. In such cases the 
following steps may be employed: 

First. — Draw a plane, cutting the web member in question and also the 
upper and lower chords. It is clearly evident that the stresses in the 
chords can have no vertical component to assist in *' taking up" the 
shear, since they are at right angle with the shear, hence it must be taken 
up by the web or shear member. 

Second. — Find the algebraic sum of the vertical outer forces to the left of 
the cutting plane, and this will be equal in magnitude and opposite in 
direction to the vertical component of the required stress in the member, 
acting at the right of the cutting plane to produce equilibrium. If the 
required stress acts away from the cutting plane the member is in ten- 
sion (-f), while if it acts toward the cutting plane the ipiember is in 
compression ( — ). 

Table 2 is based on the above rules, although the algebraic signs 
giving direction can usually be dispensed with. 



PRATT TRUSS. BOW'S NOTATION. 



309 



2. — Showing Calculation of Web Stresses — Fig. 2. 



B 


6 

4 

'■+2 
+-> 

O 


a> 


^B 
o 


Equation. 


Calculation. 


i 

C/5 


Remarks. 


6 


a— a 


IV=0 


/ 


Sq-= — Ri sec a 


(-25.0X1.325=) 
( -33.1 / 


33.1c 


(i?i=25.0; 
( seca = 1.325. 


1 


b-h 


IH=0 


- 


Si =52-0 


21.7-0=21.7 


21. 7i 


See Table 1. 


7 


b-b 


IV=0 


1 


57=P5-0 


10.0-0=10.0 


lO.Ot 


( Cutting 
< plane, 
( curved. 


8 


c—c 


IV=0 


\ 


S«=(i?i-P5)seca 


15.0X1.325=19.9 


19.9/ 




9 


f-f 


^V==0 


1 


59=-(i?i-2P) 


-25.0+20.0= -5.0 


5.0c: 




10 


d-d 


IV=0 


\ 


Sio=(i?i-2P)seca 


5.0X1.325=6.6 


6.6/ 




11 


e—e 


IV=0 


1 


5n=0 




0.0 


( Cutting 
■j plane, 
( curved. 



Symmetry. — The span and loads being symmetrical it is necessary to 
calculate the stresses to the center of the span only, the other half span being 
simply a duplication. 

Static Equilibrium. — In addition to the outer forces being in equilibrium 
and satisfying equations (1), (2) and (3), it is also true that all forces, inner 
and outer, acting at any panel point or joint of the truss are in equilibrium 
and satisfy the same equations. Fig. 4 (using above calculations of 120- 
ft. span, Fig. 2) will explain: 



^'A 




ZS.0 



B )> 







Fig. 4. 

Note. — Forces acting toward the joints denote compression', from the 
joints, tension. 

Bow's Notation. — In the above figure the inner forces or stresses in the 
members are represented as outer forces acting at the several panel points, 
and it will be found that the forces at each joint will form a closed polygon 
of forces denoting equilibrium (see Mechanics, Figs. 34, 35). Thus, the 
forces at Lq form a triangle; at Li, a rectangle; at f/i, a trapezium; etc., 
as per Figs. 5 and 6. 

The notation used, to distinguish the stress in the stress diagram corres- 
ponding to the member in the skeleton diagram of the truss, is called Bow's 
notation. For example, the line BC in the stress diagram. Fig. 6, repre- 
sents the stress in the diagonal member lying between the triangular spaces 
B and C of the skeleton diagram, Fig. 4. The space U covers the whole 
upper side of the truss from Lq to L'o, while the space L is made to cover 
the lower side of the truss between the same points; so that AU corresponds 
to LoUi\ CU to U1U2; AL to LqLi; BL to L1L2; etc. Some engineers prefer 
to use separate letters around the outside of the truss between the lines of 
outside forces, as for instance, instead of L being used universally for the 
whole lower outside space, to use L„ , Lb , L. , etc., between LqLi, L1L2, 



310 



IQ.— THEORY OF STRESSES IN STRUCTURES. 



L2L3, etc., or to use entirely different letters, as F, G, H, etc.; while some 
prefer the use of the letters X and Y to represent, respectively, the upper 
and lower outside spaces of the truss. There is, however, no real principle 
involved in these various customs. 







Fig. 5. Force polygons at joints (Fig. 4). 



Graphical Method. — The principle of force polygons at the joints of a 
structure has led to the rapid solution of stresses by the graphical method. 
It will be noticed that Figs. 5, if "fitted together," will form the com- 
plete graphical stress diagram of half the truss — Fig. 6 showing the 
complete stress diagram of the whole truss, the two halves being sym- 
metrical. 



jrLa 



+tl.7 




Fig. 6. 

Hence if the loads and reactions are laid off to proper scale, forming a 
closed polygon of outer forces called the load line, the lines drawn parallel 
with the members of the skeleton diagram of the truss and properly inter- 
secting one another, will represent by scale the actual stresses in the mem- 
bers. The following rules will be found universal in application for loads 
and reactions in any direction or applied at any points of the structure, 

GENERAL RULES FOR STRESS DIAGRAMS. 
Order of Considering Forces Around Joints. — 

1°. Draw skeleton of truss accurately to a good scale. 

2°, Show T loads at top joints and B loads at bottom joints of truss. 

Note. — In ordinary cases where T and B loads at the abutments are 
vertically downward and opposite in direction to the reactions at those 
points, these loads may be omitted — so far as the stresses in the trusses 
are concerned — and the reactions diminished accordingly: but the total 
reactions must be considered in designing the supports themselves. Where 
the loads and reactions are not in the same straight line, both must be in- 
cluded. 

3°. Find the reactions — R^ = left-hand; R2 = right-hand. 
4°. Load-line = reactions and loads forming closed polygon of outer forces. 

Note. — Begin with one of the reactions and draw, in the directions of 
the forces, the closed polygon of outer forces — reactions and loads — con- 
sidering them in right-handed (clockwise) or left-handed (unclockwise) 
order around the truss, as per the following: 



GENERAL RULES FOR STRESS DIAGRAMS. 



311 



Outer forces around truss considered in clockwise order'. 

Beginning with Rx — (mainly for T loads, at top of truss) — 

( Draw stress diagram to left of load line; 

1 Consider forces around joints in clockwise order 

Beginning with R2 — (mainly for B loads, at bottom of truss) — 

( Draw stress diagram to left of load line; 

1 Consider forces around joints in clockwise order 



}.(15) 

}.(16) 





/^T^ 




For (15) For (17) For (19) 

6°. Outer forces around truss considered in un-clockwise order'. 

Beginning with Rx — (mainly for B loads, at bottom of truss) — 

( Draw stress diagram to right of load line; ) /^yv 

1 Consider forces around joints in un-clockwise order j '^^'^ 



Beginning with R2 — (mainly for T loads, at top of truss) — 

( Draw stress diagram to right of load line; ^ ) 

\ Consider forces around joints in un-clockwise order j 



(18) 




For (16) 



For (18) 



For (20) 



For trusses loaded with T loads, at top, and B loads, at bottom chords, 
(15) and (16) may be combined in (19); while (17) and (18) may be com- 
bined in (20); as follows: 

7''. Outer forces around truss considered in clockwise order: 

[ Beginning with Rx or R2 — (for both T and B loads) — 1 

I Draw stress diagram to left of load line; [ (19) 

[ Consider forces around joints in clockwise order J 

8°. Outer forces around truss considered in un-clockwise order: 

{Beginning with Rx or R2 — (for both B and T loads) — ") 
Draw stress diagram to right of load line; [ (20) 

Consider forces around joints in un-clockwise order J 

Remarks. — The load-line diagrams above, illustrating (15) to (20) 
inclusive, simply show the order of considering the forces at the joints, but 
not their direction. Note that if the outer forces are all vertical, the load 
line will be a vertical line. 



Order of Considering Joints. — So far, we have considered the order of 
drawing the outer forces to form the load-line polygon; and the order of 
considering the forces around the joints for a stress diagram either to right 
or left of the load line. We will now consider the order of proceeding from 
one joint to another of the truss in drawing the stress diagram; bearing in 
mind that it is sometimes essential to begin at the end of the truss which 



312 



U.— THEORY OF STRESSES IN STRUCTURES. 



has the larger reaction, as a stress diagram is simply a system of triangulation 
and the larger the initial base the better. There are other reasons, how- 
ever, which must be considered, namely, the nature of the loading and the 
general convenience of always starting from the one end. In trusses which 
support moving loads the loads are made to come on from the right, and 
the left reaction is taken as the initial base of the stress diagram. In other 
words, we usually start with joint Lq of the truss. 

In following from joint to joint we must select the one which has, for 
the given loading, only two unknown stresses whose directions are known, 
because — with one force given, in magnitude and direction — 

Two unknown forces only, whose directions are known, can be] 
solved by a "triangle of forces;" and this fact determines the order [(21) 
of considering joints. J 

PRACTICAL APPLICATION OF PRECEDING GRAPHICAL RULES. 

Graphical Solution of Case A; Loads at Bottom Joints. — Data: 6 panels 
at 20-ft. = 120-ft. span; height of truss = 23ft.; load per joint= 10,000 lbs.; 
reactions, each = 25, 000 lbs. 

Draw the truss very carefully to scale, large enough so lines in the 
stress diagram can be drawn parallel with the members of the truss, with 
sufficient accuracy. 

In accordance with (17), beginning at the left-hand end of the truss, 
lay off consecutively in direction and intensity, by scale, the outer forces 
i?i = 25.0 (upward); P5, P4, P3, P2, Pi = 10.0+ 10.0+ 10.0+ 10.0+ 10.0 
(downward); and .^2=25.0 (upward). 

Draw the stress diagram as per Fig. 6 , considering the order of the 
forces in un-clockwise order around each joint and noting that when a 
stress is drawn in direction away from the joint, the member is in tension; 
while if drawn in direction toward the joint, the member is in compression. 
The arrows in Figs. 5 show the direction of the stresses of each polygon 
of forces, indicating the nature of the stress in each case. 

Table 3 shows the_ order of considering the joints, the given and 
required forces at each joint, and the nature of the stresses in the members. 



3. — Graphical Method op Solving Case A, Pigs. 4, 5 and 6. 



1 


.s 

"0 
*—> 




Forces (Fig. 4). 


1 

(U 


Direction of forces in 
force polygon. 







Given. 


To find. 


From the 
joint, denot- 
ing tension 
(+). 


Toward the 
joint, de- 
noting com- 
pression {-) 


xn 

6 

P4 


1 

2 

3 
4 
5 

6 

7 


Ln 

I 

u 


UA^AB 

CB, BL, L,, Lc 

UCCD 

UE 

EE',DE,DL,L.L\. 


LA, UA 

LB,AB 

UCBC 
LD,CD 
UE,DE 

EE', UE' 

LD',E'D' 


A 

B 

C 
D 
E 

' E' 

D' 


LA =+21.7 
JLF =+21.71 
\AB = + 10.0j 
BC =+19.9 
LD = + 34.8 
DE =+ 6.6 

EE' = ± 0.0 

D'E=+ 6.6 


UA=-S3.1 

UC =-34.8 
CD =- 5.0 
UE =-Z9.1 

UE'= -39.1 

LD'=+3i.S 


/Point 
\E=E'. 



GRAPHICAL SOLUTION OF PRATT TRUSS, 



313 



Case Aa; Loads at Top Joints. — The following is a graphical solution of 
Case A, modified — the loads at the joints acting at the top instead of at 
the bottom of the truss. Two methods are shown, namely, (15) and (18). 
Note that eacJi stress diagram is a supplement of the other, that is, revolved 
180°. 




^1 p X__/__l-H 



.^R^ 

^ / 



D^. 



D\J 



Full Lines 



Fig. 8. Fig. 9. 

(15.) Loads and reactions, clockwise. (18). Loads and reactions, unclockwise. 

Forces around joints, clockwise. Forces around joints, unclockwise. 

Remarks. — Compare the above with the diagrams obtained for bottom 

chord loading, Figs, 5 and 6, and note that the difference in the stresses is 

entirely in the vertical members. 



Case Ab; Loads at Top and Bottom Joints. — Before leaving the analysis 
of the 120-ft. span let us consider another case — top cord joints loaded with 
5,000 lbs. and bot- U 

tom chord joints T5=5.0 T4=5.0 T^=5.0 T2=5.a Tr5X> 

loaded with 10,000 
lbs. The dimen- 



sions of the truss 
are to remain the 
same, . namely, 6 
panels, 20 ft. long 
and 23 ft. deep. 




^ RrR2=^=37.5 
Stress in EU: 
=<3Z5x3- 15.0x3) 



Rr37.5 B5=»0.0 B^^IO.O 



I- 

Fig. 10. 
Note. — For this case either (19) or (20) maybe used. 
drawing the diagram to left of load line (Ri, T, R2, B). 



Br40.0 BrlOi) R^p3Z5 



We will use (19)— 



(19) Loads and reactions clock- ^, , r, '^fv 
wise. Check Sfress \\ 

Forces around joints clock- '" ^ B,/" — \ 

wise. C / 

o' 



T 



Remarks. — The stress dia- E, 



gram for the left-hand half of K 
truss is drawn complete in full E' i5 
lines; that for the right-hand 
half of truss, in dotted lines. 
The truss is symmetrical and 
symmetrically loaded, hence one 
half the stress diagram is all 
that is necessary in practice. 






-5B,7 



X 



c 



/ 



B 



'e:ft Hand 
HolfofTruss 



Fig. 11. 



x. 



?<^^ 






OQ 






314 



U.— THEORY OF STRESSES IN STRUCTURES. 



Reactions in Any Direction. — The foregoing elementary principles 
will apply to any case of loading and reactions, whether vertical or oblique; 
remembering that the load line is a polygon of forces in which the resultant 
loads and resultant reactions are respectively equal in magnitude and 
opposite in direction. There are four cases, as follows: 
Case A. — Loads and reactions vertical (these have just been considered). 

Example. — ^Truss of an ordinary span "resting" on the abutments. 

Case B. — Resultant of loads oblique; reactions oblique and parallel. 

Example. Roof truss with both ends "fastened" to supports. 
Case C. — Resultant of loads oblique; one reaction oblique, one vertical. 

Example. — Roof truss with one "roller" and one "fixed" end. 

Case D. — The loads vertical; reactions oblique. 

Example. — ^Three-hinged arch, "pressing" against abutments. 

Case B; Roof Truss; Both Ends Fixed; Wind on One Side. — A roof truss 
with both ends fastened to the supports, and wind pressure on one side, 
Fig. 12. The assumption made here is that the horizontal components 
of Ri and R2 are directly proportional to the reactions themselves; in other 
words, the end that has the greater reaction resists the greater horizontal 
thrust. 




Note. — P is the resultant of all the T loads: Ti and T5 are each one- 
half of T2, T3 or T4, the last named three being full panel loads. For height 
of truss=2 panels, the resultant P cuts the lower chord 2^ panels from 
the left-hand end, so that i?i = 5|P-^8, and i?2==2iPH-8. 




Fig. 13. 

(18). Load's and reactions, un-clockwise. 
Forces around joints, un-clockwise. 
Remarks. — ^This case is seldom used excepting perhaps for short spans. 
Case C, which will next be treated, can be applied universally. Note that 
stress diagram, Fig. 13, indicates that there is no stress in any of the dotted 
web members of the truss shown in Fig. 12, G' being made to include 
the whole triangular space comprising the right-hand half of the truss. 

Case C; Roof Truss; One Roller End; Wind on Either Side. — Roof truss 
with wind pressure on left side; two conditions as follows: 
Ca. — Left end is roller end; right end is fixed end. See Figs. 14 and 15. 
Ch. — Left end is fixed end; right end is roller end. See Figs. 14 and 16. 

Fig. 15 in the stress diagram for the truss (Fig. 14) treated as per 
Case Ca. The load line is composed of P, Ri, and R2. P, the resultant of 



REACTIONS IN ANY DIRECTION. 



315 



all the T loads, is known in intensity, direction and position, passing through 
T3 normal to the roof; Ri is known in direction and position, passing ver- 
tically through L. Let O be the point of intersection of P and Ri and 
considering it the center of moments of all the outer forces, we have, from 
IM=0, that R2 must be in the direction RO, passing through the center 
of moments O. 

The stress diagram, Fig. 15, is drawn by rule (18), page 311: 

Fig. 14 — (18). — Loads and reactions, un-clockwise. 

Forces around joints, un-clockwise. 

Note. — As the left end is the roller end the reaction Ri must be vertical. 

Fig. 16 is the stress diagram for 
the same truss, treated as per Case 
Cb by similar analysis. ^^ -. 

(15). Loads and reactions, clock- - - 
wise. 
Forces around joints, clock- 
wise. 

Note. — As the right-hand end is 
the roller end the reaction R2 must 
be vertical. 

Remarks. — ^The two conditions 
(a) and {b) of Case C are applied 
in practice to the roof truss for 
maximum wind stress in each of the .;-^ 
members. In addition to these the'^^*-\ 
members of course are designed for 
snow-, dead-, and live-loads, if any. 
For these latter cases, however, the 
reactions are vertical and the graph- 
ical methods are simple. (See Roofs, 
Section 46 j. 

Case D. Three-hinged Arch; Vertical 
Loads. — ^Three-hinged arch (Fig. 17) with a 
load P acting on the left-hand girder, one 
panel from the center of the span. The span 
is hinged in the center and at each abutment, 
and as there can be no moments at these hinges 
the lines of reactions must pass through them 
follows: 

(a.) For a load at center of span, Ri takes the direc 
tion LO, and R2 takes the direction RO. 

(b). For a load at the left of the center as P, R 
takes the direction RO, intersecting the line o 
the force P at H ; whence jRj takes the direc- 
tion LH. 

(c.) Similarly, for a load P' at the right of the center, 
Ri takes the direction LO, intersecting the line 
of the force P' at H' ; whence R2 takes the direc- 
tion RH\ 

Fig. 18 is a stress diagram of the arch, 
for the load P, Case Db, above. For 
this load the dotted members, Fig. 17, 
have no stress. 

(18). Loads and reactions, un-clockwise. 
Forces around joints, un-clockwise. 

Note.— i?2 and P de- 




termine Ri. 

Remarks. — The 4- 
and — signs for the mem- 
bers indicate respective- 
ly whether they are in 
tension or compression. 



Fig. 18. 




17.— NATURAL HISTORY OF MATERIALS. 

(For Weights and Specific Gravities, See Section 27.) 
A.— CHEMICAL. 

Composition of Matter. — If a drop of water could be magnified to the 
size of the earth it is estimated that the countless atoms of which the drop 
is composed would each appear to be about the size of a bird's egg. It 
would also be seen that all the atoms would be in groups of three, each 
group or molecule being composed of two atoms of hydrogen^ and one of 
oxygen. Subject the drop of water to a galvanic current and it will cease 
to exist as water, being decomposed into its two elements — hydrogen and 
oxygen. In other words, the molecules forming the compound, water, will 
become disintegrated, all the atoms of hydrogen passing off as hydrogen 
gas, and all the atoms of oxygen .passing off as oxygen gas. Conversely, if 
now the two evolved gases are mixed and sufficiently heated they will re- 
unite with explosive force into the same molecular composition as before, 
re-forming the drop of water. In the above processes, the decomposition 
and recomposition of the substance water, none of the atoms have been 
divided or destroyed. 

Old Atomic Theory. — Up to about the year 1898 the last sentence of the 
preceding paragraph might have been concluded about as follows, and 
generally accepted: "Because the atom is indivisible and indestructible, 
representing the smallest particle of an element of matter \ hence when we 
say that matter is indestructible we mean simply that the atoms of which 
all matter is composed are indestructible." 

Recent Discoveries. — The recent discoveries of the Hertzian ray and its 
practical use in wireless telegraphy; of the Rontgen or X-ray, by means of 
the Crooke's or vacuum tube, 1895; of the Becquerel ray, exhibiting the 
radio-activity properties of the element uranium, in 1896; and later, by 
Madame Curie and others, of similar properties in the elements thorium, 
polonium, radium and actinium, have resulted in shaking the old atomic 
theory of Dalton, which had lasted for a century. 

The Corpuscular Theory. — It is now believed that each atom, previously 
supposed to represent the smallest particle of matter, is really composed of 
a great number of smaller particles called corpuscles. For instance, the 
smallest and lightest atom known, that of hydrogen, contains about 900 
such corpuscles; while one of the heaviest atoms, that of radium, contains 
about 200,000. It is further submitted that these corpuscles are so infinitely 
small that the unoccupied space in the atom is almost infinitely great by 
comparison, and that they are vibrating or traveling through this space 
with velocities of many thousands of feet per second. This becomes all the 
more amazing when it is stated that the diameter of a molecule, which the 
atoms compose, is in the neighborhood of 2500^0000 of an inch. 

The corpuscular theory has been advanced in order to explain some of 
the recent discoveries cited above. Radium, for instance, is constantly 
giving off heat to surrounding objects without any visible source of supply. 
The heat is evidently furnished by the breaking up or disintegration of the 
atoms; i. e., by the transforiAation of part of the corpuscular energy of the 
atoms, into heat. The balance of the energy is dissipated in the forms of 
various kinds of rays. On this theory the laws of the conservation of energy 
still hold, for it has been observed that all of the radio-active substances 
gradually lose in weight. It is estimated that radium will disperse about 
i its mass in 1500 years, and x% of it in 10000 years. Thorium and uranium 
act much more slowly, requiring in the first instance about 1000 million 
years, and in the second instance 10000 million years. Moreover, all sub- 
stances are supposed to be radio-active to some extent, i. e., their atoms are 
gradually breaking up into corpuscles and the corpuscular energy is being 
transformed into other kinds of energy. In line with this argument, witness 
the scent of flowers and of many substances. 

Looking at the subject, then, from this point of view we may consider 
matter as the great storehouse of energy; and some idea of the efficiency of 

316 



THE CHEMICAL ELEMENTS. 317 

the storage may be had from the following: The hydrogen-oxygen flame is 
the hottest flame known, yet the amount of heat energy given off spon^ 
taneously by a definite quantity of radium is 30,000 times as great as that 
generated from the combustion of an equal weight of hydrogen. In the 
case of the radium the atoms burst, releasing the stored-up corpuscular 
energy, while in the case of the hydrogen the molecules burst and release the 
molecular and atomic energies only. The corpuscular energy is vastly the 
greater. It is estimated that the total work of disintegration of a small 
pinch of radium produces heat equivalent to the generation of one horse- 
power for one year! How to utilize, economically, the tremendous amount 
of stored energy in matter is a problem. We have seen that the ordinary 
process of combustion releases but an infinitesimal part of this stored energy, 
and we know that only a small percentage of even this is made available 
for useful work. Scientists are even now further speculating on the possi- 
bility of the corpuscles themselves being divisible into more minute particles 
containing vastly greater sources of energy, some of them advancing the 
theory that these corpuscles are made up of the invisible and universal 
ether of space. Others speculate on the possibility of corpuscles being 
electricity itself. 

The Electronic Theory. — Long before the recent discoveries, which led 
to the abandonment of part of the atomic theory of Dalton, chemists pre- 
dicted that, if at any future time it were found possible to transmute one 
kind of atom into another, there was little doubt that any one element of 
matter might be transmuted into any other element of a different kind. 
Profs. Ramsay and Soddy are credited with the discovery of the transmu- 
tation of radium emanations into an entirely different element, helium; 
and this wonderful discovery has brought forth new theories regarding the 
ultimate unit of matter: 

"The modern conception of the ultimate unit is the electron, and this, 
although by origin an electrical conception, is in reality a material concep- 
tion of no less than the atom of matter. The electron could be defined as 
the smallest existence known capable of isolation and of free movement 
through space. It is a definite amount or 'charge' of negative electricity; 
in a word, the smallest possible amount known to exist; for electricity, no 
less than matter, has been shown to consist of discrete particles or units, 
and not to occupy space continuously. Unlike the atoms of matter, only; 
one kind of electron is known, consisting of the same amount or charge of 
negative electricity with identical properties in all its various manifestations. 

It is certain that each atom of matter contains in the normal condition 
at least one electron, which it is capable of losing, and conversely that it may 
unite with at least one electron more than it normally possesses without deep- 
seated material change. An atom with one or more electrons less than it 
possesses in the normal state is positively charged and is often called a positive 
ion. Similarly an atom with one or more electron in excess is a negative ion.'* 

— Scientific American. 

The Elements. — At present there are about 90 known elements, that is, 
substances whose atoms are all alike in the particular element but different 
from those of any other element. Table 1 gives a fairly complete list of 
these elements, their symbols and some of their properties. The 2nd column 
represents, in a general way only, the division of the elements into three 
classes of importance, A, B, c. In the third column the names of those 
which are most common and widely distributed are in small capitals; 
while those which are rare are printed in italics. The atomic weights are 
based on two systems, both of which are frequently used: 

(a) The arbitrary weight of an atom of hydrogen = 1 (.*. 0= 15.87). 

{h) The arbitrary weight of an atom of oxygen = 16 (.'. H = 1.008). 
The first named, (a), is the most used. The 7th column shows which are 
the metallic elements; they are marked Met.; the other 13 are metalloids. 
The next two^ columns give the order in which many of the elements exist 
by quantity in the earth, air and sea combined; and opposite the nine 
leading ones are given the percentage by quantity of the total mass. Thus, 
oxygen is estimated to comprise one-half, silicon one-quarter, aluminum one- 
fourteenth, etc., of the total. The 10th column shows the state of the ele- 
ment under ordinary temperatures, as solid, liquid and gaseous. The third 
from the last column gives the molecular volumes of some of the elements; 
that is, whether the atoms are single, double, triple, etc., in the molecule. 
The last three columns are self-explanatory. 



318 



n.— NATURAL HISTORY OF MATERIALS. 



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THE CHEMICAL ELEMENTS. 



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17.--NATURAL HISTORY OF MATERIALS. 



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CHEMICAL COMPOUNDS. 321 

Compounds. — A compound is a chemical union of two or more elements, 
each molecule of the compound being a perfect likeness in miniature of 
the compound itself. It differs from a mixture in that no chemical change 
takes place in the latter, the original molecules of each of the substances 
mixed remaining intact. 

Compound substances are generally named so as 

(a) To denote the constituent elements. 

(b) To denote the kind of molecular grouping of the elements. 

(c) To denote the nature of a resulting compound by the addition or sub- 

traction of some particular kind of element. 

Simple Combinations, ides. — Many of the non-metallic elements 

when combined with a simple basic one take on the suffix ide or et, as, for 

instance, 

Iron and oxygen form iron oxide (oxide of iron). 

Lead " chlorine " lead chloride (chloride of lead). 

Potassium" bromine " potassium bromide (etc.). 

Hydrogen " sulphur " hydrogen sulphide (sulphuret of hydrogen). 

Copper " sulphur ** copper sulphide (etc.). 

But by reason of the fact that two substances often unite in different 
proportions the names of the resulting compounds are so contrived by 
Latin prefixes and suffixes as to indicate quite directly their molecular com- 
positions. Thus, 

5M6oxide of copper Cu2 O Ratio of O is < 1:1 

Protoxide of manganese ( = monoxide) Mw O ** " 1:1 

P^^ioxide of lead Ph O2 " " > 1 : 1 

Sesquioidde of manganese Mn2 O3 '* " 3:2 

jB^'woxide of silica ( = c?ioxide) 5-^02 " " 2:1 

r^roxide of chromium { = trtoxide) Cr O3 " " 3:1 

Peroxide of Nickel A^^'2 O3 " *• max. 

The same prefixes may be applied to other compounds, as chlorine, 
sulphur, etc. 

Acids, Bases and Salts. — When an acid and a base or alkali are brought 
together chemically in the proper proportions, they neutralize each other, 
and the resulting chemical products are, 1st, water and 2nd, salt. 

An acid may be defined as a substance containing hydrogen, which it 
readily exchanges for a metal when treated with a metal or metal compound 
called a base. 

A base is a substance containing a metal combined with hydrogen and 
oxygen, which metal it readily exchanges for hydrogen when treated with 
an acid. 

A salt is a neutral substance, one of the products of the action of an 
acid on a base, the other product being water. 

Oxides and Hydroxides. — An oxide is a compound of oxygen, more 
especially with a metal or metalloid. The principal classes of oxides are 
(1) basic or metallic oxides, and (2) acid oxides or acid anhydrides. 
Examples: (1), calcium oxide (CaO); (2), sulphuric anhydride (5 O3). 
A hydroxide may be formed (1) by treating oxides with water, and (2) by 
decomposing salts by the addition of soluble hydroxides to their solutions. 
Examples: (1), Ca O + H2 = Ca(0H)2; (2), Mg S04-h2 Na OH = Na2S04.+ 
Mg (0H)2. 

Acid Combinations. — An acid is a compound containing, in each mole- 
cule, one or more atoms of hydrogen which may be displaced by a metal or 
by a compound possessing metallic functions. If the acid molecule contains 
one hydrogen atom it is m^onobasic, two hydrogen atoms, bibasic; three 
hydrogen atoms, tribasic; more than one hydrogen atom, polybasic, etc. 

When acid contains oxygen the suffix — ic is usually given to the 
characteristic element of the compound, as sulphur-jc acid for H2 SO4. 
But if two acids can be formed with oxygen from the same characteristic 
element, that one which is the more highly oxidized has the suffix — ic while 
the other, the less oxidized, takes on the suffix — ous. In cases where more 
than two acids are formed iDy different proportions of oxygen the above are 



322 ir.—NATURAL HISTORY OF MATERIALS. 

further modified by the preffixes hyper =, meaning over, and hypo=, mean- 
ing under. Thus we may have according to the relative proportions of 
oxygen present, say in the sulphur acids: 

(a) /t:v^osulphurous acid, H2 OS2. (Least proportion of oxygen.) 

(b) sulphurous " H2 S O^. (Less proportion of oxygen.) 

(c) hypersnlphnrous " (Per often used instead of hyper.) 

(d) hyposnIphnTic " (This is preferable to c.) 

(e) sulphuric " H2 SO4 (Greater proportion of oxygen.) 

(f) hypersnlphuric " H2 S2 Og (Greatest proportion of oxygen.) 
Salts. — We have seen that an acid contains one or more atoms of hydro- 
gen which may be replaced by metallic atoms or basic radicals. When such 
a change takes place in an acid the resulting compound is a salt. The 
names of the salts so formed are related to the acids which produce them, as 

ic acids form ate salts. 

ous " " ite 



hypo ous " " hypo ite " 

Thus, sulphuric acid (H2 SO4) and soda (Na) form sulpha/^ of soda 
(Na2 SO4) ; hyposulphnrous acid (H2 SOs) and soda form hyposulphite of 
soda, etc. 

Avogadro's Law (of. Gases) ; sometimes called Ampere's Law.—" Equal 

volumes of all gases and vapors contain the same number of ultimate particles 
or molecules at the same temperature and pressure." 

Molecular Weight. — To determine the relative weights of molecules of sub- 
stances (which are either gaseous or can be converted into gases for this determi- 
nation) we have simply, from Avogadro's law (above), to determine the weights 
of equal volumes of the gases, at constant temperature and pressure. The system 
of molecular weights may be based on the atomic weight of hydrogen as 1, or 
oxygen as 16, the latter being more standard. 

Atomic Weights are deduced from molecular weights, and by chemical 
analysis, and other methods. 

Constitutional Formulas, expressing the composition of molecules and also 
the connecting relations existing between the atoms, are useful in chemical 
analysis. 

Valence of an element is that property by virtue of which its atom can hold 
in combination a definite number of other atoms. An atom like hydrogen, 
which can hold one atom in combination, is univalent, that is, its valency is 1; 
oxygen, which can hold two atoms, is bivalent; nitrogen, trivalent; carbon, 
quadrivalent; etc. 

Allotropy applies to two or more modifications of an element. Thus, oxygen 
has an allotropic form in ozone; carbon appears in three different forms; etc. 

Nascent'State. — When an element, as oxygen, is being evolved from a com- 
pound it will often unite with another element, as carbon; whereas, if the two 
elements are brought together at ordinary temperatures in the free state they 
will not unite. The explanation seems to be, that when the oxygen is being 
evolved its molecules are broken up into atoms, which latter readily unite with 
the carbon or other element. 

Periodic Law. — "The properties of the elements, as well as the forms and 
properties of the compounds, are in periodic dependence on, or form periodic 
functions of, the atomic weights of the elements." This law, conceived by 
Newlands and later perfected by Mendel^eff , Meyer and others, is a funda- 
mental law of Chemistry. If the elements are arranged in series on an 
ascending scale, according to their atomic weights, this series can be made 
to exhibit Family Groups showing common characteristics, in much the 
same manner as the natural divisions or classifications of the mineral, 
vegetable and animal kingdoms. 

Table 2 is such a grouping, by Meyer. The inclined lines indicate a 
spiral or continuous series if the table were wrapped around a cylinder. 
The vertical columns show the Groups, I, II, III, etc., with sub-divisions A and 
B. Each group has common characteristics, while those of the sub-divisions 
are still more in common. The vacant spaces are for other elements, most of 
which have not yet been discovered or their atomic weights determined. 
It is a significant fact that by means of this law Mendeleeff foretold the 
existence of some of the elements long before their discovery, and calculated 
their atomic weights correctly. 



PERIODIC LAW. 



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n.'-NATURAL HISTORY OF MATERIALS, 



Chemical Substances and Their Common Names. — 

(Knowledge Year Book.) 



Alum = Sulphate of aluminum and 
potassium. 

Aqua fort is = Nitric acid. 

Aqua regia = Nitro-hydrochloric acid. 

Calomel = Mercurous chloride. 

Carbolic acid = Phenol. 

Caustic potash = Potassium hydrate. 

Caustic soda = Sodium hydrate. 

Chalk = Calcium carbonate. 

Copperas = Sulphate of iron. 

Corrosive sublimate = Mercuric 
chloride. 

Cream of tartar = Bitartrate of po- 
tassium. 

Epsom salts = Magnesium sulphate. 

Fire damp = Light carburetted hy- 
drogen, methane. 

Glaubers* salt = Sodium sulphate. 

Grape sugar = Glucose. 

Goulard water = Basic acetate of 
lead. 

Iron pyrites = Sulphide of iron. 

Jewelers' putty = Oxide of tin. • 

Laughing gas = Nitrous oxide. 

Lime = Calcium oxide. 

Lunar caustic = Silver nitrate. 

Mosaic gold = Bisulphide of tin. 

Muriatic acid = Hydrochloric acid. 

Plaster of paris = Calcium sulphate. 



Realgar = Sulphide of arsenic. 

Red lead = Oxide of lead. 

Rochelle salt = Sodium potassium tar- 
trate. 

Sal ammoniac = Ammonium chloride. 

Salt, common = Sodium chloride. 

Salt of tartar (potash) = Potassium 
carbonate. 

Saltpetre = Potassium nitrate. 

Salt of lemon = Oxalic acid. 

Slacked lime = Calcium hydrate. 

Soda, washing = Sodium carbonate. 

Soda, baking = Sodium bicarbonate. 

Soda = Sodium carbonate. 

Spirits of hartshorn = Solution of Am- 
monia. 

Spirits of salt = Hydrochloric acid. 

Sugar of lead = Lead acetate. 

Tartar emetic = Potassium antimony 
tartrate. 

Verdigris = Basic acetate of copper. 

Vermillion = Sulphide of mercury. 

Vinegar=Dilute Acetic acid. 

Vitriol, blue = Copper sulphate, 
green = Ferrous sulphate, 
oil of = Sulphuric acid. 
white = Zinc sulphate. 

Volatile alkali = Ammonia. 



B.— MINERALOQICAL. 

Minerals Compose Rocks. — Mineralogical substances include all those 
natural objects which belong to the Inorganic Kingdom, whether solid, 
liquid or gaseous. A mineral may be defined as a natural, inorganic, homo- 
geneous, chemical substance; while a rock is made up of one or more mineral 
masses mixed in fairly constant proportions throughout. 

(a) Hardness. — The hardness of a mineral is its resistance to abrasion. 
There are ten recognized degrees of hardness, typified by the following 
minerals: 

1. Talc. — Soft and greasy; easily scratched by finger nail. 

2. Gypsum. — Not easily scratched by the finger nail. 

3. Calcite. — Scratches copper coin ; is scratched by same. 

4. Fluorite. — Not scratched by copper; and does not scratch glass. 

5. Apatite. — Barely scratches glass; easily scratched by a knife. 

6. Orthoclase. — Easily scratches glass; barely scratched by a file. 

7. Quartz. — Not scratched by a knife. 
8? Topaz. — Not scratched by a file. 

9. Sapphire. — About Vioo as hard as diamond. 
10. Diamond. — Hardest substance known. 



{h to /) . Other Physical Characteristics. — Minerals are often identified by 
the above characteristics of hardness and also by the kinds and degrees of 
the following: 

(b) tenacity, (c) cleavage, (d) fracture, (e) feel, (f) taste, (g) odor, 
(h) lustre, (i) color, (j) transparency, (k) translucency, (1) opaque- 
ness, etc. 

For Blowpipe Characteristics (m) see page 328. 



MINERALOGICAL. 325 

3. — Classification of Important Mineral Species.* 

I. Native Elements. 

A Series. — Non-basic or electro-positive elements. 

1. Gold Group. — Gold, silver, hydrogen, potassium, sodium, etc. 

2. Iron Group. — Platinum, palladium, mercury, copper, iron, zinc, lead, 
cobalt, nickel, chromium, manganese, calcium, magnesium, etc. 

3. Tin Group. — ^Tin, titanium, zirconium, etc. 

B Series. — Elements generally electro-negative. 

1. Arsenic Group. — Arsenic, antimony, bismuth, phosphorus, vana- 
dium, etc. 

2. Sulphur Group. — Sulphur, tellurium, selenium. 

3. Carbon-Silicon Group. — Carbon, silicon, diamond, graphite. 
C Series. — Elements always negative. 

1 Chlorine, bromine, iodine. 

2. Fluorine. 

3. Oxygen. 

II. Sulphides. — Sulphides, tellurides, selenides, arsenides, antimonides, bis- 

muthides. 

A.. Binary Compounds. — Sulphides and tellurides of metals of the 

sulphur and arsenic groups. 

(a) Realgar Group. RS. Realgar, As 5. 

(b) Orpiment Group. R2 S3. Orpiment, As2 Sz 

Stibuite, S62>53 
Bismuthinite, Bi2 S3. 

(c) Tetradymite, Biz (Te, 5)3. 

(d) Molybdenite Group. R S2. Molybdenite, Mo 52. 

B. Binary compounds. — Sulphides, tellurides, etc., of metals of the gold 
iron and tin groups. 

1. Basic Division. — Dyscrsisite, A g^ Sb; A geSb. Domey'kite, CU3 As, 

2. Proto Division. RS (or i?2 5), i? Se, R Te. 

(a.) Galenite Group. Argentite, Ag2 S. Crookesite, (Cu2, Tl, Ag) Se, 
Galenite, Pb S. 

(b) Blend Group. Sphalerite, Zn S, 

(c) Chalcocite Group. 

(Cinnabar, HgS (or Hgs S3). Millerite, NiS. 
(d) Pyrrhotite Group. •< Pyrrhotite, Fej Ss mostly. 

( Greenockite, Cd S (or Cds S3). 

3. Deuto or Pyrite Division. 

(a) Pyrite Group. Pyrite, Fe S2. Chalcopyrite, Cu Fe Sz- 

(b) Marcasite Group. Marcasite, Fe S2. 

Arsenopyrite, FeAs S = Fe S2 + Fe As2. 
Sylvanite. 

C. Ternary Compounds. — Sulpharsenites, sulphantimonites, sulphobis- 

muthites. 

(a) Group I. R (As, Sb)2S^ = RS-\-{As Sb)2 S-s. 

(b) Sub Group. Ra (As, Sb, Bi)^ So=S RS-{-2iAs, Sb, Bi)2S3, 

(c) Group II. R2 (Sb, As)2 Sc= 2i?S+ (Sb, As)2 S3. 

(d) Group III. R3(As,Sb)2S6=SRS+(As,Sb)2S3. 

(e) Group IV. R4(As, Sb, Bi)2S7 = ^RS+ (As, Sb, Bi)2S3, 

(f) Group V. R5(As, Sb)2 S8 = 5RS+ (As, 56)2 S3. 

III. Chlorides, Bromides, Iodides. 

A. Anhydrous Chlorides. R(Cl, Bi, I) ; R2(Cl, Br, I) ; R Ck. 

Halite, NaCl. Calomel, HgCl. 

Sylvite, KCl. Sal Ammoniac, NH^Cl, 

B. Hydrous Chlorides. 

C. Oxychlorides. 

IV. Fluorides. 

A. Anhydrous Fluorides. { Fl-rit|; ^-J^l F.,(or ^NaF+AlF^. 

B. Hydrous Fluorides. 



* Mostly in accordance with Dana's Classification. 



3^6 17.— NATURAL HISTORY OP MATERIALS. 

3. — Classification op Important Minerals. — Cont'd. 

V. Oxygen Compounds. 
a. Oxides. 

A. Oxides of metals of the gold, iron and tin groups. 
1. Anhydrous oxides. 

(a) Protoxides. RO or (i?20). Cuprite, Cuz O. Zincite, Zn O, 

Tenorite, Cu O. 

(b) Sesquioxides, RO3. Corundum, Al O3. Hematite, Fe O3. 

(c) Compounds of protoxides and sesquioxides, RR O4 (or RO + R O3). 

Spinel Group.— Spinel, Mg Al 0^{ = Mg + Al O3). 

Magnetite, Fe F« O^ior Fe^ O4) ^Fe + Fe O3. 
Franklinite, (Fe, Zn, Mn) {Fe, Mn) O4. 
Chromite, Fe Cr O4, or {Fe, Mg, Cr) {Al, Fe, 

Cr)04. 
Uranite, L/g Og{U O2+ 2L7 O3). 

(d) Deutoxides. RO2. 

2. Hydrous oxides. Targite, H2 F2 O7 Diapose, H2AI O4. 
Gothite, H2 Fe Oi = H^e 06+ 2 Fe O3. 
Manganite, H2 Mn 0^ = 11 eMn Oq+2 Mn O3. 
Limonite, if e Fe2 Oq = Hq Fe Oq + Fe O3. 

B. Oxides of metals of the arsenic and sulphur groups. 

Arsenolite, As2 O3. Bismite, Bt2 O3. 
Molybdite, Mo O3. Tungstite, W O3. 

C. Oxides of the carbon-silicon group. Quartz, Si O2. 

Tridymite, 5* O2. Opal, Si O2. 

fi. Ternary Oxide Compounds. 
A. Silicates. 

1. Anhydrous silicates. 
1*» Bisilicates. R Si O3. 

(a) Amphibole Group. — Wollastonite, Ca Si O3. 

Pyroxene ( = Augite) ,RSi03:R may 
be Ca, Mg, Fe, Mn, Zn, Kaz, Naz. 

(b) Amphibole Section. Amphibole, R Si O3 (var., Horn- 

Beryl, Be3 Al Sie Ois. blende) . 

2° Unisilicates, R2 Si O4. 

(a) Chrysolite Group. — Chrysolite, {Mg, Fe)2 Si O4. Olivene. 

(b) Willemite Group. — Willemite, Zw2 Si O4. 

(d) Garnet Group. — R3 RSis O12. 

(e) Vesuvianite Group. 

(f) Epidote Group. — Epidote, H2 Ca^ R3 Sio O26. 
(k) Mica Group. — Biotite. Muscovite. 

(1) Scapolite Group, 
(m) Nephelite Group. 

(n) Feldspar Group. — Anarthite. Labradorite. Andesite. 
Oligoclase. Albite. Orthoclase. 

3° Subsilicates. 

(a) Chondrodite. Tourmaline. 

(b) Andalusite. Fibrolite, Al Si O5. Cyanite, Al Si O5 
Topaz, Al Si O5. Euclase, H2 Be2 Al Si2 Oiq. 
Datolite, H2 Ca2 B2 5*2 Oio. Titamite, Ca Ti Si O5. 

(c) Staurolite. 

2. Hydrous Silicates. — I. General Section. 

Bisilicates. — Pectotite; lanmontite; okenite; crysocolla; alipite. 
Unisilicates. — Calamine; prenite. — ^Thorite, Pyrosmalite. 
Subsilicates. — Allophane. 

II. Zeolite Section. 



MINERALOGICAL. 327 

3. — Classification op Important Minerals. — Cont'd. 
Oxygen Compounds — Cont'd. 

III. Margarophyllite Section. 

Bisilicates. — Talc; pyrophyllite; sepiolite. 

Unisilicates. 

Serpentine Group. 

Kaolinite Group. 

Finite Group. 

Hydro-mica Group. 
Subsilicates. 

Chlorite Group. 

B. Tantalates, Columbates. 

C. Phosphates, Arsenates, Vanadates, etc. 

Anhydrous. 

Xenotime, Yz P2 Os' 
Apatite Group. — Apatite. Vanadinite. 
Arjtimonates. 

Hydrous. 

Antimonates. 

Nitrates. Nitre, K N O3. Soda nitre. Na NO3. 

D. Borates. Boracite. Mgj B^q Ch O3 = 2Mg-s ^g O15 + Mg CI2. 

Borax, Naz 5^ O7+ 10 aq=2{Na BOz-hHBOz) + 9ag. 

E. Tungstates (R W O4) , Molybdates (R Mo O4) . Chromates {R CiO^), 

F. Sulphates. 

Anhydrous R SO 4,. 

Barite Group. Barite, Ba SO4. Anglesite, Pb SO4. 

Hydrous. Mirabilite, Nao SO4 + 10 aq. 

Gypsum, Ca SO4+ 10 aq. var., selenite. 
Epsomite, Mg SO4+ 7 aq. 

Copperas Group. — Chalcanthite, Ca 5O4+ 5 aq. 
Alum and Halalrichite Groups. — Copiatite. 

Aluminite AIS06+ 9 aq. 
Tellurates. — Montanite, Biz Fe 0^+2 aq. 

G. Carbonates. 

Anhydrous. 

Calcite Group. R COs. Calcite, Ca CO3. 

1. Crystallized: Iceland spar, Fountainblue limestone, 
Satin spar. 

2. Massive: Granular (Saccharroidal) : Marble, Statu- 
ary marble. Shell marble, Lithographic stone, Brec- 
cia marble. Pudding stone marble, Hydraulic lime- 
stone. 

Soft Compact Limestone: Chalk, Calcareous marl. 
Concretionery, massive: Oolite. 

a. Stalactites. 

b. Stalagmites. 

c. Calc-sinter. Travertine, Calc-tufa. 
.d. Agaric mineral. 

e. Rock-meal. 

Dolomite, (Ca, Mg) CO3. 

Aukerite, Ca COs-hFe CO3+ x (Ca Mg C2 Oo). 
Magnesite, Mg CO3. 
Siderite, Fe CO3. 
Rhodochrosite, Mn CO3. 
Smithsonite, Zn CO3. 
Aragonite Group. 
Hydrous. 



328 17.— NATURAL HISTORY OF MATERIALS.- 

3. — Classification op Important Minerals. — Concrd. 
VL Hydrocarbons. 

1. Simple (no oxygen). 

(a) Marsh Gas Series. CnH2n+2' Includes liquid naphthas and 
more volatile parts of petroleum; also scheercrite and 
chrismatite. 

Petroleum. — Mineral oil, Kerosene, Bergol, Erdol. 

(b) defiant or Ethylene Series. Cn if 2n. Pittolium group of 
liquids or pettarsphalts (mineral tar) and the paraffines. 
Paraffin Group. 

(c) Camphene Series. Cn H2n-4. 

(d) Benzole Series. Cn if 2n-6. Benzole liquids. 

(e) Naphthalin Series. Cn H-za-xi- 
Naphthalin (found in Rangoon tar). 

2. Oxygenated. 

Succinate (amber). 
Appendix to Hydrocarbons. 

Asphaltum. — Bitumen, Asphalt, Mineral pitch. — Mixture of dif- 
ferent hydrocarbons, part of which are oxygenated. 

Following substances are closely related to asphaltum: 
Grahamite, Albertite, Pianzite, Wollongongite. 
Mineral Coal. — Made up of different kinds of hydrocarbons with 
perhaps, in some cases, free carbon. 

1. Anthracite. 
Bituminous. 

2. Coking. 

3. Non-coking. 

4. Cannel coal. 
6. Torbanite. 

6. Brown coal. 

7. Earth brown coal. 

(m). Blowpipe Characteristics.— The following important flame colorations 
result from the blowpipes: Carmine, from lithium compounds; scarlet, from 
strontium compounds; yellowish red, from calcium compounds; yellow, from 
sodium and all its salts; yellowish green, from barium compounds, molybde- 
num sulphide and oxide; pure green, from compounds of tellurium or thal- 
lium; emerald green, from most copper compounds with hydrochloric acid; 
bluish green, from phosphoric acid and phosphates with sulphuric acid; 
feeble green, from antimony or ammonium compounds; whitish green, from 
zinc;^ light blu£, from arsenic, selenium and lead; azure blue, from copper 
chloride; violet, from potassium compounds. (Other characteristics, p. 324.) 

Some of the Important Minerals. 

The Gold Minerals. 
Tellurides. — Sylvanite, calaverite, krennerite. 
Ores. — Pyrite, arsenopyrite, pyrrhotite, and various sulphides, etc. 

The Silver Minerals. 
Ordinary silver ores contain less than 1% of the silver compounds 
distributed through various minerals. 

The Potassium Minerals. 
Chlorides. — Sylvite, carnallite, kainite. Sulphates. — Kalinite. Nitrate. — 
Nitre. The chlorides are the natural potash salts. 
The Sodium Minerals. 
Chloride. — Halite. Sttlphate. — Mirabilite. Nitrate. — Soda nitrate. Cat' 
bonate.— From Naz CO3, H Na CO3. 2H2O. 

The Lithium Minerals. 
Used in medicine. 

The Platinum and Iridium Minerals. 
Metals. — Platinum, iridosmine. Arsenide. — Sperrylite. ^ Purified plati- 
num. — Used in incandescent lamps, for electrical contact points, and in the 
so-called "oxidizing of silver." Iridosmine. — Pointing gold pens; phosphide 
of iridium is used for pointing tools and stylographic pens and for knife 
edges in the most delicate balances. 



IMPORTANT MINERALS. 329 

The Mercury Minerals. 

Sulphide. — Cinnabar. Chloride. — Calomel. 

Uses. — Mercury is used in extraction of gold and silver from their ores; 
manufacture of Vermillion; barometers, thermometers; silvering mirrors; 
medicine. 

The Copper Minerals. 

Sulphides. — Chalcocite, bornite, chalcopyrite. Sulpho-arsenite. — Enarg- 
ite. Sulphoantimonite . — Tetrohedrite. Oxides. — Cuprite, tenorite. Basic 
chloride. — Atocamite. Sulphates. — Chalcanthite. Carbonates. — Malachite, 
azurite. Silicates. — Chrysocolla, diastase. 

Or^5.— -Chalcopyrite, bornite, native copper, cuprite, malachite, augite. 

Uses of copper. — Electrical work; alloys with zinc and tin, such as brass, 
yellow metal, bronze, bell metal. German silver. 

The Iron Minerals. 

Metal. — Iron. Sulphides and arsenides. — Pyrrhotite, pyrite, marcasite, 
arsenopyritCj leucopyrite. Oxides. — Magnetite, franklinite, hematite, menac- 
canite, turgite, goethite, melanterite. Sulphates. — Copiatite, melanterite. 
Phosphates. — Vivianite, triphylite. Arsenates. — Scorodite, pharmacosiderite. 
Carbonate. — Siderite. Chromate. — Chromite. 

Ground for paint. — Limonite, hematite. Used as iron ores. — Hematite, 
limonite, magnetite, siderite. For extracting sulphur. — Pyrite, marcasite, 
pyrrhotite. For arsenic. — Arsenopyrite. For chromium. — Potassium bi- 
chromate, potassium chromate, ferro-chromium. For tungsten. — Wolfram- 
ite, scheelite. For gold and silver. — Pyrite, arsenopyrite. For nickel. — 
Pyrrhotite. 

The Cadmium Minerals. 
Sulphide. — Greenockite, is a yellow pigment of fixed color. 

The Zinc Minerals. 
Sulphide. — Sphalerate. Oxide. — Zincite. Sulphate. — Gostarite. Car- 
bonates. — Smithsonite, hydrozincite. Silicates. — Willemite, Calamine. 

^ Ores of zinc. — Spalerite, smithsonite, calamine, willemite, zincite. Zinc 
oxide also made from franklinite. 

Uses of metallic zinc. — Galvanizing iron wire or sheets; manufacturing 
brass; sheet zinc, zinc dust. Zinc white, a paint, is zinc oxide ground in oil. 

The Lead Minerals. 

Ores. — Galenite, cerussite. Uses of lead. — Manufacture of white lead, 
preparation of red lead, litharge, shot, lead pipe, sheet lead. Alloys. — With 
antimony for type, and, friction-bearings. 

The Cobalt Minerals. 

Sulphides and arsenides. — Linnaeite, cobaltite, smaltite. Arsenates. — 
Erythrite. 

Cobalt blue. — Cobalt and alumina compound. Ainman's green. — Com- 
pound of cobalt and zinc oxide. 

The Nickel Minerals. 

Sulphides. — Millerite, pentlandite, gersdorfiite. Arsenides. — Nicolite, 
chloanthite. Arsenate. — Annabergite. Carbonate. — Zaratite. Silicate. — 
Gamierite. 

Alloyed with copper, iron and arsenic in making German silver; with 
copper (25% Ni and 75% Cu) in the five-cent piece; with steel in making 
nickel steel. 

The Manganese Minerals. 

Sulphide. — Alabandite. Oxides. — Braunite, hausmaunite. pyrolusite, 
manganite, psilomelane. Carbonate. — Rhodochrosite, wad. 

Alloys with iron, form: Spiegeleisen, ferro-manganese (for manufacture 
of steel). Used as manganese ores. — Pyrolusite, psilomelane. 

The Calcium Minerals. 
Limestone and marble are massive calcite and dolomite and are used 
largely in building construction. Limestone is used for hydraulic cements; 
and gypsum, for plaster of paris and wall plaster; also in paper making. 



I 



330 17.— NATURAL HISTORY OF MATERIALS. 

The Magnesium Minerals. 

Calcined magnesite is used as a lining for converters in the basic process 
for the manufacture of steel. 

The Tin Minerals. 
Sulphide. — Stannite. Oxide. — Cassiterite (ore of tin). 
Uses of tin. — Tin plate (sheet iron coated with tin) for making cans, 
kitchen utensils, etc. Alloy. — Bronze, bell metal, pewter, solder, tin amalgam. 

The Titanium Minerals. 
Oxide of titanium used for glazing porcelain. 

The Thorium Minerals. 
Oxide of thorium, thoria, constitutes about 99% (other 1% is cerium 
oxide) of the mantle of the Welsbach incandescent gas lamp. 

The Arsenic Minerals. 
Sulphides. — Orpiment, realgar. Sources. — From arsenides and arseno- 
sulphides of iron, cobalt and copper. The white arsenic is a poisonous 
oxide, and is used in dyeing, medicine, sheep washing, calico printing, timber 
preservative, for fly paper, rat poisons and glass manufacture. Paris green 
is an arsenate of copper. 

The Antimony Minerals. 
Sulphides. — Stibnite, kermesite. Oxide. — Valentinite. 
Uses. — ^The sulphide is used in vulcanizing rubber. 

The Bismuth Minerals. 

Uses. — Alloys with tin, lead and cadmium, expand in cooling. 

The Sulphur and Tellurium Minerals. 

Uses of Sulphur. — Manufacture of sulphuric acid, gunpowder, matches, 
rubber goods, bleaching, medicines, etc. Tellurium is closely related to 
sulphur in a chemical way. 

The Uranium Minerals. 
A small quantity of uranium in steel increases the elasticity and hard- 
ness. Ore. — Uraninite (principal source of radium). 

The Molybdenum Minerals. 

Sulphide. — Molybdenite. Oxide. — Molybdite. 

Uses. — ^The metal is becoming important as an alloy with steel. 

The Aluminum Minerals. 

Ores. — Bauxite is the principal ore. 

Uses of Aluminum. — Where lightness, strength and non-corrosiveness 
are desirable. It is replacing sheet copper and zinc and is used as bronze 
powder and alumina leaf for silvering letters and signs. It is replacing 
copper wire as electrical conductors. In metallurgy it is becoming important 
in welding pipes, rails in place, and steel castings. By adding less than 1% 
of aluminum to the melted metal it prevents blow-holes in castings of steel, 
copper and zinc. 

The Boron Minerals. 

Acid. — Sassolite. Borates. — Borax, ulexite, colemanite, boracite. Borax 
is used in welding; as a base of enamels on metal or porcelain; as a flux; 
as an antiseptic in packing meat; for powders, soaps, washing, dyeing, 
tanning. 

The Hydrogen Minerals. 
Water is the most important, one-eighth of its volume being hydrogen. 
Hydrogen is present in combination with carbon forming marsh gas, petro- 
leum, ozocerite, etc. 

The Carbon Minerals. 

Diamond and graphite are pure carbon. It is present in a large number 
of solid, liquid and gaseous compounds, as natural gas, petroleum, asphalts, 
bitumens, fossil resins, mineral waxes, coal. Carbon is present in all organic 
matter, entering with plant life from the carbon dioxide of the air; and 
exists to a large extent in natural gas. 



I 



IMPORTANT MINERALS. LITHOLOGY. 331 

The Silica Minerals. 
A. — Silica. 

B. — Anhydrous Silicates. 

1. Disilicates and polysilicatees. 
II. Metasilicates. 

III. Orthosilicates. 

IV. Subsilicates. 
C. — Hydrous silicates. 

I. Zeolite Division. 
II. Mica Division. 

III. Serpentine and Talc Division. 

IV. Kaolin Division. 
D. — ^Titano silicates. 

Some of the Most Important Silicates: — 

Granite. — Granite, syenite, gneiss, basalt, diorite, andesite, which are 
compounds of silicates (usually three or more) usually contain quartz, the 
feldspars with micas, pyroxene and amphibole. 

Sandstone. — Made up of grains, mainly quartz, with perhaps feldspar, 
mica, or other mineral: 
siliceous; ] 

ferruginous; I According to the nature of the cement which binds the 
calcareous; [ grains together, 
argillaceous; J 

Bluestone. — Hard, durable, fine-grained sandstone, cemented with sili- 
ceous material. 

Slate. — Hardened clay. Used mostly for roofing, sinks, blackboards. 

Fibrous talc and compact talc. — Used in paper manufacture. The latter, 
soapstone, is very valuable because refractory, for use in furnaces, crucibles, 
sinks, baths, hearths, electric switch-boards, cooking utensils; also used in 
paints, slate pencils, crayon, gas tips, as a lubricant, and in soap making. 

Mica. — Muscovite, phlogopite, biotite, have become of great importance 
as non-conductors in electrical apparatus; also for stove and furnace doors. 

Asbestos. — ^This is fibrous amphibole and serpentine. It is used in the 
manufacture of incombustible paper, cloth, cement, boiler and steam-pipe 
covering and packing rope or yarn for valves. 

Serpentine. — A marble. 

Feldspar. — Crushed and mixed with kaolin in the manufacture of 
porcelain. 

Quartz.- — Used in the manufacture of sandpaper, glass, porcelain, hone- 
stones, oilstones and as a flux. Ground flint is used as a scouring agent in 
soaps. ^ ^ 

Infusorial earth. — Calcined and made into water filters, polishing pow- 
ders, soap filling and boiler and steam-pipe covering. Used in dynamite. 

Kaolinite and clay. — Clay is decomposed feldspar and other silicates. 
It lies in beds composed partly of some hydrous aluminum silicates, as 
kaolinite, but is usually mixed with quartz, mica, undecomposed feldspar, 
oxides and sulphides of iron. The industry includes the manufacture of 
common brick, paving brick, fire-brick, hydraulic cement, earthenware, 
sLoneware, porcelain, terra cotta, sewer pipe, drain tile. 

Fuller's earth. — A clay used in refining and clarifying mineral oils. 

Rocks and Rock-Formations (Lithology) . 

Composition. — The chemical elements which enter chiefly into the com- 
position of the rocks may be classed as metallic and non-metallic, as follows: 
Metallic (basic) elements. — Aluminum, magnesium, calcium, iron, sodium 

potassium. 
Non-metallic (acidic) elements. — Oxygen, silicon, carbon, sulphur, phos- 
phorus, chlorine, fluorine, hydrogen. 

These are combined in various ways, forming the numerous minerals 
which compose the rocks. Table 4, following; gives a list of the principal 
rock-forming minerals, arranged as per E. S. Dana's classification. It will 
be seen that silica and the silicates play a most important part. The cal- 
careous group of minerals comprise mainly the lime and magnesium carbon- 
ates, and the lime phosphates and sulphates, so that lime is a very important 
constituent. Iron is found plentifully in most of our heavier rocks, being 
associated principally with oxygen, sulphur and carbon. The hydro-carbons 
furnish a most useful group of substances to the engineer. 



332 



17.— NATURAL HISTORY OF MATERIALS. 







wide- 
phide. 
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and 

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THE ROCK-FORMING MINERALS. 



333 



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334 



17.— NATURAL HISTORY OF MATERIALS. 



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THE COMMON ROCKS. 339 

Notes on Preceding Table. 

Class I. — ^This group of rocks illustrates the formation, by various 
cementing processes, of pudding stone and conglomerate from ordinary 
beach pebbles; limestone breccia from limestone pebbles; conglomerate 
from old broken rocks; and sandstone from sand. Gravel consists of pebbles 
worn by the action of water to the size not smaller than a pea, while sand 
is the result of greater wear, the grains being minute. Shingle is larger 
than gravel. Quartz is a large constituent of gravel owing to its resistence 
to abrasion, hence old gravel is a durable material for building operations. 
Clay is mainly decomposed feldspathic rock in place. It is made up princi- 
pally of silica and alumina, and contains also lime, magneisa, iron oxides, 
etc. As_a cement it is called argillaceous. 

The rock cements are mostly argillaceous (clayey), carbonate of lime 
(calcite), ferruginous (iron compounds), silica, arenaceous (sandy-quartz 
sand), iron oxide, etc. Very often they are a combination of these or per- 
haps consist of certain of their elements. 

Cement gravel is partially formed rock composed of cemented gravel 
and sand. It occurs in various degrees of hardness and is often difficult to 
classify by the engineer, either as earth or rock. A separate classification is 
often preferred, as it is usually more easily broken up by blasting than by 
plowing; that is, powder is cheaper in some cases than the specified "six- 
mule team." 

Class II. — ^This is a distinct group — sandstones. It is composed of strat- 
ified grains of sand cemented together. The nature of the cementing material 
determines largely the hardness or crushing strength of the sandstone, silica 
being superior to carbonate of lime or iron oxide in both strength and dura- 
bility. Quartz is the principal constituent of sandstone; but mica, feldspar, 
hornblende and augite are often present. The sandstones lie midway between 
the conglomerates (pebble base) and the shales (indurated clay base). 
They may also be said to occupy a structural position between almost loose 
sand, in which there is little or no cementing material, and the consolidated, 
metamorphosed varieties which resemble granite. If the cementing material 
is carbonite of lime and very abundant, it gradates into limestone. 

Of the varieties, quartzite is the hardest; arkose contains much feldspar; 
freestone, as its name implies, is easily worked ; brownstone is an impure 
sandstone, rather soft, and found in Connecticut and New Jersey; flagstone 
is shalely and used for pavements, and includes the bluestone variety. 

Class III. — Slate is a metamorphic clay or shale, produced by great 
heat and pressure. The old cleavage planes of the shale are changed to 
new cleavage planes in the slate, and when these are perfect it forms the 
typical roofing slate. Slate is quarried in Maine, Vermont, New York, 
Pennsylvania, Maryland, Georgia, Minnesota, California. 

Classes IV. and V. — These two groups comprise mainly the limestones, 
marbles and dolomites. Under Class III. it was seen that a sandstone con- 
taining an abundance of calcareous (carbonate of lime) cement grades into 
limestone when the cementing material increases to such an extent that it 
really forms the base of the rock, the sand diminishing in quantity and 
becoming merely accessory, forming the impurities. Pure limestone is 
carbonate of lime, the cementing material of many other rocks. Limestone 
may contain also a small amount of carbonate of magnesia and such im- 
purities as silica, alumina, ferric oxide, etc. When, however, the propor- 
tion of carbonate of magnesia increases to a considerable extent the 
limestone grades into dolomite. Further, if it contains a large amount of 
the above so-called impurities — silica and alumina — it becomes hydraulic 
limestone. Coquina and tufacious limestones in Class III., and crinoidal 
limestone in Class IV, represent different sources of formation or origin. 

Marble is a name formerly applied to all limestones of crystal and granu- 
lar texture, capable of taking a polish; but it is now commercially used 
to apply to all limestones (even the non-crystalline) which can be polished. 
_ Gypsum is composed of one-third lime {Ca O) , nearly one-half sulphuric 
acid {H2SO4), and one-fifth water. Alabaster is a pure white variety, while 
gypsite contains more or less earthy impurities. Plaster of paris is obtained 
by heating gypsum to above 127° C. (261° F.), driving off about one-seventh 
of the water of crystallization, and then grinding to powder. 



340 17.— NATURAL HISTORY OF MATERIALS. M 

Halite, No. 60, is sodium chloride (Na CI) in the crystal form. Sodium 
chloride, common salt, plays a very important part in the manufacture of 
some high explosives. 

Classes VI. and VII. — Gneiss is a name given to a class of metamorphic 
rocks composed essentially of the same elements as granite, namely, ortho- 
clase, quartz and mica, with perhaps pyroxene, hornblende, etc. They 
may be either of igneous origin, like the granites and syenites, or of sedi- 
mentary origin, as the schists, presenting all graduations between these 
two classes. As they merge from the granitic to the schistose textvue the 
feldspar disappears. Gneisses may be granitic (granite-gneiss), syenic 
(syenite-gneiss), homblendic (hornblende-gneiss), etc., according to its 
composition. 

Schist is a term applied to a metamorphic rock of slaty appearance, 
and of imperfect crystallization. Thus we have mica schist, hornblende 
schist, staurolitic schist, albite schist, etc. A slate is a schist of argillaceous 
character, forming a rather distinct class. 

Soapstone, a magnesian silicate, is a talc schist and is very useful for 
hearthstones, sinks, etc. 

Greensand or marl is composed mostly of glauconite (a silicate of iron 
and potash) and is used as a fertilizer. 

Classes VIII. , IX. and X. — These are distinctly igneous (eruptive) 
rocks, with more or less change in composition due to age of weathering^ 

Lava from an eruptive volcano is molten rock which subsequently 
hardens in cooling. Many attempts have been made to classify lava ac- 
cording to its composition or texture, but no satisfactory results have yet 
been attained. In density it varies from pumice, an extremely light, 
frothy, spongy form of obsidian (rhyolite) usually, to the heavy, compact 
basalt, which is usually the last of the lava flow. 

Obsidian is a volcanic glass consisting approximately of silica (70 per 
cent), aluminum (12), calcium, potassium, sodium, iron, etc. The prin- 
cipal kinds of obsidian are rhyolite and trachyte. 

Rhyolite has about the same chemical composition as granite, and 
from it, through weathering, etc., the latter is formed. Soda-rhyolites con- 
tain much sodium oxide; porphyritic-rhyolites are massive and compact. 
Felsite is a rhyolite which is not porphyritic. 

Trachyte has about the same chemical composition as syenite, which, 
like granite, is formed through the action of weathering, changing slightly 
the chemical composition and, to a marked degree, the texture. 

Basalt is a porphyritic rock of basaltic or columnar form. It is com- 
posed of hornblende, feldspar, augite, iron, and various other minerals. 
As it weathers very slowly it is synonymous with greenstone, trap, " basalt," 
etc., which comprise the so-called trap rocks. Melaphyr is a Tertiary 
basalt. 

C— BOTANICAL. 

About three-eighths of the area of the United States, or 700,000,000 
acres, is wooded; and of this, about 14 per cent comes under the United 
States Forest Service. The rapid depletion of our timber is a matter of 
grave concern, calling for abundant National and State reservations. The 
National forest reserves are situated usually at the heads of our tributary 
streams; serving, at the same time, to equalize the " run-off " of the pre- 
cipitation, by retarding the melting of the snows in the spring of the year. 

The following is a classification of the most important lumber trees of 
the United States and Canada: 



Arbor Vitaes.... 


..pp.342 


Cypresses pp. 342-3 


Oaks 


.pp 344-5 


Ashes 


346 


Elms, 345. Firs. 342 
Gums (Sweet) . . 345 


Pines 

Poplars . . . . 


.. 341 


Beeches 


.. 344 


.. 343 


Birches 


.. 344 


Hemlocks 342 


Redwoods . 


.. 342 


Cedars 


.. 342-3 


Hickories 343 


Spruces 

Walnuts... 


.. 341-2 


Chestnuts 


.. 344 


Larches 341 


... 343 


Cottonwoods.. 


... 343-4 


Maples 346 


Willows.... 


.. 344 



BOTANICAL— WOOD. 



341 



6. — Classification op Important Trees. 



Species. 



Height. Dia. Principal Localities. 



Uses. 



The Pines. Genus, Pinus. 
Soft Pines. 



White P. (P. strobus.) . . 



Silver P., Mountain P. . . 

(P. monticola.) 
Sugar P. (P. lambertiana) 



Rocky Mountain White 
P. (P. flexilis.) 



60'-120' 3'-4' Atlantic States. 



100' 

200'-220' 
-300' 

40'-75' 



4'-5 
6'-10' 

2'-5' 



Pacific Northwest. 

Oreg. to So. Calif., 
along W. Slopes Cas- 
cades, Sierra Nev. 
E. slope Rocky Mts. 
from B.C. to Tex- 
as. Ariz and N. 
M. 



Construction ; cabi- 
net work ; masts ; 
boxes. 

Lumber in Mon- 
tana and Idaho. 

Carpentry, doors, 
blinds, shingles. 
Sap yields sugar. 

Lumber in Eastern 
Montana, Ariz., 
and N. M. 



Pitch Pines. 



Western Yellow P., Bull 
P. (P. ponderosa.) . . , 

Longleaf P.. Southern P 
(P. paliistris.) 



Loblolly P., Old-Field P. 
(P. taeda.) 



Norway P., Red P. 
(P. resinosa.) 



Shortleaf P 

(P. echinata.) 



Cuban P., Swamp P. 
Slash P 

(P. heterophylla.) 
Spruce P., Cedar P. . 

(P. glabra.) 



150'-230' 


5'-8' 


90'-120' 


2'-2^' 
-3' 


80'-100' 


abt.2' 
-3' 


75'-120' 


2'-3' 


80'- 100' 
-120' 


3'-4' 


100' 


2F-3' 


-120' 





Mountain slopes. 
Neb. to Pacific; B. 
C. to Lower Cal. 

Belt 125 m. wide, 
Va. to Fla., W. to 
Miss. R. Isolated, 
Ala., La., Tex. 



Coast: Cape May to 
Texas. Inland: 
Carolinas to Ark., 
La. 

Nor. U. S. and Can- 
ada, Me. to Minn., 
south to Penn. 

Conn, to Fla. , W. 
Va., E. Tenn.;Gulf 
States to E. La. ; 
So. Missouri to E. 
Texas. Largest 
size W. of Miss. R. 

So. Car. to So. Fla., 
and along Gulf to 
Pearl R., La. 

So. Carolina to Miss. 



Lumber, ties, fen- 
cing, fuel. 

Most widely used In 

the East for heavy 

construction ; 

bridges, buildings, 

docks, masts, 

spars. 

Lumber, h'vy cons. ; 
docks, ships, cars, 

houses; ties. 

Cons. ; buildings, 
bridges, docks; 
masts, spars. 

Inferior to Longleaf 
P. for heavy con- 
struction. Wood 
is softer and more 
easily worked for 
carpentry. 

Same uses as Long- 
leaf pine. 

Ties and fueh 



The Larches. Genus, Larix. 



Western L., Tamarack 
(L. occidentalis.) 



,00'-250 



6'-8' 



Columbia R. Basin ; 
E'n Oreg., N'n 
Mont, and Idaho. 



Ties, fence posts; 
durable; furniture 
and interior finish. 



The Spruces (1). Genus, Picea. 



Red S 


70'-80' 


2'-3' 


P. E. Island, N. Eng, 
N. Y.. Alleg. Mts. 


Lumber, paper pulp, 
fiooring; sound'g 


(P. rubens.) 


-100' 










to N. Carolina. 


bds. for musical 
instruments. 


White S 


60'-70' 


-2' 


N'n U. S., Canada 


Inferior as lumber; 


(P. canadensis.) 


-150' 




and Alaska. 


superior for paper 
pulp. 


Engelmann S 


-150' 


4'-5' 


Alberta. B. C, 


Lumber, buildings. 


(P. Engelmanni.) 






Mont., Idaho, Cas- 
cade Mts., Wash., 
Oreg. 


fuel, charcoal; 
bark tans leather. 



342 



n.— NATURAL HISTORY OF MATERIALS. 



6. — Classification of Important Trees. — Cont'd. 



Species. 



Height. Dia. Principal Localities. 



Uses. 



The Spruces (1). Genus, Piceo. — (Continued.) 



Sitka S.. Tideland S. 
(P. sitchensis.) 



100'-200' 



3'-4' 
-16' 



Alaska, B.C., Wash., 
Oreg., to Mendoci- 
no Co., Cal. 



Most Imp. lum. In 
Alaska; bidgs., 
boats, fences, 
boxes, fuel. 



The Hemlocks. Genus, Tsuga. 



Hemlock 

(T. canadensis.) 



Western H 

(r. heterophylla.) 



Mountain H 

(T. mertensiana.) 



60'-70' 
-100' 


2'-4' 


100'-200' 
-250' 


6'-10' 


70'-100' 
-150' 


4'-5' 



N. Scotia to E'n 
Minn.; N. Y., N. 
Eng., Pa. to Ala.; 
Mich., Wis. 

S. E'n Alaska to C. 
Mendocino ; east to 
Mont, and Idaho. 

Alaska, B. C, Olym- 
pic Mts., Wash.; 
Cascades and Sier- 
ras. 



Lumber, building, 
ties; bark tans 
leather. 

Lumber for build- 
ings ; bark for tan- 
ning. 

Sometimes used for 
lumber; bark for 
tanning. 



The Spruces (2). Genus, Pseudotsuga. 



Douglas S., Red Fir. 
(P. mucronata.) 



100'-150 
-300' 



2'-3' 
-10' 



S'n B. C. W'nWash. 
and Oreg., N'n 
Cal., N. M., Ariz., 
Texas. 



Known as Greg. 
Pine, Wash. Fir. . 
Superior for con- 
struction; lumber. 



The Firs. Genus, AMes. 



White F 

(A. amabilis.) 



White F 

(A. grandis.) 



Red F. " Larch.' 
(A. nohilis.) 



70'-100' 
-200' 


4'-6' 


100'-250' 
-300' 


2' 
-4' 


150'-200' 
-250' 


6'-8' 



B. C. Olympic Mts.; 
Cascades, Oreg., 
Wash. 

Vancouver Id., 
Wash., N'n Oreg,, 
Idaho, Mont. ; N'n 
Cal. 

W'n Wash., Oreg.; 
N'n Cal. Cascades 
and Sierras. 



Largest and most 
com. Fir In Olym- 
pic Mts. Interior 
finish. 

Interior finish, 
packing boxes. 



Interior finish of 
buildings, and 
packing cases. 



The Redwoods. Genus, Sequoia. 



Redwood 

{S. sempervirens.) 



Big Tree 

(S. wellingtonia.) 



200'-300' 
-325' 

275'-325' 
-3 50' 



10'-15 
-2 8' 

20'-35 



Coast, S'n Oreg. to 
Monterey Co., Cal. 

Narrow area, W'n 
slope of Sierras. 
King's RIv. 



Lumber, buildings, 
shingles; posts, 
ties, wine-butts. 

Lumber, construc- 
tion; shingles, 
fencing. 



The Arbor Vitaes. Genus, Thuya. 



White Cedar, Arbor VI- 

tae 

(T. occidentalis.) 
Giant A.-V., Red Ced., 

Canoe Ced 

(T. plicata.) 



50'-60' 
150'-200' 



2'-3' 
-6' 
bot- 
tom 

-15' 



N. Scotia to Manito- 
ba; central N.West 
Allegheny Mts. 

Coast. Alaska to C. 
Mendocino. ; Mts. 
E'ly to Idaho, 
Mont. 



Fencing, ties, Shin- 
gles, tel. poles. 

Finishing, houses, 
doors, sashes; 
cooperage. 



The Cypresses. (1). Genus, Cypressvs. 



Monterey C | 60' 

(C. marocarpa.) 



■70' 



2'-3' I Coast of Cal., south 
-6' 1 of Monterey Bay. . 



Hedges, wind 
brakes; shingles. 



BOTANICAL— WOOD. 343 

6. — Classification op Important Trees. — Cont'd. 



Species. 


Height. 


Dia. 


Principal Localities. 


Uses. 


TheC 

" White Cedar." 

(C. thyoides.) 

Sitka C, Yellow C. ..... . 

(C. nootkatensis.) 

Lawson Cypress 

(C. lawsoniana.) 

The 
Bald Cypress 


Cypresses 
70'-80' 

100'-120' 

150'-200' 

Cypress 
75'-150' 

rhe Wal] 

50'-100' 

75'-150' 

rhe Hick 

70'-90' 
-120' 

120' 
80'-90' 

50'-80' 

-100' 

80'-100' 

100'-170' 

60'-100' 

50'-80' 
-100' 

The POPL 

60'-100' 
-150' 

100' 


(2). 

2'-2i' 
-4' 

5'-6' 
-12' 

ES (3) 

4'-5' 
-12' 

NTUTS. 
2'-3' 

4'-6' 

DRIES. 

3'-4' 

-3' 

3'-4' 

-3' 

-2' 

4'-6' 

2'-3' 

-2' 

ARS. 

7'-8' 

5'-6' 


Genus, Chamaecypar 

Coast, Mass. to Fla., 
and west to Pearl 
Riv., Miss. 

S. W'n Alaska, B. 
C, Cascade Mts., 
Wash., Oreg. 

Coast Mts. of Ore- 
gon and Cal. 

. Genus, Taxodium. 

Del. to Fla. ; Gulf 
Coast to Tex; Miss. 
Val, to Mo., Ind., 
111. 

Genus, Juglans. 

New Brunswick to 

Del. and to Dak.; 

Mid. West; Ga., 

Ala. 
Ontario to Fla. ; 

west to Nebr. and 

Texas. 

Genus. Hicoria. 

Me. to Del. to Fla.; 

N'n Ala.. Miss.; 

west to Min., Neb.; 

south to Texas. 
Ohio Basin, Middle 

West, and South 

East. 
Me. to Fla., west 

thro Ont., Mich., 

Neb. South to Tex- 
as. 
Ontario to Fla. ; 

west to Kan. and 

Tex. 
E'n S. Car.; Cen. Ala. 

Miss., to S'n Kan. 
S'n 111., Ind., la.; 

Miss. Riv. States 

to Cen. Ala. 
Me. to Fla. ; Ontario 

to Minn., Neb. and 

Tex. 
Swamp regions, Va. 

to Tex. ; north 

along Miss. R. to 

111. 

Genus, Populus. 

Quebec to N.W. Ter- 
ritory; south to 
Fla.; west to N.M. 

California, from Sac- 
ramento south; 
E'ly to Col., Tex. 


is. 

Finishing ; shingles, 
cooperage, fenc- 
ing, ties. 

Shipbuilding; In- 
terior finish; fur- 
niture. 

Int. finish and floor- 
ing; ties, fencing; 
shipbuilding. 

Lumber, flooring. 


(r. distichum.) 

Butternut, White W . . . 
iJ. cinerea.) 

Black W 


shingles, cooper- 
age, ties, fencing. 

Cabinet work; Inte- 
rior finish. 

Cabinet work ; in- 


(/. nigra.) 

Shellbark H.. Shagbark 
H 


terior finish; ship 
building. 

Agric. Imp.; wagons 
and ax-handles. 


{H. ovata.) 
Big Shellbark H 


Agric. Imp. ; wag- 
ons; ax-handles. 

Agric. Imp. ; wag- 
ons; tools. 

Agric. Imp.; wag- 
ons; ax-handles. 


(H. laciniosa.) 

Pignut, White H 

(H. glabra.) 

Mockernut H., Big Bud 
H 


(H. alba.) 
Nutmeg H 


Lumber and fuel. 


(H. myristicaeformis.) 
Pecan H 


Fuel; valuable nut 


(H. pecan.) 

Blttemut H., Swamp H. 
(H. minima.) 

Water H., Bitter Pecan. . 
(H. aquatica.) 

Cottonwood 


tree. 
Hoops and fuel. 

Fencing, fuel. 
Shelter and shade 


(P. deltoidea.) 
Cottonwood 


in prairie States. 
Shade tree; fuel. 


(P. fremontii.) 





344 17.— NATURAL HISTORY OF MATERIALS. 

6. — Classification op Important Trees. — Cont'd. 



m 



Species. 



Height. Dia. Principal Localities. 



Uses. 



The Poplars. Genus, Populus. — Continued. 



Aspen, Quaking Asp. 
(P. tremuloides.) 



Balm of Gilead 

(P. balsamifera.) 

Black Cottonwood. 
(P. trichocarpa.) 



Black W 

(S. nigra.) 

Golden Osier. 

iS. alba.) 



Canoe B., Paper B. 
iB. papyrilera.) 



Yellow B., Grey B. 
iB. lutea.) 



Beech , 

(P. Americana.) 



Chestnut 

(C dentata.) 



Chinquapin . . . 
(C. pumHa.) 



Newfoundland to 
Alaska; south to 
N. J. : west to 
Penn., Ky., Neb., 
Rocky Mts., Coast 
Range. 

Newf'dl'd to Alaska; 
S. to N. Y. ; W. to 
Mich., Neb., Idaho. 

West coast, Alaska 
to S'n Cal. 



The Willows. Genus, Salix, 



40'-100' 


-3' 


60'-100' 


6'-7' 


-200' 


7'-8' 



40'-60' 


-3' 


-120' 




40'-60' 


2' 



Me. to Fla. ; Rocky 

Mts.; Cal. 
Eastern N. America, 



The Birches. Genus. Betula. 

60'-70' 2'-3' Labrador to Alaska; 
south to Long Id. ; 
Penn., Cent. Mich. 
Minn., Neb., Black 
Hills. Mont., N.W. 
TVash. 
■100' 3'-4' Gulf of St. L. to Del., 
to N. Car., to Tenn. 
to Minn. 

The Beeches. Genus, Fagus. 



70'-80' 
-120' 



3'-4' 



N. Scotia to L. Hur- 
on; N'n Wis. ; south 
to Fla., Mo. and 
Tex. 



The Chestnuts. Genus, Castanea. 



;0'-100 
-50' 



2'-3' 



S'n Me. to Mich. ; 

south to Ind., Del.; 

to Ala., Miss. 
Penn. to Fla. ; west 

to Ark. and Texas. 



Shelter and shade; 
now rapidly spread 
over Rocky Mt. 
areas swept by 
fires. 

Shade and shelter. 



Wooden ware ; staves 
of sugar barrels. 



Borders of streams. 



Letter paper; bark 
for canoes, and 
various camp 
articles. 



Furniture, boxes, 
wheel hubs, fuel. 



Chairs, shoe-lasts, 
plane-stocks, tool 
handles; fuel. 



Int. finish, furni- 
ture, ties, fencing, 
fuel. 

Large sizes for ties 
and fencing. 



The Oaks. Genus, Quercus. 
White Oaks. 



Pacific Post O., Oregon 


60'-70' 


2'-3' 


Vancouver Id. ; W'n 


Cabinet work, wag- 


White O 


-100' 




Wash., Oreg., and 
Cal. 


ons, ship building. 


(Q. garryana.) 


cooperage, fuel. 


Live O 


40'-75' 


3'-4' 


Coast and Islands, 
Va. to Fla.; Gulf 


Excellent lumber; 


(Q. virginiana.) 


ship building. 








and Lower Cal. 




White O 


60'-100' 
-150' 


3'-4' 
-8' 


S'n Me. to Fla. ; west 
to Minn., Kan. and 


Construction; ship- 


(Q. alba.) 


building, ties. Int. 








Tex. 


finish, cabinet 
work. 


Burr O., Mossy-Cup O. . . 


70'-180' 


6'-7' 


Nova Scotia to Mon- 


Cons. ; shipbuilding, 


(Q. macrocarpa.) 






tana ;s'th to Penn., 
Tenn. and Tex. 


ties, Int. finish, 
cabinet work. 


Swamp O., Overcup O. . . 


70'-100' 


2'-3' 


Maryland to Fla; 


Confounded com- 


(Q. lyrata.) 






west to Missouri 
and Tex. 


mercially with 
Q. alba, above. 



BOTANICAL-^WOOD. 345 

6. — Classification op Important Trees. — Cont'd. 



Species. 



Height. DIa. Principal Localities. 



Uses. 



White Oaks. — Continued. 



Post O., Iron O 

(Q. minor.) 

Chestnut O., Tan-bark O 
(Q priniis.) 

Yellow O 

(Q. acuminata.) 
Swamp White O 

(Q. plantanoides.) 

Cow O.. Basket O... 
(Q. michauxii.) 



Live O., Maul O., Gold- 
cup O 

(Q. chrysolepis.) 

Pin O., Swamp Spanish 
O 

(Q. paliistris.) 
Red O , 

(Q. rubra.) 

Scarlet O 

(Q. coccinea.) 

Texan (Red) O , 

(Q. Texana.) 

Black O.. Yellow O.... 
(Q. velutina.) 

Spanish O 

(Q. digetata.) 

Water O 

(Q. nigra.) 

Willow O 

(Q. phellos.) 



Laurel O.. Shingle O. 
(Q. imbricaria.) 



White E., American E 
iU. americana.) 



Slippery E., RedE. 
(U. luLva.) 



Cedar E 

(t/. crassifolia.) 



40'-50' 
-100' 


2'-3' 


60'-70' 
-100' 


3'-4' 

-7' 


80'-100' 
-160' 
60'-70' 
-100' 


3'-4' 

2'-3' 
-9' 


60'-100' 


3'-7' 



Mass. to Fla. ; west 

to Missouri and 

Tex. 
Maine to Tenn. ; 

Mts. In Ga. and 

Ala. 
Vt. to Minn. ; south 

to Ala. and Tex. 
Me. to Great Lakes 

and la. ; south to 

Md., Ga., Ky., Ark. 
N'n Del. to Fla.; 

west to 111., Mo. 

and Texas. 



Black Oaks. 



40'-50' 



70'-80' 
-120' 
70'-80' 
-150' 

70'-80' 



50'-100' 
-200' 

70'-80' 
-150' 

70'-80' 
60'-80' 

70'-80' 



50'-60' 
-100' 



3'-5' 



2'-3' 

-5' 

3'-4' 



2'-3' 
7'-8' 
3'-4' 
2'-3' 
2'-Zi 



2'-3' 
-4' 



W'n slopes Sierras 
and Coast Mts., 
Oreg., and Cal. ; 
Ariz., N. M. 

Mass. to Del.; west 
to Wis. and Ark. 

N. Scotia to Minn. ; 

south to Ga., Tenn. 

and Kan. 
Me. to Fla. ; west to 

Ohio Val., Minn., 

Neb., Mo. 
Iowa to Ind. ; south 

to Fla. and Texas. 

Me. to Fla. ; west to 
Minn., Kan., E'n 
Tex. 

N. J. to Fla. ; west to 
Mo. and Tex. 

Del. to Fla. ; west 

thro Gulf States. 

Ky., Tenn., Ark. to 

Tex. 
N. Y.'to Fla.; Gulf 

States to Tex. ; 

N'ly in Mo., Ky., 

Tenn. 
Penn. to Ga. ; west 

to Nebr. and Ark. 



The Elms. Genus, Ulmus. 



75'-125' 


6'-ll' 


60'-70' 


2' 


-80' 


2'-3' 



Newfoundland to 
Fla. ; west to Rocky 
Mts. 

Ontario to Dak., 
Nebr. ; south to 
Fla.; west to Tex. 

S'n Ark., Miss, and 
Tex, 



Ties, fencing, fuel; 
cooperage. 

Cooperage, wheels, 
fencing, ties. 

Cooperape, wheels, 
fencing, ties. 

Cooperage, boat- 
building, fencing, 
ties, fuel. 

Fencing, ties, fuel; 
bark for tanning. 



Wagons and agrlc. 
imp. ; most val. 
Oak on Pac. 
Coast. 

Cooperage; int. fin- 
ish; shingles, clap- 
boards. 

Interior finish; fur- 
niture; bark for 
tanning. 

Interior finish; fur- 
niture. 

Lumber, Miss. Val., 
better than Q. ru- 
bra. 

Fuel. Bark used 
in medicine, dye- 
ing, tanning. 

Light construction, 
and fuel ; bark for 
tanning. 

Fuel. 



Clapboards; fellies 
for wheels ; some 
construction. 

Clapboards and 
shingles; some 
construction. 



Wheel hubs, saddle 
trees, fiooring, 
cooperage. 

Fencing, ties, wheel 
hubs, agrlc. im- 
plements. 

Fencing and fuel. 



Sweet G., Bllsted. 


The S\ 


7EET GUI 
80'-140' 


IS. 
4'- 


G 

-5' 


enus, Liguidambar. 
Conn, to Mo. • south 


Furniture. Int. fin- 


{L. styracifluxi.) 








to Fla. and Tex. 


Ish, shingles, fruit 
boxes, paving 
blocks, ties. 



346 17.— NATURAL HISTORY OF MATERIALS. 

6. — Classification of Important Trees. — Concl'd. 



Specials. 



Height. Dia. Principal Localities. 



Uses. 



Red M., Scarlet M.. 

Swamp M 

(A. rubrum.) 
Silver M., Soft M 

{A. saccharinum.) 

Oregon M., Broadleaf M 
{A. machrophyllum.) 

Sugar M., Rock M., Hard 
M 

(A. saccharum.) 
Black M., Black SugarM. 

(A. nigrum.) 



White A 

(F. americana.) 

Black A 

(F. 7iigra.) 

Blue A 

(F. quadrangidaia.) 
Oregon A 

{F. oregona.) 

Green A 

{F. lanceolata.) 



The Maples. Genus, Acer. 



Eastern and Middle 
States, Lower 
Mississippi, Texas, 

New Brunswick to 
Dak. ; south to Fla. 
and Oklohoma. 

South Coast Alaska 
to San Diego, Cal. 

Great Lakes to New- 
foundland, to Fla., 
to Neb., to Tex. 

Dak. to Vt., to Va.. 
Ky., Mo., Kan. 



The Ashes. Genus, Fraxinus. 



N. Scotia to Fla. ; 
west to Minn., Tex- 
as. Ohio R. Val. 

Va. to Del. to Mani- 
toba, III., Mo., Kan. 

Mich, to Mo., to 

Kan. to Ark. 
Coast, Puget Sound 

to San Francisco: 

Sierras. 
L. Champl'n to Fla.; 

west to Ariz., Utah, 

Tex. 



80'-] 20' 


3'-4i' 


90'-120' 


3'-4' 


SO'-lOO' 


2'-3' 


75'-120' 


2'-3' 


-80' 


-3' 



-120' 


5'-6' 


80'-90' 


H' 


60'-70' 

120' 
70'-80' 


2'-3' 
-4' 


50'-60' 


2' 



Tool handles, oars, 
furniture, wooden 
ware; fuel. 

Cheap furniture 
and flooring. 

Interior finish, fur- 
niture, ax- and 
broom-handles. 

Buildings, flooring, 
furniture, boats, 
fuel. 

Buildings, flooring, 
furniture, boats, 
fuel. 



Int. finish, stairs; 
tool handles, oars, 
furniture. 

Furniture, fencing, 
barrel hoops, cabi- 
net work. 

Flooring, tool han- 
dles, vehicles. 

Int. finish, furni- 
ture, wagons, 
cooperage, fuel. 

Int. finish, stairs; 
tool handles, oars, 
furniture. 



Interesting Facts About Trees. 

Some of the Tallest Trees. — Big Tree (350 ft.), Redwood (325 ft.). 
Sugar Pine (300 ft.), Douglas Spruce (300 ft.), Western Hemlock (250 ft.), 
Western Larch (250 ft.). 

Some of the Best Timber Trees. — Longleaf Pine, Douglas Spruce, 
White Pine, Norway Pine, Shortleaf Pine, Live Oak {Q. virginiana). 

Some of the jSest Lumber Trees. — All the timber trees make good 
lumber. In addition we may name: Most of the Pines, Spruces, Hem- 
locks, White Fir, Redwoods, Sitka Cypress, Lawson Cypress, some of the 
Walnuts and Hickories, many of the Oaks. Some of the Oaks, Ashes, 
Chestnuts, Maples, and Walnuts are used for interior finish. 

Ages of Trees at Maturity. — White Pine, 250 yrs.; White Oak, 200 yrs.; 
Chestnut, 175 yrs.; Beech, 100 yrs.; Elms, 90 yrs.; White Ash, 80 yrs.; 
Birches, 50 yrs. 

Relative Rapidity of Growth. — Silver Maple, White Elm, Red Maple, 
Sugar Maple, Chestnut, Red Oak, Pin Oak, Scarlet Oak, White Ash, White 
Oak. 

Some Important Tree Products — Tar, Rosin and Turpentine are ob- 
tained from the resin or sap of the longleaf pine. Pine tar is also obtained 
by the burning, or rather smouldering, of the limbs and knots of the longleaf 
pine. The dry distillation of woods produces wood vinegar (used for 
dyeing), acetic acid (which is made into vinegar), and wood alcohol. The 
latter may also be obtained from sawdust, and is destined to become a most 
valuable product for heating and power. Wood charcoal is also a valuable 
product. All are familiar with the products of the Balsam and the Sugar 
Maple. Wood pulp is used in making paper. 



CLASSIFICATION OF ANIMALS, 847 

D.-ZOOLOQICAL. 

Forenote. — So far as known there are about 350,000 living species of 
animals, and 50,000 extinct specimens, making a grand total of about 
400,000. The latest and most logical classification of the animal world, 
by Parker and Haswell, comprises 12 grand divisions or Phyllums, arranged 
according to structure — beginning with the lowest forms, the Protozoans or 
unicellular animals, and ending with the highest forms, the Mammals with 
Man at the head of the scale. In the evolution of the species the lower 
forms of life have played an important part in the development of the 
higher, just as today the whole animal kingdom forms a valued adjunct to 
the development and uses of man. 

Among the insects we have the honey-hee', the silk-woTm; the cochineal, 
found on certain species of cactus, for useful and harmless dye; the lac for 
shellac; the galls of the gall-fly for ink, etc. 

Many fishes, including the sword-fish, codfish, sunfish, etc., yield oil. 
Oil is also yielded by many reptiles, by the whale, walrus, manatee, etc. 

The whale, narwhal and elephant furnish us with ivory; the seal, beaver, 
and other fur-bearing animals, with furs; the alligator, goat, and many 
quadrupeds, with skins for leather; the latter, also, with horns and tallow; 
the sheep, with wool; sea-fowl, with guano; etc., etc. 

A considerable percentage of the animal world supplies us with food; 
and a few animals produce useful work. 

The following classification is now recognized by scientists as the most 
logical that has ever been submitted. 

7.— CLASSIFICATION OF ANIMALS: 
(After Parker and Haswell.) 

^ 3 ^ 

O C/3 PM 

a. Protozoa. (Unicellular animals.) 

I. Protozoa (Protozoans). One cell; or several cells of 

same kind. 

1. Rhizopoda (Amoeba, etc.). W^th retractile pseudopodia. 

2. Mycetozoa (Slimes). Terrestrial protozoa, plasmoidal. 

3. Mastigophora. Without cilia, or sucking tentacles. 

4. Sporozoa. Without appendages; internal parasites. 

5. Infusoria. With sucking tentacles. 

/?. Metazoa, (Multicellular animals.) 

II. Porifera (Sponges). Fixed; body-wall perforated, 

pores. 
1. Porifera. 

(a) . Calcarea. With skeleton or calcareous spicules. 
(6). Non-Calcarea. Where skeleton exists, composed of siliceous 
spicules. 

III. Coelenterata (Polyps, etc.). Radial structure; diges- 

tive cavity lined. 

1. Hydrozoa (Hydroids, etc.). More than two rays; single cavity. 

2. Scyphozoa ( Jellyfishes) . Many radii ; cavity divided by radial partitions. 

3. Actinozoa. Colonies or attached individuals. 

(a). Zoaw^^aWa (Sea- Anemones, Corals, etc.). Numerous tentacles, 

usually in 5's. 
(b). Alcyonaria (Sea-Fans, Red Corals, etc.). Eight tentacles. 

4. Ctenophora. With two radii, and rows of sucking tentacles. 

IV. Platyhelminthes (Flatworms). Body composed of 

loose cells. 

1. Turbellaria (Planarians) . Body covered with celia; one opening to ali- 

mentary tract. 

2. Trematoda (Flukes). Parasitic, unsegmented; adult without celia. 

3. Ce^/oda (Tapeworms). Without mouth or alimentary canal. 

4. Nemertinea. Carnivorous and aquatic; with mouth, arms, and food canal. 

(a) . Palaeonemertinea. 
(b). Schizonemertea. 
(c). Hoplonemertea. 



348 17.— NATURAL HISTORY OF MATERIALS. 

7. — Classification op Animals. — Cont'd. 



J •§ ^ 

O CO P. 

V. Nemathelminthes (Round worms). Usually — mouth, 

arms, alimentary tract. 

1. Nematoda (Threadworms). With intestinal canal; parasitic. 

2. Acanthocephala. Parasitic; no mouth or intestines. 

3. Chaetognatha (Arrow worms). Developed nervous system; spiny. 

VI. Trochelminthes (Wheel-animalcules). Larva in tro- 

chosphere form. 

1. Rotifera. Microscopic. 

2. Dinophilea. Minute; with 5 to 8 segments, usually ciliated. 

3. Gastrotricha. Minute; ciliated on ventral surface. 

VII. Molluscoida (Sea-Mats and Brachiopods) . Aquatic. 

1. Polyzoa (Bryozoans). Form colonies connected by one organic sub- 

stance, 
(a). Ectoprocta. Anus, external. 
(b). Endoprocta. Anus, internal; bud, forming colonies. 

2. Phoronida. Worm-like; bom from ova, not by buds. 

3. Brachiopoda (Lamp-shells). Body enc. in shell of two valves, usually 

on a stalk. 

VIII. Echinodermata (Echinoderms) . With intestinal 
walls. 

1. Asteroidea {^tax^sh). Star-shaped; furrows under the arms. 

2. Ophiuroidea (Brittle Stars). Arms not grooved. 

3. Echinoidea (Sea-Urchins) . Body, globular; armless. 

4. Holothuroidea (Trepangs). Worm-like; with tentacles about mouth. 

5. Crinoidea (Crinoides). Sessile; with cup-shaped body. 

6. CysLoidea (Fossil). Globular and stalked (sessile). 

7. Blastoidea (Fossil). Ovate; stalked. 

IX. Annulata (Worms). Bilaterally segmented; without 

jointed legs. 

1. Chaetopoda (Annelids). Composed of series of metameres, bearing cirri. 

(a) Polychaeta (Marine Annelids). Sexes distinct; ovaries and 

testes, many. 
(6) Oligochaeta (Fresh water and terrestrial). Sexes united; ovaries 

and testes, few. 

2. Myzostomida (Crinoides). Unsegmented. 

3. Gephyrea (Marine). Sessile; adult, without external segmentation, 

4. Archi-Annelida (Oiten Parasitic). Minute; marine, segmented (faintly). 
6. Hirundinea (Leeches). With ventral suckers. 

X. Arthropoda (Insects, Crustaceans, etc.). Symmetrical, 

segmented. 

1. Crustacea (Crustaceans). Aquatic, gill bearing; two pairs antennae, 

usually. 

(o) £n/owo5^raca (Water-Fleas, etc.). Varied number of appendages. 

{b). Malacostraca (Crabs, Crayfish, etc.). With 19 pairs of appen- 
dages. 

2. Trilobita (Extinct Trilobites). With head, thorax and abdomen. 

3. Onychophora (Peripatus). With series of short walking appendages. 

4. Myriapoda (Centipedes and MilHpedes). Segments, each with 1 or 2 

pairs of legs. 

5. Insecta (Insects). With six thoracic legs, and usually with wings. 

6. Arachnida (Spiders, Scorpions, etc). Air-breathing; without antennae. 



ZOOLOGICAL, 349 

7. — Classification op Animals. — Concluded, 

to >» G 

O C/3 CO ^ 

XI. Mollusca (Mollusks). Unsegmented ; muscular loco- 

motion. 

1. Pelecypoda (Bivalves). Gills leaf-like; two-valved shell. 

2. Amphineura (Chitons). Symmetrical bi-laterally; anus at end of body. 

3. Gastropoda (Gastropods). Unsym. body; shell (if any) univalve. 

(a). 5^r^i'iowe^ra (Limpets, Whelks, etc.). Visceral commissures like 

a figure 8. 
(6). Enthyneura (Pulmonates Nudibranchs, etc.). Vis. com. not like 

a figure 8. 

4. Scaphopoda (Marine) . Mouth -lobes formed into a tube. 

(a). Scaphopoda (Tusk-shells.) 

(6). Rhodope. Minute; no shell; with sucking tentacles. 

5. Cephalopoda (Cuttle-fish). Mouth surrounded by arms; foot funnel- 

shaped, 
(a). Dibranchiata (Squids, Octopods). Two sym. branchiae; tubular 

funnel. 
(6). Tetrabranchiata (Nautilus, Ammonites). Four branchiae; multi- 

locular shell. 

XII. Chordata (Chordates). Animals having a notochord. 

1. (A). Adelochorda (Balanoglossus, etc.). Marine; with noto- 

chord as larvae. 

2. (B). Urochorda (Ascidians). Marine; having a notochord when 

larvae, 
(C). Vertebrata (Vertebrates). Sym. bilaterally; having a back- 
bone. 

(/). Acrania (Branchiostomidae, Amphioxus, etc.). Without a 
head. 

(77). Craniata (Fishes, Reptiles, Birds, Mammals). 

1. Cyclostomata (Lampreys). Eel-like, without lower jaw; mouth,- suctorial. 

2. Pisces (Fishes). Aquatic; persistent gills, paired fins. 

(a). Elasmobranchii (Sharks and Rays). Cartilaginous skeleton; 

cranium not ossified. 
(6). Holocephali (Chimaeras). Cartilaginous; four pairs of gill slits, 
(c) . Teleostomi (Sturgeons and ordinary Fish). 
{d). Dipnoi (Lungfishes). Fish-like animals with apparatus for 

breathing. 
{e). Ostracodermi (Cephalaspis, etc.). 

3. Amphibia (Amphibians). With gills when leavae, lungs when adult. 

5-toed feet. 

4. Reptilia (Reptiles). Air-breathing; homy epidermal skeleton of scales. 

6. Aves (Birds). Clothed with feathers. 

(a). Archaeornithes (Archaeopteryx) . With prolonged tail of many 

vertebrae. 
(b). Neornithes (Modern Birds). Tail vertebrae compacted. 
6. Mammalia (Mammals). Suckle their young; clothed with hair, more or 
less, 
(a). Prototheria (Didelphia). Mammals with oviducts separated. 
(b). Theria (Monodelphia). Mammals with oviducts more or less 
united. 
(bi). M etatheria (Ma.rs\ipia\s). Rudimentary birth; shel- 
tered in pouch. 
(62). Eutheria (higher Mammals). Bom in uterus; no 
pouch. 



18.— EXPLOSIVES. 



An explosive is a mixture Avhich, under certain disturbing influences, 
enters into rapid chemical reaction, forming expansive gases and evolving 
much heat. The substances mixed may be solid or liquid. Explosives 
may be classified according to the nature of the mixture, whether mechan- 
ical or chemical. 

(a). MECHANICAL MIXTURES. 

In mechanical mixtures the component elements remain intact until 
a high temperature is reached, when they chemically react and pass off as 
gases (mostly), causing the explosion. 

Nitrate Mixtures. — This class comprises the mixture of a nitrate, 
embodying oxygen, with some base yielding carbon and usually containing 
sulphur. The explosion occurs when the oxygen leaves the nitrate and 
combines with the carbon, having a greater affinity for the latter. Gun- 
powder is typical of this class. 

Gunpowder.— This is the oldest and one of the most common of explo- 
sives. It consists of a mixture of saltpetre ( = nitre = potassium nitrate = 
KNO3), charcoal ( = carbon) and sulphur in various proportions according 
to the nature of the explosive desired. The black powder, formerly used 
almost exclusively in U. S. ordnance, was composed of 75 parts saltpetre, 
15 parts charcoal and 10 parts sulphur; while the brown or sporting powder 
contains much more charcoal and considerably less sulphur, the proportions 
being, 79 parts saltpetre, 18 parts charcoal and 3 parts sulphur. These are 
now being superseded by the smokeless powder. 

Blasting Powder. — This is made slower acting than the gunpowder by 
reducing the proportion of saltpetre to 66 parts, charcoal forming 15 parts, 
and sulphur 19 parts. Sodium nitrate or Chili saltpetre is now commonly 
used instead of the more expensive potassium nitrate. 

Assuming the weight of powder at 62.5 lbs. per cu. ft. (spec. grav. = 
unity), the following table gives the weight of powder in a hole one ft. deep 
and of various diameters: 

1. — Weight of Powder in a Hole One Foot Deep. 



Diam. 


Weight. 


Weight (Avoir.) 


Diam. 


Weight. 


Weight 


(Avoir.) 


Ins. 


Lbs. 


Lbs. Ozs. 


Ins. 


Lbs. 


Lbs. 


Ozs. 


i 


.0852 


— 1.4 


21 


2.131 


2 — 


2.1 


. 


.1332 


— 2.1 


3 


3.068 


3 — 


1.1 




.1917 


— 3.1 


3i 


4.176 


4 — 


2.8 


s 


.2610 


— 4.2 


4 


5.454 


5 — 


7.3 


1 


.3409 


— 5.5 


4i 


6.903 


6 — 


14.5 


u 


.5326 


— 8.5 


5 


8.522 


8 ~ 


8.4 


1 


.7670 


12.3 


5^ 


10.312 


10 — 


5.0 


If 


1.044 


1 — 0.7 


6 


12.27 


12 — 


4.4 


2 


1.364 


1 — 5.8 


6^ 


14.40 


14 — 


6.4 



Note. — Weight in lbs. = 0. 3409 X (diam. of hole in ins.)^; therefore it 
is proportional to the square of the diameter of the hole. 

Other Nitrate Explosive Mixtures. — The following mixtures are analo- 
gous to gunpowder in exploding at high temperatures: 
Amide. — Ammonium nitrate, potassium nitrate, charcoal, 
Azotine. — Sodium nitrate (69), carbon (15), sulphur (12), petroleum (4). 
Carbo-azotine. — Potassium nitrate (61), soot (26), sulphur (14). 

350 



MECHANICAL AND CHEMICAL MIXTURES. 351 

Diorexine. — Potassium nitrate (50), sodium nitrate (25), sulphur (12), hard 
sawdust (13). 

Johnite. — Potassium nitrate (75), sulphur (10), lignite (19), sodium picrate 
(3), potassium chlorate (2). 

Petralite. — Potassium nitrate (64), charcoal (30), crude antimony (6). 

Pyrolite. — Potassium nitrate (51), sodium nitrate (16), sulphur (20), saw- 
dust (11), charcoal (2). 
Chlorate Explosive Mixtures. — Potassium chlorate is a constituent in 

each of the following explosives, which are considered rather unsafe on 

account of spontaneous or premature reaction: 

Asphaline. — Potassium chlorate (54), potassium nitrate and sulphate (4), 
bran (42). 

Ehrhardt P. — Potassium chlorate, cream of tartar, powdered nutgalls, 
tannin. 

Fontaine P. — Potassium chlorate, potassium picrate. 

Horseley P. — Potassium chlorate (6), nutgalls (1), charcoal (1), nitro- 
glycerin (72). (This is a dynamite.) 

Michalowoski Blasting P. — Potassium chlorate (50), manganese dioxide 
(5), bran (45). 

Oriental P. — Potassium nitrate and crude gamboze, potassium chlorate. 

Pyronome. — Potassium nitrate (69), sulphur (9), charcoal (10), antimony 
(8), potassium chlorate (5), rye flour (4), potassium chromate. 

Rackarock. — Potassium chlorate (79), mono-nitrobenzine (21). Fresh 
mixed. 

lb). CHEMICAL COMPOUNDS. 
Nitro=substitution Explosive Mixtures. — The following are examples 
of nitric acid treatment of the hydrocarbons, producing compounds which 
at high temperatures become unstable: 

Ammonites. — Ammonium nitrate (88), di-nitro napthalin (12). 

Bellite. — Ammonium nitrate (5), meta-di-nitrobenzine (1), potassium 

nitrate. 
Borlinetto P. — Picric acid (10), sodium (10), potassium chromate (8). 
Extralite .-^-Kmmomvim. nitrate, potassium chlorate, naphthalin. 
/oz;^i>.— -Nitro-naphthalin, nitro-phenol, sodium nitrate. 
Rohurite.-—hm.monmvci nitrate, chlorinated di-nitrobenzine. 
i^omtV^. —^Ammonium nitrate, potassium chlorate, naphthalin. 
Securite. — Ammonium nitrate, di-nitrobenzine. 
Volney P. — Potassium nitrate, sulphur, nitro-naphthalin. 

Nitric Acid Compounds. — When certain vegetable tissues, called cellu- 
lose, notably cotton, are treated with nitric acid the resulting cellulose 
nitrate is a very high explosive. The stronger treatments produce gun- 
cotton and the weaker, pyroxylin. Similarly, when the animal compounds 
gelatin, or glycerine, are treated with nitric acid there results nitroglycerin, 
a most powerful ebcplosive. Dynam,ite consists of some absorbent soaked 
with nitroglycerin to protect the latter from decomposition and premature 
explosion. 

Guncotton. — ^This is made by soaking cotton, or other form of cellulose, 
in nitric acid (one part) and sulphuric acid (three parts) for about a day, 
and then thoroughly washing. The resulting product is from 60 to 85 per 
cent heavier than the cellulose, depending upon the proportion and strengths 
of the acids and the general treatment. It is the safest explosive known 
in general handling and shipment. It is exploded by percussion when con- 
fined and highly compressed, but if ignited burns quietly without explosion. 

Guncotton is used as an agent in various mixtures containing nitrates 
of which the following are examples: 

Glyoxiline. — Guncotton, saltpetre, nitroglycerin, — in form of peilets. 
Potentite. — Guncotton, saltpetre,— in form of cartridges. 
Tonite. — Guncotton, barium nitrate, — in form of cartridges. 



352 IS.—EXPLOSIVES. 

Detonation. — In our classification of the various explosives under the 
two headings, Mechanical Mixtures and Chemical Compounds, we are con- 
fronted with the great natural law of gradation, which is universal. For 
instance, there are some animate objects which the naturalist is unable to 
classify distinctly as plants or animals, bordering on the division line and 
having some of the common attributes of both great orders in nature. So 
with explosives^ Up to 1864, when Alfred Nobel, a Swedish engineer, 
began to put some of the higher explosives to practical use, the great prin- 
ciple of " detonation " or instantaneous explosion by " shock " was but 
vaguely known. His investigations with nitroglycerin led to the discovery 
that many explosives heretofore treated by ordinary combustion were far 
more powerful when subjected to "percussion, ' 'which converted the substance 
immediately into gas. In his experiments with nitroglycerin he used a 
percussion cap charged with fulminate of mercury as an initiatory explosive. 
Detonation is essential in the higher class of explosives as guncotton, nitro- 
glycerin, dynamite, etc. 

The following are some of the guncotton preparations resulting from 
Nobel's discovery: 

Blasting gelatin. — Guncotton absorbed in nitroglycerin. 
Gelatin dynamite. — Blasting gelatin and an absorbent. 
For cite. — Blasting gelatin (50), absorbent (50). 
Gelignite. — Blasting gelatin (65), absorbent (35). 

Smokeless Powders contain guncotton, nitrate, carbonate, nitrogly- 
cerin, and various other substances of like nature. The product is a horn- 
like substance, which is cut into chords or grains. 

Nitroglycerin. — ^To the engineer, this is one of the most useful and most 
powerful explosive agents. It is prepared by treating glycerin with strong 
nitric and sulphuric acids, producing the chemical compound, C3 H5 Ns Og. 
In its pure liquid form it bums quietly, producing carbon dioxide, hydrogen, 
nitrogen and water. But when gradually heated to 180° C. or when sub- 
jected to violent percussion it explodes, developing gases from 1200 to 1400 
times the volume of the liquid, which in turn are further expanded by the 
great amount of heat evolved, to about 10,000 times the original bulk. 
Nitroglycerin when _ mixed with infusorial earth as an (inert) absorbent 
forms dynamite^ which is sold under various trade names. 

Dynamite. — As the liquid nitroglycerin is liable to explode by heat or 
decomposition, it is rendered safer by being combined with some protecting 
absorbent. The absorbent may be inert or active, naturally dividing the 
dynamite into two classes. The second class comprises by far the more 
important, as follows: 

Atlas P. — Nitroglycerin (75), wood fibre (21), sodium nitrate (2), magne- 
sium carbonate (2), Also lower grades. 

Carhonite. — Nitroglycerin (25), wood dust (40), sodium nitrate (34), sodium 
carbonate (1). 

Dualin. — Nitroglycerin (40), wood dust (30), potassium nitrate (20). 

Giant P. — Nitroglycerin (40), resin (8), kieselguhr (8), sulphur (6), sodium 
nitrate (40). 

Hercules P. — Nitroglycerin (40), wood fibre (11), sodium nitrate (45), 
sodium chloride (1), magnesium carbonate (1). 

Judson P. — Nitroglycerin (5), cannel coal (15), sulphur (16),- sodium ni- 
trate (64). 

Lithofracteur. — Nitroglycerin (54), kieselguhr (17), sulphur (7), barium ni- 
trate (15), wood dust (2), manganese (2), soda (2), bran (1). 

Meganit^. — Nitroglycerin (60), nitrated wool and vegetable ivory (20), 
sodium nitrate (20). 

Rhexite. — Nitroglycerin (64), decomposed wood (11), wood dust (7), sodium 
nitrate (18). 

Safety nitro P. — Nitroglycerin (69), wood fibre (13), sodium nitrate (18). 

Stonite. — Nitroglycerin (68), wood dust (4), kieselguhr (20). potassium 
nitrate (8). 



DYNAMITE. 353 

Vigorite. — Nitroglycerin (68), kieselguhr (20), potassium nitrate (7), car- 
bonate, etc. (5). 

Vulcan P. — Nitroglycerin (30), charcoal (11), sulphur (7), sodium nitrate 
(52). 

Unmixed Explosives. — ^There is a certain class of explosives called Pan- 
clastites, consisting of two ingredients which separately are inexplosive but 
when mixed are considered fully as powerful as dynamite. They are 
shipped separately and mixed at the site when needed for use. The prin- 
cipal ingredients of the mixture are nitrogen tetroxide and carbon disulphide. 
They are not generally recommended for ordinary use by the class of men 
usually employed in blasting. 

Percussion Caps. — Fulminate of mercury enters into the composition 
used for percussion caps and electric fuses employed in detonating charges 
of dynamite in blasting operations. It is prepared by adding alcohol to a 
nitric acid solution of mercury. 

(c). The Handling and Use of Dynamite. 

Dynamite as invented by Noebel in 1867 consisted of nitroglycerin 
absorbed by a porous, inert solid. The best absorbent was found to be a 
siliceous, infusorial earth known as kieselguhr, obtained in Hanover, Ger- 
many. When dried it is an impalpable, white powder of cellular structure, 
and is capable of absorbing three times its weight of nitroglycerin, giving 
the resultant dynamite the appearance and consistency of heavy brown 
sugar. Coupled with its absorbent, nitroglycerin is thus, in the form of 
dynamite, free from the danger of spontaneous explosion, and detonation 
from shocks of a moderate nature. In the loose powdered form, dynamite 
loses none of its explosive properties when exposed to natural temperatures; 
i.e., it explodes readily by the action of the fulminating mercury primer 
or cap. On the other hand, when the powder is compressed into cartridges 
and the temperature is reduced to between 50° and 42° F. it freezes and, 
like frozen nitroglycerin, is inexplosive. It loses only a small percentage 
of its explosive power on being saturated with water, hence its great value 
in submarine work. If unconfined and ignited by a flame (350° F) it bums 
freely and quietly without explosion. .... 

Dynamite is usually compressed in sticks or cylinders called cartridges, 
varying from | to 2 ins. in dia. and about 8 ins. long, more or less ; or they 
are sold in any size and length ordered. The sticks are wrapped separately 
in paper, and packed in boxes, in layers cushioned with sawdust; usually 
50 lbs., or 25 lbs., to the box. 

In " freezing " weather these cartridges have to be " thawed " out. 
The terni " freezing " applies to any temperature that chills the dynamite, 
usually in the neighborhood of 45° F, sometimes lower and sometimes 
higher, depending upon the grade, the quality of the absorbent, and the 
nature of the exposure. The " thawing " of dynamite consists in slightly 
warming it to take off the chill. It sometimes requires a temperature of 
60° or 60° F to do this effectively, and it must not be allowed to chill again 
while being carried to the drill holes and loaded. The usual method of 
thawing dynamite is by a small out -door fire, but the disadvantages of this 
plan are: (1) waste of time in thawing, (2) greater or less danger attached 
to it, (3) inefficiency of thorough thawing so that the dynamite will explode 
with the greatest effect. In a pamphlet " Thawing Dynamite " published 
by the Aetna Powder Co., of Chicago, is an illustration of a Thaw House 
about 7x10 ft., fitted with steam coils and shelves for thawing. The shelves 
have a capacity of about 500 lbs. of dynamite, and about 1000 lbs. more 
can be stored in the house in boxes. The special caution for sweeping 
and cleaning merits careful attention. 

In preparing the charge, the fulminating cap, to which the safety fuse 
has been attached, is inserted in the top of the cartridge, the neck of which 
is tied around the fuse with a string, and the charge placed in the hole and 
fired. If an electric blasting machine is used, a special cap is required. 

Many of the dynamites are put up in two grades, No. 1, and No. 2, 
the former containing as high as 75% N.-G. Some of the powders are fur- 
nished in several grades, ranging from 75 down to 20% N.-G. In ordering 
dynamite it is necessary to state the percentage of nitroglycerin required, 
whether 30, 40, 60, 60, 75%, unless the name of the desired brand is known. 



354 18.~EXPL0SIVES. 

2. — Some of the Most Common Commercial Dynamites. 

(Note. — ^The percentages of nitroglycerin are shown in parentheses.) 
Aetna powder, No. 1 (65), No. 2 XX (50), No. 2 (40), No. 3X (35), 
No. 4 X (25), No. 5 (15). 

Atlas powder, A (75), B + (60), B (50), C + (45), C (40), D + (35), 
D (30), E + (25). 

E (20), F + (15). I Carbonite (25). 

Colonia powder (40). | Dualin (40 to 50). 

Dynamite— Nobel's Kieselgiihr— Old No. 1 (75), Old No. 2, (40), Old No. 

3, (25). 

Forcite (49). I Gelignite (62|). 

Giant powder. No. 1 (75), New No. 1 (50), No. 2 extra (45), No. 2 (40), 
No. 2 c (33), No. XXX (27), No. M (20). 

Giant powder— Nobel's— No. 2 (20). | Hecla powder. No. 1 XX (75). 
Hercules powder, No. 1 XX (75), No. 1 (65), No. 2 SSS (55), No. 2 SS (50), 
No 2 S (45), No. 2 (40), No. 3 S (35). No. 3 (30), No. 4 S (25), No. 4 
(20). 

Jisidson giant powder, No. 2 (40). 
Lithofracteur (54). 



Horsley powder (72). 

Judson powder, FFF (20), FF (15), 

Rendrock (40). 

Vigorite (30 to 68). 



Stonite (68). 
Vulcanite (30). 



A List of Permissible Explosives for Use in Coal Mines. — Following 
brands, tested prior to Oct. 1, 1909: Aetna coal powder A, AA, B, C; Bitu- 
minite No. 1; Black Diamond Nos. 3, 4; Carbonite Nos. 1, 2, 3, 1-L. F., 
2-L. F.; Coalite Nos. 1, 2-D; Coal Special Nos. 1, 2; Collier dynamite Nos. 
2, 4, 5; Giant A low-flame dynamite, C low -flame dynamite; Masurite 
M. L. F.; Meteor dynamite; Mine-ite A, B; Manobel; Tunelite Nos. 5, 6. 7, 8. 

Lists of Permissible Explosives, up to date, are very extensive and may be 
procured from the Bureau of Mines, Washington, D. C. 

Preparation of Primers (Miners' Circular 13, Bureau of Mines, 1913.)— In- 
struct men in proper way of preparing a primer, for many misfires result from 
use of improper primers. Teach them to cut off and throw away an inch or two 
of the fuse before inserting it in the detonator, for gunpowder (which forms the 
core of the fuse) easily gathers moisture, and the end of the fuse may have become 
damp enough to quench the burning powder or prevent the ignition of the deto- 
nator. Insist that this cut be made squarely across the fuse with a sharp cutting 
tool; if the cut is diagonal the point may curl over the end of the fuse when it is 
inserted in the detonator and thus prevent the spit of the powder train from 
reaching the mercury fulminate in the detonator, and if the tool is dull the 
powder grains in the end of the fuse may be spilled during the cutting, thus 
weakening the force of the spit into the detonator and possibly preventing its 
ignition. Have the men crimp the free end of the detonator around the fuse 
tight enough to hold the detonator and the fuse together, but not tight enough 
to cut off the powder train in the fuse. Insist that they use for this purpose 
nothing but the proper crimping tool. After crimping, the detonator should be 
buried in the end of the stick of dynamite with its axis parallel to that of the 
stick and its top flush with the top of the dynamite. If the detonator is buried 
deeper, or if the fuse is laced through the cartridge, the explosive is liable to be- 
come ignited from the side-spitting of the fuse before it is properly exploded by 
the detonator, which not only reduces the efficiency of the explosive, but creates 
a larger volume of gases, especially of those gases most dangerous to the men 
who must breathe them. See, also, that detonators of sufficient strength are 
used. Although No. 5 detonators were considered strong enough for " straight " 
nitroglycerin dynamite, the less sensitive gelatin dynamite requires a much 
stronger detonator to explode it properly. For this reason you should never 
use anything weaker than No. 6 detonators with gelatin dynamite; the uni- 
versal experience is that better results have been obtained with all dynamites 
when strong detonators are used. 

Nitroglycerin Dynamite Specifications for Panama Canal (1912) limit the 
amount of moisture to H% by weight and permit a variation of not more than 
3% from the full contract strength; a reduction of 1% being accepted without 
reduction in price, but if between 1 and 3% a deduction of 15 c. per 100 lb. for 
each 1% is made. 



19.— PRESERVATIVES. 

PAINTS. 

A paint consists of a pigment, vehicle, drier, and (usually) solvent. 
Sometimes the pigment is ground alone and then mixed with the vehicle 
and drier but usually they are ground together, and sealed in kegs or cans 
for shipment. 

Pigments. — ^The following are the most important to the engineer: 

White lead, consisting of a mixture of lead hydrate and lead carbonate, is 
universally employed for white paint. It is often adulterated, however, 
with heavyspar, gypsum, chalk, clay, sulphate of lead, etc., which renders 
it more or less inferior. 

White zinc or zinc oxide is used for white paint; often adulterated with 
heavyspar. 

Lampblack (mainly carbon) is one of the best pigments for black paint 
and for printer's ink. It is obtained by burning substances rich in carbon, 
with a low flame, that is, to incomplete combustion, so the carbon will not 
be burned. For this purpose resin, ozokerite, the by-product hydrocarbons 
from oil refineries, various kinds of oils gas, etc., are used. 

Boneblack and ivory black are obtained by heating bones with exclusion 
of air. 

Graphite is becoming one of our most useful paint pigments, especially 
as a protection to iron and steel from rust. 

Iron oxides give brownish colors varying to red, yellow, etc., depending 
upon the natural ores, as ochres, bole, pozzuole earth, sienna earth, etc. 
The ground ores are often adulterated with heavyspar, gypsum, clay, etc., 
for cheapness and shading of colors. 

Red lead (red oxide or minium) is more highly oxidized than white 
lead or litharge and is obtained by properly heating the latter in a furnace 
or oven, in contact with the air, to a certain temperature (about 400° to 
500° C). It is used as a red pigment and is also a very durable cement for 
glass and for water pipes. Commercially, as a pigment, it is often adulter- 
ated with brick dust, red bole, heavyspar, oxide of iron, etc. — the last 
named being particularly objectionable. 

Verdigris is a pigment usually mixed with white lead to produce a green 
paint. It is made by exposing thin copper plates to the air in contact with 
acetic acid. 

Ochre is a clay containing oxide of iron, and when dried and ground is 
used as a pigment. It is usually of a yellowish color but by burning it 
gradually changes to red by the oxidation of the iron present. 

Umber differs from ochre in containing a quantity of oxide of manganese 
in the clay in addition to the oxide of iron. The unbumed or " raw " umber 
is dried and ground as a pigment for brown paint; while the calcined or 
" burnt " umber yields a reddish brown color. 

Sienna is a ferruginous earth like ochre, the unbumed of " raw " sienna 
producing a yellowish-brown pigment; if " burnt " (before being ground) 
it yields a reddish-brown pigment. 

Asbestos is a fibrous hornblende (amphibole) or crysotile (serpentine) 
of various colors as white, gray, red, green and black. Its value in paint 
lies in its fireproofing qualities when applied to wood, or other inflammable 
material. 

Vehicles-— A vehicle in paint corresponds to lime paste in common 
mortar, forming the binder when dry. In calcimine, the pigment is zinc- 
white and the vehicle glue-sizing diluted with water; tinting is produced 
by adding the various coloring matters. For common paints linseed oil is 
universally r referred as a vehicle, although any kind of varnish may be used. 

355 



356 



19.—PRESERVA TIVES. 



Linseed oil is obtained from the seed of commoii flax, by (1) cold pressiire, 
(2) hot pressure, (3) extraction. The first named process produces the 
food oil; the second and third, the article of commerce. The method of 
extraction is by the use of some solvent as benzine, naphtha, etc. Carbon 
disulphide yields fully 50% more oil than the cold-drawn process, but is 
objectionable when used with a lead pigment because the retained sulphiir 
turns it dark. " Boiled " linseed oil is used as a vehicle for paints because 
drying is facilitated and it acts better with the pigment. Linseed oil is 
often adulterated with other oils as those of mustard seed, rape seed, hemp 
seed, cotton seed, etc. The engineer usually specifies " pure " raw or 
boiled linseed oil, the former being often preferred alone as a coating for 
steel when it leaves the shop. 

Driers, such as compounds of lead, manganese and zinc, are simply 
oxidizing agents for converting the oils from " vehicles " into " binders." 
Among the most common are litharge, red lead, lead acetate, oxides of 
manganese, borate of manganese, oxide of zinc, etc., or their various mix- 
tures. These act as agents in conveying oxygen from the air to the oils, 
thus drying them. 

Solvents. — A good solvent is slow to evaporate and flows sufficiently 
to eft'ace the brush marks. 

Turpentine is the best solvent. It is a repeated distillation of the 
resinous sap of certain coniferous trees, and dissolves readily the various 
substances used in paints. Only the pure turpentine should be accept- 
ed. It is composed entirely of carbon and hydrogen, but when exposed 
to the air it absorbs oxygen and turns yellow. It should be kept air-tight 
and away from the light. Turpentine has no substitute. Benzine, kero- 
sene and some other oils are sometimes used either alone or as adulter- 
ants, but should be shunned. 

House Paints. — White is obtained from white lead; black, from lamp- 
black; red, from red lead (50) and red ochre (50); green, from verdigris (75) 
and white lead (25) ; chocolate, from Spanish brown (96) and lampblack (4); 
stone color, from white lead (99) and burnt umber (1); lead color, from 
white lead (98) and lampblack. (2). 

The following table shows how various colors may be produced by a 
simple combination of two other colors. To produce " shades " of any 
color add black; to produce " tints " add white: 





Red. 


Yellow. 


Blue. 


Orange. 


Purple. 


Green. 


Red 


















Yellow 


Orange. 
















Blue 


Purple. 


Green. 














Orange 


















Purple 








Russet. 






Green 








Olive. 


Citrine 









Note. — Yellow and red produce orange; yellow and blue, green; purple 
and orange, russet; etc. 

Special Paints. — These are made by mixing substances having special 
properties, with the pigments: 

Aluminum paint is made from powdered aluminum and contains about 
91% metallic aluminum, 6% aluminum oxide, li% silica, 1% water. 
Gas or air is forced under pressure into the molten metal which is vigorously 
stirred, and forms a powder in setting. This powder is crushed and run 
through sieves and polished. About 2 lbs. of the powder is mixed with 
one gallon of varnish composed of 1.5 galls, turpentine, i gall, palest copal 
varnish, 4 oz. palest terebine, 4 oz. carbonate of magnesia. The latter is 
allowed to settle and the clear varnish is drawn off. 



PAINTS. VARNISHES. PLATING. 357 

Bronze paint is made by mixing filings of copper, brass, etc., with the 
pigment. It is used in painting iron and other materials. 

Copper paint, merciiry paint, arsenic paint and paris-green paint are 
poisonous to marine life and are used in painting ships bottoms. The 
copper paint is formed by mixing salts of copper with the pigment. The 
others are formed by mixing mercury, arsenic, and paris green, respectively, 
with the pigments used. 

VARNISHES, LACQUERS, ETC. 

Varnishes. — A varnish consists of a gum or resin dissolved in " spirit " 
or in oil, with perhaps some coloring matter added. The resinous sub- 
stances are amber, anime, copal, mastic, resin, sandarac, shellac, for the 
lighter colors; and asphaltum and pitch for the dark colors. The solvents 
are " spirits," as ethyl alcohol, wood (methyl) alcohol, ether, benzol, chloro- 
form, carbon disulphide, coal oils (light), turpentine; or " oils," as linseed, 
poppy, nut, hemp, castor, walnut, cotton-seed. The coloring matter is 
aloes, dragon's blood (a resin), tumeric, sanders-wood, saffron, anotto, 
indigo. The name of the varnish usually takes its name from the solvent, 
as spirit varnishes and oil varnishes. 

Lacquering is the varnishing of polished metal surfaces, as brass. The 
lacquer is an alcoholic solution af shellac with some coloring matter added 
to give the desired tint. Being transparent or nearly so, the brass effect 
is visible while the metal is at the same time preserved from discoloring 
or oxidizing. 

Japanning is varnishing with japan, a black varnish made with asphal- 
tum (mostly) dissolved in turpentine. 

GALVANIZING AND TINNING. 

Galvanizing consists in dipping the iron or steel sheets, rods, bolts, 
wire, etc., in a bath of molten zinc. The coat is naturally very durable 
unless exposed to sulphurous smoke or other similarly destructive chemical 
agents. 

Tinning consists in immersing the iron or steel sheets in molten tin and 
tallow. Tin plate, so called, is merely the thin metal sheet coated with tin. 
Terne plate is made by dipping the iron or steel sheets into a bath of tin 
and lead, being less expensive than tin plate and much inferior to it. It 
is used largely for roofing. 

ELECTRO=PLATING. 

This branch of Electro-Chemistry was evolved from discoveries made 
by Jacobi in 1838. By its use we are enabled to plate the cheaper and less 
durable metals with durable metal coatings. The agent employed is the 
electric current. 

Electro-chemistry is that branch of chemistry which treats of the 
chemical changes, produced by, or producing, electricity or electrical energy. 
It embraces the subjects of (1) Electrolysis, the decomposition of a chemical 
compound called an electrolyte into its constituent parts by an electric 
current; and (2) Electro-Metallurgy, the deposition of certain metals, as 
gold, silver, copper, etc., from their solutions by means of the slow action 
of an electric current. All electrolytes are either acids, bases or salts, and 
all electro-chemical reactions produce, or are produced by, these three 
classes of compounds. The transformation of chemical energy into elec- 
trical energy is through a system termed a voltaic or galvanic cell or battery, 
two or more cells in combination forming a battery. 

Electrolysis — ^When an electrolytic substance is subjected to the action 
of an electric current it decomposes into ions, the cations (electropositive 
ions) seeking the " cathode " or negative pole, and the anions (electro- 
negative ions) seeking the " anode " or positive pole. Thus, when water 
is decomposed the hydrogen atoms (cations) are attracted to the negative 
pole, while the oxygen atoms (anions) collect at the positive pole. 

Electro-Metallurgy is an electrolytic process by which the metal in an 
ore is separated from its impurities. It is one of the common processes, the 
others being: Smelting or heating of the ore; Amalgamation by the use of 
mercury; and Extraction by chemical solutions. 



35S 1^.— PRESERVATIVES. 

Electro-Plating embodies all the preceding principles. The article to 
be plated must first be cleaned thoroughly. It is dipped in a cleansing 
solution and then thoroughly rinsed with water so that none of the solution 
will remain. If necessary it is then placed in a scouring tray before going 
to the plating vat, where it is suspended in the plating solution from copper 
rods by means of short " slinging " wires of copper. ^ The electric current 
used for depositing the metal from the plating solution may be obtained 
from any source which is convenient, either from a battery or dynamo. 
The following are some of the solutions used for electro-plating: For gold 
plating, a solution of gold cyanide and potassium cyanide; for silver, a 
solution of silver cyanide and potassium cyanide ; for copper, an ammoniacal 
solution of copper and potassium cyanide; for nickel, the double sulphate 
of nickel and ammonia, or the double salt of nickel and ammonia; ioT brass, 
liquid ammonia and potassium cyanide added to a nitric acid solution of 
brass. Various other metals may be used for plating, as iron, tin, lead, 
platinum, bronze, etc. 

Gold and silver may be deposited in excellent condition on a large 
number of the metals and alloys. Copper may be deposited on steel, iron, 
tinned iron, zinc; also on lead and tin, and their alloys; it is to be noted 
also that when any of these metals are to be gold-, silver-, or nickel-plated 
it is best to plate them with copper first, as copper plating adheres more 
firmly to them, and in turn is adhered to more firmly by the desired coat- 
ings. Nickel may be deposited on most of the common metals, including 
German silver, brass, and alloys of the soft metals. 

PRESERVATION OF STEEL AND IRON. 

Oiling, Painting, Asphalting, etc. — Many engineers specify that structural 
material on leaving the shop shall simply be coated with pure boiled linseed 
oil instead of being painted. This is doubtless good practice, as the oil 
penetrates the skin of the metal sufficiently to preserve it temporarily from 
rust and does not cover up any possible imperfections, which may be de- 
tected readily before erection. Among the best paints are the lead paints 
and the carbon paints. On ordinary structures they should last at least 
ten years if properly applied. Some of the best paint companies guarantee 
their paints for that period. Graphite paint and asphalt paint are also 
largely used. One gallon of paint will cover 600 sq. ft. and upward of 
metal, two coats; costing about x% to ^o ct. per sq. ft.; or say, 2 to 3 cents 
per 100 lbs. of metal, for ordinary structures. For light bridges the cost 
per 100 lbs. would be greater; for heavy bridges and buildings, about 2 
cents. This is exclusive of labor in painting. 

Iron or steel thoroughly imbedded in cement or cement concrete is 
permanently protected from rust, but not from electrolysis. 

For water pipes the following are used: Asphalt, coal tar (inferior). 
Smith's durable metal coating, Sabin process. The last named is a process 
of applying asphalt varnish, the pipe, after dipping, being baked for several 
hours at a temp, of 400° to 600°, thus producing an enamel coating. 

Mill Scale on structural steel, can effectively be removed at the mill: 
(1) by " pickling " or cold rolling, at an expense of say 80 cents to $2.00 
per ton; (2) by the sand blast at a cost of about i less; (3) by steel scrapers 
and wire brushes at considerably less expense if only ordinary cleaning is 
desired. Engineers differ as to the advisability of using the sand blast. 
The following opinions are expressed in Report of Committee E on Pre- 
servative Coatings for Iron and Steel, Proceedings A. S. T. M., Vol. VI: 
Mr. Phelps Johnson, Eng'r Dominion Br. Co., Limited, says: " In my ex- 
perience I have found that an exposure to the weather for three or four 
months will ordinarily loosen nearly all the mill scale adhering to rolled 
material, and have considered that the subsequent removal by steel brushes 
of the slight coating of rust powder that is formed leaves the material in 
the best practicable shape for receiving the paint. I am of the opinion 
that the exposure to the weather may be prolonged to say 24 months without 
appreciable waste of metal or injurious pitting or roughening of the surfaces, 
and that there is but slight increase in the labor necessary to brush the sur- 
face clean for painting." Mr. Gustav Lindenthal, Consulting Engineer, 
says: " Mill scale is simply the magnetic iron oxide, which is insoluble in 
most acids, and an excellent preservative of the metal underneath, always 
provided that it adheres tightly to the surface of the metal. ... I 
consider the use of the sand-blast for the cleaning of metal surfaces not only 



PRESERVATION OF IRON AND TIMBER. S5d 

useless, but detrimental. . . . The problem of painting iron and steel 
is not yet solved, in spite of the volumes of papers, discussions and 
reports on investigations which encumber the bookshelves." Mr. A. H. 
Sabin, Chemist, says; " For best work, the cleaning of structural steel 
work by the use of the sand blast is probably the simplest and most satis- 
factory way to have it done. The great objection to this, as to all such 

work, is the cost For ordinary work, the wire brush is an efficient 

means of getting rid of loose scale and dirt; but it is practically worthless 
for removing thick rust or anything which adheres closely. Much of such 
material may be removed by steel scrapers, but deeply corrugated spots 
should be cleaned out thoroughly with a chisel, and then well brushed." 
Mr. George E. Thackray, Structural Engineer, Cambria Steel Co., says: 
" Sand blasting is almost useless and impracticable, as I have seen sand 
blasting done, exposing the metal surface, which was soon covered with a 
thick coat of rust before the painters could reach it, although both the 
sand blasters and painters were working with expedition." 

Mr. Leonard M. Cox, Civil Eng'r. U. S. Navy, in describing the pro- 
tective coating on the floating dry dock " Dewey," says:* " The selection 
of a protective coating for the dock was made the subject of careful study. 
Samples of a number of the best known paints on the market, exclusive 
of the oxide paints, were applied to test -plates and subjected to different 
conditions. Three plates were coated with each sample, one was exposed 
to the weather at the company's works, a second was suspended half in air 
and half in water, and a third was submerged in the water of Chesapeake 
Bay. The tests extended over a period of 2 years, during the construction 
of the dock, and resulted in the choice of a mixture of red lead, white zinc 
and linseed oil in the following proportions: 100 lbs. of red lead, 15 lbs. of 
zinc ground in oil, and 5 galls, of linseed oil. It is only fair to state that in 
these tests a graphite paint manufactured in Detroit showed as good results 
as the red lead, but was not used because of the lack of available data 
bearing on its behavior in salt water. 

" As the pontoons have more or less water in their bottoms at all times, 
except when self-docked, it was very necessary to provide adequate pro- 
tection for their floors. Experiments with Bitumastic Enamel led to its 
application to the whole of the interior floors, and to all vertical bulkheads, 
braces, etc., to a height of 12 ins. The process of applying this mixture 
consists in a careful cleaning and drying of the metal, one coating of a solu- 
tion, the function of which is to provide a surface to which the enamel will 
adhere, and a final heavy coating of enamel, from iV to i in. in thickness, 
applied hot. 

'* Great care was taken to rid the hull plating of mill scale. The speci- 
fication provided that all loose scale should be removed by hammering, 
scraping, and brushing with wire brushes, and to make its removal easier 
and surer, no paint was applied until a short time before launching. Nearly 
all the material, therefore, was exposed to the weather in the yard for periods 
ranging from 12 to 24 months. The existence of mill scale on the com- 
pleted structure has such an important bearing on the subject of corrosion 
that the additional expense entailed by pickling would be a good investment 
in the way of insurance, and it is recommended that specifications for future 
docks include this requirement. It was also noticed that, in the process 
of weathering, the mill scale, where covered with paint, adhered closely 
and could not be removed without the use of the hammer and chisel. This 
scale, of course, will come off in time, and for this reason the requirement 
that all contact surfaces be given one coat of paint before assembling, 
should be limited, so as to apply to rolled shapes only." 

PRESERVATION OF TIMBER. 

Sources of Decay. — Decay may be due to (1) "wet rot," caused 
by the attack of certain large fungi which may enter the pores of standing 
timber or of timber used in damp places; (2) " fermentation," or decompo- 
sition of the cellulose tissues by the putrifying sap, which takes place in 
unseasoned timber; (3) " dry rot," which occurs in so-called seasoned 
timber, caused by the fungus Merulius lachrymans which can operate only 
in the presence of some moisture; (4) insect larvae and forest worms; 

* Page 145, Vol. LVIII, Trans. A. S. C. E., in Paper No. 1042, entitled 
" The Naval Floating Dock — its advantages, design and construction." 



360 l^.— PRESERVATIVES. 

(5) sea worms, as the teredo (Teredo navalis), limnoria and other marine 
wood borers which inhabit pure salt water and attack woodwork exposed 
to same. These sea worms were formerly confined to southern waters, but 
have gradually worked their way northward. (Not present in fresh water.) 

Piling and timber grillage for submarine work need not be seasoned 
or treated with preservatives when below the action of the teredo, as timber 
will last thus for centuries. Piling in ordinary foundation work is prac- 
tically preserved from contact with the atmosphere and the destructive 
elements, by being sealed in the surrounding moisture. It is customary 
to leave the bark on piles when driven in the water and to peel them when 
land driven. If exposed to teredo the piles are peeled and treated with 
some preservative. They may be copper sheeted, creosoted, tarred, kyan- 
ized, wrapped with tarred paper or other similar preparation secured by 
vertical strips of wood. Specially constructed piles are sometimes used, 
made up of strips of scantling, on the principle that teredo will not cross 
a seam between two strips; the composite pile thus made being sometimes 
treated also with a surface preservative. The process of encircling piles 
with tight casings, pumping out the water, and then applying steam, has 
proved to be expensive. 

Creosoting consists in impregnating the wood with creosote oil (con- 
taining carbolic acid, cresylic acid, etc.) which is obtained^ from coal-tar 
distillations. The timber to be treated is placed in large iron cylinders, 
hermetically sealed and filled with steam at a low pressure, which softens 
and warms up the sap and woody fibres. The steam is then expelled and 
a partial vacuum produced (one-half atmosphere) with the air pumps, 
thus expunging the sap in the cells of the wood. The creosote oil is then 
turned into the cylinders and by means of hydraulic pressure is forced into 
the wood. There are many variations in the process. Creosoting is uni- 
versally recognized as one of the best and most practical preservatives 
known. Timber should be framed before treatment as the surface satura- 
tion is strongest. Ordinary ties, 6x8 — 8, usually absorb about 25 pounds 
of creosote, costing from 40 to 60 cents apiece. Piling or timber exposed 
to the teredo require much more oil. The cost or creosoting piling may 
be assumed at 20 cents upward per lin. ft.; and timber $16. upward per 
M. B. M. Creosoting probably ranks first, and bumettizing second, as a 
timber preservative, on a commercial scale. 

One valuable point to be noted in creosoting is that the hard and expen- 
sive woods like oak and longleaf pine do not absorb the creosote oil as 
rapidly nor as thoroughly as do the softer and less expensive woods like 
beech; so that the soft woods, thoroughly impregnated, really outlast the 
harder ones. 

For a good description of a creosoting plant see the article entitled 
" New Tie and Timber Preserving Plant of the A. T. & S. F. Ry, at Somer- 
ville, Texas," Eng. News, May 3, 1906. 

Burnettlzing is similar in method to creosoting. The solution used is 
chloride of zinc, a compound of zinc and hydrochloric acid. The results 
are inferior to creosoting, especially for bridge timbers. For ties, the 
comparison is not so marked. The cost of bumettizing is about i to i that 
of creosoting, but much depends upon the uses for which the timber is 
required. As water leaches out the zinc chloride in a comparatively short 
time when the treated timber is constantly submerged, it is evident that 
bumettizing will not do for piling. It even loses its strength in posi- 
tions exposed to heavy rains. 

Miscellaneous Notes. — Where timber is used in construction, the first 
proper guard against the actions of decay is the selection of the right kind 
of wood. Cedar, redwood, juniper, cypress, etc., are best able to resist 
ordinary decay, due to alternate wetness and dryness, hence their special 
usefulness for fencing, telegraph poles, ties, shingles, etc. Decay is some- 
thing that enters the wood at some time or other, either during the period 
of growth or subsequently. There is little doubt that if a timber is cut 
from a perfectly healthy tree and its surface hermetically sealed so as to 
keep the interior in a constant state of either extreme moisture or dryness, 
it will last for thousands of years. The earliest and most primitive method 
of preserving wood was that of charring the surface, practiced before the 
dawn of history. Subsequently, it was discovered that piling kept con- 
itantlv moist would last for centuries: and that the life of timber would 



TIMBER PRESERVATION, 361 

be prolonged greatly if kept dry from the influences of the weather, etc. 
Witness some of our old covered wooden bridges and the wooden joists of 
our colonial houses. 

In the West where wooden bridges are common there are four principal 
methods employed to prolong the lite of the structure: (1) Wooden covering 
(not common); (2) seasoning the timber to 12 or 15 per cent of moisture, 
and painting; (3) covering the upper surfaces of chords and end posts 
with galvanized iron (# 22 to # 27), allowing the sheets to lap a few inches 
down the sides; (4) treating the timber before erection, or painting it after 
erection, with a liquid preparation known as carbolineum avenarius. This 
preparation is also much used in preserving flooring and wooden paving 
blocks. 

A coating of coal tar forms a good preservative on any material, wood, 
iron or stone, but its appearance is often objectionable. Note the use of 
tarred paper for roofing. 

Copper sheeting for piles and ships' bottoms is very effective against 
the action of sea worms. 

It is claimed that the life of wooden posts can be lengthened by boring 
an auger hole in the top, filling same with salt, and sealing with a wooden 
plug. Probably the more effective method is to bevel the tops of the posts, 
and plane and paint them. Posts of all kinds which set in the ground may 
be painted with tar for a short distance above and below the ground level, 
and an additional covering of metal may also be used. Telegraph poles 
are frequently treated in this manner, and their life greatly lengthened. 

Chloride of zinc is often used for the preservation of timber, and as a 
disinfectant. 

Creosote oil tends, if anything, to preserve railway spikes in ties; it also 
has a deterrent effect on the ravages of the teredo on piling. 

Creosote and zinc chloride together forms Rutgen's process. 

Barshall or Hesselmann process — zinc chloride and glue. 

Kyanizing — corrosive sublimate. 

SEASONING OF TIMBER, 

The following is a digest of Bulletin No. 41, Bureau of Forestry, U. 
S. Department of Agriculture, issued in 1903, comprising an article entitled 
" Seasoning of Timber," by Hermann von Schrenk, in charge of Mississippi 
Valley Laboratory, Bureau of Plant Industry: 

Distribution of Water in Timber. — (a) Local Distribution. — Water may 
occur in wood in three conditions: (1) It forms the greater part (over 90 
per cent) of the protoplasmic contents of the living cells; (2) it saturates 
the walls of all cells; and (3) it entirely or at least partly fills the cavities 
of the lifeless cells, fibers, and vessels; in the sapwood of pine it occurs in 
all three forms; in the heart wood only in the second form, it merely satur- 
ates the walls. Of 100 pounds of water associated with 100 pounds ot dry- 
wood substance taken from 200 pounds of fresh sapwood of white pine, 
about 35 pounds are needed to saturate the cell walls, less than 5 pounds 
are contained in living cells, and the remaining 60 pounds partly fill the 
cavities of the wood fibers. This latter forms the sap as ordinarily under- 
stood. It is water brought from the soil, containing small quantities of 
mineral salts, and in certain species (maple, birch, etc.), it also contains 
at certain times a small percentage of sugar and other organic matter. 
These organic substances are the dissolved reserve food, stored during 
winter in the pith rays, etc., of the wood and bark; generally but a mere 
trace of them is to be found. From this it appears that the solids contained 
in the sap, such as albumen, gum, sugar, etc., can not exercise the influence 
on the strength of the wood which is so commonly claimed for them. 

The wood next to the bark contains the most water. In the species 
which do not form heartwood the decrease toward the pith is gradual, but 
where this is formed the change from a more moist to a drier condition is 
usually quite abrupt at the sapwood limit. In longleaf pine, the wood of 
the outer 1 inch of a disk may contain 50 per cent of water, that of the 
next, or second inch, only 35 per cent, and that of the heartwood only 20 
per cent. In such a tree the amount of water in any one section varies 
with the amount of sapwood, and is therefore greater for the upper than 
the lower cuts, greater for the limbs than stems, and greatest of all in the 
roots. 



362 V^.— PRESERVATIVES. 

Different trees, even of the same kind and from the same place, differ 
as to the amount of water they contain. A thrifty tree contains more water 
than a stunted one, and a young tree more than an old one, while the wood 
of all trees varies in its moisture relations with the season of the year. — 
Timber. By Filibert Roth (Bull. 10, Division of Forestry, U. S. Dept. 
of Agriculture, 1895.) 

Q)) Seasonal Distribution. — It is generally supposed that trees contain 
less water in winter than in summer. This is evidenced by the popular 
saying that " the sap is down in the winter." This is not always the case. 
Some trees contain as much water in winter as in summer, if not more. 
The average weight of lodgepole pine ties of the same size cut at Bozeman, 
Mont., in June, 1902, was 157 lbs.; in July, 144 lbs.; in Aug., 150 lbs.; in 
Sept., 157 lbs.; in Oct., 164 lbs. It is probable that this increase would 
keep up throughout the winter. 

Relation of Water to Decay in Timber. — Low forms of plant life called 
fungi grow in wood, and by so doing disintegrate and dissolve portions of 
the wood fiber. As a result of this, the wood changes in its physical prop- 
erties and is called decayed. When the fungus has extracted a sufficient 
amount of material, it forms, on the outside of the wood, fruiting bodies 
known as punks or toadstools, containing spores, which are blown about 
and infect sound wood. One of the most common of these wood-destroying 
fungi is called the Lentinus lepidens. The conditions necessary for the 
growth of these fungi are (1) water, (2) air, (3) organic food material, and 
(4) a certain amount of heat. The wood fiber and the organic substances 
found in the living cells of sapwood, such as albuminous substances, starch, 
sugar and oils, form the food supply necessary to start the growth of the 
fungus threads. A further requirement is oxygen; no growth will take 
place under water or in the ground at depths of 2 feet or more, the depth 
varying with the character of the soil. The best examples of this necessity 
for oxygen can be found in the way in which fence posts and telegraph and 
telephone poles decay at points just at or just below the surface of the 
ground, where there is a balance between the supply of air and of water. 

For practical purposes water is the most important factor. Without 
water no fungus growth, and consequently no decay, is possible. " Dry 
rot," a form of decay in which the wood turns to a dry, brittle, charcoal- 
like substance, is commonly supposed to take place without any water. 
But such is not the case. The atmospheric moisture is sufficient to permit 
growth of the dry-rot fungus even if no moisture is contained in the wood. 
Too much water will prevent fungus growth, because it shuts off the air 
supply. The amount of water necessary for fungi growth is very small; 
and wood freshly cut contains more than enough, at all seasons of the year. 

What Seasoning Is. — (a) Difference Between Seasoned and Unseasoned 
Timber. — Seasoning implies other changes besides the evaporation of water. 
Although we have as yet only a vague conception as to the exact nature of 
the difference between seasoned and unseasoned wood, it is very probable 
that one of these consists in changes in the albuminous substances in the 
wood fiber, and possibly also in the tannins, resins, and other incrusting 
substances. Whether the change in these substances is merely a drying 
out, or whether it consists in a partial decomposition, is as yet undetermined. 
Exposure to the wind and air, however, brings about changes in the wood 
which are of a nature that the wood becomes drier and more permeable. 
When seasoned by exposure to live steam, similar changes may take place. 
The water leaves the wood in the form of steam, while the organic com- 
pounds in the walls probably coagulate or disintegrate under the high 
temperature. 

{b) Manner of Evaporatian of Water. — The evaporation of water from 
timber takes place largely through the ends, i.e., in the direction of the 
longitudinal axis of the wood fibers. From the other surfaces it takes 
place very slowly out of doors, but with great rapidity in a kiln. The rate 
of evap. differs with the kind of timber and its shape. Thin boards and 
beams dry faster than thick ones; sapwood, faster than heartwood; and 
pine, faster than oak. Recent tests (not altogether conclusive) showed 
little difference in rate of evap. from sawed and hewn ties. Air-drying 
out of door takes from two months to a year, depending on kind of timber 
and climate. Wood which has been air-dried will absorb water in small 
quantities after a rain, or during damp weather, and lose most of it again 



SEASONING OF TIMBER. 



363 



when a few warm, dry days follow. "When soaked in water, seasoned 
timber absorbs it rapidly. It first enters the wood through the cell walls, 
and when these are soaked it will fill the cell lumen completely. 

Seasoning and Preservative Treatment. — (a) Seasoning and the Leach- 
ing of Salts. — Where timber is chemically treated with salts dissolved in 
water, it is absolutely necessary to season it after the treating process, for 
two reasons: First, to prevent the rapid leaching out of the salts pressed 
into the wood; second, to prevent subsequent decay. In the case of ties, 
the leaching out takes place very rapidly when they are laid immediately 
after treatment. 

(b) Seasoning and the Processes of Preservation.~Th.e object of timber 
treatment is to get certain chemical compounds into the wood with as 
much thoroughness as possible. Because of its peculiar structure, wood 
will not allow of the penetration of liquids into its mass as does a sponge. 
The solution must work its way into the wood fibers through walls of wood 
substance. If a water solution is used for the impregnating material, it 
ought to fill every cell and permeate every wall, at least of the sapwood. 
The most successful method for timber treatment (excepting the boiling 
process) so far used consists in pressing the solution into the wood. If the 
wood cells and the walls are already full of water, it is evident that there 
will be great difficulty in making the water already in place give way to the 
solution. When the walls and cell cavities are free from water the process 
of absorption of a solution is facilitated. Besides this, prior seasoning 
not only brings about a reduction in the amount of water, but also results 
in a partial disintegration of the albuminous substances which offer more 
or less resistance to the entrance of solutions. The steaming of wood 
before treatment with solutions can never replace seasoning, as it can do 
no more than drive off part of the water, unless the temp, of the steam is 
sufficiently high to injure certain portions of the wood fiber itself. 

Advantages of Seasoning. — The four principal advantages of seasoning 
timber may be enumerated as follows: (1) Seasoned timber lasts much longer 
than unseasoned; (2) Seasoning before chemical treatment greatly increases 
its effectiveness, and seasoning after treatment prevents the rapid leaching 
out of the salts introduced to preserve the timber; (3) The saving of freight 
due to a decrease in weight of 35 to 40% is sometimes a considerable item, 
even when compared with the total cost of the timber; (4) Seasoning allows 
the use of lower grade or softer timbers, so that red and swamp oak and 
beech are now being substituted for white oak, loblolly pine for longleaf 
pine, hemlock and tamarack for oak and pine, etc.; besides, the softer 
woods, being more porous, are more easily treated chemically. 

To prevent checking and splitting of timbers while seasoning, many 
English roads use S irons, which are driven into the ends of timbers that 
begin to show such tendencies. Fig. 1 shows one of these irons of meduim 
size, and Fig. 2 shows the method of applying same. The effect is a great 
saving. 



Length of Piece 8.15 





ohd-: 



/Diam.^US 



aJ2'i^ 




Fig. 1. 



Fig. 2. 



How Timber is Seasoned. — (a) Kiln Drying. — ^The French Eastern Ry. 
maintains at Amague a plant for completing the drying of its ties after 
they have been seasoned in the open air, which consists of four kilns. The 
structures are about 50 ft. long by 46 ft. wide, and contain two pairs of 
hot-air galleries, each pair of which is provided with an independent furnace 
and can be operated as a separate kiln. The air enters, at first cold, from 
the outside into a lower chamber of the furnace, and becomes gradually 



364 19.— PRESERVATIVES. 

heated in its upward progress. It is at last discharged into a hot-air cham- 
ber which occupies all the upper part of the kiln. From this it is carried 
down to the galleries in which the ties are dried by four vertical pipes, 
having a cross section of 18 by 18 ins. Two pipes open into each gallery, 
at the end of which the trams bearing the ties pass out after the drying 
has been completed. Each tram carries about 40 ties slightly separated 
from each other, so that all the faces may be in direct contact with the hot 
air in the galleries of the dry rooms and with the tar oil in the cylinders. 
The four kilns in all contain 16 galleries, with a capacity of 5 trams each, 
in all 80 small trams. It is thus possible to drv about 3200 ties at once. 
With an annual output of 400,000 ties, seventy-two hours would be allowed 
for the average drying period. The temperature of the galleries is at the 
maximum of 30° to 35° C. at the entrance, and 70° to 80° C. at the delivery. 
As the trams are taken from the cylinders one at a time, the drying is pro- 
gressive, and the wood, for this reason, is less liable to split or warp. To 
turn out 400,000 ties the furnaces of the four dry rooms consumed about 
200 tons of fine coal and 250 tons of the trimmings, from show machines, 
and wood trash and chips from the wood yard. This mixture develops a 
sufficient heat and offers the additional advantage of not wearing out the 
fire boxes by a two intense heat. The expense for fuel is about one-fifth 
of a cent for each tie. 

(b) Seasoning. — Seasoning of timber has been carried on in a practical 
way for many years in Europe. Most of the European railroads season 
their ties for many months before they treat them. The Eastern French 
Railway piles its ties in open piles 11.4 ft. high, 8.8 ft. wide, and 8.8 to 
65.6 ft. long. The ties are about 5 ft. apart. The ties are laid grillage 
fashion and spaced 4 ins. apart, except the two top tiers which are inclined 
(in the same direction) to shed water. At Amague, where kiln drying is 
subsequently used, oak ties are allowed to remain in piles for 15 to 20 months; 
and beech ties, 6 months. They are then kiln dried for from 60 to 80 hours 
at a temp, beginning at 35° and gradually brought to 75° C. Finally, they 
are treated with tar oil. 

(c) Seasoning by Steaming. — Steaming is at best a makeshift and unless 
modified materially it can never replace open-air seasoning, supplemented 
possibly by kiln-drying. However, it ought to last longer than unsteamed, 
and where it is necessary to secure partially seasoned wood the steaming 
may do. The use of the vacuum pump does not materially improve matters, 
for it is not possible to maintain a sufficiently high temperature in a cylinder 
in which enough of a vacuum exists to insure the complete removal of all 
the water. [With the present state of our knowledge the injurious effect 
upon timber from high-temperature steam is not definitely known.] 

(d) Seasoning by Immersion in Water. — It is very probable that im- 
mersion of timber for long periods in water materially hastens subsequent 
seasoning. The tannins, resins, albuminous materials, etc., which are 
deposited in the cells of the fibers of green wood, and which prevent rapid 
evaporation of the water, undergo changes when under water, probably 
due to the action of bacteria which can live without air, and in the course 
of time many of these substances are leached out of the wood. The cells 
thereby become more and more permeable to water, and when the wood is 
finally brought into the air the water escapes very rapidly and very evenly. 

(e) Seasoning by Boiling in Oil. — It is sometimes claimed that all season- 
ing preparatory to treatment with a substance like tar oil might be done 
away with by putting the green wood into a cylinder with the oil and heat- 
ing to 225° F., thus driving off the water in the form of steam, after which 
the tar oil would penetrate readily into the wood. This is the basis of the 
so-called " Curtiss process " of timber treatment. But the same objection 
made for steaming holds here, i.e., in order to get a temperature of 212° F. 
in the center of the treated wood the outside temperature would have to 
be raised so high that the strength of the wood might be injured seriously. 
A company on the Pacific coast which treats red fir piling asserts that it 
avoids this danger by leaving the green timber in the tar oil at a temp, 
which never-exceeds 225° F. for from 5 to 12 hours, until there is no further 
evidence of water vapor coming out of the wood. The tar oil is then run 
out, and a vacuum is created for about an hour, after which the oil is run 
in again and is kept in the cylinders under 100 lbs. pressure for from 10 to 
12 hours, until the required amount of absorption has been reached (about 
12 lbs. per cu. ft.). 



TIMBER SEASONING. CREOSOTE OIL. 365 

Conclusions and Recommendations. — Timber seasoning is a practical 
method for increasing the length of life of both untreated and treated tim- 
ber. At the same time it forms the most important preliminary step to 
successful chemical treatment. The cost of seasoning is insignificant, 
while the returns amount to a considerable sum in the end. With the 
increased cost and scarcity of timber, every step leading toward a more 
economic use of our supply ought to receive attention. It is perhaps too 
soon to draw final conclusions, but the following general recommenda- 
tions can confidently be made: 

(1) Green timber should be piled in as open piles as possible as soon as 
it is cut, and so kept until it is dry. In the case of ties the 7 by 2 form of 
pile (tiers with 7 and 2 ties alternating) is the best. No timber should be 
treated chemically until it is dry. 

(2) Timber treated with a preservative dissolved in water should be 
piled after treatment for several months at least to allow the water pressed 
into the wood with the salt to evaporate. Under no circumstances should 
timber freshly treated with a water solution be exposed to weathering 
influences. 

CREOSOTE IN WELL-PRESERVED TIMBERS. 

The following is a digest of Forest Service Circular 98, U. S. Dept. of 
Agriculture, issued May 9. 1907, comprising an article entitled " Quality 
and Character of Creosote in Weil-Preserved Timbers," by Gellert Alleman, 
Prof, of Chem. in Swarthmore College: 

" Of the various preservative processes for timber, those using coal- 
tar creosote are the most efficient, and, in the long run, are frequently the 
most economical as compared with the less expensive metallic-salts pro- 
cesses. Moreover, creosoted wood can be used for some purposes, as for 
salt-water piles, for which wood treated with metallic salts is but slightly 
more durable than untreated timber. Recent reports on the service of 
creosoted railroad _ ties, and of salt-water piles, have shown that, while 
proper treatment gives excellent results, much of this timber was not treated 
properly and has not lasted as it should. It is imperative that we should 
know, as completely as possible, just what constitutes efficient creosote 
treatment. This depends on three things — ^the amount of creosote, its 
character, and the thoroughness with which it penetrates the timber. The 
proper amount of creosote will depend upon the intended use of the timber. 
For instance, piles which must resist the attacks of marine borers need more 
creosote than telephone poles, and those in warm waters require more than 
those in cooler waters. 

*' The sort of creosote best suited to prevent decay and the inroads of 
marine borers can be ascertained only by many careful experiments. The 
best rneans for securing a maximum penetration of the oil is a problem 
complicated by many factors such as wood structure, moisture, etc. One 
way of approaching this problem involves a study of the nature of the 
creosotes present in timbers which have given long service. The results of 
a series of analyses of the oils present in such timbers forms the most im- 
portant part of this paper. A brief account of the source and composition 
of coal-tar creosote precedes the description and discussion of the experi- 
ments. 

Manufacture and Composition of Creosote. — (a) Source and Composition 
of Coal Tar. — When certain varieties of coal are heated in an oven or retort, 
in the absence of sufficient air for their combustion, the coal is decomposed 
and gas, tar, and coke are formed. The gas and tar rise from the heated 
mass and the coke remains in the retort. Coke and illuminating gas are 
manufactured in this way. , Where coke is the main product desired the 
" beehive " oven is used and the gas and tar are not collected, but when 
the volatile materials are to be collected the " by-product " oven is used. 
In making illuminating gas the coke and tar are regarded as by-products, 
and one of the problems of managament is how to dispose of these by- 
products to advantage. 

Coal tar is an extremely complex mixture of organic compounds, vary- 
ing with different coals and with different treatments of the same coal, 
which will yield at the same plant various qualities of coke, gas and tar, 
depending on the amount of heat applied, the quantity of air admitted, 
and the season of the year. With a low heat a relatively small amount of 
gas and tar is evolved and the tar contains large quantities of compounds 
of the paraffin series; but with a high temp, much larger amounts of gas 



19.—PRESERVA TIVES. 



and tar are obtained and the predominant compounds of the tar, in nearly 
all cases, are those of the aromatic series, as benzine, toluene, phenol, naphtha- 
lin, anthracene, etc. 

(b) Production of Creosote from Coal Tar. — The first distillation of 
crude tar, in which several separate fractions are usually taken, is made 
in large iron retorts holding from 10 to 30 tons. The forms of the retorts 
and the manner of controlling the distillation vary more or less in different 
work. In some cases the still is provided with a thermorneter enclosed in 
an iron tube srcewed into the still head; in other cases the time for changing 
the receiver for various fractions is judged solely by the specific gravity 
and other properties of the distillates. 

In Germany the fractions are frequently taken as follows: The temp, 
is that registered by the thermometer in the tar at the beginning of the 
distillation, but free from the oil and indicating the temp, of the vapor 
passing over when anthracene oil begins to distil: "First light running up to 
110° C; light oils, 110° to 210° C; carbolic oils, 210° to 240° C; heavy or 
creosote oils, 240° to 270° C; anthracene oils, 270° to 400° C. 

At many English works the following fractions are taken with the 
thermometer placed as in the German procedure just cited: Light naphtha 
up to 110° C; light oil, 110° to 170° C; carbolic oils, 170° to 225° C; creosote 
oils, 225° to 270° C; anthracene oils, 270° to 360° C. 

These temperatures are by no means universally accepted in the res- 
pective countries, and one or more fractions are often omitted; when, for 
example, it does not pay to extract carbolic acid, or when the demand for 
anthracene is limited. 

Owing to the variable constitution of the tar and to the different tem- 
peratures between which fractions are taken, the products of this prelim- 
inary separation are ferquently widely different in physical character and 
chemical composition. In distilling according to the German method 
given above, the " first runnings " and " light oils " contain, among other 
things, benzene, toluene, and the xylenes; the " carbolic oils " contain 
phenol, the creosotes, and some naphthalin; the "creosote oils," small 
quantities of phenols, naphthalin, anthracene, and many other hydro- 
carbons; the " anthracene oils," anthracene, acridene, etc. The residue 
in the still is either soft or hard pitch, according to the point at which the 
distillation is stopped. When the anthracene oil is completely distilled, 
the residue is largely hard pitch or carbon, and this is used as a briquette 
binder and in the manufacture of electric light carbons. When the dis- 
tillation is stopped at an earlier stage, soft pitch is obtained which contains 
a considerable quantity of the high-boiling tar constituents; and is used 
for roofing and for builders' paper. At present there is almost no market 
in America for hard pitch, whereas the demand for soft pitch for roofing 
is very great, which explains why the distillation of tars is not carried so 
far here as in some foreign works. 

(c) Statistics of Production and Importation of Creosote. — The following 
table gives, approximately, the amount of Coal Tar produced in the U. S., 
also the amount of Creosote Oil produced and imported, in millions of 
gallons, for the years named: 





Coal Tar. 


Creosote Oil. 




> 


From 

Gas 

Works. 


From 
By- 
pro- 
duct 
Ovens. 


Total. 
Mill- 
ions 
of 
Gals. 


Price 

per 

Gallon. 


Pro- 
duced 
in U.S. 


Im- 
por- 
ted. 


Total. 
Mill- 
ions 

of 
Gals. 


Price. 

per 

Gallon. 


in 


1898 


24.38 
40.80 
41.73 
43.64 


4.02 
22.15 

27.77 
36.38 


28.40 
62.95 
69.50 
80.02 


3.7cts. 
3.49" 
3.04" 
2.73" 










o o 
o GO 

|1 

£ o 


1903 
1904 
1905 


4.00 
4.85 
5.80 


3.71 
3.78 
7.75 


7.71 

8.63 

13.55 


5.8cts. 

6.3 " 

5.4 " 



CREOSOTE OILS. 367 

The estimate for 1903 includes the tar produced at 1956 " by-product " 
coke ovens, assuming 8 . 5 gallons of tar per ton of coal coked. It is generally- 
estimated that 12.5 gallons per ton of coal may be obtained from gas works, 
and that the general average from gas works and from ovens is a little over 
10 gallons. Moreover, it is usually assumed that the average coal tar pro- 
duced in this country contains at least 10% of oils which can be used as, 
or added to, creosote oil. 

(d) Composition of Commercial Creosote. — ^Technically speaking, the 
fraction of oil passing over between 240° and 270° C. during the first dis- 
tillation of the crude coal tar is known as " creosote oil," " heavy oil," or 
" dead oil of coal tar." In practice, however,_ the oily residues which 
remain after extracting carbolic acid, naphthalin, and anthracene from 
the various distillates in which they occur are added to the creosote oil, 
and, in consequence, many of the creosote oils of commerce contain con- 
siderable amounts of materials having boiling points higher than 270° C, 
and lower than 240° C. As a matter of fact, it is the practice at nearly 
all distilling plants to add to the " creosote well " or tank all those oils and 
residues which cannot profitably be worked over and used to greater com- 
mercial advantage. The solvents which are used in the purification of 
naphthalin and of anthracene are sometimes added to the " creosote well," 
and this accounts for the occasional presence of paraffin oil in creosote. 

The " creosote well " or tank is usually constructed of steel plates, 
and is fitted with inclosed steam coils at the bottom, in order that the solid 
materials crystallizing out can be melted before the oil is, delivered to tank 
cars, tank steamers, or barrels. A stirring device is also frequently used 
to secure uniformity in quality of supply. 

The creosote oil of commerce contains phenol (carbolic acid) , the ortho, 
meta, and para cresols, naphthalin, the oc and methyl-naphthalins (the 
former being a liquid, the latter a solid melting at 33° C), anthracene, 
phenanthracene, arcidene, and small quantities of certain high-boiling 
bases and acids. ^ When first distilled, creosote has a distinctly fluorescent 
appearance, and is. light green in cloor. There is strong evidence for the 
belief that some of the individual constituents in creosote oil combine with 
each other and probably form new products. 

At certain works, carbolic acid is extracted and the creosote oil coming 
from such places is low in "tar acids;" at other places, naphthalin is of 
considerable commercial importance and the creosote oils obtained from 
these works ^ contain _ little naphthalin. Usually anthracene separates 
with napthalin, and in the event that the latter is frozen out, the former 
is also lacking in the oil which is placed on the market. In America the 
creosote oils usually contain large amounts of naphthalin, very small 
amounts (about 5%) of phenols or cresols, and practically no anthracene 
(fpr the reason previously mentioned). It is evident that these variations 
in manufacture result in creosotes differing greatly in physical and chemical 
properties; some are rather thin oils, some are almost entirely solid with 
naphthalin, and some are heavy oils with a large proportion of high-boiling 
constituents. 

The different sorts^ of oils are believed to have different preservative 
values when injected into timber, but there is, unfortunately, a lack of 
uniformity of opinion. Some investigators have advocated oils rich in 
phenols, some those containing much naphthalin, some those containing 
a maximum of the high-boiling compounds. But little has been published 
on the subject. 

PROLONGING THE LIFE OF CROSSTIES. 

The following is a digest of Forest Service Bulletin 118, U. S. Dept. of Agri- 
culture, issued Nov. 9, 1912, prepared by Howard F. Weiss, Asst. Director, Forest 
Products Laboratory. In 1909 the steam and electric railroads of the U. S. 
purchased 123,751,000 wooden crOssties at a cost of $60,320,700. Of these ties, 
16,437,000, or about 13%, were furnished for new construction; the remainder, 
107,314,000, were used for renewals. Thus, about 358 ties were removed from 
every mile of track, which, at an average price of 49 cents per tie, necessitated, 
exclusive of all labor, a charge to the railroads of about $175 per mile, or $52,500,000 
for all lines in operation. To produce the ties used for renewals it was necessary 
to cut about 710,000 acres of timberland, averaging about 5000 board feet or 150 
ties per acre. The amount of wood so cut is equivalent under present conditions 
to the annual growth on about 55 million acres of forest. 



368 19.— PRBSER VA TI VES. 

PREPARING THE TIES FOR PRESERVATIVE TREATMENT. - 

Ties must be thoroughly seasoned before treatment with preservatives if the 
best results are to be secured. The first step in the process is to remove all bark 
from the ties. 

Peelinfi;. — The removal of bark hastens seasoning and permits uniform 
drying. Where an oil like creosote is used as a preservative, the presence of bark 
on the ties may result in such erratic penetration and absorption as to make 
the efficiency of the preservative practically zero. 

Some pine ties treated at Birmingham, Ala., were simply slabbed on two 
sides and permitted to season with the bark on. After three months' exposure 
to the air the ties were peeled and treated. The creosote penetrated only about 
V from the sides of the ties where the bark had adhered and more than 2W 
from the slabbed faces, thus forming two V-shaped zones of treated wood. Had 
all the bark been removed when the ties were cut, seasoning would have been 
uniform and impregnation complete, except for a core of heartwood in the center 
about 1" in diam. A thin strip of bark, which was about %" wide and as thin 
as paper, absolutely prevented the entrance of creosote and resulted in a strip of 
untreated wood from the surface to the center. As a rule, penetration is much 
greater with the grain than across it, and in some kinds of ties, therefore, the 
presence of a small amount of bark is of little importance. The same is true 
when salt solutions, such as zinc chloride, are used as preservatives, since not 
only are comparatively large quantities of these usually injected into the wood, 
but they permeate the wood fibers better than oils. 

Ties peeled in summer sometimes dry too rapidly and so " case-harden," 
which increases the difficulty of securing penetration of the preservative. Of a 
number of hemlock ties tested at Escanaba, Mich., in co-operation with the 
C. & N. W. R.R., most were peeled after seasoning. With the Burnett treat- 
ment these absorbed 6 lb. more of the solution than those peeled when cut. 

The best time to peel ties, however, is an economic question which must be 
determined for each particular case. As a general rule, it is good practice to re- 
move the bark as soon as possible after the ties are cut, and to regulate the rate 
of seasoning by methods of piling. In this way peeling will be easier, there will 
be less danger from insects, and seasoning will be more rapid. 

Seasoning. — Of the three common methods of seasoning, namely, by air, by 
steam, and by oil, the first is the best, if conditions will permit its use. Often, 
however, a treating plant is called upon to fill a rush order when its stock is 
insufficiently seasoned for treatment , or the plant may be so located that it can 
not keep a large stock of air-seasoned material on hand. In such cases artificial 
seasoning must be practiced. For a more thorough discussion of seasoning, see 
page 361. Air Seasoning. — In air seasoning the following considerations are im- 
portant: (1) Kind of wood, (2) temperature and circulation of air, and (3) manner 
of piling-. (1) Kind of Wood. — Coniferous woods, such as pines, firs, and spruces, 
season faster than many hardwoods, and because of their more uniform structure 
are easier to dry and can be piled more openly without danger of serious check- 
ing. The rates at which diflerent kinds of ties season at different periods of the 
year are shown in figures 2 to 12 (not reproduced here). The data on which 
these curves were based were gathered during several years' investigation and 
include thousands of weighings. The figures show how marked is the differ- 
ence in the rate of seasoning of such species as loblolly pine and red gum in com- 
parison with red and white oaks. Loblolly pine and red gum ties cut in sum- 
mer lost 62 and 63 lb., respectively, in about 3 mo., while red oak ties lost 
only 25 lb. and white oak 10 lb. during the same period. In other words, 
gum and loblolly ties lost from 2y2 to 6 times as much water as those of red and 
white oak. (2) Temperature and Circulation of Air. — The warmer and drier the 
air and the greater its circulation, the more rapid will be the loss of moisture 
from the ties. This is illustrated by the different rates of seasoning at differ- 
ent times of year. Red gum ties cut in summer lost 75 lb. each in 3^^ mo., 
while those cut in the fall lost only 40 lb. in same length of time. In the case 
of denser woods like the oaks this difference is less pronounced. Red gum ties 
cut in Tennessee in summer lost 26 lb. in 4J'i mo., while those cut in winter, and 
seasoned for same length of time, lost 22 lb. per tie. The retarding effect of 
winter upon the drying of partially seasoned ties is sometimes so pronounced 
as to cause the ties to actually gain in weight. Douglas fir ties seasoned during 
six spring and summer months contained 7 lb. less water per tie at the end of that 
period than after seasoning for 3 mo. longer. Because of the more rapid drying 
when the air is warm, summer-cut ties should have less circulation of air than 
those cut in winter. Coniferous woods can be given a much freer circulation 
than hardwoods, but too close piling will almost invariably bring about decay. 



TREATMENT OF TIES. 



369 



(3) Method of Piling. — Different kinds of wood and climate require different 
methods of piling. The closer the ties are piled the slower will be their loss in 
weight. In no case should more than two ties in a pile come in contact with the 
ground. The most open form is the triangular one, which can be rapidly made 
and is well adapted for use along the right of way. It should not, however, be 
used for hardwoods cut in summer, since these will check badly. Good forms 
of piles are the 7 by 2, 7 hy 1, and 8 by 1. These are well adapted for softwoods 
and most hardwoods cut in summer. They are easy to build and permit of free 
circulation of air. When it is desired somewhat to retard the rate of drying, the 

8 by 2 or the 10 by 1 form should be used, or if these are still too open, the 7 by 7 
form. An advantage of the 7 by 1, 8 by 1, 10 by 1, and similar forms, is that no 
tie lies flat on another, thus giving an easy run-off for rain water and a free cir- 
culation of air. In practically no case should untreated ties be piled solidly 

9 by 9, since such forms are exceedingly inefficient in regard to seasoning and in- 
vite decay. Though the Forest Service has made many tests to determine the 
effect of different "forms of roofs on the seasoning of ties, the data secured are not 
conclusive. However, a slanting roof of ties is fairly efficient in shedding water, 
and when not requiring too much additional labor can be employed advanta- 
geously. Steam Seasoning. — Seasoning by steam is not as common to-day as it 
was a decade ago. Perhaps the chief reason for this is the better knowledge that 
now exists as to the causes of decay and the effect of steaming on the strength of 
wood. Pioneers in wood preservation held that to prevent rot it was necessary 
to coagulate the albumen in the wood, and that steaming did this. Later in- 
vestigation has shown that the amount of albumen in wood is inappreciable. 
Moreover, the poor heat conductivity of ties makes it necessary to subject them 
to a steam bath for at least 4 to 8 hours before the interior temperature reaches 
that of the outside. This means, of course, a large consumption of steam. 
Steaming does not, as then believed, dry out the sap and moisture of the ties 
and render them thoroughly dry. On the contrary, it increases their moisture 
content. This may hold true for green as well as air-seasoned ties, as shown in 
Table 1. In order to decrease the weight of ties a final vacuum must be drawn. 

Table 1.— Increase in Weight of Loblolly-pine Ties Due to Steaming. 



Conditions of 
steaming. 


Gain in weight per tie, 
due to steaming. 


Period. 


Pressures 

per square 

inch. 


Green ties. 


Air- 
seasoned 
ties. 


Hours. 


Pounds. 


Pounds. 


Pounds. 


6 
10 


10 
20 
30 
40 
50 
20 
20 


2.13 


5.1 
6.9 
6.3 
8.1 
4.3 
10.8 
10.7 


.62 

1.12 

.62 


1.00 



This lowers the boiling point of the water in the wood, causing it to evaporate 
quickly. If the steam pressures used are too high, or of too long duration, or if 
the evaporation of the water from the wood is too rapid, serious decrease in 
strength will result. Steaming undoubtedly carries with it a danger of injury 
to the wood, but when some quick artificial means of seasoning ties must be 
used, steaming at low pressure not to exceed 30 lb., followed by a vacuum of 
25 to 26 inches, may be employed. Oil Seasoning. — While steam seasoning in- 
creases the weight of ties and necessitates the drawing of a vacuum to get the 
sap and water out of them, seasoning in oil produces the opposite effect, since the 
ties constantly lose moisture while in the hot bath, and no vacuum is required. 
Tests made on 2" X 2" X 24" specimens showed that this method of drying is 



370 19— PRESERVATIVES, 

likely to cause internal checking. Similar results were secured in drying 
8" X 16" X 16-ft. Douglas fir stringers. It is likely that the greater the moist- 
ure content of the ties boiled in oil, the greater will be the decrease in strength. 
Ties partially air seasoned, therefore, will check less during treatment than those 
thoroughly green. 

Immersion in Water. — After freshly cut ties are immersed in water, more or 
less of their sap will be leached out, and the cell walls left in a more porous con- 
dition. It might be expected that such ties after removal from the water would 
season faster than those not soaked, and that they would absorb more preserva- 
tive when treated. Of a number of hemlock ties tested at Escanaba, Mich., 
some were soaked in water for various periods. These absorbed considerable 
water, and when piled to season at first lost weight faster than the ties not soaked, 
but ultimately failed to reach a lower weight per cu. ft. However, when treated 
with chloride of zinp they absorbed slightly more (0.6%) of the solution than 
the unsoaked ties. This difference is so slight that soaking ties in water simply 
to increase their absorption of preservatives is not recommenHed, unless it can 
be done without extra expense. 

GROUPING TIES TO SECURE UNIFORM TREATMENT.-The im- 

portance'of properly grouping ties before placing them in the treating cylinder 
can not be overemphasized. If ties offering unequal resistance to penetration 
are treated at the same time, those offering the greatest resistance will take 
practically no preservative, while the others will get it all. Consequently, if ties 
so treated are placed in a roadbed, the ones heavily injected will outlast the 
others and the wear on the track will not be uniform. Furthermore, when the 
ties inadequately treated decay, the load which should be borne by them will 
be transferred to the sound ones, hastening their mechanical destruction. Thus 
much of the preservative will be wasted, since there is no economy in preserving 
ties from decay after they have been worn out. The aim, therefore, should be 
to have the ties depreciate uniformly, and this can largely be brought about by 
grouping them in such a manner that they will receive equal amounts of the 
preservative uniformly diffused. Many tie plants already realize the importance 
of grouping, and consider the added expense more than justified by the results 
secured. The chief factors which determine the ease with which wood may be 
impregnated are (1) the species of wood, (2) per cent of sap and heartwood, (3) 
moisture content. The less important are conditions under which seasoned, 
time of cutting, conditions of growth, and the character of the treatment. The 
characteristics of wood structure which apparently exert a marked influence on 
the absorption of preservative are (1) the " pores " or vessels, (2) tyloses, (3) resin 
ducts and cells, and (4) the composition and density of the cell wall. 

Species. — Pores "or vessels are characteristic of hardwoods, and do not occur 
in any of the conifers. They may be likened to pipes running through the wood, 
furnishing the channels by which food can readily be transported. It is custom- 
ary to group hardwoods, according to the distribution of these pores, into two 
classes: (a) diffuse-porous woods, like beech, gum, etc., in which the vessels are 
scattered throughout the annual ring, and (b) ring-porous woods, like oak and 
ash, in which they are confined to concentric rings. When these vessels are 
open, it is through them that the preservative first gains entrance into the wood. 
In red oak, for example, it is possible to blow through 3 or 4 feet of wood and to 
drive a preservative this divStance in one or two minutes. Such woods are as a 
rule easy to inject, but the preservative is very prone to confine itself to the 
vessels, leaving layers of untreated wood in between. When the vessels are 
scattered through the wood, the preservative is much more evenly diffused. 

Interesting experiments on 20,000 ties to ascertain the absorptive properties 
were made by F. J. Angier in 1908-9 at the treating plant of the C, B. & Q. R.R., 
using the zinc-creosote process. The results are summarized in Table 2. 

Class A includes ties absorbing less than 22 per cent of their volume; class 
B, ties absorbing between 23 and 30 per cent; class C, ties absorbing more than 
30 per cent. In all cases the ties were kept in the cylinder until no more preserva- 
tive could be forced into them. With hardwoods, such as oak, hickory, ash, 
beech, etc., the pressure was 175 pounds, but with softwoods this was reduced 
to from 125 to 150 pounds per square inch. In both cases the pressure was held 
for from two to five hours. No^separation of the ties was made on the basis of 
their proportion of sapwood and heartwood, but about 75 per cent of them ware 
hewed. 



TREATMENT OF TIES, 



371 



Table 2.— Absorption of Preservative by Various Species of Crossties. 




Class A. 



Beech 

Oak, red. . 
Hemlock . . 
Oak, pin... 
Hickory . . . 
Tamarack . 
Oak, white 



2481 


21.8 


3112 


20.9 


1364 


20.7 


671 


19.5 


414 


18.8 


2329 


17.1 


731 


14.2 



15 
6-15 
8-15 

10 

2- 8 

6- 8 

7 



Class B. 



Hard maple. 

Poplar 

Sycamore . . . 



691 


28.3 


1348 


26.8 


364 


26.6 



15 

7- 9 

7 



g O OT P^ 

I 6 Is 



Class B {continued) . 


Ash 


318 
928 
345 


23.0 
23.0 
22.6 


2-6 


Sweet gum 

Chestnut 


5- 9 
12 


Class C. 



Shortleaf pine . 

White elm 

Cypress, white 

Red elm 

Soft maple 

Red birch 

Tupelo gum . . . 



2192 


36.9 


872 


36.6 


662 


35.4 


626 


34.9 


599 


33.1 


775 


33.0 


790 


30.7 



5- 9 
7-15 
7- 8 
6-9 

6 
6-9 

8 



Proceedings of Wood Preservers' Assn., 1911. 



PROPORTION OF SAPWOOD.— The sapwood of practically all woods 
native to the U. S. readily absorb preservatives. The heartwood, however, is 
much more resistant, so much so in some cases that no effective treatment is 
possible. Two hundred maple ties thoroughly air-seasoned (the moisture con- 
tent ranging from 24 to 39%) were treated by the full-cell and Burnett processes 
at the Forest Products Laboratory. It was found that of the ties given a full- 
cell treatment, those containing 43.7% sapwood absorbed 10.48 lb. of creosote 
per cu. ft., while those containing 82.7% of sapwood absorbed 17.4 lb. per cu. ft. 
In the Burnett treatments the difference was not so pronounced. Ties containing 
46.2% sapwood absorbed 19.7 lb. of solution, while those which contained 80.5% 
absorbed 22.5 lb. Because of this difference in the absorption by sapwood and 
heartwood, ties which contain large amounts of sapwood should be treated 
separately. Differences in absorption by ties of the same species, but with 
varying proportions of sapwood, are often greater than in the case of widely 
different species. 

Moisture Content. — When ties are green the cell walls and many of the cell 
spaces are filled with water. To properly inject a preservative this water must 
be removed, and the extent to which it has been removed governs the amount 
of preservative which can be injected. It is important, therefore, that all of the 
ties to be treated at one time should have approximately the same moisture 
content. 

Cutting Season. — It is probably safe to say that, aside from moisture con- 
tent and case-hardening, the effect of the time of cutting on the penetration and 
absorption of preservative can be neglected in commercial work. Good practice 
consists in cutting all ties in winter or late fall whenever possible. 

Conditions of Growth. — The arrangement of the cells in the same species 
of wood grown under different conditions may differ to such an extent as to 
cause variability in the results of the treatment. When this is the case, the ties 
should be grouped separately. 

Length of Time in Preservative. — In some treating plants the time required 
to fill the cylinder with the preservative and empty it amounts to an hour or 
more. This means that the ties on the bottom of the trucks are just that much 
longer in the preservative than those on top, and hence absorb a larger amount 
of it. The most practical means of overcoming this objection is to provide for 
a rapid filling and emptying of the cylinder, or, if this cannot be done, to pile 
the ties most resistant to treatment on the bottom of the trucks. 



372 



IQ.—PRESERVA TIVES. 



PRESERVATIVE PROCESSES.— A brief synopsis of the various processes 
is given in Table 3. 

Table 3. — Synopsis of Wood-preserving Processes Used in the Treatment 

OF Ties. 









Treatment in cylinder. 


Approxi- 




Condition 


Pre- 




mate final 




1 




absorption 


Process. 


when 


servative 








of preserv- 




treated. 


used. 


Pre- 
liminary. 


Injec- 
tion. 


Final. 


ative per 
cubic foot 
(pounds). 


Full cell.. 


Seasoned or 
green. 


Creosote . . . 


Steam and 
vacuum. 1 




^None2 


8-12. 


Burnett . . 




ZnCl2 


" 


<L) 


<( 


0.25-0.5. 


Boiling . . . 


" 


Creosote . . . 


Heating in 
oil. 


> 


" 


8-12. 


Buehler.. 


Seasoned... 




Vacuum .... 


> 

D 


Air pres- 
sure. 


8-12. 




Green 




Heating in 
oil. 


Q< 




8-12. 


A.C.W... 


Seasoned or 


«< 


Steam, vac- 




« (( 


8-12. 




green. 




uum, and 












air pressure 








Riieping . 


Seasoned . . . 


" 


Air pressure 


^ 


Vacuum . 


4-7. 


Lowry . . . 


" 


<( 


None 


aJ 


" 


8+. 


Card 


Seasoned or 


Creosote 


Steam and 


u 


None2.... 


0.5 ZnCla, 




green. 


and ZnCl2. 


vacuum. 1 


1 




3-4 creo- 

sote.3 


Allardyce 




" 


(< 


£ 


(( 




Wellhouse 




ZnCls, glue, 
tannin. 






\ 


O.SZnCla. 



1 Steaming sometimes omitted when ties are air-seasoned. 

2 Final vacuum sometimes used to dry the ties. 

3 ZnCl2 = zinc chloride. 

Note. — Some engineers say that they disfavor the treating of green timber 
by the Burnett, Card, or Wellhouse processes. 



It will be seen from the table that a wide difference at present exists in the 
manner of injecting preservatives into ties, especially in regard to their prepa- 
ration for receiving the preservative. 

In the full-cell, Card, and Allardyce processes, if the final vacuum is drawn, 
it is used simply to dry the ties. In the Riieping and Lowry processes the final 
vacuum is drawn to remove some of the oil which has been injected into the 
wood, constituting what is commonly known as an " empty-cell " treatment. 

EXCERPTS AND REFERENCES. 

The Protection of Ferric Structures from Corrosion (By M. P. Wood. 
Trans. A. S. M. E., 1901; Eng. News, Sept. 16, 1901.— (1) Iron Oxide 
Pigments. — To neutralize the sulphur element natural to the ore or devel- 
oped in roasting, it is the common practice to add carbonate of lime (co- 
mon chalk) to the amount of 5 to 10% by weight of the iron oxide. 
(2) Boiled Oil vs. Pigment Coatings. — Many engineers have abandoned the 
use of iron oxide and other uncertain patent paint compounds, but still 
adhere to the use of oil for the first coating. The writer believes that a 
good pigment paint is much better than oil coating. Asphaltum Coatings. — 
The so-called asphaltum paints in general have thus far proved to be quite 
as ineffective protective coatings as any of the iron oxide or miscellaneoiis 



MISCELLANEOUS PRESERVATION. COSTS. 373 

compound paints. Linseed Oil. — Discusses the processes o£ extracting the 
oil, etc. Painting at the Mill. — Does not advocate painting the iron or 
steel just after it has left the rolls or hammer, and while hot; but just 
before assembling. 

The Painting and Sand=BIast Cleaning of Steel Bridges and Viaducts 

(By Geo. W. Lilly. Engrs. Club, Columbus, O., Feb. 1, 1902; Eng. News, 
April 24, 1902). — Raw Linseed Oil is said to make a better binder than 
boiled linseed oil, but it sets so slowly that in certain locations, such as 
viaducts subject to the blast, smoke and steam from locomotives, its use 
is inadvisable, because it will be filled with cinders and otherwise seriously 
injured before it is dry. The Pigments most commonly used for anti-rust 
paints on steel may be classed under the names red lead, iron oxide, carbon 
and graphite. Each of these has had its champion among men who have 
had considerable experience in the use of paints, while the experience of 
many experts has led to the conclusion that "red lead, oxide of iron, carbon 
and graphite all give results which average about the same." Mixing and 
Applying the paint. — All paints should be thoroughly mixed by machinery 
and subsequent grinding with a burr-stone mill if possible. But red lead, 
which is inclined to settle and harden somewhat in the mixture and make 
it difficult to spread and also diminish the coherence of the coating made 
by it, can be quite thoroughly mixed by hand when the facilities for ma- 
chine mixing are not at hand; and this is advisable in most cases, for the 
reason that it is usually impossible to have it mixed at the time it is needed 
except it be done by hand. All painting should be done, if possible, when 
the temperature of the atmosphere is above 55° F., when little trouble will 
be experienced in spreading red lead or any other of the commonly used 
paints. If painting is done when the temperature is lower than this, the 
paint should be warmed up by placing it in a vessel, which is set in water 
heated to a temperature of 130° to 150° F., and each painter should be 
frequently supplied with warm paint when the paint in his bucket becomes 
cool. Cleaning the Steel Before Painting. — Before any paint or other coat- 
ing of any kind is permitted to be applied to the iron or steel of a bridge or 
viaduct, all the scale, rust, dirt, grease and other foreign substances, as 
well as dead paint, should be removed from its surface, so that the coating 
may come into intimate contact with the clean surface of metal, and thus 
give the best condition for firm adhesion of the coating to the metal. The 
sand-blast has been given sufficient trial to make it reasonable to say that 
such cleaning as is necessary on new work at the shops— that is, removal 
of mill scale and some rust and grease — can be done at about ^ct. per 
sq. ft. of steel surface cleaned, and possibly a little less. On this basis the 
cost per ton for cleaning steel plates would be: For plates V thick, 49 cts.; 
y' thick, 98 cts.; i'Hhick, $1.96. For shapes: 7'' I-beams, weighing 17.5 
lbs. per ft., $1.35 per ton; 12'' I-beams at 50 lbs. per ft., 80 cts.; the heavier 
sections costing less and the lighter sections costing more per ton. The 
average cost for cleaning most plate girder bridges would probably be 
about $1 per ton; and the cost for a truss bridge might vary from $1 
per ton for heavy bridges to $1.75 per ton for light bridges. This article 
contains an ' illustrated description of the Newhouse sand-blast machine 
used in cleaning viaducts at Columbus, O. 

Creosoting Wooden Poles for Electric Line Work (By W. E. Moore. 
Read before Nat'l Elec. Lt. Ass'n, at Cincinnati, O., May 20, 1902; Eng. 
News, May 29, 1902). — Iron Poles or creosote poles are as yet seldom used 
by lighting companies, though iron poles are extensively used for street 
railway purposes. Wooden Poles. — Wooden poles are usually of cedar, 
heart-sawed pine, cypress, juniper or redwood, and are generally used on 
account of low cost and the comparative safety with which workmen may 
handle the live wires when standing on the cross-arms, as wood is a fair 
insulator. Red-cedar and white-cedar poles, while they have a compara- 
tively long life, have now become so scarce that it is extremely difficult to 
secure them in sufficient numbers of suitable sizes for electric light lines at 
any price in the Eastern or Southern market, though there is yet a con- 
siderable supply of white cedar in the Northwest. Heart-sawed pine poles 
have a somewhat longer life than cypress, ranging from 8 to 9 years; but 
sap pine, though readily secured in sticks of suitable size and necessary 
length, is never used, on account of its rapid decay. Study of Preservation 
of Poles. — ^The Augusta Ry. and Elec. Co. began about 9 years ago to study 
the problem of treating poles. The first experiment consisted in charring 



374 1^.— PRESERVATIVES. 

the butts of the poles, up to about 1 ft. above the earth Hne, and then satu- 
rating them with a coal-tar paint; but this was found to be of little service. 
Painting the poles with various brands of preservative compounds sold 
under various trade names was then tried, but with little or no beneficia.1 
results. In the meantime the poles then in use, almost entirely of cypress, 
continued to rot out after an average life of about 5 or 6 years. Creosoting 
Plant.— Consists of a steel cylinder 6' dia. and 102' long, with heavy cast- 
iron heads, securely supported on hinges and arranged to be clamped 
against a fibrous gasket on the head of a cylinder so as to resist a hydro- 
static pressure of 150 lbs. There is a narrow gage railway through it, with 
tracks continuing beyond ends of cylinder, which has a series of V pipes 
laid from end to end and covering the bottom, and supplied with steam 
from an 80 H. P. return tubular boiler, the steam being superheated to a 
temperature of 400° to 500°. To the cylinder is also connected a direct- 
acting vacuum pump, 14^^ x 24'', and, again, a direct -acting oil-pressure 
pump, 10"xl8". Method of Treatment. — Is fully described. Cost oi 
Treatment. — <^ost of average size (say 34 ft. long, 8" dia. at top) cypress 
pole, from $1.75 to $2 each; and cost of creosoting, about $20 per M. B. M., 
or about twice the first cost of pole. Assumed that life of treated pole will 
be prolonged 4 to 6 times that of untreated. Effect of Creosoting, on Line<- 
men. — Disagreeable in handling; dangerous to linemen when handling 
wires carrying 1000 to 3000 volts, as the creosoting lowers the electric 
resistivity of the timber. 

Sand=BIast Cleaning of Structural Steel (By Geo. W. Lilly. Trans. 
A. S. C. E., Vol. L). 

Creosoting Works of the Western Ry. of France (By J. M. Merklen. 
May number of "Revue Generale des Chemins de Fer;" Eng. News, July 27, 
1905). — Full description of Plant, Method of Treatment, and illustrated 
Method of Piling Ties for Seasoning. 

Tie and Timber Preserving Plant of the A. T. & S. F. Ry. at Somer- 
ville, Texas (Eng. News, May 3, 1906). — Description of Creosoting Process, 
Seasoning, Inspection and Marking, Creosoting Plant, Experimental Plant, 
Tanks, Gages and Pipe System, etc. Illustrated. 

The Inspection of Treatment for the Protection of Timber by the 
Injection of Creosote Oil (By H. R. Stanford. Trans. A. S. C. E.,.Vol. LVI). 

Coal Tar Paints (Eng. News, Aug. 16, 1906). — Discussions and 
references to other articles. 

Corrosion of Steel in Reinforced Cinder Concrete (By W. H. Fox. 
Eng. News, May 23, 1907). — Following conclusions from results of experi- 
ments: In no case was. any evidence found underneath the collars of neat 
cement which siu-roimded a portion of each steel specimen. To secure a dense 
homogeneous cinder concrete, a thorough tamping is necessary. A rich 
mixture, either a 1:1:3 or one in which the proportion of cement to aggre- 
gate is larger, should be used in all cases. The greatest care should be taken 
in mixing the materials, and it may be necessary to resort to the seemingly 
impractical method of coating the reinforcement with grout before placing 
in the concrete. 

Cleaning Steelwork by Sand=Blast and Painting by Compressed Air 

(By De Witt C. Webb. Eng. News, Sept. 19, 1907). — Plant.— Following 
outfit, purchased at cost of $2090, delivered at U. S. Naval Station, Key 
West, Fla.: 1 hor. gasoline engine, 20 H. P.; 1 air compressor, capac. 90 ft. 
of free air per min. compressed to pres. of 30 lbs. per sq. in. in one stage, 
belt connected to engine; 1 rotary circulating pump, belt connec.to engine; 
1 galv. steel water tank; 1 air receiver, 18" x 54"; (above all mounted on 
steel framed wagon with wooden housing.) 2 sand-blast machines, at capac. 
of 2 cu. ft. of sand each; 2 paint spraying machines, one a hand machine 
of §-gal. capacity for one operator, the other of 10-gal. capac. for two 
operators; 100 lin. ft. of sand-blast hose; 200 lin. ft. of pneumatic hose 
for sand-blast machines; 400 lin. ft. of pneu. hose for painting machines; 
4 khaki helmets, with mica-covered openings for the eyes; 200 lin. ft. of 
2" galv. pipe. Cost of Cleaning. — 2000 sq. ft. of previously untouched 
surface was thoroughly cleaned and 7000 sq. ft. of hand cleaning was all 
gone over and much improved at a total cost for labor of $97.68, and for 
gasoline (at 19 cts. per gal.) of $16.15. Cost of Painting.— The coal-tar 



MISCELLANEOUS PRESERVATION. COSTS, 



375 



paint originated by A. C. Cunningham was used (see Eng. News, July 12, 
1906), prepared with the following proportions (by volume): coal tar 
(4 parts), kerosene oil (1), Portland cement (1); cost, 15 cts. per gallon. 
Shed "A" required 64^ gals, for 9000 sq. ft. or 1 gal. to 140 sq. ft., at cost 
for labor of $28.16, and for gasoline of $3.80. Shed "B" required 86 gals, 
for 12500 sq. ft. or 1 gal. to 145 sq. ft., at cost for labor (including cleaning, 
painting, moving, setting up and removing) of $460, and for gasoline, $81. 

Paints for Concrete (By G. D. White. Proc. A. S. T. M.. Vol. IX. 
1909) . — With discussions. 

Comparison of Various Processes of Preserving Timber (By G. B. 

Shipley, Eng. News, Oct. 14, 1909). — Approximate cost of treating ties, exclud- 
ing royalty: — Burnetizing process: about ^-Ib. dry zinc per cu. tt., $0.12 per 
tie. Wellhouse process: about ^-Ib. zinc per cu. ft. plus glue and tannin, 
$0.16 per tie. Card process: about 1^-lbs. creosote and ^-Ib. dry zinc per cu. 
it., $0.18 per tie. Rueping process: about 6-lbs. creosote per cu, ft., $0,225 
per tie. Lowry process: about 6-lbs. creosote per cu. ft., $0,225 per tie. 
Absorption process: about 6-lbs creosote per cu. ft., $0.23 per tie. Full cell 
process: about 10-lbs. creosote per cu. ft., $0,335 per tie. The above costs 
are based on creosote oil at $0.07 per gal. and dry zinc chloride at $0.04 per 
lb. It costs about ^-cent per tie to handle in the yard. Timber. — The cost 
of creosoting timber with 10 lbs. of creosote per cu. ft. will be about $8 per 
1000 ft. B. M., and for each additional pound of creosote used add about 
$0.75 per 1000 ft. B. M. based on creosote oil at $0.07 per gal. Burnetizing, 
i-lb. dry zinc chloride per cu. ft., $4 per M. B. M. 

Development and Status of the Wood Preserving Industry of America 

(By E. A. SterHng. Paper, Eighth Int. Cong, of App. Chem., Sept., 1912; R. A. 
Gaz., Oct. 18, 1912).— At the present time only two standard preservatives are 
in general use in the U.S., namely, creosote and zinc chloride. Of these creosote 
seems to be gaining ground steadily, while zinc chloride is used mainly in the 
arid regions in the Middle West or in combination with creosote. It should be 
mentioned that the Santa Fe is making very extensive experiments with a crude 
oil which carries a very high percentage of asphalt um. The following table 
summarizes the processes now in use in America. The high pressure processes 
are most generally used, and while the so-called open tank or atmospheric pres- 
sure and the low pressure treatments have been used quite extensively by small 
concerns which could not afford expensive plants, it may be expected that the 
pressure treatment will prevail almost universally within a short time. 



High artificial pres- 
sure processes , . . . 



Atmospheric pres- 
sure processes . . . . 



r Full cell. 



Empty cell . 



Full cell. 



Empty cell. 



Bethell — creosote. 

Burnett — zinc chloride. 

Wellhouse— zinc chloride, glue and tannin. 

Card — zinc chloride and creosote. 

* Crude oil — natural asphaltic oil. 

* B. & M.— Zinc chloride and aluminum 
salts. 

r Riieping — creosote. 
\ Lowry — creosote. 

{Soaking in cold preservatives. 
Soaking in hot preservatives. 
Alternate hot and cold treatments. 

f Hot, cold and hot treatments. 

\ Hot and graded cooling treatments. 



Low artificial pres- f Full cell, 
sure processes . . . . \ Empty cell. 



* May be considered as still in the experimental stage. 

In addition to the above, kyanizing and vulcanizing plants are still oper- 
ating in New England, the latter being a rapid drying or baking process without 
the use of a solution. Zinc chloride, being a mineral salt, can be manufactured 
to meet definite specifications. Since crude oil is a natural product it is neces- 
sary to procure the supply from oil wells which produce the quality desired, 



375a 19,—PRESERVA TIVES, 

namely, that with a very high percentage of asphaltum. The oil which has been 
found most suitable for preservative treatment is that known as Bakersfield oil 
from southern California, also from districts in Mexico. 

Waterproofing Engineering Structures (By W. H. Finley. Paper before 
Western Soc. of C. E.; R. A. Gaz., Apr. 19, 1912).— I am not an advocate of felts 
or burlaps for ordinary waterproofing. I believe that an ordinary concrete 
surface, whether in a slab bridge or an arch, needs nothing but an application of 
a primer coat of asphalt and then a coat of liquid asphalt, after -^hich a hot sand 
mastic, composed of one part of asphalt to four of sand, can be applied with hot 
smoothing irons. On top of this it has been my practice to put on a swabbing 
coat of hot asphalt, and then to cover the whole surface with washed gravel, 
particularly where rock ballast was to be used. Over joints and ends of bridges 
it may be necessary to use burlaps for the added strength they give the asphalt 
in taking care of any slight movement. However, in cases of expansion joints, 
where any defined amount of movement is to take place, I believe that special 
means, such as copper flash joints, should be used. Burlap is a vegetable fiber 
and if used should be thoroughly saturated in a bath of hot asphalt of such tem- 
perature that it will not char or destroy the fiber. To apply asphalt to raw bur- 
lap in the field I think is wrong. If the asphalt is hot enough to penetrate the 
fiber it is Ukely to be so hot that it will char or destroy it, and if the asphalt is 
not hot enough one gets only a surface coat. 

Structures should be so designed that the water can be disposed of as di- 
rectly as possible. If more care were taken in designing, elaborate or expensive 
forms of water-proofing would be unnecessary. Asphalt has no affinity for 
water, and all surfaces that are to be coated with asphalt should be as dry and 
clean as possible. 

The methods I have used in preparing the asphalts are as follows: The asphalt 
should be heated in a suitable kettle to a temperature not exceeding 450" F. 
If this is exceeded it may result in " pitching " the asphalt. Before the " pitch- 
ing" point is reached the vapor from the kettle is of a bluish tinge which changes 
to a yellowish tinge after the danger point is passed. If this occurs the material 
should be tempered by the addition of free asphalt. The asphalt has been cooked 
sufficiently when a piece of wood can be put in and withdrawn without the 
asphalt clinging to it. Care should be taken not to prolong the heat to such an 
extent as to pitch the asphalt. Should it become necessary to hold the kettle 
for any length of time, bank or draw the fire, and introduce into the kettle a 
quantity of fresh asphalt to reduce the temperature. (Illustrated.) 

Waterproofing Processes (Eng. and Contr.). — (1) Sylvester Process of water 
proofing masonry surfaces: Alternate coats of soap solution and alum solution 
applied at intervals of 24 hr. with a soft flat brush, to the masonry surfaces; the 
soap solution boiling hot, and the alum solution at about 65° F. The soap solu- 
tion: made by dissolving % to 2H lb. of hard soap per gal. of water; the alum 
solution: made by dissolving 2 oz. to 1 lb. of alum per gal. of water. (2) Par a fine 
Process of waterproofing masonry surfaces: Pure paraffine wax, specially hard- 
ened, applied hot with a brush to the masonry surfaces; the latter preferably 
heated. (3) Alum-Lye-Cement Process of waterproofing concrete surfaces: Ap- 
plicable to fresh laid concrete. Dissolve 1 lb. of concentrated lye and 5 lb. of 
alum in 2 gal. of water. Then stir 1 pint of this solution in a 12-gal. pail con- 
taining 10 lb. of cement, adding enough water "to fill the pail. Apply thin, rub- 
bing well, on cloudy day. (4) Membrane Process of waterproofing: Consists of 
a covering composed of several (say 4 to 6, usually) layers of waterproofing ma- 
terial (paper, burlap, felt, etc.) cemented together with asphalt, bitupien or tar. 
The surface of the masonry must be clean, dry and smooth. Waterproof-felts 
are better than paper or burlap. They should be of non-perishable material, as 
asbestos, weighing from 5 to 7 lb. per 100 sq. ft. untreated, and double the amount 
when treated. 

Timber Preservation (By Walter Buehler. Trans. A.S.C.E., Mar., 1911),— 
(1) Full-cell Process: Enough creosote used to fill the cell structures. (2) Empty- 
cell Process: Cell structure is completely filled with oil, and then most of it with- 
drawn leaving the cell walls virtually painted. (3) Rutgers Process: Zinc chlo- 
ride and creosote injected in a mixed condition, the object being to maintain a 
uniform mixture by regulating the gravities of the two liquids so that one will 
remain practically suspended in the other. (4) Allardyce Process: Zinc chloride 
is injected first, and then the creosote oil, being a two-action process. (5) Card 
Process: Mixture of zinc chloride and creosote injected and maintained under 
pressure by a centrifugal pump, being a one-action process. (6) Wellhouse ProC' 



WATERPROOFING, MISCELLANEOUS. 375b 

ess: a combination of zinc chloride, tannin and glue. (7) Burnettizing = zinc 
chloride process. (7) Kyanizing = bichloride of mercury process. (8) Thil- 
many = sulphate of copper process. (9) Bethell = creosote-in-hermetically-sealed- 
cylinder process. (10) Lowry is an empty-cell process. 

New Process of Timber Preservation (Patented by R. A. Marr. Eng. News, 
Nov. 21, 1912). — Treatment consists in immersing the timber for a maximum 
period of about 4 hrs. (but varying with dia. of timber) in a hot bath of melted 
paraffin, naphthalin and silica. The cost of the preservative material is about 
3 c. per lb.; specific gravity, 0.84. For complete impregnation of railway ties, 
oak and pine require 1.6'5 to 2 lb. per cu. ft.; gum and ash require more. Further 
tests will be awaited with interest. 

Experiments in the Preservative Treatment of Crossties, by the U. S. For- 
est Service through the co-operation of the C. M. & St. P. Ry., using experimental 
track, laid in Aug., 1911, are reported in Forest Service Bulletin 126. Processes 
Used. — (1) Full-cell, using 12 lb. of creosote per cu. ft. of wood; (2) Riieping, 
using 5 lb. of creosote per cu. ft.; (3) Burnett, using 5 lb. of dry zinc chloride per 
cu. ft.; (4a) Card, using 0.5 lb. of zinc chloride and 4 lb. of creosote per cu. ft.; 
(4b) Two-movement creosote-zinc-chloride, using 4 lb. of creosote, followed by 
an impregnation of 0.5 lb. of dry zinc chloride per cu. ft.; (5) Semi-refined oil 
with paraffine base (gas-house oil) injected into the ties until refusal to absorb 
more at a pressure of 175 lb. per sq. in. and a temp, of 180° F. (There are 15 con- 
clusions regarding treatment of red-oak and hard-maple ties.) 

Creosoting Plant of the P. R. R., Phila. (By Grant B. Shipley. Eng. News, 
July 6, 1911). — The cylinder track is 24-in. gage, made of heavy angles, and be- 
tween the angles are fitted the heating coils for maintaining a temperature of 
180° to 225° F. The air, vacuum and general service pump connections are 
8 in. dia. and those of the pressure pump 3 in. dia. The piping is very simple 
as it can be operated from one floor; it is arranged for both the full-cell creosote 
and partial-cell Riieping processes. 

Asphaltic Oils for Preserving Railway Ties (By F. W. Cherrington. Paper, 
Wood Preserv. Assn., Jan., 1911). — For red oak, beech, elm, gum, etc., good 
penetration was secured by about 8 lb. of waterproofing asphaltic crude oils. 

Comparison of Zinc Chloride with Coal=Tar Creosote for Cross=Ties (By 

H. F. Weiss. Paper, A. W. P. A.; Eng. Rec, Jan. 25, 1913) .—Conclusions: (1) Zinc 
chloride and coal-tar creosote, when used under normal conditions, are both 
effective preservatives, and there is little choice between them so far as annual 
charges are concerned. (2) Creosoted ties generally cost initially more than 
Burnettized ties, the cost of treatment being two or three times as great. (3) Cre- 
osoted ties last, on the average, longer in the track than Burnettized ties, hence 
require less frequent renewals and changes in the roadbed. (4) If creosote ad- 
vances appreciably in price it will very probably result in stimulating the num- 
ber of ties treated with zinc chloride. 

Preservative Treatment of Wooden Poles (By R. A. Griffin. Paper, Natl. 
Elec. Lt. Assn., June 3-6, 1913; Eng. News, July 10, 1913).— The nost perfect 
method of timber preservation is the injection of coal-tar creosote or dead-oil of 
coal tar. This is done in three ways: (1) by the closed-tank pressure method, 
(2) by the open-tank method, and (3) by application with a brush. (The three 
methods are described and discussed.) 



20.— LUMBER AND LUMBERING. 

Stumpage. — The following is a digest of Forest Service Circular 97, 
U. S. Dept. of Agriculture, issued April 24, 1907, and entitled " The Timber 
Supply of the United States," by R. S. Kellogg, Forest Inspector: 

The percentage of the total lumber cut furnished by the principal regions 
since 1850, according to census figures, is as follows: 

1. — Geographical Distribution of Total Lumber Product. 



Year. 


North- 
eastern 
States. 


Lake 

States. 


Southern 
States. 


Pacific 
States. 


1850 


Per cent. 
54.5 
36.2 
36.8 
24.8 
18.4 
16.0 


Per cent. 
6.4 
13.6 
24.4 
33.4 
36.3 
27.4 


Per cent. 
13.8 
16.5 
9.4 
11.9 
15.9 
25.2 


Per cent. 
3.9 


I860 


6.2 


1870 


3.8 


1880 


3.5 


1890 


7.3 


1900 


9.6 







The principal estimates of the stumpage of the U. S. made since 1880 
are given in Table 2. The first, by Sargent ^1880), in addition to being 
too low for almost every species considered, with the possible exception of 
the hardwoods, is notable for its omission of Douglas spruce — ^which exists 
today in greater quantity than any other of our valuable timbers — and 
yellow pine, another important species. The next estimate, that of Hotch- 
kiss (1898), does not go into details. The next estimate, by Gannett (1900), 
was most carefully prepared. That by Fernow (1902) is a regional esti- 
mate. Long's estimate (1903) does not cover cypress, sugar pine, or hard- 
woods. Its principal point of interest is that it differs so radically — about 
38% — from that of the census of 1003 upon the stumpage of yellow pine. 
The last estimate given in the table is that published in the "American 
Lumberman," Sept. 23, 1905. _ It is based primarily upon census data, 
with the addition of some species and with increased figures for others: 
2. — Estimates of Stumpage of the United States. 



Kind of Timber. 


Census, 
1880. 


Hotchkiss, 
1898. 


Census, 
1900. 


Fernow, 
1902. 


Long, 
1903. 


American 
Lumber- 
man, 1905. 


White pine 

Eastern and 
northern pine. 


M b'rd ft. 
87,755.000 


M board ft. 


M board ft. 
50.000.000 


M board ft. 


M b'rd ft. 

60.000,000 


M board ft. 




55,000,000 


Southern yellow 
plae 


237,141,500 
12,265,000 
20,165,000 




300.000.000 

50.000.000 

100.000.000 

300,000.000 

125.000.000 

65.000.000 

75,000.000 




187.250.000 

18,22i,0uO 

56.571,000 

260.000.000 

138.000.000 


300,000,000 


Eastern spruce . . 







75,000.000 


Ea^^srs hemlocla 






100.000.000 


Douglas flr 






350.000.000 


Western yel.plne 
Cypress 








250,000.000 


*2.i53.666 
25,825.000 
22.800.000 






65,000,000 


Redwood 






75.000.000 
27,640.000 


75.000.000 


Cedar 








Sugar pine 




25,000.000 




50.000.000 


Other conifers. .. 


12.500.000 








250,000.000 














Total conifers . 


420.605.100 
435,685,000 




1,090,000,000 
300.000.000 




822,682.000 


1.570.000.000 


Total hardw'ds 




400.000,000 


Region: 
Northern States 


100.000.000 
300.000.000 


500,000,000 
700,000,000 
800.000.000 






Southern States 








W^estern States. 










Pacific States... 




1.000.000.000 




















Total 


856.290.100 


1,400.000,000 


1.390,000,000 


2,000,000,000 


822.682.000 


1.970.000.000 



* Florida and Alabama only. 



376 



STUMPAGE, LUMBER PRICES. 



Z71 



The " Pacific Lumber Trade Journal," in the issue of January, 1907, 
gave the following estimate of the stumpage of the Pacific Coast, including 
Idaho, Montana, and British Columbia: 

3. — Estimated Stumpage op California, Oregon, Washington, Idaho, 
Montana, and British Columbia. 



Kind of timber. 


M board feet. 


Kind of timber. 


M board feet. 


Douglas flr 


374,064.102 
175,586,520 
78.961,383 
75.000,000 
60,848,259 
50,000,000 


Spruce 


25,419,215 


Western and yellow pine. 
Red cedar 


Larch 


5,078,601 


Miscellaneous and hard- 
woods 




Redwood 


5,700,000 




Total 




Sugar pine 


850,658,080 









This total is credited by States as follows: 

M board feet. M board feet. 

Oregon 225,000,000 British Columbia 150,000,000 

Washington 195,658,080 Idaho and Montana 100,000,000 

California 180,000.000 

The present annual cut of some of the principal woods is as follows: 
White Pine — about 3 billion feet in the Lake States and 1 billion feet in 
other States; Yellow Pine — about 12 billion feet, or a little more than 
one-third the total cut of all species, and it is evident that within 10 to 15 
years there will be a most serious shortage; Spruce — about 1| billion feet, 
of which Maine furnishes about one-third; Hemlock — about 3 billion feet, 
of which Penn., Mich., and Wis. furnish about three-fourths (the cut of 
both eastern spruce and eastern himlock is decreasing, while that of western 
spruce and hemlock is increasing); Douglas Spruce — about 4^ billion feet 
(If billion feet in 1900); Western Yellow Pine — about 1 billion feet, two- 
thirds of which is in the Pacific Coast States; Redwood — about 450 million 
feet, and increasing; Cypress — about 750 million feet, with Louisiana 
supplying about 65%; Hardwoods — about 5 billion feet, consisting of 
approximately 43% oak, 12% poplar, 9% maple, and lesser amounts of 
numerous other species. 

Standard Names for Structural Timbers (Am. See. for Test. Materials, 
Year Book, 1913): 

I. Southern Yellow Pine. — Under this heading two classes of timber are 
used: {a) Longleaf Pine, {b) Shortleaf Pine. 

It is understood that these two terms are descriptive of quality rather than 
of botanical species. Thus, shortleaf pine would cover such species as are now 
known as North Carolina pine, loblolly pine, and shortleaf pine. " Longleaf 
Pine " is descriptive of qualit3% and if Cuban, shortleaf, or loblolly pine is grown 
under such conditions that it produces a large percentage of hard summer wood, 
so as to be equivalent to the wood produced by the true longleaf, it would be 
covered by the term " Longleaf Pine." 

,., ^•. Douglas Fir.— The term " Douglas Fir " is to cover the timber known 
hkewise as yellow fir, red fir, western fir, Washington fir, Oregon or Puget Sound 
nr or pine, northwest and west coast fir. 

3. Norway Pine, to cover what is known as " Red Pine." 

4. Hemlock, to cover Southern or Eastern hemlock; that is, hemlock from 
all States east of and including Minnesota. 

5. Weslern Hemlock, to cover hemlock from the Pacific coast. ■ 

6. Spruce, to cover Eastern Spruce; that is, the spruce timber coming from 
pomts east of Minnesota. 

7. Western Spruce, to cover the spruce timber from the Pacific coast. 

^ 8. White Pine, to cover the timber which has hitherto been known as white 
pme, from Maine, Michigan, Wisconsin, and Minnesota. 

9. Idaho White Pine, the variety of white pine from western Montana, North- 
ern Idaho, and Eastern Washington. 

n -i^' ^^^^^^ Pine, to cover the timber sold as white pine coming from Arizona, 
California, New Mexico, Colorado, Oregon, and Washington. This is the timber 
sometimes known as " Western Yellow Pine," or " Ponderosa Pine," or " Cali- 
fornia White Pine," or " Western White Pine." 

II. Western Larch, to cover the species of larch or tamarack from the Rocky 
Mountain and Pacific coast regions. 



378 20 —LUMBER AND LUMBERING. 

12. Tamarack, to cover the timber known as " Tamarack," or " Eastern 
Tamarack," from States east of and including Minnesota. 

13. Redwood, to include the California wood usually known by that name. 

Logging. — (a) How Trees Grow. — ^The section of an ordinary tree, 
through the trunk, discloses first the heart wood at the center, next the 
sapwood, and between the sapwood and the bark we find the cambium. 
If we call the bark the " coat " of the tree, we may call the cambium the 
(continuous) " undergarment," for it clothes every portion of the woody 
fiber from tip of root to end of stem, terminating in the leaves which are 
really an extension of the cambium itself. This slimy covering carries 
the life blood or sap of the tree. It has no definite thickness as it grad- 
ually merges into the bark on the one hand and into the woody fiber (sap- 
wood) on the other. If the trunk or limb of a tree is completely girdled, 
exposing the cambium to the air, that portion of the tree above the girdle 
will die, as if amputated. 

A tree breathes mainly through its leaves, and partly through the pores 
of the bark. The sap, containing various mineral substances as potassium, 
calcium, iron, sulphur, magnesium, phosphorous and nitrogen, ascends 
from the roots to the leaves, and is here met by the free oxygen and carbonic 
acid (CO2) which the leaves breathe from the air. Various compounds, 
mainly starch (CqHiqO^), are formed in these chemical laboratories through 
the agency of the sun, and these new products, in solution, circulate back 
with the sap, build up the woody fiber of the tree and produce growth of 
trunk, roots, branches, leaves, and buds, generally. When the .carbon 
dioxide is breathed into the leaves, much of the water of the sap is thrown 
off into the air, and this exhalation is called transpiration, a process similar 
to the perspiration of animals. Some trees will transpire considerably 
more than a hundred gallons of water in a day. When daylight ends, 
starch-making ceases, but the building up of the woody fibers goes on 
throughout the night. 

Trees grow radially outward. Drive two spikes, spaced vertically, 
into the trunk of a tree and note that the space does not increase percep- 
tibly as the tree grows taller. The trunk expands in diameter as the delicate 
cells near the cambium become thickened with starch from the down-flow 
sap into the woody fiber, and the up-flowing sap is forced outward through 
newer tubes. Finally the walls of the sapvv^ood become hardened with 
mineral deposits and form heartwood. The alternate semi-annular rings 
are due to summer and winter growth. With the fall of the leaves the 
breathing takes place through the pores of the bark, and in winter the tree 
practically sleeps or hibernates. ^ 

(b) Best Tifne for Cutting Timber. — We usually speak of " winter cut " 
timber as the best, and some give as a reason, that it contains less moisture. 
This is hardly the case. On the contrary, some authorities claim that many 
woods contain quite as much if not more moisture in winter than in summer. 
The principal reasons why winter cut timber should be preferred are that 
it is harder and denser; not so susceptible to the attacks of forest fungi; 
and capable of being more perfectly seasoned. The most rapid growth 
of timber is in the early summer, and this is the poorest time for cutting. 
Winter and late fall cutting seasons are the best. 

(c) Volume of Standing Timber. — The amount of standing timber on a 
certain tract is usually estimated by a " cruiser." This is done in various 
ways depending upon the accuracy required. For any individual tree, 
the contents is assumed equal to the area of base multiplied by one-half 
the height. 

(d) Transportation of Logs. — The " felling " of trees is a very important 
operation, often producing splits and cracks which reduce the grade of the 
timber. After felling, the bark is removed and it is cut into the desired 
lengths for mill logs, spars, poles, piles, ties, etc. Mill Iocs are usually in 
even lengths, from 12 ft. upward, generally from 16 to 34 ft. They are 
dragged, rafted, flumed, or shipped (by logging trains) to the mill. (3cean 
rafts are sometimes made up at an expense of many thousands of dollars, 
and towed hundreds of miles to save the cost of rail transportation. These 
rafts are cigar shaped, composed of the longest obtainable logs, arranged 
scientifically with " broken joints," and well bound with heavy chains 
for strength in resisting the action of the heavy seas. If this method is 
contemplated, it is wise to estimate, ordinarily, that one raft in every two 
or three is liable to be " broken up " and lost. Successful rafting of this 
kind has been performed from Astoria, Oregon, to San Francisco, and from 
Nova Scotia to Jersey City, N. J. 



SAWING. SEASONING. BOARD MEASURE. 379 

(e) Scaling Logs. — ^The determination of the number of thousand feet 
of lumber in logs, as a basis for selling, is usually made at the mill by an 
official scaler. The diam of the small end is measured in ins. by a scale 
rule, 4 ins. deducted, and the balance squared. The result is the number 
of ft. B. M. for a log 16 ft. in length — proportionate for logs of other lengths. 
Where logs are defective a reduction is made depending upon the judg- 
ment of the scaler. 

Sawing the logs up into rough lumber, such as " sticks " (general term 
for large dimension timber), posts, beams, ^ joists, ties, scantling (small 
dimension stuff), boards, shingles and laths, is done with the various saws, 
as band, circular (single or double), slab, gang, shingle, and lath saws. 
Rough lumber should be furnished " full-dimension " but not necessarily 
to exact dimension, as ordered. When txact dimensions are required the 
lumber should be ordered " sized." 

**Sizing" consists in running the rough lumber through the planer or 
" sticker," so gaged that when planed it shall be exactly to ordered dimen- 
sions. In ordering say 12'' x 12" to be planed on all sides, we generally 
say 12'' X 12" 5 4 5; if on one side, 12" x 12" j 1 s] etc. 

Planing costs the purchaser, ordinarily, from $1.50 to $2.00 per M, 
but it is often desirable to order planed lumber where rough lumber might 
answer; thus, with so-called " permanent " wooden structures like Howe 
truss bridges, for instance, the cost of framing with planed lumber is less 
than with rough, and the life greater, to say nothing of the appearance. 
Rough dimension stuff which is to be planed is usually ordered about i" 
large for each dimension, depending upon size of piece; less for smaller 
pieces. 

Seasoning. — After being sawed, lumber should be seasoned more 
or less thoroughly before being used. If " open-stacked " under roof 
shelter for three months or longer it is in fairly good condition for 
ordinary outside construction, but for so-called thoroughly dried lumber 
a year or two is required. Heavy timbers of course require longer periods. 
Kiln-dried lumber is the common practice and is very satisfactory if properly 
done, and the temperature of the kiln is not too high. If steam is admitted 
the temperature may be from 160° to 170° F. for the harder woods and 
from 170° to 180° for the softer kinds, as pine. In dry kilns the temper- 
ature should be lower. The harder woods are preferably stacked in the 
open air for some months before being placed in the kilns, or they may be 
immediately kiln dried at low temperatures. (See also "Seasoning of 
Timber," under Preservatives, page 361). 

Board Measure. — One ft. B. M. of lumber is equivalent to a board 1 in. 
thick, 12 ins. wide, and 1 ft. long, or i^ cu. ft.; hence 1000 ft. B. M. (= 1 M. 
B. M.) contains 83^ cu. ft. 

The following Table (4) of board measure embraces all the sizes ordin- 
arily used in construction, and gives the ft. B. M. for lengths from 1 to 9, 
which may be used decimally for any lengths. Interpolation may be 
resorted to for dimensions not in the table, but this will rarely be necessary. 
Another method is to find the ft. B. M. for dimensions 2 or more times as 
large or small, and then factor the results accordingly. Note that for this 
purpose the list of 1-inch stuff is very comprehensive. 

Where (exact) results are required to decimal places beyond those in 
the table — ^which will rarely be the case — a casual inspection will in most 
cases suffice. Thus, 1.083=1.083^3, 3.167=3.16^6, 1 .042= 1 .0416"'6, 
etc. The " character " of the decimal for any particular length may gen- 
erally be determined by examining other decimals in the same line, showing 
whether the decimal represents 6ths, 8ths, 12ths, 16ths, etc. 



^.—LUMBER AND LUMBERING, 
4. — Feet Board Measure — Engineers' Table.* 



g.as 






Length in Feet. 






g.2s 


sg« 












sg-S 


s-i5 


1 


2 


3 


4 


5 


6 


7 


8 


9 


S'535 


Mxl 


.0417 


.0833 


.1250 


.1667 


.2083 


.2500 


.2917 


.3333 


.3750 


Mxl 


Hxl 


.0521 


.1042 


.1563 


.2083 


.2604 


.3125 


.3646 


.4167 


.4688 


5/gxl 


^xlM 


.0651 


.1302 


.1953 


.2604 


.3255 


.3906 


.4557 


.5208 


.5859 


HxiM 


Mxi 


.0625 


.1250 


.1875 


.2500 


.3125 


.3750 


.4375 


.5000 


.5625 


Mxi 


MxiM 


.0781 


.1563 


.2344 


.3125 


, .3906 


.4688 


.5469 


.6250 


.7031 


%^\H 


^xlH 


.0938 


.1875 


.2813 


.3750 


.4688 


.5625 


.6563 


.7500 


.8438 


Mxi^ 


Kxl 


.0729 


.1458 


.2187 


.2917 


.3646 


.4375 


.5104 


.5833 


.6563 


^xl 


KxiM 


.0911 


.1823 


.2734 


.3646 


.4557 


.5469 


.6380 


.7292 


.8203 


3^x1 K 


KxlH 


.1094 


.2188 


.3281 


.4375 


.5469 


.6562 


.7656 


.8750 


.9844 


%^iy2 


KxiM 


.1276 


.2552 


.3828 


.5104 


.6380 


.7656 


.8932 


1.0208 


1.1484 


VsxiH 


1x1 


.0833 


.1667 


.2500 


.3333 


.4167 


.5000 


.5833 


.6667 


.7500 


1x1 


1x1 K 


.1042 


.2083 


.3125 


.4167 


.5208 


.6250 


.7292 


.8333 


.9375 


1x1 M 


1x1 M 


.1250 


.2500 


.3750 


.5000 


.6250 


.7500 


.8750 


1.0000 


1.1250 


ixiH 


1x1 M 


.1458 


.2917 


.4375 


.5833 


.7292 


.8750 


1.0208 


1.167 


1.313 


1x15^ 


1x2 


.1667 


.3333 


.5000 


.6667 


.8333 


1.0000 


1.167 


1.333 


1.500 


1x2 


1x2 M 


.1875 


.3750 


.5625 


.7500 


.9375 


1.125 


1.313 


1.500 


1.688 


1x2 Ji 


1x2 H 


.2083 


.4167 


.6250 


.8333 


1.0417 


1.250 


1.458 


1.667 


1.875 


1x2 H 


1x2 M 


.2292 


.4583 


.6875 


.9167 


1.146 


1.375 


1.604 


1.833 


2.063 


lx2M 


1x3 


.2500 


.5000 


.7500 


1.000 


1.250 


1.500 


1.750 


2.000 


2.250 


1x3 


1x3 M 


.2708 


.5417 


.8125 


1.083 


1.364 


1.625 


1.896 


2.167 


2.437 


lx3M 


1x3 H 


.2917 


.5833 


.8750 


1.167 


1.458 


1.750 


2.042 


2.333 


2.625 


1x31^ 


lx3M 


.3125 


.6250 


.9375 


1.250 


1.563 


1.875 


2.188 


2.500 


2.813 


1x3 M 


1x4 


.3333 


.6667 


1.0000 


1.333 


1.667 


2.000 


2.333 


2.667 


3.000 


1x4 


1x4^ 


.3750 


.7500 


1.125 


1.500 


1.875 


2.250 


2.625 


3.000 


3.375 


lx4>i 


1x5 


.4167 


.8333 


1.250 


1.667 


2.083 


2.500 


2.917 


3.333 


3.750 


1x5 


1x53^ 


.4583 


.9167 


1.375 


1.833 


2.292 


2.750 


3.208 


3.667 


4.125 


lx5M 


1x6 


.5000 


1.0000 


1.500 


2.000 


2.500 


3.000 


3.500 


4.000 


4.500 


1x6 


1x6 K 


.5417 


1.083 


1.625 


2.167 


2.708 


3.250 


3.792 


4.333 


4.875 


lx6H 


1x7 


.5833 


1.167 


1.750 


2.333 


2.917 


3.500 


4.083 


4.667 


5.250 


1x7 


1x7 J^' 


.6250 


1.250 


1.875 


2.500 


3.125 


3.750 


4.375 


5.000 


5.625 


lx7H 


1x8 


'.6667 


1.333 


2.000 


2.667 


3.333 


4.000 


4.667 


5.333 


6.000 


1x8 


1x8 H 


.7083 


1.417 


2.125 


2.833 


3.542 


4.250 


4.958 


5.667 


6.375 


lx8H 


1x9 


.7500 


1.500 


2.250 


3.000 


3.750 


4.500 


5.250 


6.000 


6.750 


1x9 


1x9 M 


.7917 


1.583 


2.375 


3.167 


3.908 


4.750 


5.542 


6.333 


7.125 


1x9 >i 


1x10 


.8333 


1.667 


2.500 


3.333 


4.167 


5.000 


5.833 


6.667 


7.500 


1x10 


1x10 >^ 


.8750 


1.750 


2.625 


3.500 


4.375 


5.250 


6.125 


7.000 


7.875 


lxl0>4 


1x11 


.9167 


1.833 


2.750 


3.667 


4.583 


5.500 


6.417 


7.333 


8.250 


1x11 


1x1 IJ^ 


.9583 


1.917 


2.875 


3.833 


4.792 


5.750 


6.708 


7.667 


8.625 


1x11 H 


1x12 


1.0000 


2.000 


3.000 


4.000^ 


5.000 


6.000 


7.000 


8.000 


9.000 


1x12 


1x12 M 


1.042 


2.083 


3.125 


4.167 


5.208 


6.250 


7.292 


8.333 


9.375 


1x123^ 


1x13 


1.083 


2.167 


3.250 


4.333 


5.417 


6.500 


7.583 


8.667 


9.750 


1x13 


1x13 M 


1.125 


2.250 


3.375 


4.500 


5.625 


6.750 


7.875 


9.000 


10.125 


1x133^ 


1x14 


1.167 


2.333 


3.500 


4.667 


5.833 


7.000 


8.167 


9.333 


10.50 


1x14 


1x14 J^ 


1.208 


2.417 


3.625 


4.833 


6.042 


7.250 


8.458 


9.667 


10.88 


1x14 H 


1x15 


1.250 


2.500 


3.750 


5.000 


6.250 


7.500 


8.750 


10.000 


11.25 


1x15 


1x151^ 


1.292 


2.583 


3.875 


5.167 


6.458 


7.750 


9.042 


10.33 


11.63 


1x15 H 


1x16 


1.333 


2.667 


4.000 


5.333 


6.667 


8.000 


9.333 


10.67 


12.00 


1x16 


1x17 


1.417 


2.833 


4.250 


5.667 


7.083 


8.500 


9.917 


11.33 


12.75 


1x17 


1x18 


1.500 


3.000 


4.500 


6.000 


7.500 


9.000 


10.500 


12.00 


13.50 


1x18 


1x19 


1.583 


3.167 


4.750 


6.333 


7.917 


9.500 


11.083 


12.67 


14.25 


1x19 



* Ex. — Find the number of ft. B. M. in 
2438 lin. ft. of 1" X OJ^i"? 



Solution.— 2000 = 1083.3 
400= 216.7 
30= 16.25 
8= 4.333 
Ans— 1321 ft. B. M. 



BOARD MEASURE. 381 

4.— Feet Board Measure— Engineers' Table.— Continued. 



g.9S 




Length in Feet. 


g.as 


sg-^ 






S So 


.^4 o ii 


I 


2 


3 


4 


5 


6 


7 


8 


9 


5-i^ 


1 x20 


1.667 


3.333 


5.000 


6.667 


8.333 


10.00 


11.67 


13.33 


15.00 


1 x20 


1 x21 


1.750 


3.500 


5.250 


7.000 


8.750 


10.50 


12.25 


14.00 


15.75 


1 x21 


1 x22 


1.833 


3.667 


5.500 


7.333 


9.167 


11.00 


12.83 


14.67 


16.60 


1 x22 


1 x23 


1.917 


3.833 


6.750 


7.667 


9.583 


11.50 


13.42 


15.33 


17.25 


1 x23 


1 x24 


2.000 


4.000 


6.000 


8.000 


10.000 


12.00 


14.00 


16.00 


18.00 


1 x24 


IMxlM 


.1302 


.2604 


.3906 


.6208 


.6510 


.7812 


.9115 


1.042 


1.172 


1^x1^ 


IMxl^ 


.1563 


.3125 


.4688 


.6250 


.7813 


.9375 


1.0938 


1.250 


1.406 


IH^IH 


IH^IH 


.1823 


.3646 


.5409 


.7292 


.9115 


1.0937 


1.276 


1.458 


1.641 


IH^IH 


lMx2* 


.2083 


.4167 


.6250 


.8333 


1.0417 


1.250 


1.458 


1.667 


1.876 


1^x2 


lMx2>i 


.2344 


.4688 


.7031 


.9375 


1.172 


1.406 


1.641 


1.875 


2.109 


1MX2M 


1^x23^ 


.2604 


.6208 


.7813 


1.0417 


1.302 


1.663 


1.823 


2.083 


2.344 


iMx23^ 


l>ix2M 


.2866 


.5729 


. 8594 


1.146 


1.432 


1.719 


2.005 


2.292 


2.578 


iMx25^ 


1^x3 


.3125 


.6250 


.9375 


1.250 


1.563 


1.875 


2.188 


2.500 


2.813 


lMx3 


IMxBH 


.3646 


.7292 


1.0937 


1.458 


1.823 


2.187 


2.652 


2.917 


3.281 


iMx33^ 


lMx4 


.4167 


.8333 


1.250 


1.667 


2.083 


2.500 


2.917 


3.333 


3.760 


lMx4 


IH^^H 


.4688 


.9375 


1.406 


1.875 


2.344 


2.813 


3.281 


3.750 


4.219 


IH^^H 


lMx5 


.5208 


1.0417 


1.563 


2.083 


2.604 


3.125 


3.646 


4.167 


4.688 


IHxS 


lMx5H 


.5729 


1.146 


1.719 


2.292 


2.865 


3.438 


4.010 


4.583 


6.156 


1^x534 


l>ix6 


.6250 


1.250 


1.875 


2.500 


3.125 


3.750 


4.375 


5.000 


6.625 


lMx6 


lMx7 


.7292 


1.458 


2.188 


2.917 


3.646 


4.378 


5.104 


5.833 


6.662 


l>ix7 


1^x8 


.8333 


1.667 


2.500 


3.333 


4.167 


6.000 


6.833 


6.667 


7.600 


1^x8 


lMx9 


.9375 


1.875 


2.813 


3.750 


4.688 


6.625 


6.563 


7.500 


8.438 


lMx9 


iMxio 


1.0417 


2.083 


3.128 


4.167 


5.208 


6.250 


7.292 


8.333 


9.376 


13^x10 


1^x11 


1.146 


2.292 


3.438 


4.683 


6.729 


6.878 


8.021 


9.167 


10 313 


IMxll 


lMxl2 


1.250 


2.500 


3.760 


5.000 


6.250 


7.500 


8.760 


10.00 


11.26 


1^x12 


1^x13^ 


.1875 


.3750 


.5626 


.7500 


.9375 


1.126 


1.313 


1.500 


1.688 


IHxlH 


iMxlM 


.2188 


.4376 


.6663 


.8760 


1.0938 


1.313 


1.531 


1.750 


1.969 


13^xlM 


13^x2 


.2500 


.5000 


.7600 


1.0000 


1.250 


1.500 


1.760 


2.000 


2.250 


13^x2 


1^x2 >i 


.2813 


.6625 


.8438 


1.1250 


1.406 


1.688 


1.969 


2.250 


2.631 


1}4^2H 


lKx2H 


.3125 


.6250 


.9375 


1.250 


1.563 


1.875 


2.188 


2.500 


2.813 


13^x2H 


lKx2M 


.3438 


.6876 


1.0313 


1.378 


1.719 


2.063 


2.406 


2.750 


3.094 


13^x2M 


13^x3 


.3750 


.7600 


1.125 


1.600 


1.875 


2.260 


2.626 


3.000 


3.376 


13^x3 


1 3^x3 3^ 


.4375 


.8760 


1.313 


1.760 


2.188 


2.626 


3.063 


3.600 


3.938 


13^x33^ 


1^x4 


.5000 


1.0000 


1.500 


2.000 


2.500 


3.000 


3.600 


4.000 


4.600 


13^x4 


13^x43^ 


.5625 


1.125 


1.688 


2.250 


2.813 


3.375 


3.938 


4.600 


6.063 


lHx4>4 


13^x5 


.6250 


1.250 


1.875 


2.500 


3.126 


3.760 


4.376 


6.000 


6.625 


13^x8 


13^x53^ 


.6875 


1.375 


2.063 


2.750 


3.438 


4.125 


4.813 


6.600 


6.188 


1^x53^ 


13^x6 


.7500 


1.500 


2.250 


3.000 


3.760 


4.600 


6.250 


6.000 


6.760 


13^x6 


13^x7 


.8750 


1.750 


2.625 


3.500 


4.376 


6.260 


6.125 


7.000 


7.876 


13^x7 


13^x8 


1.0000 


2.000 


3.000 


4.000 


6.000 


6.000 


7.000 


8.000 


9.000 


13^x8 


13^x9 


1.126 


2.250 


3.375 


4.600 


6.626 


6.760 


7.875 


9.000 


10.125 


13^x9 


13^x10 


1.260 


2.500 


3.750 


5.000 


6.250 


7.500 


8.750 


10.000 


11.25 


13^x10 


13^x12 


1.600 


3.000 


4.500 


6.000 


7.500 


9.000 


iO.500 


12.00 


13.50 


13^x12 


2 x2 


.3333 


.6667 


1.000 


1.333 


1.667 


2.000 


2.333 


2.667 


3.000 


2 x2 


2 x2Ji 


.3750 


.7500 


1.125 


1.500 


1.875 


2.250 


2.626 


3.000 


3.375 


2 x2H 


2 x2^ 


.4167 


.8333 


1.250 


1.667 


2.083 


2.500 


2.917 


3.333 


3.750 


2 x2>^ 


2 x2^ 


.4583 


.9167 


1.375 


1.833 


2.292 


2.750 


3.208 


3.667 


4.125 


2 x2M 


2 x3 


.5000 


l.OOOO 


1.500 


2.000 


2.500 


3.000 


3.600 


4.000 


4.600 


2 x3 


2 x3H 


.5833 


1.167 


1.750 


2.333 


2.917 


3.500 


4.083 


4.667 


5.250 


2 x334 


2 x4 


.6667 


1.333 


2.000 


2.667 


3.333 


4.000 


4.667 


6.333 


6.000 


2 x4 


2 x4H 


.7600 


1.500 


2.250 


3.000 


3.760 


4.600 


6.260 


6.000 


6.760 


2 x4H 


2 x6 


.8333 


1.667 


2.600 


3.333 


4.167 


8.000 


5.833 


6.667 


7.600 


2 x6 


2 x5H 


.9167 


1.833 


2.750 


3.667 


4.583 


6.500 


6.417 


7.333 


8.260 


2 x5H 


2 x6 


1.0000 


2.000 


3.000 


4.000 


5.000 


6.000 


7.000 


8.000 


9.000 


2 x6 


2 x7 


1.167 


2.333 


3.500 


4.667 


6.833 


7.000 8.167 


9.333 


10.600 


2 x7 



^.—LUMBER AND LUMBERING. 



4. — Feet Board Measure— Engineers' Table.— Continued. 



iBi 




Length in Feet. 


g.2s 


s s-^ 






sg-^ 


.303 


1 


2 


3 


4 


5 


6 


7 


8 


9 


s-il- 


2 x8 


1.333 


2.667 


4.000 


5.333 


6.667 


8.000 


9.333 


10.667 


12.00 


2 x8 1 


2 x9 


1.500 


3.000 


4.550 


6.000 


7.500 


9.000 


10.500 


12.000 


13.50 


2 x9 4 


2 xlO 


1.667 


3.333 


5.000 


6.667 


8.333 


10.00 


11.67 


13.33 


15.00 


2 xlC* 


2 xll 


1.833 


3.667 


5.500 


7.333 


9.167 


11.00 


12.83 


14.67 


16.50 


2 xll 


2 xl2 


2.000 


4.000 


6.000 


8.000 


10.00 


12.00 


14.00 


16.00 


18.00 


2 xl2 


2 xl4 


2.333 


4.667 


7.000 


9.333 


11.67 


14.00 


16.33 


18.67 


21.00 


2 xl4 


2 xl5 


2.500 


5.000 


7.500 


10.000 


12.50 


15.00 


17.50 


20.00 


22.50 


2 xl5 


2 xl6 


2.667 


5.333 


8.000 


10.67 


13.33 


16.00 


18.67 


21.33 


24.00 


2 xl6 


2Kx2M 


.5208 


1.0417 


1.563 


2.083 


2.604 


3.125 


3.646 


4.167 


4.688 


2Hx23^ 


2 3^x2 M 


.5729 


1.146 


1.719 


2.292 


2.865 


3.438 


4.010 


4.583 


5.156 


2>^x2M 


2Mx3 


.6250 


1.250 


1.875 


2.500 


3.125 


3.750 


4.375 


5.000 


5.625 


23^x3 


2 1^x3 H 


.7292 


1.458 


2.188 


2.917 


3.646 


4.375 


5.104 


5.833 


6.562 


23^x33^ 


2^x4 


.8333 


1.667 


2.500 


3.333 


4.167 


5.000 


5.833 


6.667 


7.500 


2 3^x4 


2^x41^ 


.9375 


1.875 


2.813 


3.750 


4.688 


5.625 


6.563 


7.500 


8.438 


2 3^x43^ 


23^x5 


1.042 


2.083 


3.125 


4.167 


5.208 


6.250 


7.292 


8.333 


9.375 


23^x5 


2}.ix5}4 


1.146 


2.292 


3.438 


4.583 


5.729 


6.875 


8.021 


9.167 


10.313 


2 3^x5 >^ 


23^x6 


1.250 


2.500 


3.750 


5.000 


6.250 


7.500 


8.750 


10.000 


11.25 


2^x6 


23^x7 


1.458 


2.917 


4.375 


5.833 


7.292 


8.750 


10.208 


11.67 


13.13 


2 3^x7 


2Kx8 


1.667 


3.333 


5.000 


6.667 


8.333 


10.00 


11.67 


13.33 


15.00 


2Mx8 


23^x9 


1.875 


3.750 


5.625 


7.500 


9.375 


11.25 


13.13 


15.00 


16.88 


23^x9 


23^x10 


2.083 


4.167 


6.250 


8.333 


10.417 


12.50 


14.58 


16.67 


18.75 


23^x10 


23^x12 


2.500 


5.000 


7.500 


10.000 


12.50 


15.00 


17.50 


20.00 


22.50 


2 3^^x12 


2>^xl4 


2.917 


5.833 


8.750 


11.67 


14.58 


17.50 


20.42 


23.33 


26.25 


2 3^x14 


2j^xl5 


3.125 


6.250 


9.375 


12.50 


15.63 


18.75 


21.88 


25.00 


28.13 


2 3^x15 


2Kxl6 


3.333 


6.667 


10.000 


13.33 


16.67 


20.00 


23.33 


26.67 


30.00 


23^x16 


3 x3 


.7500 


1.500 


2.250 


3.000 


3.750 


4.500 


5.250 


6.000 


6.750 


3 x3 


3 x3H 


.8750 


1.750 


2.625 


3.500 


4.375 


5.250 


6.125 


7.000 


7.875 


3 x33^ 


3 x4 


1.0000 


2.000 


3.000 


4.000 


5.000 


6.000 


7.000 


8.000 


9.000 


3 x4 


3 x4M 


1.125 


2.250 


3.375 


4.500 


5.625 


6.750 


7.875 


9.000 


10.125 


3 x43^ 


3 x5 


1.250 


2.500 


3.750 


5.000 


6.250 


7.500 


8.750 


10.000 


11.25 


3 x5 


3 x5K 


1.375 


2.750 


4.125 


5.500 


6.875 


8.250 


9.625 


11.00 


12.38 


3 x5H 


3 x6 


1.500 


3.000 


4.500 


6.000 


7.500 


9.0C0 


10.500 


12.00 


x3.50 


3 x6 


3 x7 


1.750 


3.500 


5.250 


7.000 


8.750 


10.500 


12.25 


14.00 


15.75 


3 x7 


3 x8 


2.000 


4.000 


6.000 


8.000 


10.000 


12.00 


14.00 


16. CO 


18.00 


3 x8 


3 x9 


2.250 


4.500 


6.750 


9.000 


11.25 


13.50 


15.75 


18.00 


20.25 


3 x9 


3 xlO 


2.500 


5.000 


7.500 


10.000 


12.50 


15.00 


17.50 


20.00 


22.50 


3 XlO 


3 xll 


2.750 


5.500 


8.250 


11.00 


13.75 


16.50 


19.25 


22.00 


24.75 


3 xll 


3 xl2 


3.000 


6.000 


9.000 


12.00 


15.00 


18.00 


21.00 


24.00 


27.00 


3 xl2 


3 xl3 


3.250 


6.500 


9.750 


13.00 


16.25 


19.50 


22.75 


26.00 


29.25 


3 xl3 


3 xl4 


3.500 


7.000 


10.500 


14.00 


17.50 


21.00 


24.50 


28.00 


31.50 


3 xl4 


3 xl5 


3.750 


7.500 


11.25 


15.00 


18.75 


22.50 


26.25 


30.00 


33.75 


3 xl5 


3 xl6 


4.000 


8.000 


12.00 


16.00 


20.00 


24.00 


28.00 


32.00 


36.00 


3 xl6 


3 xlS 


4.500 


9.000 


13.50 


18.00 


22.50 


27.00 


31.50 


36.00 


40.50 


3 xl8 


3 3^x3 K 


1.021 


2.042 


3.063 


4.083 


5.104 


6.125 


7.146 


8.167 


9.188 


33^x33^ 


33^x4 


1.167 


2.333 


3.500 


4.667 


5.833 


7.000 


8.167 


9.333 


10.667 


33^x4 


3^x43^ 


1.313 


2.625 


3.938 


5.250 


6.563 


7.875 


9.188 


10.500 


11.813 


33^x43^ 


33^x5 


1.458 


2.917 


4.375 


5.833 


7.292 


8.750 


10.208 


11.67 


13.13 


33^x5 


33^x53^ 


1.604 


3.208 


4 813 


6.417 


8.021 


9.625 


11.23 


12.83 


14.44 


SHx5H 


3Kx6 


1.750 


3.500 


5.250 


7.000 


8.750 


10.500 


12.25 


14.00 


15.75 


33^x6 


33^x7 


2.042 


4.083 


6.125 


8.167 


10.208 


12.25 


, 14.29 


16.33 


18.38 


3>ix7 


33^x8 


2.333 


4.667 


7.000 


9.333 


11.67 


14.00 


16.33 


18.67 


21.00 


33^x8 


33^x9 


2.625 


5.250 


7.875 


10.500 


13.13 


15.75 


18.38 


21.00 


23.63 


3 3^x9 


33^x10 


2.917 


5.833 


8.750 


11.67 


14.58 


17.50 


20.42 


23.33 


26.25 


33^x10 


33^x11 


3.208 


6.417 


9.625 


12.83 


16.04 


19.25 


22.46 


25.67 


28.88 


33^x11 


33^x12 


3.500 


7.000 


10.500 


14.00 


17.50 


21.00 


24.50 


28.00 


31.50 


33-^x12 



BOARD MEASURE. 



4.— Feet Board Measure— Engineers' Table. — Continued. 



iai 








Length in 


Feet. 






g.ss 


Sgo 


















Sgo 


i3-i5 


1 


2 


3 


4 


5 


6 


7 


8 


9 


5'i^ 


3Hxl4 


4.083 


8.167 


12.25 


16.33 


20.42 


24.50 


28.58 


32.67 


36.75 


33^x14 


33^x15 


4.375 


8.750 


13.13 


17.50 


21.88 


26.25 


30.63 


35.00 


39.38 


33^x15 


3>^xl6 


4.667 


9.333 


14.00 


18.67 


23.33 


28.00 


32.67 


37.33 


42.00 


3 3^x16 


31^x17 


4.958 


9.917 


14.88 


19.83 


24.79 


29.75 


34.71 


39.67 


44.63 


33^x17 


3Kxl8 


5.250 


10.500 


16.75 


21.00 


26.25 


31.50 


36.75 


42.00 


47.25 


33^x18 


4 x4 


1.33« 


2.667 


4.000 


5.333 


6.667 


8.000 


9.333 


10.67 


12.00 


4 x4 


4 x43^ 


1.500 


3.000 


4.500 


6.000 


7.500 


9.000 


10.500 


12.00 


13.50 


4 x43^ 


4 x5 


1.667 


3.333 


5.000 


6.667 


8.333 


10.00 


11.67 


13.33 


15.00 


4 x5 


4 x53^ 


1.833 


3.667 


5.500 


7.333 


9.167 


11.00 


12.83 


14.67 


16.50 


4 x5^ 


4 x6 


2.000 


4.000 


6.000 


8.000 


10.000 


12.00 


14.00 


16.00 


18.00 


4 x6 


4 x63^ 


2.167 


4.333 


6.500 


8.667 


10.83 


13.00 


15.17 


17.33 


19.50 


4 x63^ 


4 x7 


2.333 


4.667 


7.000 


9.333 


11.67 


14.00 


16.33 


18.67 


21.00 


4 x7 


4 x8 


2.667 


5.333 


8.000 


10.667 


13.33 


16.00 


18.67 


21.33 


24.00 


4 x8 


4 x9 


3.000 


6.000 


9.000 


12.00 


15.00 


18.00 


21.00 


24,00 


27.00 


4 x9 


4 xlO 


3.333 


6.667 


10.000 


13.33 


16.67 


20.00 


23.33 


26.67 


30.00 


4 xlO 


4 xll 


3.667 


7.333 


11.00 


14.67 


18.33 


22.00 


25.67 


29.33 


33.00 


4 xll 


4 xl2 


4.000 


8.000 


12.00 


16.00 


20.00 


24.00 


28.00 


32.00 


36.00 


4 xl2 


4 xl4 


4.667 


9.333 


14.00 


18.67 


23.33 


28.00 


32.67 


37.33 


42.00 


4 xl4 


4 xl5 


6.000 


10.000 


15.00 


20.00 


25.00 


30.00 


35.00 


40.00 


45.00 


4 xl5 


4 xl6 


6.333 


10.67 


16.00 


21.33 


26.67 


32.00 


37.33 


42.67 


48.00 


4 xie 


4 xl8 


6.000 


12.00 


18.00 


24.00 


30.00 


36.00 


42.00 


48.00 


54.00 


4 xl8 


4Jix43^ 


1.688 


3.375 


5.063 


6.750 


8.438 


10.13 


11.81 


13.50 


15.19 


43^x43^ 


4^x5 


1.875 


3.750 


6.625 


7.500 


9.375 


11.25 


13.13 


15.00 


16.88 


43^x5 


4Hx5>^ 


2.063 


4.125 


6.188 


8.250 


10.313 


12.38 


14.44 


16.50 


18.56 


43^x53^ 


4^x6 


2.250 


4.500 


6.750 


9.000 


11.25 


13.50 


15.75 


18.00 


20.25 


43^x6 


4>^x6i^ 


2.438 


4.875 


7.313 


9.750 


12.19 


14.63 


17.06 


19.50 


21.94 


4Mx63^ 


43^x7 


2.625 


5.250 


7.875 


10.500 


13.13 


15.75 


18.38 


21.00 


23.63 


43^x7 


43^x8 


3.000 


6.000 


9.000 


12.00 


15.00 


18.00 


21.00 


24.00 


27.00 


4 3^x8 


43^x9 


3.375 


6.750 


10.125 


13.50 


16.88 


20.25 


23.63 


27.00 


30.38 


4>^x9 


43^x10 


3.750 


7.500 


11.25 


15.00 


18.75 


22.50 


26.25 


30.00 


33.75 


43^x10 


43^x11 


4.125 


8.250 


12.38 


16.50 


20.63 


24.75 


28.88 


33.00 


37.13 


4>^xll 


43^x12 


4.500 


9.000 


13.50 


18.00 


22.50 


27.00 


31.50 


36.00 


40.50 


4 3^x12 


43^x14 


6.250 


10.500 


15.75 


21.00 


26.25 


31.50 


36.75 


42.00 


47.25 


4Hxl4 


4^x15 


5.625 


11.25 


16.88 


22.50 


28.13 


33.75 


39.38 


45.00 


50.63 


4^x15 


43^x16 


6.000 


12.00 


18.00 


24.00 


30.00 


36.00 


42.00 


48.00 


54.00 


43^x16 


43^x18 


6.750 


13.50 


20.25 


27.00 


33.75 


40.50 


47.25 


54.00 


60.75 


4Hxl8 


fl x5 


2.083 


4.167 


6.250 


8.333 


10.42 


12.50 


14.58 


16.67 


18.75 


5 x5 


6 x53^ 


2.292 


4.583 


6.875 


9.167 


11.46 


13.75 


16.04 


18.33 


20.63 


5 x53^ 


6 x6 


2.500 


5.000 


7.500 


10.000 


12.50 


15.00 


17.50 


20.00 


22.50 


5 x6 


6 x63^ 


2.708 


5.417 


8.125 


10.83 


13.54 


16.25 


18.96 


21.67 


24.37 


5 x63^ 


5 x7 


2.917 


5.833 


8.750 


11.67 


14.58 


17.50 


20.42 


23.33 


26.25 


5 x7 


6 x7>^ 


3.125 


6.250 


9.375 


12.50 


15.63 


18.75 


21.88 


25.00 


28.13 


5 x73^ 


6 x8 


3.333 


6.667 


10.000 


13.33 


16.67 


20.00 


23.33 


26.67 


30.00 


5 x8 


6 x9 


3.750 


7.500 


11.25 


15.00 


18.75 


22.50 


26.25 


30.00 


33.75 


5 x9 


5 xlO 


4.167 


8.333 


12.50 


16.67 


20.83 


25.00 


29.17 


33.33 


37.50 


5 XlO 


6 xll 


4.583 


9.167 


13.75 


18.33 


22.92 


27.50 


32.08' 


36.67 


41.25 


5 xll 


6 xl2 


6.000 


10.000 


15.00 


20.00 


25.00 


30.00 


35.00 


40.00 


45.00 


5 xl2 


6 xl4 


6.833 


11.67 


17.50 


23.33 


29.17 


35.00 


40.83 


46.67 


52.50 


5 xl4 


6 xl5 


6.250 


12.50 


18.75 


25.00 


31.25 


37.50 


43.75 


50.00 


56.25 


5 xl5 


5 xl6 


6.667 


13.33 


20.00 


26.67 


33.33 


40.00 


46.67 


53.33 


60.00 


5 xl6 


6 xl8 


7.500 


15.00 


22.50 


30.00 


37.50 


45.00 


52.50 


60.00 


67.50 


5 xl8 


53^x53^ 


2.521 


5.042 


7.563 


10.08 


12.60 


15.13 


17.65 


20.17 


22.69 


5 3^x5 >^ 


6^x6 


2.750 


5.500 


8.250 


11.00 


13.75 


10.50 


19.25 


22.00 


24.75 


5 3^x6 


53^x63^ 


2.979 


5.958 


8.938 


11.92 


14.90 


17.88 


20.85 


23.83 


26.81 


53^x63^ 


63^x7 


3.208 


6.417 


9.625 


12.83 


16.04 


19.25 


22.46 


25.67 


28.88 


53^x7 



384 



20— LUMBER AND LUMBERING. 



4.— Feet Board Measure— Engineers' Table.— Continued. 



g.2S 






Length in Feet. 






g.9s 


as-g 












S^f. 


s-is 


1 


2 


3 


4 


5 


6 


7 


8 


9 


s-il 


5^x73^ 


3.438 


6.875 


10.313 


13.75 


17.19 


20.63 


24.06 


27.50 


30.94 


!^S^ 


5Kx8 


3.667 


7.333 


11.00 


14.67 


18.33 


22.00 


25.67 


29.33 


33.00 


5^x9 


4.125 


8.250 


12.38 


16.50 


20.63 


24.75 


28.88 


33.00 


37.13 


5Kx9 


5KxlO 


4.583 


9.167 


13.75 


18.33 


22.92 


27.50 


32.08 


36.67 


41.25 


53^x10 


53^x11 


5.042 


10.083 


15.13 


20.17 


25.21 


30.25 


35.29 


40.33 


45.38 


5^x11 


5^x12 


5.500 


11.00 


16.50 


22.00 


27.50 


33.00 


38.50 


44.00 


49.50 


53^x12 


5>^xl4 


6.417 


12.83 


19.25 


25.67 


32.08 


38.50 


44.92 


51.33 


57.75 


5>ixl4 


5^x15 


6.875 


13.75 


20.63 


27.50 


34.38 


41.25 


48.13 


55.00 


61.88 


53^x15 


5J^xl6 


7.333 


14.67 


22.00 


29.33 


36.67 


44.00 


51.33 


58.67 


66.00 


53^x16 


53^x18 


8.250 


16.50 


24.75 


33.00 


41.25 


49.50 


57.75 


66.00 


74.25 


53^x18 


6 x6 


3.000 


6.000 


9.000 


12.00 


15.00 


18.00 


21.00 


24.00 


27.00 


6 x6 


6 x63^ 


3.250 


6.500 


9.750 


13.00 


16.25 


19.50 


22.75 


26.00 


29.25 


6 x63^ 


6 x7 


3.500 


7.000 


10.500 


14.00 


17.50 


21.00 


24.50 


28.00 


31.50 


6 x7 


6 x7M 


3.750 


7.500 


11.25 


15.00 


18.75 


22.50 


26.25 


30.00 


33.75 


6 x73^ 


6 x8 


4.000 


8.000 


12.00 


16.00 


20.00 


24.00 


28.00 


32.00 


36.00 


6 x8 


6 x9 • 


4.500 


9.000 


13.50 


18.00 


22.50 


27.00 


31.50 


36.00 


40.50 


6 x9 


6 xlO 


5.000 


10.000 


15.00 


20.00 


25.00 


30.00 


35.00 


40.00 


45.00 


6 XlO 


6 xll 


5.500 


11.00 


16.50 


22.00 


27.50 


33.00 


38.50 


44.00 


49.50 


6 xll 


6 xl2 


6.000 


12.00 


18.00 


24.00 


30.00 


36.00 


42.00 


48.00 


54.00 


6 xl2 


6 xU 


7.000 


14.00 


21.00 


28.00 


35.00 


42.00 


49.00 


56.00 


63.00 


6 xl4 


6 xl5 


7.600 


15.00 


22.50 


30.00 


37.50 


45.00 


52.50 


60.00 


67.50 


6 xl5 


6 xl6 


8.000 


16.00 


24.00 


32.00 


40.00 


48.00 


56.00 


64.00 


72.00 


6 xl6 


6 xl8 


9.000 


18.00 


27.00 


36.00 


45.00 


54.00 


63.00 


72.00 


81.00 


6 xl8 


6Mx6M 


3.521 


7.042 


10.56 


14.08 


17.60 


21.13 


24.65 


28.17 


31.69 


63^x6>4 


63^x7 


3.792 


7.583 


11.38 


15.17 


18.96 


22.75 


26.54 


30.33 


34.13 


6Hx7 


63^x73^ 


4.063 


8.125 


12.19 


16.25 


20.31 


24.38 


28.44 


32.50 


36.66 


63^x73^ 


6Mx8 


4.333 


8.667 


13.00 


17.33 


21.67 


26.00 


30.33 


34.67 


39.00 


6Kx8 


63^x9 


4.875 


9.750 


14.63 


19.50 


24.38 


29.25 


34.13 


39.00 


43.88 


6^x9 


63^x10 


5.417 


10.833 


16.25 


21.67 


27.08 


32.50 


37.92 


43.33 


48.75 


63^x10 


63^x11 


6. "958 


11.92 


17.88 


23.83 


29.79 


35.75 


41.71 


47.67 


53.63 


6>^xll 


6>^xl2 


6.500 


13.00 


19. 6p 


26.00 


32.50 


39.00 


45.50 


62.00 


58.50 


63^x12 


63^x14 


7.583 


15.17 


22.75 


30.33 


37.92 


45.50 


53.08 


60.67 


68.25 


63^x14 


63^x15 


8.125 


16.25 


24.38 


32.50 


40.63 


48.75 


56.88 


65.00 


73.13 


63^x15 


6^x16 


8.667 


17.33 


26.00 


34.67 


43.33 


52.00 


60.67 


69.33 


78.00 


63^x16 


63^x18 


9.750 


19.50 


29.25 


39.00 


48.75 


68.50 


68.25 


78.00 


87.75 


63^x18 


7 x7 


4.083 


8.167 


12.25 


16.33 


20.42 


24.50 


28.58 


32.67 


36.75 


7 x7 


7 x7)^ 


4.375 


8.750 


13.13 


17.50 


21.88 


26.25 


30.63 


35.00 


39.38 


7 x73^ 


7 x8 


4.667 


9.333 


14.00 


18.67 


23.33 


28.00 


32.67 


37.33 


42.00 


7 x8 


7 x9 


5.250 


10.500 


15.75 


21.00 


26.25 


31.50 


36.75 


42.00 


47.25 


7 x9 


7 xlO 


5.833 


11.67 


17.50 


23.33 


29.17 


35.00 


40.83 


46.67 


52.60 


7 XlO 


7 xll 


6.417 


12.83 


19.25 


25.67 


32.08 


38.50 


44.92 


61.33 


57.75 


7 xll 


7 xl2 


7.000 


14.00 


21.00 


28.00 


35.00 


42.00 


49.00 


66.00 


63.00 


7 xl2 


7 xl3 


7.583 


15.17 


22.75 


30.33 


37.92 


45.50 


53.08 


60.67 


68.25 


7 xl3 


7 xl4 


8.167 


16.33 


24.50 


32.67 


40.83 


49.00 


57.17 


65.33 


73.50 


7 xl4 


7 xl5 


8.750 


17.50 


26.25 


35.00 


43.75 


52.50 


61.25 . 


70.00 


78.75 


7 xl6 


7 xl6 


9.333 


18.67 


28.00 


37.33 


46.67 


56.00 


65.33 


74.67 


84.00 


7 xl6 


7 xl8 


10.500 


21.00 


31.50 


42.00 


52.50 


63.00 


73.50 


84.00 


94.50 


7 xl8 


7^x73^ 


4.688 


9.375 


14.06 


18.75 


23.44 


28.13 


32.81 


37.50 


42.19 


7Hx7>4 


73^x8 


5.000 


10.000 


15.00 


20.00 


25.00 


30.00 


35.00 


40.00 


45.00 


73^x8 


73^x83^ 


5.313 


10.63 


15.94 


21.25 


26.56 


31.88 


37.19 


42.60 


47.81 


73^x83^ 


73^x9 


6.625 


11.25 


16.88 


22.50 


28.13 


33.75 


39.38 


45.00 


50.63 


73^x9 


73^x10 


6.250 


12.50 


18.75 


25.00 


31.25 


37.50 


43.75 


50.00 


56.25 


7^x10 


73^x11 


6.875 


13.75 


20.63 


27.50 


34.38 


41.25 


48.13 


65.00 


61.88 


7 3^x11 


73^x12 


7.500 


15.00 


22.50 


30.00 


37.50 


45.00 


52.50 


60.00 


67.50 


7Hxl2 


73^x13 


8.125 


16.25 


24.38 


32.50 


40.63 


48.75 


56.88 


65.00 


73.13 


73^x13 



BOARD MEASURE. 385 

4.— Feet Board Measure— Engineers' Table.— Continued. 



S-2S 




Length in Feet. 


iat 


iSo 






sg-S 


s-il 


1 


2 


3 


4 


5 


6 


7 


8 


9 


S-i5 


7^x14 


8.750 


17.50 


26.25 


35.00 


43.75 


52.50 


61.25 


70.00 


78.75 


73^x14 


7)^x15 


9.375 


18.75 


28.13 


37.50 


46.88 


56.25 


65.63 


75.00 


84.38 


7 3^x15 


73^x16 


10.000 


20.00 


30.00 


40.00 


50.00 


60.00 


70.00 


80.00 


90.00 


7 3^x16 


73^x18 


11.25 


22.50 


33.75 


45.00 


56.25 


67.50 


78.75 


90.00 


101.25 


73^x18 


8 x8 


5.333 


10.67 


16.00 


21.33 


26.67 


32.00 


37.33 


42.67 


48.00 


8 x8 


8 x8>^ 


5.667 


11.33 


17.00 


22.67 


28.33 


34.00 


39.67 


45.33 


5V00 


8 x8K 


8 x9 


6.000 


12.00 


18.00 


24.00 


30.00 


36.00 


42.00 


48.00 


54.00 


8 x9 


8 xlO 


6.667 


13.33 


20.00 


26.67 


33.33 


40.00 


46.67 


53.33 


60.00 


8 xlO 


8 xll 


7.333 


14.67 


22.00 


29.33 


36.67 


44.00 


51.33 


58.67 


66.00 


8 xll 


8 xl2 


8.000 


16.00 


24.00 


32.00 


40.00 


48.00 


56.00 


64.00 


72.00 


8 xl2 


8 xl3 


8.667 


17.33 


26.00 


34.67 


43.33 


52.00 


60.67 


69.33 


78.00 


8 xl3 


8 xl4 


9.333 


18.67 


28.00 


37.33 


46.67 


56.00 


65.33 


74.67 


84.00 


8 xl4 


8 xl5 


10.000 


20.00 


30.00 


40.00 


50.00 


60.00 


70.00 


80.00 


90.00 


8 xl5 


8 xl6 


10.67 


21.33 


32.00 


42.67 


53.33 


64.00 


74.67 


85.33 


96.00 


8 xl6 


8 xl8 


12.00 


24.00 


36.00 


48.00 


60.00 


72.00 


84.00 


96.00 


108.00 


8 xl8 


8 3^x8 K 


6.021 


12.04 


18.06 


24.08 


30.10 


36.13 


42.15 


48.17 


54.19 


8Mx83^ 


8>^x9 


6.375 


12.75 


19.13 


25.50 


31.88 


38.25 


44.63 


51.00 


57.38 


sy,x9 


8Hx9J^ 


6.729 


13.46 


20.19 


26.92 


33.65 


40.38 


47.10 


53.83 


60.56 


SHxQH 


83^x10 


7.0«3 


14.17 


21.25 


28.33 


35.42 


42.50 


49.58 


56.67 


63.75 


8^x10 


8>ixll 


7.792 


15.58 


23.38 


31.17 


38.96 


46.75 


54.54 


62.34 


70.13 


83^x11 


8Kxl2 


8.500 


17.00 


25.50 


34.00 


42.50 


51.00 


59.50 


68.00 


76.50 


83^x12 


8^x13 


9.208 


18.42 


27.63 


36.83 


46.04 


55.25 


64.46 


73.67 


82 88 


8 3^X13 


83^x14 


9.917 


19.83 


29.75 


39.67 


49.58 


59.50 


69.42 


79.33 


89.25 


8 3^x14 


8Hxl5 


10.63 


21.25 


31.88 


42.50 


53.13 


63.75 


74.38 


85.00 


95.63 


8 3^xx5 


8Hxl6 


11.33 


22.67 


34.00 


45.33 


56.67 


68.00 


79.33 


90.67 


102.00 


8)ixl6 


83^^x17 


12.04 


24.08 


36.13 


48.17 


60.21 


72.25 


84.29 


96.33 


108.38 


8Hxl7 


83^x18 


12.75 


25.50 


38.25 


61.00 


63.75 


76.50 


89.25 


102.00 


114.75 


8^x18 


9 x9 


6.750 


13.50 


20.25 


27.00 


33.75 


40.50 


47.25 


54.00 


60.75 


9 x9 


9 x93^ 


7.125 


14.25 


21.38 


28.50 


35.63 


42.75 


49.88 


57.00 


64.13 


9 x934 


9 xlO 


7.500 


15.00 


22.50 


30.00 


37.50 


45.00 


52.50 


60.00 


67.50 


9 XlO 


9 xll 


8.250 


16.50 


24.75 


33.00 


41.25 


49.50 


,57.75 


66.00 


74.25 


9 xll 


9 xl2 


9.000 


18.00 


27.00 


36.00 


45.00 


54.00 


63.00 


72.00 


81.00 


9 xl2 


9 xl3 


9.750 


19.50 


29.25 


39.00 


48.75 


58.50 


68.25 


78.00 


87.75 


9 xl3 


9 xl4 


10.50 


21.00 


31.50 


42.00 


52.50 


63.00 


73.50 


84.00 


94.50 


9 xl4 


9 xl5 


11.25 


22.50 


33.75 


45.00 


56.50 


67.50 


78.75 


90.00 


101.25 


9 xl5 


9 xl6 


12.00 


24.00 


36.00 


48.00 


60.00 


72.00 


84.00 


96.00 


108.00 


9 xl6 


9 xl7 


12.75 


25.50 


38.25 


51.00 


63.75 


76.50 


89.25 


102.00 


li4.75 


9 xl7 


9 xl8 


13.50 


27.00 


40.50 


54.00 


67.50 


81.00 


94.50 


luS.OO 


121.50 


a Xl8 


9 x20 


15.00 


30.00 


45.00 


60.00- 


75.00 


90.00 


105.00 


1^0.00 


135.00 


9 x20 


93^x93^ 


7.521 


15.04 


22.56 


30.08 


37.60 


45.13 


52.65 


60.17 


67.69 


9 3^x93^ 


93^x10 


7.917 


15.83 


23.75 


31.67 


39.58 


47.50 


55.42 


63.33 


71.25 


9 3^x10 


93^x11 


8.708 


17.42 


26.13 


34.83 


43.54 


52.25 


60.96 


69.67 


78.38 


9 3^x11 


93^x12 


9.500 


19.00 


28.50 


38.00 


47.50 


57.00 


66.50 


76.00 


85.50 


9^x12 


93^x13 


10.29 


20.58 


30.88 


41.17 


51.46 


61.75 


72.04 


82.34 


92 63 


9^x13 


93^x14 


11.08 


22.17 


33.25 


44.33 


55.42 


66.50 


77.58 


88.67 


99.75 


93^x14 


9^x15 


11.88 


23.75 


35.63 


47.50 


59.38 


71.25 


83.13 


95.00 


106.88 


9^x15 


93^x16 


12.67 


25.33 


38.00 


50.67 


63.33 


76.00 


88.67 


101.33 


1x4.00 


9 3^x16 


9Mxl7 


13.46 


26.92 


40.38 


53.83 


67.29 


80.75 


94.21 


1U7.67 


121.13 


9 3^x17 


93^x18 


14.25 


28.50 


42.75 


57.00 


71.25 


85.50 


99.75 


114.00 


128.25 


9^x18 


93^x20 


15.83 


31.67 


47.50 


63.33 


79.17 


95.00 


110.83 


126.67 


1*2.50 


9 3^x20 


10x10 


8.333 


16.67 


25.00 


33.33 


41.67 


50.00 


58.33 


66.67 


75.00 


10x10 


10x11 


9.167 


18.33 


27.50 


36.67 


45.83 


55.00 


64.17 


73 33 


82.50 


10x11 


10x12 


10.00 


20.00 


30.00 


40.00 


50.00 


60.00 


70.00 


80.00 


90.00 


1^x12 
Wxl3 


10x13 


10.83 


21.67 


32.50 


43.33 


54.17 


65.00 


75.83 


86.67 


97.50 


10x14 


11.67 


23 33 


35.00 


46.67 


58 33 


70.00 


81.67 


93.33 


105.00 


11x14 



20.— LUMBER AND LUMBERING. 
4. — Feet Board Measure— Engineers' Table.— Continued. 



1 



isi 








Length ] 


in Feet. 








g.Sg 


Sgs 


















s^l 


• fh O rt 




















.ph o 2 


Q-3^5 


1 


2 


3 


4 


5 


6 


7 
87.50 


8 


9 


Q-s^ 


10x15 


12.50 


25.00 


37.50 


50.00 


62.50 


75.00 


100.00 


112.50 


10x15 


10x16 


13.33 


26.67 


40.00 


53.33 


66.67 


80.00 


93.33 


106.67 


120.00 


10x16 


10x17 


14.17 


28.33 


42.50 


56.67 


70.83 


85.00 


99.17 


113.33 


127.50 


10x17 


10x18 


15.00 


30.00 


45.00 


60.00 


75.00 


90.00 


105.00 


120.00 


135. OU 


10x18 


10x20 


16.67 


33.33 


50.00 


66.67 


83.33 


100.00 


li6.67 


133.33 


150.00 


10x20 


11x11 


10.08 


20.17 


30.25 


40.33 


50.42 


60.50 


70.58 


80.67 


90.75 


11x11 


11x12 


11.00 


22.00 


33.00 


44.00 


55.00 


66.00 


77.00 


88.00 


99.00 


11x12 


11x13 


11.92 


23.83 


35.75 


47.67 


59.58 


71.50 


83.42 


95 33 


107 26 


11x13 


11x14 


12.83 


25.67 


38.50 


51.33 


64.17 


77.00 


89.83 


102.67 


115.50 


11x14 


11x15 


13.75 


27.50 


41.25 


55.00 


68.75 


82.50 


96.25 


110.00 


123.75 


11x15 


11x16 


14.67 


29.33 


44.00 


58.67 


73.33 


88.00 


102.67 


117.33 


132.00 


11x16 


11x17 


15.58 


31.17 


46.75 


62.33 


77.92 


93.50 


109.08 


124.67 


140 26 


11x17 


11x18 


16.50 


33.00 


49.50 


66.00 


82.50 


99.00 


115 50 


132 00 


148.50 


11x18 


11x20 


18.33 


36.67 


55.00 


73.33 


91.67 


110.00 


128.33 


146.67 


165 00 


11x20 


12x12 


12.00 


24.00 


36.00 


48.00 


60.00 


72.00 


84.00 


96.00 


108.00 


12x12 


12x13 


13.00 


26.00 


39.00 


52.00 


65.00 


78.00 


91.00 


104.00 


117.00 


12x13 


12x14 


14.00 


28.00 


42.00 


56.00 


70.00 


84.00 


98.00 


112.00 


126.00 


12x14 


12x15 


15.00 


30 00 


45.00 


60.00 


75.00 


90.00 


105.00 


120.00 


135.00 


12x15 


12x16 


16.00 


32.00 


48 00 


64.00 


80.00 


96.00 


112.00 


128.00 


144.00 


12x16 


12x17 


17.00 


34.00 


51.00 


68.00 


85.00 


102.00 


119.00 


136.00 


153.00 


12x17 


12x18 


18.00 


36.00 


54.00 


72.00 


90.00 


108.00 


126.00 


144.00 


162.00 


12x18 


12x20 


20.00 


40.00 


60.00 


80.00 


100.00 


120.00 


140.00 


160.00 


180. 00 


12x20 


12x22 


22.00 


44.00 


66.00 


88.00 


110.00 


132.00 


154.00 


176.00 


198 00 


12x22 


12x24 


24.00 


48.00 


72.00 


96.00 


120.00 


144.00 


168.00 


192.00 


216.00 


12x24 


13x13 


14.08 


28.17 


42.25 


56.33 


70.42 


84.50 


98.58 


112.67 


126.75 


13x13 


13x14 


15.17 


30.33 


45.50 


60.67 


75.83 


91.00 


106.17 


121.33 


136.50 


13x14 


13x15 


16.25 


32.50 


48.75 


65.00 


81.25 


97.50 


113.75 


130.00 


146.25 


13x15 


13x16 


17.33 


34.67 


52.00 


69.33 


86.67 


104.00 


121.33 


138.67 


156.00 


13x16 


13x17 


18.42 


36.83 


55.25 


73.67 


92.08 


110.50 


128.92 


147.33 


165.75 


13x17 


13x18 


19.50 


39.00 


58.50 


78.00 


97.50 


117.00 


136.50 


156.00 


175.50 


13x18 


13x20 


21.67 


43.33 


65.00 


86.67 


108.33 


130.00 


151.67 


173.33 


195.00 


13x20 


13x22 


23.83 


47.67 


71.50 


95.33 


119.17 


143.00 


166.83 


190.67 


214.50 


13x22 


13x24 


26.00 


52.00 


78.00 


104.00 


130.00 


156.00 


182.00 


208.00 


234.00 


13x24 


14x14 


16.33 


32.67 


49.00 


65.33 


81.67 


98.00 


114.33 


130.67 


147.00 


14x14 


14x15 


17.50 


35.00 


52.50 


70.00 


87.50 


105 00 


122.50 


140.00 


157.50 


14x15 


14x16 


18.67 


37.33 


56.00 


74.67 


93.33 


112.00 


130.67 


149.33 


168.00 


14x16 


14x17 


19.08 


38.17 


57.25 


76.33 


95.42 


114.50 


133.58 


152.67 


171.75 


14x17 


14x18 


21.00 


42.00 


63.00 


84.00 


105.00 


126.00 


147.00 


168.00 


189.00 


14x18 


14x20 


23.33 


46.67 


70.00 


93.33 


106.67 


140.00 


163.33 


186.67 


210.00 


14x20 


14x22 


25.67 


51.33 


77.00 


102.67 


128.33 


154.00 


179.67 


205.33 


231.00 


14x22 


14x24 


28.00 


56.00 


84.00 


112.00 


140.00 


168.00 


196.00 


224.00 


252.00 


14x24 


15x15 


18.75 


37.50 


56.25 


75.00 


93.75 


112.50 


131.25 


150.00 


168.75 


15x13 


15x16 


20.00 


40.00 


60.00 


80.00 


100.00 


120.00 


140.00 


160.00 


180.00 


15x16 


15x17 


21.25 


42.50 


63.75 


85.00 


106 25 


127.50 


148.75 


170.00 


191.25 


15x17 


15x18 


22.50 


45.00 


67.50 


90.00 


112 50 


135.00 


157.50 


180.00 


202.50 


15x18 


15x19 


23.76 


47.50 


71.25 


95.00 


118.75 


142.50 


166.25 


190.00 


213.75 


15x19 


15x20 


25.00 


50.00 


75.00 


100.00 


125.00 


150.00 


175.00 


200.00 ' 


225.00 


15x20 


15x22 


27.50 


55.00 


82.50 


110.00 


137.50 


165.00 


192.50 


220.00 


247.50 


15x22 


15x24 


30.00 


60.00 


90.00 


120.00 


150.00 


180.00 


210.00 


240.00 


270.00 


15x24 


16x16 


21 33 


42,67 


64.00 


85.33 


106.67 


128.00 


149.33 


170.67 


192.00 


16x16 


16x17 


22.67 


45.33 


68.00 


90.67 


113.33 


136.00 


158.67 


181.33 


204.00 


16x17 


16x18 


24.00 


48.00 


72.00 


96.00 


120.00 


144.00 


168.00 


192.00 


216.00 


16x18 


16x20 


26.67 


53.33 


80.00 


106.67 


133.33 


160.00 


186.67 


213.33 


240.00 


16x20 


16x22 


29.33 


58.67 


88.00 


117.33 


146.67 


176.00 


205.33 


234.67 


264.00 


16x22 


16x24 


32.00 


64.00 


96.00 


128.00 


160.00 


192.00 


224.00 


256.00 


288.00 


16x24 



GRADING OF LUMBER— YELLOW PINE, 



387 



4. — Feet Board Measure — Engineers* Table. — Concluded. 





Length in Feet. 


Dimen- 
sion in 
Inches. 


5^£ 


1 


2 


3 


4 


5 


6 


7 


8 


9 


17x17 
17x18 
17x19 
17x20 
17x22 

17x24 

18x18 
18x19 
18x20 
18x22 

18x24 
20x20 
20x22 
20x24 
22x22 

22x24 
24x24 


24.08 
25.50 
26.92 
28.33 
31.17 

34.00 
27.00 
28.50 
30.00 
33.00 

36.00 
33.33 
36.67 
40.00 
40.33 

44.00 
48.00 


48.17 
51.00 
53.83 
56.67 
62.33 

68.00 
54.00 
57.00 
60.00 
66.00 

72.00 
66.67 
73.33 
80.00 
80.67 

88.00 
96.00 


72.25 
76.50 
80.75 
85.-00 
93.50 

102.00 
81.00 
85.50 
90.00 
99.00 

108.00 
100.00 
110.00 
120.00 
121.00 

132.00 
144.00 


96.33 
102.00 
107.67 
113.33 
124.67 

136.00 
108.00 
114.00 
120.00 
132.00 

144.00 
133.33 
146.67 
160.00 
161.33 

176.00 
192.00 


120.42 
127.50 
134.58 
141.67 
155.83 

170.00 
135.00 
142.50 
150.00 
165.00 

180.00 
166.67 
183.33 
200.00 
201.67 

220.00 
240.00 


144.50 
153.00 
161.50 
170.00 
187.00 

204.00 
162.00 
171.00 
180.00 
198.00 

216.00 
200.00 
220.00 
240.00 
242.00 , 

264.00 
288.00 


168.58 
178.50 
138.42 
1CS.33 
218.17 

23G.00 
189.00 
199.50 
210.00 
231.00 

252.00 
233.33 
256.67 
280.00 
282.33 

308.00 
336.00 


192.67 
204.00 
215.33 
226.67 
249.33 

272.00 
216.00 
228.00 
240.00 
264.00 

288.00 
266.67 
293.33 
320.00 
322.67 

352.00 
384.00 


216.75 
229.50 
242.25 
255.00 
280.50 

306.00 
243.00 
256.50 
270.00 
297.00 

324.00 
300.00 
330.00 
360.00 
363.00 

396.00 
432.00 


17x17 
17x18 
17x19 
17x20 
17x22 

17x24 
18x18 
18x19 
18x20 
18x22 

18x24 
20x20 
20x22 
20x24 
22x22 

22x24 
24x24 



Standard Classification of Structural Timber (Am. Soc. Test. Materials, 
Year Book, 1913). — Definitions. — Following is list of products recommended for 
consideration as structural timbers: Trestle timbers — stringers, caps, posts, mud 
sills, bracing, bridge ties, guard rails; car timbers — car framing (including upper 
framing), car sills; framing for buildings — posts, mud sills, girders, framing, joists; 
ship timbers — ship timbers, ship decking; cross arms for poles. Standard Defects. 
— Sound knot — solid across its face, and as hard as the surrounding wood; loose 
knot — not firmly held in place by growth or position; pith knot — sound knot with 
pith hole not more than H in. in dia. in the center; encased knot — one surrounded 
wholly or in -part by bark or pitch, but where encasement is less than y% in. in 
width on both sides, not > circum. of knot, it shall be considered a sound knot; 
rotten knot — not as hard as the surrounding wood; pin knot — sound knot not over 
\^ in. in dia.; standard knot — sound knot not over y2 in. in dia.; large knot — sound 
knot more than IJ^i ins. in dia.; round knot — oval or circular; spike knot — one sawn 
in a lengthwise direction; pitch pockets — openings in grain of wood containing more 
or less pitch or bark {small when not over y% in. wide, standard when not over 
% in. wide or 3 ins. long, large when over % in. wide or 3 ins. long); pitch streak — 
well-defined accumulation of pitch at one point, but shall not be considered a 
defect when not sufficient to develop a well-defined streak, etc.; wane — bark or 
lack of wood on edge of timber; shakes — splits or checks between annular rings; 
rot, dote, and red heart — white or red rotten spots, dark discolorations not found in 
sound wood, etc. 

Classification and Inspection of Yellow Pine Lumber.* — General Rules. — 
All lumber must be sound, commercial longleaf yellow pine (pine combining 
large coarse knots, with coarse grain, is excluded under these rules), well 
manufactured, full to size and saw butted, and shall be free from the follow- 
ing defects: Unsound, loose, and hollow knots, wormholes and knot holes, 
through shakes or round shakes that show on the surface; and shall be square 
edge unless otherwise specified. A through shake is hereby defined to be 
through or connected from side to side, or edge to edge, or edge to side. 
In the measurement of dressed lumber the width and thickness of the lumber 
before dressing must be taken-^less than one inch thick shall be measured 
as one inch. The measurement of wane shall always apply to the lumber 
in the rough. Where terms one-half and two-thirds heart are used they 
shall be construed as referring to the area of the face on which measured. 

* Interstate rules of 1905. Adopted by the Georgia-Florida Sawmill 
Association, Georgia Interstate Sawmill Association, South Carolina Lum- 
ber Association, New York Lumber Trade Association of New York City, 
Yellow Pine Exchange of New York City, The Lumbermen's Exchange 
of Philadelphia, The Lumbermen's Exchange of Baltimore. 



388 20.-^LUMBER AND LUMBERING. 

In the dressing of lumber, when not otherwise specified, one-eighth inch 
shall be taken off by each planer cut. All lumber grading higher than the 
grade for which it is sold shall be accepted as of the grade sold. 

Classification. — Flooring shall embrace fotir, five and six quarter inches 
in thickness by three to six inches in width, excluding 1^x6. For example, 
1x3, 4, 5, and 6; lix3, 4, 5, and 6; 1^x3, 4, and 5. Boards shall embrace 
all thicknesses under IJ ins. by over 6 ins. wide. For axample: f, 1, li, 
and If ins. thick by over 6 ins. wide. Plank shall embrace all sizes from 
IJ to imder 6 ins. in thickness by over 6 ins. in width. For example: li, 
2, 2J, 3, 3i, 4, 4^, 5, 5i, 51, by 6 and over in width. Scantling shall embrace 
all sizes exceeding 1 J ins. and under 6 ins. in thickness, and from 2 to under 
6 ins. in width. For example: 2x2, 2x3, 2x4, 2x5, 3x3, 3x4, 3x5, 4x4, 4x5, 
and 5x5. Dimension sizes shall embrace all sizes 6 ins. and up in thickness 
by 6 ins. and up in width. For example: 6x6, 6x7, 7x7, 7x8, 8x9, and up. 
Stepping shall embrace 1 to 2^ ins. in thickness by 7 ins. and up in width. 
For example: 1, 11, 1^, 2, and 2J x7, and up in width. Rough Edge or 
Flitch shall embrace all sizes 1 in. and up in thickness by 8 ins. and up in 
width, sawed on two sides only. For example: 1, li, 2, 3, 4, and up thick 
by 8 ins. and up wide, sawed on two sides only. 

Inspection. — Standard lumber shall be sound, sap no objection. Wane 
may be allowed \ of the width of the piece measured across face of wane, 
extending \ of the length on one comer, or its equivalent on two or more 
comers, provided that not over 10% of the pieces of any one size shall show 
such wane. Merchantable sizes under 9 ins. shall show some heart the entire 
length on one side; sizes 9 ins. and over shall show some heart the entire 
length on two opposite sides. Wane may be allowed \ of the width of the 
piece measured across face or wane, and extending \ of the length of the 
piece on one corner or its equivalent on two or more comers, provided that 
not over 10% of the pieces of any one size shall show such wane. Prime 
lumber. — Flooring shall show one heart face, free from through or round 
shakes or knots exceeding one inch in dia or more than four in a board on 
the face side. Boards 7 ins. and under wide shall show one heart face; 
over 7 ins. wide shall show f heart on both sides; all free from round or 
through shakes, large or unsound knots. Plank 7 ins. and under wide 
shall show one heart face; over 7 ins. wide shall show f heart on both sides; 
all free from round or through shakes, large or unsound knots. Scantling 
shall show three comers heart, free from through or round shakes or un- 
sound knots. Dimension sizes of Prime lumber. — AU square lumber shall 
show § heart on two sides and not less than ^ heart on two other sides. 
Other sizes shall show f heart on face and show heart f of length on edges, 
excepting when the width exceeds the thickness by 3 ins. or over; then it 
shall show heart on the edges for \ the length. Stepping shall show 3 comers 
heart, free from shakes and all knots exceeding ^ in. in dia, and not more 
than 6 in a board. Rough Edge or Flitch shall be sawed from good heart 
timber, and shall be measured in the middle on the narrow face, free from 
injurious shakes or unsound knots. Wane on not over 5% of the pieces 
jn any one size, shall be allowed as on merchantable quality. 

Rules for Grading Fir, Spruce, Cedar, and Hemlock Lumber.* — General 

Instructions. — All lumber graded with special reference to its suitability 
for the use intended; therefore each piece is considered and its grade deter- 
mined by its general character, including the sum of all its defects. "Yard 
Lumber " (dimension, common boards, finish, etc.) is graded from the face 
(best) side, except that when lumber which is dressed one side only is graded 
from the dressed side. Factory lumber (for doors, sashes, etc.) which must 
show on both sides, is always graded from the poorest side. Defects are 
taken in connection with the size of the piece, wider and longer pieces 
carrying more defects than smaller pieces in the same grade. Grade is 
determined at time of shipment and cannot be reconsidered after further 
working; a shipment of any grade to consist of a fair average of that grade. 
Material not conforming to standard sizes shall be governed by special 
contract. Standard lengths for all lumber are multiples of two feet, except 
that the standard lengths for flooring, ceiling, siding, rustic, and finish are 



* Rail shipments. Digest of rules adopted March 30, 1906, by the 
Pacific Coast Lumber Manufacturers' Association, Southwestern Wash- 
ington Lumber Manufacturers' Association, Oregon and Washington 
Lumber Manufacturers' Association. 



GRADING OF LUMBER— FIR, SPRUCE, ETC. 8^ 

multiples of one foot. Odd and fractional lengths shall be counted as next 
higher standard length. All dressed lumber shall be measured and sold 
at the full size of rough material used in its manufacture; and lumber one 
inch or less in thickness shall be counted as one inch thick. 

Defects. — Recognized defects are knots (classed as sound or loose, 
large or small), knot holes, splits and checks (considered as to length and 
direction), wane (bark or lack of wood from any cause on edges of timber), 
rot, rot streaks, worm holes, pitch seams (clearly defined openings between 
the grains of the wood, are generally filled with granulated pitch, and a 
serious defect and not admissible in any grade above No. 2 Common), 
pitch pockets (openings between the grains of the wood containing more or 
less pitch, and surrounded by sound grain wood), torn or chipped grain 
(usually caused by dressing against the grain and is more or less of a defect 
according to its depth or extent), discoloration (the result of various causes, 
and will only be considered a defect when the piece is damaged for the 
use intended). 

Miscellaneous. — Defects in rough stock caused by improper manufac- 
ture and drying will reduce the grade, unless they may be removed by dress- 
ing such stock to standard sizes. Imperfect manufacture in dressing 
stocks, such as torn grain, broken knots, mismatching, insufficient tongue 
or groove on flooring, ceiling, drop siding, etc., shall be considered defects 
and will reduce the grade accordingly as they are slight or serious in their 
effect on the use of the stock. 

Fir. 

Fir is sawn and sorted with reference to the direction of the grain. 
The E.G. (edge grain) has the grain at or nearly at right angle with face 
of board, and is adapted to flooring, stepping, etc., because it does not 
" sliver " with wear. F. G. (flat grain) means that the grain is parallel or 
nearly parallel with the face of the board; it is selected for finish because of 
the beauty in this form. 

Flooring. — No. 1 Clear E. G., 3, 4 and 6 in.; free from all defects; angle 
of grain not less than 45°. No. 2 Clear E. G., 3, 4 and 6_in.; angle of grain 
not less than 45°; will admit of slight roughness in dressing, and from 1 to 

3 small close pitch pockets, or equivalent defects. No. 3 Clear E. G., 3, 

4 and 6 in.; angle of grain not less than 45°; such defects admitted as will 
not impair its utility for cheap floors. No. 1 Clear F. G., 3, 4 and 6 in.; 
free from all defects, and all flat grain. No. 2 Clear F. G., 3, 4 and 6 in.; 
shall be flat grain; will admit of slight roughness in dressing, three close 
pitch pockets not to exceed 2 ins. in length, and 3 small tight smooth knots 
i in. dia., or their equivalent of combined defects. No. 3 Clear F. G., 3, 
4 and 6 in.; such defects admitted as will not impair its utility for cheap 
floors and sheathing. 

Ceiling. — 4 or 6 inch, Classed as No. 1 Clear, No. 2 Clear, No. 3 Clear; 
both E. G. and F. G. admissible. 

Partition.- — 4 or 6 inch. Graded same as Ceiling on the face side with 
the reverse side not more than one grade lower. 

Porch Decking^ Drop Siding, and Rustic. — Classed as No. 1 Clear (6 or 
8 in.), No. 2 Clear (4, 6 or 8 in.), No. 3 Clear (4, 6 or 8 in.). 

Bevel Siding. — 4 and 6 inch. Classed as No. 1 Clear, No. 2 Clear, No. 3 
Clear. 

Stepping. — Classed as No. 1 Clear (8 to 14 inch Clear), No. 2 Clear 
(8 to 14 inch.). 

Joist and Scantling. — Generally speaking, there should be no imper- 
fections that seriously impair the strength of the piece. 

Rough Timber. — 4x4 and larger shall not be more than i inch scant 
when green, and be evenly manufactiured from sound stock, and must be 
free from knots that would materially weaken the piece. Timbers 10x10 
in size may have a 2 -inch wane on one comer, or its equivalent on two or 
more comers, one-fourth the length of the piece. Other sizes may have 
proportionate defects. Season checks and checks extending not over 
one-eighth the length of the piece admissible. 




390 20.— LUMBER AND LUMBERING 

Spruce. 

Names and Grades. — Flooring is classed as Clear, "A,'* and " B;'* Fin- 
ish, as First and Second Clear, Third Clear, Selects; Ceiling, as Clear, "A," 
and " B;" Partition, as Clear, " A," and " B;" Porch Decking, as Clear, 
"A," and " B;" Bevel Siding, as Clear, "A," " B," and " C;" Factory 
Stock, as Select and Better, No. 1 Shop, No. 2 Shop. 

Red Cedar. 

Names and Grades. — Bevel Siding is classed as Clear, ** A," and " B;** 
Ceiling, as No. 1, No. 2, No. 3; Finish, as No. 1, No. 2, No. 3; Corrugated 
Decking, as No. 2 and Better; Flooring, as No. 1, No. 2, No. 3. 

Hemlock. 
In a general way the riiles for grading Fir and Spruce lumber are applied 
to Hemlock. 

Shingles.f — Perfections. — 18 inch, random widths, five butts must 
measure 2^q inch plump in thickness when green, or 2^ inches after drying. 
Must be well manufactured, strictly clear in every resect, and 90% vertical 
grain. Will not admit any shingle narrower than 3 inches. 

Puget A. — Same thickness as Perfections. Must be well maniif actured ; 
will admit sound knots 8 inches from butt, 16-inch shims; also admits 
slash-grain shingles, otherwise must be clear. Will not admit shingles 
narrower than 2 inches. 

Eureka. — 18 inch, random widths, five butts must measure 2x^6 inches 
in thickness when green, or 2 inches after drying. Must be well manu- 
factured, strictly clear in every respect, and 90 per cent vertical grain. 
Will not admit any shingles narrower than 3 inches. 

Skagit ^ A. — Same thickness as Eureka. Must be well manufactured. 
Will admit sound knots 8 inches from butt; 16-inch shims. Also admit 
slash-grain shingles; otherwise must be clear. Will not admit shingles 
narrower than 2 inches. 

Extra Clear. — 16 inch, random widths, five butts must measure 2^ 
inches in thickness when green, or 2 inches after drying. Must be well 
manufactured, strictly clear in every respect, and 90 per cent vertical 
grain. Will not admit any shingles narrower than 2 inches. 

Choice A. — Same width and thickness as Extra Clear. Must be well 
manufactured. Will admit sound knots 6 inches from butt; also slash- 
grain shingles, wane edge, sap, 14-inch shims, ^-inch knotholes or worm 
holes 6 inches from butt; otherwise must be clear. 

Extra *A* — 16-inch ramdom widths, six butts must measure 2/q inches 
green, or 2 inches after drying; must be well maniifactured. Will admit 
sound knots 10 inches from butt; otherwise must be strictly clear, and 90 
per cent, vertical grain. Will not admit any shingles narrower than 2 inches. 

Standard A, — Same width and thickness as Extra *A*; must be well 
manufactured. Will admit sound knots 6 inches from butt, slash-grain 
shingles, wane edge, sap, 14-inch shims, ^-inch knot holes or more holes 6 
inches from butt; otherwise must be clear. 

Shingles with the following defects are culls, and must not be put in 
any of the above grades: Rot, worm holes, except as above provided, check, 
shake, stub comers, tapering edges, rough, waney, or unevenly sawn. 

Eighteen inch, 5 to 2\ inch shingles, must be packed 20 courses per 
bunch, 5 bunches to the M. Eighteen inch, 5 to 2 inch, and 16-inch shin- 
gles, must be packed 25 courses to the bunch, 4 bunches to the M. ^ All shin- 
gles must be packed in the regulation frame, full 20 inches in width, and 
no opening of more than 1^ inches is admissible in any one course. 

All shingles to be packed as closely as possible. Bands should not be 
shorter than 19| inches in length. Every bunch of shingles must be branded. 

Dimension shingles are packed 24 courses in each bunch. 

t Oregon and Washington Lumber Manufacturers' Association. 



LOG RULES. SHIPPING WEIGHTS. 



391 



P 



EXCERPTS AND REFERENCES. 

Graphical Comparison of Various Log Rules (By A. H. Morse. Eng. 
News, April 9, 1908). — According to The Woodman's Handbook, by Prof. 
H. S. Graves, published as Bulletin 36 of the Bureau of Forestry, U. S. 
Dept. of Agric, Wash., D. C, more than 30 different "log rules" are in use 
in the U. S. All these rules profess to accomplish the same thing, viz., to 
provide means for ascertaining the nurnber of board feet of 1-in. lumber 
which can be sawn from a log of given dia. and length; that is, to give the 
"scale" of a log in board feet. This article compares the principal log rules. 

A Table of Lumber Weights (Eng. News, May 5, 1910). — The following 
table of weights of lumber represents the commercial classification, taken 
from the reports of the secretaries of the various lumber associations, and 
are accepted by the trade in place of actual weights. 

Shipping Weights in lbs. per 1000 ft. B. M. 



Ash, black. . 
Ash, white . . 
Basswood . . 

Beech 

Birch 

Buckeye. . . . 
Butternut . . 
Chestnut — 

Cherry 

Cottonwood. 

Cypress 

Elm, rock . . 
Elm, soft. . . 

Gum 

Gum, sap. .. 
Hemlo'k.Pa. 
Hemlock, 
Nor 



Green 


Ship- 


Well 


Kiln 


from 


ping 


Sea- 


Dried 


Saw. 


Dry. 


soned. 


3000 


4600 




3200 


4600 




3800 


3300 


4200 


2800 


2500 


2100 


5750 




4000 


. . . • 


5500 




4000 


.... 




2600 


. . . 


.... 


4000 


2800 


2500 


.... 


5000 


3500 


2800 


2450 


5000 


4000 






4600 


3100 


2800 


2400 


5000 


3500 


2900 


. . 


5400 


4300 


4000 


3500 


4750 


3300 


3100 


2900 


5400 


3600 


3300 


3030 


5000 


3300 


3000 


2750 


40u0 





2500 





4500 


3000 


.... 


.... 



Hickory. . . . 

Mahogany . .. 

Maple, hard 

Maple, soft. 

Oak, red. . . 

Oak, white. 

Pine, long If. 

Pine, white . 

Poplar 

Poplar Bay 
(Tap) 

Spruce, Ad- 
irondack. . 

Spruce, W. 
Va 

Sycamore . . 

Walnut, 
Black 



Green 


Ship- 


Well 


from 


ping 


Sea- 


Saw. 


Dry. 


soned. 


6000 


5000 


4500 


4500 




3500 


5400 


4150 


3900 


5000 


3650 


3300 


5500 


4250 


4000 


5700 


4500 


4100 


4500 


3500 




3500 


2500 


2400 


3900 


3000 


2800 


4200 


.... 


3000 


3300 


2700 


2300 


3000 


2700 


2300 


4750 


3200 


3000 


4900 


4000 


3800 



Kiln 
Dried. 



3400 
3000 
3400 
3600 

2*200 
2400 



2200 



Design of Log Flumes (By J. P. Martin, U. S. Forest Service, 1912).— Dia- 
grams for economic design. Illustration of V-shape flume, Deerlodge National 
Forest, Mont., 16 miles long, with grades varying from 0.5 to 12.5% and supplied 
with water at several places throughout its length. 



21.— METALLURGY. 



^ 



Iron Ore. — Hematite {Fe20z), an ore abundant around Lake Superior, 
in Alabama, etc., furnishes five-sixths of the iron manufactured in the 
United States; with limonite (one-ninth), and magnetite (one-sixteenth), 
next in order. Most of the ore has to be prepared for the blast-furnace. 
Thus, " sorting " the ore from foreign fragments; " washing " away the 
earthy matter; " concentrating " the crushed ore by magnetic " separators;" 
'* calcinating " or driving off the volatile matter by heat; etc., are methods 
usually employed. Iron ore contains from 35 to 65% of iron, the balance 
being oxygen, phosphorus, sulphur, silica (sand), and other impurities. 

Pig Iron is the cast iron product from a blast furnace. The furnace is 
shaped like a lamp chimney, and the charge is introduced at the top: layers 
of ore, limestone, fuel, ore, limestone, fuel*, etc., in regular sequence. By 
forcing a current of hot air at the bottom of the furnace into the heated 
mass a series of chemical reactions take place, producing molten iron, 
molten slag, and gases. The molten iron is drawn off after the slag and 
run into moulds or (cast) pigs.f From pig iron all kinds of iron and steel 
can be made. Among the uses to which pig-iron may be put are the follow- 
ing: 

(a) Directly without refining, for foundry work. 

{b) Refined by elimination of silica and phosphorus, then removal of 
carbon by dry-puddling, for wrought iron (not used in U. S.). 

{c) Wet puddling, in renioval of silica, phosphorus and carbon, for 
soft structural and blacksmith iron. 

{d) Acid Bessemer, in removal of silica, manganese and carbon, for 
rails and structural steels (mild). 

{e) Basic Bessemer, in removal of silica, manganese, carbon and phos- 
phorus, for rails and structural steels (mild) . 

(/) Krupp washing, and Siemens-Martin process, for structural steels 
(mild). 

(g) Charcoal hearths, for wroiight iron; cementation furnace, for 
shear steel; crucible, for high grade tool steel. 

Cast Iron. — In foundry work, for ordinary castings, the melt from 
the cupola is much more certain in composition than the direct casting 
from the unrefined pig. It gives a closer grain and somewhat increased 
strength. A good casting contains about 3.5 per cent, carbon, 1.5 per 
cent, silicon, not over . 7 per cent, phosphorus, . 6 per cent manganese, 
and a trace of sulphur. It shows a gray fracture, can be worked easily 
with chisel and file, completely fills the mold, chills with a smooth surface, 
has no blow holes, shrinks little in cooling. Large castings should contain 
less silicon than small; shrinkage may be decreased by lowering the man- 
ganese; to increase the strength the carbon, silicon and manganese are 
lowered and phosphorus eliminated as far as possible. 

The above chemical properties may be varied to suit the special pur- 
poses for which the castings are intended. Mr. Henry Souther, t in Vol. V, 
Proc. A. S. T. M., page 218, says with reference to Hard Cast Iron which 
had given complaint in the machine shop: " Cast iron that chills may be 
called hard in the truest sense of the v^ord. ... In the last five or six 
years three separate complaints of hard iron have reached the writer and 
proved of so baffling a character that in each case visits were made to the 
machine _ shops. ... It developed [citing one particular case] that 
small drills, i-in. or thereabout, were standing up with this iron just as 
well as any other, but the larger drills in the neighborhood of ^-in. and 
f-in. were dulling exactly as though the iron were charged with emery. 



* Charcoal, coke or anthracite coal. 

t Pig iron contains about 93% of pure iron, 3 to 5% of carbon (pure 
coal), also some silicon, phosphorus, sulphur, etc. . 
X State Chemist, Hartford, Conn. 

392 



IRON AND STEEL, ^n 

The edges were being ground off and would last only a fraction of the time 
usual for the same drills in the same machine. Here was an unusual C9n- 
dition, thin iron working easily, thick iron on the same castings working 
with difficulty. The chemical results were normal, except manganese: 
Silicon 2.50, phosphorus .70, sulphur about .08, total carbon 3.50 and 
manganese .16. . . . Inasmuch as the only abnormal part in the anal- 
ysis was shown in the manganese, that element was suspected, although 
there seemed to be no metallurgical reason for doing so. Means were taken 
to raise it to the neighborhood of .50, and as soon as this was done the 
difficulty disappeared in the machine shop and has not reappeared after 
some months. . . . This leads to the belief that there must be some 
carbide of iron or carbide of silicon that forms in the absence of a reasonable 
amount of manganese, and that does not form with manganese present." 

Cast Steel. — *Steel for castings may be made by the open-hearth, 
crucible or Bessemer process. Castings to be annealed unless otherwise 
specified. Ordinary castings, those in which no physical requirements 
are specified, shall not contain over 0.40% carbon, nor over 0.08% phos- 
phorus. Castings which are subjected to physical test shall not contain 
over 0.05% phosphorus, nor over 0.05% sulphur. 

Tested castings shall be of three classes: Hard, Medium, and Soft, with 
minimum physical requirements as follows: 

Hard. Medium. Soft. 

Tensile strength, lbs. per sq. in 85000 . 70000 . . 60000. 

Yield point, " " '* " 38250. 31500. 27000. 

Elongation, per cent in two ins 15 . 18 . 22 . 

Contraction of area, per cent 20 . 25 . 30 . 

For determining the above physical properties, standard turned specimens 
Y in dia. and 2" gauged length shall be used. 

For bending test, a specimen l^'xl'' shall bend cold around a dia. of 1" 
without fracture on outside of bent portion, through an angle of 90° for 
" Medium " or 120° for " Soft " castings. 

Malleable Castings. — An ordinary casting, unless it is too large, may 
be made malleable by decarburization. The original casting is preferably 
".white," indicating that is is low in carbon, and should practically be free 
from sulphur and phosphorus. The castings are packed in a pot (or retort) 
with oxide of iron, usually hematite, placed in a furnace, and kept at an 
orange or red heat for about a week, and then cooled very slowly. The 
carbon in the casting mostly unites with the oxygen of the ore, passing off 
as carbonic oxide. The casting is thus rendered malleable like crude wrought 
iron. Malleable castings are used for pipe specials which have to be 
threaded; for wood-stave (water) pipe shoes; and in general for delicate 
castings which are liable to be subjected to various kinds of stress. 

Wrought Iron. — Wrought iron is purified pig iron. Although it is 
made "directly " from the ore the " indirect " process is generally used. 
The pig iron is oxidized in a puddling furnace whence the silicon, mangan- 
ese, phosphorus and carbon impurities are removed. As with malleable 
cast iron the light grades, containing the least carbon and those practically 
free from sulphiu- and phosphorus, are selected. The wet-puddling process, 
in a reverboratory furnace, is the practice in the United States. A gray 
unrefined iron, containing more silica than the white, is used, mixed with 
a large quantity of hematite or magnetite of high grade used as reagents 
for oxidizing the impurities — the carbon passing off in the resulting carbon 
dioxide, and the other impurities, in the slag. The following shows the 
per cent, of composition of the pig and of the resulting muck-bar (puddle 
bar), in a typical case: 

Carbon. ^^^ Silicon. ^?^°^; Sulphur, 
ganese. pnorus. ^ 

Pig (before melting). 2.55% 5.15% 3.20% 0.95% 0.05% 

Muck-bariron 0.65" 0.15" 0.10" 0.10" 0.01" 



* Synopsis of Specifications adopted by letter-ballot of the A. S. T. M. 
on Sept. 1, 190$, 



394 



2l.'-METALLURGY, 



% 



The objection to this process is the enormous cost of fuel. The muck 
bars are re-heated and rolled into merchant bar — principally demanded 
by the blacksmith. (Soft or mild steels are generally replacing wrought 
iron.) 

Mr. George Schuhmann, in the " Pilot " (official publication of the 
P. 8c R. Ry. Dept.), April, 1906, says: 

" The purer the iron the higher is its melting point. Pig iron melts at 
about 2100 degrees F., steel at about 2500 degrees, and wrought iron at 
about 2800 degrees.^ The temperature in the puddling furnace is high 
enough to melt pig iron, but not high enough to keep wrought iron in a 
liquid state; therefore, as soon as the small particles of iron become purified 
they partly congeal (come to nature), forming a spongy mass in which 
small globules of iron are in a semi-plastic state, feebly cohering with fluid 
cinder filling the cavities between them. This sponge is divided by the 
puddler into lumps of about 200 lbs. each; these lumps or balls are taken 
to a steam hammer or a squeezer, where they are hammered or squeezed 
into elongated blocks (blooms), and while still hot, rolled out between the 
puddle rolls into bars 3 to 6 inches wide, about f-inch thick, 15 to 30 ft. 
long. These bars are called puddle bars or muck bars, and, owing to the 
large amount of cinder still contained therein, they have rather rough 
surfaces. The muck bars are cut up into pieces from 2 to 4 ft. long and piled 
on top of each other in so-called " piles " varying from 100 to 2000 lbs., 
according to the size product desired. These piles are heated in heating 
furnaces, and when white hot, are taken to the rolls to be welded together 
and rolled out into merchant iron in the shape of either sheets, plates, 
bars, or structural shapes as desired. When cold this material is sheared 
and straightened, and is then ready for the market. 

After leaving the puddle furnace, wrought iron does not undergo any 
material change in its chemical composition, and the only physical change 
is an expulsion of a large portion of the cinder; the small cinder-coated 
globules of iron are welded together and the subsequent rolling back and 
forth will elongate these globules, giving the iron a fibrous structure, and 
reheating and rerolling will drive these fibers closer together, thus increas- 
ing the strength and ductility of the metal. 

Steel — For structural work, steel has almost entirely superseded wrought 
iron, being stronger and more cheaply manufactured. The latter, however, 
is still used largely for underground pipes, because better able to resist 
decomposition by the natural elements and also electrolysis. The following 
typical analyses of the products of the foiu: principal processes of structural 
steel manufacture are here shown for approximate comparison, being in 
no way absolute: 



Combined carbon 

Silicon 

Sulphur 

Phosphorus. 
Manganese . . 
Arsenic 



Acid Bessemer. 


Basic Bessemer 


Acid Open 
Hearth. 


Basic 

Open 

Hearth. 


Soft S. 


Rail S. 


Soft S. 


Med. 
H'dS. 


Soft S. 


Rail S. 




.12% 

.04 

.05 

.06 

.06 

.04 


.43% 

.06 

.06 

.07 

.50 

.05 


.06 
.04 
.04 
.50 
.04 


.75% 

.20 

.04 

.04 

.70 

.04 


.12% 

".OS*" 
.06 
.60 
.04 


.43% 

.01 

.06 

.07 

.80 

.04 


.14% 

■'.04*" 

.05 
1.20 

.04 



These processes consist in decarburizing and purifying the pig iron to 
a practically pure liquid wrought iron; recharging the liquid with carbon 
and manganese where necessary; and casting it into ingots. They will be 
described as briefly as possible. 

Acid Bessemer Process. — The molten iron or pig from the blast furnace 
(direct process) or the cupola (indirect process) is poured into an " acid " 
Hned " converter," in the bottom of which are a number of small holes 



STEEL— PROCESSES OF MANUFACTURE. 395 

through which air is forced into the metal. The air oxidizes to impurities — 
carbon, siHcon, manganese, etc. — the heat of combustion from the carbon 
and sulphur maintaining the metal in a fluid condition. The " acid " 
lining is a highly refractory, siliceous (about 93 per cent, silica) material, 
which removes the carbon, silicon and manganese, but is incapable of taking 
the phosphorus and sulphur away from the metal; hence the pig iron used 
must practically be free from these two latter elements, as they are not 
allowed to enter largely into steel. The impurities being thus removed it 
is now liquid wrought iron. Carbon, for strength and hardness, and man- 
ganese, for malleability in rolling, are added — the latter in the form of 
spiegeleisen or ferro-manganese. It is now steel and is poured into molds 
for cooling down into ingots. These ingots are charged into the soaking 
pit or vertical furnace to equalize before being rolled into blooms. Later, 
the blooms are (re-heated and) rolled into rails or into the various struc- 
tural shapes desired. 

Basic Bessemer Process. — Pig iron that is too high in phosphorus (2.5 
to 3.0 per cent.) for the acid Bessemer process may be converted into steel 
by the basic Bessemer process provided it is low (below . 50 per cent.) in 
silicon. The " basic " lining of the "converter " is made from a pure 
magnesian limestone (dolomite) containing about 20 per cent, of magnesia, 
but not over 1 or 2 per cent, of silica. The pig is usually remelted in the 
cupola and transferred to the converter — hard-burned limestone low in 
5^02 being added as a flux — where air is introduced as in the acid process. 
The impurities in the metal are burned out, the phospliorus furnishing the 
heat and uniting with the lime to form^ phosphate of linie. The liquid is 
now almost a chemically pure iron, and is recarburized as in the acid Besse- 
mer process. The floating slag when cooled and ground is used as a fertil- 
izer or for cement. 

This process is not used much in the U. S. 

Acid Open Hearth Process. — This process, like the acid Bessemer, 
calls for a pig low in sulphur and phosphorus because neither is eliminated. 
The Siemens furnace is a large hearth built of refractory siliceous material 
on which the metal is melted, by the combustion of gas admitted into the 
furnace alternately with air. No solid fuel is used. A typical pig iron 
would contain about the following: Carbon 3.5 per cent., silicon 2.2, sul- 
phur 0.03, phosphorus 0.035, manganese 0.75. 

This process is particularly adapted to producing steels high in carbon 
and to the exact percentages specified. 

Basic Open Hearth Process. — In this process the lining of the hearth 
is made from the " basic " material — magnesian limestone — and like the 
basic Bessemer, a high phosphoric pig is used, say about as follows: Phos- 
phorus, 1.80 per cent.; silicon, not over 1.50; sulphur, not over 0.50; 
manganese, 1.75; carbon, 3.50. Iron and steel scrap is also added to the 
melt. The method is about the same as for the acid 0-H process, the im- 
purities being separated from the bath partly by the air admitted and 
partly by the addition of iron oxides. The product is an excellent steel. 

Cementation Process. — This process is still employed in making high 
grade carbon steels for tools, cutlery, etc. " Cement bars " are made from 
the purest and best grade iron (usually the Swedish) heated in contact with 
carbon (charcoal) to a red heat maintained for a week or more. The product 
is called " cement steel " or " blister steel." " Shear steel " is made from 
cement steel by cutting, piling, heating and rolling the cement bars — 
" double shear steel " being simply a duplication of the process. " Crucible 
steel " or " cast steel " is made from blister steel by cutting up and melting 
the bars in a black-lead pot, and cooling in the form of ingots. 

Shear steel is used for plows and cheap edge tools, but not for high 
grade tools. 

Crucible cast steel, made by melting the blister steel of the cementa- 
tion process in crucibles, is a high grade steel, used for the best tools. In 
order to avoid the permanent set of internal stresses in castings, due to 
unequal shrmkage in cooling, they are ** annealed " by being gradually 



2\.— METALLURGY. 



heated to temperattires between 800° (for high carbon steel) and 900° (for 
low carbon steels). Crucible steel is "tempered" for different uses, de- 
pending upon the percentage of carbon present, as follows: 



Name 

of 

Temper. 


Per cent. 

of 
Carbon. 


Burning . 


Welding. 


Temper. 


Remarks. 


Razor . . 

Saw-file 


1.5 

1.375 

1.25 

1.125 

1. 

0.875 
0.75 


Easy. 


Very difficult. 


Extremely hard. 
Hard. 


Razors, etc. 
Saws, etc. 


Tool 





Possible. 


Extreme care re- 


Spindle . 


Medium. 


quired in weld- 
ing. 
Machine turn- 


Chisel . . 





Quite easy. 


ing tools, cut- 
ters, etc. 
Cold chisels, hot 


Set 


Little. 
Very little. 


sets, etc. 
Cold sets, etc. 


Die 


Not easy. 


Easy. 


For hard surface 
tools as dies. 



Steels very low in carbon can easily be welded but not tempered. Steel 
with carbon above f per cent, can be tempered by heating to a high heat, 
and then quenching in water or other liquid. 

Open Hearth Cast Steel. — ^The acid open hearth is generally preferred. 
The cast steel produced is adapted to machinery castings for steel vessels, 
cars, etc.; it contains from one-sixth to one-half of one per cent, carbon. 

Harveyized steel plates for armor are mild steel plates, face-hardened 
on one side by carburization and sudden chilling; the other side may remain 
as mild steel or be decarburized into a softer metal. 

Manganese steel is a steel containing about 14 per cent, manganese, 
and is made by melting ferro-manganese with carbon steel. It is very 
tough and hard, capable of being forged. It is used principally for car 
wheels, stamp heads and jaws for crushing machines; also for crossing frogs. 

Vanadium steel is used for crusher jaws, gear wheels, etc. (See also 
page 399.) 

Chrome steel 'is made by melting ferro-chrome with bar iron, in a- cru- 
cible. It is very hard and can be forged readily. Contains about 1 . 8 per 
cent, chromium and 0.7 per cent, carbon. 

Nickel steel is produced by adding nickel ore to the carbon steel in the 
open hearth bath, thus increasing the strength and elasticity. It is used in 
armor plate manufacture; and, with the probable future reduction in cost, 
will doubtless be used in steel bridges, especially of long spans, in the near 
future. (See, also, pages 398 and 399.) 

Tungsten steel is very hard, difficult to forge, and is used for cutting- 
tools. 

Alloys. — An alloy is a mixture of two or more metals by fusion. The 
number of possible mixtures, taking into consideration the varying pro- 
portions, is infinite, but the principal alloys useful to the engineer, are to 
a certain extent limited and well known. From a mechanical point of 
view the most useful metals for alloys are: copper (Cu), tin (Sn), zinc (Zn), 
lead (Pb), antimony (Sb), nickel (Ni), bismuth (Bi), aluminum (Al), iron 
(Fe), etc., about in the order mentioned. Cu is by far the most useful and 
may be classed as the primary metal; with Sn, Zn, Pb, and Sb, secondary; 
and Ni, Bi, Al, Fe, etc., tertiary, as follows: 

Fe Sn Al 
Pb Cu Sb 
Ni Zn Bi 



ALLOYS— STEEL, BRONZE, BRASS, ETC. 



397 



Bronze (Cu, Sn, X) is the name generally given to those alloys con- 
sisting chiefly of copper and tin, with the former metal largely in excess. 
The following are some of the principal bronzes and their uses: 



Name. 


Composition (parts). 


Remarks. . 




Copper. 


Tin. 


Other metals. 




Medal B 


95 
91 
90 

89 
88 
82 
68 


5 
9 
10 
11 
12 
18 
32 






Coin B 






Gun B 




For navy ordnance. 


Statuary B 

Valve-metal . ... 






Zinc (2). 




Bell-metal 




Speculum-metal. . . 













Phosphor bronze is a bronze containing a small percentage (3±) of 
phosphorus properly introduced during fusion. It materially increases the 
strength and hardness of the metal and renders it less susceptible to oxida- 
tion and the action of the elements generally. Phosphor bronze is used for 
heavy journal bearings in the best machinery, and for wire. 

Manganese bronze (Cu, Sn, Mn) is a bronze containing a small per- 
centage (1 + ) of manganese, rendering the material tough and non-corro- 
sive. It is largely used for propeller blades. A good mixture is Cu (88), 
Sn (10), Mn (2). 

Aluminum bronze (Cu, Al), so-called, is usually composed of Cu (90) 
and A I (10) although these proportions may be varied greatly. It has a 
high tensile strength and is not easily corroded. 

Silicon bronze (Cu, Sn, Si) is used for castings and for telegraph wires. 
It is an excellent conductor of electricity and has great tensile strength. 
The proportion of silicon is from 3 to 5 per cent. Tin may or may not be 
used with the copper. 

« Brass (Cu, Zn, X) includes the alloys which are composed mainly of 
copper and zinc. The principal ones are tabularized as follows: 



Name. 


Composition (parts). 


Remarks. 




Copper. 


Zinc. 


Other metals. 




Valve-metal 


83 
75 

70 

67 
60 
58 
58 
56 

50 

60 


15 
25 

30 

33 
40 
40 
40 
42 

50 

20 


Tin (2). 




Brazing-metal 


Flanges for copper 






pipe. 
Works well under ham- 


Common B 




mer. 


Muntz-metal 






Delta-metal 


Iron, 

Tin, lead, iron. 




Tobin " Bronze " . 




Thurston-metal . . . 
Brazing-metal 


Tin (2). 


Thurston's maximum 

is 57, 42, 1. 
Solder for copper pipe 


German silver 


Nickel (20). 


flanges. 







398 



21.— METALLURGY. 



Tin=base Alloys. — 






Composition (parts). 




Name. 


Tin. 


Zinc. 


Lead. 


Other 
Metals. 


Remarks. | 


Babbitt-m.etal . . 


89 

85 

67 
50 






\Sh, 8 
\Cu, 3 

15 


Used for machinery 


Pewter 






bearings. 

Contains antimony, cop- 
per and bismuth. 


Solder 


33 






50 













Lead=base Alloys. — 





Composition (parts). 




Name. 


Lead. 


Tin. 


Anti- 
mony. 


Bis- 
muth. 


Remarks. 


White-metal.. . . 


88 
86 


.. .^^.. 


12 






Fusible plug.. . . 


4 


For steam boilers 



Alzene, a new metal alloy composed of aluminum and zinc, is said to be 
as strong as cast iron, much more elastic, does not rust easily, and takes a 
very high polish. It is a German product. Future results of investigations 
are awaited with interest. The present proportions of the alloy are 2 parts 
aluminum and 1 part zinc. It is capable of filling out the most delicate 
lines and figures of forms in casting. 

EXCERPTS AND REFERENCES. 

Malleable Cast Iron (By H. E. Diller. Jl. of the Am. Foundrymen's 
Assn.; Eng. News, Dec. 11, 1902). — "Malleable" can be made up to 60 000 
lbs. per sq. in., though this is not advisable as the shock-resisting qualities 
are sacrificed. Yet as specifications become more severe the general quality 
of this class of castings will be improved until we get a more reliable article, 
and which can better resist the encroachments of the steel casting. The 
article discusses the methods of manufacture. 

Manufacture and Properties of Nickel=Steel (By A. L. Colby. Proc. 
A. S. T. M., 1903; Eng. News, Aug. 20, 1903.)— Modulus of Elasticity.— . 

Young's modulus is practically the same for both tool steel containing 
1.40% carbon and the mildest steel used in boilers. Their modulus of elas- 
ticity is in fact rarely found to be below 29,000,000 or above 31,000,000 
and is generally taken at 29,500,000 or 30,000,000 in engineering calcula- 
tions. The high nickel-steels, especially those containing 20% nickel or 
over, have a lower mod. of elas. than carbon-steel; but nickel-steels con- 
taining say 4% of nickel or less, such as are applicable for shafting, forgings, 
bridge construction, rails, etc., have the same mod. of elas. as carbon steels, 
viz., in the neighborhood of 29,000,000 lbs. per sq. in., many authorities 
claiming that this is true of even 5% nickel-steel. Tensile Strength and 
Elastic Limit. — Nickel-steel is chiefly distinguished from carbon-steel by 
its proportionately high elastic limit. If 3% nickel is alloyed with an open- 
hearth steel of 0.25% carbon, it produces a metal equal in tensile strength 
to a simple carbon-steel of 0.45% carbon, but having the advantageous 
ductility of the lower carbon-steel. On low carbon-steels not annealed, 
the addition of each 1% of nickel up to 5% causes approx., an increase of 
6000 lbs. per sq. in. in the elastic limit, and 4000 lbs. in the ultimate or 



MISCELLANEOUS— NICKEL STEEL, ETC, 399 

tensile strength. The influence of nickel on the elastic limit and ultimate 
strength increases with the percentage of carbon present; high carbon 
nickel-steels showing a greater gain than low carbon nickel steels. Other 
Properties Discussed. — Effect of compression; rigidity; cold and quench 
bending tests; hardness; resistance to torsion; resistance to wear or 
abrasion; expansion; effect of punching and shearing; segregation. 

Notes on the Metallurgy of Steel (By Bradley Stoughton. Trans. 
A. S. C. E., Vol. LIV, Part E). — See pages 398 to 406 of Transactions for 
112 references to this subject. 

Vanadium Steel Alloys (By J. Kent Smith. Soc. of Chemical Ind- 
dustry, Liverpool; Eng. News, May 24, 1906). — ^The vanadium steel 
industry is altogether an English industry, 80% of the production being 
used for the construction of motor cars and omnibuses. The chrome- 
vanadium steels containing 10 to 20% vanadium show the most remarkable 
properties, the highest test yet obtained, after special treatment, being a 
maximum breaking strength of 103 tons per sq. in. The nickel-vanadium 
steels are also of great strength, but show a lower resistance to dynamic 
and torsional tests. 

Manganese Bronze (By C. R. Spare. Proc. A. S. T. M., Vol. XIII, 
1908) . — Properties of, with discussions. 

Vanadium Structural Steel (By G. L. Norric. Eng. Rec, June 4, 1910). 
— Table of results of tests of angles made from three different heats of 
chrome-vanadium steel. "The best chrome- vanadium steels for rolled 
shapes have from 0.18 to 0.25% carbon, 0.25 to 0.40% manganese, 0.4 to 0.6% 
chrome, and 0.15 to 0.20% vanadium.'* 



22.— BUILDING STONES AND CEMENTS. 

(For Weights and Specific Gravities, See Section 27.) 
This class of materials comprises all the non-m,etallic minerals of con- 
struction, including (I) Natural stones, (II) Cements, and (III) Artificial 
stones: 

I. NATURAL BUILDING SFONES. 

These may be classified in three divisions, as follows: 

A. Crystalline siliceous rocks, or those of the igneous types containing 

much silica, as granites and sienites. 

B. Calcareous rocks, or those consisting mainly of lime, as limestones 

and marbles. 

G. Fragmentary rocks, as sandstones and slates. 

A. Crystalline Siliceous Rocks. 

Granite. — Granite is an igneous rock of granular and acid composition 
and, contrary to former belief, is now supposed to be forming today, from 
fusion, deep in the earth's crust. It is composed mainly of orthoclase, 
quartz, mica and usually feldspar, the following chemical analysis being 
typical: Silica (70), alumina (15), iron oxides (4), sodium oxide (4), potas- 
sium oxide (4), lime (2), magnesia (1). 

Depending upon the material presence in quantity, with quartz, of 
muscovite, hornblende or augite, we have respectively, 
Muscovite granite. (Muscovite is a kind of mica.) 
Hornblende granite^ 
Augite granite. 

If the granite loses its. quartz it grades into Sienite. 

If potash feldspar is laigely replaced by lime-soda feldspar it further 
grades into Diorite. 

Granite is one of the most useful of building stones, but it is hard to 
work and cracks when subjected to great heat as evidenced by the Boston 
fire of 1871 on the Boston post office building, and by the recent Baltimore 
fire. 

Basalt.— This term is applied to rocks of basaltic lava origin, compris- 
ing the so-called trap and the common greenstone. The basalts are very 
hard, heavy, and durable as building stone, the principal objection to their 
general use being the difficulty of working them, — quarrying and cutting. 
Basalt blocks are used principally as paving stone, resisting wear and tear 
to a remarkable degree. 

Trap. — This includes a very wide range of extremely hard, crystalline 
rocks, and is sometimes applied outside the range of basalt. They are 
fine grained and usually dark-green in color. Broken trap rock makes an 
excellent base for concrete. 

Greenstone. — When trap is altered by the presence of hornblende, 
chlorite, epidote, etc., it merges into greenstone, the green color being im- 
parted by the hornblende. 

B. Calcareous Rocks. 

Limestone. — Pure limestone is carbonate of lime (CaCOs), but there 
is sometimes present also carbonate of magnesia (MgCos) and certain so- 
called impurities, as alumina, silica, iron, etc. Carbonate of lime (CaO, 
CO2) is a compound of lime (CaO) and carbonic acid (CO2), the lime being 
an oxide of calcium, — an alkaline earth of specific gravity 3. 18. Lime- 
stone and chalk are examples of carbonate of lime in the amorphous condi- 
tion, while marble, aragonite and Iceland spar are varieties in the crystalline 
form, 

400 



NATURAL BUILDING STONES. 401 

As carbonate of lime is abundantly present in both the organic and inor- 
ganic kingdoms, the great beds of limestone are thus being provided with 
these two material sources in their formation. The following are important: 

Crinoidal L. — Formed of the fossil remains of the crinoid species of 
little sea animals, as indicated by fragments of the coral stems. These 
limestones occur in Indiana, Iowa and Kansas. 

Coquina. — Composed of cemented fragments of shells and corals found 
at the bottom of the sea on the coast of Florida. 

Chalk. — A cretaceous rock composed mainly of sea shells of the order 
foraminifera. Found extensively on the coast of England and seems to 
be partially formed limestone; not used as a building stone. 

Dolomite. — A limestone containing carbonate of lime (CaCO^) and car- 
bonate of magnesia (MgCOs) in large proportions. It may be either crys- 
talline or amorphous (massive). The massive varieties containing iron 
are called brown spar. Dolomite is found in Vermont, Rhode Island, 
New York, New Jersey, Missouri. 

Hydraulic L. — A limestone containing carbonate of lime (CaCOz), 
carbonate of magnesia (MgCOs), silicon dioxide {Si02) and alumina, having 
the property of hardening under water after it has been burned. 

Marl. — A mixture of calcium carbonate (45%) and clay, with perhaps 
sand. It is unsuitable as a building stone. Occurs in New Jersey, Ken- 
tucky, Maryland, Virginia and the Carolinas. 

Travertine. — ^A limestone deposit formed at the mouths of springs and 
in some rivers. 

Marble. — ^Marble may be considered as a high grade limestone, crys- 
talline in structure and capable of being polished. Those composed wholly 
of carbonate of lime are white, while the various colorings in most of the 
marbles are due to the presence of foreign matter. Most of our marble is 
quarried in Vermont, although it is obtained in many other states as Massa- 
chusetts, New York, Tennessee, Maryland, Georgia, Pennsylvania, Arizona, 
Colorado, California. 

Among the foreign marbles imported to this country are the following: 

Italian. — Black and gold, Carrara, Landscape, Nero antico. 

Pyrennees. — Brocatelle, Griotte. 

Grecian. — Giallo antico, Parian, Pentellic, Rosso antico. 

African. — Numidian marble. 

C. Fragmentary Rocks. 

Sandstone.-;— Sandstone may be called a *' sand conglomerate," formed 
by^ sedimentation. As a building stone the cementing material is quiie 
uniformly dispersed, the whole mixture having been subjected to great 
heat and pressure. Sandstones when thoroughly dried will absorb about 
3 per cent, of their weight of water; and in cold countries where buildings 
are subjected to the action of the frost it is best to present the foliated 
edges of the stone to the weather, in order to prevent flaking or frost chipping. 
Sandstone is easily quarried, usually lying in situ in layers of greater or 
less thickness, but uniform. A channelling machine is used in cutting out 
the area required, the thickness being determined by the strata. 

Berea sandstone, obtained from Berea, Ohio, is a gritty stone well 
suited for building and general masonry construction. It is a grayish stone. 

Medina sandstone, quarried in New York (Medina) is largely used 
throughout the East. It is reddish and argillaceous. 

Potsdam (New York) sandstone is very hard and durable (sometimes 
quartzitic). It is a reddish yellow stone, found in New York, Virginia, 
Wisconsin and Michigan. 

Connecticut Valley sandstone is a Triassic brownstone, and a very im- 
portant building stone. 

New Jersey sandstone is also Triassic. 

In general, a sandstone composed mainly of quartz sand with siliceous 
cement and of old formation is the best. If metamorphosed into quartzite 
it is even much harder and more durable. 



402 22.— BUILDING STONES AND CEMENTS. 

The best test of a sandstone is to expose it to the action of frost as in 
actual construction. This is the case with all the foregoing, which have 
been well tried and tested practically. An artificial freezing is sometimes 
applied to specimens from new quarries, which tests their power of re- 
sisting frost action. It consists in boiling the specimen 10 or 15 times 
in a strong solution of soda sulphate, exposing it to the action of the 
air for some hours after each boiling. The absorbed salt expands in crys- 
tallizing, similar to frost action, flaking the specimen to a greater or less 
extent. 

Flagstones are thinly bedded sandstones, the cleavage being parallel 
with the beds (usually). Bluestone is a variety found in Pennsylvania, 
New York and New Jersey. 

Slate. — Slate is formed from shale under great pressure and heat. The 
cleavage planes may or may not be identical with the original shale folia- 
tion, but usually crosses them at different angles, and even at right angle 
to the bedding as in that at Slatington, Penn. 

Roofing slate is a true slate, being a hard, compact rock apparently not 
affected by the weather. It may be laid on boarding or on terra cotta 
roofing, the lower edge of each third layer overlapping the upper edge of 
the first by an inch or more. Japanned^ malleable iron nails are used to 
avoid rust. Vermont and Pennsylvania furnish most of the slate quarried 
in the United States. It is also distributed throughout the South, North- 
west and California. 

II. CEMENTS. 

Materials with Cementing: Properties may be mineral, vegetable or 
animal substances, either pure, mixed, or transmuted. The following 
substances have cementing qualities: Mineral. — Quartz, calcite and the 
iron ores; specifically, silica, lime, plaster of paris, sal ammoniac, sulphur, 
iron borings, brick dust, isinglass, clay, red lead, white lead, asphaltum, 
alum, copal, chalk, paraffin, gypsum, etc. Vegetable.— Gum, resin, wax 
and vegetable albumen; specifically, balsarn, india rubber, rosin, starch, 
rice flour, wheat flour, mastic, etc. Animal.- — Albumen, gelatin and 
glycerin; specifically, white of egg, lac, shellac, skim milk, cheese, beeswax, 
stearin, dextrin, etc. Gelatin is obtained by boiling animal substances as 
skins, hoofs, etc., in water. 

The Solvents most common in the mixing of cements are water, alcohol, 
naphtha, vinegar, turpentine, linseed oil, benzine, petroleum, glycerin, 
ammonia, etc. 

In the present discussion, cements are classed under Miscellaneous 
Cements, and (Builders) Cements. 

MISCELLANEOUS CEMENTS. 

Boiler cement. — ^To stop cracks and leaks in boilers and stoves. Dry 
powdered clay (6 parts), iron filings (1 part); mix to a paste with pure 
boiled linseed oil. 

Coppersmith's cement. — ^To mend leaky joints in copper boilers. Bul- 
locks blood thickened with quicklime; use immediately. 

Fireproof cement. — To mend stone. Fine river sand (20), litharge (2), 
quicklime (1) ; mix to a thin paste with linseed oil. 

Flour cement. — For general paste. To i pint water add 1 tablespoonful 
wheat flour slowly, stirring rapidly; heat until it boils, stirring. Adding 
a little powdered alum to the water strengthens the paste; a little brown 
sugar and corrosive sublimate will preserve it from turning mouldy. 

Gas Fitters* cement. — Resin (4^), beeswax (1); melt, and stir in Vene- 
tian red (3); pour into iron- or oiled paper molds. 

Iron cement. — For closing joints of iron pipes. Mix cast-iron borings 
or turnings (80) with sal-ammoniac (2), and flowers of sulphur (1). In 
using, add enough water to moisten; stir, and ram into joints. The sulphur 
is sometimes omitted. 

Gli4£ cement. — GTue (1), melted with water (least possible), and mixed 
with black resin (1) and red ochre (i). 

Steam-Pipe cement. — Grind good linseed oil varnish with equal weights 
of white lead, manganese oxide, and pipeclay. 






CEMENTS. 403 

Keene's Marble cement. — For stucco work; will not stand weather. 
Baked gypsum or plaster of paris steeped in a saturated solution of alum 
and then recalcined and reduced to powder. To use, mix with water the 
same as plaster of paris. 

For a complete list of cements and their preparation see The Scientific 
American Cyclopedia of Receipts, Notes and Queries; also Coolies' Cyclo- 
pedia of Practical Receipts. 

{BUILDERS) CEMENTS. 
Calcium {Co) is a light-yellow metal whose specific gravity is 1 . 58, 
and which oxidizes at ordinary temperatures, hence it does not occur pure 
in nature, but is found abundantly as a component of calcite {CaCOz), 
gypsum, dolomite, selenite, aragonite. Calcite or carbonate of lime is the 
principal constituent of limestone, marble and chalk. Calcium as a base 
plays a most important part in the limes, mortars and cements used in con- 
struction. 

Lime (CaO), as its symbol implies, is an oxide of calcium, and may be 
obtained by placing calcium in contact with water, whence the latter is 
decomposed, forming lime and hydrogen gas (which latter escapes). Lime 
is a white, alkaline powder, of specific gravity 3.16. It may be obtained 
also by heating pure carbonate of lime or calcite (CaCOs = CaO 4- CO2) , 
whence the carbonic acid (C02) is driven ofi^, leaving the lime. This is a 
ptire lime and not the more or less impure, commercial article known to 
the engineer. 

Common Lime is a more or less impure lime obtained by calcinating 
common limestone, composed mainly of CaCOs, in kilns or furnaces, thereby 
driving off the carbonic acid and organic impurities. Its purity is dependent 
mainly upon the mineral purity of the carbonate which is burned. Quick' 
lime or burned lime (CaO + impurities) is the name given to it as it comes 
from the kilns. In this state it is white and has a specific gravity of 3.16. 
Slacked lime is quicklime which has been slacked by exposure to the air, 
whence the term " air slacked " in contradistinction to " water slacked." 
In the former case it absorbs carbonic acid and _ moisture from the air. 
Water slacked or simply common slacked lime is a calcium hydroxide 
(CaH202). Its specific gravity is 2.1 or about two-thirds that of quick- 
lime, showing that slacking increases its bulk about one-half. " Fat lime " 
is obtained from limestone containing 5 per cent or less of impurities, and 
when these latter amount to 6 per cent the lime is poor. 

Common Lime Mortar. — This is composed of common lime mixed with 
the required amount of sand, and freshly slacked to a smooth paste. The 
proportions of the mixture of lime, water and sand vary according to the 
quality of the lime and the class of work for which it is intended. An aver- 
age proportion for brickwork is, by bulk, 1 of lime, 2 to 3 of sand. It will 
not set under water and is used only where in contact with air it can set 
slowly by absorbing carbonic acid (and some moisture). Lime mortar may 
contain any proportion of sand or even no sand. 

For a fat lime and a good quality of cement the adhesive properties of 
the resulting mortar, after being set, are about as follows for various mix- 
tures, calling that of pure lime paste unity: 

Lime. Sand. Cohesion. .^yj 

1.00 '^ 

0.5 .905 

1. .82 

1.5 .745 

2 .68 
2.5 .625 

3 .58 
3.5 .545 

4 .52 



7 



0^2 



Lime Plaster. — ^This consists approximately of 1 part quicklime, 2 parts 
sand, and to each 100 lbs. of the resulting mortar about f bushel of cow- 
hair or other fine fibre is sometimes added to give it coherent strength. 
It is applied on wooden or metal laths. Many patent plasters are manu- 
factured in slabs at the factories and shipped ready to be put in place. 
They are often compounds of gypsum. 



404 22.— BUILDING STONES AND CEMENTS, 

Plaster of Paris. — When gypsum, a hydrated sulphate of lime (CaSO^-i- 
2H2O), is heated sufficiently, part of its water of crystallization is driven 
off, leaving the resulting composition {CaS04)2+H20, called plaster of 
paris. This is one of the simplest of the mineral cements, and in addition 
to its value in the arts it is used for cementing slabs of marble in building 
construction. This consists in simply mixing the plaster of paris to a creamy 
paste and applying it. Its setting consists in taking on water, again crys- 
tallizing into the hydrate, gypsum. Plaster of paris is a useful constituent 
in many of the delicate cements. Its strength may be increased by mixing 
with it a solution of thin glue, albumen (white of egg), or vegetable gum. 
Many architectural ornaments are made of plaster of paris mixed with 
about an equal amount of paper pulp and a solution of size. 

Hydraulic Lime. — As common lime is made by burning common lime- 
stone, so is hydraulic lime obtained in a similar way from hydraulic lime- 
stone, which contains a large amount of silica and alumina. In burning, 
these latter combine with part of the lime, forming lime silicates and alu- 
minates, the other portion of the lime remaining free. The silicates and 
aluminates possess the power of hardening under water. It is used in 
high-class masonry construction in Europe, being replaced by cements 
in the United States. 

Hydraulic Cement. — ^These cements are capable of hardening undei 
water — containing a large amount, 25 to 50 per cent., of silica and alumina. 
They may be classed as natural, Portland (semi-natural), and slag (arti- 
ficial or puzzolanic) cements. 

Natural Cement is obtained by simply burning the natural hydraulic 
limestone at a low temperature, and grinding the clinker very fine. It is 
manufactured in Ulster County, N. Y.; Louisville, Ky.; Cumberland, Md.; 
Utica, 111., and Milwaukee, Wis. The limestone contains much clay, which 
supplies the required silica and alumina. The product as shipped is com.- 
posed, approximately, of lime (42 parts), silica (28), alumina (10), iron oxide, 
magnesia and impurities (20). This cement is usually called Roman 
cement in Europe, and Rosendale cement in the U. S. It is cheaper than 
Portland cement, gains strength more slowly, and sets more quickly. 

Portland Cement. — Good Portland cement contains the usual hydraulic 
cement ingredients about as follows, namely, lime (62), silica (23), alumina 
(8), and other impurities including iron oxide, magnesia, sulphuric acid (7). 
It is prepared by selecting such natural materials that when mixed, ground, 
and calcined, the product will be the required compound as above. Thus, 
the proper silicates and aluminates of lime are obtained from a mixture of 
argillaceous limestones of different chemical composition; from relatively 
pure limestone and clay or chalk and clay; and from marl and clay. The 
resulting clinker is then ground to a powder. Sand-ground cement, as its 
name indicates, is a mixture of sand and Portland cement ground together. 
It increases the bulk of the " cement " considerably without reducing its 
strength, provided the ratio of sand to cement is not greater than 1 to 1, 
and the grinding is done properly, and with a good quality of sharp, siliceous 
sand. Portland cement is stronger than Rosendale, sets more slowly, but 
acquires its strength more rapidly. 

Slag Cement. — This is made by grinding furnace slag with lime, the slag 
containing the other necessary ingredients — silica and alumina (also a 
small percentage of impurities). It is not employed to any great extent in 
the United States. 

Bitumen. — Bitumen is a mineral pitch comprising the various class of 
substances known as asphalt, maltha, petroleum, naphtha, natural gas, 
etc. Bitumen is decomposed vegetable matter comprising mainly carbon 
and hydrogen, but containing also oxygen, sulphur and nitrogen in small 
proportions. Bituminous substances are found associated with the carbon- 
iferous rocks in pre-volcanic regions. Thus we have bituminous coal, 
-limestone, -sandstone, -shale, etc. 

Asphalt. — ^This is one of the principal bitumens and is supposed to 
have been formed by the hardening of its allied liquid substances, maltha 
(mineral tar) and pretroleum. Pitch Lake is a lake of asphalt on the island 
of Trinidad, and is controlled by the Trinidad Asphalt Co. of Philadelphia. 
Its product is of the finest quality, and supply apparently inexhaustible. 
Asphalt is a natural cement; its composition is practically unalterable 



HYDRAULIC CEMENT S-^NATURAL, PORTLAND. 405 

when exposed to the natural elements; and it is quite plastic and practically 
water-proof. Thus, it forms an excellent road, pavement and roofing ma> 
terial, and as a coating preservative for water pipes it is unexcelled. Asphalt 
cement is simply the refined asphalt tempered generally with the residue 
from petroleum oil. Asphalt mastic is prepared by mixing asphalt cement 
with sand and perhaps crushed limestone, or by mixing maltha, the poorer 
quality of bitumen, with natural rock asphalt. The mixture is heated, and 
cooled in molds ready for use. Asphalt concrete consists of broken stone 
or gravel with asphalt mastic used as a binder. 

Manufacture of Portland Cement. 

Raw Materials. — ^The materials used in the manufacture of Portland 
cement are carbonate of lime and clay: occurring either more or less naturally 
mixed, as in the argillaceous limestones; or separate in natural beds, as 
the chalk deposits of England and the various clay deposits common in 
many countries. In England, chalk is the principal form of carbonate of 
lime employed, and this is mechanically mixed with estuary mud. In 
Germany, the chief material is " mergel " (marl), a limestone rock of greater 
or less hardness, containing clay; also " weisenkalk," a pure soft marl 
composed mainly of carbonate of lime. In the United States the marls 
similar to those of Germany are found in Ohio, Indiana, Michigan and New 
York, and, mixed with clay, are largely used in the manufacture of Portland 
cement; but most of our cement is made from certain limestones containing 
sufficient clay and nearly free from magnesia, and found in certain localities, 
as Phillipsburg, N. J., and Lehigh Co., Penn. 

The three distinct operations in the manufacture of Portland cement 
are (1) Pulverizing and mixing the raw materials, (2) Calcination or burning, 
(3) Grinding the clinker. These are discussed as follows: 

Pulverizing and Mixing. — ^There are two processes in use for mixing 
the raw materials preparatory to calcination. These are known as the 
*' wet " process and the " dry " process, each naturally adapted to the 
class of materials to be mixed. 

Wet Process. — ^This process is used for such materials as chalk, soft 
marl and clay, which, by the admixture of a large quantity of water in a 
" wash mill," can be reduced to a homogeneous creamy condition of " slip " 
or " slurry. "^ The most recent practice in the United States is to grind 
the mixture in a liquid state, run it into tanks where it is kept continually 
stirred, and thence to the rotary tubular furnaces where it is calcined. 

Dry Process. — ^This process is particularly adapted to the treatment of 
argillaceous limestones or limestones containing sufficient clay so that very 
little mechanical mixing is necessary; notably the limestones of Phillips- 
burg, N. J., Lehigh Co., Penn., and of some localities in the West. Such 
material is run into a " grinding machine " where it is ground to a greater 
or less degree of fineness, depending upon the relative admixture of car- 
bonate of lime and clay; i.e., if they occur intimately mixed in the same 
rock in the right proportion, very little grinding is required. The process 
becomes very expensive, however, when an almost pure limestone rock 
and a clayey shale form the ingredients to be mixed, in which case they 
must first be crushed to the size of small pebbles, then mixed and ground 
to an impalpable powder to produce an intimate blending. 

Calcination. — After pulverizing and mixing, the material is conducted 
to the rotary tubular furnace, where it is burned at a temperature of about 
1600° F. Such a furnace consists of a steel tube usually from 60 to 100 ft. 
long, and 6 to 10 ft. in diameter, and lined with fire brick. It is arranged 
to rotate on rollers and is placed on a grade of about 5%. The raw mixture 
to be calcined is introduced at the higher end of the tubular furnace and. as 
the latter revolves, the heated material forms into small balls which slowly 
gravitate to the lower end, fall into a conveyer, and are carried to the 
clinker storage-room. The degree of burning is one of the most important 
considerations in cement manufacture. 

Grinding the Clinker. — After calcination, the resulting clinker passes 
to the " ball mills " where it is broken up into sand, with a small proportion 
of dust. It then goes to the " pulverizer," where it is reduced to the re- 
quired fineness of powder — cement. 




406 22.— BUILDING STONES AND CEMENTS. 

Cement Testing. 

Requisites of a Cement, — The principal requisites of a cement are 
when combined with other materials of construction it shall possess the 
necessary resistance to disintegration, that is, (1) strength to resist safely 
the loads which may come upon it, and (2) endurance to withstand safely 
the continuous or repeated stresses to which it may be subjected, with the 
element " time " included. Certain factors affecting these qualities in a 
cement will now be considered. 

Chemical Composition is one of the most important factors entering 
into both the strength and endurance of cement; hence, the raw materials 
should be selected in the proper proportions to give the required chemical 
mix in the finished product. ^ The elements calcium, silicon, aluminum, 
carbon (and hydrogen) all unite with oxygen, forming oxides, the exact 
chemical composition being very complex, and varying greatly with different 
cements. Any foreign matter thus becomes an adulterant, tending to pre- 
vent " set " or hardening of the cement. 

Set or Hardening of cement takes place when water is introduced into 
the anhydrous cement powder. It then becomes a hydrated compound in 
the form of an artificial stone. In Trans. A. S. C. E., Vol. LIV, Part F, 
pp. 48-52, Mr. R. L. Humphrey says: " This hydration or crystallization 
begins with the addition of water, and it is doubtful whether it ever ceases. 
. . . . Upon the addition of water, it begins immediately to hydrate 
in the from of fine needle-like crystals, and it is the intermeshing of these 
crystals that produces the hardening. They can be seen forming under a 
microscope immediately upon the addition of water to cement. The action 
is analogous to the formation of crystals from a saturated salt solution. 
If water is added to a cement that has commenced to set, and it is again 
mixed, the crystals already formed will be broken up, and the mass weak- 
ened by this retempering, Successive repetitions will gradually destroy all 
bond, and the cement having set or crystallized, the mass will become like 
so much inert sand." The phenomenon of hardening, then, " is purely a 
chemical one and is the result of the mechanical intermeshing of the crystals 
of silicate of lime, etc., which are formed by the hydration of the cement 
powder." 

It will thus be seen, from the above, that cement, cement mortar, and 
concrete should be placed in the work immediately after mixing, and should 
not subsequently be disturbed. 

Rate of Setting or time of setting in an important feature in construction 
work. It is necessary that the cement should not be so " quick-setting " 
as to prevent proper manipulation of the mix and placing it in the structure, 
nor so " slow-setting " as to require extra time to set, thereby retarding 
the work. The term_ " set " is here used in the sense of *' initial set " or 
that point of crystallization where any disturbance or displacement of the 
material. would effectually weaken it. Both " initial set " and " final set " 
or " hard set " are indefinite terms because crystallization or hardening is 
a continuous process which goes on indefinitely. Cements which do not 
set in two hours are considered slow-setting. On the other hand, 10 to 30 
minutes is usually allowed any cement for manipulation and placing, before 
initial set should appear. The rate of setting of cement may be increased 
by the addition of lime, or plaster of paris or gypsum, but the resultant 
strength is thereby decreased. 

Fineness is absolutely necessary to the thorough crystallization and 
consequent hardening, previously described. If the grains are coarse the 
crystals and their intermeshing will be imperfect, which will also be the 
case if the cement powder does not contain the proper mix of the raw ma- 
terials, and the right degree or calcination. An ideal cement is one in 
which the clinker has been pulverized to a "flour" or impalpable powder, 
thoroughly anhydrous, and containing elements in such proportion that 
when water is added the hydrated compound will form into innumerable 
perfect crystals, with no free matter to obstruct their formation. Of coiirse 
this condition can never be realized because (1) the grinding or pulver- 
izing is never carried to such an extreme degree, and (2) all cements contain 
more or less free matter, as excess of lime, magnesia, iron, etc. These two 
classes of imperfections affect the strength of the cement, and the second 
glass also affects its " soundness " and consequently its endurance, 



CEMENT TESTING. 



407 



Soundness is a negative term used to express the property which a 
cement has of not unduly expanding, contracting, checking or cracking 
during setting, hardening or crystallization. Such deformations are due 
to " active " impurities in the cement, such as free magnesia, lime, sulphiu* 
trioxide, etc., which produce internal stresses and thereby impair both its 
strength and durability. The class of impurities above mentioned should 
not be confused with certain " inactive " or inert impurities or adultera- 
tions which affect only the strength of the cement and not its endurance. 

Inactive Adulterants are present to a greater or less extent in nearly all 
cement, the effect being simply to reduce its strength and commercial 
value. Other inert adulterants, as sand, in sand-cement mortar, and sand, 
broken stone, gravel, etc., in concrete, are added to or mixed with 
cement purely on the grounds of economy, in order to increase the bulk of 
the cementing material, allowing its strength to be impaired to a point 
consistent with necessary safety. 

Method of Testing Cement. — ^The following is a digest of the standard 
method of testing cements, submitted as a progress report by a Committee 
of the Am, Soc. of Civil Engineers in 1908 and 1904, and adopted by the 
Am. Soc. for Testing Materials in 1904: 

Selection of Sample. — Generally, one barrel in every ten to be sampled, 
the sample to be a fair average of contents of package; it shall be passed 
through a sieve having 20 meshes per lin. in. to remove lumps before testing. 
In obtaining sample from barrels or bags, an auger or a sampling iron 
should be used, reaching from side to center. 

A chemical analysis, if required, may be made in accordance with the 
method outlined in the Journal of the Society of Chemical Industry, pub- 
Ished Jan. 15, 1902. The determination of the principal constituents of 
cement — silica, alumina, iron oxide and lime — is not conclusive as an 
indication of quality. 

Specific Gravity. — ^This is most conveniently made with Le Chatelier's 
apparatus, which consists of a flask (D), Fig. 
1, of 120 cu. cm. (7.32 cu. ins.) capacity, 
the neck of which is about 20 cm-. (7.87 ins.) 
long. In the middle of this neck is a bulb (C), 
above and below which are two marks (F) 
and (E) . The volume between these marks is 
20 cu. cm. (1.22 cu. ins,). The neck has a 
dia. of about 9 mm. (0.35 in,), and is gradu- 
ated to tenths of cu. centimeters above 
the mark (F). Benzine (62° Baum6 naph- 
tha) , or kerosene free from water, should be 
used in making the determination. 

The specific gravity can be determined in 
two ways: 1st, the flask is filled with either of 
these liquids to the lower mark (E), and 64 gr. 
(2.25 oz.) of powder, previously dried at 100° C. (212° F.) and cooled to the temp, 
of the liquid, is gradually introduced through the funnel (B) [the stem of 
which extends into the flask to the top of the bulb (C)], until the upper 
mark (F) is reached. The difference in weight between the cement remain- 
ing and the original quantity (64 gr.) is the weight which has displaced 
20 cu. cm. 2ndy the whole quantity of powder is introduced, and the level 
of the liquid rises to some division of the graduated neck. This reading 
plus 20 cu. cm, is the volume displaced by 64 gr. of powder. 

The specific gravity is then obtained by the formula: Specific gravity = 
weight of cement -f- displaced volume. 

The flask, during the operation, is kept immersed in water in a jar (A), 
in order to avoid variations in the temperature of the liquid. Results 
from different trials should agree within 0.01. The apparatus is conven- 
iently cleaned by inverting the flask over a glass jar, and shaking it vertically 
until the liquid starts to flow freely, repeating the operation several times. 

More accurate determinations may be made with the picnometer. 

Fineness. — ^The fineness is determined by measuring the residue retained 
on certain sieves, those known as No. 100 and No. 200 being recommended 
for this purpose. The sieves should be circular, about 20 cm, (7.87 ins.) 
in dia., 6 cm. (2.36 ins.) high, and provided with a pan 5 cm, (1,97 ins.) 
deep, and a cover. 




408 



22.— BUILDING STONES AND CEMENTS, 




The wire cloth should be woven from brass wire having a dia. of 0.0045 
in. for No. 100 sieve, and . 0024 in. for No. 200 sieve. It should be mounted 
on the frames without distortion; the mesh should be regular in spacing and 
be within the limits: 96 to 100 meshes per lin. in. for No. 100, and 188 to 
200 for No. 200. For the test, 50 to 100 grams (1 . 76 to 3.52 oz.) dried at 
a temperature of 212° F. prior to sieving, should be used. 

The coarsely screened (and dried) sample is weighed and placed on the 
No. 200 sieve, which, with pan and cover attached is held in one hand in a 
slightly inclined position, and moved forward and backward, at the same 
time striking the side gently with the palm of the other hand, at the rate of 
about 200 strokes per min., and continued until not more than ^^o of 1% 
passes through after 1 minute of continuous sieving. The residue is 
weighed, then placed on the No. 100 sieve and the operation repeated. 
The results should be reported to the nearest ^^ of 1%. The work may 
be expedited by placing in the sieve a small quantity of large shot. 

Normal Consistency. — ^The use of the proper percentage of water in 
making the pastes* from which pats, tests of setting and briquettes are 
made, is exceedingly important, affecting the results vitally. No method 
is entirely satisfactory, but the following is recommended: 

VicAT Needle Test. 

The apparatus used is known as the Vicat needle, v/hich consists of a 
frame (K), Fig, 2, bearing a movable rod (L), with a cap (A) at one end, 
and at the other the cylinder (B), 1 cm. (0.39 in.) in dia.; the cap, rod 
and cylinder weighing 300 gr. (10.58 oz.). The rod, which can be held 




Fig. 2. 

in any desired position by a screw (F) , carries an indicator, which moves 
over a scale (graduated to centimeters) attached to the fram.e (K)._ The 
paste is held by a conical, hard-rubber ring (I), 7 cm. (2.76 ins.) in dia. 
at the base, 4 cm. (1 .57 ins.) high, resting on a glass plate (J), about 10 
cm. (3.94 ins.) square. 

In making the determination, the same quantity of cement as will 
subsequently be used for each batch in making the briquettes (but not less 
than 500 grains) is kneaded into a paste, as described under " Mixing," 
and quickly formed into a ball with the hands, completing the operation 
by tossing it six times from one hand to the other, maintained 6 ins. apart; 
the ball is then pressed into the rubber ring through the larger opening, 
smoothed off, and placed (on its large end) on a glass plate and the smaller 
end smoothed off .with a trowel; the paste, confined in the ring, resting on 
the plate, is placed under the rod bearing the cylinder, which is brought in 
contact with the surface and quickly released. 

The paste is of normal consistency when the cylinder penetrates to a 
point in the mass 10 mm. (1 . 39 in.) below the top of the ring. Great care 
must be taken to fill the ring exactly to the top. 

The trial pastes are made with varying percentages of water until the 
correct consistency is obtained. 

The Committee has recommended, as normal, a paste _ the consistency 
of which is rather wet, because it believes that variations in the amount of 
compression to which the briquette is subjected in moulding are likely to 
be less with such a paste. 

* The term " paste " is here used to designate a mixture of cement and 
water, and the word " mortar " a mixture of cement, sand and water. 



CEMENT TESTING. 



409 



Having determined in this manner the proper percentage of water 
required to produce a paste of normal consistency, the proper percentage 
for the mortars may be obtained from an empirical formula, which the 
Committee hopes to devise; but temporarily the following Table* may be 
used: 

Percentage op Water for Standard Sand Mortars. 



Neat. 



One Cement, 

Three Standard 

Ottawa Sand. 



Neat. 



One Cement, 

Three Standard 

Ottawa Sand. 



Neat. 



One Cement, 

Three Standard 

Ottawa Sand. 



15 
16 
17 
18 
19 
20 
21 
22 



8.0 
8.2 
8.3 
8.5 
8.7 
8.8 
9.0 
9.2 



23 
24 
25 
26 

27 
28 
29 
30 



9.3 

9.5 

9.7 

9.8 

10.0 

10.2 

10.3 

10.5 



31 
32 
33 
34 
35 
36 
37 
38 



10.7 
10.8 
11.0 



11 
11 
11 
11 



11.8 





1 to 1 


1 to 2 


1 to 3 


1 to 4 


1 to 5 


Cement 


500 
500 


333 
666 


250 
750 


200 
800 


167 


Sand 


833 







Time of Setting. — The object of this test is to determine the time which 
has elapsed from the moment water is added until the paste ceases to be 
fluid and plastic (called the " initial set "), and also the time required for 
it to acquire a certain degree of hardness (called the " final " or " hard set "). 
The former of these is the most important, since, with the commencement 
of setting, the process of crystallization or hardening is said to begin. As 
a disturbance of this process may produce a loss of strength, it is desirable 
to complete the operation of mixing and moulding, or incorporating the 
mortar into the work, before the cement begins to set. 

It is usual to measure arbitrarily the beginning and end of the setting 
by the penetration of weighted, wires of given diameters. For this purpose 
the Vicat needle, already described, should be used. In making the test, 
a paste of normal consistency is moulded and placed under the rod (L), 
Fig. 2, as described under " Normal Consistency;" this rod, bearing the 
cap (D) at one end and the needle (H), 1 mm. (0.039 in.) in dia., at the other, 
weighing 300 gr. (10 . 58 oz.). The needle is then carefully brought in contact 
with the surface of the paste and quickly released. The setting is said to 
have commenced when the needle ceases to pass a point 5 mm. (0.20 in.) 
above the upper surface of the glass plate, and is said to have terminated 
the moment the needle does not sink visibly into the mass. 

The test pieces should be stored in moist air during the test; this is 
accomplished by placing them on a rack over water contained in a pan and 
covered with a damp cloth, the cloth to be kept away from them by means 
of a wire screen; or they may be stored in a moist box or closet. Care should 
be taken to keep the needle clean, as the collection of cement on the sides 
of the needle retards the penetration, while cement on the point reduces the 
area and tends to increase the penetration. 

The determination of the time of setting is only approximate, being 
materially affected by the temperature of the mixing water, the temper- 
ature and humidity of the air during the test, the percentage of water used, 
and the amount of moulding the paste receives. 

Standard Sand. — The Committee recognizes the grave objections to the 
standard quartz now generally used, especially on account of its high per- 
centage of voids, the difficulty of compacting in the moulds, and its lack 
of uniformity. It recommends, for the present, the natural sand from 
Ottawa, 111., screened to pass a sieve having 20 meshes per lin. in., and re- 

* Prepared by the Committee on Standard Specifications for Cements 
as a temporary expedient. 




410 22.— BUILDING STONES AND CEMENTS. 

tained on a sieve having 30 meshes per lin, in.; the wires to have diameters 
of 0,0165 and 0.0112 in., respectively, i.e., half the width of the opening 
in each case. Sand having passed the No. 20 sieve shall be considered 
standard when not more than 1% passes a No. 30 sieve after one minute 
continuous sifting of a 500-gram sample. The Sandusky Portland Cement 
Co., of Sandusky, O., has agreed to undertake the preparation of this sand, 
and to furnish it at a price only sufficient to cover the actual cost of prep- 
aration. 

Form of Briquette. — While the form of the 
briquette recommended by a former Commnttee of 
the Society is not wholly satisfactory, this Com- 
mittee is not prepared to suggest any change, 
other than rounding off the corners by curves of 
J-in. radius, Fig. 3. 

Moulds. — The moulds should be made of brass, 
bronze, or some equally non-corrodible material, 
having sufficient metal in the sides to prevent 
spreading during moulding. Gang moulds, as 
shown in Fig. 4, are preferred to single moulds, 
since the greater quantity of mortar that can be 
mixed for simultaneous moulding tends to pro- 
duce greater uniformity in the results. They 
should be wiped out with an oily cloth before 
using. 

Mixing. — All proportions should be stated by Fig- 3. 

weight; the quantity of water to be used should 

be stated as a percentage of the dry material. The n a 

metric system is recommended because of the vu^^^XT'^-^^ m 

convenient relation of the gram and the cubic ^^ ^^--->^<^---J^-^-J^<^~-jp ^xs^ 
centimeter. The temperature of the room and the 
mixing water should be as near 70° F. as it is _. 

practicable to maintain it. -^^S* ^• 

The sand and cement should be thoroughly mixed dry, and on some 
non-absorbing surface, preferably plate glass. If an absorbing surface is 
used it should previously be dampened. The quantity of material to be 
mixed at one time depends on the number of test pieces to be made; about 
1000 gr. (35.28 oz.) makes a convenient quantity to be mixed, especially 
by hand methods. The material is weighed and placed on the mixing table, 
and a crater formed in the center, into which the proper percentage of clean 
water is poured ; the material on the outer edge is turned into the crater by 
the aid of a trowel. As soon as the water has been absorbed, which should 
not require more than one minute, the operation is completed by vigorously 
kneading with the hands for an additional 1| minutes, the process being 
similar to that used in kneading dough. A sand-glass affords a convenient 
guide for the time of kneading. During the operation of mixing, the 
hands should be protected by gloves, preferably of rubber. 

Moulding. — Having worked the paste or mortar to the proper con- 
sistency it is at once placed in the moulds by hand, being pressed in firmly 
with the fingers and smoothed off with a trowel without ramming. It should 
be heaped up on the upper surface of the mould, and, in smoothing off, 
the trowel should be drawn over the mould in such a manner as to exert 
a moderate pressure on the excess material. The mould should be turned 
over and the operation repeated. A check upon the uniformity of the mix- 
ing and moulding is afforded by weighing the briquettes just prior to im- 
mersion, or upon removal from the moist closet. Those varying in weight 
more than 3% from the average should be rejected. 

Storage of the Test Pieces. — During the first 24 hours after moulding, 
the test pieces should be kept in moist air to prevent them from drying out. 
A moist closet or chamber is so easily devised that the use of the damp 
cloth should be abandoned if possible. Covering the test pieces with a 
damp cloth is objectionable, as commonly used, because the cloth may dry 
out unequally, and, in consequence, the test pieces are not all maintained 
under the same conditions. Where a moist closet is not available, a cloth 
may be used and kept uniformly wet by immersing the ends in water, and 
being kept from direct contact with the test pieces by means of a wire 
screen or some similar arrangement. 




• CEMENT— TESTING, SPECIFICATIONS. 411 

A moist closet consists of a soapstone or slate box, or a metal lined 
wooden box — the metal lining being covered with felt and this felt kept 
wet. The bottom of the box is so constructed as to hold water, and the 
sides are provided with cleats for holding glass shelves on which to place 
the briquettes. Care should be taken to keep the air in the closet uniformly 
moist. . . 

After 24 hours in moist air, the test pieces for longer periods of time 
should be immersed in water maintained as near 70° F. as practicable; 
they may be stored in tanks or pans, which should be of non-corrodible 
material. 

Tensile Strength. — ^The tests may be made on any standard machine. 
A solid metal clip, as shown in Fig. 5, is recommended. 
This cHp is to be used without cushioning at the points of 
contact with the test specimen. The bearing at each point 
of contact should be i in. wide, and the distance between 
the centers of contact on the same clip should be H ins. 

Test pieces should be broken as soon as they are re- 
moved from the water. Care should be observed in cen- 
tering the briquettes in the testing machine, as cross- 
stains, produced by improper centering, tend to lower 
the breaking strength. The load should not be applied 
too suddenly, as it may produce vibration, the shock from 
which often breaks the briquette before the ultimate 
strength is reached. Care must be taken that the clips and 
the sides of the briquette be clean and free from grains Fig. 5. 

of sand or dirt, which would prevent a good bearing. The load should be 
applied at the rate of 600 lbs. per minute. The average of the briquettes 
of each sample tested should be taken as the test, 'excluding any results 
which are manifestly faulty. 

Constancy of Volume. — ^The object is to develop those qualities which 
tend to destroy the strength and durability of a cement. As it is highly 
essential to determine such qualities at once, tests of this character are for 
the most part made in a very short time, and are known, therefore, as 
accelerated tests. Failure is revealed by. cracking, checking, swelling or 
disintegration, or all of these phenomena. A cement which remains per- 
fectly sound is said to be of constant volume. 

Tests for constancy of volume are divided into two classes: (1) normal 
tests, or those made in either air or water maintained at about 70° F , and (2) 
accelerated tests, or those made in air, steam or water at a temperature of 
115° F. and upward. The test pieces should be allowed to remain 24 hours 
in moist air before immersion in water or steam, or preservation in air. 

For these tests, pats, about 7^ cm. (2.95 ins.) in dia., \\ cm. (0.49- in.) 
thick at the center, and tapering to a thin edge, should be made, upon a 
clean glass plate [about 10 cm. (3.94 ins.) square], from cement paste of 
normal consistency. 

Normal Test. — ^A pat is immersed in water maintained as near 70° F. 
as possible for 28 days, and observed at intervals. A similar pat is main- 
tained in air at ordinary temperature and observed at intervals. 

Accelerated Test. — A pat is exposed in any convenient way in an atmos- 
phere of steam, above boiling water, in a loosely closed vessel, for 3 hours. 

To pass these tests satisfactorily, the pats should remain firm and 
hard, and show no signs of cracking, distortion or disintegration. Should 
the pat leave the plate, distortion may be detected best with a straight-edge 
applied to the surface which was in contact with the plate. In the present 
state of our knowledge it cannot be said that cement should necessarilj'' be 
condemned simply for failure to pass the accelerated tests; nor can a cement 
be considered entirely satisfactory simply because it has passed these tests. 

Specifications for Cement (A. S. T. M.). 

The following specifications were adopted by the American Society 
for Testing Materials. Nov. 14. 1904: 

General Conditions. — (1) All cement shall be inspected. (2) Cement 
may be inspected either at the place of manufacture or on the work. (3) 
In order to allow ample time for inspecting and testing, the cement should 
be stored in a suitable weather-tight building having the floor properly 
blocked or raised from the ground. (4) The cement shall be stored in such 



412 22.— BUILDING STONES AND CEMENTS. ' 

a manner as to permit easy access for proper Inspection and Identification 
of each shipment. (5) Every facility shall be provided by the contractor 
and a period of at least twelve days allowed for the inspection and necessary 
tests. (6) Cement shall be delivered in suitable packages with the brand 
and name of manufacture plainly marked thereon (7) A bag of cement 
shall contain 94 pounds of cement net. Each barrel of Portland cement 
shall contain 4 bags, and each barrel of natural cement shall contain 3 bags 
of the above net weight. (8) Cement failing to meet the 7-day require- 
ments may be held awaiting the results of the 28-day tests before rejection. 
(9) All^ tests shall be made in accordance with the methods proposed by the 
Committee on Uniform Tests of Cement of the American Society of Civil 
Engineers, presented to the Society January 21, 1903, and amended January 
20, 1904, with all subsequent amendments thereto. [See " Method of 
Testing Cement," page 407.] 

(10) The acceptance or rejection shall be based on the following re- 
quirements: 

(11) Natural Cemento — Definition. — ^This term shall be applied to the 
finely pulverized product resulting from the calcination of an argillaceous 
limestone at a temperature only sufficient to drive off the carbonic acid gas. 

(12) Specific Gravity. — ^The specific gravity of the cement thoroughly 
dried at 100° C, shall not be less than 2.8. 

(13) Fineness. — It shall leave by weight a residue of not more than 
10% on the No. 100, and 30% on the No. 200 sieve. 

(14) Time of Setting. — It shall develop initial set in not less than 10 
minutes, and hard set in not less than 30 minutes nor more than 3 hours. 

(15) Tensile Strength. — ^The minimum requirements for tensile strength 
for briquettes one inch in cross section shall be within the following limits, 
and shall show no retrogression in strength within the periods specified:* 

Age. Neat Cement. Strength. 

24 hours In moist air 50-100 lbs. 

7 days (1 day in moist air, 6 days in water) 100-200 " 

28 days (1 day in moist air, 27 days in water) 200-300 " 

One Part Cement, Three Parts Standard Sand. 

7 days (1 day in moist air, 6 days in water) 25- 75 " 

28 days (1 day in moist air, 27 days in water) 75-150 " 

(16) Constancy of Volume. — Pats of the neat cement about 3 ins. in 
diameter, |-in. thick at center, tapering to a. thin edge, shall be kept in 
moist air for a period of 24 hours, (a) A pat is then kept in air at normal 
temperature, (b) Another is kept in water maintained as near 70° F. as 
practicable. 

(17) These pats are observed at intervals for at least 28 days, and, to 
satisfactorily pass the test, should remain firm and hard and show no signs 
of distortion, checking, cracking or disintegrating. 

(18) Portland Cement. — Definition. — ^Thls term is applied to the finely 
pulverized product resulting from the calcination to incipient fusion of an 
intimate mixture of properly proportioned argillaceous and calcareous 
materials, and to which no addition greater than 3% has been made sub- 
sequent to calcination. 

(19) Specific Gravity. — ^The specific gravity of the cement, thoroughly 
dried at 100° C, shall not be less than 3.10. 

(20) Fineness. — It shall leave by weight a residue of not more than 
8% on the No. 100, and not more than 25% on the No. 200 sieve. 

(21) Time of Setting. — It shall develop initial set In not less than 30 
minutes, but must develop hard set in not less than one hour nor more 
than ten hours. 



* For example, the minimum requirement for the 24-hour neat cement 
test should be some specified value within the limits of 50 and 100 lbs., and 
so on for each period stated. [The consumer, when ordering, may fix the 
min. value.] 



CEMENT— SPECIFICATIONS, 413 

(22) Tensile Sttength. — ^The minimum requirements for tensile strength 
for briquettes one inch square in section shall be within the following limits, 
and shall show no retrogression in strength within the periods specified;* 

Age. Neat Cement. Strength. 

24 hours in moist air 150-200 lbs. 

7 days (1 day in moist air, 6 days in water) 450-550 ** 

28 days (1 day in moist air, 27 days in water) 550-650 " 

One Part Cement, Three Parts Sand. 

7 days (1 day in moist air, 6 days in water) 150-200 ** 

28 days (1 day in moist air, 27 days in water) 200-300 " 

(23) Constancy of Volume. — Pats of neat cement about 3 ins. in diam- 
eter, i-in. thick at the center, and tapering to a thin edge, shall be kept in 
moist air for a period of 24 hours, (a) -A pat is then kept in air at normal 
temperature and observed at intervals for at least 28 days, (b) Another 
pat is kept in water maintained as near 70° F. as practicable, and observed 
at intervals of at least 28 days, (c) A third pat is exposed in any con- 
venient way in an atmosphere of steam, above boiling water, in a loosely 
closed vessel for five hours. 

(24) These pats, to satisfactorily pass the requirements, shall remain 
firm and hard and show no signs of distortion, checking, cracking or dis- 
integrating. 

(25) Sulphuric Acid and Magnesia. — ^The cement shall not contain 
more than 1.75% of anhydrous sulphuric acid (5O3), nor more than 4% 
of magnesia (MgO). 

Specifications for Cement (Engrs. U. S. A.). 

The following specificationsf are from Professional Papers, No. 28, 
Corps of Engineers, U. S. A. 

(a) American Portland Cement. — Shall be dry and free from lumps; 
the calcined product to contain at least 1.7 times as much lime, by weight, 
as of the materials which give the lime its hydraulic properties; and to be 
finely pulverized after said calcination, with subsequent additions or sub- 
stitutions for regulating certain properties of technical importance not to 
exceed 2% of the calcined product. 

The cement to be put up in strong, sound barrels, well lined with paper, 
or in stout cloth or canvas sacks; each package to be plainly labeled with 
name of brand and of manufacturer. Bidders will state the brand which 
they propose to furnish. The average weight per barrel shall not be less 
than 375 lbs. net. Four sacks shall contain one barrel of cement. 

Tests may be made of the fineness, specific gravity, soundness, time of 
setting, and tensile strength of the cement. 

(7) Fineness. — Ninety-two per cent of the cement must pass through 
a sieve made of No. 40 wire, Stubb's gauge, having 10,000 openings per 
square inch. 

(8) Specific Gravity. — ^The specific gravity of the cement, as determined 
from a sample which has been carefully dried, shall be between 3.10 and 
3.25. 

(9) Soundness. — To test the soundness of the cement, at least two pats 
of neat cement, as taken from the package, mixed for five minutes with 
about 20 per cent of water by weight shall be made on glass, each pat about 
3 inches in diameter and one-half inch thick at the center, tapering thence 
to a thin edge. The pats are to be kept under a wet cloth until finally set, 
when one is to be placed in fresh water for twenty-eight days. The second 
pat will be placed in water which will be raised to the boiling point for six 

* For example, the minimum requirement for the 24-hour neat cement 
test should be some specified value within the lim.its of 150 and 200 lbs., 
and so on for each period stated. [The consumer, when ordering, may fix 
the minimum values.] 

tA digest from the Report of Majors^ W. L. Marshall and S. S. Leach, 
and Capt. Spencer Crosby, Board of Engineer Officers, on testing Hydraulic 
Cements, with specifications for the several classes; Second Edition, with 
Corrections, 1902. 



414 22.—BVILDING STONES AND CEMENTS. J 

hours, then allowed to cool. Neither should show distortion or cracks. 
The boiling test may or may not reject at the option of the engineer officer 
in charge. 

(10) Time of Setting. — ^The cement shall not acquire its initial set in 
less than forty-five minutes and must have acquired its final set in ten 
hours. 

The following paragraph will be substituted for the above in case a 

quick-setting cement is desired: 

The cement shall not acquire its initial set in less than twenty nor more 
than thirty minutes, and must have acquired its final set in not less than 
forty-five minutes nor in more than two and one-half hours.) 

The pats made to test the soundness may be used in determining the 
time of setting. ^ The cement is considered tohave acquired its initial set 
when the pat will bear, without being appreciably indented, a wire one- 
twelfth inch in diameter loaded to weigh one-fourth pound. The final 
set has been acquired when the pat will bear, without being appreciably 
indented, a wire one twenty-fourth inch in diameter loaded to weigh 1 
pound. 

(11) Tensile Strength. — Briquettes made of neat cement, after being 
kept in air for twenty-four hours under a wet cloth and the balance of the 
time in water, shall develop tensile strength per sq, in. as follows: 7 days, 
450 lbs; 28 days, 540 lbs. Briquettes made of 1 part cement and 3 parts 
standard sand, by weight, shall develop tensile strength per sq. in. as follows: 
7 days, 140 lbs; 28 dayc, 220 lbs. (In case quick-setting cement is desired, 
the following tensile strengths shall be substituted for the above: Neat 
briquettes: 7 days, 400 lbs,: 28 days, 480 lbs. Briquettes of 1 part cement 
to 3 parts standard sand: 7 days, 120 lbs.; 28 days, 180 lbs.) 

The highest result from each set of briquettes made at any one time, is 
to be considered the governing test. Any cement not showing an increase 
of strength in the 28-day tests over the 7-day tests will be rejected. 

(b) Natural Cement. — The average net weight per barrel shall not be 
less than 300 lbs. (West of the Allegheny Mountains this may be 265 lbs.) 

(7) Fineness. — At least 80 per cent of the cement must pass through 
a sieve made of No. 40 wire, Stubb's gauge, having 10,000 openings per 
square inch. 

(8) Time of Setting. — ^The cement shall not acquire its initial set in less 
than twenty minutes and must have acquired its final set in four hours. 

(9) The time of setting is to be determined from a pat of neat cement 
mixed for five minutes with 30 per cent of water by weight and kept under 
a wet cloth until finally set. The cement is considered to have acquired 
its initial set when the pat will bear, without being appreciably indented, 
a wire one-twelfth inch in diameter loaded to weigh one-fourth pound. 
The final set has been acquired when the pat will bear, without being appre- 
ciably indented, a wire one twenty-fourth inch in diameter loaded to weigh 
1 pound. 

(10) Tensile Strength. — Briquettes made of neat cement shall develop 
the following tensile strengths per square inch, after having been kept in 
air for twenty-four hours under a wet cloth and the balance of the time in 
water: at the end of 7 days, 90 lbs.; 28 days, 200 lbs. Briquettes made of 
1 part cement and 1 p^rt standard sand, by weight, shall develop the follow- 
ing tensile strengths pr sq. in.: 7 days, 60 lbs.; 28 days, 150 lbs. 

(c) Puzzolan Cement. — The average weight per barrel shall not be less 
than 330 lbs. net. Four sacks shall contain one barrel of cement. 

(7) Fineness. — Ninety-seven per cent of the cement must pass through 
a sieve made of No. 40 wire, Stubb's gauge, having 10,000 openings per 
square inch. 

(8) Specific Gravity. — ^The specific gravity of the cement, as determined 
from a sample which has been carefully dried, shall be between 2 . 7 and 2 . 8. 

(9) Soundness. — ^To test the soundness of cement, pats of neat cement 
mixed for five minutes with 18 per cent, of water by weight shall be made 
on glass, each pat about 3 inches in diameter and one-half inch thick at the 
center, tapering thence to a thin edge. The pats are to be kept under wet 
cloths until finally set, when they are to be placed in fresh water. They 
should not show distortion or cracks at the end of twenty-eight days. 



ARTIFICIAL BUILDING STONES. 415 

(10) Time of Setting. — ^The cement shall not acquire its initial set in less 
than forty-five minutes and shall acquire its final set in ten hours. 
The pats made to test the soundness may be used in determining the time 
of setting. The cement is considered to have acquired its initial set when 
the pat will bear, without being appreciably indented, a wire one-twelfth 
inch in diameter loaded to one-fourth pound weight. The final set has been 
acquired when the pat will bear, without being appreciably indented, a 
wire one twenty-fourth inch in diameter loaded to 1 pound weight. 

(11) Tensile Strength. — Briquettes made of neat cement, after being 
kept in air under a wet cloth for twenty-four hours and the balance of the 
time in water, shall develop tensile strengths per square inch as follows: 
After 7 days, 350 lbs.; 28 days, 500 lbs. Briquettes made of 1 part cement 
and 3 parts standard sand, by weight, shall develop tensile strength per sq. 
in. as follows: 7 days, 140 lbs., 28 days, 220 lbs. 

III. ARTIFICIAL BUILDING STONES. 

The following classification, which includes brick, is convenient in the 
present discussion: 

(1) Brick, usually an argillaceous material, molded to the required form, 
and baked. 

(2) Concrete, consisting of a matrix of cement and sand, properly mixed 
with a stone aggregate, in situ, 

(3) Block stone, a hydraulic cement concrete specially treated and formed 
in moulds at the factory. 

(1) Brick. 

A good brick clay consists of a fairly pure hydrated silicate of alumina, 
but contains also more or less so-called impurities as iron, manganese, free 
silica (sand), potash^ lime, etc. In making bricks the clay is well kneaded, 
molded, dried, and then burned in kilns. 

Common brick is designated as " clinker," " hard " or " soft," denoting 
the degree of burning in the kiln. Clinker bricks have been overburned 
and usually have a blueish cast; they are partly vitrified, and brittle. Hard 
bricks are the best burned, red brick. Soft bricks are those which are 
under-burned, and are paler than the hard. In addition to the amount 
of burning, the color of brick is much affected by the quantity of iron or 
lime in the clay; the presence of iron peroxide produces a red brick while 
lime has a tendency to make it yellow. The common standard of brick 
in the United States is 8F'x4''x2r; in England, 8rx4rx2r. 

Face brick is made from one of more specially selected clays, mixed if 
necessary to give the desired chemical properties and color. The bricks 
are pressed in the molding machine, hence the name " pressed brick." 
They are also sometimes re-pressed. The standard size in the United States 
is 8rx4''x2i''. 

Glazed brick is a brick coated with enarnel, a fusible salt. A common 
liquid glass for glazing is composed of silicic acid (23.2 per cent.), soda 
(8.9), potash (2.5), water (65.4). The glazing may be in various colors 
and consequently pretty patterns may be obtained. Glazed bricks are 
used in buildings and subways; they are hygienic as well as ornamental. 

Vitrified Jbrick. — Vitrification is obtained from the thorough burning 
of a highly siliceous clay or of a clay mixed with a large amount of siliceous 
sand. 

Terra Cotta, meaning baked earth, is made of a specially graded clay or 
clays mixed with a certain amount of siliceous sand, if necessary, to secure 
the desired amount of vitrification. After molding and drying, the mix- 
ture is baked. The surface of terra cotta may also be glazed like that of 
the common glazed brick and tile. 

Fire brick is used for the lining of furnaces, chimneys, etc. It is made 
from the natural fire-clay of which large beds are found in New Jersey and 
elsewhere. It is a high graded hydrated silicate of alumina, containing 
preferably not over 4 per cent, of impurities. The clay contains a large 

f)roportion of silica (sand), and sometimes more' is added to insure its re- 
ractory quality. The manufacture is similar to that of common brick, 
but with the fire-clay the product is vitrified. 

Paving brick is not so much vitrified as fire brick. 
Sewer brick is ordinary, hard brick. 



416 22.-'BmLDING STONES AND CEMENTS. 

(2) Concrete (En= masse). 

(See, also, Concrete Masonry, page 439.) 
Concrete* is composed of a matrix possessing cementing properties, 
thoroughly mixed and united with an aggregate or base composed of frag- 
ments of hard, imperishable, mineral substances, the whole forming a solid, 
compact, enduring mass. The matrix is the cement-sand-water mortar, 
while the aggregate is the crushed rock or gravel base. 

Mixture. — The cement and sand are thoroughly mixed dry in the proper 
proportion, enough water being added subsequently to form a paste like 
wet sand. After this matrix has been carefully kneaded, the aggregate, 
which has previously been washed of all dirt and other injurious matter, 
is added, and the whole mass thoroughly mixed. It should be put in 
place immediately and tamped, with the water just flushing the surface. 

An ideal concrete is one in which the entire surface of every grain of 
sand is covered with the mortar paste, with the grains nearly touching each 
other but with no voids; and in which also all interstices of the aggregate 
are completely filled with the matrix, pressing firmly against every surface. 
An unduly large proportion of cement may make the mass unnecessarily 
strong and expensive, while too little cement will render it porous and 
possibly too weak. Where great strength is not essential a cheaper brand 
of cement may be used instead of the most expensive, perhaps used in 
greater proportion, sufficient to fill all^ sand voids, thus rendering the mass 
non-porous at no greater cost. This is an essential consideration in local- 
ities where frosts are liable to work injuriously upon the mass of concrete, 
if porous. As with cement, in regard to voids in the sand, so with the matrix, 
regarding voids in the aggregate. An excess of m.atrix means excessive 
cost, while a deficiency gives a porous concrete. The latter is avoided by 
thorough tamping and proper flushing, and by varying the proportions 
if necessary. 

Proportions. — The proportions, by bulk, in mixing concrete are usually 
based on 1 of cement, 2 to 3 of sand, and 4 to 7 of aggregate, depending 
upon the nature and class of the work as well as the quality and kind of 
materials available. Portland cement is the best, and costs more than the 
other brands. It is usually specified in first class work. In ordinary work 
it often gives better results than the cheaper brands, at no greater cost, 
because it " goes further "; bearing in mind of course that if scantily used 
the matrix will be more or less porous. A greater proportion of sand can 
be used in the matrix if it is fine and siliceous, hence a little money spent 
on sand will sometimes save a much greater sum on cerxient. Lastly, much 
less matrix and consequently less cement is required if the aggregate is 
graded to different sizes, lessening the voids. 

The voids in sand amount, generally, to about 33 or 34% of the gross 
bulk; in gravel, to about the same; in broken sandstone, to about 40 to 42%; 
in broken trap, to about 45 to 50%. Generally, the percentage of voids is 
greater in the freshly broken, hard, flint-like rocks, and less ifl the softer 
rock which leave the crusher with less-defined edges. With gravel, where 
the edges have worn away, the percentage of voids is the least, hence gravel 
is the most economical aggregate to use for concrete because less cement, 
or mortar, is required. For the same reason, basalt or trap are the most 
expensive, although far superior for many purposes. 

Economy in cement may be practiced by grading the size of materials 
as previoUvSly noted; thus, a coarse and a fine sand may be used together 
in forming the matrix, while the aggregate may consist of graded gravel, 
or gravel and broken stone. 



* Before hydraulic cement became recognized as an essential ingre- 
dient in concrete, lime of greater or less hydraulic properties was used, 
sometimes mixed with hydraulic cement or asphalt, etc. Thus, 

Kind of concrete. Matrix. Aggregate. 

Hydraulic cement c. Hydraulic cement mortar. 1 Broken stone, 

lime " " lime " 1 crushed brick, slag. 

Asphalt cement " Asphalt cement " [cinders, pebbles, 

J gravel, etc. 

" Concrete " is now universally recognized as hydraulic cement concrete. 



CONCRETE, BLOCK STONE, 417 

Cement^Sand Mix. — ^To get the proper proportion of cement to sand 
in the matrix: Determine the percentage of voids in the sand by noting 
the quantity c of water required to " flush " to the level in a water-tight 
box filled with an s quantity of the sand to be used; then the proportion 
of cement to sand should be about 1.1 c '. s. It is to be noted that the 
amount of cement required is thus about 10% in excess of the voids in the 
sand, which is as it should be, as the sand grains will.be coated with the 
cement and separated somewhat from each other. For example, using sand 
with 34% voids, the ratio of cement to sand would be about 1 : 2^. In 
practice this ratio is sometimes increased to 1 : 2, 1 : 1^, or 1 : 1, depending 
upon the character and importance of the work, and also upon the character 
and quality of the materials. A 1 : 2 mix is generally considered good 
practice for a cement mortar in first class masonry work; while a 1 : 1 mix 
makes an excellent wearing surface when used as a finishing coat for concrete 
walks. 

Cement=Sand=Broken=stone Mix. — ^To get the proper proportion of 
matrix to aggregate'. Determine the percentage of voids in the aggregate or 
broken stone by noting the quantity m of water required to " flush " to 
the level in a water-tight box filled with an a quantity of broken stone; 
then the proportion of matrix to aggregate should be about 1.1 w : a. 
For example, with 45% voids in the aggregate, the ratio of mortar to broken 
stone should be about 1:2. 

Fiu-ther, if we assume the sand voids at 34%, as above, the cement- 
sand-broken stone ratio would be about 1 : 2^ : 5. Note that the quantity 
of sand in the matrix, slightly increased, really determines the bulk of the 
rnatrix itself in estimating the above ratip. In practice, this ratio is some" 
times increased to 1:2:4 mix for first class building walls, piers and 
buttresses; to 1 : H : 3 mix for columns and heavy beams of concrete- 
steel ; and to 1 : 1 : 2 mix for lighter reinforced members. 

Size of Broken Stone. — The size to which the stone must be crushed 
or broken will be governed somewhat by the size of the concrete mass. 
For large bridge piers, stone which will pass through a 2^-in. ring in any 
way is often allowed. The most common specification, however, for such 
foundations stipulates the 2-in. ring. For certain classes of work both 
an upper and a lower limit are specified; as for instance, rock that will pass 
through a 2Hn. but not through a 1-in. ring, called the 2Y — 1" grade; 
similarly, we may have a 2" — i" grade, a If" — Y grade, etc. Much refine- 
ment in this respect is, however, unnecessarily expensive, excepting perhaps 
in building construction and notably in concrete -steel work. For the latter, 
sizes as small as the pea-grade size are used for reinforced floors. 

. (3) Block Stone. 

By this is meant the commonly named ** artificial stone " proper. It 
is manufactured usually at the factory and shipped to the site for erection. 
Among the most important kinds are the following: 

Beton=Coignet. — ^This was invented by Coignet , a Frenchman, and 
consisted of Portland cement, siliceous hydraulic cement, and clean sharp 
sand, thoroughly mixed with a small quantity of water, and placed in molds 
to set, for use. 

Sand bricks are manufactured from lime and sand mixed with water 
to the consistency of a mortar. This mortar is molded into bricks or blocks 
and then hardened by heating. 

Portland stone is composed of a mixture of Portland cement with water 
and sand or gravel, placed in molds and thoroughly rammed for setting. 
It is made plain and also ornamental. 

McMurtrie stone is made of Portland cement concrete to which is added 
a solution of castile soap and s, solution of alum, forming compounds of 
alumina potash and certain acids, which tend to " set " the concrete quickly. 
It is not attacked by the weather. 

Ransome stone.— -The aggregate consists of broken granite, gravel or 
sand ', the matrix, a silicate of soda and sand immersed in a hot solution of 
chloride of calcium. It makes a very good building stone. 

Sorel stone. — ^The matrix consists of a solution of magnesium chloride 
added to the oxide of magnesium. The aggregate is a finely crushed or 
powdered stone of good quality. It is extremely hard and is used for 
imitation marble, emery wheels, etc. 



418 22.— BUILDING STONES AND CEMENTS. 

EXCERPTS AND REFERENCES. 

Lutes and Cements Useful to Engineers (By S. S. Sadtler. Paper, 
Phila. Engrs. Club. June 4, 1904; Eng. News, Jan. 5, 1905).— Water=Proof 
Compositions. — Asphalt fluid coatings are useful for reservoir walls, con- 
crete foundations, brick, wood, etc. Asphalt only partly dissolves in 
petroleum naphtha, but heated in a steam- jacketed kettle and not thinned 
out too much, a mixture of the two may be obtained in which the part of 
the asphalt not dissolved is held in suspension. Asphalt is entirely soluble 
in benzol or tuluol which are about the cheapest solvents for all the con- 
stituents of asphalt. Tar and pitch are sometimes used in this connection, 
but tar contains water, light oils, and free carbon, and does not wear as 
well as good refined asphalt; ^ and pitch contains free carbon, which is some- 
times objectionable when thinned out with a solvent. The asphalt alone is 
somewhat pervious to water, and this is improved by adding about one- 
fourth its weight of paraffin, and made better if in addition a little boiled 
linseed oil is added also. For thicker compositions, where body is required, 
asbestos stone powder, cement, etc., may be added as fillers. Lutes of 
linseed oil thickened with clay, asbestos, red or white lead, etc., are water- 
proof if made thick enough. These are much used for steam joints. Flax- 
seed meal made into a paste with water is often serviceable, the oil con- 
tained serving as a binder as the water evaporates. Oil=Proof Composi' 
tions.— The most useful lute for small leaks, etc., is the well-known "hekta 
graph composition," as follows: Good glue or gelatin (2 parts), glycerine (1), 
water (7); this is applied warm and stiffens quickly on cooling. Another 
composition: a stiff paste of molasses and flour. Another preparation 
impervious to oil vapors is the "flaxseed poultice," mentioned above, which 
is proof to oil vapors. A stiff paste of glycerine and litharge makes one of 
the strongest cements (oil-proof, water-proof, acid-proof, etc.), forming a 
chemical combination and setting in a few minutes; if a little water is 
added it sets more slowly, often an advantage; it is mixed when required 
for use. Plaster of Paris wetted, by itself, or mixed with asbestos, straw, 
hair, etc., is useful. A solution of silicate of soda made into a stiff paste 
with carbonate of lime, hardens in 6 to 8 hours. Acid=Proof Compositions. — 
Several kinds given. Other Proof Compositions. — For hydrocarbon gases; 
chlorine; general purposes. Elastic cements; marine glue; gasket com- 
positions; leather cements, stone, iron and crucible cements; etc. 

Cost of Portland Cement. — Portland cement, in and around New York, 
costs from $1.10 to $1.25 per barrel, depending upon the quality and quan- 
tity required. The price varies greatly according to locality. 

A Cement Which is Proof Against Sea=Water (By S. B. Newberry. 
Cement Age, Jan., 1907; Eng. News, Dec. 12, 1907). — This cement is an 
iron-ore cement and has been manufactured for several years in Germany. 
Its composition, as compared with an average commercial Portland is as 
follows: 



Portland Ore-Port. 
Cement. Cement. 
Silica 22.0 20.5 

Alumina 8.0 1.5 

IronOxid 2.5 11.0 

Lime 63.5 63.5 



Port. Ore- 

Cem. Port. 

Magnesia . . 1.5 1.5 

Sulph. Anhyd. 1.5 1.0 

Alkalies, etc. 1.0 1.0 



100.0 100.0 



Important Illustrations — 

Description. Eng. Rec. 

A briquette storage tank, St. Loixis testing laboratory Jul. 24, '09. 



23.— QUARRYING. 



When a quarry, of whatever nature, is proposed to be opened, the site 
should be studied carefully with two main points in view, namely, (1) the 
most economic methods of mining the material, and (2) the best and cheap- 
est means of transporting the product when mined. It is well to remember 
that of the two methods of transportation — water and rail — the former is 
the great " leveler " in the equalization or adjustment of freight rates. 
Consideration must also be had to water-supply and drainage, whether the 
machinery is operated by electric- or steam power. On opening the quarry, 
the surface stripping, consisting of organic and decomposed mineral matter, 
should be removed ahead of the quarrying. The cost of stripping and 
removing the weathered rock is usually a very considerable item, and in 
new quarries opened for local work this cost must be proportioned to the 
total yardage required. 

Sand and Gravel. — An ideal sand and gravel plant is one in which the 
bank is sufficiently elevated to allow of hydraulic operations. The stream 
or streams of water may be furnished by gravity or by pumping. The 
gravel and sand are sluiced into revolving screens of two or more sizes 
which sort the material, and from which it passes into the bunkers, ready 
to be loaded on scows or cars. The revolving screens are not very durable 
and probably the best are common steel plate, perforated. The holes are 
punched the required size for sorting out the sand and gravel. The material 
as it passes into the bunkers is thoroughly washed and hence clean and ready 
for use. 

The C. O. Bartlett and Snow Co., of Cleveland, Ohio, manufacture a 
movable combined gravel digging and screening plant, operated on a track. 
The particular machine illustrated in Eng. News, May 12, 1904, weighs 25 
tons, is equipped with a 25-HP. engine and boiler, and has a capacity of 
30 to 40 cu. yds. per hour. " The gravel and sand are discharged from 
the elevator buckets into a belt conveyor, which delivers to a rotary screen. 
The screen discharges into bins, from which the material is drawn off into 
gondola cars or wagons." The fine dirt, or dust, is rejected by a belt con- 
veyor to a waste dump. The excavating buckets have a capacity of about 
three-fourths of a cubic yard. 

Rip=rap. — This class includes all rock which may be irregular in size 
and shape, and especially used for " filling " in breakwater construction, 
rock-fill dams, slope walls, paving, and crushing into broken stone for 
concrete, etc. The idea in blasting this material is thoroughly to loosen 
as much of it as possible for the labor of drilling and cost of explosives, 
hence the blasts are usually heavy and deep seated. Dynamite high in 
nitroglycerin, say 75%, is best. 

Block Stone. — Building stone is quarried in sizes considerably larger 
than the finished dimensions required, in order to allow for necessary waste 
in squaring and dressing. There are four methods employed, each one 
of which is economically suited to the special nature of the quarry, the 
material, and the product desired. 

Hand Tools are employed where the stone is bedded in thin layers and 
can be worked with pick, sledge and bar, and by hand-drilling and wedging. 
Flagstones, for sidewalks, are easily hand-quarried, being a thin-bedded 
sandstone; also some limestones, shales, etc. 

Channeling Machines are used in quarrying blocks of sandstone, lime- 
stone and marble. In many quarries the rock is bedded in various com- 
mercial thicknesses, the channels are cut to the required depth, and the 
blocks are wedged off in dimensions as ordered. 

The first method used was that of " broach channeling " which consists 
of drilling rows of holes spaced about an inch apart, and then breaking 
down these spaces by means of a tool called a " broach." 

419 



420 



2Z— QUARRYING. 



The " Steel Gang " channeler consists of a gang of tools or chisels ar- 
ranged side by side and forming a narrow cutter several inches in length. 
Fig. 1 shows a " swivel head " channeler, size Z, with boiler, manufactured 
by the Sullivan Machinery Co., of Chicago. This company also manu- 
factures the " rigid-head " channeler, used for vertical cutting, and the 
" imdercutting " channeler, used for horizontal cutting; also various special 
types. 




Fig. 1. 



The many recent improvements in channeling machines have ren- 
dered them valuable not only in quarrying dimension stone but also 
in certain classes of rock excavation, notably in the excavation of chan- 
nels or waterways through solid rock, sinking of large pits or shafts, 
etc., for water power, and in general where the sides of the finished work 
are required to be fairly smooth or regular. 

The Z machine, together with all the Sullivan channelers of other types, 
may be operated by compressed air as well as steam. Air is preferable 
under conditions necessitating the use of a number of machines at points 
distant from the central power plant and from each other. In this case a 
suitable reheater is mounted on the machine to secure the utmost efficiency. 
The Z channeler used in constructing the wheel pits at Niagara Falls and 
the great canal of the Lake Superior Power Co., at Sault Ste. Marie, were 
driven by air as above described. 

The approximate amount of air at 80 pounds pressure necessary to 
operate the Y and Z channelers is, without reheating, about 350 feet per 
minute. The size 6^ machine will use about 300 feet under the same con- 
ditions, while the VX channeler will use approximately 200 feet. 



CHANNELING MACHINES. 



421 









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422 



2^-OUARRYlNG. 



Explosives are employed principally in quarrying the harder rocks, 
as granite, syenite, trap, etc.; but they are used extensively also in quarry- 
ing sandstone, limestone and marble, in the smaller quarries where channeling 
machines have not been introduced. In blasting, the explosive must not 
be so violent as to shatter the detached portions too much, hence a coarse 
gunpowder is _ mostly preferred instead of the instantaneous dynamite. 
Where dynamite is used it is generally of the lower grades, containing say 
30 to 40% nitroglycerin for ordinary work, unless large masses of very 
heavy rock, like granite, are to be opened up, in which case 60% dynamite 
or even 75% dynamite is sometimes used. The liquid nitroglycerin so 
useful in submarine blasting is seldom employed in quarrying, being too 
violent, as well as dangerous to handle. Large blocks, detached from the 
quarry by blasting, are subsequently broken up into the required dimension, 
by drilling holes in line and wedging, that is, by plug and feather. 

Rock Drills used in quarrying may be classed under three heads, as follows: 

The Hammer- or " Jumper " Drill, which includes the smaller sizes such 
as may be operated by one man who also wields the hammer, weighing 
say 4 J lbs.; and also the larger sizes or jumpers proper, which are operated 
by one man turning and with two men striking (4i-lb. and 10-lb. hammers 
for two hand drilling, and two 10-lb. hammers for three-hand drilling). 

The '' Churn^' Drill, a heavy drill 6 to 8 ft. in length, operated usually 
by two or three, sometimes six, men who raise the drill, let it fall into the 
hole, catch it on the rebound, etc. — the most economical hand drill for 
moderately deep holes, when nearly vertical. 




Fig. 2. 

The " Percussion " Drill, which is simply a chum drill attached to the 
piston of a steam- or compressed air cylinder, and by far the most effective 
drill for quarrying, mining, tunneling and general rock excavation. Fig 2 
illustrates a percussion drill of the Ingersoll-Rand type mounted on a "quar- 
ry bar," which arrangement is specially useful for drilling " plug and 
feather " holes, for "broaching" in granite and similar materials, for taking 
out "key blocks," for "lofting" work in quarrying material with a well 
defined cleavage, and for general contract work. The " Light 3-inch " 
bars have a length (over all) of 10 ft., a cutting length of 8 ft. -4 ins., are 
suitable for 2-2i in. cyl. dia. of drill, and weigh, with weights but without 
drill, 945 lbs. The " Standard 4^-inch " bars have a length (over all) of 
12 ft., a cutting length of 10 ft., are suitable for 2|-3t in. cyl. dia. of drill, 
and weigh, with weights but without drill, 1625 lbs. The price of the 
Light bars is $175.00 and of the Standard bars $250.00. 

Table 2, page 424, gives dimensions and weights of the Rand "Little 
Giant" percussion drill (Fig. 3). 



ROCK DRILLS. 423 

♦Percussion drills may be mounted on the "tripod," which is a 
most common method; on the " column " or vertical bar; and on the " gad- 
der " or adjustable carriage designed especially for quarry work where 
parallel holes are to be drilled in any plane. " Used in connection with the 
channeler it is applied in ' lofting,' or drilling the horizontal undercutting 
holes in material which has been channeled. Used in ' plug and feather 
work,' it breaks the large blocks cut free by the channelers." (See page 
426.) 

Compressed Air, used directly in splitting granite and creating working 
beds in the quarry, is a recent novelty practised successfully by the North 
Carolina Granite Corporation at Mt. Airy, N. C, and described fully in 
*' Mine and Quarry "f of May, 1905. The quarry consists of a solid mass 
of granite without cleavage planes or beds in any direction and the com- 
pressed air is used to create artificial beds to which to work: 

In the center of the sheet or area to be lifted, a drill hole two or three 
inches in diameter is sunk six or eight feet in depth, depending on the 
greatest thickness of stone required. The bottom of the hole is enlarged 
into a pocket by exploding half a stick of dynamite. A small charge of 
powder, about a handful, is then exploded m the pocket, thus starting a 
horizontal crack or cleavage across its greater diameter. Charges increas- 
ing in size are now exploded in the cavity, the drill hole being plugged at 
each blast, to confine the powder gases and thus exert a more or less con- 
stant force upon the stone. After the cleavage has extended to a radius 
of 75 or 100 feet in all directions, a pipe is cemented into the hole and con- 
nected by means of a globe valve, to the air pipe line from an air compressor. 
Compressed air at 70 to 80 pounds pressure is gradually admitted and the 
cleavage rapidly extended until it comes out upon the hillside in a thin 
edge. A sheet of several acres in extent may be raised in this manner, 
affording a bed plane approximately horizontal, to which the quarrymen 
can work, thus securing stone of any required thickness. . . . The 
time [formerly] required for extending the cleavage by powder for 100 feet 
was between two and three weeks, while to split the larger area, between 
100 and 225 feet radius, required only half an hour when compressed air 
was used. . . . Its equipment is modern in every respect, and includes 
35 plugj drills, 3 Sullivan tripod drills, 4 Sullivan quarry bars, 15 surfacing 
machines, and 60 small hand tools. These are all operated by air power 
from a Sullivan Corliss air compressor of the two stage type, with a piston 
displacement of 2000 cubic feet of free air per minute at 78 r.p.m., against 
80 to 100 pounds terminal pressure. The dimensions of the compressor 
are as follows: Steam cylinders, 16 and 28 in. by 42 in. stroke; air cylinders, 
26 and 16 in. by 42-in. stroke. 

Cost of Quarrying Rubble and Dimension Stone for the Buffalo, N. Y., 
Breakwater (By Emile Low. Eng. News, Oct. 20, 1904. — Tabulated costs, 
method of quarrying, and plant used are given. The cost for explosives, 
which includes powder, dynamite and fuses, per ton of stone, all kinds, 
quarried was: For May (1903), 1.3 cts.; July (1903), 2.0 cts.; August (1903), 
2.1 cts. About 1 lb. of powder was used for every 7 tons of stone quarried, 
and 1 lb. of dynamite for every 67 tons of stone. One fuse was used for 
every 5| tons of stone quarried. Total cost of quarrying 1 ton of stone, 
loading and placing same aboard of scows, as follows: Labor, 33 cts.; coal, 
4 cts.; explosives, 2 cts.; miscellaneous, 5 cts.; total, 44 cts. This is exclu- 
sive of cost of plant and deterioration. 



* Pneumatic drills for hand service, like pneumatic riveters, are the 
latest development in rock drilling by machinery. They go by the various 
names of " Little Jap," " Little Imp, " and " Plug " drills, and do remark- 
able work. 

t Published by the Sullivan Machinery Co., Chicago. 

jThe plug drill is a pneumatic power drill for hand service similar to 
other pneumatic hand tools, as the pneumatic riveter, etc. 



424 



23— QUARRYING, 






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PERCUSSION ROCK DRILLS. 



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24.— STONE CUTTING. 



The following is from the report of the American Society's Com- 
mittee* to secure _ uniformity of terms. The matter is re-arranged, 
for convenience, in parallel columns. 



STONES CLASSED ACCORDING 
TO FINISH. 

All stones used in building are 
divided into three classes, accor- 
ding to the finish of the surface, 
viz.: 



I. Rough stones that are used as 
they come from the quarry. 



11. Stones roughly squared and 
dressed. 



III. Stones accurately squared and 
finely dressed. 



In practice, the line of separ- 
ation between them is not very 
distinctly marked, as one class 
gradually merges into the next. 



I. Unsquared Stones. — This 
class covers all stones which are 
used as they come from^ the quarry, 
without other preparation than the 
removal of very acute angles and 
excessive projections from the 
general figure. The term " back- 
ing " which is frequently applied to 
this class of stone, is inappropriate, 
as it properly designates material 
used in a certain relative position 
in a wall, whereas stones of this kind 
may be used in any position. 



TOOLS EMPLOYED. 

The Hand Hammer, weighing 
from 2 to 5 pounds is used in dril- 



Hand Hammer 





Fig. 1. 
ling holes, and in pointing and 
chiseling the harder rocks. 

The Plug, a truncated wedge 
of steel, and the Feathers of half- 



Plug and 

Fig. 2. 

round malleable iron are used for 
splitting unstratified stone. A row 
of holes is made with the Brill on 
the line on which the fracture is 
to be made; in each of these holes 
two feathers are inserted, and the 
plugs lightly driven between them. 
The plugs are then gradually driven 
home by light blows of the hand 
hammer on each in succession until 
the stone splits. 



Drills. 
Fig. 3. 
The Double Face Hammer is a 
heavy tool weighing from 20 to 30 
pounds, used for roughly shaping 



U3> 



Double Face Hammer 



Fig. 4. 

stones as they come from the 
quarry and for knocking off pro- 
jections. This is used only for the 
roughest work. 



* Trans. Am. Soc. C. E., vol. vi., p. 297. 

426 



CLASSIFIED FINISH, 



TOOLS EMPLOYED. 427 



II. Squared Stones. — This class 
covers all stones that are roughly 
squared and roughly dressed on beds 
and joints. The dressing is usually 
done with the face hammer or ax, 
or in soft stones with the tooth-ax. 
The distinction between this class 
and the third lies in the degree of 
closeness of the joints which is de- 
manded. Where the dressing on 
the joints is such that the distance 
between the general planes of the 
surfaces of adjoining stones is one- 
half inch or more, the stones proper- 
ly belong to this class. 



(a) Quarry=faced stones are 
those whose faces are left untouched 
as they come from the quarry. 



(b) Pitch=faced stones are those 
on which the arris is clearly defined 
by a line beyond which the rock is 
cut away by the pitching chisel, 
so as to give edges that are approxi- 
mately true. 



(c) Drafted Stones ase those on 
which the face is surrounded by a 
chisel draft, the space inside the 
draft being left rough. Ordinarily, 
however, this is done only on stones 
in which the cutting of the joints 
is such as to exclude them from 
this class. 

In ordering stones of this class 
the specifications should always 
state the width of the bed and end 
joints which are expected, and also 
how far the surface of the face may 
project beyond the plane of the 
edge. In practice, the projection 
varies between 1 inch and 6 inches. 
It should also be specified whether 
ox not the faces are to be drafted. 



The Face Hammer has one 
blunt and one cutting end, and is 
used for the same purpose as the 




Face Hammer. 



Fig. 5. 

double face hammer where less 
weight is required. The cutting 
end is used for roughly squaring 
stones, preparatory to the use of 
finer tools. 

The Cavil has one blunt and 
one pyramidal, or pointed, end, 
and weighs from 15 to 20 pounds. 





\>. 


<-- 


■■- ir- X 


1 


°J^nvil 



Fig. 6. 

It is used in quarries for roughly 
shaping stone for transportation. 
The Pitching Chisel is usually 
of li-inch octagonal steel, spread 
on the cutting edge to a rectangle of 
ix2| inches. It is used to make a 



Pitching Chisel. • Q\\s^\. 

Fig. 7. Fig. 8. 

well-defined edge to the face of a 
stone, a line being marked on the 
joint surface to which the chisel is 
applied and the portion of the stone 
outside of the line broken off by a 
blow with the hand-hammer on the 
head of the chisel. 

The Chisel, of round steel of 
i to f inch in diameter and about 
1 inches long, with one end brought 
to a cutting edge from \ inch to 2 
inches wide, is used for cutting 
drafts or margins on the face of 
stones. 

The Tooth Chisel is the same as 
the chisel, except that the cutting 
edge is divided into teeth. It is 
used only on marbles and sand- 
stones. 

n 3 ?=^=^ 



Splitting Chlael. 
Fig. 10. 



ToofVVi Chisel 
Fig. 9. 

The Splitting Chisel is used 
chiefly on the softer, stratified 
stones, and sometimes on fine archi- 
tectural carvings in granite. 



42S 



2i.--ST0NE CUTTING, 



III. Cut Stones.— This class 
covers all squared stones with 
smoothly dressed beds and joints. 
As a rule, all the edges of cut stones 
are drafted, and between the drafts 
the stone is smoothly dressed. The 
face, however, is often left rough 
when the constructions are massive. 

Rough=pomted. — When it is 
necessary to remove an inch or 
more from the face of a stone, it is 




Rough-Poin+ed. 

Fig. 12. 
done by the pick_ or heavy point 
until the projections vary from 
i inch to 1 inch. The stone is then 
said to be rough-pointed. In dress- 
ing limestone and granite, this 
operation precedes all others. 

Fine=pointed. — If a smoother 
finish is desired, rough pointing is 
followed by fine pointing. It is 
done with a fine point. Fine 




fine-Poin+ed. 



Fig. 14. 
pointing is used ^ only where the 
finish made by it is to be final, and 
never* as a preparation for a final 
finish by another tool. 



Crandalled. — ^This is only a 
speedy method of pointing, the effect 
being the same as fine pointing, 




Fig. 16. 

except that the dots on the stone 
are more regular. The variations 
of level are about i inch, and the 
rows are made parallel. When 
other rows at right angles to the 
first are introduced, the stone is 
said to be cross-crandalled. 



The Mallet is used [instead of 
the hand hammer, in pointing, 




Moilfit. 
Fig. 11. 
chiseling, etc.] where the softer 
limestones are to be cut. 

The Pick somewhat resembles 
the pick used in digging, and is 
used for rough dressing, mostly on 



Pick 



Fig. 13. 

limestone and sandstone. Its 
length varies from 15 to 24 inches, 
the thickness of the eye being 
about 2 inches. 

The Point is made of round or 
octagonal rods of steel, frorn \ inch 
to 1 inch in diameter. It is made 



Point. 
Fig. 15. 
about 12 inches long with one end 
brought to a point. It is used 
until its length is reduced to about 
5 inches. It is employed for dress- 
ing off the irregular surface of 
stones, either for a permanent fin- 
ish or preparatory to the use of the 
ax. According to the hardness of 
the stone, either the hand-hammer 
or the mallet is used with it. 

The Crandall is a malleable- 
iron bar about 2 feet long, flattened 
at one end. In this end is a slot, 






Js-X. 



0\vSt 



-y« 



Crandall. 



Fig. 17. 

3 inches long and f inch wide. 
Through this slot are passed ten 
double-headed points of J-inch 
squared steel, 9 inches long, which 
are held in place by a key. 



CLASSIFIED FINISH, 



TOOLS EMPLOYED. 429 



Axed, or Pean»hammered, and 
Patent=hammered. — These two vary 
only in the degree or smoothness of 
the siirface which is produced. The 



I. 




AxecL 



Fig. 18. 
number of blades in a patent ham- 
mer varies from 6 to 12 to the inch ; 
and in precise specifications the 
number of cuts to the inch must be 
stated; such as 6-cut, 8-cut, 10-cut, 
12-cut. The effect of axing is to 
cover the surface with chisel marks, 
which are made parallel as far as 
practicable. Axing is a final finish. 



The Ax, or Pean Hammer, has 
two opposite cutting edges. It is 
used for making drafts around the 
arris, or edge, of stones, and in 



<-,3-> 



Fig. 19. 



reducing faces, and sometimes 
joints, to a level. Its length is 
about 10 inches, and the cutting 
edge about 4 inches. It is used 
after the point and before the 
patent hammer. 



The Patent Hammer is a double- 
headed tool so formed as to hold at 
each end a set of wide thin chisels. 
The tool is in two parts, which are 
held together by the bolts which 
hold the chisels. Lateral motion 







Cfni 



(::: 



i: 



Patent Hammer. 



Tooth^axed. — The tooth-ax is 
practically a number of points, and 
it leaves the sufrace of a stone in 
the same condition as fine pointing. 
It is usually, however, only a prep- 
aration for bush-hammering, and 
the work is then done without 
regard to effect so long as the surface 
of the stone is sufficiently leyeled. 



Fig. 20. 

is prevented by four guards on one 
of the pieces. The tool without 
the teeth is 5Jx2fxli- inches. The 
teeth are 2f inches wide. Their 
thickness varies from t^ to i of an 
inch. This tool is used for giving 
a finish to the surface of stones. 



The Tooth Ax is like the ax, 
except that its cutting edges are 
divided into teeth, the number of 



Fig. 21. 

which varies with the kind of work 
required. This tool is not used in 
granite and gneiss cutting. 



430 



2L—ST0NE CUTTING. 



Bush=hamniered. — The rough- 
nesses of a stone are pounded off by 
the bush hammer, and the stone is 
then said to be " bushed." This 




I 



Bush Hammered. 

Fig. 22. 

kind of finish is dangerous on sand- 
stone, as experience has shown 
that sandstone thus treated is very- 
apt to scale. In dressing lime- 
stone which is to have a bush- 
hammered finish, the usual sequence 
of operation is (1) rough-pointing, 
(2) tooth-axing, and (3) bush- 
hammering. 

Rubbed. — In dressing sand- 
stone and marble, it is very common 
to give the stone a plane surface 
at once by the use of the stone-saw. 
Any roughnesses left by the saw are 
removed by rubbing with grit or 
sandstone. Such stones, therefore, 
have no margins. They are fre- 
quently used in architecture for 
string-courses, lintels, door-jambs, 
etc.; and they are also well adapted 
for use in facing the walls of lock- 
chambers and in other localities 
where a stone surface is liable to 
be rubbed by vessels or other 
moving bodies. 

Diamond Panels. — Sometimes 
the space between the margins is 
sunk immediately adjoining them, 
and then rises gradually until the 
four planes form an apex at the 
middle of the panel. In general, 
such panels are callled diamond 
panels, and the one just described 
is called a sunk diamond panel. 
When the surface of the stone rises 
gradually from the inner lines of 
the margins to the middle of the 
panel, it is called a raised diamond 
panel. Both kinds of finish are 
common on bridge quoins and 
similar work. The details of this 
method should be given in the 
specifications. 



The Bush Hammer is a square 
prism of steel whose ends are cut 
into a number of pyramidal points. 
The length of the hammer is from 



^-3'^ 



<-3''> 





k3^ 



Bush Hammer. 
Fi^. 23. 



4 to 8 inches, and the cutting face 
from 2 to 4 inches square. The 
points vary in number and in size 
with the work to be done. One 
end is sometimes made with a 
cutting edge like that of the ax. 



The Machine Tools used chiefly 
are the saws, planers, grinders, 
polishers, etc. 



EXCERPTS AND REFERENCES. 

Pneumatic Stone=Dressing Machines at the Wachusett Dam, Clinton, 
Mass. (Eng. News, June 30, 1904). — Illustrations of the Kotten and the 
Dallett pneumatic stone-dressing machines. No costs are given. 



25.— MASONRY. 



Kinds of Masonry. — I. Stone masonry; II. Brick masonry; III. Concrete 
masonry; IV. Reinforced-concrete masonry. In addition to the above 
we may also have: V. Mixed masonry; VI. Concrete-block masonry; etc. 
(For Masonry Arches, see Sec. 44, Arches, page 763.) 

Classification of Railroad Masonry. 

(By Committee* of Am. Ry. Eng. & M. W. Assn. — See Proceedings, Vol. 7, 

1906.) 



Kind. 


Material. 


Description. 


Manner 
of Work. 


Dressing. 


Joints or 
Beds. 


Face of 
Surface. 


Bridge, 
■ and Re- 
taining- 
Wall 

Arch 

Culvert... 
Dry 


Stone 

Concrete 

Stone 

Concrete 

1 

Brick 
f Stone 

Concrete 
Stone 


Dimension 
■ Ashlar 

Rubble 

( Reinforced 
i Plain 
( Rubble^ 

Ashlar 

Rubble 

Reinforced 
I Plain 

No. 1 

( Rubble 
(Dry 

( Reinforced 
i Plain 
l Rubble 

Rubble 


Coursed 

(Coursed ) 
-^Broken- [ 
i coursed ) 

Uncoursed 

Coursed 
Uncoursed 

f English 
1 Bond 
1 Flemish 
[ Bond 

Uncoursed 
Uncoursed 


Smooth 

( Smooth 
< Fine p'ted 
( R'gh p'ted 

( R'gh-p'ted 
I Scabbled 

( Smooth 
1 Fine-p'ted 

( Rough-p't 
\ Scabbled 

( R'gh-p'ted 
1 Scabbled 


' Smooth 
( Rock-f'ced 

' Smooth 
1 Rock-f'ced 

Rock-faced 

( Smooth 
( Rock-f'ced 

Rock-faced 
Rock -faced 



I. STONE MASONRY. 
Definitions of Parts of Wall.f — Face, the front surface of a wall; Back, 
the inside surface. Facing, the stones which form the fa€e or outside of 
the wall; Backing, the stones which form the back of the wall; Filling, the 
interior of the wall. Batter, the slope of the surface of the wall ; as, 1 on 12 = 
1 inch horizontal to 1 foot vertical. Course, a horizontal layer of stone in 

* Progress report. Published by permission. 

t See Trans. Am. Soc. C. E., Vol. VI., as previously referred to under 
Sec. 24, Stone Cutting. 

431 



432 



25.— MASONRY. 



the wall. If the stones of each layer are of equal thickness throughout it 
is termed "regular coursing;" if the thicknesses are unequal, the term 
** random " or " unequal coursing " is applied. Joints, the mortar layer 
between the stones. The horizontal joints are called " bed-joints," or sim- 
ply '* beds;" the vertical joints are sometimes called " builds." Usually 
the horizontal joints are called " beds " and the vertical ones " joints." 
Coping, a projecting course of heavy stones on the top of the wall to protect 
it. (A " weathered " coping is one whose top is beveled so as to act as a 
roof) . Pointing, a better quality of mortar put in the face of the joints 
to help them resist weathering. Bond, the arrangement of stones in ad- 
jacent courses. Stretcher, a stone whose greatest dimension lies parallel to 
the face of the wall. Header, a stone whose greatest dimension lies per- 
pendicular to the face of the wall. Quoin, a comer stone; a header for one 
face and a stretcher for the other. Dowels, straight bars or pins of iron 
which enter a hole in the upper side of one stone and also a hole in the lower 
side of the stone next above (for lateral stability). Cramps, bars of iron 
having the ends turned at right angles to the body of the bar, which ends 
enter holes in the upper side of adjacent stones. 

Definitions of Kinds of Masonry.* — (For classification of Stones, see 
Sec. 24, Stone Cutting). 

Rubble Masonry. — Masonry composed of un- 
squared stone, Fig. 1- 

Uncoursed rubble. — Masonry composed of un-'°~ 
squared stones laid without any attempt at regular 
courses. Fig. 1. a- 

Coursed rubble. — Unsquared-stone masonry which is 

leveled off at specified heights to an approximately 

horizontal surface (as at a, b, etc., Fig. 1). pig^ 1, 

(Squared rubble. — Rubble masonry in which the stone may be required 

to be roughly shaped with the hammer, so as to fit approximately.) 

Squared-Stone Masonry. — Masonry ih which the stones are roughly 
squared and roughly dressed on beds and joints. (If the joints are as small 
as about one-half inch the classification might come under that of Ashlar). 

There are five kinds. 

Regarding character of face: 
Quarry-faced masonry. — Face of stone is left as it comes from the quarry. 
Pitch-faced masonry. — Face of stone is roughly dressed. 




p?v? 



I I I I 



a 



rm 



^M 



Fig. 2. Fig. 3. Fig. 4. M 

Regarding character of course: ^ 

Range-work (Fig. 2). — Masonry laid in continuous courses throughout. 
Broken Range-work (Fig. 3) .—Masonry laid incotirses non-continuous or broken. 
Random-work (Fig. 4). — Masonry not laid in courses at all. 

Ashlar Masonry. — " Cut-stone masonry," or masonry composed of any 
of the various kinds of cut stone mentioned under Stone Cutting, Sec. 24. 
From its derivation ashlar apparently means large, square blocks; but 
practice seems to have made it synonymous with " cut -stone," and this 
secondary meaning has been retained for convenience. 

Fig. 5 represents the face of an ashlar wall, coursed throughout. 
Broken Ashlar (Fig. 6). — Cut-stone masonry in which the joints are not 
continuous. 



* See Trans. Am. Soc. C. E., Vol. VI. 
Sec. 24, Stone Cutting. 



as previously referred to under 



STONE MASONRY— SPECIFICATIONS, 



433 



r I ! I 



rrTTL T 






Fig. 5. Fig. 6. 

Small Ashlar. — Cut -stone masonry in which the stones are less than one 
foot in height; seldom used. 

Rough Ashlar. — A term sometimes given to squared stone masonry, either 
" quarry-faced " or " pitch-faced " when laid as range-work, but it is 
more logical and more expressive to call such masonry squared range-work. 
Dimension Stones. — Cut-stones, all of whose dimensions have been fixed 
in advance. If the specifications for ashlar masonry are so written as to 
prescribe the dimensions to be used, it will not be necessary to make a new 
class of such stones. 

Range-work, whether of squared stones or of ashlar, is usually backed 
up with rubble masonry, which in such cases is specified as Coursed Rubble. 

Whatever terms are employed in common use to various classes of 
masonry, it is not safe to trust to them alone in preparing specifications 
for construction, but every specification should contain an accurate des- 
cription of the character and quality of the work desired. Whenever 
practicable, samples of such kind of cutting and masonry should be pre- 
pared beforehand, and exhibited to the persons who propose to undertake 
the work. 

Specifications for Stone Masonry. — ^The following is a Committee Report* 
(amendments included) submitted to the Am, Ry. Eng. and M. W. Assn., 
March 1906, but has not formally been approved by the Association. — See 
Proceedings, Vol. 7. 

General. 

Stone.— 1. Stone masonry shall be built of the kinds specially designated, 
with such arrangements of courses and bond as shown on the drawings or as 
directed. ^ The stone shall be hard and durable, free from seams or other 
imperfections, of approved quality and shape, and in no case have less bed 
than rise, and shall be laid on their broadest beds, well bonded and solidly 
bedded. When liable to be affected by freezing, no unseasoned stone shall 
be laid. 

Dressing. — 2. Dressing shall be the best of the kind specified for each 
class of work. 3. Beds and joints or builds shall be square with each other, 
and dressed true and out of wind. Hollow beds will not be allowed. 4. 
All stones shall be dressed for laying on natural bed. 5. Marginal drafts 
should be neat and accurate. 6. Pitching shall be done to true lines and 
exact batter. 

Mortar. — 7. The sand and cement shall be mixed dry and in small 
batches in proportions as directed, on a suitable platform, which must be 
kept clean and free from all foreign matter; then water is to be added, and 
the whole remixed until the mass of mortar is thoroughly homogeneous, 
and leaves the hoe clean when drawn from it. It shall not be retempered 
after it has begun to set. Mechanical mixing to produce the same results 
may be permitted. 

Laying. — 8. All stones shall be laid on natural beds. Each stone shall 
be settled into place in full bed of mortar. 9. No stone shall be dropped 
or slid over the wall, but shall be placed without jarring the stones already 
laid._ 10. No heavy hammering shall be allowed on the wall after a course 
is laid. 11. If a stone becomes loose after the mortar is set, it shall be 
relaid with fresh mortar. 12. Each stone shall be cleansed and dampened 
before laying. 1 3. Stones shall not be laid in freezing weather unless 
allowed by the engineer. If allowed, they shall be free from ice, snow or 
frost by warming, and laid in mortar made of heated sand and water, or, 
with proper precautions, mixed with brine in proportions of one lb. of salt 
to 18 galls, of water, when the temperature is 32° F. Add one ounce of 
salt for every degree of temperature below 32° F. 14. Stones shall be laid 
to exact lines and levels so as to give the required bond and thickness of 
mortar in beds and joints. 

* Published by permission of the Executive Committee, through the 
courtesy of Mr. E. H. Fritch, Secretary. 



434 25.— MASONRY. 

Pointing. — 15. Mortar in beds and joints of exposed faces shall be 
removed to a depth of not less than one in. before it has set. No pointing 
shall be done until the wall is complete and mortar set, nor when frost is 
in the stone. Wet the joints and fill again with mortar made of equal 
parts sand and Portland cement. It shall be pounded in with " set-in " 
or calking tool, and finished with a beading tool the width of the joint, 
used with a straight-edge. 

Classification. 

16. Stone masonry shall be classified under the following heads: Bridge 
and Retaining Wall Masonry, Arch Masonry, Culvert Masonry, and Dry 
Masonry. 

Bridge and Retaining Wall Masonry, 

17. Bridge and Retaining Wall masonry shall consist of two classes: 
(a) Ashlar (either coursed or broken-coursed), and (6) Rubble. 

(o) Ashlar Masonry in Bridges and Retaining Walls. 

18. In Ashlar masonry in bridges and retaining walls (either coursed 
or broken-coursed), the stone shall be large and well-proportioned. 19. No 
course shall be less than 14 ins. nor more than 30 ins. thick; the thickness 
of courses to diminish regularly from bottom to top. 

Dressing. — 20. The beds and joints or builds of face stones shall be fine- 
pointed, so that the mortar layer shall not exceed j in. in thickness when 
the stones are laid. 21. Joints in face stones shall be full to the square 
for a depth equal to at least i the height of the course, but in no case less 
than 12 ins. 

Facing or Surface Finish. — 22. The exposed surface of each face stone 
will be rock -faced, and the edges pitched to true lines and exact batter; 
the face to have no projections over 3 ins. beyond the pitch lines. 23. A 
chisel draft 1^ ins. wide shall be cut at each exterior comer. 24. No holes 
for stone hooks shall be permitted to show in exposed surfaces. They 
must be handled with clamps, keys, lewis or dowels. 

Stretchers. — 25. Stretchers shall be not less than 4 ft. long, and to have 
at least 13^ times as much bed as thickness of the course. 

Headers. — 26. Headers shall be not less than 4 ft. in length. They 
shall occupy one-fifth of the face of the wall, and no header shall have less than 
18 ins. width of face, and when the course exceeds 18 ins. height, the width 
of face shall be not less than the height of course. Headers shall hold the 
size in the heart of the wall that they show on the face, and be so arranged 
that a header in a superior course shall be placed between two headers in 
a course below; but no header shall be laid over a joint, and no joint shall 
cover a header. They shall be similarly disposed in the back of the wall, 
interlocking with those in the face when the thickness of the wall will admit. 
When the wall is too thick to admit of such arrangement, stones of not less 
than 4 ft. in length shall be placed transversely in the heart of the wall to 
bond the two opposite sides of it. 

Backing. — 27. Backing shall be large, well-shaped stone, roughly 
bedded and jointed; the bed joints not to exceed 1 in., and vertical joints 
generally not to exceed 2 ins. No part or portion of vertical joints shall 
have a greater dimension than 6 ins., which void shall be thoroughly filled 
with spalls full bedded in cement mortar or filled with concrete. At least 
one-half of the backing stones shall be of the same size and character as 
the face stone and with parallel beds. 28. When face stone is backed 
with two courses, neither course shall be less than 8 ins. thick. 29. When 
the wall is 3 ft. thick or less, the face stone shall pass entirely through and 
no backing be allowed. 30. If the engineer so directs, the backing may be 
entirely of concrete, or back laid with headers and stretchers, as specified 
above, and heart of wall filled with concrete. 

Bond. — 31. The bond of stone on face, back and heart of wall shall not 
be less than 12 ins. Backing shall be laid to break joints with the face stone 
and with one another. 

Coping. — 32. Coping shall be dimension stone, holding full size through^ 
out, proportioned for its loading, as marked on the drawings. 33. The beds, 
joints and top shall be fine-pointed. 34. The location of joints shall be deter- 
mined by the position of the bed plates, and must be shown on the drawings. 

Locks. — 35. When required, in the judgment of the engineer, coping 
stones, stones in the wings and abutments, and stones on piers shall be 
secured together with iron clamps or dowels, their position being indicated 
by the engineer. 



STONE MASONRY—SPECIFICATIONS, 435 

(b) Rubble Masonry in Bridges and Retaining Walls. 

36. Rubble bridge- and retaining-wall masonry shall consist of stones 
roughly squared, laid in irregular courses. The beds shall be parallel; 
roughly dressed, and lie horizontally in the wall. The bottom stones shall 
be large selected flat stones. The wall shall be compactly laid, having at 
least one-fifth the surface of the back and face headers, so arrangedas to 
interlock, having all the spaces in the heart of the wall filled with suitable 
stones and spalls, thoroughly bedded in cement mortar _ or filled with 
concrete. The face joints must not be more than 1 in. in thickness. 

Arch Masonry. 

37. Arch masonry shall consist of the arch ring only, or that portion 
between the intrados and extrados, and shall be of two classes: (a) Ashlar 
Arch Masonry, and (b) Rubble Arch Masonry. 

(a) Ashlar Arch Masonry. 

38. The voussoirs shall be of full size throughout, and must haye bond 
not less than thickness of the stone and dressed true to templet. 39. The 
number of courses and depth of voussoirs shall be shown on the drawings. 
40. The joints of the voussoirs and the intrados shall be fine-pointed. 
Mortar joints shall not exceed | in. 41. The exposed surface of the ring 
stone shall be smooth or rock-faced, with a marginal draft. 42. Voussoirs 
shall be carried up simultaneously from both bench walls. 43. Backing 
shall consist of large stones shaped to fit the arch bonded to the spandrel 
and laid in full beds of mortar. Concrete may also be used for backing. 
44. If waterproofing is required, a thin coat of mortar or grout shall be ap- 
plied for a finishing coat upon which shall be placed a covering of suitable 
waterproofing material. 45. Centers shall not be struck until directed. 
46. Bench walls, piers, spandrels, parapets, wing walls and copings shall 
be built under the specifications for Ashlar Masonry in Bridges and Re- 
taining Walls. 

(b) Rubble Arch Masonry. 

47. The voussoirs shall be full size throughout, and must have bond 
not less than thickness of stone. 48. The depth of voussoirs shall be shown 
on drawings. 49. The beds need only be roughly dressed so as to bring 
them to radial planes. 50. Mortar joints shall not exceed 1 in. 51. The 
exposed surface of the ring stones shall be rock-faced, and the edges pitched 
to true lines. 52. Voussoirs shall be carried up simultaneou.sly from both 
bench walls. 53. Backing shall consist of large stones shaped to fit the 
arch bonded to the spandrel and laid in full beds of mortar. Concrete 
may also be used for backing. 54. If waterproofing is required, a thin 
coat of mortar or grout shall be applied for a finishing coat upon which 
shall be placed a covering of suitable waterproofing material. 56. Bench 
walls, piers, spandrels, parapet and wing walls shall be built under the 
specifications for Rubble Masonry in Bridges and Retaining Walls. 

Culvert Masonry. 
57. Culvert masonry shall be laid in cement mortar. The character 
of stone used and quality of work shall be similar to that specified for Rub- 
ble Masonry in Bridges and Retaining Walls. 58. One-half the top stones 
of the side walls shall extend entirely across the wall. 59. The covering 
must be of sound, strong stone at least 12 ins. thick, or as shown on draw- 
ings. They shall be roughly dressed so as to make close joints with each 
other, and must lap their whole width at least 12 ins. over the side of walls. 
They shall be doubled under high embankments, as directed by the engineer 
or shown on drawings. 60. End walls shall be covered with suitable coping. 

Dry Masonry. 

61. Dry Masonry shall include: (a) Dry Ret'g Walls, and (b) Slope Walls. 

(a) Dry Retaining Walls. 

62. Dry retaining walls shall include all dry rubble work for retaining 
embankments or similar work. ^ 63. Flat stone at least twice as wide as 
thick shall be used. Beds and joints to be roughly dressed square to each 
other and to face of stone. 64. Joints not to exceed f in. 65. The different 
sizes of stone shall be evenly distributed over the whole face of wall, gener- 
ally keeping the largest stone in the lower part of wall. 66. The work 
shall be well bonded and present a reasonably true and smooth surface, 
free from holes or projections. The wall is double-faced and self-sustaining. 



436 



25.'^MASONRY. 
(6) Slope Walls. 



1 



67. Slope walls shall be built of such thickness and slope as may be 
required by the engineer. No stones shall be used which do not reach 
through the wall. Stones shall be placed at right angles to the slopes. 
This wall is single-faced and built with steep slope simultaneously with 
the embankment which it is to protect. 

Quantity of Masonry in Abutments. — ^Tables 1 and 2, following, were cal- 
culated from Fig. 7. Table 1 gives the quantities of masonry and steel in 
one abutment of various heights, and Table 2 is a supplementary table to 
facilitate the reduction of these quantities to weights. 



1. — Quantities in One R. R. Masonry *Right Abutments op 
Various Heights h. (See Fig. 7.) 



I 



Height h 


Ai 


A2 


A3 


^4 


Fy 


F2 


F3 


F4 


Ri 


Ro 


Rz 


Rk 


of Base of 
Rail 


























M 


asonr: 


^ Abot 


e 


Masonry In 






Steel Railln 


above 




Found 


ation. 




Foundation. 






Foundation. 


Top of 
Founda- 






















^ .i2 


^ . to* 


^ .aj 


-^ .««' 


^ . «? 


^ . m 


M . K 


J4 . 03 


iA .J, 


^ .*i* 


iA .4i 


^ .43 


tion. 






21? 


lis 


1-Trac 

Road 

Cu. Yd 








If' 


IT- 


IT 


11^ 


Ft. 


S^s 






n 




f*i o 




"iH 


%H 




m 


8 


31.2 


48.4 


64.8 


78.8 


29.6 


44.2 


57.8 


68.4 


288. 


394. 


509. 


608. 


9 


38.8 


58.8 


78.6 


94.5 


33.2 


48.5 


62.8 


74.1 


308. 


421. 


541. 


643. 


10 


47.6 


70.8 


94.0 


112. 


37.0 


53.0 


68.0 


80.0 


330. 


450. 


575. 


680. 


11 


57.7 


84.2 


111. 


131. 


41.0 


57.7 


73.4 


86.1 


354. 


481. 


611. 


719. 


12 


69.0 


99.2 


130. 


152. 


45.2 


62.6 


79.0 


92.4 


380. 


514. 


649. 


760. 


13 


81.6 


116. 


150. 


175. 


49.6 


67.7 


84.8 


98.9 


408. 


549. 


689. 


803. 


14 


95.4 


134. 


172. 


200. 


54.2 


73.0 


90.8 


'106. 


438. 


586. 


731. 


848. 


15 


111. 


153. 


195. 


227. 


59.0 


78.5 


97.0 


113. 


470. 


625. 


775. 


895. 


16 


127. 


174. 


220. 


255. 


64.0 


84.2 


103. 


120. 


504. 


666. 


821. 


944. 


17 


144. 


196. 


247. 


285. 


69.2 


90.1 


110. 


127. 


540. 


709. 


869. 


995. 


18 


163. 


220. 


275. 


317. 


74.6 


96.2 


117. 


134. 


578. 


754. 


919. 


1048. 


19 


183. 


246. 


305. 


351. 


80.2 


103. 


124. 


142. 


618. 


801. 


971. 


1103. 


20 


205. 


273. 


336. 


386. 


86.0 


109. 


131. 


150. 


660. 


850. 


1025. 


1160. 


21 


227. 


301. 


369. 


423. 


92.0 


116. 


138. 


158. 


704. 


901. 


1081. 


1219. 


22 


251. 


331. 


404. 


462. 


98.2 


123. 


146. 


166. 


750. 


954. 


1139. 


1280. 


23 


276. 


363. 


440. 


503. 


105. 


130. 


154. 


175. 


798. 


1009. 


1199. 


1343. 


24 


302. 


396. 


478. 


546. 


111. 


137. 


162. 


184. 


848. 


1066. 


1261. 


1408. 


25 


330. 


430. 


517. 


591. 


118. 


145. 


170. 


193. 


900. 


1125. 


1325. 


1475. 


26 


359. 


466. 


558. 


637. 


125. 


152. 


178. 


202. 


954. 


1186. 


1391. 


1544. 


27 


389. 


503. 


601. 


685. 


132. 


160. 


187. 


211. 


1010. 


1249. 


1459. 


1615. 


28 


420. 


542. 


645. 


735. 


140. 


168. 


196. 


220. 


1068. 


1314. 


1529. 


1688. 


29 


453. 


583. 


691. 


787. 


147. 


177. 


205. 


230. 


1128. 


1381. 


1601. 


1763. 


30 


487. 


625. 


738. 


840. 


155. 


185. 


214. 


240. 


1190. 


1450. 


1675. 


1840. 


31 


522. 


668. 


787. 


895. 


163. 


194. 


223. 


250. 


1254. 


1521. 


1751'. 


1919. 


32 


558. 


713. 


838. 


952. 


171. 


203. 


233. 


260. 


1320. 


1594. 


1829. 


2000. 


33 


596. 


760. 


890. 


1011. 


180. 


212. 


243. 


271. 


1388. 


1669. 


1909. 


2083. 


34 


634. 


808. 


944. 


1072. 


188. 


221. 


253. 


282. 


1458. 


1746. 


1991. 


2168. 


35 


675. 


857. 


999. 


1135. 


197. 


231. 


263. 


293. 


1530. 


1825. 


2075. 


2255. 


36 


716. 


908. 


1056. 


1199. 


206. 


240. 


273. 


304. 


1604. 


1906. 


2161. 


2344. 



Above [Ax 

'-+5.2/1. 
For- lAa = 0.8/^2+0.2/^+12. 
mulas: U* = 0.9/j2 + 0.4/t+18. 



|(/j-5)2+3.2/j. 
f(/j-5)2- - 



Fi = 0.1/^2+ 1.9/J+ 8. i?i = ;j2 + s/t + 200. 
F2 = 0.1/^2+ 2.6/^+17. i?2 = /J2 + 10/t + 250. 
F3 = 0.1/t2+ 3.3/t+25c i?3 = /i2 + \hh + 325. 
F4 = 0.1/j2+ 4.0/1+30. i?4 = /J2 + 18/t + 400. 
* For "skew" abutments, multiply quantities in table by secant of "skew- 
angle," i.e., angle which face of abut, makes with a right angle to center line. 



MASONRY IN ABUTMENTS. BRICK MASONRY, 



437 



2. — Table for Finding Weights of Quantities in Table I. 



{X .S 



10 
100 
200 

300 
400 
500 

1000 



Weight of Masonry, 
m 1000 Lbs. 



Mas. at 
145 Lbs. 

per 
cu.ft. 



3.915 

7.830 

11.745 

15.660 
19.575 
23,490 

27.405 
31.320 
35.235 

39.150 
391.5 

783. 

1174.5 

1566. 

1957.5 

3915. 



Mas. at 
155 Lbs 

per 
cu. ft. 



4.185 

8.370 

12.555 

16.740 
20.925 
25.110 

29.295 
33.480 
37.665 

41.850 
418.5 



1255.5 

1674. 

2092.5 

4185. 



Mas. at 
160 Lbs, 

per 
cu. ft. 



4.32 

8.64 

12.96 

17.28 
21.60 
25.92 

30.24 
34.56 
38.88 

43.20 
432. 



1296. 
1728. 
2160. 

4320. 



Weight of Steel, 
in Lbs. 



Rails at 
60 Lbs. 
per yd. 



20. 
40. 
60. 



100. 
120. 

140. 
160. 
180. 

200. 
2000. 
4000. 

6000. 

8000. 

10000. 

20000. 



Rails at 
65 Lbs. 
per yd 



21.67 
43.33 
65. 

86.67 
108.33 
130. 

151.67 
173.33 
195. 

216.67 
2167. 
4333. 

6500. 

8667. 

10833. 

21667. 



Rails at 
70 Lbs. 
per yd. 



23.33 
46.67 
70. 

93.33 
116.67 
140. 

163.33 
186.67 
210. 

233.33 
2333. 
4667. 

7000. 



*Ex. — A masonry abut- 
ment contains 384 cu. yds. 
masonry at 155 lbs. per cu. 
ft., and 822 lin. ft. steel 
rails at 65 lbs per (lin) yd. 
Find the weight or load 
which the abutment alone 
produces on the pile founda- 
tion. 




9333 t • Top of FourKhtion^g* 
11667: ^|^^^^£]i 



23333. 



Section A-B. 
Fig. 7. 



Plan. 



*Ans. — From columns 1. 3 and 6 In above table we have: Weight = (masonry) 
1607.04X 1000+ (steel) 18460=1.625.500 lbs. = 812.75 tons. 



II. BRICK MASONRY. 



Bond. — The terms " header " and " stretcher " are used in brickwork 
as in stonework. The *'bond " of a brick wall depends upon the arrange- 
ment of headers and stretchers, in the same or in 
adjacent courses. 

English Bond (Fig. 8). — Alternate entire courses 
of headers and stretchers. 

Modified English Bond. — Each course consisting 
entirely of headers or of stretchers, but not alter- 
nating as above; usually one course of headers to 
every four to six courses of stretchers. (Where wall 
is laid with one course of headers to every two or 
three courses of stretchers it is generally classed as 
English bond). 

Flemish Bond (Fig. 9). — Each course made up of 
alternate headers and stretchers. 

English bond is considered the strongest. 

Brickwork is well adapted to all kinds of 
masonry construction where excessive strength and 
massive weight (as bridge abutments, piers, dams, 
etc., which call for concrete and stonework) are not 
particularly essential. Hence its use in building- 
walls (including the upper stories in tall buildings) , Fig. 9. 
tunnel linings, small arches, culverts, (street paving), sewers, etc. The 
convenience in handHng and laying brick, in forming arches and rounding " 
corners, makes it particularly useful in these classes of construction. In 
fire-resisting qualities it is superior to any natural building stone, with the 
exception possibly of sandstone, and is equal to the latter in this respect, 



■I "1 ■ 1 ' 1 1 




1 ' ' 1 ' 1 1 1 


1,1 ' 1 ' 1 1 ' ' 1 ' ' l- 






1 1 1 1 r 




' 1 1 1 1 1 




Fig. 8. 


1 1 ' 1 1 ■ II', 


■ 1 1 II 1 1 


'l 1 ' ' 1 1 ' ' 1 l' 


1 ' ' 1 1 II 1 1 


■ 1 1 II II 


1 ' ' 1 1 II II 


' 1 1 ' 1 1 11 


III II II 


' 1 1 1 1 II 


1 1 '1 1 -1-i 



438 



25.— MASONRY, 



and infinitely superior to it (i.e., to most varieties) as regards frost action 
and absorption of moisture. Generally speaking, when compared with 
stonework, brick masonry lies between ashlar and common rubble masonry, 
both in cost and strength. Regarding cost, the writer has witnessed the 
price of common building brick fluctuate during the past year, in Brooklyn, 
between $16 and $7 per thousand. It is perhaps essential to note here that 
while the higher price was maintained the builders resorted largely to 
rubble and concrete construction, which finally brought the trust price to 
the lower figure. This may illustrate also that the " cost of work '' obtained 
from any source must be used with judgment in connection with the details 
furnished therewith, and with the prevailing local conditions. 

The Mortar used in brick masonry may vary considerably, depending 
upon the class of work. For instance, the best mortar usually specified is 
composed of 1 part Portland cement, and 2 parts clean, sharp sand, where 
the structure is exposed to considerable stress; while for the most ordinary 
brickwork, 1 part fresh, well^slaked lime and 2^ to 3 parts sand, will answer. 
Between these limits we may have various mixtures of Portland cement, 
natural cement, common lime, and sand. Thus: 



Class A 
B 

c 

D 
D2 



1 Port. Cem. 
1 " " 
1 " " 



Nat. Cem. Com. Lime. 



E etc. 1 " " 

F etc 1 



1 Lime paste 
1 " " 



2 Clean, sharp Sand. 
2i 



2i 

3 

2 

2h 

3 

4 

5 

6 

2 

2 



Class A is used in superior building construction, for railroad masonry 
in general, tunnel lining, and sewers; class E, for building-work of the very 
highest class; class C, for common brickwork as in buildings. Where cement 
and lime come in barrels, " barrel " measure may be used in determining 
the " parts " of cement, lime and sand. Fire brick should be laid in fire- 
clay mortar. Colored mortar is used for pressed brick facing. 

Pressed-brick masonry, for facing, requires less mortar per cubic yard 
of masonry than the common-brick backing, because pressed bricks are a 
little larger than the common (which are 8^x4x2^ standard) to allow for 
thinner joints. 

For any kind of masonry, the bricks should be wet before laying (except 
perhaps in freezing weather — unless special care is taken to warm the water 
and sand), and thoroughly bedded in mortar, and pressed or shoved into 
place. 

3. — Quantities op Brick and Mortar in Brick Masonry. 
(Brick, standard size — 8^x4x2^.) 





No. of 


No. of 


Cu. Ft. of 


Cu. Yds. of 


Cu. Yds. of 


Cu. Yds. of 


^ 


Brick per 


Brick per 


Masonry. 


Masonry 


Mortar per 


Mortar per 


ot-» 


Cu. Ft. of 


Cu. Yc . of 


per 1000 


per 1000 


Yd. of 


1000 


r° 


Masonry. 


Masonry. 


Brick. 


Brick. 


Masonry. 


Brick. 


zero. 


23.3 


628.4 


43.0 


1.59 


.000 


.000 


v» 


21.1 


568.6 


47.5 


1.76 


.095 


.167 


M 


19.1 


516.6 


52.3 


1.94 


.180 


.350 


A 


17.4 


471.0 


57.3 


2.12 


.250 


.530 


A 


16.0 


430.9 


62.6 


2.32 


.314 


.728 


Vs 


14.6 


395.4 


68.3 


2.53 


.371 


.939 



This table is calculated for massive masonry construction; allowance 
should be made for thin walls — bricks to increase and mortar to diminish 
slightly per volume of masonry. 



BRICK MASONRY. CONCRETE MASONRY. 439 

III. CONCRETE MASONRY. 

No other class of masonry is so generally employed, especially for mas- 
sive construction, as concrete. It is almost universally used for footings 
of heavy structures such as abutments, walls and piers for bridges, revet- 
ments (breakwaters), dams, and buildings; while it often enters largely, 
sometimes wholly, into these structures themselves. 

Concrete footings are often reinforced with steel rails or I-beams, in 
one or more tiers, to distribute the loads (either above or below) more 
uniformly. Such construction may be said to form the connecting link 
between plain- and reinforced concrete. See Sec. 50, Foundations. 

Rock Crushers are now almost wholly employed in breaking stone for 
concrete, in place of hand-breaking. By the last-named method a laborer 
is usually counted upon to break about one cu. yd, of trap or from 2 to 3 
cu. yds. of limestone, in sizes to pass through a 2-in. ring, in a 10-hr. day. 
Much depends upon the size and shape of the rip-rap stone received for 
breaking. On the other hand, the capacities of machines run up to several 
hundred tons per day. A cubic yard of broken trap rock, assuming 45% 
voids, will weigh about 1 8 5x2 7x .55 = about 2750 lbs. or If tons; while a 
cubic yard of limestone, assuming 371% voids, will weigh about the same. 
Unscreened broken stone, especially of the softer kinds, has less percentage 
of voids than the screened. The softer rocks in crushing assume a more 
rounded form, and break into more variable sizes, both of these conditions 
tending to reduce the percentage of voids. Spheres of uniform size, as 
cannon balls, may be piled in large piles in pyramidal form so that the 
percentage of voids will approach the lower limit of 25.963% voids. 

Permanent crushers, often with extensive plants, are frequently installed 
at quarries. For this purpose, the gyratory type of crusher is preferred, 
a single machine being able to turn out as high as 2000 tons or more per 
10-hr. day. Such a machine would weigh in the neighborhood of 50 tons, 
have a receiving opening of say 1^x5 ft., and require about 150 H. P. to 
operate. The smaller machines are less economical in the use of power. 
A machine of ^ the above capacity would require about f the power to oper- 
ate, and would weigh about 30 tons; a machine of i the above capacity 
would require about f the power, and weigh about 20 tons; i the capacity, 
i the power, and weigh about 10 tons, etc. The above H. P.'s include 
power required to operate elevators and screens. 

Portable crushers are furnished as low as about 4 tons in weight, with 
openings for 7x10 in. stone, capacity 50 to 60 tons per 10-hr. day, and re- 
quiring about 8 H. P. to operate. Small machines cost about $8.50 to 
$9 per capacity in tons per 10 hr. day. Large machines, from $7 to $8. 

Hand crushers, costing about $30, will receive stone up to about l^x 
8 ins. 

The best crushers have jaws of manganese-steel, or other steel of equal 
hardness. 

Concrete Mixers, or machines for mixing concrete, are indispensable 
at the present day on work of any magnitude. They may be divided into 
two classes, namely, " fixed " mixers, and " mechanical " mixers. 

Fixed or Gravity Mixer consists of a steel trough fixed on an incline of 
say 4 to 4i ins. horizontal to 1 2 ins. vertical, and some provided with internal 
projections in the form of steel pins or baffle plates, which deflect and " mix " 
the material, fed at the upper end, as it descends. These mixers are econ- 
omically used in lining the bottoms and sides of reservoirs and in work of 
similar character where the mixing can be done at an elevation above the 
place for depositing the concrete. 

Mechanical Mixers are of two types: ** continuous " mixers, and " batch " 
mixers. The continuous mixers are provided with plows, shovels, or pad- 
dles which mix the material as fast as delivered, and discharge the mixture 
continuously. Portable machines of this type are used economically for 
street work. 

Batch mixers of the rotary type are the most generally used. These 
machines with engine (gasoline) , or engine and boiler (steam) , are mounted 
on skids or on wheels, are very compact, and have a capacity of about 200 
batches per 10-hr. day, each batch ordinarily containing \ yd., f yd., or 
1 yd., depending on the " size '' of the mixer. They are, however, made 



440 25.— MASONRY. | 

in sizes ranging from 2 cu. ft. up to 2 cu. yds. The number of H. P. re- 
quired is equal to size of batch in cu. yds. multipHed by 15, about. Many 
of the machines measure the proportions of the ingredients. 

The Proportions of Portland Cement, Sand and Broken Stone are, 1 : 2to 
4 : 4 to 8, depending upon the quaHty of materials used and of the resulting 
concrete required. A good mix is 1 : 2^ : 5, while 1:3:6 is perhaps 
most common for ordinary work. In the construction of the Mississippi 
jetties, the block concrete was made with the following proportion: Portland 
cement, 1, sand 2|, clean gravel IJ, broken stone 5; the resulting mass 
was about li yds. of concrete per yd. of broken stone. Common lime 
may be added to cement to increase the bulk of the ** cement " paste, 
and consequently of the mortar, where concrete is not to be placed under 
water and where it is not subjected to excessive stress; but as common 
lime has no hydraulic properties, its addition to cement in mortar is simply 
the addition of so much inert matter, like sand, when the concrete is depos- 
ited under water. 

A common custom of late is to designate concrete by the proportion of 
cement to sand assuming that the volume of broken stone shall be twice 
that of the sand. Thus. 1 to 2 matrix would mean 1 cement, 2 sand, and 
4 broken stone, or 1 : 2 : 4 mix; 1 to 2J matrix, a 1 : 2^ : 5 mix, etc. This 
abbreviation, unless explained (which explanation would tend to nullify 
the advantage of abbreviation itself) is apt to lead to error. The propor- 
tion of broken stone to sand need not necessarily be in the ratio 2:1. In 
the case of broken limestone, where the crushed rock is graded, and also 
where gravel is used, especially with broken stone, the ratio may be, say, 
21 : 1, 2^ : 1, etc. In all large work the percentage of voids, and conse- 
quently the proportions, should be determined by experiment for the 
classes of materials used. See page 417, where the method of determining 
voids, etc., is given. 

In Mixing by hand, the cement and sand are thoroughly mixed dry on 
a clean platform, enough water is added to make a stiff paste, the broken 
stone (wet and previously washed clean) is then spread over the mortar 
and the whole thoroughly mixed by shoveling into another pile, and re- 
shoveling as often as necessary.* Concrete should be mixed in small 
batches and placed immediately, as the cement may set appreciably within 
30 to 40 minutes after water is applied, and all subsequent disturbance 
may tend to weaken the mass. 

Placing, Spreading and Ramming are operations which should closely 
follow one another rapidly. The concrete may be delivered in barrows, 
cable buckets, carts or chutes, the last named being the cheapest under 
favorable conditions. Spreading consists in pushing over the top of each 
batch dumped, so as to present a fairly level bed for the next layer. The 
layers are usually specified to be from 6 to 9 ins. in thickness. Ramming 
compacts concrete from 5 to 25% and makes it stronger by bringing the 
ingredients more intimately together, so that crystallization takes place 
more firmly. Rammers are round wooden blocks or logs about 4 ft. long, 
ring-shod at the lower end and provided at the upper end with handles 
for one-man or two-man manipulation. 

Ramming should bring the concrete to a rather firm state with a flush 
of water at the top, but should not continue for an undue length of time so 
as to weaken its setting- "Dry" concrete naturally requires more ramming 
than a " wet " mixture. Before depositing a new layer, the top of the 
preceding layer should be wetted, after allowing not less than 12 hrs. for 
it to set. Where a layer is not wholly completed at the end of a day's 
work, its edge should be left rough (not smooth) for the next day's joining. 
'* Medium " concrete is between " dry " and " wet." The advantage of 
medium concrete is that it can be rammed better than " wet," while at 
the same time possessing some of the good qualities of the latter. 

Subaqueous Concrete has seldom been deposited with entire satisfaction 
even in still water to any great depth, but in moderate depths this has been 
accomplished with quite favorable results. The essential principle in 
depositing concrete under water is to convey it properly through the water 
so that the mix will not be disturbed by *' wash." Examinations by divers 



* Some engineers and most builders prefer to spread the (wet) gravel 
and broken stone on the dry cement-sand mix before water is added. — See 
German Specifications, pages 442, 443. 



CONCRETE— MIXING, PLACING. 



441 



have in many cases disclosed " streaky concrete," with cement, sand and 
matrix more or less separated by wash, owing to their different specific 
gravities and fineness. The result of this lack of homogeneity is a reduction 
of strength, hence great care should be taken to insure as little disturbance 
as possible in placing the material. Under no circumstances should con- 
crete be dumped loosely into water and allowed to settle in place. The 
three principal methods most commonly used are: 

(1) Depositing through closed chutes or tubes. 

(2) Depositing by means of specially arranged buckets. 

(3) Depositing in sacks or in bags. 

The first method, by chute, is seldom used. The tube may be of any 
section, gradually enlarging toward the bottom to avoid clogging of the 
concrete fed continuously at top of tube. The tube is suspended vertically 
with the lower end at the bottom, and is moved laterally as the concrete 
falls into place. The principal^ objection to this method is the washing 
which the concrete receives in its descent. 

The second method, by buckets, has been used with greater or less suc- 
cess. One of the most recent uses of this method was in the construction 
of a 3000-ft. pier at Superior Entry, Wis., by the Government, in about 
23 ft. of water.* The steel buckets (Fig. 10) about 4 ft. cubes, open at the 
top, were provided with canvas covers, quilted with strips of sheet lead, 
to fold over the concrete and prevent wash. The buckets when lowered 
were tripped by a latch. Mr. Clarence Coleman, Asst. Engr. in charge, 
states that the examinations of concrete lowered 23 ft. and raised again in 




IVe"Bar 
Side cff Bucket- 
with Leaf Hanging down. 



Side Off Budget. 



Fig. 10. 



the bucket showed the concrete to be in good condition; and that dis- 
coloration of the water from cement was seldom noticed during the 
descent of the bucket. 

The third method, by bags, has been used extensively in both this 
country and in Europe. Mr. W. M. Patton, in his "Treatise on Foundations,"! 
page 102, says: "Perhaps the best mode of depositing concrete under water 
is to fill open sacks or gunny sacks about two-thirds to three-fourths full 
of the concrete or mortar, and deposit these in place, arranging them in 
courses, where practicable, header and stretcher system, and ramming 
each course as laid; the bagging is close enough not to allow the cement to 
be washed out, but at the same time open enough to allow the whole mass 
to be united and to become as compact as concrete itself. The writer used 



* See Vol. 8, Part 4, of Report of Chief of Engineers, U. S. A. 
t Published by John Wiley & Sons, New York. 



442 25.— MASONRY. 

this method in the foundation of a pier over 100 feet high, and has also 
adopted this plan in other works of less magnitude, but never has the result 
been satisfactory when deposited under water in any other manner." 

Sub=Foundations are prepared by dredging, if necessary, and driving 
piles and cutting them off near the river bottom. Before concrete is de- 
posited, molds are constructed of timber and sunk in place to give form to 
the concrete pier. The inside faces of the molds are of course smooth 
lined, the timber frames being outside. Long adjustable bolts or turn- 
buckle rods are convenient to use with collapsible sides, where the same 
molds are to be used over again. See Sec. 50, Foundations. 

Cement Grout (either pure cement, or 1 cement and say 1 sand) may be 
injected into a quick-sand or gravel foundation bed to form a sub founda- 
tion. The grout is pumped into vertical iron pipes perforated with holes 
at the bottom to allow it to ooze through into the natural bed material. 
The pipes should extend downward through the soft material to bed rock or 
other firm sub-stratum. 

German Specifications for Concrete. — ^The following is a digest of portions 
of the report of standing committee of the German Concrete Society as 
adopted by that Society and entitled " Specifications for Designing, Con- 
structing and Testing Concrete Structures." These specifications were 
brought to the attention of Eng. News by L. S. Moisseiff, and published 
under date of Nov. 9, 1905. 

/. General. — Specifications apply to concrete construction in general, 
and to use of Portland cement in particular. Concrete is termed " wet " 
or " dry;" shall be capable of being tamped to acquire the necessary den- 
sity to develop the required resistance. //. Planning and Designing. 
III. Construction. {A) General. {B) ^ Superintendence and workmen. 
(C) Building Materials and their Working, (a) Materials: Cement which 
fulfils the requirements of the standard specifications for Portland cement 
shall be used exclusively. Quick setting cement shall not be used for 
concrete except in special cases. (The setting of the concrete is affected 
by the temperature and moisture of the air and the temperature of the water 
used. High temperature accelerates setting; low temperature retards it. 
In the presence of water pressure the use of quick setting cement is fre- 
quently necessary.) " Sand " includes land-, river- and sea-sand, as well 
as broken or crushed products, from fine grains to 0.28 in. in diameter 
(includes granulated furnace slag of proper consistency). " Gravel," from 
0.28 in. up. "Gravel-sand," the natural mixture from excavations or 
beds of streams. _ The sand, gravel and broken stone shall be suitable 
(loam, clay and similar admixtures have an injurious effect if they adhere 
to the sand and stone; but if they are finely distributed in the sand, without 
adhering to the grains, they are, as a rule, harmless and may even sometimes 
increase the resistance), and shall not contain vegetable matter or other 
impurities. (The stone or gravel used shall, as a rule, have at least the 
same resistance as the hardened mortar. Stone not weather-resisting, 
soft sandstone, underbumt brick shall not be used for concrete. Slag is 
variable and should be tested). According to the thickness of the concrete 
body, gravel up to 2 ins. in dia. may be used. (Clean stones of large size 
of good compressive resistances and weathering qualities may be embedded 
in the concrete up to 40% of the total volume if the character and dimen- 
sions of the member allow it and provision be made for the proper distri- 
bution of such stones in the concrete as well as for the use of a sufficiently 
wet concrete to surround them completely.) For ' broken stone," only 
hard rocks, unaffected by the weather, shall be used. As a rule, the broken 
stone shall also be of different sizes of stones as the concrete mass can then 
be worked easier and better, and gives a denser and stronger concrete. 
The sizes of the stones vary with the thickness of the concrete mass. The 
largest stones shall, according the their use, pass in any direction through 
a ring of 2 . 4 to 2 . 8 ins. diameter, or through a square of 2 to 2 . 4 on a side. 
Particles of sizes smaller than . 28 in. down to stone dust shall be considered 
as sand. The " water " used shall be clean, etc. Marshy water is in- 
jurious, (h) Preparing the concrete: ^ In proportioning the materials, if 
the cement be measured by volume it is understood that it is emptied into 
the measuring vessel without dropping and the latter is not shaken. To 
convert volumes into weights, a cubic foot of Portland cement shall be taken 
as 87.5 lbs. Gravel-sand and mixed broken stone may in many cases be 
used without being separated. Tests shall then be made by sieving to 
determine the proportions of the sand in the gravel or the stone dust in 



SPEC NS FOR CONCRETE. REIN. -CONCRETE. 443 

the broken stone. " Mixing by hand " shall be in a proper mixing box. 
The sand and gravel-sand shall first be mixed dry; with the cement to a 
mixture of uniform color, then only shall the previously wetted gravel or 
stone be added, and the whole, together with the water added to it, shall 
be mixed further until a uniformly wet mass is produced. " Mixing by 
machine " is generally superior to hand -mixing. The time required varies 
with the type of machine and the character and quantity of the mix. The 
mixture is first mixed dry i to 1 minute, and the mixing is then continued 
with a gradual addition of water until a thoroughly mixed, uniformly wet 
mass is produced. Generally, the mixing is complete when the stones are 
thoroughly coated with the mortar. The " quantity of water " to be added 
depends on whether a ** dry " or " wet '.' mix is required, the character 
of materials, proportions, moisture, and absorptive capacity. (More water 
is required in dry and hot weather. When laying concrete in wet excava- 
tions, the quantity of water will have to be reduced, while a porous, absorb- 
ing ground should be wetted before laying the concrete.) When making 
" dry " concrete, only enough water should be added that the concrete 
can just be squeezed into a ball by the closing of the hand, leaving moisture 
on the skin; when making " wet " concrete the proportion of water shall 
be so far increased as to make the concrete wet during the tamping, (c) 
Laying the concrete: The concrete shall be laid in place (excavation or 
forms) in layers only and in such depths that the thickness of the completed 
layers after tamping shall not, as a rule, exceed the following amounts: 
For " dry " concrete, according to the stress, p to 8 ins.; for " wet " concrete, 
according to the stress, 8 to 12 ins. The individual layers shall be laid, 
wherever possible, at right angles to the direction of pressure, and where- 
ever this is impossible the layers shall be laid parallel with the direction 
of pressure. (Tamping in the direction of the pressure insures a uniform 
deformation of the concrete in the whole section, and is to be desired pri- 
marily. Where this is impracticable or where the quality of the tamping 
in the desired direction would suffer, as for instance in a flat arch, care shall 
be taken that the tamping shall not tend to displace the individual layers.) 
The layers shall, as a rule, be laid on fresh concrete to obtain a sufficiently 
strong adhesion. Smooth surfaces after tamping shall be roughened; and 
in all cases the surface shall be roughened by sweeping with steel brooms. 
For laying concrete rich in stone, spread on a thin layer of wet mortar. 
" Tampers," square or rectangular, 4 to 6J ins. on a side and 22 to 27 lbs. 
weight shall be used. The " amount of tamping " required depends on 
the resistance to be obtained and the character of the concrete, " dry " or 
"wet." ("Dry" concrete requires more effort in tamping and greater 
care by the workman, but it generally produces for the same amount of 
cement a higher resistance than the " wet " concrete. Wet concrete may 
be tamped so long as to be injured by separation of the ingredients). Con- 
crete may be laid in " freezing weather " if the injurious effect of freezing 
be prevented by proper means. Frozen materials shall not be used. 



IV. REINFORCED CONCRETE. 

Reinforced concrete is now used in various classes of construction of 
which the following is a partial list: Abutments, aqueducts, arches, bins 
(for coal, grain, ore, etc.), bridges, buildings (foundations, walls, columns, 
floors and roofs), chimneys, columns, cisterns, cross-ties, dams, docks, footings, 
fortifications, foundations, girders, graving docks, piers, piles, pipes, poles, 
posts, reservoirs, retaining walls, sewers, sidewalks (supported), stand- 
pipes, subways, tanks, ties (railroad), tunnels, etc., etc. The cheapening 
of Portland cement manufacture, due to the improved rotary kiln, the 
high efficiency of the recent types of machines for preparing broken stone 
and concrete, the low cost of manufacture of the simple forms of steel used, 
and the simplicity with which the materials may be fabricated at the site, 
mostly with common labor, are factors which have entered largely into 
the cause of the recent rapid development of this class of construction, 
placing it in active competition with what are ordinarily known as steel- 
and fireproof constructions. The added strength due to the steel rein- 
forcement, allowing ^ more scientific and economical distribution of material, 
makes it an easy competitor in many structures formerly built of plain- 
concrete or other masonry, as bridges, building walls, retaining walls, etc. 
But not the least of its merits are its preservative- and fire-resisting quali- 
ties, making it a practically permanent construction. 



444 



25.— MASONRY. 



The Preservative Qualities of Cement, with regard to iron or steel 
imbedded in the concrete, are pretty clearly established. Iron exposed 
to ptire, dry air, or in moist air free from carbonic oxide, and at ordinary 
temperatures, will not rust. Rust is formed through the combined agency 
of carbonic acid and moisture, and as carbonic acid has a greater affinity 
for the cement mortar, the iron will not be attacked while the former is 
present, and encases it in a thorough manner. The few isolated cases 
recorded where rust has been found must be explained on the^ theory of 
some defect in workmanship or materials, as they stand out in marked 
contrast with practically all the accepted results of investigation on this 
subject. Perhaps the most notable instance of what may indirectly be 
called a long-time test was the finding of a small piece of " bright " iron in 
mortar taken from the base of the Egyptian obelisk (of granite, 70 ft. long, 
weighing 200 tons,) re-erected in Central Park, New York City, 1881. 
Numerous instances may be cited of iron, being found in perfect condition 
after from 10 to 25 years' service in mortar or concrete (both stone and cin- 
der) entering into various kinds of construction, both in and out of water 
(salt and fresh). To insure absolute protection, the bright rods may be 
coated with fresh cement when placed ; and it is well to remember that they 
should be embedded thoroughly in the concrete, the latter to be a wet 
mixture, or a medium, well rammed, (the former preferred) to guard against 
voids. 

The Fire=Resisting Qualities of Concrete and concrete-steel construction 
in buildings were illustrated in the Baltimore fire, in 1904. Although no 
construction was found to be really fireproof, it was noticeable that con- 
crete made from steam-boiler cinders and Portland cement seemed to act 
the best; and that stone concrete stood fully as well as, if not better than, 
terra cotta. Results of other tests, specially made for comparison, point 
to the same conclusion. We may safely say, then, that reinforced concrete 
has all the fire-resisting qualities which may be expected in the best materials 
of construction of the present day. 

The Proportions used in mixing concrete for reinforced-concrete con- 
struction almost invariably range from 1 cement, 2 sand, and 4 broken 
stone or gravel or cinders, to a 1 : 3 : 6 mix. Portland cement should be 
used. Clean, sharp sand is usually specified, but some engineers claim 
that sand well worn and rounded is best, and also cheapest as the voids are 
less, requiring less cement. A 1 : 3 : 6 mix is suitable for column- and wall 
construction; a 1 : 2^ : 5 mix, for girders; and a 1 : 2 : 4 mix for the lighter 
construction as floor slabs and small, shallow beams subject to direct impact 
from live load. The broken stone is also graded, say, from J in. upward 
to conform somewhat with the above proportions, the finer stone being 
used with the 1:2:4 mix, for the thinner masses. 

Calculation of Beams.— The following analysis assumes that the resist- 
ing moment of the beam is made up of 
two moments: (1) that above the neutral 
axis, due to the concrete area in com- 
pression; and (2) that below the neutral 
axis, due to the steel area intension, no 
account being taken of the concrete inx 
tension. This is in accordance with the 
best practice at the present time. 

Notation. 

Let h = height of beam, in ins. (Fig. 11); Fig. 11. 

6 = breadth of beam, in ins.; 
X — X = neutral axis; 

d = depth to center of reinforcement, below top of beam, in ins. ; 
/f — d = distance from bottom of beam to center of reinforcement, in ins. 
(either d or h — d may be fixed arbitrarily) ; _ 
/ = allowable compressive stress in lbs. per sq. in. on concrete to be 
used with the straight-line equivalent (instead of the stress ^* 
fo due to the actual stress-strain diagram — shaded): o " 



rm 




^-b ->j 


li^ 






, , i Steel irods^ 




-•-«-»- 


' 





*/2mos. = 720 lbs. 
*/2mos. = 660 
*/2mos. = 600 



per sq. in. for good 1:2 : 4 
" " " " " 1 : 2i : 5 
...... .. .. 1:3:6 



concrete, o -m 



* These values are about 10% greater than the values of fo which would 
be used with the exact stress-strain curve. 



KEIN. -CONCRETE BEAM FORMULAS. 445 

F= allowable tensile stress in lbs. per sq. in. on the steel rods 

(15000 for buildings); 
o = sectional area of the steel rods, in sq. ins.; 
^ = ratio of moduli of elasticity of steel and concrete: 

^ = 13 for steel and good 1:2 : 4 concrete. 

k=U " " " " 1 : 2J : 5 

yfe=15 •• •• " •• 1:3:6 

3; = distance, in ins., from top of beam to neutral axis. 

M" = bending moment = resisting moment, inch-lbs. 

M'= " " = " " ft.-lbs. 

General Formulas'. 
Resisting moment, M"==\fh y^-^aF {d-y) (1) 

For conditions of equilibrium, the algebraic sum of the horizontal forces at 
any section must equal zero {2 H = 0), hence we have, neglecting the 
tension in concrete, below the neutral axis, a F=i f b y (2) 

From the ratio k of the moduli of elasticity of the materials, and the relative 

kf 
position of the neutral axis, we have, k f (d — y) =Fy .'. y = d ^ , ux " (3) 

r + Kf 

Combining (2) and (3), we have: 

Distance to neutral axis, y = -j- ( -y/ 1 + — -r 1 1 (4) 

9 rth 

Breadth of beam, b = — g"^^""^) (^^) 

byi 
Area of steel rods, a = _, , , 7 (46) 

J/? (a — y) 

Also, from (2), a=y-^ (ic) 

Depth to center of rods, d=y (l -\- »-r I (4d) 

Combining (1) and (2) we have: 

6M" 

Stress in the concrete, / = , ,__, r lbs. per sq. in (6) 

DyiSd — y) 

Stress in the steel, F^ ^.^^ , " ' (6) 

a{M—y) 

Also, from (2), F = ^ " ' (6a) 

The following values of /f may be assumed for the three standard con- 
crete mixes: (See Table 4, next page.) 

Concrete. 

Mix. Time. Ultimate. Factor 4. Factor 5. Factor 6. 

1:2:4 1 mo. /=2400. /=600. /=480. /=400. 

2mos. 2880. 720. 576. 480. 

3mos. 3000. 750. 600. 500. 

6mos. 3600. 900. 720. 600. 

1 : 2i : 5 1 mo. 2200. 550. 440. 367. 

2mos. 2640. 660. 528. 440. 

3mos. 2750. 688. 550. 458. 

6mos. 3300. 825. 660. 550. 

1:3:6 1 mo. 2000. 500. 400. 333. 

2mos. 2400. 600. 480. 400. 

3mos. 2500. 625. 500. 417. 

6mos. 3000. 750. 600. 500. 



* Equation (3) gives y when a is unknown; it is directly proportional to d. 

t For use with straight-line formulas adopted above. The values of 
/o, or actual maximum compression on the outer element of the concrete, 
will be about 9% less. 



446 



25.— MASONRY, 



I 



4. — Properties of Rein.-Conc. Beams I'' Wide (b=l) and op Various 
Depths, d or h. (Fig. 11, page 444.) 

Data: Concrete, 1:3:6; age, 2 mos.; /= 600. Steel, F= 15000. ife= 15. 
Factor of safety, 4. For factor of 5. mult. M' of table by i«o ; for 6, by | . 



Depth d 
(Fig. 11.) 


Resisting 


Dist. y 


Area a 




Maxi- 


Approx. 


Wt. per 
Lin. Ft. 


Moment 


to Neu- 


of Steel 


Ratio 


mum 
Depth 
h. 
Ins. 


Ratio 


of Beam, 


M\ 


tral Axis. 


Rods. 


a 


a-^h. 


at 150 


Ft.-Lbs. 


Ins. 


Sq. Ins. 


d 


Ins. 


Lbs. per 
Cu. Ft. 
















Lbs. 


1 


8.2 


.375 


.0075 


.0075 


1.5 


.0050 


1.56 


2 


32.8 


.75 


.0150 






2.71 


.0055 


2.82 


3 


73.8 


1.125 


.0225 






3.87 


.0058 


4.03 


4 


131.2 


1.50 


.03 






5. 


.0060 


5.21 


5 


205.8 


1.875 


.0375 






6.12 


.0061 


6.38 


6 


295. 


2.25 


.045 






7.22 


.0062 


7.52 


7 


402. 


^ 2.625 


.0525 






8.32 


.0063 


8.67 


8 


525. 


3.00 


.06 






9.41 


.0064 


9.80 


9 


664. 


3.375 


.0675 






10.5 


.0064 


10.94 


10 


820. 


3.75 


.075 






11.58 


.0065 


12.06 


11 


902. 


4.125 


.0825 






12.66 


.0065 


13.19 


12 


1181. 


4.50 


.09 






13.73 


.0066 


14.30 


14 


1608. 


5.25 


.105 






15.87 


.0066 


16.53 


16 


2100. 


6.00 


.12 






18. 


.0067 


18.75 


18 


2658. 


6.75 


.135 






20.12 


.0067 


20.96 


20 


3281. 


7.50 


.15 






22,24 


.0067 


23.17 


22 


3970. 


8.25 


.165 






24.35 


.0068 


25.36 


24 


4725. 


9.00 


.18 


<( 


26.45 


.0068 


27.55 



Formulas for above table, reduced from General Formulas, preceding, 
are: y-=id\ a= .02xy= .0075t/; M' = 8.203 d^. These values will vary, 
of course, with variations in the values of /, F, and k. 

Suggested Formulas for Reinforced Concrete Construction (From 
Majority Report of Special Committee of Am. Soc. C. E. on Concrete and 
Reinforced Concrete. Proc. Am. Soc. C. E., Feb., 1909). — ^These formulas 
are based upon the assumptions and principles given in the chapter on De- 
sign (see Trans. A. S. C. E., Vol. LXVI). For Working Stresses, see: 
Sec. 31, Beams, page 585; Sec. 32, Columns, page 609. 

A. Standard Notation. 

a. Rectangular Beams. 

fg = tensile unit stress in steel. 

JPc = compressive unit stress in concrete. 
£9 = modulus of elasticity of steel. 
£0 = modulus of elasticity of concrete. 

iW = moment of resistance, or bending moment in general. 
A = steel area. 

6 = breadth of beam. 

(i = depth of beam to center of steel. 

yfe = ratio of depth of neutral axis to effective depth, d. 

0= depth of resultant compression below top. 

7 = ratio of lever arm of resisting couple to depth, d, 
jd = d — z = aTm of resisting couple. 

^ = steel ratio (not percentage). 

T«Beams. 

6 = width of flange. 
6'= width of stem. 
^ = thickness of flange. 



FORMULAS FOR REINFORCED CONCRETE. 



447 



Beams Reinforced for Compression. 

A' = area, of compressive steel. 
^' = steel ratio for compressive steel, 
/s' = unit compressive stress in steel. 
C'= total compressive stress in concrete. 
C = total compressive stress in steel. 
(i' = depth to center of compressive steel. 
2 = depth to resultant of C and C . 
Shear and Bond. 
1/ = total shear. 
z; = shearing unit stress. 
ii = bond stress per unit area of bar. 
o = circumference or perimeter of bar. 
2*0= sum of the perimeters of all bars. 
Columns. 

^ = total net area. 
.4s = area of longitudinal steel. 
.Ac = area of concrete. 
P = total safe load. 

B. Formulas. 
Rectangular Beams. 

,^ '- 



a. 




—4 



Fig. 12. 
Position of neutral axis, 



Arm of resisting couple, 



/=i-4-^ 



For /s= 15 000 to 16 000 and /e= 600 to 650, / may be taken 

M ^ M . . ^_2M_^2ph 
Ajd pjbd^' '' 

Steel ratio, P = 



atf.] 



Fiber stresses, /s 



jkhd^ k 



b, T>Beams. 



7c \nfc / 



kf. 


b.. 


4 


1 


4- 






• •• 






r 




^—lf-->. 


» 




Fig. 13. 



448 



2b.— MASONRY. 




Case I, When the neutral axis lies in the flange, use the formulas for 
rectangular beams. 

Case II. When the neutral axis lies in the stem. 

The following formulas neglect the compression in the stem: 

Position of neutral axis, 

2nA+2ht ' 
Position of resultant compression, 

_ Skd-2t t_ 

^~ 2kd-t ' 3* 

Arm of resisting couple. 



Fiber stresses, 



/s = 



M 



U = 



jd = d — z. 
Mkd 



fs 



Ajd '" ht{kd-\i)jd n l-k 

(For approximate results, the formulas for rectangular beams may be 
used.) 

The following formulas take into account the compression in the stem; 
they are recommended where the flange is small compared with the stem. 

Position of neutral axis, 



kd = 



V 



2ndA+{h-h')t^ ^ (nA-\-{h-b')t\^ nA + {h-U)t 



Y- 



h' \ h' 

Position of resultant compression, 

^ {kdt^ - f ^3) b-\-{kd- 02 [t + \ {kd- t)W 

^ t{2kd-'t)b+{kd-tW 

Arm of resisting couple, 

jd = d — z. 

Fiber stresses, 

^ M ^ 2 Mkd 

'" Ajd' "" [{2kd-t)ht+{kd-tYb']id 
c. Beams Reinforced for Compression. 

fc 



cz^^-SE 



I 

I m 



^■ 



I 
I 




Fig. 14. 



Position of neutral axis, 



k=J2n(^p-\-p'^^ ^nHp^p'y-n{p + p'), 
Position of resultant compression, 

Wd + 2p'nd' (k-j) 

k^-\-2p'n(k-^) 

id = d — z. 



Arm of resisting couple, 



REIN. -CONG. FORMULAS. MIXED MASONRY. 449 

Fiber stresses ' 

. eMk 



, _ M 1-k 

d. Shear, Bond, and Web Reinforcement. 

In the following formula, Iq refers only to the bars constituting the 
tension reinforcement at the section in question and / d is the lever arm of 
the resisting couple at the section. 

For rectangular beams, 

V 



u=- 



jd.Io 

[For approximate results, / may be taken at |.] 

The stresses in web reinforcement may be estimated by using the 
following formulas: 

Vertical reinforcement, 

jd 
Reinforcement inclined at 45°, 

in which P = stress in single reinforcing member, F = proportion of total 
shear assumed as carried by the reinforcement, and s = horizontal spacing 
of the reinforcing members. 

The same formulas apply to beams reinforced for compression as re- 
gards shear and bond stress for tensile steel. 
For T-beams, 

V V 



b'jd' jd . Jo 

[For approximate results, / may be taken at |.] 

e. Columns. 

Total safe load, 

P=f.(A. + nAs)==fcAa + {n-l)p). 
Unit stresses, 

f = P 

^^ A{l + (n-l)p) 

V. MIXED MASONRY. 

^ The object of using mixed masonry is to give a substantial looking, 
finished face, using a better class of masonry for this purpose, and using a 
cheaper quality for interior and back of wall. Strictly speaking, most 
masonry is more or less mixed — ashlar backed with rubble, face-brick 
backed with common brick, etc.; but the term " mixed masonry " is gen- 
erally restricted to stone facing with brick backing. 

The bond between stone and brick, or stone and concrete, is often a 
source of weakness. When interior brick walls are joined to stone face- 
walls, iron " cramps " are generally employed. 



450 25.— MASONRY. 

VI.— CONCRETE=BLOCK MASONRY. 

Solid concrete blocks are often used in submarine work, as for instance 
in breakwater construction, in preference to laying the concrete in situ. 

Hollow concrete blocks are made in many shapes and are used in build- 
ing construction. The completed wall should generally be not less than 
two-thirds solid, although in some places as much as 40- to 50 per cent, of 
voids is allowed. The surface of the blocks should be rich in cement and 
have a finer aggregate than the interior. The best blocks are made by 
machine under heavy (usually hydraulic) pressure. The writer has seen 
many blocks turned out by the small, portable machines in which hand 
tamping is required, and the results were invariably inferior. The blocks 
are laid in the wall in cement mortar, and sometimes iron cramps are used 
for bonding them together. Outside cement plaster or stucco of excellent 
quality may be laid directly on the blocks, and gives a finished appearance. 
Inside plaster should not be laid without furring and lathing, if the blocks 
are porous or inferior in quality on account of moisture and frost, unless 
the blocks have been waterproofed. 

Specifications for Hollow Concrete Building Blocks. — ^The following is 
from the Rules and Regulations governing the use and manufacture of 
Hollow Concrete Building Blocks in the City of Philadelphia — Bureau of 
Building Inspection. 

Rules and Regulations. 
f^' 1. Hollow concrete building blocks may be used for buildings 6 stories 
or less in height, where said use is approved by the Bureau of Building 
Inspection; provided, however, that such blocks shall be composed of at 
least 1 part standard Portland cement, and not to exceed 5 parts clean, 
coarse, sharp sand or gravel, or a mixture of at least 1 part Portland cement 
to 5 parts crushed rock or other suitable aggregate. Provided, further, 
that this section shall not permit the use of hollow blocks iii party walls. 
Said party walls must be built solid. 

2. All material to be of such fineness as to pass a J-in. ring and be free 
from dirt or foreign matter. The material composing such blocks shall 
be properly mixed and manipulated, and the hollow space in said blocks 
shall not exceed the percentage given in the following table for different 
height walls, and in no case shall the walls or wefbs of the block be less in 
thickness than i of the [height. The figures given in the table below, 
represent the percentage of such hollow space for different height walls: 

Stories. 1st. 2nd. 3rd. 4th. 5th. 6th. 

land 2 33 33 

3 and 4... 25 33 33 33 

5 and 6 . . . 20 25 25 33 33 33 

3. The thickness of walls for any building where hollow concrete blocks 
are used shall not be less than is required by law for brick walls. 

4. Where the face only is of hollow concrete building block, and the 
backing is of brick, the facing of hollow concrete blocks must be strongly 
bonded to the brick either with headers projecting 4 ins. into the brick- 
work, every fourth course being a heading course, or with approved ties; 
no brick backing to be less than 8 ins. Where the walls are made entirely 
of hollow concrete blocks, but where said blocks have not the same width 
as the wall, every fifth course shall extend through the wall, forming a 
secure bond. All walls, where blocks are used, shall be laid up in Portland 
cement mortar. 

5. All hollow concrete building blocks, before being used in the con- 
struction of any building in the City of Philadelphia, shall have attained 
the age of at least 3 weeks. 

6. Wherever girders or joists rest upon walls so that there is a concen- 
trated load on the block of over 2 tons, the blocks supporting the girder or 
joists must be made solid. Where such concentrated load shall exceed 5 
tons, the blocks for 2 courses below, and for a distance extending at least 
18 ins. each side of said girder, shall be made sohd. Where the load on the 
wall from the girder exceeds 5 tons, the blocks for 3 courses beneath it shall 
be made solid with similar material as in the blocks. Wherever walls are 
decreased in thickness, the top course of the thicker wall to be solid. 



SPEC'NS FOR HOLLOW CONCRETE BLOCKS. 451 

7. Provided always, that no wall, or any part thereof, composed of 
hollow concrete blocks shall be loaded to an excess of 8 tons per superficial 
foot of the area of such blocks, including the weight of the wall, and no blocks 
shall be used that have an average crushing at less than 1000 pounds per 
square inch of area at the age of 28 days; no deduction to be made in figuring 
the area for the hollow spaces. 

8. All piers and buttresses that support loads in excess of 5 tons, shall 
be built of solid concrete blocks for such distance below as may be required 
by the Bureau of Building Inspection. Concrete lintels and sills shall be 
reinforced by iron or steel rods in a manner satisfactory to the Bureau of 
Building Inspection, and any lintels spanning over 4 feet six inches in the 
clear shall rest on solid concrete blocks. 

9. Provided, that no hollow concrete building blocks shall be used in 
the construction of any building in the City of Philadelphia, unless the maker 
of said blocks has submitted his product to the full test required by the 
Bureau of Building Inspection, and placed on file with said B. of B. I. a 
certificate from a reliable testing laboratory showing that samples from the 
lot of blocks to be used have successfully passed the requirements of the 
B. of B. I., and filing a full copy of the test with the Bureau. 

10. A brand or mark of identification must be impressed in, or otherwise 
permanently attached to, each block for purpose of identification. 

11. No certificate of approval shall be considered in force for more than 
four months, unless there be filed with the B. of B. I., in the City of Phila., 
at least once every four months following, a certificate from some reliable phys- 
ical testing laboratory showing that the average of three (3) specimens 
tested for compression, and three (3) specimens tested for transverse strength, 
comply with the requirements of the B. of B. I.; samples to be selected 
either by a Building Inspector or by the laboratory, from blocks actually 
going into construction work. Samples must not be furnished by the 
contractors or builders. 

12. The manufacturer and user of any such hollow concrete blocks as 
are mentioned in this regulation, or either of them, shall, at any and all 
times, have made such tests of the cements used in making such blocks, or 
such further tests of the completed blocks, or of each of these, at their own 
expense, and under the supervision of the B. of B. I., as the Chief of said 
Bureau shall require. 

13. The cement used in making said blocks shall be Portland cement, 
and must be capable of passing the minimum requirements as set forth in 
the *' Standard Specifications for Cement " by the American Society for 
Testing Materials. 

14. Any and all blocks, samples of which, on being tested imder the 
direction of the B. of B. I., fail to stand at 28 days the tests required by this 
regulation, shall be marked " condemned " by the manufacturer or user, 
and shall be destroyed. 

15. No concrete blocks shall be used in the construction of any building 
within the City of Phila. until they shall have been inspected, and average 
samples of the lot tested, approved and accepted by the Chief of Building 
Inspectors. 

Method of Testing Hollow Blocks. 

1. These regulations shall apply to all such new materials as are used 
in building construction, in the same manner and for the same purposes 
as stones, brick, concrete are now authorized by the Building Laws, when 
said new material to be substituted departs from the general shape and 
dimensions of ordinary building brick, and more particularly to that form 
of building material known as " Hollow Concrete Block," manufactured 
from cement and a certain addition of sand, crushed stone, or similar 
material. 

2. Before any such material is used in buildings, an application for 
its use and for a test of the same must be filed with the Chief of the B. of B. I. 
A description of the material and a brief outline of its manufacture and 
proportions of the materials used must be embodied in the application. 

3. The material must be subjected to the following tests: Transverse, 
Compression, Absorption, Freezing, and Fire. Additional tests may be 
called for when, in the judgment of the Chief of the B. of B. I., the same 



452 25.— MASONRY. 

may be necessary. All such tests must be made in some laboratory of 
recognized standing, under the supervision of the Engineer of the B. of B. 1. 
The tests will be made at the expense of the applicant. 

4. The results of the tests, whether satisfactory or not, must be placed 
on file in the B. of B. I. They shall be open to inspection upon application 
to the Chief of the Bureau, but need not necessarily be published. 

5. For the purposes of the tests, at least 20 samples of test pieces 
must be provided. Such samples must represent the ordinary commercial 
product. They may be selected from stock by the Chief of the B. of B. 
I., or his representative, or may be made in his presence, at his discretion. 
The samples must be of the regular size and shape used in construction. 
In cases where the material is made and used in special shapes and forms, 
too large for testing in the ordinary machines, smaller sized specimens 
shall be used as may be directed by the Chief of Building Inspection, to 
determine the physical characteristic specified in Section 3. 

6. The samples may be tested as soon as desired by the applicant, but 
in no case later than 60 days after manufacture. 

7. The weight per cubic foot of the material must be determined. 

8. Tests shall be made in series of at least five, except that in the fire 
tests a series of two (four samples) are sufficient. Transverse tests shall 
be made on full sized samples. Half samples may be used for the crushing, 
freezing, and fire tests. The remaining samples are kept in reserve, in case 
unusual flaws or exceptional or abnormal conditions make it necessary to 
discard certain of the tests. All samples must be marked for identification 
and comparison. 

9. The Transverse test shall be made as follows: The samples shall be 
placed flatwise on two rounded knife edge bearings set parallel seven inches 
apart. A load is then applied on top, midway between the supports, and 
transmitted through a similar rounded knife edge, until the sample is 
ruptured. The modulus of rupture shall then be determined by multi- 
plying the total breaking load in pounds by twenty-one (three times the 
distance between supports in inches) and then dividing the result thus 
obtained by twice the product of the width in inckes by the square of the 

depth in inches: R= ^ , „ . No allowance should be made in figuring the 
Z o a' 

modulus of ruptiu-e for the hollow spaces. 

10. The Compression test shall be made as follows: Samples must be 
cut from blocks so as to contain a full web section. The sample must be 
carefully measured, then bedded flat-wise in Plaster of Paris, to secure a 
uniform bearing in the testing machine, and crushed. The total breaking 
load is then divided by the area in compression in square inches. No 
deduction to be made for hollow spaces; the area will be considered as the 
product of the width by the length. 

11. The Absorption test must be made as follows: The sample is first 
thoroughly dried to a constant weight. The weight must be carefully 
recorded. It is then placed in a pan or tray of water, face downward, 
immersing it to a depth of not more than one-half inch. It is again carefully 
weighed at the following periods: Thirty minutes, four hours, and forty- 
eight hours, respectively, from the time of immersion, being replaced in 
the water in each case as soon as the weight is taken. Its compressive 
strength, while still wet, is then determined at the end of the forty-eight 
hours* period, in the manner specified in section 10. 

12. The Freezing test is made as follows: The sample is immersed, as 
described in section 11, for at least four hours, and then weighed. ^ It is 
then placed in a freezing mixture or a refrigerator, or otherwise subjected 
to a temperature of less than 15 degrees F. for at least .12 hours. It is then 
removed and placed in water, where it must remain for at least one hour, 
the temperature of which is at least 150 degrees F. This operation is 
repeated ten (10) times, after which the sample is again weighed while 
still wet from the last thawing. Its crushing strength should then be 
determined as called for in section 10. 

13. The Fire test must be made as follows: — Two samples are placed 
in a cold furnace in which the temperature is gradually raised to 1 700 degrees 
F. The test piece must be subjected to this temperature for at least 30 
minutes. One of the samples is then plunged in cold water (about 50 



SPEC'NS FOR HOLLOW CONCRETE BLOCKS. 453 

degrees to 60 degrees F.) and the results noted. The second sample is 
permitted to cool gradually in air, and the results noted. 

14. The following requirements must be met to secure the acceptance 
of the materials: The Modulus of Rupture for concrete blocks at 28 days 
old must average 150 and must not fall below 100 in any case. The ultimate 
compressive strength at 28 days must average 1000 lbs. per sq. in., and 
must not fall below 700 in any case. The percentage of absorption (being 
the weight of water absorbed divided by the weight of the dry sample), 
must not average higher than 15% and must not exceed 25% in any case. 
The reduction of compressive strength must not be more than 33i%, except 
that when the lower figure is still above 1000 lbs. per sq. in., the loss in 
strength may be neglected. The freezing and thawing process must not 
cause a loss in weight greater than 10%, nor a loss in strength of more than 
33|%; except that when the lower figure is still above 1000 lbs. per sq. in., 
the loss in strength may be neglected. The fire test must cause the material 
to disintegrate. 

15. The approval of any material is given only under the following 
conditions: a. A brand mark for identification must be impressed on, or 
otherwise attached to, the material, b. A plant for the production of the 
material must be in full operation when the official tests are made. c. The 
name of the firm or corporation and the responsible officers must be placed 
on file with the Chief of B. of B. I., and changes in same promptly reported. 
d. The chief of the B. of B. I. may require full tests to be repeated on sam- 
ples selected from the open market when, in his opinion, there is any doubt 
as to whether the product is up to the standard of these regulations, and 
the manufacturer must submit to the B. of B. I., once in at least every 4 
months, a certificate of tests showing that the average resistance of 3 speci- 
mens to cross breaking and crushing are not below the requirements of these 
regulations. Such tests must be made by some laboratory of recognized 
standing, on samples selected either by a Building Inspector or the labora- 
tory, from material actually going into construction, and not on ones 
furnished by the manufacturer, e. In case the results of tests made under 
this condition (d.) should show that the standard of these regulations is 
not maintained, the approval of this bureau to the manufacturer of said 
blocks will at once be suspended or revoked. 

EXCERPTS AND REFERENCES. 

Permeability of Concrete Under High Water Pressures (By J. B. 

Mclntyre and A. L. True. Thesis, Thayer School of Civ. Engr., April, 1902; 
Eng. News, June 26, 1902). — Extensive tables of tests, using different pro- 
portions of concrete, and pressures of 20, 40, and 80 lbs., time 2 hotirs, 
"Of the various mixtures we may safely choose either 1:2:4 or 1:2.5:4, on 
account of their simplicity and the ease with which they may be propor- 
tioned, either for hand or machine mixing. In extreme cases however, 
it might be advisable to use one of the richer mixtures." 

Notes on Concrete Construction in Government Fortifications — With 
Waterproofing Data (Report of Chief of Engrs. of U. S. A. for 1902; Eng. 
News, April 2, 1903). — Subjects treated are: "Tests to show suitability of 
various kinds of sand for use in concrete," by Capt. Harry Taylor; "Damp 
proofings for ceilings of gun emplacements," by Maj. G. W. Goethals; 
"Damp proofing sunken magazines and rooms," by Col. P. C. Hains; 
"Stoppage of leaks in concrete with linseed oil," by Capt. E. W. Van C. Lucas; 
"Stoppage of leaks in concrete with asphaltum and oil," by. Maj. W. T. 
Rossell and Capt. S. Crosby; "Asphaltum and alum and lye waterproofing," 
by Capt. W. C. Langfitt; "A sliding rupture caused by tarred paper water- 
proofing," by Maj. W. T. Rossell and Capt. S. Crosby. 

The Efficiency of Concrete=Mixing Machines (By Clarence Coleman. 
Eng. News, Aug. 27, 1903).— The Necessity of Thorough Mixing.— The 

amount of cement as determined for any concrete should always be weighed, 
not measured, as the volume of cement is a variable. The entire amount 
of cement should be added before mixing is commenced, and then mixed 
dry before water is added. The proper amount of water to be added re- 
quires judgment after visual inspection, as the hygrometric condition of 
the sand is a variable; hence the entire mass of the materials should be 
plainly visible to the person adding the water. Proportion of Water in 



454 



25.^MASONRY. 



I 



Concrete. — Gives a table showing the strength of mortar due to varjring 
proportions of water. Classification of Machines. — Batch Mixers (7 types 
described); Continuous Mixers (5 types described). Comparison of Con- 
crete Mixers. — Descriptions with illustrations. 

The Method of Finishing the Concrete Surfaces of Philadelphia 
Bridges (By H. H. Quimby. Eng. News, Feb. 4, 1904). — Remove the forms 
while the concrete is still "green," and simply wash the surface with water, 
squirting it on with a nozzle if the work is soft enough, or, if harder, using 
a scrubbing brush. If the cement is too hard tr wash off it must be cut by 
hard rubbing with a brick or a wooden float and sand, using plenty of water. 
The removal of the cement exposes the sand and grit or pebbles or stone — 
whatever the aggregate may be — leaving a surface that is mostly stone, 
and is probably as little subject to discoloration and cracks as stone, and 
as diirable as any plastic material can be made. 

Tests of Adhesion and Initial Stress of Steel in Concrete (By S. W. 

Emerson. Eng. News, Mar. 10, 1904). — Diagrams showing results of tests. 

Materials Required to Make Different Classes of Concrete for Con- 
neticut Ave. Bridge, Wash., D. C. (By W. J. Douglas and A. W. Dow. Eng. 
News. Mar. 10, 1904).— 



Class 


A. 
l:2:0:4i 

4.5 

9.0 

0.0 
20.25 
21.4 


B. 
1:2^:0:6 

4.5 
11.25 

0.0 
27.0 
27.66 


B. 

1:2|:3:3 

d.5 

11.25 
1*3.5 
13.5 
27.66 


C. 


Mixture (cem., sand, gravel, stone) 

Cement, cu. ft 


1:3:10:0 
<i.5 


Sand, cu. ft 

Gravel, cu. ft 

Broken Stone, cu. ft 

Yielded, rammed concrete, cu. ft 


IS. 5 

45.0 

0.0 

45.0 



Note that 4.5 cu. ft. (Vulcanite) cement = 1 bbl. = 4 bags =378.25 lbs. 

French Government Rules for the Design and Construction of Re- 
inforced Concrete (Eng. News, Mar. 21, 1907). — Discussion of formulas, etc. 

Expansion Joints in Concrete Structures, with Special Reference to 
Block Construction in Drydocks and Reservoirs (L. F. Bellinger. Eng. News, 
May 2, 1897). — Brightwood Reservoir, Wash., D. C. — Dimensions, 415 x 
300 X 20 ft. deep; plain concrete with a vertical outer face and a sloping 
inner face of concrete ; the vertical outer face having a heavy earth em- 
bankment against it. The concrete wall was built in alternate sections not 
exceeding 50 ft. long. Vertical key ways 6 ins. square, * of which was in 
each section, were built for expansion joints and first filled with aphalt. 
The concrete proportions were 1:2^:5. In placing the fresh concrete on 
that which was already set, the surface was carefully washed and rich 
mortar was placed in order to make a joint. The forms were left on from 
one to two days to harden; surfaces which were to be exposed to the air 
or water had a face of 1:2 mortar, 6 ins. thick, placed at the same time the 
concrete backing was placed. The concrete floor was laid in two layers, 
each 5 ins. thick, and laid in blocks 15 ft. square. In the joints was placed 
a 3-ply felt, up to bevels, which topped off each joint. After the shrinkage 
cracks appeared, these bevels were run full of asphalt and then 2 ins. of 1:2 
mortar was troweled over everything. The experience with this reservoir 
is interesting since the asphalt in the key ways cracked and leaked in cold 
weather, and in warm weather ran out beneath the water surface. The 
asphalt was finally taken out and replaced with carefully selected puddling 
clay and rammed into the key ways. The contraction of the concrete 
permitted this material also to run out through the expansion joints into 
the reservoir and caused leaks to appear through the embankment. After 
this occurred, the puddling clay was taken out and plain loam, with clay, 
sand, grass roots, dirt and fibrous material, were rammed into the key 
ways. This material has successfully prevented leaks up to the present 
time. Other. Data. — About a dozen other works described; also Expansion 
Joints in Drydocks (illus.); Block Construction; Key Ways, etc. 



MISCELLANEOUS DATA. 455 

The Use of Reinforced Concrete in Engineering Structures (Trans. 
A. S. C. E., Vol. LXI). — Discussions by E. P. Goodrich, Edwin Thacher, 
S. E. Thompson, W. H. Burr, T. K. Thompson, and others. 

Traveling Concrete Mixers (Eng. News, Aug. 5, 1909). — Illustrated: U. 
S. Steel Mixer Co. 

The Bonding of New to Old Concrete (By E. P. Goodrich. Trans. A. S. 
C. E., Vol. LXIV., Sept., 1909). — Article contains: Literature on the sub- 
ject; patented processes; published experiments; Goodrich's experiments. 

Making Concrete Waterproof (By Ira O. Baker. The Univ. of 111. 
•• Technogragh," No. 23, 1908-09; Eng. News, Oct. 7, 1909).— Description 
of alum-and-soap waterproofing compound, etc. 

The Compressive Strength of Coke Concrete (By J. M. Lewis. Eng. 
Rec, Oct. 30, 1909). — Comparison of tests of concrete columns made with 
stone and with coke cinder. Weight of coke-cinder concrete, 97 lbs. per 
cu. ft.; stone concrete, 150 lbs. per cu. ft. 

Impurities in Sand for Concrete (Informal Discussion — ^Trans. A. S. C. 
E., Vol. LXV., Dec, 19 09). — Sand tests, washing, etc. 

Office Methods in a Concrete Designing Office (Eng. Rec, June 11, 1910). 
— Illustrations: Beam design sheet, half of column design sheet, typical 
details for T-stirrups, upper part of splice rod detail sheet, upper part of 
beam detail sheet. 

The Effect of Alkali on Concrete (By G. G. Anderson. Trans. A. S. C. 
E., Vol. LXVIL, June, 1910). 

Corrosion of Iron Embedded in Concrete (By G. F. Shaffer. Eng. Rec, 
July 30, 1910). — Gives results of tests at Mass. Inst. Tech., obtaining some 
data on the effect of currents of low potential on embedded steel. 

Computation of Reinforced Concrete Flat Slabs (By L. F. Bray ton. Eng. 
Rec, Aug. 27, 1910). — Comparison with the McMillan method. 

Exterior Treatment of Concrete Surfaces : Committee Report to the Nat'I 
Assn. Cement Users (Eng. News, Sept. 15, 1910). — (a) Effect of material and 
workmanship on surface, (b) Removal of surface in various ways, (c) 
Coating surfaces in various ways, (d) Defects, blemishes of various sorts, 
and remedies, (e) Costs. 

Investigations on the Slip of Rods Imbedded in Concrete Beams ("Armitier 
Beton," Sept., 1910; Eng. Rec, Nov. 12, 1910). — Gives the slip at four points 
of the beams under incremental twin loadings. Beams are 12 x 12-ins., 
74-in. span. Rod, 0.94 in. dia. 

Oil=Mixed Concrete as a Waterproofing Material (By T. W. Symons, 
Eng. News, Dec 15, 1910). — "The peculiarly waterproof and water-repellant 
quality of this oil-concrete, combined with its strength and greater density 
renders it a material remarkably adapted to such canal structures as culverts, 
locks, the canal prism troughs crossing the Irondequoit Valley and the Me- 
dina Gorge, to the core walls of earthen dams like that at Hinckley, etc" 
Oil to the amount of about 10% of the weight of the cement gives very 
satisfactory results. The oil costs about 6 to 7 cents per gallon, or about 
40 to 50 cents per cu. yd. of concrete; or about 60 to 70 cents more per cu. 
yd. of concrete (including handling and incorporation) than plain concrete, 
in place. 

Specifications for Scrubbed Concrete Surface (By H. H. Quimby. Paper 
before Annual Conv. of Nat'I Cement Users, New York, Dec 10, 1910; Eng. 
News, Dec. 22, 1910). — Sent by that Convention to a letter ballot of the 
Association to be adopted as a standard of the Association. 



456 25.^MASONRY. 

Some Important Illustrations. 

Description. Eng. News. 
Spec, for plain and rein. -cone, and steel rein., A. R. E. & M.W.AApr. 14, '10 

Stand. Spec. (Assn. of Am. Steel Mf'rs) cone. rein. bars. June 16, '10 

Eng. Rec. 

Standard forms for a reinforced-concrete viaduct Feb. 13, '09 

Plant for washing and screening concrete aggregates June 26, '09 

A traveling shed for cold-weather viaduct concreting July 10, '09 

Tests of bond between steel and concrete. — H. C. Berry Sept. 4, '09 

Analysis of concrete-bridge failures. — C. R. Young Apr. 16, '10 

Clips and rods for tying concrete to steel members Sept. 3, '10 

Diagram of bending moments in concrete columns and beams Oct. 15, *10 



26.— STEREOTOMY. 

This term in its broadest sense includes the subject of Stone Cutting 
(see page 426.). but it is here restricted to the preparation of the drawings 
and sketches of the dimension stones, which enter into the masonry structure, 
previously designed. 

The drawings of the simpler shapes may be ordinary sketches in pro- 
jection, showing the principal faces with dimensions thereon, or descriptions 
of same; but isometric views should be shown of the more complex shapes. 
The dimensions of the shapes may be obtained (1) by analytic calculation, 
(2) by projection and development, (3) by Descriptive Geometry. All these 
methods commonly enter into the case of a single structure. Only a few 
hints can be given here, but these, it is hoped, will serve to illustrate a more 
extended application. 

Wall of Building. — Fig. 1 illustrates the simple method of showing the 
dimensions of stones on the Elevation plan. These dimensions are usually 
given on the drawing to center of joint with a note on plan to that effect, as 



48C 


) 


^9.- 1 


284 !^ 


.ll't 


286 


27^" . 
167 -te 


168 ^i^ 





Fig. 1. 

"Allow for ^^ joints." The thickness of joint may vary from about J^" up- 
ward, depending upon the quality of the masonry; -i^" is common for public 
buildings. In engineering structures i'^ is about the minimum, and from 
that up to Y for dimension work. Note that in Fig. 1 each stone is num- 




Fig. 2. 

bered. Fig. 2 is an isometric view of stone No. 63 (not shown in elevation) 
of the water-table. All the dimensions L, H, h, W and w should be 
placed directly on the drawing. L and H may be "finished dimensions" 
like the others, or they may be distances to center of joints. Proper notes 
should be made showing which system is used. 

Stone Arch. — Fig. 3 illustrates, briefly, two methods of showing the 
dimensions of the voussoirs for the stone cutter. Those shown in the end 
view (face of arch) are in planes perp to axis of arch, whether for right- 
or skew arch. The measurements shown may be (1) to center of joints as 
per the "numbered" vouissoirs, or (2) actual dimensions as per the "lettered" 
voussoirs (Fig. 3). Isometric views of stones "4" and "I" assume the arch 
to be skewed or oblique, with coursing joints parallel with axis of arch. 
(Such a construction is allowable only for slight skew, for small arches, or 
culverts. When the skew is considerable, especially for large spans, the 

457 



458 



2^.—STERE0T0MY, 



construction should be as per Figs. 6 or 7, page 764, Section 44.) The 
obliquity of faces of end voussoirs may be shown directly on the end view of 
arch by measurements in the corners o, h, c, showing the projection of t^tese 
corners beyond a right section passing through the comers o, assumed as 
zero. These projections may be obtamed graphically as illustrated, and 



RJghfSection 



RfghfSecfion 




JZI—Lg: i RacLfo"-' 



i 



Fig. 3. 

should appear also on the isometric views. A plan of soffit and also of back 
of arch should also be submitted showing the coursing joints, lengths of 
voussoirs (at least near ends of arch), and other information. The kind of 
finish desired should be marked plainly on plans. 



27.— WEIGHTS AND SPECIFIC GRAVITIES 
OF MATERIALS. 

(For Strength and Resistance of Materials, see Sec. 28.) 

DEFINITIONS. 

Mass (M) = Matter. — If two separate quantities of matter will balance 
each other in vacuuo they are said to have equal masses; and this regard- 
less of kind of matter, volume, or temperature. The only stipulation is that 
they shall be affected by the same gravity acceleration g, and this neces- 
sarily obtains when they are counterpoised at the same tirne, at the same 
elevation above sea level, and at the same parallel of latitude or place. 

weight ^iV 

Mass= T- :; -: — •', or M = — , the usual formula. 

gravity acceleration g 

The "unit^ of mass"_ (Mi) may be a definite and standard quantity of 
matter. For instance, if a certain quantity of a particular metal occupies 
one cubic inch (Ci) , and weighs one standard pound (T^i) at a point on the 
earth where the gravity acceleration (gi) corresponding to Wi, is equal to, 

W-, 1 

say, 32.16, — then its mass (M) is equal to — = ■ lb. Hence, the unit 

gi oZ.io 

of mass (Ml) of the metal would be equal to 32.16 cubic inches, or 32.16 lbs. 
at that particular locality. 

The "unit of mass" is equal to g pounds. 

Gravity Acceleration (g). — ^The value of g increases with the latitude of 
the place, and decreases with the elevation above sea level. The following 
formula* gives the value of g for any latitude and elevation: 

g= 32.172-0.082 cos 2A- 0.000003 /t (1) 

where g = acceleration in ft. per sec, per sec; 
A = latitude of the place in degrees; 
/j = elevation in feet above sea level. 
From this formula the values of g and Vg (used in Hydraulics) are as 
follows, for various latitudes, at sea level: 

Latitude 0° 10° 20° 30° 40° 45° 

_g 32.090 32^95 32.109 32.131 32.158 32.172 
\/2g 8.011 8.012 8.014 8.016 8.020 8.021 
The value of g may be designated as the intensity of gravity. 

Weight {W)=Mg. — ^The weight of a body depends upon its mass (M) 
and upon the intensity of gravity (g). If the mass is constant the weight 
will vary directly with g. From the preceding table we see that a mass 
weighing 32,131 lbs. in latitude 30°, would weigh 32,172 lbs. in latitude 45°; 
that is, the weight of the same mass would increase 41 lbs. in 15° of latitude, 
or a little more than 2\ lbs. per ton. This is so small as to be practically 
negligible in engineering calculations, and we generally assume for g its 
value in latitude 40°, namely, g= 32.16. It is to be noted, then, that the 
wieghts per cubic foot of substances given in the subjoined tables are practi- 
cally constant, in so far as the intensity of gravity alone is concerned, in all 
portions of the country. 

* In C. G. S. (Centimeter-Gram-Second) System, adopted by the British 
Association, the value of g in centimeters is sometimes given: g' = 980.6056 — 
2.5028 cos 2 ^-.000003/^'. See also formula for g under Simple Circular 
Pendulum, Mechanics, page 287. 

459 



460 21'-WEIGHTS AND SPECIFIC GRA VITIES OF MA TERIALS. 

Volume (V). — ^The unit of volume generally used in the United States 
is the cubic foot, and weights are given in pounds per cubic foot. The 
effect of temperature upon any mass is to increase its volume (ice excepted) ; 
hence, in giving the weights of those substances which expand materially 
with heat, the temperature should be stated. This is true especially with 
gases and, to a much less extent, with liquids. But with ordinary materials 
of engineering the temperature effect on volume is so slight, within the 
natiural range of the thermometer, as to be negligible. 

M 

The volume of any mass is inversely as its density; thus, V= j^. 

Then for any unit mass the volume and density are reciprocals of each other. 

Density (D). — ^The density of any body is the mass of a unit of its 
volume; thus in the equation M= VD, ii V=l, M = D. If, now, this unit 
mass is increased, say by temperature, to two units of volume, then will 
its density be equal to ^M, or 50 per cent of what it was. If, on the other 
hand, the unit mass is decreased, say by pressure, to | its original volume, 
then will its density be 1.25 M. 

The relative density of a substance is called its specific gravity, when 
referred to water (at maximum density — 4° Centigrade) . 

Specific Gravity (5. g.). — ^The specific gravity of a substance is its rela- 
tive density, to a unit standard. More definitely, it is the ratio of the 
weight of a given volume of the substance to the weight of the same volume 
of distilled water at 4°C (39.1 F.), its maximum density, equal to 62.424 
pounds per cubic foot, and considered as unity. Moreover, the specific 
gravities of other substances than water are assumed to be taken at 0°C., 
or if not they are reduced by corrections to that temperature. These are 
the standards of physicists. 

Experimenters, especially in the field of engineering, have not always 
reduced their results to the above standards, but have variously employed 
as standard units, uncorrected, 

Water at 0° C. or 32° F., equal to 62.416 lbs. per cu. ft. 

( •• •' 4°C. *• 39.1° F., " " 62.424 " " " ) 

" '• 15.56° C. " 60° F., " " 62.366 " " 
" " 16° C. " 60.8° F., " " 62.361 " " 
" " 16.67° C. " 62° F.. " " 62.355 " " 
And for the substances experimented with, the temperatures, when stated, 
have been as varied as the above. Hence, in the subjoined tables of specific 
gravities, extreme accuracy cannot be expected in many cases, but they are 
sufficiently exact for all engineering purposes. 

Specific gravity = weight in grams per cubic centimeter. 

METHODS FOR DETERMINING SPECIFIC GRAVITY. 

Solids Heavier than Water.— A common method for determining the 

specific gravity of a solid heavier than water is to weigh it in air and then 

weigh it in water; that is, while still in the scale it is immersed in water, 

where its weight will be found to be less. Tne specific gravity may then 

be expressed by the following formula, when temperature reduction is not 

considered: 

W 
Specific gravity = rr^ (2) 

W — w 

In which W = weight of the substance in air; 

w= " " " " ^ when immersed in water; 

W — If = "loss of weight" by immersion. 

Solids Lighter than Water maybe determined by finding the weight {W) 

in air, as above, and then the buoyancy or minus weight {w') in water, 

when completely immersed. Formula (2) then reduced to (3), as follows: 

W 
Specific gravity = .^_^^, (3) 

In which w' = —w = force required to immerse the body. 

Displacement Method. — This consists in immersing the substance to be 
determined, in a vessel full of water. Hence, 

j^ .-j . _ weight of the substance ... 

weight of water displaced 

Porous substances whose specific gravities are to be determined should 
be painted with a thin coat of varnish before being immersed, in order to 
exclude all moisture. 



METHODS FOR DETERMINING SPECIFIC GRAVITY. 461 



Granular substances may have two specific gravities, namely, in bulk 
and in granule. Substances which are affected by water should be weighed 
in some other liquid which will not affect them and whose specific gravity is 
known. Alcohol, turpentine and benzine are often used for this purpose. 
Then, the specific gravity obtained with respect to the particular liquid 
must be multiplied by the specific gravity of the liquid itself, to find the 
true specific gravity. 

For determining the specific gravity of cement, see description of 
method, page 407, under Building Stones and Cements, Sec. 22. 

Liquids.'*' — The practical determination of the specific gravities of 
liquids may be made with instruments called hydrometers. They consist 
usually of a glass tube (or wire) so arranged that it will stand vertically 
when partly immersed in a liquid. The depth of immersion, registered 
by a scale on the tube, or the weight required to immerse the tube to a 
certain fixed mark upon it, with reference to the surface of the liquid in 
which it is imm.ersed, is the basis on which the specific gravity is deter- 
mined. "Scale" hydrometers are of variable immersion and constant 
weight; "fixed-mark" hydrometers are of constant im- 
mersion and variable weight. Some hydrometers are 
specially adapted to determining liquids heavier than 
water; and some to determine liquids lighter _ than 
water. Again, some hydrometers are immersed in the 
liquid whose specific gravity is to be determined; while 
others are immersed in a standard liquid, and are pro- 
vided with a cup at the top of the tube to receive the 
liquid which is to be determined. Other forms of instru- 
ments, perhaps not strictly hydrometers, are used to de- 
termine the purity or adulteration of various liquids, as 
spirits, solutions, milk, urine, etc. The relative density to 
some standard of purity is the basis of the determination. 



Beaume's hydrometer, Fig. 1, is a "scale" hydrometer 
of variable immersion, which is immersed in the liquid 
to be determined. The graduation of the scale for 
liquids lighter than water is different than for those 
heavier than water. The graduation is standardized by 
the depth of immersion (1) in pure water for one point on 
the scale, and (2) in a saline solution of known strength 
for another point. The distance between the two points 
is then graduated, and the graduation extended be- 
yond either point when necessary. The lower end of the 
tube is loaded with mercury, and a bulb is blown 
above it. 

The following relations exist between Beaume's Hydrometer 
the corresponding Specific Gravity desired: 




. 1. 

scale and 



Beaum^ (deg.) 





10 


20 


30 


40 


50 


60 






Liquid heavier than water 

Liquid lighter than water 


1.000 


1.070 
1.000 


1.152 
.936 


1.246 
.880 


1.357 
.830 


1.490 

.785 


1.652 
.745 









Many hydrometers now record the specific gravities directly. 

Tweddell's hydrometer is a scale hydrometer for determining liquids 

heavier than water. It is graduated in degrees D° such that 

„ .^ .^ 5D°+1000 ,_, 

Specific gravity = :^^ (5) 

Rousseau's densimeter is for variable immersion in a standard liquid. 
It is constructed somewhat similar to Beaume's (Fig. 1) but has in addi- 
tion a tube or cup at the top of the stem which contains the liquid to be 
determined. Hence, it is specially adapted to determining specific gravities 
of small quantities of liquids. 

* The specific gravity of a liquid may be obtained directly by weighing 
equal volumes of the liquid and of water, dividing the weight of the former 
by that of the latter. 



462 21— WEIGHTS AND SPECIFIC GRA VITIES OF MA TERIALS. 

Nicholson's and Fahrenheit's hydrometers are hydrometers of constant 
immersion and variable weight. They differ from the above. For instance, 
Nicholson's hydrometer consists of a hollow metal float, always submerged, 
below which is suspended a dish loaded with weights. Above the float is 
supported a shallow dish on a thin vertical wire. On this wire is a mark 
which is brought to the surface of the liquid by the weights in the upper 
and lower dishes. These weights determine the specific gravity of the 
liquid in which the hydrometer is immersed. 

Refinements. — In laboratory work where great refinement is necessary 
the exact determination of specific gravities involves quite intricate formulas 
and extremely careful observations. The formulas include reduction of 
weighings to vacuuo, and various temperature reductions. 

Gases. — When we speak of the specific gravity of a gas we must have 
clearly in mind its pressure and temperature, and to what standard it is 
referred. The specific gravity of a unit mass of gas varies directly as its 
density and inversely as its volume. For constant temperature the density 
varies directly as the pressure, for constant pressure the density varies 
inversely as the temperattire, meaning of course the temperature above 
absolute zero.* 

The "standard" pressure of the gas determined is taken at, or reduced 
to, one "standard" atmosphere. Now a standard atmospheric pressure is 
about 14.7 lbs. per square inch, near enough for all practical engineering 
work. But it is not a fixed quantity. In French units it is assumed as a 
pressure equivalent to a column of mercury at 0° C. (32° F.), 760 mm ( = 0.76 
meter= 29.921 ins.) in height, and acted upon by an intensity of gravity, g, 
equal to that of Paris. This value of g has been determined as equal to an 
acceleration of 980.94 c m ( = 9.8094 meters = 32.183 ft.) per sec. Now the 
specific gravity of mercury at 0° C. is taken at 13.596 (Regnault's determin- 
ation, 13.5959; commonly assumed at 13.6); hence the pressure of a stand- 
ard atmosphere at Paris= 76X 13.596= 1033.3 grams per sq. centimeter = 
10.333 kilograms per sq. meter (=14.697 lbs. per sq. in. = 2116.37 lbs. per 
sq. ft.). It is thus seen that the atmospheric pressure may be stated in 
terms of a column of mercury, or as a pressure per unit of area; and that these 
values are dependent on the temperature (mainly affecting the density of 
the mercury, and very slightly the air), the specific gravity of the mercury 
(affected by temperature and value of g), the value of g (affected by latitude, 
and elevation obove sea level), the latitude, and the distance above sea 
level. In view of this, English units are often used, with round imits of 
16° C, 30 ins. of mercury, and 14.7 lbs. per sq. ft. Thus: 

In English units the "standard" pressure is often assumed equivalent to 
a column of mercury at 60° to 62° F. (16° C), 30 inches (762 milli- 
meters) in height, and acted upon by an intensity of gravity g equal 
to that at 45° latitude. This gives a pressure practically the same as 
that derived from the French standard, namely, 14.7 lbs. per sq. in. 

The standard temperature of the gas whose specific gravity is determined 
should be taken at or reduced to 0° C. (32° F.) for French units, or 60° to 
62° F. (16° C.) for English xmits. The temperature reduction should be 
stated in all cases. 

The standard substance referred to in determining the specific gravities 
of gases is air at 0° C. (32° F.), with the barometer at 760 m m (29.921 ins.) 
or one standard atmospheric pressure. Sometimes air at 60° or 62° F. is 
used. The specific gravity of the air multiplied by the sp grav of the gas 
referred to it = the sp grav of the gas with reference to water. Water at its 
maximum density, 4° C, is the standard, but sometimes 60° or 62° F. is 
used. 

One cubic foot of air at 0° C. ( 32° F.) weighs 0.0807 lb. 

" " 4°C. (39.1° F.) " 0.0796 " 

" " 15.56° C.( 60° F.) " 0.0764" 

" " 16° C. (60.8° F.) " 0.0763 " 

" " 16.67° C. ( 62° F.) " 0.0761 " 

" water" 0° C. ( 32° F.) " 62.416 " 

" •• 4°C. (39.1° F.) " 62.424 " 

" " 15.56° C.( 60° F.) " 62.366 " 

" " 16° C. (60.8° F.) " 62.361 " 

" " 16.67° C. ( 62° F.) " 62.355 " 

* Absolute zero (no heat) is 273.7° C. below 0° C, or 460.7° F. below 0° F. 



I 



GASES— AIR, 



463 



GASES. 

1. — Weight of a Cubic Foot of Dry Air at Various Temperatures.* 
(At Atmospheric Pressure— 14.7 lbs. per sq. in.) 



Temper- 
ature 
(Deg.F.) 


Weight 
(Lbs.) 


Temper- 
ature 
(Deg. F.) 


Weight 
(Lbs.) 


Temper- 
ature 
(Deg. F.) 


Weight 
(Lbs.) 


Temper- 
ature 
(Deg. F.) 


Weight 
(Lbs.) 





.0863 


50 


.0779 


100 


.0710 


15D 


.0652 


1 


.0862 


51 


.0777 


101 


.0708 


151 


.0650 


2 


.0860 


52 


.0776 


102 


.0707 


152 


.0649 


3 


.0858 


53 


.0774 


103 


.0706 


153 


.0648 


4 


.0856 


54 


.0773 


104 


.0705 


154 


.0647 


5 


.0854 


55 


.0771 


105 


.0703 


155 


.0646 


6 


.0852 


56 


.0770 


106 


.0702 


156 


.0645 


7 


.0851 


57 


.0768 


107 


.0701 


157 


.0644 


8 


.0849 


58 


.0767 


108 


.0700 


158 


.0643 


9 


.0847 


59 


.0766 


109 


.0698 


159 


.0642 


10 


.0845 


60 


.0764 


110 


.0697 


160 


.0641 


11 


.0843 


61 


.0763 


HI 


.0696 


161 


.0640 


12 


.0842 


62 


.0761 


112 


.0695 


162 


.0639 


13 


.0840 


63 


.0760 


113 


.0694 


163 


.0638 


14 


.0838 


64 


.0758 


114 


.0692 


164 


.0637 


15 


.0836 


65 


.0757 


115 


.0691 


165 


.0636 


16 


.0834 


66 


.0755 


116 


.0690 


166 


.0635 


17 


.0833 


67 


.0754 


117 . 


.0689 


167 


.0634 


18 


.0831 


68 


.0752 


118 


.0688 


168 


.0633 


19 


.0829 


69 


.0751 


119 


.0686 


169 


.0632 


20 


.0828 


70 


.0750 


120 


.0685 


170 


.0631 


21 


.0826 


71 


.0748 


121 


.0684 


171 


.0630 


22 


.0824 


72 


.0747 


122 


.0683 


172 


.0629 


23 


.0822 


73 


.0745 


123 


.0682 


173 


.0628 


24 


.0821 


74 


.0744 


124 


.0680 


174 


.0627 


25 


.0819 


75 


.0743 


125 


.0679 


175 


.0626 


26 


.0817 


76 


.0741 


126 


.0678 


176 


.0625 


27 


.0816 


77 


.0740 


127 


.0677 


177 


, 0624 


28 


.0814 


78 


.0739 


128 


.0676 


178 


.0623 


29 


.0812 


79 


.0737 


129 


.0675 


179 


,0622 


30 


.0811 


80 


.0736 


130 


.0674 


180 


.0621 


31 


.0809 


81 


.0734 


131 


.0672 


181 


.0620 


32 


.0807 


82 


.0733 


132 


.0671 


182 


.0619 


33 


.0806 


83 


. 0732 


133 


.0670 


183 


.0618 


34 


.0804 


84 


.0730 


134 


.0669 


184 


.0617 


35 


.0803 


85 


.0729 


135 


.0668 


185 


.0616 


36 


.0801 


86 


.0728 


136 


.0667 


186 


.0615 


37 


.0799 


87 


.0726 


137 


.0666 


187 


.0614 


38 


.0798 


88 


.0725 


138 


.0665 


188 


.0613 


39 


.0796 


89 


.0724 


139 


.0663 


189 


.0612 


40 


.0795 


90 


.0722 


140 


.0662 


190 


.0612 


41 


.0793 


91 


.0721 


141 


.0661 


191 


.0611 


42 


.0791 


92 


.0720 


142 


.0660 


192 


.0610 


43 


.0790 


93 


.0719 


143 


.0659 


193 


.0609 


44 


.0788 


94 


.0717 


144 


.0658 


194 


.0608 


45 


.0787 


95 


.0716 


145 


.0657 


195 H 


.0607 


46 


.0785 


96 


.0715 


146 


.0656 


196 


.0606 


47 


.0784 


97 


.0713 


147 


.0655 


197 


.0605 


48 


.0782 


98 


.0712 


148 


.0654 


198 


.0604 


49 


.0781 


99 


.0711 


149 


.0653 


199 


.0603 


50 


.0779 


100 


.0710 


150 


.0652 


200 


.0602 



* For comparison of Fahrenheit with Centigrade scale see Table 3. 



464 27 —WEIGHTS AND SPECIFIC GRAVITIES OF MATERIALS. 



2. — ^Weights and Specific Gravities of Gases. 



Substance under 1 atmosphere. 



Name. 



Temp. 



Relative 
Density 
to Air. 



Relative 
Density 

to Water. 

tSp. Grav. 



Weight 

per 

Cubic 

Foot. 

Lbs. 



Coef. of 
Expan- 
sion per 
Degree 
J Cent. 



Air, dry (see Table 1). 

Ammonia , 

Binoxide of Nitrogen. 

Carbonic acid 

Carbonic oxide 

Chlorine 

Cyanogen 

Hydrogen 

Marsh gas 

Nitrogen 

defiant gas 

Oxygen 

Protoxide of nitrogen. 

Steam (ideal) 

Sulphurous acid 



0°C. 



o°c. 



1.0000 

.5967 
1.0388 
1.52901 

.9569 
2.4216 
1.8064 

.06926 

.559 

.97137 

.985 
1.10563 
1.5269 

.6221 
2.1930 



.001,293,2 

.000,769,7 

.001,343,4 

.001,977,4 

.001,234,4 

.003,132,8 

.002,330,2 

.000,089,57 

.000,727,0 

.001,256,15 

.001,274,0 

.001,429,8 

.001,969,7 

.000,804,5 

.002,728.9 



.08071 
.04805 
.08387 
.12345 
.07706 
.19558 
.14547 
.00559 
.04539 
.07842 
.07953 
.08926 
.12297 
.05022 
.17036 



,003666 



003665 



003900 



* Air at 0° C. under 760 mm of mercury at Paris, unless otherwise 
stated. 

t Water at 4** C. under 760 w m of mercury at Paris, unless otherwise 
stated. 

Specific gravity of a substance is its weight in grams per cubic centi- 
meter. 

% Coef. of expansion for low temperatures approaches ^ per degree C, 
or ^02^ per degree F. 



LIQUIDS— WATER. 



465 



LIQUIDS. 

3. — Weight op a Cubic Foot of Water at Various Temperatures. 
(Also comparison of Fahr. and Cent. Scales.) 



Deg.C 


. Add: 


.0 


.06 


.11 


.17 


.22 


.28 


.33 


.39 


.44 


.50 




Deg. F. 


.0 


.1 


.2 


.3 


.4 


.5 


.6 


.7 


.8 


.9 





32 


62.416 


62.416 


62.416 


62.417 


62.417 


62.417 


62.417 


62.417 


62.418 


62.418 


.56 


33 


.418 


.418 


.418 


.419 


.419 


.419 


.419 


.419 


.420 


.420 


1.11 


34 


.420 


.420 


.420 


.420 


.420 


.421 


.421 


.421 


.421 


.421 


1.67 


35 


62.421 


62.421 


62.421 


62.421 


62.421 


62.422 


62.422 


62.422 


62.422 


62.422 


2.22 


36 


.422 


.422 


.422 


.422 


.422 


.423 


.423 


.423 


.423 


.423 


2.78 


37 


.423 


.423 


.423 


.423 


.423 


.423 


.423 


.423 


.423 


.423 


3.33 


38 


.423 


.423 


.423 


.423 


.423 


.423 


.423 


.424 


.424 


.424 


3.89 


39 


.424 


*.424 


*.424 


.424 


.424 


.424 


.424 


.424 


.424 


423 


4.44 


40 


62.423 


62.423 


62.423 


62.423 


62.423 


62.423 


62.423 


62.423 


62.423 


62.423 


5.00 


41 


.423 


.423 


.423 


.423 


.423 


.423 


.423 


.423 


.423 


.423 


5.56 


42 


.423 


.423 


.423 


.423 


.423 


.423 


. 422 


.422 


.422 


.422 


6.11 


43 


.422 


.422 


.422 


.421 


.421 


.421 


.421. 


.421 


.420 


.420 


6.67 


44 


.420 


.420 


.420 


.420 


.420 


.420 


.419 


.419 


.419 


.419 


7.22 


45 


62.419 


62.419 


62.419 


62.419 


62.419 


62.419 


62.418 


62.418 


62.418 


62.418 


7.78 


46 


.418 


.418 


.418 


.417 


.417 


.417 


.417 


.417 


.416 


.416 


8.33 


47 


.416 


.416 


.415 


.415 


.415 


.415 


.414 


.414 


.414 


.413 


8.89 


48 


.413 


.413 


.413 


.412 


.412 


.412 


. 412 


.412 


.411 


.411 


9.44 


49 


.411 


.411 


.410 


.410 


.410 


.410 


.409 


.409 


.409 


.408 


10.00 


50 


62.408 


62.408 


62.407 


62.407 


62.407 


62.407 


62.406 


62.406 


62.406 


62.405 


10.56 


51 


.405 


.405 


.404 


.404 


.404 


.404 


.403 


.403 


.403 


.402 


11.11 


52 


.402 


.402 


.401 


.401 


.400 


.400 


.400 


.399 


.399 


.398 


11.67 


53 


.398 


.398 


.397 


.397 


.396 


.396 


.396 


.395 


.395 


.394 


12.22 


54 


.394 


.394 


.393 


.393 


.392 


.392 


.392 


.391 


.391 


.390 


12.78 


55 


62.390 


62.390 


62.389 


62.389 


62.388 


62.388 


62.388 


62.387 


62.387 


62.386 


13.33 


56 


.386 


.386 


.385 


.385 


.384 


.384 


.383 


.383 


.382 


.382 


13.89 


57 


.381 


.381 


.380 


.380 


.379 


.379 


.379 


.378 


.378 


.377 


14.44 


58 


.377 


.377 


.376 


.376 


.375 


.375 


.374 


.374 


.373 


.373 


15.00 


59 


.372 


.371 


.371 


.370 


.370 


.369 


.368 


.368 


.367 


.367 


15.56 


t60 


62.366 


62.365 


62.365 


62.364 


62.364 


62.363 


62.362 


62.362 


62.361 


62.361 


tl6.11 


61 


.360 


.360 


.359 


.359 


.358 


.358 


.358 


.358 


.357 


.357 


16.67 


t62 


.355 


.354 


.354 


.353 


.353 


.352 


.351 


.351 


.350 


.350 


17.22 


63 


.349 


.348 


.348 


.347 


.346 


.346 


.345 


.344 


.343 


.343 


17.78 


64 


.342 


.341 


.341 


.340 


.340 


.339 


.338 


.338 


.337 


.337 


18.33 


65 


62.336 


62.335 


62.335 


62.334 


62.333 


62.333 


62.332 


62.331 


62.330 


62.330 


18.89 


66 


.329 


.328 


.328 


.327 


.326 


.326 


.325 


.324 


.323 


.323 


19.44 


67 


.322 


.321 


.321 


.320 


.319 


.319 


.318 


.317 


.316 


.316 


20.00 


68 


.315 


.314 


.314 


.313 


.312 


.312 


.311 


.310 


.309 


.309 


20.56 


69 


.308 


.307 


.306 


.306 


.305 


.304 


.303 


.302 


.302 


.301 


21.11 


70 


62.300 


62.299 


62.299 


62.298 


62.297 


62.297 


62.296 


62.295 


62.294 


62.294 


21.67 


71 


.293 


.292 


.291 


.291 


.290 


.289 


.288 


.287 


.287 


.286 


22.22 


72 


.285 


.284 


.283 


.283 


.282 


.281 


.280 


279 


.279 


.278 


22.78 


73 


.277 


.276 


.275 


.275 


.274 


.273 


.272 


.271 


.271 


.270 


23.33 


74 


.269 


.268 


.267 


.267 


.266 


.265 


.264 


.263 


.263 


.262 


23.89 


75 


62.261 


62.260 


62.259 


62.259 


62.258 


62.257 


62.256 


62.255 


62.255 


62.254 


24.44 


76 


.253 


.252 


.251 


.250 


.249 


.249 


.248 


.247 


.246 


.245 


25.00 


77 


.244 


.243 


.242 


.241 


.240 


.240 


.239 


.238 


.237 


.236 


25.56 


78 


.235 


.234 


.233 


.233 


.232 


.231 


.230 


.229 


.229 


.228 


26.11 


79 


.227 


.226 


.225 


.224 


.223 


.222 


.221 


.220 


.219 


.218 


26.67 


80 


62.217 


62.216 


62.215 


62.214 


62.213 


62.213 


62.212 


62.211 


62.210 


62.209 


27.22 


81 


.208 


.207 


.206 


.205 


.204 


.204 


.203 


.202 


.201 


.200 


27.78 


82 


.199 


.198 


.197 


.196 


.195 


.194 


.193 


.192 


.191 


.190 


28.33 


83 


.189 


.188 


.187 


.186 


.185 


.184 


.183 


.182 


.181 


.180 


28.89 


84 


.179 


.178 


.177 


.176 


.175 


.174 


.173 


.172 


.171 


.170 



♦ Maximum density, at about 39.1 F. or 4° C. 
t 'Ordinary" temperatures used for determining specific gravities, about 60° to 
62° F. or 16° C. For scientific determinations, 0° C. is referred to as standard. 



466 27— WEIGHTS AND SPECIFIC GRA VITIES OF MA TERIALS, 


3. — ^Weight of a Cubic Foot of Water at Various Temperatures. 






— Continued. 


Deg.C 


. Add: 


.0 


.06 


.11 


.17 


.22 


.28 


.33 


.39 


.44 


.50 




Deg. F. 


.0 


.1 


.2 


.3 


.4 


.5 


.6 


.7 


.8 


.9 


29.44 


85 


62.169 


62.168 


62.167 


62.166 


62.165 


62.164 


62.163 


62.162 


62.161 


62.160 


30.00 


86 


.159 


.158 


.157 


.156 


.155 


.154 


.153 


.152 


.151 


.150 


30.56 


87 


.149 


.148 


.147 


.146 


,145 


.144 


.143 


.142 


.141 


.140 


31.11 


88 


.139 


.138 


137 


.136 


.135 


.134 


.133 


.132 


.131 


.130 


31.67 


89 


.129 


.128 


.127 


.126 


.125 


.124 


.122 


.121 


.120 


.119 


32.22 


90 


62.118 


62.117 


62.116 


62.115 


62.114 


62.113 


62.111 


62.110 


62.109 


62.108 


32.78 


91 


.107 


.106 


.105 


.104 


.103 


.102 


.100 


.099 


.098 


.097 


33.33 


92 


.096 


.095 


.094 


.092 


.091 


.090 


.089 


.088 


.086 


.085 


33.89 


93 


.084 


.083 


.082 


.081 


.080 


.079 


.077 


.076 


.075 


.074 


34.44 


94 


.073 


.072 


.071 


.069 


.068 


.067 


.066 


.065 


.063 


.062 


35.00 


95 


62.061 


62.060 


62.059 


62.057 


62.056 


62.055 


62.054 


62.053 


62.051 


62.050 


35.56 


96 


.049 


.048 


.046 


.045 


.044 


.043 


.041 


.040 


.038 


.03^ 


36.67 


97 


.036 


.035 


.034 


.032 


.031 


.030 


.029 


.027 


.026 


.025 


36.22 


98 


.024 


.023 


.021 


.020 


.019 


.018 


.016 


.015 


.013 


.012 


37.11 


99 


.011 


.010 


.008 


.007 


.006 


.005 


.003 


.002 


.000 


61.999 


37.78 


100 


61.998 


61.997 


61.996 


61.994 


61.993 


61.992 


61.991 


61.989 


61.988 


61.987 


38.33 


101 


.986 


.985 


.983 


.982 


.981 


.980 


.978 


.977 


.975 


.974 


38.89 


102 


.973 


.972 


.970 


.969 


.968 


.967 


.965 


.964 


.962 


.961 


39.44 


103 


.960 


.959 


.957 


.956 


.955 


.954 


.952 


.951 


.949 


.948 


40.00 


104 


.947 


.946 


.944 


.943 


.941 


.940 


.939 


.937 


.936 


.934 


40.56 


105 


61.933 


61.932 


61.930 


61.929 


61.928 


61.926 


61.925 


61.924 


61.922 


61.921 


41.11 


106 


.920 


.919 


.917 


.916 


.915 


.913 


.912 


.911 


.910 


.908 


41.67 


107 


.907 


.906 


.904 


.903 


.901 


.900 


.899 


.897 


.896 


.894 


42.22 


108 


.893 


.892 


.890 


.889 


.887 


.886 


.885 


.883 


.882 


.880 


42.78 


109 


.879 


.878 


.876 


.875 


.873 


.872 


871 


.869 


.868 


.866 


43.33 


110 


61.865 


61.864 


61.862 


61.861 


61.859 


61.858 


61.857 


61.855 


61.854 


61.852 


43.89 


111 


.851 


.850 


.848 


.847 


.845 


.844 


.843 


.841 


.840 


.838 


44.44 


112 


.837 


.836 


.834 


.833 


.831 


.830 


.829 


.827 


.826 


.824 


45.00 


113 


.823 


.822 


.820 


.819 


.817 


.816 


.815 


.813 


.812 


.810 


45.56 


114 


.809 


.808 


806 


.805 


.803 


.802 


.800 


.799 


.797 


.796 


46.11 


115 


61.794 


61.793 


64.791 


61.790 


61.788 


61.787 


61.786 


61.784 


61.783 


61.781 


46.67 


116 


.780 


.779 


.777 


.776 


.774 


.773 


.771 


.770 


.768 


.767 


47.22 


117 


.765 


.764 


762 


.761 


.759 


.758 


.756 


.755 


.753 


.752 


47.78 


118 


.750 


.748 


.747 


.745 


.744 


.742 


.740 


.739 


.737 


.736 


48.33 


119 


.734 


.733 


.731 


.730 


.728 


.727 


.725 


.724 


.722 


.721 


48.89 


120 


61.719 


61.717 


61.716 


61.714 


61.713 


61.711 


61.709 


61.708 


61.706 


61.705 


49.44 


121 


.703 


.701 


.700 


.698 


.697 


.695 


.693 


.692 


.690 


.689 


50.00 


122 


.687 


.685 


.684 


.682 


.681 


.679 


.677 


.676 


.674 


.673 


50.56 


123 


.671 


.669 


.668 


.666 


.665 


.663 


.661 


.660 


.658 


.657 


51.11 


124 


.655 


.653 


.652 


.650 


.648 


.646 


.645 


.643 


.641 


.640 


51.67 


125 


61.638 


61.636 


61.635 


61.633 


61.631 


61.629 


61.628 


61.626 


61.624 


61.623 


52.22 


126 


.621 


.619 


.618 


.616 


.615 


.613 


.611 


.610 


.608 


.607 


52.78 


127 


.605 


.603 


.602 


.600 


.598 


.596 


.595 


.593 


.591 


.590 


53.33 


128 


.588 


.586 


.585 


.583 


.581 


.579 


.578 


.576 


.574 


.573 


53.89 


129 


.571 


.569 


.568 


.566 


.565 


.563 


.561 


.560 


.558 


.557 


54.44 


130 


61.555 


61.553 


61.552 


61.550 


61.549 


61.547 


61.545 


61.544 


61.542 


61.541 


55.00 


131 


.539 


.537 


.536 


.534 


.533 


.531 


.529 


.528 


.526 


.525 


55.56 


132 


.523 


.521 


.520 


.518 


.516 


.514 


.513 


.511 


.509 


.508 


56.11 


133 


.506 


.504 


.503 


.501 


.500 


.498 


.496 


.495 


.493 


.492 


56.67 


134 


.490 


.488 


.487 


.485 


.483 


.481 


.480 


.478 


.476 


.475 


57.22 


135 


61.473 


61.471 


61.470 


61.468 


61.466 


61.464 


61.463 


61.461 


61.459 


61.458 


57.78 


136 


.456 


.454 


.453 


451 


.449 


.447 


.446 


.444 


.442 


.441 


58.33 


137 


.439 


.437 


.436 


.434 


.432 


.430 


.429 


.427 


.425 


.424 


58.89 


138 


.422 


.420 


.418 


.417 


.415 


.413 


.411 


.409 


.408 


.406 


59.44 


139 


.404 


.402 


.400 


.399 


.397 


.395 


.393 


.391 


.390 


.388 



























LIQUIDS— WATER. 



467 



3. — Weight of a Cubic Foot of Water at Various Temperatures. 

— Concluded. 



Deg. C 


. Add: 
Deg.F. 


.0 




.0 


60.00 


140 


61.386 


60.56 


141 


.368 


61.11 


142 


.350 


61.67 


143 


.332 


62,22 


144 


.314 


62.78 


145 


61.296 


63.33 


146 


.278 


63.89 


147 


.259 


64.44 


148 


.241 


65.00 


149 


.222 


65.56 


150 


61.203 


66.11 


151 


.184 


66.67 


152 


.165 


67.22 


153 


.146 


67.78 


154 


.126 


68.33 


155 


61.106 


68.89 


156 


.086 


69.44 


157 


.067 


70.00 


158 


.047 


70.56 


159 


.027 


71.11 


160 


61.006 


71.67 


161 


60.986 


72.22 


162 


.966 


72.78 


163 


.945 


73.33 


164 


.925 


73.89 


165 


60.904 


74.44 


166 


.882 


75.00 


167 


.862 


75.56 


168 


.841 


76.11 


169 


.821 



61.384 
.366 
.348 
.330 
.312 

61.294 
.276 
.257 
.239 
.220 

61.201 
.182 
.163 
.144 
.124 

61.104 
.084 
.065 
.045 
.025 

61.004 

60.984 

.964 

.943 

.923 

60.902 
.880 
.860 
.839 
.819 



11 



61.382 
.364 
.346 
.328 
.310 

61.292 
,274 
,255 
,237 
,218 

61.199 
.180 
.161 
.142 
.122 

61.102 
.082 
.063 
.043 
.023 

61.002 

60.982 

.962 

.941 

.921 

60.900 

.878 
.858 
.837 
.817 



61.381 
.363 
.345 
.327 
.309 

61.291 
.272 
.254 
.235 
.216 

61.197 

.178 
.159 
.140 
.120 

61.100 
.080 
.061 
.041 
.021 

61.000 

60.980 

.960 

.939 

.919 

60.898 
.876 
.856 
.835 
.815 



,22 



61.379 
.361 
.343 
.325 
.307 

61.289 
.270 
.252 
.233 
.214 

61.195 
.176 
.157 
.138 
.118 

61.098 
.078 
.059 
.039 
.019 

60.998 
.978 
.958 
.937 
.917 

60.896 
.874 
.854 
.833 
.813 



,28 



61.377 
.359 
.341 
.323 
.305 

61.287 
.268 
.250 
.231 
.212 

61.193 
.174 
.155 
.136 
.116 

61.096 
.076 
.057 
.037 
.017 

60.996 
.976 
.956 
.935 
.915 

60.894 
.872 
.852 
.831 
.811 



,33 



61.375 
.357 
.339 
.321 
.303 

61.285 
.267 
.248 
.230 
.211 

61.192 
.173 
.154 
.134 
.114 

61.094 
.075 
.055 
.035 
.014 

60.994 
.974 
.953 
.933 
.912 

60.891 
.870 
.850 
.829 
.809 



61.373 
.355 
.337 
.319 
.301 

61.283 
.265 
.246 
.228 
.209 

61.190 
.171 
.152 
.132 
.112 

61.092 
.073 
.053 
.033 
.012 

60.992 
.972 
.951 
.931 
.910 



60. 



.848 
.827 
.807 



.44 



61.372 
.354 
.336 
.318 
.300 

61.282 
.263 
.245 
.226 
.207 

61.188 
.169 
.150 
.130 
.110 

61.090 
.071 
.051 
.031 
.010 

60.990 
.970 
.949 
.929 
.908 

60. 887 
.868 
.846 
.825 
.805 



,50 



61.370 
.352 
.334 
.316 
.298 

61.280 
.261 
.243 
.224 
.205 

61.186 
.167 
.148 
.128 
.108 

61.088 
.069 
.049 
.029 
.008 

60.888 
.968 
.947 
.927 
.906 

60.885 
.864 
.843 
.823 
.802 



76.67 


170 


60.800 


82.22 


180 


60.587 


87.78 


190 


60.366 


93.33 


200 


60.136 


98.89 


210 


59.894 


104.44 


220 


59.641 


110.00 


230 


59.372 


115.56 


240 


59.096 


121.11 


250 


58.812 


126.67 


260 


58.517 


132.22 


270 


.58.214 


137.78 


280 


57.903 


143.33 


290 


57.585 


148.89 


300 


57.259 


154.44 


310 


56.925 


160.00 


320 


56.584 


165.56 


330 


56.236 


171.11 


340 


55. 883 


176.67 


350 


55.523 


182.22 


360 


55.158 


187.78 


370 


54.787 


193.33 


380 


54.411 


198.89 


390 


54.030 


204.44 


400 


53.635 



Difference for 
1 degree F. 



,0213 
.0221 
.0230 
.0242 
.0253 
.0269 
.0276 
.0284 
.0295 
.0303 
.0311 
.0318 
.0326 
.0334 
.0341 
.0348 
.0353 
.0360 
.0365 
.0371 
.0376 
.0381 
.0395 



References: 

Water. Table. Page. 

Critical temperature .... 11 514 

Critical pressure 11 514 

Boiling point 11 514 

Boiling point 12 514 

Freezing point 11 514 

Melting point (ice) 13 515 

Coef . of expansion (ice) . . 14 516 

Specific gravity 4 468 

Specific gravity (sea-) ... 4 468 

Weight in pipes 19 246 

See, also. 

Hydrostatics Section 61. 

Hydraulics *' 62. 



468 27— WEIGHTS AND SPECIFIC GRAVITIES OF MATERIALS, 



4 — Weights and Specific Gravities op Liquids. 
(Authoritative. See also Table 5.) 



Substance under 1 Atmosphere. 


Relative 
Density 
to Water 
at 4° C. 
Sp. Gr. 


Relative 
Density 
to Water 
at 60° to 

62° F. 
(=16° C). 


Weight 

per Cubic 

Foot. 

Lbs. 


* Coef. of 

Expansion 

per Degree. 

Cent. 


Name. 


Temp. 


Acid, acetic 


60° F. 
4°C. 
0°C. 
60° F. 
0°C. 
60° F. 
0°C. 
60° F. 
0°C. 
0°C. 
60° F. 
60° F. 
60° F. 
60° F. 
60° F. 
60° F. 
0°C. 
0°C. 
0°C. 

o°c. 

0°C. 
60° F. 
0°C. 
0°C. 
0°C. 




1.065 


. 66.41 




arsenous 






carbonic (liquid) . 
fluoric 


0.830 
***i!246*' 
***i.*426** 




51.810 
93.54 
77.406 
74,83 
88.642 
97.16 
114.923 
49.502 
50.82 
51.95 
57.37 
58.24 
55.81 




1.500 




hydrochloric... . 
muriatic . ... 




1.200 




nitric 


.00110 


phosphoric 

sulphuric 

Alcohol, pure (absolute) . 

95 per cent 

commercial .... 


1.558 




1.841 
0.793 


.00063 




.00104 


.815 
.833 
.920 
.934 
.894 








proof spirits 

50 per cent 

Ammonia 














Benzine 


.69 
.90 
1.060 
2.960 
1.293 
1.525 




Benzole. 








Blood 




66.169 
184.775 
80.714 
95.197 
63.48 
62.049 
44.508 
121.477 




Bromine . . 




.00104 


Carbon bisulphide 

Chloroform 




.00114 





.00111 


Cider 


1.018 




Claret 


0.994 
0.713 
1.946 




Ether 




.00015 


Ethvllc Iodide 






Gasoline 






Glvcerine 


6° C. 

0°C. 

o°c. 
o°c. 

60° F. 
60° F. 
60° F. 
0°C. 
60° F. 
60° F. 
0°C. 
0°C. 
60° F. 
60° F. 
60° F. 
60° F. 
0°C. 
60° F. 
60° F. 
60° F. 
0°C. 
0°C. 
0°C. 
0°C. 
4°C. 
0°C. 
60° F 
60° F. 


1.260 

13.598 

3.342 

1.029 




78.654 
848.84 
208.62 
64.234 
52.88 
59.93 
57.87 
53.185 
58.31 
52.88 
57.118 
52.186 
57.62 
56.93 
58.12 
58.37 
54.309 
58.12 
58.37 
57.37 
56.119 
39.077 
52.186 
63.672 
62.424 
62.416 
63.98 
77.33 




Mercury . . 




.00018 


Methylene iodide 

Milk 










Naohtha foil) 


.848 
.961 
.928 




Oil, castor 












lemon 


0.852 




linseed 


.935 
.848 










olive 


0.915 
0.836 




Dptroleum 








.924 
.913 
.932 
.936 




rape-seed 

sesame . 










spindle 








0.870 


.00090 


walnut ... . . 


.926 
.936 
.920 




■wood 






whale (sperm) 

Oxygen Giquid) 

Pentane 






0.899 
0.626 
0.836 
1.020 
1.000 
0.999 








Petrol piim foID 






Urine 






Water distilled . 






distilled 






sea 


1.026 
1.240 

















♦ For moderate temperatures. 



LIQUIDS— MISCELLANEOUS. 



469 



5. — Weights and Specific Gravities op Miscellaneous Liquids.* 

(See also Table 4.) 



Name. 



Specific 


Wt. Lbs. 


Gravity. 


Cu. Ft. 


.667 


41.59 


1.034 


64.48 


1.521 


94.85 


2.800 


174.61 


.794 


49.51 


.863 


53.82 


.951 


59.30 


.970 


60.49 


.986 


61.49 


.992 


61.86 


.891 


55.56 


1.300 


81.07 


1.200 


74.83 


1.034 


64.48 


.848 


52.88 


.924 


57.62 



Name. 



Specific 
Gravity. 



Wt. Lbs. 

per 
Cu. Ft. 



Acid, Benzoic 

Citric 

Concentrated . . 
Phosphoric 

(solid) 

Alcohol, Pure. 60° 

80% 
40% 

25% 

10% 

5% 

Ammonia, 28% 

Aqua fortis, double . . . 

single 

Beer 

Bitumen, liquid 

Brandy 



Ether, Acetic 

Muriatic 

Sulphuric... 

Honey 

Oil, Anise-seed 

Codfish 

Palm 

Sunflower 

Spirit, rectified 

Tar 

Vinegar 

Water (See Table 3) 

Wine, Burgundy . . . 

Champagne.. 

Madeira 

Port 



.866 

.845 

.715 

1.450 

.986 

.923 

.969 

.926 

.824 

1.015 

1.080 

1.000 

.992 

.997 

1.038 

.997 



54.00 
52.69 
44.59 
90.42 
61.49 
57.56 
60.43 
57.75 
51.38 
63.30 
67.35 
62.36 
61.86 
62.17 
64.73 
62.17 



* Based on weight of water at 62 . 36 lbs. per cubic foot, i.e., at 16° C. c= 60.8° F.) 



470 21— WEIGHTS AND SPECIFIC GRA VITIES OF MATERIALS. 



O li 

O4 o 
Q d 






s 

s 



!2; 





Vh 




(Tt 




3 
^ 






1 


u 


«o 


a 









<H 




f^ 



^5 ^ 









O) J, ^ 



a?s 



a ft- 



-a 

o 

ft 
• 3 



i2.>» « 



03 ^-^ 



a a KM 

COS- 

^ft e3 
43 02 o 

"E^ >> 



li 

Is 

CO 



^■^" 

TS 43 ^ 
g 03 « 

ft«e3 
rs ^^ 

ftO " 

o3'Oi2^o3 c3h »3^^ 

'i^i^G'^'^^'^ft^'S 



ft-o a- 

52 «« . 
• Pfti 




OS 



^O^ W03 (U-O 

d C"0 C) -^ 1=^ 



ile£li 

03 13 o m ti 



^ fe c: c 
'if oS c 



«ft - • 

e3 o, ^ o3 

sf as 

o3 w 03 o3 
X2 20.0 .0 
rt iS e3 o3 



>-^5 «> 5 ?; o;:; 



<< 



irs 03" 2 03 a 

i^a|at 

i > c3 S 03 w 
*a gX! 5,0 ro^ 
o3^ 03-^ e^ m e^ 



03 M<^ T3— O3^oaif-io3^c3c3.:::io3'f;(:3w<:3 



.03 



»« inos 10 OS 



e>a «o «o -^ ifi o> ^ <o »« osoioqevaos 

1— I 0> l£5 M O »^ 10 ev5 05 t^ e<l CO -^ 

•«J< «o 00 cocq^oicqos cm -■*< c<i i-h «o 



d 

^02 



t^ 00 »o "<1< M 



PhcqPQ 






^.^ 



gig^sl 5s;sgg 



O c ^ *- ^ 



"2^ 



cot«oo o>^ — NCO-^ 



»o«o ^^ ooo» 



SOLIDS. MISCELLANEOUS. 



471 



^ 



S 



S 



.1^ 



la 
P 



■ ee Pt A >4 

■SftftOOOOOftcS 



fe. CQ 03 



0,0 

1« 



^ §§ 



1^^ 



lDOia^«0e01f5t-'^Ht>.C^00 



Oi005005t>.OOOS<Ot^'tt<0 

t>. CO 1-H 1-H cq »-( oa ■>* »o 



«o-*e>si"<*'coeqci3C>qeocoT-«c~ 







o i-H CQ CO -<*« »o «o t>. 00 o» o .-H esi 
M oa CO cq CM c>q oq c^ e>a CM CO CO CO 



o 

511 

C - 



1) 



.2 

o 









«D 



r>. 



If^ 



O O C 

O O 0^ <y 



cd a to 
oj rt 2 

d o ^ 

w C "^ 

'^'^^ 

«+-( TO rt 



tH O U ^ 

<u w w w 

o ^ 
+-> 



" 043 

Vh 2^ « 



tt! c3 

V.JB 

^^ O CO 

<2 ^ g 

£^ t> a 
> ^ :i 

6 « ?^ 






472 21— WEIGHTS AND SPECIFIC GRA VITIES OF MA TERIALS. 

7. — ^Average Weights and Specific Gravities op Woods. 
General Table. 
[See also Table 6.] 



Name of Species. 



Alder 42—1.01 

Apple 66—1.25 

Ash, Green 

Mountain 

White 

Bamboo 3 1 — 

Beech 65—1.12 

Birch 52—1.08 

Box, Brazilian 

Dutch 92—1.25 

French 

Cedar, Red! 48— 

White 32— 

Cherry 61 — 

Chestnut 53 — 

Corlj 

Cypress, Bald 

Douglas " Fir " (See Spruce, Douglas) . 

Ebony 

Elm, Cedar 

White 

" Fir " Washington (See Spruce, Doug- 
las) 

Gum, Sweet 

Hackmatack 

Hemlock 

Hickory, Bitternut ^. 

Mockernut 
Nutmeg 
Pecan 
Pignut 
Shagbark 
Water ^ Q 

(Average of above Hickories) 

Juniper 

Larch 

Lignum- Vitae 1.18—1.38 

Logwood 

Mahogany, Honduras 

Spanish 56 — 

St. Domingo 

(Average of above Mahoganies) 

Maple 53 — 1.05 

Oak, Cow 



Si«5|^ 






fc o 



S^oJ 



Overcup 
Post 
Red 
Spanish 
Texan 
Water 
White 
Willow 
Yellow 
(Average of above Oaks, say) 

African 

Canadian 

Dantzlc 



B 
'6 



•OC03 

t H 



Dry. 



Spec. 
Grav. 



.55 
.75 
.62 
.55 
.62 
.36 
.74 
.65 
1.03 
1.04 
1.33 
.53 
.37 
.66 
.63 
.24 
.46 

1.24 

.74 
.54 



.59 
.61 
.42 
.77 
.85 
.78 
.78 
.89 
.81 
.73 
.80 
.57 
.55 
1.28 
.91 
.56 
.85 
.72 
.71 
.75 
.74 
.74 
.80 
.73 
.73 
.73 
.73 
.80 
.72 
.72 
.75 
.82 
.87 
.76 



Wt. In 
Lbs. per 



Cu. 
Ft. 



34.3 
46.8 
38.7 
34.3 
38.7 
22.5 
46.2 
40.6 
64'. 3 
64.9 
83.0 
33.1 
23.1 
41.2 
39.3 
15.0 
28.7 

77.4 
46.2 
33.7 



36.8 
38.1 
26.2 
48.1 



53.1 
48.7 
48.7 
55.6 
50.6 
45.6 
49.9 
35.6 
34.3 
79.9 
56.8 
35.0 
53.1 
44.9 
44.3 
46.8 
46.2 
46.2 
49.9 
45.6 
45.6 
45.6 
45.6 
49.9 
44.9 
44.9 
46.8 
51.2 
54.3 
47.4 



Ft. 
B. M. 



2.86 
3.90 
3.23 
2.86 
3.23 
1.87 
3.85 
3.38 
5.36 
5.41 
6.92 
2.76 
1.92 
3.43 
3.28 
1.25 
2.39 

6.45 
3.85 
2.81 



3.07 
3.17 
2.19 
4.01 
4.42 
4.06 
4.06 
4.63 
4.21 
3.79 



Green. 



Spec. 
Grav. 



16 

97 

86 

66 

73 

91 

42 

3.75 

3.69 

3.90 

3.85 

3.85 

4.16 

3.79 

3.79 

3.79 

3.79 

4.16 

3.75 

3.75 

3.90 

4.27 

4.53 

3.95 



1.10 



.84 



1.22 
*!75 



.81 



06 



94 



1.10 



Wt. m 
Lbs. per 



Cu. 
Ft. 



51.2 
68.7 



52.4 



61.2 
58.7 



76.2 
46.* 8 



M 

go 
o 



^ 









'O A ^ 



50.6 



66.2 



58.7 



68.7 



4.21 



* A wood is considered "dry" when it contains not more than 15 per 
cent of moisture. 



WOODS, 



473 



7. — Average Weights and Specific Gravities op Woods — Concluded. 

General Table. 
[See also Table 6.] 





Dry. 


Green, 


Name of Species. 


Spec. 
Grav. 


Wt 
Lbs 


.in 
. per 


Spec. 
Grav. 


Wt. in 
Lbs. per 




Cu. 
Ft. 


Ft. 
B. M. 


Cu. 
Ft. 


Ft. 
B. M. 


Oak — Continued. 

English 


.93 
1.17 

.76 
1.07 

.67 
.63 
.53 
.61 
.50 
.51 
.44 
.38 
.52 
.52 
.45 
.38 
.53 
.44 
.40 

.51 

.59 
.70 
.58 
.49 


58.1 
73.0 

47.4 
66.8 

41.8 
39.3 
33.1 
38.1 
31.2 
31.8 
27.5 
23.7 
32.5 
32.5 
28.1 
23.7 
33.1 
27.5 
25.0 

31.8 

36.8 
43.7 
36.2 
30.6 


4.84 
6.09 
3.95 
5.57 

3.49 
3.28 
2.76 
3.17 
2.60 
2.65 
2.29 
1.98 
2.71 
2.71 
2.34 
1.98 
2.76 
2.29 
2.08 

2.65 

3.07 
3.64 
3.02 
2.55 


1.15 


71.8 


5.98 


" (heart of oak) 




James River 








Live 


1.26 
1.01 


78.7 
63.0 


6 55 


Oregon " Pine " (See Spruce. Douglas) 

Pear 61—1.07 

Pine, Cuban ^ 


5.25 


Loblolly 12 -C ^ 
Longleaf (Yellow) fl j- «5 5? o ^ 
Red (Norway) o^.^,^^ 




















Shortleaf £StJ*=^'i 








Spruce -^ a H 








White ^ Q 

(Average of above Pines, say) 

Pitch 


.60 
.75 


37.5 
46.8 


3.12 
3.90 


Sugar 


.58 


36.2 


3.02 


Poplar 




White 








Redwood 41 — . 87 


.82 


51.1 


4 27 


Spruce, California 




■nmiP-ifls?-! Oregon " Pine " I 

^^^^^^®1 Washington "Fir" i 

Sycamore 


.63 


39.3 


3.28 


Teak 








Walnut, Black 








Willow 


















Note that treated timbers will weigh from 5 to 20 lbs. per cu. ft. more 
than the untreated timbers, well seasoned. Beech, well treated, will receive 
18.5 lbs. per cu. ft. of zinc chloride solution or about x% that quantity of 
creosote oil; pine, about the same quantities, usually a little less; and oaks 
about I the above quantities. In general, the harder woods increase in 
weight less than the softer; exact quantities depend upon the specifications. 



474 27— WEIGHTS AND SPECIFIC GRA VITIES OF MA TERIALS. 



8. — Weights and Specific Gravities of Building Stones, 

Masonry and Cements. 
. (Average Values.) 



Name of Material 


Specific 
Gravity. 


Wt. per 

Cu. Ft. 

Lbs. 


Wt. in Lbs. per - 




Cu. Yd. 


Cu. In. 


Brick, Chrome 


2.803 
2.643 
2.403 
2.403 
2.163 
2.163 
1.922 
1.602 
1.442 

(1.04) 
(1.36) 

2.70 
2.79 
2.88 
2.88 
3.15 
2.87 
3.12 
3.04 
2.98 
2.95 


175. 
165. 
150. 
150. 
135. 
135. 
120. 
100. 
90. 

65. 

85. 

100. 


4725. 
4455. 
4050. 
4050. 
3645. 
3645. 
3240. 
2700. 
2430. 

1755. 
2295. 


1013 


Magnesia 160—170 

Pressed (hard) 


.0955 
.0868 


Paving and Fire 


0868 


Lime — sand 130 — 140 

Pressed 


.0781 
.0781 


Common, building '. 


0694 


Soft, building 


0579 


Light, Inferior 85— 95 

Cement (loose or granular) 

Natural (Rosendale) ... 55—75 

Portland 75— 95 

" shaken, usually taken at. . 


.0521 


Cement (solid) 

Natural, Illinois, Utica 






Kansas, Fort Scott 








Maryland, Cumberland 








Round Top 








Minnesota, Austin 








Mankato 








New York, Akron 








Rosendale 








Pennsylvania, Lehigh 








fOeneral average, sav) 


(184.1) 


(4971.) 




(Specifications A. S. T. M., 1904)2 . 8 min. 




(Report U. S. Engrs., 1 902) 2.5—2.8 
(Extreme limits for good) 2 . 7 — 3 . 2 


















Portland 

(General average, say) : 

(Specifications A. S. T. M., 1 904) 3 . 1 min. 


3.15 


(196.6) 


(5308.) 




(Extreme limits for good) 3 . — 3 . 2 










Puzzolan or slag 

(General average, say) 


2.85 


(177.9) 


(4803.) 




(Report U. S. Engrs., 1902) 2.7—2.8 
(Extreme limits for good) 2 . 7—2 . 9 
Blended, Natural (50) and Portland (50), 
say 












3.05 


(190.4) 


(5141.) 




Cement (In barrels), weight per bbl., net: 
Spec. U. S. Engineers, 1902: 
Natural 300 lbs., min. 




West of Allegheny 

Mts. may be . . .265 ' 










Portland 4sacks @ 93.751bs.37 5 ' " 










Puzzolan 4sacks@ 82.50 " 333 * " 










Spec. Am. Soc. T. M., 1904: 
Natural 3 bags @ 94 lbs.. . . 282 lbs. min. 










Portland 4bags@94 " ..376 " 

Foreign: 
English Portland 400 to 430 lbs 














' 




German Portland, gross 400 to 440 " 

net 37 5 " 





























♦Specific Gravity Equivalent for any Weight. 
Water at 4° C. 



Weight. 


Spec. Grav. 


Weight. 


Spec. Grav. 


Weight. 


Spec. Grav. 


1 
2 
3 


.016 02 
.032 04 
.048 06 


4 
5 
6 


.064 08 
.080 10 
.096 12 


7 
8 
9 


.112 14 
.128 16 
.144 18 



BUILDING STONES. MASONRY, CEMENTS, 



475 



8. — Weights and Specific Gravities op Building Stones, 

Masonry and Cements — Continued. 
(Average Values.) 



Name of Material. 


Specific 
Gravity. 


Wt. per 

Cu. Ft. 

Lbs. 


Wt. in Lbs. per- 


Cu. Yd. 


Cu. In. 


Cement Mortar (cement-sand-water-mix) 
Portland, say 


1.68 

1.68 

2.33 

(2.00) 

(1.76) 

(1.36) 

(1.79) 

(1.2) 

(1.6) 

2.77 
2.68 
2.63 
2.84 
2.66 

2.69 
2.63 
2.65 
2.72 
2.70 
2.61 
2.68 
2.65 
2.66 
2.74 
2.67 
2.65 
2.70 
2.64 
2.683 
(1.79) 
(2.00) 

(.96) 
(2.08) 
3.12 
1.60 
2.56 
2.58 
2.57 
2.48 
2.51 
2.67 
2.70 
2.34 
2.54 

2.53 
2.76 
2.67 
2.32 

2.71 
2.71 

2.71 


105. 

105. 
145. 
125. 
110. 

85. 
112. 

75. 
100. 

172.9 
167.3 
164.2 
177.3 
166.0 

167.9 

164.2 

165.4 

171.0 

168.5 

162.9 

167.3 

165.4 

166.0 

171.0 

166.7 

1G5.4 

168.5 

164.8 

167.5 

112. 

125. 

60. 
130. 
(195.) 
100. 
159,8 
161.1 
160.4 
154.8 
156.7 
166.7 
168.5 
146.1 
158.6 

157.9 
172.3 
166.7 
144.8 

169.2 
169.2 

169.2 


2835. 

2835. 
3915. 
3375. 
2970. 
2295. 
3024. 


.0608 


Resulting Volume of Mix : 
1 cement, 1 sand =1.5; 1 cement, 2 

sand=2.3; 
1 cement, 3 sand =3.1; 1 cement; 4 
sand=3.8. 

Concrete, Cinder 100 — 110 

Stone 130 — 150 

Earth, Clayey, moistened and rolled 

dry 100—120 

Loose, dry 80 — 90 

Muddy, say 105 — 120 


.0608 
.0839 


Loam, dry, loose 




packed 






Gneiss and 

Granite, California, Penryn (hornblende) . . . 






Rocklin (muscovite) 






Colorado, Georgetown (biotite) 






Connecticut, Greenwich 






New London 






Georgia, Lithornla and Stone Moun- 
tain 






Maine, Fox Island 






Hallowell 






Maryland, Port Deposit 






Massachusetts, Quincy (hornblende) 






Rockport 






Minnesota 2.66—2.69 






New Hampshire, Concord 






Keene 






New York 2.71—2.76 






Rhode Island, Westerley 






Vermont, Barre 






Wisconsin, Athelstane 






Montello 







(Average of above Granites, say) 


4523. 
3024. 
3375. 

1620. 

3510. 

(5265.) 

2700. 


0969 


Gravel, Dry say 




Wet, say 




Lime Goose or granular) 

Freshly burned (quicklime) 




Slaked 




Lime (solid) 3.09—3.15 

Lime Mortar 


0579 


Limestone, Illinois, Joliet 




Lemont... 2.51 — 2.65 






Quincy 




* 


Indiana, Bedford 







Salem 






Kentucky, Bardstown 






Bowling Green 






Michigan, Marquette 






Lime Island 






Minnesota, Fron- 

tenac 2.42— 2. G3 






Stillwater 






Winona 






Missouri, Canton 







New Yqrk, Canajoharie 

2.69—2.73 






Cobelskill 






Glens Falls 

2.70—2.72 













476 27— WEIGHTS AND SPECIFIC GRA VITIES OF MA TERIALS. 

8. — Weights and Specific Gravities of Building Stones, 

Masonry and Cements — Continued. 

(Average Values.) 





Specific 
Gravity. 


Wt. per 

Cu. Ft. 

Lbs. 


Wt. in Lbs. per - 




Cu. Yd. 


Cu. In. 


Limestone— Continued. 

New York — Continued. 

Kingston 


2.69 
2.75 
2.6 

2.75 
2.73 
2.76 
2.87 
2.87 
2.66 
2.77 
2.80 
2.71 
2.71 
2.72 
2.67 
2.65 
2.71 
2.84 
2.24 
2.00 
1.60 
1.68 
2.33 
2.64 
2.56 

2.53 
2.48 
2.24 
2.60 
2.56 

2.50 
2.72 
2.40 
(1.60) 
(1.98) 
(1.52) 
(1.85) 
2.73 
2.43 
2.23 
2.34 
2.50 
2.49 
2.17 
2.54 
2.25 
2.41 
2.60 
2.49 
2.41 
2.60 
2.62 
2.71 
2.70 
2.68 
2.34 
2.21 
2.11 


167.9 
171.7 
162.5 

171.7 

170.4 

172.3 

179.2 

179.2 

166.0 

173. 

174.8 

169.2 

169.2 

171.0 

166.7 

165.4 

169.2 

•177.3 

140. 

125. 

100. 

105. 

145. 

165. 

160. 

158. 
155. 
140. 
162. 
160. 

156. 
170. 
150. 
100. 
124. 
95. 
115. 
170.4 
151.7 
139.2 
146.1 
156-1 
155.4 
135.5 
158.6 
140.5 
150.4 
163.3 
155.4 
150.4 
162.3 
163.6 
169.2 
168.5 
167.3 
146.1 
138.0 
131.7 




i 


Lake Champlaln 






(Average of above Limestones, say) 

Marble, California, Colton 


4388. 


.0940 


Georgia, Tate 






New York, Gouverneur 






Pleasantville 






Tuckahoe 






Vermont, Dorset 






(Average of above Marbles, say) 


4671. 


.1001 


African 




Biscayan 






British 






Carrara 






Egyptian 






French 






Italian (white) 






Parian 






Masonry, Brick, Pressed 


3780 

3375. 

2700. 

2835. 

3915. 

4455. 

4320. 

4266. 
4185. 
3780. 
4374. 
4320. 

4212. 
4590. 
4050. 
2700. 
3348. 
2565. 
3105. 


.0810 


Medium 


.0723 


Soft 


.0579 


Concrete, Cinder 


.0608 


Stone 130—150 

Granite. Dressed, for Buildings 

Bridges 

Dams 
2.50—2.56 

Rubble In cement 

dry, say 

Limestone, Dressed, for Buildings . 
Bridges . . . 
Dams 
2.47—2.53 

Marble, Dressed, for Buildings 

Sandstone, Dressed, for Buildings . 
Sand, Fine, dry 1 . 40— 1 . 70 


.0839 
.0955 

■"!6938^ 

"*!6984 
.0868 


wet 1.90 — 2.05 




Coarse 1.40—1.65 

Mixed, coarse and fine 




Sandstone, California, Angel Island 




Colorado, Fort Collins 






Manitou 






Trinidad 






Connecticut, Portland 2.36 — 2.63 






Massachusetts. Long Meadow 

Michigan Marquette . . 










Portafife Entrv 






Minnesota, Fond du Lac 






New Jersey, Belleville 2 . 26—2 . 56 
New York Albion 










Medina 2 40 — 2 58 






Hulberton 






Potsdam 






Oswego 






Oxford 






Fortage 






Warsaw 






Ohio. Berea 2 11 — 2.57 






Cleveland 






Massillon 













BUILDING STONES, MASONRY. CEMENTS. 



477 



8. — Weights and Specific Gravities of Building Stones, 

Masonry and Cements — Concluded. 

(Average Values.) 



Name of Material. 


Specific 
Gravity. 


Wt. per 

Cu. Ft. 

Lbs. 


Wt. in Lbs. per - 


Cu. Yd. 


Cu. In. 


Sandstone^Continued. 

Pennsylvania, Hummelstown .... 


2.66 

2.60 

2.22 

2.61 

2.47 

2.81 

2.80 

2.78 

2.8 

2.75 

2.81 

2.88 

2.95 

3.00 

3.03 

2.86 

2.96 


166.0 

162.3 

138.6 

162.9 

154. 

175.4 

174.8 

173.5 

175. 

171.7 

175.4 

180. 

184.2 

187.3 

189.1 

178.5 

185. 






Lumberville 






Wisconsin, Fond du Lac 






Virginia, Brlstow 






(Average of above Sandstones, say) 

Slate, New York, Granville 2.78—2.84 


4158. 


.0891 


Pennsylvania, Slatington 






Vermont, Rutland 2.7 6 — 2 . 80 






(Average of above Slates, say) 2 . 7 5 — 2 . 85 
Austria, Silesia 


4725. 


.1013 


England, Cornwall 






Welsh 






Trap, Minnesota, Duluth 






Taylors' Falls 






New Jersey, Jersey City 






New York, Staten Island 






(Average of above Trap rocks, say) 


4995. 


.0171 



478 27— WEIGHTS AND SPECIFIC GRAVITIES OF MATERIALS. 

9. — General Table op Weights and Specific Gravities op 
Materials. 
(Average Values.) 



Name of Substance- 



Spec. 
Gray. 



Wt. per 

Cu. Ft. 

Lbs. 



Name of Substance. 



Spec. 
Grav. 



Wt. per 

Cu. Ft. 

Lbs. 



Acid (See Table 4) 

Agate 2.5 — 2. 

Air (See Table 2) 

Alabaster, Calcareous 
—2. 
Gypseous 
2 3 

Alcohol (See Table 4) . . 

Alder (See Table 7) 

Alloys (See Brass, Bronze, 

etc.) 

Alum 

Aluminum, Cast . . 

Hammered . . 

Drawn wire . . 

Pure 

Sheet 

Aluminum bronze 

Amalgam... 13.7 — 14.1 

Amber 

Ambergris 

Ammonia (See Table 4) . . 
Anthracite (solid) 

Antimony, Cast 

6.67—6.74 

Pure 

Apatite 3.16—3.22 

Apple-tree (See Table 7) . . 
Aqua fortis (See Table 

5) 

Aragonite 

Arsenic. ... 5. 7 — 5. 8 

Asbestos 2.1 — 3.1 

paper 

Ash (See Table 7) 

Ashes, Coal packed . 6 — . 8 

Asphalt, Paving 

Asphaltum, Natural 

1.1— 1.8 
Atmospheric air (See 

Table 2) 

Ballast, brick and gravel 
Bamboo (See Table 7) . . . 

Barium 

Bary tes 

Basalt (See Trap) 

Beech (See Table 7) . ... 

Beef fat 

Beeswax 

Benzine 

Beer 

Beton (See Concrete) 

Birch (See Table 7) 

Bismuth, Cast 

9.76—9.90 
Bitumen (See Ashphal- 

tum) 

Blood 

Bone 1.8 — 2.0 

Borax 1.7—1.8 

Boxwood (See Table 7) . . 
Cast 7.8 — 8.8 
Rolled 



2.59 



161.7 



2.76 
2.61 
".*55 



172.3 
162.9 
'34!3 



1.72 
2.56 
2.75 
2.68 
2.67 
2.67 
7.7 
13.9 
1.08 
.87 
.894 

1.55 

6.71 

6.80 

3.19 

.75 



107.4 
160. 
171.7 
167.3 
166.7 
166.7 
480. 
868. 
67.4 
54.3 
55.8 



419. 
424.5 
199. 
46.8 



3.0 
5.76 
2.8 
1.2 



18.7 
360. 
175. 

75. 



(.7) 
1.6 



1.44 



44. 
100. 



(1.79) 

.36 

.47 

4.45 

2.96 

.74 

.92 

.96 

.69 

1.034 



112. 
22.5 
29.3 

278. 

185. 
46.2 
57.4 
60. 
43.1 
64. 4i 



.65 
9.82 



40.6 



13. 



1.06 

1.9 

1.75 



66.2 
118.6 
109.2 



Brass, Sheet 

Wire 

Brick (See Table 8) . . . 
Brickwork (See Masonry, 

Table 8) 

Bromine 

Bronze, Aluminum.... 

Coinage 

Gun Metal 

8.15—8.95 

Ordinary 

Bushel of Produce, etc. 
1 pound per U. S. bu.= 



8.5 
8.&4 



530. 
533. 



2.96 

7.7 
8.66 

8.6 
8.4 



do 
do 
do 
do 
do 
do 
.94—. 



95 



3— 8.7 



8.4 
8.5 



524. 
530. 



1.24445 
24 
32 
40 
48 
56 

Butter 

Butternut tree 
Cadmium... i 

Calclte 2.6—2 

Calcium 

Camphor 

Caoutchouc 

Carbon (See Diamond) . . 

Carbon bisulphide 

Carbonic acid (liquid) . . 

Castor oil 

Cedar (See Table 7) 

Cement (See Table 8) . . . 

Chalk 2.2—2.8 

Champagne 

Charcoal, Birch 

Oak and Fir . . . 

Pine 

Powdered 

Cherry (See Table 7) . . . 
Chestnut (See Table 7) . 

Chloroform 

Chromium 

Cider 

Cinnabar 

Claret 

Clay, Dry 

Moist, loose 1.7 — 2. 
Moist, packed 

2.0—2.4 

with gravel 

Coal, Anthracite (solid) 

Lump 

Broken 

Egg 

Stove (average) . 

Nut : 

Pea 

Buckwheat 

Bituminous (solid). 

Loose . . 

Cobalt. . . 8.51—8.54 

Coke, Solid. Natural . . 

Pressed 

Loose 28 — 36 



.94 

.38 

8.65 

2.7 

1.58 

.99 

.93 

i!293 
.830 
.96 



185. 
480. 
540.6 

537. 
524. 

.80356 
1.00000 
19.29 
25.71 
32.14 
38.57 
45.00 
58.7 
23.7 

540. 

168.5 
98.6 
61.8 
58. 

"so!?' 

51.8 
59.9 



2.6 

.997 
(.54) 
(.45) 
(.30) 
(1.38) 

.68 

.63 
1.525 
5. 

1.01 
8.1 

.994 
1.52 
1.9 

2.2 

2.5 
1.55 
(1.04) 
(1.02) 
(.99) 
(.96) 
(.93) 
(.90) 
(.88) 
i:33 
(.83) 
8.52 
(1.0) 
1.4 
.5 



162.3 

62.2 

34. 

28. 

19. 

86. 

41.2 

39.3 

95.2 
312. 

63.5 
506. 

62.2 

95. 
119. 

137. 
156. 

96.8 

65. 

64. 

62. 

60. 

58. 

56. 

55. 

83. 

52. 
532. 

62. 

87. 

32. 



GENERAL TABLE. 



479 



9. — General Table of Weights and Specific Gravities of 

Materials — Continued. 

(Average Values.) 



Name of Substance. 



Concrete (See Table 8) . . .. 
Copper, Cast. 8.6 — 8.9 
Drawn wire 

8.8—9.0 
Hammered 

8.9—9.0 

Melted 

Rolled 8.9—9.0 

Sheet 

Cork 

Creosote oil. 1.04—1.10 

Cypress (See Table 7) 

Delta Metal (Copper 60, 
zinc, 34-44, iron 2-4, 

tin 1-2) 

Diamond... 3.45 — 3.60 

Dogwood 

Dolomite (See Limestone) 
Douglas •' Fir " (See Table 

7) 

Earth (See Table 8) 

Ebony (See Table 7) 

Egg 



Spec. 
Grav. 



8.89 

8.95 
8.22 
8.95 
8.89 

.24 
1.07 

.46 



3.52 
.75 



.51 



1.24 
1.09 



Wt. per 

Cu. Ft. 

Lbs. 



550. 
555. 

559. 

513. 

559. 

555. 
15.0 
66.8 
28.7 



537. 
(220.) 
46.8 



31.8 



77.4 
68.0 



Name of Substance. 



Elder pith 

Elm (See Table 8) 

Emerald 

Emery 

Ether (See Table 5) . . . 
Ethylic iodide (Table 4) . 

Fat, Beef 

Hog 

Mutton 

Feldspar 

Filbert tree 

Fir (See Table 7) 

Flint 

Gallon, of liquid 

1 pound per U.S. gal- 
lon= , 

0.13368 pound per 

U. S. gallon= 

Gamboge , 

Garnet 3.75—4.20 

German silver 8. 4 — 8. 7 
Glass, Crown , 

Common Window . . . 



Spec. 
Grav. 



076 



2.7 

4.0 
.713 

1.946 
.92 
.94 
.93 

2.60 
.60 



2.59 



.120 

.016 
1.2 
4.2 
8.55 
2.50 
2.50 



Wt. per 

Cu. Ft. 

Lbs. 



4.7 



(168.5) 
250. 

44.5 
121.5 

57.4 

58.7 

58. 
(162.) 

37.5 



(162.) 



7.4805 

1.0000 

75. 
(262.) 
(534.) 
156. 
1.56 



Window Glass. 

(a.) Official Prices Current American Pittsburg Plate Glass Co. 

Prices are per box of about 50 square feet of glass. 



i 


Sizes and No. of Lights per Box. 


Double. 


Single. 


u 


(No.) 


Sizes. 


(No.) 


AA 

$42.75 

46.75 

52.00 

56.00 

57.50 
58.75 

62.75 

6.800 

69.50 

73.50 

74.75 

88.00 

94.75 

105.50 

118.75 

140.00 

153.50 

167.00 


A 

$37.50 

41.50 

45.50 

49.50 

50.75 
52.00 

56.00 

61.50 

62.75 

66.75 

68.00 

80.00 

86.75 

97.50 

108.00 

126.75 

140.25 

153.7 5 


B 

$35.50 

38.75 

41.50 

46.00 

46.75 
47.50 

50.75 

55.50 

56.75 

61.50 

62.75 

74.7 5 

80.00 

90.75 

101.50 

120.00 

133.50 

147.00 


AA 

$32.00 

33.50 

36.00 

37.50 

38.75 
40.00 

42.75 

48.75 
52.00 


A 

$26.75 

28.00 

30.00 

3L75 

32.75 
34.75 

38.50 

44.50 
47.50 


B 

$25.50 

26.75 

28.00 

29.50 

30.00 
31.00 

33.75 

38.50 
41.75 


C 


25 

34 

40 

50 

54 
60 

70 

80 

84 
90 


(150) 

[ (47) 

I (46) 

(28) 

[ (18) 

I (18) 

(13) 

(10) 

f (8) 

(8) 

\ (8) 

(6) 

I (6) 

(5) 

(4) 

(4) 

(3) 

(3) 

(3) 

(3) 

(3) 

(3) 


6x 8 to lOx 15 

J}^J^}tol4x20 

10x26 to 16x24 

20x2o}^« 20x30 

15x36 to 24x30 
26x28 to 24x36 
26 X 341 

28x32 [to 30x 40 
30x30 

34X36 1^-30x50 

30x52 to 30x54 
30 X 56 to 34 X 56 
34x58 to 34x60 
36 X 60 to 40 X 60 
40x62 to 40x64 
40x66 to 40x70 
40x72 to 40x74 
40x76 to 40x80 
40 X 82 to 40 X 84 
40 X 86 to 40 X 90 


(48) 

(26) 

(19) 

(12) 

(10) 
( 9) 

(6) 

(5) 

(4) 
(4) 
(4) 
(3) 
(3) 
(3) 
(3) 
(3) 
(3) 


$25.00 
26.00 
27.00 


94 










100 










105 










110 










115 










120 










125 










130 













Note. — An additional 10 per cent will be charged for all glass more than 
40 inches wide. All sizes over 52 inches in length, and not making more 
than 81 united inches, willbe charged in the 84 united inches bracket. All glass 
54 inches or wider, not making more than 116 united inches, will be charged in 
the 120 united inches bracket. Prices are subject to discount in quantities. 



480 27— WEIGHTS AND SPECIFIC GRAVITIES OF MATERIALS. 

9. — General Table op Weights and Specific Gravities op 

Materials — Continued. 

(Average Values.) 



Name of Substance. 



Spec. 
Grav. 



Wt. per 

Cu. Ft. 

Lbs. 



Name of Substance. 



Spec. 
Grav. 



Wt. per 

Cu. Ft. 

Lbs. 



Glass — Continued. 

Flint 

Flooring, thick . . 

Green 

Optical 

Plate 

White 

Window 

Gneiss (See Table 8). ... 

Gold, Cast 

Native, hammered 
Pure 
Granite (See Table 8) . . . 

Graphite 

Gravel (See Table 8) 

Greenstone trap 

Grindstone 

Gum Arabic 1.32—1.44 

goods 

Raw (Caoutchouc = 

India rubber) 

Gun Metal (bronze) 

Gunpowder (granular) . . 

Gutta percha 

Gypsum, Pure, unburned 
Calcined, 1 lump 
Powder, solid 
loose 
shaken 
Plaster of paris 

2.1—2.4 
(See Plaster) . . . 

Hackmatack 

Hemlock 

Hickory (See Table 7) . . . 

Holly 

Honey 

Horn 

Hornblende ... 3 . — 3 . 5 

Human body 

Ice 88 — .92 

melting 

India rubber 

Iodine 

Iridium, pure 

Iron, Cast 

Wrought, purest . . . 
average . . 

Molten 

Ivory 

Juniper tree 

Kaolin 

Lava, Basaltic 

Trachytic 

Larch 

Lard 

Lead, Commercial, Cast . 
Sheet. 

Pure 

Molten 

Lignite, perfect 

Llgnum-Vitae .1.18—1.38 
Lime (See Table 8) 



187. 
158. 
167. 
215. 
175. 
180. 
156. 



258 

4 

50 



1202. 
1211. 
1217. 



2.26 



.41. 



.14 
.38 
.50 
93 
.93 
.6 
.0 
.98 
.30 
.80 
.55 
97) 
.03) 



2.25 



185. 
134. 

86. 

94. 

58. 

58. 
537. 

62.4 

61. 
144. 
112. 
159. 

60. 

64. 

140. 



38.1 
26.2 



47.4 
90.5 

105.5 

203. 
66.8 
57.4 
57.4 
58. 

309. 
1320. 

450. 

485. 

480. 

433. 

117. 
35.6 

137. 

181. 

150. 
34.3 
58.7 

710. 

712. 

713. 

649. 
80.5 
79.9 



Limestone (See Table 8) . , 

Linden tree 

Linseed oil (See Table 5) . 

Lithium 

Locust 

Logwood 

Magnesia, solid 

loose 

Magnesite 

Magnesium, pure 

Magnetic iron ore 

Mahogany (See Table 7) 

Manganese, pure 

ore, black . . 
red.. . 
Maple (See Table 7) . . . . 
Marble (See Table 8) . . . 

Marl 1.7—2.5 

Masonry (See Table 8) . . 

Mastic (resin) 

Mercury, Solid. — 40° F. 

Liquid +32° F. 

+ 60° F. 

212° F. 

Methylene iodide 

Mica 2.65 — 3.15 

Milk 

Molybdenum, pure 

Mortar, Cement 

Lime 

Mud 105—120 

Mulberry tree 

Nickel 8.3—9.2 

Nitric acid (See Table 4). 
Commercial . . . 

Oak (See Table 7) 

Ochre 

Oil (See Table 4) 

Olive oil (See Table 4) . . 
Oolitic stones 1.9—2.6 

Opal 2.1—2.2 

Opium 

Orange tree 

Ore, Brown iron 

Chrome dust, shaken 

Magnetic iron 

Red iron (specular) . . 

Spathic 

Palladium 

Paper 75 — 1.15 

Paraffin 

Pear tree (See Table 7) . 
Peat, pressed . 6 — . 8 5 

Pentane 

Petroleum (See Table 4) . 

Pewter 

Phosphorus 

Pine (See Table 7) 

Pitch 

Plaster (burned Gypsum) 

Cement 

Keene's cement . . 
of Paris 



.60 
.935 
.585 
.71 
.91 
3.2 
(1.74) 
3.0 
1.75 
(5.0) 



8 00 
3.45 
4.0 

.75 
2.77 
2.1 



.85 
15.632 
13.598 
13.580 
13.370 

3.342 

2. 

1. 



90 
029 
8.63 
1.68 
1.60 
(1 7 9) 
.75 
8.8 
1.420 
1.22 



3.5 



.915 
2.25 
2.15 
1.34 
.71 
3.9 
2.56 
5.0 
5.2 
3.9 
.1.8 
.95 
.88 
.67 
.72 
.621 
.836 
1.6 
1.77 



1.12 



2.25 



GENERAL TABLE, 



481 



9. — General Table of Weights and Specific Gravities op 

Materials — Continued. 

(Average Values.) 



Name of SuDStance. 



Spec. Wt. per 
Grav. Cu. Ft. 
Lbs. 



Name of Substance. 



Spec. 
Grav. 



Wt. per 

Cu. Ft. 

Lbs. 



Plaster — Continued. 

Parian cement . . 

Stucco 

Average J or above: 
Gypsum, unburned 
Calcined, 
lump . . 
powder, 
loose, 
shaken 
Sand (1) plaster (2), 

dry 

Ordinary plaster .... 
(See Gypsum) 
Platinum, Cast 

19.6—20.3 

Native 

Pure 

Rolled 

(Average, use) 

Plum tree 

Plumbago 

Poplar 

Poppy oil (See Table 4) . . 

Porcelain China 

Porphyry 

Portland cement (See Ce- 
ment) 

Potash 

Potassium 

Pumice stone 

Quartz crystal, pure 

2.61—2.71 

Quince tree 

Rape-seed oil 

Red lead 

*Resin 

Rock crystal (halite) .... 

Rock Elm 

Rosendale cement (See 

Cement) 

Rosewood 

*Rosin 

Ruby... 

Salt, Common, solid 

loose, 
packed . 
Coarse, Syracuse . . . 
Turk's Island 
7 8—80 

Fine, Liverpool 

Saltpetre, Chili 

Kali 

Sand (See Table 8) 

Sandstone (See Table 8) . . 

Sapphire 

Selenium 



2.30 
1.80 



.97 
1.03 



1.52 
1.76 



19.95 
16.0 
21.50 
22.07 
21.50 
.78 
.1 
.38 
.924 
.3 
,76 



2.05 
.865 
.92 

2.66 

.71 

.913 

8.94 

1.089 

2.69 

.80 



.73 
1.1 
3.9 
2.20 

(.96) 
(.72) 

(1.25) 

(.78) 
2.26 
2.05 



3.95 
4.5 



2.5—2, 



.44. 

12. 

60. 
64. 

95. 
10. 



1245. 

lOOO. 

1342. 

1378. 

1342. 

48. 

131. 

23. 

57.' 

244. 

172.; 



128. 
54. 
57.4 

166. 

44.3 

56.9 
558. 

68. 
168. 

50. 



45.6 
68.7 

(243.) 

137. 

60. 
45. 

78. 

49. 
141. 
128. 



(246.6) 
281. 



Serpentine 

Shale 

Silica. . 

Silicic acid, crystalline . . 

powder 

Silver 

Slag 

Slate (See Table 8) 

Smalt 

Snow, freshly fallen 

wet 

Soapstone 

Sodium 

Spar, Calcareous 

Feld- 

Fluor 

Heavy.. 4.4 — 4.5 

Spelter 

Spindle oil 

Spirit, rectified 

Spruce (See Table 7) 

Steel, average 

(Handbook calcula- 

tions)t 

Wire 

Strontium 

Sulphur 

Sulphuric Acid 

Sycamore 

Talc (steatite) 

Black 

Tallow 

Tamarack 

Tar, average 

Teak 

Tellurium, pure 

Tiles, solid . . 1 . 9—2 . 5 

hollow (variable) . 

Tin, Cast 

Rolled. . . 7 . 3 — 7 . 5 

Molten 

Topaz 

Trap rock (See Table 8) . 

Tungsten 

Turpentine 

Type metal, cast 

Uranium 

Urine 

Vinegar 

Walnut, Black 

Walnut oil 

Water (See Tables 3 
and 4) 

Dead Sea 

Distilled 4° C 

0° C 

Mediterranean . . . . 



2.65 
2.6 
2.66 
2.6 
(2.2) 
10.5 



165, 
162. 
166. 
162. 
137. 
655. 
40. 



2.44 



2.73 
.97 
2.74 
2.70 
3.40 
4.43 
7.1 
.936 
.824 



152. 
8. 

50. 
170. 

61. 
171. 
168.5 
212. 
277. 
443. 

58.4 

51.4 



7.85 

7.843 

7.856 

2.54 

2.0 

1.841 

.59 
2.73 
2.9 

.93 

.38 
1. 

.70 
6.11 
2.2 



490. 

489.6 

490.4 

158.6 

125. 

114.9 
36.8 

170. 

181. 
58. 
23.7 
62.4 
43.7 

381. 

137. 



17.62 

.87 

10.45 

8.3 

1.02 

1.08 

.58 

.926 



456. 
462. 
439. 
(220.) 
185. 
1100. 

54.3 
652. 
1142. 

63.7 

67.4 
36.2 

58.1 



1.24 
1.000 
.999 
1.029 



77.33 
62.42 
62.42 
64.23 



* Resin is the liquid sap of the Longleaf Yellow Pine; rosin is the hard, 
brittle substance which remains after the turpentine has been extracted. 

t Based on bar 1 in. sq. and 1 ft. long weighing 3.4 lbs.; or plate 12 ins. 
sq. and 1 in. thick weighing 40.8 lbs.; that is, 2 per cent heavier than iron 
at 480 lbs. per cu. ft. 



482 27— WEIGHTS AND SPECIFIC GRA VITIES OF MA TERIALS. 



9. — General Table of Weights and Specific Gravities of 

Materials — Concluded. 

(Average Values.) 



Name of Substance. 


Spec. 
Grav. 


Wt. per 

Cu. Ft. 

Lbs. 


Name of Substance. 


Spec. 
Grav. 


Wt. per 

Cu. Ft. 

Lbs. 


Water — Continued. 


1.000 

1.026 

.96 

.49 

**!992 
.997 


62.42 
63.98 
60. 
30.6 

"6i!92 
62.24 


Wine, Madeira 


1.038 
.997 

7.12 
.936 

6.86 

7.1 

7.24 

7.1 


64 80 


Rain 4° C 


Port 


62.24 


Sea (ocean) 

Wax, Bees 


Wolfram 


445 


Wood Oil 

Zinc, Cast 


58 4 


Willow 


428 


Wine 99 — 1.04 


Pure 


443 


Burgundy 


Sheet 


452 


Champagne 


(Average, say) 


443. 



•< 



I 



10. — Weights op Produce.* 
The following are minimum weights according to the laws of the United 
States, and adopted by a majority of States.: 



Per bushel. 

a Apples (dried) 26 lbs. 

6 Barley 48 ' 

c Beans (white) 60 

Beans (castor) 46 

Blue grass seed 44 

Bran 20 

d Buckwheat 48 

e Clover seed 60 

Coal 80 

/Corn (on cob) 70 

g Com meal 48 

hCovn (shelled) 56 

i Flaxseed, 56 

Hair (plastering) 8 

Hemp seed 44 

Hungarian grass seed. ... 50 



Lime (unslacked) 

Malt 38 

Millet seed 50 

j Oats 32 

k Onions 57 

Peaches (dried) 33 

/ Peas 60 

Peas (ground) 24 

Potatoes (sweet) 55 

m Potatoes (white) 60 

wRye 56 

oSalt (fine) 55 

o Salt (coarse) 50 

p Timothy seed 45 

<7 Turnips 55 

r Wheat 60 



Per bushel, 
. . 30 lbs. 



* The following are the greatest and least minimum weights adopted by 
the various States: Apples {not dried), 24 to 57 lbs. Anthracite coal, 76 to 
80 lbs. a, 22 to 28 lbs. b, 47 to 50 lbs. c, 60 to 62 lbs. d, 42 to 56 lbs. 
e, 60 to 64. /, 68 to 70. g, 46 to 50. h, 52 to 58. i, 44 to 56. /, 26 to 32. 
k, 48 to 57. /, 46 to 60. w, 56 to 60. n, 54 to 56. o, from 50 to 80 lbs. 
Coarse salt, in Penn., 80 lbs.; in 111., 50 lbs. Fine salt, in Penn., 62 lbs.; 
in Ky., and 111., 55 lbs. p, 42 to 60. q, 42 to 60. r, 60 lbs. per bu. in all 
the States. 



SPECIFIC GRAVITY REDUCED TO WEIGHT. 



483 



REDUCTION TABLES. 

11.— Weight Equivalent for any Specific Gravity. 
Water at 4° C. and 16° C. 



Specific 


Weight of any Substance Compared with Distilled Water at — 


Gravity 
o.f the 
Sub- 
stance. 


4°C.= 39.1° F. 


16°C. = 60.8° 


F. 


Per 

Cu. Ft. 

Lbs. 


Per 

Ft. B. M. 

Lbs. 


Per 

Cu. In. 

Lbs. 


Per 

Cu. Ft. 

Lbs. 


Per 

Ft. B. M. 

Lbs. 


Per 

Cu. In. 

Lbs. 


.1 
.2 
.3 


6.2424 

12.4848 
18.7272 


.5202 
1.0404 
1.5606 


.0036 125 
.0072 250 
.0108 375 


6.236 
12.472 
18.708 


.5197 
1.0393 
1.5590 


.0036 089 
.0072 178 
.0108 267 


.4 
.5 

.6 


24.9696 
31.2120 
37.4544 


2.0808 
2.6010 
3.1212 


.0144 500 
.0180 625 
.0216 750 


24.944 
31.180 
37.416 


2.0787 

2.5983 
3.1180 


.0144 356 
.0180 444 
.0216 533 


.7 
.8 
.9 


43.6968 
49.9392 
56.1816 


3.6414 
4.1616 
4.6818 


.0252 875 
.0289 000 
.0325 125 


43.652 
49.888 
56.124 


3.6377 
4.1573 
4.6770 


.0252 622 
.0288 711 
.0324 800 


1.0 


62.4240 


5.2020 


.0361 250 


62.360 


5.1967 


.0360 889 



Example. — What are the weights per cu. ft., per ft. B. M., and per cu. in., 
of a wood whose specific gravity is 0.61, water at 16° C. ? 

Solution.— Cu. Ft. Ft. B. M. Cu. In. 

.60 37.416 3.118 .02165 

,01 .624 .052 .00036 



61 



38.040 lbs. 



3. 170 lbs. 



.022011b. 



Note that for the same specific gravity compared with water at 4° C. 
the resultant weight is 1 part in 1000 greater than when compared with 
water at 1 6° C. ; that is, a volume of water at the lower temperature is ^q of 
1% heavier than an equal volume at the higher temperature. As the 
specific gravity of many substances varies greatly, depending upon the 
particular specimens selected for test, it will be noted that water at either 
temperature may generally be used as a basis for calculating the weight 
of the material. This is especially allowable for woods, building stones and 
other natural materials subject to variable composition, texture, and degrees 
of moisture. With liquids the variation of the substance is less marked; 
and with gases the temperature of the water with which it is compared 
should always be stated, as well as the temperature and pressure of the gas. 



484 21— WEIGHTS AND SPECIFIC GRAVITIES OF MATERIALS. 



12. — ^Weight of Sheets, Bars and Wire, of any Material 
From its Specific Gravity. 
(Water at 4° C.) 



Specific 

Gravity 

of the 

Materia]. 


Weight of 

12''xl2" 

Sheet 

■h' Thick. 

Lbs. 


Weight of 
1" Square 

Bar 
1 Ft. Long. 

Lbs. 


Weight of 
V Round 

Bar 

1 Ft. Long. 

Lbs. 


Weight of 1000 Lineal Ft. of 


Wire 

. ft' 
m Dia. 

Lbs. 


Wire 

.or 

in Dia. 
Lbs. 


Wire 

.oor 

in Dia. 

(1 Mil.) 

Lbs. 


.1 
.2 
.3 

.4 
.5 

.6 

.7 
.8 
.9 

1.0 


.0325 125 
.0650 250 
.0975 375 

.1300 500 
.1625 625 
.1950 750 

.2275 875 
.2601 000 
.2926 125 

.3251 250 


.04335 
.08670 
.13005 

.17340 
.21675 
.26010 

.30345 
.34680 
.39015 

.43350 


.034 047 
.068 094 
.102 141 

.136 188 
.170 235 
.204 282 

.238 329 
.272 376 
.306 423 

.340 470 


.133 
.266 
.399 

.532 
.665 
.798 

.931 
1.064 
1.197 

1.330 


. 00340 
.00681 
.01021 

.01362 
.01702 
.02043 

.02383 
.02724 
.03064 

.03405 


.0000 340 
.0000 681 
.0001 021 

.0001 362 
.0001 702 
.0002 043 

.0002 383 
.0002 724 
.0003 064 

.0003 405 



II 



Example. — What is the weight of 
1000 ft. of wire .02 in. in dia., if the 
specific gravity of the drawn metal is 
8.9? 



Solution: 
Wt. is proportional to (diam.)2 
For wire .or dia., 8.0=. 2724 
.9= .03064 



8.9 = 



30304 
4 



For wire .02" dia, wt. = 1 .21216 lbs. 



13. — Weight per Cubic Yard of any Material 
From its Specific Gravity. 

Specific gravity of material X 1685.4 =weight in pounds per cu. yd, 

X .8427= " " " " 



.7524 = 



short tons 
long tons 



REDUCTION TABLES, 



485 



14. — Comparison of Various Weights, Capacities and Volumes. 
From 1 to 9 Units. 





♦Weight In pounds per — 




tNumber of Cu. Ft. per - 


Cubic Foot. 
Lbs. 


U. S. Liquid 


U. S. Bushel 


Square Yard 


Short Ton 


Long Ton 


Gallon 


(2150.42 


1 In. Thick 


(2000 Lbs.) 


(2240 Lbs.) 


(231 Cu. Ins.) 


Cu. Ins.) 


(1296 Cu. Ins.) 


No. 


No. 




Lbs. 


Lbs. 


Lbs. 






.803 564 


.107 421 


1 


.602 673 


2 488.912 


2 787.581 


1 


.133 681 


1.244 456 


.75 


2 COO. 


2 240. 


1.333 333 


.178 241 


1.659 275 




1 5C9. 


1 680. 


1.607 128 


.214 842 


2 


1 205 346 


1 244.456 


1 393.791 


2 


.267 362 


2.488 912 


1.50 


1 000 


1 120. 


2.410 692 


.322 263 


3 


1.808 019 


829.637 


929.194 


2 666 667 


.356 482 


3.318 550 


2 


750 


840. 


3 


.401 043 


3.733 368 


2,25 


666.667 


746.667 


3.214 256 


.429 684 


4 


2.410 692 


622.228 


696.895 


4 


.534 724 


4.977 824 


3 


500. 


560. 


4.017 820 


.537 105 


5 


3.013 365 


497.782 


557.516 


4.821 384 


.644 526 


6 


3.616 038 


414.819 


464.597 


5 


.668 405 


6.222 280 


3.75 


400. 


448. 


5:333 333 


.712 964 


6.637 100 


4 


375. 


420. 


5.624 948 


.751 947 


7 


4.218 711 


355. 559 


398.226 


6 


.802 086 


7.466 736 


4.50 


333.333 


373.333 


6.428 512 


.859 368 


8 


4.821 384 


311.114 


348.448 


6.666 667 


.891 205 


8.296 375 


5 


300. 


336. 


7 


.935 767 


8.711 192 


5.25 


285.714 


320. 


7.232 076 


.966 789 


9 


5.424 057 


276.546 


309.731 


7.480 519 


1 


9.309 178 


5.610 39 


267.362 


299.445 


8 


1.069 448 


9.955 648 


6 


250. 


280. 


9 


1.203 129 


11.200 


6.75 


222.222 


248.889 


9.333 333 


1.247 687 


11.615 


7 


214.286 


240. 


10.667 


1.425 928 


13.274 


8 


187.500 


210. 


12.000 


1.604 169 


14.933 


9 


166.667 


186.667 


14.961 


2 


18.618 


11.221 


133.681 


149.722 


22.442 


3 


27.928 


16.831 


89.121 


99.815 


29.922 


4 


37.237 


22.442 


66.840 


74.861 


37.403 


5 


46.546 


28.052 


53.472 


59.889 


44.883 


6 


55.855 


33.662 


44.560 


49.908 


52.364 


7 


65.164 


39.273 


38.195 


42.778 


59.844 


8 


74.473 


44.883 


33.420 


37.431 


62.360 


8.336 328 


77.604 


46.770 


32.072 


35.920 


62.424 


8.344 875 


77.684 


46.818 


32.039 


35.884 


62.500 


8.355 035 


77.779 


46.875 


32. 


35.840 


67.325 


9 


83.783 


50.494 


29.707 


33.272 


75. 


10.026 


93.334 


56.25 


26.667 


29.867 


100. 


13.368 


124.446 


75. 


20. 


22.400 


125. 


16.710 


155.557 


93.75 


16. 


17.920 


150. 


20.052 


186.668 


112.50 


13.333 


14.933 


175. 


23.394 


217.780 


131.25 


11.429 


12.800 


200. 


26.736 


248.891 


150. 


10. 


11.200 


222.222 


29.707 


276.546 


166.667 


9 


10.080 


248.889 


33.272 


309.731 


186.667 


8.036 


9 


250. 


33.420 


311.114 


187.50 


8 


8.960 


280. 


37.431 


348.448 


210. 


7.143 


8 


285.714 


38.195 


355.559 


214.286 


7 


7.840 


320. 


42.778 


398.226 


240. 


6.25 


7 


333.333 


44.560 


414.819 


250. 


6 


6.720 


373.333 


49.908 


464.597 


280. 


5.357 


6 


400. 


53.472 


497.782 


300. 


5 


5.600 


448 


59.889 


557.516 


336. 


4.464 


5 


500. 


66.840 


622.228 


375. 


4 


4.480 


560. 


74.861 


696.895 


420. 


3.571 


4 


666.667 


89.121 


829.637 


500. 


3 


3.360 


746.667 


99.815 


929.194 


560. 


2.679 


3 


1 000. 


133.681 


1 244.456 


750. 


2 


2.240 


1 120. 


149.722 


1 393.791 


840. 


1.786 


2 


2 000. 


267.362 


2 488.912 


1 500. 




1.120 


2 240. 


299.445 


2 737.581 


1 680. 


.893 




Weight per 


Above value 


3S are directly i 


Droportional 


Above values, 


inversely pro- 


Cubic Foot 


tow 


eight per cubic 


foot. 


portional to ^ 


vt. per cu. ft. 



* Inversely proportional to Capacity or Volume per given Weight. 
t Inversely proportional to Weight per given Capacity or Volume. 



28.— STRENGTH AND RESISTANCE OF 

MATERIALS. 

I. GENERAL PRINCIPLES. 

The_ theory of the resistance of beams is explained in Sec. 15, 
Mechanics, page 298. The following discussion is pertinent to the subjoined 
tables, and to general working formulas. 

a. Stresses and Resistance. 

Stress /= force acting on any plane section of a material, produced 
either directly or by transmission; as, for instance, by leverage. It is 
measured usually in lbs. per sq. in. (or lbs. per sq. ft.), that is, in units of 
force per unit of area. The three primary stresses are tension, compression 
and shear. Transverse bending is accompanied by all three of the primary 
stresses. Torsion or twisting is usually treated as shear. 

Strain e = percentage of distortion produced by stress. All strains can be 
reduced primarily to tension, compression or shearing, although torsion 
(mainly shear) is sometimes classed separately. 

Modulus of elasticity E=^^ — -. — ^(within the "elasticlimit") =a constant. 
Stram e 

This is equivalent to stating that "Stress is proportional to strain" (Hooke's 
law), the stress / being equal to the applied load per square inch of cross- 
section, and the strain e, the percentage of resulting deformation of the 
material in the direction in which the force acts. If the material is in ten- 
sion there obtains Et = — , in which ft equals tensile stress per sq. in.; if in 
e 

compression, Ec =— ; if in shear, E^ = —\ if in torsion, £^tor=-^^ Within 
e e e 

practical limits, the moduli of elasticity for tension and compression are 

regarded by most engineers as equal for the same class of material.* If 

this is true the neutral axis in the beam under flexure (and also in the ideal 

column "under concentric loading) will pass through the center of gravity 

of the section and remain stationary for all safe loadings. This is funda-^ , 

mental to the present theories of beams and columns. « i 

Modulus of elasticity E is assumed to mean the tension or compression 
modulus unless otherwise stated or characterized, and will be so considered 
hereafter. If a weight is suspended at the lower end of a rod one inch in 

cross-section, producing an equal stress / in the rod, then will / = T77^ when 

1 E 

the rod is extended tttt^ part of its length.f i. e., when ^ = 0.001; / '^-ftcr 
lUUU OUU 

when the extension is -zj:;: or ^ = 0.002; etc. In general, f = Ee. It is easy 
ouu 

to see, then, that if these ideal conditions of elasticity could continue to the 

point where ^ = unity, / would equal E. Hence, the modulus or "coefficient" 

of elasticity is sometimes defined as that force which will stretch a rod of 

unit cross-section to double its length, based, of course, on the original 

length of rod and on the cross-section remaining constant. Long before 

this 100 per cent deformation can obtain, however, we reach the "elastic 

limit" or limit of elasticity of the material. 



* The modulus of elasticity for most materials varies more or less with 
the intensity of stress, and is different for tension and compression of the same 
material, the difference becoming more apparent as the stresses increase. 
Hence, the neutral axis of beams varies in position with the amount of loading. 

t The original length is used by the engineer for simplicity, although 
the final, or some distorted, length might be more applicable, 

486 




KINDS OF STRESSES. 487 

Elastic limit, and yield point. — Within the elastic limit of the material 
the strain is proportional to the stress or load, and when the latter is re- 
moved the mateVial will return to its original length, form or position. 
(In other words, a perfectly elastic material or a material strained withm 
the elastic limit is capable of transmitting all the energy that it receives, 
none of it being absorbed in internal work.) If, however, the material has 
been strained (by stress) in excess of or "beyond" I 
the elastic limit it will begin to "yield" and the re- at y^—'rldT^''^^' 
suiting deformation will increase more rapidly than % /e/^l'icistiliurTjif ' 
the increase of stress or load. Hence the term "yield ^ ' ' 
point" used to denote a point just beyond the elastic tf) 
limit of the material. In Fig. 1, which illustrates 

the method of plotting the relation between stress"- ,^ . 

and strain for any material, y is the yield point; oTravj 

o — e.l. is a straight line showing "stress pro- Fig. 1. 

portional to strain" within the elastic limit; and u is the position of ulti- 
mate strength, showing the relation between stress and strain at the "break- 
ing point." The yield point is often almost coincident with the elastic 
limit. 

The elastic limit is affected more or less by static-, repeated-, and 
alternating stresses. 

Ultimate strength — ultimate stress. — ^The ultimate strength of material 
is the ultimate stress per sq. in. which the material is capable of resisting, 
up to the point of breaking. This is shown in Fig. 1 as point u. In 
testing material the stress is increased gradually by incremental loading, 
and the breaking point is always above the yield point. 

The ultimate strength of material may be affected by the kind of stress 
or stresses to which it is subjected. 

Static stress, or stress in one direction, tends to raise the elastic limit. 

Repeated stresses, or stresses varying in intensity in one direction 
(either in tension or in compression, etc., and never reversing or passing 
through zero), tend to raise the elastic limit, sometimes materially, even 
when the stresses are well below it; and at the same time they tend to 
lower the ultimate strength. The latter will be the more marked the greater 
the number of repetitions and the wider their range. 

Alternating stresses, or stresses passing through zero from tension to 
compression or vice versa, or from positive to negative shear, torsion, etc., 
in a vibratory manner, as often occurs in actual structures, tend to act 
injuriously in lowering both the elastic limit and the ultimate strength. 
Hence, working alternating stresses should be kept low. 

Working stress and factor of safety. — The working stress is the maxi- 
mum (safe) stress allowed by the designing engineer in his calculations, 
and from the previous discussion it will be seen that its value may vary 
greatly. In selecting the proper working stresses we must consider: (1) the 
kind of material, its physical qualities, and its durability; (2) the kind of 
loading, and the nature of the stresses; (3) the probable life of the struc- 
ture, whether permanent or temporary. In general, the working stresses 
must be well within the elastic limit of the material, for any possible loading, 
and during any age of the completed* structure. The ratio of the ultimate 
strength to the working stress is called the factor of safety; thus, 

T^ ^ r r ^ ultimate strength 

Factor of safety = -^r-. . 

working stress 

The reason for basing the safety factor on the ultimate strength instead 
of on the elastic limit is that the former is more definitely ascertained; 
but the elastic limit is also considered to a certain extent, perhaps indirectly, 
in fixing the working stress. 

With reference to the same kinds of loading or stresses, we select larger 
factors of safety for natural materials, as wood and stone, than we do for 



* Concrete and reinforced concrete structures are not considered "com- 
pleted" until the cement has set to that degree of hardness consistent with 
the working stresses adopted and based on the "age" of the concrete. 
Concrete structures grow stronger with age; wooden structures, weaker. 



488 28.— STRENGTH AND RESISTANCE OF MATERIALS. 

manufactured materials, as steel and iron. The obvious reason for this is 
that the ultimate strength of the manufactured material may reasonably 
be expected to vary but little from the specification, say 8 to 10% for 
metals, whereas the ultimate strength of wood and stone, even of selected 
quality, may vary 100% or more; furthermore, tests of metals from the 
same heat will determine the general character of the material from that 
heat, while the physical properties of natural materials are not easily de- 
termined, excepting for the particular piece under test. Again, granular 
substances, as stone, are more uncertain than fibrous substances, as wood; 
weaknesses are less easily detected; and hence stone should have larger 
factors of safety than wood. 

Factors of safety may be considered to range from 3, for static loads 
on steel, under favorable conditions, up to 20 or 30, for impact loads on 
masonry, due to moving machinery. 

Resilience = work. — If a weight W is applied gradually to a rod of 
sectional area A, and length /, equal to volume Al=V, the elongation s 

W I 
will be equal to "T'^'e^' ^^^ ^^ ^ ^^ applied gradually, the work performed 

W WH 

in stretching the rod = — X 5 = -^-13 • K / represents the stress per sq. in., 

W P 

then-T-=/, and the work = -^XA/; but A/=the volume V, hence the 

ny 
resilience or work = ^r=r . This is called the elastic resilience when f equals 
Ah, 

the elastic limit of the material. 

Similarly, if a beam is loaded in the middle with a concentrated load 

Wl^ * 
W applied gradually, the deflection at the center will equal .„„- , and as 

W W Wl^ 

the average load — -jr-, the total work performed = -^r X T^-^q^. But from the 

Z 2i 'koiLi 

theory of beams, the bending moment Mb = "irX-^, equals Mr = — , or W = 

Afl 

-7— ; and, substituting this value of W in the above, we have 

ly 

1^2/3 f2ji f^ArH /2 y2 
Resilience or work = ^^ = ^^^ = -^^ = — . — . A/; 

or, in terms of volume it is equal to „^ „ • V" = 7^7^ • 7r~i • F. 

QEy^ 2E dy^ 

By the last equation it can be compared directly with the resilience of 
a rod. The above equations will represent the ^/as/^*c resilience of the beam, 
for concentrated loading at the middle, when / (the outer fiber stress per 
sq. in.) is equal to the elastic limit of the material. In general, the resilience 
of the beam is obtained by multiplying the greatest allowable gradually 
applied load, by one-half the deflection. 

The resilience t of a material may be defined as the capacity of that 
material to absorb and give out energy, measured in units of work, or 
inch-pounds. 

For different kinds of stress we have the following values for resilience 
or work of deformation of any material. By proper substitution of values 
for the particular kind and shape of the material the resilience may be 
obtained in definite units; and the result will be the elastic resilience if the 
value of / represents the elastic limit of the material: 

Stress. Resilience. 

/2 

Tension, direct, -------- — — ; V 

P 
Compression, direct, ------- ■— V 



I 



* See Table 1, Sec. 31, Properties and Tables of Beams and Girders, p. 562. 

t The modulus of resilience is the resilience of a unit volume (one cubic 

inch) of the material. It is obtained by making 1^= unity, in the equations. 



RESILIENCE SUDDEN LOADING. 489 

Stress. Resilience. 

Bending, in beam, from concentrated load,. - • ^^-^ V 

" " " uniform load, - - - t^^ V 

P 
•* ** rectangular beam (bXh), concen. load, - -r^ V 

uniform " - -^V 

^bE 

P 
•• " beam with unif. longitudinal bending moment, ^ V 

Torsion, in solid rod, ------- V 

oiitor 

where y = volume ot material under action, in cubic inches; 
jE = modulus of elasticity of the material, 
r = radius of gyration, in inches; 

3; = distance from neutral axis to extreme outer fiber, in inches; 
/= extreme outer fiber stress, in lbs. per sq. in. 

b. Loading and Impact. 
Sudden Loading. — In the discussion of resilience, the loading is con- 
sidered as applied gradually, the resultant being a static load; and at no 
time is the distortion greater than that due to the load statically applied. 
The intensity of stress increases constantly from zero to a maximum, and 
consequently the average stress equals one-half the maximum stress. If, 
however, a load just "touching" a rod or beam, so as to produce the least 
amount of stress, is applied suddenly but without impact, that is, "let go," 
the distortion will practically be doubled, and hence the fiber stress, deflec- 
tion, and work performed will be double that produced by resilience as when 
the load is applied gradually. For this reason "live loads" on structures, 
and especially train loads on bridges, are considered as "sudden loads," 
the working stresses for which are usually but one-half the working stresses 
for dead or static loads. 

Impact is the energy of the blow when one mass, possessing momentum, 
comes in contact with another mass. At the instant of contact each gives 
energy to and absorbs energy from the other, and if they were perfectly 
elastic (an ideal condition never realized in practice) none of the energy 
would be dissipated into heat. 

The mathematical analyses of many operations, as pile driving, water 
ram in pipes, etc., are dependent upon reducing the energy of the blow to an 
equivalent static force; but no formula yet devised correctly represents the 
relation. To do so, it must take into consideration the element of distor- 
tion or distance (energy is a measure of force X distance) , the inertia of the 
resisting mass, the loss of energy in heat, in deformation of material, etc. 
Partial analyses of some of these conditions have been attempted, upon 
certain assumptions, but they are of doubtful utility unless accompanied by 
data from practical experiments. The term "impact" is sometimes wrongly 
applied to "sudden loading." 

For discussion of pure Impact, see Sec. 15, Mechanics, pages 303, 304. 

II. TABLES OF STRENGTH OF MATERIALS. 

Comprising also Standard Specifications, 
Treated under the following heads: 

A. Woods, page 490. C. Building Stones, Cements, etc., page 507. 

B. Metals, page 496. D. Miscellaneous Materials, page 512. 



490 28,— STRENGTH AND RESISTANCE OF MATERIALS. 



A. Woods. 

1. — Compression (End) Tests op Timber. 

(From Circular No. 15, U. S. Dep't of Agric. — Div. of Forestry. 

[Pounds per square inch.] 



No 

* 



Species. 



Reduced to 15 per cent 

moisture. 

(See Table 2). 

Longleaf Pine 

Cuban Pine 

Shortleaf Pine 

Loblolly Pine 

Reduced to 1 2 per cent 
moisture. 

White Pine , 

Red Pine 

Spruce Pine 

Bald Cypress , 

White Cedar 

Douglas Spruce a — 

White Oak 

O vercup Oak 

Post Oak 

Cow Oak , 

Red Oak 

Texan Oak 

Yellow Oak 

Water Oak 

Willow Oak 

Spanish Oak 

Shagbark Hickory . . 
Mockernut Hickory. 

Water Hickory 

Bitternut Hickory. . 
Nutmeg Hickory . . . 

Pecan Hickory 

Pignut Hickory 

White Elm 

Cedar Elm 

White Ash 

Green Ash 

Sweet Gum 









<=> ri; 


o ^ 




m 4^ 








-is 


-=§ 




■^ fl 


Num- 


High- 


Low- 


S^ 




Aver- 


Is 
o 'O 


ber 


est 


est 


W'O 


age 


Z ftoj 


of 


single 


single 


fcJDCU 


bed) 


of all 


2-^ 


tests. 


test. 


test. 


tests. 


tafe 








^1 


1^^ 


(8) 
















Per 














cent. 


1.230 


11.900 


■ 3.400 


8,600 


5,700 


6,900 


53 


410 


10,600 


2.800 


9,500 


6,500 


7.900 


61 


330 


8,500 


4.500 


7.600 


4.800 


5,900 


47 


660 


11.200 


3.900 


8.700 


5,400 


6,500 


49 


130 


8.500 


3,200 


6.800 


4.000 


5.400 


49 


100 


8.200 


4,300 


8.100 


4,900 


6.700 


54 


170 


10.000 


4.400 


8.800 


5.600 


7,300 


66 


655 


9.900 


2.900 


8.500 


4.200 


6,000 


31 


87 


6,200 


3.200 


6.000 


4.400 


5,200 


79 


41 


8,900 


4.100 


8.100 


4.200 


5,700 


28 


218 


12,500 


5.100 


11.300 


6.300 


8.500 


40 


216 


9,100 


3,700 


8.600 


6.000 


7.300 


70 


49 


8,200 


5.900 


8,100 


6.000 


7.100 


58 


256 


11,500 


4,600 


9,800 


5.600 


7.400 


51 


57 


9,700 


5,400 


9,200 


5.500 


7.200 


36 


117 


11,300 


5,800 


9,800 


6.900 


8.100 


62 


40 


8,600 


5,500 


8,300 


5.800 


7.300 


58 


31 


9,200 


6,200 


9.000 


6.300 


7.800 


75 


153 


11,000 


4.200 


8.7 00 


5.500 


7.200 


51 


251 


10,600 


3,700 


9,500 


5,100 


7.700 


61 


137 


13,700 


5,800 


10.900 


7,500 


9.500 


79 


75 


12.200 


6,200 


11,600 


8,000 


10.100 


65 


14 


10.000 


6,700 


9,600 


7,000 


8.400 


71 


25 


11.500 


7,300 


11,200 


7,800 


9.600 


60 


72 


12.300 


6,400 


11,000 


7,100 


8,800 


79 


37 


10,500 


5.800 


10,400 


7,300 


9,100 


51 


30 


13,000 


8,700 


12,700 


8,900 


10,900 


72 


18 


8.800 


4.900 


8,^00 


5,000 


6,500 


28 


44 


10.600 


6,200 


10,100 


6,500 


8,000 


66 


87 


9,600 


5,000 


8,700 


5,7 00 


7,200 


48 


10 


9 800 


6,600 


9,800 


6.600 


8,000 


29 


118 


8.900 


4,600 


8.500 


5,600 


7,100 


60 



m +3 
Sio bfl 



Per 
cent. 



90 
93 
90 
84 



93 
96 
95 
74 
99 
65 
81 
95 

100 
89 
94 
98 

100 

100 
88 
94 
97 
99 

100 

100 
97 
95 

100 
88 
95 
96 

100 
97 



* Nos. correspond with similar numbers in Sec. 27, Weights and Specific 
Gravities of Materials, Table 6, page 470; also in the five following tables. 
a Actual tests on"dry" material not reduced for moisture. 



TIMBER— COMPRESSION TESTS. 



491 



2. — Factors to be Added to the strength factors o£ Southern Pines 
at 15 per cent moisture in order to reduce them to 12 per cent: 



No. 







Bending — 


Modulus 


Crush- 




Crush- 
ing end- 




of 
elas- 


ing 
across 


Species. 


At 


At 




wise. 


elastic 
limit. 


mp- 
ture. 


ticity. 


grain. 




Lbs. per 


Lbs. per 


Lbs. per 


Lbs. per 


Lbs. per 




sq. in. 


sq. in. 


sq. m. 


sq. in. 


sq. in. 


I.ongleaf Pine (Pinus palusiris) . 


1.100 


1,500 


1,700 


180.000 


180 


Cuban Pine (Pinus heterophylla) 


800 


1.500 


1.700 


70,000 


220 


Sliortleaf Pine (Pinus echinata) . 


600 


600 


900 


80.000 


60 


Loblolly Pine (Pinus toeda) 


900 


1,000 


1,200 


100,000 


150 


Reference to Table 


Tab. 1. 
Col. (8). 


Tab. 5. 
Col. (8). 


Tab. 4. 
Col. (8). 


Tab. 5. 
Col. (11). 




and name of Column 









3. — Compression (End) Tests op, Green Timber. 

(Above 40 per cent moisture; not reduced.) 

[Pounds per square inch.] 



No. 



Species. 



Number 
of tests. 



Highest 
single, 
test. 



Lowest 
single 
test. 



Average 
of all 
tests. 



Longleaf Pine 

Cuban Pine 

Shortleaf Pine 

Loblolly Pine 

Spruce Pine 

Bald Cypress 

White Cedar 

White Oalt 

Overcup Oak 

Cow Oak 

Texan Oak 

Willow Oak 

Spanish Oak 

Shagbark Hickory. 
Mockernut Hickory 

Water Hickory 

Nutmeg Hickory. . . 
Pecan Hickory. . . . 
Pignut Hickory.... 
Sweet Gum 



86 
38 



71 

280 

34 

25 

45 

58 

39 

49 

52 

22 

18 

4 

26 

4 

5 



7.300 
6,100 
4,000 
5,500 
4,700 
8.200 
3,400 
7,000 
4,900 
4,900 
6,000 
5,500 
5,100 
6,900 
7,200 
5.600 
5,500 
3,800 
6.200 
3,600 



2.800 
3,500 
3,000 
2.600 
2.800 
1,800 
2.300 
3,200 
2,800 
2,300 
3.100 
2.300 
2.500 
3.500 
4.500 
4.700 
3.700 
3.300 
4,700 
3.000 



4.300 
4.800 
3.300 
4.100 
3,900 
4,200 
2.900 
.5.300 
3.800 
3.800 
5.200 
3.800 
3.900 
5.700 
6.100 
5.200 
4.500 
3.600 
5,400 
3,300 



492 28.—STRENGTH AND RESISTANCE OF MATERIALS. 



4. — Bending Tests op Timber at Rupture. 
[Pounds per square inch.] 



No. 



Species. 



Num- 


High- 


Low- 


ber 


est 


est 


of 


single 


single 


tests. 


test. 


test. 


1.160 


17,800 


3,300 


390 


17,000 


2,900 


330 


15,300 


5,000 


650 


14,800 


3,900 


120 


11.100 


4.600 


95 


12,900 


3,100 


170 


16,300 


3,100 


655 


14,800 


2,300 


87 


9,100 


3,500 


4] 


13,000 


3.800 


218 


20.300 


5,700 


216 


19.600 


4.900 


49 


1G.400 


5.100 


256 


23.000 


3.300 


57 


16,500 


5.700 


117 


19.500 


8.200 


40 


15,000 


5,100 


31 


16.000 


5,800 


153 


16,000 


3,200 


257 


17.300 


5,000 


187 


23,300 


5,700 


75 


20,700 


5,300 


14 


18,000 


5,300 


25 


19,500 


7,000 


72 


16,600 


G,700 


37 


18,300 


5.600 


30 


25,000 


11,100 


18 


14,000 


7.300 


44 


19,200 


6,600 


87 


15,000 


5,000 


10 


16,000 


5,100 


ua 


14,400 


5,100 



-2 

■4.3 CC 
03 Ol 



lo 

Q 



Aver- 
age 
of all 
tests. 

(8) 



o « 



tH C O) 

©2 > 



^o 



si 

a-i t-» 



fc- fl CJ 



9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 



Reduced to 15 per 
cent moisture. 
(See Table 2). 

Longleaf Pine 

Cuban Pine 

Shortleaf Pine 

Loblolly Pine 

Reduced to 12 ver 
cent moisture. 

White Pine 

Red Pine 

Spruce Pine 

Bald Cypress 

White Cedar 

Douglas Spruce a . 

White Oak 

Overcup Oak 

Post Oak 

Cow Oak 

Red Oak 

Texan Oak 

Yellow Oak 

Water Oak 

Willow Oak 

Spanish Oak 

Shagbark Hickory. 
Mockernut Hickory 

Water Hickory 

Bitternut Hickory . 
Nutmeg Hickory , . 
Pecan Hickory .... 
Pignut Hickory . . , 

White Elm 

Cedar Elm 

White Ash 

Green Ash 

Sweet Gum 



Per 
cent. 



Per 
cent. 



14,200 
14,600 
12,400 
13,100 



10,100 
12,300 
13,600 
11,700 
8,400 
12.000 
18,500 
14,900 
15,300 
12.500 
15.400 
16.900 
14,600 
15,700 
13,800 
15.600 
20,300 
19.700 
17,300 
19,300 
15,600 
18,100 
24,300 
13,600 
17,300 
14.200 
16.000 
12,700 



8,800 
8,800 
7,000 
8.100 



5,000 
4,900 
5,800 
5,000 
4,000 
4,100 
7,600 
6,300 
7,400 
6.500 
9,100 
10,000 
5.700 
7,200 
5,400 
6,900 
9,400 
7,900 
5,400 
8,700 
8,100 
10.300 
11,500 
7.300 
8,500 
6,300 
5,100 
6.000 



10.900 

11.900 

9.200 

10,100 



7,900 
9.100 
10.000 
7,900 
6,300 
7.900 
13.100 
11.300 
12.300 
11,500 
11,400 
13.100 
10.800 
12.400 
10.400 
12.000 
16.000 
15.200 
12.500 
15.000 
12,500 
15.300 
18.700 
10.300 
13.500 
10.800 
11,000 
9,500 



84 
83 
79 
84 



81 
60 
81 
69 
78 
58 
75 
81 
92 
68 
84 
86 
65 
76 
70 
72 
84 
78 
64 
60 
88 
95 
77 
72 
86 
77 
60 
79 



a Actual tests on "dry" material not reduced for moisture. 



TIMBER—BENDING TESTS, 



493 



6.— Bending Tests op Timber at Relative Elastic Limit. 
[Pounds per square inch.] 



Species. 



Reduced to 15 per 
cent moisture. 
(See Table 2.) 

Longleaf Pine 

Cuban Pine 

Shortleaf Pine 

Loblolly Pine 

Reduced to 12 per 
cent moisture. 

White Pine 

Red Pine 

Spruce Pine 

Bald Cypress 

White Cedar 

Douglas Spruce a . . 

White Oak 

Overcup Oak 

Post Oak 

Cow Oak 

Red Oak 

Texan Oak 

Yellow Oak 

Water Oak 

Willow Oak 

Spanish Oak 

Shagbark Hickory. 
Mockernut Hickory 

Water Hickory 

Bitternut Hickory . 
Nutmeg Hickory . . 

Pecan Hickory 

Pignut Hickory.. . . 

White Elm 

Cedar Elm 

White Ash 

Green Ash 

Sweet Gum 



Num- 
ber 
of 
tests 



160 
390 
330 
650 



130 

95 

170 

655 

87 

41 

218 

216 

49 

256 

57 

117 

40 

31 

153 

257 

187 

75 

14 

25 

72 

37 

30 

18 

44 

87 

10 

118 



High- 
est 

single 
test. 



13.500 
12,900 
11,900 
12.700 



Low- 
est 
single 
test. 



10.000 
11.300 
13.700 
12.000 
8.200 
13.700 
15.700 
11.600 
10,600 
14.200 
14,500 
12.000 
11.800 
11,800 
13.100 
13.500 
16,100 
15,400 
11.900 
14.300 
12.200 
15.000 
17.500 
700 
10,700 
11,500 
13,200 
11.000 



2,400 
2.200 
2,900 
3,100 



4,100 
3,100 
3.000 
2,200 
3,400 
2,800 
4,400 
4.000 
5.100 
3.400 
5,100 
5.900 
4,900 
4,500 
2.700 
5.100 
5,400 
4,300 
4,100 
7,500 
4,200 
5 800 
7.400 
5.300 
4,700 
3,600 
3.200 
3.500 



ja '^ 
o 

Si +^ 

>2^ 



11,100 

11,500 

9,700 

10,800 



8.200 

10.300 

11.200 

9,900 

7,390 

9,600 

14.100 

9,500 

9.600 

11.600 

13.600 

11,400 

11,100 

11,400 

10,000 

11,600 

14.200 

14.600 

11.800 

14.000 

11,200 

14,400 

16,400 

9,600 

10,100 

10.400 

13,200 

10.100 



M O 
^^ 

2, a 
o 

o 



bjDO 

>2l 



5,400 
5.600 
4.800 
5,400 



4.500 
4,500 
5,000 
4,200 
4.000 
3.400 
6.100 
5.400 
6.000 
5.000 
5,600 
7.800 
5.100 
5,500 
4,300 
6,600 
7.700 
7.800 
4.800 
7,600 
6.400 
7.900 
8.300 
5.400 
5.800 
5.200 
3.200 
5.100 



Aver- 
age 
of all 
tests. 

(8) 



8,500 
9,500 
7.200 
8,200 



6.400 
7.700 
8.400 
6,600 
5.800 
6.400 
9.600 
7.500 
8.400 
7.600 
9,200 
9,400 
8.100 
8,800 
7,400 
8,600 
11,200 
11.700 
9,800 
11,100 
300 
11,500 
12,600 
7,300 
8,000 
7.900 
8.900 
7,800 



OQ-M 


02 4i 


■^ d 




ss 


Ss 


„_. (h 


«-, M 


iono 
10 pe 
age. 


iono 
25 pe 
age. 


■»-= ^ ^ 


•^ ^ ^ 


t-, ti Qi 


tH fl <U 


as§ 


ss§ 


£^ = 


r^"o 


Per 


Per 


cent. 


cent. 


43 


81 


42 


83 


48 


81 


46 


85 


58 


85 


38 


73 


51 


82 


25 


66 


44 


86 


32 


56 


37 


73 


47 


91 


34 


76 


50 


95 


15 


49 


62 


94 


35 


75 


40 


84 


42 


81 


41 


80 


50 


89 


39 


83 


21 


86 


44 


84 


46 


93 


65 


89 


40 


83 


33 


71 


57 


91 


43 


83 


40 


70 


46 


82 



Modulus 

of 
elasticity 
(average 

of all 

tests) 

(11) 



1,890.000 
2.300.000 
1,600.000 
1,950,000 



1,390.000 
1,620.000 
1,640,000 
1,290,000 
910,000 
1,680,000 
2,090,000 
1,620.000 
2.030,000 
1.610.000 
1,970,000 
1,860,000 
1,740,000 
2.000,000 
1.750.000 
1,930.000 
2,390.000 
2,32,0.000 
2.080,000 
2.280.000 
1.940.000 
2.530,000 
2.730.000 
1.540.000 
1.700.000 
1,640.000 
2.050.000 
1.700.000 



a Actual tests on "dry" material not reduced for moisture. 



494 28.— STRENGTH AND RESISTANCE OF MATERIALS. 

6. — Compression Tests of Timber Across Grain,* and Shearing 

Tests of Timber with Grain. 

[Pounds per square inch.] 



No. 



Species. 



Num- 
ber of 
tests. 



Compres- 
sion 
across 
grain. 

(4) 



Shearing 

with 
grain not 
reduced 

for 
moisture. 



Reduced to 15 per cent, moisture. 
{See Table 2). 

Longleaf Pine 

Cuban Pine 

Shortleaf Pine 

Loblolly Pine 

Reduced to 12 per cent, moisture. 

White Pine 

Red Pine '. 

Spruce Pine 

Bald Cypress 

White Cedar 

Douglas Spruce f 

White Oalj 

Overcup Oals 

Post Oak 

Cow Oak 

Red Oak 

Southern Red Oak 

Black Oak 

Water Oak 

Willow Oak. 

Spanish Oak 

Shagbark Hickory 

White Hickory 

Water Hickory 

Bitternut Hickory 

Nutmeg Hickory 

Pecan Hickory ...:.. 

Pignut Hickory 

White Elm 

Cedar Elm 

White Ash 

Green Ash 

Sweet Gum 



1.210 
400 
330 
690 



130 

100 

175 

650 

87 

41 

218 

216 

49 

256 

57 

117 

40 

30 

153 

255 

135 

75 

14 

25 

72 

37 

30 

18 

44 

87 

10 

118 



1.000 

1.000 

900 

1,000 



700 
1.000 
1.200 
8 00 
700 
800 
2.200 
1.900 
3.000 
1.900 
2.300 
2.000 
1.800 
2.000 
1.600 
1.800 
2.700 
3.100 
2.400 
2.200 
2.700 
2.800 
3.200 
1.200 
2.100 
1.900 
1,700 
1,400 



700 
700 
700 
700 



400 

500 

800 

500 

400 

500 

1.000 

1.000 

1.100 

900 

1,100 

900 

1.100 

1.100 

900 

900 

1.100 

1.100 

1,000 

1.000 

1.100 

1,200 

1,200 

800 

1.300 

1,100 

1,000 

800 



* To an indentation of 3 per cent of the height of the specimen, 
t Actual tests on "dry" material not reduced for moisture. 

The above tests made under the direction of J. B. Johnson, for the 
Division of Forestry, are given in full because, aside from the actual value 
of the results, they show clearly the wide variation which may be expected 
in submitting timber to actual test. 

Relation Between Weight and Strength of Timber. — For want of space the 
author omits two diagrams which exhibit close relations of weight to 
breaking strength of timber, but the relations may be shown analytically, 
as follows: 

Compression, endwise; pounds per square inch: 

For pines and hickories = weight (dry) in pounds per cu. ft. X 195. 

For oaks = weight (dry) in pounds per cu. ft. X 170. 

Bending, transverse; pounds per square inch (fiber stress): 

For pines and hickories = weight (dry) in pounds per cu. ft. X 300. 

For oaks = weight (dry) in pounds per cu. ft. X 255. 

It is to be noted that the dry weights per cubic foot are to be used. (See 
Foot-note to Table 6, Sec. 27, Weights and Specific Gravities, pages 470, 471.) 



TIMBER— TENSION, COMPRESSION, ETC. 



495 



7. — ^Timber in Tension, Compression, Bearing, Bending and 

Shear. 

Practical Working Units for Columns, Beams, etc.* 

(See Author's Column Formula, Sec. 32, Columns, page 590.) 

[Note. — Values in table are in thousand pounds per square inch; hence, 
multiply by 1000 to reduce to pounds per square inch. Thus 12,0= 12000.1 





y 


Tens. 

d 
O 


Compression. 


Bearing. 


Bending. 


Shear 


Timber Safet 
Facte 


Zero 
Length. 


10 


d 

a 



< 


1 

is 


1 

1 

r 




4.^ 

Is 




a 


(1) 

Douglas Spruce (Oreg. and 
Wash. "Pine" or "Fir"), 

Longleaf Pine • 


1 

5 
16 

1 
5 

16 
1 
5 

^6 

5 

,6 
1 
5 
.6 
1 
5 

.6 
1 
5 

:6 

1 

5 

.6 

1 

5 

.6 

1 

5 

.6 

1 

5 

.6 

' 1 

5 

'? 

5 
6 
1 
5 

.6 
1 
5 

,6 

I 

= 6 


(2) 

12,0 
2.4 
2.0 

12.0 
2,4 
2.0 

10.0 
2.0 
1,7 

10.0 
2.0 
1.7 
8.0 
1.6 
1.3 
9.0 
1,8 
1.5 
9.0 
1.8 
1.5 

10.0 
2.0 
1.7 

10.0 
2.0 
1.7 
8.0 
1.6 
1.3 
8.0 

1:1 

1.2 
6.0 
1.2 
1.0 

1:1 

1.2 
6.0 
1.2 
1.0 
9.0 
1.8 
1.5 


(3) 

7.5 
1.5 

'i.'o 

1,4 

'5.5 
1.1 

'6.5* 
1.3 

'5.5 
1.1 

*5.V 
1,1 

'5,*5* 
1.1 

'7.0' 
1,4 

"7.0' 
1.4 

'5,5 
1.1 

■5.5 
1.1 


(4) 

7.0 
1.4 

'6.5' 
1.3 

•5.b- 
1.0 

'6.0 
1.2 

'5.0* 
1.0 

*5.0 
1.0 

*5.0 
1,0 

'6,5* 
1.3 

'e.'s' 

1.3 

'5,'0* 
1.0 

*5.*0* 
1.0 


(5) 

5.5 
1.1 

*5.'0* 
1.0 

'4.0 
0.8 

*4.'5* 
0,9 

'4.*0' 
0,8 

*4.'0* 
0.8 

'4,'o' 
0,8 

'5,*o" 
1.0 

'5,'0* 
1.0 

4,0* 
0.8 

4,0* 
0.8 


(6) 

8.0 
1.6 
1.3 
8,0 
1.6 
1.3 
6,0 
1.2 
1.0 
7.0 
1.4 
1,2 
5.5 
1.1 
0.9 
6.0 
1.2 
1.0 
6,0 
1.2 
1.0 
7.0 
1,4 
1.2 
7.0 
1.4 
1.2 
6.0 
1,2 
1.0 


(7) 

1.2 

'i.V 
*i.o 

*2,0 

*6,V 

'6,*8' 

'is 
'6,'s 

'6.*8* 
'6.*7* 

'o.V 


(8) 

6.0 
1.2 
1.0 
7.0 
1.4 
1.2 
6.0 
1.2 
1.0 
6,5 
1,3 
1.1 
4.5 
0.9 
0.7 
5.0 
1.0 
0.8 
4.5 
0.9 
0.7 
5.0 
1.0 
0.8 
5.0 
1.0 
0.8 
4.0 
0.8 
0.7 
5.0 
1.0 
0.8 
4.5 
0.9 
0.7 
3,5 
0,7 
0,6 
5,0 
1.0 
0.8 
5.0 
1.0 
0.8 
5,0 
1.0 
0.8 


(9) 
1500, 
1600, 
1100. 
1400, 
1000, 
1100, 
1100. 
1400. 
1400, 
1100. 
1100. 
1000. 
1000, 
900, 
1000. 
1200. 


(10) 
0.6 

'0,6* 

* 0.4' 
'0.8' 

* 0.4* 
"0,4* 
"0,4* 

* 0.4* 
**0.4* 
"0,4* 
"0*4* 
"0.4* 
'0.4' 
*'0*.4* 

*m' 

"0.6* 


Shortleaf Pine 


White Oak 


White Pine 


Red Pine 


Norway Pine 


Canadian White Pine 

(Ottawa) ^ 

Canadian Red Pine 

(Ontario) ^ 


Spruce and Eastern Fir 

California Spruce i 








California Redwood i 


5.5 
1.1 


5,0 
1.0 


4,0 
0,8 


.... 


0,8 








Hemlock , . . . i 


5,5 
1,1 


5.0 
1.0 


4.0 
0.8 


.... 


0,6 








Wliite Cedar 


5.5 
1.1 

'5,5' 
1,1 

*7.0 
1.4 


5.0 
1.0 

'5,'0' 
1.0 

*6.'5' 
1.3 


4.0 
0.8 

*4.'o' 
0.8 

'5.*0* 
1.0 


6,0 
1.2 
1.0 
6.0 
1,2 
1.0 


0,7 
"6.7* 

*6.*9' 


Bald Cypress i 


Chestnut 























*See Recommendations and Remarks on following page. 



496 2S.'-STRENGTH AND RESISTANCE OF MATERIALS, 



Recommendations and Remarks on Preceding Table: 

Tension. — Use safety factor of 8 for tension, in bridges and buildings. 

Compression. — Use safety factor of 5 for compression, in bridges and build- 
ings, under ordinary circumstances. 

Bearing. — Use safety factor of 5 for end bearing, with grain; and safety 
factor of 4 for side bearing, across grain. 

Bending. — Use safety factor of 6 for extreme fiber of railroad stringers and 
for joists; use factor of 5 for floorbeams and for large girders. 

Shear. — Use safety factor of 5 for shearing, with grain. Note that the 
longitudinal shearing force H per unit of length of beam is equal to the 
total vertical shear V multiplied by the statistical moment Q of area 
above plane of shear, about neutral axis, and divided by the moment of 

VQ 



inertia / of total section. Thus, H = 
Girders, page 565.) 



(See Sec. 31, Beams and 



B. Metals. 

-Tension, Compression, Bending, Shearing, etc., op Metals. 
[Pounds per square inch.] 



Metal. 


Tension 
(Ult.) 


Elastic 
Limit. 


Com- 
pression 
(Ult.) 


Bend- 
ing 
(Extr. 
Fiber.) 


Shear- 
ing 
(Ult.) 


Modu- 
lus of 
Elas- 
ticity. 


♦Aluminum, Commercial 

Castings 


15 000 


6 500 


12 000 





12000 
16000 


11 000 000 


Forgings 




Sheets 


24 000 
28 000 
(30-65) 
48 000 
(20-35) 
28 000 
(70-80) 

75 000 

100 000 
(40-50) 
45 000 

1 000 
6 000 

24 000 

80 000 
50 000 


12 000 
14 000 
(16-3 0) 
23 000 








Bars 




















Wire (hard) . . . | 










Wire (Elec. j 










cond.) ] 


14 000 










♦Aluminum-bronze I 










5% to 7i% Al. with cop- ■! 
per L 


40 000 
60 000 


120 000 


. 






10% Al. with copper, J 

forged eye-bars \ 

♦Aluminum, " Nickel," 2 to f 














1 


7% of nickel, copper, Iron, -j 
etc I 


25 000 


















Antimony, cast 




Bismuth 












Brass, cast • 


6 000 
16 000 


(say) 
30 000 


(say) 
20 000 


36 000 




wire, unannealed 

(hard) 

annealed . ... 


9 000 00 
14 000 000 























* Mainly on the authority of Alfred E. Hunt, and the Pittsburg Reduc- 
tion Co., principal manufacturers of aluminum in the United States. Assum- 
ing the conductivity of copper wire at .97 and aluminum wire at .62, the 
area of aluminum wire must be 1.565 times that of copper to carry the same 
electrical current; and on this basis the aluminum wire, for any given 
length, say per mile, will weigh .47 as much as copper wire. Hence, when 
the price of aluminum wire per pound is .47 that of copper, or when copper 
is 2.13 that of aluminum, they will be on about an equal footing for economy, 
on transmission lines. The relative cross-sections of wire of different con- 
ductivities, for equal electric transmission, are about as follows: Copper at 
.97 conductivity equals area 1.00; aluminum at .60 conductivity = area 
1.617, at .61 cond = area. 1.59, at 62 cowc/ = area 1.565, at .63 cond = a,Tea 1.54. 
The physical properties of pure aluminum are increased by cold rolling and 
forging. There are two principal alloys with copper, viz., aluminum-bronze 
and copper-hardened aluminum, the latter containing from 2 to 15 per cent 
copper. 



METALS— TENSION, COMPRESSION, ETC. 



497 



8. — ^Tension, Compression, Bending, Shearing, etc. op Metals — Cont'd. 
[Pounds per square inch.] 



Metal. 


Tension 
(Ult.) 


Elastic 
Limit. 


Com- 
pression 
(Ult.) 


Bend- 
ing 
(Extr. 
Fiber.) 


Shear- 
ing 
(Ult.) 


Mod- 
ulus of 
Elas- 
ticity. 


Bronze, Aluminum (See j 














Aluminum-bronze) \ 

♦Gun (metal). U.S. Ord-J 
nance, cop. 9, tin 1 .. I 
" same greatly compres- ' 
sed 














25-55 












40 000 

72-78 

75 000 

60 000 

100 000 
50 000 

100 000 
55 000 
75 000 

108 000 
66 000 
80 000 

100 000 
32 000 
25 000 
32 000 

35 000 
(55-65) 
60 000 

36 000 
45 000 
68 000 
85 000 

100 000 
20 000 
30 000 
50 000 


10 000 




52 UOO 




10 000 000 












Manganese, cast 


30 000 
80 000 
24 000 


125 666 








rolled 








Phosphor . , 










" wire 










Silicon, cast, 3% SI 













" 5 % " 












" hard wire t 












Tobin, cast 












rolled 


40 666 








4 500 066 


cold rolled 










Copper bolts 


"6*666 

10 000 


32 000 
40 000 








cast 


22 000 




30 000 


10 000 000 


plates 

rods (drawn) 












wire, unannealed f 

(hard) 1 

" annealed 












34 500 








18 000 666 








15 000 000 


X Delta metal, cast 












rolled plates .... 












small bars 












wire (hard) 












Gold, cast 


4 000 








8 000 000 


wire (hard) 










Gold (5), copper (1) part 












Gun metal (s&e Bronze) 


















(say) 
80 000 








Iron, cast (ordinary) | 

aGray cast 


15 000 


6 000 


30 000 


18 000 


12 000 000 


"Malleable Cast / 


(27-35) 

32 000 

18000 + 












Common 1^ 

Staybolts 


26 666 


















^Wrought (see Steel) .... 



























* The physical properties of gun-metal vary greatly with the method of 
casting, and also with the position of the metal in the cast. The tenacity 
increases with the specific gravity and pressure; it is greater at the bottom 
of the melt than at the top. 

t The relative conductivity of silicon-bronze wire to pure copper at 1.00 
varies from .95 for the soft wire down to .35 for the hard. 

t Composed of about 60 parts copper, 38 to 40 parts zinc, 2 to 4 iron, 
1 to 2 tin. 

a See Proposed Standard Specifications for Gray Iron Castings, p. 498. 

° The Am. Soc. for Testing Materials adopted (by letter-ballot on Nov. 
15, 1904 — Vol. IV, page 96) the following tests for tensile and transverse 
strength of malleable castings: Tensile Test. The tensile strength of a 
standard test bar (a bar 1 in. square and 14 ins. long, without chills and 
with ends perfectly free in the mold) for castings under specification shall 
not be less than 40000 lbs. per sq. in. ; and the elongation measured in 2 ins. 
shall be not less than 2^ per cent. Transverse Test. The transverse strength 
of a standard test bar, on supports 12 ins. apart, pressure being applied at 
center, shall be not less than 3000 lbs., deflection being at least i inch. 

^ Wrought iron is now seldom manufactured except for blacksmith iron 
and water pipes. Steel which often goes under the double name of "Iron 
and Steel," in specifications, has entirely superseded iron for structural 
work because it is stronger and more cheaply manufactured. 



498 28.— STRENGTH AND RESISTANCE OF MATERIALS. 

8. — Tension, Compression, Bending, Shearing, etc. op Metals — Cont'd. 
[Pounds per square inch.] 



Metal. 


Tension 
(Ult.) 


Elastic 
Limit. 


Com- 
pression 
(Ult.) 


Bend- 
ing 
(Extr. 
Fiber.) 


Shear- 
ing 
(Ult.) 


Modu- 
lus of 
Elas- 
ticity 


Iron — Cont'd. 

Wrought shapes 


48 000 
50 000 
52 000 
60 000 
80 000 

1 800 
3 200 

2 500 


26 000 

27 000 


46 000 
48 000 


44 000 
48 000 


40 000 
40 000 




" bars 


28 000 000 


bolts 




Wire, annealed 










15 000 000 


" unannealed 


27 000 








25 000 000 


Lead, cast 








1 000 000 


milled 












wire 












Malleable castings (see Iron) . . . 












* Nickel 














Platinum wire, unannealed 


53 000 
32 000 
40 000 












annealed 












Silver, cast 

























* Alloys with copper and steel. (See Steel, Nickel.) 



Gray Iron Castings. — Proposed Standard Specifications, adopted by 
letter-ballot of the Am. Soc. for Testing Materials, September 1, 1905. 
Digest: — 1. Unless furnace iron is specified, all gray castings are to be 
made by the cupola process. 2. The sulphur contents to be as follows: 
Light castings, not >.08%; medium castings, not >.10%; heavy castings, 
not >.12%. 3. "Light" castings are those having any section less than 
i in. thick; "heavy" castings are those in which no section is less than 2 ins. 
thick; "medium" castings are those not included under light or heavy, 
4. Transverse Test. The minimum breaking strength of the "Arbitration 
Bar" (a round bar H ins. dia. and 15 ins. long) under transverse load shall 
not be under: Light castings, 2500 lbs.; medium castings, 2900 lbs.; 
heavy castings, 3300 lbs. In no case shall the deflection be under .10 inch. 
(The loads to be applied at middle of bars resting on supports 12 ins. apart. 
The deflection at rupture. Two sets of 2 bars each from each heat; one set 
from the first, and the other set from the last iron going into the casting. 
Where the heat exceeds 20 tons, an additional set of two bars shall be cast 
for each 20 tons or fraction thereof 
above this amount. In case of change ^ 

of mixture during the heat, one set of f§j 

two bars shall also be cast for every 



-/i' 



i£> 



mixture other than the regular one. 
Each set of 2 bars is to go into a single 
mold. The bars shall not be rum- 
bled or otherwise treated, being sim- 
ply bushed off before testing.) Tensile 
Test. Where specified, this shall not ^ 
run less than: Light castings, 18000 ^ 
lbs. per sq. in. ; medium castings, « 
21000 lbs. per sq. in.; heavy castings, ^ 
24000 lbs. per sq. in. The tensile test 
shall be made on the "Tensile Test 
Piece*" (3^ ins. long with sectional 
diameter 1 in. round). 5. The quality 
of the iron going into castings under 
specification shall be determined by 
means of the "Arbitration Bar." The 

tensile test is not recommended, but _,. „ n*- u r ..* i.-^ ^. n *% 
in case it is called for, the "Ten- F^g- 2.— Mold for Arbitration Bar. 
sile Test Piece" turned up from any of the pieces of the transverse test 
shall be used. The expense of the tensile test shall fall on the purchaser. 



V-Zi 



> 


y Pouring Basin j 




5^ 


\ / 






ir-n/ 




1 
5> 


> 


// 


< > 


li 


< 














^ 














"^ 




^2i'> 




<3"> 




<?i'i 






























•^ 














^^ 






L/ ^-^ 1 


'^ 


J 


1 




•f ^- 


«' A 


..V 








,_, ft 



* See Fig. 3, page 601. 



STEEL— WROUGHT, CAST, FORGED, ETC, 



499 



8.— Tension, Compression, Bending, Shearing, etc., op Metals— Cont'd 
[Pounds per square inch.] 



Metal. 


Tension 
(Ult.) 


Elastic 
Limit. 


Com- 
pression 
(Ult.) 


Bend- 
ing 
(Extr. 
Fiber.) 


Shear- 
ing 
(Ult.) 


Modu- 
lus of 

Elas- 
ticity. 


Steel * 

Steel, castings, max. 


142 000 
85 000 
70 000 
60 000 












hard 












*' medium 

•• soft 


40 000 


70 000 


70 000 


60 000 


30 000 000 


tforglngs 












springs, (untempered) . | 
wire, annealed 


(65-115) 
90 000 
80 000 
120 000 
180 000 
200 000 
280 000 

3 500 

11 000 

(4-6) 

5 000 

16 000 


(40-70) 
55 000 
40 000 
60 000 
80 000 
95 000 


























" unannealed 










" crucible 











" susp. bridges 










" extra, tempered . . . 










Tin, cast 


1 800 


(say) 
6 000 


4 000 






Tin 10, antimony 1 

Zinc, cast ■, 

rolled : 


4 000 000 


"4*666 


(say) 
18 000 








7 000 





13 000 000 



* High ult tens str of steel was employed in the following structures: 
St. Louis Br., 100000; Plattsmouth Br. and Niagara Centilever, 80 000; 
Bismarck Br., 80-90000 comp.. 70-80000 tens; Susquehanna Br. (B. & O.), 
Ky. and Ind. Cantilever, and Van Buren Br., 80000 comp, 70000 tens. 
The Manufacturers' Standard (1903) employs three grades: Rivet (48- 
58000), Railway Bridge (55-65000), Medium (60-70000); elastic limit not 
less than \ ult str; percentage of elongation, 1400000-^ ultimate strength. 

t The Bethlehem Steel Co. guarantees the following physical properties 
in solid and hollow steel forgings of different sizes; the tensile test specimen 
to be used being the U. S. Standard as adopted by the Am. Soc. for Testing 
Materials (see Fig. 5, page 501). 







Tensile 


Elastic 




Contrac- 


Dimensions of solid and 


Class of Steel 


Strength. 


Limit. 


Elonga- 


tion of 


hollow forgings In which 


Forging 


, 


Lbs. per 


Lbs. per 


tion. 


Area. 


the physical qualities 






Sq. In. 


Sq. In. 


Per Cent. 


Per Cent. 


mentioned to the left are 
guaranteed. 














[Solid or Hollow forgings. 




(1) 


95 000 


65 000 


21 


50 


- no diameter or thickness 
. of section to exceed 3 ins. 

r Solid forgings of rectan- 
gular sections not exceed- 


Nickel 


(2) 


90 000 


60 000 


22 


50 


ing 6 ins. thick; or Hollow 
forgings, the walls of 


Steel 












which do not exceed 6 


Oil- 












Ins. In thickness. 


Tempered. 












Solid forgings of rectan- 
gular sections not exceed- 




(3) 


85 000 


55 000 


24 


45 


ing 10 ins. thick, or Hol- 
low forgings, the walls of 
which do not exceed 10 
ins. In thickness. 














[Solid or Hollow forgings. 




(1) 


80 000 


50 000 


25 


45 


no diameter or thickness 
of section to exceed 1 Ins. 


Nickel 












[Solid forgings, no dlam- 


Steel 


(2) 


80 000 


45 000 


25 


45 


] eter to exceed 20 Ins., or 


Annealed. 












thickness of section 1 5 Ins. 




(3) 


80 000 


45 000 


24 


40 


Solid forgings over 20 
Jns. diam. 



500 28.— STRENGTH AND RESISTANCE OF MATERIALS, 



Class of Steel 
Forging. 


Tensile 
Strength 
Lbs. per 

Sq. In. 


Elastic 
Limit. 
Lbs. per 
Sq. In. 


Elonga- 
tion. 
Per Cent. 


Contrac- 
tion of 
Area. 

Per Cent. 


Dimensions of solid and 
hollow forgings in which 
the physical qualities 
mentioned to the left are 
guaranteed. 


Carbon 
Steel 
on- 
Tempered. 


■ 
(1) 

(2) 
(3) 


90 000 
85 000 

80 000 


55 000 
50 000 

45 000 


20 

22 

23 


45 
45 

40 


Solid or Hollow forgings, 
no diameter or thickness 

.. of section to exceed 3 ins. 
Solid forgings of rectan- 
gular sections not exceed- 
ing 6 ins. in thickness; or 
Hollow forgings, the walls 
of which do not exceed 

: 6 ins. in thickness. 
Solid forgings or rectan- 
gular section not exceed- 
ing 10 ins. in thickness; or 
Hollow forgings, the walls 
of which do not exceed 

, 1 ins. in thickness. 


Carbon 

Steel 

Annealed. 


(1) 

(2) 
(3) 


80 000 

75 000 
70 000 


40 000 

37 500 
35 000 ^ 


22 

23 
24 


35 

35 
30 


Solid or Hollow forgings, 
no diam. or thickness of 
section to exceed 10 ins. 
Solid forgings, no diam. 
to exceed 20 Ins., or 

. thickness of section 1 5 Ins. 
Solid forgings over 20 

1 ins. diam. 



Digest of Standard Specifications for Steel, adopted by the Am. Soc. 
for Testing Materials. (See Proceedings, Vols. I, V, etc.) 

Ref. Adopted, 

i. Stand. Spec, for Structtiral Steel for Bridges. Vol. V. Sept. 1, 1905. 
ii. Stand. Spec, for Open-hearth Boiler Plate and 

Rivet Steel. Vol. I. 

iii. Standard Specifications for Steel Rails. - Vol. VI. Proposed, 

iv. Standard Specifications for Steel Castings. - Vol. V. Sept. 1, 190 

V. Standard Specifications for Steel Axles. - Vol. V. Sept. 1, 190 

vi. Standard Specifications for Steel Forgings. - Vol. V. Sept. 1, 190 



i. Structural Steel for Bridges 
Steel shall be made by the open-hearth process. 2. The chemical and 



f 



physical properties shall conform to the following limits : 



Elements Considered. 



Phosphorus Max. ( 
Sulphur Max 



Basic 
Acid 



Ult. tensile strength 
Lbs. per sq. in. 

Elong.: Min. per cent 
in 8 ins. (Fig. 3) 

Elong. : Min. per cent 
in 2 ins. (Fig. 4) 

Character of fracture 



Cold bend without 
fracture 



Structural Steel. 



0.04 per cent. 
0.06 " 
0.05 " 



Desired 
60 000 
1 500 000* 

Ult. tens. str. 

22 

Silky 



180 degrees flat f 



Rivet Steel. 



0.04 per cent. 

0.06 

0.04 " 



Desired / 
50 000 1 
1 500 000 

Ult. tens. str. 



Silky 



180 degrees flat t 



* See par. 11. f See par. 12, 13 and 14. % See par. 15 



Steel Castings. 



0.05 per cent. 
0.08 " 
0.05 " 



1 



Not less than 
65 000 



18 

Silky or fine gran- 
ular 

90° on dia.=to 3 
X the thickness 



{d=Zt) 



I 



STEEL— SPECIFIC A TIONS. 



501 



.'•Sfancfcrrd Thread-^. 




used 



Fig. 3. 



{U4. 



:<.. — ziJ. 



-about IS- 
Fig. 4. 



The yield point, as indicated by the drop of beam, shall be recorded in 
the test reports. 3. If the ult str varies more than 4000 lbs. from that de- 
sired, a re-test may be made, at the discretion of the inspector, on the same 
gauge, which, to be acceptable, shall be within 5000 lbs. of the desired 
ultimate. 4. Chemical determinations for percentages of carbon, phos- 
phorus, sulphur and managnese to be made from test ingot at time of pour- 
mg; check analyses from finished material may show 25 per cent above 
required limits. 5. Specimens for tensile and bending tests for plates, 
shapes and bars shall be made by cutting coupons from the finished product, 
which shall have both faces rolled and both edges milled to the form shown 
by Fig. 4; or with both edges parallel; or they may be turned to a dia of 
\ in. for a length of at least 9 ins., with enlarged ends. 6. Rivet rods shall 
be tested as rolled. 7. For pins and rollers, specimens shall be cut from the 
finished rolled or forged bars in such a manner that the center of the speci- 
men shall be 1 inch from surface of bar. Specimen for tensile test shall be 
turned to the form shown by Fig. 5; specimens for bending test shall' be 
1 X i in. in section. 8. Steel Castings. 
The number of tests will depend on 
the character and importance of the K' 
castings. Specimens shall be cut cold ^ 
from coupons molded and cast on .^ 
some portion of one or more castings «J^ 
from each melt or from the sink-heads, ''•^ 
if the heads are of sufficient size. i.. 
The coupon or sink-head, so used, 
shall be annealed with the casting be- 
fore it is cut off. Test specimens to be 
of the form prescribed for pins and 
rollers. 9. Material which is to be 
without annealing or further treatment shall 
be tested in the condition in which it comes 
from the rolls. When material is to be an- 
nealed or otherwise treated before use, the 
specimens for tensile test, representing such 
material, shall be cut from properly annealed 
or similarly treated short lengths of the full 
section of the bar. 10. Number of tests. At 
least one tensile and one bending test shall be 

made from each melt of steel as rolled. In, ^ __. 

case steel differing | inch and more in thick-'' i<r|^.'V"i<~-- 
ness is rolled from^ one melt, a test shall be ' 
made from the thickest and thinnest mate-' 
rial rolled. 11. Elongation. For material 
less than -^ inch and more than f inch in 
thickness the following modifications will be 
allowed in the requirements for elongation: 

(a) For each re in. in thickness below i^ in. 

specified percentage. 

(b) For each i in. in thickness above f in. 

specified percentage. 
12. Bending tests may be made by pressure or by blows. Plates, shapes 
and bars less than 1 in. thick shall bend as called for in par. 2. 13. Full- 
sized material for eye-bars and other steel 1 inch thick and over, tested as 
rolled, shall bend cold 180° around a pin whose diam is twice the thickness 
of the bar, without fracture on outside of bend. 14. Angles f inch and less 
in thickness shall open flat, and angles \ inch and less in thickness shall 
bend shut, cold, under blows of a hammer, without sign of fracture. This 
test to be made only when required by inspector. 15. Rivet steel, when 
nicked and bent around a bar of the same diam as the rivet rod, shall give 
a gradual break and a fine, silky, uniform fracture. 

ii. Open-Hearth Boiler Plate and Rivet Steel. 

1. Steel shall be made by the open-hearth process. 2. Chemical proper- 
ties of the three classes shall conform to the following limits: 

Flange or boiler steel. Fire-box steel. Extra soft steel. 
Phosphorus shall /Acid 0.06 per cent. 0.04 per cent. 0.04 per cent. 

not exceed \Basic 0.04 " 0.03 " 0.04 

Sulphur shall not exceed 0.05 " 0.04 " 0.04 

Manganese 0.30 to 0.60 0.30 to 0.50 0.30 to 0.50 



r — ^ 







Fig. 5. 
a deduction of 2| from the 

a deduction of 1 from the 



502 28.— STRENGTH AND RESISTANCE OF MATERIALS. 

3. Steel for boiler rivets shall be of the extra soft class as specified in par. 2 
and 4. 4. Physical properties of the three classes shall be as follows: 

Flange or boiler steel. Fire-box steel. Extra soft steel. 

^^^^^^t fJ^f}?^^' ^^^' ]55000 to 65000 52000 to 62000 45000 to 55000 

per SQ. m. j 

Yield point, in lbs. per sq. 1 

in. shall not be less > i tens str i tens str i tens str 

than ^ J 

Elongation, per cent in 8 1 

ins., shall not be less [ 25 26 28 

than J (see par. 5) (see par. 5) (see par. 5) 

5. Elongation. For material less than ^ inch and more than | inch in 
thickness the following modifications will be allowed in the requirements 
for .elongation: 

(a) For each J in. in thickness above f in., a deduction of 1 from the 

specified percentage. 

(b) For each ^ in. in thickness below -^q in., a deduction of 2^ from the 

specified percentage. 

6. Bending tests. The three classes shall conform to the following bending 
tests; the bending-test specimen to be H ins. wide if possible, and for all 
material f iii. in thickness or less, it shall be of the same thickness as that 
of the material from which it is cut; but it may be i in. thick for material 
over i in. thick. (Round rods shall be tested of full size as rolled.): (c) Test 
specimens cut from the rolled material as specified above, shall be subjected 
to a cold bending test, and also to a quenched bending test. The cold bend- 
ing test shall be made on the material in the condition in which it is to be 
used; and prior to the quenched bending test, the specimen shall be heated 
to a light cherry-red as seen in the dark and quenched in water, the tem- 
perature of which is between 80 and 90° F. (d) Flange or boiler steel, fire- 
box steel and rivet steel (all three classes) , both before and after quenching, 
shall bend cold 180° flat on itself without fracture on outside of bend. 7. Homo- 
geneity. For fire-box steel a sample taken from a broken tensile test speci- 
men shall not show any single seam or cavity more than i in. long in either 
of the three fractures obtained on the test for homogeneity as described 
below in paragraph 12. 8. Test pieces and testing. The standard test 
specimen of 8 in. gauged length, chall be used to determine the physical 
properties specified in par. 4 and 5. The standard shape of the test speci- 
men for sheared plates shall be as shown in Fig. 4, preceding, the piece to be 
of same thickness as the plate. For other material the test specimen may 
be the same as for sheared plates, or it may be planed or turned parallel 
throiighout its entire length, and in all cases where possible two opposite 
sides of the test specimens shall be the rolled surfaces. Rivet rounds and 
small rolled bars shall be tested of full size as rolled. 9. One tensile test 
specimen will be furnished from each plate as it is rolled, and two tensile 
test specimens will be furnished from each melt of rivet rounds. In case 
any one of these develops flaws or breaks outside of the middle third of its 
gauged length, it may be discarded and another test specimen substituted 
therefor. 10. For material f in. or less in thickness the bending test specimen 
shall have the natural rolled surface on two opposite sides. The bending 
test specimens cut from plates shall be 1^ ins. wide, and for material more 
than f in. thick they may be i in. thick; the sheared edges may be milled 
or planed. The bending test specimens for rivet rounds shall be of full size 
as rolled. The bending test may be made by pressure or by blows. 11. One 
cold and one quenched bending specimen will be furnished from each 
plate as it is rolled. Two cold and two quenched bending specimens will 
be furnished from each melt of rivet rounds. The homogeneity test for fire- 
box steel shall be made on one of the broken tensile test specimens. 12. The 
homogeneity test for fire-box steel is made as follows: A portion of the 
broken tensile test specimen is either nicked with a chisel or grooved on a 
machine, transversely about -^ in. deep, in three places about 2 ins. 
apart. The first groove should be made on one side, 2 ins. from the square 
end of the specimen; the second, 2 ins. from it on the opposite side; and 
the third, 2 ins. from the last, and on the opposite side from it. The test 
specimen is then put in a vice, with the first groove about i in. above the 
jaws, care being taken to hold it firmly. The projecting end of the test 
specimen is then broken off by means of a hammer, a number of light blows 
being used, and the bending being away from the groove. The specimen is 



It 



STEELSPECIFICATrONS—RAILS, ETC. 



503 



broken at the other two grooves in the same way. The object of this treat- 
ment is to open and render visible any seams due to failure to weld up, or 
to foreign interposed matter, or cavities due to gas bubbles in the ingot. 
After rupture, one side of each fracture is examined, a pocket lens being 
used if necessary, and the lengths of seams and cavities determined. 

iii. Steel Rails. 

1. Manufacture, (a) The entire process of manufacture and testing 
shall be in accordance with the best current practice, and conform carefully 
to the following instructions: (b) Ingots shall be kept in a vertical position 
in the pit heating furnaces until ready to be rolled, or until the metal in 
the interior has time to solidify, (c) No bled ingots shall be used, (d) Suffi- 
cient material shall be discarded from the top of ingot to insure sound rails. 
2. Chemical composition. Rails of the various weights per yard specified 
below shall conform to the following limits in chemical composition: ' 



Carbon 

Phosphorus, shall not ex- 
ceed 

Silicon, shall not exceed 

Manganese 



50 to 59 

pounds. 
Per cent, 



.35-. 45 

.10 

.20 

70-1.00 



60 to 69 

pounds. 

Per cent. 



.38-48. 

.10 

.20 

70-1.00 



70 to 79 

pounds. 
Per cent. 



.40-50 

.10 

.20 

75-1.05 



80 to 8 9 

pounds. 

Per cent 



.43-53 

.10 

.20 

,80-1.10 



90 to 100 

pounds. 
Per cent. 



.45-55 

.10 

.20 

.80-1.10 



3. One drop test shall be made on a piece of rail not less than 4 ft. and 
not more than 6 ft. long, selected from every fifth blow of steel. The test 
shall be taken from the top of the ingot. The rail shall be placed head 
upwards on the supports, and the various sections shall be subjected to the 
following impact tests under a free falling weight: 

Weight of Rail — lbs. per yard, 45 to 55; height of drop, 15 ft. 

554- to 65 •' " 16 " 

65+ to 75 " " 17 •• 

75+ to 85 " " 18 " 

85+ to 100 " " 19 " 

If any rail break when subject to the drop test, two additional tests 
taken from the top of the ingot will be made of other rails from the same 
blow of steel, and if either of these latter tests fail, all the rails of the blow 
which they represent will be rejected, but if both of these additional test 
pieces meet the requirements, all the rails of the blow which they represent 
will be accepted. 4. Finishing temperature. The number of passes and 
speed of train shall be so regulated that on leaving the rolls at the final pass 
the temperature of the rail will not exceed that which requires a shrinkage 
allowance at the hot-saws, for a 30-ft. rail of 100-lb. section, of 6-^ ins., and 
^ in. less for each 5-lb. decrease of section. These allowances to be decreased 
at the rate of x^n in. for each second of time elapsed between the rail leaving 
the finishing rolls and being sawn. No artificial means of cooling the rails 
shall be used between the finishing pass and the hot-saws. 5. The drop 
testing machine shall have a tup of 2000 lbs. weight, the striking face of 
which shall have a radius of not more than 5 ins., and the test rail shall be 
placed head upwards on solid supports 3 ft. apart. The anvil block shall 
weigh at least 20000 lbs., and the supports shall be part of, or firmly secured 
to, the anvil. The report of the drop test shall state the atmospheric tem- 
perature at the time the test was made. (The Am. Soc. C. E. standard 
rail section is recommended. See Sec. 30, page 560, and Sec 59, page 1060.) 

iv. Steel Castings. 
1. Steel for castings may be made by the open-hearth, crucible or 
Bessemer process. Castings to be annealed unless otherwise specified. 
2. Ordinary castings, those in which no physical requirements are specified, 
shall not contain over 0.40 per cent of carbon, nor over 0.08 per cent of 
phosphorus. 3. Castings which are subjected to physical test shall not 
contain over 0.05 per cent of phosphorus, nor over 0.05 per cent of sulphur. 



Hard Castings. 


Medium C. 


SoftC 


85 000 


70 000 


60 000 


38 250 


31 500 


27 000 


15 


18 


22 


20 


25 


30 



604 28.— STRENGTH AND RESISTANCE OF MATERIALS. 

4. Tested castings shall be of three classes: "Hard," "Medium," and 
"Soft." The minimum physical qualities required in each class shall be as 
follows; 

Tensile strength, lbs. per sq. in 
Yield point, lbs. per sq. in. [.45 T.] 
Elongation, per cent in two ins. 
Contraction of area, per cent - 

5. Drop test. A test to destruction may be substituted for the tensile test, 
in the case of small or unimportant castings, by selecting three castings 
from a lot. This test shall show the material to be ductile and free from 
injurious defects, and suitable for the purposes intended. A lot shall consist 
of all castings from the same melt or blow.- annealed in the same furnace- 
charge. 6. Percussive test. Large castings are to be suspended and ham- 
mered all over. No cracks, flaws, defects, nor weakness shall appear after 
such treatment. 7. Bending test. A specimen iXi in. shall bend cold 
around a diam. of 1 in. without fracture on outside of bent portion, through 
an angle of 120° for "soft" castings, and 90° for "medium" castings. ^ 8. Ten- 
sile test piece. The standard turned test specimen (Fig. 5, p. 501), i in. diam 
and 2-in. gauged length, shall be used to determine the physical properties 
specified in par. 4. 9. The number of standard test specimens shall depend 
upon the character and importance of the castings. A test piece shall be cut 
cold from a coupon to be molded and cast on some portion of one or more 
castings from each melt or blow or from the sink heads (in case heads of 
sufficient size are used). The coupon or sink-head must receive the same 
treatment as the casting or castings, before the specimen is cut out, and 
before the coupon or sink-head is removed from the casting. 10. Test piece 
for bending. One specimen for-bending test, VX¥', shall be cut cold from 
the coupon or sink-head of the casting or castings as specified in par. 9. 
The bending test may be made by pressure or by blows. 

V. Steel Axles. 

1. Manufacture. Steel for axles shall be made by the open-hearth proc- 
ess. 2. Chemical properties. There will be three classes of steel axles 
which shall conform to the following limits in chemical composition: 





Car and 

Tender Truck 

Axles. 

Per cent. 


Driving and Engine 
Truck Axles. 




(CarbonSteel) 
Per cent. 


(NickelSteel) 
Per cent. 


Phosphorus shall not exceed .... 

Sulphur " " " 

Manganese " " ** .... 


0.06 
0.06 


0.06 
0.06 
0.06 


0.04 
0.04 


Nickel 




3. to 4. 











3. Physical properties. For car and tender truck axles no tensile test shall 
be required. 4. The minimum physical qualities required in the two classes 
of driving and engine truck axles shall be as follows: 



Driving and Engine Truck Axles 



(Carbon Steel) 



(Nickel Steel) 






Tensile strength, pounds per sq. in 
Yield point, pounds per sq. in ... . 

Elongation.. per cent in 2 ins 

Concentration of area, per cent . . . 



80 000 

40 000 

20 

25 



80 000 

50 000 

25 

45 



STEEL— SPECIFICATIONS— FORGINGS, ETC. 



505 



5. Drop test. One axle selected from each melt, when tested by the drop 
test described in par. 9, shall stand the number of blows at the height 
specified in the following table without rupture and without exceeding, 
as the result of the first blow, the deflection given. Any melt failing to meet 
these requirements will be rejected: 



Diam. of axle at center, inches. . . 


4i 


4f 


4^ 


4f 


41 


51 


5i 


Number of blows 


5 

24 

8i 


5 

26 

8i 


5 

28i 
8i 


5 
31 

8 


5 

34 

8 


5 
43 

7 


7 


Height of drop, feet 


43 


Deflection (max., 1st blow), inches. 


5i 



6. Carbon steel and nickel steel driving and engine truck axles shall not be 
subject to the above drop test. 7. Test pieces and testing. The standard 
turned test specimen (Fig. 5, page 501), \ in. diam. and 2-in. gauged 
length, shall be used to determine the physical properties specified in par. 4. 
8. For driving and engine truck axles one longitudinal test specimen shall 
be cut from one axle of each melt. The center of this test specimen shall 
be half-way between the center and outside of the axle. 9. The points of 
supports on which the axle rests during tests must be 3 ft. c. to c, the tup 
must weigh 1640 lbs.; the anvil, which is supported on springs, must weigh 
17500 lbs.; it must be free to move in a vertical direction; the springs upon 
which it rests must be 12 in number, of the kind described on drawing; and 
the radius of supports and of the striking face on the tup in the direction of 
the axis of the axle must be 5 ins. When an axle is tested it must be so 
placed in the machine that the tup will strike it midway between the ends, 
and it must be turned over after the first and third blows, and when required, 
after the fifth blow. To measure the deflection after the first blow prepare 
a straight edge as long as the axle, by reinforcing it on one side, equally at 
each end, so that when it is laid on the axle, the reinforced parts will rest 
on the collars or ends of the axle, and the balance of the straight edge not 
touch the axle at any place. Next place the axle in position for test, lay 
the straight edge on it, and measure the distance from the straight edge to 
the axle at the middle point of the latter. Then after the first blow, place 
the straight edge on the now bent axle in the same manner as before, and 
measure the distance from it to that side of the axle next to the straight 
edge at the point farthest away from the latter. The difference between 
the two measurements is the deflection. The report of the drop test shall 
state the atmospheric temperature at the time the tests were made. 

vi. Steel Forcings. 

1. Steel forgings may be made by the open-hearth, crucible or Bessemer 
process. 2. Chemical properties. There will be four classes of steel forgings 
which shall conform to the following limits in chemical composition: 





Forgings 

of Soft 

or Low 

Carbon 

Steel. 

Per cent. 


Forgings 
of Carbon 
Steel not 
An- 
nealed. 

Per cent. 


Forgings 
of Carbon 
Steel,Oil 

Tem- 
pered or 
Aneal'd. 

Per cent. 


Loco- 
motive 
Forgings 

Per cent. 


Forgings 
of Nickel 
Steel, Oil 

Tem- 
pered or 
Aneal'd. 

Per cent. 


Phosphorus shall not exceed 
Sulphur " " " 
Manganese " " " 


0.10 
0.10 


0.06 
0.06 


0.04 
0.04 


0.05 
0.05 
0.60 


0.04 
0.04 


Nickel 








3 to 4 















506 2S.STRENGTH AND RESISTANCE OF MATERIALS, 

3. Physical properties. The minimum physical qualities required of the 
different sized forgings of each class shall be as follows; 

i I iih 



^S 






H >< W O 

5of^ Sfe^/ or Low Carbon Steel. Lbs. per sq. in. Per cent. 

For solid or hollow forgings, no diam. or thick-"! cq nnn on nnn oo or 

ness of section to exceed 10" j58 000 29 000 28 35 

Carbon Steel Not Annealed. 

For solid or hollow forgings, no diam. or thick-ly^ f.f.f. on nnn ic on 

ness of section to exceed 10" '. ]75 000 37 500 18 30 

Elastic 

Carbon Steel Annealed. Limit. 

For solid or hollow forgings, no diam. or thick-Ion mn m nnn oo qk 

ness of section to exceed 10" j80 000 40 000 22 35 

For solid forgings, no diam. to exceed 20" or thick-l^f. ^nn 07 kaa 00 oc 

ness of section 15" j^^ 0"^ 37 500 23 35 

For solid Forgings, over 20" diam 70 000 35 000 24 30 

Carbon Steel, Oil Tempered. 

For solid or hollow forgings, no diam. or thick-lo^ ^nn kk nnn on ak. 

ness of section to exceed 3" j 90 000 55 000 20 45 

For solid forgings of rect. sections not exceeding 6"] 

in thickness; or hollow forgings the walls of [85 000 50 000 22 45 

which do not exceed 6" in thickness J 

For solid forgings of rect. sections not exceeding] 

. 10" in thickness; or hollow forgings, the walls [80 000 45 000 23 40 

of which do not exceed 10" in thickness J 

Locomotive Forgings ' 80 000 40 000 20 25 

Nickel Steel Annealed. 

For solid or hollow forgings, no diam. or thick-\Q« /v^n Kn (\f\t\ ok ak. 

ness of section to exceed 10" r 80 000 50 000 25 45 



For solid forgings, no diam. to exceed 20" orlo^ ^nn ak nnn oc ak 

thickness of section 15" ]80 000 45 000 25 45 

For solid forgings, over 20" diam 80 000 45 000 24 40 

Nickel Steel, Oil Tempered. 

For solid or hollow forgings, no diam. or thick-loR nnn ak nnn 01 sn 

ness of section to exceed 3" j95 000 b5 000 21 50 

For solid forgings of rect. sections not exceeding] 

6" in thickness; or hollow forgings, the walls [90 000 60 000 22 50 

of which do not exceed 6" in thickness I 

For solid forgings of rect. sections not exceeding] 

10" in thickness; or hollow forgings, the walls [85 000 55 000 24 45 

of which do not exceed 10" in thickness J 

4. Bending test. A specimen IX ^ in. shall bend cold 180® around the 
following diameters without fracture on outside of bent portion: ^" diam. for 
forgings of soft steel, and for forgings of nickel steel annealed ; 1" diam. for 
forgings of carbon steel annealed, if under 20" diam. and for forgings of 
carbon steel oil-tempered, and for forgings of nickel steel oil-tempered; 
1^" diam. for forgings of carbon steel not annealed, and for forgings of carbon 
steel annealed, if 20" in diam. or over. For locomotive forgings no bending 
test will be required. 5. The standard turned test specimen (Fig. 5, page 
501), i in. diam. and 2-in. gauged length, shall be used to determine the 
physical properties specified in par. 3. 6. The number and location of test 
specimens to be taken from a melt, blow, or a forging shall depend upon its 
character and importance and must therefore be regulated by individual 
cases. The test specimens shall be cut cold from the forging or full-sized 
prolongation of same parallel to the axis of the forging and half-way between 
the center and outside, the specimens to be longitudinal, i. e., the length of 
the specimen to correspond with the direction in which the metal is most 
drawn out or worked. When forgings have large ends or collars, the test 



STEEL FORCINGS. BRICK. CEMENT, 



507 



specimens shall be taken from a prolongation of the same diameter or section 
as that of the forging back of the large end or collar. In the case of hollow- 
shafting, either forged or bored, the specimen shall be taken within the 
finished section prolonged, half-way between the inner and outer surface of 
the wall of the forging. 7. The specimen for bending test, 1 X i in., shall be 
cut as specified in par. 6. The bending test may be made by pressure or by 
blows. 

C. Building Stones, Cements, Etc. 

9. — Compression, Tension, Bending, etc., op Above Materials. 
[Pounds per square inch.] 



Material. 


Compression. 
(Ult.) 


Tension. 
(Ult.) 


Bending. 
(Extr. Fiber.) 


♦Bluestone (New York) Use 

Brick soft, inferior 


12 000 
1 000 

10 000 

11 000 

13 000 
15 000 

6 000 

6 000 

38 000 


1200 

60 

200 


2500 


sfood comniOQ 


600 


t Bay State, medium burned 

t Face, hard burned 








t Common, hard burned 






t Paving, 5000 to 7000 






Pressed 4000 to 11000 






Vitrified, tests on 2" cubes 


Extra high record. 



*High Records for Compressive Strength of Bluestone (Hudson R.). — 
41422 lbs. per sq. in.; 38000 lbs. per sq. in., at capacity of machine without 
failing. Test made on 2-in. cube. 

t Tests at Watertown Arsenal (1883) : Bay State, medium burned, 10390 
to 12709; Face, hard burned, 11050 to 16734; Common, hard burned, 12995 
to 22351 lbs. per sq. in. The direction of pressure was at right angle to the 
largest face. (For tests of Brick Piers, see page 522.) 

$ A commission appointed by the Nat'l Brick Man'f'rs' Ass'n (1895) to 
study the best methods for testing paving brick concluded that the "Rattler" 
test was the most valuable; that the absorption and cross-bearing tests were 
of little use ; and the crushing test was useless. A standard rattler test was 
formulated as follows: The rattler barrel is 28" diam. X 20'' long ; it is a 
14-sided polygon, supported on trunnions, the shaft not being continuous 
through the barrel; it is built of cast iron heads and either cast or wrought 
iron staves 6" wide; the space between the staves, for the escape of dust, 
not to exceed f" in width; rate of rotation 30 revs, per min., with 28 and 32 
as limits; each test requiring 1800 revs., or 60 mins. at standard speed; 
number of brick per test charge, 9 to 12; an official test to be the average 
loss in per cent of initial charge from two separate charges; the brick to be 
absolutely dry when tested; each charge to contain, in addition to the brick, 
225 lbs. of li in. cast iron cubes at 0.88 lb. each when new, and also 75 lbs. 
cast iron shot 2iX 2^X4^ ins. at about 7 lbs. each (to be renewed when the 
loss in weight is 10%). The losses in the brick vary from 12 to 25%. Mr. 
Edward Orton, Jr., (see Proc. Am. Soc. for Test'g Mat 'Is, Vol. V, page 296) 
considers 17% loss for heavy traffic streets and 20% loss for light residence 
streets to constitute reasonable limits for standard tests. 

[Pounds per square inch.] 



Material. 



Cement, Natural, neat • ^ 

1 cement, 3 sand o g 

Portland, neat ^ -o 

1 cement, 3 sand ... a § 

^ ■** 



Com- 
pression. 
(Ult.) 



Tension. 
(Ult.) 



Bending, 
(Extr. 
Fiber.) 



Refer- 
ence. 



Tensile Strength. 



1 day. 

50-100 



50-200 



7 days. 


28 da3rs. 




100-200 


200-300 




25- 75 


75-150 




450-550 


550-650 




150-200 


200-300 





See 
pages 
412. 
413. 



508 28.— STRENGTH AND RESISTANCE OF MATERIAL. 
9. — Strength op Building Stones, Cements, Etc. — Continued. 



* Tests by W. Purves Taylor. 
(Rate of stress, 600 lbs. per min.) 

Cement, Natural, neat. (24 specimens) 

1 cem., 2 std. quartz sand 

(24 specimens) 

Portland, neat. (24 specimens) 

1 cem., 3 st'd quartz 
sand. (96 specimens) . 
1 cem., 3 Ottawa sand. 
(24 specimens) 



Cement, Natural, neat. (24 specimens) 

1 cem., 2 st'd quartz 
sand. (24 specimens) . 

Portland, neat. (24 specimens) 

1 cem., 3 st'd quartz 
sand. (96 specimens) ; 
" 1 cem., 3 Ottawa sand. 
(24 specimens) 



253 


355 


441 


196 
703 


301 
741 


426 

744 


238 


316 


343 


320 


374 


435 



I I I I 

I 7 days. | 28 days.l 3 mos. | 6 mos. 
Tensile Strength, t, 

453 

508 
705 

349 

400 

5052 

2012 
9760 

1529 

2181 
Ratios of t to c, and c to t. 
t c t c t c t 



Compressive Strength, c. 



2010 


2689 


3643 


940 
5915 


1390 
7041 


1730 
7347 


941 


1290 


1490 


1199 


1796 


-1887 





t 


c 


t 


c 


t 


c 


t 


c 


Cement, Natural, neat. (24 specimens)... 


7.9 


.13 


7.6 


.13 


8.5 


.12 


11.1 


.09 


1 cem., 2 st'd quartz 


















sand. (24 specimens). 


4.7 


.21 


4.6 


.19 


4.1 


.25 


4.0 


.25 


Portland, neat. (24 specimens) . . . 


8.4 


.12 


9.5 


.11 


9.9 


.10 


13.8 


.07 


1 cem., 3 st'd quartz 


















sand. (96 specimens) . 


3.9 


.25 


4.1 


.24 


4.3 


.23 


4.4 


.23 


1 cem., 3 Ottawa sand. 


















(24 specimens) 


3.7 


.27 


4.8 


.21 


4.3 


.23 


5.5 


.18 



* Compression tests made on cylindrical specimens one inch in height 
and with a diam. (1^4- in.) that will give an area of one sq. in. Note 
the dropping off in tensile strength of neat Portland, from 3 to 6 mos.; 
also the same characteristic when used with Ottawa sand. Long time tests, 
extending up to a year or more, frequently show a dropping off in tensile 
strength in a few weeks or months, and later increased strength. This is 
doubtless due to some changed condition in the process of hardening. At 
age of two years the tensile strength of neat Portland may be assumed at 
14 to 18% greater than 6 mos.; and the tensile strength of neat Natural 
18 to 22% greater. Portland mortars increase about 18%, and Natural 
mortars about 6%, in the period between 6 mos. and two years. 

Concrete, Portland. Compression. 

Varying Mixtures, Columns, 60 days. 

The following formula for strength of short, round, plain (unrein- 
forced) stone concrete columns, is deduced from tests reported by Prof. 
A. N. Talbot (see Eng. News, Sept. 26, 1907). The columns were 
generally 12 ins. in diam and 10 ft. long. The concrete was a wet mix- 
ture of proportions 1 : U : 3, 1 : 2 : 4, 1 : 3 : 6, and 1:4:8. The forms 
were removed in 10 days and the columns were tested in 60 days. The 
straight-line formula is: 

Ultimate compressive strength, lbs. per sq. in. = „ ^-. — 400 (1) 

in which 5« = ratio of sand to cement, ) Thus, in 1 : 2 : 4 concrete, 
and St = ratio of stone to cement. ] 5a = 2, and St = 4. 

The following values are deduced from equation (1): 

Mixture 1:1:2 1:U:3 1:2:4 1:2^:5 1:3:6 l:3i:7 

^Ibs^pTr^sq^in } (3600) 2267 1600 1200 933 743 

The value 3600, for 1:1:2 concrete, is beyond the range of the tests, 
but it is believed to be fairly reliable. 



1:4:8 
600 



CEMENT, CONCRETE, 



509 



6" Stone Concrete Cubes, 1:2:4 mix, 30 days. 
Compression tests on 6-inch cubes, by Prof. Edgar Marburg (Proc. 
Am. Soc. Test'g Mat 'Is, 1904), give mean average values as follows: 
Wet, or rather medium, mixtures (amount of water 16% of combined 
weight of cement and sand, the water flushing freely to top during tamp- 
ing), av. 1650 lbs. per sq. in., with 1364 min. and 2503 max.; two cubes 
mixed "dry" with 10% water, instead of 16%, developed a compressive 
strength, without crushing, of 2778 lbs. per sq. in., the limit of the 
testing machine being 100000 lbs. 

8" and 12" Trap-rock Concrete Cubes, 1:2.3:4.6 mix. 
(Watertown Arsenal Tests.) 

These were wet, or rather medium, mixtures (amount of water 15.2% 
of combined weight of cement and sand): 

2- 8" cubes developed (2630 and 2440), av. 2535, lbs. per sq. in. at 30dys 

2-8'' " •' (3830 and 3300), av. 3565, lbs. " " 60" 

2-12" cubes " (3360 and 3100), av. 3230, lbs. " " 30" 

2-12" " " (3940 and 4780), av. 4360, lbs. " " 60 " 

1-12" cube " 5170 lbs. " " 6 mos 



Age, months. 



2, 6. 1, 



1, 2, 6, 



Ratio of strength, 8" cubes. 
12" " . 



1.0 
1.0 



1.40 
1.35 



0.71 
0.74 



1.0 ... 
1.0 1.19 



0.62 0.84 1.0 



Note that the size of the cube seems to be a function of the crushing 
strength of concrete, as it is with natural stone — granite, limestone, etc. 

The elastic limit of concrete under compression is about ^ to f (say |) 
the ultimate strength. 

Concrete, Portland. Tension. 

(Tests by Prof. W. K. Hatt, 1901-2.) 



No. 

1 
2 
3 
4 


Kind 
Stone. 
1:2:4 
1:2:4 
1:2:4 
1:2:4 


Age in 
Days. 

35 
33 

28 
26 


Modulus of 
Elasticity, lbs. 
per sq. in. 
2 700 000 
2 400 000 
1 400 000 
1 900 000 


Elongation Strength 

at Rupture lbs. per 

1 part in. sq. in. 

11 660 300 
8 750 305 
4 400 360 
7 700 280 


Where 
Broken. 
At pin. 


5 


Average. . . . 
1:2:4 


""28" 


. 2 100 000 


7 000 
910 


311 
281 


Bodv 



The tension specimens were 4" square in cross-section, with gauged 
length of 24", and 30" between centers of pins, through heads 8" diam. 
The concrete was fairly dry concrete intended to be plastic after a 
thorough ramming. About 5.5% of water by weight was added to the 
mortar. Prof. Hatt says: These tension tests were not satisfactory for 
determination, since the heads of the bars, with some exceptions, pulled 
off; but it is believed that the strength of the body of the bars did not 
differ greatly from the loads recorded at the point of rupture of the 
heads. They served, however, very well for the determination of modu- 
lus of elasticity. 

The elastic limit of concrete under tension is very near the ult. str. 

Concrete, Portland. Compression, Tension, Bending, and Shearing.* 

That some broad relations exist between the compressive strength, 
tensile strength, modulus of rupture (extreme fiber stress due to bending) 
and shearing strength of concrete, is quite probable; but it is certain 
that no constant ratios can be assigned. Even for the same proportions 
of mixture the above mentioned ratios would vary with the consistency 
of the mix and the age of the concrete. We have seen also that com- 
pressive strength per sq. in. increases with the size of the cube or mass, 



*For Shearing values of stone and concrete, 
VIII.. 1908.— By H. H. Quimby. 



see Proc. A. S. T. M., Vol. 



510 2S.^STRENGTH AND RESISTANCE OF MATERIALS. 

but we have every reason to believe that a like increment would not hold 

for tension, bending and shearing, although the compressive strength of 

cubes is influenced greatly by the shearing resistence of the concrete. 

The following ratios are approximate only: 

Compressive Tensile Mod. of Shearing 

Strength. Strength. Rupture. Strength. 

1.00 0.08to0.12 

1.00 1.4tol.8 1.2tbl.5 



I 



Concrete, Portland. Modulus of Elasticity (E) of concrete increases with the 
richness of the mixture and with the age; it decreases (usually) as the 
stress increases; and is greater (usually) when the concrete is under 
compression than when under tension, although for practical purposes 
of design the latter distinction is seldom recogniz.ed. Some German 
authorities assume the ratio of the modulus of elasticity of concrete to 
that of steel as 1:9 or 230 000 (kilograms per sq. meter), making E in 
English units =3 270 000 (lbs. per sq. in.), this value being used in the 
calculations of reinforced concrete beams. In American practice this 
high value of E would be assigned to a rich mixture of concrete at age of 
3 to 6 months. (See Sec. 25, Masonry, page 445.) 

Concrete, Portland. Heat Effect. The coefflcient of linear expansion of 
concrete may be assumed, for all practical purposes, to equal that of 
steel, namely, .OOOOOGf per degree Fahrenheit, or a change in length of 
.01 ft. per 100 ft. in length for each 15 degrees variation in temperature, 
Fahr. To this fact, the use of reinforced-concrete is made possible. 

Heating tests on concrete prisms, by Prof. Ira H. Woolson, Columbia 
University, 1905-6, indicate a marked decrease in strength and modulus 
of elasticity after the specimens had been subjected to a temp, of 500 to 
750° F., and a decided decrease in both of these physical characteristics 
after exposure to 1500° F. It is to be noted that broken limestone 
concrete stood the heating effect far better than the smooth-surfaced 
gravel concrete, which latter practically disintegrated at the higher 
temperature, maintained for 2 to 3 hours. The specimens were from 
4^^ to 7'' in size which naturally exposed, in the furnace, a proportion- 
ately large "surface" to "mass," and hence no direct conclusions can be 
drawn from these tests as to the fire-resisting qualities of concrete in 
actual construction. The tests, however, showed the inferiority of 
gravel concrete under the conditions imposed. 

Concrete,^ Natural. The compression, tension, modulus of rupture and 
shearing of natural (Rosendale) cement concrete may be assumed at 
about 50% of the values for Portland cement concrete, unless partic- 
ular brands of known value are used. 

Concrete, Cinder. 

Watertown Arsenal Tests (1903) for Eastern Expanded Metal Co. 
(12-in. cubes, Lehigh Portland Cement.) 

Proportions. Age, Modulus of Elasticity — Comp. Str. 

Cem. Sand. Cinder. Days. At 500 to 1000 lbs. At ult. str. Lbs.persq.in 

f 38 1 786 000 1 136 000 1950 

• 1 : 2 : 4 J 38 1 923 000 1 136 000 2050 

1 224 1 471 000 1 087 000 2600 

I 224 1 563 000 463 000 2500 

1400 

2i : 5 j 38 893 000 1400 

893 000 1980 

694 000 2020 



38 

38 

224 

224 


1 250 000 

893 000 

1 136 000 

1 250 000 


34 

34 

220 

220 


781 000 
1 000 000 
1 000 000 

735 000 



1200 

1330 

694 000 1720 

463 000 1560 

Glass. (See D. Miscellaneous Materials, Table 10, page 612.) 



CONCRETE. GRANITE. MARBLE. MASONRY. 511 

-Strength op Building Stones, Cements, Etc. — Continued. 



Material. Color & Grain. 




Compression. 
(Ult.) 


Tension. 
(Ult.) 


Bending. 
(Extr. Fiber.) 


Gneiss and 

Granite [tested In small cubes]. 

California, Penryn, Gray; fine. 
Rocklin, White; fine. 


6 000 
5 200 

17 500 

16 000 

18 000 

15 000 

23 000 
21 000 

17 000 

19 000 
27 000 

24 000 
23 000 
13 000 

16 000 

25 000 

20 000 
13 500 

17 733 

12 000 










Colorado, Georgetown, Gray; fine. 
Connecticut, Stony Creek, 

Pink; fine. 
Georgia, Llthonla, Gray; fine. 
Maine, Fox Island, Gray; coarse. 

Waldo County, Gray; fine. 
Maryland. Port Deposit, Bluish-gray. 
Massachusetts, Quincy, BI.-Gr.; coarse. 

Rockport. Greenish. 
Minnesota, E. St. Cloud. Gray. 

E. St. Cloud. Red. 

Sauk Rapids, Gray, 
Missouri, Granitevllle, Red; coarse. 
N. Hampshire, Concord, White; fine. 
Virginia, Richmond, Gray; fine. 
Wisconsin, Athelstane, 






(Note.— High Rec- 
ords for Compressive 
Strength of Granite 
(Conn. Val.).— 43300 and 
40840 lbs. persq. in. ; aver- 
age of 23 tests from same 
quarries gave 35000 lbs. 
per sq. in. Tests made on 
nominally 2-ln. cubes, 
prepared by sawing and 
rubbing, but not polish- 
ed.) 






Montello, Rd.-Gr. ; fine . 






(Average of above Gneisses and Gran- 
ites) 






Unless tested, use, for good stone 




1 600 








Material. 


Compression. 
(Ult.) 


Tension. 
(Ult.) 


Bending. 
(Extr. Fiber.) 


Limestone (L) and 
Marble {.M) [tested In small cubes]. 
California, Colton (M) 


17 000 
5 800 

13 000 
16 500 
11 500 
15 500 

22 000 
20 000 

7 500 
11 500 
25 000 

7 000 
20 000 

11 000 

12 000 

23 000 
12 000 

15 000 
10 500 

8 000 

18 000 

16 000 

14 445 
8 000 

2 500 
2 000 

1 500 

2 000 
1 500 
1 000 






Connecticut, Canaan, (M) 






Illinois, Lemont, (Dolomite IS) 






Indiana, Greensborough, (Z/) 






Putnamville, (Z/) 






Kentucky, Bardstown, (JJ) 






Massachusetts, Lee, (M) 






Michigan, Lime Island. (L) 
Marquette. (L) 

Minnesota, Frontenac, (Dolomite Li) 
Stillwater, (Dolomite L) 

Missouri, Canton, (2>) 

New York, Canajoharle, (Z/) 
Glens Falls. (L) 
Kingston, (Z/) 


(NOTE.- 

Strengths o 
Limestones 
tests in "Ba 
kunde,"Au 
News, Oct. 


—For Tensile 
f Granites and 
, see table of 
Lumateriallen- 
g.l,1905;Eng. 
12, 1905.) 


Lake Champlain, \h) 






Ohio, Marblehead, (L) 






Pennsylvania, Conshohocken, \lS) 






Montgomery Co., (Af) 






Vermont, Dorset, (ilf ) 






Wisconsin, Big Sturgeon Bay, (JS) 






Door County, (L) 






(Av. of above Limestones and Marbles) 






Unless tested, use, for good limestone 
Masonry, Brickwork, Cement mortar 
Small Brick Piers. 
Face brick; Port. cem. 1, sand 2 


800 


1 500 


Nat. cem., 1 sand 2 






Lime 1, sand 3 






Com. brick; Port. cem. 1, sand 2 






Nat. cem. 1, sand 2 






Lime 1, sand 3 






Concrete (see Concrete) 

Stonework. Squared stone work 
from i to i the strength of 
the stone. Use about 4- 1 for 
good ashlar In cement 















512 28.—STRENGTH AND RESISTANCE OF MATERIALS, 
9. — Strength op Building Stones, Cements, Etc. — Concluded. 



Material. Color & Grain. 


Compression. 
(Ult.) 


Tension, 
(irit.) 


Bending. 
(Extr. Fiber.) 


Sandstone [tested In small cubes]. 
California, Angel Island, 

Gr'n-gr'y; fine. 


4 500 
9 500 

13 000. 

8 500 
6 000 

6 500 
10 000 
15 000 

8 000 

9 000 

12 500 

5 500 
9 000 

5 000 

10 000 

5 000 






Colorado, Manltou, L't red. 
Connecticut, Portland, R'd-brown. 
Massachusetts, Long Meadow. 

Red; fine. 
Michigan, Marquette, Br.-red. 
Minnesota, Fond du Lac, 

Red. 
New Jersey, Belleville, R'd-brown. 
New York, Medina, R'd-gr'y; fine 
North Carolina, Wadesborough, 

R'd-br.; fine. 
Ohio, Berea, Gray; fine. 
Pennsylvania, Hummelstown, 

R'd.-br.i med. 


(Note.— High Rec- 
ords for Compressive 
Strength of Sandstone 
(Pottsdam, N.Y.).— 42804 
lbs. persq. in., at capacity 
of machine without fail- 
ing. Tests made on 
nominally 2-in. cubes, 
prepared by sawing and 
rubbing, but not polish- 
ed.) 


Wisconsin, FondduLac, Brownstone. 







(Average of above Sandstones) 






Unless tested, use, for good stone 

Slate 


150 
3 000 


1 200 
5 000 


Terra Cotta 











D. Miscellaneous Materials. 

10. — Ultimate Tension, Compression, etc., op Miscellaneous 

Materials. 

[Pounds per square inch.] 

Canvas. Tensile strength: lengthwise, 260; crosswise, 330. 

Cotton. Tensile strength: belting, solid, woven, 7260; belting, folded, 
stitched, 6160. 

Flax. Tensile strength: yam fiber, 25000; belting, solid, woven, 9950; belt- 
ing, folded, stitched, 6400. 

Glass, Common green. Tension: thin plates, 480.0; bars Y' diam., 2900. 
Compression: small cubes, 20000; small cylinders, 40000. Elastic limit, 
3000. Mod. of rupture (extreme fiber) 4000. Mod. of elasticity, 8000000. 

Glass, Flint (best). Tension: thin plates, 4200; bars V'diam., 2400. Com- 
pression: small cubes, 13000; small cylinders, 27000. 

Glass, Flooring. Tension, 3000; compression, 10000; mod. of rupture, 3000. 

Ice, Hard. Compression, 250. 

Leather, Ox. Tension, 3600. Modulus of elasticity, 250000. 

Plaster of Paris. Tension, 70; compression, 700. 

III. HEAT EFFECT ON VARIOUS SUBSTANCES. 

General Discussion. — Substances are usually classed from a physical 
standpoint as gases, liquids and solids, according to their constitution at 
natural atmospheres (with regard to both temperature and pressure) . 

A Gas may be defined as a substance which when enclosed in a small 
vessel will assume the shape and volume of the vessel; a Liquid will assume 
the shape of that portion of the vessel which corresponds to the volume of the 
liquid itself; while a Solid will assume neither the shape nor the volume 
of an arbitrary vessel. 

All gases may be reduced to liquids by lowering the temperature of the 
gas sufficiently, and subjecting it to the required pressure. 

The Critical Point or Critical Temperature of a gas may be defined as 
that temperature above which it cannot be liquefied, no matter how great 
the pressure exerted upon it. If the temperature of a gas is already below 
the critical temperature it is known as vapor, and pressure alone will suffice 
to reduce it to a liquid. Every gas has its critical temperature, and below 
this point a sudden change from gas to liquid is accompanied by a sudden 
change of volume — with a clear line of demarcation when only a portion 
of the gas is so transformed. 



SANDSTONE, MISCELLANEOUS. HEAT EFFECT. 513 

The Critical Pressure is that pressure at which condensation takes place 
at the critical temperature; i. e., it is the vapor tension of a liquid at its 
critical temperature. The Critical Volume is the specific volume of the 
saturated vapor at the critical temperature, and under the critical pressure. 

The Boiling Point depends upon the temperature and pressure, and may 
be defined as that temperature above which the body of the liquid passes 
into a gaseous state. The Absolute Boiling Point of a liquid is identical with 
its critical temperature; i. e., it is that temperature above which the body 
of the liquid passes into a gaseous state no matter what pressure is exerted 
upon it. 

The Latent Heat of Vaporization is the quantity of heat absorbed in boiling, 
i. e., transforming the liquid to a gaseous condition merely, without increase 
in temperature; and, conversely, it is the quantity of heat given out in 
passing from a state of vapor to that of a liquid merely, without decrease 
in temperature. The latent heat of vapor may be expressed (1) in British 
Thermal Units (B. T. U.), one B. T. U. being equivalent to 778 ft. -lbs. of 
work, or the quantity of heat required to raise the temperature of one pound 
of water 1° F. at or near its temperature of maximum density (39.1° F.), 
or, according to Peabody, at 62° F. ; (2) in calories, one calorie being 
equivalent to 426.9 meter-kilograms of work, or the quantity of heat re- 
quired to raise the temperature of one kilogram of water 1° C. at or near 
4° C. (39.1° F.); (3) in pounds of the substance which can be raised, by the 
latent heat, through 1° C. from 0°C.; (4), equivalent to (3), the tempera- 
ture attained by applying the latent heat to one pound of the substance 
from 0° C. (See, also. Section 69, Steam and Gas Power.) 

The Melting Point of a solid ( = practically the freezing point of the liquid) 
is termed the teniperature of fusion. In some substances, as glass and iron, 
there is no definite melting point, while in others it is quite definite. The 
melting point of a body may be raised by increase of pressure, but in most 
cases the change is relatively small and may be neglected. The laws of 
fusion are: (1) Every substance begins to fuse at a certain temperature 
which is constant for constant pressure; and (2) during fusion the tempera- 
ture of the body remains constant. 

Latent Heat of Fusion is analogous to latent heat of vaporization, and 
may be defined as the quantity of heat absorbed in liquefaction, i. e., trans- 
forming the solid to a liquid condition merely, without increase in tempera- 
ture; and conversely, since the laws of solidification are similar to those of 
fusion in the reverse process. 

Liquefaction of Gases may be accomplished : 

(1) By Expansion. If a gas under pressure is allowed to expand sud- 
denly it cools; and if under sufficiently high pressure and allowed to ex- 
pand its temperature will fall to such an extent that it will liquefy. The ice 
machine, operated by cylinder and piston, produces cold by the expansion 
of gases. 

(2) By Evaporation. The process of evaporation requires heat which 
naturally comes from the substance being evaporated and from its sur- 
roundings, hence a lowering of temperature. 

(3) By Freezing-mixtures. Certain substances when properly mixed in 
the right^ proportions and to the required fineness will produce a decided 
lowering in temperature. The lowest temperature ever attained by mixture 
is — 110°C., from sulphuric ether and solid carbon dioxide, known _ as 
"Philorier's mixture." The most common mixture is that of salt and ice, 
by which a temperature of — 22° C. may be attained. 

(4) By Regenerative Method. This method is by far the most general 
and effective. The gas, which is under pressure, is allowed to expand and, 
thus cooled, is made to circulate around the vessel containing the compound 
gas. Hence the temperature of the latter is lowered incrementally to the 
point of liquefaction. 

The Absolute Zero of Temperature is that temperature at which heat 
ceases to exist. On the centigrade scale it is about — 273.7° C, usually 
assumed at — 273° C. All the known gases except helium have been lique- 
-fied, and it has been found that this gas does not liquefy at the low tempera- 
ture of — 258° C. The absolute temperature scale is based on the absolute 
zero of temperature. 



514 28.'-STRENGTH AND RESISTANCE OF MATERIALS, 

11. — Critical Temperature, Critical Pressure, 

Boiling Point and Freezing Point. 

Gases and Liquids. 



Substance. 


Critical 
Temperature 


Critical 

Pressure 

in 

Atmosph's 


Boiling Point 
at Atmos. 
Pressure. 


Freezing 
Point. 




Cent. 


Fahr. 


Cent. 


Fahr. 


Cent. 


Fahr. 


Acetic acid 


+ 322° 
+ 37 
-140 
+ 244 
+ 130 
-121 
+ 31 
-141 
+ 146 
+ 268 
+ 194 
-120 
+ 52 
-235 
+ 36 
- 82 
-146 
-115 
+ 155 
+ 365 
+ 233 


+ 612° 
+ 99 
-220 
+ 471 
+ 266 
-186 
+ 88 
-222 
+ 295 
+ 514 
+ 381 
-184 
+ 126 
-391 
+ 97 
-116 
-231 
-175 
+ 311 
+ 689 
+ 451 


57 
68 
39 
64 
115. 


+ 117° 


+ 243° 


+ 17° 


+ 63° 


Acetylene 




Air 










Alcohol 


+ 78 

- 34 
-187 

- 80 


+ 172 
- 29 
-305 
-112 


-130 

- 75 
-190 

- 78 


-202 


Ammonia 


-103 


Argon . ... 


-310 


Carbonic acid 

Carbonic oxide 


77 
35 
94 
55 
36 


-108 


Chlorine 










Chloroform 


+ 60 
+ 37 
-187 


+ 140 
+ 99 
-305 






Ether 






Fluorine 






Hydrochloric acid. . . 
Hydrogen 


86 
20 
73 
55 
35 
50 
79 
196 
73 






-253 


-423 


-258 


-432 


Laughing Gas 




Marsh gas 










Nitrogen 


-192 
-181 
- 10 
+ 100 
+ 66 


-314 
-294 
+ 14 
+ 212 
+ 150 


-214 


-353 


Oxygen 




Sulphurous acid .... 
Water 


-100 



-148 
+ 32 


Wood alcohol 











Note.— F°=-|-C° + 32°; C°=4(^° " ^2°). 



12. — Boiling Points at Atmospheric Pressure. 



Substance. 



Acetic acid 

Acetone 

Alcohol. 

Ammonia 

Amylic alcohol ...... 

Aldehyde ." . . 

Aniline 

Arsenic 

Benzine 

Benzoic acid 

Benzole 

Bisulphide of carbon. 

Bromine 

Cadmium 

Carbon bisulphide 

Carbonic acid 

Chlorform 

Cyanogen 

Ether 

Fluorine 

Hydrogen 

Iodine 

Linseed oil 



Cent. 


Fahr. 


+ 117° 


+ 243° 


+ 56 


+ 133 


+ 78 


+ 172 


- 34 


- 29 


+ 131 


+ 268 


+ 21 


+ 70 


+ 182 


+ 360 


+ 437 


+ 819 


+ 80 


+ 176 


+ 261 


+ 502 


+ 80 


+ 176 


+ 47 


+ 117 


+ 60 


+ 140 


+ 746 


+ 1375 


+ 48 


+ 118 


- 80 


-112 


+ 60 


+ 140 


- 20 


- 4 


+ 37 


+ 99 


-187 


-305 


-253 


-423 


+ 200 


+ 392 


+ 314 


+ 597 



Substance. 



Mercury 

Methylic alcohol 

Naphthalin 

Nitric acid 

Nitrogen 

Nitrous oxide 

Oxygen 

Phenanthrene 

Phosphate of phenyl . . 

Phosphorus 

Propionic acid 

Saturated brine 

Selenium 

Sulphur 

Sulphuric acid 

" (strong) 

Sulphurous acid 

Turpentine (oil) 

Water (distilled) 

(sea) 

Wood alcohol 

Zinc 



Cent. 



+ 358* 
+ 66 
+ 217 
+ 120 
-192 

- 92 
-181 
+ 340 
+ 407 
+ 290 
+ 137 
+ 108 
+ 665 
+ 448 
+ 310 
+ 318 

- 10 
+ 157 
+ 100 
+ 101 
+ 66 
+ 940 



FREEZING AND BOILING POINTS, ETC. 



515 



13. — Melting Points op Various Substances. 



Substance. 

Acetic acid 

Alcohol 

Aluminum 

Ammonia 

Antimony 

Argon 

Arsenic. 

Benzoic acid . . . . 

Bismuth 

Brass 

Bromine 

Bronze 

Butter 

Cadmium 

Calcium 

Carbonic acid. .. . 

Cast iron, white. . 

gray.. 

Copper 

Ethylene 

Formic acid. . . . . 

Gold 

Hydrogen. ...... 

Hyponitric acid.. 

Ice 

Indium 

Iodine 

Iridium 

Iron, wrought. . . 
Lard 



Cent. 



17° 
130 
625 

75 
500 
190 
500 
120 
262 
1025 

12 
910 

33 
275 
red 

78 
113, 
1220 
1054 
169 

13 
1045 
258 
9 

176 
107 
1950 
1550 

33 



Fahr. 



+ 63° 
-202 
+ 1157 
-103 
+ 932 
-310 
+ 9C2 
+ 248 
+ 504 
+ 1377 
+ 10 
+ 1670 
+ 91 
+ 525 
heat. 
-108 
+ 2075 
+ 2228 
+ 1929 
-272 
+ 9 
+ 1913 
-432 
+ 16 
+ S2 
+ 349 
+ 225 
+ 3542 
+ 2822 
+ 91 



Substance. 

Lead 

Magnesium 

Margaric acid 

Mercury 

Nitrobenzine 

Nitrogen 

Nitroglycerin 

Palladium 

Phosphorus 

Platinum 

Potassium 

Potassium sulphate . . . 
Rose's fusible metal. . . 

Rubidium 

Silver 

Sodium 

Spermaceti 

Stearic acid 

Stearine. 

Steel 

Sulphur 

Sulphurous acid 

Tallow 

Tin 

Turpentine, oil of 

Wax 

Wood's fusible metal . 

Wrought iron 

Zinc 



Cent. 



325° 

700 

57 



Fahr. 



+ 617*» 
+ 1292 
+ 135 



39 - 38 



3 
214 
7 
1500 

44 
1775 

60 
1015 

94 

39 
982 

90 

49 

70 

55 
1350 
114 
100 

33 
229 

27 

65 

68 
1550 
412 



+ 37 ■ 
-353 
+ 45 
+ 2732 
+ 111 
+ 3227 
+ 140 
+ 1859 
+ 201 
+ 102 
+ 1800 
+ 194 
+ 120 
+ 158 
+ 131 
+ 2462 
+ 237 
-148 
+ 92 
+ 444 
- 17 
+ 149 
+ 154 
+ 2822 
+ 774 



^ Note. — Some substances, as glass and iron, have no definite melting 
point, passing gradually from the solid to the liquid state, by what is known 
as vitreous fusion. 

Thermodynamics — References. 
In Sec. 69, Steam and Gas Power, the following siubjects are defined and 
discussed, and maybe considered somewhat pertinent to the present matter 
in this Section; 



British Thermal Unit (B.T.U.) 

Density. 

Entropy. 

Entropy Diagrams. 

External and Internal Work. 

First Law of Thermodynamics. 

Heat. 

Heat Equivalent of External Work. 

Heat Equivalent of Internal Work. 

Heat of Combustion. 

Heat of the Liquid. 

Heat of Vaporization. 

Heating Power of Fuels. 

Mechanical Equivalent of Heat. 



Mechanical Equivalents of Heat 

(Table). 
Pressure of Saturated Steam. 
Specific Heat of the Liquid. 
Specific Volume. 

Specific Volume of Saturated Steam. 
Specific Volume of Water. 
Steam. 

Steam Tables. 
Thermal Energy. 
Total Heat. 
Thermal Units. 
Thermal Unit Equivalents (Table). 



616 28.'-STRENGTH AND RESISTANCE OF MATERIALS. 



Coefficient of Expansion. — The coefficient of expansion of a body is the 
rate of increase per deg . of temperature , usually based on the Fahrenheit scale. 
Let us take a cube of metal, and denote the length of each edge by L, 



when the temperature is t°. 
each edge expands to L+x, 

Length of each edge= 
Surface of each face = 
Volume of the cube == 



At the higher temperature of T° the length ^ 
Then we have — 

AtT°, Att°. Expansion for 7°- 

L+x; =L. X 

(L+^)2; =L2. 2Lx -hx^ 

(L+^)3; =L3. ZL^x + 3Lx^+x?. 



Hence (for one degree) , we have — 



1 



(Exact.) 



Coef . of Linear Expansion = ^<r~;3 X -f 




1 



2Lx+x^ 

L2 
SL^x-hSLx^+x^ 



(i) 

T°-t° \ l) 

r°-i° V l) 



Coef. of Surface Expansion = '^^~~^ X 

Coef. of Volumetric Expansion = ^o_fp X t 3 

The approximate results are obtained from the assumption that, as x 
itself is so extremely small, any power of x greater than unity (as x^ and 
o(^) would be practically zero. 

It is thus seen that the Surface expansion and the Volumetric expansion 
are, respectively, two and three times the Linear expansion. 

14. — Coefficients of Linear Expansion of Solids.* ^H 

Average Values. ^B 



Substance. 


*Coef. of Linear 
Expansion. 


Substance. 


*Coef. of Linear 
Expansion. 


For 1° 
Cent. 


For 1° 
Fahr. 


For 1° 
Cent. 


For 1° 

Fahr. 


Aluminum, cast.. . 

wro'ht 

Antimony 


.0000222 
.0000231 
.0000113 
.0000175 
.0000188 
.0000055 
.0000181 
.0000079 
.0000106 

.0000110 

.ooeoi07 

.0000143 
.0000168 
.0000012 
.0000770 
.0000085 
.0000150 
.0000084 
.0000079 
.0005980 
.0000640 
.0000064 

.0000120 
.0000286 
.0000100 


.0000123 
.0000128 
.0000063 
.0000097 
.0000104 
.0000031 
.0000101 
.0000044 
.0000059 

.0000061 

.0000059 
.0000079 
.0000093 
.0000007 
.0000428 
.0000047 
.0000083 
.0000047 
.0000044 
.0003322 
.0000356 
.0000036 

.0000067 
.0000159 
.0000056 


Masonry, brick 

concrete.. 

" granite. . . 

" limestone 

sandstone 

Nickel 


.0000067 
.0000120 

.0000126 

.0000100 
.0002785 
.0000203 
.0000050 
.0000166 
.0000090 

.0000081 
.0000036 
.0000830 
.0000630 
.0000110 
.0000192 
.0000104 
.0000720 
.0000404 
.0000110 
.0000132 
.0000120 
.0000110 
.0000124 
.0000210 
.0000290 


.0000037 
.0000067 


Bismuth 




Brass 




Brick 


.0000070 


Bronze. . . 


Oak 




Carbon, graphite. . 

Cast iron, gray 

white... 


Palladium 


.0000056 


Paraffin 

Pewter 


.0001547 
.0000113 


Cast steel 


Pine 


.0000028 




Plaster 


.0000092 


Cement, Portland, 


Platinum 


.0000050 


neat 


Platinum- Iridium 

(85-15%).... 

Porcelain 




Concrete 


.0000045 


CoDDer 


.0000020 


Diamond 


Potassium 


.0000461 


Ebonite 


Sal ammoniac. . . . 


.0000350 


Glass 


Sandstone 


0000061 


Gold 


Silver 


.0000107 


Granite > 


Slate 


.0000058 


Graphite 


Sodium 


.0000400 


Guttapercha 

Ice 


Sodium chloride. . 
Steel, cast . ... 


.0000224 
0000061 


Iridium 


" hard 


.0000073 


Iron, cast (see 

above) 


" medium 

" soft 


.0000067 
.0000061 


wrought .... 
Lead 


" tempered. . . 
Tin 


.0000069 
.0000117 


Marble 


Zinc 


.0000161 



* Values in table are for Linear expansion. For Surface expansion and 
Volumetric expansion, multiply by 2 and 3, respectively. To change from 
Cent, to Fahr., mult by i; from Fahr. to Cent., mult by | = 1.8. Note that 
a range of temperature of 180° F= 100° C. 



EXPANSION BY HEAT. FRICTION. 



617 



IV. FRICTIONAL RESISTANCE OF MATERIALS. 

In 1831-4 General Arthur Morin, of the French Artillery, conducted a 
series of experiments on friction, which were published in his "Lecons de 
Mecanique Pratique." The results of these experiments have been 
reprinted extensively in various works because of the care with which the 
tests were conducted. Tables 15, 16 and 17 are reproduced here from 
Joseph Bennett's translation of the above work.* Table 16, from Rankine's 
Applied Mechanics, is also added in order to show that author's version of 
Morin's tests. Later experiments, notably those of Beauchamp Tower, f 
have added to our knowledge of the laws of friction. 

The three fundamental laws of friction deduced by Morin, who experi- 
mented with pressure up to about 30 lbs. per sq. in., are as follows: 

1st. Friction is directly proportional to the pressure; the coefficient of 
friction [natural tang, of the angle of repose] being constant at all pressures. 

2nd. The total friction and the coefficient of friction are independent 
of the areas in contact; the total pressure remaining the same. [This 
naturally follows from the 1st law.] 

3rd. The coefficient of friction is independent of the velocity of one 
surface on the other. 

It is perhaps well to suggest here that Table 15 be not used with undue 
reliability, as any jarring of the structure, the conditions of the atmos- 
phere, etc., might affect the tabular results. Table 16 would be safer to 
use where stability is desired. 

Mr. Tower found in experimenting with revolving journals at high 
speeds and under heavy pressures, that the coefficient of friction, /, varied 
directly with the square root of the linear velocity, and inversely with the 
pressure. With good oil the value of / wit h revolving journals may be 

, , . \/ Velocity in ft. per sec. 

assumed at: 7= tttt> ■ — ^ — • 

4 X Pres. m lbs. per sq. m. 

* " Fundamental Ideas of Mechanics," revised and translated by Joseph 
Bennett; D. Appleton & Co., New York, 1860. Published by permission, 
t See Proceedings of the Institution of Mechanical Engineers, 1883. 



15. — Friction op Plane Surfaces Which Have Been Some Time 

IN Contact. 
(Morin.) 



Kind of Surfaces In 
Contact. 



Disposition of 
the Fibres. 



Condition of the 
Surfaces. 



I I 

U 03 






Oak on oak. 

Oak on elm. 
Elm on oak. 



Ash, pine, beech, sorb, "I 
on oak / 

Tanned leather on oak 



Black curried 
leather or belt 



On 
plane 
oak 
sur- 
face . . 
On 
oak 
. drum. 



Parallel 

Perpendicular 

Wood upright on 

wood flatwise 

Parallel 

<< 

Perpendicular 

Parallel 

The leather flatwise 
The leather on edge. 

Parallel 

Perpendicular 



Without unguent. 
Rubbed with dry soap 
Without unguent. 
Moistened with water. 

Without unguent. 



Rubbed with dry soap 
Without unguent. 



Moistened with water. 



Without unguent. 



0.62 
0.44 
0.54 
0.71 

0.43 
0.38 
0.69 
0.41 
0.57 

53 

0.61 
0.43 
0.79 



0.74 
0.47 



31-48 
23-45 
28-22 
35-22 

23-16 
20-48 
34-36 
22-18 
29-41 

27-55 

31-23 
23-16 
38-19 



36-30 
25-10 



518 2S.STRENGTH AND RESISTANCE OF MATERIALS. 

15. — Friction op Plane Surfaces Which Have Been Some Time in 
Contact. — Concl'd. 



Kind of Surfaces in 
Contact. 



Disposition of 
the Fibres. 



Condition of the 
Surfaces. 



O 03 

el 



•^•2 3 






Hemp matting on oak . j 

Hemp cord on oak 

Iron on oak i 



Cast iron on oak 

Brass on oak 

Ox hide for piston f 
packing on cast iron, i 

Block curried leather, [ 
or belt upon cast iron [ 
pulley I 

Cast iron upon cast] 
iron J 

Iron upon cast iron ..... 

Oak, elm, yoke elm, 
iron, cast iron, and 
brass, sliding two 
and two one upon 
the other 

Calcareous oolite upon' 
oolite limestone / 

Hard calcareous stone 1 
called Muschelkalk } 
upon oolite limestone J 

Brick on calcareous oolite 

Oak 

Iron 

Hard muschelkalk on 
muschelkalk 

Calcareous oolite upon 
muschelkalk 

Brick on muschelkalk . . . 

Iron upon " 

Oak on 

Calcareous oolite on cal- 
careous oolite 



Parallel 

Parallel.'..!'.! 
<< 

<« 

Flatwise 

On edge 

Flatwise 

(< 
(< 

f • 
<( 

« 

Wood upright 



Without unguent. 
Moistened with water. 
Without unguent. 

Moistened with water. 

Without unguent. 
Moistened with water. 
With oil, lard, tallow. 

Without unguent. 
Moistened with water. 

Without unguent. 



Spread with tallow. 
With oil or lard. 



Without unguent. 



With mortar: hyd. 1 
sand, 3 J 



cem., 1: fine sand. 



0.50 
0.87 
0.80 
0.62 
0.65 
0.65 
0.62 
0.62 
0.12 

0.28 
0.38 

0.16* 
0.1 



O.lOt 
0.15t 

0.74 

0.75 

0.67 
0.63 
0.49 

0.70 

0.75 
0.67 
0.42 
0.64 

0.741 



26-34 
41-01 
38-40 
31-48 
33-01 
33-01 
31-48 
31-48 
6-51 

15-39 
20-48 

9-05 
10-45 



5-43 
8-32 



■30 



36-52 

33-49 
32-13 
26-06 

35-00 

36-52 
33-49 
22-47 
32-37 

36-30 



* The surfaces being somewhat unctuous (oily or greasy). 

t When the contact had not been long enough to press out the unguent 
(ointment) . 

tWhen the contact had been long enough to press out the unguent 
and bring the surfaces to an unctuous state. 
H After a contact of from 10 to 15 minutes. 



FRICTION. 



519 



16. — Friction op Plane Surfaces in Motion upon Each Other. 

(Morin.) 



Surfaces In Contact. 



Position of Fibres. 



State of Surfaces. 



:s2 



S' 



at: 



Oak on oak. 



Elm on oak. 



Ash, pine, beach, wild ] 
pear and sorb, on oak J 



Iron on oak . 



Cast Iron on oak . 



Copper on oak 

Iron on elm 

Cast iron on elm 

Black curried leather 1 
on oak j 



Tanned leather on oak 



Tanned leather on cast ] 
iron and brass j 



Hemp strips or cords 

upon oak 

Oak and elm on cast] 

iron / 

Wild pear on cast iron . . 

Iron upon iron 

Iron upon cast Iron and ] 

brass j 

Cast Iron upon cast! 

Iron and brass j 

Cast Iron on cast Iron 

ion brass 
on cast Iron 
on iron 

Oak, elm, yoke elm, 
wild pear, cast iron, 
Iron, steel, steel brass 
sliding upon each 
other or themselves 



on 



Calcareous oolite 
calc. oolite 



Muschelkalk on calc. 

oolite 

Common brick on calc. 

oolite 



Parallel 

Perpendicular. 



Upright on Flatwise 

Parallel 

Perpendicular . . . 
Parallel 



Flatwise on edge. . 



Flatwise and on 
edge 



Parallel 

Perpendicular. 

Parallel 



Without unguent 
Rubbed with dry 
Without unguent 
Wet with water. 
Without unguent 



Wet with water. 
Rubbed with dry 
Without unguent, 
Wet with water. 
Rubbed with dry 
Without unguent, 



soap 



soap 
soap 



Wet with water. 
Without unguent. 
Wet with water. 
Unctuous and wet 

with water. 
Spread with oil. 
Without unguent. 
Wet with water. 

Without unguent. 



Wet with water. 
Without unguent. 



Lubricated In the 
usual way with tal- 
low, lard, soft coom, 
etc. 

Slightly unctuous to 

the touch. 
Without unguent. 



0.48 

0.1 

0.34 

0.25 

0.19 

0.43 

0.45 

0.25 

0.36to 

0.40 

0.62 

0.26 

0.21 

0.49 

0.22 

0.19 

0.62 

0.25 

0.20 

0.27 

.30 to 
.35 

0.29 
0.56 
0.36 

0.23 
0.15 
0.52 
0.33 

0.38 

0.44 



O.lSf 

0.15t 

0.31 
0.20 
0.22 
0.16t 

.7 
to 

.81 



15 



0.64 
0.67 

0.65 



25-38 
9-05 
18-47 
14-02 
10-45 
23-16 
24-14 
14-02 
19-48to 
21-48 
31-48 
14-34 
11-52 
26-06 
12-24 
10-45 
31-48 
14-02 
11-19 

15-07 

16-42 
to 

19-17 
16-10 
29-15 
19-48 

12-57 

8-32 

27-28 

18-16 

20-48 

23-45 



10-12 

8-32 

17-13 

11-19 

12-24 

9-05 

35° 

to 

38-40 



8-32 
32-37 
33-49 

33-01 



* Surfaces worn when there was no unguent (ointment), 
t The surfaces still being slightly unctuous (oily or greasy). 
t The surfaces slightly unctuous. 

H When the unguent is constantly supplied, and uniformly laid oh, this 
ratio may be lowered to 0.05. 



520 28.— STRENGTH AND RESISTANCE OF MATERIALS. 

16. — Friction of Plane Surfaces in Motion upon Each Other — Concl'd. 



Surfaces In Contact. 



Position of Fibres. 



State of Surfaces. 



O OS 

T. ^ 



oa«i. 



03 -M 
P5 






41 



Oak on oolitic lime-1 
stone i 

Forged iron on oolitic! 
limestone J 

Muschelkalk upon mus- 1 
chelkalk J 

Oolitic limestone upon 
muschelkalk 

Common brick on mus- 
chelkalk 

Oak on muschelkalk 

Iron on muschelkaJk 



Wood upright 
Parallel 



Without unguent. 



••{ 



Wood upright. 
Parallel 



Wet with water. 



0.38 
0.69 
0.38 
0.65 

0.60 

0.38 
0.24 
0.30 



20-48 
34-36 
20-48 
33-01 

30-58 

20-48 
13-30 
16-42 



17. — Friction of Journals in Motion Upon Their Pillows. 
^(MorinO 



Surfaces In Contact. 



Cast Iron journals in 
cast Iron bearings . 



Cast Iron journals on 
brass cushions . . . 



Cast Iron journals on 
lignum vitae bear- 
ings 

Wrought Iron jour- 
nals on cast iron 
bearings 



Wrought Iron jour- 
nals on brass bear- 
ings 

Iron journals on lig- f 
num vitae bearings 1 

Brass journals on f 
brass bearings,. . . i 

Brass journals on "I 
cast Iron cushons J 

Lignum vitae jour- ] 
nals on cast iron } 
cushions J 

Lignum vitae jour-l 
nals on lignum j- 
vitae cushions .... J 



State of Surfaces. 



Unguents of olive oil, of lard, 

of tallow, or of soft coom. 
With the same unguents 

and moistened with water. 

Asphalt 

Unctuous 

Unctuous and wet with water 
Unguents of olive oil, of lard, 

of tallow, and of soft coom. 

Unctuous 

Unctuous and wet with water 

Slightly unctuous 

Without unguent 

Unguents of oil or lard 

Unctuous with oil or lard . . . 
Unctuous, with a mixture of 

lard and black lead 



Unguents of olive oil, tal- \ 
low, lard, or soft coom . . j 

Unguents of olive oil, tallow, 

lard 

Unguents of soft coom 

Unctuous, and wet with water 

Slightly unctuous 

Unguents of oil or lard 

Unctuous 

Unguents of oil 

Unguents of lard 



Unguents of oil or tallow 



Unguents of lard 
Unctuous 



Unguents of lard 



Ratio of Friction to Pres- 
sure when the Unguent 
Is Renewed — 



In the Com- 
mon way. 



Contin- 
uously. 



0.07 to 0.08 

0.08 
0.054 
0.14 
0.14 

0.07 to 0.08 
0.16 
0.16 
0.19 
0.18 



0.10 

0.14 

0.07 to 0.08 



07 to 0.08 
0.09 
0.19 
0.25 
0.11 
0.19 
O.IO 
0.09 



0.12 
0.15 



0.030 to 0.054 



0.030 to 0.054 



t 
0.090 



0.030 to 0.054 



0.030 to 0.054 



0.030 to 0.052" 



0.07 



I 



* The surfaces began to wear. t The surfaces began to wear away, 
t The wood being slightly unctuous. 



FRICTION. 



521 



18. — Coefficients of Friction and Angles of Repose or Friction. 
(Condensed and adapted from Morin and others, by Rankine.*) 



Materials. 



Ratio of 

Friction to 

Pressure. 



Angle of 

Repose or 

Friction. 





Cot. (ft 
1 



Structures'. 



Dry masonry and brickwork 

Masonry and brickwork, witli damp mortar. 

Timber on stone 

Iron on stone 

Timber on timber 

Timber on metals 

Metals on metals 

Masonry on dry clay 

Masonry on moist clay 

Earth on earth 

Earth on earth, dry sand, clay, and mixed 

earth 

Earth on earth, damp clay 

Earth on earth, wet clay 

Earth on earth, shingle and gravel 



Machines: 



Wood on wood, dry 

" " soapy 

Metals on oak, dry 

" " wet 

" " soapy 

Metals on elm, dry 

Hemp on oak, dry 

" " wet 

Leather on oak 

Leather on metals, dry 

wet 

greasy 

oily 

Metals on metals, dry 

" " wet 

Smooth surfaces, occasionally greased . . 

continually greased 

best results 



0.6 to 0.7 

0.74 

abt. 0.4 

0.7 to 0.3 

0.5 to 0.2 

0.6 to 0.2 

0.25to 0.15 

0.51 

0.33 
0.25 to 1.0 

0.38to 0.75 
1.0 
0.31 
0.81 to 1.11 



.25 to .5 

.2 
. 5 to .6 
.24 to .26 
.2 
.2 to .25 
.53 
.33 
.27 to .38 
.56 
.36 
.23 
.15 
. 1 5 to .2 

.3 
07 to .08 

.05 
.03 to .036 



31° to 35° 

36F 

22° 

35° to 16f° 

36F to lli° 

31° to m° 

14° to 8i° 

27° 

18i° 

14° to 45° 

21° to 37° 

45° 

17° 
39° to 48° 



14° to 26i° 

m° 

26i° to 31° 
13i° to 14i° 

1H° 
UF to 14° 
28° 
18i-° 
1 5° to 1 9i° 
29^ 
20° 
13° 
8i° 
8F to 1 U° 

16^° 
4° to 41° 

3° 
1|° to 2° 



67 to 1.43 

1.35 

2.5 

43 to 3.33 

2 to 5 

67 to 5 

4 to 6.67 

1.96 

3 

4 to 1 

63 to 1.33 

3.23 
23 to 0.9 



4 to 2 
5 

2 to 1. 
17 to 3. 
5 

5 to 4 
1.89 



3.7 to 2.86 

1.79 

2.78 

4.35 

6.67 
6.67 to 5 

3 33 
14.3 to 12.5 

20 
33.3 to 27.6 



* Applied Mechanics, published by Charles Griffin & Co., London. 

Rolling Friction (Dry). — In the above discussion we have dealt with 
sliding friction only, which of course includes journal friction. — (See Tables 
15 to 18). 

Rolling friction is the resistance at the line of contact only, when a 
wheel or cylinder is rolled over a "smooth," (horizontal) surface, by a force 
P (acting parallel to the surface) applied at the axle of the wheel or roller 
of radius r inches, in moving the total load {W). 

The general equation of rolling friction is, 

F=^»^ (1) 

r 

in which C, of equation (1), is a constant determined by experiment, for 
each kind of material rolling on each kind of material. 

For iron or steel, rolling on iron or steel, C = 0.02, about. 

Example. — A train weighing 1200 tons, with steel car-wheels 24 ins. in 
dia., is drawn on a level track of steel rails, by an engine exerting a pulling 
force P. What part of P is required to overcome the rolling friction of the 
train ? 

Solution. — From equation (1), we have, by substitution, 

Rolling friction F= (0.02X1200) ^12 = 2 tons. Ans. 

Rolling friction is a factor in estimating the power required to swing 
turntables and drawbridges. 



522 2S.— STRENGTH AND RESISTANCE OF MATERIALS. 

EXCERPTS AND REFERENCES. 
Results of Tests of Materials Subjected to Combined Stresses (By E. L- 

Hancock. Proc. A. S. T. M., Vol. VIII., - 1908).— Maximum stress and 
shear theories. Diagrams of tests. 

Crushing Tests on Water=Soaked Timbers (By E. C. Sherman. Eng. 
News, July 1, 1909). — Diagrams of tests on water-soaked oak and yellow 
pine blocks, transverse to grain, proposed for supporting tracks for rolling 
caisson lock gates for Charles River dam. Result of tests showed trans- 
verse tests so low that cast iron chairs were substituted. 

Crushing Tests of Brick and Terra=Cotta Piers (By A. N. Talbot and 
D. A. Abrams. Bulletin No. 27, 111. Eng'g Exper. Sta.; Eng. News, Aug. 12, 
1909). — Summaries of the brick and terra-cotta tests are given below: 

Summary of *Brick Column Tests. 



Column Characteristics. 

(Mortar, Age, Loading, Workmanship, 

Kind of Brick.) 



: 3 Portland, well laid (3 tests) 

: 3 Portland, hurriedly laid (2 tests) . . 
: 3 Portland, six monthsf (2 tests) . . . . 
: 3 Portland, eccentric loading (2 tests) 

: 3 Portland, soft brick (2 tests) 

: 5 Portland, (2 tests) 

: 3 Natural (1 test) 

: 2 Lime (2 tests) 



Ultimate Crushing Strength. «: 






Ratio M 


Lbs. 


Ratio to 
first of 
series. 


to strength of 


per 
sq. in. 


Single 


Mortar 






brick. 


cubes. 


3365 


St'd 


0.31 


1.17 


2920 


0.87 


0.27 


1.05 _ 


3950 


1.18 


0.37 


1 


2800 


0.83 


0.26 


1060 


0.31 


0.27 


0.37 


2225 


0.66 


0.21 


1.30 


1750 


0.52 


0.16 


5.75 


1450 


0.43 


0.14 





*Hard bricks: size 2.35x3.95x8.25 ins.; crush, strength, flatwise, 
bedded in plaster, av. 10700 lbs. per sq. in.; cross-breaking, edgewise, 6-in. 
span, av. mod. of rupture, 1670 lbs. per sq. in. Soft bricks: 2.40 x 4.05 
X 8.20 ins.; crush, strength, as above, 3900; mod. of rupture, as above, 490. 
Size of piers: 12§" x 12^'' x 10 ft. high, with a li'' C. I. plate at bott. and top.. 

tOne column failed at 7th repetition of 4110-lb. load. d 

Summary op Terra-Cotta Column* Load. 



Column Characteristics. 

(Mortar, Loading, Workmanship, Kind 

of Blocks.) 



1 : 3 Portland, well laid (2 tests) 

1 : 3 Portland, hurriedly laid (1 test) 

1 : 3 Portland, eccentric loading^ well laid 

(1 test) 

1 : 3 Portland, hurriedly laid (1 test) 

1 : 3 Portland, underhurned hlocks% (1 

test) 

1 : 5 Portland (2 tests) 

Concave-end blocks, 1 : 2 Port. (6 tests). . 



Ultimate Crushing Strength. 



Lbs. 

per 

sq. in. 



4300t 
3305 

3470 
3110 

3050 
3350 
2970 



Ratio to 
first of 
series. 



St'd 
0.76 

0.81 
0.75 

0.71 
0.78 
0.69 



Ratio 
to strength of 



Single 
block. 



0.83 
0.64 

0.65 
0.60 

0.59 
0.65 
0.86 



Mortar 
cubes. 



1.26 
1.05 

1.12 
1.06 

0.88 



*A11 columns 12^ x 12^ ins. nominal, about 9f ft. long; made from blocks 
of second shipment except where noted "Concave-end blocks;" all tested 
in two months. 

tEstimated. 

JBlocks from second shipment, all of which were of good quality; enough 
blocks that appeared somewhat underbumed or otherwise inferior were 
selected to make up one column. 



EXCERPTS AND REFERENCES. 



523 



The Effect of Fire on Building Materials (By Jas. E. Howard. Paper 
read before the Natl. Assn. of Mutual Insurance Go's.; Eng. Rec, Dec. 3, 
1910). — ^The following stresses in lbs. per sq. in. for various materials, when 
confined, correspond to a rise in temperature of 160° P.: Steel, 30000 lbs. 
per sq. in.; cast iron, 17000 to 13590 lbs. per sq. in.; granite, 6270 to 3200 
lbs.; marble, 10800 to 2880 lbs.; slate,, 13480 lbs.; sandstone, 3200 to 960 
lbs.; hard burned brick, 4960 lbs.; light hard brick, 2230 lbs.; soft (salmon) 
brick, 248 lbs.; neat portland cement, 3710 lbs. per sq. in. Other experi- 
ments show that stones when even moderately heated do not return exactly 
to their primitive dimensions, but retain as a permanent set some of the ex- 
pansion which they acquired when hot, although the permanent sets are 
comparatively small. Experiments on 12'' x 12'' sticks of Douglas fir, 
burned over a wood fire for 2^ hours, at about 1380° F., until they were from 
6 to 7 ins., showed increased compressive strength in lbs. per sq. in. for the 
unbumed portion. 

Gypsum as a Fireproof ing Material (By H. G. Perring. Paper, Natl. 
Fire Protection Assn.; Eng. Rec, Dec. 3, 1910). — Advantages in the use of 
the plaster: (1) Low heat conductivity: the heat penetrates the plaster at 
such a slow rate that in fires of ordinary duration the metal would hardly 
get warm. (2) No appreciable expansion of plaster under heat action. 
(3) Hence, no contraction and destructive cracking when water is applied 
in a fire. (4) Does not bum or support combustion. (5) Plaster is the 
lightest practical fireproofiing material. (6) Plaster has the required 
strength for use in partitions and column coverings, and against fire streams. 
(7) It is adaptable to any form of construction: even when used in slabs they 
are readily cut or sawed to fit any desired location. (8) Plaster is low in 
cost. (9) Trim can be nailed to plaster fireproofing without the use of wood 
blocks or grounds. (10) Plaster blocks or boards, made of pure gypsum 
and wood fiber, are true and even and reduce the cost of plastering to a 
minimum. 

Some Tests of Old Timber (By C. P. Buchanan. Eng. News, Dec. 8, 
1910). — Crushing strength of 25-year-old and of new white pine, compared: 



Area, 


Crushing load. 


Remarks. 


sq. ins. 


lbs. per sq. in. 








Specimens from Bridge. 


29.25 


5780 


Clear, bolt-hole mark i" deep on narrow side. 


38.50 


4795 


Clear. Hart. Slightly decayed. 


46.31 


5950 


Sound. One sap corner. 

Clear. Hart. Well defined season cracks. 


70.00 


4920 


29.25 


5330 


Large black knot in longer side and end. 


50.00 


5100 


Clear. 


66.00 


5440 


Clear and slightly decayed. 


99.00 


5060 


Clear. Hart. Well defined season crack. 




5300 


Average of above 8 tests. 
Specimens from Stock. 


30.00 ( 


4510 


Clear. 


28.50 


3210 


Clear. 


47.50 


4570 


Clear. Slight season cracks. 


70.00 


3900 


Clear. Hart. Well defined season cracks. 


30.00 


5420 


Clear. Slight season cracks. 


50.00 


4420 


Clear. Hart. Slight season cracks. 


63.89 


4070 


Clear. Hart. Large season cracks. 


96.52 


3970 


Clear. Hart. Large season cracks. 




4260 


Average of above 8 tests. 



29.— PROPERTIES AND TABLES OF PLANE 

SURFACES. 

I.— GEOMETRICAL FIGURES. 

Center of gravity indicated by heavy dot. 
(Values are exact.) 

Notation. 
A = area of figure. 7 = i4r2 = moment of inertia about axis X — X; r = 

\// ^ A = radius of gyration about axis X — X. S= — = section modiilus; 

y 

5i = — ; 52= — . aoTyo= distance from axis considered to parallel axis 

yi y2 

through center of gravity. ; x = length of strip of infinitesimal width dy, 
distant y from neutral axis. All in inches. 




Fig. 1. — Any figure; neutral axis 
X—X through center of gravity. 



xdy 1= I xy 

-2/2 J -y 



r^ =1 -^A 



yi 



xy^dy 
2/2 

A 



y2 




Fig. 2. — Any figure; axis at base. 






J*2/i 

I xy 

Jo 



xy^dy 



•/ Jo 



ydy 



Vo 

xdy 



s=-L 

y\ 







-X 



Fig. 3. — Any figure; axis X' -X 
parallel with neutral axis X—X. 



V (about X'-X') 



l->r a^A 
xy'^dy + a\ 



■■ I xy^ 
J 



I 



(7 is for neutral axis through 
cen of grav as in Fig. 1; a = perp 
dist between the two axes.) 




I 



Fig. 4. — Triangle; 
cen of grav. 



axis through 



A = 



hd 



2 

yi = \d 



•^^-24 



7 =^ 



5'2 = J 

e hd^ 



Vl8 



0.236 fi 



524 



GEOMETRICAL FIGURES, 



525 




Fig. 5. — ^Triangle; axis at base. 



12 

d 



e hd'^ 



12 



\/6 



= 0.408 d 



■B77 

Fig. 6 — ^Triangle; axis through apex. 




/ = 



hd^ 



5i = 



W2 

4 



r = -^= 0.707 d 

V2 






y. 



Fig. 7. — Rectangle or parallelo- 
gram ; yi = y2. 

A = bd ' 

12 
5i=52=-y 

r = -^ ==0.289d 
Vl2 




Fig. 8. — Hollow rectangle; yt = y2- 
A = bd-bidi 

12 

c c bd^-bidi^ 

•Jl = ->2 = -iT-, 



Qd 



V 



bd^ - fcidi3 
12 (W - b^di) 



I 

d 



« 

y. 

I 



Fig. 9. — Rectangle; axis at base. 



b^ 
3 



Si = 



bd^ 



r ^ = 0.577 (i 

V3 




Fig. 10. — Rectangle; diagonal axis. 
bd 



y\ = y2 



Vb^-hd^ 



I = 



b^d^ 



6 (62+fi2) 



5i=52 = 



62,^2 



6V624-£i2 
bd 



\/6(62+d2) 




Fig. 11. — Rectangle; angular axis. 
d cos a + 6 sin a 



yi — y2 — ' 2 


/ = ^ (^2 cos2 a 4- 62 sin2 a) 


^ ^ 6i /d^ cos2 a+62 sin2 a\ 
^^-^2- 6 Wcosa + 6sina/ 


/d2 cos2 a -h b- sin2 a 


'■ ->/ 12 


<i; 




-^-x 


*J 


b=d\ 


> 


Fig. 12.— S qua 


ire or 


rhomboid 


yi = y2' 


r=i 


5.= 5.= f 


r = 


-4= =0.289 ci 
Vl2 



526 29.— PROPERTIES AND TABLES OF PLANE SURFACES. 



X-oh 

I 









Fig. 
A = 

/ = 

5i = 



13. — Hollow square; yi = ^2- 
^2 - d,^ 
d^ - dt^ 
12 



Qd 



\f"l2~ 




Fig. 14. — Square; diagonal axis. 



= ^2= —=: =0.707 (i 

V2 

= — (same as for 12) 



— = 0.118 (i3 




Fig. 15. — Hollow square; diag axis. 

yi = 
A = 

I 



■.y2= -4= = 0.707 cf 

\/2 
d^ - di2 



^4 - di4 



12 



Si =52 = 



(i3 _ di3 



6VT 

= 0.118 (d3 - di3) 

^^ = 0.289 (d-di) 
Vl2 



I 



X- 



b»d 



-^-X 



Fig. 16. — Square; axis at base. 
7=^ S.= 



d3 



vy 



= 0.577 d 




Fig. 17. — Trapezoid; 
cen of grav. 

d /bi + 2b2\ 

^^ = "3"VferTT7/ 

£/2bi±b2\ 
S\bi-h 62/ 



axis througl 



^2 

A 
I 

5i 



36 \ 



(&i + 62) 

&l2+4&,&2 + &22^ 



61 + 62 
d2 /bJ-\. 46162+62^ 

12 



(V 



61 + 262 



^ / 6l2+ 46ib2+622 \ 

12 V 26i + 62 / 



6(61 + 62) 



\/2(6i2+ 46162 + 632) 



Fig. 18. — Regular hexagon. 

d 6\/"3" 
3^1 = :V2 = y = — ^ 

A = ^^= 2.5986^ 

= 1^2 tan 30° = 0.866 ci2 
(hence, d^ = 362) 

, =5^1^^=0.5416^ 



—^=0.8666 



Si 



16 

= 0.060^4 

563 
S2=^ =0.120^3 




Vi 



0.4566 
0.264 cf 




Fig. 19. — Regular hexagon, 

yt =y2 = 0.433 (i =6 
A (same as for 18) 
I (same as for 18) 

5, =52 = 5^^ =0.54163 



16 
r (same as for 18) 



0.104 d3 



GEOMETRICAL FIGURES, 



627 




Fig. 20. — Hollow hexagon; 

d 
2 






A = 
I = 



2 
5 V~3 



26i =di, 
(b^-bi^) =2.598 (62- &i2) 



16 



(54-5^4) =:0.541(&4-6i4) 



Si =52 = 



0.541 (64 -&i4) 



= y|;C6-W = 



0.456 (6-6i) 




Fig. 21 



^1=^2 = 



-Regular octagon, 
J b b 
"^ V~2 



= 1.2076 
(hence d -=2.4146) 
A =2(^2 tan 22^° = 0.828 J2 

= 2c2 VY = 2.828 c2 
= 262 cot 22^ = 4.82862 

(hence, (2 VT-^2) (i2 = 
2 VYc^ = (2V T+ 2) 62; 
or, c2 = 0.2929 ci2 = 1.70762) 

/ = £^(i + 2\/y) =0.638c4 
6 

= 0.055 £f 4 

= 1.85964 
5i=52=0.109fi3 

r =0.475t7 =0.257 ci =0.6216 









Lk^^l^ 



Fig. 22. — Analysis of properties of 
Fig. 21. Thus, 
7o (for octagon) =78 (for square) 
— 4/t (for corner triangles, 
about axis X—X. 
Continued in next column. 



Fig. 22. — Continued, 

Zs (Fig 12) = ^ = 0.0833C/4 

It about X'-X' (see Fig. 4) 



36 



and 



Zt about X- X (see Fig. 3) 

4 rr2c2 

Hence, 



36"^ 



4Zt =-Q+ 2a2s2 

= y(3+2V'Y) =0.971464 
= 0,0287^^4 
Zo = Z« - 4Zt 

= (0.0833 - 0.0287) d^ = 0.055 ci4 
as given for Fig. 21, 




Fig. 23. — Hollow octagon. 

A = 0.828 (ci2-c?i2) 

= 2.828 (c2-ci2) 

= 4.828 (62 -6i2) 
Z = 0.055 (J4-cii4) 

= 0.638 (c4-(7i4) 

= 1.859 (64 -6i4) 



yi 



So^- 



a 



^-"•3^ 













Fi 


g. 24.- 


—Regular octagon. 




yi 


= ^2 


= 6 (sin 22i° 
= 1.30666 
= 0.5412i 


+ cos 


22^*^) 


A (same as for 21) 






Z 


(same as for 21) 






S^ 


_ Z^ 

yi 


y2 






r 


(same as for 21) 







528 29.'-PROPERTIES AND TABLES OF PLANE SURFACES, 




Fig. 25.— Circle. 



yi 
A 



y2 



^^=2" 



nri^=~^0.7S5idi 



itT-i 



rd^ 



I =111^ = 1^= 0.0491 d* 



Si 



4 

52 



64 
32 



0.0982 d* 



^ = f = 0.25<i 




Fig. 26. — Hollow circle. 

A = ;r(ri2 - r22) = 0.7854 {d'^ - dt^) 

T TT 



4 

Si=S2 



(ri4-r24) = 0.0491(£i4 - cf i4) 

yi y2 

\/ri2 + rz^ _ Vcf^ + cJi2 
2 4 



Fig. 27. — Semicircle; axis at base. 



Trri 



^ = '^ = 1.5708ri2 



TtTx* 



= 0.3927 ri* 



Si=^= 0.3927 fi» 



0.5fi 




Fig. 28. — Semicircle; axis through 
cen of grav. 

At I, S and r same as for 27. 




Ffg. 29. — Semicircle; axis through 
cen of grav. 

(3;r-4)ri ^ „,^ 



IVl 






3. -"-^ 


foori 


y2 


= 


4ri 

37r 


= 0.4244 ri 




I 


= 


(97 


-2-64)ri4 

72;r 


0.1098 ri* 


St 


= 


yx 


52 


y2 


r 


= 


riV97r2-64 
6;r 


0.264 ri 










5|x, 








X-\— f-- 








/ >Ol?l&^ 


i 



Fig. 30. — Circular sector; 
through cen of grav. 



ixis 



y2 



» sm a 

I ri 

a 

r\~y2 



(a in 
degrees.) 



yx 

A = ri2a 
[Note. 180° = ;r = 3.1416.] 




Vxtxi 



Fig. 31. — Circular half-segment; 
axes through cen of grav. 



y2 = \rx 



4 sin2 \a — sin2 « cos cr 
a — sin a cos a 



yx = n sm a — ^2 

sin3 a 
Xt = 



fi cos a 



a — sin a cos a 
^2 = ^1 (1 — cos a) — Xi 

A = -^ (a — sm a cos a) 



Note. — For transposed axis, see 
Fig. 3. For skeleton figures, see 
Figs. 49 to 54. For Mensuration, see 
Sec. 11, page 214. 



GEOMETRICAL FIGURES. SKELETON FIGURES. 



529 




Fig. 32. — Ellipse. 

a = semi major axis 
b = semi minor axis 
d 

A=7tab = '^=0.7mwd 
4 

^ - 4 - 



5i = 52 = 



^, = 0.049 W(i3 
64 
Tta^) _ Ttwd'^ 

~ir~ 32 

= 0.098 wd2 



r = -^ = -r 



i^'V^ 




Fig. 33. — Hollow ellipse. 

a and h = outer semi-axes 
at and h\ = inner semi-axes 
A = Tt {ah — ath\) 

/ = J (a3&-ai36i) 



Vi 



Note.— For other properties of 
the Ellipse, see Sec. 11, Mensuration, 
page 238. 




Fig. 34. — Parabolic half -segment; 
axes X'—X' and Y'— V passing 
through cen of grav. 
3 I. 2 , 

yi ==-^d\ y\ = 3- 

A =4-6d. 



Yy' 







Fig. 35. — Parabolic spandril; axes 
X'-X' and Y'-Y' passing 
through cen of grav. 



^1 



y2 



10 



7, 



3 ^ 1 ^ 



Note. — For other properties of 
the Parabola, see Sec. 11, Mensura- 
tion, page 237. 

2.-SKELET0N FIGURES WITH THIN LINES OF WIDTH t. 

For Notation, See Page 524. 

Center of gravity indicated by heavy dot. 

(Values are generally more or less approx., depending on thickness of lines.) 



i 



rv'i' 






X2 



Fig. 36. — Vertical web plate. 

A =dt 

J dH Ad^ 

12 12 

V12 
(Values exact.) 



= 0.289£i 





.37.- 

lelaj 

-Ay, 


j<---b--»» 


Fig 

A = 
/ = 


X *.._.x 

-Straight line about paral 
cis. 

2 = hty^'^ 



Vi- 



yi 



(I and r practically correct for 
small values oft). 



II 



630 ^.—PROPERTIES AND TABLES OF PLANE SURFACES. 



cf 



i^-J:^ 



^J 



-K'i 



t. 



Xt »-t, 



% 



Fig. 38. — Angle. Tee. 
A =dh + bt2 

^* ~ 2 (dh + btz) 



y2 



2bt2-\- dh J 
2 (bt2 4- dh) 



^ 12 \ &/2 + dh I 



r = 



(Errors in :yi, :V2. -? and r decrease 
as b approaches zero) . 






Fig. 



-Cross; yi = :^2 



2* 



i4. = <i?i + ^^2 

/ = ^ (line b neglected) 
12 



-a/5 



(Note that line b is included in A 
and not in I) . 






l-x 






Fig. 40. — H -section; yi = ^2 =■ -g 

A = 2dri + bt2 

/ = ^ (line 6 neglected) 

D 



V? 



1 n, ^'^ 



+,■^•1^X2 



A 



i^-b-v 



Fig. 41. — Rectangular cell. 



j=y 



y\ = :V2 

A = 2 (ci^i + 6^2) 

Note. — For square cell, 
^2 = ^1 = ^, we have, 

A = idt 

I ==ldH 

d 



b ^d. 



r = 



v^ 



= 0.408i 



tt t. 



•1- 



Fig. 42. — Channel. I-beam. 
d 

yi = ^2 = Y = y 

A 

/ 



= dh + 26^2 



VJ 



b--^. 






Fig. 43. — Channel. 

A =2dix + bt2 
d% 



yi = 



3^2 



2dh + ^^2 
. I_dh±bt2 



) 



bt2j 
bht2 



^d^(h, bht2 \ 
2 \3 2dh + W 



SKELETON FIGURES. 



5iiL 






f% 



_a^ 



l^-k 



L%k^.f 



Fig. 44. — I-beam; unequal flanges. 
A = dh + bt2 + ct-i 

_ ^ / 2bt2+dh \ 
y^~ 2 \dh + bt2 + chl 

__ d ( 2ch + dh \ 



_ dybhch , dH^ 
^ - A ^ \2A 



dHx ( bh + ct3 \ 
3 V A / 



Vi 



Fig. 45. — Inclined 
touching axis. * 

r =-^:^= 0.577 j'l 



straight line 



V3 

V_ >r ^^^^'' 

Fig. 46. — Line. * Angle. * 

^1=^2 = ^ 

A = lt 
I =llty^ 

y 



r = 



VI 



= 0.5773? 




Fig. 47. — Angle; unequal legs.* 
bd 



yi=y2='i 



V62 4- d^ 



A = {b + d)t 
J _ b Wt (b + d) 

12(62+(i2) 



* Thickness of lines = t. 



Fig. 47.— Continued. 






Mote. — For angle 


with ^g«a/ legs 


b 


= d, and 




yi 


b VT . 


3.3535(i 


A 


= 2cff 




I 

r 


12 
^ 204/^ 





V24 




Fig. 48. — Triangular cell. * 

d 
yi = 3^2 = Y 

A== (d-^2b)t 

I =^^(d + 2b) 

0.289 d 



Vl2 




Fig. 49. — Circular cell.* 

d 
yi =y2^ ri = -^ 

A =2 7rri< = 6.2832 n^ 
= S.UlQdt 
I = TtnH = 3.1416ri3f 

= o.zmdh 

"L- = 0.707 ri 



V^ 



= 0.354ci 






Fig. 50. — Semi-circular arc* 
A = jznt = 3.1416 rii 

/ = 



"illl = l.5708ri3i 
o 



r = 



VT 



0.707 ri 



532 29.— PROPERTIES AND TABLES OF PLANE SURFACES, 




Fig. 51. — Semi-circular arc. ' 
A, I and r same as for 50. 



— * 



y^ 
Xz 



Fig. 52. — Semi-circular arc ; 
through cen of grav. * 

yi = ''1 (l - 7) = 0.3634ri 

y2= — = 0.6366 ri 

A = Tzrit = Z.UlQrit 

I = riH (y - -i) = 0.2976ri3< 



axis 



'^j^~v^^ 



0.308 ri 



(Note that I52 can be obtained 
from /50 by method explained under 
Fig. 3; using the minus sign, as 

/52</50). 




Fig. 53. — Circular arc* 
/ . sin a\ 

(sin a \ 
cos a) 

A =2riat 

, / . 2sin2a\ 
/ = n^t la-hsm a cos a j 

A 

(Note. — ^The angle a may be ex- 
pressed in terms otn] thus, 180° = n, 

900 = y, etc.) 




Fig. 54. — Circular arc* 

- sin^^a 

^2 = 2ri yi = ri sm a — y2 

a 



•Fig. 54r— Continued. 

(sin a \ 
cos a I 

X2 = ri (1 — cos a) — xi 

A = riat 

J. , a /« ~ sin a cos a 4 sin* ^ a\ 

I =,,,3,j _ ^ 

Special Case, 

For axis X' at base: 

r-i^t 
I = -r— (a — sin a cos a) 



= n^ 



smacosa 



1-x 



Fig. 55. — Corrugated sheet. Assum- 
ing corrugations to be cycloidal 
curves, we have,* 

b = Breadth of sheet before corru- 
gating. 

d 

A=ht 

I =3%(i% = 0.1333(^26/ 

r = d VS = 0.365i 

(Compare this with 49, 50, 51; note 
that semi-circular arcs would give 
r = 0.354ci.) 



t:Xzf-W 

Fig. 56. — Corrugated sheet. 

yi=y2= y. Let 
F = area top flanges, 

= area bottom flanges, 
W = area of webs. Then 
A =2F+W 



f(-f) 



r = 



(Note. — The above value of I 
holds true for any inclination of the 
webs, from vertical to horizontal; 
the upper and lower flanges to have 
equal areas) . 



*Thickness of lines = t. 



SKELETON FIGURES. BLOCK SHAPES, 



533 



Fig. 57. — Flanges only. 



3.— BLOCK SHAPES. 

For Notation, see page 524. 

Center of gravity indicated by heavy dot. 
(Values are exact.) 

4MZO 

Fig. 61.— Tee. 
Properties same as for 58. 




A=2bt = b (d-di) 



/=J,«3 



di') 



S,= S.= ^«3-d,3) 




Fig. 58.— Cross. 
d 

yi=y2= -TT 



dh + /2 (& - h) 
hd^ + (& ~ h) i 
12 



5i = S2 = 



Qd 



j hd^ + (b - 
^j 12 [tid-hib 






*— b— >, 



Fig. 59. — I-beam. 
Properties same as for 58, 



■1 






LZOvli^ 



)fX 



Fig. 60. — Z-bar. 
Properties same as for 58, 



" V !- 



-^^|lx 



1 



Fig. 62.— Angle. 

A =dti-\- 12 (b - h) 
dHi + t2^ (b - ti) 

^^ = 2 A 

y2 = d-yi 



I = 



byi^ + ti y2^ 



I_ 

y\ 



3 



■\Ja 






Fig. 63.— Tee. 
Properties same as for 62. 



«<— -"b --->i 












Fig. 64. — Channel. 
Properties same as for 62. 



5U 29.— PROPERTIES AND TABLES OF PLANE SURFACES. 



<--b--^ 

Fig. 65. — ^I-beam. 
d 

A =^dh-\-2t2{b-h) 

bd^-ib-h)(d-2t2y 
^ - 12 

^ ^ I -L 



I 



i<-b-->! 












Fig. 66. — Channel. 
Properties same as for 65. 



f^ 






I 
t 
« 



Fig. 67. — Continued. 



-4'x 

Fig. 67. — Z-bar. 
Properties same as for 65. 

i #_ :^ 

Fig. 68. — ^Tee; tapered stem. 
^1 = 2A 

(fl-^o) (d-i2) {d + 2t2) 

■^ 6A 

T W , (c^-^2)^ (3/0 +^i) 

^ ~ 3 "^ 12 

-A (3/1-/2)2 

Continued in next column. 



5.= i 



^2 



VJ 



^-fct — 






h\ ^X 



^-^ 



Fig. 69. — I-beam; unequal flangej 
with unequal thicknesses. 

A = 6^2 + (.d — t2 — to)ti 4- 60/0 

y^^^k^+t,(d-t2-t0) {t2+'^-^ 

+ boto(d-fj'^-^A 

y2 = d-yi 

^ by^^ + 60^2^ 



;^\ 



(b-t,) (yi-t2y 



(bp-tx) {yr-toY 
3 



5. = ^ 

yi 



52=- 

3^2 



.j^v-""^— ^ 



ft^ 



4-4fc 



^x 



k-b,--> 

Fig. 70. — I-beam; unequal flanges 
with equal thicknesses. 

A = Q) + bo) t2 + (d - 2/2) h 
yi = [y (&-&o) + t2fi ibo-h) 



m- 



y2 -- 


= d- 


■yi 










I = 


^byr 


3 + boy2^ 

3 

ib-t,) (yy 


-/2)3 


(&0- 


-/l) (5/2-/2)3 


s,- 


yi 


3 


52 = 


2 
3^2 


3 





-V? 



POLAR MOMENT OF INERTIA 635 

4.— ROLLED SHAPES. 

(See, also. Sections 30, 31, 32, following.) 

In the following illustrations the center of gravity is indicated by a 
heavy dot. 

Values are exact for shapes as outlined. These shapes do not show 
the actual rounded ' ' fillets ' ' because the latter are disregarded in structural 
calculations, except in special cases. 

The flanges of I-beams and standard channels slope at the rate of 2 : 12 
on their inner faces, equivalent to an angle of 9° 28'. The heavier sections 
of I-beams, channels and Z-bars are made by spreading the rolls, thereby 
increasing the width of flange; but many angles are rolled in " finishing ** 
grooves, which maintain a standard width of flange for various thicknesses 
of metal. 

Moment of inertia (la) about inclined axis. ^HS^Q. ; \^Q. 

— In the preceding illustrations (Figs. 1-70) we v^"""! — ^->v ^ 

have confined our attention to moments of y^ k x d v'' 

inertia about the coordinate axes X — X and / J ^"A 

y—y, one of which is usually drawn parallel ^_J__ ,2?i-ijcsCv._A _v 

with a principal line of the figure, and the other X"" I ^"•io ~ f ~ ^ 

at right angle to the first. It very often hap- \ ^^^ } / 

pens, however, that we wish to know the value ,^\ { / 

/a, the moment of inertia about an inclined axis <f \^^ } 7 

making an angle a with the axis X — X, Fig. 71, -2rdn^^^"""''''^ithrN 

about which latter axis Ix is the moment of 3 — Q. { A — G(. 

inertia. This may be obtained from the fol- Y 

lowing formula: Fig. 71. 

Ia = Ix cos2 a + Iy sin2 a—2K cos a sin a (1) 

In which Ix= l \y"dxdy = moment of inertia about axis X — X; 



-If' 
-If' 



Iy= i ix^dx dy = moment of inertia about axis Y—Y; 

7 a = moment of inertia about axis a— a; 
a = angle between the axes a— a and X — X, the functions 
of a to be considered algebraically; thus, sin a is + in 
the 1st and 2nd quadrants, and — in the 3rd and 4th; 
while cos a is -f- in the 1st and 4th, and negative in the 
2nd and 3rd. Hence, cos^a and sin^o; are always 
positive ( + ), while sin a cos a becomes positive (-I-) 
when the axis a— a lies in the 1st and 3rd quadrants, 
and negative ( — ) when it lies in the 2nd and 4th. 



-If- 



= I ixy dx dy = double integration (summation) of dA 

( = dx dy, or D) multiplied by its axial distances x and 
y (see Fig. 71). Those portions of the figure or surface 
considered which lie in the 1st and 3rd quadrants tend 
to make K positive ( + ), while those portions which 
lie in the 2nd and 4th quadrants tend to make K nega- 
tive (-). 

In solving equation (1) we find the values of ix, ly and K from the 
shape of the plane figure and with the coordinate axes X — X and Y—Y 
intersecting at the origin O. Then by assuming the proper value for the 
angle a we may solve equation (1) for /a. Values of Ix and ly for many 
figures are given under the preceding illustrations. We will now explain 
how K is derived. 

Values of K in Equation (1). — ^The value of K is 

K:= I I xydxdy (2) 

bearing in mind what has previously been said regarding positive and 
negative values. 



536 29.— PROPERTIES AND TABLES OF PLANE SURFACES. 



Secondly, the preceding formula (2) may also be expressed: 

K=A xoyo (3) 

In wkich A = area of the figure or plane surface, 

Xq==+ or — distance from axis of Y—Y to cen of grav of i4, 
yo=+ or — distance from axis of X^X to cen of grav of A, 
all in inches. It is evident that when the cen of grav lies at the origin O 
(Fig. 71) the value of K is zero; when in the 1st or 3rd quadrants, it is + ; 
and when in the 2nd and 4th, it is — . But the minus sign in equation (1) 
must also be considered, as well as the algebraic value of cos a sin a, in 
solving for la. 

Thirdly, the figure may be cut up into smaller areas, each area being 
multiplied by its respective coordinate distances x and y to the cen of grav 
of each area; thus. 



K=2 A^ xi yi+A2X2y2+A3X3yz-\ etc. 



(4) 

The four following examples will illustrate the methods of finding the 
value of K in Equation (1): — • 

"'-/."J?— :[tX*-t:[¥-¥4'-T- 

Or, K = hd • Y • Y ^ ~T~ ^^^^^^^ method) . 



=-^nT+ — 2 — ) 



lid,= Q, K=-bd[^+^-fj, 



If , also, 6i = 0, K=- 



b^d^ 



(Compare with Fig. 72) 



(3) K 



1st Q. 

fe2£f2 



2nd Q. 
4 



3rdQ. 
■*" 4 



4th Q. 
4 



If bt = b and di = d, K = 0. 

But if 6 = and di = Ot there will remain for the 2nd ^ 



quadrant, K 



4 * 



(4) K = Kt-hK2+K2 

= t (d-t) xiyi + btXO + t(d-t) (-xs) (-y^). 

But Xi = X3= — o"-. and 3^1=^3=^; hence 
K==i (bt-fi) (d^-dt). 




^ Fig. 75. 



ROLLED SHAPES. 



537 



Problem. — (1) Find the moment of inertia la, about the axis ai— ai 
of a square beam of cross-section dXd, ai making 
an upward angle of 45'' with the axis X — X; 
(2) find Ia2 about the axis a2— a2 making a down- 
ward angle of 45° with the axis X — X. 

Solution.— Ix=Iy = -^ (Fig. 9). 



K = Y (Fig. 72). 



(n/ai 



3 
3" 

rf4 



Then from equation (1): 
2 



(cos2 a + sin2 a) 



cos a sm a 



2 * =^"12 



(See Fig. 14). 
d^ 



(2°)Ia2 = -o (cos^a + sin^a) — Y cos a (^sin a) 




3 2 



i = 



12* 



Maximum and minimum values of la. — If a Z-bar, angle or other shape 
is used as a column or strut it is essential that we know the least radius of 
gyration of the section, and this can be obtained from the minimum moment 



of inertia, as r = 



~yJA ' 



also / min and / max are valuable in connection with 



the resisting moments of beams. The following formulas give the value of 
a for minimum or maximum value of /«: 

Tan 2a = -tr^-^ (5) 

jy — Ix 

Moreover, it can be stated that the axis say ai— at giving lai a, minimum 
value will be at right angle to the axis say a2— 0:2, giving /a2 a maximum 
value. That is, " maximum " and " minimum " axes intersect at the 
origin, O, at an angle of 90°. In practice it is easy to distinguish one from 
the other; thus, in Fig. 76, ai— ai is the " minimum " axis, and a2— «2 
the " maximum," The maxima and minima axes are called the " principal " 
axes, and in this particular case they happen to lie at an angle of 45° with 
the axes of X and Y. Figs. 80 and 81, following, show positions of axes 
for minimum values of / for the Z-bar and angle. 

For a more complete discussion of moments of inertia about inclined 
axes see Lanza's " Applied Mechanics "; also Paper No. 1020, Trans. Am. 
Soc. C. E., Vol. LVI, p. 169. See also Handbook of Cambria Steel Co. 





■:■ 

s 


Fig. 77.—: 
A=^td + 2v{m + n 

Slopes * '^~' 


r 

r-beam. 

) 

n h-l 

b-t 









1 I 



«« ; i I 



X 



/x=:^[6d3+^(/l4_/4)] 



ly = i^[63 (d-h) + Ifi +-j(b^- tm 
2/x 2Iy 



o .. ^ , e 2/x 2Iy 

Section modulus b = —7- or -7- 
a o 



Rad of gyration r 



Fig. 78.— Channel. 
id + Q)-t) (m-n) 
h-l 



A 

Slope s = i = 



2ib-t) 
/x =- Mbd^-^(h^-l^)] 



ly 



8s 



i [2&3» + lt^ + -^ (Jb^-t^)^-Axt^ 



538 29.— PROPERTIES AND TABLES OF' PLANE SURFACES. 

Fig. 80. — Continued, 
ly 



;^W^^ 




Fig. 79.— Tee. 
A = ^{k+h) + V (m+n) + mh 

y\ = [^v (m—n) {m + 2n) 
+ 3w2(6-^i) + 3^2 
-Ikh-to) {'id-m-^^A 

-2v{m- »)3] - A (yi- >n)2 
nZ>3 4- {m-n)h^+ lt(? 



ly 



12 

V {m-n){2v'^+ {2v+Zhy 

36 
/(/i-^)[ai-/o)^+2(ti + 2^o)2] 
144 
Y 



F-— -V. 



— 4.>r: 









I 

I 

Fig. 80. — Z-bar; uniform thickness /. 
il=i(6+2c^i) 



/x 



12 

Continued in next column. 



la 



db^'-di(b-2t)^ 

12 
7x cos2 a — ly sin2 a 



cos 2 a 



In which tan 2 a = 



(bt-t^) (d^-dt) 



ly-Ix 
(Note. — la = minimum/.) 









Fig. 81.— Angle. 



A =t(b + di) 

t(2bi-hd) + bi^ 



Xi = 

yi = 

/y= 

7a = 



2 (6i + d) 
t(2di + b) + di^ 

2 (di + b) 
i {d-yi)^ + byi3 ~ {b-t) {y^- 1)^ 

3 
t {b-XtY + ffa:x3 - {d-t) (xi -ty 

3 
/x cos^ a — ly sin^ a 
cos 2 a 

In which tan 2a = t[(2yi-t) {b2-2bxi) 
+ i2xi-t) (d-t) id-ht-2yi)]^2(lylx) 

(Note. — I a = minimum /.) 



Reference. 

The Polar Moment of Inertia, and Its Graphical Application to 
Riveted Joints (C. F. Blake and R. W. Runge. Eng. News, May 21. 1903).— 
Formulas and illustrations. 



ROLLED SHAPES. RECTANGLES, 



539 



5.—* MOMENTS OF INERTIA (/) OF B^ RECTANGLES. 



Depth 
in 




6.- 


-Width of Rectangle 


in Inches. 






Inches. 
d 


i 


ft 


f 


/^ 


i 


A 


f 


2 


.17 


.21 


.25 


.29 


.33 


.38 


.42 


3 


.56 


.70 


.84 


.98 


1.13 


1.27 


1.41 


4 


1.33 


1.67 


2.00 


2.33 


2.67 


3.00 


3.33 


5 


2.60 


3.26 


3.91 


4.56 


5.21 


5.86 


6.51 


6 


4.50 


5.63 


6.75 


7.88 


9.00 


10.13 


11.25 


7 


7.15 


8.93 


10.72 


12.51 


14.29 


16.08 


17.86 


8 


10.67 


13.33 


16.00 


18.67 


21.33 


24.00 


26.67 


9 


15.19 


18.98 


22.78 


26.58 


30.38 


34.17 


37.97 


10 


20.83 


26.04 


31.25 


36.46 


41.67 


46.87 


52.08 


11 


27.73 


34.66 


41.59 


48.53 


55.46 


62.39 


69.32 


12 


36.00 


45.00 


54.00 


63.00 


72.00 


81.00 


90.00 


13 


45.77 


57.21 


68.66 


80.10 


91.54 


102.98 


114.43 


14 


57.17 


71.46 


85.75 


100.04 . 


114.33 


128.63 


142.92 


15 


70.31 


87.89 


105.47 


123.05 


140.63 


158.20 


175.78 


16 


85.33 


106.67 


128.00 


149.33 


170.67 


192.00 


213.^3 


17 


102.35 


127.94 


153.53 


179.12 


204.71 


230.30 


255.89 


18 


121.50 


151.88 


182.25 


212.63 


243.00 


273.38 


303.75 


19 


142.90 


178.62 


214.34 


250.07 


285.79 


321.52 


357.24 


20 


166.67 


208.33 


250.00 


291.67 


333.33 


375.00 


416.67 


21 


192.94 


241.17 


289.41 


337.64 


385.88 


434.11 


482.34 


22 


221.83 


277.29 


332.75 


388.21 


443.67 


499.13 


554.58 


23 


253.48 


316.85 


380.22 


443.59 


506.96 


570.33 


633.70 


24 


288.00 


360.00 


432.00 


504.00 


576.00 


648.00 


720.00 


25 


325.52 


406.90 


488.28 


569.66 


651.04 


732.42 


813.80 


26 


366.17 


457.71 


549.25 


640.79 


732.33 


823.88 


915.42 


27 


410.06 


512.58 


615.09 


717.61 


820.13 


922.64 


1025.16 


28 


457.33 


571.67 


686.00 


800.33 


914.67 


1029.00 


1143.33 


29 


508.10 


635.13 


762.16 


889.18 


1016.21 


1143.23 


1270.26 


30 


562.50 


703.13 


843.75 


984.38 


1125.00 


1265.63 


1406.25 


32 


682.67 


853.33 


1024.00 


1194.67 


1365.33 


1536.00 


1706.67 


34 


818.83 


1023.54 


1228.25 


1432.96 


1637.67 


1842.38 


2047.08 


36 


972.00 


1215.00 


1458.00 


1701.00 


1944.00 


2187.00 


2430.00 


38 


1143.17 


1428.96 


1714.75 


2000. 54 


2286.33 


2572.13 


2857.92 


40 


1333.33 


1666.67 


2000.00 


2333.33 


2666.67 


3000. 00 


3333.33 


42 


1543.50 


1929.38 


2315.25 


2701.13 


3087.00 


3472.88 


3858.75 


44 


1774.67 


2218.33 


2662.00 


3105.67 


3549.33 


3993.00 


4436.67 


46 


2027.83 


2534.79 


3041.75 


3548.71 


4055.67 


4562.63 


5069.58 


48 


2304.00 


2880.00 


3456.00 


4032.00 


4608.00 


5184.00 


5760.00 


50 


2604.17 


3255.21 


3906.25 


4557.29 


5208.33 


5859.38 


6510.42 


52 


2929.33 


3661.67 


4394.00 


5126.33 


5858.67 


6591.00 


7323.33 


54 


3280.50 


4100.63 


4920.75 


5740.88 


6561.00 


7381.13 


8201.25 


56 


3658.67 


4573.33 


5488.00 


6402.67 


7317.33 


8232.00 


9146.67 


58 


4064.83 


5081.04 


6097.25 


7113.46 


8129.67 


9145.87 


10162.08 


60 


4500.00 


5625.00 


6750.00 


7875.00 


9000.00 


10125.00 


11250.00 



*/= 



12" 



Section modulus 



-¥ 



ibdK 



Resisting moment M"= — = ^ =fS=i fbd^, in which 
y a 

/=the value in above table; 

/ = the outer fiber stress in beam, in lbs. per sq. in.; 
M" = moment in inch-lbs. ; M' = moment in ft.-lbs. ==M"-i-12. 



540 2^.— PROPERTIES AND TABLES OF PLANE SURFACES. 



5. — * Moments op Inertia (/) OP 1^ Rectangles. — Concluded. 



Depth 

In 
Inches. 



&. — ^Width of Rectangle in Inches. 



IJL 
16 



.46 
1.55 
3.67 

7.16 
12.38 
19.65 
29.33 
41.77 

57.29 

76.26 

99.00 

125. 87 

157.21 

193.36 
234.67 
281.47 
334.13 
392.96 

458.33 
530.58 
610.04 
697.07 
792.00 

895.18 
1006.96 
1127.67 
1257.67 
1397.29 

1546.88 
1877.33 
2251.79 
2673.00 
3143.71 

3666.67 
4244.63 
4880.33 
5576.54 
6336.00 

7161.46 

8055. 57 

9021.38 

10061.33 

11178.00 

12375.00 



.50 
1.69 
4.00 

7.81 
13.50 
21.44 
32.00 
45.56 

62.50 

83.19 

108.00 

137.31 

171.50 

210.94 
256.00 
307.06 
364.50 
428.69 

500.00 
578.81 
665.50 
760.44 
864.00 

976.56 
1098.50 
1230.19 
1372.00 
1524.31 

1687.50 
2048.00 
2456.50 
2916.00 
3429.50 

4000.00 
4630.50 
5324.00 
6083.50 
6912.00 

7812.50 

8788.00 

9841.50 

10976.00 

12194.50 

13500.00 



-13 
16 



.54 
1.83 
4.33 

8.46 
14.63 
23.22 
34.67 
49.36 

67.71 

90.12 

117.00 

148.75 

185.79 

228.52 
277.33 
332.65 
394.88 
464.41 

541.67 
627.05 
720.96 
823.81 
936.00 

1057.94 
1190.04 
1332.70 
1486.33 
1651.34 

1828.13 
2218.67 
2661.21 
3159.00 
3715.29 

4333.33 
5016.38 
5767.67 
6590.46 
7488.00 

8463.54 

9520.33 

10661.63 

11890.67 

13210.71 

14625.00 



i 



.58 
1.97 
4.67 

9.11 
15.75 
25.01 
37.33 
53.16 

72.92 

97.05 

126.00 

160.20 

200.08 

246.09 
298.67 
358.24 
425.25 
500.14 

583.33 
675.28 
776.42 
887.18 
1008.00 

1139.32 

1281.58 
1435.22 
1600.67 
1778.36 

1968.75 
2389.33 
2865.92 
3402.00 
4001.08 

4666.67 
5402.25 
6211.33 
7097.42 
8064.00 

9114.58 
10252.67 
11481.75 
12805.33 
14226.92 

15750.00 



15 
16 



.63 
2.11 
5.00 

9.77 
16.88 
26.80 
40.00 
56.95 

78.13 
103.98 
135.00 
171.64 
214.38 

263.67 
320.00 
383.83 
455.63 
535.86 

625.00 
723.52 
831.87 
950.55 
1080.00 

1220.70 
1373.13 
1537.73 
1715.00 
1905.39 

2109.38 
2560.00 
3070.63 
3645.00 
4286.88 

5000.00 
5788.13 
6655.00 
7604.38 
8640.00 

9765.63 
10985.00 
12301.88 
13720.00 
15243.12 

16875.00 



1 



.67 
2.25 
5.33 

10.42 
18.00 
28.58 
42.67 
60.75 

83.33 
110.92 
144.00 
183.08 
228.67 

281.25 
341.33 
409.42 
486.00 
571.58 

666.67 

771.75 

887.33 

1013.92 

1152.00 

1302.08 
1464.67 
1640.25 
1829.33 
2032.42 

2250.00 
2730.67 
3275.33 
3888.00 
4572.67 

5333.33 
6174.00 
7098.67 
8111.33 
9216.00 

10416.67 
11717.33 
13122.00 
14634.67 
16259.33 

18000.00 



Ex. — What resisting moment has a beam 4^ ins. wide by 15 ins. in 
depth ? 

Ans.— 7=281.25X4+140.63=1265.63; hence, 

M = ^ /= 168.75 / inch-lbs. = 14.06 /ft.-lbs. 

If /= 1000. M'= 14060 ft.-lbs. 



30.— PROPERTIES AND TABLES OF STEEL 

SHAPES. 

List op Tables in This Section (30). 

Table 1.— Steel Rods— Weights and Areas Pages 542-543 

2.— Steel Plates— Weights and Areas " 544-547 

" 3.— Properties of Angles (Steel)— Unequal Legs " 548-551 

4.— Properties of Angles (Steel)— Equal Legs " 552-553 

5.— Properties of I-Beams (Steel) " 554-555 

6.— Properties of Channels (Steel) " 556 

7.— Properties of Z-Bars (Steel) " 557 

8.— Properties of T-Shapes (Steel) " 558-559 

9 —Standard Rail Sections— Cambria and A. S. C. E. . . " 560 

List op Relevant Tables in Other Sections. 

Section 2. 

Table 9.— Squares of Numbers 1 to 1600 Pages 31-43 

Section 4. 

Table 5. — Fractions of an Inch Reduced to Millimeters Page 69 

" 6.— Hundredths of an Inch Reduced to Millimeters .... " 69 

" 7. — Millimeters Reduced to Decimals of an Inch " 70 

" 13. — Square Inches and Square Millimeters — Equivalents " 80 

" 28.— Pounds and Kilograms— Equivalents " 85 

Section 11. 

Table 11. — Circumferences of Circles for Given Diameters — 

Decimals Pages 224-225 

" 12. — Circumferences of Circles for Given Diameters in 

Inches " 226-229 

" 13.— Areas of Circles for Given Diameters in Inches " 230-231 

" 19.— Properties of Hollow Cylinders— Dia. to Circum., etc. " 246-247 

" 20. — Spheres — Areas of Surf ace for Given Diameters ... . " 251 

" 21.— Spheres— Volumes for Given Diameters " 252 

Section 27. 

Table 9.— Weights and Specific Gravities of Metals Pages 478-482 

" 11.— Weight Equivalent for Any Specific Gravity " 483 

" 12.— Weight of Sheets, Bars and Wire— from the Specific 

Gravity " 484 

14. — Comparison of Various Weights, Capacities and 

Volumes " 485 

Section 29. 

Table 1.— Properties and Tables of Geometrical Figures Pages 524-529 

" 2.— Properties and Tables of Skeleton Figures " 529-532 

" 3.— Properties and Tables of Block Shapes " 533-534 

" 4.— Properties and Tables of Rolled Shapes " 535-538 

5.— Moments of Inertia of Rectangles " 639-540 

Section 31. 

Table 8.— Bethlehem Girder I-Beams Page 583 

" 9. — Bethlehem Special I-Beams *' 584 

Section 32. 

Table 14.— Rolled Steel H-Columns Page 608 

Section 59. 
Table 43.— Standard Dimensions of Rails Pages 1060-1062 

641 



542 dO.— PROPERTIES AND TABLES OF STEEL SHAPES. 



1. — Steel Rods — Weights and Areas. 
Weights at 489 . 6 lbs, per cu. ft., or 2% more than Iron at 480 lbs. 



O g 


Weight 


Weight 


Area 


Area 


U 


Weight 


Weight 


Area 


Area 


10 •«-! 


of 


of 


of 


C/3 •!-< 


of 


of 


of 


■ Bar 


• Bar 


D Bar 


Bar 


g.S 


■ Bar 
IFoot 


• Bar 


D Bar 


Bar 


-^6 


1 Foot 


1 Foot 


in 


in 


-36 


1 Foot 


in 


in 


•^.5 


Long. 


Long. 


Square 
Ins. 


Square 
Ins. 


'^■^ 


Long. 


Long. 


Square 
Ins. 


Square 
Ins. 


HP 


Lbs. 


Lbs. 


HQ 


Lbs. 


Lbs. 













3 


50.60 


24.03 


9.0000 


7.0686 


1^ 


.013 


.010 


.0039 


.0031 


^ 


31.89 


25.04 


9.3789 


7.3662 


Vs 


.053 


.042 


.0156 


.0123 


Vs 


33.20 


26.08 


9.7656 


7.6699 


-h 


.119 


.094 


.0352 


.0276 


■h 


34.55 


27.13 


10.160 


7.9798 


H 


.212 


.167 


.0625 


.0491 


K 


35.92 


28.20 


10.563 


8.2958 


TS 


.333 


.261 


.0977 


.0767 


■h 


37.31 


29.30 


10.973 


8.6179 


Vs 


.478 


.375 


.1406 


.1104 


Vs 


38.73 


30.42 


11.391 


8.9462 


^ 


.651 


.511 


.1914 


.1503 


^ 


40.18 


31.56 


11.816 


9.2806 


¥ 


.850 


.667 


.2500 


.1963 


¥ 


41.65 


32.71 


12.250 


9.6211 




1.076 


.845 


.3164 


.2485 




43.14 


33.90 


12.691 


9.9678 


/^ 


1.328 


1.043 


.3906 


.3068 


Vs 


44.68 


35.09 


13.141 


10.321 


H 


1.608 


1.262 


.4727 


.3712 


tt 


46.24 


36.31 


13.598 


10.680 


H 


1.913 


1.502 


.5625 


.4418 


M 


47.82 


37.56 


14.063 


11.045 


xa. 


2.245 


1.763 


.6602 


.5185 


it 


49.42 


38.81 


14.535 


11.416 


/4 


2.603 


2.044 


.7656 


.6013 


Vs 


51.05 


40.10 


15.016 


11.793 


if 


2.989 


2.347 


.8789 


.6903 


H 


52.71 


41.40 


15.504 


12.177 


1 


3.400 


2.670 


1.0000 


.7854 


4 


54.40 


42.73 


16.000 


12.566 


^ 


3.838 


3.014 


1.1289 


.8866 


^ 


56.11 


44.07 


16.504 


12.962 


3^ 


4.303 


3.379 


1.2656 


.9940 


Vs 


57.85 


45.44 


17.016 


13.364 


^ 


4.795 


3.766 


1.4102 


1.1075 


A 


59.62 


46.83 


17.535 


13.772 


^ 


5.312 


4.173 


1.5625 


1.2272 


K 


61.41 


48.24 


18.063 


14.186 


A 


5.857 


4.600 


1.7227 


1.3530 


^ 


63.23 


49.66 


18.598 


14.607 


Vs 


6.428 


5.049 


1.8908 


1.4849 


Vs 


65.08 


51.11 


19.141 


15.033 


A 


7.026 


5.518 


2.0664 


1.6230 


A 


66.95 


52.58 


19.691 


15.466 


¥ 


7.650 


6.008 


2.2500 


1.7671 


V2 


68.85 


54.07 


20.250 


15.904 




8.301 


6.520 


2.4414 


1.9175 


A 


70.78 


55.59 


20.816 


16.349 


^ 


8.978 


7.051 


2.6406 


2.0739 


Vs 


72.73 


57.12 


21.391 


16.800 


ii 


9.682 


7.604 


2.8477 


2.2365 


H 


74.70 


58.67 


21.973 


17.257 


34 


10.41 


8.178 


3.0625 


2.4053 


H 


76.71 


60.25 


22.563 


17.721 




11.17 


8.773 


3.2852 


2.5802 


a 


78.74 


61.84 


23.160 


18.190 


j^ 


11.95 


9.388 


3.5156 


2.7612 


Vs 


80.81 


63.46 


23.766 


18.665 


H 


12.76 


10.02 


3.7539 


2.9483 


H 


82.89 


65.10 


24.379 


19.147 


2 


13.60 


10.68 


4.0000 


3.1416 


5 


85.00 


66.76 


25.000 


19.635 




14.46 


11.36 


4.2539 


3.3410 


^ 


87.14 


68.44 


25.629 


20.129 


lA 


15.35 


12.06 


4.5156 


3.5466 


Vs 


89.30 


70.14 


26.266 


20.629 


-h 


16.27 


12.78 


4.7852 


3.7583 


^ 


91.49 


71.86 


26.910 


21.135 


yi 


17.22 


13.52 


5.0625 


3.9761 


H 


93.72 


73.60 


27.563 


21.648 


^ 


18.19 


14.28 


5.3477 


4.2000 


A 


95.96 


75.37 


28.223 


22.166 


^ 


19.18 


15.07 


5.6406 


4.4301 


Vs 


98.23 


77.15 


28.891 


22.691 


1^ 


20.20 


15.86 


5.9414 


4.6664 


^ 


100.5 


78.95 


29.566 


23.221 


y2 


21.25 


16.69 


6.2500 


4.9087 


V2 


102.8 


80.77 


30.250 


23.758 


P. 


22.33 


17.53 


6.5664 


5.1572 


^ 


105.2 


82.62 


30.941 


24.301 


23.43 


18.40 


6.8906 


5.4119 


Vs 


107.6 


84.49 


31.641 


24.850 


H 


24.56 


19.29 


'7.2227 


5.6727 


i^ 


110.0 


86.38 


32.348 


25.406 


^ 


25.71 


20.20 


7.5625 


5.9396 


H 


112.4 


88.29 


33.063 


25.967 


4I 


26.90 


21.12 


7.9102 


6.2126 


H 


114.9 


90.22 


33.785 


26.535 


Vs 


28.10 


22.07 


8.2656 


6.4918 


Vs 


117.4 


92.17 


34.516 


27.109 


il 


29.34 


23.04 


8.6289 


6.7771 


^ 


119.9 


94.14 


35.254 


27.688 


3 


30.60 


24.03 


9.0000 


7.0686 


6 


122.4 


96.14 


36.000 


28.274 



STEEL RODS— SQUARE AND ROUND. 



543 



1. — Steel Rods — Weights and Areas. — Concluded. 
Weights at 489 . 6 lbs. per cu. ft., or 2% more than Iron at 480 lbs. 



II 


Weight 

■ Bar 

1 Foot 

Long. 

Lbs. 


Weight 

of 

• Bar 

1 Foot 

Long. 

Lbs. 


Area 

of 
D Bar 

in 

Square 

Ins. 


Area 

of 
O Bar 

in 

Square 

Ins. 




Weighl 

H^ar 

1 Foot 

Long. 

Lbs. 


.Weighl 

of 

• Bar 

1 Foot 

Long. 

Lbs. 


: Area 
of 
n Bar 

in 

Square 

Ins. 


Area 

of 
O Bar 

in 

Square 

Ins. 


6 

! 


122.4 
125.0 
127.6 
130.2 


96.14 
98.14 
100.2 
102.2 


36.000 
36.754 
37.516 

38.285 


28.274 
28.866 
29.465 
30.069 


9 

! 


275.4 
279.3 
283.2 
287.0 


216.3 
219.3 
222.4 
225.4 


8L000 
82.129 
83.266 
84.410 


63.617 
64.504 
65.397 
66.296 


1 


132.8 
135.5 
138.2 
140.9 


104.3 
106.4 
108.5 
110.7 


39.063 
39.848 
40.641 
41.441 


30.680 
31.296 
31.919 
32.548 


% 


290.9 
294.9 
298.9 
302.8 


228.5 
231.5 
234.7 
237.9 


85.563 
86.723 
87.891 
89.066 


67.201 
68.112 
69.029 
69.953 


i 

ii 


143.6 
146.5 
149.2 
152.1 


112.8 
114.9 
117.2 
119.4 


42.250 
43.066 
43.891 
44.723 


33.183 
33.824 
34.472 
35.125 


i 

ii 


306.8 
310.9 
315.0 
319.1 


241.0 
244.2 
247.4 
250.6 


90.250 
91.441 
92.641 
93.848 


70.882 
71.818 
72.760 
73.708 


M 


154.9 
157.8 
160.8 
163.6 


121.7 
123.9 
126.2 
128.5 


45.563 
46.410 
47.266 
48.129 


35.785 
36.450 
37.122 
37.800 


i 


323.2 
327.4 
331.6 
335.8 


253.9 
257.1 
260.4 
263.7 


95.063 
96.285 
97.516 
98.754 


74.662 
75.622 
76.589 
77.561 


7 


166.6 
169.6 
172.6 
175.6 


130.9 
133.2 
135.6 
137.9 


49.000 
49.879 
50.766 
51.660 


38.485 
39.175 
39.871 
40.574 


10 


340.0 
344.3 
348.5 
352.9 


267.0 
270.4 
273.8 
277.1 


100.00 
101.25 
102.52 
103.79 


78.540 

79.525 
80.516 
81.513 


¥ 

1 


178.7 
181.8 
184.9 
188.1 


140.4 
142.8 
145.3 
147.7 


52.563 
53.473 
54.391 
55,316 


41.282 
41.997 
42.718 
43.445 


i 


357.2 
361.6 
366.0 
370.4 


280.6 
284.0 

287.4 
290.9 


105.06 
106.35 
107.64 
108.94 


82.516 
83.525 
84.541 
85.562 


ii 


191.3 
194.4 
197.7 
200.9 


150.2 
152.7 
155.2 
157.8 


56.250 
57.191 
58.141 
59.098 


44.179 
44.918 
45.664 
46.415 




374.9 
379.4 
383.8 
388.3 


294.4 
297.9 
301.4 
305.0 


110.25 
111.57 

112.89 
114.22 


86.590 
87.624 
88.664 
89.710 


i 


204.2 
207.6 
210.8 
214.2 


160.3 
163.0 
165.6 
168.2 


60.063 
61.035 
62.016 
63.004 


47.173 
47.937 
48.707 
49.483 


1 

8 


392.9 
397.5 
402.1 
406.8 


308.6 
312.2 
315.8 
319.5 


115.56 
116.91 
118.27 
119.63 


90.763 
91.821 

92.886 
93.956 


8 

1 


217.6 
221.0 
224.5 
228.0 


171.0 
173.6 
176.3 
179.0 


64.000 
65.004 
66.016 
67.035 


50.265 
51.054 

51.849 
52.649 


11 


411.4 
416.1 
420.9 
425.5 


323.1 
326.8 
330.5 
334.3 


121.00 

122.38 
123.77 
125.16 


95.033 
96.116 
97.205 
98.301 


A 


231.4 
234.9 
238.5 
242.0 


181.8 
184.5 
187.3 
190.1 


68.063 
69.098 
70.141 
71.191 


53.456 
54.269 
55.088 
55.914 


1 


430.3 
435.1 
439.9 
444.8 


337.9 
341.7 
345.5 
349.4 


126.56 
127.97 
129.39 
130.82 


99.402 
100.51 
101.62 
102.74 


if 


245.6 
249.3 
252.9 
256.6 


193.0 
195.7 
198.7 
201.6 


72.250 
73.316 
74.391 
75.473 


56.745 
57.583 
58.426 
59.276 




449.6 
454.5 
459.5 
464.4 


353.1 
357.0 
360.9 
364.8 


132.25 
133.69 
135.14 
136.60 


103.87 
105.00 
106.14 
107.28 


i 


260.3 
264.1 
267.9 
271.6 


204.4 
207.4 
210.3 
213.3 


76.363 
77.660 
78.766 
79.879 


60.132 
60.994 
61.862 
62.737 


1 


469.4 
474.4 
479.5 
484.5 


368.6 
372.6 
376.6 
380.6 


138.06 
139.54 
141.02 
142.50 


108.43 
109.59 
110.75 
111.92 


9 275.4 1 


216.3 


81.000 


63.617 


12 


489.6 


384.6 


144.00 


113.10 



544 20.— PROPERTIES AND TABLES OF STEEL SHAPES, 

2. — Steel Plates — Weights and Areas. 
(Weights at 489.6 lbs. per cubic foot.) 













Width of Plate in 


Inches. 






» 


1" 


IM" 


IK" 


m" 


r 


2H" 


23^'' 


2%" 


r 


m' 


SH'' 


m" 






WEIGHT lbs. per lin. 


ft. 








^^ 


i 


HVHHHB 




H 








^ 


.212 


.266 


.319 


.372 


.425 


.48 


.53 


.58 


.64 


.69 


.74 


.80 


Vs 


.425 


.531 


.637 


.744 


.850 


.96 


1.06 


1.17 


1.28 


1.38 


1.49 


1.59 


^ 


,638 


.797 


.957 


1.11 


1.28 


1.44 


1.59 


1.75 


1.91 


2.07 


2.23 


2.39 


H 


.850 


1.06 


1.28 


1.49 


1.70 


1.91 


2.12 


2.34 


2.55 


2.76 


2.98 


3.19 


■TS 


1.06 


1.33 


1.59 


1.86 


2.12 


2.39 


2.65 


2.92 


3.19 


3.45 


3.72 


3.99 


% 


1.28 


1.59 


1.92 


2.23 


2.55 


2.87 


3.19 


3.51 


3.83 


4.15 


4.47 


4.78 


"IE 


1.49 


1.86 


2.23 


2.60 


2.98 


3.35 


3.72 


4.09 


4.46 


4.83 


5.20 


5.58 


3^ 


1.70 


2.12 


2.55 


2.98 


3.40 


3.83 


4.25 


4.67 


5.10 


5.53 


5.95 


6.38 


A 


1.92 


2.39 


2.87 


3.35 


3.83 


4.30 


4.78 


5.26 


5.74 


6.22 


6.70 


7.17 


H 


2.12 


2.65 


3.19 


3.72 


4.25 


4.78 


5.31 


5.84 


6.38 


6.91 


7.44 


7.97 


ii 


2.34 


2.92 


3.51 


4.09 


4.67 


5.26 


5.84 


6.43 


7.02 


7.60 


8.18 


8.76 


M 


2.55 


3.19 


3.83 


4.47 


5.10 


5.75 


6.38 


7.02 


7.65 


8.29 


8.93 


9.57 


H 


2.76 


3.45 


4.14 


4.84 


5.53 


6.21 


6.90 


7.60 


8.29 


8.98 


9.67 


10.36 


^ 


2.98 


3.72 


4.47 


5.20 


5.95 


6.69 


7.44 


8.18 


8.93 


9.67 


10.41 


11.16 


if 


3.19 


3.99 


4.78 


5.58 


6.38 


7.18 


7.97 


8.77 


9.57 


10.36 


11.16 


11.95 


1 


3.40 


4.25 


5.10 


5.95 


6.80 


7.65 


8.50 


9.35 


10.20 


11.05 


11.90 


12.75 


^ 


3.61 


4.52 


5.42 


6.32 


7.22 


8.13 


9.03 


9.93 


10.84 


11.74 


12.65 


13.55 


J^ 


3.83 


4.78 


5.74 


6.70 


7.65 


8.61 


9.57 


10.52 


11.48 


12.43 


13.39 


14.34 


A 


4.04 


5.05 


6.06 


7.07 


8.08 


9.09 


10.10 


11.11 


12.12 


13.12 


14.13 


15.14 


M 


4.25 


5.31 


6.38 


7.44 


8.50 


9.57 


10.63 


11.69 


12.75 


13.81 


14.87 


15.94 


A 


4.46 


5.58 


6.69 


7.81 


8.93 


10.04 


11.16 


12.27 


13.39 


14.50 


15.62 


16.74 


^ 


4.67 


5.84 


7.02 


8.18 


9.35 


10.52 


11.69 


12.85 


14.03 


15.20 


16.36 


17.53 


^ 


4.89 


6.11 


7.34 


8.56 


9.78 


11.10 


12.22 


13.44 


14.66 


15.88 


17.10 


18.33 


M 


5.10 


6.38 


7.65 


8.93 


10.20 


11.48 


12.75 


14.03 


15.30 


16.68 


17.85 


19.13 








AREA of sect 


ion in sq. ins 


3. 










'C 


• 




A 


.063 


.078 


.094 


.109 


.125 


.141 


.156 


.172 


.188 


.203 


.219 


.234 


3^ 


.125 


.156 


.188 


.219 


.250 


.281 


.313 


.344 


.375 


.406 


.438 


.469 


A 


.188 


.234 


.281 


.328 


.375 


.422 


.469 


.516 


.563 


.609 


.656 


.703 


^ 


.250 


.313 


.375 


.438 


.500 


.563 


.625 


.688 


.750 


.813 


.875 


.938 


A 


.313 


.391 


.469 


.547 


.625 


.703 


.781 


.859 


.938 


1.02 


1.09 


1.17 


^ 


.375 


.469 


.563 


.656 


.750 


.844 


.938 


1.03 


1.13 


1.22 


1.31 


1.41 


^ 


.438 


.547 


.656 


.766 


.875 


.984 


1.09 


1.20 


1.31 


1.42 


1.53 


1.64 


y2 


.500 


.625 


.750 


.875 


1.00 


1.13 


1.25 


1.38 


1.50 


1.63 


1.75 


1.88 


■h 


.563 


.703 


.844 


.984 


1.13 


1.27 


1.41 


1.55 


1.69 


1.83 


1.97 


2.11 


% 


.625 


.781 


.938 


1.09 


1.25 


1.41 


1.56 


1.72 


1.88 


2.03 


2.19 


2.34 


H 


.688 


.859 


1.03 


1.20 


1.38 


1.55 


1.72 


1.89 


2.06 


2.23 


2.41 


2.58 


.750 


.938 


1.13 


1.31 


1.50 


1.69 


1.88 


2.06 


2.25 


2.44 


2.63 


2.81 


i 


.813 


1.02 


1.22 


1.42 


1.63 


1.83 


2.03 


2.23 


2.44 


2.64 


2.84 


3.05 


.875 


1.09 


1.31 


1.53 


1.75 


1.97 


2.19 


2.41 


2.63 


2.84 


3.06 


3.28 


H 


.938 


1.17 


1.41 


1.64 


1.88 


2.11 


2.34 


2.58 


2.81 


3.05 


3.28 


3.52 


1 


1.00 


1.25 


1.50 


1.75 


2.00 


2.25 


2.50 


2.75 


3.00 


3.25 


3.50 


3.75 


Jig. 


1.06 


1.33 


1.59 


1.86 


2.13 


2.39 


2.66 


2.92 


3.19 


3.45 


3.72 


3.98 


1^ 


1.13 


1.41 


1.69 


1.97 


2.25 


2.53 


2.81 


3.09 


3.38 


3.66 


3.94 


4.22 


A 


1.19 


1.48 


1.78 


2.08 


2.38 


2.67 


2.97 


3.27 


3.56 


3.86 


4.16 


4.45 


3 


1.25 


1.56 


1.88 


2.19 


2.50 


2.81 


3.13 


3.44 


3.75 


4.06 


4.38 


4.69 


A 


1.31 


1.64 


1.97 


2.30 


2.63 


2.95 


3.28 


3.61 


3.94 


4.27 


4.59 


4.92 


^ 


1.38 


1.72 


2.06 


2.41 


2.75 


3.09 


3.44 


3.78 


4.13 


4.47 


4.81 


5.16 


X. 


1.44 


f.80 


2.16 


2.52 


2.88 


3.23 


3.59 


3.95 


4.31 


4.67 


5.03 


5.39 


H 


1.50 


1.88 


2.25 


2.63 


3.00 


3.38 


3.75 


4.13 


4.50 


4.88 


5.25 


5.63 



STEEL PLATES—WEIGHTS AND AREAS. 



545 



2. — Steel Plates — Weights and Areas. — Continued. 
(Weights at 489.6 lbs. per cubic foot.) 

Width of Plate in inches. 



4M'' 


43^'' 


m' 


h" 


5M" 


5^" 


5M" 


6" 


6M" 


63^" 



WEIGHT lbs. per lin. ft. 



.85 
1.70 
2.55 
3.40 

4.25 
5.10 

5.95 
6.80 

7.65 

8.50 

9.35 

10.20 

11.05 
11.90 
12.75 
13.60 
14.45 
15.30 
16.15 
17.00 

17.85 
18.70 
19.55 
20.40 



.90 
1.81 
2.71 
3.61 

4.52 
5.42 
6.32 

7.22 

8.13 

9.03 

9.93 

10.84 

11.74 
12.65 
13.55 
14.45 
15.35 
16.26 
17.16 
18.06 
18.96 
19.87 
20.77 
21.68 



.96 
1.91 

2.87 
3.83 

4.78 
5.74 
6.70 
7.65 
8.61 
9.57 
10.52 
11.48 

12.43 
13.39 
14.34 
15.30 

16.26 
17.22 
18.17 
19.13 

20.08 
21.04 
21.99 
22.95 



1.01 
2.02 
3.03 
4.04 

5.05 

6.06 

7.07 

8.08 

9.09 

10.10 

11.11 

12.12 

13.12 
14.13 
15.14 
16.15 
17.16 
18.17 
19.18 
20.19 

21.20 
22.21 
23.22 
24.23 



1.06 
2.13 
3.19 
4.25 

5.31 

6.38 
7.44 
8.50 

9.57 
10.63 
11.69 
12.75 

13.81 
14.87 
15.94 
17.00 
18.06 
19.13 
20.19 
21.25 
22.32 
23.38 
24.44 
25.50 



1.12 
2.23 
3.35 
4.46 

5.58 

6.69 

7.81 

8.93 

10.04 

11.16 

12.27 

13.39 

14.50 
15.62 
16.74 
17.85 

18.96 
20.08 
21.20 
22.32 

23.43 
24.54 

25.66 
26.78 



1.17 

2.34 

3.51 

4.67 

5.84 

7.02 

8.18 

9.35 

10.52 

11.69 

12.85 

14.03 

15.19 

16.36 

17.53 

18.70 

19.87 
21.04 
22.21 
23.38 
24.54 
25.71 
26.88 
28.05 



1.22 
2.44 
3.67 
4.89 

6.11 
7.34 
8.56 
9.77 

11.00 
12.22 
13.44 
14.67 
15.88 
17.10 
18.33 
19.55 

20.77 
21.99 
23.22 
24.44 
25.66 
26.88 
28.10 
29.33 



1.28 
2.55 
3.83 
5.10 

6.38 
7.65 
8.93 
10.20 
11.48 
12.75 
14.03 
15.30 

16.58 
17.85 
19.13 
20.40 

21.68 
22.95 
24.23 
25.50 
26.78 
28.05 
29.33 
30.60 



1.33 
2.66 
3.99 
5.31 
6.64 
7.97 
9.29 
10.63 

11.95 
13.28 
14.61 
15.94 

17.27 
18.60 
19.92 
21.25 

22.58 
23.91 
25.23 
26.56 

27.90 

29.22 
30.55 
31.88 



1.38 
2.76 
4.14 
5.53 

6.90 

8.29 

9.67 

11.05 

12.43 
13.81 
15.20 
16.58 

17.95 
19.34 
20.72 
22.10 

23.48 

24.87 
26.24 
27.62 

29.01 
30.39 
31.77 
33.15 



1.43 

2.87 
4.30 
5.74 

7.17 

8.61 

10.04 

11.48 

12.91 
14.34 
15.78 
17.22 

18.65 
20.08 
21.51 
22.95 

24.39 
25.82 
27.25 
28.69 
30.12 
31.56 
32.99 
34.43 



AREA of section in sq. ins. 



.250 


.266 


.281 


.297 


.313 


.328 


.344 


.359 


.375 


.391 


.406 


.500 


.531 


.563 


.594 


.625 


.656 


.688 


.719 


.751 


.781 


.813 


.750 


.797 


.844 


.891 


.938 


.984 


1.03 


1.08 


1.13 


.117 


1.22 


1.00 


1.06 


1.13 


1.19 


1.25 


1.31 


1.38 


1.44 


1.50 


1.56 


1.63 


1.25 


1.33 


1.41 


1.48 


1.56 


1.64 


1.72 


1.80 


1.88 


1.95 


2.03 


1.50 


1.59 


1.69 


1.78 


1.88 


1.97 


2.06 


2.16 


2.25 


2 84 


2.44 


1.75 


1.86 


1.97 


2.08 


2.19 


2.30 


2.41 


2.52 


2.63 


2.73 


2.84 


2.00 


2.13 


2.25 


2.38 


2.50 


2.63 


2.75 


2.88 


3.00 


3.13 


3.25 


2.25 


2.39 


2.53 


2.67 


2.81 


2.95 


3.09 


3.23 


3.38 


3.52 


3.66 


2.50 


2.66 


2.81 


2.97 


3.13 


3.28 


3.44 


3.59 


3.75 


3.91 


4.06 


2.75 


2.92 


3.09 


3.27 


3.44 


3.61 


3.78 


3.95 


4.13 


4.80 


4.47 


3.00 


3.19 


3.38 


3.56 


3.75 


3.94 


4.13 


4.31 


4.50 


4.69 


4.88 


3.25 


3.45 


3.66 


3.86 


4.06 


4.27 


4.47 


4.67 


4.88 


5.08 


5.28 


3.50 


3.72 


3.94 


4.16 


4.38 


4.59 


4.81 


5.03 


5.25 


5.47 


5.69 


3.75 


3.98 


4.22 


4.45 


4.69 


4.92 


5.16 


5.39 


5.63 


5.86 


6.09 


4.00 


4.25 


4.50 


4.75 


5.00 


5.25 


5.50 


5.75 


6.00 


6.25 


6.50 


4.25 


4.52 


4.78 


5.05 


5.31 


5.58 


5.84 


6.11 


6.38 


6.64 


6.91 


4.50 


4.78 


5.06 


5.34 


5.63 


5.91 


6.19 


6.47 


6.75 


7.03 


7.31 


4.75 


5.05 


5.34 


5.64 


5.94 


6.23 


6.53 


6.83 


7.13 


7.42 


7.72 


5.00 


5.31 


5.63 


5.94 


6.25 


6.56 


6.88 


7.19 


7.50 


7.81 


8.13 


5.25 


5.58 


5.91 


6.23 


6.56 


6.89 


7.22 


5.55 


7.88 


8.20 


8.53 


5.50 


5.84 


6.19 


6.53 


6.88 


7.22 


7.56 


7.91 


8.25 


8.59 


8.94 


5.75 


6.11 


6.47 


6.83 


7.19 


7.55 


7.91 


8.27 


8.63 


8.98 


9.34 


6.00 


6.38 


6.75 


7.13 


7.50 


7.88 


8.25 


8.63 


9.00 


9.38 


9.75 



.422 
.844 
1.27 
1.69 

2.11 
2.53 
2.95 
3.38 

3.80 

4.22 
4.64 
5.06 
5.48 
5.91 
6.33 
6.75 

7.17 
7.59 
8.02 
8.44 
8.86 
9.28 
9.70 
10.13 



646 '^.—PROPERTIES AND TABLES OF STEEL SHAPES. 



2. — Steel Plates — Weights and Areas. — Continued. 
(Weights at 489.6 lbs. per cubic foot.) 





Width of Plate in Inches. 


r 


7M" 


7y/ 


7M" 


8" 


SH" 


83^" 


SH" 


r 


m' 


m'' 


m" 








WEIGHT lb. 




X!'-' 


B 


■■■■■m 


3. per lin. ft. 


H 






1 


1.49 
2.98 
4.46 
5.95 


1.54 
3.08 
4.62 
6.16 


1.59 
3.19 
4.78 
6.36 


1.65 
3.29 
4.94 
6.58 


1.70 
3.40 
5.10 
6.80 


1.75 
3.51 
5.26 
7.01 


1.81 
3.61 
5.42 

7.22 


1.86 
3.72 
5.58 
7.43 


1.91 
3.83 
5.74 
7.65 


1.97 
3.93 
5.90 
8.86 


2.02 
4.04 
6.06 
8.08 


2.07 
4.14 
6.22 
8.29 


^ 
% 


7.44 

8.93 

10.41 

11.90 


7.70 

9.25 

10.78 

12.32 


7.97 

9.57 

11.16 

12.75 


8.23 

9.88 

11.53 

13.18 


8.50 
10.20 
11.90 
13.60 


8.76 
10.52 
12.27 
14.03 


9.03 
10.84 
12.64 
14.44 


9.29 
11.16 
13.02 
14.87 


9.56 
11.48 
13.40 
15.30 


9.83 
11.80 
13.76 
15.73 


10.10 
12.12 
14.14 
16.16 


10.36 
12.44 
14.51 
16.58 


1 


13.39 
14.87 
16.36 
17.85 


13.86 
15.40 
16.94 
18.49 


14.34 
15.94 
17.53 
19.13 


14.82 
16.47 
18.12 
19.77 


15.30 
17.00 
18.70 
20.40 


15.78 
17.53 
19.28 
21.04 


16.26 
18.06 
19.86 
21.68 


16.74 
18.59 
20.45 
22.32 


17.22 
19.13 
21.04 
22.96 


17.69 
19.65 
21.62 
23.59 


18.18 
20.19 
22.21 
24.23 


18.65 
20.72 
22.79 
24.86 


i 


19.34 
20.83 
22.32 
23.80 


20.03 
21.57 
23.11 
24.65 


20.72 
22.32 
23.91 
25.50 


21.41 
23.05 
24.70 
26.35 


22.10 
23.80 
25.50 
27.20 


22.79 
24.55 
26.30 
28.05 


23.48 
25.30 
27.10 
28.90 


24.17 
26.04 
27.89 
29.75 


24.86 
26.78 
28.69 
30.60 


25.55 
27.52 
29.49 
31.45 


26.24 
28.26 
30.28 
32.30 


26.94 
29.01 
31.08 
33.15 


1 


25.29 
26.78 
28.26 
29.75 


26.19 
27.73 
29.27 
30.81 


27.10 

28.68 
30.28 
31.88 


28.00 
29.64 
31.29 
32.94 


28.90 
30.60 
32.30 
34.00 


29.80 
31.56 
33.31 
35.06 


30.70 
32.52 
34.32 
36.12 


31.61 
33.47 
35.33 
37.20 


32.52 
34.43 
3Q.34 
38.26 


33.41 
35.38 
37.35 
39.31 


34.32 
36.34 
38.36 
40.37 


35.22 
37.29 
39.37 
41.44 


i 

3^ 


31.23 
32.72 
34.21 
35.70 


32.35 
33.89 
35.44 
36.98 


33.48 
35.06 
36.66 
38.26 


34.59 
36.23 
37.88 
39.53. 


35.70 
37.40 
39.10 
40.80 


36.81 
38.57 
40.32 
42.08 


37.93 
39.74 
41.54 
43.35 


39.05 
40.91 

42.77 
44.63 


40.16 
42.08 
44.00 
45.90 


41.28 
43.25 
45.22 
47.18 


42.40 
44.41 
46.44 
48.45 


43.52 
45.58 
47.66 
49.73 








A] 


REA of secti 






c 


1 


on m sq. ins. 








1 


.438 
.875 
1.31 
1.75 


.453 
.906 
1.36 
1.81 


.469 
.938 
1.41 

1.88 


.484 
.969 
1.45 
1.94 


.500 
1.00 
1.50 
2.00 


.516 
1.03 
1.55 
2.06 


.531 
1.06 
1.59 
2.13 


.547 
1.09 
1.64 
2.19 


.563 
1.13 
1.69 
2.25 


.578 
1.16 
1.73 
2.31 


.594 
1.19 

1.78 
2.38 


.609 
1.22 
1.83 
2.44 


1 


2.19 
2.63 
3.06 
3.50 


2.27 
2.72 
3.17 
3.63 


2.34 
2.81 
3.28 
3.75 


2.42 
2.91 
3.39 
3.88 


2.50 
3.00 
3.50 
4.00 


2.58 
3.09 
3.60 
4.13 


2.66 
3.19 
3.72 
4.25 


2.73 
3.28 
3.83 
4.38 


2.81 
3.38 
3.94 
4.50 


2.89 
3.47 
4.05 
4.63 


2.97 
3.56 
4.16 
4.75 


3.05 

3.66 
4.27 
4.88 


i 


3.94 
4.38 
4.81 
5.25 


4.08 
4.53 
4.98 
5.44 


4.22 
4.69 
5.16 
5.63 


4.36 
4.84 
5.33 
5.81 


4.50 
5.00 
5.50 
6.00 


4.64 
5.16 
5.67 
6.19 


4.78 
5.31 
5.84 
6.38 


4.92 
5.47 
6.02 
6.56 


5.06 
5.63 
6.19 
6.75 


5.20 

5.78 
6.36 
6.94 


3.34 
5.94 
6.53 
7.13 


5.48 
6.09 
6.70 
7.31 


.i 


5.69 
6.13 
6.56 
7.00 


5.89 
6.34 
6.80 
7.25 


6.09 
6.56 
7.03 
7.50 


6.30 

6.78 

7.27 
7.75 


6.50 
7.00 
7.50 
8.00 


6.70 

7.22 
7.73 
8.25 


6.91 
7.44 
7.97 
8.50 


7.11 

7.66 
8.20 
8.75 


7.31 
7.88 
8.44 
9.00 


7.52 
8.09 
8.67 
9.25 


7.72 
8.31 
8.91 
9.50 


7.92 
8.53 
9.14 
9.75 


1 


7.44 

7.88 
8.31 
8.75 


7.70 
8.16 
8.61 
9.06 


7.97 
8.44 
8.91 
9.38 


8.23 
8.72 
9.20 
9.69 


8.50 

9.00 

9.50 

10.00 


8.77 

9.28 

9.80 

10.31 


9.03 

9.56 

10.09 

10.63 


9.30 

9.84 

10.39 

10.94 


9.56 
10.13 
10.69 
11.25 


9.83 
10.41 
10.98 
11.56 


10.09 
10.69 
11.28 

11.88 


10.36 
10.97 
11.58 
12.19 


1 


9.19 

9.63 

10.06 

10.50 


9.52 

9.97 

10.42 

10.88 


9.84 
10.31 
10.78 
11.25 


10.17 
10.66 
11.14 
11.63 


10.50 
11.00 
11.50 
12.00 


10.83 
11.34 

11.86 
12.38 


11.16 
11.69 
12.22 
12.75 


11.48 
12.03 
12.58 
13.13 


11.81 
12.38 
12.94 
13.50 


12.14 
12.72 
13.30 

13.88 


12.47 
13.06 
13.66 
14.25 


12.80 
13.41 
14.02 
14.63 



STEEL PLATES—WEIGHTS AND AREAS. 



647 



2. — Steel Plates — Weights and Areas. — Concluded. 
(Weights at 489.6 lbs. per cubic foot.) 









• 


Width of Plate in 


[nches 










10" 


lOM'' 


10^'' 


lOM" 


11'' 


llM'' 


ny2'^ 


IIM" 


12'' 


12M'' 


i2y2" 


12^'' 






WEIGHT lbs. per lin. f1 


'• 






^•-' 


■ 


HHHHHB 




^ 








1 


2.13 
4.25 
6.38 
8.50 


2.18 
4.36 
6.54 
8.71 


2.23 
4.46 
6.70 
8.92 


2.28 
4.57 
6.86 
9.14 


2.34 
4.68 
7.02 
9.34 


2.39 
4.78 
7.17 
9.57 


2.44 

4.89 
7.32 
9.78 


2.50 

4.99 

7.49 

10.00 


2.55 

5.10 

7.65 

10.20 


2.60 
5.21 

7.82 
10.42 


2.66 

5.31 

7.98 

10.63 


2.71 

5.42 

8.13 

10.84 




10.62 
12.75 

14.88 
17.00 


10.89 
13.07 
15 25 
17.42 


11.16 
13.39 
15.62 
17.85 


11.42 
13.71 
15.99 

18.28 


11.68 
14.03 
16.36 
18.70 


11.95 
14.35 
16.74 
19.13 


12.22 
14 68 
17 12 
19.55 


12.49 
14 99 
17.49 
19.97 


12 75 
15.30 
17.85 
20.40 


13.01 
15.62 
18.23 
20.82 


13.28 
15.94 
18.60 
21.25 


13.55 
16.26 
18.97 
21.67 




19.14 
21.25 
23.38 
25.50 


19.61 
21.78 
23.96 
26.14 


20.08 
22.32 
24.54 
26.78 


20.56 
22.85 
25.13 
27.42 


21.02 
23.38 
25.70 
28.05 


21.51 
23.91 
26.30 
28.68 


22.00 
24.44 

26.88 
29.33 


22.48 
24.97 
27.47 
29.97 


22 95 
25.50 
28.05 
30.60 


23.43 
26.03 
28.64 
31.25 


23.90 
26.56 
29.22 
31.88 


24.39 
27.09 

29.80 
32.52 


"i 


27.62 
29.75 
31.88 
34.00 


28.32 
30.50 
32.67 
34.85 


29.00 
31.24 
33.48 
35.70 


29.69 
31.98 
34.28 
36.55 


30.40 
32.72 
35.06 
37.40 


31.08 
33.47 
35.86 
38.25 


31.76 
34.21 
36.66 
39.10 


32.46 
34.95 
37.46 
39.95 


33.15 
35.70 

38.25 
40.80 


33.83 
36.44 
39.05 
41.65 


34.53 
37.19 
39.84 
42.50 


35.22 
37.93 
40.64 
43.35 


i 


36.12 
38.25 
40.38 
42.50 


37.03 
39.21 
41.39 
43.56 


37.92 
40.17 
42.40 
44.63 


38.83 
41.12 
43.40 
45.69 


39.74 
42.08 
44.42 
46.76 


40.64 
43.04 
45.42 

47.82 


41.54 
44.00 

46.44 
48.88 


42.45 
44.94 
47.45 
49.94 


43.35 
45.90 
48.45 
51.00 


44.25 
46.86 
49.46 
52.06 


45.16 
47.82 
50.46 
53.12 


46.06 
48.77 
51.48 
54.19 


i 


44.64 
46.75 
48.88 
51.00 


45.75 

47.92 
50.10 
52.28 


46.86 
49.08 
51.32 
53.55 


47.97 
50.25 
52.54 
54.83. 


49.08 
51.42 
53.76 
56.10 


50.20 
52.59 
54.99 
57.37 


51.32 
53.76 
56.21 
58.65 


52.44 
54.93 
57.43 
59.93 


53.55 
56.10 
58.65 
61.20 


54.67 
57.27 
59.87 
62.48 


55.78 
58.44 
61.10 
63.75 


56.90 
59.60 
62.32 
65.03 






- 


AREA 


of sect 


ion in sq. in 


LS. 








c 


^ 












S 

^ 
M 


.625 
1.25 
1.88 
2.50 


.641 
1.28 
1.92 
2.56 


.656 
1.31 
1.97 
2.63 


.672 
1.34 
2.02 
2.69 


.688 
1.38 
2.06 
2.75 


-.703 
1.41 
2.11 
2.81 


.719 
1.44 
2.16 
2.88 


.734 
1.47 
2.20 
2.94 


.750 
1.50 
2.25 
3.00 


.766 
1.53 
2.30 
3.06 


.781 
1.56 
2.34 
3.13 


.797 
1.59 
2.39 
3.19 


1 


3.13 
3.75 
4.38 
5.00 


3.20 
3.84 
4.48 
5.13 


3.28 
3.94 
4.59 
5.25 


3.36 
4.03 
4.70 
5.38 


3.44 
4.13 
4.81 
5.50 


3.52 

4.22 
4.92 
5.63 


3.59 
4.31 
5.03 
5.75 


3.67 
4.41 
5.14 

5.88 


3.75 
4.50 
5.25 
6.00 


3.83 
4.59 
5.36 
6.13 


3.91 
4.69 
5.47 
6.25 


3.98 
4.78 
5.58 
6.38 


ft 


5.63 
6.25 
6.88 
7.50 


5.77 
6.41 
7.05 

7.69 


5.91 
6.56 
7.22 
7.88 


6.05 

6.72 
7.39 
8.06 


6.19 

6.88 
7.56 
^.2b 


6.33 
7.03 

7.73 
8.44 


6.47 
7.19 
7.91 
8.63 


6.61 
7.34 
8.08 
8.81 


6.75 
7.50 
8.25 
9.00 


6.89 
7.66 
8.42 
9.19 


7.03 

7.81 
8.59 
9.38 


7.17 
7.97 
8.77 
9.56 


if 

,8 


8.13 

8.75 

9.38 

10.00 


8.33 

8.97 

9.61 

10.25 


8.53 

9.19 

9.84 

10.50 


8.73 

9.41 

10.08 

10.75 


8.94 

9.63 

10.31 

11.00 


9.14 

9.84 

10.55 

11.25 


9.34 
10.06 
10.78 
11.50 


9.55 
10.28 
11.02 
11.75 


9.75 
10.50 
11.25 
12.00 


9.95 
10.72 
11.48 
12.25 


10.16 
10.94 
11.72 
12.50 


10.36 
11.16 
11.95 
12.75 


1 


10.63 
11.25 
11.88 
12.50 


10.89 
11.53 
12.17 
12.81 


11.16 
11.81 
12.47 
13.13 


11.42 
12.09 
12.77 
13.44 


11.69 
12.38 
13.06 
13.75 


11.95 
12.66 
13.36 
14.06 


12.22 
12.94 
13.66 
14.38 


12.48 
13.22 
13.95 
14.69 


12.75 
13.50 
14.25 
15.00 


13.02 
13.78 
14.55 
15.31 


13.28 
14.06 
14.84 
15.63 


13.55 
14.34 
15.14 
15.94 


1 


13.13 
13.75 
14.38 
15.00 


13.45 
14.09 
14.73 
15.38 


13.78 
14.44 
15.09 
15.75 


14.11 
14.78 
15.45 
16.13 


14.44 
15.13 
15.81 
16.50 


14.77 
15.47 
16.17 

16.88 


15.09 
15.81 
16.53 
17.25 


15.42 
16.16 
16.89 
17.63 


15.75 
16.50 
17.25 
18.00 


16.08 
16.84 
17.61 
18.38 


16.41 
17.19 
17.97 

18.75 


16.73 
17.53 
18.33 
19.13 



548 20.— PROPERTIES AND TABLES OF STEEL SHAPES. 



-xZ ,> 3. — Properties of Angles (Steel). 






.ofe." 

Y 

Fig. 1. 



Unequal Legs. 

(Weights at 489.6 lbs. per 
cubic foot.) 



r 




<- 


— s — 


^^-_ 


T 



Fig."2 



1^ 



4^ 



Size. 



Ins. 



Ins. 



LIL 



Wt. 

per 

Foot, 

W 
Lbs. 



Area 
of 

Sec. 
A 
Sq. 

Ins. 



Dist. from 

b of Angle 

to c of g. 

Fig. 1. 



Xl 



yi 



Moment 

of 

Inertia /, 

(Fig. 1) 

about 



X 

I 

X 
Ix 



Radius of Gyration r, 



Single Angle, 

(Fig. 1) 
about axis 



Hr, 



2 Angles 
about axis 
(Fig. (Fig. 

2) 3) 



I 2 



*8x3^ 


U 


20.5 


6.02 


*7x3^ 


1 


32.3 


9.50 


* ' * 


H 


30.5 


8.97 


• ' ' 


V, 


28.7 


8.42 


• ' ' 


H 


26.8 


7.87 


• « ' 


H 


24.9 


7.31 


• ' ' 


H 


23.0 


6.75 


• ' ' 


V, 


21.0 


6.17 


• « « 




19.1 


5.59 


• ' ' 


^2 


17.0 


5.00 


• ♦ • 


^ 


15.0 


4.40 


6x4 


1 


30.6 


9.00 




H 


28.9 


8.50 


i « 


1^ 


27.2 


7.99 


' ' 


Tfi 


25.4 


7.47 


' ' 


H 


23.6 


6.94 


' ' 


H 


21.8 


6.41 


' ' 


% 


20.0 


5.86 


( < 




18.1 


5.31 


• ' 


L^ 


16.2 


4.75 


( ( 


T6 


14.3 


4.18 


' ' 


Vs 


12.3 


3.61 


6x3^ 


1 


28.9 


8.50 




^ 


27.3 


8.03 


* ' 


v^ 


25.7 


7.55 


' ' 


■H 


24.0 


7.06 


I > 


¥ 


22.4 


6.56 


♦ ' 


H 


20.6 


6.06 


< < 


^ 


18.9 


5.55 


< < 


1^ 


17.1 


5.03 


( ( 




15.3 


4.50 


< ( 


JL. 


13.5 


3.97 


« « 


^ 


11.7 


3.42 


*5x4 


K 


24.2 


7.11 


* «« 


J-3. 


22.7 


6.65 


• • ' 


¥ 


21.1 


6.19 


* « ' 


ii 


19.5 


5.72 



3.00 



0.75 4.92 



2.17 
2.14 
2.12 
2.10 
2.08 
2.06 
2.03 
2.01 
1.99 
1.96 
1.94 

2.26 
2.24 
2.22 
2.20 
2.18 
2.15 
2.13 
2.11 
2.08 
2.06 
2.04 

1.71 

1.68 
1.66 
1.64 



0.96 
0.94 
0.92 
0.89 
0.87 
0.85 
0.82 
0.80 
0.78 
0.75 

1.17 
1.14 
1.12 
1.10 
1.08 
1.06 
1.03 
1.01 
0.99 
0.96 
0.94 

1.01 
0.99 
0.97 
0.95 
0.93 
0.90 
0.88 
0.86 
0.83 
0.81 
0.79 

1.21 
1.18 
1.16 
1.14 



7.53 
7.18 
6.83 
6.46 
6.08 
5.69 
5.28 
4.86 
4.41 
3.95 

10.75 
10.26 
9.75 
9.23 
8.68 
8.11 
7.52 
6.91 
6.27 
5.60 
4.90 

7.21 
6.88 
6.55 
6.20 
5.84 
5.47 
5.08 
4.67 
4.25 
3.81 
3.34 

9.23 
8.74 
8.23 
7.70 



39.96 

45.37 
43.13 
40.82 
38.45 
35.99 
33.47 
30.86 
28.18 
25.41 
22.56 

30.75 

29.26 
27.73 
26.15 
24.51 
22.82 
21.07 
19.26 
17.40 
15.46 
13.47 

29.24 
27.84 
26.38 
24.89 
23.34 
21.74 
20.08 
18.37 
16.59 
14.76 
12.86 

16.42 
15.54 
14.60 
13.62 



0.90 

0.89 
0.89 
0.90 
0.91 
0.91 
0.92 
0.93 
0.93 
0.94 
0.95 

1.09 
1.10 
1.11 
1.11 
1.12 
1.13 
1.13 
1.14 
1.15 
1.16 
1.17 

0.92 
0.93 
0.93 
0.94 
0.94 
0.95 
0.96 
0.96 
0.97 
0.98 
0.99 

1.14 
1.15 
1.15 
1.16 



2.58 

2.19 
2.19 
2.20 
2.21 
2.22 
2.23 
2.24 
2.25 
2.25 
2.26 

1.85 

1.86 

1.86 

1.87 

1.88 

1.89 

1 

1 

1 

1 



90 
90 
91 
92 
1.93 



1.85 
1.86 
1.87 
1.88 
1.89 
1.89 
1.90 
1.91 
1.92 
1.93 
1.94 



1.52 
1.53 
1.54 
1.54 



0.74 


1.35 


0.88 


1.50 


0.88 


1.49 


0.88 


1.47 


0.88 


1.46 


0.88 


1.44 


0.89 


1.43 


0.89 


1.41 


0.89 


1.40 


0.89 


1.39 


0.89 


1.39 


0.85 


1.79 


0.85 


1.78 


0.86 


1.77 


0.86 


1.75 


0.86 


1.74 


0.86 


1.73 


0.86 


1.72 


0.87 


1.71 


0.87 


1.69 


0.87 


1.68 


0.88 


1.67 


0.74 


1.56 


0.74 


1.55 


0.75 


1.54 


0.75 


1.52 


0.75 


1.51 


0.75 


1.50 


0.75 


1.48 


0.75 


1.47 


0.76 


1.46 


0.76 


1.44 


0.77 


1.43 


0.84 


1.85 


0.84 


1.84 


0.84 


1.83 


0.84 


1.81 



*Camegie special angles 



Ura = minimum radius. 



yrx2+ (3^1 + y) ' tr2=^ry2+ (^i + |) 



Note. 



■rx' 
ry2 



ZxH- A. 
/y-e- A. 



STEEL ANGLES— UNEQUAL LEGS, 



549 



Xo r--t" " ^'-Xo 



-Properties op Angles (Steel). 
Unequal Legs. 
— Continued. 

(Weights at 489.6 lbs. per 
cubic foot.) 



Long 



1 



Legs 



Fig. 2. 



Fig. 3. 



Ins. 



LIL 



Wt. 

per 

Foot. 

W 
Lbs. 



Area 
of 

Sec. 
A 
Sq. 

Ins. 



Dist. from 

h of Angle 

to c of g. 

Fig. L 



Xx 



yi 



Moment 

of 

Inertia 7, 

(Fig. 1) 

about 



X 

I 

X 

Ix 



I 



Radius of Gyration r. 



Single Angle, 

(Fig. 1) 
about axis 



X 

I 

X 

rx 



H^a 



2 Angles 
about axis 
(Fig. (Fig. 

2) 3) 



I ;: 

x& 



16 

il 



17.8 


5.23 


16.2 


4.75 


14.5 


4.25 


12.8 


3.75 


11.0 


3.23 


22.7 


6.67 


21.3 


6.25 


19.8 


5.81 


18.3 


5.37 


16.8 


4.92 


15.2 


4.47 


13.6 


4.00 


12.0 


3.53 


10.4 


3.05 


8.7 


2.56 


19.9 


5.84 


18.5 


5.44 


17.1 


5.03 


15.7 


4.61 


14.3 


4.18 


12.8 


3.75 


11.3 


3.31 


9.8 


2.86 


8.2 


2.40 


18.5 


5.43 


17.3 


5.06 


16.0 


4.68 


14.7 


4.30 


13.3 


3.90 


11.9 


3.50 


10.6 


3.09 


9.1 


2.67 


7.7 


2.25 


18.5 


5.43 


17.3 


5.06 


16.0 


4.68 


14.7 


4.30 



1.62 
1.60 
1.57 
1.55 
1.53 

1.79 
1.77 
1.75 
1.72 



1.86 

1.84 

1.82 

80 



1 

1.77 

1.75 

1.73 

1.70 

1.68 

1.66 
1.63 
1.60 
1.58 
1.56 
1.54 
1.51 
1.49 
1.47 

1.36 
1.34 
1.32 
1.29 



1.12 
1.10 
1.07 
1.05 
1.03 

1.04 
1.02 
1.00 
0.97 
0.95 
0.93 
0.91 
0.88 
0.86 
0.84 

0.86 
0.84 
0.82 
0.80 
0.77 
0.75 
0.73 
0.70 
0.68 

0.90 
0.88 
0.85 
0.83 
0.81 
0.79 
0.76 
0.74 
0.72 

1.11 
1.09 
1.07 
1.04 



7.14 
6.56 
5.96 
5.32 
4.67 

6.21 
5.89 
5.55 
5.20 
4.83 
4.45 
4.05 
3.63 
3.18 
2.72 

3.71 
3.51 
3.29 
3.06 
2.83 
2.58 
2.32 
2.04 
1.75 

3.60 
40 
19 
98 
75 
51 
25 
98 
73 



3. 
3. 
2. 
2. 
2. 
2. 
1. 
1. 

5.49 
5.18 
4.86 
4.52 



12.61 

11.55 

10.46 

9.32 

8.14 

15.67 

14.81 

13.92 

12.99 

12.03 

11.03 

9.99 

8.90 

7.78 

6.60 

13.98 

13.15 

12.28 

11.37 

10.43 

9.45 

8.43 

7.37 

6.26 

10.33 
9.73 
9.10 
8.44 
7.75 
7.04 
6.29 
5.50 
4.69 

7.77 
7.32 
6.86 
6.37 



1.17 


1.55 


1.18 


1.56 


1.18 


1.57 


1.19 


1.58 


1.20 


1.59 


0.96 


1.53 


0.97 


1.54 


0.98 


1.55 


0.98 


1.56 


0.99 


1.56 


1.00 


1.57 


1.01 


1.58 


1.01 


1.59 


1.02 


1.60 


1.03 


1.61 


0.80 


1.55 


0.80 


1.55 


0.81 


1.56 


0.82 


1.57 


0.82 


1.58 


0.83 


1.59 


0.84 


1.60 


0.84 


1.61 


0.85 


1.61 


0.81 


1.38 


0.82 


1.39 


0.83 


1.39 


0.83 


1.40 


0.85 


1.41 


0.85 


1.42 


0.85 


1.43 


0.86 


1.44 


0.88 


1.44 


1.01 


1.19 


1.01 


1.20 


1.02 


1.21 


1.03 


1.22 



0.84 
0.85 
0.85 
0.85 
0.86 

0.75 
0.75 
0.75 
0.75 
0.75 
0.75 
0.75 
0.76 
0.76 
0.76 

0.64 
0.64 
0.64 
0.64 
0.65 
0.65 
0.65 
0.65 
0.66 

0.64 
0.64 
0.64 
0.64 
0.64 
0.65 
0.65 
0.66 
0.66 

0.72 
0.72 
0.72 
0.72 



1.80 
1.79 
1.78 
1.76 
1.75 

1.61 



1.37 
1.36 
1.34 



1.46 
1.44 
1.42 
1.40 
1.38 
1.37 
1.35 
1.33 
1.31 

1.69 
1.68 
1.67 
1.66 



*Camegie special angles. Hr^ = minimum radius. 



550 20.— PROPERTIES AND TABLES OF STEEL SHAPES. 



[LongiL 



X-^' t'tc-.5f6-->^ 



Y 

Fig. 



3. — Properties op Angles (Steel). 

Unequal Legs. 

— Continued. 



-r« > 



Long 



(Weights at 489.6 lbs. 
cubic foot.) 



per 



1 



Fig. 3. 



Legs 



Size. 



Ins. 



Ins. 



LIL 



Wt. 

per 

Foot. 

W 
Lbs. 



Area 

of 
Sec. 

A 

Ins. 



Dist. from 

b of Angle 

to c of g. 

Fig. 1. 



Xl 



yi 



Moment 

of 

Inertia I, 

(Fig. 1) 

about 



I 



Radius of Gyration r. 



Single Angle, 

(Fig. 1) 
about axis 



X 

1 

X 



I 



2 Angles 
about axis 
(Fig. (Fig. 

2) .3) 



o o 



*4x3i 


^ 


13.3 


3.90 


1.27 


1.02 


* ' ' 


V?, 


11.9 


3.50 


1.25 


1.00 


• « < 


^ 


10.6 


3.09 


1.23 


0.98 


• « « 


Vh 


9.1 


2.67 


1.21 


0.96 


• ' » 


^ 


7.7 


2.25 


1.18 


0.93 


4x3 


la 


17.1 


5.03 


1.44 


0.94 


* ' 


/<£ 


16.0 


4.69 


1.42 


0.92 


' ' 


H 


14.8 


4.34 


1.39 


0.89 


' ' 


% 


13.6 


3.98 


1.37 


0.87 


< { 


^ 


12.4 


3.62 


1.35 


0.85 


' ' 




11.1 


3.25 


1.33 


0.83 


* ' 


X. 


9.8 


2.87 


1.30 


0.80 


' • 


H 


8.5 


2.48 


1.28 


0.78 


' ' 


^ 


7.2 


2.09 


1.26 


0.76 


31x3 


H 


15.8 


4.62 


1.23 


0.98 


' ' 


H 


14.7 


4.31 


1.21 


0.96 


' ' 




13.6 


4.00 


1.19 


0.94 


1 ( 


5^ 


12.5 


3.67 


1.17 


0.92 


• • 


A 


11.4 


3.34 


1.15 


0.90 


( < 


L^ 


10.2 


3.00 


1.13 


0.88 


• ' 


O" 


9.1 


2.65 


1.10 


0.85 


< 1 


% 


7.9 


2.30 


1.08 


0.83 


' ' 


^ 


6.6 


1.93 


1.06 


0.81 


3^x2^ 


H 


12.5 


3.65 


1.27 


0.77 




% 


11.5 


3.36 


1.25 


0.75 


1 1 


■h 


10.4 


3.06 


1.23 


0.73 


' ' 




9.4 


2.75 


1.20 


0.70 


( ( 


"lE 


8.3 


2.43 


1.18 


0.68 


' ' 


3/g 


7.2 


2.11 


1.16 


0.66 


' ' 


■A 


6.1 


1.78 


1.14 


0.64 


c < 


M 


4.9 


1.44 


1.11 


0.61 


*3ix2 


■^ 


9.0 


2.64 


1.21 


0.59 


• " 


^2 


8.1 


2.38 


1.19 


0.57 


• ' « 


\E 


7.2 


2.11 


1.17 


0.54 


• ' ' 


Vh 


6.3 


1.83 


1.15 


0.52 


• ' « 


■h 


5.3 


1.54 


1.12 


0.50 


• ' ' 


M 


4.3 


1.25 


1.09 


0.48 



4.17 
3.79 
3.40 
2.99 
2.59 

3.47 
3.28 
3.08 
2.87 
2.66 
2.42 
2.18 
1.92 
1.65 



1.72 
1.61 
1.49 
1.36 
1.23 
1.09 
0.94 
0.78 

0.75 
0.69 
0.62 
0.55 
0.48 
0.40 



5.86 


1.08 


1.23 


0.72 


1.64 


5.32 


1.04 


1.23 


0.72 


1.63 


4.76 


1.05 


1.24 


0.72 


1.62 


4.18 


1.06 


1.25 


0.73 


1.61 


3.56 


1.07 


1.26 


0.73 


1.60 


7.34 


0.^3 


1.21 


0.64 


1.45 


6.93 


0.84 


1.22 


0.64 


1.44 


6.49 


0.84 


1.22 


0.64 


1.43 


6.03 


0.85 


1.23 


0.64 


1.41 


5.55 


0.86 


1.24 


0.64 


1.40 


5.05 


0.86 


1.25 


0.64 


1.39 


4.52 


0.87 


1.25 


0.64 


1.38 


3.96 


0.88 


1.26 


0.64 


1.36 


3.38 


0.89 


1.27 


0.65 


1.35 


4.98 


0.85 


1.04 


0.62 


1.50 


4.70 


0.85 


1.04 


0.62 


1.49 


4.41 


0.86 


1.05 


0.62 


1.48 


4.11 


0.87 


1.06 


0.62 


1.46 


3.79 


0.87 


1.07 


0.62 


1.45 


3.45 


0.88 


1.07 


0.62 


1.44 


3.10 


0.89 


1.08 


0.62 


1.43 


2.72 


0.90 


1.09 


0.62 


1.41 


2.33 


0.90 


1.10 


0.63 


1.40 


4.13 


0.67 


1.06 


0.53 


1.23 


3.85 


0.69 


1.07 


0.53 


1.22 


3.55 


0.70 


1.08 


0.53 


1.20 


3.24 


0.70 


1.09 


0.53 


1.19 


2.91 


0.71 


1.09 


0.54 


1.17 


2.56 


0.72 


1.10 


0.54 


1.16 


2.19 


0.73 


1.11 


0.54 


1.14 


1.80 


0.74 


1.12 


0.54 


1.13 


2.64 


0.53 


1.00 


0.44 


0.99 


2.42 


0.54 


].01 


0.44 


0.98 


2.18 


0.54 


1.02 


0.44 


0.96 


1.92 


0.55 


1.02 


0.44 


0.95 


1.65 


0.56 


1.03 


0.45 


0.93 


1.36 


0.57 


1.04 


0.45 


0.92 



♦Camegie special angles, ^r^ = minimum radius. 



tri = Jrx2 + (:Vi + y) ' trz^Jry^-^ ( Xi + j ) 



2 Note.— fx2 = 7x->. A. 
ry2 = ly ■*• A. 



STEEL ANGLES— UNEQUAL LEGS. 



i51 



n 



Xp^E^^i; 



>^ 3.- 



Fig. 2. 



-Properties of Angles (Steel). 

Unequal Legs. 

— Concluded. 

(Weight at 489.6 lbs. per 
cubic foot.) 



Long 



1 



Fig. 3. 



Legs 





i 






Dist. from 

b of Angle 

to c of g. 

Fig. 1. 


Moment 

of 

Inertia 7, 

(Fig. 1) 

about 


Radius of Gyration r, 


Size. 


LIL 


Single Angle, 

(Fig. 1) 
about axis 


2 Angles 
about axis 
(Fig. (Fig. 

2) 3) 




Wt. 
per 


Area 

of 
Sec. 

A 
Sq. 






XI 


>H 


X 


^ 




ft 






^ 


Foot. 
W 


^1 


yi 


1 
X 


1 


X 


1 


1 


1 u 


^0 


Ins. 


Ins. 


Lbs. 


Ins. 






Ix 


h 


rx 


^y 


^^a 


Ui 


tr2 


3x2| 


^ 


9.5 


2.78 


1.02 


0.77 


1.42 


2.28 


0.72 


0.91 


0.52 


1.25 


1.56 




i 


8.5 


2.50 


1.00 


0.75 


1.30 


2.08 


0.72 


0.91 


0.52 


1.24 


1.55 


< i 


7.6 


2.22 


0.98 


0.73 


1.18 


1.88 


0.73 


0.92 


0.52 


1.23 


1.54 


( ( 


3^ 


6.6 


1.92 


0.96 


0.71 


1.04 


1.66 


0.74 


0.93 


0.52 


1.21 


1.52 


( ( 


A 


5.6 


1.62 


0.93 


0.68 


0.90 


1.42 


0.74 


0.94 


0.52 


1.19 


1.51 


' ' 


M 


4.5 


1.31 


0,91 


0.66 


0.74 


1.17 


0.75 


0.95 


0.53 


1.18 


1.50 


*3x2 


3^ 


7.7 


2.25 


1.08 


0.58 


0.67 


1.92 


0.55 


0.92 


0.43 


1.00 


1.62 


• « ' 


-7 


6.8 


2.00 


1.06 


0.56 


0.61 


1.73 


0.55 


0.93 


0.43 


0.98 


1.60 


• ' « 


^ 


5.9 


1.73 


1.04 


0.54 


0.54 


1.53 


0.56 


0.94 


0.43 


0.96 


1.59 


• < ' 


1^ 


5.0 


1.47 


1.02 


0.52 


0.47 


1.32 


0.57 


0.95 


0.43 


0.95 


1.57 


• ' ' 


K 


4.1 


1.19 


0.99 


0.49 


0.39 


1.09 


0.57 


0.95 


0.43 


0.93 


1.56 


2ix2 


^2 


6.8 


2.00 


0.88 


0.63 


0.64 


1.14 


0.56 


0.75 


0.42 


1.04 


1.35 




7 


6.1 


1.78 


0.85 


0.60 


0.58 


1.03 


0.57 


0.76 


0.42 


1.03 


1.34 


' * 


/4 


5.3 


1.55 


0.83 


0.58 


0.51 


0.91 


0.58 


0.77 


0.42 


1.01 


1.32 


♦ ' 


■^ 


4.5 


1.31 


0.81 


0.56 


0.45 


0.79 


0.58 


0.78 


0.42 


1.00 


1.31 


* ' 


H 


3.7 


1.06 


0.79 


0.54 


0.37 


0.65 


0.59 


0.78 


0.42 


0.98 


1.29 


' ' 


A 


2.8 


0.81 


0.76 


0.51 


0.29 


0.51 


0.60 


0.79 


0.43 


0.97 


1.28 


*2ixU 


3^ 


5.6 


1.63 


0.86 


0.48 


0.26 


0.75 


0.40 


0.68 


0.39 


0.83 


1.30 


• * « 


J^ 


5.0 


1.45 


0.83 


0.46 


0.24 


0.68 


0.41 


0.69 


0.39 


0.82 


1.29 


• ' « 


H 


4.4 


1.27 


0.81 


0.44 


0.21 


0.61 


0.41 


0.69 


0.39 


0.80 


1.27 


• ' « 


A 


3.7 


1.07 


0.79 


0.42 


0.19 


0.53 


0.42 


0.70 


0.40 


0.79 


1.26 


• « ♦ 


M 


3.0 


0.88 


0.77 


0.39 


0.16 


0.44 


0.42 


0.71 


0.40 


0.77 


1.24 


• * « 


^ 


2.3 


0.67 


0.75 


0.37 


0.12 


0.34 


0.43 


0.72 


0.40 


0.76 


1.23 


*2xl| 


M 


2.7 


0.78 


0.69 


0.37 


0.12 


0.37 


0.39 


0.63 


0.30 


0.73 


1.13 


• '« 


^- 


2.1 


0.60 


0.66 


0.35 


0.09 


0.24 


0.40 


0.63 


0.31 


0.72 


1.11 


nf xi 


H 


1.9 


0.53 


0.48 


0.29 


0.04 


0.09 


0.27 


0.41 


0.22 


0.60 


0.84 


• ' ' 


H 


1.0 


0.28 


0.44 


0.26 


0.02 


0.05 


0.29 


0.44 


0.22 


0.59 


0.82 



♦Carnegie special angles. Ura 
tri = ^/rx2+ (yi + ^) ' Xr2 



minimum radius. 



yrx2+ (yi + j)' tr2==^ry2^ (^^"^t) 



2 Note. 



-rx 

ry 



2 = 7x -J- A. 
= 7y + A. 



552 20.— PROPERTIES AND TABLES OF STEEL SHAPES. 



4. — Properties of Angles (Steel). 

Equal Legs. 

(Weights at 489.6 lbs. per cubic foot.) 













"p . 


Radius of 












"2 . 


Radius of 






u 


IL 


J 


it 


Gyration. 






L 


L 


J 




Gyration. 




















"o 


■u 














'o 


•T-J TO 












<j 


^ 


^ ^ 










, 


o 


,6 


^ r. 






o 






CO 

o 

< 


S5 


£5 

-0- 

1- 


i 


►-5 




i 

IS 


4J 


CO 

i 
< 




AS 

. 

O 

^6 




1 




^ 


w 


A 


^ 


Ks" 


v^ 






t^ 


W 


A 


^ 


< 


c 




w 


en 




Sq. 


11^ 


II 






to 


w 




Sq. 


II 


II 


II 








Lbs. 


Ins. 


H 


t-HI 


%- 


Ta 


& 


»— 1 


Lbs. 


Ins. 


« 


Ks- 


vT 


ra 


8^ 


13^ 


56.9 


16.73 


2.41 


97.97 


2.42 


1.55 


4 


H 


9.8 


2.86 


1.14 


4.36 


1.23 


0.79 




1t6 


54.0 


15.87 


2.39 


93.53 


2.43 


1.56 


' ♦ 


A 


8.2 


2.40 


1.12 


3.71 


1.24 


0.79 


*' 


1 


51.0 


15.00 2.37 


88.98 


2.44 


1.56 


















' * 


H 


48.1 


14.12 


2.34 


84.33 


2.44 


1.56 


31 


it 


17.1 


5.03 


1.17 


5.25 


1.02 


0.67 


• ' 


Vs 


45.0 


13.23 


2.32 


79.58 


2.45 


1.57 




% 


16.0 


4.69 


1.15 


4.96 


1.03 


0.67 


' ' 


it 


42.0 


12.34 


2.30 


74.71 


2.46 


1.57 


' ' 


ii 


14.8 


4.34 


1.12 


4.65 


1.04 


0.67 


( ( 


M 


38.9 


11.44 


2.28 


69.74 


2.47 


1.57 


' ' 


H 


13.6 


3.98 


1.10 


4.33 


1.04 


0.67 


" 


ii 


35.8 


10.53 


2.25 


64.64 


2.48 


1.58 


' ' 


■h 


12.4 


3.62 


1.08 


3.99 


1.05 


0.68 


< < 


^ 


32.7 


9.61 


2.23 


59.42 


2.49 


1.58 


' ' 


¥2 


11.1 


3.25 


1.06 


3.64 


1.06 


0.68 


' ' 


^ 


29.6 


8.68 


2.21 


54.09 


2.50 


1.58 


' ' 


^ 


9.8 


2.87 


1.04 


3.26 


1.07 


0.68 


t i 


H 


26.4 


7.75 


2.19 


48.63 


2.50 


1.58 


.. 


i 


8.5 
7.2 


2.48 
2.09 


1.01 
0.99 


2.87 
2.45 


1.07 
1.08 


0.69 
0.69 


6^ 


1 


37.4 


11.00 


1.86 


35.46 


1.80 


1.16 




















H 


35.3 


10.37 


1.84 


33.72 


1.80 


1.16 


3 


Vs 


11.5 


3.36 


0.98 


2.62 


0.88 


0.57 


( < 


K 


33.1 


9.74 


1.82 


31.92 


1.81 


1.17 


' ' 


P 


10.4 


3.06 


0.95 


2.43 


0.89 


0.58 


* ' 


"16 


31.0 


9.09 


1.80 


30.06 


1.82 


1.17 


• ' 




9.4 


2.75 


0.93 


2.22 


0.90 


0.58 


• ' 


M 


28.7 


8.44 


1.78 


28.15 


1.83 


1.17 


' ' 




8.3 


2.43 


0.91 


1.99 


0.91 


0.58 


t ( 


ii 


26.5 


7.78 


1.75 


26.19 


1.83 


1.17 


' ' 


Vs 


7.2 


2.11 


0.89 


1.76 


0.91 


0.58 


♦ ' 


^8 


24.2 


7.11 


1.73 


24.16 


1.84 


1.18 


' ' 


^ 


6.1 


1.78 


0.87 


1.51 


0.92 


0.59 


t ( 




21.9 


6.43 


1.71 


22.07 


1^5 


1.18 


* ' 


H 


4.9 


1.44 


0.84 


1.24 


0.93 


0.59 


I ( 


H 


19.6 


5.75 


1.68 


19.91 


1.86 


1.18 


















• ' 


'TS 


17.2 


5.06 


1.66 


17.68 


1.87 


1.19 


*2| 


y2 


8.5 


2.50 


0.87 


1.67 


0.82 


0.52 


C ( 


Vs 


14.9 


4.36 


1.64 


15.39 


1.88 


1.19 


• « « 


A 


7.6 


2.22 


0.85 


1.51 


0.82 


0.53 


















• ' ' 


^ 


6.6 


1.92 


0.82 


1.33 


0.83 


0.53 


*5 


1 


30.6 


9.00 


1.61 


19.64 


1.48 


0.96 


• « « 


■h 


5.6 


1.62 


0.80 


1.15 


0.84 


0.54 


• « « 


H 


28.9 


8.50 


1.59 


18.71 


1.48 


0.96 


• ' « 


H 


4.5 


1.31 


0.78 


0.93 


0.85 


0.55 


• « ' 


K 


27.2 


7.99 


1.57 


17.75 


1.49 


0.96 


















• ' ' 


16 


25.4 


7.46 


1.55 


16.77 


1.50 


0.97 


21 




7.7 


2.25 


0.81 


1.23 


0.74 


0.47 


• « ' 


M 


23.6 


6.94 


1.52 


15.74 


1.51 


0.97 






6.8 


2.00 


0.78 


1.11 


0.74 


0.48 


• ' • 


i 


21.8 


6.42 


1.50 


14.68 


1.51 


0.97 


' • 


H 


5.9 


1.73 


0.76 


0.98 


0.75 


0.48 


• « ' 


20.0 


5.86 


1.48 


13.58 


1.52 


0.97 


* ' 


^ 


5.0 


1.47 


0.74 


0.85 


0.76 


0.49 


• «' 




18.1 


5.31 


1.46 


12.44 


1.53 


0.98 


' ' 


H 


4.1 


1.19 


0.72 


0.70 


0.77 


0.49 


• * ' 


16.2 


4.75 


1.43 


11.25 


1.54 


0.98 


' • 


3 


3.1 


0.90 


0.69 


0.55 


0.78 


0.49 


• ' ' 


-7 


14.3 


4.18 


1.41 


10.02 


1.55 


0.98 


















• ' ' 


3^ 


12.3 


3.61 


1.39 


8.74 


1.56 


0.99 


*2i 




6.8 


2.00 


0.74 


0.87 


0.66 


0.43 


















• ' ' 




6.1 


1.78 


0.72 


0.79 


0.67 


0.43 


4 


ie 


19.9 


5.84 


1.29 


8.14 


1.18 


0.77 


• ' « 




5.3 


1.55 


0.70 


0.70 


0.67 


0.43 


' ' 


M 


18.5 


5.44 


1.27 


7.67 


1.19 


0.77 


• ' ' 




4.5 


1.31 


0.68 


0.61 


0.68 


0.44 


• ' 


i 


17.1 


5.03 


1.25 


7.17 


1.19 


0.77 


• « ' 




3.7 


1.06 


0.66 


0.51 


0.69 


0.44 


• ' 


15.7 


4.61 


1.23 


6.66 


1.20 


0.77 


• ' ' 




2.8 


0.81 


0.63 


0.39 


0.70 


0.44 


' ' 




14.3 


4.18 


1.21 


6.12 


1.21 


0.78 


















' ' 


3^ 


12.8 


3.75 


1.18 


5.56 


1.22 


0.78 


2^ 


■h 


5.3 


1.56 


0.66 


0.54 


0.59 


0.39 


1 < 


A 


11.3 


3.31 


1.16 


4.97 


1.23 


0.78 




Vs 


4.7 


1.36 


0.64 


0.48 


0.59 


0.39 



* Carnegie Special Angles, 
foot-notes to same. 



Note. — See Figs. 1, 2 and 3, Table 3; also 



STEEL ANGLES— EQUAL LEGS. 



553 



4. — Properties op Angles (Steel). — Concluded. 

Equal Legs. 

(Weights at 489.6 lbs. per cubic foot.) 













P . 


Radius of 












"p . 


Radius of 






L 


L 


J 


la 

"3 § 


Gyration. 






L 


L 


J 

«4-l 




Gyration. 




















o 


'Ss 














O 














cj 


^' 


2? o 












o 


^ 


S^n 






i 


^ 
g 


+3 " 

0) 


CO 

'o 


2" 


55 
o . 


i 




i 


i 




o 


o ^ 


55 

M-l 

o . 


t 






IS 


ft 

1 


< 


5^ 




c 
^ 
O 


i 


J3 


IS 


ft 


< 


•^ o 


O 


.s 


1 




/ 


PF 


A 


^ 


»*h" 


t^ 






/ 


T^ 


A 


^ 


K^" 


v."- 




^ 


M 




Sq. 




11^ 






w 


M 




Sq. 


II 


II 


11 




t— 1 


1— I 


Lbs. 


Ins. 


H 


t*^' 


K 


ra 


t— 1 


& 


Lbs. 


Ins. 


H* 


•^ 


vT 


ra 


2 


^ 


4.0 


1.15 


0.61 


0.42 


0.60 


0.39 


u 


A 


2.4 


0.69 


0.42 


0.09 


0.36 


0.23 


;; 


M 


3.2 


0.94 


0.59 


0.35 


0.61 


0.39 




J^ 


2.0 


0.56 


0.40 


.077 


0.37 


0.24 




A 


2.5 


0.72 


0.57 


0.28 


0.62 


0.40 


<< 


s 


1.5 
1.1 


0.43 
0,30 


0.38 
0.35 


.061 
.044 


0.38 
0.38 


0.24 
0.25 


11 


^ 


4.6 


1.30 


0.59 


0.35 


0.51 


0.33 


















( < 


^ 


4.0 


1.17 


0.57 


0.31 


0.51 


0.34 


1 


M 


1.5 


0.44 


0.34 


.037 


0.29 


0.19 


* ' 


A 


3.4 


1.00 


0.55 


0.27 


0.52 


0.34 


' ' 


A 


1.2 


0.34 


0.32 


.030 


0.30 


0.19 


( ( 


M 


2.8 


0.81 


0.53 


0.23 


0.53 


0.34 


' ' 


3^ 


0.8 


0.24 


0.30 


.022 


0.31 


0.20 


♦' 


^ 


2.2 


0.62 


0.51 


0.18 


0.54 


0.35 


































n 


1^ 


1.0 


0.29 


0.29 


.019 


0.26 


0.18 


It 


t 


3.4 

2.9 


0.99 
0.84 


0.51 
0.49 


0.19 
0.16 


0.44 
0.44 


0.29 
0.29 


• * ' 


H 


0.7 


0.21 


0.26 


.014 


0.26 


0.19 


' ' 


M 


2.4 


0.69 


0.47 


0.14 


0.45 


0.29 


^f 


1^ 


0.9 


0.25 


0.26 


.012 


0.22 


0.16 


( < 




1.8 
1.3 


0.53 
0.36 


0.44 
0.42 


0.11 
0.08 


0.46 
0.46 


0.29 
0.30 




Vs 


0.6 


0.17 


0.23 


.009 


0.23 


0.17 



* Carnegie Special Angles. 
foot-notes to same. 



Note. — See Figs. 1, 2 and 3, Table 3; also 



554 dO.— PROPERTIES AND TABLES OF STEEL SHAPES. 

5. — Properties of I-Beams (Steel). 
(Weights at 489 . 6 lbs. per cubic foot.) 



B 
a 
a> 


I 

1 


] 

1 


Si 


■g 


W) 


nertia Neutral 
pendicular to 
enter. 


nertia Neutral 
ncident with 
ne of Web. 


Gyration Neu- 
Perpendicular 
t Center. 


Gyration Neu- 
Coincident 
er Line of Web. 




1! 




u 




m. ofl 
is Per 
bate 


Mom. of I 

Axis Coi 
Center Li 


diusof 
lAxis 
Web a 


diusof 
1 Axis 
hCent 




^ ■ 


S^' 




1^^ 


c3 c3 ^ 


ce e3.-S 


« 


^ 







H 


? 


I 


r 


r 


r' 



I— I +i MH 



24 


100.00 
95.00 
90.00 
85.00 
80.00 


29.41 
27.94 
26.47 
25.00 
23.32 


0.754 
0.692 
0.631 
0.570 
0.500 


7.254 
7.192 
7.131 
7.070 
7.000 


2380.3 
2309.6 
2239.1 
2168.5 
2087.9 


48.56 
47.10 
45.70 
44.35 
42.86 


9.00 
9.09 
9.20 
9.31 
9.46 


1.28 
1.30 
1.31 
1.33 
1.36 


17.82 
17.99 
18.21 
18.43 
18.72 


20 


100.00 
95.00 
90.00 
85.00 
80.00 


29.41 
27.94 
26.47 
25.00 
23.73 


0.884 
0.810 
0.737 
0.663 
0.600 


7.284 
7.210 
7.137 
7.063 
7.000 


1655.8 
1606.8 
1557.8 
1508.7 
1466.5 


52.65 
50.78 
48.98 
47.25 
45.81 


7.50 

7.58 
7.67 
7.77 
7.86 


1.34 
1.35 
1.36 
1.37 
1.39 


14.76 
14.92 
15.10 
15.30 
15.47 


20 


75.00 
70.00 
65.00 


22.06 
20.59 
19.08 


0.649 
0.575 
0.500 


6.399 
6.325 
6.250 


1268.9 
1219.9 
1169.6 


30.25 
29.04 
27.86 


7.58 
7.70 
7.83 


1.17 
1.19 
1.21 


14.98 
15.21 
15.47 


18 


70.00 
65.00 
60.00 
55.00 


20.59 
19.12 
17.65 
15.93 


0.719 
0.637 
0.555 
0.460 


6.259 
6.177 
6.095 
6.000 


921.3 
881.5 
841.8 
795.6 


24.62 
23.47 
22.38 
21.19 


6.69 
6.79 
6.91 
7.07 


1.09 
1.11 
1.13 
1.15 


13.20 
13.40 
13.63 
13.95 


15 


100.00 
95.00 
90.00 
85.00 
80.00 


29.41 
27.94 
26.47 
25.00 
23.81 


1.184 
1.085 
0.987 
0.889 
0.810 


6.774 
6.675 
6.577 
6.479 
6.400 


900.5 

872.9 
845.4 
817.8 
795.5 


50.98 
48.37 
45.91 
43.57 
41.76 


5.53 
5.59 
5.65 
5.72 
5.78 


1.31 
1.32 
1.32 
1.32 
1.32 


10.75 
10.86 
10.99 
11.13 
11.25 


15 


75.00 
70.00 
65.00 
60.00 


22.06 
20.59 
19.12 
17.67 


0.882 
0.784 
0.686 
0.590 


6.292 
6.194 
6.096 
6.000 


691.2 
663.6 
636.0 
609.0 


30.68 
29.00 
27.42 
25.96 


5.60 

5.68 
5.77 
5.87 


1.18 
1.19 
1.20 
1.21 


10.95 
11.11 
11.29 
11.49 


15 


55.00 
50.00 
45.00 
42.00 


16.18 
14.71 
13.24 
12.48 


0.656 
0.558 
0.460 
0.410 


5.746 
5.648 
5.550 
5.500 


511.0 
483.4 
455.8 
441.7 


17.06 
16.04 
15.00 
14.62 


5.62 
5.73 
5.87 
5.95 


1.02 
1.04 
1.07 
1.08 


11.05 
11.27 
11.54 
11.70 


12 


55.00 
50.00 
45.00 
40.00 


16.18 
14.71 
13.24 
11.84 


0.822 
0.699 
0.576 
0.460 


5.612 
5.489 
5.366 
5.250 


321.0 
303.3 
2«5.7 
268.9 


17.46 
16.12 
14.89 
13.81 


4.45 
4.54 
4.65 
4.77 


1.04 
1.05 
1.06 
1.08 


8.65 
8.83 
9.06 
9.29 


12 


35.00 
31.50 


10.29 
9.26 


0.436 
0.350 


5.086 
5.000 


228.3 
215.8 


10.07 
9.50 


4.71 
4.83 


0.99 
1.01 


9.21 
9.45 


10 


40.00 
35.00 


11.76 
10.29 


0.749 
0.602 


5.099 
4.952 


158.7 
146.4 


9.50 

8.52 


3.67 
3.77 


0.90 
0.91 


7.12 
7.32 



Note. — Weights in heavy type are standard; others are special. 



STEEL I-BEAMS. 



555 



-Properties of I-Beams (Steel). — Concluded. 
(Weights at 489.6 lbs. per cubic foot.) 



, 


I 

4i 

1 


it 


4 




nertia Neutral 
pendicular to 
enter. 


nertia Neutral 
ncident with 
ne of Web. 


Gyration Neu- 
Perpendicular 
,t Center. 


Gyration Neu- 
Coincident 
er Line of Web. 


[Stance Center to Cen- 
r Required to make 
idii of Gyration equal. 




2| 


o 1 


o 

1" 




1-1 t^ o 


?63 

o ^ g 


■S.2^ 


'3 — -S 


43 c 






S5 


if a 
-dtH 




03 cS ^ 


c3 C3.-S 


"T" "T" 


^ 


^ 


< 


H 


^ 


^ 


If 


r 


r' 


H 


10 


30.00 


8.82 


0.455 


4.805 


134.2 


7.65 


3.90 


0.93 


7.57 




25.00 


7.37 


0.310 


4.660 


122.1 


6.89 


4.07 


0.97 


7.91 




35.00 


10.29 


0.732 


4.772 


111.8 


7.31 


3.29 


0.84 


6.36 


9 


30.00 


8.82 


0.569 


4.609 


101.9 


6.42 


3.40 


0.85 


7.58 


25.00 


7.35 


0.406 


4.446 


91.9 


5.65 


3.54 


0.88 


6.86 




21.00 


6.31 


0.290 


4.330 


84.9 


5.16 


3.67 


0.90 


7.12 




25.50 


7.50 


0.541 


4.271 


68.4 


4.75 


3.02 


0.80 


5.82 


8 


23.00 


6.76 


0.449 


4.179 


64.5 


4.39 


3.09 


0.81 


5.96 


20.50 


6.03 


0.357 


4.087 


60.6 


4.07 


3.17 


0.82 


6.12 




18.00 


5.33 


0.270 


4.000 


56.9 


3.78 


3.27 


0.84 


6.32 




20.00 


5.88 


0.458 


3.868 


42.2 


3.24 


2.68 


0.74 


5.15 


7 


17.50 


5.15 


0.353 


3.763 


39.2 


2.94 


2.76 


0.76 


5.31 




15.00 


4.42 


0.250 


3.660 


36.2 


2.67 


2.86 


0.78 


5.50 




17.25 


5.07 


0.475 


3.575 


26.2 


2.36 


2.27 


0.68 


4.33 


6 


14.75 


4.34 


0.352 


3.452 


24.0 


2.09 


2.35 


0.69 


4.49 




12.25 


3.61 


0.230 


3.330 


21.8 


1.85 


2.46 


0.72 


4.70 




14.75 


4.34 


0.504 


3.294 


15.2 


1.70 


1.87 


0.63 




5 


12.25 


3.60 


0.357 


3.147 


13.6 


1.45 


1.94 


0.63 






9.75 


2.87 


0.210 


3.000 


12.1 


1.23 


2.05 


0.65 






10.50 


3.09 


0.410 


2.880 


7.1 


1.01 


1.52 


0.57 




4 


9.50 


2.79 


0.337 


2.807 


6.7 


0.93 


1.55 


0.58 




8.50 


2.50 


0.263 


2.733 


6.4 


0.85 


1.59 


0.58 






7.50 


2.21 


0.190 


2.660 


6.0 


0.77 


1.64 


0.59 






7.50 


2.21 


0.361 


2.521 


2.9 


0.60 


1.15 


0.52 




3 


6.50 


1.91 


0.263 


2.423 


2.7 


0.53 


1.19 


0.52 






5.50 


1.63 


0.170 


2.330 


2.5 


0.46 


1.23 


0.53 





Note. — ^Weights in heavy type are standard; others are special. 
For Bethlehem girder (single I) beams and Bethlehem special I-beams, 
see Tables 8 and 9, Sec. 31, pages 583 and 584. 



556 m.— PROPERTIES AND TABLES OF STEEL SHAPES. 

6. — Properties of Channels (Steel). 
(Weights at 489.6 lbs. per cubic foot.) 



"1 


== 








•§3 


J 


, 










4^ 


> 


■s 


6 


ce-2 t: 


o 




"a 


03 


nerti 
ente 


si 


O 3 


o 

is 




m.of I 
IS Per 
b at C 


,fi; 


i^ 




S5 


14^ 


^ 


< 




5 


I 






1^ 



5 


Sk 




|§ 


^'^ 


lo 


t-i o 








2?> 


ft as 


i3Q 


•1 r* 


r' 


>l 


.868 


8.53 


.873 


8.71 


.882 


8.92 


.893 


9.15 


.908 


9.43 


.912 


9.50 


.751 


6.60 


.757 


6.81 


.768 


7.07 


.785 


7.36 


.805 


7.67 


.672 


5.17 


.672 


5.40 


.680 


5.67 


.696 


5.97 


.718 


6.33 


.637 


4.84 


.646 


5.12 


.665 


5.49 


.674 


5.63 


.600 


4.23 


.603 


4.38 


.610 


4.54 


.619 


4.72 


.630 


4.94 


.565 


3.48 


.564 


3.64 


.568 


3.80 


.5'^5 


3.99 


.586 


4.22 


.529 


2.91 


.529 


3.09 


.534 


3.28 


.542 


3.52 


.493 


2.34 


.493 


2.56 


.498 


2.79 


.455 


1.85 


.454 


1.96 


.453 


2.06 


.421 


1.07 


.415 


1.19 


.409 


1.31 



55.00 


16.18 


0.818 


3.818 


430.2 


50.00 


14.71 


0.720 


3.720 


402.7 


45.00 


13.24 


0.622 


3.62? 


375.1 


40.00 


11.76 


0.524 


3.524 


347.5 


35.00 


10.29 


0.426 


3.426 


320.0 


33.00 


9.90 


0.400 


3.400 


312.6 


40.00 


11.76 


0.758 


3.418 


197.0 


35.00 


10.29 


0.636 


3.296 


179.3 


30.00 


8.82 


0.513 


3.173 


161.7 


25.00 


7.35 


0.390 


3.050 


144.0 


20.50 


6.03 


0.280 


2.940 


128.1 


35.00 


10.29 


0.823 


3.183 


115.5 


30.00 


8.82 


0.676 


3.036 


103.2 


25.00 


7.35 


0.529 


2.889 


91.0 


20.00 


5.88 


0.382 


2.742 


78.7 


15.00 


4.46 


0.240 


2.600 


66.9 


25.00 


7.35 


0.615 


2.815 


70.7 


20.00 


5.88 


0.452 


2.652 


60.8 


15.00 


4.41 


0.288 


2.488 


50.9 


13.25 


3.89 


0.230 


2.430 


47.3 


21.25 


6.25 


0.582 


2.622 


47.8 


18.75 


5.51 


0.490 


2.530 


43.8 


16.25 


4.78 


0.399 


2.439 


39.9 


13.75 


4.04 


0.307 


2.347 


36.0 


11.25 


3.35 


0.220 


2.260 


32.3 


19.75 


5.81 


0.633 


2.513 


33.2 


17.25 


5.07 


0.528 


2.408 


30.2 


14.75 


4.34 


0.423 


2.303 


27.2 


12.25 


3.60 


0.318 


2.198 


24.2 


9.75 


2.85 


0.210 


2.090 


21.1 


15.50 


4.56 


0.563 


2.283 


19.5 


13.00 


3.82 


0.440 


2.160 


17.3 


10.50 


3.09 


0.318 


2.038 


15.1 


8.00 


2.38 


0.200 


1.920 


13.0 


11.50 


3.38 


0.477 


2.037 


10.4 


9.00 


, 2.65 


0.330 


1.890 


8.9 


6.50 


1.95 


0.190 


1.750 


7.4 


7.25 


2.13 


0.325 


1.725 


4.6 


6.25 


1.84 


0.252 


1.652 


4.2 


5.25 


1.55 


0.180 


1.580 


3.8 


6.00 


1.76 


0.362 


1.602 


2.1 


5.00 


1.47 


0.264 


1.504 


1.8 


4.00 


1.19 


0.170 


1.410 


1.6 



12.19 

11.22 

10.29 

9.39 

8.48 

8.23 

6.63 
5.90 
5.21 
4.53 
3.91 



45 
95 
1.77 

2.25 
2.01 
1.78 
1.55 
1.33 

1.85 
1.62 
1.40 
1.19 
0.98 

1.28 
1.07 

0.70 

0.82 
0.64 
0.48 

0.44 
0.38 
0.32 

0.31 
0.25 
0.20 



5.16 
5.23 
5.32 
5.43 
5.58 
5.62 

4.09 
4.17 
4.28 
4.43 
4.61 

3.35 
3.42 
3.52 
3.66 
3.87 

3.10 
3.21 
3.40 
3.49 

2.77 
2.82 
2.89 
2.98 
3.11 

2.39 
2.44 
2.50 
2.59 
2.72 

2.07 
2.13 
2.21 
2.34 

1.75 
1.83 
1.95 



.46 
.51 
56 

.08 
.12 



1.17 



CHANNELS. Z-BARS, 



557 



7. — ^Properties op Z-Bars (Steel). 
(Weights at 489 . 6 lbs. per cubic foot.) 









1 


1 


Moment of 
Inertia. 


Radii 


of Gyration. 

r 




^ rO 


r^ ^ 


Xi ^ 


-^ r^• 


'cS 












bo <u 


bO r^ 


bfl (U 


bO rQ 


u 






13 




t/3 


s-^ 




s-^ 


iS 


•+J 


1. 


! 


1^ 


o 


ot— 1 


^1^ 

^ ot3 

g »H C 


,— 1 "^ 


rj -M 

m 

h <i> 9^ 


T-1 <1^ 

^11 


II 


p 


g 




^ 


u 

< 


0^ S CD 


^ S 




^68 




6 


33^ 


J^ 


15.6 


4.59 


25.32 


9.11 


2.35 


1.41 


0.83 


6i^ 


3i^ 
3^ 


^ff 


18.3 


5.39 


29.80 


10.95 


2.35 


1.43 


0.84 


m 


^ 


21.0 


6.19 


34.36 


12.87 


2.36 


1.44 


0.84 


6 


SH 


S 


22.7 


6.68 


34.64 


12.59 


2.28 


1.37 


0.81 


Q^ 


3A 


25.4 


7.46 


38.86 


14.42 


2.28 


1.39 


0.82 


6H 


3^ 


ii 


28.0 


8.25 


43.18 


16.34 


2.29 


1.41 


0.84 


6 


3H 


M 


29.3 


8.63 


42.12 


15.44 


2.21 


1.34 


0.81 


6j2g- 


3A 


if 


31.9 


9.40 


46.13 


17.27 


2.22 


1.36 


0.82 


m 


3^ 


Vs 


34.6 


10.17 


50.22 


19.18 


2.22 


1.37 


0.83 


5 


3M 


A 


11.6 


3.40 


13.36 


6.18 


1.98 


1.35 


0.75 


5t^ 


3A 


^ 


13.9 


4.10 


16.18 


7.65 


1.99 


1.37 


0.76 


SVs 


3^ 


^ 


16.4 


4.81 


19.07 


9.20 


1.99 


1.38 


0.77 


5 


33i 


^ 


17.9 


5.25 


19.19 


9.05 


1.91 


1.31 


0.74 


5i'5' 


3t^ 


^ 


20.2 


5.94 


21.83 


10.51 


1.91 


1.33 


0.75 


6H 


SH 


^ 


22.6 


6.64 


24.53 


12.06 


1.92 


1.35 


0.76 


5 


3A 


i* 


23.7 


6.96 


23.68 


11.37 


1.84 


1.28 


0.73 


5^ 


M 


26.0 


7.64 


26.16 


12.83 


1.85 


1.30 


0.75 


63^ 


3^ 


if 


28.3 


8.33 


28.70 


14.36 


1.86 


1.31 


0.76 




3i^ 


¥ 


8.2 


2.41 


6.28 


4.23 


1.62 


1.33 


0.67 


4i^ 


3>g 




10.3 


3.03 


7.94 


5.46 


1.62 


1.34 


0.68 


43^ 


3A 


^ 


12.4 


3.66 


9.63 


6.77 


1.62 


1.36 


0.69 




3i^ 


A 


13.8 


4.05 


9.66 


6.73 


1.55 


1.29 


0.66 


4i^ 


3H 


H 


15.8 


4.66 


11.18 


7.96 


1.55 


1.31 


0.67 


iVs 


3i^ 


A 


17.9 


5.27 


12.74 


9.26 


1.55 


1.33 


0.69 




3t^ 


ys 


18.9 


5.55 


12.11 


8.73 


1.48 


1.25 


0.66 


4i^ 


33^ 


H 


20.9 


6.14 


13.52 


'9.95 


1.48 


1.27 


0.67 


43^ 


3i^ 


H 


23.0 


6.75 


14.97 


11.24 


1.49 


1.29 


0.69 


3 


2H 


K 


6.7 


1.97 


2.87 


2.81 


1.21 


1.19 


0.55 


3fjr 


2M 


A 


8.4 


2.48 


3.64 


3.64 


1.21 


1.21 


0.56 


3 


2tt 


Vs 


9.7 


2.86 


3.85 


3.92 


1.16 


1.17 


0.55 


3fj 


2M 


A 


11.4 


3.36 


4.57 


4.75 


1.17 


1.19 


0.56 


3 


2H 


3^ 


12.5 


3.69 


4.59 


4.85 


1.12 


1.15 


0.55 


3i^ 


2M 


A 


14.2 


4.18 


5.26 


5.70 


1.12 


1.17 


0.56 



558 m.— PROPERTIES AND TABLES OF STEEL SHAPES. 



-Properties op Carnegie T-Shapes (Steel). 
(Weights at 489 . 6 lbs. per cubic foot.) 



Size, 
Flange 

by 
Stem. 
Inches. 


T 

Weight 
per Foot 
Pounds. 


T 

Area 

of 

Section. 

Square 

Inches. 


Distance 

of 
Center of 
Gravity 

from 
Outside 

of 
Flange. 
Inches. 


Moment of 

Inertia, 
Neutral Axis 
through 
Center of 
Gravity 
Parallel to 
Flange. 


Radius 

of 

Gyration, 

Neutral 

Axis 

as 
before. 


Moment of 
Inertia, 

Neutral Axis 

through 

Center of 

Gravity 

Coincident 

with Center 

Line of 

Stem. 


Radius 

of 

Gyration, 

Neutral 

Axis 

as 

before. 










I 


r 


F 


r' 


5 x3 


13.6 


3.99 


0.75 


2.6 


0.82 


5.6 


1.19 


5x2^ 


11.0 


3.24 


0.65 


1.6 


0.71 


4.3 


1.16 


4|x3| 


15.9 


4.65 


1.11 


5.1 


1.04 


3.7 


0.90 


4^x3 
4ix3 


8.6 
10.0 


2.55 
3.00 


0.73 
0.75 


1.8 
2.1 


0.87 
0.86 


2.6 
3.1 


1.03 
1.04 


4ix2| 
42x2| 


8.0 
9.3 


2.40 
2,79 


0.58 
0.60 


1.1 
1.2 


0.69 
0.68 


2.6 
• 3.1 


1.07 
1.08 


4 x5 
4 x5 


15.7 
12.3 


4.56 
3.54 


1.56 
1.51 


10.7 

8.5 


1.54 
1.56 


2.8 
2.1 


0.79 
0.78 


4 x4i 
4 x4i 


14.8 
11.6 


4.29 
3».36 


1.37 
1.31 


8.0 
6.3 


1.37 
1.38 


2.8 
2.1 


0.81 
0.80 


4 x4 
4 x4 


13.9 
10.9 


4.02 
3.21 


1.18 
1.15 


5.7 

4.7 


1.20 
1.23 


2.8 
2.2 


0.84 
0.84 


4 x3 


9.3 


2.73 


0.78 


2.0 


0.86 


2.1 


0.88 


4 x2| 
4 x2| 


8.7 
7.4 


2.52 
2.16 


0.63 
0.60 


1.2 
1.0 


0.69 
0.70 


2.1 

1.8 


0.92 
0.91 


4 x2 
4 x2 


7.9 
6.7 


2.31 
1.95 


0.48 
0.51 


0.6 
0.54 


0.52 
0.51 


2.1 
1.8 


0.96 
0.95 


31x4 
3^x4 


12.8 
10.0 


3.75 
2.91 


1.25 
1.19 


5.5 

4.3 


1.21 
1.22 


1.89 
1.42 


0.72 
0.70 


3^x3^ 
3ix3i 


11.9 
9.3 


3.45 
2.70 


1.06 
1.01 


3.7 
3.0 


1.04 
1.05 


1.89 
1.42 


0.74 
0.73 


3^x3 
3ix3 
3ix3 


11.0 

8.7 
7.7 


3.21 
2.49 
2.28 


0.88 
0.83 
0.78 


2.4 
1.9 
1.6 


0.87 
0.88 
0.89 


1.88 
1.41 
1.18 


0.77 
0.75 
0.76 


3 x4 
3 x4 
3 x4 


11.9 

10.6 

9.3 


4.38 
3.12 
2.73 


1.32 
1.32 
1.29 


5.2 

4.8 
4.3 


1.23 
1.25 
1.26 


1.21 
1.09 
0.93 


0.59 
0.60 
0.59 


3 x3i 
3 x3i 
3 x3i 


11.0 

9.8 
8.6 


3.21 
2.88 
2.49 


1.12 
1.11 
1.09 


3.5 
3.3 

2.9 


1.06 
1.08 . 
1.09 


1.20 
1.31 
0.93 


0.62 
0.68 
0.61 



CARNEGIE T-SHAPES. 



559 



8. — Properties of Carnegie T-Shapes (Steel). — Concluded. 
(Weights at 489.6 lbs. per cubic foot.) 



Size. 
Flange 

by 
Stem. 
Inches. 


T 

Weight 
per Foot 
Pounds. 


T 

Area 

of 

Section. 

Square 

Inches. 


Distance 

of 

Center of 

Gravity 

from 
Outside 

of 
Flange. 
Inches. 


Moment of 

Inertia, 
Neutral Axis 

through 
Center of 

Gravity 
Parallel to 

Flange. 


Radius 

of 

Gyration. 

Neutral 

Axis 

as 
before. 


Moment of 

Inertia, 
Neutral Axis 
through 
Center of 
Gravity- 
Coincident 
with Center 
Line of 
Stem. 


Radius 

of 

Gyration, 

Neutral 

Axis 

as 

before. 










I 


r 




r' 


3 x3 
3 x3 
3 x3 
3 x3 


10.1 
9.0 

7.9 
6.8 


2.94 
2.67 
2.28 
1.95 


0.93 
0.92 

0.88 
0.86 


2.3 
2.1 
1.8 
1.6 


0.88 
0.90 
0.90 
0.90 


1.20 
1.08 
0.90 
0.75 


0.64 
0.64 
0.63 
0.62 


3 x2i 
3 x2i 


7.2 
6.2 


2.10 
1.80 


0.71 
0.68 


1.1 
0.94 


0.72 
0.73 


0.89 
0.75 


0.66 
0.65 


2fx2 


7.4 


2.16 


0.53 


1.1 


0.71 


0.62 


0.54 


24x3 
2ix3 


7.2 
6.2 


2.10 
1.80 


0.97 
0.92 


1.8 
1.6 


0.92 
0.94 


0.54 
0.44 


0.51 
0.51 


2ix2f 
2ix2i 


6.8 
5.9 


1.98 
1.71 


0.87 
0.83 


1.4 
1.2 


0.84 
0.83 


0.66 
0.44 


0.58 
0.51 


2ix2i 
2ix2i 


6.5 
5.6 


1.89 
1.62 


0.76 
0.74 


1.0 
0.87 


0.74 
0.74 


0.52 
0.44 


0.53 
0.52 


2ixU 


3.0 


0.84 


0.29 


0.094 


0.31 


0.29 


0.58 


2ix2i 
2ix2i 


5.0 

4.2 


1.44 
1.20 


0.69 
0.66 


0.66 
0.51 


0.68 
0.67 


0.33 
0.25 


0.48 
0.47 


2 x2 
2 x2 


4.4 
3.7 


1.26 
1.08 


0.63 
0.59 


0.45 
0.36 


0.60 
0.60 


0.23 
0.18 


0.43 
0.42 


2 xU 


3.2 


0.90 


0.42 


0.16 


0.42 


0.18 


0.45 • 


lixlf 


3.2 


0.90 


0.54 


0.23 


0.51 


0.12 


0.37 


HxU 
lixU 


2.6 
2.0 


0.75 
0.54 


0.42 
0.44 


0.15 
0.11 


0.49 
0.45 


0.08 
0.06 


0.34 
0.31 


UxU 
Uxli 


2.1 
1.7 


0.60 
0.45 


0.40 
0.38 


0.08 
0.06 


0.36 
0.37 


0.05 
0.03 


0.27 
0.26 


1 xl 
1 xl 


1.3 
1.0 


0.36 
0.26 


0.32 
0.29 


0.03 
0.02 


0.29 
0.29 


0.02 
0.01 


0.21 
0.21 



560 20.— PROPERTIES AND TABLES OF STEEL SHAPES. 



9. — * Standard Steel Rail Sections. 
(Cambria Steel Co.). 




Weight 


Area. 


b 


d 


w 


t 


yi 


AxisX-X 


per 
Yard. 


Moment 
of Inertia 


Section 
Modulus 


Pounds. 


Sq. Ins. 


Inches 


Inches 


Inches 


Inch 


Inches 


Ix 


S 


8 


0.78 


iy2 


IM 


il 


^2 


0.75 


0.23 


0.31 


12 


1.18 


VA 


VA 


1^ 


^ 


0.92 


0.55 


0.58 


16 


1.57 


2H 


V/4. 


m 


II 


1.1 


1.1 


0.95 


20 


2.00 


2y2 


2y2 


Ws 


il 


1.2 


1.7 


1.3 


25 


2.5 


2% 


2% 


VA 


n 


1.3 


2.6 


1.8 


30 


2.9 


3 


3 


m 


21 
64 


1.4 


3.6 


2.3 


35 


3.4 


m 


. m 


VA 


II 


1.6 


4.9 


2.9 


40 


3.9 


3H 


m 


VA 


II 


1.7 


6.6 


3.6 


45 


4.4 


31i 


3H 


2 


15 


1.8 


8.1 


4.2 


50 


4.9 


m 


m 


2A 


fe 


1.9 


9.8 


4.9 


55 


5.4 


4i^ 


4^ 


2K 


M 


2.0 


12.2 


5.9 


60 


5.9 


4^ 


^K 


2A 


M 


2.1 


14.7 


6.7 


65 


6.4 


4^ 


4i^ 


2il 


H 


2.2 


17.0 


7.4 


70 


6.9 


4^ 


m 


2^ 


if 


2.2 


20.0 


8.4 


75 


7.4 


m 


m 


211 


H 


2.3 


23.0 


9.1 


80 


7.8 


5 


5 


23^ 


If 


2.4 


26.7 


10.1 


85 


8.3 


6^ 


5^ 


2A 


^ 


2.5 


30.5 


11.2 


90 


8.8 


hVs 


m 


2ys 


A 


2.6 


34.4 


12.3 


95 


9.3 


5^ 


5A 


2ii 


A 


2.7 


38.6 


13.3 


100 


9.8 


m 


hK 


2H 


^ 


2.8 


43.4 


14.7 


150 


14.7 


6 


6 


iH 


1 


3.0 


69.3 


23.1 



* Sections at 40 to 100 lbs. per yard are Am. Soc. C. 
Note. — See page 541, for reference to Tables in Sec. 5! 



E. Standard. 



STANDARD STEEL RAIL SECTIONS. 561 

For Table of 
Bethlehem H-Shapes (Steel) 
See Table 14, Sec. 32. Columns, page 608. 

EXCERPTS AND REFERENCES. 

The Strength of 'Corrugated Sheeting (By G. H. Blakely. Eng. News, 
Dec. 12, 1907). — Discussion of formulas. 

READERS' MEMORANDA. 

The following skeleton arrangement is for the use of the reader in making 
memoranda of tables and items of special interest which may be found in 
catalogs and other works and to which reference may be convenient. 

Standard Specifications for Reinforced-Concrete Bars is printed in a 
little pamphlet published by the Steel Manufacturers' Association and may 
be procured free upon application. 

Standard Steel Shapes. 

Page 
Page 
Page 
Page 
Page 
Special Steel Shapes. 

Page 
Page 
Page 
Page 
Page 
Plain Reinforced-Concrete Bars. 

Page 
Page 
Page 
Special Reinforced-Concrete Bars. 

Page 
Page 
Page 
Page 
Page 

Miscellaneous, 

Page 

Page 



1. 


See 


2. 


See 


3. 


See 


4. 


See 


5. 


See 


6. 


See 


7. 


See 


8. 


See 


9. 


See 


10. 


See 


11. 


See 


12. 


See 


13. 


See 


14. 


See 


15. 


See 


16. 


See 


17. 


See 


18. 


See 


19. 


See 


20. 


See 







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lit ^ 


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q" 


c 


Q 


o 


a "i 


^ 


»J 


i-^ ^ 


<1 


^ 

H 


a (u ..0) 


M 


^a <uc«a 


1— ( 


Pi 
O 

1 


rH c o"' hzs 


o 


1. 


1— 1 


T-J 


•^ § w m 0-43 ;3.y.j5 C 


CO 




fill gillie 

fe^aaSi-xi^'Sati 










tf -i^ C rt w s^ 






g^aa^a^aa 


^ 



**^ki^*** ^ ^I-t^*^'^ 



s 


1 




X II 


g 


1 


c^ 


t3 


s 


a; 

8 



CO 



it a. 

II g 



<M to 



CO 



00 



bj s. 






tt5 






^ 


?s 


?% 


taj^l 


i^:;t 


t^:;t 


C<1 


CO 


-* 



S 









5| 



00 



^:;: 



I 



O) 



il I 

II I 



II 

*l rr^l 

II II 



£2» ijq 






»o 



^ 



562 



BEAM FORMULAS. 



563 











kJ 


.^S 


^.-J 


>.Si 


on 


i^:;: 


t^::? 


"^ 


CQ 


l-H 


y-i 





1 


II II 


S 


CO 









§£ 


3S: 


S^ 


C^l 


CO 


<M 


O lO 



^3 «) 



1^ 



^ iO 



c 


., 


^ 


c» 


»^ 


-r^ -^ 


•r,'^ 


-r^^ 


^ 


'^E 


C<J 


'^i 



II II 



II II 



« 


CO CO 


CO 


Kj 


J^^Kj 


5?,^ 


<M 


-O <M ^ (M 

^ g tq ^ 


-^ o 


53 


^^ 


CO 


(M 





?\ 


?S 


5fe 


-J^ 


^ K, 


(M '+^ 


CO '+^ 


CO (M 


'-' 









1 1 


1 


1 


n 


« 


m 


« 


. -4 


^^ 


kJ 


k4 


2 ^ 

.1. CD 


5^ 




s^ 


J=. t- 


r-t 


CO 


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V "^ 


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e^ 


c» 


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CO tS. 




jS. 


CD -O 





1 




II 


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^2?. 


^ ^ 


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g-t- 


^^ 


S ^ 



II 


1 


1 1 


CO 


e>9 


eo 


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kJ 


^ 


K^ 


kJ 


^ 


S^ 


S^ 


2^ 


T— 1 


^ g 


^g^ 


t^g 


Oi 


lO 


rtl 


<M 


CD 


C^ 







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►5i 



It) 



.5^ 



^"^ 





I 


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1 1 


M 


*j 


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N 


kJ 


^ kJ 


^ *-] 


.^ "-J 


^ 


5^ 


:^^ 


3^ 


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CD 


l> 


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t3 

CO 



2^ 

^co 






g^ 



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3:^ 



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1 


1 


1 1 






i-l 


r. ^ 


rr, Kj 


rn "-J 


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2fe 


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1^ on 


ti^ <M 


CD 




o 


—1 JPJ 


UO 


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rH 





f 











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Q ^ 


M-* 


*+^ 


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t:^ l:^ i::^ "-^ 
^ fe ^ ^ 

Cq CD CO CO 



^§ ^§ ^ ^ I 



II II II 



I II II n II 

. :^ § § § 






to 
8 



k k 






I. iiJ 






2- 






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b^ 



^1 









ll^ 



<N 



II II 






CO <^' 





M 


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94 


N t3 


-S 


'T^ 


-^ -ra ^ ^ 


^O 


CD .O CO -K. 


CO M^ 


't^ 


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"Tt* 



^ 


•-^ s^ ^ ^ 


-^^ 


►o 


Cq rO CO ^0 


oo .o 


«+^ 


t^ M~. CO H^ 


T-l «+^ 



564 31— PROPERTIES AND TABLES OF BEAMS AND GIRDERS, 



Examples in the Use of Table 1. 

Ex. 1. (Cases A and B combined). — A 15-in. steel I-beam weighing 60 lbs. 
per ft., and 10 ft. long, is fixed at one end and loaded with 10000 lbs. at 
the other. Find the max. fiber stress /, max. moment M'\ max. slope i, and 
max. deflection v ? 

Solution. — From "Properties of I-Beams," Table 5, page 554, /==609. 
15 
Also, £ = 29 000 000; y=ir- Then, from preceding table. 



Case Ab 
(W= 10000) 



Case Bb 
{W= 600) 



Totals 



/ 



(3) 



(3) 



\f- 



12 WLy 



If- 
1 = 



I 

14778 
QWLy 



I 
443 



/ = 15221 

lbs. per sq. in. 



(3) 



(3) 



= 12 WL 
200 000 
M''= QWL 
= 36000 



I =1: 



M" 



12 WU 



(4) 



(4) 



u 
u 



EI 

.00408 
2i WL^ 



EI 
.00008 



.00416 




z;= 0.333 in. 



'=1236 000 
in.-lbs. 

Ex. 2. (Cases C and D combined). — A wooden beam 10" x 16", and 15-ft. 
span, is loaded at the middle so as to produce a total max. outer fiber stress 
/=1000 lbs. per sq. in. Assuming the weight of the beam at 4 lbs. per ft. 
B. M., find the superimposed load, max. moment M'\ max. slope i, and 
max. deflection v? 

Solution. — ^Weight of beam = 800 lbs. Assume E=l 400 000. 

M" 



Case Dd 
(W= 800) 



Case Cd 
(/= 1000-42) 



Totals 



(3)[ = 
(3)!^^ = 



QWL 



bd^ 

42 

fbd^ 



18L 
9083 



(3) 



(3) 



IM" 



iir = 



SWL 
2 

18000 

3WL 

408735 



(4)< 



(4) 



_8fL 
Ed 
[ = .00022 

r. ^fL 

}'='-Ed 
[ = .00385 



i = mm 



(5)< 



(5)< 



V 

30/L2 
Ed 

■- .013 
24fL2 
Ed 
- .230 



z;= 0.243 in. 



M" ;= 426735 
in.-lbs. 

The weight of beam produces a max. fiber stress of 42 lbs. per sq. in., 
and this deducted from 1000= 958, produced by the center load of 9083 lbs. 
Note that twice the center load or 18166 lbs. if uniformly distributed (mak- 
ing about 1211 lbs. per lin. ft.) would produce the same fiber stress of 958 
lbs. per sq. in. 

Ex. 3. (Span and Deflection for Plastered Ceiling). — ^The max. allowable 
deflection v of beams for plastered ceiling is tJ\j of an in. per ft. of span, or 

v = -prr,= ir;77,' What general formulas may be used for wooden beams 

30 360 
(£=1000 000) and for steel girders (£ = 29 000 000) to represent this 
limiting relation? (See Note (*) for Wooden Beams on page 565.) 
Solution (from Table 1). 



Loading . 
I. Uniform load W 

II. 

III. Center load W 
IV. 



Wooden Beams.* 

10000 bd^ 



Case Dd, (2) : W= 
" (5):/ = 

Case Cd, (2): W= 
" (5):/ = 



81 L2 
10000 d 



9L 
6250 bd^ 



81 L2 
12500 d 



9L 



Steel Girders. 

11600007 



Case Db, (2): W= 

•* (5):/ = 

Case Cb, (2): W= 

" (5):/ = 



27 L2 
580 000 y 

9L 
725000/ 

27 L2 

725 000 y 

9L 



DEFLECTION. LONGITUDINAL SHEAR, 565 

Furthermore, if the allowable outer fiber stress / for timber is 1000 lbs. 
per sq. in., and / for steel is 15000, we have, from II and IV: — 



Loading. 
V. Uniform load W 

VI. Center load W 



Wooden Beams.* 
L=jd==l.n'^ld 

L=?|ci = 1.38V8d 



Steel Girders, 
L=^^y^i.29Qy 

14'? 

L=~^y=^5.370y 



which are the limiting spans for plastered ceiling, and which enable us to 
design these spans economically. 

Longitudinal Shear in Beams. — While long beams, even of stifficient 
strength to resist bending, are often limited by the deflection, as previously 
noted, short beams may fail by horizontal (longitudinal) shear. The general 
formula for longitudinal shear is 

rr VQ _ VA,y^ 

^^T-T- ^^^ 

In which i? = the longitudinal shearing force in lbs. per unit (lin. in.) length 
of beam; 
F = the total vertical shear in lbs. at section considered; 
j2 = the statical moment in in. -lbs. of the area Ax above the plane 
of shear = yli:Vi, in which :vi = distance from neutral axis to 
cen. of grav. of A^', 
I = the moment of inertia of beam section. 
For a rectangular beam, the longitudinal shear at the neutral axis 
would be, if & = breadth and d = depth in inches: 

^hd ^d ,hd^_Z V 
and the intensity of shear per square inch would be 

b-2 bd ^^ 

Note that the vertical shear V at any point of a beam is equal to the 
differential coefficient of the bending moment at that point; thus, 

y=^ (4) 

ax 
X being the distance from left-hand end of beam to section considered. 

Problem: — A wooden beam 12 x 12 ins., weighing 600 lbs. (=w) , supports 
a uniform load of W lbs. What will be the maximum value of W, so that 
the maximum intensity of shear, H -^ b, (at the neutral axis,) shall not 
exceed 50 lbs. per sq. in. ? 

Solution: — ^The maximum vertical shear (at the end of the beam) equals 
W+w 
y = — _ — , Substituting this value of V in equation (3) , we have, 

rr o 

Intensity of shear = v-=Y ' ~i7rj^"=^ 




Hence, by substitution, W= 9600-600=9000 lbs. Ans. (See also. Example 
1, bottom of page 567.) 

Note that the length of span, in the above problem, is not a factor in 
longitudinal shear. 



* Note that W and / are directly proportional to the modulus of elastic- 
ity, E, of the material, which, in the present instance, is assumed at 
1 000 000. For any other modulus, as Ei, multiply above values by 

rooO^OOO* ^^^ ^^^' ^^' S^^^^S^^ ^^ Materials, Table 7, page 495. 



566 dl— PROPERTIES AND TABLES OF BEAMS AND GIRDERS. 



2. — Uniformly Distributed Loads W in Pounds 

On Rectangular Beams 1 Inch Wide 

Producing Extreme Fiber Stress /= 1000 Lbs. per Sq. In. 

(By Formula. Case Od (2): 1^=1^=1^) . 
For any other fiber stress, as fi, multiply values in table by t^^. 







[Total load in Pounds, including ^ 


weight of beam.] 








Depth of Beam. 


r 


2" 


3^ 


4// 


5" 


6" 


7„ 


8^^ 


9^^ 


10^ 


IV 


12" 


1 


111 

56 
37 

28 

22 
19 
16 
14 
12 

11 
10 


444 
222 
148 
111 

89 
74 
64 
56 
49 

44 
40 
37 
34 
32 

30 

28 
26 
25 
23 


1000 
500 
333 
250 

200 
167 
143 
125 
111 

100 

91 
83 
77 
71 

67 
62 
59 
56 
53 

50 

48 
45 
43 
42 


1778 
889 
593 
444 

356 
296 
254 
222 
198 

178 
162 

148 
137 
127 

119 
111 
105 

99 
94 

89 
85 
81 
77 
74 


2778 

1389 

926 

694 

556 
463 
397 
347 
309 

278 
253 
231 
214 
198 

185 
174 
163 
154 
146 

139 
132 
126 
121 
116 


4000 
2000 
1333 
1000 

800 
667 
571 
500 
444 

400 

364 
333 
308 
286 

267 
250 
235 
222 
211 

200 
190 
182 
174 
162 















9, 


2722 
1815 
1361 

1089 
907 

778 
681 
605 

544 

495 
454 
419 
389 

363 
340 
320 
302 

287 

272 
259 
247 
237 
227 


3556 
2370 

1778 

1422 
1185 
1016 

889 
790 

711 

646 
593 
547 
508 

474 
444 
418 
395 
374 

356 
339 
323 
309 
296 










8 


3000 
2250 

1800 
1500 
1286 
1125 
1000 

900 
818 
750 
692 
643 

600 
563 
529 
500 
474 

450 
429 
409 
391 
375 


3704 

2778 

2222 
1852 
1587 
1389 
1235 

1111 

1010 

926 

855 

794 

741 
694 
654 
617 

585 

556 

529 
505 
483 
463 






4 

5 
6 

7 
8 
9 

10 
11 
12 
13 
14 

15 
16 
17 
18 
19 

?0 


3361 

2689 
2241 
1921 
1681 
1494 

1344 
1222 
1120 
1034 
960 

896 
840 
791 
747 
708 

672 
640 
611 
585 
560 


4000 

3200 
2667 
2286 
2000 
1778 

1600 
1455 
1333 
1231 
1143 

1067 

1000 

941 

889 
842 

800 


?1 






762 


9:?, 






727 


?I3 






696 


94 






667 





















Depth of Beam. 










13^ 


14// 


15" 


16" 


17" 


18" 


19" 


20" 


21" 


22" 


23" 


24" 


8 


2347 


2722 


3125 


3556 


4014 


4500 


5014 


5556 


6125 


6722 


7347 


8000 


9 


2086 


2420 


2778 


3160 


3568 


4000 


4457 


4938 


5444 


5975 


6531 


7111 


10 


1878 


2178 


2500 


2844 


3211 


3600 


4011 


4444 


4900 


5378 


5878 


6400 


11 


1707 


1980 


2273 


2586 


2919 


3273 


3646 


4040 


4455 


4889 


5343 


5818 


12 


1565 


1815 


2083 


2370 


2676 


3000 


3343 


3704 


4083 


4481 


4898 


5333 


13 


1444 


1675 


1923 


2188 


2470 


2769 


3085 


3419 


3769 


4137 


4521 


4923 


14 


1341 


1556 


1786 


2032 


2294 


2571 


2865 


3175 


3500 


3841 


4198 


4571 


15 


1252 


1452 


1667 


1896 


2141 


2400 


2674 


2963 


3267 


3585 


3919 


4267 


16 


1174 


1361 


1563 


1778 


2007 


2250 


2507 


2778 


3062 


3361 


3674 


4000 


17 


1105 


1281 


1471 


1673 


1889 


2118 


2359 


2614 


2882 


3163 


3458 


3765 


18 


1043 


1210 


1389 


1580 


1789 


2000 


2228 


2469 


2722 


2988 


3265 


3556 


19 


988 


1146 


1316 


1497 


1690 


1895 


2111 


2339 


2579 


2830 


3094 


3368 



Note. — For allowable fiber stresses in Wooden Beams, see Sec. 28, 
Strength of Materials, Table 7, column 8, page 495, For bridge stringers 
use safety factor 6; for floorbeams, 5, 



LOADS ON RECTANGULAR BEAMS. 



667 



2. — Uniformly Distributed Loads W in Pounds 

On Rectangular Beams 1 Inch Wide. 

— Concluded. 

[Total load in Pounds, including weight of beam.] 



^^ 










Depth of Beam. 












13'' 


14" 


15" 


16" 


17" 


18" 


19" 


20" 


21" 


22" 


23" 


24" 


20 


939 


1089 


1250 


1422 


1606 


1800 


2006 


2222 


2450 


2689 


2939 


3200 


21 


894 


1037 


1190 


1354 


1529 


1714 


1910 


2116 


2333 


2561 


2799 


3048 


22 


854 


990 


1136 


1293 


1460 


1636 


1823 


2020 


2227 


2444 


2672 


2909 


23 


816 


947 


1087 


1237 


1396 


1565 


1744 


1932 


2130 


2338 


2556 


2783 


24 


782 


907 


1042 


1185 


1338 


1500 


1671 


1852 


2042 


2241 


2449 


2667 


25 


751 


871 


1000 


1138 


1284 


1440 


1604 


1778 


1960 


2131 


2351 


2560 


26 


722 


838 


962 


1094 


1235 


.1385 


1543 


1709 


1885 


2068 


2261 


2462 


27 


695 


807 


926 


1053 


1189 


1333 


1486 


1646 


1815 


1992 


2177 


2370 


28 


671 


778 


893 


1016 


1147 


1286 


1433 


1587 


1750 


1921 


2099 


2286 


29 


648 


751 


862 


981 


1107 


1241 


1383 


1533 


1690 


1854 


2027 


2207 


30 


626 


726 


833 


948 


1070 


1200 


1337 


1481 


1633 


1793 


1959 


2133 


31 


606 


703 


806 


918 


1036 


1161 


1294 


1434 


1581 


1735 


1896 


2065 


32 


587 


681 


781 


889 


1003 


1125 


1253 


1389 


1531 


1681 


1837 


2000 


33 


569 


660 


758 


862 


973 


1091 


1215 


1347 


1485 


1630 


1781 


1939 


34 


552 


641 


735 


837 


944 


1059 


1180 


1307 


1441 


1582 


1728 


1882 


35 


537 


622 


714 


813 


917 


1029 


1146 


1270 


1400 


1537 


1677 


1829 


36 


522 


605 


694 


790 


894 


1000 


1114 


1235 


1361 


1494 


1633 


1778 


37 


507 


589 


676 


769 


868 


973 


1084 


1201 


1324 


1453 


1589 


1730 


38 


494 


573 


658 


749 


845 


947 


1056 


1169 


1289 


1415 


1547 


1684 


39 


481 


558 


641 


729 


823 


923 


1028 


1140 


1256 


1379 


1507 


1641 


40 


469 


544 


625 


711 


803 


900 


1003 


1111 


1225 


1344 


1469 


1600 



Note. — For allowable fiber stresses in Wooden Beams, see Sec. 28, 
Strength of Materials, Table 7, column 8, page 495. For bridge stringers 
use safety factor 6; for floor beams, 5. 

Remember that Wis proportional to d^ and inversely proportional to L; 
thus the range .of the table may be extended greatly. 

Examples in Use of Table 2. 

Example 1. — In the problem at bottom of page 565, longitudinal shear, we 
find that a 12 x 12-in. wooden beam will support a total load (including 
weight of beam = 600 lbs.) of 9600 lbs. Find the maximum length of span 
so that the outer fiber stress / shall not exceed 1000 lbs. per square inch. 

Solution. — From the data given, abeam 12 ins. wide and 12 ins. deep 
will support a total load of 9600 lbs.; hence, for the same depth of iDeam, 
each one inch in width will support 9600-^12=800 lbs. Now, in the last 
column (the 12" column) on preceding page we find that the load 800 corre- 
sponds to a span of 20 ft. Ans. 

Example 2. — In the preceding Example, find the maximum length of 
span so that the outer fiber stress / shall not exceed 600 lbs. per square inch. 

Solution. — Corresponding length of span, 



^=4=''^=^'"- 



Ans. 



Example 3. — A beam supports a load W= 9600 lbs., 
fiber stress /=1000. What load Wi will it support if 
stress /i is increased to 1200 lbs. per square inch ? 

Solution. — W is proportional to /; hence 



producing an outer 
the allowable fiber 



Wi= 1^^=9600^^=11520 lbs. 



Ans. 



568 Zl— PROPERTIES AND TABLES OF BEAMS AND GIRDERS. 




3. — Beam Box Girders (Steel). 
(Adapted from Carnegie and Cambria.) 

Note. — I-beams are the sections as rolled by the 
principal mills. Finished weight includes rivet heads. 
Section modulus, 5, allows for W' rivet holes deducted, 
both flanges. Safe load W includes weight of girder; 
therefore, the allowable superimposed load will be less 
than W. For Deflection, see Table 1, preceding, and 



Fig 


:. 1. 


Example c 


1. 
















Sec- 


*Safe 








Sec- 


*Safe 


Section. 


Fin- 


tion 


total 


Sect] 


on. 


Fin- 


tion 


total 




ished 


modu- 


loadP^ 






ished 


modu- 


load W 




weight 
per 


lus 
7x 


Unif. 
Dis- 






weight 
per 


lus 

73C 


Unif. 










Dis- 


Two 
I-Beams 


Two 
Plates 


foot. 


s=- 

y 


trib't'd 

forlO-ft 

span. 


Two 
I-Beams 


Two 
Plates 


foot. 


5=- 

y 


trib't 'd 

forlO-ft 

span. 






Lbs. 


Ins. 


Lbs. 






Lbs. 


Ins. 


Lbs. 


10''-25# 


12x^ 


94.6 


90.1 


90,100 


15''-60# 


14x ys 


180.8 


246.8 


246,800 




X A 

X H 


99.8 


96.3 


96,300 


" 


xii 


187.2 


258.4 


258,400 


104.8 


102.4 


102 ,400 


« 


xM 


193.6 


270.0 


270,000 


°? « II 
II « ^ 




110.0 
115.0 
120.1 
125.2 


108.6 
114.8 
121.0 
127.2 


108,600 
114,800 
121 ,000 
127,200 


1 " 


xfl 

xK 

xM 


199.9 
206.3 
212.7 


281.7 
293.4 
305.1 


281 ,700 
293,400 
305,100 


II " 1 


xM 


130.3 


133.5 


133,500 


:^ " 


xl 


219.0 


316.9 


316,900 


« " 


xl 


135.4 


139.8 


139,800 


o « 


xli^ 


225.4 


328.7 


328 ,700 


+^ 










II « 


xlH 


231.8 


340.6 


340,600 




14xH 

X ^ 


114.4 
120.4 
126.3 


132.1 
141.0 

149.7 


132,100 
141 ,000 
149.700 


u 


xlA 

xlM 


238.2 
244.6 


352.5 
364.4 


352,500 
364,400 


T " S 


xii 


132.3 


158.5 


158 ,500 












II *^ II 

CO « rH\ 

II *^ 

II (( II 


xM 
xlt 

x3^ 

xif 
xl 


138.3 
144.2 
150.1 
156.1 
162.0 


167.4 
176.3 
185.3 
194.2 
203.2 


167 ,400 
176,300 
185.300 
194,200 
203,200 


15"-60# 


I5x|| 

ii 


187.6 
194.0 
200.4 
206.7 


259.2 
270.8 
282.4 
294.1 


259.200 
270,800 
282,400 
294,100 


o 










S « 


xK 


213.1 


305.8 


305,800 


ir-m . 


14x3^ 


131.4 


146.6 


146,600 


II " 


xH 


219.5 


317.5 


317,500 


" :^ 


X 1% 
X ^ 

X ii 

xi 

xK 


137.4 


155.3 


155,300 


rO tt 


xl 


225.8 


329.3 


329,300 




143.3 
149.3 
155.3 
161.2 
167.1 


164.0 
172.7 
181.4 
190.2 
199.0 


164 ,000 
172 ,700 
181 ,400 
190,200 
199,000 


a 


xl^ 
xl3^ 

XlM 


232.2 
238.6 
245.0 
251.4 


341.1 
353.0 
365.0 
376.8 


341,100 
353,000 
365,000 
376,800 


Q M II 


xM 


173.1 


207.8 


207 ,800 












u ^ 

O 
4j 


xl 


179.0 


216.7 


216,700 


15''-80# 


14x^s 


220.8 


287.5 


287,500 


15'H12# 


14x^ 
xH 


147.3 
153.3 


212.1 
223.0 


212 ,100 
2'>3,000 


M 


xH 
xM 


227.2 
233.6 


299.8 
310.2 


299,800 
310.200 


a 


x% 


159.3 


234.0 


234,000 


:i" " 


xlf 


239.9 


321.5 


321,500 


<^" " s 


xii 


165.2 


245.0 


245,000 


o « 


^ys 


246.3 


332.9 


332,900 


1 " IT 


xK 


171.1 


256.0 


256,000 


II U 


X U 


252.7 


344.4 


344,400 


II u " 

i ' i 


X xl 
xl 

xliV 


177.1 
183.0 
189.0 


267.0 
278.0 
289.3 


267,000 
278,000 
289,300 




xl 

xli^ 


259.0 
265.4 


355.9 
367.4 


355,900 
367,400 


« « e 


xl>^ 


194.9 


300.5 


300,500 


II « 


xlH 


271.8 


379.2 


379,200 


« o 


xlA 


200.9 


311.6 


311 ,600 


Q « 


xIt^ 


278.2 


390.6 


390,600 


M 


xlM 


206.8 


322.8 


322,800 




XlM 


284.6 


402.2 


402,200 



♦ Based on allowable fiber stress of 15000 lbs. per sq. in. For 16-ft. span, 
divide total load by 1.6; 20-ft. span, divide by 2; 30-ft. span, divide by 3; etc. 

Note that total load W includes weight of girder, which must be de- 
ducted to obtain superimposed load. 



BEAM BOX GIRDERS. 



569 



3. — Beam Box Girders (Steel) — Concluded. 
(Adapted from Carnegie and Cambria.) 









Sec- 


*Safe 






■ 


Sec- 


*Safe 


Section. 


Fin- 


tion 


total 


Section. 


Fin- 


tion 


total 




ished 


modu- 


loadW 




ished 


modu- 


loadW 




weight 
per 


lus 


Unif. 
Dis- 




weight 
per 


lus 
7v 


Unif. 










Dis- 


Two 
I-Beams 


Two 
Plates 


foot. 


5=^ 

y 


trib't'd 

forlO-ft 

span. 


Two 
I-Beams 


Two 
Plates 


foot. 


y 


trib't'd 

for 10-ft 

span. 






Lbs. 


Ins. 


Lbs. 






Lbs. 


Ins. 


Lbs. 


15''-80# 


15x^ 


227.6 


299.7 


299,700 


20"-65# 


16xlJi 


269.8 


548.1 


548,100 


u 


xH 


234.0 


311.0 


311,000 


« 


xlA 


276.7 


565.3 


565,300 


^ " 


240.4 


322.4 


322 ,400 


M 


xl^ 


283.4 


582.5 


582,500 


O M 

1—i 


xi 


246.7 


333.7 


333,700 












II « 


253.1 


345.1 


345,100 












rO a 


X T6 


259.5 


356.6 


356 ,600 


20''-80# 


16x% 


245.5 


463.8 


463,800 




X 1 


265.8 


368.1 


368,100 


M 


xH 


252.2 


480.4 


480 ,400 


xli^ 


272.2 


379.6 


379 ,600 


•- " ^ 


259.0 


497.1 


497,100 


II " 


nvs 


278.6 


391.2 


391 ,200 


S^ « ^ 


X M 


265.8 


513.8 


513,800 




xlA 
xlM 


285.0 
291.4 


402.8 
414.4 


402,800 
414,400 


CO r; 


xl 

xli^ 


272.6 
279.4 


530.6 
547.3 


530,600 
547,300 


18"-55# 


16x 3^ 

xH 


195.5 
202.2 


340.5 
355.8 


340,500 
355,800 




xli^ 


286.2 
293.1 


564.1 
581.2 


564,100 
581 ,200 


« ^ 


x>^ 


209.0 


371.2 


371 ,200 


II « Q 


xlM 


299.8 


597.8 


597,800 


o " ;^ 


xM 


215.8 


386.6 


386,600 


■^ «t 


xlA 


306.7 


614.7 


614,700 


V « II 


xl 


222.6 


402.0 


402,000 


« 


xl^ 


313.4 


631.7 


631,700 


^0 « '^. 


xli^ 


299.4 


417.5 


417,500 












^^ " ^ 


xl3^ 


236.2 


433.0 


433,000 












^ « °f 


xlA 


243.1 


448.6 


448,600 


24''-80# 


18xM 


255.7 


593.7 


593,700 


II " e 


xlM 


249.8 


464.2 


464,200 


<j ' 


xH 


263.3 


616.9 


616,900 


^ " 2 


xli^ 


256.7 


479.8 


479,800 


u 


x^ 


271.0 


640.1 


640,100 


U •4J 


xl^ 


263.4 


495.4 


495,400 


CO 


xM 


278.6 


663.4 


636,400 


20''-65# 

i-H U II 
II il 


16x3^ 

X T6 
X p8 

xif 


215.5 
222.2 
229.0 
235.8 


411.8 

428.7 
445.7 
462.7 


411 ,800 
428,700 
445,700 
462 ,700 


II « i 

II ^ 
II II 


xl 

Xln^ 

xl3^ 

xlA 


286.2 
293.9 
301.5 
309.2 


686.7 
710.0 
733.3 
757.1 


686,700 
710,000 
733,300 
757,100 


J) " ^. 


xl 


242.6 


479.7 


479,700 


d " I 


xlM 


316.8 


780.2 


780,200 


t.- " ^ 


xli^ 


249.4 


496.7 


496,700 


o 


xIt^ 


324.5 


803.6 


803,600 


11 " II 


xl^- 


256.2 


513.8 


513,800 


« 4J 


xl^ 


332.1 


827.1 


827,100 


« « 1 


xli^ 


263.1 


531.2 


531 ,200 


a 


xli^ 


339.8 


850.5 


850,500 



* Based on allowable fiber stress of 15 000 lbs. per sq. in. For 16-ft. span, 
divide total load by 1.6; 20-ft. span, divide by 2; 30-ft. span, divide by 3; etc. 

Note that total load W includes weight of girder, which must be de- 
ducted to obtain superimposed load. 

Problem. — Required to design a beam box girder of 16-ft. clear span, 
for a superimposed load which produces a maximum bending moment M"= 
3,150,000 in .-lbs . ; with a total allowable fiber stress /= 14400 lbs. per square inch. 

Solution. — (See Table 1, page 562, for formulas.) From the general 
formulas (3) we nave, 

(1.) For the superimposed load: Sec. mod. 5 = - = ^ = ?j|^ = 218.75 

From the above table this value of 5 calls for 15''— 42# beams with 

/O N T^' .-U • U^ f.-U ' ^ C, ^ ^ I M'' Wl Wl^ 

(2.) For the weight of the girder: Sec. mod. 5 = — = — r- 

y 
in which w — the weight in lbs. per lin. ft. of girder. 

. ,, ,Q„ ^. e 183X256X12X12 

table, we may assume w = 183; then 5 = 



/ 8/ 8/' 

From the above 



(1 and 2.) Ans. 
required 5 = 



8 X 14400 
-Use two 15" - 42# beams and two 14 X V plates: For 



58.55 



277.30 



570 31— PROPERTIES AND TABLES OF BEAMS AND GIRDERS, 

4. — Plate-Girder Tables* (Steel). 
Properties of Plate Girders Complete. 

(Compiled from Tables 5, 6 and 7, following.) 
[Resisting moment ( = bending moment) in 1000 Ft. -Lbs.] 



6 



9 



10 



11 



12 



13 



14 



15 



16 



17 



6xM 
6xM 



6xM 
6xM 



7xM 
7x>^ 



7xM 
7x3^ 



7xM 
7x5^ 



7xM 
7x^8 



8x3^ 
8x^ 



8xA 
SxVs 



._o 

u o 
w 

Lbs. 



30.0 
37.2 
44.4 
40.2 
64.8 



35 

42 

49.5 
45.3 
69.9 



33 

41 

49.3 

45.2 

73.1 

38.4 
46.8 
54.4 
50.3 

78.2 

34.9 
52.9 
65.3 
45.9 
95.1 

38.7 
56.7 
69.1 
50.6 
98.9 



38.7 
56.7 
69.1 
52.3 
103.1 



45 

69 

62 

103 



Resisting 
moments in 
1000 ft.-lbs. 

including 
in flanges: 



Vh 


T^n 


of 


of 


web. 


web. 


30.6 


29.3 


41.0 


39.7 


50.9 


49.6 


45.2 


43.9 


80.7 


79.3 


46.3 


44.4 


61.0 


59.2 


75.0 


73.2 


m.i 


64.4 


115.4 


113.5 


39 


38 


53 


51 


66 


64 


59 


58 


107 


105 


57 


54 


76 


73 


93 


90 


83 


80 


148 


145 


42 


40 


72 


70 


85 


83 


62 


61 


136 


135 


44 


42 


74 


72 


87 


85 


64 


62 


139 


136 


45 


43 


77 


73 


89 


87 


69 


67 


151 


149 


56 


54 


96 


94 


87 
158 


84 
15.^ 



No 
part 

of 
web. 



23.9 
34.4 
44.3 
38.5 
74.0 

33.6 
48.4 
62.4 
53.5 
102.8 

31 
45 
58 
51 
99 

42 
61 

78 

68 

133 

34 
63 
76 
54 

128 

34 
63 
76 
54 
128 

35 
65 
79 
59 
140 

46 

85 

76 

147 



..2 

^ ,^ 

^' 2 
w 

Lbs, 



31.7 
38.9 
46.1 
41.9 
66.5 



44.0 
51.2 
47.0 
71.6 

35.0 
43.4 
51.0 

46.9 
74.8 

40.1 

48.5 
56.1 
52.0 
79.9 

40.0 
58.0 
70.4 
51.9 
100.2 

45.1 
63.1 
75.5 
57.0 
105.3 

45.1 
63.1 
75.5 
58.7 
109.5 

51.9 

75.5 

68.9 

109.5 



Resisting 

moments in 

1000 ft.-lbs. 

including 

in flanges: 



Vs 


T^TT 


of 


of 


web. 


web. 


35.6 


33.9 


47.5 


45.8 


58.7 


57.1 


52.0 


50.3 


92.0 


90.3 


51.9 


48.9 


68.0 


65.0 


83.5 


80.5 


73.5 


70.5 


127.4 


124.4 


45 


43 


60 


58 


75 


7S 


67 


65 


120 


118 


63 


59 


83 


80 


103 


99 


92 


88 


161 


158 


61 


58 


101 


98 


120 


117 


88 


85 


188 


185 


65 


61 


105 


101 


123 


120 


91 


88 


191 


188 


66 


62 


107 


103 


125 


122 


97 


94 


206 


203 


81 


77 


135 


132 


121 


117 


217 


213 



No 
part 

of 
web. 



27.2 
39.0 
50.3 
43.5 
83.6 

36 

53.0 

68.5 

58.5 

112.4 

34 
50 
65 
56 
110 

45 
66 
85 
74 
144 

46 
86 

105 
73 

173 

46 
86 

105 
73 

173 

47 

88 
107 

79 
188 

62 
117 
102 

198 



..2 

U <U 

a w 

w 

W 
Lbs, 



33.4 
40.6 

47.8 
43.6 
68.2 

38.5 
45.7 
52.9 
48.7 
73.3 

36.7 
45.1 
52.7 
48.6 
76.5 

41.8 
50.2 
57.8 
53.7 
81.6 

45.1 
63.1 
75.5 
57.0 
105.3 

51.5 
69.5 
81.9 
63.4 
111.6 

51.5 
69.5 
81.9 
65.1 
115.9 

58.3 

81.9 

75.3 

115.9 



Resisting 

moments in 

1000 ft.-lbs. 

including 

in flanges: 



Vh 


tV 


of 


of 


web. 


web. 


40.8 


38.7 


54.1 


52.0 


66.8 


64.7 


58.9 


56.9 


103.6 


101.5 


57.7 


54.2 


75.3 


71.8 


92.1 


88.6 


81.1 


77.6 


139.6 


136.1 


51 


49 


68 


66 


84 


82 


75 


73 


134 


132 


69 


65 


92 


87 


112 


108 


100 


96 


175 


171 


82 


77 


133 


128 


156 


151 


118 


113 


241 


236 


87 


82 


139 


133 


162 


156 


124 


118 


246 


241 


88 


82 


142 


135 


164 


158 


128 


122 


265 


259 


108 


102 


177 


171 


158 


152 


278 


272 



No 

part 

of 
web. 



30.4 
43.7 
56.4 

48.5 
93.2 

40.1 
57.7 
74.5 
63.5 
122.0 

38 
55 
71 
63 
121 

49 
71 
92 
80 
155 

58 
110 
133 

94 
217 

58 
110 
133 

94 
217 

59 
111 
135 

99 
235 

79 
148 
129 
249 



* Table 4 will cover most cases in practice ; but Tables 5, 6 and 7 may 
further be combined to meet almost any requirement. 
t One top and one bottom. 



PLATE GIRDERS— COMPLETE. 



671 



4. — Plate-Girder Tables (Steel) — Continued. 
Properties of Plate Girders Complete. 

(Compiled from Tables 5, 6 and 7, following.) 
[Resisting moment (== bending moment) in 1000 ft.-lbs.] 



2 3 4 



6 



7 8 9 10 11 12 



13 14 15 16 



Four 

Angles, 

Each 



0) "■' 



w 

Lbs. 



Resisting 

moments in 

1000 ft.-lbs. 

including 

in flanges: 



H 


tV 


of 


of 


web. 


web. 


88 


84 


148 


144 


134 


130 


242 


238 


97 


93 


166 


16S 


143 


140 


260 


257 


119 


114 


203 


198 


190 


186 


347 


343 


125 


120 


214 


209 


196 


191 


358 


354 


127 


122 


246 


242 


214 


209 


452 


448 


133 


128 


259 


255 


219 


215 


466 


461 


355 


342 


550 


538 


558 


546 


963 


950 


753 


708 


1110 


1066 


1107 


1062 


1823 


1778 


816 


779 


1251 


1214 


1221 


1184 


2486 


2449 


1388 


1304 


2060 


1975 


1993 


1909 


3895 


3810 



No 
part 

of 
web. 



..2 

^-^ o 

W 
Lbs. 



Resisting 

moments in 

1000 ft.-lbs. 

including 

in flanges: 



Vh 


T^n 


of 


of 


web. 


web. 


116 


110 


193 


187 


174 


168 


309 


303 


129 


123 


218 


212 


186 


181 


334 


328 


198 


187 


326 


316 


302 


292 


540 


530 


207 


197 


345 


335 


31S 


303 


559 


549 


210 


200 


395 


385 


339 


329 


699 


689 


220 


210 


417 


407 


349 


339 


721 


711 


469 


449 


718 


699 


722 


703 


1231 


1211 


900 


842 


1312 


1254 


1304 


1246 


2124 


2066 


996 


945 


1511 


1459 


1468 


1417 


2945 


2894 


1599 


1495 


2350 


2246 


2271 


2167 


4385 


4281 



No 

part 

of 
web. 



.2 
ft w 



w 

Lbs. 



Resisting 

moments in 

1000 ft.-lbs. 

including 

in flanges: 



V^ 


^ 


of 


of 


web. 


web. 


147 


139 


240 


231 


216 


208 


379 


371 


163 


154 


271 


263 


232 


224 


410 


402 


288 


270 


461 


443 


428 


410 


745 


727 


301 


283 


487 


469 


442 


424 


771 


753 


305 


287 


555 


537 


476 


458 


957 


939 


318 


300 


585 


567 


489 


471 


988 


970 


591 


563 


894 


866 


895 


866 


1507 


1479 


1057 


983 


1523 


1449 


1511 


1437 


2435 


2361 


1187 


1120 


1780 


1713 


1725 


1659 


3415 


3348 


2054 


1904 


2962 


2812 


2859 


2709 


5397 


5247 



4x3 x^ 

" ^H 

- x^ 
4x4 xA 

" X5/, 
« X^ 

" xK 



9x^ 
9x^ 



9Xi6 
9xH 



5x3 X 

" X3^ 
« X" 

'♦ xMllxM 



^llx^ 



6x3ix^ 
" xM 



Xf^llx3^ 

' llxM 



6x3ix; 

" X 

" ^% 

6x4 x^ 

" ^% 

" X3^ 

" ^% 

6x6 x3^ 

" xK 
" xK 
" xj^ 

6x6 x3^ 



13x3^ 



13x^ 
13xK 



14xH 
14x1 



xM14xi/^ 



xj^ 



14x1 



8x8 -s^A 

" xH18xH 
** x7^18xU 



8x8x3^ 



xM18xH 
xJ^18xU 



54.3 

79.9 

73.5 

118.2 

58.3 

88.3 

77.5 

126.6 

70 
105 

98 
161 

72 
110 
100 
166 

77 
133 
111 
211 

80 
139 
113 
217 

129 
183 
177 
279 

183 
237 
230 
332 

208 

282 
269 
466 

259 
333 
320 
517 



129 
115 
223 

78 
148 
125 
242 

96 
180 
168 
325 

102 

191 
173 
336 

104 
224 
191 
430 

110 

237 
197 
443 

292 
488 
496 
900 

530 

887 

883 

1600 

629 
1064 
1034 
2299 

966 
1638 
1571 
3474 



60.7 

86.3 

79.8 

124.6 

64.7 
94.7 
83 
133.0 

85 
120 
113 
176 



125 
116 
181 

93 
149 
126 
226 

95 
155 
128 
232 

142 
196 
190 
291 

197 
251 
245 
347 

225 

299 
286 
483 

276 
350 
337 
534 



87 
163 
145 
280 

99 
188 
157 
305 

147 
276 
253 
490 

157 

294 
262 
508 

160 
344 

289 
649 

169 
366 
298 
671 

371 

621 

625 

1133 

609 
1020 
1012 
1833 

741 
1255 
1213 
2690 

1079 
1829 
1750 
3864 



67.1 

92.7 

86.2 

130.9 

71.1 
101.1 

90.2 
139.3 

100 
135 

128 
191 

103 
140 
131 
197 

108 
164 
141 
241 

110 
170 
144 
247 

155 
209 
203 
304 

212 
266 
260 
362 

242 
316 
303 
500 

310 

384 
371 
568 



t One top and one bottom. 



572 Zl— PROPERTIES AND TABLES OF BEAMS AND GIRDERS, 




5. — Plate Girders (Steel). 
Properties of Flange Angles Only. 



1 


2 


3 


4 


5 


6 


7 


8 


9 


4 Flange 
Angles. 


-t-> rj 

%^ 


o 


(3^ 


NetArea 

Each 

Flange 

(2 
Angles) 


■B « 1 


§1^ 
w ^! n 


Resist- 
ing 
Moment 

12y 

for 
/=10,000. 


c/i a> 


Size. 


w 


D 


d 


A 




^^s 




^.aw"s 


Ins. 


Lbs. 


Ins. 


Ins. 


Sq. Ins. 


Ins. 


3 


Ft.-Lbs. 


5 



(a) Use with or without Cover Plates. Rivets M"". 


One W' rivet-hole deducted 








from'Each Angle for A. 






2ix2ixA 


12.4 


12^ 


10.87 


1.47 


15.98 


17.64 


13 320 


14 700 


" xM 


16.4 


u 


10.81 


1.94 


20.97 


23.28 


17 480 


19 400 


" X]^ 


20.0 


u 


10.77 


2.39 


25.74 


28.68 


21 450 


23 900 


" X3^ 


23.6 


u 


10.73 


2.80 


30.04 


33.60 


25 040 


28 000 


" ^^ 


27.2 


u 


10.69 


3.23 


34.53 


38.76 


28 770 


32 300 


** x3^ 


30.8 


" 


10.63 


3.63 


38.59 


43.56 


32 160 


36 300 


3x2ixJ^ 


18.0 


12M 


10.93 


2.18 


23.83 


26.16 


19 860 


21 800 


" x^ 


22.4 


u 


10.89 


2.69 


29.29 


32.28 


24 410 


26 900 


" x^ 


26.4 


« 


10.83 


3.18 


34.44 


38.16 


28 700 


31 800 


" x^ 


30.4 


« 


10.79 


3.67 


39.60 


44.04 


33 000 


36 700 


" x3^ 


34.0 


« 


10.75 


4.13 


44.40 


49.56 


37 000 


41 300 


3 x3 xj^ 


19.6 


18K 


16.57 


2.44 


40.43 


29.28 


33 690 


24 400 


" xA 


24.4 




16.51 


3.01 


49.70 


36.12 


41 410 


30 100 


" x^ 


28.8 


tl 


16.47 


3.56 


58.63 


42.72 


48 860 


35 600 


" x5 


33.2 


" 


16.43 


4.09 


67.20 


49.08 


56 000 


40 900 


" xj^ 


37.6 


« 


16.39 


4.63 


75.89 


55.56 


63 240 


46 300 


" x^ 


41.6 


« 


16.35 


5.14 


84.04 


61.68 


70 030 


51 400 


" xH 


50.0 


** 


16.29 


5.63 


91.71 


67.56 


76 430 


56 300 


3^x2ixM 


19.6 


18M 


17.03 


2.44 


41.55 


29.28 


34 630 


24 400 


■" x^ 


24.4 


<( 


16.97 


3.01 


51.08 


36.12 


42 570 


30 100 


" x^ 


28.8 


« 


16.93 


3.56 


60.27 


42.72 


50 230 


35 600 


" X3^ 


33.2 


u 


16.89 


4.09 


69.08 


49.08 


57 570 


40 900 


" XH 


37.6 


u 


16.85 


4.63 


78.02 


55.56 


65 010 


46 300 


" xA 


41.6 


u 


16.79 


5.14 


86.30 


61.68 


71 920 


51 400 


" X54 


50.0 


u 


16.75 


5.63 


94.30 


67.56 


78 590 


56 300 


3^x3 x^ 


26.4 


18M 


16.63 


3.31 


55.05 


39.72 


45 870 


33 100 


xi^ 


31.6 




16.59 


3.94 


65.36 


47.28 


54 470 


39 400 


" x^ 


36.4 


« 


16.55 


4.53 


74.97 


54.36 


62 480 


45 300 


" 4>2 


40.8 


" 


16.49 


5.13 


84.59 


61.56 


70 490 


51 300 


" Xt^ 


45.6 


« 


16.45 


5.70 


93.77 


68.40 


78 140 


57 000 


" x^ 


50.0 


<< 


16.41 


6.25 


102.56 


75.00 


85 470 


62 500 


" Xxi 


54.4 


" 


16.37 


6.80 


111.32 


81.60 


92 760 


68 000 


" xM 


58.8 


<t 


16.33 


7.31 


119.37 


87.72 


99 480 


73 100 



PLATE GIRDERS—FLANGE ANGLES ONLY. 



573 



5. — Plate Girders (Steel) — Continued. 
Properties of Flange Angles Only. 



4 Flange 
Angles. 



Size. 
Ins. 



Ph 5 



<D o 

•a 



Vi 



w 

Lbs. 



^0^ 

P o 

D 

Ins. 



. > 

. u 

d 

Ins. 



NetArea 
Each 
Flange 

(2 
Angles) 

A 
Sq. Ins. 



.2 II II 
^ ^ ^ 

Ins. 



O oC^ 

w ^s a 



Resist- 
ing 
Moment 



M' 



for 
f =10,000. 



Ft.-Lbs. 



• U 4> 



(U 



V) 



(a) Use with or without Cover Plates. Rivets %''. 


One W rivet-hole deducted 








from Each Angle for A. 






3ix3ix^ 


28.8 


18M 


16.27 


3.63 


59.06 


43.56 


49 220 


36 300 


" X3^ 


34.0 


« 


16.23 


4.30 


69.79 


51.60 


58 160 


43 000 


" X,3^ 


39.2 


<( 


16.17 


4.97 


80.36 


59.64 


66 970 


49 700 


" x3^ 


44.4 


u 


16.13 


5.63 


90.81 


67.56 


75 680 


56 300 


" x^ 


49.6 


u 


16.09 


6.26 


100.72 


75.12 


83 940 


62 600 


" xH 


54.4 


<( 


16.05 


6.87 


110.26 


82.44 


91 890 


68 700 


" xH 


59.2 


(( 


16.01 


7.48 


119.75 


89.76 


99 800 


74 800 


« xM 


64.0 


(( 


15.95 


8.07 


128.72 


96.84 


107 260 


80 700 


4x3x^ 


28.8 


24M 


22.73 


3.63 


82.51 


43.56 


68 760 


36 300 


" x^ 


34.0 




22.69 


4.30 


97.57 


51.60 


81 310 


43 000 


" xA 


39.2 


<t 


22.65 


4.97 


112.57 


59.64 


93 810 


49 700 


" x3^ 


44.4 


u 


22.59 


5.63 


127.18 


67.56 


105 980 


56 300 


" x^ 


49.6 


(( 


22.55 


6.26 


141.16 


75.12 


117 640 


62 600 


" x^ 


54.4 


<( 


22.51 


6.87 


154.64 


82.44 


128 870 


68 700 


" xH 


59.2 


a 


22.47 


7.48 


168.08 


89.76 


140 060 


74 800 


" xM 


64.0 


<( 


22.41 


8.07 


180.85 


96.84 


150 710 


80 700 


4 x4 x-i^ 


32.8 


24M 


22.01 


4.26 


93.76 


51.12 


78 140 


42 600 


" x^ 


39.2 




21.97 


5.06 


111.17 


60.72 


92 640 


50 600 


" x^ 


45.2 


(( 


21.93 


5.85 


128.29 


70.20 


106 910 


58 500 


" x3^ 


51.2 


« 


21.89 


6.63 


145.13 


79.56 


120 940 


66 300 


" xA 


57.2 


(( 


21.83 


7.38 


161.11 


88.56 


134 260 


73 800 


« x5/8 


62.8 


« 


21.79 


8.13 


177.15 


97.56 


147 630 


81 300 


" xH 


68.4 


a 


21.75 


8.86 


192.71 


106.32 


160 590 


88 600 


« xM 


74.0 


« 


21.71 


9.57 


207.76 


114.84 


173 140 


95 700 


6x3x^ 


39.2 


24M 


22.85 


5.06 


115.62 


60.72 


96 350 


50 600 


" x,V 


45.2 


« 


22.79 


5.85 


133.32 


70.20 


111 100 


58 500 


" x3^ 


51.2 


« 


22.75 


6.63 


150.83 


79.56 


125 690 


66 300 


" xA 


57.2 


« 


22.71 


7.38 


167.60 


88.56 


139 670 


73 800 


** xKs 


62.8 


« 


22.65 


8.13 


184.14 


97.56 


153 450 


81 300 


« xtt 


68.4 


« 


22.61 


8.86 


200.32 


106.32 


166 940 


88 600 


« xM 


74.0 


<( 


22.57 


9.57 


215.99 


114.84 


180 000 


95 700 


5 x3^x^ 


41.6 


24M 


22.53 


5.44 


122.56 


65.28 


102 140 


54 400 


" x^ 


48.0 


M 


22.49 


6.29 


141.46 


75.48 


117 890 


62 900 


" x^ 


54.4 


« 


22.43 


7.13 


159.93 


85.56 


133 270 


71 300 


" x^ 


60.8 


<( 


22.39 


7.96 


178.22 


95.52 


148 520 


79 600 


" xVs 


67.2 


<( 


22.35 


8.75 


195.56 


105.00 


162 970 


87 500 


" xH 


73.2 


<( 


22.31 


9.54 


212.84 


114.48 


177 360 


95 400 


« xM 


79.2 


M 


22.25 


10.31 


229.40 


123.72 


191 160 


103 100 



574 dl—PROPERTIES AND TABLES OF BEAMS AND GIRDERS. 

5. — Plate Girders (Steel) — Concluded. 
Properties of Flange Angles Only.* 



4 Flange 
Angles. 



Size. 
Ins. 



Ph o 

<u o 
o. <u 

Lbs. 



^0 
O 

o 

D 
Ins. 



Q o 
Ins. 



NetArea 

Each 

Flange 

(2 
Angles) 

A 
Sq. Ins. 



I II ^ 

•2 II II 
c/2 :j 



Ins. 



■gW o 
c/2 o w 

w ;j C 



8 



Resist- 
ing 
moment 

fl_ 
12y 

for 
f=10,000. 



M' 



Ft.- Lbs. 



I V4 (U 

"MOW 



(b) Use with or without Cover Plates. Rivets H"- Two J4" rivet-holes deducted 
from Each Angle for A, 



6 x3ix3^ 


46.8 


SOH 


28.67 


5.53 


158.55 


66.36 


132 120 


55 300 


" x^ 


54.0 


u 


28.63 


6.41 


183.52 


76.92 


152 930 


64 100 


" xH 


61.2 


u 


28.59 


7.25 


207.28 


87.00 


172 730 


72 500 


" x^ 


68.4 


u 


28.53 


8.09 


230.81 


97.08 


192 340 


80 900 


" xVs 


75.6 


a 


28.49 


8.91 


253.85 


106.92 


211 540 


89 100 


" xii 


82.4 


M 


28.45 


9.71 


276.25 


116.52 


230 210 


97 100 


« xM 


89.6 


a 


28.39 


10.50 


298.10 


126.00 


248 410 


105 000 


« xH 


96.0 


u 


28.35 


11.28 


319.79 


135.36 


266 660 


112 800 


" xj^ 


102.8 


u 


28.31 


12.04 


340.85 


144.48 


284 040 


120 400 


6x4x^ 


49.2 


ZOH 


28.37 


5.91 


167.67 


70.82 


139 720 


59 100 


" xJj 


57.2 




28.33 


6.83 


193.49 


81.96 


161 240 


68 300 


" xj^ 


64.8 


« 


28.27 


7.75 


219.10 


93.00 


182 580 


77 500 


" xA 


72.4 


M 


28.23 


8.65 


244.19 


103.80 


203 490 


86 500 


U -yS/ 


80.0 


" 


28.19 


9.53 


268.65 


114.36 


223 880 


95 300 


" xii 


87.2 


U 


28.13 


10.41 


292.83 


124.92 


244 030 


104 100 


" xM 


94.4 


" 


28.09 


11.26 


316.29 


135.12 


263 580 


112 600 


" xH 


101.6 


u 


28.05 


12.10 


339.41 


145.20 


282 840 


121 000 


" xyg 


108.8 




28.01 


12.92 


361.89 


155.04 


301 570 


129 200 



(c) Use with or without Cover Plates. Rivets K''- Two 1" rivet-holes deducted 
from Each Angle for A. 



6 x6 x^ 

" xi^ 

" x^ 

" X64 

" xH 

" xM 

" xH 



8 x8 



xH 

x^ 
xH 
xH 
xH 



68.8 


3014 


26.93 


8.37 


225.40 


100.44 


187 840 


78.4 




26.89 


9.50 


255.46 


114.00 


212 880 


87.6 


M 


26.83 


10.61 


284.67 


127.32 


237 220 


96.8 


U 


26.79 


11.72 


313.98 


140.64 


261 650 


106.0 


U 


26.75 


12.81 


342.67 


153.72 


285 560 


114.8 


U 


26.69 


13.88 


370.46 


166.56 


308 710 


124.0 


U 


26.65 


14.93 


397.88 


179.16 


331 570 


132.4 


u 


26.61 


15.98 


425.23 


191.76 


354 360 


105.6 


36K 


31.87 


13.50 


430.25 


162.00 


358 540 


118.4 




31.83 


15.11 


480.95 


181.32 


400 790 


130.8 




31.79 


16.72 


531.53 


200.64 


442 940 


143.2 


« 


31.75 


18.31 


581.34 


219.72 


484 450 


155.6 


" 


31.69 


19.88 


630.00 


238.56 


525 000 


168.0 


" 


31.65 


21.43 


678.26 


257.16 


565 220 


180.0 




31.61 


22.96 


725.77 


275.52 


604 800 



83 700 
95 000 
106 100 
117 200 
128 100 
138 800 
149 300 
159 800 

135 000 
151 100 
167 200 
183 100 
198 800 
214 300 
229 600 



*See Fig. 2, page 572. 



r 

Fig. 3. 



PLATE GIRDERS— WEB PLATES ONLY, 

6. — Plate Girders (Steel) 

Properties of Web Plates Only. 

(M' = moment in ft.-lbs.; M" = moment in in.-lbs.) 



575 



2 


3 


4 


5 


6 



8 



d 













■4J 




o 




(l> 




CO 




^ 


d 


y 


o 


O 


4-> 

o 


+3 

o 


c^ 


£ 


^ 




2 




«4-< 


A 


O 


.2? 


d 


(U 


?^ 


^ 


< 


T^ 


A 


Lbs. 


Sq. Ins 



Full Efficiency of 
Web; 3^ Area Con- 
sidered at Upper 
and Lower Edges ; 
No Deduction for 25% 
Rivet Holes, etc 



75% Efficiency of 
Web;3^AreaCon 
sidered at Upper 
and Lower Edges; 
' ''o Deduct'nfor40% 
Rivet Holes, etc 



Section 
Modulus 



5 = 



Ad 



Moment 
of Re- 
sistance 

12 

(f = 
10000) 

Ft.- Lbs. 



60% Efficiency of 
Web; tV Area Con- 
sidered at Upper 
and Lower Edges; 
^Deduct'n for 
Rivet Holes, etc. 



Section 
Modulus 

r. Ad 



Moment 
of Re- 
sistance 

^ 12 

(/ = 
10000) 

Ft. -Lbs. 



Section 
Modulus 

^_Ad 
'^~ 10 



Moment 
of Re- 
sistance 

12 

(/ = 
10000) 



M' 



Ft. -Lbs. 



.212 
.85 
3.40 


.062 
.25 
1.00 


.425 
1.70 
6.80 


.125 
.50 
2.00 


.637 
2.55 
10.20 


.187 
.75 
3.00 


.85 

3.40 

13.60 


.25 
1.00 
4.00 


1.062 
4.25 
17.00 


.312 
1.25 
5.00 


1.275 
5.10 
20.40 


.375 
1.50 
6.00 


1.487 
5.95 
23.80 


.437 
1.75 
7.00 


1.70 

6.80 

27.20 


.50 
2.00 
8.00 


1.912 
7.65 
30.60 


.562 
2.25 
9.00 



.010 
.042 
.167 

.042 
.167 
.667 

.094 

.375 

1.500 

.167 

.667 

2.667 

.260 
1.042 
4.167 

.375 
1.500 
6.000 

.510 
2.042 
8.167 

.667 

2.667 

10.667 

.844 

3.375 

13.500 



35 
139 

35 

139 
556 

78 

312 

1250 

139 

556 

2222 

217 

868 

3472 

312 
1250 
5000 

425 
1701 
6806 

556 
2222 



008 


7 


031 


26 


125 


104 


031 


26 


125 


104 


500 


417 



.006 
.025 
.100 

.025 
.100 
.400 



5 

21 

83 

21 

83 

333 



Note. — Columns 4 and 5 are useful 
also in the calculation of wooden 
beams; 

Problem. -:— What bending 
moment in ft.-lbs, will a beam 
3"x8'' sustain safely at an allow- 
able fibre stress of 1100 lbs. per 
sq.in.? 1100 5 

Solution 1. — M' = ^^^ = 
llOfl 1^ 

i^ X 3 X 10.667 = 2933 ft. lbs. 

I 1 .• 9 ^J^^ 3X8889X1100 
Solution 2.-M'= ^^^^^^— 

= 3 X 8889 X .11 = 2933 ft.-lbs. 
Hence, the problem can be solved 
by the use of either S or M'. 



703 

2812 

11250 



.383 
1.531 
6.125 


319 
1276 
5104 


.306 
1.225 
4.900 


.500 
2.000 
8.000 


417 
1667 
6667 


.400 
1.600 
6.400 


.633 

2.531 

10.125 


527 
2109 
8437 


.506 
2.025 
8.100 



255 
1021 
4083 

333 
1333 
5333 

422 
1687 
6750 



Note that properties in table are directly proportional to width 6; and 
that the Sectional Moduli and Moments of Resistance are proportional to 
the square of depth d. Hence the range of the table may be increased 
indefinitely. See first column on following page. 



576 U— PROPERTIES AND TABLES OF BEAMS AND GIRDERS. 

6. — Plate Girders (Steel) — Continued. 

Properties of Web Plates Only.* 

(M' = moment in ft.-lbs.; M" = moment in in .-lbs.) 



2 


3 


4 


5 


6 


' 


8 



Web 
Plate. 






d b 
Ins. 



d 




o 








-M 




O 




CD 




(n 




^ 


d 


o 


o 






O 


o 




^ 


o 


m 





tn 


Ph 


xA 


u 


2 


■4^ 


M-- 


^ 


O 


bfl 


<A 


*S 


? 


^ 


< 


w 


A 


Lbs. 


Sq.Ins 



Pull Efficiency of 
Web; 3^Area Con- 
sidered at Upper 
and Lower Edges : 
No Deduction for 25% 
Rivet Holes, etc 



Section 
Modulus 



5 = 



Ad 



Moment 
of Re- 
sistance 

12 

(/ = 
10000) 

Ft.-Lbs. 



W 



75% Efficiency of 



Web ;H Area Con 
sidered at Upper 
and Lower Edges ; 

^Deduct'n 
Rivet Holes, etc 



60% Efficiency of 
Web; T^o Area Con- 
sidered at Upper 
and Lower Edges; 
for 40% Deduct'n for 
Rivet Holes, etc. 



Section 
Modulus 



S = 



Ad 



Moment 
of Re- 
sistance 

M'J^ 
12 

(f = 
10000) 

Ft.-Lbs. 



Section 

Modulus 



5 = 



Ad 
10 



Moment 
9f Re- 
sistance 

12 

(f = 
10000) 



M^ 



Ft.-Lbs. 



10 X 
t -x 



a'<:»X Ye" 

OJ SX 3^ 
OS .X fg- 

'^ ox 1^ 

l-^xil 
a3gxK 

■*^.2x M 



o. 
12; 



'xl 



12 xJ^ 



.X 



o.gx J^ 
a^'x A 
o^x^ 
cs^x fj 

^OXt^ 
•^ (TX ^ 

t/^+jX 



ii 



6 ex 



■g.Sx 



xl 



^4 
if 

if 



14 



X ^ 
X A 
x^ 

X T6 

X J^ 

xl 



2.125 


.625 


1.042 


868 


.781 


651 


4.25 


1.25 


2.083 


1736 


1.562 


1302 


6.375 


1.875 


3.125 


2604 


2.344 


1953 


8.50 


2.50 


4.167 


3472 


3.125 


2604 


10.625 


3.125 


5.208 


4340 


3.906 


3255 


12.75 


3.75 


6.250 


5208 


4.687 


3906 


14.875 


4.375 


7.292 


6076 


5.469 


4557 


17.00 


5.00 


8.333 


6944 


6.250 


5208 


19.125 


5.625 


9.375 


7812 


7.031 


5859 


21.25 


6.25 


10.417 


8681 


7.812 


6510 


23.375 


6.875 


11.458 


9549 


8.594 


7161 


25.50 


7.50 


12.500 


10417 


9.375 


7812 


27.625 


8.125 


13.542 


11285 


10.156 


8463 


29.75 


8.75 


14.583 


12153 


10.937 


9115 


31.875 


9.375 


15.625 


13021 


11.719 


9766 


34.00 


10.00 


16.667 


13889 


12.500 


10417 


2.55 


.750 


1.500 


1250 


1.125 


937 


5.10 


1.50 


3.000 


2500 


2.250 


1875 


7.65 


2.25 


4.500 


3750 


3.375 


2812 


10.20 


3.00 


6.000 


5000 


4.500 


3750 


12.75 


3.75 


7.500 


6250 


5.625 


4687 


15.30 


4.50 


9.000 


7500 


6.750 


5625 


17.85 


5.25 


10.500 


8750 


7.875 


6562 


20.40 


6.00 


12.000 


10000 


9.000 


7500 


22.95 


6.75 


13.500 


11250 


10.125 


8437 


25.50 


7.50 


15.000 


12500 


11.250 


9375 


28.05 


8.25 


16.500 


13750 


12.375 


10312 


30.60 


9.00 


18.000 


15000 


13.500 


11250 


33.15 


9.75 


19.500 


16250 


14.625 


12187 


35.70 


10.50 


21.000 


17500 


15.750 


13125 


38.25 


11.25 


22.500 


18750 


16.875 


14062 


40.80 


12.00 


24.000 


20000 


18.000 


15000 


11.90 


3.50 


8.167 


6806 


6.125 


5104 


14.88 


4.38 


10.208 


8507 


7.656 


6380 


17.85 


5.25 


12.250 


10208 


9.187 


7656 


20.83 


6.13 


14.292 


11910 


10.719 


8932 


23.80 


7.00 


16.333 


13611 


12.250 


10208 


47.60 


14.00 


32.667 


27222 


24.500 


20417 



.625 
1.25 
1.875 
2.50 
3.125 
3.75 
4.375 
5.00 
5.625 
6.25 
6.875 
7.50 
8.125 
8.75 
9.375 
10.00 

.9 
1.8 
2.7 
3.6 



4.5 

5.4 

6.3 

7.2 

8.1 

9.0 

9.9 

10.8 

11.7 

12.6 

13.5 

14.4 



4.90 
6.125 
7.35 
8.575 
9.80 
19.600 



521 
1042 
1562 
2083 
2604 
3125 
3646 
4167 
4687 
5208 
5729 
6250 
6771 
7292 
7812 



750 
1500 
2250 
3000 
3750 
4500 
5250 
6000 
6750 
7500 
8250 
9000 
9750 
10500 
11250 
12000 

4083 
5104 
6125 
7146 
8167 
16333 



*See Fig. 3, page 575. 



PLATE GIRDERS— WEB PLATES ONLY. 



577 



6. — Plate Girders (Steel) — Continued. 

Properties of Web Plates Only.* 

(M' = moment in ft.-lbs.; M" = moment in in.-lbs.) 



2 


3 


4 


5 


« 1 



8 



d 




o 








-l-> 




o 




& 




m 


. 


m 


c 


^ 


o 


o 


4-> 

o 




9^ 


o 


tfx 


£ 




u 

0^ 


O 




<4-4 


.cj 


o 


B 


CtJ 


(U 


9 


^ 


< 


w 


A 


Lbs. 


Sq. Ins 



Full Efficiency of 
Web; K Area Con- 
sidered at Upper 
and Lower Edges, 
No Deduction for 
Rivet Holes, etc 



75% Efficiency of 
Web; 3^ Area Con- 
sidered at Upper 
and Lower Edges 
25%Deduct'n for 
Rivet Holes, etc 



Section 
Modulus 

^_Ad 
^~ 6 



Moment 
of Re- 
sistance 

12 

(/ = 
10000) 

Ft. -Lbs. 



60% Efficiency of 
Web ;t\ Area Con- 
sidered at Upper 
and Lower Edges; 
40% Deduct 'n for 
Rivet Holes, etc. 



Section 
Modulus 

^~ 8 



Moment 
of Re- 
sistance 

12 

(/ = 
10000) 

Ft. -Lbs. 



Section 
Modulus 

^_Ad 
'^~ 10 



Moment 
of Re- 
sistance 

12 

(f = 
10000) 

Ft.-Lbs. 



12.75 


3.75 


9.375 


7812 


7.031 


5859 


5.625 


15.94 


4.687 


11.719 


9766 


8.789 


7324 


7.031 


19.13 


5.625 


14.062 


11719 


10.547 


8789 


8.437 


22.31 


6.562 


16.406 


13672 


12.305 


10254 


9.844 


25.50 


7.500 


18.75 


15625 


14.062 


11719 


11.250 


51.00 


15.000 


37.50 


31250 


28.125 


23438 


22.500 


13.60 


4.00 


10.667 


8889 


8.000 


6667 


6.400 


17.00 


5.00 


13.333 


11111 


10.000 


8333 


8.000 


20.40 


6.00 


16.000 


13333 


12.000 


10000 


9.600 


23.80 


7.00 


18.667 


15556 


14.000 


11667 


11.200 


27.20 


8.00 


21.333 


17778 


16.000 


13333 


12.800 


54.40 


16.00 


42.667 


35556 


32.000 


26667 


25.600 


15.30 


4.50 


13.500 


11250 


10.125 


8437 


8.100 


19.13 


5.625 


16.875 


14062 


12.656 


10547 


10.125 


22.95 


6.75 


20.250 


16875 


15.187 


12656 


12.150 


26.78 


7.875 


23.625 


19687 


17.719 


14766 


14.175 


30.60 


9.00 


27.000 


22500 


20.250 


16875 


16.200 


61.20 


18.00 


54.000 


45000 


40.500 


33750 


32.400 


17.00 


5.00 


16.667 


13889 


12.500 


10417 


10.000 


21.25 


6.25 


20.833 


17361 


15.625 


13021 


12.500 


25.50 


7.50 


25.000 


20833 


18.750 


15625 


15.000 


29.75 


8.75 


29.167 


24306 


21.875 


18229 


17.500 


34.00 


10.00 


33.333 


27778 


25.000 


20833 


20.000 


68.00 


20.00 


66.667 


55556 


50.000 


41667 


40.000 


18.70 


5.50 


20.167 


16806 


15.125 


12604 


12.100 


23.38 


6.875 


25.208 


21007 


18.906 


15755 


15.125 


28.05 


8.25 


30.250 


25208 


22.688 


18906 


18.150 


32.73 


9.625 


35.291 


29410 


26.469 


22057 


21.175 


37.40 


11.000 


40.333 


33611 


30.250 


25208 


24.200 


74.80 


22.000 


80.667 


67222 


60.500 


50417 


48.400 


20.40 


6.00 


24.000 


20000 


18.000 


15000 


14.400 


25.50 


7.50 


30.000 


25000 


22.500 


18750 


18.000 


30.60 


9.00 


36.000 


30000 


27.000 


22500 


21.600 


35.70 


10.50 


42.000 


35000 


31.500 


26250 


25.200 


40.80 


12.00 


48.000 


40000 


36.000 


30000 


28.800 


81.60 


24.00 


96.000 


80000 


72.000 


60000 


57.600 



4687 
5859 
7031 
8203 
9375 
18750 

5333 
6667 
8000 
9333 
10667 
21333 

6750 
8437 
10125 
11812 
13500 
27000 

8333 
10417 
12500 
14583 
16667 
33333 

10833 
13542 
16250 
18958 
21667 
43333 

12000 
15000 
18000 
21000 
24000 
48000 



* See Fig. 3. page 575. 



678 Zl— PROPERTIES AND TABLES OF BEAMS AND GIRDERS. 



6. — Plate Girders (Steel) — Continued. 

Properties of Web Plates Only.* 

(M' = moment in ft.-lbs. ; M" = moment in in, -lbs.) 



1 


2 


3 


4 


5 


6 


7 


8 


9 




1 




Full Efficiency of 


75% Efficiency of 


60% Efficiency of 




1 




Web; 3^ Area Con- 


Web; 3^ Area Con- 


Web ;xin Area Con- 






sidered at Upper 


sidered at Upper 


sidered at Upper 




w 


. 


and Lower Edges; 


and Lower. Edges 


and Lower Edges; 




O 




o 


no Deduction for 


25% Ded 


uct'n for 40% Deduct'nfor 


Web 


'•Jj 


Rivet Holes, etc. 


Rivet Holes, etc. 


Rivet Holes, etc. 


Plate. 


^ 






























& 


^ 




Moment 




Moment 




Moment 




U 


s 




of Re- 




of Re- 




of Re- 




o 
a 


o 




sistance 




sistance 




sistance 


-' t 


^-3 

be 


o 


Section 


^'=i 


Section 


^'=i 


Section 


^^'=i 


cd 


Modulus 


Modulus 


Modulus 


& ^ 


'S 


S 




12 




12 




12 


P ^ 


^ 


< 


s^^ 


(f = 


S=^ 


(/ = 


Ad 


(f = 








6 


10000) 


8 


10000) 


10 


10000) 


d b 


17 


A 














Ins. 


Lbs. 


Sq.Ins 




Ft.-Lbs. 




Ft.-Lbs. 




Ft.-Lbs. 


26 X J^ 


22.10 


6.50 


28.167 


23472 


21.125 


17604 


16.900 


14083 


" X ^ 


27.63 


8.125 


35.208 


29340 


26.406 


22005 


21.125 


17604 


" X ^ 


33.15 


9.75 


42.250 


35208 


31.687 


26406 


25.350 


21125 


"x A 


38.68 


11.375 


49.292 


41076 


36.969 


30807 


29.575 


24646 


«x3^ 


44.20 


13.00 


56.333 


46944 


42.250 


35208 


33.800 


28167 


28xJi 


23.80 


7.00 


32.667 


27222 


24.500 


20417 


19.600 


16333 


"xA 


29.75 


8.75 


40.833 


34028 


30.625 


25521 


24.500 


20417 


-x^ 


35.70 


10.50 


49.000 


40833 


36.750 


30625 


29.400 


24500 


"x^ 


41.65 


12.25 


57.167 


47639 


42.875 


35729 


34.300 


28583 


"xK 


47.60 


14.00 


65.333 


54444 


49.000 


40833 


39.200 


32667 


30 x^ 


25.50 


7.50 


37.500 


31250 


28.125 


23437 


22.500 


18750 


" X ^ 


31.88 


9.375 


46.875 


39062 


35.156 


29297 


28.125 


23437 


"X3^ 


38.25 


11.25 


56.250 


46875 


42.187 


35156 


33.750 


28125 


"x^ 


44.63 


13.125 


65.625 


54687 


49.219 


41016 


39.375 


32812 


''xH 


51.00 


15.00 


75.000 


62500 


56.250 


46875 


45.000 


37500 


32 X J^ 


27.20 


8.00 


42.667 


35556 


32.000 


26667 


25.600 


21333 


"x^ 


34.00 


10.00 


53.333 


44444 


40.000 


33333 


32.000 


26667 


" x^ 


40.80 


12.00 


64.000 


53333 


48.000 


40000 


38.400 


32000 


" X ^ 


47.60 


14.00 


74.667 


62222 


56.000 


46667 


44.800 


37333 


" X ^ 


54.40 


16.00 


85.333 


71111 


64.000 


53333 


51.200 


42667 


SQxH 


30.60 


9.00 


54.000 


45000 


40.500 


33750 


32.400 


27000 


"X A 


38.25 


11.25 


67.500 


56250 


50.625 


42187 


40.500 


33750 


"x^ 


45.90 


13.50 


81.000 


67500 


60.750 


50625 


48.600 


40500 


" X T^ 


53.55 


15.75 


94.500 


78750 


70.875 


59062 


56.700 


47250 


. " X 3^ 


61.20 


18.00 


108.000 


90000 


81.000 


67500 


64.800 


54000 


40 xM 


34.00 


10.00 


66.667 


55556 


50.000 


41667 


40.000 


33333 


"x^ 


42.50 


12.50 


83.333 


69444 


62.500 


52083 


50.000 


41667 


" x^ 


51.00 


15.00 


100.000 


83333 


75.000 


62500 


60.000 


50000 


"xt^ 


59.50 


17.50 


116.667 


97222 


87.500 


72917 


70.000 


58333 


"x3^ 


68.00 


20.00 


133.333 


111111 


100.000 


83333 


80.000 


66667 


42 xM 


35.70 


10.50 


73.500 


61250 


55.125 


45937 


44.100 


36750 


"x^ 


44.63 


13.125 


91.875 


76562 


68.906 


57422 


55.125 


45937 


"x^ 


53.55 


15.75 


110.250 


91875 


82.687 


68906 


66.150 


55125 


"Xt^ 


62.48 


18.375 


128.625 


107187 


96.469 


80391 


77.175 


64312 


"xi^ 


71.40 


21.00 


147.000 


122500 


110.250 


91875 


88.200 


73500 



* See Fig. 3, page 675. 



PLATE GIRDERS— WEB PLATES ONLY, 



619 



6. — Plate Girders (Steel) — Concluded. 

Properties op Web Plates Only.* 

(M' = moment in ft.-lbs.; M" = moment in in.-lbs.) 



1 


2 


3 


4 


5 


6 


7 


8 


9 




1 




Full Efficiency of 


75% Efficiency of 


60% Efficiency of 








Web; 3^ Area Con- 


web ; ^8 Area Con- 


Web ;x^tT Area Con- 




02 




sidered at Upperlsidered at Upper 


sidered at Upper 




u 




and Lower Edges; 


and Lower Edges; 


and Lower Edges; 




O 


o 


No Deduction for 


25% Deduct'nfor 


40% Deduct'nfor 


Web 






Rivet-Holes, etc. 


Rivet Holes, etc. 


Rivet Holes, etc. 


Plate. 


^ 






























£ 


^ 




Moment 




Moment 




Moment 




fe' 


2 




of Re- 




of Re- 




of Re- 




£ 


o 




sistance 




sistance 




sistance 




1 


"o 


Section 


^'=i 


Section 


^'=i 


Section 


M'Ji 




Modulus 


Modulus 


Modulus 


ft ^ 


(U 


S 




12 




12 




12 


n g 


^ 


< 


s.- 


10000) 


-f 


10000) 


^ Ad 
•^"10 


(/ = 
10000) 


d b 


w 


A 














Ins. 


Lbs. 


Sq. Ins 




Ft.-Lbs. 




Ft.-Lbs. 




Ft.-Lbs. 


44 X A 


46.75 


13.75 


100.83 


84028 


75.62 


63021 


60.50 


50417 


" x^ 


56.10 


16.50 


121.00 


100833 


90.75 


75625 


72.60 


60500 


"x^ 


65.45 


19.25 


141.17 


117639 


105.87 


88229 


84.70 


70583 


"x3^ 


74.80 


22.00 


161.33 


134444 


121.00 


100833 


96.80 


80667 


48 X A 


51.00 


15.00 


120.00 


100000 


90.00 


75000 


72.00 


60000 


"x3^ 


61.20 


18.00 


144.00 


120000 


108.00 


90000 


86.40 


72000 


"x^^ 


71.40 


21.00 


168.00 


140000 


126.00 


105000 


100.80 


84000 


"x3^ 


81.60 


24.00 


192.00 


160000 


144.00 


120000 


115.20 


96000 


50xt^ 


53.13 


15.625 


130.21 


108507 


97.53 


81276 


78.12 


65104 


" X 3^ 


63.75 


18.75 


156.25 


130208 


117.04 


97531 


93.75 


78125 


"X 3^ 


74.38 


21.875 


182.29 


151910 


136.54 


113786 


109.37 


91146 


"xK 


85.00 


25.00 


208.33 


173611 


156.05 


130042 


125.00 


104167 


54 X A 


57.38 


16.875 


151.87 


126562 


113.91 


94922 


91.12 


75937 


"x^ 


68.85 


20.25 


182.25 


151875 


136.69 


113906 


109.35 


91125 


"x^ 


80.33 


23.625 


212.62 


177187 


159.47 


132891 


127.57 


106312 


"xi^ 


91.80 


27.00 


243.00 


202500 


182.25 


151875 


145.80 


121500 


60 x^ 


76.50 


22.50 


225.00 


187500 


168.75 


140625 


135.00 


112500 


"x^ 


89.25 


26.25 


262.50 


218750 


196.87 


164062 


157.50 


131250 


"xH 


102.00 


30.00 


300.00 


250000 


225.00 


187500 


180.00 


150000 


70 x^ 


89.25 


26.25 


306.25 


255208 


229.69 


191406 


183.75 


153125 


"x^ 


104.13 


30.625 


357.29 


297743 


267.97 


223307 


214.37 


178646 


"xj^ 


119.00 


35.00 


408.33 


340278 


306.25 


255208 


245.00 


204167 


80 x^ 


119.00 


35.00 


466.67 


388889 


350.00 


291667 


280.00 


233333 


" X 1^ 


136.00 


40.00 


533.33 


444444 


400.00 


333333 


320.00 


266667 


90 x^ 


133.88 


39.375 


590.62 


492187 


442.97 


369141 


354.37 


295312 


"x>^ 


153.00 


45.00 


675.01 


562500 


506.25 


421875 


405.00 


337500 


ioox ^ 


148.75 


43.75 


729.17 


607639 


546.87 


455729 


437.50 


364583 


" X 1^ 


170.00 


50.00 


833.33 


694444 


625.00 


520833 


500.00 


416667 


120x3^ 


204.00 


60.00 


1200.00 


1000000 


900.00 


750000 


720.00 


600000 



* See Fig. 3, page 575. 



580 Zl— PROPERTIES AND TABLES OF BEAMS AND GIRDERS. 



*" 



5 I 



I 
1 



Fig. 4. 



7. — Plate Girders (Steel). 
Properties op Flange Plates only. 

M'=moment in ft. -lbs.; 
M''=moment in in.-lbs. 



1 


2 


3 


4 


5 


6 


7 


8 


9 


10 1 


11 


(0 

1 

bp u 


Lbs. 


o 

&< 

Q"S 

D 
Ins. 


O 

•SO 

d 

Ins. 


Net 

Area 

Each 

Flange 

Plate) 

A 
Sq. Ins 


3 II 

CO CO 

Ins. 


Increase of 
S for Each 
Additional 


Resist- 
ing 
Moment 

M'-// 

12y 

(For/ 
= 10000). 

Ft.-Lbs. 


Increase of 
M' for Each 
Additional 


rt > 

EcS 

Size. 
Ins. 


lin. 
Width 

of 
Plate. 


12 ins. 

of 
Depth 

*(D). 


lin. 
Width 

of 
Plate. 


12 ins. 

of 
Depth 

t(D). 



(a) Use with smaller Flange Angles. Rivets, %". Two J^" rivet-holes de- 
ducted from Each Plate for A. 



X 



H 



7 X 

•* X 

" X 

8xM 
"xA 
" x^ 
"x^ 
"x3^ 

•* X ' 
*♦ X 

"xii 

"xM 
"xit 
"xK 

"xH 

« xl 



9 

« X 

" X 

" X 

- X 



xH 
3^ 



lOx H 

"x^ 



^ 



llxM 
"x^ 
"xH 

« X5^ 



10.20 


12H 


12H 


1.06 


13.25 


3.12 


12.72 


11 040 


2 600 


15.30 


u 


12H 


1.59 


20.07 


4.73 


19.08 


16 730 


3 950 


20.40 


a 


12H 


2.13 


27.16 


6.37 


25.56 


22 630 


5 310 


25.50 


u 


UVs 


2.66 


34.25 


8.05 


31.92 


28 540 


6 710 


11.90 


12M 


12H 


1.31 


16.37 


3.12 


15.72 


13 650 


2 600 


17.85 


u 


12^8 


1.97 


24.87 


4.73 


23.64 


20 730 


3 950 


23.80 


u 


12M 


2.63 


33.53 


6.37 


31.56 


27 940 


5 310 


29.75 


u 


123^ 


3.28 


42.23 


8.05 


39.36 


35 190 


6 710 


13.60 


18M 


183^ 


1.56 


28.86 


4.62 


18.72 


24 050 


3 850 


17.00 


u* 


183^ 


1.95 


36.20 


5.80 


23.40 


30 160 


4 830 


20.40 


u 


18^ 


2.34 


43.58 


6.98 


28.08 


36 320 


5 820 


23.80 


u 


18tt 


2.73 


51.02 


8.18 


32.76 


42 520 


6 810 


27.20 


u 


ISH 


3.13 


58.69 


9.37 


37.56 


48 910 


7 810 


30.60 


u 


1811 


3.52 


66.22 


10.58 


42.24 


55 180 


8 820 


34.00 


u 


18K 


3.91 


73.80 


11.80 


46.92 


61 500 


9 830 


37.40 


u 


18i| 


4.30 


81.43 


13.02 


51.60 


67 860 


10 850 


40.80 


u 


19 


4.69 


89.11 


14.25 


56.28 


74 260 


11 870 


44.20 


u 


19^ 


5.08 


96.84 


15.49 


60.96 


80 700 


12 910 


47.60 


u 


193^ 


5.47 


104.61 


16.73 


65.64 


87 180 


13 950 


51.00 


u 


19A 


5.86 


112.44 


17.99 


70.32 


93 700 


14 990 


54.40 


" 


19M 


6.25 


120.31 


19.25 


75.00 


100 260 


16 040 


15.30 


2iH 


24K 


1.81 


44.35 


6.12 


21.72 


36 950 


5 100 


22.95 


u 


24^ 


2.72 


66.98 


9.23 


32.64 


55 820 


7 700 


30.60 


u 


24% 


3.63 


89.84 


12.37 


43.56 


74 870 


10 310 


38.25 


u 


243^ 


4.53 


112.68 


15.55 


54.36 


93 900 


12 960 


45.90 


u 


25 


5.44 


136.00 


18.75 


65.28 


113 330 


15 620 


17.00 


2iH 


24K 


2.06 


50.47 


6.12 


24.72 


42 060 


5 100 


25.50 


u* 


245^ 


3.09 


76.09 


9.23 


37.08 


63 410 


7 700 


34.00 


u 


24% 


4.13 


102.22 


12.37 


49.56 


85 180 


10 310 


42.50 


" 


24^ 


5.16 


128.36 


15.55 


61.92 


106 960 


12 960 


51.00 


" 


25 


6.19 


154.75 


18.75 


74.28 


129 000 


15 620 


18.70 


24K 


24H 


2.31 


56.60 


6.12 


27.72 


47 160 


5 100 


28.05 




24^ 


3.47 


85.45 


9.23 


41.64 


71 210 


7 700 


37.40 


" 


24% 


4.63 


114.59 


12.37 


55.56 


95 490 


10 310 


46.75 


u 


24% 


5.78 


143.78 


15.55 


69.36 


119 810 


12 960 


36.10 


u 


25 


6.94 


173.50 


18.75 


83.28 


144 580 


15 620 



96.0^ 

95.0 " 

94.1 " 

93.2 " 

96.0^ 

95.0 " 

94.1 « 

93.2 " 

64.9^ 
64.6 - 
64.4 " 
64.2 « 
64.0 « 
63.8 " 

63.6 " 

63.4 « 

63.2 " 
63.0 « 

62.7 " 

62.5 " 

62.3 " 

49.0^ 
48.7 " 
48.5 " 
48.2 « 
48.0 " 

49.0^ 

48.7 " 
48.5 « 
48.2 « 
48.0 " 

49.0^ 
48.7 " 
48.5 « 
48.2 « 
48.0 « 



* Percent of increase is same as for column 1 1. 
t Actual increase in ft. -lbs. = 10 000 A (col. 5). 



PLATE GIRDER^— FLANGE PLATES ONLY. 



S8i 



7. — Plate Girders (Steel) — Continued. 
Properties of Flange Plates only. 

M'=moment in ft. -lbs.; 
iW" =moment in in. -lbs. 



1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


11 




fee 
Lbs. 


o 

-•^ 
&< 

Ins. 


2 

Ins. 


Net 
Area 
Each 

Flange 
(1 

Plate) 

A 

Sq. Ins 


1^ 

::s II 

(=i II 
O ' 

^ II 

C/3 CO 

Ins. 


Increase of 
5 for jEach 
Additional 


Resist- 
ing 
Moment 

Ft.-Lbs. 


Increase of 
M' for Each 
Additional 


Size. 
Ins. 


lin. 

Width 

of 
Plate. 


12 in. 

of 

Depth 


lin. 
Width 

of 
Plate. 


12 ins. 

of 
Depth 

t(^). 



Two J^" rivet-holes de- 









ducted from Each Plate for A 


. 






12x 14 


20.40 


2414 


^iV?. 


2.56 


62.72 


6.12 


30.72 


52 270 


5 100 


49.0/0 


"X3/^ 


30.60 




24H 


3.84 


94.56 


9.23 


46.08 


78 800 


7 700 


48.7 " 


"xH 


40.80 


w 


243^ 


5.13 


126.97 


12.37 


61.56 


105 810 


10 310 


48.5 •* 


" X H 


51.00 


u 


247/, 


6.41 


159.45 


15.55 


76.92 


132 870 


12 960 


48.2 " 


«xM 


61.20 


u 


25 


7.69 


192.25 


18.75 


92.28 


160 210 


15 620 


48.0 " 



(b) Use with 6x3i and 6x4 Angles. Rivets, M'', TwoJi 
ducted from Each Plate for A. 



rivet- holes de- 



3^ 



13x 
" X 

« X 

«X3^ 
"X J 

" xl 

14x 3.^ 

X ,0 

" X 
« xl 



/4 
V8 



33.15 
44.20 
55.25 
66.30 
77.35 
88.40 

35.70 
47.60 
59.49 
71.40 
83.30 
95.20 



30M 



30M 



30^8 

30M 

303^ 

31 

31^ 

31M 

30^ 



30J^ 
31 

313^ 
31M 



4.22 
5.63 
7.03 
8.44 
9.84 
11.25 

4.59 
6.13 
7.66 
9.19 
10.72 
12.25 



129.24 
173.12 
217.05 
261.64 
306.27 
351.56 

140.57 
188.50 
236.50 

284.89 
333.66 
382.81 



11.48 
15.37 
19.30 
23.25 
27.23 
31.25 

11.48 
15.37 
19.30 
23.25 
27.23 
31.25 



50.64 

67.56 

84.36 

101.28 

118.08 

135.00 

55.08 

73.56 

91.92 

110.28 

128.64 

147.00 



107 700 
144 270 
180 880 
218 030 
255 230 
292 970 

117 140 
157 080 
197 090 
237 410 
278 050 
319 010 



9 570 
12 810 
16 080 
19 370 
22 700 
26 040 

9 570 
12 810 
16 080 
19 370 
22 700 
26 040 



39.2fo 
39.0 " 
38.9 " 
38.7 " 

38.6 " 
38.4 « 

39.2fo 
39.0 " 
38.9 " 

38.7 " 
38.6 " 
38.4 " 



(c) Use with 6 x 6 and 8x8 Angles. Rivets, J^". Twol'' rivet-holes de- 
ducted from Each Plate for A . 



u^y? 


47.60 


30K 


mi 


6.00 


184.50 


15.37 


72.00 


153 750 


12 810 


39.0/0 


" X H 


59.50 




80^/8 


7.50 


231.56 


19.30 


90.00 


192 970 


16 080 


38.9 " 


"x34 


71.42 


" 


31 


9.00 


279.00 


23.25 


108.00 


232 500 


19 370 


38.7" 


"x7/« 


83.30 


« 


3m 


10.50 


326.81 


27.23 


126.00 


272 340 


22 700 


38.6 " 


" Xl 


95.20 


" 


31M 


12.00 


375.00 


31.25 


144.00 


312 500 


26 040 


38.4 « 


15x ^ 


51.00 


30K 


303^ 


6.50 


199.88 


15.37 


78.00 


166 560 


12 810 


39.0/0 


" X % 


63.75 




30'^8 


8.13 


251.01 


19.30 


97.56 


209 180 


16 080 


38.9 " 


"X34 


76.50 


<( 


31 


9.75 


302.25 


23.25 


117.00 


251 870 


19 370 


38.7 " 


"X V, 


89.25 


" 


3m 


11.38 


354.20 


27.23 


136.56 


295 170 


22 700 


38.6 " 


" xl 


102.00 


« 


3114 


13.00 


406.25 


31.25 


156.00 


338 540 


26 040 


38.4 " 


"xm 


114.75 


" 


3134 


14.63 


459.02 


35.30 


175.56 


382 510 


29 410 


38.2 " 


"xlM 


127.50 


ti 


31M 


16.25 


511.88 


39.37 


195.00 


426 560 


32 810 


38.1 " 



* Percent of increase is same as for column 11. 
t Actual increase in ft. -lbs. = 10 000 A (col. 5). 



582 Zl— PROPERTIES AND TABLES OF BEAMS AND GIRDERS. 

7. — Plate Girders (Steel) — Concluded. 

Properties of Flange Plates only. 

(See Fig. 4, page 580.) 

M'=momentinft.-lbs.; 
M''=momentinin -lbs. 



1 


2 


3 


4 


5 


6 


1 


f§ 


rib 







Net 


It 

o ( 






+-> . 

Qo 




Area 

Each 

Flange 

(1 
Plate) 


Size. 


W 


D 


d 


A 


02 in 


Ins. 


Lbs. 


Ins, 


Ins. 


Sq.Ins 


Ins. 



9 



10 



11 



Increase of 
S for Each 
Additional 



lin. 
Width 

of 
Plate. 



12 ins. 

of 
Depth 

*(D). 



Resist- 
ing 
Moment 

12 :v 

(For/ 
=10000). 

Ft. -Lbs. 



Increase of 
M' for Each 
Additional 



lin. 
Width 

of 
Plate. 



12 ins. 

of 
Depth 

HD). 



(c) Use with 6x6 and 8x8 Angles. Rivets, J^". Two 1" rivet-holes de- 
ducted from Each Plate for A. 



16x H 

llu 

«xl 

"xl3^ 

"xlM 
"xl^ 

"xl3^ 

18x1^ 
"xs/g 
«x^ 

:^^ 

" xl 

"xl3^ 
"xlM 

"Xl3^ 

"xli^ 



54.40 

68.00 

81.60 

95.20 

108.80 

122.40 

136.00 

149.60 

163.20 

61.20 
76.50 
91.80 
107.10 
122.40 
137.70 
153.00 
168.30 
183.60 



30M 



36M 



30M 

303^ 

31 

313^ 

31^ 

31^ 

31M 

31^8 

31M 



37 

373^ 
37K 
37^ 
37^ 
37K 
37M 



7.00 
8.75 
10.50 
12.25 
14.00 
15.75 
17.50 
19.25 
21.00 

8.00 
10.00 
12.00 
14.00 
16.00 
18.00 
20.00 
22.00 
24.00 



215.25 
270.16 
325.50 
381.28 
437.50 
494.16 
551.25 
608.78 
666.75 

294.00 

368.75 
444.00 
519.75 
596.00 
672.75 
750.00 
827.75 
906.00 



15.37 
19.30 
23.25 
27.23 
31.25 
35.30 
39.37 
43.48 
47.62 

18.37 
23.05 
27.75 
32.48 
37.25 
42.05 
46.87 
51.73 
56.62 



84.00 
105.00 
126.00 
147.00 
168.00 
189.00 
210.00 
231.00 
252.00 

96.00 
120.00 
144.00 
168.00 
192.00 
216.00 
240.00 
264.00 
288.00 



179 370 
225 130 
271 250 
317 730 
364 580 
411 800 
459 380 
507 320 
555 620 

245 000 
307 290 
370 000 
433 120 
496 670 
560 620 
625 000 
689 790 
755 000 



12 810 
16 080 
19 370 

22 700 

26 040 
29 410 
32 810 
36 240 
39 690 

15 310 
19 210 

23 120 

27 070 
31 040 
35 040 
39 060 
43 110 
47 190 



39.0^ 
38.9 ** 

38.7 •* 
38.6 " 
38.4 " 
38.2 « 

38.1 " 
37.9 « 

37.8 " 

32.79& 
32.6 " 
32.4 " 
32 3 " 

32.2 « 
32.1 « 
32.0 " 

31.9 « 
31.8 " 



* Percent of increase is same as for column 11. 
t Actual increase in ft.-lbs. = 10 000 A (col. 5). 



BETHLEHEM GIRDER BEAMS, 



• u 

m o 

W O 

PQ o 



•S3[J«UI8H 



I o ^ 5l <» 43 



•-g 03 



*H "■' —. 

Ilia 






&: 






^ rt O 03 
'^ m 









^: 






^02 
o 



<o o 



II X^MM 



go 



ft^ 



w a 5° 
11 ^ M wi 



<u o 



^ 



as 

o 



(^ 






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584 31— PROPERTIES AND TABLES OF BE A MS AND GIRDERS. 



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REINFORCED CONCRETE BEAMS. 585 

Reinforced Concrete Beams. — The following Working Stresses for Static 
Loads are recommended by the Special Committee of the Am. Soc. C. E., 
on Concrete and Reinforced Concrete. (See Trans. A. S. C. E., Vol. LXVII., 
page 452.) For Notation and Formulas, see Sec. 25, Masonry, page 446. For 
working stresses for columns, see Sec. 32, Columns, page 609. 

Compressive Strength of Stone Concrete.— Average, 2000 lbs. per sq. in. 
at 28 days, when tested in cylinders 8 ins in dia. and 16 ins. long, under 
laboratory conditions of manufacture and storage. 

Bearing. — 650 lbs. per sq. in. on 2000-lb. concrete. 

Bending. — Compression on extreme fiber of beam, 650 lbs. per sq. in. for 
2000-lb. concrete ; adjacent to support of continuous beams, 750 lbs. per sq. in. 

Pure Shear. — When uncombined with compression normal to the shear- 
ing surface, and with all tension normal to the shearing plane provided for 
by metal reinforcement, 120 lbs. per sq. in. on 2000-lb. concrete. 

Shear with Diagonal Tension.— In beams without diagonal reinforcement, 
40 lbs. per sq. in. for 2000-lb. concrete. 

Bond. — Between concrete and plain reinforcing bars, 80 lbs. per sq. in. 
for 2000-lb. concrete; between concrete and drawn wire, 40 lbs. per sq. in. 

Reinforcement. — Tensile stress in steel shall not exceed 16000 lbs. per 
sq. in. 

EXCERPTS AND REFERENCES. 

Tests of Reinforced Concrete Beams (By W. K. Hatt. Proc. A. S. 
T. M., 1902; Eng. News, July 17, 1902). — This is a very complete Paper 
and contains numerous diagrams and tables of the tests, with discussion 
of same; also includes Hatt's formula for reinforced concrete beams. 

Computing the Strength of Reinforced Concrete Beams (By Edwin 
Thacher. Eng. News, Feb. 12, 1903). — Discussion of above. "It is the 
practice of the writer to give the concrete a certain factor of safety at the 
end of one month, and to give the steel the same factor of safety as the 
concrete at the end of six months, as it is evident that if there was not an 
excess of steel in one month, there would be a deficiency after the concrete 
gained its full strength. For example, to design a slab of length /=6 ft., 
using 60000-lb. steel, for a total load w of 400 lbs. per sq. ft., that shall 
have a factor of safety of 4 in one month, but in which the tensile strength 
of the steel shall be equal to the compressive strength of the concrete after 
6 months: — ^Then, the distance from top of beam to center of reinforcement 

= (i= v//2^--883= ^6X6X400-^883= 3.92 ins.; or, if the center of bars is 
1.08 ins. from bottom of concrete, the total height /j= 3.92-1- 1.08 = 5 ins., 
and the area of metal for I" width of beam = A=£/-^90= 3.92-r-90 = .0435 
sq. in. Using Thacher bars, having original diam. of f in., area=.276, 
weight per ft. = .94 lb., dist. c. to c. bars = . 276 -i-. 0435= 6.35 ins." Gives 
tabulated formulas for calculating rein. cone, beams (£s= 30,000,000). 

Table of Tests on the Union Between Concrete and Steel (At Mass. 
Inst. Tech. Eng. News, May 5, 1904). — ^Types of rods used in the tests 
were: Thacher, Ransome, Johnson, and plain round and square rods. 

Tests of Reinforced Concrete Beams (By Arthur N. Talbot. Proc. 
A. S. T. M., 1904. Eng. News, Aug. 11, 1904).— Types of rods used in the 
tests were: Thacher, Ransome, Johnson, Kahn, dnd plain round and square 
rods. Table and Diagrams. 

Tests of Reinforced Concrete Beams and Floors (At St. Louis Expo- 
sition. By R. L. Humphrey. Eng. News, Sept. 21, 1905). — Illustrated 
diagrams of beams tested. 

The Design of Reinforced Concrete Structures (By F. W. Keyser and 
E. L. Heidenreich; and C. A. P. Turner). — Discussion. 

Notes on the Stress-Deformation Curve in Concrete Beams (By 
N. Werenskiold. Eng. News, April 5, 1906.) — Diagrams and formulas. 

Economic Design of Reinforced Concrete (By F. W. Hanna. Eng. 
News, Feb. 21, 1907). — Gives a table of the economical working stresses 
and percentages of steel reinforcement for varying relative costs of steel 
and concrete — calculated for unit compressive working stress in outer fiber 
of concrete = 600 lbs. per sq. in., and for r = £8^£^o= 10. (See, also, Eng. 
News, June 20, 1907.) 



586 Zl— PROPERTIES AND TABLES OF BEAMS AND GIRDERS, 

Diagram for Proportioning Reinforced Concrete Beams (By A. H. 

Perkins. Eng. News, April 25, 1907). — It is a compound diagram, com- 
prising (1) a set of lines giving the bending moment in a simple beam for 
any span and any load per foot of length and per inch of width; (2) a set 
of lines giving the corresponding values of breadth and depth of reinforced 
concrete beam to resist this moment. This latter part of the diagram is 
based on the following unit values: — Max. comp. in concrete c = 600 lbs. 
per sq. in.; tension in steel s= 12,000 lbs. per sq. in.; Es-^Ec = n== 10; 
percentage of steel ^ = 0.95%; moment capacity of beam b inches wide 
and d inches deep = M= 100 bd^. The stress-strain curve of concrete is 
assumed to be between the straight line and the parabola. (See, also, Tiech- 
man's formula and diagrams in Eng. News, July 11, 1907.) 

Reinforced Concrete T-=Beam and Column Tests (At Univ. of HI. 
By A. N. Talbot. Bulletins Nos. 10 and 12 of the Eng. Exper. Station, 
Univ. of 111.; Eng. News, July 11, 1907). — Diagrams of beams and columns 
tested, and tables of tests. 

Tests of Adhesion of Steel to Concrete in Beams (By L. J. Johnson. 
Jl. of Assn. of Eng. Soc. for June, 1907; Eng. News, Aug. 15, 1907).— Tables 
and diagrams. 

Effect of Time Element in Loading Reinforced Concrete Beams (By 

W. K. Hatt. Proc. A. S. T. M., 1907; Eng. News, Oct. 24, 1907).— Tables 
and diagrams. 

Section Modulus Diagram for Plate and Lattice Girders (By L. R. 

Shellenberger. Eng. News, Nov. 26, 1908). — ^The diagram is for flange 
angles 6''x 6"; and for section moduli up to 3400 ins., and depth of web up 
to 85 ins. 

Tests of Standard I Beams and Special I Beams and Girder Beams (By 

Edgar Marburg. Proc. A. S. T. M., Vol. IX., 1909; Eng. News, Aug. 12, 
1909). — Illustrations, diagrams and tables. Generally speaking, the modu- 
lus of rupture in lbs. per sa. in. decreases as the depth of the girder increases. 
In actual practice, however, the beams would be supported laterally, at 
intervals, which might affect the comparative results of the experiments. 

Moments in Continuous Concrete Beams Under Uniform Loading (By 

R. E. Spaulding. Eng. News, Sept. 30, 1909). — Numerous formulas and 
diagrams. 

Slide=°Rule for Reinforced-Concrete Slabs (By A. W. French. Eng. 
News, Feb. 3, 1910). — Illustrated. 

Table of Moments of Inertia of Flat Rectangles (By H. Loewenhern. 
Eng. News, Feb. 24, 1910). — For widths b (advancing by one inch from 6 to 
40 ins.) and depths d (advancing by sixteenths of an inch from J to 3 inches). 

Some Deductions from Warburg's I-Beam Tests (By C. J. Tilden. Eng. 
News, Feb. 24, 1910). — With discussion by Edgar Marbvirg. 

Comparison of Methods of Computing the Strength of Flat Reinforced 
Concrete Plates (By A. B. MacMillan. Paper, Natl. Assn. Cement Users, 
Feb. 21-25, 1910; Eng. News, Mar. 31, 1910). — Illustrated. Discussions: 
Cantilever method; Tumeaure and Maurer method; Grashof's Analysis; 
Mensch's method; Turner's method; Macmillan method; Mushroom floor. 

Interesting Illustrations. 

Description. Eng. Rec. 

Diagrams for reinforced-concrete beams. — Schermerhom Mar. 6, '09 

Diagrams for disigning reinforced-concrete beams — Carter June 18, '10 

Instruction sheet for placing floor reinforcement June 25, '10 



32.— PROPERTIES AND TABLES OP 

COLUMNS. 

General Stresses.— A column is a pillar or strut acting mainly in com- 
pression; but in addition to the direct compressive stress, it has also to 
resist, to a greater or less extent^ shearing and bending. To what degree 
the shearing stresses enter into and affect a column we are unable to 
say, and no column formula has yet been devised which includes the shear- 
ing effect of the stresses. Indeed, omitting the effect of shear altogether, 
it is a mooted question whether we are able even to combine rationally in a 
single formula the effects of direct compression and bending. 

Shearing Effect. — Although the effect of shear is seldom noticeable in an 
ordinary column, yet it becomes quite apparent in a short block or cube of 
stone, cement, or other granular material when tested in compression. 
The sides of the block flake away, often leaving the broken specimen like a 
shattered pyramid or cone. The same shearing stresses which rupture the 
block of stone are undoubtedly present in complex form in any column, 
but our means of analysis are limited. If, instead of the stone, we select a 
short block of wood, with the grain in the direction of the pressure, the 
shearing will be along the grain, or planes of least resistance, although the 
tendency to shear will be along sloping, pyramidal planes. 

If, now, a sheet of lead is interposed between the y///'/////////,v'///^///////y<y 
stone cube and one of the platforms of the testing \m! /'//// '//////'/'/A ' '-^cicj 

machine the flaking of the stone will be more pro- 
nounced, due to the spreading or lateral expansion 
of the lead as it is flattened against the stone.* This 
illustrates in an exaggerated way how the material in 
any column tends to spread out under pressure, and 
weaken the resistance to shear. 

Fig. 1. 
Notation for Column Formulas : 

A = area of column section, in sq. ins.; 
/ = moment of inertia of section in ins. =A r'^', 
r = radius of gyration of section in ins. = \//t->1; 

c? = least diam. of rectangular section, in ins.; 

(io = diameter of round column, in ins.; 

P-= total load on column, in lbs.; 

/? = load per sq. in. on column = Ph- A; 

3; = distance, in ins., of most strained fiber, from neutral axis; 

^; = lateral deflection of column in ins., under load P\ 
e == eccentricity of loading on columns, in ins. ; 

/c = compressive stress in lbs. per sq. 'ni. = P-^A=p\ 

/b=bending stress, outer fiber, in lbs. per sq. in. = Pvy -^ I '=pvy-^r^', 
f = max. outer fiber stress, comp. 4- bending, in lbs. per sq. in. =/c+/b; 

JQ — f at the elastic limit of the material ; 

/u = f at the ultimate strength of the material; 

/' = / at any assignable value, for distinction; 

£ = modulus of elasticity of material, in lbs. per sq. in.; 
/ = effective length of column, in inches; 

L = effective length of column, in feet; 

nit^ E r^ nn^ E r^ 

na = value of p in Euler's formula for long columns = tz — = ^.. r„ : 

l^ 144 L,^ 

In which w = 1 for 2 pivot ends — established by Euler, 

» = V3for 2 pin ends — established by Johnson, 
w = V for 1 pivot, 1 fixed . — estabhshed by Euler, 
w = ft for 1 pin, 1 flat — established by Johnson, 
n = 6/2 for 2 flat ends — established by Johnson, 
w =- 4 for 2 fixed ends — established by Euler. 




* For this reason sheets of lead or other soft material are now considered 
objectionable in testing machines; and, by some, on masonry abutments 
under bridge shoes or pedestals. 

587 



5SS 



^.—PROPERTIES AND TABLES OF COLUMNS. 



Short Strut — Direct Compression Only. — By a short strut is meant one 
tvhose length is not over, say, 12 or 15 times its diameter or least dimension. 
For such a column we have the formula: 



'=1 



Whence P = fA and A 



(1) 
.(2) 



Short Strut — Eccentric Loading — ^The usual formula for the outer fiber 
stress / in a short column eccentrically loaded, is 



^^P_^Piy^P(^^ey\ 
' A^ I AV W 



(3) 



it being assumed that the column is braced laterally, as in a 
building, against falling over; and furthermore that there is no 
appreciable lateral deflection v. If lateral deflection were also 
taken into account we would have 

P . P{e + v)y P[, , (efvh^l ^^^ 



^-Z+""^T^=l[i + 



Long Columns — Bending Only. — ^The following formulas for 
end conditions of long columns, bending only, are adapted 
from Eiiler and T. H. Johnson. 

Euler's Formulas'. 

Ttmi TZ^EI 



2 Pivot Ends 



P=- 



/2 

7r2Er2 



144 L2 



--^=(T)-(iii:)'^ 



f ^ IQn^EI 
1 Pivot, I ^= 9/2 



1 Fixed i 



j^EI_ 
81 L2 



167r2£:r2 /4;rr\2^ / ^rr \ 2_ 



2 Fixed Ends 



P = 



47r2£:/ n^EI 



36 L2 




U-^^-CfY^-iW^ 



Fig. 3. 



2 Pin Ends 
O O 

1 Pin, 1 Flat 
O D 

2 Flat Ends 

D a 



5;r2£:/ 



Johnson's Formulas: 
5tz^EI 



L - 


3/2 


432 L2 


P 


Sn^Er^ 
3/2 


4(t) 


P 


257r2£:/ 
12/2 


257r2£/ 
1728 L2 


P 


25:r2£:r2 
12/2 


25 /7rr\ 
"12 U/ 


P 


57r2£:/ 

2/2 


Sn^EI 

~288L2 


P 


57r2£;r2 

12 


4(t) 



b 



I 

I 
I 
I 
I 

Fig. 2. 
(5) 

(5a) 

(6) 

(6a) 

(7) 

(7a) 

(8) 



5 

432 



(!) 



25 

1728 



(f) 




COLUMN FORMULAS, 



5ad 



There is one peculiar feature about the preceding formulas, namely, that 
they do not take into consideration the compressive (nor the shearing) 
resistance of the material, but simply the bending resistance or stiffness of 
the column. Hence they give excessive values for the loads P or p. Euler 
attempted to remedy this defect by placing an upper limit to the value of p, 
equal to the compressive resistance fc of the material. This is illustrated in 
Fig. 5, a diagram showing the (elastic) strength of steel columns with 
pivoted ends. 



65,000 
60,000 
55,000 
50,000 
45,000 
p. 40,000 
o 35.000 

2 30,000 

^ 25,000 

eo,ooo 

15,000 

10,000 

5,000 











\ 


















\ 

1 


































































T 








\ 










Eulenp-:Q 












"^^ 


p^^i. 




% 












^4 




-\^ 












-V.^ 


;^:]%^ 


















'^^^ 




.^ 






















'"-^^ 


::-^ 


5^ 


==s 






















:ii 



20 40 60 80 100 120 140 160 l80 200 
Values of p. 



Fig. 5. 



Ritter's Formula for Columns. — ^The following formula was proposed by 
Ritter in 1873 and has been worked out by others later. Although very 
ingenious and possessing considerable merit the reasoning by which it is 
deduced is not considered strictly rational and it takes its place with all 
other formulas in this respect. The equation of the elastic strength of 
columns is 

^ = A = .^-7^=— 77 ^^^^ 



14- 



nn'^Er^ 



na 



The curve of this formula, for pivoted ends, is platted in Fig. 5. 
method of applying a factor of safety to formula (11) is 



1 + 



na 



The 



(12) 



in which f is considered to be the working stress of the material, and /' any 
value between the working stress and the ultimate stress. It is to be noted 
that with / constant, the factor of safety increases as /' increases, but not 
proportionately. Strictly speaking, the formula applies only within the 
elastic limit of the material, i. e., when /' (and /) do not exceed /e. The 
value of f should be above any possible value of f, and should always be 
greater than 2/ for maximum static loading, or its equivalent. Again, with 
f'>f the factor of safety is greater for long columns than for short. 



590 d2.— PROPERTIES AND TABLES OF COLUMNS. 

Author's Formula for Columns. — From mechanics, /=fc+/b=-T-H — 7^= 

p {l+~o) . whence 
V rV P f 

'"'A-TT^ (13) 

Again, the bending moment at center of column is 

M = Pv (14) 

nn^ EI 
But according to Euler, P = — ^ — , whence (14) reduces to 

M MI2 

^=p=;^^^ (15) 

Substituting this value of v in equation (13) we have 

^^ ^4.My (-I—\ ^TIMi ^^^^ 

From the theory of flexure, we have, 

^=7^''=^('-^<=)=7 0-|)=i(^-^) (") 

Whence, by substitution, equation (16) reduces to 

P- 7—T=^ A T- (18) 



na \ na/ na 



So far, our reasoning seems to be correct, but when we attempt to 
solve equation (18) we find that p = f or na. From an examination of the 
last form of equation (18) it will be seen that the first term p, equal to the 
load per sq. in. on column, would be decreased by substituting unity for 

the quantity ( 1 — — 1 in the denominator, and hence would be on the side of 

safety. Moreover, it would reduce to Ritter's formula (12), by making f=f; 

thus, p = 7- . But from the above discussion, we are certain that, for any 

na 

column of given length I, p< na, and 1 1 — — ) < unity. Assuming tenta- 

\ na/ 

tively the value of Ritter, p= j-, in order to arrive at an approximate 

na 



value of 1 1 ) in the second form of equation (18), we have p + — =i\ 

\ na] ^ "^ na ' 

whence, by transposi 

equation (18) gives*'' 



fp p p 

whence, by transposition, p=f , or 1 =-t\ and this substituted in 

na na f 



^=rfT (i»> 

/ na 

This reduces still further to 

p=f (sec ^-tan d) (20) 

/e 
the desired formula; in which tan 6 = -^ — . 

2na 

In using formula (20), fi nd 
1st. The value of Vtan 6, from Table 1, on the following page. 
2nd. The corresponding value of sec ^ — tan 6, from Table 2. 
3rd. Multiply this last value by / to obtain the allowable load />persq. in. 
on column ; / being the allowable working stress per square inch 
for a short column. 
Thus, by the use of Tables 1 and 2, the value (sec. 0— tan 6) in equation 
(20) can be transformed into a numerical quantity, or decimal factor. See 
Examples, page 592. 



COLUMN FORMULAS. 



591 



1. — Values op Vtan d in Column Formula, Equation (20) 

For Various Materials and End Conditions op Columns. 

[Use this table in connection with Table 2.] 



Material. 



/. 

With 

Factor of 

Safety. 



Values of Vtan d. 



End Conditions of Column. 



Pivot Ends. Pivot&Fix'd Fixed Ends. 



Any Material 
Wrought Iron. I 
Soft Steel.. . . 
Medium Steel 
Hard Steel I 



Cast Iron 
(hollow). 



2.70 



8 000(5) 
10 000(4) 



25 000 



28 000 000 



10 000(5)1 30 000 
12 500(4) /29 000 000 



12 000(5) 
15 000(4) 

14 000(5) 
17 500(4) 



10 000(8) 



Timber. 



Any Material. 
Wrought Iron . < 

Soft Steel 

Medium Steel. . 
Hard Steel [ 



Cast Iron 
(hollow) . 



Timber. 



8 000(5) 
10 000(4) 

10 000(5) 
12 500(4) 

12 000(5) 
15 000(4) 

14 000(5) 
17 500(4) 



10 000(8) 



35 000 



30 000 000 
40 000 



31 000 000 



\250 



50/ 



1 
333 



/ 1120 
1 

967 
1 

857 



1 

775 



\250/ 



1 
333 



.081- 
r 



.087 



.092- 

r 

.097- 
r 



2.025^ /| . - 



^■^^!i 



.061 



.065 



Red. 



CO 



0.51- 



Round 



0.59 4^ 

do 



Pin Ends. 



W^-7 



.062 



.067 



.071- 
r 



.075 



Red. 



0.40-^ 



Round 



0.46^ 



.073- 
r 



.040 



.043 



.046- 
r 



Red. 



CO 

o 



0.38^ 
d 



Round 
o 



0.44^ 



do 



Pin & Flat. 



^•«^V& • 7 



.056 



.060 



.064- 
r 



.067 



Red. 
o 



0.35^ 

a 



Round 



0.41 X 



.049 



Red. 

CO 
O 



0.26- 



Round 



0.30: 



do 



Flat Ends. 



-VS-7 



.051 



.055 



.058 



.061- 
r 



Red. 
o 



0.32^ 

Oo 



Round 
o 



0.37^ 

Oo 



* See column 4, Table 7, Strength and Resistance of Materials, page 495, 
for values of /. See also Table 3, following. 



592 



l—PROPERTIES AND TABLES OF COLUMNS. 



2. — Values op (Sec ^ — tan 6) in Column Formula, Equation(20) 

For Successive Values op Vtan ^.Table 1. 

[Values of V^tan 6 may be derived from Table 1.] 





1 


1' 






«5i 

1 


<5S 






> 


<5> 






1 


<55 

C/3 




0.00 
0.10 
0.20 
0.30 
0.40 
0.50 


1.000 
.990 
.961 
.914 
.853 
.781 


.010 
.029 
.047 
.061 
.072 


0.50 
0.60 
0.70 
0.80 
0.90 
1.00 


.781 
.703 
.624 
.547 
.477 
.414 


.078 
.079 
.077 
.070 
.063 


1.00 
1.10 
1.20 
1.30 
1.40 
1.50 


.414 
.360 
.313 
.274 
.240 
.212 


.054 
.047 
.039 
.034 
.028 


1.50 
1.60 
1.70 
1.80 
1.90 
2.00 


.212 
.188 
.168 
.151 
.136 
.123 


.024 
.020 
.017 
.015 
.013 



Examples in the Use op Tables 1 and 2. 
[From Column Formula, p = f (sec ^ — tan 6), equation (20).] 

Ex. 1. — Find the safe load, factor 4, of a medium-steel z-bar column 
with flat ends, whose length is 26 ft., sectional area 24.8 sq. ins., and radius 
of gyration 2.6. 

Solution.— From Table 1, Vtan 5 = .058 — = .58; and from Table 2, the 

r 
corresponding value of (sec ^ — tan ^) = .72; hence the safe working load 
^ = .72X 15 000=10,800 lbs. per sq. in., and the total safe load=10 800 X 
24.8=267 840 lbs. 

Ex. 2. — Find the safe load, factor 5, of a vertical post of Douglas spruce 
in a Pratt steel combination highway bridge, said post being 12X14 ins., 
28 ft. long, and with pin end bearings. 

Solution. — ^The value of f for Douglas spruce (col. 4, Table 7, page 495) 
is 1400. From Table 1, Vtan 6==^% • fi =.933; and from Table 2, the cor- 
respondmg value of (sec ^-tan 6) is .456; hence total allowable load 
P = .456X 1400X12X14= 107 251 lbs. (See also Table 3.) 

Ex. 3. — What load would be carried by a column similar to that in 
Ex. 2, but with flat ends? 

Solution.— Table 1: t%% • f i =.747; Table 2 equivalent = .5 88; hence 
P = .588X 1400X12X14= 138 298 lbs. 



Gordon's Formula for Columns. — Probably no other formula has been 
so universally accepted as that of Gordon — sometimes called Rankine's 
formula. Adopting the previous notation, it is deduced as follows: The 
maximum outer fiber stress due to both compression and bending is 
P 
A 

P 
whence — = p= — '- — (22) 






1 + 



vy 



Now the deflection v of the column is an unknown quantity, but Gordon 

assumed it to be proportional to — , which assumption would be allowable 

if the total fiber stress f were due to bending only. Substituting this propor- 
tional equivalent of V in equation (21) and supplying the coefficient c whose 
value is to be determined by experiment, we have the general form: 



P = 






(23) 



GORDON'S FORMULA. STRAIGHT-LINE FORMULA. 593 

For the working formula, the value assigned to fis usually the ultimate 
strength per sq. in. of a short column of the material, divided by factor of 
safety, as 4, 5, '6, etc. The coefficient c is an arbitrary constant determined 
by trials in fitting the curves of the formula to plattings from actual tests 
of strengths of columns up to the points of failure, and_ rnaking / = ult. 
strength -^ factor of safety. (See Sees, on Bridges and Buildings for special 
application of Gordon's formula; also the tables following under this 
section.) 

C. Shaler Smith's Formula for Wooden Columns. — ^This formula gives 
values much too low for long columns. It is reproduced here more for its 
historical, than for its actual working value. The formula is as follows: 

/ 



P = 



i-f-.oo4f; 



(24) 



the ultimate strength of white pine being assumed at /„= 5000. The ends 
of the column are to be flat and firmly fixed; with concentric loading. 

Straight=Line Formulas. — Curved-line formulas, previously explained, 
may be reduced to straight-line formulas by finding the equation of the 
tangent to the curve at the point of contra-flexure. Thus, in Fig. 5, page 589, 
c is the point of contra-flexure of the curve, and its tangent at that point cuts 
the coordinate axes of the diagram at T and t', the resulting straight-line 
formula for the elastic strength of the steel column with pivot ends thus 
reducing to 

^=35 400- 169— (25) 

r 

in which 35 400= 1.18 fe= 1.18X 30 000; E being assumed at 30 000 000 in 
the present instance. If now this tangent is swung slightly on the point c 
so that the point T is lowered to 34 000, it will practically fit the curve for 
a long distance, with the resulting equation 

^=34 000- 150— (26) 

r 

Straight -Line Formulas for Steel and Wrought Iron Columns. 

(Reduced from data in Tables 1 and 2.) 



Material. 


Ulti- 
mate 
Stren'th 
/u 


Elastic 

Limit. 

/e 


Elastic Strength of Column. 
Pounds per sq. in. 


Pin Ends. 


Pin & Flat. 


Flat Ends. 


Wrought iron 

Soft steel 

Medium steel. . . . 
Hai-d steel 


40000 
50000 
60000 
70000 


25000 
30000 
35000 
40000 


28500- 95- 
r 

35000-1 30-- 

r 

41000-160-^ 
r 

46500-185-^ 
r 


28500- 85-^ 
r 

35000-115-^ 

41000-145-^ 
r 

46500-170-^ 
r 


28500- 75y 

35000-100-^ 
r 

41000-130y 
46500- 155j 



594 



■PROPERTIES AND TABLES OF COLUMNS, 



z 


C3 


>i 


^-' 


!3 

o 


o 


O 


1^1 


z 


:3 


w 

Q 


W 


n 




o 


<3i 


^ 


a 




CTt 


Pi 


+j 


o 


1 


(K 


<I5 


lO 


() 




<v 


(Vi 


m 


o 




H 


«+~. 


O 


II 




^ 



^ a 

< 

O 

w 



T 



■§ 



^ o 

- & 
c (u 

II 









C3 K2 r^ 






23^ 









eooocoooeooocoooco05lr^ot»fooc^■^■l-^05eor»<e^aoost»^o 



ooojo>oocot^t^«o«o«oiOirt->*i-^T}<Tj<eocoeocr?ooc<ic>ac3pq 



e^q^Oieoe<lOOlr5coe^3■*oo^c^ooocoQO«o«•^«oo-*o«oeo 
♦ •♦- * 



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CO /-s 









cviooin>'^oo-«^oeocooiir5e»ciooicie>qo5t^Tj<c<ioii>.mooevic>oo 



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coot>.-^oc^coot^coot:^Ti<r^ooiir3ooooo«o-^cci»-i<3»t~«o 
cjo»oooooot>.t^i>.cotc>5omiaiOTj<'^-^Tj<cooocioe»3co«MesiM 






i,-i»HiFH»-n-<T-«i-i^HcvjeapqeQcsiesiesieac>acsicococoeomco 



WOODEN COLUMNS— SAFE LOADS. 



595 



e^e<ic<jc<ie^i-i,-H.-<^i-ii-(i-Hi-iT- 



NC>aeVH-HT-(»-(rHi-H,-Hr-l,-l^,-Hi-li-l 



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eNqC>ai-C1r-(T-(,-li-li-4T-(.-li-(l-Hi-lTH 



cqooJt>i«okO->i<eoc>qi-ioajoot>.t~ 
eocooacQcsiesic>aNcsic^c>a>-iT-H,-H»-( 



Ncacv]cqcQ«Me^acQeo»-ti-Hi-i^H^,-H 



mC<lOO«0«D«OCOOOOT-irJ*l>iO'^00 



m'^i>qT-iC)050000t--«omif5Tj<T^cr5 

c<ieMe^qpac>qi-H.-ii-Hi-i»-<i-iT-(»-iT-<,-i 



CC1^00SOOOOl>.COl010-^}<COCOC<l(M 



<ot>iOoa)Oi-4cqeo'^»«<oi>.oooso 



On 

— 98~0^SI 
I 

Op 
T 



Op 

— Q'28— 08^1 
1 

Op 

— 068—08^1 
T 



08—0281 



098—0281 



S'Zg— 0121 
088— 0T21 



Op 

— 22- OOIT 
1 



-008— OOTT 



82—0^1-1 



— 988— OZl^I 



— 92—5981 



— 2T8— S98I 



1^2-0921 



—882—0921 



— 22— SQTI 



■^92- 6QTI 



— 02— OQOT 



■OK— OQOI 






(V 

a 

ii 
§1 



d 

2 H 

o o 






p. 2 
o d 

eS « ^ § 

""•"do 
d d 03 o 

ail's 

55o2 

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o o^ o 

d d c3 ce 



SBiSnoQ 



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UBip-BU^O 



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596 



^.—PROPERTIES AND TABLES OF COLUMNS. 



Steel Columns. — From the preceding discussion of Column Formulas we 
might readily conclude that, for a column of given sectional area A and 
length /, that column is the strongest which has the greatest radius of 
gyration r. But this is not always true. To illustrate: On page 528, 
Fig. 26, we find the properties of th e Hollow Circle, of outside diameter d, 

and inside diameter c?i, giving r = — i- and A = 0.7854 (<i2 — d^^). Now 

if we let A remain constant and gradually increase d^ from zero upward 
until it approach d, which must also gradually increase, but less rapidly, — 
then r likewise increases in value from r =-0.25 d (when the column is a solid 
circular cylinder) up to its maximum limit r = 0.3535 d (when the column 
becomes a cylindrical shell whose thickness approaches zero). But it is 
quite apparent that before the maximum limit of ris reached, the loaded 
column will fail through auxiliary or secondary stresses, set up in the thin 
metal shell, causing it to buckle and collapse. 

The Secondary Stresses, above described, may be provided against by 
either (1) Making the metal thick enough, throughout, to withstand them; 

(2) Providing longitudinal ribs, as in the case with the Phoenix column (see 
Table 11, page 604) and other standard built-up columns, in general; 

(3) Inserting (transverse) diaphrams at intervals along the column, as is 
done with very large sections. 

Where Columns are Latticed, or where stay-plates are simply used, the 
independent (unsupported) portion should be calculated as a separate 
column with its own radius of gyration. 

Column Sections may be made up as follows: — 
(A). — Two angles riveted back to back, forming a T-section. 
(Aa). — Same as (A), but with round fillers or a plate riveted between the 

angles, giving a greater radius of gyration. 
(B). — Four angles and a web plate, forming an H-section, similar to the 

rolled H-Column of Table 14, page 608. The distance back to back 

of angles should be in whole or half inches, and from %'^ to H" 

greater than width of web plate. 
(Ba). — Same as (B), but with the addition of two channels riveted to the 

backs of the angles, with flanges of channels projecting inward. 
(Bb). — Same as (B), but with lattice bars instead of web plate. 
(C). — ^Two channels and two plates, as per Fig. 6, below. See also Tables 

8, 9, and 10, pages 601, etc. 
(Ca). — Same as (C), but with latticing instead of plates — one or both. 
(Cb). — Same as (C), but with flanges of channels projecting inward, and 

latticing or stay plates instead of cover plates. 
(D). — Same as (C), (Ca) and (Cb), but with plate and two angles instead 

of channel. 
(E). — Z-bar column. See Tables 6 and 7, pages 598, etc. 
Also other sections made up of a combination of these shapes. 

To find the moment of inertia, /, and the radius of gyration, r, of any 
column section, see Example, page 637, in connection with Tables 31 and 32, 
pages 639, etc. 

The Channel Column, Fig. 6, is standard for all classes of construction. 
The following table gives the standard dimensions: 

4. — Channel Columns — Standard Dimensions in Inches. 



Depth of 

Channel 

C 


Width of 

Plate 

P 


W 


R 


W-[-R 


E 


>iRI<-W->i<-W+R>1E|< 
r-^ . c^^ 


15 
15 


18 
16 


43^ 




7H 

m 




^=n 




r t 


12 
12 


16 
14 


5 

4 


M 


in 


m 
m 




•( 


c 


10 
10 


14 
12 

12 
10 


41^ 

3H 
2Vs 


1/^ 


in 

IP! 


13^ 

1^ 

IM 


A 




^ V 


9 
9 


Hit ' ex" 

.k- P ->l 


(X>CC 


12 
10 


3^ 
2M 




n 


IK 


Fig. 6. 



STEEL COLUMNS, 



597 






5. — Ultimate Strength op Steel Columns.* 



FLAT ends 



PIN AND PLAT 



PIN ENDS 



50000 



50000 



60000 



50000 



50 000 



50000 



(12L)2 
36000r2 
45000 



1 + 



L2 



l+.004^ 1 + 



45000 



(12L)2 
24000r2 
45000 



1+.006 



L2 



1 + 



45000 



(12L)2 
18000r2 
45000 



1+.008 



L2 



1 + 



(12L)2 
36000r2 



L2 



l+.004^ 1 + 



(12L)2 
24000r2 



14-.006 



L2 



1 + 



(12L)2 
18000r2 



45000 
1+.008-! 





tUltimate Str'gth in 1000 lbs. pr sq. in. 


L 
r 


tUItimate Str'gth in 1000 lbs. pr sq.ln. 


r 


Medium Steel. 


Soft Steel. 


Medium Steel. 


Soft Steel. 


Sqr. 


Pin& 
Sqr. 


Pin. 


Sqr. 


Pin& 
Sqr. 


Pin. 


Sqr. 


Pln& 
Sqr. 


Pin. 


Sqr. 


Pin& 
Sqr. 


Pin. 


3.0 


48.26 


47.44 


46.64 


43.44 


42.69 


41.98 


12.0 


31.73 


26.82 


23.23 


28.55 


24.14 


20.91 


3.2 


48.03 


47.11 


46.21 


43.23 


42.39 


41.59 


12.2 


31.34 


26.41 


22.82 


28.21 


23.77 


20.54 


3.4 


47.79 


46.76 


45.77 


43.01 


42.08 


41.19 


12.4 


30.96 


26.01 


24.42 


27.86 


23.41 


20.18 


3.6 


47.54 


46.39 


45.30 


42.78 


41.75 


40.77 


12.6 


30.58 


25.61 


22.03 


27.52 


23.05 


19.82 


3.8 


47.27 


46.01 


44.82 


42.54 


41.41 


40.34 


12.8 


30.21 


25.21 


21.64 


27.18 


22.69 


19.47 


4.0 


46.99 


45.62 


44.33 


42.29 


41.06 


39.89 


13.0 


29.83 


24.83 


21.26 


26.85 


22.34 


19.13 


4.2 


46.71 


45.21 


43.82 


42.03 


40.69 


39.43 


13.2 


29.47 


24.45 


20.89 


26.52 


22.00 


18.80 


4.4 


46.41 


44.80 


43.30 


41.76 


40.32 


38.97 


13.4 


29.10 


24.07 


20.52 


26.19 


21.66 


18.47 


4.6 


46.10 


44.37 


42.76 


41.49 


39.93 


38.48 


13.6 


28.74 


23.70 


20.16 


25.86 


21.33 


18.15 


4.8 


45.78 


43.93 


42.22 


41.20 


39.53 


38.00 


13.8 


28.38 


23.34 


19.81 


25.54 


21.00 


17.83 


5.0 


45.46 


43.48 


41.67 


40.91 


39.13 


37.50 


14.0 


28.03 


22.98 


19.47 


25.22 


20.68 


17.52 


5.2 


45.12 


43.12 


41.11 


40.61 


38.81 


37.00 


14.2 


27.68 


22.63 


19.13 


24.91 


20.36 


17.22 


5.4 


44.78 


42.56 


40.54 


40.30 


38.30 


36.49 


14.4 


27.33 


22.28 


18.81 


24.60 


20.05 


16.92 


5.6 


44.43 


42.08 


39.97 


39.98 


37.87 


35.98 


14.6 


26.99 


21.94 


18.48 


24.29 


19.75 


16.63 


5.8 


44.07 


41.60 


39.40 


39.66 


37.44 


35.46 


14.8 


26.65 


21.60 


18.17 


23.98 


19.44 


16.35 


6.0 


43.71 


41.12 


38.82 


39.33 


37.01 


34.94 


15.0 


26.32 


21.28 


17.86 


23.68 


19.15 


16.07 


6.2 


43.34 


40.63 


38.24 


39.00 


36.57 


34.42 


15.2 


25.98 


20.95 


17.55 


23.39 


18.86 


15.80 


6.4 


42.96 


40.14 


37.66 


38.66 


36.12 


33.89 


15.4 


25.66 


20.64 


17.26 


23.09 


18.57 


15.53 


6.6 


42.58 


39.64 


37.08 


38.32 


35.68 


33.37 


15.6 


25.34 


20.32 


16.97 


22.80 


18.29 


15.27 


6.8 


42.20 


39.13 


36.50 


37.98 


35.22 


32.85 


15.8 


25.02 


20.02 


16.68 


22.52 


18.01 


15.10 


7.0 


41.80 


38.64 


35.92 


37.62 


34.78 


32.33 


16.0 


24.70 


19.72 


16.40 


22.23 


17.74 


14.76 


7.2 


41.41 


38.14 


35.34 


37.27 


34.32 


31.81 


16.2 


24.39 


19.42 


16.13 


21.95 


17.48 


14.52 


7.4 


41.02 


37.64 


34.77 


36.91 


33.87 


31.29 


16.4 


24.09 


19.13 


15.86 


21.68 


17.22 


14.28 


7.6 


40.62 


37.13 


34.20 


36.55 


33.42 


30.78 


16.6 


23.78 


18.84 


15.60 


21.41 


16.96 


14.04 


7.8 


40.21 


36.63 


33.63 


36.19 


32.97 


30.27 


16.8 


23.49 


18.56 


15.35 


21.14 


16.71 


13.81 


8.0 


39.81 


36.13 


33.07 


35.83 


32.51 


29.76 


17.0 


23.19 


18.29 


15.09 


20.87 


16.46 


13.58 


8.2 


39.40 


35.63 


32.51 


35.46 


32.06 


29.26 


17.2 


22.90 


18.02 


14.85 


20.61 


16.22 


13.37 


8.4 


38.99 


35.13 


31.96 


35.09 


31.61 


28.76. 


17.4 


22.61 


17.75 


14.61 


20.35 


15.98 


13.15 


8.6 


38.59 


34.63 


31.41 


34.73 


31.17 


28.27 


17.6 


22.33 


17.49 


14.38 


20.10 


15.74 


12.94 


8.8 


38.18 


34.14 


30.87 


34.36 


30.72 


27.79 


17.8 


22.05 


17.23 


14.14 


19.85 


15.51 


12.73 


9.0 


37.76 


33.65 


30.34 


33.99 


30.28 


27.31 


18.0 


21.78 


16.98 


13.92 


19.60 


15.29 


12.53 


9.2 


37.35 


33.16 


29.81 


33.61 


29.84 


26.83 


18.2 


21.50 


16.74 


13.70 


19.35 


15.06 


12.33 


9.4 


36.94 


32.68 


29.29 


32.25 


29.41 


26.36 


18.4 


21.24 


16.49 


13.48 


19.11 


14.84 


12.13 


9.6 


36.53 


32.20 


28.78 


32.88 


28.98 


25.90 


18.6 


20.97 


16.26 


13.27 


18.88 


14.63 


11.94 


9.8 


36.12 


31.72 


28.28 


32.51 


28.55 


25.45 


18.8 


20.71 


16.02 


13.06 


18.64 


14.42 


11.76 


10.0 


35.71 


31.25 


27.78 


32.14 


28.12 


25.00 


19.0 


20.46 


15.80 


12.86 


18.42 


14.22 


11.58 


10.2 


35.31 


30.78 


27.29 


31.78 


27.71 


24.56 


19.2 


20.21 


15.57 


12.66 


18.18 


14.01 


11.39 


10.4 


34.90 


30.32 


26.81 


31.41 


27.29 


24.12 


19.4 


19.96 


15.35 


12.47 


17.96 


13.81 


11.22 


10.6 


34.50 


29.87 


26.33 


31.05 


26.88 


23.70 


19.6 


19.71 


15.13 


12.27 


17.74 


13.62 


11.05 


10.8 


34.09 


29.42 


25.87 


30.68 


26.47 


23.28 


19.8 


19.47 


14.91 


12.09 


17.52 


13.42 


10.88 


11.0 


33.69 


28.97 


25.41 


30.32 


26.07 


22.87 


20.0 


19.23 


14.71 


11.90 


17.31 


13.23 


10.71 


11.2 


33.29 


28.53 


24.96 


29.96 


25.67 


22.46 


20.2 


19.00 


14,50 


11.72 


17.10 


13.05 


10.55 


11.4 


32.90 


28.09 


24.51 


29.61 


25.28 


22.06 


20.4 


18.76 


14.30 


11.55 


16.89 


12.87 


10.43 


11.6 


32.50 


27.67 


24.08 


29.25 


24.90 


21.67 


20.6 


18.54 


14.10 


11.38 


16.68 


12.69 


10.25 


11.8 


32.11 


27.24 


23.65 


28.90 


24.52 


21.29 


20.8 


18.31 


13.90 


11.21 


16.48 


12.51 


10.09 



*The following notation is used in the above formulas: ^ = ultimate 
strength in lbs. per sq. in.; L = length of column in ft.; r = radius of gyra- 
tion in inches. 

t For loads as in buildings, divide by 4; for loads as in bridges, divide by 5. 



598 



l.—PROPERTIES AND TABLES OF COLUMNS. 



6. — ^*DiMENSiONS OF Carnegie Z-Bar Columns — and — 

(Without Side Plates.) 



Note. — Diameter of Bolt or Rivet, ^ in. Thick- 
ness of Web Plate = thickness of Z-Bar metal. All 
dimensions in inches. See opposite page for areas, 
weights, safe loads, etc. 




-^i.. 

Fig. 7. 



(For Columns with Side Plates, see Table 7.) 



Column. 


Thick- 
ness of 
Metal. 

t 


Notation in Cut, above. 




A 


B 


c 


D 


E 


F 


G 


H 


I 


6 -inch Col. 
4 Z-bars 3- 
3iVin.deep. 
Web pi. 6'' 
by thick- 
ness of bars 


8 


12M 
12% 
125^ 
12H 
12^ 
12^ 


33^ 

33\ 

m 


5A 
5^ 

is 


33^ 
3>^ 
33^ 
33^ 
33^ 
33^ 


3 
3 
3 
3 
3 
3 


1^8 

in 
in 

\y% 


18 

in 


8^ 
83^ 
8^ 
83^ 
83^ 
8H 


3% 
3^ 


8-in. Colum,n. 
4 Z-bars 4-4H in. 
deep. Web pi. 7'' 
by thickness of 
bars. 


¥ 

1 


15A 

153^ 

15^ 

15A 

15A 

143^ 

15 

15H 


43^ 

i 

til 


i 
^4 


3^ 
3^ 
3^ 

3^ 
3^ 

3^8 

3^ 
3^ 


33^ 
33^ 

33J 
3M 


m 
•in 

IK 
13^ 

IVs 


3i^ 
33^ 

33^ 

is 


10 
10 
10 
10 
10 
10 
10 
10 
10 


ti 

4iV 
4^ 
4ii 
4^ 
43^ 
43^ 


10-tn. Colum-n. 
4 Z-bars S-bVs in. 
deep. Web pi. 7" 
by thickness of 
bars. 


1 


16i| 

16ll 
163^ 
16^ 
16M 
163^ 
163^ 
16^ 


5ii 

51^ 

5y 

5|| 


61^ 


3^ 
3^ 
3^ 
3^ 
3^ 
3H 
3^ 
3^ 
3^ 


33^ 

3M 

33^ 
33^ 
33^ 
33^ 
33^ 
33^ 
33^ 


VA 
VA 
VA 
VA 
VA 
VA 
VA 
VA 
VA 


3A 

3A 
3% 


10 
10 
10 
10 
10 
10 
10 
10 
10 


5tf 


l£-in Column. 
4 Z-bars 6-63^ in. 
deep. Web pi. 8'' 
by thickness of 
bars. 


1 


19t^ 
19A 

im 

19 

19^ 
18^ 
183^ 
19 


6A 
ell 


It 
1% 

m 


43^ 
4% 
43^ 
43^ 
4% 
4>^ 
43^ 
43^ 
43^ 




2A 
2A 
2A 
2A 
2% 
2A 
2H 
2A 
2A 


33^ 
3^ 
3^ 
3H 

3K 
3A 
3^ 


11 
11 
11 
11 
11 
11 
11 
11 
11 


6% 
6j| 



* All dimensions in inches. 



STEEL Z-BAR COLUMNS. 



699 



— ^ToTAL Safe Load in Thousand Pounds for Z-Bar Columns 

(Without Side Plates.) 
^ = thickness of metal; 
A = area of cross-section of column; 

m; = weight of column in lbs. per lin. ft., not including weight of rivets; 
r = least radius of gyration. 

Allowable stresses per^ 1^.000 lbs. for lengths of 90 radii or under; 
sq. in.: safety factor 4: | 17,100- 57— for lengths over 90 radii. 

(See opposite page for Dimensions.) 



t 


A 


w* 


r 


Length of Columns in Feet. 


Ins. 


Sq. 


Lbs. 


(Min) 








ins. 


12 


14 


18 


22 


26 


30 


34 


38 


42 


46 50 


H 


9.31 


31.7 


1.86 


111 


111 


98 


84 


70 


57 


ooeci o tn 


Seo^-o^ii^S 




11.7 
13.6 


39.8 
46.2 


1.90 

1.88 


141 
163 


141 
163 


125 
143 


103 
124 


91 
104 


74 
84 


^ll^e 




A 


16.0 


54.3 


1.93 


192 


192 


171 


149 


126 


103 


.sSSga^ 


^ 


17.6 


59.9 


1.90 


211 


211 


187 


162 


136 


111 


^lldi 


^g^'^gSfl^S 


^ 


20.0 


67.9 


1.95 


240 


240 


216 


188 


160 


132 




¥ 


11.3 


38.3 


2.47 


.III 


135 


124 


111 


99 


86 


74 




4 


14.1 


48.1 


2.52 


170 


157 


142 


127 


111 


96 


•So-^g^Sc-ls 


H 


17.1 


58.0 


2.57 


ig--2 


205 


192 


174 


155 


137 


119 






19.0 


64.7 


2.49 




228 


211 


190 


169 


148 


127 


y^ 


21.9 


74.4 


2.55 


262 


245 


221 


198 


174 


151 




Tfi 


24.8 


84.1 


2.60 


297 


280 


254 


228 


202 


176 


if 


26.3 
29.0 


89.2 
98.8 


2.52 

2.58 


00 O J5 r^ ^ 


315 
349 


292 
327 


264 
296 


235 

265 


207 
235 


179 
204 




% 


31.9 


108.4 


2.63 


^•5^^ 


382 


363 


329 


296 


263 


230 


_5_ 


15.8 


53.7 


3.08 


. ^M 




189 


179 


165 


151 


137 


123 


108 


94 


yk 


19.0 


64.7 


3.13 


i-^gg 




228 


217 


200 


184 


167 


151 


134 


117 


■h 


22.3 


75.8 


3.18 






268 


257 


237 


218 


199 


180 


161 


141 


Vo 


24.5 


83.2 


3.10 




294 


278 


257 


235 


214 


192 


170 


149 


u 


27.7 
30.9 


94.2 
105.2 


3.15 
3.21 




332 
371 


317 
357 


293 
331 


269 
305 


245 

278 


221 
252 


197 
225 


178 





199 


H 


32.7 


111.0 


3.13 


^ o JS S^ ^- 




392 


373 


344 


316 


287 


259 


230 


202 


V, 


35.8 


121.8 


3.18 




430 


412 


382 


351 


320 


289 


258 


228 


« 


39.0 


132.6 


3.25 


>-«i fV ^ ^ o 




468 


453 


420 


388 


355 


322 


289 


256 


1 


21.4 
25.0 
28.8 
31.2 
34.8 
38.5 


72.7 

85.2 

97.8 

106.2 

118.5 

130.9 


3.67 
3.72 
3.77 
3.70 
3.75 
3.73 








257 
301 
345 
375 

418 
462 


246 
290 
335 
360 
405 
447 


230 
272 
314 
337 
380 
418 


214 
253 
294 
314 
354 
390 


198 
235 
273 
291 
329 
362 


182 
217 
252 
268 
303 
334 


166 






198 






?31 






?!45 


% 






?78 


H 







305 


H 


40.5 
44.1 

47.7 


137.8 
149.9 
162.1 


3.68 
3.66 
3.64 








486 
529 
572 


466 
506 
546 


436 
473 
510 


406 
440 
475 


376 
407 
439 


346 
374 
403 


316 


8 






341 






367 













Add weight of rivet heads. 



600 



,^PROPERTIES AND TABLES OF COLUMNS, 






dsiiiSiL^^f 



M 









Fig. 8. 



7. — Carnegie 14-inch Z-Bar Columns. 
With Side Plates. 
Note. — Diameter of Bolt or Rivet, % in. 
For notation, A, w, r, safety factor, working formula, 
etc., see top of preceding page. 

For increase of safe load over tabulated values, the 
area A may be increased proportionately by increasing 
metal in side plates. 



Dimensions and Properties 




Safe Load in Thousand Pounds. 




Two 


U 0) 


Dim'n's 


A 


,«,• 


p3 


Length of Column in Feet. 


Side 
Pl'ts 






Sq. 

Ins. 








A 


B 


Lbs. 


"a 


26t 


28 


30 


34 


38 


42 


46 


50 


14x3^ 


pj fi 


19t^ 


61^ 


49.0 


166.6 


3.80 


588 


588 


573 


538 


503 


467 


432 


397 


14x,^ 




19H 


6^? 


50.8 


172.6 


3.81 


609 


609 


594 


558 


521 


485 


449 


412 


14x1/^ 


^r* 


1934 


6^* 


52.5 


178.5 


3.82 


630 


630 


615 


578 


540 


503 


465 


427 


14xt^ 


^^ 


19% 


7nV 


54.3 


184.5 


3.82 


651 


651 


637 


598 


559 


520 


481 


443 


14x5^ 


«o . 


19y| 


7?. 


56.0 


190.4 


3.83 


672 


672 


658 


618 


578 


538 


498 


458 


14xH 


ii'd 


20tV 


7^^. 


57.8 


196.4 


3.84 


693 


693 


679 


638 


597 


556 


514 


473 


14x34 


^^ 


20% 


7^^ 


59.5 


202.3 


3.85 


714 


714 


700 


658 


615 


573 


530 


488 


14xi* 


tS3 ^ 


20M 
20^ 


7^. 


61.3 


208.4 


3.85 


735 


735 


721 


677 


634 


590 


547 


503 


14x% 


rJHi-l 


7H 


63.0 


214.2 


3.85 


756 


756 


742 


697 


652 


608 


563 


518 


UxH 


. 


19,^ 


QH 


51.0 


173.4 


3.75 


612 


612 


593 


556 


519 


482 


445 


407 


14xt^ 


•S-S 


19% 


6H 


52.8 


179.4 


3.76 


633 


633 


614 


576 


538 


499 


461 


423 


14xH 


rT 


19^/^ 


6% 


54.5 


185.3 


3.77 


654 


654 


636 


596 


557 


617 


477 


438 


14xt^ 


^ci; 


1934 


6H 


56.3 


191.4 


3.78 


675 


675 


657 


616 


575 


535 


494 


453 


UxVh 


P'li 


19i* 


7 


58.0 


197.2 


3.79 


696 


696 


678 


636 


594 


552 


510 


468 


U^U 


^^ 


19% 


7iV 


59.8 


203.2 


3.80 


717 


717 


699 


656 


613 


570 


527 


484 


14x3^- 


^-^ 


20 


7% 


61.5 


209.1 


3.80 


738 


738 


720 


676 


631 


587 


543 


499 


14xit 


sj^ 


20tV 


7t^ 


63.3 


215.1 


3.81 


759 


759 


741 


696 


650 


605 


559 


514 


14xJ^ 


^.H 


201^ 


7H 


65.0 


221.0 


3.82 


780 


780 


762 


716 


669 


622 


576 


529 


UxVh 


d d 


19t% 


Q^ 


54.6 


185.6 


3.73 


655 


653 


633 


593 


553 


513 


473 


433 


Ux^ 


p 

Hsoo 


195/^ 


6P 


56.3 


191.5 


3.74 


676 


675 


654 


613 


572 


531 


490 


449 


14x1^ 


1934 


6^i 


58.1 


197.5 


3.75 


697 


697 


675 


633 


591 


549 


506 


464 


14xt^ 


19% 


7?^, 


59.8 


203.4 


3.76 


718 


718 


697 


653 


610 


566 


523 


479 


14xH 


19H 


7,3. 


61.6 


209.4 


3.77 


739 


739 


718 


673 


628 


584 


539 


494 


14xH 


U ^ 


20tV 


7,^. 


63.3 


215.3 


3.78 


760 


760 


739 


693 


647 


601 


555 


510 


14x3^ 




20% 


7^, 


65.1 


221.3 


3.78 


781 


781 


760 


713 


666 


619 


572 


525 


14xi| 


20t^ 


7^n. 


66.8 


227.2 


3.79 


802 


802 


781 


733 


685 


636 


588 


540 


14x3^ 


r*4^ 


20M 


7-^^ 


68.6 


233.2 


3.80 


823 


823 


802 


753 


703 


654 


605 


555 


14x3^ 


C d 


1934 


6H 


58.2 


197.8 


3.71 


698 


695 


673 


631 


588 


545 


502 


459 


14xtV 




19H 


7 


59.9 


203.8 


3.72 


719 


717 


694 


650 


606 


562 


518 


474 


UxVo 


« >< 


19% 


7iV 


61.7 


209.7 


3.73 


740 


738 


716 


670 


625 


580 


535 


490 


14xt^ 


^^ 


20 


7% 


63.4 


215.7 


3.74 


761 


760 


737 


690 


644 


598 


551 


505 


HxH 




20tV 


7t^ 


65.2 


221.6 


3.75 


782 


782 


758 


710 


663 


615 


568 


520 


14xH 


20% 


7¥ 


66.9 


227.6 


3.76 


803 


803 


779 


730 


682 


633 


584 


535 


14x3^ 


20K 


7t^ 


68.7 


233.5 


3.77 


824 


824 


800 


750 


700 


650 


601 


551 


14xH 


tSl 5? 


20t^ 


73/^ 


70.4 


239.5 


3.77 


845 


845 


821 


770 


719 


668 


617 


566 


14x>g 


■^1-H 




7^ 


72.2 


245.4 


3.78 


866 


866 


842 


790 


738 


686 


633 


581 



* Add weight of rivet heads. 



t 26 feet or less. 



tB-H 
c 



s 



Z-BAR COLUMNS. CHANNEL COLUMNS, 

" 8 — Carnegie Channel Columns — Flat Ends. 

Safe Loads* and Properties. 
(Safe Loads are in Thousand Pounds.) 



601 

r 



Fig. 9. 



2 6-inch Chan. Lat.f or W3 


th\B 


=SVs'' 


2 7-inch Chan. Lat.t or with ) E 


'=4^" 


2 Side Plates 8 inches wideiC=5H" 


2 Side Plates 9 inches wide ) C=6M" 


si 


si 




i 

o 


Rad 

of 

Gyr 


Short Col. 


tH <u t: 

III 


+3 • 

«2« 


1^ 


S8 


i 


Rad 

of 

Gyr 

'a 
"1 


Short Col. 




9. 


IS 




m5 


1^ 

^3 








Sq. 




L 


S 










Sq. 


L 


s 




Lbs. 


Ins. 


Lbs. 


Ins. 


Ft. 


1000# 


1000# 


Lbs. 


Ins. 


Lbs. 


Ins. 


Ft. 


1000# 


1000# 




Lat. 


16.0 


4.76 


2.33 


17.4 


57.1 


1.40 




Lat. 


19.5 


5.70 


2.72 


21.3 


68.4 


1.43 




¥ 


29.6 


8.76 


2.32 


17 3 


105.1 


2.58 




H 


34.8 


10.20 


2.67 


19.9 


122.4 


2.61 






33.0 


9.76 


2.32 


17.3 


117.1 


2.88 


(A 


^ 


38.6 


11.32 


2.67 


19.9 


135.8 


2.90 


t/i 


3^ 


36.4 


10.76 


2.32 


17.3 


129.1 


3.17 


£ 


Vs 


42.5 


12.45 


2.66 


19.8 


149.4 


3.20 


£ 


^ 


39.8 


11.76 


2.32 


17.3 


141.1 


3.47 


U5 


^ 


46.3 


13.58 


2.66 


19.8 


163.0 


3.49 


00 


3^ 


43.2 


12.76 


2.32 


17.3 


153.1 


3.76 


t^ 


y2 


50.1 


14.70 


2.65 


19.7 


1.76.4 


3.79 




■ft 


46.6 


13.76 


2.32 


17.3 


165.1 


4.06 


• ai 


A 


53.9 


15.82 


2.65 


19.7 


189.8 


4.08 




50.0 


14.76 


2.32 


17.3 


177.1 


4.35 




H 


57.8 


16.95 


2.64 


19.7 


203.4 


4.39 




ii 


53.4 


15.76 


2.32 


17.3 


189.1 


4.65 




H 


61.6 


18.07 


2.64 


19.7 


216.8 


4.68 




Lat. 


21.0 


6.18 


2.20 


16.4 


74.2 


1.92 




Lat. 


24.5 


7.20 


2.59 


19.3 


86.4 


1.90 




M 


34.6 


10.18 


2.24 


16.7 


122.2 


3.11 




/€ 


39.8 


11.70 


2.60 


19.4 


140.4 


3.08 




A 


38.0 


11.18 


2.25 


16.8 


134.2 


3.40 


w 


TQ 


43.6 


12.82 


2.60 


19.4 


153.8 


3.37 


1 


¥ 


41.4 


12.18 


2.25 


16.8 


146.2 


3.70 


rQ 


Vs 


47.5 


13.95 


2.60 


19.4 


167.4 


3.67 




44.8 


13.18 


2.26 


16.9 


158.2 


3.99 


»o 


^ 


51.3 


15.08 


2.60 


19.4 


181.0 


3.97 


»o 


3^ 


48.2 


14.18 


2.26 


16.9 


170.2 


4.29 


C5 


34 


55.1 


16.20 


2.60 


19.4 


194.4 


4.26 


s 


1^ 


51.6 


15.18 


2.27 


16.9 


182.2 


4.57 


y—^ 


A 


58.9 


17.32 


2.60 


19.4 


207.8 


4.56 




^8 


55.0 


16.18 


2.27 


16.9 


194.2 


4.87 




ys 


62.8 


18.45 


2.60 


19.4 


221.4 


4.85 




H 


58.4 


17.18 


2.27 


16.9 


206.2 


5.17 




ii 


66.6 


19.57 


2.60 


19.4 


234.8 


5.15 




Lat. 


26.0 


7.64 


2.09 


15.6 


91.7 


2.50 




Lat. 


29.5 


8.68 


2.50 


18.6 


104.2 


2.38 




^ 


43.0 


12.64 


2.18 


16.3 


151.7 


3.96 




H 


44.8 


13.18 


2.54 


18.9 


158.2 


3.55 




^ 


46.4 


13.64 


2.19 


16.3 


163.7 


4.26 


5^ 


^ 


48.6 


14.30 


2.55 


19.0 


171.6 


3.84 


CO 


1^ 


49.8 


14.64 


2.20 


16.4 


175.7 


4.55 


J^ 


H 


52.5 


15.43 


2.55 


19.0 


185.2 


4.14 


£ 


J^ 


53.2 


15.64 


2.21 


16.5 


187.7 


4.84 


^ 




56.3 


16.56 


2.55 


19.0 


198.7 


4.44 


CO 




56.6 


16.64 


2.21 


16.5 


199.7 


5.15 




/4 


60.1 


17.68 


2.56 


19.1 


213.2 


4.72 


1—1 


¥ 


60.0 


17.64 


2.22 


16.6 


211.7 


5.44 


1—) 


^ 


63.9 


18.80 


2.56 


19.1 


225.6 


5.02 




T¥ 


63.4 


18.64 


2.22 


16.6 


223.7 


5.75 




Vs 


57.8 


19.93 


2.56 


19.1 


239.2 


5.33 




M 


66.8 


19.64 


2.23 


16.6 


235.7 


6.03 




H 


71.6 


21.05 


2.56 


19.1 


252.6 


5.62 




Lat. 


31.0 


9.12 


2.00 


14.9 


109.4 


3.12 




Lat. 


34.5 


10.14 


2.43 


18.1 


121.7 


2.85 




^ 


48.0 


14.12 


2.12 


15.8 


169.4 


4.56 




A 


53.6 


15.76 


2.49 


18.6 


189.1 


4.33 


. 


H 


51.4 


15.12 


2.13 


15.9 


181.4 


4.86 


w 


Vs 


57.5 


16.89 


2.50 


18.6 


202.7 


4.62 


1 


i 


54.8 


16.12 


2.14 


16.0 


193.4 


5.15 


^ 


^ 


51.3 


18.02 


2.50 


18.6 


216.2 


4.93 


U5 


3^ 


58.2 


17.12 


2.15 


16.0 


205.4 


5.45 


^ 


V2 


55.1 


19.14 


2.51 


18.7 


229.7 


5.22 


A- 


61.6 


18.12 


2.16 


16.1 


217.4 


5.74 


<M 


^ 


58.9 


20.26 


2.51 


18.7 


243.1 


5.52 


rH 


5^ 


65.0 


19.12 


2.17 


16.2 


229.4 


6.03 




% 


72.8 


21.39 


2.52 


18.8 


256.7 


5.81 




U 


68.4 


20.12 


2.18 


16.2 


241.4 


6.31 




ii 


76.6 


22.51 


2.52 


18.8 


270.1 


6.11 




H 


71.8 


21.12 


2.18 


16.2 


253.4 


6.63 




% 


30.4 


23.64 


2.53 


18.9 


283.7 


6.39 


* Allowed stresses per sq. in. ; safety 




Lat. 


39.5 


11.62 


2.35 


17.5 


139.4 


3.38 


factor 4: 




A 


58.6 


17.24 


2.44 


18.2 


206.9 


4.82 


12,000 lbs. for length of 90 radii or 


W 


Vs 


52.5 


18.37 


2.45 


18.3 


220.4 


5.13 


under; , 
17,100-57 ;;rfor lengths over 90 radii 


jQ 


^ 


66.3 


19.50 


2.45 


18.3 


234.0 


5.44 




^ 


70.1 
73.9 


20.62 
21.74 


2.46 
2.47 


18.3 
18.4 


247.4 
260.9 


5.73 
6.02 


and up to 125. land r in inches. 


T-H 


% 


77.8 


22.87 


2.48 


18.5 


274.4 


6.31 


Note. — Above weights do not in- 




H 


B1.6 


23.99 


2.48 


18.5 


287.9 


6.62 


clud 


e ri\ 


^et h 


eads 


or la 


ttice 


bars 


1 




V 


B5.4 


25.12 


2.49 


18.6 


301.4 


6.90 



t Lattice bars not less than 1>^" X ^". t Lattice bars not less than 1%" X ^'^ 



602 



i< -C- 



32.— PROPERTIES AND TABLES OF COLUMNS, 
= 8' 



Fig. 9. 



h 



9. — Carnegie Channel Columns — Flat Ends. 

Safe Loads and Properties. 

(Safe Loads are in Thousand Pounds.) 



9^ 



28-in. Channels Lat.f or with ) B=5H" 
2 Side Plates 10 inches wide, j C=7}4" 


2 9-in. Channels Lat .J or with \JB=^M" 
2 Side Plates 11 inches wide. /C=8jl" 


O CD 


k 

Is 


li 


0-1 

O 


Rad 

of 

Gyr 


Short Col. 




IS 


Ho 


Id 

ra 


k 

m 

4-1 

c 


Rad 

of 

Gyr 

c 


Short Col. 




fi 


m5 

c3bC 


§1 


ii 




Qa5 








Sq. 


i 


L 


S 










Sq. 


i 


L 


s 




Lbs. 


Ids. 


Lbs. 


Ins. 


Ft. 


1000# 


1000# 


Lbs. 


Ins. 


Lbs. 


Ins. 


Ft. 


1000# 


J000# 




Lat. 


22.5 


6.70 


3.11 


23.2 


80.4 


1.47 




Lat. 


26.5 


7.78 


3.49 


26.0 


93.4 


1.53 




¥ 


39.5 


11.70 


3.03 


22.6 


140.4 


2.64 




K 


45.2 


13.28 


3.40 


25.4 


159.4 


2.67 






43.7 


12.95 


3.02 


22.5 


155.4 


2.93 




■h 


49.9 


14.66 


3.38 


25.2 


175.9 


2.97 


1 


3^ 


48.0 


14.20 


3.01 


22.4 


170.4 


3.23 


% 


% 


54.6 


16.03 


3.36 


25.1 


192.4 


3.27 


to 


^ 


52.3 


15.45 


3.00 


22.4 


185.4 


3.52 


U5 


^ 


59.2 


17.40 


3.35 


25.0 


208.8 


3.56 


c4 


y^ 


56.5 


16.70 


2.99 


22.3 


200.4 


3.82 


C<J 


^ 


63.9 


18.78 


3.33 


24.8 


225.4 


3.86 


1-H 


^e 


60.8 


17.95 


2.98 


22.2 


215.4 


4.12 


CO 

1-t 


A 


68.5 


20.16 


3.32 


24.8 


241.9 


4.15 




H 


65.0 


19.20 


2.98 


22.2 


230.4 


4.42 




ys 


73.3 


21.53 


3.31 


24.7 


258.4 


4.45 




ii 


69.2 


20.45 


2.97 


22.2 


245.4 


4.71 




ii 


77.9 


22.90 


3.31 


24.7 


274.8 


4.73 




Lat. 


27.5 


8.08 


2.98 


22.2 


97.0 


1.86 




Lat. 


30.0 


8.82 


3.40 


25.4 


105.8 


1.78 




A 


48.7 


14.33 


2.97 


22.2 


172.0 


3.30 




H 


48.7 


14.32 


3.36 


25.1 


171.8 


2.92 


. 


ft/g 


53.0 


15.58 


2.96 


22.1 


187.0 


3.60 




^ 


53.4 


15.70 


3.34 


24.9 


188.4 


3.22 


^ 


^ 


57.3 


16.83 


2.96 


22.1 


202.0 


3.89 


w 


ys 


58.1 


17.07 


3.33 


24.8 


204.8 


3.51 


iO 


\/L 


61.5 


18.08 


2.95 


22.0 


217.0 


4.19 


rQ 


^ 


62.7 


18.44 


3.32 


24.8 


2^1.3 


3.80 


t>: 


i\ 


65.8 


19.33 


2.95 


22.0 


232.0 


4.48 


lO 


y2 


67.4 


19.82 


3.31 


24.7 


237.8 


4.09 


CO 


H 


70.0 


20.58 


2.95 


22.0 


247.0 


4.78 




1% 


72.0 


21.20 


3.30 


24.6 


254.4 


4.39 




ii 


74.2 


21.83 


2.94 


21.9 


262.0 


5.08 




ys 


76.8 


22.57 


3.29 


24.5 


270.8 


4.69 




^ 


78.5 


23.08 


2.94 


21.9 


277.0 


5.37 




Lat. 


81.4 


23.94 


3.29 


24.5 


287.3 


4.98 




Lat. 


32.5 


9.56 


2.89 


21.6 


114.7 


2.26 


40.0 


11.76 


3.21 


23.9 


141.1 


2.51 




.^ 


58.0 


17.06 


2.92 


21.8 


204.7 


3.99 




ys 


68.1 


20.01 


3.25 


24.2 


240.1 


4.22 




^ 


62.3 


18.31 


2.91 


21.7 


219.7 


4.30 




^ 


72.7 


21.38 


3.25 


24.2 


256.6 


4.50 


1 


3^ 


66.5 


19.56 


2.91 


21.7 


234.7 


4.60 


"k 


y^ 


77.4 


22.76 


3.24 


24.2 


273.1 


4.81 


40 


^ 


70.8 


20.81 


2.91 


21.7 


249.7 


4.89 


jQ 


A 


82.0 


24.14 


3.24 


24.2 


289.7 


5.10 


<N 


^ 


75.0 


22.06 


2.91 


21.7 


264.7 


5.18 


i 


% 


86.8 


25.51 


3.23 


24.1 


306.1 


5.41 


«d 




79.2 


23.31 


2.91 


21.7 


279.7 


5.48 




H 


91.4 


26.88 


3.23 


24.1 


322.6 


5.69 




zx 


83.5 


24.56 


2.91 


21.7 


294.7 


5.78 




^ 


96.1 


28.26 


3.23 


24.1 


339.1 


5.99 




if 


87.8 


25.81 


2.91 


21.7 


309.7 


6.07 




H 


100.8 


29.63 


3.23 


24.1 


355.6 


6.28 




Lat. 


37.5 


11.02 


2.82 


21.0 


132.2 


2.67 




Lat. 


50.0 


14.70 


3.10 


23.1 


176.4 


3.24 




1^ 


67.3 


19.77 


2.87 


21.4 


237.2 


4.71 




A 


92.0 


27.08 


3.17 


23.6 


325.0 


5.84 


w 


3 


71.5 


21.02 


2.87 


21.4 


252.2 


5.01 




ys 


96.8 


28.45 


3.17 


23.6 


341.4 


6.13 


£ 


A 


75.8 


22.27 


2.87 


21.4 


267.2 


5.31 


o* 


ii 


101.4 


29.82 


3.17 


23.6 


357.8 


6.43 


U3 


II 


80.0 


23.52 


2.87 


21.4 


282.2 


5.61 


■^ 


% 


106.1 


31.20 


3.17 


23.6 


374.4 


6.73 


b-; 


^ 


84.2 


24.77 


2.87 


21.4 


297.2 


5.91 


s 


if 


110.8 


32.57 


3.17 


23.6 


390.8 


7.03 


CO 


M 


88.5 


26.02 


2.88 


21.5 


312.2 


6.18 




K 


115.4 


33.95 


3.17 


23.6 


407.4 


7.33 




4 


92.8 


27.27 


2.88 


21.5 


327.2 


6.48 




M 


120.1 


35.32 


3.17 


23.6 


i23.8 


7.63 




y. 


97.0 


28.52 


2.88 


21.5 


342.2 


6.78 




1 


124.8 


36.70 


3.17 


23.6 


440.4 


7.92 




Lat. 


42.5 


12.50 


2.77 


20.7 


150.0 


3.09 


Problem. — A column composed 




J^ 


76.5 


22.50 


2.83 


21.1 


270.0 


5.44 


of 2 8-in. channels at 16.25 lbs. and 






80.8 


23.75 


2.84 


21.2 


285.0 


5.73 


2 plates IOXtt ins. will support a 


1 


41 


85.0 


25.00 


2.84 


21.2 


300.0 


6.02 


load of 249,700 lbs. up to 21.7 ft. 


»0 


II 


89.2 


26.25 


2.84 


21.2 


315.0 


6.33 


in length. Find the load that it 


c5 


93.5 


27.50 


2.84 


21.2 


330.0 


6.63 


will support if 30 ft. long? 


?5 


i 


97.8 
102.0 


28.75 
30.00 


2.84 
2.85 


21.2 
21.3 


345.0 
360.0 


6.93 
7.21 


Solution. — 249,700 lbs. 
4,890X (30.0- 21.7)= 40,600 " 




M 


106.2 


31.252.851 


21.3 


375.0 7.50 ' 


Answer 209,100 '* 



t Lattice bars not less than IM'' X i^''. X Lattice bars not less than 1" X -h". 



CHANNEL COLUMNS. 



603 






ids 



10" 10. — Carnegie Channel Columns — Flat Ends. 12" 
Safe Loads and Properties. 
(Safe Loads are in Thousand Pounds.) 



Fig. 9. 



210-in. Chan 
2 Side Plates 



Lat. t or with )B 
12 inches wide. ( C 



=7 '' 



2 12-in. Chan.Lat.t or with. ) 5=8 M" 
2 Side Plates 14 inches wide ] C=n\i" 



o o 
Lbs, 



Ins 



Lat 



Lat. 

¥2 



+3 



Lbs. 



Lat. 

9 

T6 

% 



Lat. 

H 

JL3 
16 

M 
1 

Ws 
IM 



Lat. 

13 

l(i 

% 

1 
IK 

IK 



30.0 
55.5 
60.6 
65.7 
70.8 
75.9 
81.0 
86.1 
91.2 

40.0 

75.7 

80.8 

85.9 

91.0 

96 

101.2 

106.3 

111.4 



u< o 

Sq. 

Ins. 



50.0 
95.9 
101 
106 
111.2 
116.3 
121.4 
126.5 
131.6 



.029 
1 



60.0 
116.1 
121.2 
126.3 
131.4 
136. 
141.6 
151.8 
162.0 



70.020 
136.3 
141.4 
146.5 
151.644 
161.847 
172.050. 
182.2 
192.4 



8.92 
16.42 
17.92 
19.42 
20.92 
22.42 
23.92 
25.42 
26.92 



11.76 
22.26 
23.76 
25.26 
26.76 
28.26 
29.76 
31.26 
32.76 



Rad 
of 
Gyr 



3.87 
3.74 
3.72 
3.70 
3.68 
3.67 
3.65 
3.64 



Short Col. 



L 

Ft. 



3.6327.0 



3.6026.8 



14.70 
28.20 
70 
31.20 
32.70 
34.20 
35.70 
37.20 
38.70 



17.64 
34.14 
35.64 
37.14 
38.64 
.14 
41.64 
44.64 
47.64 



540 



59 
3.59 

58 



5526, 
5526. 



3.3525. 

3.4525. 

3.45: 

3.45 

3.4525 

45 

45 

4525 

4525 



S 
1000# 



.8107.0 
.8197.0 
.7215.0 
,6233.0 
4251.0 
.4269.0 
287.0 
305.0 
1323.0 



141.1 

267.1 



27 
27 
27.0285.1 

27 
26 



&Q O . 

Qd-£ 



1000# 



1.57 

3.00 

3.30 

3.59 

3.89 

4.18 

4 

4.78 

5.07 



26.! 
26.! 
26.' 



176.4 

338.4 

356.4 

374.4 

392.4 

410.4 

4428.4 

4446.4 

3464.4 



303 

321 

339.1 

357 

375.1 

393.1 



25. 

26. 

26. 

26. 

26. 

26. 

26.0499.7 

26.0535.7 

26.0571.7 



211 

409 

427 

0445 

0463 

0481. 



0247.0 

481.0 

499.0 

7517.0 

7535.0 

7571.0 

7607.0 

643.0 

079. 



2.20 

4.18 
4.48 
4.77 
5.07 
5.37 
5.67 
5.96 
6.26 



2. 

5.42 

5.72 

6.02 

6.32 

6.62 

6.91 

7.20 

7.50 



3.53 
6.67 
6.97 

7.28 
7.58 
7.88 
8.17 
8.75 
9.33 



4.20 
7.94 
8.24 
8.54 
8.83 
9.43 
10.0 
10.6 
11.2 






^t 



Lbs 



Ins, 



Lat 
_§_ 

16 

T6 

9 
16 

'A 
H 



Lat. 

1^ 



Vs 



u 



Lbs. 



41.0 
70.8 
76.7 
82.6 
88.6 
94.6 

100. 

106. 

112. 



Sq. 
Ins. 



Lat 



Lat. 

H 

13 
16 

1 



Lat. 

1 

1^ 

m 
m 



50 

91 

97.6 
103 
109 
115.4 
121.4 
127 
133 



70 

135 

141 

147 

153 

159 

165.2 

177.1 

189.0 



80.0 



157 
163. 
169, 
175 

187. 
199. 
210. 



13^222.8 



12.06 

20.81 

22.56 

24.31 

26.064 

27.81 

29.564 

31.31 

33.06 



Rad 

of 

Gyr 



14.704 

26.95 

28.704 

30.454 

32.204 

33.954 

35.70 

37.45 

39.204 



60.0 
113. 
119.5 
125.4 
131.4 
137.3 
143.342.14 
149.343.894, 
155.2 



17.64 

33.394 

35.14 

36.89 

38.64 

40.394 



45.64 



20.584 
39.83 
41.584 
43.334 
08 
834 
48.584 
52.08 
55.58 



.345 
.346 



23.524 



346 

348 



274 
024 

774 



524 
024 



349. 

51.5 

55 
058.52 
962.02 

65.52 



.43 33 
32 



Short Col. 



03 fa» 

L 

Ft. 



.09 
.12 
.11 
.11 
,11 
,10 
.10 
.10 
.09 



^1 

S 

1000# 



144 

249, 
6270. 
4291 
3 



:312 

333 

354 

.0375.7 

;396 









1000# 



211 

400 

421.7 

442.7 
3463.7 
3484.7 

505.7 



2526 
547 



30.8 



30.5 
30.7 
30.6 
30.6 
30.6 



30.5 
30.5 
30.5 



176.4 

323. 

344.4 

365.4 

386.4 

407.4 

428.4 

449.4 

470.4 



247.0 

478.0 

0499.0 

0520.0 

9541.0 

562.0 

583.0 

625.0 

667.0 



282.2 
555.2 
576.2 
597.2 
618.2 



30.5660.2 



702.2 
744.2 
786.2 



1.79 
3.23 
3.52 
3.82 
4.12 
4.41 
4.71 
4.98 
5.30 



.27 
28 
.58 
.88 
18 
.47 
.77 
6.06 
6.36 



2.82 
6.39 
5.69 
6.00 
6.29 
6.57 
6.87 
7.17 
7.47 



3.37 
6.54 
6.84 
7.13 
7.43 
7.72 
8.02 
8.61 
9.20 



3.93 

7.68 
7.98 
8.28 
8.58 
9.18 
9.76 
10.3 
10.9 



t Lattice bars not less than 2''k^". J Lattice bars not less than 2"x3 



604 



22.— PROPERTIES AND TABLES OF COLUMNS. 



11. — * Phcenix Steel Columns. 
Dimensions, Properties and Loads. 

"The dimensions given in the following table are subject to such slight 
variations as are unavoidable in the manufacture of these shapes. 

The weights given are those of the segments composing the columns, 
and from 2 to 5 per cent must be added for weight of the rivet heads. 

The safb loads specified are computed as being one-fourth of the ulti- 
mate, or breaking loads, and as producing a strain, or pressure, in an axial 
direction on square-end columns, of not more than 12,000 lbs. per square 
inch for lengths of 90 radii and under. 

The A, Bl, B2, and C columns have each 4 segments, the E have 6, 
and the G have 8 segments. 

Any desired thickness between the minimum and maximum can be 
furnished. 

Least Radius of Gyration = Do X 0.3636. 



One Segment. 


Diameters, in Ins. 


One Column. 


Section. 




II 


Wt. 

Foot 

in 

Lbs. 


D 

Inside 


Out- 
side. 


rP' 

Over 

Flan- 


Area 

of 
Cross 
Sec- 


Wt. 

Foot 

in 

Lbs. 


Least 
Rad. 

of 
Gyra- 


0) V'-H 






S-s 




ges. 


tion. 
Sq.In. 


tion 
in Ins. 


'Ss.s 


,f^^ 


w" 






















A 


3.2 




4 


^V 


3.8 


12.9 


1.45 


36 




(U 




4.1 


A 


Ws 




4.8 


16.3 


1.50 


47 


TS 


4.9 


3^ 


m 


6A 


5.8 


19.7 


1.55 


58 




Vs 


5.8 




Ws 


6^ 


6.8 


23.1 


1.59 


69 


^^ 




^ 


5.4 




h% 


83^ 


6.4 


21.8 


1.95 


74 


^w 


to 

4J 


"lE 


6.6 




6^ 


8A 


7.8 


26.5 


2.00 


90 


!8 /// - 


g 


% 


7.8 




hVs 


8A 


9.2 


31.3 


2.04 


108 




a 




9.0 


Bl 


b% 


83^ 


10.6 


36.0 


2.09 


126 




3^ 


10.2 


43^ 


SVs 


8H 


12.0 


40.8 


2.13 


144 


P?\\\^<Sp 


w 


^ 


11.4 




6 


8^ 


13.4 


45.6 


2.18 


161 


/i<t' 


Tt< 


^ 


12.6 




QVs 


8ii 


14.8 


50.3 


2.23 


178 


&f 




M 


6.3 




6^ 


9M 


7.4 


25.2 


2.39 


89 


(0 

if 


A 


7.6 






9Vs 


9.0 


30.6 


2.43 


108 


ifi J 


g 


ys 


9.0 




6i|. 


9^ 


10.6 


36.0 


2.48 


127 


i Wr"'a> 


a 


^ 


10.4 


B2 


6fi 


W2 


12.2 


41.5 


2.52 


146 


^ n 


^ 




11.7 


6i^ 


7^ 


m 


13.8 


46.9 


2.57 


166 




^ 


j> 


13.1 




7A 


m 


15.4 


52.4 


2.61 


185 


-^ 


% 


14.4 




7A 


m 


17.0 


57.8 


2.66 


204 






¥ 


8.5 




7H 


im 


10.0 


34.0 


2.84 


120 


^ 






10.3 




7H 


11% 


12.1 


41.3 


2.88 


145 


\%^ 




ZA 


12.0 




8^ 


Hit 


14.1 


48.0 


2.93 


169 




A 


13.7 




8i^ 


113^ 


16.0 


54.6 


2.97 


192 


1 /)/ 


M 


H 


15.3 




8A 


iiM 


18.0 


61.3 


3.01 


216 


f« / // 


■4-> 


^ 


17.0 




8,^ 


12 


20.0 


68.0 


3.06 


239 




a 


fi 


18.3 


c 


8^ 


123^ 


21.9 


74.6 


3.11 


263 


ii 


20.7 


7^ 


8ii 


12^ 


24.3 


82.6 


3.16 


292 


•F W 


H 


22.7 




811 


12^ 


26.6 


90.6 


3.20 


319 


f^ \ \\ 




if 


24.3 




8H 


12A 


28.6 


97.3 


3.24 


343 


00 \ ^\ 




j| 


26.0 




9iV 


12H 


30.6 


104.0 


3.29 


367 


c y^'^ 




1 


29.7 




9A 


12^ 


34.9 


118.6 


3.34 


418 






Ws 


33.0 




9A 


12i| 


38.8 


132.0 


3.48 


466 


Figs. 10-13. 




IM 


36.3 




9f| 


13 


42.7 


145.3 


3.57 


512 


*By permission 


of I 


^r.D. ^ 


N. Bow 


man.C 


hiefEn 


gineer ] 


Phcenix 


:IronV 


^orks.^ 



PHCENIX COLUMNS. 605 

11. — Phcenix Steel Columns. — Concluded. 





One Segment. 




Diameters, in Ins. 




One Column. 




Section. 


II 


Wt. 

Foot 

in 

Lbs. 


D 
Inside 


Do 
Out- 
side 


rP' 
Over 

Flan- 
ges. 


Area 

of 
Cross 
Sec- 
tion. 
Sq.In. 


Wt. 

Foot 

in 

Lbs. 


Least 

Rad. 

of 

Gyra- 
tion 

in Ins. 


SafeLd.for 
16-ft. Col. 
in 1000 lbs. 1 






¥ 


9.3 




llA 


153^ 


16.5 


56.0 


4.20 


198 


A:* 






10.8 




IIH 


15^ 


19.1 


65.0 


4.25 


229 


\? 


^^ 


3^ 


12.3 




llil 


15M 


21.7 


74.0 


4.29 


260 


1/ 


y 


1^ 


14.0 




UtI 


15>g 


24.7 


84.0 


4.34 


296 


f 


3^ 


15.7 




12i^ 


1511 


27.6 


94.0 


4.38 


331 


R / 




A 


17.3 




12A 


16i^ 


30.6 


104.0 


4.43 


367 


^ «J 


6^ 


19.0 


E 


12A 


16i^ 


33.5 


114.0 


4.48 


402 


1 \ 


"H 


20.7 


lli^ 


12i^ 


16j^ 


36.4 


124.0 


4.52 


437 


W i^ 


M 


22.7 




12A 


16t^ 


40.0 


136.0 


4.56 


480 


^X 


^^ o 


n 


24.3 




12ii 


16^ 


43.0 


146.0 


4.61 


516 


(y-^ 


^^ 


Vs 


26.0 




1211 


161^ 


45.9 


156.0 


4.66 


551 


\y<i: 


1 


29.3 




13^ 


1611 


51.7 


176.0 


4.73 


620 


Fig. 


14. 


1^ 


32.7 




13A 


17i^ 


57.6 


196.0 


4.84 


691 






IM 


36.0 




13^ 


17A 


63.5 


216.0 


4.93 


762 






A 


10.3 




15M 


193^ 


24.3 


82.6 


5.54 


290 






Vs 


12.0 




15^ 


193^ 


28.2 


96.0 


5.59 


337 


rrlc 




^ 


13.7 




15K 


19^ 


32.1 


109.3 


5.64 


384 


(^ 


^7- 


¥ 


15.3 




15^ 


191i 


36.0 


122.6 


5.68 


432 


^ / 


}7 i 




17.0 




15M 


19^ 


40.0 


136.0 


5.73 


479 


R 7 


' « 


/^ 


18.7 




153^ 


193^ 


43.9 


149.3 


5.77 


526 


^ ,0 


a ' hi) 


■li 


20.3 


G 


16 


20 


47.8 


162.6 


5.82 


572- 


to ly 
o 


H 


22.0 


14^ 


16^ 


203/g 


51.7 


176.0 


5.88 


620 


s \ 


A ^ 


\i 


23.7 




16M 


20M 


55.7 


189.3 


5.91 


667 


- J. 


)\ cS 


Vs 


25.3 




16^ 


20^ 


59.6 


202.6 


5.95 


715 


Yc^ 


<A 


1 


28.7 




im 


20^ 


67.4 


229.3 


6.04 


809 


l>c^ 


% 


IH 


32.0 




IQVs 


20^ 


75.3 


256.0 


6.13 


904 






IM 


35.3 




nvs 


21 


83.1 


282.6 


6.27 


997 


Fii 


^l5. 


1^ 


38.7 




nvs 


21H 


90.9 


309.3 


6.32 


1091 



606 



32.— PROPERTIES AND TABLES OF COLUMNS, 



12. — Ultimate and Safe C4) Strengths of Hollow Round and 
Hollow Rectangular Cast Iron Columns. 

In the following formulas, U = 80 000 and 5= 10 000 lbs. per sq. in.: 

Round Columns. Rectangular Columns. 

Square Square Pin Square Square Pin 

Ends. & Pin. Ends. Ends. & Pin. Ends. 

UorS UotS UotS U ot S U or S U or S 



1 + 



(12L)' 



1 + 



3(12L)2 



1 + 



(12L)^ 



3(12L)f 
3200^2 



9(12102 . 3(12L)2 
6400£i2 "*■ 1600(^2 



800(i2 1600^/2 400 d^ 

L = length of column in feet; d = (least) outside diameter in inches; 
^ = load in lbs. per square inch on column to produce U or S. 







Round Columns. 






Rectangular Columns. 




L 

d 


Loads 


in 1000 lbs. 


per square 


inch. 


Loads 


in 1000 lbs. per square Inch. 


Square Ends 


Sq. and Pin 


Pin Ends 


Squar 
Ult. 


eEnds 


Sq. and Pin 


Pin Ends 




Ult. 


Safe 


Ult. 


Safe 


Ult. 


Safe 


Safe 


Ult. 


Safe 


Ult. 


Safe 




Load. 


Load. 


Load. 


Load. 


Load. 


Load. 


Load. 


Load. 


Load. 


Load. 


Load. 


Load. 


1.0 


67.80 


8.47 


62.99 


7.87 


58.82 


7.35 


70.48 


8.81 


66.52 


8.31 


62.99 


7.87 


1.1 


65.69 


8.21 


60.30 


7.54 


55.7c 


6.97 


68.79 


8.60 


64.26 


8.03 


60.30 


7.54 


1.2 


63.53 


7.94 


57.60 


7.20 


52.69 


6.59 


67.00 


8.37 


61.94 


7.74' 57.60 


7.20 


1.3 


61.34 


7.67 


54.93 


6.87 


49.74 


6.22 


65.14 


8.14 


59.60 


7.45 


54.96 


6.87 


1.4 


59.14 


7.39 


52.31 


6.54 


46. 9C 


5.86 


63.26 


7.91 


57.27 


7.16 


52.32 


6.54 


1.5 


56.94 


7.12 


49.77 


6.22 


44.20 


5.52 


61.35 


7.67 


54.96 


6.87 


49.76 


6.22 


1.6 


54.76 


6.84 


47.30 


5.91 


41.63 


5.20 


59.45 


7.43 


52.68 


6.58 


47.30 


6.91 


1.7 


52.62 


6.58 


44.94 


5.62 


39.21 


4.90 


57.55 


7.19 


50.46 


6.31 


44.96 


5.62 


1.8 


50.53 


6.32 


42.67 


5.33 


36.93 


4.62 


55.67 


6.96 


48.30 


6.04 


42.67 


5.33 


1.9 


48.49 


6.06 


40.51 


5.06 


34.79 


4.35 


53.80 


6.72 


46.23 


5.78 


40.51 


5.06 


2.0 


46.51 


5.81 


38.46 


4.81 


32.79 


4.10 


51.94 


6.49 


44.20 


5.52 


38.46 


4.81 


2.1 


44.60 


5.57 


36.52 


4.56 


30.92 


3.86 


50.16 


6.27 


42.26 


5.28 


36.52 


4.56 


2.2 


42.75 


5.34 


34.68 


4.33 


29.18 


3.65 


48.40 


6.05 


40.40 


5.05 


34.68 


4.33 


2.3 


40.98 


5.12 


32.94 


4.12 


27.54 


3.44 


46.67 


5.83 


38.63 


4.83 


32.95 


4.12 


2.4 


39.28 


4.91 


31.31 


3.91 


26.03 


3.25 


44.99 


5.62 


36.93 


4.62 


31.31 


3.91 


2.5 


37.65 


4.71 


29.77 


3.72 


24.62 


3.08 


43.39 


5.42 


35.31 


4.41 


29.76 


3.72 


2.6 


36.09 


4.51 


28.32 


3.54 


23.30 


2.91 


41.82 


5.23 


33.77 


4.22 


28.32 


3.54 


2.7 


34.60 


4.32 


26.95 


3.37 


22.07 


2.76 


40.32 


5.04 


32.31 


4.04 


26.95 


3.37 


2.8 


33.18 


4.15 


25.67 


3.21 


20.93 


2.62 


38.87 


4.86 


30.92 


3.86 


25.67 


3.21 


2.9 


31.82 


3.98 


24.46 


3.06 


19.86 


2.48 


37.47 


4.68 


29.60 


3.70 


24.46 


3.06 


3.0 


30.53 


3.82 


23.32 


2.91 


18.87 


2.36 


36.12 


4.51 


28.34 


3.54 


23.32 


2.91 


3.1 


29.31 


3.66 


22.25 


2.78 


17.94 


2.24 


34.83 


4.35 


27.15 


3.39 


22.25 


2.78 


3.2 


28.14 


3.52 


21.25 


2.66 


17.07 


2.13 


33.58 


4.20 


26.03 


3.25 


21.25 


2.66 


3.3 


27.03 


3.38 


20.30 


2.54 


16.26 


2.03 


32.39 


4.05 


24.96 


3.12 


20.30 


2.54 


3.4 


25.97 


3.25 


19.41 


2.43 


15.50 


1.94 


31.25 


3.91 


23.94 


2.99 


19.41 


2.43 


3 5 


24.96 
24.00 
23.09 
22.23 
21.40 


3.12 
3.00 
2.89 
2.78 
2.67 


18.58 
17.71 


2.32 
2.21 






30.15 
29.10 
28.09 
27.13 
26.21 


3.77 
3.64 
3.51 
3.39 
3.28 


22.97 
22.05 


2.87 
2.76 






3 6 










3 7 










3 8 


















3 9 





































Safe loads given in the table are equal to one-eighth the ultimate loads, 
that is, using safety factor 8. If the safety factor 10 is preferred the new 
safe load may be found from the ultimate load by moving the decimal point 
one place to the left. 



CAST IRON COLUMNS. 



607 



13.— Safe 04) Loads on Hollow Round Cast Iron Columns with Flat Ends 
10 OOOA f P = total load on column in 1000 lbs. 

(12L)^' \ L = length of column in feet. 



By Formula P = 



1 + 



800J2 



; in which] A = sectional area of column in sq.in. 
j L = length of column in feet. 
[ c^ = outside diam. of column in ins. 



la 




li 


lit of 


Length of Column In Feet 






IS 


•^3.a 


6 


8 


10 


12 


14 


16 


18 


20 


22 


24 


OP 


Ho 
Ins. 


Ins. 


Lbs. 






















Ins. 




Total *Safe Load on 


Column In 1000 lbs. 




6 


^ 


8.6 


27.0 


73 


65 


57 


50 


44 


38 


33 


29 


25 


22 




H 


10.6 


33.0 


90 


80 


71 


62 


54 


46 


40 


35 


31 


27 




H 


12.4 


38.7 


105 


94 


82 


72 


62 


54 


47 


41 


36 


32 




% 


14.1 


44.0 


119 


107 


94 


82 


71 


62 


54 


47 


41 


36 




1 


15.7 


49.1 


133 


119 


105 


91 


79 


69 


60 


52 


46 


40 


7 


% 


12.5 


39.1 


111 


101 


91 


82 


73 


64 


57 


51 


45 


40 




M 


14.7 


46.0 


130 


119 


108 


96 


86 


76 


67 


60 


53 


47 




% 


16.8 


52.6 


149 


136 


123 


110 


98 


87 


77 


68 


61 


54 




1 


18.9 


58.9 


167 


153 


138 


123 


109 


97 


86 


76 


68 


60 




1^ 


20.8 


64.9 


184 


168 


152 


136 


121 


107 


95 


84 


75 


67 


8 


H 


17.1 


53.4 


155 


145 


133 


122 


110 


99 


89 


80 


72 


65 




% 


19.6 


61.2 


178 


166 


153 


139 


126 


114 


104 


92 


83 


75 




1 


22.0 


68.7 


200 


186 


172 


158 


142 


128 


115 


103 


93 


84 




IH 


24.3 


75.9 


220 


206 


190 


173 


157 


142 


127 


114 


103 


93 




m 


26.5 


82.8 


239 


225 


207 


189 


171 


154 


139 


125 


112 


101 


9 


% 


22.3 


69.8 


207 


196 


183 


169 


159 


142 


130 


118 


108 


98 




1 


25.1 


78.5 


233 


220 


206 


190 


179 


160 


146 


133 


121 


110 




m 


27.8 


87.0 


258 


244 


228 


211 


198 


177 


162 


147 


134 


122 




VA 


30.4 


95.1 


281 


266 


249 


230 


212 


194 


177 


161 


147 


133 




m 


32.9 


102.9 


304 


288 


269 


249 


229 


210 


191 


174 


159 


145 


10 


14 


25.1 


78.4 


235 


225 


212 


199 


185 


172 


158 


146 


134 


123 




1 


28.3 


88.4 


265 


254 


240 


224 


209 


194 


178 


164 


151 


139 






»4.4 


107.4 


323 


308 


291 


273 


254 


235 


217 


200 


184 


169 




1^ 


41.1 


125.2 


376 


359 


339 


318 


296 


274 


253 


233 


214 


197 




m 


45.4 


141.7 


426 


407 


384 


360 


335 


310 


286 


264 


242 


223 


11 


1 


31.4 


98.2 


298 


287 


273 


259 


243 


227 


212 


197 


183 


169 




IH 


38.3 


119.7 


363 


350 


333 


315 


296 


277 


258 


240 


223 


206 




IH 


44.8 


139.9 


425 


409 


390 


369 


346 


322 


302 


286 


260 


241 




m 


50.9 


158.9 


483 


464 


443 


419 


394 


368 


343 


324 


296 


274 




2 


56.6 


176.7 


537 


516 


492 


466 


438 


409 


381 


354 


329 


306 


12 


1 


34.6 


108.0 


331 


320 


307 


293 


277 


262 


246 


230 


215 


201 




134 


42.2 


131.9 


404 


391 


375 


358 


339 


320 


300 


281 


263 


245 




1^ 


49.5 


154.6 


473 


458 


440 


419 


397 


375 


352 


330 


308 


288 




IM 


56.4 


176.1 


540 


522 


501 


477 


453 


427 


401 


376 


351 


328 




2 


62.8 


196.4 


602 


582 


558 


532 


505 


476 


447 


419 


391 


365 


13 


1 


37.7 


117.8 


363 


353 


341 


327 


312 


296 


280 


264 


249 


234 




IM 


46.1 


144.2 


444 


432 


417 


400 


382 


363 


343 


323 


304 


286 




1^ 


54.2 


169.4 


522 


507 


490 


470 


448 


426 


403 


380 


358 


336 




1^4 


61.9 


193.3 


596 


579 


559 


536 


512 


486 


460 


434 


408 


383 




2 


69.1 


216.0 


665 


647 


625 


599 


572 


543 


514 


485 


456 


428 


14 


1 


40.8 


127.6 


395 


386 


374 


361 


346 


331 


315 


299 


283 


267 




IM 


50.1 


156.5 


485 


473 


459 


442 


424 


405 


386 


366 


347 


327 




1^ 


58.9 


184.1 


570 


556 


540 


520 


499 


477 


454 


431 


408 


385 




m 


67.4 


210.5 


652 


636 


617 


595 


571 


545 


519 


492 


466 


440 




2 


75.4 


235.6 


730 


712 


690 


666 


639 


610 


581 


551 


522 


493 


15 


1 


44.0 


137.4 


427 


418 


407 


394 


380 


365 


349 


333 


317 


301 




m 


54.0 


168.7 


525 


514 


500 


484 


467 


448 


429 


409 


389 


370 




1^ 


63.6 


203.4 


618 


605 


589 


570 


550 


528 


505 


482 


459 


439 




1?^ 


72.9 


227.6 


708 


694 


675 


653 


630 


605 


577 


552 


525 


502 




2 


81.7 


255.2 


795 


777 


756 


732 


706 


678 


649 


619 


589 


559 



* This table is for safety factor 8; for safety factor 10 multiply the tabu- 
lar safe loads by /g. 



608 



32.— PROPERTIES AND TABLES OF COLUMNS. 



14. — Rolled Steel H Columns. 

(Bethlehem Steel Co.) 



For all 


sections. 






• 




,f T t A 


t-'"^r-i-' 


-^ \ 




W) 


c I D 






E ^ 


-A. ; 






A| "i, 1 y 


K— -B— -H 




Fig. 16. 




W = wt. of section in lbs. per lin. ft. | 


A =area of section in sq. ins. 




r' = least radius of gyration. 




8" H Columns. 




(Section Number = H8.) 








Dimen. In Ins.i 




W 


A 




T' 


D 


t 


w 


31.5 


9.17 




7% 


^ 


.31 


1.98 


34.5 


10.17 


8 


1 


.31 


2.01 


39.0 


11.50 


8M 


.35 


2.03 


43.5 


12.83 


SH 


.39 


2.04 


48.0 


14.18 


8% 


ii 


.43 


2.05 


53.0 


15.53 


m 


H 


.47 


2.07 


57.5 


16.90 


8^A 


i| 


.51 


2.08 


62.0 


18.27 


SU 


% 


.55 


2.09 


67.0 


19.66 


W8 


16 


.59 


2.11 


71.5 


21.05 


9 


1 


.63 


2.12 


76.5 


22.46 


9^ 


jJL 


.67 


2.13 


81.0 


23.78 


9M 


1/^ 


.70 


2.14 


85.5 


25.20 


9% 


Ij^ 


.74 


2.16 


90.5 ' 26.64 


9^ 


IM 


.78 


2.17 


1 / = 6.14=tang. dist. bet. fillel 


:s; r = 


0.40 =rad. of fillets; 5 = 7.69+ w 


;; w = 


i-0.038;« = i+0.038. 




10'' H Columns. 




(Section Number = HIO.) 








Dlmen. In Ins. 2 




W 


A 




r' 


D 


t 


w 


49.0 


14.37 


9% 


■h 


.36 


2.49 


54.0 


15.91 


10 


% 


.39 


2.51 


59.5 


17.57 


10^ 


il 


.43 


2.53 


65.5 


19.23 


1014 


M 


.47 


2.54 


71.0 


20.91 


109^ 


if 


.51 


2.56 


77.0 


22.59 


10^ 


% 


.55 


2.57 


82.5 


24.29 


105^ 


\^ 


.59 


2.58 


88.5 


25.99 


10^^ 


1 


.63 


2.60 


94.0 


27.71 


10% 


li^ 


.67 


2.61 


99.5 


29.32 


11 


m 


.70 


2.62 


105.5 


31.06 


IIV^ 


1^ 


.74 


2.64 


111.5 


32.80 


UM 


m 


.78 


2.65 


117.5 


34.55 


im 


1^ 


.82 


2.66 


123.5 


36.32 


n^ 


m 


.86 


2.67 


2 / = 7.67 = tang. dist. bet. fillet 


s; ? = 


0.50 = rad. of fillets; B=9M+w 


; m = 


t-OMS 


\n=t+( 


104 


8. 









12" H Columns. 
(Section Number = H12.) 



w 


A 


Dlmen. In Ins.3 


D 


t 


w 


64.5 
71.5 


19.00 
20.96 


iiM 
11% 


I 


.39 
.43 


78.0 
84.5 
91.5 
98.5 


22.94 
24.92 
26.92 
28.92 


12 

1234 




.47 
.51 
.55 
.59 


105.0 
112.0 
118.5 
125.5 


30.94 
32.96 
34.87 
36.91 


12J^ 
125^ 
12^ 
12% 


1 

li^ 

^¥ 
1^ 


.63 
.67 
.70 
.74 


132.5 
139.5 
146.5 
153.5 


38.97 
41.03 
43.10 
45.19 


13 
13% 
13^ 
13% 




.78 
.82 
.86 
.90 


161.0 


47.28 


1334 


1^ 


.94 



2.98 
3.00 
3.01 
3.03 
3.04 
3.06 

3.07 
3.08 
3.10 
3.11 

3.13 
3.14 
3.15 



3.18 



3 / = 9.21= tang. dist. bet. fillets; r = 
0.60 = rad. of fillets; B = 11.53+m;; w = 
i-0.058;w = ^+0.058. 

14" H Columns. 
(Section Number = H14.) 







Dimen. In Ins.^ 




w 


A 




1- 


r' 






D 


t 


w 




83.5 
91.0 


24.46 
26.76 


13% 


n 


.43 

.47 


3.47 
3.49 


99.0 
106.5 
114.5 
122.5 


29.06 
31.38 
33.70 
36.04 


14 

14% 

14>4 


1^* 


.51 
.55 
.59 
.63 


3.50 
3.52 
3.53 
3.55 


130.5 
138.0 
146.0 
154.0 


38.38 
40.59 
42.95 
45.33 


14^ 

14% 

im 

14% 


i 


.67 
.70 
.74 
.78 


3.56 
3.58 
3.59 
3.61 


162.0 
170.5 
178.5 
186.5 


47.71 
50.11 
52.51 
54.92 


15 
15% 

15% 


1 


.82 
.86 
.90 
.94 


3.62 
3.64 
3.65 
3.66 


195.0 
203.5 
211.0 
219.5 


57.35 
59.78 
62.07 
64.52 


15^ 
15% 

15% 


i 


.98 
1.02 
1.05 
1.09 


3.68 
3.69 
3.70 
3.71 


227.5 
236.0 
244.5 
253.0 


66.98 
69.45 
71.94 
74.43 


16 
16% 
16M 
16% 


2^^ 


1.13 
1.17 
1.21 
1.25 


3.72 
3.74 
3.75 
3.76 


261.5 
270.0 
278.5 
287.5 


76.93 
79.44 
81.97 
84.50 


16^ 
16% 

16% 


2^ 
2% 

2^ 
2k 


1.29 
1.33 
1.37 
1.41 


3.77 
3.79 
3.80 
3.81 



4 / = 11.06 =tang. dist. bet. fillets; r = 
0.60 = rad. of fillets; B = 13.49+ w; m = 
0.067; » = i+0.067. 



STEEL H'COLUMNS. REIN.-CONC, COLUMNS, 609 

Reinforced Concrete Columns. — The following working stresses for static 
loads are recommended by the Special Committee of the Am. Soc. C. E., on 
Concrete and Reinforced Concrete. See Trans. A. S. C. E., Vol. LXVI., 
page 452. For Notation and Formulas, see Sec. 25, Masonry, page 446. For 
working stresses for Beams, see Sec. 31, page 585. 

Average compressive strength of concrete — 2000 lbs. per sq. in. at 28 days, 
when tested in cylinders 8 ins. in dia. and 16 ins. long, under laboratory con- 
ditions of manufacture and storage. 

Bearing. — See page 585. 

Axial Compressions. — (A). For concentric compression on a plain con- 
crete column or pier, when the length does not exceed 12 diameters, 450 lbs. 
per sq. in. on2000-lb. concrete may be allowed. (B). Columns with longi- 
tudinal reinforcement only, 450 lbs. per sq. in. on 2000-lb, concrete. (C). Col- 
umns with reinforcement of bands or hoops, 540 lbs. per sq, in. on 2000-lb. 
concrete may be allowed. (D). Columns reinforced with not less than 1% 
and not more than 4% of longitudinal bars and with bands or hoops, 650 lbs. 
per sq. in. for 2000-lb. concrete. (E). Columns reinforced with structural 
steel column units which thoroughly encase the concrete core, 650 lbs. per 
sq. in. for 2000-lb. concrete. 

Reinforcement. — In all cases, longitudinal steel is assumed to carry its 
proportion of stress; and the compressive stress shall not exceed 16000 lbs. per 
sq. in. or 15 times the working compressive stress in the concrete. Hoops or 
bands are not to be counted upon directly as adding to the strength of the 
column. Bars composing longitudinal reinforcement shall be straight, and 
shall have sufficent lateral support to be securely held in place until the con- 
crete has set. When bands or hoops are used, the total amount of reinf orce- 
ment shall not be less than 1 % of the volume of the column enclosed. The 
clear spacing of such bands or hoops shall not be greater than one-fourth the 
diameter of the enclosed column. Adequate means must be provided to 
hold bands or hoops in place so as to form a column with a straight and well- 
centered core. Bending stresses due to eccentric loads must be provided for 
by increasing the section until the maximum stress does not exceed the 
values above specified. 

EXCERPTS AND REFERENCES. 

Reinforcement of Concrete Columns (By E. P. Goodrich. Eng. News, 
July 19, 1906). — "Considere reports that spiral steel is 2.4 times as effective 
as longitudinal reinforcement of equal weight, and this figure was closely 
checked by v. Bach, of Stuttgart." 

Table of Weights of Lacing for Steel Compression Members (By 
C. T. Lewis. Eng. News, Aug. 2, 1906). — Weights are in lbs. per lin. ft. of 
single- or double-laced member on one 6ide; the depths of member ranging 
from 5'' to 23'', and the size of lacing bars from U''xi" to 3''xr. Rivets, 

r to r. 

Table of the Various Column Formulas in Use (Eng. News, Jan. 3, 
1907). — Comprises formulas of the Rankine type for steel and cast iron; of 
the straight-line type for steel, cast iron and timber; and of miscellaneous 
type for timber: with an equivalent reduction, for all formulas, to a formula 
having a factor of eccentricity e. Interesting as a study. 

Detachable Form for Concrete Columns (By W. S. Coulter. Eng. 
News, Mar. 28, 1907). — Blustration and description. When building 
reinforced concrete columns, having flexure rods united at intervals by ties, 
forms are often used having one side open, which is built up in sections as 
the concrete is deposited. These forms usually consist of four or more 
uprights enclosed on three sides, the fourth being open to allow access to 
the interior, and closed as the work proceeds by horizontal boards nailed 
to the uprights. Through the open side all the operations of depositing, 
spading and tying are conducted. The present device is intended to facili- 
tate the work of erection and expedite the placing of the concrete. 

Table of Tests of Carbon=Steel and Nickel=Steel Columns, and Com- 
parison with Formulas (By C. P. Buchanan. Eng. News, Feb. 13, 1908). — 
Table gives actual strength and computed strength of columns. The com- 
puted strengths are from the following formulas: 

(1) Dagron's tests (steel cols.) compared with P = 61000— 263 l/r 

(2) Waddell's tests (nickel-steel) compared with P = 47000 - 1 78 l/r 

(3) Buchanan's tests (steel) compared with . .P= 79000- 388 l/r 

In which P = computed strength in lbs. per sq. in.; and / and r the 

length and radius of gyration, in ins. (See, also, Eng. News of April 9, 1908.) 



610 32.— PROPERTIES AND TABLES OF COLUMNS. 

Safe Stresses in Steel Columns (By T.R.Worcester. Trans A. S. C. E., 
Vol. LXI). 

Tests of Reinforced Concrete Columns at Minneapolis, Minn. (By 

J. G. Houghton and W. P. Cowles. Eng.News, Dec. 3, 1908) . — The columns 
tested had the following kinds of reinforcement: Spiral wire hooped; 
circular fiat-bar bands; wire bands, square; wire bands, circular. The test 
columns were made of 1:2: 3^ concrete, using bank sand and blue limestone, 
one-half of the stone being of |-in. size and one-half.of pea size. The re- 
inforced columns averaged about 25% stronger than the plain columns, 
though 5 of the 17 failed at lower loads than the plain columns. 

Preliminary Program of Tests of Steel Columns (Proc. A. S. T. M., Vol. 
VIII. , 1908). — ^Types of sections selected for testing, at Watertown Arsenal: 
(a) Annular section (welded tubes); (b) New wide flange H sections; (c) I 
section of four angles and central web-plate; (d) Double-channel section, 
latticed in two planes. Also other shapes. Illustrated. "Some results of 
the tests," by J. E. Howard, may be found on pages 336 to 344, same Vol. 
of Proc; also Vol. IX., 1909, page 413. 

Tests of Plain and Reinforced Concrete Columns (By M. O. Withey; 
Proc. A. S. T. M., Vol. IX., 1909).— Tables and diagrams. 

Tests of Plain and Reinforced Concrete Columns (By M. O. Withey. 
Paper A. S. T. M., July 1, 1909; Eng. Rec, July 10, 1909).— Conclusions 
derived from the tests: 1. A small amount, ^ to 1%, of closely spaced lateral 
reinforcement, such as the spirals used, will greatly increase the toughness 
and ultimate strength of a concrete column, but does not materially affect 
the yield point. More than 1% of lateral reinforcement does not appear to 
be necessary. The use of lateral reinforcement alone does not seem advis- 
able. 2. Vertical steel in combination with such a lateral reinforcement 
raises the yield point and ultimate strength of the column and increases its 
stiffness. Columns reinforced with vertical steel only are brittle and fail 
suddenly when the yield point of the steel is reached, but are considerably 
stronger than plain columns made from^ the same grade of concrete. 3. 
Increasing the amount of cement in a spirally reinforced column increases 
the strength and stiffness of the column. A column made of rich concrete or 
mortar and containing small percentages of longitudinal and lateral rein- 
forcement is without doubt fully as stiff and strong and more economical 
than one made from a leaner mux reinforced with considerably more steel. 
In these tests doubling the amount of cement increased the yield point and 
ultimate strength of the columns without vertical steel about 100% and added 
about 50% to the strength of those with 6.1% vertical steel. 4. From the 
behavior of the columns reinforced with spirals and vertical steel, under test 
and the results computed, it would seem that a static load equal to 35 to 40% 
of the yield point would be a safe working load. As the ultimate strength 
of the concrete and the yield point of the steel are generally known, or can 
be assumed with fair accuracy, formula (A) can be readily used to determine 
the working load. In Fig. 7 the dotted lines represent working values of 
P-7-A equal to 40% of the yield point load. (See original article for formula 
A and Fig. 7). 5. The results obtained from tests of columns reinforced 
with structural steel indicate that such columns have considerable strength 
and toughness and that the steel and concrete core act in unison up to the 
yield point of the former. The shell concrete will remain intact until the 
yield point of the steel is reached, but no allowance should be made for its 
strength or stiffness. 6. As many of the blotters on the tops and bottoms 
of columns bore imprints of the vertical steel after failure, it would seem a 
safe precaution to use bed plates at the foundations for such columns and 
thus prevent any possibility of the steel punching through the concrete under 
an excessive load. 

Tests of NickeUSteel Models of Compression Members in the Official 
Design of the New Quebec Bridge (Eng. Rec, Nov. 19, 1910). — Illustrated 
description with table of results of tests. 

Illustrations and Diagrams. 

Description. Eng. Rec. 

Rein.-terra-cotta col. tested to 4109 lbs. per sq. in., uninjured Feb. 20, '09 

Economy diagrams of plain and reinforced concrete columns Jan. 8, *10 

Repeated and eccentric load tests on rein. -cone, columns July 9, *10 



33.— STRUCTURAL DETAILS, 



The handbooks published by the various steel manufacturers are now 
pretty well standardized, and are indispensable to constant designers and 
detailers of structural work. The writer aims to keep this volimie abreast 
of the most approved practice in the design of ordinary structures. 

Rivets. — In the '80's, iron rivets began to give way to steel rivets in 
engineering structures, and now the latter are universally employed. The 
best rivet steel is a soft steel whose tensile strength varies not more than a 
few thousand pounds from 53 000 lbs. per sq. in.; the manufacturer's stan- 
dard specifying 48 000 and 58 000 as the lower and upper limits. Higher 
grades of steel are more liable to fracture both in driving and afterward 
when subjected to repeated stresses in the structure. 



Shop. 



-1- 
O 



Q 



High High 

a 



# 

Shop. 



Conventional Rivet Signs. 
(Osbom Code.) 



Field. 



Full Heads 



{ Both Sides. 
Figs. 1. 



Countersunk and 
Chipped. When 
No Chipping is ^ 
Required Mark 
"Not Chipped" 
(Or See Below). 



This Side (Outside). 
Other Side (Inside). 
Both Sides. 

Figs. 2. 



® 



( 


r 


o 


1 


< 


) 





1 


i] 







1 



Head Flattened 
To Vs'' High, or J 
C ount ers u nk and 
Not Chipped. 



This Side (Outside). 
Other Side (Inside). 
Both Sides. 
Figs. 3. 



. Head 
Flattened. 



This Side (Outside). 
Other Side (Inside). 
Both Sides. 
Figs. 4. 

6H 



ToV To\" 
Hiqh HigTi 

Field. 



612 



SS.—STRUCTURAL DETAILS. 



1. — ^Rivets — Shearing and Bearing Value 

In Pounds. 

(Dimensions are in Inches.) 



Di^m. 
of 


Area. 
Sq. 
Ins. 


Single 
Shear 




Bearing Value for Plates of Thickness — • 


Rivet. 




K 


A 


H 


A 


H 


^ 


% 


H 


H 


H 


Vs 


.1104 
.1963 
.3068 
.4418 
.6013 


03 


660 
1180 
1840 
2650 
3610 


i 
§ 

q 

< 


1130 
1500 


'1410 


1690 




3000 


4220 


4690 
5630 


6190 


6750 
7880 




y2 


1880 


2250 
2810 


[2630 




% 


I88O1 2340 

^250|2810 

26301 3280 


3280 


|3750 




K 


3380 3940 


4500 


5910 




% 


3940 4590 5250 


6560 


7220 


8530 


% 


.1104 
.1963 
.3068 
.4418 
.6013 


trl 

i 

♦J 
< 


830 
1470 
2300 
3310 

4510 


i 

< 


1410 


1760 


2110 1 


5280 


5860 
|7030 


7720 
9030 


8440 
9850 




H 


1880 
2340 


2340 2810 
2930 3520 


13280 


3750 

|4690 




% 


4io9 




M 


2810 
3280 


3520 


4220 4920 


5630 


6330 
7380 




'A 


4100 


4920 57401 6560 


8200 


10670 


% 


.1104 
.1963 
.3068 
.4418 
.6013 


00 

O 


1100 
1960 
3070 
4420 
6010 


oj 

1 

< 


1880 
2500 
3130 


2340 


2810 




7030 


7810 
9380 


10310 
12030 


11250 
13130 




V2 


3130 
3910 


3750 
4690 


4380 


5000 
|6250 




H 


5470 
6560 




% 


3750 
4380 


4690| 5630 
5470 6570 


7500 


8440 
9840 




'A 


7660l 8750 


10940 


14220 


=1= 


.1104 
.1963 
.3068 
.4418 
.6013 


en 
< 


1320 
2360 
3680 
5300 
7220 


i 

5 


2350 
3130 
3910 


2930 3520 




8790 


9770 




14060 
16410 




y2 


3910 4690 
4880 5860 


5470 


6250 
7810 




% 


6840 
8210 
9580 




K 


4690 
5470 


5860 


7030 
8210 


9380 
10940 


10550 
12310 


U720}l2890 




K 


6840 


13670 


15040 


17770 



_ Note. — Bearing values above or to right of upper zigzag line of each 
pair are greater than double shear. Values between upper and lower 
zigzag lines of each pair are less than double and greater than single shear. 
Values below or to left of lower zigzag line of each pair are less than single 
shear. 

Problem in Riveted Joints. 
(Reference to Tables 1, 2, 6; and to Figs. 5, 6, 29.) 

Example. — Let it be required to design a fiat steel bar, spliced at middle 
with a chain-riveted joint (Fig. 29, page 617), to resist a tensile stress of 
69600 lbs. ; as per following data. 

Data. — Assume rivets in single shear at 10000 lbs. per sq. in. and bearing 
value for plates at 20000 lbs. per sq. in. (see Table 1, above). Assume allow- 
able tension on net area of splice plates and bars, at 17500 lbs. persq. in. 
(see Figs. 5 and 6, in Table 2, for pitch, etc., of rivets.) 

Solution. — 
Allowable total stress on bar and joint =69600 lbs. 

Rivets in shear (Table 1) : Eight ^-in. rivets in double shear 

=8X2X4420=70720 lbs. 
Rivets in bearing on splice plates (Table 1) : Two A-in. plates 

= 8 X 2 X 4690 = 75040 lbs. 
Rivets in bearing on main bar (Table 1) : One ^-in. bar =8X9380 = 75040 lbs. 
Tension on main bar: HXl7500X6^i (net width of bar in ins.) =69730 lbs. 
Transverse width in bar occupied by holes (see Note to Table 6): 

3(M+H)=2^ins. 



RIVETS AND RIVETING. 



613 



Total width of bar (or splice plate) required for stress 

=6^(net)+2^(holes) =9 ins. 
Natural spacing of rivets on transverse section of bar and splice 

=9-i-3 = 3ins. 
(This pitch is allowable, from Table 2, second column.) 
Distance from center of rivet to edge of plate = (9-^2) — 3 = 13^ ins. 

(This is allowable, from Table 2, and Fig. 6.) 
Distance between centers of rows of rivets (Table 2) : 3X cos 30° =2^ ins. 
Distance from centers of rivets to ends of bars or plates: 

not < }4 pitch -r- = 13^ ins. 



Length of splice plates: lJ^+2^+2^+li4+13^+2^8+2^+13^ 



—163^ ins. 



2. — General Rivet Spacing, Clearances, 
(All dimensions in Inches.) 



ETC. 





i ^ 

9-! K 

Fig. 5. 

Min. 

Pitch. 

P 


w . 


u 
o^ 


w 

a 

'o 
o . 

P 

o 


Min. Dist. ^ to Edge li;0 o 

of Plate. ^^^ ' 

Fig. 6. 


Min. Clear. 

From Center 

of Rivet. 






2 


e 


Plates Over 
^/ Thick. 


Plates Not 

Over 
H" Thick. 




:di 


O 
u 


CO 

e 


e 


^3 • 
C/D 


e 


Fig. 7. 
A 


Fig. 8. 
C 


H 


Best 
Ml 

W2 IM 

2)4: 

23^2^ 
3 










Not including Fillers 
or Lattice Bars. 


A (min 


) 














V, 












Vo 


















1^ 

8 

1 
ii^ 


Xi 


y. 


4 
4 
4 

4 


23^ 
3 

33^ 
4 


6 


18 












13^ 

IM 


IM 
1>^ 


IM 

13^ 




l}4 















614 



^3.— STRUCTURAL DETAILS. 



-Rivet Gages for Standard Shapes. 
(All dimensions in Inches.) 



Angles. 


I-Beams. 


Channels. 


Z-Bars. 


T-Bars. 


■r^^'^ 


1^\. 


°© 




^ 




c^i^ 




f 


nr^ 


*— . 








H* 












■~::i 


D 


Fig. 9. 


Fig. 10. 


Fig. 11. 


Fig. 12. 


Fig. 13. 






<u 




M 






"S 


w 












S)6 




1 


• 




bo 


• 


^ 






Sh 


^ 












op q; 








1 




1 


■4-> 




4-5 


d 


3 

4-5 






1 






^ 






o 


o 


^ 


<+-! 


fe 




> 


«+-< 


fe 


^ 


> 


tM 


^ 


> 


'ar^ 


aj 


1 


^ 

1 
^ 


C 

6^ 




G2 


O 

a 

(0 

Q 


a 

i 


O 
G 


s 

>< 

OS 


o 
Q 


I. 


o 

G 


5 


o 

A 

a 
Q 


o 

G 


5 




G 


'5^ 


6 

bo 


a 

<u 


8 


4H 


3 


3 


24 


100 


4 


3^ 


15 


55 


V/4. 


Ya 








Ada 


pt 


ed 


1^ 


om 


7 


4 


2J^ 


3 


24 


80 


4 


3^ 


15 


45 


2M 


K 


Car 


neg 


ie 




Ca 


m 


br 


ia 


6 


3K 


2H 


2M 


20 


100 


4 


Vs 


15 


40 


Ws 


% 


6to6K 


2H 




5 


2K 




Y 




5 


3 


2 


IM 


20 


80 


4 


Vs 


15 


33 


Ws 


M 


5to5K 


IK 




4K 


2M 




Y 




m 


21^ 
2M 


2 


\y 


20 
20 


75 

65 


3H 
3^ 


1 


12 
12 


40 
30 


2 
2 


M 
M 


4to4K 
3to3ii6 


Ws 
lYs 


•• 


4 
3K 


2 

IM 


f 


Ys 
Ys 


i" 


4 


Max. 


1 


31^ 


2 


Rivet. 


18 


70 


3M 


K 


12 


25 


1^ 


M 








3 


IK 


m 




K 


3M 


IH 


Vs 


18 


55 


3M 


3^ 


12 


20.5 


IM 


^ 


Ca 


mbr 


ia 


2M 


IK 




K 




3 


IH 


Vs 


15 


100 


3% 


Vs 


10 


35 


2 


M 


6to6K 


2 


1 


2K 


IM 


m 


K 


M 


2M 


m 


H 


15 


80 


3^ 


Vs 


10 


25 


2 


^ 


5to5K 


IK 


Vs 


2H 


IK 


IH 


K 


M 


2H 


m 


H 


15 


75 


3^ 


M 


10 


20 


13^ 


M 


4to4H 


lY 


Vs 


2 


1 


IK 


K 


K 


2M 


IH 


H 


15 


60 


3M 


M 


10 


15 


IH 


M 


3to3j^ 


lYs 


Y 


IM 




1 




/^ 


2 


m 


Vs 


15 


55 


3 


M 


9 


25 


1^ 


^ 








IK 


K 


Y 


K 


K 


1^ 


1 


Vs 


15 


42 


3 


^ 


9 


20 


IM 


M 








IM 


K 


H 


K 


K 


IH 


Vs 


V2 


12 


55 


3 


M 


9 


15 


1^ 


M 










K 


Ys 


K 


K 


1^ 


Va 


y^ 


12 


40 


3 


M 


9 


13.25 


1^ 


M 








IK 


K 


A 


K 


K 


IM 


M 


y^ 


12 


35 


2^ 


^ 


8 


21.25 


IK 


M 








lA 




^ 




K 


IVs 


% 


y2 


12 


31.5 


234 


M 


8 


16.25 


IK 


M 








1 


K 


1% 


M 


^ 


^ 


¥ 


Vs 


10 




25/^ 


% 


8 


13.75 


IM 


^ 


















1 




Vs 


9 




23^ 


% 


8 


11.25 


IM 


M 


















J^ 


1^ 


Vs 


8 




2M 


H 


7 


19.75 


IK 


Ys 


















74 


re 




7 




2M 


Vs 


7 


17.25 


IK 


Ys 


















% 


^ 




6 
5 
4 
3 




2 

IM 
IH 
lA 


Vs 

y^ 

Vs 


7 
7 
6 
6 
6 
6 
5 
5 
5 
4 
3 


14.75 

9.75 
15.5 
13 
10.5 

8 
11.5 

9 

6.5 


1 


Ys 
K 



















Remarks on Tables 3 and 4. 

The advantage of using standard rivet gages and connections is the sav- 
ing of time in both the drafting room and in the shop. In detailing a steel 
building, the first step is to get out the erection drawings, showing diagrams 
of columns, floor plans, sectional elevations of complicated and special con- 
nections, and all sizes and dimensions^ from previous calculations of the 
design. Each member of the structure is numbered for the convenience of 
the detailer, the shop and the erection gangs. The detailing begins with the 
columns and then proceeds with the girders, beams, etc. 

Caution. — For special heavy and short beams the erection drawings 
should show special connections, if required. 



RIVETS AND RIVETING. 



615 



4. — Standard Connection Angles for I-Beams and Channels. 



For 24" Beams. 

Ls rx^xj^ff^'xl' 6" Ls 4"x4"x^''xl' 6" 



>fl 


< > 


-k 


'< 


^ 


2!( 


It 

< > 


^ 


t 

< 






to 

c 

i 
















\ 








> 


' 




-i 








< 








^ 








f 


-1 


-( 


- 




-( 


> 


1 




-(: 


^ 




f 


s 


^ 










m 










'J 







Fig. 14. 



For 2(f Beams. 

Ls 4"x4''x^''xl' r Ls 4"x4"x^"xl' 3'' 



tf LI. 





t ? 


^ 

t \ 


y 































I 
I 

I 



^ 



Fig. 15. 



For 18" Beams. 



Ls 4:''xA''xy8''xr 1" 



For 18" Beams 
The Carnegie Steel 
Company usessame 
connections as for 
20" Beams. 



=f2-^ >r£^ 



<> 






K> 



Fig. 16. 



For 15" Beams and Channels. 

L5 6"x4"x^"x0' 10" L56"x4"x^"x0'10'^ 



■» 


ft 




Hi 


ti^ 




^ 


^ /■^ 


- 




z 


c 


1^* 


7 


r 
















Fig. 17. 



For 12" Beams and Channels. 

L5 6"x4"x^"x0'73^" 

L5 6"x4"x^"x0'7i^" 



'-^rti^k^ 




y=K 



"^y ^%2V 



:<> 



1^ 



K> 



^^ 



<)■ 



<)■• 






Fig. 18. 



For 10," 9", 8" and 7" Beams and 
Channels. 

L5 6"x4"x^"x0' 5" Ls 6"x4"x^"x0' 5" 



Rivet Spacing 2J^". 
i" I* I* 







Fig. 19. 



For 6" and 5" Beams and Channels. 
6 in.: 

Ls 6"x4"xi^"x0' 3" L5 6"x4"x3^"x0' 2^" 
5 in.: do. xO'23^" 



Fig. 20.' 



For 4" and 3" Beams and Channels. 
Ls 6"x4"x^"x0' 2" L5 6"x4"x^"x0' 1^" 



Fig. 21. 



Note. — In the above illustra- 
tions, two systems of connection 
angles are shown, that of the Car- 
negie Steel Co. being on the left, 
and Cambria on the right, in each 
case. Legs showing shop rivets 
are riveted directly to ends of 
I-beam or channel, while those show- 
ing field rivets are for field connec- 
tion. Angles are riveted on in 
pairs. A play of tV of an inch 
is allowed at each end of the built 
member between the back of angles 
and the girder to which it is to be 
connected in the field. 



616 



B3.— STRUCTURAL DETAILS. 



5. — Weights and Dimensions of Rivets. 



. XX l'L5— 6rip->L K— 6rip— l^.i^-Jy 



Countersunk 



•Length— > •<— Length' 

Dimensions (in inches) of Heads after Driving. 



Head. 



Kind of Rivet. 


Diameter of Rivet, in Inches. 


H 


H 


^ 


% 


% 


H 


1 


m 


Button [Values of H( = j% diam.) 
Head { Values of i?( =f dicim.+^) 
Rivets. [Values oi W 


.15 

t 


.225 
U 

I 


.3 

11 


.375 


.45 


.525 


.6 

■I 


.675 


Countersunk/ Values of h 


si 


Rivets. \ Values of w 


W 











Lengths 


and 


Weights 


of Rivets. 








♦Lengths of Rivets for Various 


J Weight 


oi 100 Steel Rivetsf of Various 


Grips. 


Lengths 


. (Weight in Lbs 


; Dimen- 


(Dimensions in Inches.) 




sions in Inches.) 


a 


Diameter of Rivets. 




^ 

1 
^ 


Diameter of Rivets. 


u 
O 


M 


^ 


¥2 


ys 


% 


ys 


1 


H 


3^ 


ys 


H 


ys 


1 


y?, 


1 


1¥ 


ly?. 


m 


m 


2 


214 


l¥ 


5.5 


12.8 


22.0 


29.3 


43.9 


66.6 


/^ 


IVh 


13/8 


m 


m 


2 


2y 


214 


IH 


6.3 


14.2 


24.1 


32.4 


48.2 


72.1 


H 


1^ 


IH 


m 


2 


2y 


2H 


234 


1^ 


7.0 


15.5 


26.3 


35.5 


52.5 


77.7 


Vh 


13^ 


1^8 


m 


2% 


2H 


234 


2y 


2 


7.9 


16.9 


28.5 


38.7 


56.7 


83.3 


1 


IH 


134 


2 


214 


234 


214 


2/8 


214 


8.7 


18.3 


30.7 


41.8 


61.0 


88.8 


IVh 


m 


1^/^ 


214 


234 


214 


2/8 


234 


214 


9.4 


19.7 


32.8 


44.9 


65.2 


94.4 


IH 


m 


2 


214 


214 


2^4 


234 


274 


234 


10.2 


21.1 


35.0 


48.0 


69.5 


100. 


IH 


VA 


2^ 


234 


2^4 


2^4 


3 


3 


3 


11.0 


22.5 


37.2 


51.1 


73.7 


105. 


IV?. 


2 


214 


214 


234 


3 


314 


314 


3^4 


11.7 


23.9 


39.3 


54.3 


78.0 


111. 


IH 


21/^ 


2-^ 


2^4 


2^4 


314 


314 


314 


3H 


12.6 


25.3 


41.5 


57.4 


82.3 


116. 


IH 


2K 


21/^ 


234 


3 


314 


334 


314 


m 


13.4 


26.7 


43.7 


60.5 


86.5 


122. 


V/h 


2Vh 


2y^ 


2^4 


314 


334 


314 


3/8 


4 


14.1 


28.1 


45.9 


63.6 


90.8 


128. 


2 


2H 


2% 


314 


334 


314 


3/8 


334 


4H 


14.9 


29.4 


48.0 


66.7 


95.0 


134. 


2% 


2H 


2V^ 


314 


314 


3^4 


334 


'sy 


4H 


15.7 


30.8 


50.2 


69.9 


99.3 


139. 


2H 


2H 


3 


334 


3^/8 


334 


3/8 


4 


m 


16.5 


32.2 


52.4 


73.0 


104. 


145. 


2H 


2V^ 


314 


314 


3H 


3/8 


4 


4^4 


5 


17.2 


33.6 


54.5 


76.1 


108. 


150. 


2^ 


3 


314 


3^4 


374 


4 


414 


414 


514 


18.1 


35.0 


56.7 


79.2 


112. 


156. 


234 


314 


3^4 


374 


414 


414 


434 


414 


514 


18.8 


36.4 


58.9 


82.3 


116. 


161. 


3 


3H 


374 


414 


434 


414 


m 


434 


534 


19.6 


37.8 


61.1 


85.5 


120. 


166. 


3^ 


334 


414 


434 


434 


474 


5 


514 


6 


20.4 


39.2 


63.2 


88.6 


124. 


172. 


3H 


4 


434 


454 


5 


514 


514 


534 


6^4 


21.9 


42.0 


67.6 


95.1 


133. 


184. 


334 


414 


454 


474 


514 


534 


5y 


by 


7 


23.5 


44.7 


71.9 


101. 


142. 


195. 


4 


4^ 


474 


514 


514 


5^ 


534 


by 


7/2 


25.1 


47.5 


76.1 


108. 


150. 


206. 


4M 


434 


514 


514 


534 


5% 


6 


m 


8 


26.6 


50.3 


80.6 


114. 


159. 


217. 


4^ 


5 


534 


534 


6 


m 


614 


6/8 


8/2 


28.2 


53.1 


85.0 


120. 


167. 


227. 


434 


514 


5^ 


6 


614 


614 


6/8 


634 


9 


29.8 


55.9 


89.3 


126. 


176. 


239. 


5 


hV? 


574 


614 


6^4 


634 


674 


7 


W. 


31.3 


58.7 


93.7 


133. 


185. 


250. 


5K 


5% 
6 

6M 
6V^ 


6J^ 
6^ 
6^ 


6M 

7 
7H 


6% 

m 

7^ 
1% 


7 
7M 

7/8 

774 


lys 
7^ 


7H 
8 


10 


32.8 


61.4 


98.0 


139. 


193. 


261. 


63/2 
53^ 


Weiglit of One Button Head. 


6 


*• 


.018 


.058 


.111 


.136 


.226 


.390 


















11. . . 


.016 


.042 


.090 


.125 


.215 


.330 


Deduction in Inches from Above 












Lengths if One Head is 


Weig 


jhtof 


OneL 


ichof 


Shank. 


Countersunk. 














H 


^ 


y2 


^ 


^ 


H 


ysl 




.031 


.056 


.087 


.125 


.170 


.222 



* For /s'' to r diam., L= 1 .08 (Grip 4- diam. + 3^). 

For W rivets, L= 1.08 (Grip 4- J^). t Also round-headed bolts without nuts. 
X For iron rivets deduct 2 per cent. ** Before driving. || After driving. 



RIVETS AND RIVETING. 



617 



Lap Joints. 



6. — Riveted Joints. 
Kinds of Riveted Joints, 



Butt Joints. 





oj 
o| 



Single Riveted. 



o{<? 
o|o 

0|0 



Fig. 24. 



Fig. 25. 



Double Riveted. 





O ;0 
o o 

o!o 












>^ V^ V— > \^ 





Fig. 26. 



Fig. 27. 




i-\ ,r~» !=>_ 



'w' «^ V^ V-/ 1 ^ 



Chain Riveted, 



^o^l'^o ° 

^o^i°o° 
o ojo o 













Fig. 28. 



Fig. 29. 



Table for Finding Net Areas of Riveted Joints. 

Areas in square inches to be deducted for rivet holes in plates of various 
thicknesses to obtain net area of joint for tension. 



lick- 
ss of 
ate. 










* Size of Hole. 


hS^ 


M 


^ 


H 


1^ 


V2 


T^ 


% 


\h 


M 


it 


% 


^ 


V 


Ih 


IH 


lA 


m 


w 


.06 


.08 


.09 


.11 


.12 


.14 


.16 


.17 


.19 


.20 


'.22 


.23 


.25 


.26 


.28 


.30 


.31 


■h" 


.08 


.10 


.12 


.14 


.16 


.17 


.18 


.21 


.23 


.25 


.27 


.29 


.31 


.33 


35 


.37 


.39 


Vh" 


.09 


.12 


.14 


.16 


.19 


.21 


.23 


.26 


.28 


.30 


.33 


.35 


.37 


.40 


.42 


.44 


.47 


^" 


.11 


.14 


.16 


.19 


.22 


.25 


.27 


.30 


.33 


.35 


.38 


.41 


.44 


.46 


.49 


.52 


.55 


V?!' 


.12 


.16 


.19 


.22 


.25 


.28 


.31 


.34 


.37 


.41 


.44 


.47 


.50 


.53 


.56 


.59 


.62 


1%'' 


.14 


.17 


.21 


.25 


.28 


.32 


.35 


.39 


.42 


.46 


.49 


.53 


.56 


.60 


.63 


.66 


.70 


Vf^" 


.16 


.19 


.23 


.27 


.31 


.35 


.39 


.43 


.47 


.51 


.55 


.58 


.62 


.66 


.70 


.74 


.78 


W 


.17 


.21 


.26 


.30 


.34 


.39 


.43 


.47 


.51 


.55 


.60 


.64 


.69 


.73 


.77 


.81 


.86 


H" 


.19 


.23 


.28 


.33 


.37 


.42 


.47 


.51 


.56 


.61 


.66 


.70 


.75 


.80 


.84 


.89 


.94 


W 


.20 


.25 


.30 


.35 


.41 


.46 


.51 


.55 


.61 


.66 


.71 


.76 


.81 


.86 


.91 


.96 


1.01 


%" 


.22 


.27 


.33 


.38 


.44 


.49 


.55 


.60 


.66 


.71 


.76 


.82 


.87 


.93 


.98 


1.04 


1.09 


w 


.23 


.29 


.35 


.41 


.47 


.53 


.58 


.64 


.70 


.76 


.82 


.88 


.94 


1.00 


1.05 


1.13 


1.17 


1 " 


.25 


.31 


.37 


.44 


.50 


.56 


.62 


.69 


.75 


.81 


.87 


.94 


1.00 


1.06 


1.12 


1.19 


1.26 


liV'' 


.26 


.33 


.40 


.46 


.53 


.60 


.66 


.73 


.80 


.86 


.93 


1.00 


1.06 


1.13 


1.19 


1.26 


1.33 


\y^" 


.28 


.35 


.42 


.47 


.56 


.63 


.70 


.77 


.84 


.91 


.98 


1.05 


1.12 


1.19 


1.26 


1.33 


1.41 


i^." 


.30 


.37 


.44 


.52 


.59 


.66 


.74 


.81 


.89 


.96 


1.04 


1.13 


1.19 


1.26 


1.33 


1.41 


1.48 


IK" 


.31 


.39 


.47 


.55 


.62 


.70 


.78 


.86 


.94 


1.01 


1.09 


1.17 


1.25 


1.33 


1.41 


1.48 


1.56 



* Note that diam. of hole is greater than diam. of rivet; usually assumed 
at ^ to >^ in. greater. 

For Problem in Riveted Joints see page 612. 



618 



33.— STRUCTURAL DETAILS, 



7. — Standard Bolts for Fastenings. 



Drift Bolts for Timber. 

41^ 



Headed 

and 
Pointed. 



Plain. 



Fig. 30. Fig. 31. 



Screw Bolts for Timber and Metal. 



Square Wr f? Hexagonal 



Head 
and Nut. 



Head 
and Nut. 



Figs. 34, 35. 



Expans'n Bolts for Timber and Stone. 



Before 
Expansion. 




In Place. 



Hook Bolt for Bridge Work. 
HMutLockg^ 

"^^^^^ Fastening 

g^^j_^ Guard Rail and Tie 
d"'W^'\A to Girder. 



Fig. 36. 




Stone Bolts for Bridge Work. 



Diam. of hole = diam. 
of bolt + M". 

Length of wedge for 
split bolt = 3c?; width = (i; 
thickness at head = ^d; 
point rounded. 




Swedge 

Bolts. 

(Fig. 37.) 



M^'x 9",Wt.=2#. 

K''xl2",Wt. = 3#. 
1 ''xl2''. Wt.=4#. 
lM''xl5^Wt. = 7#. 



A u^^ f Screw bolts (Figs. 34, 

^ A A^y. c 1^1 ?^ 35) with flat washers; 

Swedged. SpM. Screw. Bolts for ^ middle of bolt de- 
Concrete fleered 1 dia. before 
Figs. 37, 38. 39. ^ setting. 



8. — Standard Screw Threads. 
(Sellers; Franklin Institute. Dec. 1864; United States, 1868.) 









tn 




tfl 




w" 




CO 




<-;:D, 


Diam. 


1^ 


Diam. 




Diam. 


1-0 

u q 


Diam. 


^■S 




















cl a 








.rJl-H 






J31-H 






r^HH 






.15 HH 








*o 


^fe 




TJ 


^fe 




^ 


^fc 




13 


Hw. 


f-:^^-j 


«h 


"•1 




a 

1^' 


^1 






4 




o a 


C/2 




6> 




H 


.185 


20 


1 


.837 


8 


2 


1.712 


iH 


4 


3.567 


3 




T^ 


.240 


18 


IVf^ 


.940 


7 


2K 


1.962 


4H 


4¥ 


3.798 


2Vh 




^^ 


H 


.294 


16 


m 


1.065 


7 


2y,, 


2.175 


4 


4H 


4.028 


2H 




^ 


.344 


14 


I'H 


1.160 


6 


2H 


2.425 


4 


4% 


4.255 


2% 


f 


^ 




.400 


13 


m 


1.284 


6 


3 


2.629 


3H 


5 


4.480 


2y. 




^ 


.454 


12 


m 


1.389 


sy?. 


3H 


2.879 


3H 


514 


4.730 


2H 




5/^ 


.507 


11 


Wa 


1.490 


5 


3V?. 


3.100 


3H 


5H 


4.953 


2H 




ZX 


.620 


10 


VA 


1.615 


5 


3H 


3.317 


3 


5«4 


5.203 


2% 


p 


ig.40. 


% 


.731 


9 














6 


5.423 


2^ 


















Note. — ^The Pitch is -rj-; and the fiat /, at top and bottom, is ^. 

In the Whitworth or English standard the angle of thread is 55° instead 
of 60°; the top and bottom of threads are rounded; and N is 12 for D=}^, 
otherwise N is the same as ^bove for Z? up to 3 inches. 



BOLTS AND NUTS, 



619 



9. — Dimensions and Weights of Hot Pressed Nuts. 

The sizes are the usual manufacturers', not the Franklin Institute 
Standard. Both weights and sizes are for the unfinished nut. 
(Dimensions in ins.; weights in lbs.) 



"o 


"o 


♦Square Nuts. 


t Hexagon Nuts. 


pq 

<4-l 


"o 






No. of 


V ti 








No. of 


o 


M 




Diag- 


Wt.of 


Nuts 




Short 


Long 


Wt.of 


Nuts 




§ 


QJ r-J 


onal. 


100 


in 


Diam. 


Diam. 


100 


in 


CO 


(5 


aDc/3 




Nuts. 


100 lbs. 


^""^ 






Nuts. 


100 lbs. 


H 


/2 


y2 


.71 


1.5 


6800 


M 


3^ 


.58 


1.3 


8000 


A 


1^2 


H 


.88 


2.9 


3480 


A 


H 


.72 


2.4 


4170 


Vs 


M 




1.06 


4.9 


2050 


Vs 


^ 


.87 


4.1 


2410 


^ 


il 


Vs 


1.24 


7.7 


1290 


^ 


Vs 


1.01 


6.8 


1460 


H 


^ 


% 


1.24 


8.6 


1170 


y2 


% 


1.01 


7.1 


1410 




"YE 


1 


1.41 


11.8 


850 


^ 


1 


1.15 


9.8 


1020 


1^ 


V2 


IVs 


1.59 


16.7 


600 


■h 


ly^ 


1.30 


14.0 


710 


H 


■h 


m 


1.59 


17.7 


570 


% 


lys 


1.30 


14.7 


680 


5A 


■h 


IH 


1.77 


22.8 


440 


H 


IM 


1.44 


19.1 


520 




^ 














1.44 


22.9 


440 


16 


IVs 


1.94 


32.3 


310 


1.59 


27.2 


370 


M 


%l 


iy2 


2.12 


39.8 


251 


% 


IK 


1.73 


39. 


256 


y% 


§i 


IVs 


2.30 


53. 


190 


% 


m 


1.88 


44. 


226 


Vs 


11 


IH 


2.47 


63. 


159 


1 


lys 


1.88 


50. 


198 




^ 


m 


2.47 


68. 


146 


1 


m 


2.02 


57. 


176 




Vi 


2 


2.83 


94. 


106 


13^ 


IH 


2.02 


64. 


156 


ii/g 




2 


2.83 


103. 


97 


IM 


2 


2.31 


96. 


104 


11^ 


14 


2M 

2K 
2K 


3.18 

3.18 
3.54 


137. 

145. 

186. 


73 

69 
54 












IM 












13i 


"m 


"iii" 


*'2;66* 


"m." 


"75" 


1^ 


lA 


2^ 


3.89 


247. 


41 


m 


23^ 


2.89 


180. 


56 


V4 


1^ 


3 


4.24 


319. 


31.3 


iVs 


2H 


3.18 


235. 


42 


1/4 


1^ 


m 


4.60 


400. 


24.8 


m 


3 


3.46 


300. 


33.4 


1% 


Ij^ 


^V2 


4.95 


500. 


19.9 


VA 


3M 


3.75 


370. 


26.7 


IH 


lii 


SH 


5.30 


620. 


16.2 


2 


33^ 


4.04 


460. 


21.5 


2 


m 


4 


5.66 


750. 


13.4 


2 


33^ 


4.04 


450. 


22.4 


2^ 


1% 


4 


5.66 


780. 


12.8 


23^ 


SH 


4.33 


560. 


18.0 


2H 


2 


4M 


6.01 


930. 


10.7 


2H 


3M 


4.33 


560. 


17.7 


2^ 


2H 


4M 


6.01 


960. 


10.4 


2Vs 


4 


4.62 


680. 


14.7 


23^ 


23^ 


43^ 


6.36 


1130. 


8.9 


2^ 


43<^ 


4.91 


810. 


12.3 


2M 


2i^ 


4M 


6.72 


1370. 


7.3 


2M 


4>^ 


5.20 


980. 


10.2 


3 


2H 


5 


7.07 


1610. 


6.2 


3 


4M 


5.48 


1150. 


8.7 


3M 


2y| 


5H 


7.78 


2110. 


4.7 


3M 


5 


5.77 


1340. 


7.5 


3M 


SVs 


6 


8.49 


2750. 


3.6 


W2 


5H 


6.06 


1580. 


6.3 



♦Thickness of square nut is equal to diameter of bolt. 
tThickness of hexagon nut not always equal to diameter of bolt. 



620 



3S,— STRUCTURAL DETAILS. 



10. — Bolts. 

Dimensions for Heads, Nuts, etc., etc. 

(Am. Bridge Co, Standard.) 







Round. 






Head 










Nut 


, 










Hexag- 


a; 






°i 


Hexag- 








onal. 


sis 


Square . 


Square. 


iJ n* 


onal. 












Kco 






fficS^ 








r"-- ■■• 


o©n 


.M Gm€> 






Fig. 41. 




+3 




Figs. 42. 


Figs. 43. 


4J 


W 




-M 


4JM-4 


u 






















PQ 






c3 

SI'S 




a 

xn 


s 


ct5 
S 


-(-> 


a 


s 




B 

5 


S 


+j 


5 


a 




i 

S 


1 


s 






bo 

§ 
v-3 


? 

cH 


0) 


4J 

eg 


J 


3 


4-> 


^ 
.^ 

^ 




g 
3 


a 

OJ 

S 


Ins. 


Sq. Ins. 


Ins. 


Ins. 


No. 


Ins. 


ins. 


Ins. 


Ins. 


Ins. 


Ins. 


Ins. 


Ins. 


Ins. 


Ins. Ins. 


¥ 


.049 


.185 


.027 


20 


■h 


Vh 


.... 


^ 


^ 


P 


3^ 


K 


K 


\% 


K 


H 


.110 


.294 


.068 


16 


Ik 


■h 


.... 




^ 


1 


•^ 


H 


•H 


^, 


/4 




.196 


.400 


.126 


13 


% 


'% 


.... 


f 1 


1 


114 


A 


y?, 


K 


1 


L^ 


.307 


.507 


.202 


11 






, 


«^ 


ln%- 


ly? 


ItV 


% 


I1V 


U^ 


5^ 


M 


.442 


.620 


.302 


10 


\^ 


V/^ 


H 


US 


ni 


m 


IH 


% 


IK 


IH 


z/ 


Vs 


.601 


.731 


.420 


9 


HI 


Ih 


H 


u\ 


IH 


2^\ 


1^ 


Vs 


Ifa 


Uh 


Vb 


1 


.785 


.837 


.550 


8 


US. 


ly?. 


% 


1^ 


2tV 


2^ 


1^/^ 


1 


1^4 


V/h 


1 


IVh 


.994 


.940 


.694 


7 


yih 


m 


1 


2t^ 


2fk 


m 


ly 


in 


2^ 


11/^ 


1¥ 


1.227 


1.065 


.893 


7 


2t^ 


ry. 


1^8 


m 


2~^ 


2^ 


2 


IH 


2 


2^ 


IM 


14 


1.485 


1.160 


1.057 


6 


2t^ 


2 


ly^r 


2 


2-4 


3f. 


2A 


IVh 


2^ 


2\l 
2H 


\Z/^ 


11^ 


1.767 


1.284 


1.295 


6 


2H 


214 


m 


2t^ 


3tV 


334 


2^ 


VA 


234 


m 


1^/^ 


2.074 


1.389 


1.515 


by?. 


214 


2t^«- 


1^2 


2«/^ 


3A^ 


35/8 


2t^ 


IVh 


2^ 


2U 


1^4 


IH 


2.405 


1.491 


1.746 


5 


3,^, 


2^/8 


iH 


2^^ 


3^^ 


3^^ 


2% 


IH 


23^ 
2H 


3^ 


134 


VA 


2.761 


1.616 


2.051 


5 


m 


2H 


iM 


2M 


3H 


4A 


2H 


VA 


3M 


1:^ 


2 


3.142 


1.712 


2.302 


4H 


3-^^ 


3 


13^ 


2^ 


iH 


4t^ 


31/^ 


2 


314 


354 


2 


2/^ 


3.976 


1.962 


3.023 


4H 


3P 


SVh 


2^8 


314 


4^ 


4i^ 


31^ 


214 


3H 


4f« 


2^ 


24 


4.909 


2.176 


3.719 


4 


4^^ 


334 


2«/^ 


3^ 


5K8 


51^ 


37/8 


2y 


374 


414 


2H 


2M 


5.940 


2.426 


4.620 


4 


m 


43^ 


2H 


4 


5§i 


6 


4M 


2% 


4M 


411 


2M 


3 


7.069 


2.629 


5.428 


3H 


5Tf. 


41/^ 


2% 


4^^ 


6Tf. 


61^ 


4^ 


3 


454 


534 


3 


31/f 


8.296 


2.879 


6.510 


'^y?. 


5^8 


4'^ 


3j/8 


m 


im 


7^ 


5 


314 


5 


5^1 


3;^ 


3H 


9.621 


3.100 


7.548 


m 


6tV 


5H 


3«/^ 




7^ 


7H 


53/^ 


3H 


534 


614 


3H 


3M 


11.045 


3.317 


8.641 


3 


6H 


5H 


3^8 


5i^ 


7H 


m 


5M 


3M 


5M 


6§i 


m 


4 


12.566 


3.567 


9.963 


3 


6M 


6 


3^ 


5il 


s^\ 


m 


m 


4 


614 


7f. 


4 


414 


14.186 


3.798 


11.329 


'm 


7AA 


6«/^ 


4^ 


6tt^. 


8r. 


9t^ 


m 


414 


614 


7t% 


414 


4H 


15.904 


4.028 


12.753 


2^4 


7§i 


6«4 


4«/^ 


6^;^ 


9^S 


9^4 


eyn 


4H 


6% 


7^^ 


4H 


4M 


17.721 


4.256 


14.226 


2^8 


8^^. 


7^8 


4H 


6^8 


9H 


lOM 


7H 


4M 


7M 


SH 


4M 


6 


19.635 


4.480 


15.763 


21/^ 


8^^ 


7H 


4^ 


7H 


lOH 


10^§ 


7H 


5 


754 


8^? 


5 


5K 


21.648 


4.730 


17.572 


2H 


Uo 


7'^ 


53^ 


n% 


lOH 


11^4 


8 


514 


8 


9^^ 




6^ 


23.758 


4.953 


19.267 


2«/^ 


9H 


8K 


5«/^ 


m 


ll'f'. 


11^/^ 


83^ 


514 


834 


n^ 


53^ 


534: 


25.967 


5.203 


21.262 


2«/^ 


91^ 


8^-8 


5H 


^u 


11^,^ 


123^ 


8^4 


534 


834 


m. 


53^ 


6 


28.274 


5.423 


23.098 


2M 


10^8 


9 


5^8 


8!§ 




12H 


93^ 


6 


93^ 


lOJi 


6 



BOLTS AND NUTS. 



621 



11. — "Weight op 100 Bolts with Square Heads and Nuts. 



Length 






Diameter of Bolts. 




head 
to point. 












Min. 


i^in. 


Vsin. 


i^in. 


^in. 


^in. 


Hin. 


J^in. 


lin. 




lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


ji^ 


4.0 

4.4 
4.8 


7.0 
7.5 
8.0 


10.5 
11.3 
12.0 


15.2 
16.3 
17.4 


22.5 
23.8 
25.2 


39.5 
41.6 
43.8 


63.0 
66.0 
69.0 






IM 






2^ 


109.0 


163 


2H 


5.2 


8.5 


12.8 


18.5 


26.5 


45.8 


72.0 


113.3 


169 


2^2 


5.5 


9.0 


13.5 


19.6 


27.8 


48.0 


75.0 


117.5 


174 


2H 


5.8 


9.5 


14.3 


20.7 


29.1 


50.1 


78.0 


121.8 


180 


3 


6.3 


10.0 


15.0 


21.8 


30.5 


52.3 


81.0 


126.0 


185 


3K 


7.0 


11.0 


16.5 


24.0 


33.1 


56.5 


87.0 


134.3 


196 


4 


7.8 


12.0 


18.0 


26.2 


35.8 


60.8 


93.1 


142.5 


207 


4)^ 


8.5 


13.0 


19.5 


28.4 


38.4 


65.0 


99.1 


151.0 


218 


5 


9.3 


14.0 


21.0 


30.6 


41.1 


69.3 


105.2 


159.6 


229 


5^ 


10.0 


15.0 


22.5 


32.8 


43.7 


73.5 


111.3 


168.0 


240 


6 


10.8 


16.0 


24.0 


35.0 


46.4 


77.8 


117.3 


176.6 


251 


QH 






25.5 
27.0 
28.5 
30.0 


37.2 
39.4 
41.6 
43.8 
46.0 
48.2 
50.4 
52.6 


49.0 

51.7 

54.3 

59.6 

64.9 

70.2 

75 5 

80.8 

86.1 

91.4 

96.7 

102.0 

107.3 

112.6 

117.9 

123.2 


82.0 

86.3 
90.5 
94.8 
103.3 
111.8 
120.3 
128.8 
137.3 
145.8 
154.3 
162.8 
171.0 
179.5 
188.0 
206.5 


123.4 
129.4 
135.0 
141.5 
153.6 
165.7 
177.8 
189.9 
202.0 
214.1 
226.2 
238.3 
250.4 
262.6 
274.7 
286.8 


185.0 
193.7 
202.0 
210.7 
227.8 
244.8 
261.9 
278.9 
296.0 
313.0 
330.1 
347.1 
364.2 
381.2 
398.3 
415.3 


262 


7 






273 


73^ 






284 


8 






295 


9 






317 


10 








339 


11 








360 


12 








382 


13 








404 


14 










426 


15 










448 


16 










470 


17 










492 


18 










514 


19 










536 


20 










558 














Per Inch 


1.4 


2.1 


3.1 


4.2 


5.5 


8.5 


12.3 


16.7 


21.8 


additional. 





















Weights of Nuts and Bolt-Heads in Pounds. 
For calculating the weight of longer Bolts. 



Diameter of Bolt 
in Inches. 




K 


Vs 


^ 


m 


H 


Ji 


Weight of Hexagon Nut 
and Head 




.017 
.021 


.057 
.069 


.128 
.164 


.267 
.320 


.43 
.55 


.73 


Weight of Square Nut 
and Head 




.88 








Diameter of Bolt 
in Inches. 


1 


1^ 


m 


IH 


2 


2y2 


3 


Weight of Hexagon Nut 
and Head 


1.10 
1.31 


2.14 
2.56 


3.78 
4.42 


5.6 
7.0 


8.75 
10.5 


17.0 
21.0 


28.8 


Weight of Square Nut 
and Head 


36.4 







622 



33.— STRUCTURAL DETAILS. 



12. — Lag Screws — Weight in Lbs. op 100. 







K—'Lerrgrffr ->H. 

Fig. 44. 



si 












Length of Screw, In Inches. 












iVz 


m 


2 


2H 


2H 


3 


33^ 


4 


4^i 


5 


53^ 


6 


7 


8 


9 


10 


4 
% 


6.88 


7.50 
11.75 
16.88 


8.25 
12.62 
17.18 


9.25 
12.88 
18.07 


9.6 
13.28 
19.18 


10.8 
16.62 
22. 
34.1 


11.5 
18.18 
24. 
35.9 


13.3 

18.9 

26.82 

39.3 

6i. 


14.8 

19.5 

28.25 

42.6 

67.9 


16.5 

21.3 

30.37 

47.8 

71.4 


17.4 
23.6 
33.8 
51.6 
79.4 


18.8 
25.3 
35.4 
55.1 
86.6 


38! 9 
61.9 
92.8 


44!4 
68.8 
97.5 


'77!* 
108.8 


96!* 












124.8 



















The Use op Lag Screws. 

The principal use of lag screws is for bridge work — ^in fastening one 
timber down on another. They are better than spikes, but not as good as 
screw bolts (with nut and head.) 

The lag screw has a square head and is screwed in place with a wrench. 
A plate washer is used under the head to distribute the bearing stress on 
the timber. 

A hole is first bored in the timber, somewhat smaller in diameter than 
the screw, the lag screw is inserted in the hole and tapped on the head a few 
times with a sledge hammer, and then screwed to a firm bearing with the 
wrench. 

Caution.The writer has used lag screws for fastening small wooden guard 
rails (4''x6'0 to wooden ties. When such work is being done, especially by 
contract, there should be an inspector on the work to see that the lag screws 
are not hammered into the timber too far with the^ sledge hammer before 
the wrench is used. To the contractor, hammering is much the cheaper and 
is a great temptation. 



13. — Wood Screws — Sizes. 
(Diameter in Inches = Number x . 01 325 + .056.) 





H 




fi 




g 




B 




B 




B 




a 




g 


0' 


.^ 


6 


03 





d 





03 





03 





OS 





oj 





5 


^ 


Q 


iz; 


Q 


^ 


Q 


^ 


Q 


iz; 


P 


^ 


Q 


^ 


Q 


^ 


P 





0,56 


4 


.109 


8 


.162 


12 


.215 


16 


.268 


20 


.321 


24 


.374 


28 


.427 


1 


069 


5 


.122 


9 


.175 


13 


.228 


17 


.281 


21 


.334 


25 


.387 


29 


.440 


?, 


082 


6 


.135 


10 


.188 


14 


.241 


18 


.293 


22 


.347 


26 


.401 


30 


.453 


3 


.096 


7 


.149 


11 


.201 


15 


.255 


19 


.308 


23 


.361 


27 


.414 











LAG SCREWS. CAST IRON SEPARATORS. 



623 






J4. — Standard Cast Iron Separators 
POR I-Beams. 



CE) 



Fig. 45. 


















Fig. 46. 




Beam. 


Distances. 


Bolts. 


Weights. 


•4-3 

a 

a 


1 


Out to Out of 

Flanges of Beams, 

Inches. 


<L) C3 
O cfl 

(3^ 


to 

1 


n 

II 

(32 




•SI 

'o 


Increase in Weight 

of Separator Bolts 

for 1 in. Additional 

Spread of Beams, 

Pounds. 


a 


Increase in Weight 

of Separator for 

lin. Additional 

Spread of 

Beams, Pounds. 



Separators With Two Bolts. (Fig. 45.) 



24 


80 


1434 


1% 


H 


12 


9K 


3.41 


.250 


32 


5.50 


20 


80 


UH 


7H 


H 


12 


m 


3.41 


.250 


28 


3.10 


20 


65 


UH 


7 


H 


12 


sy?. 


3.23 


.250 


25 


3.10 


18 


55 


12M 


m 


H 


9 


m 


■ 3.16 


.250 


16 


2.75 


15 


80 


13H 


7H 


H 


7y?. 


9 


3.55 


.250 


15 


1.75 


15 


60 


12M 


6H 


H 


7H 


sy 


3.23 


.250 


15 


1.75 


15 


42 


UH 


m 


H 


7^ 


7y?. 


2.98 


.250 


15 


1.75 


12 


40 


nn 


6 


H 


5 


7y?. 


2.98 


.250 


11 


1.50 


12 


31.5 


mi 


5H 


H 


5 


7H 


2.92 


.250 


11 


1.50 



Separators With One Bolt. (Fig. 46.) 



12 
12 
10 



40.0 
31.5 
25.0 
21.0 
18.0 

15.0 

12.25 

9.75 

7.50 

5.50 



IIM 
10^ 

lOH 
9A 



73^ 
7A 
6H 

5A 



6 

5h 

5M 

5 

4M 

4M 

4 

3^ 

m 
3 



3^ 



H 



73^ 
7H 



5H 
5M 
4M 
4H 
4M 



1.49 
1.46 
1.40 
1.34 
1.28 

1.25 
1.22 
1.16 
1.13 
0.70 



.125 
.125 
.125 
.125 
.125 

.125 
.125 
.125 
.125 
.09 



10 
10 

8 
7 
6 

4 
4 
3 
3 
2 



1.50 
1.50 
1.25 
1.20 
1.00 

.76 
.60 
.60 
.40 
.25 



Separators for 18, 20 and 24 in. beams are made of ^ in. metal. 

Separators for 6 to 15 in, beams are made of 3^ in. metal. 

Separators for 5 in. beams and under are made of ^ in. metal. 
Remarks on Cast Iron Separators. 

The use of cast iron separators and bolts for fastening two or more 
I-beams together, side by side, is gradually giving place to steel diaphrams, 
composed of angles riveted to a connecting web plate. The latter is much 
the better, and is now preferred in important steel building construction. 
Cast iron separators, however, are still used in connecting steel I-beams in 
grillage foundations, where spaces between the beams are filled with concrete. 

Cast separators for timber work are round, of cylindrical or spool shape, 
with a hole through which the screw bolt passes. They are used principally 
for packing, between lines of wooden bridge-stringers, spacing the wooden 
stringers about one inch apart. This spacing provides for the circulation of 
air, which dries the moisture from rain and retards rotting of the wood. Cast 
spool separators are shaped like a spool, while the cylindrical separators are 
perfectly cylindrical on the outside. In order to save metal, in the separator, 
the hole expands towards the ends, being a little larger than the bolt, only, 
at middle of separator. 



624 



'STRUCTURAL DETAILS. 



15. — Dimensions and Weights op Cast Iron Washers. 
(Dimensions in inches; weight in lbs. each.) 



!<-T-^ 




Fig. 47. — Diametric sec- 
tion of roiind washer. 



-♦J 

'o 


w 


w 


h 


H 





R 


r 


T 


y?. 


2H 


IH 


% 


H 


« 


H 


yn 


^ 


% 


3 


IH 


% 


^ 


J€ 


K2 


^% 


T^ 


% 


3H 


\y^ 


yf^ 


Te 




tk 


tv 


U 


1/2, 


m 


VA 


1 


ItV 


•5^ 


li 


^y. 


H 


1 


m 


2yH 


ly^ 




•j^ 


'^ 


H 


% 


\y^ 


m 


2V, 


IH 


1^ 


^ 


i* 




1 


IH 


5H 


2% 


m 


1^ 


"iS 


^8 


"TS 


1^8 


m 


5% 


2«4 


1K2 


1t^ 


tv 


J^ 


H 


1^ 


ly? 


614 


3 


m 


IH 


■^ 


y 


l«/8 


iH 


m 


314 


IH 




T6 


% 




IK2 


m 


1% 


3H 


lys 


■l^Te 


•^ 




A 


1^8 


V/h 


r% 


-m 


2 


2^ 


H 




U 


P4 


2 


m 


4 


2y 


2-h 


% 


l^ 


H 


1^8 


m 


m 


4^ 


2H 


2^ 


% 


m 


^ 


2 


'2H 


m 


m 


m 


2^ 


Vh 


IH 


^ 


2K8 


'2Vh 


m 


m 


2y 


2t^ 


% 


m 


M 


2^ 


m 


m 


5 


2^ 


2i* 


% 


IH 


ys 


2^ 



Wt. 



# 

0.32 

0.61 

0.78 

0.89 

1.75 

2.30 

3.00 

4.20 

5.20 

7.00 

8.30 

10.40 

12.40 

13.40 

15.80 

17.50 

20.00 



16. — Plate (Flat) Washers. 
Dimensions, and number per pound. 



i^-i 


Diam. 

Washer 
in Inches. 


Thickness 
of Washer. 


No. 

per 

Pound. 


Diam. 
Bolt. 
Ins. 


Diam. 
^ Washer 
in Inches. 


Thickness 
of Washer. 


??t c 


.<5 c 




'0 


6 


1— 1 




'0 


PQ 


t— 1 




^ 


H 


^ 


18 


.05 


450 


Vs 


ly?, 


i* 


11 


.125 


23. 


H 


5/^ 


16 


.062 


293 


ii 


m 




11 


y% 


15. 




H 


T^ 


16 


.062 


138 


H 


2 


y^ 


10 


.141 


11. 


"TE 


ys 


3/^ 


16 


tV 


110 


% 


2H 


1 


9 


.156 


8.5 


H 


1 


iV 


14 


.078 


76 


1 


2y, 


\y^ 


9 


/z 


6.3 




IH 




14 


^4 


43 


Ws 


2% 


1¥ 


8 


.172 


4.7 


y^ 


m 


ft 


12 


.109 


26 


m 


3 


1«^ 


7 


■h 


3.6 


y. 


iM 


12 


i'z 


23 


m 


W2 


IH 


6 


.203 





Remarks on Cast and Plate Washers. 

Cast iron washers are used principally with screw bolts in connecton with 
timber work. They are round and have a diametric section as shown in 
Fig. 47. These washers are sometimes called O. G. washers, or 5 washers, 
because the outline curve has an O.G. (architectural term) or 5, shape, being 
a reversed curve. 

Large, square, cast iron washers are often used for anchorage at the ends 
of rods in concrete. 

Plate, or flat, washers are stamped from sheet metal, being round with a 
hole in the center for the bolt. Large bolts require the thicker washers, that 
will not bend when the bolt is tightened. 



WASHERS. STEEL WIRE NAILS. 



625 



17. — Miscellaneous Steel Wire Nails. 
Approximate Number per Pound. 



Iso 


1 i 




Lengths — Inches. 
































«« 


n "^ 


^ 


H 


Vs 


^ 


Vs 


H 


Vs 


1 


VA 


IH 


13^ 


00 


.380 
.375 




















33 
33 


27 


0^ 




















27 


.340 




















34 


29 


A 


.313 
















57 


50 


45 


38 


2 


.284 
.259 
.238 
.220 
.203 
.180 
.165 
.148 
.134 
.120 
.109 
.095 
.083 
















65 
76 
90 
106 
123 
149 
172 
207 
248 
314 
411 
536 
710 


58 

67 

80 

94 

111 

133 

153 

184 

220 

279 

365 

476 

631 


52 

60 

72 

85 

99 

120 

137 

165 

198 

251 

329 

429 

568 


44 


3 












100 

120 

1 141 

' 164 
200 
229 
276 
333 
418 
548 
714 
947 


87 
104 
121 
141 
171 
197 
236 
283 
359 
469 
613 
811 


50 


4 












60 


5 








2ii 

247 
299 
345 
414 
496 
628 
822 
1072 
1420 


m 

197 
23c 
275 
331 
397 
502 
658 
857 
1136 


71 


6 








82 


7 








100 


8 








115 


9 








138 


10 






663 

837 

1096 

1429 

1893 


165 


11 






?m 


12 






274 


13 






357 


14 




2840 


473 


15 


.072 




3504 


2336 


1752 


1402 


1168 


1001 


876 


778 


701 


584 


16 


.065 




4571 


3048 


2280 


1828 


1523 


1305 


1143 


1015 


913 


761 


17 


.058 




6233 


4156 


3116 


2495 


2077 


1781 


1558 


1385 


1246 


1038 


18 


.049 




8276 


5517 


4138 


331C 


2758 


2364 


2069 


1839 


1655 


1379 


19 


.042 




10668 


7112 


5334 


4267 


3556 


2933 


2667 


2370 


2133 


1778 


20 


.035 


2000C 


15000 


10000 


7500 


600C 


5000 


4400 


3750 


3333 


3000 




21 


.032 


23702 


17777 


11850 


8888 


711] 


5926 


5079 


444^ 








22 


.028 


30476 


22856 


15237 


11428 


914c 


7618 












^2 . 


1 s 

OS P 


Lengths — Inches. 


^so 






























PQ c3 


P " 


IH 


2 


2M 


23^ 


2H 


3 


3H 


4 


43^ 


5 


6 


7 


8 


9 


10 


00 


.380 


23 


20 


18 


16 


15 


14 


12 


10 


9 


8 


7 


6 


5 


41/ 


^ 4 


Vs 


.375 


23 


20 


18 


16 


15 


14 


12 


10 


9 


8 


7 


6 


5 


41/ 


'? 4 





.340 


25 


21 


19 


17 


16 


15 


13 


11 


10 


9 


8 


7 


51/^ 


5 


41^ 


A 


.313 


32 


28 


25 


23 


21 


19 


16 


14 


13 


11 


10 


8 


7 


6 


51^ 


2 


.284 


37 


32 


29 


26 


24 


22 


19 


16 


14 


13 


11 


9 


8 


7 


6H 


3 


.259 


43 


2< 


34 


30 


28 


25 


22 


19 


17 


15 


13 


11 


10 


8 


7^ 


4 


.238 


51 


45 


40 


36 


33 


3C 


26 


23 


20 


18 


15 


13 


11 


10 


9 


5 


.220 


60 


5^ 


47 


42 


39 


3f 


30 


26 


24 


21 


18 


15 








6 


.203 


71 


62 


55 


50 


45 


4] 


35 


31 


28 


25 


21 


18 








7 


.180 


85 


75 


67 


60 


54 


5C 


43 


37 


33 


30 


25 










8 


.165 


98 


86 


76 


69 


62 


57 


49 


43 


39 


35 


29 










9 


.148 


118 


103 


92 


82 


75 


m 


59 


52 


46 


41 












10 


.134 


142 


L24 


lie 


99 


90 


HI- 


) 71 


62 


55 


bO 






11 


.120 


179 


157 


139 


125 


114 


10£ 


90 


79 


70 






Lengths. 


12 


.109 


235 


204 


182 


164 


149 


13/ 


117 


103 






^B 






13 


.095 


306 


268 


238 


214 


195 


m 


153 








11 


12 


14 
15 


.083 
.072 


406 
500 


350 

438 


315 

389 


284 
350 


258 


236 
























00 


3M 


^H 


16 


.065 


653 


571 


508 
















Vs 


334 


m 


17 


.058 


890 


779 





















4 


3H 


18 


.049 


1182 




















2^ 


5 

6 


43^ 
5H 



These approximate numbers are an average only, and the figures given 
may be varied either way, by changes in the dimensions of heads or points. 
Brads and no-head nails will run more to the pound than table shows, and 
large or thickheaded nails will run less. 



626 



ZZ.'^STRUCTURAL DETAILS. 



18. — Standard Steel Wire Nails. 
Sizes, Lengths and Approximate Number per Pound. 





a 
3 


Common. 


a 

pq 


i 
6 


a 


th 
a 

1 

a 


be 

a 


q3 


J 


be 

a 


1 

.£3 

§ 
s 

CQ 


ffl 


1 
o 


Barbed 

Oval Head 

Car Nails. 


bi) 

a 


be 

c 

•s 

648 
413 
384 
339 
231 


6 
be 
a 

CQ 


8 

1 






DIam. 


3 

i 








1 

a 


4i 


> 

w 


a 

a 
5 




2M 
2% 

l« 

3^ 

4 

4^ 

5 

5^ 


















940 
804 
620 
590 
542 
365 
332 








1980 




































IfifiO 


2d 


15 


.072 


900 


860 


622 


... 


1558 


1558 


1440 
810 


1140 


1000 


1000 








385 


1440 


*3d 










m 


14 


.083 


615 


594 


412 




884 


884 


675 


660 


660 








230 


380 


















4d 


12^ 
l2^ 
IIH 
11^ 
lOM 
lOM 

9 

9 

8 

6 

5 

4 

3 

2 


.102 
.102 

.in 

.115 
.124 
.124 
.148 
.148 
.165 
.203 
.220 
.238 
.259 
.284 


322 

250 

200 

154 

106 

85 

74 

57 

46 

29 

23 

17 

13^ 

10^ 


339 

230 

205 

135 

96 

92 

63 

52 

38 

30 

23 

17 

13^ 
10^ 


267 

230 

156 

110 

98 

86 

66 

57 

46 

35 


127 
114 
88 
74 
58 
42 
36 
28 
22 


767 
491 
359 
317 
214 
195 
134 
120 
91 
61 


767 
491 
359 
317 
214 
195 
134 
120 
91 
61 


550 


567 

396 

260 

239 

160 

148 

108 

99 

69 

50 

45 

35 


550 

366 

250 

236 

157 

145 

107 

98 

65 

45 

40 

30 


550 

366 

250 

236 

157 

145 

107 

98 

65 

45 

40 

30 


151 
136 
98 
86 
66 
51 
40 
29 


260 
134 
119 
85 
75 
58 
55 
43 
39 
31 
27 
21 
18 
15 


164 
103 
91 
73 
65 
51 
45 
38 
34 
26 
23 
17 
14 
13 


198 
125 
112 


154 

135 

90 


256 
226 
200 
130 
120 
115 
79 


256 
226 
145 
130 
100 
85 
65 




fid 








7d 
M 


.... 






9d 








t?d 








ind 








Ifid 
















?,0<] 


. . . . 














30d 
















40d 








. . . . 


. .. . 














fiOd 
























60d 






























. ... 



































* Fine, t Common. 



19. — Standard Steel Wire Spikes. 



Diameter {g;X;G 

Length, Inches. . . . 
No. per Pound 



6 
.203 


5 
.22t) 


4 
.238 


3 

.259 


2 

.284 


1 
.300 


1 
.300 












^ 


^ 


Vs 


H 


H 


3 


31/^ 


4 


4H 


5 


5H 


6 


m 


7 


8 


9 


10 


37 


29 


23 


18 


13 


10 


9 


7^ 


63^ 


iH 


3% 


SH 



12 

2H 



Remarks on Spikes and Nails. 

Steel wire spikes and nails are made from steel wire, being cut, headed 
and pointed by machine. They may be either plain (smooth) or barbed (for 
added holding power). 

Cut spikes and nails have greater holding power than the smooth steel 
wire, because they have a rougher surface to give the greater friction. 

Tests on the holding power of railway spikes were made by Mr. Roy I. 
Webber, Instructor of Civ. Eng., Univ. of 111., and the results are given in 
Bulletin No. 6, issued by the Experiment Station. The tests were on screw 
spikes and plain spikes: direct pull and lateral displacement. 

It is to be noted that the variation in weight of spikes and nails is con- 
siderable, even for the same sizes. 



SPIKES. NAILS. TACKS. 



627 



20. — Spikes and Nails. 





"WfrMiarVif ?soitrf»<5 


lbs. 


St 


eel Wire 


Standard Steel Wire Nails. 


Number to a keg of 150 


Spikes. 


to 


1 

t— 1 


Common. 


Finishing. 




No. 


t: 


No. 


to. 


lo. 




si 




Q C 


u . 

o o 




1.2 


3 


2250 
1890 
1650 
1464 
1380 
1292 
1161 










3 
4 

5 

5M 

6 

63^ 

8 
9 


.1620 
.1819 
.2043 
.2294 
.2576 
.2893 
.2893 
.2249 
.2249 
.3648 
.3648 


41 
30 
23 
17 
13 
11 
10 
7H 

5 

4H 


2d 

3d 

4d 

5d 

6d 

7d 

8d 

9d 

lOd 

12d 

16d 

20d 

30d 

40d 

50d 

60d 


1 

\% 

fi 

3H 

4 

4^ 

5 

5H 

6 


.0524 
.0588 
.0720 
.0764 
.0808 
.0858 
.0935 
.0963 
.1082 
.1144 
.1285 
.1620 
.1819 
.2043 
.2294 
.2576 


1060 

640 

380 

275 

210 

160 

115 

93 

77 

60 

48 

31 

22 

17 

13 

11 


.0453 
.0508 
.0508 
.0571 
.0641 
.0641 
.0720 
.0720 
.0808 
.0808 
.0907 
.1019 


1558 


3H 
4 


1208 
1135 
1064 
930 
868 
662 
635 
573 








913 








761 


43^ 

5 

6 








500 


742 
570 
482 
455 
424 
391 






350 






315 


7 

8 

9 

10 


445 
834 
300 
270 
249 
236 


306 
256 
240 
222 
203 
180 


214 
195 
137 
127 


11 






90 


12 








62 













21.— Tacks. 



^i 


^1 


S s^ ^ 




-5^- 
^^. 


mber 

er 

und. 






mber 

)er 

und. 




^1 


mber 

er 

und. 


h5 


^^ 








j3 wo 


^^ 


^^ 




c ^ 

^O 


^^ 


5 ^^ 


1 


v^ 


16000 


3 


^ 


5333 


10 


1* 


1600 


18 


H 


888 


m 


A 


10666 


4 


A 


4000 


12 


H 


1333 


20 


1 


800 


2 


il 


8000 


6 


1^ 


2666 


14 


H 


1143 


22 


liV 


727 


23^ 


A 


6400 


8 


^ 


2000 


16 


% 


1000 


24 


13^ 


666 



22. — Miscellaneous Spikes. 
Railroad Spikes. 







Quantity of Spikes per 




Size Measured 


Average 


Mile of Single Track. 


Rail Used. 


Under Head. 


Number per 


Ties 2 Feet c. to c. 


Weight per Yard. 




Keg of 200 


4 Spikes per Tie. 




Inches. 


Pounds. 




Pounds. 










Pounds. 


Kegs. 




53^x^ 


300 


7040 


351/5 


75 to 100 


5Kx^ 


375 


5870 


29M 


45 " 75 


5 x^ 


400 


5170 


26 


40 " 56 


6 x3^ 


450 


4660 


"mi 


35 " 40 


4Hx)^ 


530 


3960 


20 


30 " 35 


4 x3^ 


600 


3520 


17H 


25 " 35 


4^X1^ 


680 


3110 


153^ 


20 " 30 


4 xA 


720 


2910 


14M 


20 " 30 


33^x^ 


900 


2350 


11 


16 " 25 


4 x^ 


1000 


2090 


103^ 


16 " 25 


3Hx3^ 


1190 


1780 


9 


16 " 20 


3 x^ 


1240 


1710 


83^ 


16 " 20 


23^x^ 


1342 


1575 


VA 


8 " 16 


2KxA 


1600 


1292 


5 


8 " 10 



628 



.STRUCTURAL DETAILS. 



22. — Miscellaneous Spikes. — Concluded. 

Square Boat Spikes. 
Approximate Number in a Keg of 200 Pounds. 



Size, 
Inches. 


Length of Spike, Inches. 


3 


4 


5 


6 


7 


8 


9 


10 


11 


12 


14 


16 


A 


3000 
1660 
1320 


2375 
1360 
1140 


2050 

1230 

940 


1825 

1175 

800 

600 

450 


990 
650 
590 
375 


880 
600 
510 
335 
260 


525 
400 
300 
240 


475 
360 
275 
220 


320 
230 
205 


280 
240 
190 


175 














5J 








160 

















Street Railway Spikes. 



Spikes. 


Number per Keg of 
200 Pounds. 


Kegs per Mile. 
Ties 2 feet c. to c. 


51/^x^ 
5 x3^ 
43^x^ 


400 
575 
800 


30 
19 
13 



23. — Cut Steel Nails and Spikes. 
Sizes, Lengths, and Approximate Number per Pound. 



J 

CO 


is 


§ 

a 
a 
c3 





a 


X 

c3g 


.a 
f2 




2 
^ 

m 


^1 


i 


i 




d 


d 




13^ 

IK 

¥ 

m 

4M 

43^ 

5 

53^ 

6 

63^ 














750 
600 
500 
450 
310 
280 
210 
190 








H 

1 


1462 

1300 

1100 

^00 

650 













































2d 
3d 
4d 


960 


?,c\ 


740 


400 


1100 









■466 
304 


. 340 
"280" 


750 










600 


3rl 


460 


260 


880 








Tobacco 


Brads 












Shl'gle 


4d 

5rl 


280 
210 
160 
120 

88 

73 

60 

46 

33 

23 

20 

163^ 

12 

10 
8 


180 
125 
100 
80 
68 
52 
48 
40 
34 
24 


530 

350 

300 

210 

168 

130 

104 

96 

86 

76 


420 

300 

210 

180 

130 

107 

88 

70 

52 

38 






224 


220 
. 180 




ioo' 

80 
60 
52 
38 
26 
20 
18 
16 




130 

97 
. 85 
68 
68 
48 






6rl 








120 
94 
74 
62 
50 
40 
27 




7d 












8d 










90 


9d 










72 


lOfl 










60 


}?r\ 












16rl 


17. 
14 












































30 
26 
20 
16 





11 

9 

53^ 
5 









































































































































SPIKES. NAILS, PIN NUTS. PILOT NUTS. 



629 



24. — Carnegie Standard Pin-Nuts. 
With Weight of Pins per Lineal Inch. 



Pins. 


Pin-Nuts. 


Pins. 


Pin-Nuts. 








4 


i 


0) 






u 

<u . 

Is 


i>5 
is 


as. 


wo 


i 

.a 


0) 




4-> 


VA 
2% 


.782 
1.005 


Ig 


8 
8 


2K 
2H 


2^ 

2% 


3^ 


0.85 
1.03 


m 
m 


3.342 
3.787 


3M 
3H 


6 
6 


5 

53^ 


5M 
6^ 


ig 


4.74 
6.19 


ti 


1.127 
1.2,55 


1^ 
1% 


8 
8 


2^ 
3 


2% 
3K 


1 
1 


0.97 
1.50 




4.259 
4.760 


3M 


6 
6 


5H 
53^ 


6^8 

6^ 


IM 
IM 


6.19 
5.37 


fi 


1.391 
1 .583 


r^ 


8 
8 


3 

3K 


3)^ 
4 


1 
1 


1.37 
2.06 




5.288 
5.845 


4 
4 


6 
6 


6 
6 


i 


IM 

IM 


6.63 
6.63 


2M 

2K 


1.683 
1.839 


2H 

2y4. 


8 
8 


3^ 
4 


4 

4^ 


1 

IM 


1.96 
3.38 




6.429 
7.041 


4^ 


6 
6 


6 

6M 


i 


IM 
IM 


6.82 
8.53 


3 

3K 


2.003 
2.173 


2H 
2H 


8 
8 


4 

4M 


4^ 
43^ 


IS 


3.22 
3.63 




7.681 
8.483 


4^ 
4M 


6 
6 


6^ 

m 


?i 


IM 


7.59 
7.59 


3Ji 


2.351 
2.535 


2% 
2M 


8 
6 


4K 
4J^ 


4^ 
5A 


IM 
IM 


3.41 
4.09 




9.042 
9.767 


5 

5H 


6 
6 


8 
8 


9J€ 
9M 


1^ 


13.06 
14.86 


33^ 

3^ 


2.726 
2.924 


2K 
3 


6 
6 


4M 
5 


5K 
6M 


IM 
IM 


4.63 
5.25 


QV8 

7K 


10.517 
11.300 


53^ 

5M 


6 
6 


8 

8 


9M 
9M 


IH 
13^ 


14.00 
13.10 



All dimensions given above are in inches. 

* Weights refer to untapped nuts. 

t Weight is for normal diameter before being turned. 

Phoenix Standard Pin Nuts and Pilots. 

Addition to diam. of pins for turning: 
Up to 43^'' inclusive allow 3^'' to ^" to 2' 0" 
long, and allow }4" to ^" above 2' 0'' long. 
Over 43^'' allow not less than i^" to 2' 0" long, 
and allow %" above 2' 0'' long. 

Weight of pilots: 4 lbs. for 2^" diam. of 
pin; 10 lbs. for 3i^"; 18 lbs. for 4^''; 34 lbs. 
for 5t^"; 65 lbs. for 6^''; 116 lbs. for 8^"; 
267 lbs. for 11" pin. 




^^■4' far Pins up to 7,| inc 



Fig. 48 



ab^e 



Cambria Bridge Pins, Nuts and Pilot Nuts. 

^ ^ D.F 




Fig. 49. 

All Threads 8 per inch. 
Allow 3^" excess for each eye bar packed on the pin. 



Cold Rolled Steel Cotter Pins. 



Lateral Pins. 





Figf. 51, 



630 



23.— STRUCTURAL DETAILS, 



25. — Pins— Bending Moments ; and Bearing Values on Plates 1 In. Thick. 
(Bending moment = .0982 X fiber stressX(dia.)3=3^Xfiber stress X area Xdia.) 



S m 


Area of 

Pin. 
Sq. Ins. 


Moments in Inch-Pounds for Fiber Stresses of 


Bearing In lbs. on 
Plate 1 In. thick, at 


S3 


15,000 lbs. 
per sq. in. 


18.000 lbs. 
per sq. in. 


20.000 lbs. 
per sq. in. 


22.500 lbs. 
per sq. in. 


25.000 lbs. 
per sq. in. 


12,000 lbs. 
per sq. In. 


15,000 lbs. 
per sq. in. 


m 


0.785 
0.994 
1.227 
1.485 


1 470 

2 100 

2 880 

3 830 


1 770 

2 520 

3 450 

4 590 


1 960 

2 800 

3 830 
5 100 


2 210 

3 140 

4 310 

5 740 


2 450 

3 500 

4 790 
6 380 


12 000 

13 500 

15 000 

16 500 


15 000 

16 900 
18 800 
20 600 


1^ 


1.767 
2.074 
2.>405 
2.761 


4 970 

6 320 

7 890 
9 710 


5 960 

7 580 

9 470 

11 600 


6 630 

8 430 

10 500 

12 900 


7 460 

9 480 

11 800 

14 600 


8 280 
10 500 
13 200 
16 200 


18 000 

19 500 

21 000 

22 500 


22 500 
24 400 
26 300 
28 100 


2 

in 

2% 


3.142 
3.547 
3.976 
4.430 


11 800 
14 100 
16 800 
19 700 


14 100 
17 000 
20 100 
23 700 


15 700 
18 800 
22 400 
26 300 


17 700 
.21 200 
25 200 
29 600 


19 600 
23 600 
28 000 
32 900 


24 000 

25 500 

27 000 

28 500 


30 000 

31 900 
33 800 
35 600 


2^ 


4.909 
5.412 
5.940 
6.492 


23 000 
26 600 
30 600 
35 000 


27 600 
32 000 
36 800 
42 000 


30 700 
35 500 
40 800 
46 700 


34 500 
40 000 
45 900 
52 500 


38 400 
44 400 
51 000 
58 300 


30 000 

31 500 

33 000 

34 500 


37 500 
39 400 
41 300 
43 100 


3 

3M 
3% 


7.069 
7.670 
8.296 
8.946 


39 800 
44 900 
50 600 
56 600 


47 700 
53 900 
60 700 
67 900 


53 000 
59 900 
67 400 
75 500 


59 600 
67 400 
75 800 
84 900 


66 300 
74 900 
84 300 
94 400 


36 000 

37 500 

39 000 

40 500 


45 000 

46 900 
48 800 
50 600 


3^ 


9.621 
10.321 
11.045 
11.793 


63 100 
70 100 
77 700 
85 700 


75 800 

84 200 

93 200 

102 800 


84 200 

93 500 

103 500 

114 200 


94 700 
105 200 
116 500 
128 500 


105 200 
116 900 
129 400 
142 800 


42 000 

43 500 

45 000 

46 500 


52 500 
54 400 
56 300 
58 100 


4^ 


12.566 
13.364 
14.186 
15.033 


94 200 
103 400 
113 000 
123 300 


113 100 
124 000 
135 700 
148 000 


125 700 
137 800 
150 700 
164 400 


141 400 
155 000 
169 600 
185 000 


157 100 
172 300 
188 400 
205 500 


48 000 

49 500 

51 000 

52 500 


60 000 

61 900 
63 800 
65 600 


4M 

4^ 


15.904 
16.800 
17.721 

18.665 


134 200 
145 700 
157 800 
170 600 


161 000 
174 800 
189 400 
204 700 


178 900 
194 300 
210 400 
227 500 


201 300 
218 500 
236 700 
255 900 


223 700 

242 800 
263 000 
284 400 


54 000 

55 500 

57 000 

58 500 


67 500 
69 400 
71 300 
73 100 


5 

5M 
5% 


19.635 
20.629 
21.648 
22.691 


184 100 
198 200 
213 100 
228 700 


220 900 
237 900 
255 700 
274 400 


245 400 
264 300 
284 100 
304 900 


276 100 
297 300 
319 600 
343 000 


306 800 
330 400 
355 200 
381 100 


60 000 

61 500 

63 000 

64 500 


75 000 

76 900 
78 800 
80 600 


5^ 

5% 

m 

m 


23.758 
24.850 
25.967 
27.109 


245 000 

262 100 
280 000 
298 600 


294 000 
314 500 
335 900 
358 300 


326 700 
349 500 
373 300 
398 200 


367 500 
393 100 
419 900 
447 900 


408 300 
436 800 
466 600 
497 700 


66 000 

67 500 

69 000 

70 500 


82 500 
84 400 
86 300 
88 100 


6 


28.274 
29.465 
30.680 
31.919 


318 100 
338 400 
359 500 
381 500 


381 700 
406 100 
431 400 
457 800 


424 100 
451 200 
479 400 
508 700 


477 100 
507 600 
539 300 
572 300 


530 200 
564 000 
599 200 
635 900 


72 000 

73 500 

75 000 

76 500 


90 000 

91 900 
93 800 
95 600 


6^ 

6% 

6% 


33.183 
34.472 
35.786 
37.122 


404 400 
428 200 
452 900 
478 500 


485 300 
513 800 
543 500 
574 200 


539 200 
570 900 
603 900 
638 000 


606 600 
642 300 
679 400 
717 800 


674 000 
713 700 
754 800 
797 500 


78 000 

79 500 

81 000 

82 500 


97 500 

99 400 

101 300 

103 100 


7 
8 


38.485 
44.179 
50.265 


505 100 
621 300 
754 000 


606 100 
745 500 
904 800 


673 500 

828 400 

1 005 300 


757 700 

931 900 

1 131 000 


841 900 
1 035 400 
1 256 600 


84 000 
90 000 
96 000 


105 000 
112 500 
120 000 



BRIDGE PINS. STEEL EYE BARS. 



631 



26. — Eye Bars. 
American Bridge Company's 



Standards. 





Ordinary. 






Adjustable. 


' 


I 


^ 1 


t^.X 1 ^„^*h f -tn CnW 


Mg. 52. 


'6"e:pr7ferabfy7''o' 

Fig. 53. 




•Spq 


Head. 


Screw End. 


i 

II 




i 


P 












•o c 


2 

3 

4 

5 
6 

7 


H 
H 

H 
K 

*1 

1 

1 

iH 

13^ 

^¥ 
iy2 


43^ 
63^ 

5H 
63^ 

7 
8 

93^ 
103^ 
IIH 

113^ 
13 

133^ 
143^ 

163^ 
17>^ 

173^ 
18M 
19K 

22 

233^ 

243^ 

26 

27H 

29 

303^ 
32>i^ 
33H 


2li 
3M 

3 
4 

4K 
5M 
6M 

5 

63^ 

ig 

7^ 

7 
8 
9 

9 

lOM 
IIM 

10 

IIM 
13 

12 
14 
15 


O-lO 
1— 2 

0-11 
1— 2 

1— 1 
1-5 

1-6 

1— 9 

2— 3 

1—8 
2- 3 

1-10 
2-4 

2— 4 
2-8 
■ 

2— 3 
2-8 
3-0 

2-11 
3—4 

3— 8 

a- 3 
3-9 

4— 2 

3-10 
4-5 
4— 8 


0— 7 

1— 1 

1-5 
1—5 

1— 8 
1— 8 

1— 9 
1-9 

1—11 
1—11 

2— 3 
2— 3 


2 

23^ 

23^ 

2% 

3 

33€ 

3M 
4 

43^ 


5 

5 

53^ 
6 

6 

63^ 

63^ 

8 
8 

9 
9 


H to 1x^6 

1 tolA 
13^ to IM 

1 to 13^ 
lA to 1% 

1 tol^ 
13^ to \K 

IVs to 1^ 
IM to IVs 

m to lA 

m to 13^ 


2 

23^ 
3 
4 

5 
6 

7 


8 










8 


























10 










10 


























12 










12 
















- 










14 










14 



























Bars marked * should only be used when absolutely unavoidable. 

Note. — Eye bars are hydraulic forged and are guaranteed to develop 
the full strength of the bar, under conditions given in the above table, 
when tested to destruction. 



632 



d3.--STRUCTURAL DETAILS. 



27. — Clevises. 
American Bridge Company's Standards. 



All dimensions in 
inches. 




GTg 



Grip G can be made 
to suit connections. 



Diameter 


Max. 
Pin P. 








Clevis 








of Clevis 
















D. 


Fork F 


NutN 


width W 


Thickness T 


A 


B 


Ap'rx.Wt. 


3 


13^ 


13^ 


IH 


13^ 


% 


6 


5 


5 


4 


2H 


IH 


IH 


IM 


3^ 


9 


8 


9 


5 


3 


2H 


2H 


2H 




9 


8 


14 


6 


33^ 


2H 


2H 


2H 


% 


9 


8 


25 


7 


4 


SH 


3M 


m 


% 


9 


8 


39 



Table giving Diameter of Clevis for given Rod and Pin. 



Rod. 


Pins. 


Rod. 


'd 
q 


i 


•4-> 


1 1^ 13^ m 


2 2M 2J^ 2K 


3 3>i 3^ ZK 4 




§ 




& 


s 


^ 










t^ 


c^ 


& 


Va 


Vs 


1 


3 1 3 3 














A 


^ 


K 




Ya 




3 3 


3 


4 


4 4 
4 4 










13^ 
IK 


A 




% 


4 


4 


4 


A 


1 




IH 


4 


4 4 


4 4 










VA 




1 


1^ 


1 

13^ 


1^ 


4 4 

4 


4 


4 4 


5 5 


5 
5 
5 






13^ 

13/< 


1 
IH 


13^ 


m 


4 


5 5 


5 5 


IM 


m 


5 5 


5 5 5 5 


P/^ 




IM 


VA 


5 5 


5 5 5 5 


5 






IK 


IM 




m 


VA 


2 


5 


5 |5 5 5 


6 6 6 




2 


lA 


13^ 


m 


V/2 


2A 
214 


5 


5 15 5(6 


6 6 6 

6 6( 7 


7 7 


2>^ 
2M 


m 


1^ 


IH 




6 6 6 6 


1^ 


VA 


IH 


2A 




6 6 6 6 


7 7 7 


7 7 


2K 


1^8 


IJ^ 


2 


IM 


2^ 

2% 




6 6| 7 


7 7 7 
7 7 7 


7 7 


23^ 
2Vh 


IM 


2 


2% 




7 7 




23^ 




2 


2A 




7 7 
7 


7 7 7 




2M 
2K 


13^ 
2 




2M 


7| 


2H 


i 


i 




1 IM 13^ 1^ 


2 2M 2K 2M 


3 31i 33^ 


3M 4 


1 


q 


n* 


a 










a 


rr" 


o 


^ 


S 


^ 










P 


CO 


P^ 


Rod. 




Pins. 








Rod 





Clevises above and to right of heavy zigzag line may be used with forks 
straight. 

Clevises below and to left of same line should have forks closed in until 
pin is not overstrained. 



CLEVISES, SLEEVE NUTS. TURNBUCKLES. 



633 



28. — Sleeve Nuts and Turnbuckles. 

American Bridge Company's Standards. 

All Dimensions in Inches. 



Sleeve Nuts. 




It 

Q 



1 



\H 

2 

2}^ 
2M 
2^ 

2M 
23^ 



33^ 



33^ 
3^ 
3^ 



4 

43^ 



Fig. 55. 



rM (1) 



1^ 

1% 
m 

2 

2 

2M 

2M 
2H 

2^ 

2^ 
2M 
3 

3 

3M 
33^ 

3H 
SH 
SH 
4 

4 
4M 

4K 

4K 
4M 
5 



bori 



7 

7 

7^ 

7^ 

8 



10 

10 

lOK 
103^ 
11 

11 

11^ 
113^ 
12 

12 

123^ 
123^ 
13 

13 

13K 

14 



a 

S 

O 



\y% 



2y8 2H 
2y2H 

2H ' 



2H 



33^ 



a 

Q 

bopQ 
o 

►J 



2A 



3x6 
3i^ 



3K3^ 
33^35^ 



4^ 



3K4J^ 
3K43^ 

3K43^ 
43^411 
4^4- 
4^5^ 

4^53^ 

5 

5 



5H 



5^6H 

5^63^ 
5%6H 
5^61i 
6)^73^ 

63^73^ 
6H7^ 

6K" 






IH 

13^ 
13-^ 
1^ 

lys 
m 

VA 
VA 
2A 

2A 
2% 
2^ 
2% 

2y 

2% 
2% 
33^ 

3K 



3^8 

3^ 
3K 
3K 
43^ 

43^ 
4^ 
4^ 



to 
m 

0) 



^ 



1 



3 w 

Ah w 



2^ 

3 

3>^ 

4 

43^ 

8 

11 
14 
15 

18 

19 
22 
23 
27 

28 
34 
35 
39 

40 
45 

47 
52 

55 
65 
75 



Turnbuckles 
Cleveland City Forge and Iron Co. 




^ L 

Fig. 56. 
Standard Length, X = 6 in. 
Extra Lengths, 9, 12, 18, 24, 
48 and 72 in. (Special Prices.) 



36, 



a CO 


Standard Dimensions. 














Bo 


t 


A 


B 


c 


L 


T 
















23^ 


y 


'iH 


2H 


IH 


8^ 


1t% 


3H 


^ 


IH 


2^-, 


1t^ 


9 


ly 


4 


H 


Wa 


2^ 


1t« 


9^8 


ly 


5K 


y?. 


ly, 


2y4 


1^ 


9^ 


6 


^ 


ly 


'S^ 


Its 


lOK 


2iV 


7 


y 


p-^ 


'S^ 


1^4 


1014 


214 


81/$ 


y 


IH 


33/2 


2 


107/^ 


2iV 


10 


y. 


2 


'6H 


2y 


1134 


2^8 


113^ 


ii 


2 


3^8 


2^ 


11^/^ 


2i^ 


13 


lA 


214 


4M 


2y 


12 


3 


15 


3I 


2H 


4^2 


2y 


1234 


3f^ 


18 


"16 


2H 


4K 


2t^ 


1234 


33/^ 


20 


xt 


2M 


4^8 


2y 


133^ 


3i^ 


24 


27 


3 


5^ 


3t\ 


1314 


334 


28 


If 


3 


?^ 


33/8 


137/8 


m 


30 


i-| 


314 


5^ 


3H 


1414 


414 


34 


13^2 


3^^ 


61^ 


3i^ 


14S/8 


4A 


38 


lA 


3M 


6^ 


zy 


15 


43^ 


50 


ll^ 


4 


6M 


Ws 


15% 


Ws 


65 


1:^5 


4 


7M 


4Ji 


163^ 


5y 




lA 


5 


SH 


41^6 


18 


6 




li^ 


5 


sy 


45/^ 


18 


6 































So 

S ^ 



3^ 
I 

13^ 
IM 
1^ 

1^ 

1^ 
IM 

ly 

2 

23^ 

2M 
2^ 

23^ 
2% 

2y 
2y 

3 

sy 
m 



sy 

sy 

sy 
4 



634 



dZ.— STRUCTURAL DETAILS, 



29. — Upset Screw Ends for Round and Square Bars. 
Screw threads are the Franklin Institute Standard. 
Allow 6 inches additional length of rod for 5 in. length of thread. 



o 


oi3 

III 


III 


Diameter of 
Screw at Root 
of Thread. 
Inches. 




Length of 
Upset. 
Inches. 


e3 tn 

q-iM 
O 

OQPP 


Oe3 


III 
111 


Diameter of 
Screw at Root 
of Thread. 
Inches. 


u 

t. 


Length of 
Upset. 
Inches. 


Is 


• 


t 


• 


t 


3^ 

"I 
A 

IP. 
18 


2 
i 

1 

'in 

1^ 

i^" 
lA 

m 
m 


1 
1 

1 
ill 

13^ 

1^ 

2 

23^ 
23^ 

i^ 

2^ 
2^ 
23^ 


.620 
.620 
.731 
.837 

.837 

.940 

1.065 

1.065 

1.160 
1.160 
1.284 
1.284 

1.389 
1.389 
1.490 
1.490 

1.615 
1.615 
1.712 
1.712 

1.837 
1.837 
1.962 
1.962 

2.087 
2.087 
2.175 


10 

10 

9 

8 

8 
7 
7 

7 

6 
6 
6 
6 

53^ 
53^ 
5 
5 

5 

5 

43^ 
43^ 

4H 
4K 
4H 
43^ 

43^ 
43^ 
4 


2M 

r- 

IS 

4 

4M 

5 
5 

5H 

53^ 

53^ 

m 


4M 
4M 

43^ 

43^2 

43^ 
4^ 

5 
5 
5 
5 

5M 
53€ 
5M 

53^ 
53^ 
53^ 
53^ 

5M 

i 

6 
6 
6 


m 

13^ 

m 

2 

2i^ 
23^ 
2A 

2^ 

'i' 

2i^ 

23^ 

2ii 

i 

*3>^' 


2 

2A 

2^ 

IPs 
2ii 

2M 
2il 
2K 
2H- 

3 

'33^' 
3M 

33^ 
ZVs 


23^ 
2^ 
2H 
2% 

2K 
2K 
3 
3>^ 

33^ 
3M 
3M 
3^ 

33^ 
3^ 

3^8 

33^ 

4 

4K8 
43^ 
434 
4^ 

43^ 

4^ 
4^ 


2.175 
2.300 
2.300 
#.425 

2.550 
2.550 
2.629 
2.754 

2.754 
2.879 
2.879 
3.004 

3.004 
3.100 
3.225 
3.225 

3.317 
3.442 
3.442 
3.567 

3.692 
3.692 
3.798 
3.923 

4.028 
4.153 
4.255 


4 
4 
4 
4 

t 

33^ 
33^ 

33^ 
33^ 
33^ 
33^ 

33^ 
3M 

3 
3 
3 
3 

3 
3 

2% 

2H 

2ys 


fi 
?^ 

8 

fi 

83^ 

83^ 
9 

10 
10 
10 
10 

10^ 

103^ 
103^ 
103^ 

IIH 


6 

6Ji 
63€ 
6>^ 

6>| 
6>^ 

6M 

m 
m 

7 
7 

m 
m 

m 
73I 

8* 
8 

8 

834 

83^ 



* Phoenix Iron Company. 

t Cambria Steel Company upset lengths for use with Standard turn- 
buckles (6 inches between heads) and with clevises. Make upset one inch 
shorter for use with ordinary right and left nuts. For other uses length of 
upset will vary to suit the particular case. 

Right and Left Nuts. 
(Sleeve Nuts.) 



Counter and Lateral Rods. 
Solid or Upset Eyes. 



Fig. 57. 



Counter and Lateral Rods. 
Loop Welded Eyes. 



BoundBare. Square Bars. 
Fig. 58. 



^i 




Fig. 59. 



UPSET RODS. SEG'L ROLLERS. HOWE TRUSS. 



635 



30. — Segmental Rollers. 
(For Bridge Expansion.) 



Q 



•tHf 



if- 



r"l^ 






F+r 



I. 

Fig. 60. 



Theoretical Values. 

. , ^ 180 X arc A 

Angled = ■ ^^ ,. =: 

TT X diam. D 

w = -T- COS 6 

it: = Y sin e 



cos 6 



* Practical Values. 
(All dimensions in Inches.) 



Diam. of rollers 6'' 


Diam. of rollers 8'' 


Diam. of rollers 10" 


Diam. of rollers 12* 


Thickness T = 3" 


Thickness T = 4'' 


Thickness T = 4'' 


Thickness T = 5" 


Expan- 








Expan- 








Expan- 








Expan- 








sion of 


w 


X 


y 


sion of 


w 


X 


V 


sion of 


w 


X 


T 


sion of 


w 


X 


T 


Bridge. 








Bridge. 








Bridge. 








Bridge. 








1 tol^ 


2H 


% 


3t1. 


2 


Z% 


IH 


iH 


2 to 23^ 


iH 


IH 


4H 


3 to3M 


5^ 


m5^ 


IK to IK 




1 


3^8 


23/gto2^8 






4^ 


23^ to 2^ 




IH 




SVs 


" 


u 


bH 


2 to >^ 




\\i 


3tV 


2^ 




IH 




2Mto2J.^ 


m 




4t^ 


33^to3K 




2 




2Mto23^ 






3^ 


2^to2J^ 


'i% 




4^ 


3 




IH 




4 .to4M 


5H 


2^ 


5-h 










3 to 33^ 




IH 


4i^ 


33^ to 3^ 


" 




414 


iVs 






5H 










m 






4^ 


SH 


u 


2 




43^ 


5H 


" 


u 










Wb 


'iy?. 




" 


3^ to 4 


iV? 


" 


4^ 


4^ 




2H 












33^ to 3^ 




2 


4iV 










4^to4K 


u 




5tV 










3Mto3J^ 
4 


u 


u 


43/2 
4A 










5 






53^ 



*Allow about 1 inch expansion per 100 ft. of bridge. The rollers should 
be vertical (as shown in above cut) for mean temperatures; and hence the 
arc A must be equal to, and even a little greater than, half the expansion. 

Howe Truss Brace Problem. — The following solution of the "Howe 
truss brace problem" is one which the writer devised and has used exten- 
sively. 

Theory: In Fig. 61, let the length 
of the longer side of panel (generally 
the depth of truss between chords) = 2 fe, 
and of the shorter side (generally the 
panel length) = 2p. Let x and y- be 
co-ordinates of the center of end (point 
5) of brace of width = 2 d. With h 
and p variable and d constant the 
point s will describe a circle of radius 
s — d about the origin o. 

x^ + y'^ = d^ (1) 

With h and p constant and d varia- 
ble the point s will trace the curve oso\ 
the equation of which is derived as fol- 
lows: By geometry the angles a and ai 



fli = 



equal. 
p-x 



But tan 



and tan 



h —y 

X 

.*. hy — y^ 



Equating, 
p-x 



h—y ' 




636 



),— STRUCTURAL DETAILS. 



and if p is less than h. 



,.|_y(^)^ _.(,_,) 



(2) 



y =• 




Combining the equation of the circle (1) and of the equilateral hyper- 
bola (2) analytically is a complex 
process and unnecessary in solv- T 
ing the values of x and y. 

By inspection, Fig. 61, it will 
.be seen that as the point s ap- 
proaches the origin o (as the 
width of brace decreases) the 
angles a and a^ increase, and 
when 5 reaches o tan a = tan 

ai = ^ . That is to say, the equa- 
tion of a straight line tangent to 
the hyperbola at the origin o is 

¥^ ^^^ Fig.62. 

Plotting this line (see Fig. 62), we know very nearly where the point s 
is on the circle (1). It will be just below and close to it. Then solving 
equation (3) for two values of x, on either side of and near s will give, without 
appreciable error, the tangent to the hyperbola at s, and the intersection of 
this tangent with the circle will be the point 5. 

Example: Let height of truss = 20 ft. clear between chords; panel 
length =12 ft.; width of brace = 14 ins.; then, reducing to inches, h — 
120, p = 72, d = 7. With o as origin plot circle (1) with d as radius, full 
size; also lay off tangent, equation (3), intersecting the circle at T. The 
point 5 will lie a li ttle below T on the circle. Assume Xi = 6, then, equation 
(2). yi = 60 - Vs. 600 - 39 6 = 3.40. Assume X2 = QH, then equation (2), 
y2 = 60 -V3.600 - 410.94 = 3.52. 

Plot Xi yx and X2 y2\ connect with line intersecting circle at s. Then, 
by scale, x == 6.09 ins., y = 3.45 ins. A s B is face of angle block. Length 
of brace is 

2 \/{h - y)2 + {p - x)2 = 22 ft. 3M ins. + . Ans. 

*TabIes of Cubes and Squares. — The subjoined tables of cubes and squares 
are very useful in designing and detailing. For convenient reference they 
are listed as follows: 

Table 31. — Cubes of Inches, 0" to 9", advancing by 64ths and 32ds. 
" 31a.— " " " g^'to 29^', " '^ 16ths. 

«« QJ1-J «• •« «« 29^' to 109'' ** ** 8ths 

" 32. '—Squares of Inches, (Y' to 1 2*' (0' to 1') , advancing by 64ths. 

^ " 32a.— •• •• " 12'' to 120" (r to 100 " " 32nds. 

•* 32b.— 120"to 624" (10'to52') " " 16ths. 

Note that the last column of Table 32b, ( + 55), is the average amount 
(the amount at the center of the respective line — the half-inch column) to 
be added for each 32nd of an inch. Thus, the square for 13' Oi^" is 
24472.69; and for 13' OM" is 24472.69+ 9.78 = 24482.47. The same result 
can also be obtained by adding the squares of 13' 0^" and 13' 03^" 
together, and dividing the sum by 2; thus, (24472.69 + 24492.25)^2 = 
24482.47. Remember that this difference for (^^) is for the 3^"-column, in 
middle of line ; and that for the beginning of the line the tabular difference 
in the last column should be decreased by 0.03, and increased by 0.03 for 
end of same line. This refinement, however, of modifying the amounts 
given in the last column, to conform to different parts of the line, may 
generally be neglected. 

Explanation of Uses of Tables 31 to 32 b. — ^These tables may be used in 
finding bending moments, moments of resistance, moments or inertia and 
radii of gyration; and in the solution of right angle triangles when two sides 



*These tables are abridged from those in the writer's forthcoming 
structural handbook. 



PROBS. IN CUBES AND SQUARES— MOMENTS, 



637 



are given, and of any triangle when three sides are given. The following 
examples will illustrate. 

Finding Bending Moments. (Table 32b.) 
Example. — A girder having a span (L) of 14' 5i^" supports a uniform 
load (W) of 400 lbs. per lin. ft. What is the bending moment (M") in in.-lbs. ? 

12 WL^ 
Solution. — From the well-known formula M" = ^ , we have, if / = 

• . inr .^,/ Wr^ 400X30080.6 .^rooa- ^u a 

span m ms. = 12L. M" = = i2~x'8 "^ 125336 m.-lbs. Ans. 

Finding Resisting Moments. (Table 32a.) 
Example. — What is the resisting moment in inch-lbs. (ikf ) of a rect- 
angular beam 6" wide (b) and 12'J4" deep (d), assuming the allowable fiber 
stress per sq. in. (/) = 1000 lbs.? 

Solution. — From the well-known formula M" = }^ f b d^, we have, by 

t. ^-^ *. 7,^// 1000 X 6 X 165.766 .n^^aa • ^u A 

substitution, M'' = = 165766 m.-lbs. Ans. 



Finding Radii of Gyration. 
The square of the radius of gyration (r^) = the mome nt of inertia (/) 
divided by the area (A) of the section. Thus r = VZ-^A. 

Finding MoMfeNTS op Inertia. 



Example. — Fig. 63 
represents the section of 
a steel column latticed on 
two of its sides. Find the 
moments of inertia about 
the axes X and Y, 

Solution. — The mom- 
ent of inertia (/) of a rect- 
angle about its base, is 7 = 
bh^ -r- 3; in which b = width 
and /t = height of rectangle. 
Hence, 3I = b¥. Now, 
Fig. 63 is symmetrical 
about each of its axes, 
hence if we find the value 
of 3/ for one-quarter of the 
section and then increase 
this value by one-third, we 
obtain the value 4/ for one- 
fourth of the section = 7 
for the whole section. The 
calculation is tabulated 
below; the values of bh^ 
being made up from the 
various rectangles in one- 
quarter of the section. 



TT 



i£^ 



i.^ 



^ 



^ 



X.. 



^^-^ 



-5' 









^-S' 



r 

I 

r 



:- 9"' 



9f- 



a-9i 



> 



i_jr_"^_l J 



- Rate 30'kK 



/^. 



--•! Plate 17 fy^^ 



•^ngje dxdx.^ 



i H- 

K— 



/9i 



L" 



^(?f 



Fig. 63. 



bh^ about axis X. 



bh^ about axis Y. 



30 " plate. iiX(15 



173^" plate. 



)3 = + 2320.31 15 X(10A)3 
-15 X( 9^)3 



+ 16450.65 
-13375.00 



X( 8M)3 = + 418.70 



6 "angle. 6 X (153^)3 = +20760.48 



8^X( 9^)3 
8HX( 9 )3 
6 X( 9^)3 



= + 



= + 



5^X(14J^)3 = -16386.36 - 5^ X ( 9 )3 = - 



- ysXi 93^)2 



- 474.88 - ysXi 3^)3 = - 



7802.08 
6378.75 
5350.00 
3918.38 
29.77 



Add one-third- 



+ 6638.25 
2212.76 



For the whole section, Ix = 8851.00 



+ 5900.83 
1966.94 

ly = 7867.77 



638 2Z.— STRUCTURAL DETAILS. 

Another Method. — Instead of using the "rectangle method" (preceding 
case), we may use the method of "transferrence of neutral axis." Thus, 
to find the moment of inertia of the 17^'' X H" plate about the axis Y we 
proceed as follows: Find lo about its own neutral axis {lo = bh^ -*- 12 = 
17H X (^)3 ^ 12 = 0.356) and then use the formula ly = Jo + Aa^, in 
which A = area of plate =• 17H X Vs, and a = distance between the two 
axes (see Fig. 63). Hence, the moment of inertia of the plate about the 
axis Y = ly = lo -h Aa^ = .356 -h 17J^ X ^ X (9^)2 = 948.88. Proceed 
in like manner with the other plates, and the angles, etc. The sum of the 
several moments of inertia will be the total moment of inertia of the section. 

Corollary. — In the last method, above, we have the equation, ly =■ 
lo 4- Aa^. When the section of the column is un symmetrical, it is often 
necessary to assume an axis Y and find the value of ly about this axis, and 
then, from the said equation, to find the vaules of a and lo. 

Solving Right Angle Triangles. (Tables 32, 32a, 32b.) 

Example. — ^What is the hypo then use of a right angle triangle whose 
base is 21' 7^r and perpendicular 38' 9i^''? (h^ = l>^-{'i>^.) 

Solution. — From Table 32b we have: 

For square corresponding to 21' 73^'' = 67340.25 

16.22 



I If 
38' 9t?" 



216515.70 



= 283872.17 
Ans. 44' 4|r = 283855.9 

Corollary. — From the above equation, h^ = b^ + p^, it is evident that 
62 = ;t2 _ ^ and ^2 = ;j2 _ 52. 

Solving Any Triangle — 3 Sides Given. 

Example. — In Fig. 64 let there be given the sides H, h and B + b. 
Solve the triangle. 

Solution. — Drop the perpen- 
dicular p upon the base B -h b. 
Then, p^ == H^ - B^ = h^ - 62; 
therefore. H^-h^= B^-b^. 

Whence, B — b = „ , , , 
£> -f- o 

K A T> B +b , B ~b 

And 5 = -^-4--^-, 

. ^ , B + b B -b 
And b 2 2~- 

Problem. — Given the three sides of a triangle (Fig. 64): h = 14' 3^", 
H = 28' 7^", and B + 6 = 35' 6M". Solve for the left-hand angle at the base 
(which call 5)? 

Solution. — Using the formulas in the solution to the above Example, 
we have, 

H = 28' 7^"; H^ (ins.) = 118078.1 
/j = 14'3:^": /j2(ins.)= 29433.7 

.-. m-h^ (ins.) = 88644.4; log = 4.9476513 
B +b (ins.)= 426.50; log = 2.6289190 

diff. = 2.3177323 
.'. B -b (ins.)= 207.84 ; log = 2.3177291 

2 )634.34 

.'.B (ins.)= 317.17; log = 2.5012921 
H (ins.)= 343.625; log = 2.5360848 

diff. = 9.9652073 
Ans.— 5=22° 37' 46", from log cos 5= 9.9652076 




0" (=0' on— CUBES— r (=0' 9"). 639 

31. — Cubes of Inches, 0" to 9'\ Advancing by 64ths and 32nds. 



No. 





1/64 
1/32 
3/64 
1/16 
5/64 
3/32 
7/64 
1/8 
9/64 
5/32 
11/64 
3/16 
13/64 
7/32 
15/64 

1/4 
17/64 
9/32 
19/64 
5/16 
21/64 
11/32 
23/64 

3/8 
25/64 
13/32 
27/64 
7/16 
29/64 
15/32 
31/64 

1/2 
33/64 
17/32 
35/64 
9/16 
37/64 
19/32 
39/64 

5/8 
41/64 
21/32 
43/64 
11/16 
45/64 
23/32 
47/64 

3/4 
49/64 
25/32 
51/64 
13/16 
53/64 
27/32 
55/64 

7/8 
57/64 
29/32 
59/64 
15/16 
61/64 
31/32 
63/64 



Cube. 



.0538147 
O4305I8 
.O3IO3OO 
.O324414 
.0347684 
.O382397 
.0213084 
.0019531 
.0027809 
.0038147 
.0050774 
.0065918 
.0083809 
.0104675 
.0128746 
.0156250 
.0187416 
.0222473 
.0261650 
.0305176 
.0353279 
.0406189 
,0464134 
,0527344 
,0596046 
,0670471 
,0750847 
0837402 
0930367 
1029968 
1136436 
1250000 
1370888 
1499329 
1635551 
1779785 
1932259 
2093201 
2262840 
2441406 
2629128 
2826223 
3032951 
3249512 
3476143 
3713074 
3960533 
4218750 
4487953 
4768372 
5060234 
5363770 
5679207 
6006775 
6346703 
6699219 
7064552 
7442932 
7834587 
8239746 
8658638 
9091492 
9538536 



No. 



1/32 
1/16 
3/32 
1/8 
5/32 
3/16 
7/32 
1/4 
9/32 
5/16 
11/32 

3/8 

13/32 

7/16 

15/32 

1/2 

17/32 

9/16 

19/32 

5/8 
21/32 
11/16 
23/32 

3/4 
25/32 
13/16 
27/32 

7/8 

29/32 

15/16 

31/32 

2 

1/32 

1/16 

3/32 

1/8 

5/32 

3/16 

7/32 

1/4 

9/32 

5/16 

11/32 

3/8 

13/32 

7/16 

15/32 

1/2 

17/32 

9/16 

19/32 

5/8 
21/32 
11/16 
23/32 

3/4 
25/32 
13/16 
27/32 

7/8 
29/32 
15/16 
31/32 



Cube. 



9. 

9. 
10. 
10. 
10. 
11. 
11. 
12. 
12. 
13. 
13. 
14. 
15. 
15. 
16. 
16. 
17. 
18. 
18. 
19. 
20. 
20. 
21. 
22. 
22. 
23. 
24. 
25. 
26. 



.000000 
.096710 
.199463 
.308441 
.423828 
.545807 
.674561 
.810272 
.953125 
.103302 
.260986 
.426361 
.599609 
.780914 
.970459 
.168427 
.375000 
.590363 
.814697 
.048187 
.291016 
.543365 
.805420 
.077362 
,359375 
,651642 
,954346 
,267670 
.591797 
,926910 
,273193 
630829 
000000 
380890 
773682 
178558 
595703 
025299 
467529 
922577 
39€625 
871857 
366455 
874603 
396484 
932281 
482178 
046356 
625000 
218292 
826416 
449554 
087891 
741608 
410889 
095917 
796875 
513947 
247314 
997162 
763672 
547028 
347412 
165009 



No. 



3 

1/32 

1/16 

3/32 

1/8 

5/32 

3/16 

7/32 

1/4 

9/32 

5/16 

11/32 
3/8 

13/32 
7/16 

15/32 
1/2 

17/32 
9/16 

19/32 
5/8 

21/32 

11/1 
23/32 

3/4 
25/32 
13/16 
27/32 

7/8 

29/32 

15/16 

31/32 

4 

1/32 

1/16 

3/32 

1/8 

5/32 

3/16 

7/32 

1/4 

9/32 

5/16 

11/32 

3/8 

13/32 

7/16 

15/32 

1/2 

17/32 

9/16 

19/32 

5/8 
21/32 
11/16 
23/32 

3/4 
25/32 
13/16 
27/32 

7/8 
29/32 
15/16 
31/32 



Cube. 



27 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40, 
41, 
42, 
44. 
45, 
46, 
47. 
48. 
50. 
51. 
52. 
54. 
55. 
56. 
58. 
59. 
61. 
62. 
64. 
65. 
67. 
68. 
70. 
71. 
73. 
75. 
76. 
78. 
80. 
81. 
83. 
85. 
87. 
89. 
91. 
93. 
94. 
96. 
98. 

100. 

102. 

105. 

107. 

109. 

111. 

113. 

115. 

118. 

120. 

122. 



.00000 
.85257 
.72290 
.61118 
.51758 
.44229 
.38550 
.34738 
.32813 
.32791 
.34692 
.38535 
.44336 
.52115 
.61890 
.73679 
.87500 
.03372 
.21313 
.41342 
.63477 
.87735 
.14136 
.42697 
.73438 
.06375 
.41528 
,78915 
,18555 
,60464 
,04663 
,51169 
,00000 
51175 
04712 
6062 
18945 
7967 
42847 
08469 
76563 
47147 
20239 
95859 
74023 
54752 
38062 
23972 
12500 
03665 
97485 
93979 
93164 
95059 
99683 
07053 
17188 
30106 
45825 
64365 
85742 
09976 
37085 
67087 



No. 



5 

1/32 
1/16 
3/32 
1/8 
5/32 
3/16 
7/32 
1/4 
9/32 
5/1 
11/32 

3/8 

13/32 

7/16 

15/32 

1/2 

17/32 

9/16 

19/32 

5/8 
21/32 
11/16 
23/32 

3/4 
25/32 
13/16 
27/32 

7/8 

29/32 

15/16 

31/32 

6 

1/32 

1/16 

3/32 

1/8 

5/32 

3/16 

7/32 

1/4 

9/32 

5/16 

11/32 

3/8 

13/32 

7/16 

15/32 

1/2 

17/32 

9/16 

19/32 

5/8 
21/32 
11/16 
23/32 

3/4 
25/32 
13/16 
27/32 

7/8 
29/32 
15/16 
31/32 



Cube. 



125 
127 
129 
132 
134 
137 
139 
142 
144 
147 
149 
152 
155 
158 
160 
163 
166 
169 
172, 
175. 
177. 
180. 
183. 
187. 
190. 
193. 
196. 
199. 

202. 

206. 

209. 

212. 

216. 

219. 

222. 

226. 

229. 

233. 

236. 

240. 

244. 

247. 

251. 

255. 

259. 

262. 

266. 

270. 

274. 

278. 

282. 

286. 

290. 

294. 

299. 

303. 

307. 

311. 

316. 

320. 

324. 

329. 

333. 

338. 



00000 
.35843 
.74634 
.16391 
.61133 
.08878 
.59644 
.13449 
.70313 
.30252 
.93286 
.59433 
.28711 
.01138 
.76733 
.55515 
.37500 
.22708 
.11157 
.02866 
.97852 
.96133 
.9772 
.02658 
.10938 
.22586 
.37622 
.56064 
.77930 
.03238 
.32007 
.64255 
.00000 
.39261 
.82056 
.28403 
.78320 
31827 
88940 
49680 

14063 
82108 
53833 
29257 
08398 
91275 
77905 
68307 
62500 
60501 
62329 
68002 
77539 
90958 
08276 
29514 
54688 
83817 
16919 
54013 
95117 
40250 
89429 
42673 



No. 



7 
1/32 

1/1 
3/32 
1/8 
5/32 
3/1 
7/32 
1/4 
9/32 
5/16 
11/32 

3/8 

13/32 

7/16 

15/32 

1/2 
17/32 
9/16 
19/32 
5/8 
21/32 
11/16 
23/32 

3/4 
25/32 
13/16 
27/32 

7/8 

29/32 

15/16 

31/32 

8 

1/32 

1/16 

3/32 

1/8 

5/32 

3/16 

7/32 

1/4 

9/32 

5/16 

11/32 

3/8 

13/32 

7/16 

15/32 

1/2 

17/32 

9/16 

19/32 

5/8 
21/32 
11/16 
23/32 

3/4 
25/32 
13/16 
27/32 

7/8 
29/32 
15/16 
31/32 



Cube. 



343 

347 
352 
356 
361 
366 
371 
376 
381 
386 
391 
396 
401 
406 
411 
416, 
421, 
427, 
432, 
437, 
443, 
448. 
454. 
459. 
465. 
471. 
476. 
482. 
488. 
494. 
500. 
506. 
512. 
518. 
524. 
530. 
536. 
542. 
548. 
555. 
561. 
567. 
574. 
580. 
587. 
594. 
600. 
607. 
614. 
620. 
627. 
634. 
641. 
648. 
655. 
662. 
669. 
677. 
684. 
691. 
699. 
706. 
713. 
721. 



.00000 
.61429 
.26978 
.96664 
.70508 
.48526 
.30737 
.17160 
.07813 
.02713 
.01880 
.05331 
.13086 
.25162 
.41577 
.62350 
.87500 
.17044 
.51001 
.89389 
.32227 
.79532 
.31323 
.87619 
.48438 
.13797 
.83716 
.58212 
.37305 
.21011 
,09351 
.02341 
,00000 
,02347 
,09399 
,21176 
,37695 
58975 
85034 
15891 
51563 
92068 
37427 
87656 
42773 
02798 
67749 
37643 
12500 
92337 
77173 
67026 
61914 
61856 
66870 
76974 
92188 
12527 
38013 
68661 
04492 
45523 
91772 
43259 



♦.0638147-. 0000038147. 



640 23.— STRUCTURAL DETAILS, 

31a. — Cubes of Inches, 9" to 29", Advancing by 16ths. 



No. 



Cube. 



9 

1/16 

1/8 

3/16 

1/4 

5/16 

3/8 

7/16 

1/2 

9/16 

5/8 

11/16 
3/4 

13/1 
7/8 

15/16 

10 
1/16 
1/8 
3/16 
1/4 
5/16 
3/8 
7/16 
1/2 
9/16 
5/8 

11/16 
3/4 

13/16 
7/8 

15/16 

11 
1/16 
1/8 
3/16 
1/4 
5/16 
3/8 
7/16 
1/2 
9/16 
5/8 

11/16 
3/4 

13/16 
7/8 

15/16 

12 
1/16 
1/8 
3/16 
1/4 
5/16 
3/8 
7/16 
1/2 
9/16 
5/8 

11/16 
3/4 

13/16 
7/8 

15/16 



729, 
744, 
759, 
775, 
791, 
807, 
823, 
840, 
857, 
874. 
891. 
909. 
926. 
944. 
962. 
981. 

1000. 

1018. 

1037. 

1057. 

1076. 

1096. 

1116. 

1137. 

1157. 

1178. 

1199. 

1220. 

1242. 

1264. 

1286. 

1308. 

1331. 

1353. 

1376. 

1400. 

1423. 

1447. 

1471. 

1496. 

1520. 

1545. 

1571. 

1596. 

1622. 

1648. 

1674. 

1701. 

1728. 

1755. 

1782. 

1810. 

1838. 

1866. 

1895. 

1923. 

1953. 

1982. 

2012. 

2042. 

2072. 

2103. 

2134. 

2165. 



0000 

2932 

7988 

5183 

4531 

6047 

9746 

5642 

3750 

4084 

6660 

1491 

8594 

7981 

9668 

3669 

0000 

8674 

9707 

3113 

8906 

7102 

7715 

0759 

6250 

4202 

462 

7546 

2969 

0911 

1387 

4412 

0000 

8167 

8926 

2292 

8281 

6907 

8184 

2126 

8750 

8069 

0098 

4851 

2344 

2590 

5605 

1404 

0000 

1409 

5645 

2722 

2656 

5461 

1152 

9744 

1250 

5686 

3066 

3406 

6719 

3020 

2324 

4646 



No. 



13 

1/16 

1/8 

3/16 

1/4 

5/16 

3/8 

7/16 

1/2 

9/16 

5/8 

11/1 
3/4 

13/16 
7/8 

15/16 

14 
1/16 
1/8 
3/16 
1/4 
5/16 
3/8 
7/1 
1/2 
9/16 
5/8 

11/16 
3/4 

13/16 
7/8 

15/16 

15 
1/16 
1/8 
3/16 
1/4 
5/16 
3/8 
7/16 
1/2 
9/16 
5/8 

11/16 
3/4 

13/16 
7/8 

15/16 

16 
1/16 
1/8 
3/16 
1/4 
5/16 
3/8 
7/16 
1/2 
9/16 
5/8 

11/16 
3/4 

13/16 
7/8 

15/16 



Cube. 



2197 

2228 

2260 

2293 

2326 

2359 

2392 

2426 

2460 

2494 

2529 

2564 

2599 

2635 

2671 

2707 

2744 

2780 

2818 

2855 

2893 

2931 

2970 

3009 

3048 

3088 

3128, 

3168, 

3209 

3250 

3291, 

3332 

3375, 

3417, 

3460, 

3503, 

3546, 

3590, 

3634, 

3679, 

3723, 

3769, 

3814, 

3860, 

3906, 

3953, 

4000, 

4048, 

4096, 

4144. 

4192 

4241. 

4291, 

4340. 

4390. 

4441. 

4492. 

4543. 

4594. 

4647. 

4699. 

4752. 

4805. 

4859. 



0000 
8401 
9863 
4402 
2031 
2766 
6621 
3611 
3750 
7053 
3535 
3210 
6094 
2200 
1543 
4138 
0000 
9143 
1582 
7332 
6406 
8821 
4590 
3728 
6250 
2170 
1504 
4265 
0469 
0129 
3262 



0000 
3635 
0801 
1511 
5781 
3625 
5059 
0095 
8750 
1038 
6973 
6570 
9844 
6809 
7480 
1873 
0000 
1877 
7520 
6941 
0156 
7180 
8027 
2712 
1250 
3655 
9941 
0125 
4219 
2239 
4199 
0115 



No. 



17 

1/16 

1/8 

3/16 

1/4 

5/16 

3/8 

7/16 

1/2 

9/16 

5/8 

11/16 
3/4 

13/16 
7/8 

15/16 

18 
1/16 
1/8 
3/16 
1/4 
5/16 
3/8 
7/16 
1/2 
9/16 
5/8 

11/16 
3/4 

13/16 
7/8 

15/16 

19 
1/16 
1/8 
3/16 
1/4 
5/16 
3/8 
7/16 
1/2 
9/16 
5/8 

11/16 
3/4 

13/16 
7/8 

15/16 

20 
1/16 
1/8 
3/16 
1/4 
5/16 
3/8 
7/16 
1/2 
9/16 
5/8 

11/16 
3/4 

13/16 
7/8 

15/16 



Cube. 



4913 
4967, 
5022, 
5077 
5132, 
5188 
5245, 
5302 
5359, 
5417", 
5475, 
5533, 
5592 
5651, 
5711, 
5771, 
5832, 
5892, 
5954, 
6016, 
6078, 
6141, 
6204, 
6267, 
6331, 
6396, 
6460, 
6526, 
6591, 
6657. 
6724. 
6791. 
6859. 
6926. 
6995. 
7064. 
7133. 
7203. 
7273. 
7343. 
7414. 
7486. 
7558. 
7630. 
7703. 
7777. 
7850. 
7925. 
8000. 
8075. 
150. 
8227. 
8303. 
8380. 
8458. 
8536. 
8615. 
8694. 
8773. 
8853. 
8934. 
9015. 
9096. 
9178. 



0000 

3870 

1738 

3621 

9531 

9485 

3496 

1580 

3750 

0022 

0410 

492 

3594 

6418 

341 

4607 

0000 

9612 

3457 

1550 

3906 

0540 

1465 

6697 

6250 

0139 

8379 

0984 

7969 

9348 

5137 

5349 

0000 

9104 

2676 

0730 

3281 

0344 

1934 

8064 

8750 

4006 

3848 

8289 

7344 

1028 

9355 

2341 

0000 

2346 

9395 

1160 

7656 

8899 

4902 

5681 

1250 

1624 

6816 

6843 

1719 

1458 

6074 

5583 



No. 



21 

1/16 

1/ 

3/16 

1/4 

5/16 

3/8 

7/16 

1/2 

9/1 

5/8 

11/16 
3/4 

13/16 
7/8 

15/16 

22 
1/16 
1/8 
3/16 
1/4 
5/1 
3/8 
7/16 
1/2 
9/16 
5/8 

11/16 
3/4 

13/16 
7/8 

15/16 

23 
1/16 
1/8 
3/16 
1/4 
5/16 
3/8 
7/16 
1/2 
9/16 
5/8 

11/16 
3/4 

13/16 
7/8 

15/16 

24 
1/16 
1/8 
3/16 
1/4 
5/16 
3/8 
7/16 
1/2 
9/16 
5/8 

11/16 
3/4 

13/16 
7/8 

15/16 



Cube. 



9261 
9343 
9427 
9511 
9595 
9680 
9766 
9851 
9938 
10025 
10112 
10200 
10289 
10378 
10467 
10557 
10648 
10739 
10830 
10922 
11015 
11108 
11201 
11295 
11390 
11485 
11581 
11677 
11774 
11871 
11969 
12068 
12167 
12266 
12366 
12466 
12568 
12669 
12771 
12874 
12977 
13081 
13186 
13291 
13396 
13502 
13609 
13716 
13824 
13932 
14041 
14150 
14260 
14371 
14482 
14593 
14706 
14818 
14932 
15046 
15160 
15276 
15391 
15508 



,000 
,934 
,361 
.284 
.703 
.620 
,037 
,955 
.375 
.299 
.729 
,665 
,109 
,064 
.52 
,508 
,000 
,008 
.533 
.577 
.141 
.226 
,834 
,967 
,625 
,811 
,525 
,770 
,547 
,857 
,701 
,082 
,000 
,457 
,455 
,995 
,078 
,706 
,881 
603 
,875 
,698 
072 
.001 
,484 
,525 
123 
281 
000 
281 
127 
538 
516 
062 
178 
865 
125 
959 
369 
356 
922 
068 
795 
105 



No. 



25 

1/16 

1/8 

3/16 

1/4 

5/1 

3/8 

7/16 

1/2 

9/16 

5/8 

11/16 
3/4 

13/16 
7/8 

15/16 

26 
1/16 
1/8 
3/16 
1/4 
5/16 
3/8 
7/16 
1/2 
9/1 
5/8 

11/16 
3/4 

13/16 
7/8 

15/16 

27 
1/16 
1/8 
3/16 
1/4 
5/16 
3/8 
7/16 
1/2 
9/16 
5/8 

11/16 
3/4 

13/16 
7/8 

15/16 

28 
1/16 
1/8 
3/16 
1/4 
5/16 
3/8 
7/16 
1/2 
9/16 
5/8 

11/16 
3/4 

13/16 
7/8 

15/16 



Cube. 



15625.000 
15742.481 
15860.549 
15979.206 
16098.453 
16218.292 
16338.725 
16459.752 
16581.375 
16703.596 
16826.416 
16949.837 
17073.859 
17198.486 
17323.717 
17449.554 
17576.000 
17703.055 
17830.721 
17958.999 
18087.891 
18217.398 
18347.521 
18478.263 
18609.625 
18741.608 
18874.213 
19007.442 
19141.297 
19275.779 
19410.889 
19546.629 
19683.000 
19820.004 
19957.643 
20095.917 
20234.828 
20374.378 
20514.568 
20655.400 
20796.875 
20938.994 
21081.760 
21225.173 
21369.234 
21513.947 
21659.311 
21805.328 
21952.000 
22099.328 
22247.314 
22395.960 
22545.266 
22695.234 
22845.865 
22997.162 
23149.125 
23301.756 
23455.057 
23609.028 
23763.672 
23918.989 
24074.982 
24231.652 



9^ (=0' 9")— CUBES— 69" (=5' 9"). 641 

31b. — Cubes of Inches, 29'' to 69", Advancino by 8ths. 



No. 



29 



30 



31 



32 



33 



34 



35 



36 



1/8 
1/4 
3/8 
1/2 
5/8 
3/4 
7/8 
I 

1/8 
1/4 
3/8 
1/2 
5/8 
3/4 
7/8 

1/8 
1/4 
3/8 
1/2 

5/8 
3/4 
7/8 

> 

'l/8 
1/4 
3/8 
1/2 
5/8 
3/4 
7/8 

'l/8 
1/4 
3/8 
1/2 
5/8 
3/4 
7/8 

1/8 
1/4 

3/8 
1/2 
5/8 
3/4 
7/8 

1/8 
1/4 
3/8 
1/2 
5/8 
3/4 
7/8 

'l/8 
1/4 
3/8 
1/2 
5/8 
3/4 
7/8 



Cube. 



24389. 
24705. 
25025. 
25347. 
25672. 
26000. 
26330. 
26663. 
27000. 
27338. 
27680. 
28025. 
28372. 
28722. 
29076. 
29432. 
29791. 
30152. 
30517. 
30885. 
31255. 
31629. 
32005. 
32385. 
32768. 
33153. 
33542. 
33933. 
34328. 
34725. 
35126. 
35530. 
35937. 
36346. 
36759. 
37176. 
37595. 
38017. 
38443. 
38872. 
39304. 
39739. 
40177. 
40618. 
41063. 
41511. 
41962. 
42417, 
42875, 
43336, 
43800. 
44267 
44738, 
45213, 
45690, 
46171, 
46656, 
47143, 
47634, 
48129, 
48627, 
49128 
49633 
50141 



e. 


No. 


000 


37 


736 


1/8 


203| 


1/4 


412: 


3/8 


375 


1/2 


104 


5/8 


609 


3/4 


904 


7/8 


000 


38 


90 8 


1/8 


641 


1/4 


209 


3/8 


625 


1/2 


900 


5/8 


047 


3/4 


076 


7/8 


000 


39 


830 


1/8 


578 


1/4 


256 


3/8 


875 


1/2 


447 


5/8 


984 


3/4 


498 


7/8 


000 


40 


502 


1/8 


016 


1/4 


553 


3/8 


125 


1/2 


744 


5/8 


422 


3/4 


170 


7/8 


000 


41 


924 


1/8 


953 


1/4 


100 


3/8 


375 


1/2 


791 


5/8 


359 


3/4 


092 


7/8 


000 


42 


095 


1/8 


391 


1/4 


896 


3/8 


625 


1/2 


.588 


5/8 


.797 


3/4 


264 


7/8 


.000 


43 


018 


1/8 


.328 


1/4 


.943 


3/8 


.875 


1/2 


.135 


5/8 


.734 


3/4 


.686 


7/8 


.000 


44 


.689 


1/8 


.766 


1/4 


.240 


3/8 


.125 


1/2 


.432 


5/8 


.172 


3/4 


.357 


7/8 



Cube. 



50653. 
51168. 
55686. 
56208. 
52734. 
53263. 
53796. 
54332. 
54872. 
55415. 
55962, 
56512. 
57066. 
57624. 
58185. 
58750. 
59319. 
59891. 
60467. 
61046. 
61629. 
62216. 
62807. 
63401. 
64000. 
64601. 
65207. 
65816. 
66430. 
67047. 
67667. 
68292. 
68921. 
69553. 
70189. 
70829. 
71473. 
72121. 
72772. 
73428. 
74088. 
74751. 
75418. 
76090. 
76765. 
77444 
78128, 
78815, 
79507. 
80202. 
80901, 
81605. 
82312, 
83024, 
83740, 
84460, 
85184, 
85912, 
86644, 
87380, 
88121, 
88865, 
89614 
90367 



000 

111 

703 

787 

375 

479 

109 

279 

000 

283 

141 

584 

625 

275 

547 

451 

000 

205 

07 8 

631 

875 

822 

484 

873 

000 

877 

516 

928 

125 

119 

922 

545 

000 

299 

453 

475 

375 

166 

859 

467 

000 

471 

891 

271 

625 

963 

297 

639 

000 

393 

82 

318 

875 

510 

234 

061 

000 

064 

266 

615 

125 

807 

672 

732 



No. 



45 



46 



47 



48 



49 



50 



51 



52 



1/8 
1/4 
3/8 
1/2 
5/8 
3/4 
7/8 

1/8 
1/4 
3/8 
1/2 
5/8 
3/4 
7/8 

1/8 
1/4 
3/8 
1/2 
5/8 
3/4 
7/8 

/^ 
1/4 

3/8 

1/2 

5/8 

3/4 

7/8 

I 

1/8 

1/4 

3/8 

1/2 

5/8 

3/4 

7/8 

I 

1/8 

1/4 

3/8 

1/2 

5/8 

3/4 

7/8 

1/8 
1/4 
3/8 
1/2 

5/8 
3/4 
7/8 
» 

'l/8 
1/4 
3/8 
1/2 
5/8 
3/4 
7/8 



Cube. 



91125 
91886 
92652 
93422, 
94196 
94974 
95757, 
96544, 
97336 
98131 
98931, 
99735, 
100544, 
101357, 
102175 
102996 
103823 
104653, 
105488 
106328 
107171 
108020 
108872 
109730 
110592 
111458 
112329 
113204 
114084 
114968 
115857 
116750 
117649 
118551 
119458 
120370 
121287 
122208 
123134 
124064 
125000 
125939 
126884 
127833 
128787 
129746 
130709 
131678 
132651 
133628 
134611 
135598 
136590 
137587 
138589 
139596 
140608 
141624 
142645 
143671 
144703 
145739 
146780 
147826 



No. 



53 



54 



55 



56 



57 



58 



59 



60 



1/8 
1/4 
3/8 
1/2 
5/8 
3/4 
7/8 

1/8 
1/4 
3/8 
1/2 
5/8 
3/4 
7/8 

1/8 
1/4 
3/8 
1/2 
5/8 
3/4 
7/8 

'l/8 
1/4 
3/8 
1/2 

5/8 
3/4 
7/8 

1/8 
1/4 
3/8 
1/2 
5/8 
3/4 
7/8 

1/8 
1/4 
3/8 
1/2 
5/8 
3/4 
7/8 
I 

1/8 
1/4 
3/8 
1/2 
5/8 
3/4 
7/8 
I 

1/8 
1/4 
3/8 
1/2 
5/8 
3/4 
7/8 



Cube. 



148877. 

149932. 

150993. 

152059. 

153130. 

154206. 

155287. 

156373. 

157464. 

158560. 

159661. 

160767. 

161878. 

162995. 

164116. 

165243. 

166375. 

167511. 

168654. 

169801. 

170953. 

172111. 

173274. 

174442. 

175616. 

176794. 

177978. 

179167. 

180362. 

181561. 

182766. 

183977. 

185193, 

186414, 

187G40. 

188872, 

190109, 

191351, 

192599, 

193853, 

195112, 

196376, 

197645, 

198921, 

200201, 

201487, 

202779, 

204076, 

205379, 

206687, 

208000, 

209320, 

210644, 

211975, 

213311, 

214652, 

216000, 

217352, 

218711, 

220075, 

221445, 

222820, 

224201 

225588, 





No. 


00 


61 


86 


1/8 


70 


1/4 


54 


3/8 


38 


1/2 


23 


5/8 


11 


3/4 


03 


7/8 


00 


62 


03 


1/8 


14 


1/4 


33 


3/8 


63 


1/2 


03 


5/8 


55 


3/4 


20 


7/8 


00 


63 


96 


1/8 


08 


1/4 


38 


3/8 


88 


1/2 


57 


5/8 


48 


3/4 


62 


7/8 


OU 


64 


63 


1/8 


52 


1/4 


68 


3/8 


13 


1/2 


87 


5/8 


92 


3/4 


29 


7/8 


00 


65 


05 


1/8 


45 


1/4 


22 


3/8 


38 


1/2 


92 


5/8 


86 


3/4 


22 


.7/8 


00 


66 


22 


1/8 


89 


1/4 


02 


3/8 


63 


1/2 


71 


5/8 


30 


3/4 


39 


7/8 


00 


67 


14 


1/8 


83 


1/4 


07 


3/8 


88 


1/2 


26 


5/8 


23 


3/4 


81 


7/8 


00 


68 


81 


1/8 


27 


1/4 


37 


3/8 


13 


1/2 


56 


5/8 


67 


3/4 


48 


7/8 



Cube. 



226981.00 
228379.24 
229783.20 
231192.91 
232608.38 
234029.60 
235456.61 
236889.40 
238328.00 
239772.41 
241222.64 
242678.71 
244140.63 
245608.40 
247082.05 
248561.58 
250047,00 
251538.33 
253035.58 
254538.76 
256047.88 
257562.95 
259083.98 
260611.00 
262144.00 
263683.00 
265228.02 
266779.05 
268336.13 
269899.24 
271468.42 
273043.67 
274625.00 
276212.42 
277805.95 
275405.60 
281011.38 
282623.29 
284241.36 
285865.59 
287496.00 
289132.60 
290775.39 
292424.40 
294079.63 
295741.09 
297408.80 
299082.76 
300763.00 
302449.52 
304142.33 
305841.44 
307546.88 
309258.63 
310976.73 
312701.19 
314432.00 
316169.19 
317912.77 
319662.74 
321419.13 
323181.93 
324951.17 
326726.86 



642 3Z.— STRUCTURAL DETAILS. 

31b. — Cubes of Riches, 69" to 109", Advancing by 8ths. — Concluded. 



No. 


Cube. 


No. 


Cube. 


No. 


Cube. 


No. 


Cube. 


No. 


Cube. 


69 


328509.0 


77 


456533.0 


85 


614125.0 


93 


804357.0 


101 


1030301.0 


1/8 


330297.6 


1/8 


458760.0 


1/8 


616838.4 


1/8 


807604.7 


1/8 


1034131.1 


1/4 


332092.7 


1/4 


460994.2 


1/4 


619559.7 


1/4 


810861.2 


1/4 


1037970.7 


3/8 


333894.3 


3/8 


463235.7 


3/8 


622289.0 


3/8 


814126.4 


3/8 


1041819.8 


1/2 


335702.4 


1/2 


465484.4 


1/2 


625026.4 


1/2 


817400.4 


1/2 


1045678.4 


5/8 


337567.0 


5/8 


467740.4 


5/8 


627771.7 


5/8 


820683.1 


5/8 


1049546.5 


3/4 


339338.1 


3/4 


470003.6 


3/4 


630525.1 


3/4 


823974.6 


3/4 


1053424.1 


7/8 


341165.8 


7/8 


472274.2 


7/8 


633286.5 


7/8 


827274.9 


7/8 


1057311.3 


70 


343000.0 


78 


474552.0 


86 


636056.0 


94 


830584.0 


102 


1061208.0 


1/8 


344840.8 


1/8 


476837.2 


1/8 


638833.5 


1/8 


833901.9 


1/8 


1065114.3 


1/4 


346688.1 


1/4 


479129.6 


1/4 


641619.1 


1/4 


837228.6 


1/4 


1069030.1 


378 


348542.1 


3/8 


481429.5 


3/8 


644412.8 


3/8 


840564.2 


3/8 


1072955.6 


1/2 


350402.6 


1/2 


483736.6 


1/2 


647214.6 


1/2 


843908.6 


1/2 


1076890.6 


5/8 


352269.8 


5/8 


486051.2 


5/8 


650024.5 


5/8 


847261.9 


5/8 


1080835.3 


3/4 


354143.5 


3/4 


488373.0 


3/4 


652842.5 


3/4 


850624.0 


3/4 


1084789.5 


7/8 


356024.0 


7/8 


490702.3 


7/8 


655668.7 


7/8 


853995.1 


7/8 


1088753.5 


71 


357911.0 


79 


493039.0 


87 


658503.0 


95 


857375.0 


103 


1092727.0 


1/8 


359804.7 


1/8 


495383.1 


1/8 


661345.5 


1/8 


860763.8 


1/8 


1096710.2 


1/4 


361705.1 


1/4 


497734.6 


1/4 


664196.1 


1/4 


864161.6 


1/4 


1100703.1 


3/8 


363612.1 


3/8 


500093.5 


3/8 


667054.9 


3/8 


867568.3 


3/8 


1104705.6 


1/2 


365525.9 


1/2 


502459.9 


1/2 


669921.9 


1/2 


870983.9 


1/2 


1108717.9 


5/8 


367446.3 


5/8 


504833.7 


5/8 


672797.1 


5/8 


874408.5 


5/8 


1112739.8 


3/4 


369373.5 


3/4 


507215.0 


3/4 


675680.5 


3/4 


877842.0 


3/4 


1116771.5 


7/8 


371307.4 


7/8 


509603.8 


7/8 


678572.1 


7/8 


881284.5 


7/8 


1120812.9 


72 


373248.0 


80 


512000.0 


88 


681472.0 


96 


884736.0 


104 


1124864.0 


1/8 


375195.4 


1/8 


514403.8 


1/8 


684380.1 


1/8 


888196.5 


1/8 


1128924.9 


1/4 


377149.5 


1/4 


516815.0 


1/4 


687296.5 


1/4 


891666.0 


1/4 


1132995.5 


3/8 


379110.4 


3/8 


519233.8 


3/8 


690221.2 


3/8 


895144.6 


3/8 


1137075.9 


1/2 


381078.1 


1/2 


521660.1 


1/2 


693154.1 


1/2 


898632.1 


1/2 


1141166.1 


5/8 


383052.6 


5/8 


524094.0 


5/8 


696095.4 


5/8 


902128.8 


5/8 


1145266.1 


3/4 


385033.9 


3/4 


526535.4 


3/4 


699044.9 


3/4 


905634.4 


3/4 


1149375.9 


7/8 


387022.1 


7/8 


528984.4 


7/8 


702002,8 


7/8 


909149.2 


7/8 


1153495.5 


73 


389017.0 


81 


531441.0 


89 


704969.0 


97 


912673.0 


105 


1157625.0 


1/8 


391018.8 


1/8 


533905.2 


1/8 


707943.6 


1/8 


916205.9 


1/8 


1161764.3 


1/4 


393027.5 


1/4 


536377.0 


1/4 


710926.5 


1/4 


919748.0 


1/4 


1165913.5 


3/8 


395043.0 


3/8 


538856.4 


3/8 


713917.7 


3/8 


923299.1 


3/8 


1170072.5 


1/2 


397065.4 


1/2 


541343.4 


1/2 


716917.4 


1/2 


926859.4 


1/2 


1174241.4 


5/8 


399094.7 


5/8 


543838,0 


5/8 


719925.4 


5/8 


930428.8 


5/8 


1178420.2 


3/4 


401130.9 


3/4 


546340.4 


3/4 


722941.9 


3/4 


934007.4 


3/4 


1182608.9 


7y8 


403174-.0 


7/8 


548850.3 


7/8 


7^5966.7 


7/8 


937595.1 


7U 


1186807.5 


74 


405224.0 


82 


551368.0 


90 


729000.0 


98 


941192.0 


106 


1191016.0 


1/8 


407281.0 


1/8 


553893.4 


1/8 


732041.7 


1/8 


944798.1 


1/8 


1195234.5 


1/4 


409344.9 


1/4 


556426.4 


1/4 


735091.9 


1/4 


948413.4 


1/4 


1199462.9 


3/8 


411415.8 


3/8 


558967.2 


3/8 


738150.5 


3/8 


952037.9 


3/8 


1203701.3 


1/2 


413493.6 


1/2 


561515.6 


1/2 


741217.6 


1/2 


955671.6 


1/2 


1207949.6 


5/8 


415578.5 


5/8 


564071.8 


5/8 


744293.2 


5/8 


959314.6 


5/8 


1212208.0 


3/4 


417670.3 


3/4 


566635.8 


3/4 


747377.3 


3/4 


962966.8 


3/4 


1216476.3 


7/8 


419769.1 


7/8 


569207.5 


7/8 


750469.9 


7/8 


966628.3 


7/8 


1220754.6 


75 


421875.0 


83 


571787.0 


91 


753571.0 


99 


970299.0 


107 


1225043.0 


1/8 


423987.9 


1/8 


574374.3 


1/8 


756680.6 


1/8 


973979.0 


1/8 


1229341.4 


1/4 


426107.8 


1/4 


576969.3 


1/4 


759798.8 


1/4 


977668.3 


1/4 


1233649.8 


3/8 


428234.8 


3/8 


579572.2 


3/8 


762925.6 


3/8 


981367.0 


3/8 


1237968.3 


1/2 


430368.9 


1/2 


582182.9 


1/2 


766060.9 


1/2 


985074.9 


1/2 


1242296.9 


5/8 


432510.0 


5/8 


584801.4 


5/8 


769204.8 


5/8 


988792.1 


5/8 


1246635.5 


3/4 


434658.2 


3/4 


587427.7 


3/4 


772357.2 


3/4 


992518.7 


3/4 


1250984.2 


7/8 


436813.6 


7/8 


590061.9 


7/8 


775518.3 


7/8 


996254.7 


7/8 


1255343.1 


76 


438976.0 


84 


592704.0 


92 


778688.0 


100 


1000000.0 


108 


1259712.0 


1/8 


441145.6 


1/8 


595353.9 


1/8 


781866.3 


1/8 


1003754.7 


1/8 


1264091.1 


1/4 


443322.3 


1/4 


598011.8 


1/4 


785053.3 


.1/4 


1007518.8 


1/4 


1268480.3 


3/8 


445506.1 


3/8 


600677.5 


3/8 


788248.9 


3/8 


1011292.2 


3/8 


1272879.6 


1/2 


447697.1 


1/2 


603351.1 


1/2 


791453.1 


1/2 


1015075.1 


1/2 


1277289.1 


5/8 


449895.3 


5/8 


606032.7 


5/8 


794666.1 


5/8 


1018867.4 


5/8 


1281708.8 


3/4 


452100.7 


3/4 


608722.2 


3/4 


797887.7 


3/4 


1022669.2 


3/4 


1286138.7 


7/8 


454313.2 


7/8 


611419.6 


7/8 


801118.0 


7/8 


1026480.4 


7/8 


1290578.7 



69'' (=5' 9")— CUBES— 109" (=9' V). 



0'' (=0' 0'')'-SQUARES-—6'' (=0' 6*). 



643 



32,— Squares of Inches, 0" to 6" (0' 0" to 0' 6'Of Advancing 
BY 64ths. 



In. 





. 


2 


3 


4 


5 


In. 






1.0000000 
1.0314941 


4.0000000 
4.0627441 


9.000000 
9.093994 


16.000000 
16.125244 


25.000000 
25.156494 




*i/64* 




'"!666244i4 


i/64 




1/32 


.00097656 


1.0634766 


4.1259766 


9.188477 


16.250977 


25.313477 


1/32 


3/64 




.00219727 


1.0959473 


4.1896973 


9.283447 


16.377197 


25.470947 


3/64 




1/16 


.00390625 


1.1289063 


4.2539063 


9.378906 


16.503906 


25.628906 


1/16 


5/64 




.00610352 


1.1623535 


4.3186035 


9.474854 


16.631104 


25.787354 


5/64 




3/32 


.00878906 


1.1962891 


4.3837891 


9.571289 


16.758789 


25.946289 


3/32 


7/64 




.01196289 


1.2307129 


4.4494629 


9.668213 


16.886963 


26.105713 


7/64 




1/8 


.01562500 


1.2656250 


4.5156250 


9.765625 


17.015625 


26.265625 


1/8 


9/64 




.01977539 


1.3010254 


4.5822754 


9.863552 


17.144775 


26.426025 


9/64 




5/32 


.02441406 


1.3369141 


4.6494141 


9.961914 


17.274414 


26.586914 


5/32 


11/64 




.02954102 


1.3732910 


4.7170410 


10.060791 


17.404541 


26.748291 


11/64 




3/16 


.03515625 


1.4101563 


4.7851563 


10.160156 


17.535156 


26.910156 


3/16 


13/64 




.04125977 


1.4475098 


4.8537598 


10.260010 


17.666260 


27.072510 


13/64 




7/32 


.04785156 


1.4853516 


4.9228516 


10.360352 


17.797852 


27.235352 


7/32 


15/64 




.05493164 


1.52.%816 


4.9924316 


10.461182 


17.929932 


27.398682 


15/64 




1/4 


.06250000 


1.5625000 


5.0625000 


10.562500 


18.062500 


27.562500 


1/4 


17/64 




.07055664 


1.6018066 


5.1330566 


10.664307 


18.195557 


27.726807 


17/64 




9/32 


.07910156 


1.6416016 


5.2041016 


10.766602 


18.329102 


27.891602 


9/32 


19/64 




.08813477 


1.6818848 


5.2756348 


10.869385 


18.463135 


28.056885 


19/64 




5/16 


.09765625 


1.7226563 


5.3476563 


10.972656 


18.597656 


28.222656 


5/16 


21/6,4 




.10766602 


1.7639160 


5.4201660 


11.076416 


18.732666 


28.388916 


21/64 




11/32 


.11816406 


1.8056641 


5.4931641 


11.180664 


18.868164 


28.555664 


11/32 


23/64 




.12915039 


1.8479004 


5.5666504 


11.285400 


19.004150 


28.722900 


23/64 




3/8 


.14062500 


1.8906250 


5.6406250 


11.390625 


19.140625 


28.890625 


3/8 


25/64 




.15258789 


1.9338379 


5.7150879 


11.496338 


19.277588 


29.058838 


25/64 




13/32 


.16503906 


1.9775391 


5.7900391 


11.602539 


19.415039 


29.227539 


13/32 


27/64 




.17797852 


2.0217285 


5.8654785 


11.709229 


19.552979 


29.396729 


27/64 




7/16 


.19140625 


2.0664063 


5.9414063 


11.816406 


19.691406 


29.566406 


7/16 


29/64 




.20532227 


2.1115723 


6.0178223 


11.924072 


19.830322 


29.736572 


29/64 




15/32 


.21972656 


2.1572266 


6 0947266 


12.032227 


19.969727 


29.907227 


15/32 


31/64 




.23461914 


2.2033691 


6.1721191 


12.140869 


20.109619 


30.078369 


31/64 




1/2 


.25000000 


2.2500000 


6.2500000 


12.250000 


20.250000 


30.250000 


1/2 


33/64 




.26586914 


2.2971191 


6.3283691 


12.359619 


20.390869 


30.422119 


33/64 




17/32 


.28222656 


2.3447266 


6.4072266 


12.469727 


20.532227 


30.594727 


17/32 


35/64 




.29907227 


2.3928223 


6.4865723 


12.580322 


20.674072 


30.767822 


35/64 




9/16 


.31640625 


2.4414063 


6.5664063 


12.691406 


20.816406 


30.941406 


9/16 


37/64 




.33422852 


2.4904785 


6.6467285 


12.802979 


20.959229 


31.115479 


37/64 




19/32 


.35253906 


2.5400391 


6.7275391 


12.915039 


21.102539 


31.290039 


19/32 


39/64 




.37133789 


2.5900879 


6.8088379 


13.027588 


21.246338 


31.465088 


39/64 




5/8 


.39062500 


2.6406250 


6.8906250 


13.140625 


21.390625 


31.640625 


5/8 


41/64 




.41040039 


2.6916504 


6.9729004 


13.254150 


21.535400 


31.816650 


41/64 




21/32 


.43066406 


2.7431641 


7.0556641 


13.368164 


21.680664 


31.993164 


21/32 


43/64 




.45141602 


2.7951660 


7.1389160 


13.482666 


21.826416 


32.170166 


43/64 




11/16 


.47265625 


2.8476563 


7.2226563 


13.597656 


21.972656 


32.347656 


11/16 


45/64 




.49438477 


2.9006348 


7.3068848 


13.713135 


22.119385 


32.525635 


45/64 




23/32 


.51660156 


2.9541016 


7.3916016 


13.829102 


22.266602 


32.704102 


23/32 


47/64 




.53930664 


3.0080566 


7.4768066 


13.945557 


22.414307 


32 883057 


47/64 




3/4 


.56250000 


3,0625000 


7.5625000 


14.062500 


22.562500 


33.062500 


3/4 


49/64 




.58618164 


3.1174316 


7.6486816 


14.179932 


22.711182 


33.242432 


49/64 




25/32 


.61035156 


3.1728516 


7.7353516 


14.297852 


22.860352 


33.422852 


25/32 


51/64 




.63500977 


3.2287598 


7.8225098 


14.416260 


23.010010 


33.603760 


51/64 




13/16 


.66015625 


3.2851563 


7.9101563 


14.535156 


23.160156 


33.785156 


13/16 


53/64 




.68579102 


3.3420410 


7.9982910 


14.654541 


23.310791 


33.967041 


53/64 




27/32 


.71191406 


3.3994141 


8.0869141 


14.774414 


23.461914 


34.149414 


27/32 


55/64 




.73852539 


3.4572754 


8.1760254 


14.894775 


23.613525 


34.332275 


55/64 




7/8 


.76562500 


3.5156250 


8.2656250 


15.015625 


23.765625 


34.515625 


7/8 


57/64 




.79321289 


3.5744629 


8.3557129 


15.136963 


23.918213 


34.699463 


57/64 




29/32 


.82128906 


3.6337891 


8.4462891 


15.258789 


24.071289 


34.883789 


29/32 


59/64 




.84985352 


3.6936035 


8.5373535 


15.381104 


24.224854 


35.068G04 


59/64 




15/16 


.87890625 


3.7539063 


8.6289063 


15.503906 


24.378906 


35.253906 


15/16 


61/64 




.90844727 


3.8146973 


•8.7209473 


15.627197 


24.533447 


35.439697 


61/64 




31/32 


.93847656 


3.8759766 


8.8134766 


15.750977 


24.688477 


35.625977 


31/32 


63/64 




.96899414 


3.9377441 


8.9064941 


15.875244 


24.843994 


35.812744 


63/64 



644 



S3.^STRUCTURAL DETAILS. 



32.— Squares of Inches, 6" to 12" (0' 6" to I'O"), Advancing 

BY 64ths. — Concluded. 



In. 


6 


7 


8 


9 


10 


11 


In. 




36.000000 
36.187744 


49.000000 
49.218994 


64.000000 
64.250244 


81.000000 
81.281494 


100.00000 
100.31274 


121.00000 
121.34399 




*i/64' 




i/64 




1/32 


36.375977 


49.438477 


64.500977 


81.563477 


100.62598 


121.68848 


1/32 


3/64 




36.564697 


49.658447 


64.752197 


81.845947 


100.93970 


122.03345 


3/64 




1/16 


36.753906 


49.878906 


65.003906 


82.128906 


101.25391 


122.37891 


1/16 


5/64 




36.943604 


50.099854 


65.256104 


82.412354 


101.56860 


122.72485 


5/64 




3/32 


37.133789 


50.321289 


65.508789 


82.696289 


101.88379 


123.07129 


3/32 


7/64 




37.324463 


50.543213 


65.761963 


82.980713 


102.19946 


123.41821 


7/64 




1/8 


37.515625 


50.765625 


66.015625 


83.265625 


102.51563 


123.76563 


1/8 


9/64 




37.707275 


50.988525 


66.269775 


83.551025 


102.83228 


124.11353 


9/64 




5/32 


37.899414 


51.211914 


66.524414 


83.836914 


103.14941 


124.46191 


5/32 


11/64 




38.092041 


51.435791 


66.779541 


84.123291 


103.46704 


124.81079 


11/64 




3/16 


38.285156 


51.660156 


67.035156 


84.410156 


103.78516 


125.16016 


3/16 


13/64 




38.478760 


51.885010 


67.291260 


84.697510 


104.10376 


125.51001 


13/64 




7/32 


38.672852 


52.110352 


67.547852 


84.985352 


104.42285 


125.86035 


7/32 


15/64 




38.867432 


52.336182 


67.804932 


85.273682 


104.74243 


126.21118 


15/64 




1/4 


39.062500 


52.562500 


68.062500 


85.562500 


105.06250 


126.56250 


1/4 


17/64 




39.258057 


52.789307 


68.320557 


85.851807 


105.38306 


126.91431 


17/64 




9/32 


39.454102 


53.016602 


68.579102 


86.141602 


105.70410 


127.26660 


9/32 


19/64 




39.650635 


53.244385 


68.838135 


86.431885 


106.02563 


127.61938 


19/64 




5/16 


39.847656 


53.472656 


69.097656 


86.722656 


106.34766 


127.97266 


5/16 


21/64 




40.045166 


53.701416 


69.357666 


87.013916 


106.67017 


128.32642 


21/64 




11/32 


40.243164 


53.930664 


69.618164 


87.305664 


106.99316 


128.68066 


11/32 


23/64 




40.441650 


54.160400 


69.879150 


87.597900 


107.31665 


129.03540 


23/64 




3/8 


40.640625 


54.390625 


70.140625 


87.890625 


107.64063 


129.39063 


3/8 


25/64 




40.840088 


54.621338 


70.402588 


88.183838 


107.96509 


129.74634 


25/64 




13/32 


41.040039 


54.852539 


70.665039 


88.477539 


108.29004 


130.10254 


13/32 


27/64 




41.240479 


55.084229 


70.927979 


88.771729 


108.61548 


130.45923 


27/64 




7/16 


41.441406 


55.316406 


71.191406 


89.066406 


108.94141 


130.81641 


7/16 


29/64 




41.642822 


55.549072 


71.455322 


89.361572 


109.26782 


131.17407 


29/64 




15/32 


41.844727 


55.782227 


71.719727 


89.657227 


109.59473 


131.53223 


15/32 


31/64 




42.047119 


56.015869 


71.984619 


89.953369 


109.92212 


131.89087 


31/64 




1/2 


42.250000 


56.250000 


72.250000 


90.250000 


110.25000 


132.25000 


1/2 


33/64 




42.453369 


56.484619 


72.515869 


90.547119 


110.57837 


132.60962 


33/64 




17/32 


42.657227 


56.719727 


72.782227 


90.844727 


110.90723 


132.96973 


17/32 


35/64 




42.861572 


56.955322 


73.049072 


91.142822 


111.23657 


133.33032 


35/64 




9/16 


43.066406 


57.191406 


73.316406 


91.441406 


111.56641 


133.69141 


9/16 


37/64 




43.271729 


57.427979 


73.584229 


91.740479 


111.89673 


134.05298 


37/64 




19/32 


43.477539 


57.665039 


73.852539 


92.040039 


112.22754 


134.41504 


19/32 


39/64 




43.683838 


57.902588 


74.121338 


92.340088 


112.55884 


134.77759 


39/64 




5/8 


43.890625 


58.140625 


74.390625 


92.640625 


112.89063 


135.14063 


5/8 


41/64 




44.097900 


58.379150 


74.660400 


92.941650 


113.22290 


135.50415 


41/64 




21/32 


44.305664 


58.618164 


74.930664 


93.243164 


113.55566 


135.86816 


21/32 


43/64 




44.513916 


58.857666 


75.201416 


93.545166 


113.88892 


136.23267 


43/64 




11/16 


44.722656 


59.097656 


75.472656 


93.847656 


114.22266 


136.59766 


11/16 


45/64 




44.931885 


59.338135 


75.744385 


94.150635 


114.55688 


136.96313 


45/64 




23/32 


45.141602 


59.579102 


76.016602 


94.454102 


114.89160 


137.32910 


23/32 


47/64 




45.351807 


59.820557 


76.289307 


94.758057 


115.22681 


137.69556 


47/64 




3/4 


45.562500 


60.062500 


76.562500 


95.062500 


115.56250 


138.06250 


3/4 


49/64 




45.773682 


60.304932 


76.836182 


95.367432 


115.89868 


138.42993 


49/64 




25/32 


45.985352 


60.547852 


77.110352 


95.672852 


116.23535 


138.79785 


25/32 


51/64 




46.197510 


60.791260 


77.385010 


95.978760 


116.57251 


139.16626 


51/64 




13/16 


46.410156 


61.053156 


77.660156 


96.285156 


116.91016 


139.53516 


13/16 


53/64 




46.623291 


61.279541 


77.935791 


96.592041 


117.24829 


139.90454 


53/64 




27/32 


46.836914 


61.524414 


78.211914 


96.899414 


117.58691 


140.27441 


27/32 


55/64 




47.051025 


61.769775 


78.488525 


97.207275 


117.92603 


140.64478 


55/64 




7/8 


47.265625 


62.015625 


78.765625 


97.515625 


118.26563 


141.01563 


7/8 


57/64 




47.480713 


62.261963 


79.043213 


97.824463 


118.60571 


141.38696 


57/64 




29/32 


47.696289 


62.508789 


79.321289 


98.133789 


118.94629 


141.75879 


29/32 


59/64 




47.912354 


62.756104 


79.599854 


98.443604 


119.28735 


142.13110 


59/64 




15/16 


48.128906 


63.003906 


79.878906 


98.753906 


119.62891 


142.50391 


15/16 


61/64 




48.345947 


63.252197 


80.158447 


99.064697 


119.97095 


142.87720 


61/64 




31/32 


48.563477 


63.500977 


80.438477 


99.375977 


120.31348 


143.25098 


31/32 


63/64 




48.781494 


63.750244 


80.718994 


99.687744 


120.65649 


143.62524 


63/64 



6^ (=0' 6'')^SOUARES—30'' (=2' 6'0. 



645 



32a.— Squares of Inches, 12" to 30" (I'O" to 2' 6"), Advancing 

BY 32nds. 



In. 


12 


14 


16 


18 


20 


22 


24 


26 


28 


"i/32 

1/16 

3/32 

1/8 

5/32 

3/16 

7/32 

1/4 

9/32 

5/16 

11/32 

3/8 

13/32 

7/16 

15/32 

1/2 

17/32 

9/16 

19/32 

5/8 

21/32 

11/16 

23/32 

3/4 

25/32 

13/16 

27/32 

7/8 

29/32 

15/16 

31/32 


144.0000 
44.7510 
45.5039 
46.2588 

147.0156 
47.7744 
48.5352 
49.2979 

150.0625 
50.8291 
51.5977 
52.3682 

153.1406 
53.9150 
54.6914 
55.4697 

156.2500 
57.0322 
57.8164 
58.6025 

159.3906 
60.1807 
60.9727 
61.7666 

162.5625 
63.3604 
64.1602 
64.9619 

165.7656 
66.5713 
67.3789 
68.1885 


196.0000 
96.8760 
97.7539 
98.6338 

199.5156 

200.3994 
01.2852 
02.1729 

203.0625 
03.9541 
04.8477 
05.7432 

206.6406 
07.5400 
08.4414 
09.3447 

210.2500 
11.1572 
12.0664 
12.9775 

213.8906 
14.8057 
15.7227 
16.6416 

217.5625 
18.4854 
19.4102 
20.3369 

221.2656 
22.1963 

223.1289 
24.0635 


256.0000 
57.0010 
58.0039 
59.0088 

260.0156 
61.0244 
62.0352 
63.0479 

264.0625 
65.0791 
66 0977 
67.1182 

268.1406 
69.1650 
70.1914 
71.2197 

272.2500 
73.2822 
74.3164 
75.3525 

276.3906 
77.4307 
78.4727 
79.5166 

280.5625 
81.6104 
82.6602 
83.7119 

284.7656 
85.8213 
86.8789 
87.9385 


324.0000 
25.1260 
26.2539 
27.3838 

328.5156 
29.6494 
30.7852 
31.9229 

333.0625 
34.2041 
35.3477 
36.4932 

337.6406 
38.7900 
39.9414 
41.0947 

342.2500 
43.4072 
44.5664 
45.7275 

346.8906 
48.0557 
49.2227 
50.3916 

351.5625 
52.7354 
53.9102 
55.0869 

356.2656 
57.4463 
58.6289 
59.8135 


400.0000 
01.2510 
02.5039 
03.7588 

405.0156 
06.2744 
07.5352 
08.7979 

410.0625 
11.3291 
12.5977 
13.8682 

415.1400 
16.4150 
17.6914 
18.9697 

420.2500 
21.5322 
22 8164 
24.1025 

425.3906 
26.6807 
27.9727 
29.2666 

430.5625 
31.8604 
33.1602 
34.4619 

435.7656 
37.0713 
38.3789 
39.6885 


484.0000 
85.3760 
86.7539 
88.1338 

489.5156 
90.8994 
92.2852 
93.6729 

495.0625 
96.4541 
97.8477 
99.2432 

500.6406 
02.0400 
03.4414 
04.8447 

506.2500 
07.6572 
09.0664 
10.4775 

511.8906 
13.3057 
14.7227 
16.1416 

517.5625 
18.9854 
20.4102 
21.8369 

523.2656 
24.6963 
26.1289 
27.5635 


576.0000 
77.5100 
79.0039 
80.5088 

582.0156 
83.5244 
85.0352 
86.5479 

588.0625 
89.5791 
91.0977 
92.6182 

594.1406 
95.6650 
97.1914 
98.7197 

600.2500 
01.7822 
03.3164 
04.8525 

606.3906 
07.9307 
09.4727 
11.0166 

612.5625 
14.1104 
15.6602 
17.2119 

618.7656 
20.3213 
21.8789 
23.4385 


676.0000 
77.6260 
79.2539 
80.8838 

682.5156 
84.1494 
85.7852 
87.4229 

689.0625 
90.7041 
92.3477 
93.9932 

695.6406 
97.2900 
98.9414 

700.5947 

702.2500 
03.9072 
05.5664 
07.2275 

708.8906 
10.5557 
12.2227 
13.8916 

715.5625 
17.2354 
18.9102 
20.5869 

722.2656 
23.9463 
25.6289 
27.3135 


784.0000 
85.7510 
87.5039 
89.2588" 

791.0156 
92.7744 
94.5352 
96.2979 

798.0625 
99.8291 

801.5977 
03.3682 

805.1406 
06.9150 
08.6914 
10.4697 

812.2500 
14.0322 
15.8164 
17.6025 

819.3906 
21.1807 
22.9727 
24.7666 

826.5625 
28.3604 
30.1602 
31.9619 

833.7656 
35.5713 
37.3789 
39.1885 


In. 


13 


15 


17 


19 


21 


23 


25 


27 


29 


"i/32 

1/16 

3/32 

1/8 

5/32 

3/16 

7/32 

1/4 

9/32 

5/16 

11/32 

3/8 

13/32 

7/16 

15/32 

1/2 

17/32 

9/16 

19/32 

5/8 

21/32 

11/16 

23/32 

3/4 

25/32 

13/16 

27/32 

7/8 

29/32 

15/16 

31/32 


169.0000 
69.8135 
70.6289 
71.4463 

172.2656 
73.0869 
73.9102 
74.7354 

175.5625 
76.3916 
77.2227 
78.0557 

178.8906 
79.7275 
80.5664 
81.4072 

182.2500 
83.0947 
83.9414 
84.7900 

185.6406 
86.4932 
87.3477 
88.2041 

189.0625 
89.9229 
90.7852 
91.6494 

192.5156 
93.3838 
94.2539 
95.1260 


225.0000 
25.9385 
26.8789 
27.8213 

228.7656 
29.7119 
30.6602 
31.6104 

232.5625 
33.5166 
34.4727 
35.4307 

236.3906 
37.3525 
38.3164 
39.2822 

240.2500 
41.2197 
42.1914 
43.1650 

244.1406 
45.1182 
46.0977 
47.0791 

248.0625 
49.0479 
50.0352 
51.0244 

252.0156 
53.0088 
54.0039 
55.0010 


289.0000 
90.0635 
91.1289 
92.1963 

293.2656 
94.3369 
95.4102 
96.4854 

297.5625 
98.6416 
99.7227 

300.8057 

301.8906 
02.9775 
04.0664 
05.1572 

306.2500 
07.3447 
08.4414 
09.5400 

310.6406 
11.7432 
12.8477 
13 9541 

315.0625 
16.1729 
17.2852 
18.3994 

319.5156 
20.6338 
21.7539 
22.8760 


361.0000 
62.1885 
63.3789 
64.5713 

365.7656 
66.9619 
68.1602 
69.3604 

370.5625 
71.7666 
72.9727 
74.1807 

375.3906 
76.6025 
77.8164 
79.0322 

380.2500 
81.4697 
82.6914 
83.9150 

385.1406 
86.3682 
87.5977 
88.8291 

390.0625 
91.2979 
92.5352 
93.7744 

395.0156 
96.2588 
97.5039 
98.7510 


441.0000 
42.3135 
43.6289 
44.9463 

446.2656 
47.5869 
48.9102 
50.2354 

451.5625 
52.8916 
54.2227 
55.5557 

456.8906 
58.2275 
59.5664 
60.9072 

462.2500 
63.5947 
64.9414 
66.2900 

467.6406 
68.9932 
70.3477 
71.7041 

473.0625 
74 4229 
75.7852 
77.1494 

478.5156 
79.8838 
81.2539 
82.6260 


529.0000 
30.4385 
31.8789 
33.3213 

534.7656 
36.2119 
37.6602 
39.1104 

540.5625 
42.0166 
43.4727 
44.9307 

546.3906 
47.8525 
49.3164 
50.7822 

552.2500 
53.7197 
55.1914 
56.6650 

558.1406 
59.6182 
61.0977 
62.5791 

564. 0625 
65.5479 
67.0352 
68.5244 

570.0156 
71.5088 
73.0039 
74.5010 


625.0000 
26.5635 
28.1289 
29.6963 

631.2656 
32.8369 
34.4102 
35.9854 

637.5625 
39.1416 
40.7227 
42.3057 

643.8906 
45.4775 
47.0664 
48.6572 

650.2500 
51.8447 
53.4414 
55.0400 

656.6406 
58.2432 
59.8477 
61.4541 

663.0625 
64.6729 
66.2852 
67.8994 

669.5156 
71.1338 
72.7539 
74.3760 


729.0000 
30.6885 
32.3789 
34.0713 

735.7656 
37.4619 
39.1602 
40.8604 

742.5625 
44.2666 
45.9727 
47.6807 

749.3906 
51.1025 
52.8164 
54.5322 

756.2500 
57.9697 
59.6914 
61.4150 

763.1406 
64.8682 
66.5977 
68.3291 

770.0625 
71.7979 
73.5352 
75.2744 

777.0156 
78.7588 
80.5039 
82.2510 


841.0000 
42.8135 
44.6289 
46.4463 

848.2656 
50.0869 
51.9102 
53.7534 

855 5625 
57.3916 
59.2227 
61 0157 

862.8906 
64.7275 
66.5664 
68.4072 

870.2500 
72.0947 
73.9414 
75.7900 

877.6406 
79.4932 
81.3477 
83.2041 

885.0625 
86.9229 
88.7852 
90.6494 

892.5156 
94.3838 
96.2539 
98.1260 



646 



ZZ,^STRUCTURAL DETAILS. 



32a.— Squares of Inches, 30'' to 48'' (2' 6" to 4'0''), Advancing 
BY 32nds. — Continued. 



In. 


30 


32 


34 


36 


38 


40 


42 


44 


46 


"i/32 

1/16 

3/32 

1/8 

5/32 

3/16 

7/32 

1/4 

9/32 

5/16 

11/32 

3/8 

13/32 

7/16 

15/32 

1/2 

17/32 

9/16 

19/32 

5/8 

21/32 

11/16 

23/32 

3/4 

25/32 

13/16 

27/32 

7/8 

29/32 

15/16 

31/32 


900.000 
01.876 
03.754 
05.634 

907.516 
09.399 
11.285 
13.173 

915.063 
16.954 
18.848 
20.743 

922.641 
24.540 
26.441 
28.345 

930.250 
32.157 
34.066 
35.978 

937.891 
39.806 
41.723 
43.642 

945.563 
47.485 
49.410 
51.337 

953.266 
55.196 
57.129 
59.063 


1024.000 
026.001 
028.004 
130.009 

1032.016 
034.024 
036.035 
038.048 

1040.063 
042.079 
044.098 
046.118 

1048.141 
050.165 
052.191 
054.220 

1056.250 
058.282 
060.316 
062.353 

1064.391 
066.431 
068.473 
070.517 

1072.563 
074.610 
076.660 
078.712 

1080.766 
082.821 
084.879 
086.938 


1156.000 
158.126 
160.254 
162.384 

1164.516 
166.649 
.168.785 
170.923 

1173.063 
175.204 
177.348 
179.493 

1181.641 
183.790 
185.941 
188.095 

1190.250 
192.407 
194.566 
196.728 

1198.891 
201.056 
203.223 
205.392 

1207.563 
209.735 
211.910 
214.087 

1216.266 
218.446 
220.629 
222.813 


1296.000 
298.251 
300.504 
302.759 

1305.016 
307.274 
309.535 
311.798 

1314.063 
316.329 
318.598 
320.868 

1323.141 
325.415 
327.691 
329.970 

1332.250 
334.532 
336.816 
339.103 

1341.391 
343.681 
345.973 
348.267 

1350.563 
352.860 
355.160 
357.462 

1359.766 
362.071 
364.379 
366.688 


1444.000 
446.376 
448.754 
451.134 

1453.516 
455.899 
458.285 
460.673 

1463.063 
465.454 
467.848 
470.243 

1472.641 
475.040 
477.441 
479.845 

1482.250 
484.657 
487.066 
489.478 

1491.891 
494.306 
496.723 
499.142 

1501.563 
503.985 
506.410 
508.837 

1511.266 
513.696 
516.129 
518.563 


1600.000 
602.501 
605.004 
607.509 

1610.016 
612.524 
615.035 
617.548 

1620.063 
622.579 
625.098 
627.618 

1630.141 
632.665 
635.191 
637.720 

1640.250 
642.782 
645.316 
647.853 

1650.391 
652.931 
655.473 
658.017 

1660.563 
663.110 
665.660 
668.212 

1670.766 
673.321 
675.879 
678.438 


1764.000 
766.626 
769.254 
771.884 

1774.516 
777.149 
779.785 
782.423 

1785.063 
787.704 
790.348 
792.993 

1795.641 
798.290 
800.941 
803.595 

1806.250 
808.907 
811.566 
814.228 

1816.891 
819.556 
822.223 
824.892 

1827.563 
830.235 
832.910 
835.587 

1838.266 
840.946 
843.629 
846.313 


1936.000 
938.751 
941.504 
944.259 

1947.016 
949.774 
952.535 
955.298 

1958.063 
960.829 
963.598 
966.368 

1969.141 
971.915 
974.691 
977.470 

1980.250 
983.032 
985.816 
988.603 

1991.391 
994.181 
996.973 
999.767 

2002.563 
005.360 
008.160 
010.962 

2013.766 
016.571 
019.379 
022.188 


2116.000 
118.876 
121.754 
124.634 

2127.516 
130.399 
133.285 
136.173 

2139.063 
141.954 
144.848 
147.743 

2150.641 
153.540 
156.441 
159.345 

2162.250 
165.157 
168.066 
170.978 

2173.891 
176.806 
179.723 
182.641 

2185.563 
188.485 
191.410 
194.337 

2197.266 
200.196 
203.129 
206.063 


In. 


31 


33 


35 


37 


39 


41 


43 


45 ^ 


47 


**i/32 

1/16 

3/32 

1/8 

5/32 

3/16 

7/32 

1/4 

9/32 

5/16 

11/32 

3/8 

13/32 

7/16 

15/32 

1/2 

17/32 

9/16 

19/32 

5/8 

21/32 

11/16 

23/32 

3/4 

25/32 

13/16 

27/32 

7/8 

29/32 

15/16 

31/32 


961.000 

62.938 

64.879 

66.821 

968.766 

70.712 

72.660 

74.610 

976.563 

78.517 

80.473 

82.431 

984.391 

86.353 

88.316 

90.282 

992.250 

94.220 

96.191 

98.165 

1000.141 

002.118 

004.098 

006.079 

1008.063 

010.048 

012.035 

014.024 

1016.016 

018.009 

020.004 

022.001 


1089.000 
091.063 
093.129 
095.196 

1097.266 
099.337 
101.410 
103.485 

1105.563 
107.642 
109.723 
111.806 

1113.891 
115.978 
118.066 
120.157 

1122.250 
124.345 
126.441 
128.540 

1130.641 
132.743 
134.848 
136.954 

1139.063 
141.173 
143.285 
145.399 

1147.516 
149.634 
151.754 
153.876 


1225.000 
227.188 
229.379 
231.571 

1233.766 
235.962 
238.160 
240.360 

1242.563 
244.767 
246.972 
249.181 

1251.391 
253.603 
255.816 
258.032 

1260.250 
262.470 
264.691 
266.915 

1269.141 
271.368 
273.598 
275.829 

1278.063 
280.298 
282.535 
284.774 

1287.016 
289.259 
291.504 
293.751 


1369.000 
371.313 
373.629 
375.946 

1378.266 
380.587 
382.910 
385.235 

1387.563 
389.892 
392.223 
394.556 

1396.891 
399.228 
401.566 
403.907 

1406.250 
408.595 
410.941 
413.290 

1415.641 
417.993 
420.348 
422.704 

1425.063 
427.423 
429.785 
432.149 

1434.516 
436.884 
439.254 
441.626 


1521.000 
523.438 
525.879 
528.321 

1530.766 
533.212 
535.660 
538.110 

1540.563 
543.017 
545.473 
547.931 

1550.391 
552.853 
555.316 
557.782 

1560.250 
562.720 
565.191 
567.665 

1570.141 
572.618 
575.098 
577.579 

1580.063 
582.548 
585.035 
587.524 

1590.016 
592.509 
595.004 
597.501 


1681.000 
683.563 
686.129 
688.696 

1691.266 
693.837 
696.410 
698.985 

1701.563 
704.142 
706.723 
709.306 

1711.891 
714.478 
717.066 
719.657 

1722.250 
724.845 
727.441 
730.040 

1732.641 
735.243 
737.848 
740.454 

1743.063 
745.673 
748.285 
750.899 

1753.516 
756.134 
758.754 
761.376 


1849.000 
851.688 
854.379 
857.071 

1859.766 
862.462 
865.160 
867.860 

1870.563 
873.267 
875.9'73 
878.681 

1881.391 
884.103 
886.816 
889.532 

1892.250 
894.970 
897.691 
900.415 

1903.141 
905.868 
908.598 
911.329 

1914.063 
916.798 
919.535 
922.274 

1925.016 
927.759 
930.504 
933.251 


2025.000 
027.813 
030.629 
033.446 

2036.266 
039.087 
041.910 
044.735 

2047.563 
050.392 
053.223 
056.056 

2058.891 
061.728 
064.566 
067.407 

2070.250 
073.095 
075.941 
078.790 

2081.641 
084.493 
087.348 
090.204 

2093.063 
095.923 
098.785 
101.649 

2104.516 
107.384 
110.254 
113.126 


2209.000 
211.938 
214.879 
217.821 

2220.766 
223.712 
226.660 
229.610 

2232.563 
235.517 
238.473 
241.431 

2244.391 
247.353 
250.316 
253.282 

2256.250 
259.220 
262.191 
265.165 

2268.141 
271.118 
274.098 
277.079 

2280.063 
283.048 
286.035 
289.024 

2292.016 
295.009 
298.004 
301.001 



30" (=2' &')— SQUARES— 66'' (=5' 6"), 



647 



32a.— Squares of Inches, 48'' to 66" (4' 0" to 5' 6"), Advancing 
BY 32nds. — Continued. 



In. 


48 


50 


52 


54 


56 


58 


60 


62 


64 




2304. 00( 


2500. OOC 


2704. OOC 


2916.000 


3136.000 


3364.000 


3600. OOC 


3844. OOC 


4096.000 


"i/32 


307.001 


503.126 


707.251 


919.376 


139.501 


367.626 


603.751 


847.876 


100.001 


1/H 


310.00^ 


506.254 


710.504 


922.754 


143.004 


371.254 


607.504 


851.754 


104.004 


3/32 


313. OOc 


509.384 


713. 75£ 


926.134 


146.509 


374.884 


611.259 


855.634 


108.009 


1/8 


2316.016 


2512.516 


2717.016 


2929.516 


3150.016 


3378.516 


3615.016 


3859.516 


4112.016 


5/32 


319.02^ 


515.645 


720.274 


932.899 


153.524 


382.149 


618.774 


863.399 


116.024 


3/16 


322. 03E 


518.785 


723.532 


936.285 


157.035 


385.785 


622.535 


867.285 


120.035 


7/32 


325. 04J 


521.923 


726.798 


939.673 


160.548 


389.423 


626.298 


871.173 


124.048 


1/4 


2328. 06S 


2525.063 


2730.063 


2943.063 


3164.063 


3393.063 


3630.063 


3875.063 


4128.063 


9/32 


331. 07S 


528.204 


733.329 


946.454 


167.579 


396.704 


633.829 


878.954 


132.079 


5/16 


334.098 


531.348 


736.598 


949.848 


171.098 


400.348 


637.598 


882.848 


136.098 


11/32 


337.118 


534.493 


739.868 


953.243 


174.618 


403.993 


641.368 


886.743 


140.118 


3/8 


2340.141 


2537.641 


2743.141 


2956.641 


3178.141 


3407.641 


3645.141 


3890.641 


4144.141 


13/32 


343.156 


540.790 


746.415 


960.040 


181.665 


411.290 


648.915 


894.540 


148.165 


7/16 


346.191 


543.941 


749.691 


963.441 


185.191 


414.941 


652.691 


898.441 


152.191 


15/32 


349.220 


547.095 


752.970 


966.845 


188.720 


418.595 


656.470 


902.345 


156.220 


1/2 


2352.250 


2550.250 


2756.250 


2970.250 


3192.250 


3422.250 


3660.250 


3906.250 


4160.250 


17/32 


355.282 


553.407 


759.532 


973.657 


195.782 


425.907 


664.032 


910.157 


164.282 


9/16 


358.316 


556.566 


762.816 


977.066 


199.316 


429.566 


667.816 


914.066 


168.316 


19/32 


361.353 


559.728 


766.103 


980.478 


202.853 


433.228 


671.603 


917.978 


172.353 


5/8 


2364.391 


2562.891 


2769.391 


2983.891 


3206.391 


3436.891 


3675.391 


3921.891 


4176.391 


21/32 


367.431 


566.056 


772.681 


987.306 


209.931 


440.556 


679.181 


925.806 


180.431 


11/16 


370.473 


569.223 


775.973 


990.723 


213.473 


444.223 


682.973 


929.723 


184.473 


23/32 


373.517 


572.392 


779.267 


994.142 


217.017 


447.892 


686.767 


933.642 


188.517 


3/4 


2376.563 


2575.563 


2782.563 


2997.563 


3220.563 


3451.563 


3690.563 


3937.563 


4192.565 


25/32 


379.610 


578.735 


785.860 


3000.985 


224.110 


455.235 


694.360 


941.485 


196.610 


13/16 


382.660 


581.910 


789.160 


004.410 


227.660 


458.910 


698.160 


945.410 


200.660 


27/32 


385.712 


585.087 


792.462 


007.837 


231.212 


462.587 


701.962 


949.337 


204.712 


7/8 


2388.766 


2588.266 


2795.766 


3011.266 


3234.766 


3466.266 


3705.766 


3953.266 


4208.766 


29/32 


391.821 


591.446 


799.071 


014.696 


238.321 


469.946 


709.571 


957.196 


212.821 


15/16 


394.879 


594.629 


802.379 


018.129 


241.879 


473.629 


713.379 


961.129 


216.879 


31/32 


397.938 


597.813 


805.688 


021.563 


245.438 


477.313 


717.188 


965.063 


220.938 


In. 


49 


51 


53 


55 


57 


59 


61 


63 


65 




2401.000 


2601.000 


2809.000 


3025.000 


3249.000 


3481.000 


3721.000 


3969.000 


4225.000 


"i/32 


404.063 


604.188 


812.313 


028.438 


252.563 


484.688 


724.813 


972.938 


229.063 


1/16 


407.129 


607.379 


815.629 


031.879 


256.129 


488.379 


728.629 


976.879 


233.129 


3/32 


410.196 


610.571 


818.946 


035.321 


259.696 


492.071 


732.446 


980.821 


237.196 


1/8 


2413.266 


2613.766 


2822.266 


3038.766 


3263.266 


3495.766 


3736.266 


3984.766 


4241.266 


5/32 


416.337 


616.962 


825.587 


042.212 


266.837 


499.462 


740.087 


988.712 


245.337 


3/16 


419.410 


620.160 


828.910 


045.660 


270.410 


503.160 


743.910 


992.660 


249.410 


7/32 


422.485 


623.360 


832.235 


049.110 


273.985 


506.860 


747.735 


996.610 


253.485 


1/4 


2425.563 


2626.563 


2835.563 


3052.563 


3277.563 


3510.563 


3751.563 


4000.563 


4257.563 


9/32 


428.642 


629.767 


838.892 


056.017 


281.142 


514.267 


755.392 


004.517 


261.642 


5/16 


431.723 


632.973 


842.223 


059.473 


284.723 


517.973 


759.223 


008.473 


265.723 


11/32 


434.806 


636.181 


845.556 


062.931 


288.306 


521.681 


763.056 


012.431 


269.806 


3/8 


2437.891 


2639.391 


2848.891 


3066.391 


3291.891 


3525.391 


3766.891 


4016.391 


4273.891 


13/32 


440.978 


642.603 


852.228 


069.853 


295.478 


529.103 


770.728 


020.353 


277.978 


7/16 


444.066 


645.816 


855.566 


073.316 


299.066 


532.816 


774.566 


024.316 


282.066 


15/32 


447.157 


649.032 


858.907 


076.782 


302.657 


536.532 


778.407 


028.282 


286.157 


1/2 


2450.250 


2652.250 


2862.250 


3080.250 


3306.250 


3540.250 


3782.250 


4032.250 


4290.250 


17/32 


453.345 


655.470 


865.595 


083.720 


309.845 


543.970 


786.095 


036.220 


294.345 


9/16 


456.441 


658.691 


868.941 


087.191 


313.441 


547.691 


789.941 


040.191 


298.441 


19/32 


459.540 


661.915 


872.290 


090.665 


317.040 


551.415 


793.790 


044.165 


302.540 


5/8 


2462.641 


2665.141 


2875.641 


3094.141 


3320.641 


3555.141 


3797.641 


4048.141 


4306.641 


21/32 


465.743 


668.368 


878.993 


097.618 


324.243 


558.868 


801.493 


052.118 


310.742 


11/16 


468.848 


671.598 


882.348 


101.098 


327.848 


562.598 


805.348 


056.098 


314.848 


23/32 


471.954 


674.829 


885.704 


104.579 


331.454 


566.329 


809.204 


060.079 


318.954 


3/4 


2475.063 


2678.063 


2889.063 


3108.063 


3335.063 


3570.063 


3813.063 


4064.063 


4323.063 


25/32 


478.173 


681.298 


892.423 


111.548 


338.673 


573.798 


816.923 


068.048 


327.173 


13/16 


481.285 


684.535 


895.785 


115.035 


342.285 


577.535 


820.785 


072.035 


331.285 


27/32 


484.399 


687.774 


899.149 


118.524 


345.899 


581.274 


824.649 


076.024 


335.399 


7/8 


2487.516 


2691.016 


2902.516 


3122.016 


3349.516 


3585.016 


3828.516 


4080.016 


4339.516 


29/32 


490.634 


694.259 


905.884 


125.509 


353.134 


588.759 


832.384 


084.009 


343.634 


15/16 


493.754 


697.504 


909.254 


129.004 


356.754 


592.504 


836.254 


088.004 


347.754 


31/32 


496.876 


700.751 


912.626 


132.501 


360.376 


596.251 


840 126 


092.001 


351.876 



648 



SS.— STRUCTURAL DETAILS. 



32a.— Squares of Inches, 66" to 84'' (5' 6" to 7' O'O, Advancing 
BY 32nds. — Continued. 



In. 


66 


68 


70 


72 


74 


76 


78 


80 


82 


"i/32 

1/16 

3/32 

1/8 

5/32 

3/16 

7/32 

1/4 

9/32 

5/16 

11/32 

3/8 

13/32 

7/16 

15/32 

1/2 

17/32 

9/16 

19/32 

5/8 

21/32 

11/16 

23/32 

3/4 

25/32 

13/16 

27/32 

7/8 

29/32 

15/16 

31/32 


4356.000 
360.126 
364.254 
368.384 

4372.516 
376.649 
380.785 
384.923 

4389.063 
393.204 
397.348 
401.493 

4405.641 
409.790 
413.941 
418.095 

4422.250 
426.407 
430.566 
434.728 

4438.891 
443.056 
447.223 
451.392 

4455.563 
459.735 
463.910 
468.087 

4472.266 
476.446 
480.629 
484.813 


4624.000 
628.251 
632.504 
636.759 

4641.016 
645 274 
649.535 
653.798 

4658.063 
662.329 
666.598 
670.868 

4675.141 
679.415 
683.691 
687.970 

4692.250 
696.532 
700.816 
705.103 

4709.391 
713.681 
717.973 
722.267 

4726.563 
730.860 
735.160 
739.462 

4743.766 
748.071 
752.379 
756.688 


4900.000 
904.376 
9T)8.754 
913.134 

4917.516 
921.899 
926.285 
930.673 

4935.063 
939.454 
943.848 
948.243 

4952.641 
957.040 
961.441 
965.845 

4970.250 
974.657 
979.066 
983.478 

4987.891 
992.306 
996.723 

5001.142 

5005.563 
009.985 
014.410 
018.837 

5023.266 
027.696 
032.129 
036.563 


5184.000 
188.501 
193.004 
197.509 

5202.016 
206.524 
211.035 
215.548 

5220.063 
224.579 
229.098 
233.618 

5238.141 
242.665 
247.191 
251.720 

5256.250 
260.782 
265.316 
269.853 

5274.391 
278.931 
283.473 
288.017 

5292.563 
297.110 
301.660 
306.212 

5310.766 
315.321 
319.879 
324.438 


5476.000 
480.626 
485.254 
489.884 

5494.516 
499.149 
503.785 
508.423 

5513.063 
517.704 
522.348 
526.993 

5531.641 
536.290 
540.941 
545.595 

5550.250 
554.907 
559.566 
564.228 

5568.891 
573.556 
578.223 
582.892 

5587.563 
592.235 
596.910 
601.587 

5606.266 
610.946 
615.629 
620.313 


5776.000 
780.751 
785.504 
790.259 

5795.016 
799 774 
804.535 
809.298 

5814.063 
818.829 
823.598 
828.368 

5833.141 
837.915 
842.691 
847.470 

5852.250 
857.032 
861.816 
866.603 

5871.391 
876.181 
880.973 
885.767 

5890.563 
895.360 
900.160 
904.962 

5909.766 
914.571 
919.379 
924.188 


6084.000 
088.876 
093.754 
098.634 

6103.516 
108.399 
113.285 
118.173 

6123.063 
127.954 
132.-848 
137.743 

6142.641 
147.540 
152.441 
157.345 

6162.250 
167.157 
172.066 
176.978 

6181.891 
186.806 
191.723 
196.642 

6201.563 
206.485 
211.410 
216.337 

6221.266 
226.196 
231.129 
236.063 


6400.000 
405.001 
410.004 
415.009 

6420.016 
425.024 
430.035 
435.048 

6440.063 
445.079 
450.098 
455.118 

6460.141 
465.165 
470.191 
475.220 

6480.250 
485.282 
490.316 
495.353 

6500.391 
505.431 
510.473 
515.517 

6520.563 
525.610 
530.660 
535.712 

6540.766 
545.821 
550.879 
555.938 


6724.000 
729.126 
734.254 
739.384 

6744.516 
749.649 
754.785 
759.923 

6765.063 
770.204 
775.348 
780.493 

6785.641 
790.790 
795.941 
801.095 

6806.250 
811.407 
816.566 
821.728 

6826.891 
832.056 
837.223 
842.392 

6847.563 
852.735 
857.910 
863.087 

6868.266 
873.446 
878.629 
883.813 


In. 


67 


69 


71 


73 


75 


77 


79 


81 


83 


"i/32 

1/16 

3/32 

1/8 

5/32 

3/16 

7/32 

1/4 

9/32 

5/16 

11/32 

3/8 

13/32 

7/16 

15/32 

1/2 

17/32 

9/16 

19/32 

5/8 

21/32 

11/16 

23/32 

3/4 

25/32 

13/16 

27/32 

7/8 

29/32 

15/16 

31/32 


4489.000 
493.188 
497.379 
501.571 

4505.766 
509.962 
514.160 
518.360 

4522.563 
526.767 
530.973 
535.181 

4539.391 
543.603 
547.816 
552.032 

4556.250 
560.470 
564.691 
668.915 

4573.141 
577.368 
581.598 
585.829 

4590.063 
594.298 
598.535 
602.774 

4607.016 
611.259 
615.504 
619.751 


4761.000 
765.313 
769.629 
773.946 

4778.266 
782.587 
786.910 
791.235 

4795.563 
799.892 
804.223 
808.556 

4812.891 
817.228 
821.566 
825.907 

4830.250 
834.595 
838.941 
843.290 

4847.641 
851.993 
856.348 
860.704 

4865.063 
869.423 
873.785 
878.149 

4882.516 
886.884 
891.254 
895.626 


5041.000 
045.438 
049.879 
054.321 

5058.766 
063.212 
067.660 
072.110 

5076.563 
081.017 
085.473 
089.931 

5094.391 
098.853 
103.316 
107.782 

5112.250 
116.720 
121.191 
125.665 

5130.141 
134.618 
139.098 
143.579 

5148.063 
152.548 
157.035 
161.524 

5166.016 
170.509 
175.004 

179.501 

■ 


5329.000 
333.563 
338.129 
342.696 

5347.266 
351.837 
356.410 
360.985 

5365.563 
370.142 
374.723 
379.306 

5383.891 
388.478 
393.066 
397.657 

5402.250 
406.845 
411.441 
416.040 

5420.641 
425.243 
429.848 
434.354 

5439.063 
443.673 
448.285 
452.899 

5457.516 
462.134 
466.754 
471.376 


5625.000 
629.688 
634.379 
639.071 

5643.766 
648.462 
653.160 
657.860 

5662.563 
667.267 
671.973 
676.681 

5681.391 
686.103 
690.816 
695.532 

5700.250 
704.970 
709.691 
714.415 

5719.141 
723.868 
728.598 
733.329 

5738.063 
742.798 
747.535 
752.274 

5757.016 
761.759 
766.504 
771.251 


5929.000 
933.813 
938.629 
943.446 

5948.266 
953.087 
957.910 
962.735 

5967.563 
972.392 
977.223 
982.056 

5986.891 
991.728 
996.566 

6001.407 

6006.250 
011.095 
015.941 
020.790 

6025.641 
030.493 
035.348 
040.204 

6045.063 
049.923 
054.785 
059.649 

6064.516 
069.384 
074.254 
079.126 


6241.000 
245.938 
250.879 
255.821 

6260.766 
265.712 
270.660 
275.610 

6280.563 
285.517 
290.473 
295.431 

6300.391 
305.353 
310.316 
315.282 

6320.250 
325.220 
330.191 
335.165 

6340.141 
345.118 
350.098 
355.079 

6360.063 
365.048 
370.035 
375.024 

6380.016 
385.009 
390.004 
395.001 


6561.000 
566.063 
571.129 
576.196 

6581.266 
586.337 
591.410 
596.485 

6601.563 
606.642 
611.723 
616.806 

6621.891 
626.978 
632.066 
637.157 

6642.250 
647.345 
652.441 
657.540 

6662.641 
667.743 
672.848 
677.954 

6683.063 
688.173 
693.285 
698.399 

6703.516 
708.634 
713.754 
718.876 


6889.000 
894.188 
899.379 
904.571 

6909.766 
914.962 
920.160 
925.360 

6930.563 
935.767 
940.973 
946.181 

6951.391 
956.603 
961.816 
967.032 

6972.250 
977.470 
982.691 
987.915 

6993.141 
998.368 

7003.598 
008.829 
014.063 
019.298 
024.535 
029.774 

7035.016 
040.259 
045.504 
050.751 



66" (=5' 6n—SQU ARES— 102" (=8' 6"). 



649 



32a.— Squares of Inches, 84" to 102" (7' 0" to 8' 6"), Advancing 

BY 32NDS. — Continued. 



In. 


84 


86 


88 


90 


92 


94 


96 


98 


100 




7056.000 


7396. OOC 


7744. OOC 


8100. OOC 


8464.000 


8836. OOC 


9216. OOC 


9604. OOC 


10000.00 


"i/32 


061.251 


401.376 


749.501 


105. 62C 


469.751 


841.876 
847.754 


222.001 


610.126 


0006.25 


1/16 


066.504 


406.754 


755.004 


111.254 


475.504 


228.004 


616.254 


0012.50 


3/32 


071. 75S 


412.134 


760. 50S 


116.884 


481.259 


853.634 


234.009 


622.384 


0018.76 


1/8 


7077.016 


7417.516 


7766.016 


8122.516 


8487.016 


8859.516 


9240.016 


9628.516 


10025.02 


5/32 


082.274 


422.899 


771.524 


128. 14S 


492.774 


865. 39S 


246.024 


634.649 


0031.27 


3/16 


087.535 


428.285 


777.035 


133.785 


498.535 


871.285 


252.035 


640.785 


0037.54 


7/32 


092.798 


433.673 


782.548 


139.423 


504.298 


877.173 


258.048 


646.923 


0043.80 


1/4 


7098.063 


7439.063 


7788.063 


8145.063 


8510.063 


8883.063 


9264.063 


9653.063 


10050.06 


9/32 


103.329 


444.454 


793.579 


150.704 


515.829 


888 954 


270.079 


659.204 


0056.33 


5/16 


108.598 


449.848 


799.098 


156.348 


521.598 


894.848 


276.098 


665.348 


0062.60 


11/32 


113.868 


455.243 


804.618 


161.993 


527.368 


900.743 


282.118 


671.493 


0068.87 


3/8 


7119.141 


7460.641 


7810.141 


8167.641 


8533.141 


8906.641 


9288.141 


9677.641 


10075.14 


13/32 


124.415 


466.040 


815.665 


173.290 


538.915 


912.540 


294.165 


683.790 


0081.42 


7/16 


129.691 


471.441 


821.191 


178.941 


544.691 


918.441 


300.191 


689.941 


0087.69 


15/32 


134.970 


476.845 


826.720 


184.595 


550.470 


924.345 


306.220 


696.095 


0093.97 


1/2 


7140.250 


7482.250 


7832.250 


8190.250 


8556.250 


8930.250 


9312.250 


9702.250 


10100.25 


17/32 


145.532 


487.657 


837.782 


195.907 


562.032 


936.157 


318.282 


708.407 


0106.53 


9/16 


150.816 


493.066 


843.316 


201.566 


567.816 


942.066 


324.316 


714.566 


0112.82 


19/32 


156.103 


498.478 


848.853 


207.228 


573.603 


947.978 


330.353 


720.728 


0119.10 


5/8 


7161.391 


7503.891 


7854.391 


8212.891 


8579.391 


8953.891 


9336.391 


9726.891 


10125.39 


21/32 


166.681 


509.306 


859.931 


218.556 


585.181 


959.806 


342.431 


733.056 


0131.68 


11/16 


171.973 


514.723 


865.473 


224.223 


590.973 


965.723 


348.473 


739.223 


0137.97 


23/32 


177.267 


520.142 


871.017 


229.892 


596.767 


971.642 


354.517 


745.392 


0144.27 


3/4 


7182.563 


7525.563 


7876.563 


8235.563 


8602.563 


8977.563 


9360.563 


9751.563 


10150.56 


25/32 


187.860 


530.985 


882.110 


241.235 


608.360 


983.485 


366.610 


757.735 


0156.86 


13/16 


193.160 


536.410 


887.660 


246.910 


614.160 


989.410 


372.660 


763.910 


0163.16 


27/32 


198.462 


541.837 


893.212 


252.587 


619.962 


995.337 


378.712 


770.087 


0169.46 


7/8 


7203.766 


7547.266 


7898.766 


8258.266 


8625.766 


9001.266 


9384.766 


9776.266 


10175.77 


29/32 


209.071 


552.696 


904.321 


263.946 


631.571 


007.196 


390.821 


782.446 


0182.07 


15/16 


214.379 


558.129 


909.879 


269.629 


637.379 


013.129 


396.879 


788.629 


0188.38 


31/32 


219.688 


563.563 


915.438 


275.313 


643.188 


019.063 


402.938 


794.813 


0194.69 


In. 


85 


87 


89 


91 


93 


95 


97 


99 


101 




7225.000 


7569.000 


7921.000 


8281.000 


8645.000 


9025.000 


9409.000 


9801.000 


10201.00 


**i/32 


230.313 


574.438 


926.563 


286.688 


654.813 


030.938 


415.063 


807.188 


0207.31 


1/16 


235.629 


579.879 


932.129 


292.379 


660.629 


036.879 


421.129 


813.379 


0213.63 


3/32 


240.946 


585.321 


937.696 


298.071 


666.446 


042.821 


427.196 


819.571 


0219.95 


3/8 


7246.266 


7590.766 


7943.266 


8303.766 


8672.266 


9048.766 


9433.266 


9825.766 


10226.27 


5/32 


251.587 


596.212 


948.837 


309.462 


678.087 


054.712 


439.337 


831.962 


0232.59 


3/16 


256.910 


601.660 


954.410 


315.160 


683.910 


060.660 


445.410 


838.160 


0238.91 


7/32 


262.235 


607.110 


959.985 


320.860 


689.735 


066.610 


451.485 


844.360 


0245.24 


1/4 


7267.563 


7612.563 


7965.563 


8326.563 


8695.563 


9072.563 


9457.563 


9850.563 


10251.56 


9/32 


272.892 


618.017 


971.142 


332.267 


701.392 


078.517 


463.642 


856.767 


0257.89 


5/16 


278.223 


623.473 


976.723 


337.973 


707.223 


084.473 


469.723 


862.973 


0264.22 


11/32 


283.556 


628.931 


982.306 


343.681 


713.056 


090.431 


475.806 


869.181 


0270.56 


3/8 


7288.891 


7634.391 


7987.891 


8349.391 


8718.891 


9096.391 


9481.891 


9875.391 


10276.89 


13/32 


294.228 


639.853 


993.478 


355.103 


724.728 


102.353 


487.978 


881.603 


0283.23 


7/16 


299.566 


645.316 


999.066 


360.816 


730.566 


108.316 


494.066 


887.816 


0289.57 


15/32 


304.907 


650.782 


8004.657 


366.532 


736.407 


114.282 


500.157 


894.032 


0295.91 


1/2 


7310.250 


7656.250 


8010.250 


8372.250 


8742.250 


9120.250 


9506.250 


9900.250 


10302.25 


17/32 


315.595 


661.720 


015.845 


377.970 


748.095 


126.220 


512.345 


906.470 


0308.59 


9/16 


320.941 


667.191 


021.441 


383.691 


753.941 


132.191 


518.441 


912.691 


0314.94 


19/32 


326.290 


672.665 


027.040 


389.415 


759.790 


138.165 


524.540 


918.915 


0321.29 


5/8 


7331.641 


7678.141 


8032.641 


8395.141 


8765.641 


9144.141 


9530.641 


9925.141 


f0327.64 


21/32 


336.993 


683.618 


038.243 


400.868 


771.493 


150.118 


536.743 


931.368 


0333.99 


11/16 


342.348 


689.098 


043.848 


.406.598 


777.348 


156.098 


542.848 


937.598 


0340.35 


23/32 


347.704 


694.579 


049.454 


412.329 


783.204 


162.079 


548.954 


943.829 


0346.70 


3/4 


7353.063 


7700.063 


8055.063 


8418.063 


8789.063 


9168.063 


9555.063 


9950.063 


10353.06 


25/32 


358.423 


705.548 


060.673 


423.798 


794.923 


174.048 


561.173 


956.298 


0359.42 


13/16 


363.785 


711.035 


066.285 


429.535 


800.785 


180.035 


567.285 


962.535 


0365.79 


27/32 


369.149 


716.524 


071.899 


435.274 


806.649 


186.024 


573.399 


968.774 


0372.15 


7/8 


7374.516 


7722.016 


8077.516 


8441.016 


8812.516 


9192.016 


9579.516 


9975.016 


10378.52 


29/32 


379.884 


727.509 


083.134 


446.759 


818.384 


198.009 


585.634 


981.259 


0384.88 


15/16 


385.254 


733.004 


088.754 


452.504 


824.254 


204.004 


591.754 


987.504 


0391.25 


31/32 


390.626 


738.501 


094.376 


458.251 


830.126 


210.001 


597.876 


993.751 


0397.63 



660 



\.— STRUCTURAL DETAILS, 



32a. — Squares of Inches, 102" to 120" (8' 6" to 10' 0")» Advancing 
BY 32nds. — Concluded. 



In. 


102 


104 


106 


108 


110 


112 


114 


116 


118 




10404.00 


10816. OC 


11236.00 


11664.00 


12100.00 


12544. OC 


12996.00 


13456.00 


13924.00 


"i/32 


0410.38 


0822. 5C 


;242.63 


1670.75 


2106.88 


2551. OC 


3003.13 


3463.25 


3931.38 


1/16 


0416.75 


0829. OC 


1249.25 


1677.50 


2113.75 


2558. OC 


3010.25 


3470.50 


3938.75 


3/32 


0423.13 


0835.51 


1255.88 


1684.26 


2120.63 


2565.01 


3017.38 


3477.76 


3946.13 


1/8 


10429.52 


10842.02 


11262.52 


11691.02 


12127.52 


12572.02 


13024.52 


13485.02 


13953.52 


5/32 


0435.90 


0848.52 


1269.15 


1697.77 


2134.40 


2579.02 


3031.65 


3492.27 


3960.90 


3/16 


0442.29 


0855.04 


1275.79 


1704.54 


2141.29 


2586.04 


3038.79 


3499.54 


3968.29 


7/32 


0448.67 


0861.55 


1282.42 


1711.30 


2148.17 


2593.05 


3045.92 


3506.80 


3975.67 


1/4 


10455.06 


10868.06 


11289.06 


11718.06 


12155.06 


12600.06 


13053.06 


13514.06 


13983.06 


9/32 


0461.45 


0874.58 


1295.70 


1724.83 


2161.95 


2607.08 


3060.20 


3521.33 


3990.45 


5/16 


0467.85 


0881.10 


1302.35 


1731.60 


2168.85 


2614.10 


3067.35 


3528.60 


3997.85 


11/32 


0474.24 


0887.62 


1308.99 


1738.37 


2175.74 


2621.12 


3074.49 


3535.87 


4005.24 


3/8 


10480.64 


10894.14 


11315.64 


11745.14 


12182.64 


12628.14 


13081.64 


13543.14 


14012.64 


13/32 


0487.04 


0900.67 


1322.29 


1751.92 


2189.54 


2635.17 


3088.79 


3550.42 


4020.04 


7/16 


0493.44 


0907.19 


1328.94 


1758.69 


2196.44 


2642.19 


3095.94 


3557.69 


4027.44 


15/32 


0499.84 


0913.72 


1335.59 


1765.47 


2203.34 


2649.22 


3103.09 


3564.97 


4034.84 


1/2 


10506.25 


10920.25 


11342.25 


11772.25 


12210.25 


12656.25 


13110.25 


13572.25 


14042.25 


17/32 


0512.66 


0926.78 


1348.91 


1779.03 


2217.16 


2663.28 


3117.41 


3579.53 


4049.66 


9/16 


0519.07 


0933.32 


1355.57 


1785.82 


2224.07 


2670,32 


3124.57 


3586.82 


4057.07 


19/32 


0525.48 


0939.85 


1362.23 


1792.60 


2230.98 


2677.35 


3131.73 


3594.10 


4064.48 


5/8 


10531.89 


10946.39 


11368.89 


11799.39 


12237.89 


12684.39 


13138.89 


13601.39 


14071.89 


21/32 


0538.31 


0952.93 


1375.56 


1806.18 


2244.81 


2691.43 


3146.06 


3608.68 


4079.31 


11/16 


0544.72 


0959.47 


1382.22 


1812.97 


2251.72 


2698.47 


3153.22 


3615.97 


4086.72 


23/32 


0551.14 


0966.02 


1388.89 


1819.77 


2258.64 


2705.52 


3160.39 


3623.27 


4094.14 


3/4 


10557.56 


10972.56 


11395.56 


11826.56 


12265.56 


12712.56 


13167.56 


13630.56 


14101.56 


25/32 


0563.99 


0979.11 


1402.24 


1833.36 


2272.49 


2719.61 


3174.74 


3637.86 


4108.99 


13/16 


0570.41 


0985.66 


1048.91 


1840.16 


2279.41 


2726.66 


3181.91 


3645.16 


4116.41 


27/32 


0576.84 


0992.21 


1415.59 


1846.96 


2286.34 


2733.71 


3189.09 


3652.46 


4123.84 


7/8 


10583.27 


10998.77 


11422.27 


11853.77 


12293.27 


12740.77 


13196.27 


13659.77 


14131.27 


29/32 


0589.70 


1005.32 


1428.95 


1860.57 


2300.20 


2747.82 


3203.45 


3667.07 


4138.70 


15/16 


0596.13 


1011.88 


1435.63 


1867.38 


2307.13 


2754.88 


3210.63 


3674.38 


4146.13 


31/32 


0602.56 


1018.44 


1442.31 


1874.19 


2314.06 


2761.94 


3217.81 


3681.69 


4153.56 


In. 


103 


105 


107 


109 


111 


113 


115 


117 


119 




10609.00 


11025.00 


11449.00 


11881.00 


12321.00 


12769.00 


13225.00 


13689.00 


14161.00 


*"i/32 


0615.44 


1031.56 


1455.69 


1887.81 


2327.94 


2776.06 


3232.19 


3696.31 


4168.44 


1/16 


0621.88 


1038.13 


1462.38 


1894.63 


2334.88 


2783.13 


3239.38 


3703.63 


4175.88 


3/32 


0628.32 


1044.70 


1469.07 


1901.45 


2341.82 


2790.20 


3246.57 


3710.95 


4183.32 


1/8 


10634.77 


11051.27 


11475.77 


11908.27 


12348.77 


12797.27 


13253.77 


13718.27 


14190.77 


5/32 


0641.21 


1057.84 


1482.46 


1915.09 


2355.71 


2804.34 


3260.96 


3725.59 


4198.21 


3/16 


0647.66 


1064.41 


1489.16 


1921.91 


2362.66 


2811.41 


3268.16 


3732.91 


4205.66 


7/32 


0654.11 


1070.99 


1495.86 


1928.74 


2369.61 


2818.49 


3275.36 


3740.24 


4213.11 


1/4 


10660.56 


11077.56 


11502.56 


11935.56 


12376.56 


12825.56 


13282.56 


13747.56 


14220.56 


9/32 


0667.02 


1084.14 


1509.27 


1942.39 


2383.52 


2832.64 


3289.77 


3754.89 


4228.02 


5/16 


0673.47 


1090.72 


1515.97 


1949.22 


2390.47 


2839.72 


3296.97 


3762.22 


4235.47 


11/32 


0679.93 


1097.31 


1522.68 


1956.06 


2397.43 


2846.81 


3304.18 


3769.56 


4242.93 


3/8 


10686.39 


11103.89 


11529.39 


11962.89 


12404.39 


12853.89 


13311.39 


13776.89 


14250.39 


13/32 


0692.85 


1110.48 


1536.10 


1969.73 


2411 35 


2860.98 


3318.60 


3784.23 


4257.85 


7/16 


0699.32 


1117.07 


1542.82 


1976.57 


2418.32 


2868.07 


3325.82 


3791.57 


4265.32 


15/32 


0705.78 


1123 66 


1549.53 


1983.41 


2425.28 


2875.16 


3333.03 


3798.91 


4272.78 


1/2 


10712.25 


11130.25 


11556.25 


11990.25 


12432.25 


12882.25 


13340.25 


13806.25 


14280.25 


17/32 


0718.72 


1136.84 


1562.97 


1997.09 


2439.22 


2889.34 


3347.47 


3813.59 


4287.72 


9/16 


0725.19 


1143.44 


1569.69 


2003.94 


2446.19 


2896.44 


3354.69 


3820.94 


4295.19 


19/32 


,0731.67 


1150.04 


1576.42 


2010.79 


2453.17 


2903.54 


3361.92 


3828.29 


4302.67 


5/8 


10738.14 


11156.64 


11583.14 


12017.64 


12460.14 


12910.64 


13369.14 


13835.64 


14310.14 


21/32 


0744.62 


1163.24 


1589.87 


2024.49 


2467.12 


2917.74 


3376.37 


3842.99 


4317.62 


11/16 


0751.10 


1169.85 


1596.60 


2031.35 


2474.10 


2924.85 


3383.60 


3850.35 


4325.10 


23/32 


0757.58 


1176.45 


1603.33 


2038.20 


2481.08 


2931.95 


3390.83 


3857.70 


4332.58 


3/4 


10764.06 


11183.06 


11610.06 


12045.06 


12488.06 


12939.06 


13398.06 


13865.06 


14340.06 


25/32 


0770.55 


1189.67 


1616.80 


2051.92 


2495.05 


2946.17 


3405.30 


3872.42 


4347.55 


13/16 


0777.04 


1196.29 


1623.54 


2058.79 


2502.04 


2953.29 


3412.54 


3879.79 


4355.04 


27/32 


0783.52 


1202.90 


1630.27 


2065.65 


2509.02 


2960.40 


3419.77 


3887.15 


4362.52 


7/8 


10790.02 


11209.52 


11637.02 


12072.52 


12516.02 


12967.52 


13427.02 


13894.52 


14370.02 


29/32 


0796.51 


1216.13 


1643.76 


2079.38 


2523.01 


2974.63 


3434.26 


3901.88 


4377.51 


15/16 


0803.00 


1222.75 


1650.50 


2086.25 


2530.00 


2981.76 


3441.50 


3909.25 4385.00 


31/32 


0809.50 


1229.38 


1657.25 


2093.13 


2537.00 


2988.88 


3448.75 


3916.63 


4392. CO 



102'^ (=8' 6n—S0UARBS—iS6'' (=13' 00* 



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c^t^t^ooooajoo>C30'-<T-(^e^cqoooo'<i«-.s<'<4«ioio«3'X)c^t^t^ooooo505ooT-<T-ii-i 
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(M^C30S0000l>-t^C^t>-00000SO^C0iOcO00— <eOCOOSCN31000(MCOC3-^OOCOOOCOOO-* 

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«>-t>.t^ooooosos05(oO'-<^— ^eQc>aooeooo-<*<'*ioiococo<oc^t^ooooosos<ooo^'-< 
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•*oooo3i-<io-.-<ooe^qcooo^»-<eooo-^ev](MiOCT>ioeo"*coocotoiot^'-HOOcocooooo 

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t^c^t— ooooosoiosoo^^'-HT-ipciesiooooeO'^'fioiocococot-.t^ooooososcsoO''— I'-i 
cococoeo'^-«i<-<*<-rf«-»t*-«i<-^-^-<*<'*-^ ■>*' to lO to 



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C)-*oocvicoo'^ooeNjcoc>'^ooc<3co<0'>*'05eot~-"^coo'^oocoi>-evjcoomos-^oocot^ 
t-c^t^ooooo50s0500'-ii-('-Hcgevacococo-^'^ioiococococ^t^ooooososo50o^»-i 

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c>'^ooevico<o-»*<oo<>acoo-^ooe>3coc5-^oocot^'— iioo'^ooooc^'— 'cooioos-^oocot- 
t^t^t^QOOoosososcDC)'-!-^'— 'e^esicococo-«t<'*»iOiOcococot^t~-ooooo3050scr)C»-^i-< 



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oscot^T-^inoicot^^Hioosoot^^HiooscooocsacocDiooicot^esicocsmos-rtioocot^cqco 
coc^t^oooooooso5c>oo^'— e<iM!Mcoeo'^-^ioioiococot^t^ooooooo30sioO'-<'-t 

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i-iCMeMCMcooo'«*<-<i<Tt<iniococot~t^ooooooososcDO— <i-i 



oococsosocoooinincoos-^Y^o^-^cDt^cot^cDinc^T-icoco^oot— oo^-<co-*co"«^i>. 
oot^co"^'*cocMcaeMc^jeMCO'*'iocoi>-os<D>eM-^t^oseMinoo^Htnooc<jcoT-(tnc>inoif5 
ooc<ico<:D'*'oocMcoo->*<ooc^acoc=)-^oocMt>-^inoscooocMcoT-itnos-*<oocot^cMco^io 
cot~tr^ooooooosos<r>ocDi-HT-(eM'— ieMoocO'^'^T><iniococot^r-~t^ooooosos<DO»-<i-t 

COCOCOCO-^-^TtiT>'-J'Tf'»rt<T^-^-^r»<totO 



itioscotocoosTfi— io^H->iHOs«oincoos-*ii— icD'^H->ij<cscoincoos->^i— (Oi— '■^oscoiocoos 
co'^cocM-^cDOCsocDcso— <cMco'*icoooo>cM-<*'cooscMioooeMcocs->*'ooeMt^eMt^eM 
cxJCMcoo-^oocMcoco-^cocMcocD-^oocMcoi— (incyscot^cMcooinos'tt<ooeMr~'— ico^io 
coc^t>-QOooooosc7scDOC>T-(-.--c>acMeMcooo-^'^-^iniocococ^t^t^oooooso5<oo»H»-i 
cocoooeo'^-<i<^">a<-«i<-<*<'<i<M'Tj<Tt<-^j<rt<inm 



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^^^^^^^,_icMCMCMCMC>ac<icMeMCMc<ieMCMCMc<jeMeMeMeMeMeMCMeMeMc^cvaeMeMeM 

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«o t^ 00 



654 



ZS.—STRUCTURAL DETAILS, 



+ 



■^■^•^•^•^•^■^■^•^•^•^•^iaia\air>ia\ci^aic>iainxc>iaiiaiciiaic>e£>(0(Ofoeo<£>(£>to 



ia\fS 



»-it>.ooOiir:ie>aov?ocOi— lO>^-)oco»-c<r>0500^^t>.<x><£>^ot^^-.ooo30^HC»5•>t^'tc>oo■^-^eo«o 
lo m lo in in in in in laxo'^ to co co co to to to to 



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00-«*<(0':OC005tOCT3C>OOtO-«J'C>qC>OOt^tOin->*->^COCCOO-^'*intOt~OOOi^COint^C>Cr3 

cooocot^pQto^HtO''— iincin<oino5'«i<03->*io5-^o>'^05'^Oi-<tio>-«!»«osTt<c>in<om'r-ito 
c<ie^ooeo-^-^inintotot>.i>'CX>ooooo505cr>o^^'^^c^e^c»3oo'^"<*'inintot^t^ooooojo» 
in in in in in in in in in to to to to to to to to to to 



lnco'*l^-l-l00tot^-o■*•-HOOCot^■*cceo«o^t^«3«^05■«*cDOocq^~coMCOlnol^- 
in^t>.oootoeooooinoo^05t-in-«i<co(Ni'-i^o<o<oO'-<<Me^q-<t<intocooe^-^t^o> 
ooooc<it^fNjtoi— ito<oin<oin05-^05'^a5-<*ioJ-^05-<t'05-«*'05rt<05-Ttio3-»j<ojinG>inc>in 
Meviecco-*-^inintotoc^t^c^ooooo>05<oO'-H— ce^q«sicoco-<i<-<*<inintotoc^cx30oojos 
inininminininininto to to to to to to to to to 



eoH 



toininooe<i05t~oooin'-<oooot^-<*<c<]00inotoininooc<iOit>.ooc>in^oooot^-<* 
eN500->*iot~co<s>t^in«sioooto'^c<ir-i<oa500ooc^t>-t-t^ooooo50e«cier5intr^05i-ieoto 
eoD-c^t^i— ito-i— iin<r>inc>'«*iOi'^05-^05CococcoocT50ocoooeooo-«t'OS'»t'05'<*<csinoin 
<Mc^ecco-«i*-<i<inintotot't~t>-ooooo50500'-ii-ic^ac>qcoec-«»<-«i<inintotot^t^ooo50i 
in in in in in in in in in to to to to to to to to to to 






c^toco05ooo5Qooooin'-HO<r)egt^eo^oq-«*05inco-«*<coot^intoooe<iOit-t^O"*^ 
o»inT-<t^->!»<c>t^'^e^05t^-incO'-^oiooi>.toin-^'^'>*|'*''«!>*inintot>.ooO'-icomoooeo 
e^at^(Nito-^toc>incD-<*05-Tj<os-<i<(»cooooooococ>occoocooooooooQOO'*o-^o>-^c3in 
eMc<icoe*5-«*<"*inintototoi>.i>.c»Qooso50(0'r-i^cqc^coeo-^'<i*inintotot-b^oooio> 
inininininininininto to to to to to to to to to 



»obo 



05t^t>.o-*oooosi-iin^ooe^jtoccT-(T-iet:ioo-«i*cQC<iinoiinoo-^tootoinint>-^oo 
toesiQoin»-ioo-*'i— iostO'^evi<oootoin'^eccsir-<^^,-i^H^H,— les^oO"<**lnt>-ooc>ccl•Tf^~os 
(rcii>.— ito^inoinos-^oi-^aiecoooQOoecooccoooo caoco oocooooooooQOO'<»<05'«*<cr>-<*< 
cqc>qcoco-^-*ininin«otot>.t^ooooo50o<=>^^cqeMroe»3-<!»<-«*<inintotot>.c^ooooo» 
in m in in in in m in intoto to to to to to to to to 



ooo05»-iin^040si-<ine>qooe^toe<iooeoi>-cc^^cot^ooe«ae>q-<*<o0'«*«>qpq-«*osin 
'^05incKicx)in^ootocoT-io:>c^incoM^oo500oooooooocx)o>o— <MCoint^05i-icoto 
cato^to<=)in<=)-fOJ-<4<Oieoooeooocooocot^<Mt>-(rcit~Mt~c^cioococoecoocooo-^aj-<i* 
cMeNjccec-*-<*ininintotot^t^ooooo5C35(oo'-H^cMesjcoeo-<j'">*inintotot^t^ooooo» 



HlN 



c<^0ocqtoe>qoo<^c^t0l^c^ooc<ltoe<l0oe<^toe>cl<3oc<^toc^00e>a«oc<l00ecltoe^a 

^t-eooiincQajtocot=>c»tO'#esic50>c»tr-toinininminintot^ooojoe>q-^toooc»eo 
(^^to^lnolno5-^05-5t^ooc«?ooeoooe^lt-<^^t^(^cIt:^l^qI^-<^qc^e^t:^c^^>.eooocr5oocoai'* 
e>Qe<iPocc-«*'"«*"^mintotot^c^oooooi05<oC3i— i^He^)c<qooeo'^'^mintotot^t-ooooo» 
in in in in in in in in intoto totototototototo 



eY3,w^ect^ecooc<itoc^o o e<i ini-iOJOiTMin-HOJooo-^oooooo'^ojt^b-oiecai 
oo-««<<otocMOJtoooot^incoi-i05t>.to-^cocoevjcvai-H^c<i«^]eoco-<i'tot-cooevj-«»<t>.oj 
»--ito»-Hin<o-^o^-^a}eoc30oocooat^esit^cqt^oat-«^at>.c<ii>.e<it>.esit~evit^coooeoooc»5 
cqe^acoe^^-^■^-«l'lnlntotoc^t^ooooo»05 o o »-i-HC<ic<icceo-*-^mintotot^t~ooooos 
in m in in m in in in in to to to to to to to to to to 



incq<>3-«i<oooO'-H,-ieotoMOio-— <in^0500o-^ot^t^05eooototooo^-it^inin«ooto 
in»-Ht^cooitDcoc5t^-^eNac3C»tO'^oo^Hooo5ajooooooo50><0'-ie<i'^int^OJi-<'>i»<to 
»-itooino5-^03'>^oocooocot^e<ir^(Mt~-cMt^^to^to-Hto^t^evjt^«sit>.e^at^eoooeo 
c^c<icococo-^-r»<inintotoc^c-ooooo50500'-i-Hes]esjeooo'4»-<i<inintotot^t^c300oos 
inininininininminto to to to to to to to to to 



to-^eoino5-*(Nj-HOot^(MOO'-iinoooooocoostotot»''-it^-<i<-*toosinpqc>a'>*<t>.co 
csjoo-^otoeocDt^-^i— iOJt^inco^-<c>ootr^totoinininmtotot^oooi<r>e>q'^toooofO 
^incDinai-^05Cooooot^cqt^Nt^cv)to^to^to»-ito-Hto— (toi-ito<Mt~«sic^(MOoeo 
c<iesie*3e»5co-^-^min<r>tot^c^cx50oosOic>0'— i»^e<ie>acoeo'<*T*<inintotot^t^oooooj 
ininmininininin intoto totototototototo 



coinintooincoe^'^t^eooc)!— iinooot^ose^ooinintooineocvi'^t^POOO^Hino 
05ln'-H^^-<*l<ot^■*'-^ootO'tt^<N^c^oo^>-ln•^cocOl^ae^qc^qcvloooo■^lntot>-OJl-leolnt^c^ 
oino-^05-<!*<ooeooocgt>.esit^c\jto»-Hto^Hto»-Hto^Hto-^tO'^HtOi— itOi— ltoe>q^^c^l^>-er5 
eMesjcococo-^-^inintotot^t^ooooajosooi— (■^c<ic<ipoeo"<*-«*'mintotot~t^ooooo 



OS to to 

to <NI 00 

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t^^tocooo-^t^coooi-H-^ot-tooor-^t^-^cioinooco^oe^inooot^ooesit^ 
•*^t^-<f'-iooincot-<o>t^in-^irg'-<C50osa30JOJOi<o>-He>aco-«*<tot^o>— (■^to 
■^a50ooocot>-c<it>-coto^^to»^to<z>to>— (in<oin<oin.— (to— HtO'Hto^-^toccit^cq 
e*3eo-«a<"<**inintotot>.t^ooooosojc30T-Hr-i«siMcooo-<j<-*inintotot^t^ooooo> 



r+O 



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— HOO-<ij<T-ioom«sjoQOto-^cKic3<3soot~t^tototototot--.t^ooo»-ic^'»ftoooc5eo 
•rfooecoooac^c^c^T-tto— itO'— cinc3inc5inoinc3inoinoto^^to^HtO'-it>.cM 
c*3oo-<i<-^inintotot>.c^ooooo30500'-i^cM(NioopQ-«*i-cttinintotoc^t~ooooo> 



-« 



pao5C005coooinTt<inooeoo<o»-iT*io5tointoo5ine^i-He^in<ot^tooo^HtococQCC)to-H 
i-<eocviooin»-HOOinpQ05t^incoi-H03t^toin-«i<coa3oooocoeo-^-^intooooii--iecint^o 
0"^dcoc»eot^e^c~-»-itOi— itO'r-Hinoin<oinoinoinc>in<omoinoin-^^to»-itoeM 
e^^<^^(^apoco•«l^'<}^lnlntDtot^t^oocoo50500^— ieNjesicoPO-«»<-^inintotot-t-ooooo> 
in in in in in in in in intoto to to to to to to to to 



■»j<i-HO-HTtiostointoc&-^i— ic>^^Tt"oitointoo5-^^H<z).-^-«j<05toin«oo5-^»-cO'^^'^o> 
co-^otocNjooine^oito-^evaocotO'^coevj^HioocDioooO'— icgco'^tooocJCQ'^to 
Oi'^05e>oooc^t^e>cito»-Hto»-Htooinoinoinc3in;=>inoincDinc>inoinotO'^Hto— < 
•^HC<iccicccc'xi<->!f<inmtotot-c^c»oooso5C3CD'-i— ^c^^e^acT3e*^■^•^mlntotot~t^ooooo> 
inioininminmin intoto totototototototo 



00050— <NC'5-^intot^oooio^«^aco'et<intot^ooosoT-<cqco«<intot^oo050'-icoeo 
e^cMeococooocofOPOooeocc'ti<'^->ii'-«f'^'>j*'^'^"^-^inininininininininintotototo 
e^leslCQev^e^(^q<^ae^e^aege^^c<^ca<^^e^^pclegeslcqoqege>c^e<^eQpae<^cqe<lcs^e<qe^e^acge^^c^^ea 



intot-oooso'-HOi 

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ic<icQ'^mtot>.ooo>0' 



iOr-ic<3oo'^>a<oc»ooo)< 



228'' (=19' on— SQUARES— 300'' (=25' 0"). 



655 



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t>.t>.t:^oooscD^Hesji*<mt^cJsc^-rt<t>.ocoioesioo-^i-Ht^-<*<<oc^eoococoascocoo5coco 
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CO 1-^ CO 1— < CO tH CO r*( CO 1-^ CO i-H CO ^-H CO v— ( CO f-H CO ▼-* CO i-H CO v^ CO i-H CO ^^ CO t-H CO ^^ CO i"^ CO tH 

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cr5-«»<-*»^>cocot^ooooOioC5'-||^a(^qoo•>*m»ococ^t^cx)0300^-H(^3«v^eo•«l<»OlOcot^c» 

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e^05coeoo5cocooi>-'^T-it^">*ii— lOoioeocst^'^'^HOoioevicst^'^'— loscocor-iookoeoo 
eoc'0•^»n»ococ^ooooosoo^cc^(^c^eo•>!:fmlOcot^t:^ooe3>o<o^Nc^lC»^■*ltomcot-oo 
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e>acoe^aooe^3co(^^ooc<^coe^aoocqcoccIO<=>cqcoc<lOOCcIco^qocDcacoe>c^ooe<l 

coc>ooco•^e>qc^osoot^col^)lOlf5lOIomcoI^~ooo5C)cq■T}^coooococoo5c^q»oo5cot-■^H 
c>qa3iocaoicocooscocoot--»t*'-ioomcg05cocooooiocgo^co-^i-iooiocoot-iocqo 
coco-*ioiocoi>-i>.ooo500'-ie<ic<ico'*-^iocot^t-ooo50s«DrHc<iccieo-^»oio«ot^oo 
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coco'*ioiocot-c^c»05<o«r>-HC<ic<ieo-«i<-^iocor-t^oooi050'-icQegco-*iomcot^t-. 
,_(^r-i,-H,-ii-iT-(,-<>-(,-Hoqc>acac^3cqcac<ioQc>qcac^cae^cMevicococococococococococo 

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coeo'*miocot^t^oooj05<c>»— ((^acqco■^■^locot^t^cooi050'— ic<ie^qco-«s<-Tfiocot^t>. 
T-(,-i»-ii-(r-<T-(i-(T-(T-i»-i<-<c<ic<icqe<icacqe>cicQccicQe^acvjc^c>acocococococooooocococo 

i-it^coococaooi£5»-Ht-coococcioomT-ic^coococaoomT-ic^coococsioo»ni»-it*coo 

coosu^coe<i'«i<t>.coi-HoeacO'i-HO>ooo-^*<;75t^t>^coc<it--ioincoocococo-^cx5'*^Hi--ico 
ot^lOCOl-lOic^colO•*cocqc^c^T-(l-(c^ac<^e^aco•^lot^-ooc^«^■*c^OicqlOOO^lOO>cot>• 
-Ht^-»*<T-Hoo-«*<i-iooioc<iojcocoot--<**^oome^aojcoco^ooioevaa>c-'«i<— <o>coco^oo 
coco'*ioiocot^t~ooosojc5'— icge^co-^'«4<mcocot^c»ososc)i— I'l— ic<jco->it*Tt<ificot^t>. 
,M,-^^,_l»-^,-^l-l,-^,-^l-H,-Hcqe<^cs^cqca{^qcclc^qc<Ic^ac<^e<^c^^cqcococococococococococo 

T-H CO ^H CO ^H CO i~H CO tH CO i~^ CO i~H CO v~H CO *— t CO ^~^ CO v-H CO i~~< CO v~H CO ^H CO T^ CO T^ CO T^ CO v~* CO 

rj<i^coc3<3»-H»oooot~oje^'(»iniocooxocoe^a-^t-^cooO'--i»j;3000t~(3>caooin)if5co 
coco^CT>t^iococ<iooiooooj>.t^t~t^oocoo5o^cvi-*coc»oe^qiot-C3cot^<=)-^oocq 
C3t^-^c^t~'^»-HOOiO'--<c»mifiKi05cocoot^'^eMOJcocoot>.ioc>cioco-^»HOOcocoooo 
coco'*ioiocot^c^ooo»o5c>?-<-H(Nico'<i<-«»*in>cDcot^ooojojo— <i-ic<jco-«i<-*iococ^t>. 
^^^T_,^^r_,^^,_i^ffge>qcqcqcgcqcqcqc^acaccj«vicQC<icococococooocococococo 

0"<i<ooc>cimo:>cot^o-*oOMioo5coc^O'*oocaio05cot~0'^oocQioo>cot^o-*ooevi 

ev3ioocot^ooe<it>.m'«i<ioo>'<«^»-Hccico^ooooos«vaooiomcoo»incci^cococqc3>ooo 
c^ao>^-.■<^(^qoo5^-com'!feocococococo->*'■^lOcoooo5^HCOlOt^ococoo5cqcoOJCOOo 
ocoeo<:3t--<*iot-">*<^ooioevj05cocoot>--*^ooiocvac3t^-<a<'i-ioacooo<r>QOiocaot~ 
coco-^ioiocot-t^oooiOicsi— iT-(cqco">*''^in)cocot^ooo>030'^Hi— icaco-^'^mcotr^t^ 
^^^^^^T-^^l-l,-l^c^^(Mc^le^c^e<^(^3e^acqc<lc<^e>q(^qcae^3Cococococococoeocococo 

ocoinooocomooocoioooocoioooocomooocoinooocoirsooocoioooocoio oo" 

ocooolnlOcoo5•^e<I»-HcalnT-^c)Ot>Iooe^qt^•<icolnoocooc>»-^"*a>^-^cot>^c>cococ<ICO 

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CvqCO'^iOiOCOt^t^OOOJOSO^H^HC^ICOCO'^lOCOCOt^OOOOOSO'-HT-HCCICO-^TflOCOCOt^ 
^^^^^^^^._i^^5vqj>qc<,5Nqjvqjvqpqi^g^C,qjv,i-qjv35vq(Y3COCOCOCOCOCOeoCOCOCO 

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oo^cocoe^acoco^c3iOOOieMt^'^cO'^c»cooo»ocoooioioco05">*»-iO»-<-^'ot^cot^ 
eo»-(ooco-><<e<icDait>.coioio-^-<*<-^'^-«<<iococoooo5<oca-<*«coooi— i'^t^C)Cot~o-^oo 
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c<icO'^-<<t'iocot-~t^oooja5c>^H---Hcacoco-^iococoi>-oooocr3C3»-i'-He<ico'^Ttiinicococ>. 

^,-(^^^^^^^^^5vjjva5s3jv3jvqi^p^pqjvjjvqjsq<^fQfsjfr3ev3j,5jy3{,5C3{,^COCOCOCO 

coosT}<^o^-*Oicoioco o"'* T-HO»H-**05coocoo5T}<-^o^'«4<o>co'iccooj'«i<-Hcr>'-4 
oco-^cvioooco-^coca^HcooocDcisocis^Hcqco'^cooooevi'^cooicqioooevicoO'^ 
ooioc<ioi<ocvj05cocoor^-^T-HOOioc^ao>cococot^-^»— I00cocool>>■^c^^03co■^•-l0sco 
evaco-^-«*'»ococot^ooas05<0'-H-HCMCOco-^mcocot^ooooo5<o»-ii— (cacoco'tfiococot^ 
^,_(T-ll-^T-lr-c^T-^T-lrt,-(eMc^a«^a<^^e^ac^^e^al^^c^cqe^acae^aoacococococococococococo 

cot>-ooa5CD-^c<ico-<*<iocot--oo(3>Oi-<cqcO'!j<iocoi>.oo<3io»-icqco-^»ocot»ooosoi-« 

COCOCOCO-«*<-*■rf■«*^-^■^-^-^^<•^t^■^tl^OLnli:51010l010mlLOlOeocOCOCOCOCOCOCOCOCOt:^^* 
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C5c»iococ300iocoi-H<»co->*ir-<o5tr^Tre^c>oococO'— lost^irjco^-toit^-inco'— lOit^-ini-^ 
05050'-HC»ci(>cico-^ioiocot^oo(X)05C>'— lesie^ico'^iciincot-.ooasoscS'-Hcsicoco-^inco 
focc-^-^-«*<"*'«*<-^-*-^-^'^-^'*'4*»oinioio»oiLOioioiomioir5iococococotocococo 



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M:JN»-te^-r}<05co»nxn(»coc>oooie^at~cocqoocoot^cot~05'^^HOC>coodineo'Tt<t^e^ 
cocxjcooocooo-^ococ^oscocKiost^'^icsicsoocoioeoesi^— iOcsc3C3<o<oc>'— ic<ico->*ico 
C3t^>oev]<ot^ioco<^ooincC'-H(»co-^e^ac5t^ioco^H05t-.iooo<— i05c^»ocO'^Ha5t>.ioco 
05crJC^l-Hc^a(^qco-^lOl^)co^^ooooo5C^^^e^l<^aco•^lOlOco^^ooo5a5c:5•^He^aeoco•«*<loco 
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588" (=49' on— SQUARES— 624'' (=52' 0"), 665 

EXCERPTS AND REFERENCES. 

Direct Method of Spacing Rivets and Finding Position, etc., of Stiffs 
eners in Plate Girders (By E. Schmitt. Trans. A. S. C. E., Vol. XLV). 

Diagrams for Determining Minimum Alternate Spacing of Rivets (By 

F. L. Batchelder. Eng. News, Oct. 31, 1901). — This diagram is very simple 
to construct for any diameter of rivets, and is based on the general equation, 
a^ = x'^ + y^. 

A Table for Pitch and Efficiency of Riveted Joints (By P. B. Hill. 
Eng. News, July 16, 1903). — For various thicknesses of plates and diameters 
of rivets. 

Standard Heads for Machine Screws (By H. G. Reist, Eng. News, 
June 29, 1905). — Illustrations and table of dimensions of screws with round 
head, fillister head, flat head and hexagon head. (Also see Eng. News, Dec. 
28, 1905, for table of standard threads for machine screws and taps, for ^" 
diameter and less; also, Eng. News, June 20, 1907, for A. S. M. E. stand- 
ards.) 

Tension-Tests of Steel Angles with Various Types of End=Connection 

(By F. P. McKibben. Proc. A. S. T. M., 1906; Eng. News, July 5, 1906).— 
'Table shows percent strength of material developed: usually from 75 to 
80%. (See, also, Eng. News, Aug. 22, 1907.) 

Cost of Shop Drawings for Structural Iron and Steel (By R. H. Gage. 
"The Technograph," Univ. of 111.. No. 21, 1906-7; Eng. News, Aug. 8. 
1907). — Condensed as follows: 

Av. Cost 
Type. Character of Building. per Ton. 

A. Entire skel. cons.; loads all carried to found'n by steel columns $1 .45 

B. Exterior supported on steel cols. ; floor loads carried by exter. walls 1.22 

C. Inter, portion sup. by cast iron cols.; fl. Ids. by exter. walls .70 

D. No cols.; floor beams resting on masonry walls throughout . 85 

E. Structure consisting mostly of roof trusses resting on columns 2.47 

F. Structure consisting mostly of roof trusses resting on mas. walls 1.25 

G. Mill buildings 2 . 56 
H. Flat one-story shop or manufacturing buildings . 74 

I. Tipples, mining- or other complicated structtires 4 . 88 
J. Malt or grain bins and hoppers 2.47 
K. Remodeling and additions where measurements are necessary before 

details can be made 1.87 

The Detailing of Skew Portals (By J. P. Davies. Eng. News, Feb. 

II, 1909). — Formulas, diagrams and shop detail drawings. 

Diagrams for Rivet Pitch in Loaded Girder Flanges (By P. L. Pratley. 
Eng. News, Feb. 18, 1909). — Formulas and diagrams. 

Tests of Nickel Steel Riveted Joints (By A. N. Talbot. Eng. Rec, Aug. 
20, 1910) . — Gives details and results of tests made for the Board of Engineers 
of the Quebec Bridge. Illustrations of the joints tested, and tables of the 
results of the tests. Table 3 (not reproduced here) gives the ultimate 
strength of the nickel steel joints, and of the carbon steel joints (tested in 
1905 by Am. Ry. Eng. & M. W. Assn.) reported in lbs. per sq. in. of the 
shearing area of the rivets. The average for the nickel steel joints is seen to 
be about 16% greater than for the carbon steel joints. It should be noted 
that the rivets of the nickel steel joints are considerably weaker than the 
plates and that the failure of these joints was due in all cases to shear of 
rivets, while in the carbon steel joints there was evidently less inequality 
between the strength of rivets and of plates, a number of joints failing by 
tearing the plate. 

Formulas for Use in Detailing Steel Structures (By H. Vance. Eng. 
News, Sept. 1, 1910). — Examples: Hoppers, towers and oblique bends in 
riveted pipe. 



34.— METAL GAGES. 

1. — Standard Gages.* 



0} 




Thickness in 


Decimals of an 


Inch. 






«-4 

o 

1 










he 


§6 

a a 

IS 


^2 

01 


d o o 


7° 






500 

.46875 
.4375 
.40625 


.500 
.464 
.432 
.400 


.49 
.46 
.43 
.3938 








6° 












5° 






".45 

.40 






4° 


■!454" 


"*.46 




".'454"'** 


3° 


.425 


.40964 


.375 


.372 


.3625 


.36 




.425 


2° 


-.380 


.3648 


.34375 


.348 


.3310 


.33 




.380 





.340 


.32486 


.3125 


.324 


.3065 


.305 




.340 


1 


.300 


.2893 


.28125 


.300 


.2830 


.285 


.227 


.300 


2 


.284 


.25763 


.265625 


.276 


.2625 


.265 


.219 


.284 


3 


.259 


.22942 


.25 


.252 


.2437 


.245 


.212 


.259 


4 


.238 


.20431 


.234375 


.232 


.2253 


-.225 


.207 


.238 


5 


.220 


.18194 


.21875 


.212 


.2070 


.205 


.204 


.220 


6 


.203 


.16202 


.203125 


.192 


.1920 


.190 


.201 


.203 


7 


.180 


.14428 


.1875 


.176 


.1770 


.175 


.199 


.180 


8 


.165 


.12849 


.171875 


.160 


.1620 


.160 


.197 


.165 


9 


.148 


.11443 


.15625 


.144 


.1483 


.145 


.194 


.148 


10 


.134 


.10189 


. 140625 


.128 


.1350 


.130 


.191 


.134 


11 


.120 


.090742 


.125 


.116 


.1205 


.1175 


.188 


.120 


12 


.109 


.080808 


.109375 


.104 


.1055 


.1050 


.185 


.109 


13 


.095 


.071961 


.09375 


.092 


.0915 


.0925 


.182 


.095 


14 


.083 


.064084 


.078125 


.080 


.0800 


.0800 


.180 


.083 


15 


.072 


.057068 


.0703125 


.072 


.0720 


.0700 


.178 


.072 


16 


.065 


.05082 


.0625 


.064 


.0625 


.0610 


.175 


.065 


17 


.058 


.045257 


.05625 


.056 


.0540 


.0525 


.172 


.058 


18 


.049 


.040303 


.05 


.048 


.0475 


.0450 


.168 


.049 


19 


.042 


.03589 


.04375 


.040 


.0410 


.0400 


.164 


.040 


20 


.035 


.031961 


.0375 


.036 


.0348 


.0350 


.161 


.035 


21 


.032 


.028462 


.034375 


.032 


.03175 


.0310 


.157 


.0315 


22 


.028 


.025347 


.03125 


.028 


.0286 


.0280 


.155 


.0295 


23 


.025 


.022571 


.028125 


.024 


.0258 


.0250 


.153 


.027 


24 


.022 


.0201 


.025 


.022 


.0230 


.0225 


.151 


.025 


25 


.020 


.0179 


.021875 


.020 


.0204 


.0200 


.148 


.023 


26 


.018 


.01594 


.01875 


.018 


.0181 


.0180 


.146 


.0205 


27 


.016 


.014195 


.0171875 


.0164 


.0173 


.0170 


.143 


.01875 


28 


.014 


.012641 


.015625 


.0148 


.0162 


.0160 


.139 


.0165 


29 


.013 


.011257 


.0140625 


.0136 


.0150 


.0150 


.134 


.0155 


30 


.012 


.010025 


.0125 


.0124 


.0140 


.0140 


.127 


.01375 


31 


.010 


.008928 


.0109375 


.0116 


.0132 


.0130 


.120 


.01225 


32 


.009 


.00795 


.01015625 


.0108 


.0128 


.0120 


.115 


.01125 


33 


.008 


.00708 


.009375 


.0100 


.0118 


.0110 


.112 


.01025 


34 


.007 


.006304 


.00859375 


.0092 


-.0104 


.0100 


.110 


.0095 


35 


.005 


.005614 


.0078125 


.0084 


.0095 


.0095 


.108 


.009 


36 


.004 


.005 


.00703125 


.0076 


.0090 


.0090 


.106 


.0075 


37 




.004453 


.006640625 


.0068 


.0085 


.0085 


.103 


.0065 


38 





.003965 


.00625 


.0060 


.008 


.0080 


.101 


. 00575 


39 




.003531 




.0052 


.0075 


.0075 


.099 


.005 


40 





...003144 . 




.0048 


.007 


. 0070 


.097 


.0045 




Stubs' 
Iron 
Wire 


American 






Roebling. 
Wash- 
burn & 
Moen 









* See also Table 3, Sec. 35, Cordage, Wire and Cables, page 671, which 
includes the Edison Gage. t 7° means 0000000, 2° means 00, etc. 

666 



STANDARD METAL GAGES. 



667 



2. — United States Standard (July 1, 1893) Gage 
For Sheet and Plate Iron and Steel. 
(Adopted as Standard by American Railway Master Mechanics Asso- 
ciation, and Association of American Steel Manufacturers.) 



1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


11 


12 


13 


o 


Approximate 1 


rhlckness 


a o 


Weight in 


Weight in 

Kilograms 

per Square 

Meter 


Weight 
in Kilo- 


Weight In 
Pounds per 






?s^ 


Pounds per 


grams per 


Square 


Inches 


Milli- 
meters 


'^8 


Square Foot 


Sq. Foot 


Meter 


O 


fl 








o 


Frac- 
tion 


Decimal 


♦Iron 


Steel 


*Iron 


Steel 


1 

M 
* 


1 

m 


*Iron 


Steel 


§ 


Iron 


7° 


1-2 


.5 


12.7 


320 


20 


20.4 


97.648 


99.601 


9.072 


9.253 


215.28 


219.58 


6° 


15-32 


.46875 


11.90625 


300 


18.75 


19.125 


91.545 


93.376 


8.505 


8.675 


201.82 


205.86 


5° 


7-16 


.4375 


11.1125 


280 


17.50 


17.85 


85.442 


87.151 


7.938 


8.097 


188.37 


192.14 


4 


13-32 


.40625 


10.31875 


260 


16.25 


16.575 


79.339 


80.926 


7.371 


7.518 


174.91 


178.41 


3 


3-8 


.375 


9.525 


240 


15 


15.3 


73.236 


74.701 


6.804 


6.940 


161.46 


167.69 


2 


11-32 


.34375 


8 73125 


220 


13.75 


14.025 


67.133 


68.476 


6.237 


6.362 


148.00 


150.96 





5-16 


.3125 


7.9375 


200 


12.50 


12.75 


61.030 


62.251 


5.670 


5.783 


134.55 


137.24 


1 


9-32 


.28125 


7.14375 


180 


11.25 


11.475 


54.927 


56.026 


5.103 


5.205 


121.09 


123.52 


2 


17-64 


.265625 


6.746875 


170 


10.625 


10.8375 


51.876 


52.913 


4.819 


4.916 


114.37 


116.65 


3 


1-4 


.25 


6.35 


160 


10. 


10.2 


48.824 


49.800 


4.536 


4.627 


107.64 


109.79 


4 


15-^64 


.234375 


5.953125 


150 


9.375 


9.5625 


45.773 


46.688 


4.252 


4.338 


100.91 


102.93 


5 


7-32 


.21875 


5.55625 


140 


8.75 


8.925 


42.721 


43.575 


3.969 


4.048 


94.18 


96.07 


6 


13-64 


.203125 


5.159375 


130 


8.125 


8.2875 


39.670 


40.463 


3.685 


3.759 


87.45 


89.21 


7 


3-16 


.1875 


4.7625 


120 


7.5 


7.65 


36.618 


37.350 


3.4U2 


3.470 


80.72 


82.34 


8 


11-64 


.171875 


4.365625 


110 


6.875 


7.0125 


33.567 


34.238 


3.118 


3.181 


74.00 


75.48 


9 


5-32 


.15625 


3.96875 


100 


6.25 


6.375 


30.515 


31.125 


2.835 


2.892 


67.27 


68.62 


10 


9-64 


.140625 


3.571875 


90 


5.625 


5.7375 


27.464 


28.013 


2.552 


2.603 


60.55 


61.76 


11 


1-8 


.125 


3.175 


80 


5. 


5.1 


24.412 


24.900 


2.268 


2.313 


53.82 


54.90 


12 


7-64 


.109375 


2.778125 


70 


4.375 


4.4625 


21.361 


21.788 


1.984 


2.024 


47.09 


48.03 


13 


3-32 


.09375 


2.38125 


60 


3.75 


3.825 


18.309 


18.675 


1.701 


1.735 


40.36 


41.17 


14 


5-64 


.078125 


1.984375 


50 


3.125 


3.1875 


15.258 


15.563 


1.417 


1.446 


33.64 


34.31 


15 


9-128 


.0703125 


1.7859375 


45 


2.8125 


2.86875 


13.732 


14.006 


1.276 


1.301 


30.27 


30.88 


16 


1-16 


.0625 


1.5875 


40 


2.5 


2.55 


12.206 


12.450 


1.134 


1.157 


26.91 


27.45 


17 


9-160 


.05625 


1.42875 


36 


2.25 


2.295 


10.985 


11.205 


1.021 


1.041 


24.22 


24.70 


18 


1.20 


.05 


1.27 


32 


2. 


2.04 


9.765 


9.960 


.907 


.925 


21.53 


21.96 


19 


7-160 


.04375 


1.11125 


28 


1.75 


1.785 


8.544 


8.715 


.794 


.810 


18.84 


19.21 


20 


3-80 


.0375 


0.9525 


24 


1.50 


1.53 


7.324 


7.470 


.680 


.694 


16.15 


16.47 


21 


11-320 


.034375 


0.873125 


22 


1.375 


1.4025 


6.713 


6.848 


.624 


.636 


14.80 


15.10 


22 


1-32 


.03125 


0.79375 


20 


1.25 


1.275 


6.103 


6.225 


.567 


.578 


13.46 


'13.72 


23 


9-320 


.028125 


0.714375 


18 


1.125 


1.1475 


5.493 


5.603 


.510 


.520 


12.11 


12.35 


24 


1-40 


.025 


0.635 


16 


1. 


1.02 


4.882 


4.980 


.454 


.463 


10.76 


10.98 


25 


7-320 


.021875 


0.555625 


14 


.875 


.8925 


4.272 


4.358 


.397 


.405 


9.42 


9.61 


26 


3-160 


.01875 


0.47625 


12 


.75 


.765 


3.662 


3.735 


.340 


.347 


8.07 


8.23 


27 


11-640 


.0171875 


0.4365625 


11 


.6875 


.70125 


3.357 


3.424 


.312 


.318 


7.40 


7.55 


28 


1-64 


.015625 


0.396875 


10 


.625 


.6375 


3.052 


3.113 


.284 


.289 


6.73 


6.86 


29 


9-640 


.0140625 


0.3571875 


9 


.5625 


.57375 


2.746 


2.801 


.255 


.260 


6.05 


6.18 


30 


1-80 


.0125 


0.3175 


8 


.5 


.51 


2.441 


2.490 


.227 


.231 


5.38 


5.49 


31 


7-640 


.0109375 


0.2778125 


7 


.4375 


.44625 


2.136 


2.179 


.198 


.202 


4.71 


4.80 


32 


13-1280 


.01015625 


0.25796875 


6.5 


.40625 


.414375 


1.983 


2.023 


.184 


.188 


4.37 


4.46 


33 


3-320 


.009375 


0.238125 


6 


.375 


3825 


1.831 


1.868 


.170 


.174 


4.04 


4.12 


34 


11-1280 


.00859375 


0.21828125 


5.5 


.34375 


.350625 


1.678 


1.712 


.156 


.159 


3.70 


3.77 


35 


5-640 


.0078125 


0.1984375 


5 


.3125 


.31875 


1.526 


1.556 


.142 


.145 


3.36 


3.43 


36 


9-1280 


.00703125 


0.17859375 


4.5 


.28125 


.286875 


1.373 


1.401 


.128 


.130 


3.03 


3.09 


37 


17-2560 


. 006640625 


0.168671875 


4.25 


.265625 


.2709375 


1.297 


1.323 


.120 


.123 


2.87 


2.92 


38 


1-160 


.00625 


0.15875 


4 


.25 1.255 


1.221 


1.245 


.113 


.116 


2.69 


2.74 



Remarks. — In column 1, 7° means 0000000, 2° moans 00, etc. The thickness of metal 
as given in columns 2, 3 and 4 is based on the weight of iron In avoirdupois ounces per 
squarefoot as given in column 5, assuming the weight of iron at 480 lbs. per cubic foot. 
Columns 6 to 13 are self-explanatory, steel being assumed 2 per cent heavier than iron. 
A variation of 2 1/2 per cent either way from the above weights is usually allowed. 

To reduce pounds per square foot to kilograms per square meter, mult, by 4.8824. 

To reduce pounds per square foot to kilograms per square foot, mult, by 0.4536. 

To reduce pounds per square foot to pounds per square meter, mult, by 10.7639. 

*For copper, multiply the tabulated weight of iron by 1.133; for brass, mult, by 1.07; 
for zinc, mult, by 0.9; for lead, mult, by 1.47; for aluminum, mult, by 0.347. All some- 
what approximate. See also Standard Gages, Table 1, preceding. 



35.— CORDAGE, WIRE AND CABLES. 



Technical Cordage Terms. 



at 



Make TJp (in Manufacture); 

Marline. — ^Two yams twisted together. 
Thread. — T w o o r ) 
more small yarns > Cord. — Several threads twisted together. 
twisted together. ) 
String. — ^Two or more slightly larger yams twisted together, 

^ ] Shroud laid. — Rope 

Rope. — Several 
strands twisted to- 
gether. 
Hawser. — Large rope 
of three strands. 



Strand. — ^Three (by 
some authorities 
two) or more large 
yarns twisted to- 
gether. 



of four strands 
(with a heart) . 
Cable. — Three haw- 
sers twisted to- 
gether (left 
handed) . 



A Rope Is: 

Laid — By twisting strands together in making the rope. 

Spliced — By joining to another rope by interweaving the strands. 

Whipped — By winding yam or small stuff around the end to prevent un- 

stranding. 
Served — When wound tightly or continuously with yam or small stuff. 
Parceled — When served or wrapped tightly with canvas. 
Seized — When two parts are bound tightly together by yam or small stuff. 
Payed — When painted, tarred or greased to resist wet. 

Practical Operation: 

Haul. — ^To pull on a rope. 
Taut'. — Drawn tight or strained. 
Bight. — A loop in the rope. 
Fall. — ^The rope in a hoisting tackle. 
Tackle. — An assemblage of ropes and 
blocks. 



Hitch. — Attaching a rope to an 
object. 

Bend. — Attaching two ropes to- 
gether or to an object. 

Knot. — A loop or fastening with a 
rope. 



Knots, Hitches, etc. — (See Manila Rope, next page.) 

(Note that Ends are whipped to prevent unstranding.) 




Fig. 1, Bight. 



Fig. 2, Simple Knot. 



Fig. 3. Figure 8 Knot. 






Fig. 4, Double Knot. Fig. 5, Boat Knot. 



Fig. 6, Bowline. 




Fig. 7, Square Knot. Fig. 8, Weaver's Knot. Fig. 9, Toggle. 

668 



CORDAGE— ROPE, 



669 





Fig. 10, Carrick Bend. Fig. 11, Stevedore Knot. Fig. 12, Slip Knot. 






Fig. 13, Half Hitch. Fig. 14, Timber Hitch. Fig. 15, Clove Hitch. 





Fig. 16, Timber Hitch and Half 
Hitch. 



Fig. 17, Round .Turn and Half 
Hitch. 




Fig. 18, Blackwall Hitch. Fig. 19, Fisherman's Bend. 

Splices. — ^To splice an ordinary transmission rope, wind twine around 
the rope the length of the proposed splice, say 6 ft. or more, from each end, 
and unlay the strands back to the twine. Then butt the ropes together so 
that the untwisted strands will meet opposite each other in pairs. Next, 
cut the twine and unlay one strand from one rope end, following it up with 
a strand from the other rope end, and leaving about 18 or 20 ins. loose end 
on each strand at the meeting point, for sub-splicing. Make the points of 
meeting of other pairs of strands staggered regularly so no two points will 
be opposite. For sub-splicing of each pair of strands, split each strand, 
unlay and interweave, passing the ends through the rope, or tie with ordi- 
nary knots. Hammer smooth. 

Manila Rope. — (Adapted from C. W. Hunt.*) Manila rope is made 
from Manila hemp fibers (inferior in strength to the Italian hemp).^ In 
manufacturing rope, the fibers are first spun into a yarn about \ in. in diam. 
this yarn being twisted in a "right-hand" direction. From 20 to 80 of these 
yarns, depending on size of rope, are then put together and twisted in a 
"left-hand" direction, into a strand. In a 3-strand rope three, and in a 
4-strand rope four, of these strands are then twisted together, again in a 
"right-hand" direction. Note that when each strand is twisted it tends to 
untwist the threads, but later when the strands are twisted together into a 
rope, each strand tends to untwist, but to twist up the threads. It is this 
opposite twist of the threads and strands that keeps the rope in its proper form . 

The durability of Manila rope is quite variable under different uses. 
Experience has shown that 4-strand rope is more serviceable than 3-strand; 
it is stronger for the Same diameter, wears rounder and smoother, and the 
section is much nearer a circle. 

The Strength and Weight of Manila Rope are given by Mr. Hunt in the 

following formulas: 

Breaking strength, in pounds = 7160 X (diam in inches)^ (IJ 

= 725 X (circum in inches)^ .... (la) 
Weight per lineal foot, in pounds = . 34 X (diam in inches)^. ...... (2) 

= 0.0344 X (circum in inches)2 (2a) 

* See Manila Rope by C. W. Hunt; also Trans. Am. Soc M. E., 
Vol. xii, p. 230, and Vol. xxiii (1901). 



670 



S5.— CORDAGE, WIRE AND CABLES. 



Hence, from (1) and (2) we have: 
Breaking strength (lbs.) = 21060 X weight per lineal foot (lbs.) (3) 

It is to be noted from formula (3) that pound for pound, manila rope is 
as strong as steel which has an ultimate tensile strength of 71600 lbs. per 
square inch: as a square inch bar of steel weighs 3 . 4 lbs. per lineal foot, and 
21060X3.4= 71600 lbs. 

1. — Weight and Strength of Manila Rope. (By Slide Rule.) 





9. 




tStrength of 




6 




t Strength of 


fi 






Rope. 


a 


c 




Rope. 


n) 




* Weight ot 
100 ft. of 






i5 


^ 
^ 


♦Weight of 
100 ft. of 




n 












g 


Rope. 


Break- 


Safe 


a 

o 


a 


Rope. 


Break- 


Safe 


6 






mg 


{.Ss) 


d 




ing 


U\) 


^ 


U 




Str'gth. 


Str'n- 
gth. 


^ 







Str'gth. 


Str'n- 
gth. 


Ins. 


Ins. 


Lbs. 


Lbs. 


Lbs. 


Ins. 


Ins. 


Lbs. 


Lbs. 


Lbs. 


^ 


■h 


1.1 + 1.0 


230 


11 


iv? 


414 


69.6- 4.5 


14600 


730 


H 


% 


1.9 + 0.8 


410 


20 


1t% 


434 


77.5- 5.0 


16300 


815 


^ 


1 


3.4 + 0.6 


730 


36 


IVh 


5 


85.9- 5.5 


18200 


910 


Vh 


11/8 


4.4+0.4 


920 


46 


IH 


514 


104.0- 6.5 


21800 


1090 


^ 


1^4 


5.4+0.2 


1130 


57 


2 


6 


124.03- 8.0 


26100 


1300 


V?, 


ly?. 


7.7±0.0 


1630 


81 


2H 


614 


145.0- 9.0 


30600 


1530 


■^ 


IH 


10.5 + 0.0 


2220 


111 


2M 


7 


168.0-10.0 


35500 


1780 


% 


2 


13.8-0.3 


2900 


145 


2V, 


714 


193.0-11.0 


40800 


2040 


% 


2H 


17.4-0.6 


3660 


183 


2Vh 


8 


220.0-12.0 


46400 


2320 


% 


2H 


21.5-0.9 


4530 


226 


2y, 


814 


247.0-13.0 


52200 


2610 


2H 


25.9-1.2 


5480 


274 


3 


9- 


278.0-14.0 


58600 


2930 


1 


3 


30.9-1.6 


6520 


326 


31/^ 


914 


310.0-15.0 


65400 


3270 


ItV 


3K 


36.3-2.0 


7630 


382 


314 


10 


344.0-16.0 


72600 


3630 


Wh 


3H 


42.1-2.5 


8880 


444 


314 


11 


416.0-18.0 


87800 


4390 


IH 


Z% 


48.4-3.0 


10200 


510 


334 


12 


495.0-20.0 


104300 


5210 


1^ 


4 


55.0-3.5 


11600 


580 


414 


13 


580.0-22.0 


122400 


6120 


IH 


4M 


60.4-4.0 


13100 


655 


4J^ 


14 


670.0-24.0 


142000 


7100 



* The left-hand figures in the column are the weights according to Mr. 
Hunt's formula (2), and the right-hand figures are corrections which give 
resulting weights in accordance with some of the manufacturers' tables. 
The true weights are somewhat approximate and probably lie within the 
two limits. 

t Calculated from Mr. Hunt's formula. See also table of Breaking 
Strength of Manila Rope by Spencer Miller in Eng. JSfews, Dec. 6, 1890. 

The safe strength of manila rope given in the above table is based on a 
safety factor of 20, which Mr. Hunt recommends for rope driving. The 
following working loads are recommended for rapid (400 to 800 feet per 
minute), medium (wharf and cargo, hoisting 150 to 300 feet per minute), 
and slow (derrick, crane and quarry, speed at 50 to 100 feet per minute) 
work: 

2. — Working Load for Manila Rope. 



Diameter 


Ultimate 


Working Load in 


Pounds. 


Minimum Diameter of 
Sheaves in Inches. 


of Rope, 


Strength, 
Pounds. 












Inches. 


















Rapid. 


Medium^ 


Slow. 


Rapid. 


Med'm. 


Slow. 




7,100 


200 


400 


1.000 


40 


12 


8 


1/^ 


9,000 


250 


500 


1,250 


45 


13 


9 


1/4 


11,000 


300 


600 


1,500 


50 


14 


10 


1^ 


13,400 


380 


750 


1,900 


55 


15 


11 


13^ 


15,800 


450 


900 


2,200 


60 


16 


12 


1^ 


18,800 


530 


1,100 


2,600 


65 


17 


13 


IM 


21,800 


620 


1,250 


3,000 


70 


18 


14 



MANILA ROPE. WIRE GAGES. 



671 



3. — Comparison of Wire Gages. 
Birmingham (B. W. G.), Browne & Sharpe (B. & S.), and Edison. 
Note. — 1 mil = .001 inch. (Diam in mils of round bar)2 = area of section 
ifi circular mils; and this multiplied by 0.78540 = area of section in square 
mils; and this divided by 1 000 000 = area of section in square inches. 







m 


Area of Cross- 


■so4 


Gages. 


m 


Area of Cross- 


i§^ 


^6 


Gages. 


d 


Section, in 


S-1 

Is'" 


:z3 

d 
B 

5 


Section, in 










1 


53» 
P 

Bo 


1 

con 






1 


go 




o 






360 


600. 


360 001 


282 743 


942.5 


11 






120. 


14400. 


11310. 


37.70 






340 


583. IC 


340 OOC 


267 035 


890.1 




9 




114.43 


13094. 


10284. 


34.28 


"80' 






320 


565.69 


320 OOC 


251 327 


837.8 






12 


109.55 


12000. 


9425. 


31.42 








300 547.73 


300 OOC 


235 619 


785.4 


12 






109. 


11881. 


9331. 


31.10 






280 529.16 


280 000 


211 911 


706.4 




10 




101.89 


10382. 


8154. 


27.18 






260 


509.91 


260 000 


204 203 


680.7 


13 






95. 


9025. 


7088. 


23.63 


6§| 






240 


489.90 


240 000 


188 496 


628.3 




11 




90.742 


8234. 


6467. 


21.56 






220 


469 05 


220 000 


172 788 


576.0 






8 


89.45 


8000. 


6283. 


20.94 




0000 




460. 


211 600 


166 190 


554.0 


14 






83. 


6889. 


5411. 


18.04 


•-^1 


0000 






454. 


206 116 


161 883 


539.6 




12 




80.808 


6529. 


5128. 


17.09 


-^3 0,," 






200 


447.22 


200 000 


157 080 


523.6 


15 






72. 


5184. 


4072. 


13.57 








190 


435.89 


190 000 


149 226 


497.4 




13 


- 


71.961 


5178. 


4067. 


13.56 


000 






425. 


180 625 


141 863 


472.9 






5 


70.72 


5000. 


3927. 


13.09 






180 


424.27 


180 000 


141 372 


471.2 


16 






65. 


4225. 


3318. 


11.06 






170 


412.32 


170 000 


133 518 


445.1 




14 




64.084 


4107. 


3225. 


10.75 


8.§t 




000 




409.64 


167 805 


131 790 


439.3 


17 






58. 


3364. 


2642. 


8.807 






160 


400. 


160 000 


125 664 


418.9 




15 




57.068 


3257. 


2558. 


8.527 






150 


387.30 


150 000 


117 810 


392.7 






3 


54.78 


3000. 


2356. 


7.853 




00 






380. 


144 400 


113 411 


378.0 




16 




50.82 


2583. 


2029. 


6.763 






140 


374.17 


140 000 


109 956 


366.5 


18 






49. 


2401. 


1886. 


6.287 




00 




364.80 


133 079 


104 518 


348.4 




17 




45.257 


2048. 


1609. 


5.363 


M 






130 


360.56 


130 000 


102 102 


340.3 


19 






42. 


1764. 


1385. 


4.617 






120 


346.42 


120 000 


94 248 


314.1 




18 




40.303 


1624. 


1276. 


4.253 









340. 


115 600 


90 792 


302.6 




19 




35.89 


1288. 


1012. 


3.373 


3 w sf 






no 


331.67 


110 000 


86 394 


288.0 


20 






35. 


1225. 


962.1 


3.207 











324.86 


105 534 


82 887 


276.3 


21 






32. 


1024. 


804.2 


2.681 






100 


316.23 


100 000 


78 540 


261.8 




20 




31.961 


1022. 


802.3 


2.674 


H"5 d 






95 


308.23 


95 000 


74 613 


248.7 




21 




28.462 


810.1 


636.2 


2.121 


^§ 


1 




90 


300. 


90 000 


70 686 


235.6 


22 






28. 


784.0 


615.8 


2.053 


^^.B 






85 


291.55 


85 000 


66 759 


222.5 




22 




25.347 


642.5 


504.6 


1.682 


8-g 




1 




289.30 


83 694 


65 732 


219.1 


23 






25. 


625.0 


490.9 


1.636 


2 






284. 


80 656 


63 347 


211.1 




23 




22.571 


509.4 


400.1 


1.334 


o3'c5 






80 


282.85 


80 000 


62 832 


209.4 


24 






22. 


484.0 


380.1 


1.267 


^s ... 






75 


273.87 


75 000 


58 905 


196.4 




24 




20.1 


404.1 


317.3 


1.058 






70 


264.58 


70 000 


54 978 


183.3 


25 






20. 


400.0 


314.2 


1.047 


3 






259. 


67 081 


52 685 


175.6 


26 






18. 


324.0 


254.5 


.848 


^^^ 




2 




257.63 


66 373 


52 128 


173.8 




25 




17.9 


320.4 


251.7 


.839 


»5- 






65 


254.96 


65 000 


51 051 


170.2 


27 






16. 


256.0 


201.1 


.670 






60 


244.95 


60 000 


47 124 


157.1 




26 




15.94 


254.1 


199.6 


.665 


rO^ ^ 


4 






238. 


56 644 


44 488 


148.3 




27 




14.195 


201.5 


158.3 


.528 








55 


234.53 


55 000 


43 197 


144.0 


28 






14. 


196.0 


153.9 


.513 


oo "^ 




3 




229.42 


52 634 


41 339 


137.8 


29 






13. 


169.0 


132.7 


.442 


^^3 






50 


223.61 


50 000 


39 270 


130.9 




28 




12.641 


159.8 


125.5 


.418 


^^B 


5 






220. 


48 400 


38 013 


126.7 


30 






12. 


144.0 


113.1 


.377 






45 


212.14 


45 000 


35 343 


117.8 




29 




11.257 


126.7 


99.51 


.332 






4 




204.31 


41 743 


32 784 


109.3 




30 




10.025 


100.5 


78.93 


.263 


6 






203. 


41 209 


32 365 


107.9 


31 






10. 


100.0 


78.54 


.262 






40 


200. 


40 000 


31 416 


104.7 


32 






9. 


81.00 


63.62 


.212 






35 


187.09 


35 000 


27 489 


91.63 




31 




8.928 


79.71 


62.60 


.209 


^rri 2 




5 




181.94 


33 102 


25 999 


86.67 


33 






8. 


64.00 


50.27 


.168 


^s^ 


7 






180. 


32 400 


25 447 


84.82 




32 




7.95 


63.20 


49.64 


.165 


s^^ 






30 


173.21 


30 000 


23 562 


78.54 




33 




7.08 


50.13 


39.37 


.131 


^6'^ 


8 






165. 


27 225 


21 382 


71.27 


34 






7. 


49.00 


33.48 


.111 


§"?o 




6 




162.02 


26 250 


20 618 


68.73 




34 




6.304 


39.74 


31.21 


.104 






25 


158.12 


25 000 


19 635 


65.45 




35 




5.614 


31.52 


24.76 


.082 


Tjd-: 


9 






148. 


21 904 


17 203 


57.34 


35 


36 




5. 


25.00 


19.64 


.065 


Is" 




7 




144.28 


20 817 


16 350 


54.50 




37 




4.453 


19.83 


15.57 


.052 


rt"^ >> 






20 


141.43 


20 000 


15 708 


52.36 


36 






4. 


16.00 


12.57 


.042( 


^b-^ 


10 






134. 


17 956 


14 103 


47.01 




38 




3.965 


15.72 


12.35 


.0411 


>c J3 >> 




8 




128.49 


16 510 


12 967 


43.22 




39 




3.531 


12.47 


9.79 


.033 


•So, 






15 


122.48 


15 000 


11 781 


39.27 





40 




3.144 


9.88 


7.76 


026 


TJ'^ 



672 



}.-^ORDAGE, WIRE AND CABLES. 



4, — Properties op Roebling Steel Wire. 



to 

a 

o 

fee 

S 

12; 



10 

11 

12 
13 
14 

15 
16 
17 
18 
19 
20 
21 
22 
23 
24 

25 

26 
27 
28 
29 

30 
31 
32 
33 
34 

35 
36 



♦English System. 



^2 
H a 



460 
430 
393 
362 
331 

307 
283 
263 
244 
.225 

207 
192 
177 
162 
148 
135 
120 
105 
092 
080 

072 
063 
054 
047 
041 

.035 
032 
028 

.025 
023 

.020 
018 
017 

.016 
015 

.014 
0135 
013 

on 

010 

0095 
009 



OS 

:3 . 

CO 'V 

.So 



.166191 
.145221 
.121304 
.102922 
.086049 
,074023 
,062902 
054325 
,046760 
,039761 
,033654 
028953 
024606 
020612 
017203 
014314 
011310 
008659 
006648 
005027 
004071 
003117 
002290 
001735 
001320 
000962 
000804 
000616 
000491 
000415 
000314 
000254 
000227 
000201 
000177 
000154 
000143 
000133 
000095 
000079 

000071 
000064 



e3 



Weight In 
pounds. 



t-^ 



16.619 

14,522 

12,130 

10,292 

8.605 

7.402 

6,290 

5.433 

4.676 

3.976 

3.365 

2.895 

2,461 

2,061 

1.720 

1,431 

1,131 
866 
665 
503 

407 
312 
229 
174 
132 

96 

80 

62 

49 

42 

31 
25 
23 
20 
18 

15 

14 

13 
9.5 
7.9 

7.1 
6.4 



558.4 
487.9 
407.6 
345.8 
289.1 

248.7 
211.4 
!.5 
157.1 
133.6 

13.1 

97.3 

82.7 

69.3 

57.8 

48.1 

38.0 

29.1 

22.3 

16.9 

13.7 

10.5 
7.70 
5.83 
4.44 

3.23 
2.70 
2.07 
1.65 
1.40 
1.06 
.855 
.763 
.676 
.594 
.517 
.481 
.446 
.319 
.264 

.238 
.214 



2,948 
2.576 
2.152 

.826 
1,527 
1,313 
1,116 

964 

830 

705 

597' 

514 

437 

366 

305 

254 

201 

154 

118 
89.2 

72.2 
55.3 
40.6 
30.8 
23.4 

17.1 
14.3 
10.9 
8.71 
7.37 
5.58 
4.51 
4.03 
3.57 
3.14 

2.73 
2.54 
2.36 
1.69 
1.39 
1.26 
1.13 



0)0 



1,791 

2,050 

2,453 

2,891 

3.458 

4,020 

4,731 

5,478 

6,365 

7.485 

8,843 

10,279 

12,095 

14,439 

17,300 

20,792 

26,315 

34,376 

44,762 

59,206 

73,099 

95,511 

129,954 

171,556 

225,428 

309,310 

370,096 

483.325 



tMetrlc System. 



t^ O) 
CD-M 

OJS 



11.683 

10.921 

9.982 

9.195 

8.407 

7.798 
7.188 
6.680 
6.198 
5.715 
5.257 
4.877 
4.496 
4.115 
3.759 

3.429 
3.048 
2.667 
2.337 
2.032 

1.829 
1.600 
1.372 
1.194 
1.041 

.8890 
.8128 
.7112 
.6350 
. 5842 

.5080 
.4572 
.4318 
.4064 
.3810 
.5556 
.3429 
.3302 
.2794 
.2540 
.2413 
.2286 



c3 no 

QQ O 

BB 

osS 



•— 'O TO — 

I- 



107.200 
93.673 
78.258 
66.404 
55.510 

47.759 
40.580 
35.046 
30.171 
25.652 
21.705 
18.668 
15.876 
13.299 
11.098 
9.2347 
7.2966 
5.5865 
4.2895 
3.2422 

2.6274 
2.0106 
1.4784 
1.1197 
0.8511 

0.6207 
0.5189 
0.3973 
0.3167 
0.2680 

0.2027 
0.1642 
0.1464 
0.1297 
0.1140 

0.0993 

0.0923 

0.0856 

0.0613 

0.0507 

0.04573 

0.04104 






7536 
6585 
5502 
4668 
3902 
3357 
2853 
2464 
2121 
1803 
1526 
1314 
1116 

935 

780 
•649 

513 

393 

302 

230 

184.7 
141.3 
103.9 
78.71 
59.83 
43.64 
36.48 
27.93 
22.26 
18.84 

14.25 

11.54 

10.29 

9.12 

8.01 

6.98 
6.49 
6.02 
4.31 
3.56 

3.21 
2.90 



S3 . 

beg H 



830.9 
726.1 
606.5 
514.7 
430.4 

370.1 
314.5 
271.7 
233.9 
198.7 
168.3 
144.9 
123.2 
103.2 
85.96 

71.59 
56.65 
43.41 
33.26 
25.14 

20.35 

15.59 

11.44 

8.68 

6.60 

4.82 
4.03 
3.07 
2.45 
2.08 
1.57 
1.27 
1.14 
1.01 
0.885 
0.769 
0.716 
0.665 
. 0.476 
0.392 

0.355 
0.318 



♦Thlstable was calculated on a basis of 483.84 lbs. per cubic foot for steel wire. 
Iron wire is a trifle lighter. 

The brealcing strains were calculated for 100,000 lbs. per sq. In. throughout, sim- 
ply for convenience, so that the breaking strains of wires of any strength per square 
inch may be quickly determined by multiplying the values given in the table by the 
ratio between the strength per square inch and 100,000. Thus, a No. 15 wire, with a 

strength per square Inch of 150,000 lbs., has a breaking strain of 407x |oo'ooq = 610.5 lbs. 

It must not be thought from this table that steel wire Invariably has a strength 
of 100,000 lbs. per sq. in. As a matter of fact it ranges from 45,000 lbs. for soft an- 
nealed to over 400,000 lbs. per sq. in. for hard wire. 

tThls table gives the strength of wire at the rate of 70.3 kilograms per square milli- 
meter, which is equivalent to 100,000 lbs. per sq. in. and was calculated on this basis 
simply for convenience. Steel wire may have a tensile strength from 30 to 300 kilo- 
grams per square millimeter, according to treatment, composition, etc. 



STEEL WIRE. WIRE ROPE, 673 

Wire Rope. — (Adapted from Roebling.) Wire rope is made of wires 
either (1) twisted together ^ which is the most common form, or (2) laid 
parallel to each other, as employed in cables of large suspension bridges. 

(1) Twisted or stranded rope may be either (a) round or (b) iiat. The 
wire strands are each commonly composed of 4, 7, 12, 19 or 37 wires twisted 
together generally in the opposite direction to the twists of the strands into 
rope. 

Round wire rope is composed of a number of wire strands twisted around 
a core of hemp or around a wire strand or wire rope. Standard wire rope is 
made of 6 wire strands and a hemp core practically of the same size. But 
for special purposes 4, 5, 7, 8, 9 or any reasonable number of strands may be 
utilized. If the strands are twisted around the core to the right the rope is 
called "right lay," and if to the left "left lay." When wires and strands are 
twisted in the same direction the resulting rope is called "lang" rope. The 
shorter twists form the more flexible rope, and for great flexibility the 
strands themselves may consist of wire ropes, as in tiller ropes. Generally, 
the core is hemp saturated with tar, providing little additional strength but 
acting as a cushion to preserve the shape of the rope and helping to lubricate 
the wires. A wire-strand or wire-rope core will usually add from 7 to 10 per 
cent to the strength of the rope, but, excepting in stationary ropes, the wear 
from core friction is as rapid as the outside wear of the rope. For ships' 
hawsers and running ropes, where great pliability is demanded, as well as 
resistance to outside friction, the strands are made of 12 or 18 wires twisted 
about a hemp center. 

Flat wire rope consists of a number of strands laid side by side in alter- 
nate right and left lay, and sewed together with annealed wire so as to form 
a band or ribbon usually from ^ to 3^ inch in thickness, and from 1 inch 
upward in width. (The sewing wires wear out and .have to be renewed 
occasionally.) This kind of wire rope is specially adapted for long and 
heavy hoisting as in deep shafts, under which conditions it possesses advan- 
tages over round rope which has a tendency to twist and untwist. 

The wires composing wire ropes and cables are usually of steel or iron; 
but for a few special purposes they are sometimes made of copper, bronze, 
etc The strength of iron wire ranges from 45,000 to 100,000 lbs. per sq. in. 
The three grades of steel employed are open -hearth steel with an ultimate 
strength of from 50,000 to 130,000 lbs. persq. in.; crucible steel from 130,000 
to 190,000 lbs. per sq. in.; and plough steel from 190,000 to 350,000 lbs. per 
sq. in. 

The strength of a wire rope is seldom more than 90 per cent of the 
aggregate strength of all of its wires in a testing machine; and the average 
strength is only about 82H P^r cent. This is due to the difficulty in making 
perfect attachments at the ends of the test piece. 

The safe strength of wire rope, that is, the working load , is usually assumed 
at Vs the ultimate strength, for ordinary work. For severe or hazardous 
work the factor of safety is increased from 5 up to even 10 or 15, as for 
elevator service. 

The sizes of drum and sheave should be large enough that the wires of 
the rope shall not be strained beyond the elastic limit in passing over them. 
In general, their diameters increase with the size of the rope and its speed. 
If the diameters are too small the life of the rope will be shortened. 

Lubricants, as linseed oil, tar or other similar materials free from acids 
or corrosive substances, should be used freely and frequently in preventing 
rust and reducing wear, thus increasing the life of the rope. In order to 
provide a protection against the action of salt air, rust, etc., wire is often 
galvanized or tinned, as for ships' rigging, etc. ; but ropes subject to constant 
bending around drums and sheaves are not usually so treated. 

Wire rope must not be coiled or uncoiled like hemp rope. When it is 
received upon a reel, the latter should be mounted upon a spindle or turn- 
table and the rope then run off. When shipped in a coil it should be rolled 
along the ground like a wheel. All untwisting and kinking must be avoided. 
When a wire rope is to be cut, soft iron wire should be served on each side 
of the place where the division is to be made to prevent the rope from 
untwisting. 

The "diameter" of a rope is the diameter of a circumscribing circle and 
hence is the longest diameter. 



674 



35.— CORDAGE, WIRE AND CABLES, 



5. — RoEBLiNG Round Wire-Rope. 

(Swedish Iron, Cast Steel, Extra Strong Crucible Cast Steel, and 

Plough Steel.) 

Table of dimensions, weight, breaking strength, safe (Vs) strength. 

Also minimum diameter of drum or sheave. 



to 


i 

l-H 

3 


_p 


Breaking Strength 


Safe (V5) Strength 


Min. Diam. of Drum 


o 


fe w 


in 1,000 lbs. 


in 1,000 Pounds. 


or Sheaves in Ft. for 








bO 


^ r 






bo 


^- 






W) 


• 


c 


^ 


fe'H 




. 


f3 ^• 

O 0) (D 


^ 






(U (U 


s^ 






f3 ^• 
<u q3 


^ 


Lj 


D 


Ss 




% 




02 




S^ 




CO 




1j 


^3^ 


xn 


a 


i 

a 
< 




^ o 

0) u 


1 


xtra S 

Cruci 

Cast S 


1 

o 


'd 




Xtra S 

Cruci 

Cast S 





^ 
<u u 


c^ 







« 


0} 


o 


W 


p^ 


C/5 


c:» 


W 


^ 


C/3 


(_) 


W 


P^ 




A 


B 


C 


D 


A 


B 


c 


D 


A 


B 


c 


D 



Rope 


Composed of 6 


Stran 


ds and a Hemp Center, 19 Wires to the Strand. 


23^ 


SH 


11.95 


228. 


456. 


532. 


610. 


45.6 


91.2 


106.4 


122.0 


16. 


10. 


10. 


11. 


2V^ 


7Vh 


9.85 


189. 


379. 


444. 


508. 


37.9 


75.8 


88.8 


101.6 


17. 


9.5 


9.5 


10. 


214 


IVh 


8.00 


156. 


312. 


364. 


416. 


31.2 


62.4 


72.8 


83.2 


13. 


8.5 


8.5 


9. 


2 


fii/f 


6.3C 


124. 


248. 


288. 


330. 


24.8 


49.6 


57.6 


66.0 


12. 


8. 


8. 


8. 


1^^ 


A 


4.85 


96. 


192. 


224. 


256. 


19.2 


38.4 


44.8 


51.2 


10. 


7.25 


7.25 


7.5 


1^/8 


5 


4.15 


84. 


168. 


194. 


222. 


16.8 


33.6 


38.8 


44.4 


8.5 


6.25 


6.25 


6. 


11/^ 


434' 


3.55 


72. 


144. 


168. 


192. 


14.4 


28.8 


33.6 


38.4 


7.5 


5.75 


5.75 


5.5 


P/^ 


414 


3.00 


62. 


124.- 


144. 


164. 


12.4 


24.8 


28.8 


32.8 


7. 


5.5 


5.5 


5.25 


1¥ 


4 


2.45 


50. 


100. 


116. 


134. 


10.0 


20.0 


23.2 


26.8 


6.5 


5. 


6. 


5. 


IV^ 


?M> 


2.00 


42. 


84. 


98. 


112. 


8.4 


16.8 


19.6 


22.4 


6. 


4.5 


4.5 


4.5 


1 


3 


1.58 


34. 


68. 


78. 


88. 


6.8 


13.6 


15.6 


17.6 


5.25 


4. 


4. 


4.25 


v^ 


234- 


1.2c 


26. 


52. 


60. 


68. 


5.2 


10.4 


12.0 


13.6 


4.5 


3.5 


3.5 


3.75 


% 


214 


0.89 


19.4 


38.8 


44. 


50. 


3.88 


7.76 


8.8 


10.0 


4. 


3. 


3. 


3.5 


% 


2 


0.62 


13.6 


27.2 


31.6 


36. 


2.72 


5.44 


6.32 


7.2 


3.5 


2.25 


2.25 


3. 


■h 


1^4 


0.5C 


11.0 


22.0 


25.4 


29. 


2.20 


4.40 


5.08 


5.8 


2.75 


1.75 


1.75 


2.5 


V9 


m 


0.3c 


8.8 


17.6 


20.2 


22.8 


1.76 


3.52 


4.04 


4.56 


2.25 


1.5 


1.5 


2. 


^ 


Wa 


0.3C 


6.8 


13.6 


15.6 


17.7 


1.36 


2.72 


3.12 


3.54 


2. 


1.25 


1.25 


1.5 


Vn 


\v^ 


0.22 


5.0 


10.0 


11.5 


13.1 


1.00 


2.00 


2.31 


2.62 


1.5 


1. 


1. 


1. 




1 


0.15 


3.4 


6.8 


8.1 


9.0 


.68 


1.36 


1.62 


1.80 


1. 


.67 


.67 


.88 


H 


M 


0.10 


2.4 


4.8 


5.4 


6.0 


.48 


.96 


1.08 


1.20 


.75 


.50 


.50 


.67 



Rope Composed of 6 Strands and a Hemp Center, 7 Wires to the Strand. 



w 


434 


3.55 


68. 


136. 


158. 


182. 


13.6 


27.2 


31.6 


36.4 


13. 


8.5 


8.5 


8.5 


Wh 


414 


3.00 


58. 


116. 


136. 


156. 


11.6 


23.2 


27.2 


31.2 


12. 


8. 


8, 


8. 


IVa 


4 


2.45 


48. 


96. 


112. 


128. 


9.6 


19.2 


22.4 


25.6 


10.75 


7.25 


7.25 


7.25 


m 


3V^ 


2.00 


40. 


80. 


92. 


106. 


8.0 


16.0 


18.4 


21.2 


9.5 


6.25 


6.25 


6.25 


1 


3 


1.58 


32. 


64. 


74. 


84. 


6.4 


12.8 


14.8 


16.8 


8.5 


5.75 


5.75 


5.5 


v^ 


23^ 


1.2c 


24. 


48. 


56. 


64. 


4.8 


9.6 


11.2 


12.8 


7.5 


5. 


5. 


5. 


% 


2H 


0.89 


08.6 


37.2 


42. 


48. 


3.72 


7.44 


8.4 


9.6 


6.75 


4.5 


4.5 


4. 


^ 


2% 


0.75 


15.8 


31.6 


36.8 


42. 


3.16 


6.32 


7.36 


8.4 


6. 


4. 


4. 


3.5 


Yh 


2 


0.62 


13.2 


26.4 


30.2 


34. 


2.64 


5.28 


6.04 


6.8 


5.25 


3.5 


3.5 


3. 


■h 


IVa 


0.5C 


10.6 


21.2 


24.6 


28. 


2.12 


4.24 


4.92 


5.6 


4.5 


3. 


3. 


2.75 


V, 


IV? 


0.39 


8.4 


16.8 


19.4 


22. 


1.68 


3.36 


3.88 


4.4 


4. 


2.5 


2.5 


2.5 


^ 


IH 


0.3C 


6.6 


13.2 


15.0 


17.1 


1.32 


2.64 


3.00 


3.42 


3.25 


2.25 


2.25 


2. 


H 


IH 


0.22 


4.8 


9.6 


11.1 


12.7 


.96 


1.92 


2.22 


2.5^ 


2.75 


2. 


2. 


1.5 


^ 


1 


0.15 


3.4 


6.8 


7.7 


8.7 


.68 


1.36 


1.54 


1.7^ 


2.5 


1.75 


1.75 


1.25 


^% 


Vs 


0.125 


2.8 


5.6 


6.4 


7.3 


.56 


1.12 


1.28 


1.46 


2.25 


1.5 


1.5 


1. 



Note. — ^The above rope is furnished either galvanized or tinned ; also with 
wire center — at an extra cost of 10 per cent for each. For standard hoisting 
rope the Swedish iron (A) and cast steel (B)with 19 wires to the strand, are 
used; while for transmission or haulage, the same, but with 7 wires to the 
strand, are used. Before ordering, consult the manufacturers in regard to 
the best size of rope, grade of steel, etc., to use, if not familiar with same. 



WIRE ROPE AND FASTENINGS. 



675 



Wire Rope Fastenings. 
(Best Forged Steel.) 



Fig. 20. Closed Socket. 





Fig, 21. Open Socket. 




Fig. 22. Socket and Swivel Hook. Fig. 23. Open Socket and Hook. 



Fig. 24. Special Swivel Hook and 
Socket. (Double Swivel). 



Fig. 25. Hook and Thimble. 




Fig. 27. Closed Cast-iron Socket for Fig. 28. Open Cast-iron Socket for 
Suspension Bridge and Cableway. Suspension Bridge and Cableway. 




Fig. 29. Eye-bolt and Open Socket. 




Fig. 30. Crosby Wire-Rope Clip. Fig. 31. Jupiter Wire-Rope Clip, 




Fig. 32. Roebling's Extra Heavy Wire-Rope Clamp with Three Bolts. 




Fig. 33. Tumbuckle. 



676 25.— CORDAGE, WIRE AND CABLES. 

EXCERPTS AND REFERENCES. 

Telephone Cable in the St. Gotthard Tunnel ("Elektrotechn Zeit- 
schrift" for June 27, 1901; Eng. News, Dec. 5, 1901).— Illustrated. A 
paper-and-air-insulated cable. It includes 7 two-wire circuits, each wire 
1.8 m m. in dia., each set being covered with paper tape to a dia. of 7 m m.; 
stranded together and covered with a triple envelope of cotton and a 
double tin -lead sheath; outside is a layer of waterproof compound, a strong 
armor of interlocking steel wires and an outer coating of jute yam soaked 
in a protecting compound. The finished external diameter is 44 m m., or 
1.7 ins. 

READER'S MEMORANDA. 

The following skeleton outline is for the use of the reader in making 
reference to tables and general items of interest which may be found in this 
book or in other works. 

Cordage. 

Page 
Page 
Page 

Steel Wire. 

Page 

Page 

Page 

Copper Wire. 

Table 1, Section 70, Electric Power and Lighting Page 

Page 
Page 
Aluminum Wire. 

Page 
Page 
Page 

Cables. 

Page 
Page 
Page 

Laying Cables. 

Page 
Page 
Miscellaneous. 
Table 1, Section 34, Metal Gages Page 

Page 
Page 



1. 

2. 
3. 


See 
See 
See 


4. 
5. 
6. 


See 

See 
See 


7. 
8. 
9. 


See 
See 
See 


10. 
11. 
12. 


See 

See 
See 


13. 
14. 
15. 


See 
See 

See 


16. 
17. 


See 
See 


18. 
19. 
20. 


See 
See 
See 



36.— PIPES AND TUBES. 

(See also various pipes, fittings and specials in Sec. 64, Water Works, page 

1207, etc.) 



1. — *Standard Wrought Iron Welded Steam, Gas and Water Pipe. 
(National Tube Works.) 

[For Weight of Seamless Brass tubing, iron pipe size (1), following page, 
multiply tabulated weight by 1 . 07.] 

(a) External Diameters and Properties. 







External. 




L'thper 


Couplings for (1), next page. 


Nom. 
Diam. 








Thre'ds 

per 

Inch. 


Sq. Ft. 

Exter'l 

Heat'g 

Surf. 






Diam. 


Circum. 


Area. 


Inside 
Diam. 


Outside 
Diam. 


Length 


Aver'ge 
Weight 


Ins. 


Ins. 


Ins. 


Sq. Ins. 


No. 


Ft. 


Ins. 


Ins. 


Ins. 


Lbs. 


H 


.405 


1.272 


.1288 


27 


9.44 


11/32 


19/32 


13/16 


.031 




.540 


1.696 


.2290 


18 


7.07 


15/32 


23/32 


15/16 


.046 


^ 


.675 


2.121 


.3578 


18 


5.66 


37/64 


27/32 


1 1/16 


.078 


^ 


.840 


2.639 


.5542 


14 


4.55 


23/32 


1 


1 5/16 


.124 


H 


1.050 


3.299 


.8659 


14 


3.64 


63/64 


1 21/64 


1 9/16 


.250 


1 


1.315 


4.131 


1.3581 


11^ 


2.90 


1 11/64 


1 9/16 


1 13/16 


.455 


1/4 


1.660 


5.215 


2.1642 


11^ 


2.30 


1 1/2 


1 61/64 


2H 


.562 


1^ 


1.900 


5.969 


2.8353 


11^ 


2.01 


1 3/4 


2 7/32 


m 


.800 


2 


2.875 


7.461 


4.4301 


11^ 


1.61 


2 7/32 


2 3/4 




1.250 


2^ 


2.875 


9.032 


6.4918 


8 


1.33 


2 21/32 


3 9/32 


2% 


1.757 


3 


3.500 


10.996 


9.6211 


8 


1.09 


3 1/4 


3 15/16 


m 


2.625 


3^ 


4.000 


12.566 


12.566 


8 


.955 


3 25/32 


4 7/16 


3% 


4.000 


4 


4.500 


14.137 


15.904 


8 


.849 


4 17/64 


5 


3% 


4.125 


m 


5.000 


15.708 


19.635 


8 


.764 


4 3/4 


5 1/2 


3^ 


4.875 


5 


5.563 


17.477 


24.306 


8 


.687 


5 9/32 


6 7/32 


m 


8.437 


6 


6.625 


20.813 


34.472 


8 


.577 


6 11/32 


7 5/16 


m 


10.625 


7 


7.625 


23.955 


45.664 


8 


.501 


7 3/8 


8 5/16 


4^ 


11.270 


8 


8.625 


27.096 


58.426 


8 


.443 


8 3/8 


9 5/16 


4% 


15.150 


9 


9.625 


30.238 


72.760 


8 


.397 


9 7/16 


10 3/8 


5^ 


17.820 


10 


10.750 


33.772 


90.763 


8 


.355 


10 7/16 


11 21/32 


QH 


27.700 


11 


11.750 


36.913 


108.43 


8 


.325 


11 15/32 


12 21/32 


QH 


33.250 


12 


12.750 


40.055 


127.68 


8 


.299 


12 7/16 


13 7/8 


QH 


43.187 



*Allow variation of 5 per cent, above and 5 per cent, below standard in 
weight per foot. Cannot cut to length closer than i^ inch. 

t Shipped threads and couplings (above) unless otherwise ordered. 

t Shipped plain ends unless otherwise ordered. Where Extra Strong 
Pipe is ordered with threads and couplings, regular line pipe couplings (not 
shown above) will be furnished, unless otherwise specified. 

il Shipped plain ends unless otherwise ordered. 



677 



678 



ZQ.— PIPES AND TUBES. 



1. — Standard Wrought Iron Welded Pipe. — Concluded, 
(b) Internal Diameters and Properties. 





Internal. 


Metal. 


Nom; 

Weight 

per 

Foot. 


L'thper 
Sq. Ft. 
Intern'l 
Heat'g 
Surf. 


L'th of 

Pipe 

Con'tg 

1 Cubic 

Foot. 


U.S. 
Gallons 


Nom. 
Diam. 


Diam. 


Circum. 


Area. 


Thick- 
ness. 


Area. 


per 
Ft. of 
Pipe. 


Ins. 


Ins. 


Ins. 


Sq. Ins. 


In. 


Sq. Ins. 


Lbs. 


Ft. 


Ft. 


Gals. 



H 


.269 


.845 


M 


.364 


1.144 


% 


.493 


1.549 


3^ 


.622 


1.954 


H 


.824 


2.589 


1 


1.047 


3.289 


IH 


1.380 


4.335 


m 


1.610 


5.058 


2 


2.067 


6.494 


2^ 


2.467 


7.750 


3 


3.066 


9.632 


3H. 


3.548 


11.146 


4 


4.026 


12.648 


4^ 


4.508 


14.162 


5 


5.045 


15.849 


6 


6.065 


19.054 


7 


7.023 


22.063 


8 


7.981 


25.073 


9 


8.937 


28.076 


10 


10.018 


31.472 


11 


11.000 


34.558 


12 


12.000 


37.699 



(1) Black or Galvanized Standard Weight Pipe.f 



.0568 


.068 


.0720 


.241 


14.2 


2535. 


.1041 


.088 


.1249 


.42 


10.5 


1383. 


.1909 


.091 


.1669 


.559 


7.76 


754.3 


.3039 


.109 


.2503 


.837 


6.15 


473.8 


.5333 


.113 


.3326 


1.115 


4.64 


270.0 


.8609 


.134 


.4972 


1.668 


3.66 


167.3 


1.4957 


.140 


.6685 


2.244 


2.77 


96.3 


2.0358 


.145 


.7995 


2.678 


2.38 


70.8 


3.3556 


.154 


1.074 


3.609 


1.85 


42.9 


4.7800 


.204 


1.712 


5.739 


1.55 


30.1 


7.3827 


.217 


2.238 


7.536 


1.25 


19.5 


9.886 


.226 


2.680 


9.001 


1.08 


14.56 


12.730 


.237 


3.174 


10 665 


.949 


11.31 


15.960 


.246 


3.675 


12.34 


.848 


9.02 


19.985 


.259 


4.321 


14.602 


.757 


7.20 


28.886 


.280 


5.586 


18.762 


.630 


4.98 


38.743 


.301 


6.921 


23.271 


.544 


3.72 


50.021 


.322 


8.405 


28.177 


.478 


2.88 


62.722 


.344 


10.04 


33.701 


.427 


2.29 


78.822 


.366 


11.94 


40.065 


.381 


1.82 


95.034 


.375 


13.40 


45.95 


.348 


1.52 


113.09 


.375 


14.59 


48.985 


.319 


1.27 



H 
M 
H 
H 

1 

2 

2^ 

3 

4 

4^ 
5 
6 

7 





(2) 


.205 


.644 


.294 


.924 


.421 


1.323 


.542 


1.703 


.736 


2.312 


.951 


2.988 


1.272 


3.996 


1.494 


4.694 


1.933 


6.073 


2.315 


7.273 


2.892 


9.086 


3.358 


10.549 


3.818 


11.995 


4.280 


13.446 


4.813 


15.120 


5.751 


18.067 


6.625 


20.813 


7.625 


23.955 


8.625 


27.096 


9.750 


30.631 


11.750 


36.914 



Standard 

.033 

.068 

.139 

.231 

.425 

.710 

1.271 

1.753 

2.935 

4.209 

6.569 

8.856 

11.449 

14.387 

18.193 

25.976 

34.472 

45.664 

58.426 

74.662 

108.43 



rd Extra Strong Pipe 


.t 




.100 ) .096 


.29 


18.63 


4364. 


.123 


.161 


.54 


12.99 


2118. 


.127 


.219 


.74 


9.07 


1035. 


.149 


.323 


1.09 


7.05 


623. 


157 


.441 


1.39 


5.11 


339. 


.182 


.648 


2.17 


4.02 


202.8 


.194 


.893 


3.00 


3.00 


113.1 


.203 


1.082 


3.63 


2.56 


82.2 


.221 


1.495 


5.02 ' 


1.97 


49.1 


.280 


2.283 


7.67 


1.65 


34.2 


.304 


3.052 


10.25 


1.33 


21.95 


.321 


3.710 


12.47 


1.14 


16.25 


.341 


4.455 


14.97 


1 00 


12.57 


.360 


5.248 


18.22 


.893 


10.01 


.375 


6.113 


20.54 


.793 


7.92 


.437 


8.496 


28.58 


.664 


5.54 


.500 


11.192 


37.67 


.598 


4.18 


.500 


12.762 


43.00 


.502 


3.15 


.500 


14.334 


48.25 


.443 


2.46 


.500 


16.101 


54.25 


.399 


1.93 


.500 


19.25 


65.00 


.325 


1.33 







(3) Standard Double 


Extra Strong Pipe. I] 






^ 


.244 


2.639 


.554 


.298 


.507 


1.7 


15.67 


260.0 


.029 


M 


.422 


3.299 


.866 


.314 


.726 


2.44 


9.05 


166.3 


.045 


1 


.587 


4.131 


1.358 


.364 


1.087 


3.65 


6.51 


106.0 


.071 


IM 


.885 


5.215 


2.164 


.388 


1.549 


5.2 


4.32 


66.5 


.113 


1^ 


1.088 


5.969 


2.835 


.406 


1.905 


6.4 


3.51 


50.8 


.148 


2 


1.491 


7.461 


4.430 


.442 


2.686 


9.02 


2.56 


32.5 


.231 


2H 


1.755 


9.032 


6.492 


.560 


4.073 


13.68 


2.18 


22.20 


.339 


3 


2.284 


10.996 


9.621 


.608 


5.524 


18.56 


1.67 


14.97 


.502 


3^ 


2.716 


12.566 


12.566 


.642 


6.772 


22.75 


1.41 


11.45 


.656 


4 


3.136 


14.137 


15.904 


.682 


8.180 


27.48 


1.22 


9.06 


.830 


4^ 


3.564 


15.708 


19.635 


.718 


9.659 


32.53 


1.07 


7.33 


1.025 


5 


4.063 


17.477 


24.306 


.750 


11.341 


38.12 


.94 


5.93 


1.270 


6 


4.875 


20.813 


34.472 


.875 


15.807 


53.11 


.78 


4.18 


1.80 


7 


5.875 


23.955 


45.664 


.875 


18.555 


62.38 


.65 


3.15 


2.38 


8 


6.875 


27.096 


58.426 


.875 


21.304 


71.62 


.55 


2.47 


3.05 



t J II Foot-notes, preceding page. 



WROUGHT IRON PIPE.. LEAD PIPE. 



679 



2. — Lead and Tin Lined Lead Pipe. 
(Tatham and Brothers, New York.) 



v>a 


'^.• 


Weight 


Jr)a 


-^ . 


Weight 


^a 


.^ 


Weight 


^>B 


M 


Weight 






per Ft. 


M 




per Ft. 


^5 




per Ft. 


^5 


M 

H c 


per Ft. 


Ins. 


Ins. 


Lbs. 


Ins. 


Ins. 


Lbs. 


Ins. 


Ins. 


Lbs. 


Ins. 


Ins. 


Lbs 


H 


.05 


0. 42^42 


6 V?. 


.25 


3. 


1 1 


.11 


2. 


ly?. 


.23 


6.5 


1 «« 


.06 


0.625 


Vh 


.08 


0.72^72 


2 " 


.14 


2.5 




.25 


7.5 


2 " 


.08 


0.75 


1 « 


.09 


1. 


3 « 


.17 


3.25 


• " 


.27 


8. 


3 « 


.12 


1. 


2 « 


.13 


1.5 


4 « 


.21 


4. 


« 


.28 


8.5 


4 « 


.16 


1.25 


3 « 


.16 


2. 


5 « 


.24 


4.75 


IH 


.13 


4. 


5 " 


.19 


1.50 


4 « 


.20 


2.5 


6 « 


.30 


6. 


14 


.17 


5. 


6 « 


.27 


1.75 


5 « 


.22 


2.75 


114 


.10 


2. 


" 


.19 


6. 


T^ 




0.8125 


6 « 


.25 


3.5 


1 « 


.12 


2.5 


• « 


.21 


6.5 


M 




1. 


34 


.08 


1. 


2 «♦ 


.14 


3. 


" 


.23 


7. 


H 


.07 


0.54^54 


1 « 


.10 


1.25 


3 «' 


.16 


3.75 


(( 


.27 


8.5 


1 « 


.09 


0.75 


2 « 


.12 


1.75 


4 « 


.19 


4.75 


w 


.30 


10. 


2 «« 


.11 


1. 


• « 


.14 


2. 


5 « 


.25 


5.75 


2 


.15 


4.75 


3 « 


.13 


1.25 


3 " 


.16 


2.25 


6 « 


.28 


6.75 


u 


.18 


6. 


• « 


.14 


1.50 


4 « 


.20 


3. 


11/^ 


.12 


3. 


« 


.22 


7. 


4 « 


.16 


1.75 


5 « 


.23 


3.5 




.14 


3.5 


u 


.25 


8. 


5 « 


.19 


2. 


6 « 


.30 


4.75 


<( 


.17 


4.25 


u 


.27 


9. 


* « 


.23 


2.5 


1 


.10 


1.5 




.19 


5. 


<( 


.30 


11.75 



^ Repeating decimal: . 42^42 lb. per ft. = 7 lbs. per rod; . 54''54 lb. 
per ft. = 9 lbs. per rod; 0.72^72 lb. per ft. = 12 lbs. per rod 
* Special. 



g decimal: U.4li'4Z lb. per tt. = 
per rod; 0.72^72 lb. per ft.= 
"" bpecial. 

1 Recommended for pressure of 1 5 lbs 

2 " « « « 25 " 



25 
" 38 
« 50 
« 75 

Weight of Lead Pipe per 
Lin. Ft.* 





Thickness. 




Inner 
Diam. 






•h" 


H" 


1^" 


^" 


Ins. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


23^ 


8. 


11. 


14. 


17. 


3 


9. 


12. 


16. 


20. 


sy2 


9.5 


15. 


18. 


22. 


4 


12.5 


16. 


21. 


25. 


43^ 


14. 
15. 


18. 
20. 






5 


25. 


31. 


6 


18. 


24.5 


30. 


37. 



per sq. in., or hydrostatic head of 30 ft. 
" " " " " " " 50 " 

« (( <( M (i U ** 7 ^ " 

u u u u M ({ 1 r\n u 

a u u u u <{ M 1 r A u 

u u u u u u u OAA « 

Lead and Tin Tubing. 
H inch. }i inch. 

Sheet Lead. 

Weight per square foot, 23^, 3, 3}4, 
4, 43^, 5, 6, 8, 9, 10 lbs. and upwards. 

Lighter weights rolled to order at 
special prices. 

Block Tin Pipe. 



* Manufactured in lengths of 
10 ft. 



Lead Waste Pipe. 



^ in., 4, 5, 6 and 8 

• oz. per ft. 

3^ in., 6, 73^ and 10 

oz. per ft. 
^in.,8andl0oz. 

per ft. 
H in., 10 and 12 

oz. per ft. 



1 in., 15 and 18 oz. 
per ft. 

l}im.,lHsindiy2 

lbs. per ft. 
13^ in., 2 and 23^ 

lbs. per ft. 

2 in., 2i and 3 lbs. 
per ft.. 



13^in.,2and31bs. 
per ft. 

2 in., 3 and 4 lbs. 
per ft. 

3 in., 33^, 5 and 6 
lbs. per ft. 

33^in.,41bs.perft. 



4 in., 5, 6 and 8 
lbs. per ft. 

43/^in.,6and81bs. 
per ft . 

5 in., 8, 10 and 12 
lbs. per ft. 

6in.,121bs. perft. 



Special sizes made to order. 
Lead: Wt. per cu. ft., about 710 lbs.; 
per sq. ft. 1 in. thick, about 59.2 
lbs. ; bar 1 in. square and 1 ft. long, 
about 4.93 lbs.; per cu. in., about 
0.411 lb. 

Cast Tin; Wt. per cu. ft., about 459 
lbs. ; per sq. ft. 1 in. thick, about 
38.25 lbs.; bar 1 in. square and 1 
ft. ■'ong, about 3.19 lbs.; percu. in. 
about 0.266 lbs. 



680 



).— PIPES AND TUBES. 



3. — Spiral Riveted Steel Pipe as Manufactured by American Spiral 
Pipe Works, Chicago. 




Fig.l. 



Joint 




Fig. 2. 



Diam- 
eter, 

Inches, 



Standard Weight Pipe. 



Use Galvanized 
Pipe for: 



Exhaust Steam, 
Pump Suction, 
Brine Circulation, 
Refrigerating 
CtoUs. Etc. 



Use Galvanized or 
Asphalted Pipe for, 



Paper and Pulp 

Mills, 
Irrigation, 
Pump Discharge, 
Water Pipe 

Lines, Etc. 



Extra Heavy Weight Pipe. 



Asphalted for: 



Intake Mains, 
Water Works, 
Hydraulic Min 

ing. 
Water Sup.Lines. 



Galvanized and 
Flanged for: 



Compressed Air, 
Pump Suction, 
Condenser Pipes, 
Vacuum Pipes, 
Etc. 



C be 



•OS'S 

o^5 



O) (-1 






<:^s 



o o 

a oj 02 






r 



OPh 



o 

o 



e8 



O 

S OJ OQ 

22g 



fr P. . 

^ v« 03 



3 


No. 20 


$0.50 


$0.35 


2.25 


1500 


No. 18 


$0.55 


$0.40 


2.60 


2000 


4 


( ( 


.70 


.45 


3.00 


1125 


♦• 


.80 


.55 


3.45 


1500 


5 


" 


1.00 


.55 


4.00 


900 


" 


1,10 


.65 


4.50 


1200 


6 


No. 18 


1.20 


.75 


5.00 


1000 


No. 16 


1.30 


.90 


6.40 


1250 


7 


" 


1.40 


.80 


6.00 


860 


( < 


1.50 


.95 


7.50 


1070 


8 


« « 


1.70 


'.95 


7.00 


750 


«' 


1.85 


1.15 


8.90 


935 


9 


*• 


2.00 


1.10 


8.00 


665 


'* 


2.20 


1.30 


10.25 


835 


10 


No. 16 


2.60 


1.45 


11.00 


750 


No. 14 


2.80 


1.65 


13.25 


935 


11 


" 


2.85 


1.55 


12.00 


680 


*• 


3.05 


1.80 


14.75 


850 


12 


•' 


3.15 


1.80 


14.00 


625 


(< 


3.40 


2.15 


17.00 


781 


13 


«« 


3.60 


1.95 


15.00 


575 


** 


3.80 


2.35 


18.25 


720 


14 


No. 14 


4.00 


2.50 


20.00 


670 


No. 12 


5.00 


3.30 


24.50 


935 


15 


( ( 


4.40 


2.75 


22.00 


625 




5.25 


3.60 


26.85 


875 


16 


" 


5.00 


3.05 


24.00 


585 


•* 


6.00 


3.80 


29.20 


820 


18 


" 


6.00 


3.50 


29.00 


520 


( < 


7.00 


4.20 


34.70 


675 


20 


< < 


7.00 


3.90 


34.00 


470 


< ( 


8.00 


4.80 


40.3.0 


655 


22 


No. 12 


9.00 


5.55 


40.00 


595 


No. 10 


10.00 


6.20 


50.10 


765 


24 


• • 


10.50 


6.00 


50 00 


540 


< i 


12.00 


7.00 


60.20 


705 


26 


< < 


11.80 


8.50 


58.00 


505 


( ( 


13.00 


9.55 


66.00 


650 


28 


No. 10 


14.60 


10.25 


27.00 


605 


No. 8 


16.60 


11.65 


83.00 


735 


30 




15.70 


11.25 


79.00 


560 




17.65 


12.60 


90.00 


685 


32 


•• 


16.70 


12.00 


85.00 


525 


« « 


19.25 


13.80 


97.00 


675 


36 


< < 


18.45 


13.20 


94.00 


469 


«« 


21.00 


15.00 


112.00 


573 


40 


« ( 


20.80 


14.90 


106.00 


420 


< < 


25.00 


17.80 


128.00 


515 



The above list is for pipe in standard lengths, with flanges attached or 
bolted joint connection. 

We recommend the use of bolted joints with asphalted pipe for all high 
pressure water works. 



SPIRAL RIVETED STEEL PIPE. 



681 



3. — Spiral Riveted Steel Pipe as Manufactured by American Spiral 
Pipe Works, Chicago. — Concluded. 
Note. — Standard Flanges (Fig. 3) are used with spiral pipe unless other- 
wise specified. When other than the standard flanges are required, be 
sure to give outside diameter of flange, number and size of bolts, and diameter 
of bolt circle. All flanges and flanged fittings are drilled in multiples of 
four so that fittings may be made to face in any quarter,' and holes straddle 
center line. 




Fig. 3. 




.^<^. 



fA^Al 



B^' r% 






Diameters and Drilling of Standard 


♦Dimensions of Standard 






Flanges. 








Fittings. 






Size, 
Inches. 


i 


O 

o 


o 


I 


«M 

o 


S' 


go 

!s 

SI 


M d 


1^ 


ft 




O 


W 


;?; 


m 


yA 


o 


O 


o 


O 


^ 


3 


6 


m 


4 


7/16 


m 


3% 


2^ 


9 


2M 




4 


7 


5 15/16 


8 


7/16 


m 


m 


2 15/16 


11 


2% 


23 


5 


8 


6 15/16 


8 


7/16 


m 


5^ 


3M 


12 


3 


23 


6 


9 


7% 


8 


^ 


m 


^H 


m 


13^ 


314 


23 


7 


10 


9 


8 


^ 


m 


7M 


m 


15 


4^ 


22 


8 


11 


10 


8 


^ 


2 


SH 


4% 


17 


5 


22 


9 


13 


nu 


8 


^ 


2 


9M 


5 1/16 


18^ 


5M 


22 


10 


14 


12M 


8 


}4 


2 


lOH 


5 7/16 


21 


5^ 


33 


11 


15 


13% 


12 


¥z 


2 


11 


5M 


22^ 


5% 


33 


12 


16 


14M 


12 


^. 


2 


12^ 


6 5/16 


24 


6 


33 


13 


17 


15^ 


12 


^ 


2 


13 


5^ 


26 


6^ 


33 


14 


18 


16^ 


12 


U. 


2H 


14 


6 


27 


6^ 


32 


15 


19 


J7 7/16 


12 


^ 


2H 


15 


5^ 


29^ 


6% 


32 


16 


21M 


19^ 


12 


^ 


2H 


16 


6 11/16 


313^ 


7 


32 


18 


23M 


21M 


16 


% 


2^ 


16^ 


8^ 


35 


7^ 


32 


20 


25^ 


23J^ 


16 


% 


2^ 


18 


9^ 


38J^ 


8 


32 


22 


28M 


26 


16 


% 


2^ 


20 


10 


41 


9 


32 


24 


30 


27H 


16 


H 


2^ 


22 


11 


44 


10 


32 


26 


34^ 

36^ 

41 

45^ 

50 


31% 

34 

36 

3SH 

i2H 

46% 


24 

28 
28 
28 
32 
32 


H 

H 

Vs 


3 

3 

3 

3 

3^ 

3^ 


23 

24 
25 
26 
28 
30 


13 

14 
15 
16 
18 
20 








28 








30 








32 








36 








40 

















♦ Face to center dimensions are not changed on reducing outlets to tees 
and crosses. On increasing outlets, the face to center dimensions are the 
same as their respective standards. 

For Pipe Details, see following page. 



682 Z'o.— PIPES AND TUBES. 

Spiral Riveted Steel Pipe Details. 




Fig. 5. Riveted Lap. 




Fig. 6. Flange Connection. 



/Rubber 
PaJii ngf' 





Fig. 7. Bolted Joint Connection. Fig. 8. Slip Joint Connection. 




Fig. 9. Bolts. 




Fig. 10. Gaskets. 




Fig. 11. Fig. 12. 

Threaded Disc. Clamp Band. 



Fig. 13. Reducer. 



EXCERPTS AND REFERENCES. 

Making Tight Joints in Vitrified Pipe at Atlantic City, N. J. (By 

Kenneth Allen. Eng. News, Oct. 8, 1903.) 

Experiments on Reinforced=Concrete Pipes Made for the U. S. Rec- 
lamation Service (By J. H. Quinton. Eng. News, Mar. 9, 1905). — Illus- 
trated 

Reinforced=Concrete Pipe with Reinforced Joint (By Lock Joint Pipe 
Co., N. Y.; Eng. News, Dec. 10, 1908).— Illustrated. 

Standard Specifications for Hard=Drawn Copper Wire (Proc. A. S. T. M., 
Vol. IX., 1909).— Adopted Aug. 16. 1909. 



Illustrations. 

Description. 
Tests of lock -bar pipe 42" dia., Springfield, Mass. 



Eng. Red. 
April 17, '09 



37.— BRIDGES. 



Economic Lengths of Spans. — If a bridge consisting of any number of 
spans (to be determined) is to be built over a stream or other crossing of 
length L, it can be shown that, aside from the (end) abutments.* the most 
economic layout of spans and piers will obtain when the cost of each pier is 
about equal to the cost of that portion of the structure which it supports 
when stripped of about one-half the floor system ; that is, equal to the cost 
of the supported trusses and laterals and about one-half the floor, the cost 
per lin. ft. of same being assumed as proportional to length of span. 
Let L = total length of crossing, in ft.; 

P = cost in dollars of one pier at any given point of profile; 
/= length of economic spans, in ft., based on P; 
ix;=number of spans / in the crossing L; 

(7 = cost in dollars per lin. ft. of trusses, laterals, etc., for span L; 
c = cost in dollars per lin. ft. of trusses, laterals, etc., for span I; 
3/ = total cost of trusses, laterals, etc., piers, and a proportionate cost of 
foundations in abutments. 



Then, since 



we have. 



= — , and / = — 

X X 

y = h Px, 

X 



(1) 



Differentiating, 



and equating with zero 
dy _ _ CJL 
dx ~ x^ 



for minimum, 
+ P = 



whence, P = ^ = c/ (2) 

x^ 

Now the value of c for span / may be obtained from any known value c' 

for span /' as follows: 

c = y (3) 

it being assumed of course that the same types of bridges are used and the 
same specifications; also that the lengths of spans / and V do not vary 
greatly. 

Equation (2) may be applied without serious error to a crossing with 
irregular profile as in Fig. 1. In such a case we would have, numbering the 
piers and spans consecutively, 

^1 = 2 ' 2 = 2 ' 3 = 2 .etc., KV 

which is the proportion stated in the opening paragraph. 



/ 


\ 


/ \/ 




A/ \ 


/ A 


t 


1, 


h 


13 




M 




^ 


p.- - 


]^2-=t 


- "3 


h- 






°^^s^ 







Fig. L 
Referring to Fig. 1 and to equation (4) it will be noted that for the 
deeper, or rather more expensive, foundations the longer spans are re- 
quired. The principles evolved above will apply to spans of plate girders, 
beams, etc., and also to many other economic problems. 

* Abutments usually perform, in part, certain "constant" functions, as 
retaining walls, etc., not connected with "supporting" the spans, and, if 
included in the problem, these functions should be considered carefully. 



683 



684 2^7.— BRIDGES. 

Equation (2) may be transformed into 



X \ c \ w p 



(5) 



an economic form for general use, in which — = ratio of length of span to 

w 
weight in lbs. perlin. ft. of trusses, laterals, etc., and will be found to be 
fairly constant within quite wide limits of /; and 
^ = price in dollars per lb. of w. 

Economic Depth of Plate Girders.— Equation (7) will usually give a 
greater depth of girder than practice warrants, but it will be useful in 
finding out how much material is sacrificed in any practical limitation of 
depth. 

Let M = total max. bending moment in ft. -lbs. on girder; 

A =area of top flange = area of bottom flange, in sq. ins. 
/ = allowable compressive stress in lbs. per sq. in. in top flange; 
rt = effective depth of girder = depth of web, in ins.; 
i = thickness of web in ins.; 

a = total horizontal section of vertical stiffeners and fillers, in sq. ins.; 
L = length of span, in ft.; 
y = total weight of girder, in lbs. 

Then, for a steel girder, asstiming no part of web to resist bending, we have, 

12M 
smce A 



fx 



3.4 X 24ML _L o. r. _._ 3.4 .«, 

y = 2 + 3.4 Ltx + ~T^ ax. (6) 



Differentiating and equating with zero, for minim imi, 
ax fx^ 12 



whence rx: = 16.97 ' 



the economic depth of girder. 

Ex. — A track stringer, 20-ft. span, is designed to support a total uniform 
load, including its own weight, of 2500 lbs. per lin. ft. Assimiing /= 10 000, 
t—i, and a= 30, 'find the economic depth. 

2500 L2 

Solution. — From equation (7) we have, since M = ^ , 



4 



20 
16.97 X 50 X 20^1 sx 10 000 C2iOXi+30r '**'"=• '"''■ 



Economic Depth of Trusses. — In the preceding case, of the plate girder, 
it is to be noted that the weight of each part of the girder is given in terms 
of the variable depth x in order that the value of x may be found for the 
minimum value of y. This will be done also in the present case, of the 
truss, but in order to introduce the working stresses per sq. in. in the com- 
pression members as "constants" into the fundamental formula (8) for total 
weight of truss it is convenient to assume that the bridge has been designed 
using a certain depth of truss (parallel chords) and find whether this depth 
is the most economic one, which will be the case when the value of x in 
equation (10) is equal to the assumed depth. If less, the depth will be de- 
creased; if greater, increased. If the difference is slight the economic 
depth may be assumed equal to the value of x found. One or two trials 
will be sufficient in most cases. The general equation for total weight of 
steel truss members is 



(8) 



Wt. Wt. 
bottom top 
chord. chord. 


Wt. 

end 

posts. 


Wt. 

int. 
posts. 


Wt. 
vert, 
susp. 


Wt. 
diag- 
onals. 


B + T + 


E + 


P + 


V 


+ D 



ECONOMIC DESIGN. ESTIMATING WEIGHTS, 



685 



M 
In which 5= 3.4Jf^(^+ e) 

M 
E= ZAS ^^^^P^iX + %) 

sJ"" 

P= ZAI ^ (1 + %) 



I 



Notation: 
(May vary for each member.) 
M = bending moment, ft. -lbs.; 
f = allowable stress per sq. in. ; 
ic = depth of truss, in ft.; 
^ = panel length, in ft.; 
t^ = added length for two heads; 
% = percentage added for details; 
5 = vertical shear, in lbs. 



V= ZAIy (1 + %) 

fx 

Now, substituting the above values in equation (8) and using summated 
constants for simplicity, we have the following form: 

~^-\-^ + ^ + E'x + P'x+V'x+^ + D'x\ (9) 

Differentiating and equating with zero, for minimimi, 

^ = 3.4 [ - ^ - ^' - ^^ - ^'f% £' + P' + F' + D' 1 = 

dx L x,^ x^ x^ x^ J 

>2 



whence, 



M 



^ y E' + P' -h V + D' 



(10) 



In which B^ = I -j- (p + e), for each bottom chord member; 



T 


I%Ul^%) ' 


top 


E' 


= I^(l + %), • 


end post; 


P' 


= 2-1(1 + %), 


' intermediate post; 


V 


= i'|(l4-%). • 


vertical suspender; 


D' 


= ^1(1 + %), • 


' diag. (incl. counters) 



Estimating Weights of Bridges. — ^The correct weight of a proposed 
structure can be estimated 'only from a complete bill of material of the fin- 
ished design, and the young engineer should use this method on all possible 
occasions. After he has become expert in bridge designing he may resort 
to quicker methods, more or less approximate, depending on the purpose 
for which the estimate is to be used. 

In designing a structure the live loads, snow load and wind loads must 
of course be known or assumed. The lengths of spans, if indefinite, may 
be fixed by the use of formulas (2) and (5); the depths of steel stringers, 
floorbeams and plate-girders, by equation (7); and the depth of trusses, by 
equation (10), bearing in mind, of course, that these values are often fixed 
arbitrarily and that any moderate variation from the economic depth will 
not add greatly to the weight or cost. For instance, the depth of plate- 
girders is usually assumed at about -^z the span, and the depth of trusses 
at about ^ to i the span, the larger ratio applying to the shorter spans. 

The steps in calculating the design and weights of an ordinary span are 
as follows: (1) That part of the roadway which directly supports the live 
load, as the paving, planking and street-car tracks of highway bridges; 
and the "track" (including rails, guard rails, ties and fastenings) of railroad 
(steam and electric) bridges, usually assumed at 400 to 450 lbs. per lin. ft. 
per track. (2) The ballast and corrugated flooring, if used. (3) The 
stringers or joists. (4) The floorbeams. (5) The top and bottom lateral 
systems. (6) The portal and vertical sway bracing. (7) The trusses. 



* (1 + %) is used instead of (p-he) to simplify equation (9). 
t See formula (11) for value of t. 



686 m.-'BRIDGES. 

Each of the above operations, excepting (5) and (6), is dependent on 

the weights obtained from the preceding operations. Although no "backing 
up" is required we have to assume in some of the operations, as (3), (4) and 
(7), the weights of the members we are calculating, and sometimes two or 
three "assumptions" are necessary before the correct one is made. 

In estimating the weights of details it is the practice with many engi- 
neers to add certain percentages to the weights of the main ribs of such 
members, or to add to their lengths, or both. The method of percentages is 
always uncertain, and may vary from a few %, for rivet heads, up to 50 or 
60 % or more for latticing both sides of light channels. For instance, lacing 
may be used instead of latticing, or, what is still cheaper, occasional tie 
plates may be used. Again, a top chord may or may not have a cover 
plate, and with this uncertainty note what a variation in percentage for 
details this would imply. Very close estimates have been made on the 
total weights of bridges by adding a certain percentage, say 25%, to the 
weights of all the main members stripped of their details. As the per- 
centage varies with the type of structure and the specification such a prac- 
tice would be dangerous for any but the most careful expert. 

A good formula for estimating the added length of eye bars to form the 
heads is the following: For forming two heads, 

The added length in ft. ^= M diam of pin in ins (11) 

this to be added to length c. to c. of pins and estimated as a plain bar. 

EXCERPTS AND REFERENCES. 

Diagrams and Formulas for Weights of Steel Bridges and Trestles 

(By H. G. Tyrrell. Eng. News, May and June, 1901).— The following 
excerpts are noted from the formulas: 

Unit stresses in all cases are 10,000 and 12,000 lbs. per sq. in. L = length 
of span in ft., c. to c. bearings. 

Railway Bridges. — Weights are per lin. ft. of single track bridge for 
steel only; 1. 1., 2 100-ton engines followed by 4,000 lbs. per lin. ft. of track: 
Deck plate girder bridge, 100-1- 9L; deck lattice girder bridge, 100+ 8L; 
half through pi. gird. br. with floor. 300-M2L; same with ties on shelf 
angle, 200-}-8iL; same with trough floor, 600-1- lOL; riveted through truss 
br., 4004- 6L; riveted deck truss bridge (ties on top chord), 2004- 7L; pin 
through truss bridge, 4004- 5^L; pin deck truss bridge with stringers, 
4004- 6L; pin deck truss bridge (ties on top chord), 3004- 6L. 

Railway Trestles. — Assumed loads same as above; weight of spans as 
above. Weight of bents and bracing is 9 lbs. per sq. ft. of side profile from 
ground to base of rail. 

Electric Railway Bridges. — Live load assumed 25-ton cars, or 2,000 lbs. 
per lin. ft. of track: Beam bridges, 304- 5iL; deck plate girder bridges, 
304- 5L; Pony truss bridges, 2004- 1.8 L; through truss bridges, 2004- 1.6L. 

Highway Bridges with Wooden Floors. — Assumed dead weight of floor 
is 40 lbs. per sq. ft.; assumed live load is 100 lbs. per sq. ft.; the weights 
are per sq. ft. of floor, and include that only with joists: Girder bridges 
and sidewalks, 3-t-L-T-4.4; same without sidewalks, 34-L-i-3.4; Truss 
bridges with sidewalks, 34-L^8; same without sidewalks, 5-^-L-^7. 

Highway Bridges with Solid Floors. — Assumed dead weight of floor is 
150 lbs. per sq.ft.: Deck plate girder bridges, 34-L-i-2.6; Half-through 
girder bridges, 34-L-5-2.4; Truss bridges, 34-L^4. 

Graphical Method for Finding Bending Stresses in Eyebars (By W. E. 

Belcher. Eng. News, July 17, 1902). — From the formula, 
4 900 000 /^ 

Pt+23 333 000^ 

Where P = stress in lbs. per sq. in. due to bending; 
Pt = working tensile stress in lbs. per sq. in.; 
^ = depth of bar in inches; 
/ = length of bar in inches. 
The formula is based on the value of E = 28 000 000 and a weight of 
metal of 0.28 lb. per cu. in. 



MISCELLANEOUS DATA, 687 

Wind Pressure to be Assumed in the Design of Long Bridge Span 

(By Theodore Cooper. Eng. News, Jan. 5, 1905). 

Nickel Steel for Bridges (By J. A. L. Waddell. Trans. A. S. C. E., 
Vol. LXIII). 

Concrete Floors for Railway Bridges (Eng. News, Feb. 16, 1905). — 
Illustrated. 

Tables and Diagrams of the 31 Bridges Over the Missouri River 

(Eng. News, April 29, 1909). 

Standard Specifications for Structural Steel for Bridges (Proc. A. S. T. M.. 
Vol. IX., 1909).— Adopted Aug. 16, 1909. 

Diagrams and Illustrations. 

Description. Eng. Rec. 

Diagram of compression formulas in long-span bridges Sept. 3, '10 



38.— RAILROAD BRIDGES, 



I.— MOMENTS AND SHEARS— BEAMS OR GIRDERS, 
(a) Uniformly Distributed Loads. 

ze; = load in lbs. per lin. foot; /=span in feet. 

Full Loading — Moments and Shears — Figs. 1 and 2. 
At any point distant x from left end: 

Moment M-s. =\wx {l—x),in ft. -lbs. 
= ^wx (l—x), in in.-lbs. 
Maximum moment (at center), 
= i «;/2, in ft. -lbs. 
=§ wl^, in in.-lbs. 






X 



.\. ^' Pull. 

..1 y' Looclin3 



p^ 



]S 



Fig. 2. Shears. 



Fig. 1. Moments. 
Shear Sx =w (-o""^) i^ 1^^., 

(left end positive; right end negative). 
Maximum shear (at ends), 

= i wl, in lbs. 
Minim imi shear (at center) =0. 

Note that maximum moment occurs at point of minimttm shear; and 
minimum moment occurs at point of maximum shear. 

To Draw a Moment Parabola, divide the span into any number of 
equal parts, odd or even; nimiber the points from one end, beginning with 
zero; and for ordinates, multiply together the numbers equidistant from 
center. Afterward, all ordinates so obtained may be multiplied by a con- 
stant (one that will give the required middle ordinate). See Figs. 3 and 4, 





Fig. 3. Even. Fig. 4. Odd. 

Partial Loading — Moments and Shears — Figs. 5 and 6. 
At any point distant x (>a) from left end : 

Moment M^ = jfj(l-a)2-(x-a)A, in ft.-lbs. 
= 6z£;[-j (l-a)2-ix-a)A 

72_U/t2 

Maximum moment is at re = 



a)2- (it;-o)2 I , in in.-lbs. 
72 + a2 



21 



r \ n ,,^ partial 



<-a 

r — X 



jMx •<- Vs^-H Loading 



Fig. 5. Moments. 




t:^.::{ J 

Fig. 6. Shears. 



688 



MOMENTS AND SHEARS— BEAMS. 689 

Note that curve of moments directly above the loading, to the right of T, 
is a parabola; and to the left of T it is a tangent to the parabola. The point 
of intersection / is directly over the center of gravity of the load, and the 

moment at T, i.e., when x=a, is M,x=^-^(l—ay, in ft.-lbs. 

= —j-{l—a)^, m m.-lbs. 

w 
Shear Sx=Yji^~'^)^~'^ (x — a), in lbs. 

(left end positive; right end negative). 
Maximum ( + ) shear (at x = a): 

= |^^(/-a)2, inlbs. 

Maximum ( — ) shear (at x = l): 

= Y^(l~a)^-w (l-a), in lbs. 

Mmmium (0) shear is at :*: = — ^7 \-a = — ^j— . 

Notice that maximum moment occurs at point of minimum shear; and 
minimum moment occurs at point of maximvim shear. 

Isolated Loading — Moments and Shears — Figs. 7 and 8, 
At any point distant x {>a and < l — c) from left end: 

Moment M^ =^ (y"^^^) -■|(^-«)'' i" ^^--Ibs. 

= — -, — I— -\-bc j —ow (x—a)^,in m.-lbs. 
Moment Ma (when x = a), at Ti 

=-r- I Y+^^l ft.-lbs. = — J — ('2'^ I ^^-"l^s- 
Moment Ma+b (when it: = a-l-Z)), at T2 

= -j- (y+<2^) ft.-lbs. = — j — ( y-f-a^j m.-lbs. 

Maximum moment is at x=-j- (~^ + ^ ) +ct. 

■a-)lt4bt-~^--c-4' Loading ^_^_Lbn;j^;,^g_^__T 



g 



— I — I ^ |f^(|+c>d ^^- ^ 

l<i— ♦ •J,-— .•— 4>^ .. 

Fig. 7. Moments. Fig. 8. Shears. 

Shear •S"x = y ( Y+^^) """^ (^— o). in ft.-lbs. 

=—p (y+^^) —120^ (ii? — o), in in.-lbs. 
Maximum (-H) shear (at ik; = a) 

=y I Y+^^j ft.-lbs. =—v- ( Y+t^ I m.-lbs. 
Maximum ( — ) shear (at x = a + b) 

^ /^^ ^ j.\ f. 1T. 12w /fc2 \ 

=y ( Y"*"^^) ft.-lbs. -=-y- ( Y"^^^) in.-lbS. 

Minimum (0) shear isatit: = -y-(Y-l-cj +a. 

By an extension of the principle, the remarks under "Partial loading" 
can be made to apply to "Isolated loading." Thus, in the above Fig. (7) 



690 



I— RAILROAD BRIDGES. 



the "curve of moments" comprises the parabola T1 — T2 and the adjacent 
tangents, their intersection at / being vertically above the center of gravity 
of the uniform load. 



(b) Concentrated (Engine, Car, Axle, Wheel, etc.) Loads. — During the past 
75 years the weight of the locomotive has increased from about 7 to over 
200 tons. In approaching the latter mark many engineers predicted that 
we were reaching the limit, but similar prophesies may be recalled for the 
100-ton engine which appeared and disappeared as a standard for bridge 
calculations during the short period of about ten years. The following 
may be considered as typical for our heaviest loading — present and pros- 
pective — for bridge calculations: 

Two engines coupled and followed by uniform load of w pounds per Hn. ft. 



Axle concentrations: Bogie axle load = bw 



Driving " 

Tender " 

Or. for floor loads. I ^n^gj.' 

as per the following diagrams: 



= 10 w, each; 
= 6to7w " 

= 14 a; •* 



1 axle. 

4, spaced 5 ft. 

4, spaced 5 and 6 ft. 

2, spaced 7 ft. 



Type R — Engine Diagrams — Axle Loads. 

Note. — Wheel* or rail loads Wi are one-half of axle or track loads w,) 
Diagram No. 1. No. 2. 



5 ^5?$ ?^55 

\h S2.2 2 vo*©^o«JP 



^ ^ ^ ^ 5?^? 

2222 vs>v9vi>\p 






tr.: 



-56- >j<— 53'- -H 

Figs. 9. 



w lbs. 



iDCb 



1. Types "R" Engine Diagrams — Axle Loads. 
[From Figs. 9.] 







Driver 


Tender 


IWt. 

ngine 

ender. 


pi 5? 


Special Pair 


<u 


Bogie 


Loads. 


Loads. 


0^^ 


of Axles, 




Loads. 






Tota 
OneE 
andT 


Unii 

Lc 

Folio 


7' Centres. 


^ 


Each. 


Total. 


Each. 


Total. 


Each. 


Both. 




Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Tons 


Lbs. 


Lbs. 


Lbs. 


R 


5 w 


10 w 


40 w 


6 w 


24 w; 


.0345 7£^ 


w 


14 w 


28 w 


R30 


15,000 


30,000 


120,000 


18,000 


72 ,000 


103.5 


3.000 


42,000 


84,000 


R35 


17,500 


35,000 


140.000 


21,000 


84,000 


120.75 


3,500 


49,000 


98,000 


R40 


20,000 


40,000 


160.000 


24,000 


96 ,000 


138. 


4,000 


56,000 


112,000 


R45 


22,500 


45,000 


180.000 


27,000 


108 ,000 


155.25 


4,500 


63.000 


126,000 


R50 


25.000 


50,000 


200,000 


30.000 


120,000 


172.5 


5,000 


70,000 


140,000 


R55 


27,500 


55,000 


220 ,000 


33,000 


132 ,000 


189.75 


5,500 


77,000 


154,000 


R60 


30.000 


60.000 


240,000 


36.000 


144.000 


207. 


6.000 


84,000 


168.000 



* By substituting Wi for w the diagrams will represent "wheel-load'* 
instead of "axle-load" diagrams. Wheel load diagrams are often used in 
calculations, but axle load diagrams are safer. 



ENGINE MOMENT-DIAGRAM, 



691 



Typical Moment Diagram — "R 50" Loading. 

Below is a Moment Diagram of special type "R 50" locomotive (172.5 
tons) followed by 5000 lbs. per lineal foot per track. The loads are given 
in thousands of pounds per "axle" and per "track" in order that the diagram 
taay be directly convenient for calculation of one truss for double track 
bridge and for safety in all cases. For practical bridge calculations it will 
be convenient to reproduce this diagram on tracing cloth to a scale of say 
5^''=r, to be used over skeleton diagrams of bridges drawn to the same 
scale. By sliding the engine diagram over the span the moments may be 
calculated readily for any position and hence the maximum moment obtained 
Eor any point on the span. 



CO- 



5r/n 0^56/11 „ M apu^v 

56'/UO^/0 su^dsjojv 
8S0^^"®^^H uinuitx^j^ JO J 




As all the diagrams in Table 1 are proportional throughout it is necessary 
only to multiply the results obtained from the use of the diagram, by the 
proper factors, which are clearly evident. Thus for "R 45," "R 55," "R 
60," multiply by 0.9, 1.1, 1 . 2, respectively. 



692 



28.^RAILROAD BRIDGES. 



Table 2. following, shows the positions of axle loads, R 50,* giving 
maximum bending moments on spans up to 55 feet; the maxim tun bending 
moments in ft. -lbs. from these positions; and the maxim lun shears. Useful 
in the design of girders, stringers, and floorbeams. 

2. — Bending Moments and Shears for "R 50" Loading. 



tPosition of Loading for 

Maximum Moment. 

(See Fig. 10.) 



Position for maximum moment 
occurs when center of span is equi- 
distant from point of moment and 
center of gravity of loading. 



0) tA M 






a^ 

TO <U 



Position of 

Loading for 

Maximum 

Shear. 



A 

70.000 



A t40 6 

_ TOLOOO; 70.000 

»<-Spans^ll55tDl4.15il.-J 
50.000 5Q00O~6Q00a 



too 

5Q000 50,000jui SQjJOO 5Q000 



k— -SDans:lSX 



-Spans: l8.eGtor7.65^. • 



^. 



::! 



Z5W0 StOOO 6(1000 [ SOOOO 5(tM0 



t^ 



Span&-.27.65to35.37A -• 



J 



17,500 

35.000 

52,500 

70,000 

87,500 

105.000 

122.500 

140.000 

157.500 

175.000 

192,500 



140.000 
70.000 
46,670 
35,000 
28.000 
23,330 
20,000 
17.500 
15 .560 
14,000 
12,727 



210.730 
242,980 
275,630 



312,500 
350,000 

387,500 
425,000 



466,450 
515,630 
564 ,880 
614 .200 
663,590 
713 ,020 
762 ,500 
812.020 
861 ,570 



913,470 
969,680 
1,025,890 
1,082,110 
1,138.320 
1.194,540 
1,250.780 
1.306,980 



11,707 
11 ,502 
11,250 



11.111 
10,938 

10.727 
10.494 



70,000 
70.000 
70,000 
70.000 
70.000 
70.000 
70,000 
78.750 
85 .560 
91.000 
95.450 

99,170 
102,310 
105,000 

107.330 
109.380 



70JOOO 



Change at 7.00 



70 70 



10,337 

10.313 

10.247 

10.152 

10.035 

9,903 

9,760 

9,610 

9,455 



111 ,760 
116,670 

121 ,050 
125,000 
128,570 
131,820 
134.780 
138 ,540 
142,000 
145,190 
148.150 



Change at 16.33 



90 W90$0 



Change at 23.00 
60 50 50 60 25 



9,321 
9,224 
9,119 
9,008 
8,893 
8,775 
8.656 
8.535 



150,890 
153 ,450 
157 ,000 
160,320 
163,440 
166,360 
169,120 
171 ,710 Change 



Change at 29.00 
50 50 50 50 30 30 

^ t 

at 35.00 



♦ The "positions" and bending moments are for "R 50". The positions 
will remain the same for any engine of Type R. Table 1; and the bending 
moments and shears will be directly proportional to the loading. Thus, 
for "R 40" mult, above tabular values by 0.8; for "R 60," by 1.2; etc. 

t The position of the loading which gives the maximum bending moment 
at center of any span / will give also the max. floor-beam reaction, or max. 

loading on a floor-beam joining two panels each -^ in length. 



MOMENTS AND SHEARS— SPANS. 693 

2. — Bending Moments and Shears for "R 50" Loading. — Concluded. 



Position of Loading for 

Maximum Moment. 

(See Fig. 10.) 



25,000 



t 



t55, 
,000 



3o.ooa' 



Spans-.35.?7to38.96£fi-- 



?0 5CtO005(y)0O5a00O SQOpO 3q000 ZOfii 



5Q0(W 3q000 30,000 



h:: 



?5,000 



Spans «38.% to48.49it. 



~1 



fe 



37 



► 50.000 50,000 50,000 \5m lojxo-iom ym 



k— Spans. -.46.49 to 51.78 iV." 
345 
JP.pOO 2^00050,00050.000 50.000 50,000 3(p?(]0(» 3(^000 



K— v-Spo,ns-.5l.78+o ''" — 



M ^ <u 



:aoa 



•h 



1.367 
1.431 
1.494 



,860 
,000 
.500 



1,558 
1,630 
1,701 
1,772 
1,843 
1,915 
1,986 
2.057 
2,128 
2,200 



,850 
,000 
,330 
.580 
,830 
,090 
,330 
,580 
,830 
.100 



2.275 
2,353 
2,432 



,050 
,590 
,150 



49 
50 
51 

522.546,500 
532,632,740 
,000 
200 



2,719, 
2,805, 



8,444 
8,362 
8.280 



8.199 
8,150 
8,097 
8,039 
7,977 
7,914 
7.847 
7.779 
7.710 
7,639 



7,580 
7,531 
7.481 



7.534 
7,498 
7,459 
7.419 



6^ 



175,000 
178,110 
181 .050 

183,850 
186.500 
189.760 
192 .860 
195.810 
198 .640 
201 .330 
203.970 
206,600 
209,220 

211 ,840 
214,450 
217,060 

219,660 
222.260 
224.860 
227,450 



Position of 

Loading for 

Maximum 

Shear. 



-Change at 35.00 

50 50 50 5030 30 30 

15151519151^^ 



T 

-Change at 40.00 
5050 50 5030305030 
i5J5J^lgi^J^J.^ 



^ 



-Change at 45.00 



SOa>SOE0KMW3O'<^g, 



^ 



n.— MOMENTS AND SHEARS— SPANS WITH FLOOR BEAMS. 
(a) Uniformly Distributed Loads. — ^The maximum moment at any panel 
point obtains with the span fully loaded from end to end. The maxim imi 
shear in any panel obtains with the head of the moving load at or in that 
panel: specifically, when the length of moving load in the panel -;- panel 
length = (the niimber of the panel from the right end of span, minus 1) -r- 
(total nimiber of panels in the span, minus 1). Thus: 
Let ^ = panel length in feet, 

w = total number of panels in span, 

it; = the niimber of the panel from right end of span; then for maxi- 

X — 1 

mum shear in panel x, the length of load in "panel x" = p z. Thus for 

n — 1 

a six-panel span the head of load for maximum shear in the 5th panel is 

z ^a — 7=T across the panel; for the 4th 

«— 1 0—1 5 

3 2 

panel, -^; for the 3rd panel, -^; etc. The maxi- 
mum shear in the 6th panel obtains with the Fig. 11. 
full panel (and bridge) loaded. 

In practice, however, the above refinement is eliminated, the panel 
loads being considered as concentrated at the panel points or joints — upper 
for "deck" and lower for "through" bridges. This applies also to dead 
loads, for short spans. For long, through spans a certain portion, say \ to i, 
of the dead load per panel is often assumed to be carried at the top chordjoints. 

In calculating live load stresses, engineers as a rule prefer to use specified 
engine diagrams, somewhat heavier and more severe, for the structure, 
than the actual engines in use or immediately contemplated. This is by 
far the safer, more scientific and more economical when we consider the 
strength of a structure to be measured by its weakest part. An approxi- 
mation to actual engine diagrams is the use of equivalent uniform loads 
which may be used with or without "engine excess." 



694 



I— RAILROAD BRIDGES. 



(b) Concentrated Loads — Maximum Floor=Beam Reactions. — ^The loading 

which gives the greatest bending moment at the center of a span two panels 
in length, will give also the greatest loading at that point when supported 
by a floor-beam, and this loading or its equivalent is called the floor-beam 
reaction — a necessary factor in the design of the floor-beam itself. 

Table No. 3 gives the floor-beam reactions due to "R 50" axle loads, 
and the positions of the loads, over two panel lengths, which produce these 
reactions. For any other loading of Type R the floor-beam reaction will 
be directly proportional to the weights on the drivers. 

3.— Maximum Floorbeam Reactions and Positions of Loading for 
Same. Type "R 50." 



Position of Loading for 

Maximiim Floor-beam 

Reaction. 






PL, 



™ f «H • 



g^I-'S 



.2 ^^ 



^ & 
^ ^ 






CO 



Floor-beam Moments 
for Single and 
Double Track. 



70000 
Change at ^ = 6.25— 



70,000 
70 ,000 
70 ,000 
70,000 
70.000 
70,000 



50,000 ^,m 50.000 
. I 5 15 X 



Change at ^ = 10.00- 



6.5 
7 

7.5 
8 

8.5 
9 

9.5 
10 



73 ,080 
78 ,570 
83,330 
87,500 
91 ,180 
94 ,440 
97 ,370 
100,000 



5Q00O ^m 5Q000 %m 
i 5 i 5 i 5 4 



+ 



Y p-—- 4t p — -4 

Change at ^ = 13.00-- 



10 

11 

11.5 

12 

12.5 

13 



104 ,760 
109 ,090 
113,040 
116 ,670 
120,000 
123,080 



e5.000 5(P) 50.000 5Q00050.00D 
J i 8 15X5 15 4. 



Change at ^ = 18.17 

15.000 5(i000 50,000 50,000 5C>000 2Q000 
i8i5i5|5i9i 

%. p J< p — -^ 

Change at ^ = 19.00 



126,850 
130,360 
133,620 
136 ,670 
139 ,520 
142 ,190 
144 .700 
147 ,060 
149 ,290 
151 ,390 



18.5 
19 



153,920 
156 ,580 



nOOO 50,l»0 5QOOO 50,000 5Q000 30,000 30,000 
■I 8 i5>i5|5>l 9 i5| 



tc— --p -—A—. 



19.5 

20 

20.5 

21 

21.5 

22 

22.5 

23 

23.5 

24 

24.5 

25 



159 ,870 
163,000 
165 ,960 
168,810 
171 ,510 
174 .090 
176,560 
178,910 
181 ,170 
183,330 
185 ,410 
187 .400 



35,000 
35 ,000 
35,000 
35,000 
35,000 
35 ,000 



36 ,540 
39 ,285 
41 ,665 
43 ,750 
45,590 
47 ,220 
48 ,685 
50,000 



52,380 
54 ,545 
56,520 
58,335 
60,000 
61 ,540 



63,425 
65 ,180 
66 ,810 
68 ,335 
69 ,760 
71 ,095 
72,350 
73,530 
74 ,645 
75.695 



76 ,960 
78.290 



79,935 
81,500 
82 ,980 
84,405 
85,755 
87 ,045 
88,280 
89 ,455 
90,585 
91 ,665 
92 ,705 
93,700 






- o 



X. 



t.. 



I 



I 



f^ 



6 



X"l^— ^l*^ 



>i 1 



■K|eJ 



to I 

II 
a 

o 

a 
a 



FLOORBEAM REACTIONS. CHORD STRESSES, 



695 



(c) Concentrated Loads — Positions for Maximum Moment. — In order to 
find the stresses in the top and bottom chords of bridge trusses and girders 
with floor-beams, it is necessary to know the maxim^um bending moments 
at the floor-beam or panel points. The main problem consists in finding 
the "position" of the loading at each panel point, the bending moment 
being then found by taking moments about that point. The chord stresses 
are obtained by dividing the bending moments in ft. -lbs. by the respective 
moment arms of the chord pieces, in feet. _ For trusses with parallel chords 
the moment arm is constant, being the height of truss. 

The position of loading for maximum bending mom.ent at any panel 
point* usually obtains: 

(1) When the heaviest loads are nearest the point. 

(2) When one of these loads is at the point. 

(3) When the average loading per Hneal foot to the left of the point = the 
average loading per lineal foot on the whole span (the load at the point 
to be applied in whole or in part to either portion of the span). 

Chord Stresses in Pratt Truss. 
Problem. — Find the live-load stresses in the top chord member T and 
in the bottom chord member B of Prattf truss. Fig. 12, of a single track 
railroad bridge, 144 ft. span; using loading "R 50" (page 691). Note that 
in all cases the moving load comes on at the right hand end of span. 




Solution. — By "cutting" sections a-a and h-h it is evident that the 
bending moment for maximum stress in T will be at the point P\ and for 
B, at the point P, directly above. Hence the same bending moment will 
do for both stresses. Applying the moment diagram and the rules just 
given, we place one of the heavy loads at P' so that the loads to the 
left of P' will equal § (or i) the total loads on the bridge. Hence, if load 
(5) is placed at P' the load to the left is 175 to 225 thousand lbs., while the 
whole load on the span is 6904-5X22 = 800 thousand lbs. As \ of 800, or 
200, is between 175 and 225 we adopt this position as the probable one for 
maximum moment at P' . By trial we find this to be true and that the 
moment at P' = 12,045,000 ft. -lbs. As there are two trusses each resisting 
one-half the moment, and further, as the moment arms are 26 ft. the stresses 
in T and B are each equal to 12, 045, 000 -^ 52 = 231,600 lbs. — compression 
in T and tension in B. If the engine loading is "R 40" instead of "R 50" 
the stresses will be | the above or 185,300 lbs., etc. 

The following are methods of calculation with various loads at P', 
using the engine diagram: 
Load (3) at P' . Load {^) at P' . 

12 ft. of uniform load. 17 ft. of uniform load. 

Mom. at J/ =40,090. Mom. at H = 40,090. 

Add 690X12= 8,280. Add 690X17=11,730. 

360. "5X17X8^= 722.5 



5X12X6 = 



48,730X36 



Load (5) at P'. 

22 ft. of uniform load. 

Mom. at IZ = 40,090; 
Add 690X22=15,180. 

" 5X22X11= 1,210. 



144 
Deduct 



48,730. 
12,182.5 
575. 



X 



J6_ 
144 



52,542.5 


56,480. 


13,135.6 


Xj= 14,120. 


1,200. 


2,075. 



11,607.5 11,935.6 Max. = 12,045. 

Hence the maximtim moment is 12,045,000 ft. -lbs., as above noted. With 
load 6 at P' the moment is only 11,870,600 ft.-lbs. 

* The moment at any floor-beam panel point for any system of loading 
is the same as if there were no floor-beams. 

t In this case the chord stresses will be the same whether the bridge is 
"deck" or "through." 



696 



I'—RAILROAD BRIDGES, 



Chord Stresses in Warren Truss. 
Triangular Web System. — For practical reasons the Post truss and the 
through triangular or Warren truss types have nearly disappeared. The 
Warren truss remains principally for deck spans and swing bridges; and 
for short, fixed, through spans. For deck spans it is economical to erect 
verticals at the lower chord panel joints to support intermediate floor-beams 
as in Fig. 13. 





J 


TV 










1 


A 


A 


A 


^-/5^ 


hl8-^ 




R 


1 i 

1 








r 



Fig. 13. 

Using the same live load, "R 50," as in the preceding case, note that 
the stress in the top chord members TT is equal to the stress in T of the 
Pratt truss. Fig. 12, namely, 231,600 lbs. It is obtained by placing load 
(5) of the engine diagram at t, a floor-beam point, and taking moments 
about ^.* 

Consider, now, the stress in the upper chord a b. Fig. 13, assiiming no 
floor-beam at t, in a vertical line with the center of moments, f . For this 
case the loading for maximum will be somewhat modified, and the following 
rule will usually apply: The average loading per lineal foot to the left of the 
center of moments, f, {including^ one-half -\ the load in the panel under con- 
sideration, ab) = the average loading per lineal foot on the whole span. After 
the position of loading for maxim imi has been fixed take moments about t\ 
using the reaction at a due to loading in panel ab, in deducting the negative 
moment. 



(d) Concentrated Loads — Positions for Maximum Shear, — On page 693, 

under imiformly distributed loads, we find that for maximum shear in any 

x—\ 
panel the length of the moving load in that panel will be p—^^ ^^ which 

:^ = panel length, a: = number of the panel from right hand end of span, 
and « = the total number of panels. Thus, the head of the moving loadj in 
any panel is such that the average load per lineal foot in the panel equals the 
average load per lineal foot on the whole span. This principle may be applied 
tentatively in fixing the position of concentrated loads for maximtun shear. 
The position of loading for maximum shear in any panel of a truss 
with simple web system|| usually obtains: 

(1) When the heaviest loads are nearest to and just to the right of the panel. 

(2) When one of these loads is at the panel point at the right hand end of 
the panel in question. 

(3) When the average loading per lineal foot in the panel = the average 
loading per lineal foot on the whole span (the load at the panel point at 
the right hand end of the panel may be applied in whole or in part to the 
panel loading). 

Problem. — Find the live-load shear in panel t b. Fig. 13, of a two-truss 
double track deck railroad bridge, 144 ft. span; using loading "R 50." 



* The stress in B of the Warren truss is greater than B of the Pratt, 
and is obtained by taking moments about b, as would natiu-ally be indicated 
by the cutting section c-c, with load (8) at the panel point. This produces a 
moment of 14,698,750 ft. -lbs. and a corresponding stress in B, for one 
triangular truss, of 282,670 lbs., equivalent to stress in B' of the Pratt truss. 

t Accurately, at-^ab, whether it is one-half or any other fraction. 

JThe head of the moving load for maximimi shear is at the "neutral 
shear point" because any concentrated load at that point produces zero shear 
in the panel. A load to the right of the neutral point will produce positive 
shear, while a load to the left will produce negative shear. 

II For special case with sub-panel system, see Sec. 40, Highway Bridges, 
page 725. 



MAXIMUM SHEAR. LATERAL BRACING, 



697 



Solution. — ^The three conditions above named for maximum shear are 
fulfilled by placing load (3) at &, whence the total load in the panel, 75 to 125, 
equals the total load on the bridge, 660, divided by the number of panels, 
8; thus, 660-5-8 = 82.5. Inspection, however, reveals to us the possibiHty 
of a maximum with load (2) at 6, hence we solve for these two positions, 
using moment diagram: 

Load (2) at h. Load (3) at b. 

R^l^ 29.560 i?i/= 33,340 

+ 630X5= 3,160 +660X4= 2,640 



Dividing by U R, 
Deduct reac- 
tion at t due to 
load (1) in pan- 
el, 25X/s 



32,710 

227,153 lbs. 

11,111 •• 



Dividing by /, Ri 
Deduct reac- 
tion at t due to 
loads (1) and 
(2) in panel 



35,980 

249,861 lbs. 

31.944 " 



Shear = 216,042 



Shear= 217.917 



Therefore, maximum shear. 217,917 lbs., is obtained with load (3) at h. With 
load (4) at h the shear is 206,944 lbs. The compressive stress in the diagonal 

t'h is 217,917 X ^ = 217,917 X 1.216 = 264,990 lbs. 
z t 

in.— LATERAL BRACING.— WIND AND CURVE PRESSURE. 

Horizontal or lateral bracing is designed to resist the lateral forces due 
to wind pressure, and the centrifugal pressure of moving trains on 
curves; to shorten the unsupported " column length " of upper chord , for 
economy of design ; and to give general rigidity to the structiure. Stiff 
bracing is preferable to rods. 

Wind Pressure. — The direct wind pressure on any exposed surface is 
about proportional to the square of the velocity.* A velocity of 100 miles 
per hour produces a normal pressure of about 50 lbs. per sq. ft.; 90 miles 
per hour, about 40 lbs. ; and 80 miles per hour, about 30 lbs. As rigidity 
is a very essential feature it is advisable to use the higher figure, 50 lbs., 
as the pressure per sq. ft. on the exposed surface of all trusses and the floor 
system when the bridge is unloaded. An alternative wind load of 30 lbs. 
per sq. ft. on the same surface and also on a moving train between elevations 
2.5 and 10 ft. above base of rail is also prescribed. Sometimes the wind 
load per lin. ft. is stated in actual amounts for each chord. Thus, a load 
(either fixed or moving) of say 150 lbs. per lin. ft. for the unloaded chord 
and a moving load or say 600 lbs. per lin. ft. for the loaded chord. See 
Wind Pressure under General Specifications for Steel Railroad Bridges, 
following. 

Problem. — Find the stresses in top lateral system of single track through 
Pratt truss span of 144 ft., as per following sketch; loading 75 lbs. per lineal 
foot for each chord ( 18 X 75= 1,350 lbs. per joint.) 




Solution. — Let the diagonals and struts be stiff members and let either 
single member in each panel be capable of "taking" all the shear, in tension 
or compression. Then with a moving load from A to B of 1350 lbs. per 
single joint or 2700 lbs. per double joint (both chords) the shears in panels 
a, h and c, due to the top lateral span of 108 ft., are 6750 lbs., 4500 lbs. and 
2700 lbs., respectively* Hence the stresses in the diagonals (ft X the 



* Pressure in lbs. per sq. ft. = 



(Velocity in miles per hour)^ 
250 



approxi- 
mately. (Recent experiments by M. Eiffel, and also by Dr. Stanton, indi- 
cate that for velocities of 40 to 90 miles per hour the denominator of the 
fraction may safely be increased from 250 to 300 or 333. — See Engineering 
DigesU March, 1908.) See also. Section 46, Roofs, pages 794, etc. 



698 



2S.^RAILR0AD BRIDGES, 



shears) are 10,100 lbs.. 6,800 lbs. and 4,100 lbs. In practice, minimum 
sections of material are specified, either direct or by formula, below which 
the design would be considered weak no matter how small the stresses.* 

IV.— PORTAL AND INTERMEDIATE VERTICAL BRACING. 

Portals should be designed to transmit all lateral-bracing stresses at 
the imloaded or far chords, directly to the_ abutments through the end 
posts. For long spans, vertical bracing is inserted at the intermediate 
posts rather for stiffness than to transmit any wind pressure from one 
lateral bracing system to another, although it is usually designed suffi- 
ciently strong to meet the "local" wind pressure. The same principles 
which apply to portal- also apply to intermediate bracing, hence our re- 
marks will be confined to the former, and for brevity of explanation, to 
"through" bridges. 

The calculations of portal stresses are much simplified by making certain 
assumptions which render the framework quite statically determinate. 
These assimiptions are that the bottoms of the end posts are hinged laterally; 
that all stiff -riveted portal connections are also hinged; and that for a double 
or multiple system of portal bracing only one simple statically determinate 
system acts at a time. These asstimptions are on the side of safety. 

The following simple illustrations are typical. They are tipped to a 
horizontal position so that the acting forces P may be vertical and perhaps 
better illustrate the cantilever principle. Note that the horizontal and 
vertical reactions at the bottom of the two end posts are ntimerically the 
same for each post and, consequently, the shears and the bending moments 
in the latter are respectively similar.f 

Diagonal d in tension (-f). Diagonal d in compression (— ). 



V^ 



— 1-- 



H 






H' 




V-4 



i< \ 



1^ 



V^ 



t 



oc' 




Fig. 16. 



Fig. 16. 



P P' PI 

Fig. 15 or 16, with either P or P' acting: H = H'=-^ or -^\ y=F'=— or 

P7 , . PI , P'l ^ . , PI P'l 
; stress m a;= H orH ; stress m x' = or ; 

W WW WW 



Fig. 15, with either P or P' acting 

Stress m o= — -jr- or — -77— . 

2y 2y 



Fig. 16, with either P or P' acting: 

PI PI 

Stress in b=+-^oT + -^. 

y w y w y w y w 

The following are some of the usual types of portals. 



i=- (P + 



Px\ P'x 



« ..^=+^.^or+^^ 



b 





Figs. 17. 



* For highway bridges adjustable rods are often used instead of stiff mem- 
bers and it is customary to specify an initial stress of, say, 10,000 lbs. to be 
added to the calculated stress in proportioning the dimensions of the laterals. 

t In the design of the posts these wind stresses have to be "considered" 
in addition to the regular stresses as truss members. 



PORTAL BRACING. SPECIFICATIONS. 



699 



Fig. a is the simplest and the one just analyzed by calculation. By 
similar methods and with reasonable assumptions the others may be calcu- 
lated readily. 

v.— GENERAL SPECIFICATIONS FOR STEEL RAILROAD BRIDGES. 

The main specifications below are adapted from those of the American 
Bridge Company (Am. Br Co.). 1900, C. C. Schneider, Vice President. 
The foot-notes are inserted by the writer as showing the practice (in devia- 
tion or otherwise) of a few of our leading railroad companies, as follows: 
Chicago, R. I. and Pacific (C. R. I. & P.), 1906; J. B. Berry, Chief Engineer. 
Del., Lack, and Western (D. L. & W.), 1903; A. E. Deal, Bridge Engineer. 
Lake Shore and Mich. Sou. (L. S.&M. 5.), 1904; Sam'l Rockwell, Chief Eng. 
* Lehigh Valley (L. V.), 1906; Walter G. Berg, Chief Engineer. 
Philadelphia and Reading (P. & R.), 1906; Wilham Hunter, Chief Engineer. 
Southern Pacific (S. P.), 1906; William Hood, Chief Engineer, 

(a.) General Description. 

Material. — 1. To be of rolled steel t as specified below. JCast iron or cast 
steel permitted only in machinery of movable bridges and in special cases 
for shoes and bearings. 

Types of Bridges Recommended. — 2. Limiting spans, in ft. to be: || 

Clearance. — 3. On straight line a clear section shall be_ provided to con- 
form to given requirements. 1[ The width must be increased so as to 
allow the same minimum clearance on curves§ and on double track. 



* "All Railroad Bridges on the Lehigh Valley Railroad System are to 
be built in accordance with the 'General Specifications for Steel Railroad 
Bridges and Viaducts; New and Revised Edition, 1901; by Theodore Cooper, 
Consulting Engineer,' modified as follows:" [Some of the modifications are 
incorporated in these foot-notes.] — F. E. Schall, Bridge Engineer. 

t P. & R. allows use of wrought iron for laterals and unimportant 
members. 

t L. S. & M. S. specifies all castings to be cast steel. 5. P. specifies 
rollers for swing bridges to be cast steel; expansion rollers to be bar steel, 
or cast steel of equal strength. 



II Type. 


Am. B. 
Co. 


C. R. I. 
& P. 


D.L. & 
W. 


L. 5. & 
M.S. 


L. V, 


S. P. 


Trough floors (long'l) . . . 


0-20 

0-20 

20-100 

100-140 








0-20 

0-20 

20-120 




Rolled beams 


0-19 
19-110 


0-20 
16-100 
90-160 


0-23 
23-100 


0-19 


Plate girders (riv.) 

Lattice girders (riv.) .... 


19-100 


Riveted trusses 


100-180 


120-150 




with inclined end posts 


iio-* 


100-200 
200-' 




100-150 


Pin con. trusses.- 


150- 


180- 


150- 




with inclined end posts 


150- 





H' 


H 


m 


t 


b 


d 


h 






i 


If Road. 


1* 


\^ 


CR.I.& P 

D.L.&W 

L. S & M.S... 


23'- 
22'- 


-6" 


22'* & 
'2V-& 


7'-0" 
7'-6'' 

7-6" 


3'-6'' 
3'-6'' 
4'-0'' 
2'-9'' 
3'-0'' 
3'-0'' 


5'-6" 
5'-6'' 
5'-3'' 
5'-3'' 
5'-0'' 
5'-6'' 


5'-0'' 
5'-0'' 
6'-0" 
4'-3'' 
5'-0" 
5'-0" 


4'-0'' 
4'-0'' 
5'-0'' 
6'-6'' 
2'-0'' 
4'-0'' 






L. V 


22'- 
22'- 


-0" 


, 


P.& R 

S. P 


r 



H'= to Top of Rail. 
H = to Base of Rail. 
§ Increase width for curvature and super-elevation for car 80' Ig., 14' 
high and 60' c.-c. trucks — C. R. I. & P. Increase m 1 in. per deg. of curva- 
ture, and increase m on inside of curve 2\ ins. additional per each inch super- 
elevation of track. Width to be increased proportionately for 2 ormore 
tracks.— L. S. & M. S. 



700 38,'-RAILROAD BRIDGES. 

Spacing of Trusses, — 4. Width between centers of trusses to be not less than 
5^jj* of the span. 

Spacing of Deck Plate Girders. — 5. Generally 6it ft. centers. 

Floor Beams. — 6. Shall be riveted between the posts, above or below the 
pin, in through bridges. 

Spacing of Stringers. — 7. Generally 6i| ft. centers, the tracks being 13 ft. 
centers, standard. 

Wooden Floor.|| — 8. Cross-ties S'^xS'' for stringers spaced 6^ ft. centers. 
For spacing over 6^ ft., ties to be proportional for fiber strain not to 
exceed 1000 lbs. per sq. in. on timber, assuming max. wheel load dis- 
tributed over three ties. Ties to be spaced 6'' or less in the clear, notched 
down i", and have full bearing on stringers. 9. Every fifth tie to be 
fastened to stringer by a f" bolt. 

Guard Rails.® — 10. Timbers 6''x8" on each side of each track, with inner 
faces not less than S'-Z" from cen. of track. To be notched 1" over 
every tie and fastened to every third tie and at each splice by a f-in. bolt. 
Sphces to be over floor timbers, with half -lap joints 6'' long. 11. Floor 
timbers and guards to be continuous over piers and abutments. 12. 
On curves, outer rails to be elevated as may be required. 

(b) Loads. 

Dead Load.# — 13. In estimating the weight of the structure for the 
purpose or calculating stresses, timber is assiimed at 4^ lbs. per ft. B.M.; 
and rails, spikes and joints at 100 lbs. per lin. ft. of track. 

Live Load.lf — 14. A moving load for each track, consisting of two 
engines coupled at the head of a uniformly distributed train load, placed 
so as to give the greatest stress in each part of the structure. The load 
will be as specified by the railroad company. Cooper's standard loading, 
however, is recommended. (See Tables 4 and 5, pages 707, 708.) 

* And preferably not less than ^^2 of the span. — L. 5. & M. S. 

t 7 ft. for spans up to 60 ft.; 8 ft. for spans 60 to 110 ft. — C. R. I. & P, 

% 7 ft.— C. R. I. & P. and S. P. 6 ft. for double track through, and 
deck plate; 8 ft. for single track through. — P. & R. 6^ ft. — D. L. & W. and 
L. 5. & M. 5. 

II 8^x8" by 10 ft., framed to 7^", for stringers spaced 6i ft. centers; 
depth of tie to be increased 1" per each 6'' increase in stringer spacing. 
Ties spaced 12" centers, every fourth tie fastened to each stringer by a V 
hook bolt. — L. S. & M. S. Ties (yellow pine or white oak) to be propor- 
tioned to an extreme fibre stress of 800 lbs. per sq. in. from the loading; 
spaced not over ^^ in clear, notched down Y, and secured to supporting 
girders by f" bolts not over 6 ft. apart. — L. V. 

° Guards to be 8" wide and 6'' deep, framed to 4'' over ties, with inner 
edges 4'- 2" from cen. of track. They shall be fastened to the ties by a 
\" bolt through every fourth tie, these bolts to be through the ties which 
are connected to the stringers by hook bolts. — L. 5. & M. S. Guards to 
be 6"x8''' southern yellow pine, notched \Y over every tie, bolted by a \'' 
bolt to every third tie and spliced over a tie by half and half joint 8" wide'' 
bolted at splice; the inner face of guard to be 4^-1^' from cen. of track; 
and all heads or nuts on upper faces of guard timber to be countersunk 
below surface of wood. — L. V. 

# Timber at 4^ lbs. per ft. B. M.; ballast, 100 lbs. per cu. ft.; rails and 
fastenings, 150 lbs. per lin. ft. of track. — C. R. I. & P. The floor, con- 
sisting of the ties, rails, guard rails, and all spikes and bolts necessary to 
fasten same, assiimed at 400 lbs. per lin. ft. of track.— D. L. & W. Ordinary 
floor at 400 lbs. per lin. ft. of track. Make due allowance when plank- or 
ballast floor is used. — L. 5. & M. S. Ordinary floor assumed at 500 lbs. 
per lin. ft. of track. For ballast floors allow following weight per cu. ft.: 
Ballast, 120 lbs., Concrete, 140 lbs., Asphalt, 90 lbs.. Lumber 54 lbs. — 
P. & R. Ordinary floor assumed at 500 lbs. per lin. ft. per track. Apply 
three-fourths of total dead load at panel points of loaded chord, and one- 
fourth at unloaded chord. — S. P 

^ The following are Railroad Co. Standards. Diagrams 1 and 2 are 
shown, and that one which will produce the greatest stress in any member 



SPECIFICATIONS FOR STEEL R, R. BRIDGES. 



701 



Effect of Impact.* — 15. The effect of impact and vibration due to live 
load shall be detennined by the following formula and added to the live 
load stress: 



/ = S {j^^^^ (See Table 6, page 709.) 



,L + 300; 
where /= impact to be added to live load strees 5. 

L= length in ft. of loaded track distance which produces the maxi- 
mum stress in member. 



is to be used, the latter usually for stringers and floor beams. Loads (lbs.) 
are axle loads and uniform loads, per lin. ft. of track. 

Diagrams'. No. 1. No. 2. 

C. R, I. & P. (1906). Two 1951 ton consolidation. Class E 55 (Cooper): 



|gg8 gjetqiqs 



Ki mtninin tototoro oj u^uninin torototo 



II 

D. L. & W. (1903). Two 167 ton consolidation engines: 



^S 



(u (u cu <u 



S <5^ o c5 S^ o^ o^ o. o. 



11 



4 cbcbcfxl) cl)c|), 6( i? 4 cl x ly txt) Hih 



4,500Tbs.perft. 



L. S. & M. 5. (1904): Two 177J ton consolidation. Class E 50 (Cooper): 



S vx> 5? 



-. I §. ? 

s s s s^ 



S S S^ S S BPfSfW & S S' S ^ SfRfSfSf 



5,0 00 Ibs^j !'. 

L. V, (1906). Cooper's Loading; to be specified for each structure. 

P. & R. (1906). Two 21 If ton consolidation engines. [For trans, trough 
floor, 1600()# p. 1. f. of track.] (Moving simultaneously on each track 
in the same direction) : 



tint liii §ii§§ 1^ 



o 

° I 

O (O 



*l 



E3" S lg 18 S' ?"$" r f Kf S- ar SB" i8~ ?■*■?■ ? ^ XX 

If 5\' 9* k-- 51' -V 



S. P. (1906). Two 192^ ton consolidated engines: 



%A 



000 

Sfll 






It 



OOOO 

8000 
000, 



^q^^5'+5'*_9'.i5'J^X5'J,.8'.J.^'.J,5'J.5'4.5'Jv-9'-'^'*keU5'A5J 



'■X^i 



*For loaded ) / 

length of track)- 7 = 5 ( 
up to 100 ft. ) \ 



300 



100 



L+300 



500 



L\ . Over ? t_o 3( 
~/ ' lOOft.J^-'^LT 



300_ —C.i?. 
300* /. (^•P. 



For swing bridges and other movable structures while in motion only, 
7 = 0.25 D. For counter stresses, 7 = L. For hip and panel suspenders 
and riveted connections of the floor system, 7=1.25 L. For all other 

cases, 7 = L ^ ^ . In which L = combined stresses from live load and cen- 
trifugal force; and D = dead load stress. — L. 5. & M. S. 



702 28.— RAILROAD BRIDGES. 

Wind Pressure.* — 16. Shall be assumed acting in either direction 
horizontally: First. At 30 lbs. per sq. ft. on exposed surface of all trusses 
and the floor as seen in elevation, in addition to a train of 10 ft. average 
height, beginning 2'-6" above base of rail, moving across the bridge. 
Second. At 50 lbs. per square foot on the exposed surface of all trusses and the 
floor system. The greatest result to be assumed in proportioning the parts. 
17. For determining the requisite anchorage for the loaded structure, the 
train shall be assumed to weigh 800 lbs. per lin. ft. 

Momentum of Train. — 18. Coefficient of sliding friction of wheels on 
rails, in stopping train, to be assumed at 0.2; this to apply to longitudinal 
bracing of trestle towers and similar structures. 

Centrifugal Force of Train, — 19. On curves, assume the centrifugal 
force C, in lbs., due to each train, to be: C = . 03 (Wt. of train in lbs. X degree 
of curvature up to 5°); the coefficient (0.03) to be reduced 0.001 for every 
degree of curvature above 5 degrees. 

(c.) Proportion of Parts. 
Least Thickness of Material. t — 20. Except for lining or filling vacant 
spaces, f" thick for main members and their connections, and -^e" thick for 
laterals and their connections. 

Permissible Tensile Stresses. J — 21. On all parts of structure, sum of 
maximum loads, together with impact: Soft steel, 15000; Medium steel, 
17000 lbs. per sq. in. 22. Same limiting stresses for wind pressure, centrif- 
.ugal force, or momentum of train. 23. For net section, deduct size of 
rivet hole i'' larger than diam. of rivet. 24. For pin connected riveted 



* "Lateral Load:" 750 lbs. per ft. of loaded chord; 200, for unloaded; 
both considered as moving. "Wind load" on viaduct towers: 50 lbs. per 
sq. ft. on li times the vertical projection of structure unloaded; or 30 lbs. 
per sq. ft. on same surface plus 400 lbs. per lin. ft. of structure applied 7 ft. 
above the rail for assumed wind on train when the structure is either fully 
loaded or loaded on either track with empty cars assumed to weigh 1200 
lbs. per lin. ft., whichever gives the greater stress.— C. R. I. & P. 

For spans, 300 lbs. per lin. ft. of bridge, acting on moving train of loaded 
or unloaded cars, for lateral system attached to the loaded chord, and 300 
lbs. per lin. ft. acting on trusses and divided equally between top and 
bottom lateral systems. Trestle towers shall be proportioned for above 
wind forces for spans, and in addition thereto a wind pressure of 100 lbs. 
per vertical ft. of tower. — D. L. & W. 

For single track bridges, 300 lbs. live load and 150 lbs. dead load per 
lin. ft. of loaded chord, and 150 lbs. dead load per lin. ft. of unloaded chord. 
For double track bridges, increase the above loads 50%. Where 30 lbs. 
per sq. ft. of exposed surface produces larger dead loads than the above in 
plate girder bridges or special structures, these shall be taken instead. — 
L. S. & M. S. 

For girder bridges, 200 lbs. per lin. ft. for each chord. Also, f9r loaded 
chord, a moving load of 400 lbs. per lin. ft., with point of application 7i ft. 
above the rail. For viaducts and trestle towers, 50 lbs. per sq. ft. of the 
projected surface of two trusses and two sides of towers on the vertical 
plane through the axis of the structure when same is unloaded; with struc- 
ture loaded, take 30 lbs. per sq. ft. of this same surface, and in addition, 
the moving wind load specified for girder bridges. For determining the 
requisite anchorage for the loaded structiire, assume the train to weigh 
600 lbs. per ft.— P. & R. 

Lateral wind pressure same as Am. Br. Co. specifications with the 
following minimum values; Bracing of loaded chord, 500 lbs. per lin. ft., 
300 of which is moving- and 200 dead load; unloaded chord, 150 lbs. per 
lin. ft., uniformly distributed. On viaduct towers, as seen in elevation, 
use 60 lbs. per sq. ft. on the loaded, and 100 lbs. per sq. ft. on the unloaded, 
structure. — 5. P. 

t I" except for fillers. — C. R. I. & P. |" except for fillers; 1 sq. in. in 
section for rods or bars; li sq. ins. for counters. — D. L. & W. |" except 
for latticing and fillers; 3X3X| angles. — L. S. & M. S. 

t 16000 for structural steel at 60000 ult.— C. R. I. & P. 

Soft Steel: — Bottom chords and main diagonals: Eye bars, 9000 /. /., 
14000 d. l\ built sections, 8500 I l.» 12500 d. l.\ counters, 8600. Hip sus- 



SPECIFICATIONS FOR STEEL R. R. BRIDGES. 



703 



tension members, make net section through pin hole li of net section 
through body; and net section back of pin hole at least f of net section 
through pin hole. 

Permissible Compressive Stresses.* — 25. Compression members: 
Soft Steel. Medium Steel. 

15000 17000 



1 + 



/2 



1 + 



/2 



^=working load in lbs. per sq. in.; 
/ = effective length of column in ins. 
r = least rad. of gyr. (See Table 7, 
page 710.) 



13500r2 - 11000r2 

26. Limit of r: 120 for wind bracing; 100 for other members. 

27. Least width of post in pin connected bridges, 10 ins. 



penders and floorbeam hangers, etc., 7500. Tension flanges of built girders 
and rolled beams, 9000. Lateral, vibration and cross bracing, 12000. All 
net section; hole i^' + diam. of rivet. Medium Steel: — Allow 10% increase 
over soft steel. Holes must be drilled, or punched and reamed. Double 
Track: — For main members of trusses, as well as flanges and webs of girders 
and floor beams designed to carry a double track, the above unit live load 
stresses for trusses, and the unit live and dead load stresses for girders and 
floorbeams will be increased 10%; but not to apply to details. — D. L. & W. 

Including impact, 16000 for 56-64000 steel.— L. 5. & M. S. 

Including impact, 15000 for 60000 steel.— P. & R. 

For rolled beams and girders, 7000 (l-\ ^) ; bars, forged ends, 

\ max./ 

( H \ ; plates or shapes (net) , 8000 ( H 1 ; floorbeam hangers. 

plates or shapes (net), 6000. Medium steel, 60-68000. — ^S. P. 

*16000-70 — for 60000 steel.— C. R. I. & P. 
r 

For Soft Steel 54-62000, p= ^ 



8500(1 + 



For end posts and vertical posts of pin spans 
and all main members with two pin ends. 
For end posts and top chords of lattice spans, 
top chords of pin spans, and all main mem- 
bers with one or both ends fixed, not other- 
wise specified J 

( Live load, 
\ Dead load 



/2 

( Live load, 
( Dead load, 

!Live load, 
Dead load, 



For main webs of lattice spans. 



8500 
12500 

8500 
12500 

8000 
12000 
12000 
12000 



h 
18000 
18000 

24000 
24000 

24000 
24000 
24000 
24000 



For lateral, vibration and cross bracing ( Live load, 

\ Dead load, 
For Medium Steel, 62-70000, above values of p increased 10%. — D. L. & W. 

Including impact, p= :;^ for 56-64000 steel 



1 + 



/2 



12000 r2 



and— not> 120 for laterals, nor 100 for main members (40-60 preferred)^ 
— L. 5. & M. 5. 

Including impact, p= — for 60000 steel; 



and — shall not exceed 
r 



100, 



l + ~ 

13500 r2 

except wind bracing, 120. — P. & R. 



Rolled beams used as girders, and compression members with — < 40, same 
as for tension (see S. P. foot-note above). For lengths where — >40 use: — 

T 

For both ends fixed, /7 = 8000 ( l-\ -^ ) —.35—; for one or both ends hinged, 

\ max. I r 

^ = 8000 ( 1 + "^^"' ) -45—. Steel, 60-68000 ult. No compression member 
\ max./ r 

shall have a length greater than 46 times its least width. — ^5. P. 



704 28.— RAILROAD BRIDGES. 

Alternate Stresses. — 28. Make total area in member equal to sum of 
areas required for each stress. 

Combined Stresses. — 29. Maximum stresses due to wind and centrifugal 
force, added to those due to vertical loading (including impact), shall not ex- 
ceed: 19000 lbs. for soft steel, or 210001bs. for medium steel, properly reduced 
for compression. 30. Reversal of stresses, if any, must be considered. 

Transverse Loading of Tension or Compression Members.* — 31. When 
the floor system rests directly on the chord, the chord member must be pro- 
portioned so that the algebraic sum of the stresses per sq. in. on outer fibre, 
resulting from the direct compression or tension, and Hoi the max. bending 
moment (considering the chord as a beam of one panel length, supported 
at ends), shall not exceed the before-mentioned limiting stresses in tension 
or compression, the proper amount of impact being added to each kind of 
loading. 32. Bending moment at panel points shall be assimied equal to 
that at center, but in opposite direction. 33. Other members similarly 
affected are to be treated likewise. 

Shearing and Bearing Stresses f. — 34. For rivets, bolts and pins, the 
shearing stress per sq. in. shall not exceed 11000 for soft steel, and 12000 for 
medium steel; and the pressure upon the bearing surface of the projected 
semi-intrados (diam. X thickness) shall not exceed 22000 for soft steel, and 
24000 for medium steel. (See Table 1, page 612.) 35. Increase nimiber of 
rivets thus foimd if field driven: by hand, 25%; by power, 10%. 

Bending Stresses on Pins.J — 36. Extreme fibre stress: Soft steel, 22000; 
medium, steel, 25000 lbs. per sq. in. Use centers of bearings of strained 
members. (See Table 25, page 630.) 

Plate Girders.ll — 37. Assume >^ gross area of web as available flange 
area. Compression flange to have same sectional area as tension flange; 

* Should the pins be out of the neutral axis, the additional stress thus 
produced shall be provided for. — P. & R. 

t Shearing: Shop rivets and pins, 12000; field rivets and turned bolts, 
10000;. plate girder webs (gross section) 10000. Bearing: Shop rivets and 
pins, 24000; field rivets and turned bolts, 20000; granite masonry and Port- 
land cement concrete, 600; sandstone and limestone, 400 lbs. per sq. in. — 
C. R. I.& P. 

Shearing: Pins and rivets, 7500; web plate, 5000 in direction of rolling, 
and 6000 across fibre. Bearing: Pins and rivets, 12000; bed plates on ma- 
sonry, 250. Decrease 25% for hand and field rivets; increase 25% for lateral 
and vibration riveted connections. — D. L. & W. 

Shearing: Rivets, bolts and pins, 11000; web plates of stringers, floor 
beams, and plate girders (net section), 11000. Bearing: Rivets, bolts and 
pins, 22000; masonry, 400. Deduct 20% for field rivets. — L. 5. & M. S. 
Same as preceding excepting, use 10000 for "shear in webs of plate girders." 
—P. & R. 

Shearing: Pins and rivets, 7500; webs of plate girders, 6000. Bearing: 
Pins and rivets, 15000. Increase stresses 50% for knee bracing; decrease 
20% for hand rivets.— 5. P. 

X 24000.— C. R. I. & P. and L. S. & M. 5. 15000.-1). L. & W. 
22000.— P. & R. 1800.— 5. P. 

1 1 Proportioned either by moment of inertia of net section : or by assuming 
}/8 of gross web section to be added to fiange area. Gross section of comp. 
flange shall not be less than that of tens, flange; nor shall working stress in 

comp. flange of any beam or girder exceed 16000— 200-r-, where /=unsup- 

o 
ported distance, and 6= width of flange. — C. R. I. & P. 

Girders and beams must have top or comp. flange braced at intervals 
of at least 20 times the width of flange. No part of web of plate girder 
considered as flange area. — D. L. & W. 

Depth of girder generally ^ to t^ of span up to 75 ft. span, and propor- 
tionately less up to a minimum of i^ for the largest practicable plate girder. 
No part of web to be considered as flange area. No cover plate shall have 
a thickness greater than the angles; M in. to be about the max. thickness. 
First cover plate to extend full length; other plates, 12'' beyond theoretical 
cut-off. For spans over 70 ft., flange members may be spliced, only one 
angle at any one point of flange. Stiff ener angles to be: 3>^x3Kx^ for 
spans up to 50 ft.; 4x3Kx^, 50 to 70 ft. spans; 5x3Kx^, 70 to 90 ft. spans; 
6x3Mx^, above 90 ft.— L. S. & M. S. 



SPECIFICATIONS FOR STEEL R. R. BRIDGES, 705 

but unsupported length shall not exceed 16 times its width. 38. In design- 
ing web rivets of plate girders, assume total shear at abutment as transferred 
into flange angles in a distance equal to depth of girder. 39. Minimum 
web,^''. Shear, 9000 for soft steel; 10000 for med. steel. 40. Stiffeners, 
both sides, close flange bearings, at points of concentrated load; also, when 
t of web is less than g^^ of unsupported distance between flange angles, stiff- 
eners to be spaced generally not farther apart than depth of full web plate, 
with max. limit of 5 ft. 

Provision for Future Increase of Live Load.* — 41. When live and 
dead load stresses are of opposite character, only 70% of the dead ^oad 
stress shall be considered as effective in counteracting the live load stress. 

(d.) Details of Construction. 

Camber.t — 42. For truss bridges, increase length o^ top chord K'' to 
every 10 ft. [Best: Shorten diagonals, without increasing the chord. — 
Author.] 

Adjustable Members. — 44. Preferably avoided. 

Truss Bridges. — 45. Stiff end vertical suspenders for through spans. 
For end panels of lower chord, preferably stiff members for single track 
spans. 

Lateral and Sway Bracing. — 48. To be compression shapes. 

Diagonal Bracing.J — 50. Deck bridges shall have diagonal braces at each 
panel, of sufficient strength to carry half the maximum strain increment 
due to wind and centrifugal force. 

Gusset Plates. — 51. At each end of pony trusses and through plate 
girders, and at fioorbeam of same. 

Temperature.il — 52. Provision for 150° F. variation. 

Bolsters and Expansion Rollers. § — 53. For bridges exceeding 80 ft. 

span, hinged bolsters at both ends, with nests of turned friction rollers at 
one end. Rollers not less than 4" dia.; and pressure p, in lbs. per lin. in. 
of roller, not to exceed 1200 \/d for steel rollers between steel surfaces 
(GJ = diam. of roller in ins.). 

Bed Plates.^ — 55. Pressure on masonry, including impact, 400 lbs. per 
sq. in. 

Rivets. — 56. In direction of strain, max. pitch to be 6" or 16 X thickness 
of thinnest outside plate, at right angles to strain, max. pitch to be 402" X 
that thickness. At ends of compression members, pitch not to exceed 4 
diameters of rivet for a length equal to 2(P times width of member. 

Tie Plates. — 60. All segments of compression members connected by 
latticing only, shall have tie plates placed as near the ends as practicable, 
with length not less than greatest depth or width of member, and thickness 
not less than ^V of the distance bet. the rivets connecting them to the com- 
pressed members. 

* If live load be increased 70% the stress per sq. in. in any member shall 
not exceed 1 . 7 X the allowed unit stress, and in case of reversal of stress 
proper provision shall be made for same. — -C. R. I. & P. 

t Camber for plate girders over 50 ft. span to be -ts'^ per 10 ft. of length. — 
C. R. I. & P. Camber for all spans, about M'' in 100 ft.— D. L. & W. Cam- 
ber for movable bridges, such that when end supports are raised to their 
exact position the base of rail will be at the same level on ends of bridge as 
at center. — L. 5. & M. S. Arc of circle; and at least Yzhxi of the span. — 
5. P. 

t Overhead diag. bracing at each panel point when height of truss 
exceeds 25 ft. — P. & R. 

J I Expansion and contraction, 1" in every 100 ft. — D, L. & W. and 
L. S. & M. 5. 150° F.— P. & R. and S. P. H" for each 10 ft.— C. R. I. & P. 

§ Rollers not less than 6" diam.— C. R. I. & P. Rollers not less than 
3H" diam.; or ^=300 d.—D. L. & W. Rollers: 1200 y/d.—P.&R. For 
bridges over 75 ft. span, segmental steel friction rollers not less than 6" 
diam; but cylindrical rollers V diam. may be used. — 5. P. 

A See, also, foot-notes to 34, preceding, for bearing on masonry. 

2 30.— P. & R, IM— C. R. I. & P. 



706 



dS.—RAILROAD BRIDGES, 



Lacing.* — 61. Single lattice bars shall have a thickness of not less than 
5^5, and double bars connected by rivet at intersection, not less than ^^, 
of dist. bet. rivets connecting them to the member; and their width shall be: 
2V' (Y' riv.) forl5" chans., or built sections with 3^' and 4" angles; 

2i" (F " ) " 12'' and 10'' " " " " " 3" 

r (f" " ) " 9" and 8" " " " " " 2^ 

Pin Plates. — 62. Must contain enough rivets to transfer the proportion 
of pressure upon them, and at least one plate on each side shall extend not 
less than 6" beyond edge of tie plate. 

(e.) Workmanship. 

Riveted Work. — 65. Hole = rivet + ^"; enlargement, by reaming. 

Planing and Reaming. — 67. In mediimi steel over ^", sheared edges 
to be planed, and holes to be drilled or reamed to diam. of rivet -f^". 

Eye=Bars. — 75. Heads of eye-bars to be made by upsetting, rolling, 
or forging. Welds not allowed. 78. All eye-bars shall be annealed. 

Machine Work. — 79. Abutting surfaces in compression members shall 
be truly faced to even bearings. 80. Ends of floor girders shall be faced 
true and square. 81. No variation of more than qV' for every 20 ft. will 
be allowed in length between centers of pin holes. 84. Clearance between 
pin and pin hole shall be 3^2'' for lateral pins; and for truss pins, 3V' for pins 
3i" diam., gradually increased to uV' for pins 6" diam. and over. 

(f.) Steel. 

Process of Manufacture. — 87. Open hearth. If by acid process, not 
over .08% phosphorous; if by basic process, not over .05% phosphorous. 

Physical Properties.! — 92. Three grades: 

Rivet Steel. 
Ultimate strength, lbs. per sq. in., 48-58000. 
Elastic limit, " *' " " i ult. 

Elongation, 26%. 

Bending test — ^without fracture on ' rmt 
outside bent portion. 180° 

Pins. — 100. Up to 7" diam., rolled. 101. Above 7" diam., forged. 

Steel Castings.! — 103. Open hearth, containing from . 25 to . 40% 
carbon, and not over .08% phosphorous. 



on 

itself. 



Soft Steel. 
52-62000. 

i ult. 

25%. 

Flat 

on 

itself. 



Medium Steel. 

60-70000. 

h ult. 

22%. 

To diam. 

= thick. 

of piece. 



* Same as Am. Br. Co., 61, above.— C. R. I. & P. 

Lacing bars shall not be less than 2ixf , and shall form an angle of not 
less than 60° with axis of member in single lacing, and 45° in double lacing. 
Double lacing must be riveted at intersections. — D. L. & W. 

Latticing shall be double, and shall preferably cross at right angles, 
and be riveted at intersections. Lacing shall be single, and be at angle of 
about 60° with axis of member. Minimiim size of lattice bars shall be as 
follows: 2ixA for 8" and 9" chans.; 2^x| for 10" and 12" chans. and 3" 
angles; 2jXte for 15" chans. and 3^" and 4" angles; connected by one rivet 
at each end. 4xi^ for built sections over 15" wide; connected by two rivets 
at each end. — L. 5. & M. S. For chords and posts the lattice bars shall 
generally be 3xi^; the width may be: 2" for lattice under 10" [long, 2\" 
under 15" long, and 2^' under 20'^long.— P. & R. 



tRoad. 


Properties. 






Steel. 






Medium . 


R'y Br. 


Struct'l. 


Soft. 


Rivet. 


Castings. 


C. R- 1. & P. 


Ult. tens. str.. 






56-64000 


54-62000 

.5 ult. 

26 


46-54000 

48-56000 

.5 ult. 

28 

48-56000 

.6 ult. 

28 


65000 




■ Ult. tens. str.. 

Elas. limit. . . . 
[Elong., %, 8". 
r Ult. tens. str.. 

Elas. limit. . . . 

Elong., %.... 

For bed plates 

For gearing. . . 


62-70000 

.5 Ult. 

22 

62-70000 

.6 ult. 

25 




66000 


D. L. & W. 






.5 ult. 












56-64000 

.6 ult. 

28 














L. S. & M. 








S. 






55-65000 














65-75000 


P. & R. 


Ult. tens str.. 






56-64000 




46-54000 
46-54000 

26000 

26 


65000 




f Ult. tens. str.. 

Elas. limit 

I Elong., %, 8". 


60-68000 
33000 






S P. 












22 












SPECIFICATIONS FOR STEEL R. R. BRIDGES. 



707 



VI.— TABLES USEFUL IN CONNECHON WITH PRECEDING 
SPECIFICATIONS. 



4. — Cooper's Standard Loading — Axle Loads. 
[Loads Are In Lbs. Per Track.] 

distances in feet. 



6 5 5 5_ 9 5 & 5 



8 ^6_5_5^ 



5 6 B 5 



^ nnnc) ooooooooo oooo^ 



Umf.toad 
>er-fr. 



E 30 



o o o o 

lO "lO "O to 

05 Oi 05 05 



o o o 

iO iO iO iO 

01 Ci OS Oi 



3000 lbs. 



E 40 



o o o o o 

S §iSS 

O O O o o 



§ 



8 



o o o o 

(M C<l (M <M 



4000 lbs. 



E 45 



III 

lO »0 lO 



lO 



lO lO »o to o 
<M (M C<l <M »0 

<M <M oq (M (M 



§88 



(M (M C^ (M 

05 05 Oi 05 
<M (M <M <M 



4500 lbs. 



£50 



88 8888 8 

So to to to to o 

O <M <M (M C<l to 

to to CO CO CO CO (M 



O O O O 

<M (M cq (M 

CO CO CO CO 



5000 lbs. 



E 55 



88 



o o o o o 

to ^ to CO o 

t^ l>- Ir^ !>. to 

to to to to b- 

CO CO CO CO <M 



t>. t^ !>. t^ 

CO CO CO CO 



5500 lbs. 



E 60 



888 
888 



8888 

o o o o 

05 Oi Oi 05 
CO CO CO CO 



6000 lbs. 



70S 



l—RAILROAD BRIDGES, 



5.--MAXIMUM Moments M, End Shears S, and Floorbeam Reactions R, 

Per Track, Produced by Cooper's Loadings* E 50 and E 40. 

[Mult. Values in Table by 1000.] 







Loading E 50. 








Loading 


E iO. 






i 


as . 


nd 


Sd. 








IH 


'^ d tA 


ia . 






^ 


. Mo: 
InlO( 
.-Lbs 


Max. En( 
Shear S 1 
1000 Lbs. 


Fl.-B 
c.R I 
) Lbs 


Equiv. Unif. Load 


1^3 


Fl.-B 

OLbs 


Equiv. Unit. Load 




in 1000 Lbs. 


^rJ 


in 1000 Lbs. 


pt 


|.e 


Max. 
Rea 

100( 








lis 


IP 


Max. 
Rea 

100 






OQ 


M 


S 


R 


M 


S 


R 


10 


141 


75 


100 


11.25 


15.00 


10.00 


113 


60 


80 


9.00 


12.00 


8.00 


11 


164 


82 


109 


10.86 


14.88 


9.92 


131 


66 


87 


8.69 


11.91 


7.94 


12 


200 


88 


117 


11.11 


14.58 


9.72 


160 


70 


93 


8.89 


11.67 


7.77 


13 


238 


92 


123 


11.24 


14.21 


9.47 


190 


74 


99 


9.00 


11.35 


7.58 


14 


275 


96 


130 


11.22 


13.78 


9.31 


220 


77 


104 


8.98 


11.03 


7.45 


15 


313 


100 


137 


11.11 


13.33 


9.11 


250 


80 


109 


8.89 


10.67 


7.29 


16 


350 


106 


142 


10.94 


13.28 


8.89 


280 


85 


114 


8.75 


10.62 


7.11 


17 


388 


112 


147 


10.73 


13.15 


8.65 


310 


90 


118 


8.58 


10.53 


6.92 


18 


425 


117 


152 


10.49 


12.96 


8.43 


340 


93 


121 


8.40 


10.38 


6.74 


19 


466 


121 


157 


10.34 


12.74 


8.28 


373 


97 


126 


8.27 


10.19 


6.62 


20 


516 


125 


164 


10.31 


12.50 


8.19 


413 


100 


131 


8.25 


10.00 


6.56 


21 


565 


129 


170 


10.25 


12.24 


8.09 


452 


103 


136 


8.20 


9.79 


6.48 


22 


614 


132 


175 


10.15 


11 98 


7.97 


491 


106 


140 


8.12 


9.59 


6.38 


23 


664 


135 


180 


10.04 


11.72 


7.84 


531 


108 


144 


8.03 


9.38 


6.27 


24 


713 


139 


185 


9.90 


11.55 


7.70 


570 


111 


148 


7.92 


9.23 


6.17 


25 


763 


142 


189 


9.76 


11.36 


7.56 


610 


114 


151 


7.81 


9.09 


6.05 


26 


812 


145 


194 


9.61 


11.18 


7.47 


650 


116 


155 


7.69 


8.93 


5.97 


27 


862 


148 


200 


9.46 


10.97 


7.41 


689 


119 


160 


7.56 


8.78 


5.93 


28 


914 


151 


206 


9.32 


10.79 


7.35 


731 


121 


165 


7.46 


8.63 


5.88 


29 


970 


154 


211 


9.23 


10.61 


7.27 


776 


123 


169 


7.37 


8.49 


5.82 


30 


1026 


158 


216 


9.12 


10.51 


7.19 


821 


126 


173 


7.30 


8.41 


5.75 


31 


1082 


161 


221 


9.01 


10.39 


7.14 


866 


129 


177 


7.21 


8.31 


5.71 


32 


1139 


164 


227 


8.90 


10.27 


7.11 


911 


132 


182 


7.12 


8.22 


5.69 


33 


1195 


167 


233 


8.78 


10.14 


7.07 


956 


134 


187 


7.02 


8.11 


5.66 


34 


1251 


170 


239 


8.66 


10.01 


7.02 


1001 


136 


191 


6.92 


8.01 


5.62 


35 


1307 


173 


244 


8.54 


9.88 


6.97 


1046 


138 


195 


6.84 


7.91 


5.57 


36 


1372 


176 


249 


8.47 


9.80 


6.91 


1097 


141 


199 


6.77 


7.84 


5.53 


37 


1436 


180 


253 


8.39 


9.71 


6.85 


1149 


144 


203 


6.71 


7.77 


5.50 


38 


1500 


183 


259 


8.31 


9.62 


6.82 


1200 


146 


208 


6.65 


7.70 


5.46 


39 


1567 


186 


265 


8.24 


9.52 


6.79 


1254 


149 


212 


6.59 


7.62 


5.43 


40 


1639 


189 


270 


8.20 


9.43 


6.75 


1311 


151 


216 


6.56 


7.54 


5.40 


42 


1784 


195 


280 


8.09 


9.30 


6.67 


1427 


156 


224 


6.48 


7.46 


5.34 


44 


1929 


201 


291 


7.97 


9.15 


6.62 


1543 


161 


233 


6.37 


7.32 


5.30 


46 


2074 


207 


302 


7.84 


9.00 


6.56 


1659 


166 


241 


6.28 


7.20 


5.24 


48 


2219 


212 


312 


7.71 


8.84 


6.49 


1776 


170 


250 


6.17 


7.07 


5.20 


50 


2377 


218 


322 


7.61 


8.71 


6.44 


1902 


174 


257 


6.09 


6.97 


5.14 


52 


2538 


223 


333 


7.51 


8.58 


6.40 


2030 


179 


267 


6.01 


6.87 


5.13 


54 


2703 


228 


345 


7.42 


8.44 


6.39 


2162 


182 


276 


5.93 


6.76 


5.12 


56 


2880 


233 


357 


7.35 


8.30 


6.38 


2304 


186 


286 


5.88 


6.64 


5.11 


58 


3058 


238 


370 


7.27 


8.22 


6.38 


2446 


191 


295 


5.82 


6.58 


5.09 


60 


3247 


244 


383 


7.22 


8.13 


6.38 


2599 


195 


305 


5.78 


6.51 


5.08 


62 


3441 


250 


395 


7.16 


8.07 


6.37 


2753 


200 


315 


5.73 


6.46 


5.08 


64 


3639 


256 


407 


7.11 


8.01 


6.36 


2911 


205 


325 


5.69 


6.41 


5.07 


66 


3849 


262 


419 


7.07 


7.95 


6.35 


3079 


210 


334 


5.66 


6.36 


5.07 


68 


4059 


270 


431 


7.02 


7.93 


6.34 


3247 


216 


344 


5.61 


6.34 


5.06 


70 


4269 


276 


443 


6.97 


7.89 


6.32 


3415 


221 


354 


5.58 


6.31 


5.06 


72 


4479 


283 


454 


6.91 


7.87 


6.30 


3584 


227 


362 


5.54 


6.30 


5.03 


74 


4699 


291 


465 


6.86 


7.86 


6.28 


3758 


233 


371 


5.49 


6.29 


5.01 


76 


4925 


298 


476 


6.82 


7.83 


6.26 


3942 


238 


379 


5.46 


6.27 


4.99 


78 


5160 


304 


487 


6.79 


7.80 


6.24 


4129 


243 


388 


5.43 


6.24 


4.97 


80 


5399 


311 


497 


6.75 


7.76 


6.21 


4321 


248 


396 


5.40 


6.21 


4.95 



* Note that values for all classes are proportional; thus, for E 55, mult, 
values for E 50 by 1 . 1; for E 45, by . 9; etc. 



MAX, M, S AND R FOR ENGINE LOADS. IMPACT. 



709 



5. — Maximum Moments M , End Shears 5, and Floorbeam Reactions R, 

Per Track, Produced by Cooper's Loadings* E 50 and E 40. — Concl'd. 

[Mult. Values in Table by 1000.] 



Loading E 50. 












Equlv. Unlf. Load 
In 1000 Lbs. 



M 



S R 



Loading E iO. 






«23 



• o3 o 



Equlv. Unlf. Load 
In 1000 Lbs. 



M 



82 


5638 


84 


5891 


86 


6145 


88 


6406 


90 


6674 


92 


6941 


94 


7209 


96 


7476 


98 


7755 


100 


8048 


125 


12491 


150 


17625 


175 


23400 


200 


29625 


250 


44025 



317 
324 
330 
337 
343 

349 
356 
362 
369 
375 

449 
522 

585 
656 

787 



507 
517 
527 
537 
546 

556 
566 
575 
584 
593 



6.71 
6.68 
6.65 
6.62 
6.59 
6.56 
6.53 
6.49 
6.46 
6.44 
6.40 
6.26 
6.11 
5.92 
5.64 



7.74 
7.71 
7.68 
7.65 
7.62 

7.60 
7.57 
7.54 
7.52 
7.50 

7.18 
6.96 
6.69 
6.55 
6.29 



6.19 
6.16 
6.13 
6.10 
6.07 
6.04 
6.02 
5.99 
5.96 
5.93 



4513 
4713 
4919 
5128 
5341 
5552 
5771 
5988 
6213 
6440 

9993 
14100 
18720 
23700 
35220 



254 
259 
264 
269 
275 
280 
285 
290 
295 
300 

359 

418 
468 
524 
629 



404 


5.37 


6.19 


412 


5.34 


6.17 


421 


5.32 


6.15 


429 


5.30 


6.12 


437 


5.28 


6.10 


444 


5.25 


6.08 


452 


5.23 


6,06 


459 


5.20 


6.03 


467 


5.18 


6.02 


474 


5.15 


6.00 




5.12 


5.74 




5.01 


5.57 




4.89 


5.35 




4.74 


5.24 




4.51 


5.03 



4.93 
4.91 
4.89 
4.87 
4.86 

4.83 
4.81 
4.78 
4.76 
4.74 



* Note that values for all classes are proportional; 
values for E 50 by 1.1; f or E 45, by 0.9; etc. 



thus, for E 55, mult. 



-Coefficients of Impact (/). 
(See page 701.) 



L. 


300 


L. 


300 


L. 


300 


L. 


300 


L. 


300 




L+300 


L+300 


L+300 


L+300 


L+300 


5 


0.984 


31 


0.906 


57 


0.840 


83 


0.783 


145 


0.674 


6 


0.980 


32 


0.904 


58 


0.838 


84 


0.781 


150 


0.667 


7 


0.977 


33 


0.901 


59 


0.836 


85 


0.779 


155 


0.659 


8 


0.974 


34 


0.898 


60 


833 


86 


0.777 


160 


0.652 


9 


0.971 


35 


0.896 


61 


0.831 


87 


0.775 


165 


0.645 


10 


0.968 


36 


0.983 


62 


0.829 


88 


0.773 


170 


0.638 


11 


0.965 


37 


0.890 


63 


0.826 


89 


0.771 


175 


0.632 


12 


0.962 


38 


0.888 


64 


0.824 


90 


0.769 


180 


0.625 


13 


0.958 


39 


0.885 


65 


0.822 


91 


0.767 


185 


0.619 


14 


0.955 


40 


0.882 


66 


0.820 


92 


0.765 


190 


0.612 


15 


0.952 


41 


0.880 


67 


0.817 


93 


0.763 


195 


0.606 


16 


0.949 


42 


0.877 


68 


0.815 


94 


0.761 


200 


0.600 


17 


0.946 


43 


0.875 


69 


0.813 


95 


0.759 


210 


0.588 


18 


0.943 


44 


0.872 


70 


0.811 


96 


0.758 


220 


0.577 


19 


0.940 


45 


0.870 


71 


0.809 


97 


0.756 


230 


0.566 


"20 


0.937 


46 


0.867 


72 


0.806 


98 


0.754 


240 


0.556 


21 


0.935 


47 


0.865 


73 


0.804 


99 


0.752 


250 


0.546 


22 


0.932 


48 


0.862 


74 


0.802 


100 


0.750 


260 


0.536 


23 


0.929 


49 


0.860 


75 


0.800 


105 


0.741 


270 


0.526 


24 


0.926 


50 


0.857 


76 


0.798 


110 


0.732 


280 


0.517 


25 


0.923 


51 


0.855 


77 


0.796 


115 


0.725 


290 


0.508 


26 


0.920 


52 


0.852 


78 


0.794 


120 


0.714 


300 


0.500 


27 


0.917 


53 


0.850 


79 


0.792 


125 


0.706 


400 


0.429 


28 


0.915 


64 


0.847 


80 


0.789 


130 


0.698 


500 


0.375 


29 


0.912 


55 


0.845 


81 


0.787 


135 


0.690 


600 


0.333 


30 


0.909 


56 


0.843 


82 


0.785 


140 


0.682 


700 


0.300 



Note. — For notation and formula, see page 701. 



710 



],— RAILROAD BRIDGES, 



7. — Permissible Compressive Stresses. 

(See page 703.) 

^ = stress allowed in lbs. per sq. in.; / = length, r = least raditis ot gyration 

(both in inches) . 





Formula. 


r 


Formula. 


r 


Formula. 


/ 


Soft Steel 
15,000 


Medium 
Steel 
17,000 


Soft Steel 
15,000 


Medium 
Steel 
17,000 


Soft Steel 
15,000 


Medium 
Steel 
17,000 

^^ /2 




1. '' 


"^13,500^2 


1." '' 




^^ll,000r2 


^'^ll,000r2 




13,500r2 


13,500r2 


^ ' ll,000r2 


10 
12 
14 
16 


14900 
14840 
14780 
14720 


16850 
16780 
16710 
16610 


50 
52 
54 

56 


12660 
12500 
12340 
12180 


13850 
13650 
13340 
13230 


90 
92 
94 • 
96 


9370 
9220 
9060 
8910 


9790 
9610 
9420 
9240 


18 
20 
22 
24 


14650 
14560 
14480 
14400 


16510 
16410 
16290 
16150 


58 
60 
62 
64 


12010 
11840 
11670 
11500 


13020 
12810 
12600 
12390 


98 
100 
102 
104 


8760 
8610 
8470 
8320 


9080 
8910 
8470 
8570 


26 
28 
30 
32 


14280 
14180 
14070 
13940 


16020 
15870 
15710 
15550 


66 
68 
70 

72 


11340 
11140 
11010 
10840 


12180 
11970 
11760 
11550 


106 
108 
110 
112 


8180 
8050 
7910 
7780 


8410 
8250 
8100 
7940 


34 
36 
38 
40 


13810 
13690 
13550 
13420 


15380 
15210 
15030 
14840 


74 
76 
78 
80 


10670 
10500 
10340 
10180 


11350 
11510 
10950 
10750 


114 
116 
118 
120 


7640 
7510 
7380 
7260 


7790 
7650 
7500 
7360 


42 
44 
46 

48 


13270 
13120 
12960 
12820 


14650 
14460 
14260 
14060 


82 
84 
86 
88 


10010 
9850 
9690 
9530 


10550 

10350 

10160 

9970 









VII.-APPROX. WEIGHT OF STEEL IN STEEL RAILROAD BRIDGES. 



By Formula, «; = (ic/+:y) ^/Loading 

In which w =weigh.t in lbs. of steel per lin. ft. of bridge (does not include 
rails, guards nor ties), 
/= length of span in ft. 



8. — Values of x and y, in 


Above 


Formula. 




Kind of Bridge. 


to 

si 


Single 
Track. 


Double 
Track. 




X 


y 


X y 


Deck Plate Girder, spans 20-100 


Extra. 
Good. 


2. 


20 

20 

70 

130 

30 

90 

100 

120 


• 1 >-t 






8 
8 
8 
2 
2 
1 



Sil- 


Through '* " " " 


m;D 03 


" " " (solid floor) 
Deck Lattice " " 90-150 


^ «3 O . 


Through " " " " 


^^-i 


" Pratt Truss " 130- 


i^«^ 


" Baltimore" " 200- 


Mu 
sing 
abou 

1.8 









WEIGHT OF STEEL BRIDGES. HOWE TRUSS. 



711 



9. — Values of VLoading, in preceding Formula. 



Loading (Class): — 


E 60 


E55 


E50 


E45 


E 40 


E35 


E 30 




7.75 


7.42 


7.07 


6.71 


6.32 


5.92 




VLoading: — 


5.48 


Relative Values of 


1.00 

1.05 
1.10 
1.15 
1.23 
1.31 
1.41 


0.96 
1.00 

1.05 
1.10 
1.17 
1.25 
1.35 


0.91 
0.95 
1.00 
1.05 
1.12 
1.19 
1.29 


0.86 
0.90 
0.95 
1.00 
1.06 
1.13 
1.22 


0.81 
0.85 
0.89 
0.94 
1.00 
1.07 
1.15 


0.76 
0.80 
0.84 
0.88 
0.94 
1.00 
1.08 


0.71 
0.74 
0.78 


VLoading, for Classes 
E 30 to E 60. 


0.82 
0.87 
0.93 
1.00 



Example. — Find the approx. weight of steel in a Through Plate Girder 
Bridge with floor beams and stringers, span 100 ft., loading E 40? 
Solution.— /Xw= 100 (180+70) 6.32= 158000 lbs. 

VIII.— PLANS AND DETAILS OF RAILROAD SPANS. 

Howe Truss Bridges are being rapidly replaced by steel and concrete 
structures on all of our American roads, but the following typical plans will 
be of interest. 

N. P. R'y Standard Plan of Howe Truss and Details. 
90 Ft. Through Span. 
Loads. — Dead Load, 1300 lbs. per lin. ft. Live Load, consolidation 
engine and tender, 225,000 lbs., followed by 3000 lbs. per lin. ft. 



Top Chord. — Pack- 
ing bolts ^'' dia. C.-I. 
Sep. for bolts, 6" and 
SH'' dia. 




\TopChorcf. 




Bottom ^ Chord. — 
Iron packing keys. 
Packing bolts, ^/i" dia. 



Figs. 25. — General Plan and Elevation. 
Timber Specifications. — Must be sound, live, straight and close grained 
red or yellow fir, cut free from wanes, shakes, pitch seams and clear of 



712 



2S,— RAILROAD BRIDGES. 



knots larger than 1" dia., nor closer than 4 ft. on any side of stick. Knots 
must be sound and clear of pitch. All timber must be cut from large live 
logs free from heart and sap, and sawed true to sizes ordered; and delivered 
f . o. b. cars, subject to inspection before delivery. 

Notes. — Use red fir cross-ties west of Helena. Frame out camber by 
sizing stringers into floor-beams. In framing, carpenters must lay out 
panels full size, make templates of angle blocks from the castings and cut 
braces to correct 1 ength and shape so determined. Cast iron separators for 
packing bolts in top chord are 5" dia. except for bolts in centers of panels 
and at ends of chords, which have separators 3M'' dia. All other separators 
for packing bolts are SW^ dia. Guard rails bolted to ties at ends and in 
middle and spiked to all other ties with 3^'' x lO'^ bridge spikes. Every 



fifth tie spiked to stringers with ^'^ x 14'' 



H-'i??^ 


H 


o o 


o 






] 



a: 



S:^/ Channel. 



^'Wrt'lF/crfe 




bridge spikes. 

Steel Channels: 

Holes spaced 7^'' centers. 

Make 8 chans. with holes iVs" dia. 
.. g .. .. .. ^y^" '' 

- 8 '• " " 2W " 

.. 4 .. .. .. 2%" " 



Top View. I Bo-H*. View. 



l<-743b;74--n|4' ^^^-z4^^'->,-.4 



Section. End View. | 






-^'->\fi\^=''-'--^ 



Tpp -View, i Bortt. View. 



t . — .\ . 

End View. I Section. 



r<-v": 



.'iM'i® Iff 






_>, 



Pfe^ 






a. 



Top View. Bottom View. 




Bills of Material. — Cast 
iron, 16,908 lbs.; wrought 
iron, 17,948 lbs.; timber, 45,- 
126 ft. B.M. 



nmr\m m i|| 



.Side Elevafion, 

Figs. 26. — Howe Truss Details. 

Reinforced Concrete Bridges. — The principal use of reinforced concrete 
in railroad bridges is in the form of trestles and arches, where, under cer- 
tain conditions, it is found most economical. Many useful hints in this 
class of construction will be found in the three following pages of "Excerpts 
and References." For formulas and working, stresses, see Sections 25, 31 
and 32. See, also, other Sections and Index. 



HOWE TRUSS DETAILS. MISCELLANEOUS. 



713 



EXCERPTS AND REFERENCES. 

A Graphical Method of Finding FIoor=Beani Concentrations Under 
Wheel=Loads (By R. H. Bulloch. Eng. News, May 21, 1903). 

Standard Plans for Bridges on the Atchison, Topeka & Santa Fe Ry. 

(By J. Dun. Eng. News, May 28, 1903).— Illustrated. 

A Comparison of the Requirements of Recent Railway Bridge Speci- 
fications (By A. H. Heller. Eng. News, Nov. 19, 1903).— Tabulated data 
from 29 specifications. 

A New Truss Design (By J. W. Schaub. Eng. News, Mar. 24, 1904).— 
Warren type, with a combination of pin and riveted connections. Adopted 
by some of the railroads. 

A Moment Table for Wheel Loads (By R. B. Ketchum. Eng. News, 
May 12, 1904). Table and diagrams. 

New Westminster Bridge over Fraser River. B. C. (By Waddell and 
Herrick. Eng. News, June 15 and 22, 1905). — Foundations, superstructure 
and erection; illustrated. 

Recent Railway Viaducts of Reinforced Concrete (Eng. News, May 
31, 1906).— Illustrated. 

Diagram Table for Cooper's E-50 Loading (By J. Gibson. Eng. 
News, June 21, 1906). — Double inset sheet. Very elaborate table. 

Wheel Loadings of Mallet Duplex Compound Locomotive for Great 
Northern Ry. (Eng. News, Nov. 22/ 1906).— Corrected diagram. 

Rail Expansion Joints on the Thebes Bridge (By Ralph Modjeski. 
Eng. News, Aug. 22, 1907).— Illustrated. 

Proportioning Steel Railway Bridge Members (By H. S. Pritchard. 
Eng. News, Sept. 19, 1907). 

Erection of Long Span Trusses by End Launching (Can. Soc. Civ. 
Engrs. April 16, 1908; Eng. News, July 23, 1908).— Illustrated. 

Safe Unit Stresses in Structural Timber (Eng. Rec, Apr. 24, 1909). — 
The safe unit stresses in structural timber recommended by the committee 
on wooden bridges and trestles of the Am. Ry. Eng. and M. of W. Assn., for 
green condition of timbers and without increasing the live-load stresses for 
impact, are as follows: — 





Bending, 


Shearing. 


Compression. 




Extreme 
fiber 
stress. 


Parallel 

to 
grain. 


Long'l 

in 
beams. 


Perp. 

to 
grain. 


Parallel 

to 
grain. 


Cols. 

under 

15 dia's. 


Douglas fir 

Longleaf pine .... 
Shortleaf pine .... 

White pine 

Spruce 

Norway pine 

Tamarack 

Western hemlock . 
Redwood 


1 200 

1 300 

1 100 

900 

1 000 

800 

900 

1 100 

900 

900 

800 

1 100 


170 
180 
170 
100 
150 
130 
170 
160 
80 
120 

"216 


110 
120 
130 
70 
70 
100 
100 
100 

iio 


310 
260 
170 
150 
180 
150 
220 
220 
150 
170 
230 
450 


1 200 
1 300 
1 100 
1 000 
1 100 

800 
1 000 
1 200 

900 
1 100 

900 
1 300 


900 
980 
830 
750 
830 
600 
750 
900 
680 


Bald cypress 

Red Cedar 

White oak 


830 
680 
980 



For long columns exceeding 15 diameters in length the safe stress given 
for compression parallel to grain are taken and decreased by 60L-r-D, 
where L=length in ins., and P=least side in ins. 

Measurement of Impact Stresses (By B. W. Dunn. Proc. A. S. T. M., 
Vol. IX.. 1909). 



714 38.— RAILROAD BRIDGES. 

Reinforced=Concrete Bridges for Track Elevation on 111. Cent R. R. ; 
Failure Test of Very Large Concrete Slabs (Eng. News, Aug. 6, 1908). — Illus- 
trated. 

Application of Spiral Hooping to a French Concrete Bridge (Eng. 
News, April 22, 1909).— Illustrated. 

Erection of New River Bridge by the Cantilever Method (By L. L. Jewel. 

Eng. News, July 8, 1909). — Single-track deck structure, 2155 ft. long, 112 
ft. above low water; spans, 125 to 140 ft.; all trusses are riveted Warren 
trusses, 26 ft. deep c.-c. chords and spaced 12 ft. apart; no floor system 
provided ; the deck of extra heavy ties being carried directly on the top chords. 
Erected by cantilever traveler; 11 illustrations. 

Guard Rail and Deck Construction for Railway Bridges (Eng. News. 
Sept. 9, 1909). 

Illustrations: 

Typical arrangement of bridge guards; Lehigh Valley R. R. Deck and 
guard rail construction (with two and three inside guards); Can. Pac. R. R. 
Bridge deck with sidewalk; Carolina, Clinchfield and Ohio R. R. Bridge 
deck construction on curves; Carolina, Clinchfield and Ohio R. R. Details 
of bridge deck and guards; Lehigh Valley R. R. Bridge deck with timber 
guards; Louisville and Nashville R. R. Bridge deck and guards (with 
tangents and curves); Penn. Lines. Bridge guards with re-railing devices; 
Southern Pacific R. R. Details of re-railing devices; Southern Pacific R. R. 
Re-railing devices at approaches to jack-knife draws; B. & M. R. R. 
Bridge guards on concrete bridge on a French railway. Deck and guard 
rail construction; So. Side Elev. Ry., Chicago. Deck and guard rail con- 
struction for curves; So. Side Elev. Ry., Chicago. Examples of bridge 
guards on English railway bridges. 

Lethbridge Viaduct over the Belly River, Canadian Pac. Ry. (By J. E. 

Schwitzer. Eng. News, Sept. 23, 1909). — Plans of floor and tower. 

Four=Track Truss Bridge with Solid Floor; Chicago & Oak Park Elev. 

Ry. (Eng. News, Dec. 9, 1909). — Numerous plans and details, including 
falsework. 

Long Span Reinforced=Concrete Girder Bridges (By F. W. Scheidenhelm. 
Eng. News, Jan. 27, 1910). — Illustrations: Girders, abutment and piers; 
with details of centering and forms used on the 75-ft. girder; also, splice 
clamp for reinforced-concrete bars. 

Reinforced=Concrete vs. Steel for Short Span Railway Bridges (Eng. 

News, Mar. 24, 1910). — Reinforced-concrete flat slabs or girders not gener- 
ally advisable for railway loads in spans exceeding 40 feet. 

450=Ft. Steel R. R. Span of the Miles Glacier Bridge, Alaska (Eng. Rec, 
Aug. 6, 1910). — Illustrations: Elevation of bridge (one 450' span, two 400' 
spans, one 300' span); truss details of 450' span; fixed and expansion end 
shoes, with segmental rollers and nest and grillage, 450' span; erection filler 
between shoes on pier; erection adjustment devices for top and bottom 
chords. 

The St. Louis Municipal Bridge Superstructure (Eng. Rec, Dec. 3, 1910). 

— Bridge composed of three 668-ft. steel spans carrying two railroad tracks 
on the lower floor at bottom chord level, and two electric car tracks, a drive- 
way, and two cantilever sidewalks one on second floor 22 ft. in the clear 
above the lower floor. The two pin -connected trusses 35 ft. apart on centers 
are 110 ft. deep, 65 ft. in the clear above high water and will be made of 
nickel steel, while the floor system and bracing will be made of carbon steel. 
(See Eng. Rec. of Oct. 30, 1909, for diagrams, stress sheets, specifications, 
etc.; also Eng. Rec. of Oct. 15, 1910, for description of design and con- 
struction of the substructure.) The main span trusses are longer than those 
of any other span with independent trusses; the panel lengths vary with the 
depths of the trusses in accordance with economical inclinations of the 
diagonal members: the top chord has a special type of cross-section ; the upper 
deck is made with intermediate floorbeams carried on longitudinal girders; 
the main truss shoes and pedestals are heavy steel castings; many of the 
compression members have half-hole pin-bearings without interlocking 



REFERENCES, 716 

pin plates; and the members and details are of standard construction. 
Illustrations: — Regular cross-section; general diagram of 668-ft. span, 
giving heights of truss; upper and lower portal bearing; roadway bent over 
pier; sections of upper and lower decks; expansion shoe and pedestal for 
same ; pedestal for fixed shoe ; top chord details of sidewalk and curb girder. 

Important Illustrations of Railroad Bridges and Details. 

Description. Eng. News 

Large rein. -cone, railway viaduct, Rotterdam, Holland June 16,' 10 

Through-truss, 517i-ft. river spans, St, Louis July 28,' 10 

Part details 420-ft. truss span and floor Aug. 25,' 10 

Stringer connection allowing flexibility Aug. 25,' 10 

Erie R. R. viaduct, Penhom Creek Oct. 13, ' 10 

Eng. Rec. 

Details of steel viaduct, B. & M. R. R Feb. 27, '09 

Ballasted-floor details and steel construction, Erie R. R Feb. 27, '09 

116-ft. railroad span Mar. 27, '09 

Erection of long span bridges across the Susquehanna Apr. 3, '09 

Falsework for replacing the Cuyahoga Val. viaduct June 5, '09 

6-track, short span, solid-floor bridge, N. Y., N. H. & H. R. R. .July 24, '09 

Strain -sheet 668-ft. span — St. Louis Municipal Bridge Oct. 30, '09 

Typical falsework for Poughkeepsie Bridge reinforcement Oct. 30, '09 

Heavy steel floor for double-track bridge, D. L. & W Jan. 15, '10 

100-ft. span plate-girder bridge, N. Y., N. H. & H. R. R Mar. 12, '10 

C. M. & St. P. R. R. bridge across Missouri River, Mobridge, 

S. Dak June 11, '10 

Standard 150-ft. Howe truss R. R. span and details Sept. 3, '10 

Standard I-beam R. R. bridges over streets, .N. Y. Cent Sept. 17,'10 

Elev. and section of 8-track R. R. bridge over streets Oct. 1, '10 

Structural details, Providence, R. I., station viaduct, N. Y., N. 

H. & H. R. R Oct. 22, '10 

Construction and reconstruction of the Coteau steel bridge Dec. 3.. '10 



39.— ELECTRIC RAILWAY BRIDGES. 

See, also, Sec. 40, Highway Bridges, page 727. 

Typical "L" Loading as follows, for electric railway bridges may be 
used for any structure, heavy or light, by assigning proper values to w.* 



(a) For calculation of floor system, use two axle concen- 
trations of 15 w each, spaced 10 ft. centers. (See Fig. 1.) 

(b) For calculation of trusses, use for each track a uniform 
moving load oi w lbs. per lin. ft. for spans up to 100 ft.; 
and 0.8 w for spans 200 ft. and over; with a reduction of 
loading for intermediate spans of 0.01 w for every 5-ft. 
increase over 100 ft. The uniform load m; per track is assumed to cover 
a surface 12 ft. wide for single track; 22 ft. wide for double track. 

Tables 1 and 2 are based on the above typical "L" loading. 



t5w i§w 

(1) ( b 

Fig. 1. 



1.— Special Type "L 


"OF 


Electric-Car Loading t — Axle 


AND 


Uniform. 






For the trusses. Uniform live load (12 ft. wide) 




* For the 


per lineal foot of track. 




floor and its 
supports. 




Name. 


Up 
























Concentrated 


to 


110' 


120' 


130' 


140' 


150' 


160' 


170' 


180' 


190' 


200' 




Axle Loads. 


100' 

span 


span 


span 


span 


span 


span 


span 


span 


span 


span 


span 




Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


L 


15 w 


w 


.98w 


.9Qw 


.Mw 


.92w 


.90w 


.88w 


.SQw 


.8Aw 


.82w 


.8w 


L 18 


18,000 


1200 


1176 


1152 


1128 


1104 


1080 


1056 


1032 


1008 


984 


960 


L 24 


24 ,000 


1600 


1568 


1536 


1504 


1472 


1440 


1408 


1376 


1344 


1312 


1280 


L 30 


30.000 


2000 


1960 


1920 


1880 


1840 


1800 


1760 


1720 


1680 


1640 


1600 



Typical "K" Loading, Fig. 2, consists of a train of electric cars with 
axle spacing, 5'-15'-5'-15', etc., continuous. The loading shown in the sub- 
joined diagram is for 100,000-lb. cars, each covering a length of 40 feet. 
Tables 3, 4 and 5, following, are calculated from this diagram. 



10 

o 
o 
o 
to 
cu 



o 

o 50-Ton 
S Car 



& 


1 




J 


M 


§ 


O 

o 


Axle 


o 
o 
o 


o 

8 


IT) 




Loac^s 


irT 


ta 



o o 

so -Ton S^ 8 

Car cu cu 



4\ A'^ 



sns 



wri^ 



5nD 



4v7? 



2n5 



^k rr 



STffi 



K5 H<-— -/5' —• ->K^'>i<- 



75' 



->j<5'>K /^' -—->}< 5'h 



Fig. 2. 



* For the Manhattan suspension bridge across the East River, New 
York, the value of w was originally assumed at 1700 lbs. per lin. ft. for 
each of four rapid transit trains; and 1000 lbs. per lin. ft. for each of four 
lines of trolley cars. The revised loading is given on page 756. 

t The concentrated loading gives maximum moments for spans under 
48 . 2 feet, and the uniform loading for spans over 48 . 2 feet. For maximum 
floorbeam reactions use concentrated loading for panels up to 23.66 feet 
in length (2 panel span of 47 . 32 feet). Beyond this length uniform loading 
gives maximum floorbeam reaction. For maximum end shear use con- 
centrated loading up to 54 . 5 ft. span, and uniform loading beyond. 



716 



MOMENTS AND SHEARS FOR BRIDGE SPANS 



717 



2. — ^Maximum Moments, End Shears, and Floorbeam Reactions R 
per track for concentrated "L 24" loading — 24,000 lbs. on each of two axles 
spaced 10 feet apart. 
Values for any other "L" loading are directly proportional to the axle loads. 





Moment 


Equivalent 


End 


Equivalent 


Floorbeam 


Equivalent 


Span 


M. 


Uniform 


Shear S. 


Uniform 


Reaction i?. 


Uniform 


Ft. 


Ft.-lbs. 


Load w for 


Lbs. 


Load w for 


Lbs. 


Load w for 




9» M 


Moment. 




End Shear. 


7A f4 


Floorbeam 




1.. 1 u 


Lbs. per 
lin. ft. 


Lbs. per 
lin. ft. 


1^ 


Reaction. 

Lbs. per 

lin. ft. 




o ^ '^ 


From 


For/>10' 


From 


FoTl>W 


Frorn 




S SH 


equation 


use 


equation 


use 


equation 






wP 




wl 




R = wl 


f 




^-T 




^=T 




we have 


» 


U Rr, 


we have 


S = "24" + 


we have 


i? = "24" + 






& f^ 


8M 


'•'24" (I-IO) 


w-^ 


"24" (/-lO) 


R 






^- I, 


I 


"'- / 


/ 


w- ^ 


10 


60,000 


4,800 


24,000 


4.800 


24,000 


2,400 


11 


66,000 


4,364 


26.180 


4,760 


26,180 


2,380 


12 


72 ,000 


4,000 


28,000 


4,667 


28,000 


2,333 


13 


78,000 


3,692 


29,540 


4,545 


29,540 


2,272 


14 


84,000 


3,429 


30,860 


4,409 


30,860 


2,204 


15 


90,000 


3,200 


32 ,000 


4,267 


32,000 


2,133 


16 


96,000 


3,000 


33,000 


4,125 


33,000 


2,062 


17 


102,000 


2,824 


33,880 


3,986 


33,880 


1,993 


18 


112,670 


2,782 


34,670 


3,852 


34,670 


1,926 


19 


123,790 


2,743 


35,370 


3,723 


35,370 


1,861 


20 


135,000 


2,700 


36,000 


3,600 


36,000 


1,800 


21 


146,290 


2,654 


36.570 


3,483 


36,570 


1,741 


22 


157,640 


2,606 


37.090 


3,372 


37,090 


1,686 


23 


169,040 


2,556 


37,570 


3,267 


37,570 


1,633 


24 


180,500 


2,507 


38,000 


3,167 






25 


192,000 


2,458 


38,400 


3,072 






26 
27 


203,540 
215,110 


2,409 
2,361 


38,770 
39,110 


2,982 
2,897 


Single Track. 


28 


226,710 


2,313 


39.430 


2,816 




^^ 


29 


238,340 


2.267 


39,720 


2,740 


P^ 


30 


250 ,000 


2,220 


40,000 


2,667 


n 


Xl 


31 


261,680 


2,178 


40,260 


2,598 


k 


' % 


32 


273,380 


2,136 


40,500 


2,531 


^ 


33 


285.090 


2,094 


40,730 


2,468 


Floorbeam 


Moment = 


34 


296.820 


2,054 


40,940 


2,408 


R (s-c) 
2 2 




35 


308.570 


2,015 


41.140 


2,351 


- ft.-lbs. 


36 


320,330 


1,977 


41,330 


2,296 




37 


332,110 


1,941 


41,510 


2.244 






38 


342,900 


1,905 


41,680 


2,194 






39 


355,700 


1,871 


41,850 


2.146 






40 


367,500 


1,838 


42 ,000 


2,100 






41 


379,320 


1,806 


42,150 


2,056 






42 
43 


391,140 
402,980 


1,774 
1,744 


42,290 
42,420 


2,014 
1.973 


Double 


Track. 


44 


414,820 


1,714 


42,550 


1,934 




^ 


45 
46 


426.670 
438.520 


1,686 
1,658 


42.670 
42,780 


1,896 
1,860 


PpT^^ ^ 


t U 


U> 1 


47 


450,380 


1,631 


42,890 


1,825 


48 


462 ,250 


1.605 


43 ,000 


1,792 


a 


A 


49 






43,100 
43,200 
43,290 


1,759 

1,728 
1,698 


Floorbean 
at a-i?W- 


TVT 1 - 


50 






CI Moment: 


51 






r^'^ft.-ibs. 


52 






43,380 
43,470 
43.560 


1,668 
1,640 
1,613 


ath = R^;- 


2 


53 






^) f4. It, 


54 







-— ft. lbs. 



718 



I— ELECTRIC RAILWAY BRIDGES. 



3. — Maximum Moments M per Track for Concentrated "K 25" Load- 
ing (Fig. 2) — A Train of 50-Ton Electric Cars.* 
Values for any other "K" loading are directly proportional to the axle loads. 



4J 
<U 




Equiva- 


-*-s 
0) 




Equiva- 






Equiva- 


(U 




lent Unif . 






lent Unif. 


<U 




lent Unif. 


f^ 


Moment 


Load w 


Moment 


Load w 


fe 


Moment 


Load w 


.B 


M. 


for 


•S 


M. 


for 


^ 


M. 


for 


c 


Ft.-Lbs. 


Moments. 


G 


Ft.-Lbs. 


Moments. 


d" 


Ft.-Lbs. 


Moments. 


CO 




Lbs. per 


a 

CO 




Lbs. per 


03 




Lbs. per 




lin. ft. 




lin. ft. 


CO 




lin. ft. 


10 


70,300 


5,625 


41 


559,300 


2,662 


72 


1,653,500 


2,550 


11 


82,100 


5,430 


42 


583,500 


2,646 


73 


1,697,100 


2,548 


12 


94,000 


5,225 


43 


607,700 


2,629 


74 


1,740,800 


2,543 


13 


106,000 


5,020 


44 


632,000 


2,611 


75 


1,787,500 


2,542 


14 


118,100 


4,820 


45 


659,000 


2,604 


76 


1,837,000 


2,544 


15 


130,200 


4,630 


46 


690,200 


2,609 


77 


1,886,500 


2,545 


16 


142,400 


4,450 


47 


721,400 


2,613 


78 


1,936,100 


2,546 


17 


154,600 


4,280 


48 


752,600 


2,613 


79 


1,986,900 


2,547 


18 


166,800 


4,120 


49 


783,800 


2,612 


80 


2,035.200 


2,544 


19 


179,100 


3,970 


50 


835,000 


2,609 


81 


2,084,700 


2,542 


20 


191,400 


3,830 


51 


854,600 


2,628 


82 


2,134,300 


2,539 


21 


203,700 


3,695 


52 


892,000 


2,639 


83 


2,183,900 


2,536 


22 


216,100 


3,570 


53 


929,400 


2,647 


84 


2,233,500 


2,532 


23 


228,400 


3.455 


54 


966,800 


2,653 


85 


2,283,100 


2,528 


24 


240,800 


3,345 


55 


1,004,300 


2,656 


86 


2.337,700 


2,529 


25 


253,100 


3,240 


56 


1,041,700 


2,657 


87 


2,393,900 


2,530 


26 


265,500 


3,140 


57 


1,079,100 


2,657 


88 


2,450,100 


2,531 


27 


277,900 


3,050 


58 


1,116,500 


2,655 


89 


2,506,300 


2,531 


28 


290,300 


2,960 


59 


1.154,000 


2,652 


90 


2,566,800 


2,535 


29 


302,700 


2,880 


60 


1,191,400 


2,648 


91 


2,629,300 


2,540 


30 


319,400 


2,840 


61 


1,228,800 


2,642 


92 


2,691,700 


2,544 


31 


338,000 


2,815 


62 


1,266,300 


2,635 


93 


2,754,200 


2,548 


32 


356,500 


2,785 


63 


1,303,700 


2,628 


94 


2,816,700 


2,550 


33 


375,100 


2,755 


64 


1,341,200 


2,619 


95 


2,879,100 


2,552 


34 


393,600 


2,725 


65 


1.378,600 


2,610 


96 


2,941,600 


2,553 


35 


415,200 


2,710 


66 


1,416,100 


2,601 


97 


3,004,000 


2,554 


36 


439,100 


2,710 


67 


1,453,500 


2,590 


98 


3,066,500 


2,554 


37 


463,000 


2,705 


68 


1,490,900 


2,579 


99 


3,128,900 


2,554 


38 


487,000 


2,700 


69 


1,528,400 


2,568 


100 


3,191,400 


2,553 


39 


511,100 


2,690 


70 


1,566,100 


2,557 


101 


3,253,900 


2,552 


40 


535,200 


2,675 


71 


1,609,800 


2,555 


102 


3,316,300 


2,551 



*For a single car on the span multiply values in table by the following 
percentages: Spans 0' to 44', mult, by 100% ; span 45', mult, by 99.6% ; span 
50',by93.2%; span 60', by 85.9%; 70',by81.1%; 80',by 74.6%; 90',by 68.8%; 
100', by 63.1%. 

4. — Maximum Floorbeam Reactions R, per Track, for Concentrated 

"K25" Loading (Fig. 2). — A Train of 50-Ton Electric Cars. 

(Span = Panel length.) 

Values for any other "K" loading are directly proportional to the axle loads. 



+3 




Equival't 


a> 




Equival't 






Equival't 


r? 


Floor- 


Unif. Load 


£ 


Floor- 


Unif. Load 




Floor- 


Unif. Load 


P^ 


beam 


wior 


beam 


•wioT 


beam 


w;for 


.s 


Reaction 


Floorbe'm 


.H 


Reaction 


Floorbe'm 


G 


Reaction 


Floorbe'm 


c 


R. 


Reaction. 


c 


R. 


Reaction. 


u 


R. 


Reaction. 


ci 


Lbs. 


Lbs. per 


U 


Lbs. 


Lbs. per 


ci 


Lbs. 


Lbs. per 
lin. ft. 


CO 




lin. ft. 


c& 




lin. ft. 


CO 




10 


37,500 


3,750 


15 


41,700 


2,780 


20 


50,000 


2,500 


11 


38,600 


3,510 


16 


42,200 


2,635 


21 


52,400 


2,495 


12 


39,600 


3,300 


17 


43,100 


2,540 


22 


55,700 


2,531 


13 


40,400 


3,110 


18 


44,900 


2,495 


23 


58,700 


2.552 


14 


41,100 


2,935 


19 


47,400 


2,490 


24 


62,800 


2,615 



MOMENTS AND SHEARS FOR BRIDGE SPANS, 



719 



5. — Maximum End Shears 5, per Track, for Concentrated •'K25" 

Loading (Fig. 2). — A Train of 50-Ton Electric Cars.* 
Values for any other "K" loading are directly proportional to the axle load. 







Equiva- 


4-> 




Equiva- 


a> 




Equiva- 


fr 




lent Unif . 


0) 




lent Unif. 


(U 




lent Unif. 


pt^ 


End 


Load w 


End 


Load w 


i^ 


End 


Load w 


C! 


Shears. 


for End 


.2 


Shear 5. 


for End 


.2 


Shear 5. 


for End 


f{ 


Lbs. 


Shear. 


c 


Lbs. 


Shear. 


cT 


Lbs. 


Shear. 


a 

C/3 




Lbs. per 
lin. ft. 






Lbs. per 
lin. ft. 






Lbs. per 
lin. ft. 


10 


37,500 


7,500 


37 


66.200 


3.590 


64 


98,000 


3,090 


11 


38,600 


7,020 


38 


67.100 


3.530 


65 


100,000 


3,075 


12 


39,500 


6,580 


39 


67.900 


3,480 


66 


101,500 


3,075 


13 


40,400 


6,220 


40 


68,800 


3 440 


67 


103,000 


3,075 


14 


41,100 


5.870 


41 


70,100 


3 420 


68 


104.400 


3,070 


15 


41,700 


5,560 


42 


71,400 


3,400 


69 


105,800 


3,065 


16 


42,200 


5,270 


43 


72.700 


3,380 


70 


107,100 


3,060 


17 


42,600 


5,020 


44 


73.600 


3 350 


71 


108,500 


3,055 


18 


43,100 


4,790 


45 


75.000 


3,335 


72 


109,700 


3,050 


19 


43,400 


4,570 


46 


76.600 


3,330 


73 


111,000 


3,040 


20 


43.800 


4,380 


47 


78,200 


3,325 


74 


112,200 


3,030 


21 


45,200 


4,310 


48 


79,700 


3.320 


75 


113,300 


3,020 


22 


46,600 


4,240 


49 


81,100 


3.310 


76 


114,500 


3,010 


23 


47,800 


4,160 


50 


82,500 


3,300 


77 


115,600 


3,000 


24 


49,000 


4,080 


51 


83,800 


3.290 


78 


116,700 


2,990 


25 


50,000 


4,000 


52 


85.100 


3,275 


79 


117,700 


2,980 


26 


51,900 


3,990 


53 


86.400 


3,260 


80 


118,800 


2,970 


27 


53,700 


3,980 


54 


87.500 


3.240 


81 


120,100 


2,965 


28 


55,400 


3.960 


55 


88.600 


3.225 


82 


121,300 


2.960 


29 


56,900 


3,920 


56 


89.700 


3.205 


83 


122,600 


2,955 


30 


58,300 


3,890 


57 


90.800 


3.185 


84 


123,800 


2,950 


31 


59,700 


3.850 


58 


91.800 


3.170 


85 


125,000 


2,940 


32 


60,900 


3,810 


59 


92.800 


3,145 


86 


126,500 


2,940 


33 


62,100 


3,760 


60 


93,800 


3,125 


87 


127,900 


2,940 


34 


63,200 


3,720 


61 


95,100 


3,115 


88 


129,300 


2,940 


35 


64,300 


3,670 


62 


96,400 


3,110 


89 


130.600 


2,935 


36 


65,300 


3,630 


63 


97,600 


3,100 


90 


131,600 


2,930 



*For a single car on the span multiply values in table by the following 
percentages: Spans 0' to 40', mult, by 100%; span 45'. mult '^^^ oft^o/.. 
span50'.by90.97o; span 60', by 84.4%; 70'. by 76.8% "' 
65.4%. 



__, by 96.3%; 

80', by 71.0%; 90'. by 



Illustrations, with Details. 

Description. Eng. Rec. 

Details, portal, floorbeam, floor, 515-ft. elec. -highway bridge . . .Jime 12,' 09 
Details, 245' deck and 517' through spans, McKinley Bridge . . .Apr. 30, '10 
187-ft. riveted truss bridge, carrying street and 2 trolley tracks .May 7, '10 
Columns, girders, stringers, Manhattan Elev. Ry May 28, '10 



40.— HIGHWAY BRIDGES. 

I.— UNIVERSAL STRESS=SHEETS. 
Explanation. 

The following are types of trusses suitable for bridge spans up to about 
260 ft. in length. They are designated by a figure, indicating the number of 
panels in the truss, and also by a distinguishing letter when more than one 
type of the same number of panels are used. Thus, Types 4A, AB and 4C 
each have 4 panels, but their truss systems are different. 

The Diagrams are proportioned by scale and show height of truss in 
terms of panel length p=\. The members of the truss are nimibered in 
each case to correspond with the numbers in the adjoining tables. The 
practical upper and lower limiting spans, with resulting panel length, are 
given for each type. The fractions at lower chord joints express the live- 
load reactions, at left abutment, in terms of a panel load = unity, of panel 
loads from the right-hand end up to and including that joint; and are useful 
for finding the shears. 

The Tables accompanying the diagrams give the unit length of each truss 
member for a panel length of unity (actual length = unit length X panel 
length) ; also the dead- and live-load unit stresses in each truss member for 
unit panel loads (actual stress = unit stress X panel load per truss) . 

Ex. 1. — What type of truss would be suitable for a 168-ft. span? Find 
the lengths of the members ? Find maximum stresses in members 1 1 and 16 ? 

Solution.— Type 8; 8 panels (^ 21 = 168. Height = 21 X U = 28. Diago- 
nals = 2lXlf=35 ft. Assimiing dead-load at 1000, and live-load at 1200 
lbs. per lin. ft. of bridge, the respective panel loads per truss are: d. 1. = 10.5 
and 1. 1. = 12.6 thousand lbs.; whence the max compressive stress in 11 = 
1.5X10.54- 1.875X12.6= 39,375 lbs., and the max tensile stress in 16 = -.625 
X10.54-. 938X12.6 = 5,256 lbs. 

Types of Trusses, and Unit Stress Sheets. 

[See Explanation, preceding]. 
c = compression ; t = tension. 



Type 2A. 
For spans 36' to 40'. 

2 (^ 18' = 36'. 

2 (^ 20' = 40'. 
Live reaction summations: 




Table 2A. 



Name of member 


1 


2 


3 


Unit length of member. . 
Dead load unit stress, D 
Live load unit stress, L. . 


1. 

.500 t 
.500 t 


1.414 
.707 c 
.707 c 


1. 

1.000 t 
1.000 t 



Type 2B. 
For spans 25' to 32'. 

2 @ 12'.5=25'. 

2 (^ 16' = 32'. 
Live reaction summations: 



Table 2B. 



Name of member '. . 


1 


2 


3 


Unit length of member. . 
Dead load unit stress, D. 
Live load unit stress, L. . 


1. 

.800 t 
.800 t 


1.179 
.943 c 
.943 c 


.625 
1.000 t 
1.000 t 



720 



UNIVERSAL STRESS-SHEETS. 



721 



Type 2C. 
For spans 25' to 32'. 

2 @ 12'.5 = 25'. 

2 @ 16' = 32'. 
Live reaction summations: 



mNO 





Table 2C. 








Name of member 


1 


2 


3 


4 


5 


Unit length of member. . 
Dead load unit stress, D . 
Live load unit stress, L. . 


1. 1 .5 
.400 t\ .800 c 
.400 t .800 c 


.8 

.640 c 
.640 c 


.8 

.640 i 
.640 t 


.625 





Type 3A. 
For spans 48' to 63'. 

3 @ 16' = 48'. 

3 @ 21' = 63'. 
Live reaction simimations: 




Table 3A. 



Name of member. 



Unit length of member. 
Dead load unit stress, D 
Live load unit stress, L. 



1, 2 



1. 

.889 i 
.889 t 



.889 T 
.889 c 



1.505 
1.338 c 
1.338 c 



1.125 

1. i 
1. t 



1.505 


.446 c 



Type 3B. 
For spans 37'.5to 48'. 

3@ 12'.5=37'.5 

3@16' =48'. 
Live reaction summations: 



II 
3 -c 



^-^T^^T^ 



Table 3B. 








Name of member 


1. 2 


3 


4 


5 


6 


Unit length of member. . 
Dead load unit stress, D . 
Live load unit stress, L. . 


1. 

1.600 i 
1.600 t 


1. 

1.600 c 
1.600 c 


1.179 
1.887 c 
1.887 c 


.625 

1. t 
1. t 


1.179 


.629 t 



Type 3C. 
For spans 3 7'. 5 to 48'. 
3 @ 12'. 5 = 37'. 5 
_ 3 @ 16' = 48'. 
Live reaction simimations: 






m^\Z:K 



Table 3C. 



8 



1.179 


.629 t 



Name of member 


1 


2 


3 


4 


5 


6 


7 


Unit length of member. . 
Dead load unit stress, D . 
Live load unit stress, L. . 


1. 

.800 t 
.800 i 


1. 

1.600 i 
1.600 t 


1. 

1.600 c 
1.600 c 


.5 

1.600 c 
1.600 c 


.8 
1.281 c 
1.281 c 


.8 

1.281 t 
1.281 / 


.625 

.333 5 



722 



'iO.— HIGHWAY BRIDGES. 



Type 4A. 
For spans 64' to 80'. 

4 @ 16' = 64'. 

4 @ 20' = 80'. 
Live reaction summations: 



>i=l? 






Table 4A. 










Name of member 


1, 2 


3 


4 


5 


6 


7 


8 


Unit length of member. . 
Dead load unit stress, D . 
Live load unit stress, L. . 


1. 

1.3125^ 

1.3125i 


1. 

1.750 c 
1.750 c 


1.519 
1.994 c 
1.994 c 


1.143 

1. t 
1. t 


1.519 
.665 t 
.997 t 


1.143 

.250 c 


1.519 
.665 c 
.332 r 



Type 4B. 
For spans 50' to 60'. 

4 @ 12'. 5 =50'. 

4 @ 15' =60'. 
Live reaction summations: 



h=5 



^^^fXT^ 



I 







Table 4B. 










Name of member 


1.2 


3 


4 


5 


6 


7 


8 


Unit length of member. . 
Dead load unit stress, D. 
Live load unit stress, L. . 


1. 

2.400 / 
2.400 i 


1. 

3.200 c 
3.200 c 


1.179 
2.831 c 
2.831 c 


.625 
1.000 t 
1.000 i 


1.179 
.943 t 
1.414 t 


.625 

.250 c 


1.179 
.943 c 
.471 t 



Type 4C. 
For spans 50' to 60'. 
4 @ 12'. 5 = 50'. 
4 @ 15' =60'. 
Live reaction summations: 



H 



/X\J^^X\/X 



Table 4C. 



1.179 
.943 t 
1.414 t 



9 



,625 

250 c 



10 



1.179 
.943 c 
.471 t 



Name of member 


1 


2 


3 


4 


5 


6 


7 


Unit length of member. . 
Dead load unit stress, D . 
Live load unit stress, L. . 


1. 

1.200 t 
1.200 t 


1. 

2.400 t 
2.400 t 


1. 

3.200 c 
3.200 c 


.5 
2.400 c 
2.400 c 


.8 
1.920 c 
1.920 c 


.8 
1.920 t 
1.920 i 


.625 
.500 c 
.750 c 



Type 5A. 
For spans 80' to 105'. 

5 @ 16'= 80'. 

6@ 21'x=i05'. 
Live reaction summations: 




Table 5A. 



10 



1.537 

.790 t 



Name of member 


1, 2 


3 


4. 5 


6 


7 


8 


9 


Unit length of member. . 
Dead load unit stress, D . 
Live load unit stress, L. . 


1. 

1.714 t 
1.714 t 


1. 

2.571 t 
2.571 / 


1. 

2.571 c 
2.571 c 


1.537 
2 634 c 
2.634 c 


1.167 
1. t 
1. t 


1.537 
1.317 / 
1.580 / 


1.167 

.600 c 



UNIVERSAL STRESS-SHEETS. 



723 



Type 5B. 
For spans 60' to 80'. 

5 @> 12' = 60'. 

5@ 16' = 80'. 
Live reaction summations: 



Table 5B. 



10 



1.179 


1.132 t 



Name of member 


1, 2 


3 


4. 5 


6 


7 


8 


9 


Unit length of member. . 
Dead load unit stress, D. 
Live load unit stress, L. . 


1. 

3.200 i 
3.200 t 


1. 

4.800 i 
4.800 t 


1. 

4.800 c 
4.800 c 


1.179 
3.774 c 
3.774 c 


.625 
1. t 
1. t 


1.179 
1.187 i 
2.264 t 


.625 

.600 c 



Type 6. 
For spans 102' to 129'. 

6@ 17' =102'. 

6@ 21'. 5= 129'. 
Live reaction summations: 





. 5 4 




h'lR 


I 


^m 


X 


/ 


\ 


'f i 1 1 i 



Table 6. 



Name of member -. . 


1.2 


3 


4 


5 


6 


7 


8 


Unit length of member. . 
Dead load unit stress, D . 
Live load unit stress, L. . 


1. 

2.083 i 
2.083 i 


1. 

3.333 t 
3.333 t 


1. 

3.750 c 
3.750 c 


1. 

3.333 c 
3.333 c 


1.562 
3.255 c 
3.255 c 


1.2 

1. i 
1. t 


1.562 
1.953 t 
2.170 t 




9 


10 


11 


12 


1vjo+- "n* 




1.2 

.500 c 
1.000 c 


1.562 
.651 t 
1.302 t 


1.2 

.500 c 


1.562 
.651 c 
.651 t 


omy 
whei 


, use type 5A 
a practicable. 



Type 7. 
For spans 126' to 150.' 5 

7@ 18' =126'. 

7@ 21'.5=150'.5 
Live reaction summations: 




Table 7. 



Name of member 


1.2 


3 


4 


5 


6 


7 


8 


Unit length of member. . 
Dead load unit stress, D . 
Live load unit stress, L. . 


1. 

2.400 t 
2.400 t 


1 

4.000 t 
4.000 t 


1. 

4.800 t 
4.800 t 


1. 

4.800 c 
4.800 c 


1. 

4.800 c 
4.800 c 


1. 

4.000 c 
4.000 c 


1.601 

3.843 c 
3.843 c 




9 


10 


11 


12 


13 


14 


15 




1.25 
1. / 

1. t 


1.601 
2.562 t 
2.745 i 


1.25 
1. c 
1.429 c 


1.601 
1.281 / 
1.830 t 


1.25 


.857 c 


1.601 


1.098 / 


1.601 
1.281 c 
.549 t 



724 



40.— HIGHWAY BRIDGES. 



Type 8. 
For spans 144' to 172'. 

8@ 18' =144'. 

8@ 21'.5=172'. 
Live reaction summations: 




Table 8. 



Name of member 




1. 2 


3 . 


4 


5 


6 


7 


8 






Unit length of member. . 
Dead load unit stress, D . 
Live load unit stress. L. . 


1. 

2.625 t 
2.625./ 


1. 

4.500 t 
4.500 t 


1. 

5.625 i 
5.625 t 


1. 

6.000 c 
6.000 c 


1. 

5.625 c 
5.625 c 


1. 

4.500 c 
4.500 c 


1^ 
4.375 c 
4.375 c 




9 


10 


11 


12 


13 


14 


15 


16 




IM 
1. t 
1. t 


3.125 t 
3.281 t 


1.500 c 
1.875 c 


1^ 
1.875 t 
2.344 t 


1^ 

.500c 

1.250 c 


.625 i 
1.563 t 


IM 

.750 c 


.625 c 
.938 / 



Type 9A. 
For spans 162' to 19 3'. 5 

9@ 18' =162'. 

9 @ 21'.5=193'.5 
Live reaction summations: 



9 6 7 6 



•h'l? 



l|]ili\l) 



36^ £1 15 |0 6 3 



m 











Table 


9A. 










Name of member 


1, 2 


3 


4 


5 


6 


7 


8 


Unit length of member. . 
Dead load unit stress, D . 
Live load unit stress, L. . 


1. 

2.667 i 
2.667 t 


1. 

4.667 t 
4.667 t 


1. 

6.000 t 
6.000 t 


1. 

6.667 t 
6.667 t 


1. 

6.667 c 
6.667 c 


1. 

6.667 c 
6.667 c 


1. 

6.000 c 
6.000 c 


9 


10 


11 


12 


13 


14 


15 


16 


17 


18 


19 


1. 

4.667c 
4.667c 


1.803 
4.808 c 
4.808 c 


1.5 
1. t 
1. t 


1.803 
3.606 t 
3.740 t 


1.5 

2.000 c 
2.333 c 


1.803 
2.404 t 
2.805 t 


1.5 
1.000 c 

1.667 c 


1.803 
1.202 t 
2.003 t 


1.5 

1.111 c 


1.803 


1.336 t 


1.803 
1.202 c 
.801 t 






Type 9B. 
For spans 162' to 19 3'. 5 
9 @ 18' =162'. 
9 @ 21'.5=193.'5 CK^---— 15p-. 

Live reaction summations: 

Table 9B. 




N t^\ _&\ 7 6 






, 1'1£SV^4' 




28 21 15 10 6 3 
9 9 9 3 9 5 5 



Name of member 


1,2 


3 


4 


• 5 


6 


7 


8 








Unit length of member. . 
Dead load unit stress, D. 
Live load unit stress, L. . 


1. 

2.667 i 
2.667 t 


1. 

4.391 / 
4.391 t 


1. 

5.333 t 
5.333 / 


1. 

5.926 t 
5.926 t 


1. 

5.926 c 
5.926 c 


1. 

5.926 c 
5.926 c 


1.004 
5.357 c 
5.357 c 


9 


10 


11 


12 


13 


14 


15 


16 


17 


18 


19 


1.004 
4.410 c 
4.410 c 


1.803 
4.808 c 
4.808 c 


1.5 
1. t 
1. t 


1.803 
3.111 t 
3.299 / 


1.594 
1.588 c 
2.059 c 


1.882 
1.771 t 
2.296 / 


1.688 
.500 c 
1.389 c 


1.962 
1.163 t 
1.938 t 


1.688 


1.111 c 


1.962 


1.292 t 


1.962 
1.163 c 
.775 t 



UNIVERSAL STRESS-SHEETS, 



726 



Type 10. 
For spans 180' to 215'. 
10 @ 18' =180'. 
10 @ 21'.5=215'. 
Live reaction summations: 
Name of loads: 




43 


36 


28 


?• 


!? 


10 


« 


li 


1 


»o 


to 


10 


10 


10 


»o 


!P 


10 


10 


P9 


Pe 


P7 


^6 


% 


^4 


•i 


Pa 


t 



= 1.562 











Table 10. 










Name of member 


1,2 


3,4 


5 


6 


7,8 


9 


10 


Unit length of member. 
Dead load unit stress, D 
Live load unit stress, L. 


1. 

5.400 t 
5.400 i 


1. 

4.800 t 
4.800 t 


1. 

7.200 t 
7.200 t 


1. 

7.800 c 
7.800 c 


1. 

7.800 c 
7.800 c 


1.302 

6.248 c 
6.248 c 


1.302 
7.029 c 
7.029 c 


11, 16, 20 


12 


13 


14 


15 


17 


18 


19 


21 


.833 
1. t 
1. t 


1.302 

.781c 
.781c 


1.667 
1.500 / 
1.500 i 


1.302 
3.905 t 
4.374 t 


1.302 

.781 t 
.781 / 


1.302 
2.343 c 
.937 t 


1.302 
3.124 t 
3.593 t 


1.667 
1.000 c 
1.700 c 


1.302 

.781 t 
2.343 i 


1.302 


1.562 t 



The following graphics illustrate the loadings which give maximum live 
stresses in members 14, 17 and 18 of Type 10. (Note the two last cases which 
do not give maxima for these members) : 





18. 



Loads Pi to P^ inclusive give maxima in members 14 and 17. 



mi 



Loads Pi to P7 inclusive, omitting load Pe, give maximum in member 



R»«2.» 



Loads Pi to Pe inclusive do not give maximum in member 17. 

5 

1-2 3-4X\ 
D o>/ X 
R|-l.5 





Load^ Pj to P5 inclusive do not give maximum in member 18. 



726 



AO,— HIGHWAY BRIDGES. 



/ 



# 



/ 



»-r-7- 



,/. v^^^ipM 



a 



ii.^«.S«.a^„s pt t f ^ y S 10 I 
toe^/'W X ^K. ^ P» P7 Pe P5 ? Pj 



%-.^ / 

Type 12. 
For spans 216' to 258'. 
12 @ 18'. = 216'. 
12 @ 21'. 5= 258'. 



3 


1 


12 


72 


\ 


? 



Fig. 22. Type 12. 











Table 12. 










Name of men 


iber 


1.2 


3,4 


5, 6 


7.8 


9, 10 






Unit length of member 

Dead load unit stress, D 

Live load unit stress, L 


I. 

3.667 i 
3.667 i 


1. 

5.333 i 
5.333 i 


1. 

7.111 i 
7.111 i 


1. 

8.444 c 
8.444 c 


1.017 

7.687 c 
7.687 c 


11 


12 


13 


14 


15 


16 


17,22 


18 


19 


20c 


1.068 

6.696c 
6.696c 


1.803 

6.611c 

6.611c 


1.6 
1. t 
1. t 


1.803 
3.005 t 
3.305 t 


1.875 
1.500 c 
2.250 c 


1.875 
1.500 c 
.600 t 


1.371 
3.046 i 
3.655 i 


1.125 




.938 
1. t 
1. t 


1.371 
2.332 i 
2.942 i 


2.25 
.795c 
1.567c 


20 t 


21 


23 


24 


25 


26 


27 


28 


2.25 

.795c 
.583^ 


1.505 
2.007 t 
3.122 t 


1.125 
1. t 
1. t 


1.505 

1.338 i 
2.453 t 


2.25 
1. c 
1.833 c 


1.505 
.669 
.669 


1.505 

i .669 ( 
a. 672 


1.505 
; 
i 


1.505 

1.338^ 

1.115 


1.65 

.628^ 
.628i 


1.65 

1.711c 

1.222i 



Notes. — Using "balanced loads," the tension of 0.6 in member 15 is ob- 
tained from live loads Pi, P2, Pio and Pn. Tension of 0.583 in member 20 is 
obtained from live loads Pi, P2, P4, Ps, Pio and Pn. Compression of 1.567 
in member 20 is obtained from live loads Pi to P9 inclusive, omitting Pg. 
In the last calculation note that the cutting plane will cut jour active 
members, 9, 28, 20 (27 inactive) and 5, with the center of moments at 0'. 
But the stress in member 28 is one-half the panel load P9 multiplied by the 
secant of its angle of inclination = . 5 X 1.257=0.628, and themoment of this 
stress about 0' is 5.138. This enables us to solve the stress in member 20 by 
cutting the jour active members. Thus, stress in member 20 = i^ ( — i?^X8 — 
5.138+P9X 11) = - 1.567. 



UNIVERSAL LOADINGS. 



727 






> 


r M 


< 


^ S 


^ 


!^-^ 


E 


tC3 ^ 


O 


^3 a 
■? 8 








w rt 


D^ 


11 


£ 


§ 


&1 


Q 


Is 


< 

S 


•SP-o 



< 






09 



f - CO o^O'd 



(U 



rt 



■Mr^ WC"' rt^-H 



CO ^ 



-ssnjx 









•ssnjx 



CO ^ 



I : • wc<ioo 



'.CO vh C 5 a 

O"^ boa)^'g 
B o c <1> ^ S 












h-5 









o '^ m 
o3'd 'XJ 
tH c3 a J 

^35 






t4 







CO C<I »-H 

^ ^ "^ 



•A'BAip'BOJ JO noi:;jod 
AuB no (aSBS -^j-Q 
'sja^nao •^j-oi'saix'B 
Z) Smp'BOT; ., — 7,, 
8Aiioadsaj aq^ ^o 



.002 o;^ ,001 en^ds 

lOJ SpBOl IBUOUIOdOJJ 






i, 









O ■* 00 04 
CO N ^ ^ 

►^ Kq ►^ ^ 



to &3 &5 tcj 

y II II II 

-^ PQ O Q 



728 



40.— HIGHWAY BRIDGES. 



14. — Live-Load Data for Designing Floor Systems and Spans 

Under 50 Ft. 

Maximum Moments M and End Shears 5 per Track for "L" Loadings. 

Note. — For maximum floor-beam reactions use the end shears S down to the 

(*), and below the (*) use the uniform load, covering the two panels. 





Class A 


aass B 


aass C 


aassD 


aass E 




"L 30" 


Load- 


"L 24 " Load- 


"L 18' 


Load- 


"L 12' 


Load- 


"L 6" 


Load- 




Ing. 




ing. 


ing. 


ing. 


ing. 




30 Tons — 2 


24 Tons — 2 


18 Tons — 2 


12 Tons — 2 


6 Tons — 2 


Span. 


Axles— 


lO'c.-c. 


Axles — lO'c.-c. 


Axles— 


lO'c.-c. 


Axles — 


lO'c-c. 


Axles — lO'c.-c. 




12' Wide— 5' 


12' Wide— 5' 


12' Wide— 5' 


12' Wide— 5' 


12' Wide— 5' 




Gage. 


Gage. 


Gage. 


Gage. 


Gage. 




M 


S 


M 


S 


M 


S 


M 


S 


M 


S 




Thou- 


Thou- 


Thou- 


Thou- 


Thou- 


Thou- 


Thou- 


Thou- 


Thou- 


Thou- 




sand 


sand 


sand 


sand 


sand 


sand 


sand 


sand 


sand 


sand 




Ft.-Lb. 


Lb. 


Ft.-Lb. 


Lb. 


Ft.-Lb. 


Lb. 


Ft.-Lb. 


Lb. 


Ft.-Lb. 


Lb. 


Ft. 


Units. 


Units. 


Units. 


Units. 


Units. 


Units. 


Units. 


Units. 


Units. 


Units. 


10 


75.0 


30.0 


60.0 


24.0 


45.0 


18.0 


30.0 


12.0 


15.0 


6.0 


11 


82.5 


32.7 


66.0 


26.2 


49.5 


19.6 


33.0 


13.1 


16.5 


6.5 


12 


90.0 


35.0 


72.0 


28.0 


54.0 


21.0 


36.0 


14.0 


18.0 


7.0 


13 


97.5 


36.9 


78.0 


29.5 


58.5 


22.2 


39.0 


14.8 


19.5 


7.4 


14 


105 


38 6 


84.0 


30. 9 


63 


23. 1 


42 


15. 4 




7 7 


15 


112!5 


40.0 


90 !o 


32!o 


67^5 


24!o 


45!0 


*16.0 




8!o 


16 


120.0 


41.3 


96.0 


33.0 


72.0 


24.8 


48.0 


16.5 




8.3 


17 


127.5 


42.4 


102.0 


33.9 


76.5 


25.4 


51.0 


16.9 




8.5 


18 


140.8 


43.3 


112.7 


34.7 


84.5 


26.0 


56.3 


17.3 


j_5 


8.7 


19 


154.7 


44.2 


123.8 


35.4 


92.9 


26.5 


61.9 


17.7 


4^ 


8.8 


20 


168.8 


45.0 


135.0 


36.0 


101.3 


27.0 


67.5 


18.0 


1 


9.0 


21 


182 9 


45 7 


146 3 


36.6 
37.1 


109 7 


27.4 
27.8 


73. 1 


18.3 




22 


19711 


46!4 


157!6 


118!2 


78^8 


18.5 


1 


i 


23 


211.3 


47.0 


169.0 


37.6 


126.8 


*28.2 
28.5 


84.5 


18.8 


M 


24 


225.6 


47.5 


180.5 


38.0 


135.4 


90.3 


19.0 


.d 


25 


240.0 


48.0 


192.0 


38.4 


144.0 


28.8 


96.0 


19.2 


s 


"^ 


26 


254.4 


48.5 


203.5 


38.8 


152.7 


29.1 


101.8 


19.4 


fe 


> 




27 


268.9 


48.9 


215.1 


39.1 


161.3 


29.3 


107.6 


19.6 


a 


28 


283.4 


49.3 


226.7 


39.4 


170.0 


29.6 


113.4 


19.7 


4^ 


29 


297.9 


49.7 


238.3. 


*39.7 


178.8 


29.8 


119.2 


19.9 


*^ 


o 


30 


312.5 


50.0 


250.0 


40.0 


187.5 


30.0 


125.0 


20.0 


a 


*t 


31 


327.1 


50.3 


261.7 


40.3 


196.3 


30.2 


130.8 


20.1 


t^ 


•" 


32 


341.7 


50.6 


273.4 


40.5 


205.0 


30.4 


136.7 


20.3 


1 


a 


33 
34 


356.4 
371.0 


50.9 
*51.2 


285.1 
296.8 


40.7 
40.9 


213.8 
222.6 


30.5 
30.7 




20.4 
20.5 






1 




35 


385.7 


*51.4 


308.6 


41.1 


231.4 


30.9 




20.6 




36 


400.4 


51.7 


320.3 


41.3 


240.2 


31.0 




20.7 


o 
o 


i 


37 


415.3 


51.9 


332.1 


41.5 


249.1 


31.1 




20.8 


o> 


38 


429.9 


52.1 


343.9 


41.7 


257.9 


31.3 


o 


20.8 


o 


o 


39 


444.6 


52.3 


355.7 


41.9 


266.8 


31.4 


■d 


20.9 


•d 


o> 


40 


459.4 


52.5 


367.5 


42.0 


275.6 


31.5 


|d 


21.0 


o 


o 


41 


474.2 


52.7 


379.3 


42.2 


284.5 


31.6 






"^ 






1 


42 


488.9 


52.9 


391.2 


42.3 


293.4 


31.7 


"^ . 


sj 


43 


503.7 


53.0 


403.0 


42.4 


302.2 


31.8 


oi5 


a 

3 


44 


518.5 


53.2 


414.8 


42.6 


311.1 


31.9 


a>£, 


^5 




a 


45 


533.3 


53.3 


426.7 


42.7 


320.0 


32.0 


^ 03 


"St 


2 


§ 


46 


548.2 


53.5 


438.5 


42.8 


328.9 


32.1 


^r-^ 


is 


_• 


47 


563.0 


53.6 


450.4 


42.9 


337.8 


32.2 


So 


a 


m 


48 


577.8 
592.7 


53.8 
53.9 


462.3 
474.1 


43.0 
43.1 


346.7 


32.3 
32.3 


Sin 

^2 


i^ 


0) 

r1 


^ 


49 


oJ? <=Q ^ 


d 


50 


607.5 


54.0 


486.0 


43.2 


d b£i2 


32.4 


o 


=s 


5 




Unlf. 


load. 


Unif.load, 


elow 1 
seunif.l 
f 1200 
er lin. 




^2 


g 


fe 




12x 


125 


12xll2i 






o 


o 


o 




= 1 


500 


= 1350 






« = 


1 










ffl ;3d a 













* See note at head of table. 



UNIVERSAL LOADINGS, 



729 



15. — Uniform Live Loads for Trusses op SI'ans over 50 Ft. 

(Each street car track loading occupies width of 12 feet for single track, 

and 11 feet for double track.) 





Class A 


Class B 


Class C 


Class D 


Class E 




Vehic- 


Each 


Vehic- 


Each 


Vehic- 


Each 


Vehic- 


Each 


Vehic- 


Each 




ular 


Street 


ular 


Street 


ular 


Street 


ular 


Street 


ular 


Street 




Road- 


car 


Road- 


car 


Road- 


car 


Road- 


car 


Road- 


car 


Span. 


way, 


Track. 


way, 


Track. 


way, 


Track. 


way, 


Track. 


way, 


Track. 




and 


Lhs. 


and 


Lbs. 


and 


Lbs. 


and 


Lbs. 


and 


Lbs. 




Walks. 


per 


Walks. 


per 


Walks. 


per 


Walks. 


per 


Walks. 


per 




Lbs. per 


Lin. 


Lbs. per 


Lin. 


Lbs. per 


Lin. 


Lbs. per 


Lin. 


Lbs. per 


Lin. 


Ft. 


Sq. Ft. 


Ft. 


Sq. Ft. 


Ft. 


Sq. Ft. 


Ft. 


Sq. Ft. 


Ft. 


Sq. Ft. 


Ft. 


50 to 






















100 


100 


2000 


90 


1600 


80 


1200 


70 




60 




105 


99 


1980 


89 


1584 


79 


1188 


69 


4J 


59 


4J 


110 


98 


1960 


88 


1568 


78 


1176 


69 


i^ 


59 




115 


97 


1940 


87 


1552 


78 


1164 


68 


O* 


58 


a 


120 ■ 


96 


1920 


86 


1536 


77 


1152 


67 




58 


m 


125 


95 


1900 


86 


1520 


76 


1140 


67 


'i^ 


57 


1-^ 


130 


94 


1880 


85 


1504 


75 


1128 


66 


^1 


56 


"3 


135 


93 


1860 


84 


1488 


74 


1116 


65 


^ ^ 


56 


•0 fe 

0-0 


140 


92 


1840 


83 


1472 


74 


1104 


64 


0^ 


55 


145 


91 


1820 


82 


1456 


73 


1092 


64 


a «^ 


55 


fl-S 


150 


90 


1800 


81 


1440 


72 


1080 


63 


5i 


54 


5i 


155 


89 


1780 


80 


1424 


71 


1068 


62 


53 


160 


88 


1760 


79 


1408 


70 


1056 


62 


w >. 


53 


^>. 


165 


87 


1740 


78 


1392 


70 


1044 


61 


11 


52 


« 03 


170 


86 


1720 


77 


1376 


69 


1032 


60 


52 


175 


85 


1700 


77 


1360 


68 


1020 


60 


51 


^0 


180 


84 


1680 


76 


1344 


67 


1008 


59 


50 


185 


83 


1660 


75 


1328 


66 


996 


58 


d 


50 


d 


190 


82 


1640 


74 


1312 


66 


984 


57 


13^ 


49 


-i^ 


195 


81 


1620 


73 


1296 


65 


972 


57 


1 


49 


?-. 


200 


80 


1600 


72 


1280 


64 


960 


56 


^ 


48 


K 


and 
















CQ 




m 


over 























III.— DETAILS OF COMBINATION BRIDGE, 230 FT. SPAN. 

TYPE 12. 




Diagram of Truss sHowing Comber and Heights of Chords. 
Span » 12 Panels (5) I9'2 =230 'C+o.C. 







■5 y ^^'^^ 

5^ )f^^"li'CotferPin 




Figs. 24. Truss Diagram. Bottom Chord and Laterals. 



730 



40.— HIGHWAY BRIDGES, 






25fick5 7ixl52\37'l0'long ^ 5 



Timber Details. 
Chord Ui-Lo, 



Cast Details. 
End shoe, 12B, 
Post shoe, 12G, 
Bed plate, 12A, 
Lateral struts, 12J, 
Lateral struts, 12K, 
Washers, 
Separators, 



Figs. 
29 

Figs. 

30 
31 
32 
34 
35 
48 
52 




5^x7^y.ie5%'^ 



r\ 



fca... 



5jx7jx/6'5^ 



/|xj| 





Fig. 25. 



5. Elevation. 




Steel Details. 


Figs. 


Stone bolt, 


33 


Lateral rods. 


36, 40. 41 


Eye-bars, 


37 


Counters, 


38 


Suspenders, 


38 


Pin fillers. 


39 


Portal rods 


41 


Nominal rods, 


42 


Sway rods. 


43 


Cotter pins. 


44, 49 


Wing plates. 


45 


Lateral plates, 


46 


Chord pins, 


47, 51 


Hangers, 


50 


Bolts, 


53 


Washers, 


54 



-5x%xl2 5tPL 

Fig. 26. Section. 



DETAILS COMBINATION BRIDGE. 



731 




'"fPin 

Fig. 27. End Panel 



Fig. 28. Top Laterals. 



zz'ioj 



•#-f 



n 

















O © f^ - O 






Inside View 



iH<----6'5|"-->i</i'f'--4jl 
41 ^ — ^ 



T^r — ; — p TT — I — 7^ n TJ 

-il C Ll U J^ Li il 



?-^A 






I ^-72 I 

End View. 



KS'H 



Top View 



O 1— 



o ^ l^^t^r 



1P 



I.JJV, ; ^-'^^■f^-S/anf/ng Hole 



-• ri ttO 



U-/5'— ->j 
End View. 



1 



Outside View. 
Fig. 29. Top Chord Framing. 



732 



iO,— HIGHWAY BRIDGES. 




Note. — For location of de- 
tails see pages 729 and 730. 



Boltom View. 
Fig. 30. End Shoe. 



i MerAe 10 Castings 

1 . I like this, Cast oyer 

%'-^ Chill for 2^' fumed 

-I- i All Mefal i thick. 



^./|V-5<~>{ 



,.r- 



-9'- 






^- 



h 



Fig. 31. Post Shoe. 



K3M<-— ^ — /j*-...^ — -*^iH 



ISA 



O-I^Ho/e '^^ 



j? y/arTteef 
for Sliding End 




Fig. 32. Bed Plate. 



UJC 



1^ 
I 



Fig. 33. 



f*~«'— n 



o 


1 

09 


12 J 


o— 


.Sit 



Fig. 34. 




Fig. 35. 



B 



y-'d 



K 



.4| 






->1 



— A 

Fig. 36. Lower Lateral Rods and Table. 



Number 
Required. 


Mark. 


Dia. 

d 


B 


A 


Ends 

Upset 

to 


Nom- 
inal 
D 


r 


X 


z 


4 


u-u 


\\"U 


23' er 


24' 41" 


iro 


2" 


ir 


4" 


ir 


4 


u-u 


1 "U 


23' 7i" 


24' 5i" 


iro 


2" 


ir 




1*1' 


4 


u-u 


\"U 


23' 61" 


24' 4|" 


iro 


ir 


ir 




1 " 


4 


u-u 


\"U 


23' 6r 


24' 4^" 


iro 


If" 


ir 


1// 


1 " 


4 


u~u 


\"o 


23' 6 J" 


24' 4t" 


1"0 


ir 


ir 


1/' 


1 '^ 


4 


u-u 


ro' 


23' W 


24' 3r 


1"0 


ir 


ir 


Y 


1 " 



DETAILS COMBINATION BRIDGE. 



733 



Steel Eye-Bars, 




w 



L — - 




Fig. 37. 







Mark. 


Dimens'ns. 


Length 

Li 


One Head 


Other Head 


Number 
Required. 


W 


T 


H 


Hole. 


h 


d 

Hole. 


48 

16 

8 
8 
8 
8 
8 


f 8 
8 

16 
16 

r 8 
I 8 


U2-M3 

U4.-M5 


3" 

3" 

3" 
3" 

2V 

2r 

2i" 


. 1" 

*;; 

# " 
1 // 


19' 2 " 
19' 2 " 

34' 2r 

26' or 
26' or 

28' 61" 
28' 61" 


7" 

7 " 

7 " 
6i" 
6" 
6" 
6 " 


2M''+5^/ 

2ir+5V' 
2m:+.v; 

2A''+^V' 
2i^"+bV' 
2i^"4-3V' 

2^" + 5V' 


6r 
6r' 

61" 

6r 
6r' 

6" 

6r 


2^''+3V' 

2i^''+5^.: 
2^"+s^/ 
2i|"4-3V' 
2T^"4-gV' 
2if"+^V' 



Counters and Suspenders. 







— H N B ->< 

— L ->1 

Fig. 38. 
Note.— M"P upset to irO ; 1"0 upset to 1|"0. 



Number 


Mark. 


Dia. 


Length 


A 


B 


Nom- 
inal 


Nom- 
inal 


. tu 


Required. 




TO 


L 






D 


d 




8 


U^-Mz 


WO 


31' 4 A" 


25' 11;V' 


5'0" 


2r 


2h" 


rn 


8 


Ue-M, 


i"o 


28' 1W' 


23' 21f" 


5'0" 


2r 


2r 


rn 


8 


Ut-Li 


i"o 


28' 6i" 
17'8|"^ 


23' 1 " 


5'0" 


2r 


2r 


8 


Ms-Ls 


i"o 


12' 3^" 


5'0" 


2r' 


2r 


rn 


8 


M5-L5 


1"0 


21' 2i" 


15' 9r' 


5'0" 


2r 


2r 


rn 


8 


Mr-U 


1"0 


28' 7f" 


23' 21" 


5'0" 


2Y 


3" 


rn 



Wrought Fillers for Bottom 
Chord Pins. 

Fig. 39. 



Number 
Required 


Mark. 


Nominal 
D 


L 


48 

8 

12 


L1L3&L6 
L2 &*L6 


2i" 
3" 
3" 


1" 
1" 
3" 



Top Lateral and Portal Rods. 

HM i^..4'3i'— ~ ^5'4< 2l'4'- >j ~<!i* 

**^ J^- L=Z6'0k' before Bending ->1 

k-— ZiSi- ■>^< ^ '-^ 

Fig. 40. 



Number 
Required 


Mark. 


d 


L 


Up- 
set to 


4 


U,-U2 


vo 


26' or 


iro 



734 



^.—HIGHWAY BRIDGES. 



Top Lateral and Portal Rods. — Concluded. 




^ 






.t?.j| 









Fig,41. 










Number 


Mark. 


d 


L 


A 


B 


T 


U 


H 


Required . 




















Portal 


iro 


21'2 " 


4" 


20' 10 " 


ir 


iro 


2U 




U2-Us 


ro 


26' er 


3" 


26' 3r 


1 " 


1 "O 


lU 




Us~U, 


ro 


26' 5r 


3" 


26' 2r 


1 " 


1 "O 


lU 




u,-u. 


ro 


26' 3 " 


3" 


26' " 


1 " 


1 "O 


IH 




U,-Ue 


ro 


26' 2i" 


3" 


26' lU" 


1 " 


1 "O 


m 



Nominal Rods. 



s'A' 



— ^^IJ. — 

Fig. 42. 



«^to 



.m 



Number 
Required. 


Mark. 


d 


L 


u 


t 


8 
8 

« it 

4 


Ms- Us 

M4-M5 j 
Ms-Mq 


ro 
ro 

ro 

ro 


22' or 
22' or 

19' 9r 

19' sr 


ro 
ro 

ro 

ro 


1" 

r 
r 



t^^..:.. 



Sway Rods. 



=^ 



— L--- 

Fig. 43. 



=«2&3^'§* 



-t?Jl 



Number 
Required. 


Mark. 


Dia. 
d 


Length 


4 
6 


U2 
U^ and Ue 


ro 
ro 


23' ir 
23' 8r 



Cotter Pins for Lower Lateral Rods, 






Fig. 44. 



Number 
Required. 


Mark. 


D 


L 


G 


p 


Remarks. 


8 
8 

16 
16 


Lo and Li 
Li and L2 
L2 L3 & L4 
L4 L5 & Le 




o'4r 
0' 3r 
0' 3r 
0' 3r 


2i" 
2i" 
2 " 

ir 


ir 
ir 
ir 
ir 


Turned 

<< 

1 1 



DETAILS COMBINATION BRIDGE, 



735 



Wing Plates for Bottom Laterals. 




''*^ WLenath far i^ouH 

Fig. 45. 



Number 


Mark. 


W 


T 


L 


a 


A 


B 


D 


;7 


h 


Nom. 
Pin 


Nom. 
Pin 


Required 






















Dia. X 


Dia. 




U 


5 " 


Iff 


0' 11 " 


" 


5 '' 


2r 


3 " 




10" 




2 " 




U 


w 


// 


V 4" 


5" 


5 " 


2V 


2V 


10" 


10" 


2 " 


2 " 




U 


5 " 


h-ff 


V 4 " 


5 " 


5 '^ 


w 


3^ 


10'^ 


10" 


2 " 


If" 




U 


W 


V 3r 


5'' 


41'^ 


2V 


2V 


10'' 


10" 


If" 


If" 




L^ 


5 " 


\H 


r 3 '^ 


4^' 


W 


2V 


3 '' 


10'' 


10" 


If" 


W 




U 


W 


1// 


V 2r 


w 


4i" 


2V' 


2^' 


10" 


10" 


ir 


ir 


2 


Ls 


5 " 


\tf 


V 2V 


w 


4t'' 


2V 


3 '' 


10" 


10" 


ir 


ir 



Top Lateral Plates, Bolts and 
Wrought Washers. 






H-5'H 



Figs. 46. 




Top Chord Pins. 



>i/<i« 



t^E^ 



Turnecf 



J^TopLatemt 



fil^- 



--© 



Fig. 47. 



Number 
Required. 


Mark. 


(? 


4 
4 
4 
2 


t/6 


211" 
18i" 
21 " 

2ir 



Also ship 10 washers |" thick for 
above pins. 



Top Lateral, Portal and Sway Brace Washers. 



5* ■ 



^J-'> 



Fig. 48 



\^^^ 



Number 
Required. 


Mark. 


A 


B 


c 


D 

w 

31" 
3" 


2i" 

ir 
i-r 


F 

lA" 


J/ 

2i" 

lIV 


7 
1 " 

r 


K 

If" 


L 


4 

16 
20 


Portal 

Top Lateral 

Sway 


2 " 
2 " 


7// 


If" 
ir 
ir 


2^' 
2 " 



736 



^.—HIGHWAY BRIDGES. 



Cotter Pins for Mid Truss 
Connections. 

Fig. 49. 



Number 
Required. 


Mark. 


G 


4 
4 


Ms 
Ms 


11 " 

i2r 



Bottom Chord Pins. 



e 



Turned #, 



a 



>f/f-K- 6 — H/f-l< ' 

Fig. 51. 



Hangers and Plates. 



Nominal ^ 







1^. -I'va 





Fig. 50. 


-->J 


Number 
Required. 


Mark. 


D 


10 
12 


Li—Ls—Ls 


3'' 

2r 



Number 


Mark. 


D 


G 


p 


Required. 










4 


Lo 


2W'0 


IW 


2" 




L, 


^^'O 


iir 


2" 




L2 


2W'0 


i4r 


2" 




U 


2^"0 


i2r 


2" 




U 


2M''0 


16" 


r 




U 


2^"0 


i3r 


2" 


2 


Le 


2W'0 


m" 


r 



Cast Separator Spools for Floor 
Beams. 




Number 
Required. 


Mark. 


90 


Floor Beam 



Bolts with Nut and 2 Washers 
Each. 



I* 



JDlgsi 



9f .^ . 

Fig. 53. 



Number 
Required. 


Mark. 


d 


L 


90 
150 

20 
100 


Floor Beam 

Chords 

Posts 

Railing 


ro 
ro 
ro 


IS^'' 

i7r 
i3r 



Wrought Packing Washers for 
Chords. 








Fig. 


54. 


Number 
Required. 


Mark. 


Remarks. 


150 


Chords 


Standard a and I> 



COMBINATION DETAILS, STEEL SPANS, 737 

IV.— NICKEL-STEEL AND CARBON-STEEL SPANS. 

Table 35, following, has been condensed from very extensive and 
specially prepared data in the author's forthcoming structural handbook. 
The table covers a range of span lengths from 40 to 200 feet, with 5 ft. 
intervals. The respective live loads for the same length of span were selected 
so that the weight of trusses would be about equal. In the nickel-steel 
spans the lateral systems are of carbon steel, the floor-beams and trusses 
being of nickel-steel. Although two different specifications were used — one 
for the carbon-steel and one for the nickel-steel spans — the comparative 
weights of the total amount of steel are not materially affected. 

The Estimated Costs are based on the following prices: 

Carbon-steel, delivered, at ii cents per pound; 

Nickel-steel, delivered, at 6 cents per pound; 

Lumber, delivered, at $35 per M. ft., B. M.; 

Erection of carbon-steel span, cost in dollars = 2 X span in feet + 0.003 

X weight of metal in lbs.; 
Erection of nickel-steel span, cost in dollars = 2. IX span in feet -H 0.003 

X weight of metal in lbs.; 
Painting, cost in dollars = i the span in feet + 0. 00 1 X weight of metal in lbs. 

The carbon-steel spans have a 3" wooden floor on wooden joists; the 
nickel-steel spans have an additional covering of flooring plank, 2 ins. thick. 
An examination of the table, below, shows that while the nickel-steel spans 
cost about 30 to 40% more than the carbon -steel spans, yet their live-load 
capacity is from 60 to 80% greater, thus showing a decided economy in 
the use of nickel-steel — based on the above assumed prices. As none of 
the rolling mills are prepared to engage in the general manufacture of 
nickel-steel for structures at the present time, however, the estimated costs 
must be considered as being more or less approximate. 

The Specifications used in proportioning the principal members of the 
spans are given as follows: 

Carbon Steel. 

Bottom chord, main diagonals, counters and long verticals, tension: 
area = (live stress + ^ dead stress) -^ 12500. 

Top chord, compression: 

area = (live stress 4- ^ dead stress) -^ (12000 —55 — ). 

T 

End posts (inclined), compression: 

area = (live stress 4- ^ dead stress) -^(11000 —50 — ). 

T 

Intermediate posts (vertical), compression: 

area = (live stress + i dead stress) -^( 10000 —45— ). 

T 

Floor-beams — bottom flange of riveted girder, tension: 
area (net) = (live stress + dead stress) -^13,000. 

Nickel-Steel, 

Bottom chord, main diagonals, etc. — eye bars — tension: 

area = [live stress (14-P) + dead stress] -J- 30000. 
Top chord, compression: 

area = [live stress ( l-f P) + dead stress] -J- ( 30000 -130—). 
End posts (inclined), compression: 

area = [live stress (l-f-P) + dead stress] -^ (30000 -140 —). 

Intermediate posts (vertical), compression: 

area = [live stress (H-P)+dead stress] h- (27000 -160— ). 

r 
Floor-beams — bottom flange of riveted girder, tension: 
area (net) = [live stress (l + P)H-dead stress] -5- 24000. 



738 



40.— HIGHWAY BRIDGES. 



Notation. 

/ and r = length and radius of gyration of member, in ins. 
P = percentage of impact for live-load stress, 
in formula, P= 10000-j- (1 50 + length of span, in feet, or portion of span cov- 
ered by live load when the member considered is subject to maximum stress.) 

35. — Comparison of Carbon-Steel and Nickel-Steel Spans, 40-200 Ft. 
Roadway, 20 Feet Wide. 



St 


S 
F& 




"Carbon Steel 


" Spans 




"Nickel Steel" Spans. 


Loads 






Esti- 


Loads 






Esti- 


fl 




Per Lin. Ft. 


Total 


Total 


mated 


Per Lin. Ft. 


Total 


Total 


mated 




^ 






Steel. 


Lumber 


Cost. 
Erected 






Steel. 


Lumber 


Cost, 
Erected 


% 










a 


£-1 


Live. 


Dead 


Lbs. 


Ft.B.M: 


and 


Live. 


Dead 


Lbs. 


Ft.B.M: 


and 






Lbs. 


Lbs. 






Painted 


Lbs. 


Lbs. 






Painted 


40 


9 


2000 


585 


7120 


4400 


$ 585 


3200 


725 


'7730 


6250 


$ 810 


45 


1900 


590 


7870 


4950 


649 


3100 


730 


8410 


7000 


892 


50 


Ph 


1800 


595 


8610 


5500 


719 


3000 


735 


8990 


8150 


983 


55 


i-r 


1700 


600 


9760 


6050 


803 


2900 


740 


10240 


9150 


1110 


60 


H^ 


1700 


615 


11940 


6600 


936 


2900 


745 


12590 


9300 


1279 


65 


fcS 


1650 


620 


13320 


7150 


1032 


2800 


750 


14210 


10050 


1420 


70 


Uh 


1600 


625 


14750 


7700 


1131 


2800 


760 


15780 


11300 


1576 


75 


>. 

^ 


1550 


630 


16100 


8250 


1225 


2800 


770 


17110 


12750 


1724 


80 


1550 


635 


18260 


8800 


1357 


2700 


780 


19450 


12400 


1872 


80 


(U 


1550 


640 


19460 


8800 


1412 


2700 


790 


20130 


12400 


1872 


85 




1550 


645 


21000 


9350 


1516 


2700 


800 


21860 


13750 


2043 


90 




1550 


650 


21610 


9900 


1577 


2700 


810 


22650 


14900 


2176 


95 




1550 


655 


23550 


10450 


1701 


2700 


820 


24040 


16100 


2290 


100 




1550 


660 


25000 


li>000 


1798 


2700 


830 


26320 


16650 


2463 


105 


t; 


1550 


665 


27290 


11950 


1950 


2700 


840 


28300 


17950 


2646 


110 


n 


1550 


670 


30750 


12000 


2125 


2700 


850 


32260 


18050 


2905 


115 




1550 


680 


32530 


12650 


2245 


2700 


860 


33840 


19450 


3060 


120 


1550 


690 


34280 


13200 


2356 


2700 


870 


36020 


19950 


3234 


125 




1550 


700 


36040 


14200 


2486 


2700 


880 


38130 


21500 


3433 


130 


n 


1500 


710 


41600 


14300 


2761 


2700 


890 


43180 


22200 


3776 


135 


c 


1500 


720 


44070 


14850 


2907 


2700 


900 


45440 


22900 


3958 


140 


1500 


730 


45700 


15400 


3014 


2700 


910 


48150 


23250 


4152 


145 


Ph 


1500 


740 


48530 


16500 


3198 


2700 


920 


51530 


24950 


4441 


150 


-P 


1500 


750 


53670 


16500 


3449 


2700 


930 


56530 


25400 


4777 


155 


r?, 


1500 


760 


57870 


17050 


3675 


2700 


940 


59540 


26250 


5007 


160 


t4 


1500 


770 


60520 


17600 


3831 


2700 


950 


63330 


26600 


5272 


165 


1500 


780 


65090 


18750 


4095 


2700 


960 


66830 


28400 


5545 


170 


,C 


1500 


790 


68560 


19300 


4290 


2700 


970 


70380 


30050 


5830 


175 




1500 


815 


77090 


19250 


4696 


2700 


985 


79450 


29150 


6378 


180 


1500 


830 


80470 


19800 


4884 


2700 


1000 


82690 


29900 


6606 


185 


^ 

H 


1500 


845 


84280 


21000 


5117 


2700 


1020 


86380 


31850 


6938 


190 


1500 


860 


86890 


21600 


5272 


2700 


1040 


89830 


33650 


7231 


195 




1500 


875 


95820 


21350 


5689 


2700 


1060 


100080 


32200 


7872 


200 




1500 


890 


101240 


22000 


5978 


2700 


1080 


104440 


33200 


8149 



v.— REINFORCED CONCRETE BRIDGES. 

Design and Cost of Reinforced=Concrete Highway Bridges (By A.N. 
Johnson. Paper, 111. Soc. Eng'rs and Surveyors, Jan. 26, 28, 1910; Eng. 
News, Feb. 10, 1910). — Summary of cost per cu. yd. of concrete is as follows: 

Cement $2 . 3 5 to $ 1 . 25 

Stone 3.23;; 1.30 

Sand (excluding gravel concrete) 1.47 " .32 

Gravel 2.43 ; .70 

Forms 3.95 .83 

Steel in place 2iS .. -SS 

Mixing and placing concrete 2.72 ^^ .72 

Excavation 3.91 .21 

Total $17.51 " $6.54 

Spans range from 7 ft. to 60 ft., roadways mostly 16 ft., abutments 
mostly 10 to 13 ft. high. Extensive cost table not reproduced here. 



NICKEL-, CARBON-, CONCRETE-STEEL SPANS. 739 



EXCERPTS AND REFERENCES. 

Bascule Bridge at Grand Ave., Milwaukee, Wis. (Eng. News, July 3, 
1902).— Illustrated. 

Page Bascule Bridge Over Chicago River at Ashland Ave. (Eng. News, 
Jan. 1, 1903. — Illustrated. 

The Wabash River Bridge at Terra Haute, Ind. (By M. A. Howe. 
Eng. News. Mar. 8, 1906).— Illustrated. 

Reinforced=Concrete Viaduct with some Structural Steel Reinforcement 
(Eng. News, July 1, 1909). — Illustrated: longitudinal section, tower bent, 
detail of floor system, detail of expansion joint. Consists of three 24-ft., 
five 30-ft. and two 24-ft. spans. Floor, 3-in. tar-macadam roadway 16-ft. 
wide, carried on a slab crowned to 9 ins. Designed for 1. 1. of 100 lbs. per 
sq. ft., with a concentrated load on the floor system of a 15-ton wagon; 
future provision for 30-ton car. Cost of viaduct, about $2.30 per sq. ft. of 
roadway. 

Reinforced=Concrete and Steel Spans, Sparkman St. Bridge, Nashville 
(By Howard M. Jones. Eng. News, Nov. 25, 1909). — Illustrations: Half 
section of roadway (macadam); reinforced-concrete retaining walls; floor 
system; typical bent of trestle approach; reinforced-concrete staircase; 
details of reinforced-concrete hand railing ; details of reinforced-concrete 
trusses; typical channel pier; stress sheet of 318-ft. truss; details of 318-ft. 
truss. 

Illustrations and Diagrams. 

Description. Eng. News. 

Truss and floor details of 194-ft. span, Waterford, N. Y July 14, '10 

Eng. Rec» 
Steel highway bridge with concrete stringers and floor slabs. . . .Mar. 20, '09 
Short span bridges, N. Y., N. H. & H. R. R., N. Y. City cross- 
ings July 10, '09 

Short span bridges, N. Y., N. H. & H. R. R., N. Y. City Cross- 
ings July 17, '09 

Umbrella-column for supporting foot-bridge over street Aug. 14, '09 

Strain-sheet 668-ft. span — St. Louis Municipal Bridge Oct. 30, '09 

Pier type of abutment for highway bridges Feb. 5, '10 

2-span (each 36 ft.) rein. -cone, bridge; cost table Feb. 19, '10 

Overhead street bridge details; concrete-protected floorbeams. .Mar. 5, '10 
Rein. -cone, girder bridge (30 ft. span), arched bottom chord... .Apr. 9, '10 

109-ft. concrete-floor plate girder bridge May 21, '10 

Floor construction, Kensington Ave. bridge, Buffalo Oct. 22, '10 



41.— CANTILEVER BRIDGES. 

In True Cantilever Bridges the stresses are statically determinate, and to 
American engineers this type offers weighty advantages over the continuous- 
girder* types generally adopted by European engineers. Let us assume an 
ordinary form of cantilever as per Fig. 1, with a free central span I suspended 




b c c' b' 

Tr; a o a' R/] 




li — + X -J* ir~ 



Fig. 1. 
at the points h and 6', introducing in the cantilever at those points the 



downward forces i?i and R^. Ri produces a downward reaction Rz = 



h 



and an upward reaction J?2= ^ ] -\ and similarly with Rz and R^l, 

h 
Reaction Rz is upward when h is loaded direct, and likewise with Rz' when 
the loading is on W . Hence to resist both kinds of reaction, upward and 
downward, at the ends of the bridge, anchorages as well as piers or supports 
are required (see Fig. 2); while at R2 and R2\ supports only are needed. 

In erection, the spans are built outward from the abutments in canti- 
lever fashion to the central point o] hence the lower chords out to the 
points a and a' must be stiff members to resist compression, while he and 
6V are introduced as tension members. For any load P on span /i the 
reactions Rz and R2 may be obtained as above, but by using Px instead of 
Rxh. Any loading on h can affect only that span. The usual problem is to 
(1) find the loading which will give the maxim tmi (-Hand — ) stresses for 
each member; (2) find the required reaction; and (3) solve as in ordinary 
trusses. For instance, the maximum compressive stress in the end post E 
obtains with I2 full loaded (/i and / unloaded) ; but for maximum tensile 
stress in that member the reverse loading obtains, Ix and / loaded {I2 unloaded) . 

The position of loading for reactions at supports, and also for maximum 
stresses in members, may sometimes be studied conveniently by the use of 
influence diagrams. Fig. 2 is such an influence diagram, and shows the 




Fig. 2. Reactions R3 for Left Support, 
reaction Rz for a load moving from the right-hand end of the central span /, 

* The writer knows of but one steel truss bridge in America built on the 
continuous girder principle, excepting of course draw bridges. This is the 
O. R. & N. R'y Co.'s bridge across the Williamette river at Portland, Oregon. 
Later (191 4) —This bridge has now been replaced by another structure. 



740 



THROUGH AND DECK SPANS. 



741 



toward the left-hand anchorage, over /, /i and I2. The reactions Rs, for 
successive positions of the concentrated load, are shown by the ordinates 
at the point where the load is applied on the structure (see Fig. 1); down- 
ward when load is on / and /i, and upward when load is on I2. 

Carbon-steel cantilever spans are economical, generally, up to 
about 1600 to 1700 feet; nickel-steel spans, up to about 1900 or 2000 feet. 
Beyond these spans, suspension bridges become more economical. Much 
depends on local conditions. 

The limiting length of carbon-steel cantilever spans is practically 
about 2000 feet; and of nickel-steel cantilever spans, about 2500 feet. 

Deck Cantilever Bridges are economical in certain localities where there 
is sufficient head-room over the street, valley or stream to be bridged. 
Fig. 3 is typical. The general outline is practically an invert of Fig. 1. 



s 




















s 




^ 


/ 


y 


\ 


\ 


\bJ^:-l^^i4^ 


/ 


/ 


V 




'^ 


* 




N 


\ 


p^ 


• 


\i 


r- 

N 


\y 


r^ 


</ 



Fig. 3. 

The stresses in the truss shown in Fig. 3 are rendered statically deter- 
minate by making slotted pin-holes at 5, inserting pins at all lettered 
points, and providing a roller end at R. F is the fixed end. 

Camber.-y-In providing for camber the simplest method, and the best 
for erection, is to raise the panel points by shortening the diagonals, main- 
taining all vertical posts mathematically parallel. 



EXCERPTS AND REFERENCES. 

The Mississippi River Cantilever Bridge at Thebes, III. (Eng. News, 
May 11, 1905). — Illustrated. Railway. 

Quebec Cantilever Bridge Disaster and Discussions (Eng. News, 
Sept., 1907, to Aug., 1908). 

Information about Great Cantilever Bridges (Eng. News, April 30, 
1908).— Tables. 

Stresses in the Blackwell's Island Cantilever Bridge (By Boiler and 
Hodge. Eng. News, Nov. 19, 1908). — ^Table of calculated stresses. 



Illustrations, 

Description. Eng. News. 

P. & L. E. cant, bridge (769-ft. span) and foundations May 5, '10 

Outline of trusses for new Quebec bridge Sept. 8, '10 

Eng. Rec. 
Board of Engineers* design for new Quebec bridge Sept. 10,' 10 



42.— MOVABLE BRIDGES. 

Movable Bridges are designed to provide temporary openings for one 
line of traffic (usually a waterway) which is crossed by another (usually a 
street or railway) . There are five distinct types, as follows: 

(1). Swing Bridge or "Drawbridge" — balanced and swinging horizon- 
tally round an arc (usually a quadrant) of a circle: (a) Equal spans or 
arms, supported by pivot pier; (b) One span, counterpoised; (c) Two spans 
counterpoised. 

Type (la) is discussed below. 

(2) . Traversing Bridge — counterpoised tail-end raised from its sup- 
ports and span drawn back on rollers along one of its approaches. Used 
principally for narrow openings. 

^ (3) . Bascule Bridge — tail-end ballasted with cast iron and lead and span 
swinging vertically: (a) One arm; (b) Two arms, either with tail-ends to- 
gether, or like a jack-knife with two blades hinged at either end. The 
bascules are raised and lowered by pinions (operated by hydraulic, electric, 
steam, gasoline or other power) working in segmental racks. 

(4). Lift Bridge — whole span raised vertically by chains, at the four 
corners, suspended from towers. Counterpoised weights are used in order 
to reduce the required power for operating. 

(5) . Floating or Pontoon Bridge — iron pontoons (usually rectangular) 
coupled in pairs for stability and moored at all corners. Bridge is opened 
to river traffic by the use of running back drawbridges. This type of bridge 
is used where foundations for piers would be difficult and excessively 
expensive. 

SWING BRIDGES. 

Drawbridges may be either rim-bearing or center-bearing. 

The rim-hearing draw, with each truss supported at two points on the 
center pier, is the more common type of the two. This is shown in Fig. 1, 
the center points of support for the trusses 
being set a b c d, irrespective of the diam- 
eter of circular drum or turntable. Usually 
the circular drum is of such diameter as to 
give support to the center panel of draw at 
8 equi-distant or nearly equi-distant points, 
as shown in the figure. In such a case heavy 
girders, acting as cantilevers, are introduced 
in the chords ab and cd and in the floor- 
beams ad and be in order to distribute the 
loads more uniformly on the table. These ^. 

cantilevers may be designed sufficiently -rig. 1. 

strong to carry the live and dead loads, or only sufficient to carry the dead 
load, in which latter case adjustable supports may be introduced at points 
a b c d to transmit the live loads directly from trusses to center pier 
when draw is closed. In either event the center pier supports are at 
a b c d so far as the calculation of trusses is concerned, and when the draw 
is closed each truss becomes a continuous girder of 3 spans over 4 supports, 
for any moving load on the draw. 

The center -bearing draw differs from the rim bearing in that the weight 
is supported at the center either (1) on a vertical, steel pivot pin, or (2) on 
a nest of conical rollers. The weight is transferred to the center from the 
trusses by means of transverse superimposed girders between the chords, 
a light turntable being used to "steady" the span in revolving, but not 
calculated to give material support. When the draw is closed each truss 
becomes a continuous girder of two spans over 3 supports, for any moving 
load on the draw; and such live load may be supported, if desirable, by adjust- 
able wedges placed at the middle of draw under the trusses. 

742 




CONTINUOUS GIRDER— FOUR SUPPORTS. 743 

For the calculation of swing bridges three cases* or conditions are usually 
employed in determining the stresses, as follows: 
I. Each arm treated as a simple span resting on two supports, with live-, 

dead- and wind loads acting. 
Ila. Bridge swinging and treated as a cantilever, with dead- and wind 

loads acting, and with reactions at center supports only. 
lib. Bridge closed and treated as a continuous girder over all supports, 
with live load acting. It is assumed in this case that the vertical re- 
actions at ends of draw are ±0 when no live load is acting. The live 
load is considered as " balanced," that is, it advances symmetrically 
on both arms from the ends toward center of span. 

For maximum stresses in the truss members use Case I, or Cases Ila 
and lib combined, whichever gives the maximum. For maximum stresses 
in the lateral systems use each of the cases separately and select that stress 
which is a maximimi for each member. In case there_ is a reversal of 
stress in any member, two maxima will be required, one in tension and one 
in compression. Some specifications make it necessary to determine mini- 
mum as well as maximum stresses, in which case it is evident that, for the 
trusses. Case lib cannot be used alone. 

Rim-bearing Draw — 4 Supports. — For Case I with the draw considered 
as two simple spans, and Case Ila as a cantilever, the reactions at the sup- 
ports are obtained easily and the stresses in the members are statically 
determinate. For Case lib, however, it is treated as a continuous girder 
which takes the form of an elastic curve when loaded. The girder is as- 
sumed to be of homogeneous material with constant modulus of elasticity, 
E; constant cross-section or rather moment of inertia, I; and resting on level 
supports. As a matter of fact none of these exact conditions actually 
obtain in practice. The necessary reactions 
at the supports due to any loading on the 

continuous girder are deduced from the ,«.— q_J=> u^ u h~ -^ 
^'theorem of three moments." k * ^ ^ £ -4 

Reactions at the Supports — Continuous ^ — -"^l 4— c— »)' — -I H 

Girder. — Let P be any concentrated load, R| Re Ra R4 

Fig. 2, on the left arm, distant a from the 
end; then will Fig. 2. 

2/ 

■'--{'-T)['-i'-j)T ,^^\^ ] 



c 

I 



(1) 

-F-1 



R^^ P l-.^\± ^'^ (2) 



^3= [R.-R.-p(i-'f)]j-R. 



(3) 



i?2 = P- (i?i + i?4) -i?3 (algebraically) (4) 

Of course it is clearly evident that if the load P is on the right arm instead 
of the left, and distant a from the right hand end, the resulting reactions 
will be interchanged — Ri with R4, and R2 with i?3. Hence, for balanced 
loads, that is, the same loading in position and amount on both arms, each 
end-support reaction will equal R1 + R4, and each center-support reaction 
will equal (algebraically) R2-hR's for the specified load on either arm. In 
drawbridge calculations P is the panel load and a the distance from end of 
draw to each consecutive panel point where the load acts. When the 
reactions are known the stresses in the structure may be solved by the or- 
dinary methods. 

^Assumptions for cheap highway drawbridges. — Various assumptions 
are made for cheap swing bridges as follows: Case A. Draw swinging and 
treated as a cantilever, as in Case Ila above. Case B. Draw closed and end 
raised to support say 34 the dead load of one panel; and acting as a continu- 
ous girder for the live load. 



744 



42.— MOVABLE BRIDGES. 



^ 



It is to be noted that the length c of the center panel (over the pier) 
affects the values of the reactions. Preferably, c is made equal to the dis- 
tance between the trusses unless this is so great as to require too large a 
turntable. ^ It is also usually equal to about one panel length of truss. 
The following Table of reactions is based on c = one panel length of truss, 
and will be useful for general reference. 

1. — Reactions Ri, R4, R3 and R2 for Drawbridge Spans of 2 to 10 Panels 

IN EACH Arm; with Central Panel c equal to One Panel Length of 

Truss; and Load P equal to Unity. 

See Fig. 1, and Formulas (1), (2), (3), and (4). 

^==One Panel Length = Unity . 



No. of 


r-i 


(M 


CO 


rj< 


»o 


«o 


t«- 


00 


OS 


Panels 


II 


II 


II- 


II 


II 




II 






in Each 


^ 


^ 


^ 


* 


^ 


d' 


.s- 


^ 


§• 


Arm / of 


II 




II 




II 






II 




Draw. 


« 


Q 


Q 


Q 





Q 


Q 


Q 





(i?i = 


+ 0.855 


+ 0.713 


+ 0.576 


+ 0.447 


+ 0.329 


+ 0.225 


+ 0.137 


+ 0.069 


+ 0.022 


10 i?4 = 


+ 0.002 


+ 0.004 


+ 0.006 


+ 0.007 


+0.008 


+ 0.008 


+0.007 


+0.006 


+ 0.004 


Rz = 


-0.473 


-0.919 


-1.307 


-1.607 


-1.794 


-1.836 


-1.707 


-1.378 


-0.818 


[R2 = 


+ 0.616 


+ 1.202 


+ 1.725 


+ 2.153 


+ 2.45? 


+ 2.603 


+ 2.563 


+ 2.303 


+ 1.792 


{Ri- 


+ 0.839 


+ 0.682 


+ 0.533 


+ 0.395 


+ 0.271 


+ 0.166 


+ 0.084 


+ 0.027 




R4 = 


+ 0.002 


+ 0.005 


+ 0.007 


+ 0.008 


+0.009 


+ 0.008 


+0.007 


+0.004 




9"i?3 = 


-0.470 


-0.905 


-1.270 


-1.531 


-1.646 


-1.588 


-1.316 


-0.799 




li^2 = 


+ 0.629 


+ 1.218 


+ 1.730 


+ 2.128 


+ 2.366 


+ 2.414 


+ 2.225 


+ 1.768 




{Ri = 


+ 0.820 


+ 0.646 


+ 0.481 


+ 0.333 


+ 0.205 


+ 0.104 


+0.034 






R.= 


+ 0.003 


+ 0.006 


+ 0.008 


+ 0.010 


+ 0.009 


+ 0.008 


+ 0.005 






8 i?3 = 


-0.466 


-0.888 


-1.222 


-1.425 


-1.444 


-1.242 


-0.777 






[R2 = 


+ 0.643 


+ 1.236 


+ 1.733 


+ 2.082 


+ 2.230 


+ 2.131 


+ 1.738 






{Ri = 


+ 0.796 


+ 0.599 


+ 0.418 


+ 0.260 


+ 0.132 


+ 0.043 








,i?4 = 


+ 0.004 


+ 0.007 


+ 0.010 


+ 0.011 


+ 0.010 


+ 0.006 








7 i?3 = 


-0.460 


-0.864 


-1.152 


-1.268 


-1.152 


-0.749 








[R2 = 


+ 0.660 


+ 1.258 


+ 1.724 


+ 1.997 


+ 2.010 


+ 1.700 








{Ri = 


+ 0.764 


+ 0.539 


+ 0.338 


+ 0.174 


+ 0.057 










r\- 


+ 0.005 


+ 0.009 


+ 0.012 


+ 0.011 


+ 0.008 










6 i?3 = 


-0.454 


-0.830 


-1.050 


-1.037 


-0.712 










U2 = 


+ 0.685 


+ 1.282 


+ 1.700 


+ 1.852 


+ 1.648 










{Ri = 


+ 0.719 


+ 0.459 


+ 0.239 


+0.079 












i?4 = 


+0.007 


+ 0.012 


+ 0.013 


+ 0.010 












5 i?3 = 


-0.443 


-0.775 


-0.886 


-0.665 












[i?2 = 


+ 0.717 


+ 1.304 


+ 1.634 


+ 1.576 












{Ri = 


+ 0.655 


+ 0.349 


+ 0.118 














Ri = 


+ 0.009 


+ 0.015 


+ 0.013 














4 i?3 = 


-0.426 


-0.682 


-0.593 














li?2 = 


+ 0.762 


+ 1.318 


+ 1.462 














{Ri = 


+ 0.554 


+ 0.192 
















r\= 


+ 0.014 


+ 0.018 
















3li?3 = 


-0.395 


-0.494 
















^2 = 


+ 0.827 


+ 1.284 
















{Ri = 


+ 0.371 


















rU 


+ 0.021 


















2]Rs = 


-0.321 


















[R2 = 


+ 0.929 



















Note — ^The reactions given in the table are for loads P=l at panel points. 
Hence, to find the actual reactions, multiply values in table by actual loads 
P at panel points. 



DRAWBRIDGE REACTIONS AND MOMENTS, 



745 



2. — Practical Data for Drawbridge Calculation, 4 Supports. 

Case lib. 

Reactions and Moments for Balanced Loads. 

Reactions are in terms of unit panel load. Moments are at foot of Tower 

Posts, and are for Unit Panel loads and Unit Panel Lengths. The Loads 

are considered as extending from the ends toward the center. 



No. of Panels 

in Each Arm 

/ of Draw. 



iM2 = M3 = 



A 

(1) 



+0.857 
+0.143 
-0.43 



A B 
(2) 



+ 1.574 
+0.426 
-1.26 



ABC 
(3) 



+2.156 
+0.844 
-2.44 



AtoD 
(4) 



+2.610 
+ 1.390 
-3.90 



AtoE 
(5) 



+2.947 
+2.053 
-5.53 



AioF 
(6) 



+ 3.180 

+2.820 

7.20 



Ato G 
(7) 



+3.324 
+ 3.676 
-8.76 



AtoH 
(8) 



+ 3.399 
+4.601 
-10.01 



A to 
(9) 



+3.425 
+5.575 
-10.75 



{Ri =i?4 = 

9]R2 =R3 ■■ 

(M2 = M3 = 



+0.841 
+0.159 
-0.43 



+ 1.528 
+0.472 
-1.25 



+2.068 
+0.932 
-2.39 



+2.471 
+1.529 
-3.76 



+2.751 
+2.249 
-5.24 



+2.925 
+3.075 
-6.68 



+3.016 
+3.984 
-7.86 



+ 3.047 
+4.953 
-8.58 



(■gl=-g4 = 

iM2 = M3 = 



+0.823 
+0.177 
-0.42 



+1.475 
+0.525 
-1.20 



+ 1.964 
+ 1.036 
-2.29 



+2.307 
+ 1.693 
-3.54 



+2.521 

+2.479 
-4.83 



+2.633 
+3.367 
-5.94 



+2.672 
+4.328 
-6.62 



7^i?2=i?3 = 
/M2 = M3 = 



+0.800 
+0.200 
-0.40 



+1.406 
+0.594 
-1.16 



+ 1.834 
+ 1.166 
-2.16 



+2.105 

+1.895 

3.27 



+2.247 
+2.753 
-4.27 



+2.296 
+3.704 
-4.93 



iR^ =i?4 = 
•j i?2 = ^3 = 
/M2 = M3 = 



+0.769 
+0.231 
-0.39 



+ 1.317 
+0.683 
-1.10 



+ 1.667 

+ 1.333 

2.00 



+ 1.852 
+2.148 
-2.89 



+1.917 
+ 3.083 
-3.50 



(^1=^4 = 
<R2 =R3 = 

iM2=M^= 



+0.726 
+0.274 
-0.37 



+1.197 
+0.803 
-1.02 



+ 1.449 

+ 1.551 

1.76 



+ 1.538 

+ 2.462 

2.31 



(^1=^4 = 

<R2— Rs = 

(M2 = M3 = 



+ 0.664 
+0.336 
-0.34 



+ 1.028 
+0.972 
-0.89 



+ 1.159 

+ 1.841 

1.36 



+0.568 
+0.432 
-0.30 



+0.778 
+ 1.222 
-0.67 




+ 0.392 
+ 0.608 
-0.22 



A B CEtc Eh:.C B A 

R, Re Ra R* 

Fig. 3. 
Example. — For /=9 (panels), and with 
Loads at A* B and C on each arm, 
the end reactions are 2.068 times a 
panel load; and center reactions, 0.932. 
Moments M2 and M3 are — 2.39 times 
a panel load times a panel length. 



Calculation op 315-Ft. Draw. (Fig. 4, page 746.) 

Figs. 5, 6, 7 and 8 are stress diagrams. 

Fig. 5 (Case I) shows one arm treated as a simple span, fully loaded. 
Similar diagrams must be drawn for live load retreating, panel by panel, 
from end of span toward center. 

Fig. 6 (Case Ila) is a dead-load stress-diagram of one arm when both 
arms are swinging clear of end supports. 

Fig. 7 (Case lib) is a stress-diagram for full live load when draw is 
closed; the condition being that ends of arms are simply touching supports 
under the dead load only. Similar diagrams must be drawn for live load 
symmetrically retreating, panel by panel, from the pier ends of arms. 

Fig. 8 (Case lib) is a stress-diagram of a retreating load last mentioned — 
with symmetrical loads at panel points A, B, C and D, for maximum stress 
in member 17. 

Wind-load stress diagram can similarly be drawn. 

See, also, page 742. 



746 



42.— MOFA^LjE bridges. 



Top Chord fbintS'A Parabola^ 49' \ h 

r 



iSlta 




CHoads at 6 for 6ubsfrucfure) 
Fig. 4. 315-Ft. Draw. 



»;/^ 




t ^^^ fa 





Fig. 5. 
Case I — Simple Span — 
Full Loaded — Live or Dead, 




24-25- 



Fig. 6. 

Case Ila — Draw Swinging — 
Cantilever — Dead Load only. 

(Stress in 24-25 = ^= Y-^2J= 11.) 

Load at end, a, is f of Panel Load. 




Fig. 7. 

Case lib — Continuous Span- 
Ftdl Loaded — Live Load. 
i?i = 2.296: i?2=3.704; 
M2 = 4.93 /. Stress in 24-25 = 
4.93-i-2i 
= 2.11. (See Table 2.) 




Fig. 8. 

Case lib — Continuous Span — 

Loaded (A, B, C, D, both arms) for 

Max. Live Load in 17. 

i?i = 2.105 (See Table 2). 



Center=bearing Draw — 3 Supports. — ^This differs from the rim-bearing 
draw in having one support at the center pier in- 
stead of two. Hence, Case lib (the only variation 
from 

continuous 
ports. 

Reactions at the Supports — Continuous girder. 
Let P be any concentrated load. Fig. 9, on the left 
arm, distant a from the end; then will 



Q Ol uwu. xxcnv;c, vyctac x±u v.^'J^xc Kjiiiy vctiicinvjn 

1 the preceding illustrations) will be treated as a <*---a«-^ JV 
:inuous girder of two equal spans over three sup- t * | — ^ 



. — «i Hz R3 



Fig. 9. 



Mo = —^ 



R^ = r. 



Pa 

4 
P^ 

4/ 

Pa 

I 



' 


02-1 


1 






/2 


' 


a2-| 


1 






/2 




Pa 


+ 


2/ 



[=R3l.l 



(5) 



a2-| 



=m 



(6) 



=^- 2R3 (algebraically) | . . (7) 



I- 



*•■ (■ -f)-t{'-S) [-''(' -f)+''.<*.™l.» 



CONTINUOUS GIRDER— THREE SUPPORTS, 



747 



The right hand half of the following table is all that is necessary in Case 
lib of drawbridge caldlilations, the first half being interesting as a study of 
the continuous girder of two equal spans for any concentrated loading. It 
is to be noted that for loading on right-hand arm, Ri and R3 will be 
interchanged. 

3. — Practical Data for Drawbridge Calculation, 3 Supports. 

Case lib. 

Reactions and Moments for Balanced Loads. 
Reactions are in Terms of Unit Panel Load. Moments are at Center Sup- 
port and are for Unit Panel Loads and a (Respective Distance from End 
Support). Balanced loads are Symmetrical Loads, Both Arms. 



Concentrated 


Loads on Left Arm 
a From End. 


, Distant 


Balanced Concentrated 
Loads on Both Arms, Dis- 
tant a From Ends. 


a 
T 










d 






Double 


M2 


Rz 


R2 


^1 


T 


M2 


Ri=Rs 


Shear. 
R2 


.04 


-0.2496 a 


-0.009984 


0.059968 


0.950016 


.04 


-0.499 a 


0.940 


0.120 


.08 


-0.2484 a 


-0.019872 


0.119744 


0.900128 


.08 


-0.497 a 


0.880 


0.239 


.12 


-0.2464 a 


-0.029568 


0.179136 


0.850432 


.12 


-0.493 a 


0.821 


0.358 


.16 


-0.2436 a 


-0.038976 


0.237952 


0.801024 


.16 


-0.487 a 


0.762 


0.476 


.20 


-0.2400 a 


-0.048000 


0.296000 


0.752000 


.20 


-0.480 a 


0.704 


0.592 


.24 


-0.2356 a 


-0.056544 


0.353088 


0.703456 


.24 


-0.471 a 


0.647 


0.706 


.28 


-0.2304 a 


-0.064512 


0.409024 


0.655488 


.28 


-0.461 a 


0.591 


v,0.818 


.32 


-0.2244 a 


-0.071808 


0.463616 


0.608192 


.32 


-0.449 a 


0.536 


^0.927 


.36 


-0.2176 a 


-0.078336 


0.516672 


0.561664 


.36 


-0.435 a 


0.483 


^1.033 


.40 


-0.2100 a 


-0. 084000 


0.568000 


0.516000 


.40 


-0.420 a 


0.432 


"1.136 


.44 


-0.2016 a 


-0.088704 


0.617408 


0.471296 


.44 


-0.403 a 


0.383 


^1.235 


.48 


-0.1924 a 


-0.092352 


0.664704 


0.427648 


.48 


-0.385 a 


0.335 


gl.329 


.52 


-0.1824 a 


-0.094848 


0.709696 


0.385152 


.52 


-0.365 a 


0.290 


c^ 1.419 


.56 


-0.1716 a 


-0.096096 


0.752192 


0.343904 


.56 


-0.343 a 


0.248 


V. 1.504 


.60 


-0.1600 a 


-0.096000' 


0.792000 


0.304000 


.60 


-0.320 a 


0.208 


^1.584 


.64 


-0.1476 a 


-0.094464 


0.828928 


0.265536 


.64 


-0.295 a 


0.171 


H.1.658 


.68 


-0.1344 a 


-0.091392 


0.862784 


0.228608 


.68 


-0.269 a 


0.137 


XI. 726 


.72 


-0.1204 a 


-0.086688 


0.893376 


0.193312 


.72 


-0.241 a 


0.107 


1.787 


.76 


-0.1056 a 


-0.080256 


0.920512 


0.159744 


.76 


-0.211 a 


0.079 


1.841 


.80 


-0.0900 a 


-0.072000 


0.944000 


0.128000 


.80 


-O.180 a 


0.056 


1.888 


.84 


-0.0736 a 


-0.061824 


0.963648 


0.098176 


.84 


-0.147 a 


0.036 


1.927 


.88 


-0.0564 a 


-0.049632 


0.979264 


0.070368 


.88 


-0.113 a 


0.021 


1.959 


.92 


-0.0384 a 


-0.035328 


0.990656 


0.044672 


.92 


-0.077 a 


0.009 


1.981 


.96 


-0.0196 a 


-0.018816 


0.997632 


0.021184 


.96 


-O.039 a 


0.002 


1.995 



f 



,^a— |P 



M* 



^.- 



^. 






^ 



Ra 



R» 



Fig. 10. 

Example. — Solve for one load 
P= 16000 lbs. on left arm; /= 45 ft.; 
o-17ft. 

Solution.— 4= H= .378; M2 = 



.214XaXP 



-.214X17X 



16,000= -58,200 ft.-lbs.; Ri= .542 
X 16,000=8,672 lbs.; i?2= .539X 
X 16,000= 8,624 lbs.; i?3= - .081X 
16,000= -1,296 lbs. (R3 acting 
downward.) 



Fig. 11. 

Example. — Solve for load P= 
16,000 lbs. on each arm; /=45 ft.; 
a=17ft. 

Solution.— M2-= - .428 X 17 X 
16,000= -116 ,400 ft.-lbs.; Rt = R3 
= . 46X16,000=7,360 lbs.;i?2=l. 08 
X 16,000= 17,280 lbs. (All reac- 
tions acting upward.) 

Note. — A minus bending mo- 
ment ( — M2) means tension in top 
flange or chord, and compression in 
bottom. 



748 



42.— MOVABLE BRIDGES. 



Deck Drawbridge — Center Bearing. 

« 




/N 


^. 


t..._ 


— i- 


— ^ 


^ — 


lx-l_J-=i.J 



Re 
{Hoad at E for aubstrucfure) 

Fig. 12. 

Hints for Calculation of Trusses. — In Fig. 12, let P = panel load per truss 
at A, B, C, D and E\ and let H P = dead load at a when draw is swinging. 

Case I — Simple Span. — Ends at a are raised so there is no stress in 
DE. For dead load and full live load the shear in panels a — A and D — E 
are equal. Maximum compressive stress in 10 occurs with live loads at 
B, C and D, 

Case II — Cantilever. — Draw swinging, assuming % panel load at a and 
full panel load at other points. Shear oX D — E which is used for i?2 in this 



case = 0i P. Stress inD—E- 



lZHPXf = lSH P. 



Case III — Continuous Girder. — Draw closed and just touching supports, 
with no live load. Balanced live loads are now applied on both arms, 
extending from ends toward center of draw. The reactions and moments 
for this loading may be obtained from the preceding table, as it now becomes 
a continuous girder of two equal spans over three "level" supports. 

^ "-^ i?i=.704;3^i?2=.296: M2=-0.48a=-0.48^ 
i?i=.432;i^i?2=.568; M2= -0.42a= -0.84 ^. 
i?i=.208;3^i?2=.792; M2= -0.32a= -0.96^. 
i?i=.056;Hi?2=.944; ^2= -0.18a= -0.72 ^ 



For loads at A, 


/ 


.20; 


For loads at B, 


a 
I " 


.40; 


For loads at C, 


a 
l~ 


.60; 


For loads at D, 


a 

7 = 


.80; 



13, let / = the span HE, 



Forloadsat A,B,C, D, Ri = lAOO; J^i?2= 2.600; *M2= =-3.00p. 

Note that all calculations can be made for any loading when Ri is 
known, for we have only to apply the methods of moments and shears from 
the outer forces acting to the left of the section considered, to obtain the 
stress in any member. 

WEIGHT OF STEEL IN MOVABLE BRIDGES. 

Steel Swing Bridges. — ^The following formula is for single track standard 
R. R. bridges calculated for live load of two 160-ton engines followed by a 
uniform load of 4600 lbs. per lineal foot: Total weight of steel in lbs. = 

7.8 L2 + 12 (L XvT')+ 100 L-l- 20000; in which L = extreme length of draw in 
feet. For double track bridges, multiply results obtained in above formula 
by 1.85. 

Counterweight Jack=knife Draw. — In Fig. 
hinged at H and connected at Di (distant 
/i from H) by the chain C + Ci running over 
the pulley P, with the cylindrical counter- 
weight Wi, suspended at the other end and 
rolling along the modified cycloidal plane 
AP. Assume the total weight of the mova- 
ble leaf to be W acting at the point G. 
Then we have the following; 

WlV 2 
Weight of counterweight Wt = — wi '» A 

Length of chain Ci = /i\^ (l+sin|-cos|) . '^'^' 13. 

*Hence, for full loading, both arms, stress in D — E = I M2 :h=' — 3 -f- 
- 3 X 1 = - 3 (panel loads) . 




MISCELLANEOUS. 749 

Steel Bascule Bridges. — ^The weight of double track bascule bridges for 
openings 50 to 120 ft. varies from about 2000 to 6000 lbs. per lin. ft. of bridge, 
depending upon the type and loac^ng. The counterweights often weigh as 
much as the leaves themselves. 

EXCERPTS AND REFERENCES. 

Bascule Bridge Over the Chicago River at Clybourne Place (Eng. 
News, Jan. 31, 1901). — Illustrated. Highway. 

Four-Track Two=Truss Swing Bridge, Chicago & Western Indiana 

R. R. (Eng. News, Sept. 12, 1901).— Illustrated. 

The Kinnickinnic River Drawbridge, C. & N.-W. Ry. (Eng. News, 
Aug. 8, 1901).— Illustrated. 

Novel Type of Drawbridge (By Ward Baldwin. Eng. News, Oct. 22, 
1903). — Illustrated. Plate-girder deck structure of 64-ft. span, the swing- 
ing end supported on a rack circle, laid on a pile foundation, at bottom of 
canal. 

Pivot Pier Caisson and Operating Machinery for a Heavy Swing 
Bridge (Eng. News, Jan. 7, 1904). Illustrated. Machinery calculations. 

Short Span (26-ft,) Bascule Bridge on the P., Ft. W. & Chi. Ry. (By 
C. L. Strobel. Eng. News, May 25,1905). — Illustrated. Double-track, 
single-leaf. 

Rails and Rail-Lifts on the Thoroughfare Draw Near Atlantic City, 
N.J. (Eng. News, Nov. 15, 1906).— Illustrated. 

A Center-Bearing Drawbridge with Pneumatic Raising Jack and 
Turning Engine (Eng. News, April 4, 1907). — Illustrated. Constructed in 
Hamburg, Germany. 

Minimum End-Lift Device of a German Swingbridge (Eng. News, 
Jan. 9, 1908). — Illustrated. 

A Gyratory Lift Bridge (By Eric Swensson. Eng. News, April 2, 
1908). — Illustrated. 

Pontoon- or Floating Drawbridge (Eng. News, April, 30, 1908). — 
Illustrated. 

Bascule Bridges Over the East Chicago River (Eng. News, Mar. 18, 
1909). 

Illustrations of Movable Bridges and Details. 

Description. Eng. News. 

152-ft. rolling-lift railway span, piers, fender, etc Apr. 21, *10 

Sliding rail connection for drawbridges Sept. 1, * 10 

Center pier and live ring of drawbridge Oct. 6, '10 

Cannon-ball turntable bearing, concrete ball race Oct. 6, * 10 

Eng. Rec. 

Counterweight and operating mechanism of bascule bridge Mar. 27, *09 

Locking device for a Scherzer rolling lift bridge Aug. 21, '09 

Special masonry pivot pier for 283-ft. drawbridge, Boston Apr. 16, *10 

Temporary 69-ft. span lift bridge, N. Y., N. H. & H. R. R Aug. 20. '10 



43.— SUSPENSION BRIDGES. 



THEORETICAL CONSIDERATIONS. 

Curve of Main Cables. — The curve assumed by the main cables of a 
suspension bridge when the latter is uniformly loaded, is called a modified 
or transformed catenary. This curve lies somewhere between the catenary 
and the parabola. When the cables are first suspended between the bridge 
towers (before they support any extraneous load) they form a true catenary, 
due to the weight of the cables alone, assuming of course that they have a 
uniform weight per lineal foot of cable. But later, when the floor of the 
bridge is in position, that is, suspended from the main cables, the curve of 
the cables is "modified" and tends to approach the parabola in form. The 
true parabola, however, is hardly ever realized, the ideal condition for this 
curve obtaining only when the horizontal combined loading per lineal foot 
on the cables (including weight of cables themselves) is uniform. 

For Short Spans, where weight of cables is small compared with weight 
of floor, we may assume for all practical purposes that the main cables 
take the form of a parabola; also bearing in mind that on this assumption 
the error decreases as the ratio of central deflection to length of span de- 
creases. Indeed, even the arc of a circle may be used in the drafting room 
and also for the purpose of making estimates, when the central deflection 
is only ^^ or even tu the span. See also Tables 1 and 2. 

In the above discussion of the modified catenary and the parabola it is 
assumed that the suspenders transmitting the loads to the cables are verti- 
cal, uniformly spaced horizontally, and very close together, so that the 
cables form true and continuous curves throughout, between points of 
supports. But in actual practice the suspenders are not always vertical 
and are usually spaced quite far apart, tending to further modify these 
curves. 

Force Polygons. — ^The following force polygons assume that the cables 
or chords of the equilibrium polygon have no weight, being acted upon by 
outside forces only. 




EquUibHum Polygfon. %/^^. 
Figs. 1. 



Equilibrfum Polygon. 
Figs. 2. 




Polygort. 



The Parabolic Cable. — Horizontal Uni- 
form Load. — Let z£; = uniform load, in lbs., per 
lineal foot of bridge; /=span, in feet; c = half 
span; s = length of cable between towers; 
d = angle of inclination of tangent at any 
point p whose coordinates are x and y\ d — 
center deflection of cable; constant if = hori- 
zontal component of tension (lbs.) of cable at 
any point p. Then with vertical suspenders, 
we have: 

General equation. 




^ 2H~ 



4dx^ 

/2 ■ 



loacf y/per Linecrl Jvar 
Fig. 3. 



Horizontal tension (when «=c, y=d), H=-s-i= -^ 



(1) 



(2) 



750 



CURVE OF MAIN CABLES. 761 

At any point p, tan 6 = -rr'^'ir '*/ 

ri l^ 

id 
At top of towers, tan 61 =-— <4) 



(5) 



Length of cable. 5=~\/i^^+^2+2.302585m*log ( c + Vm^+cA 

m \ m / 

m which w=— =-5:7 ; ^ — T' i;:='"T'=tan di. 
w 8d 2m/ 

Tension in cables at any point p=H sec d == -^ sec d (6) 

Tension in cables at top of towers=gj Vp+ IQd^ (7) 

Tension in cables at top of towers is greater than at points between, 
being the least at point of greatest deflection. Hence by the use of 
equation (7) we may determine the size of cables required; and their weight 
per lineal foot multiplied by the value of s in equation (5) will give the 
weight of suspended cable between towers. 

For mid-span, equation (6) reduces to H = ^ (8) 




D/recfr/x. 

Fig. 4. 

The Catenarian Cable — Load Uniform Along Cable. — In Fig. 4 let 

/=span, in feet; 

c = half span; 

c^ = center deflection; 

5 = length of suspended cable between towers; 
ze;i=weight of cables per lin. ft. of s; 

^ = angle of inclination of tangent at point ^ whose coordinates are ^ and :v; 
if = constant horizontal component of tension (lbs.) in cable at any point pi 

H A 

/t= maximum value of ordinate y = d + m; m= — == — ; 

Wi s 

i4x = area of shaded portion of length x, above directrix, 
A=m 5 = total area for length /, between catenary and directrix; 
t ^=2.7182818 = base of Naperian system of logarithms. 
Then for equation of the catenary, we have: 

General equation, ^~"2 \ ^°* +e"~ "m ) (1) 

- AH horizontal tension ,. , , ,^ . 

where m= — = — = — r-rr r- — zr of cable (la) 

s ivt weight per Im. ft. 

But as i4, s and H are functions of m itself we have to resort to other methods 
to find the value of m. Thus: 

The approximate value of m= ^^^^^ ^'^^^ (lb) 

which may be substituted for value of m in the second member of equa- 
tion (Ic) to obtain its more nearly correct value. 

r? ^ , . 0.4342945 c 

Exact value of m= 7 . ^ (Ic) 



log y ^ "Y V w / J 



^Common logarithm. 2.302585 = — - ; in which e is the Neparian base = 

2.7182818. Log 2.302585 is 0.3622157. 
t Log ^ = 0.4342945; log (log e) = 9.6377843- 10. Also note that 

— = ~T~» i^ fl^6 general equation of the catenary. 



762 



iZ.—SUSPENSION BRIDGES. 



wlien the exact value of m is substituted in the second member; but when m 
from equation (lb) is substituted, the result from (Ic) is too small by about 
its excess over that obtained from equation (lb) alone. In the following 
table, column Ci is obtained by adding this excess, i. e., the difference between 
values in columns b and c, to the values in column c. Column "diff." is a 
difference colimin, omitting the decimal point, between Ci and e, the latter 
containing exact values of nt; and is useful in making corrections to values 
in column Ci obtained by the approximate method as above described, for 

any values of -y-. 

1. — ^Values op Parameter m for the Catenary, based on /= unity. 
For successive values of y. 





(Multiply tabular values of w, 


below, by length of span 


/.) 




a 


b 


c 


Cl 


Diff 


e 


a 


b 


c 


Cl 


Diff 


e 










G* 








o 




<o 




73 


B + o, 


1— 1 o 

2g^ 






+5 W ^ 


o 




2§a 




o 






£«-& 

fe ^ 


^33 


3 


g 

w 


W>0 


1 


2^ 2- 




CO 


o 


W>o 


.01 


12.5 


12.50084 


12.50168 


1 


12.50167 


.11 


1.13636 


1.14541 


1.15446 


22 


1.15424 


.02 


6.25 


6.25166 


6.25333 





6.25333 


.12 


1.04167 


1.05151 


1.06135 


27 


1.06108 


.03 


4.16667 


4.16917 


4.17167 


1 


4.17166 


.13 


.96154 


.96217 


.98280 


33 


.98247 


.04 


3.125 


3.12833 


3.13166 


1 


3.13165 


.14 


.89286 


.90427 


.91569 


41 


.91528 


.05 


2.5 


2.50415 


2.50830 


2 


2.50828 


.15 


.83333 


.84553 


.85773 


50 


.85723 


.06 


2.08333 


2.08832 


2.09330 


4 


2.09326 


.16 


.78125 


.79422 


.80719 


60 


.80659 


.07 


1.78571 


1.79152 


1.79732 


6 


1.79726 


.17 


.73529 


.74902 


.76275 


70 


.76205 


.08 


1.5625 


1.56912 


1.57574 


9 


1.57565 


.18 


.69444 


.70894 


.72344 


82 


.72262 


.09 


1.38889 


1.39632 


1.40375 


11 


1.. 40364 


.19 


.65789 


.67312 


.68835 


94 


.68741 


.10 


1.25 


1.25824 


1.26648 


16 


1.26632 


.20 


.625 


.64097 


.65694 


107 


.65587 



Remarks. — ^To find values of m (column e) for values of-y intermediate 

to those in the table: Calculate for column Ci and subtract the interpo- 
lated difference in colimin "diff." 



Length of arc Sx = -^ I e 



m tan 6 



I ^ m —e m j 

above equation, 
Total length of cable s = m I e"^— e "^ ) ~ ^ wtan^i 



(2) 



Substituting c for x in above equation, we have, when towers are of equal 
height, 

(3) 



2. — Lengths s op Cable in the Catenary, based on /== unity. 

For successive values of -y- . 

(Multiply tabular values of s, below, by length of span /.) 



d 
I 


s 


d 
I 


5 


d 
I 


s 


.01 
.02 
.03 
.04 
.05 


1.000267 
1.001066 
1.002396 
1.004254 
1.006636 


.06 
.07 
.08 
.09 
.10 


1.009537 
1.012949 
1.016868 
1.021283 
1.026187 


.11 
.12 
.13 
.14 
.15 


1.031569 
1.037421 
1.043729 
1.050484 
1.057674 



* See foot-note, following page. 



THE CATENARIAN CABLE. 753 

Tangent to curve at pointy (Fig. 4) is: tan e=\l e~—e~~^\ =~ . . (4) 

Tangent to curve at top of tower is: tan dx = \{ e~^—e~~\ =— ...(4a) 

c7-7 
Total area between catenary and directrix (equa. la): A=ms= - — ■ ... .(5) 

Horizontal component of tension in cable: H = -='mwi (6) 

Actual tangential tension at any point p=H sec 6=H^li-\. ( — ) ^ (7) 



(8) 



Actual tangential tension at top of tower =H sec di=H^ I + I- — j .. 

A Linear or Skeleton Arch, acted upon by gravity, if constructed on the 
curve of the catenary (inverted) , will possess maximum stability because the 
line of resultant pressure throughout the arch will coincide with the center 
line of the arch ring. 

Graphical Solution of the Catenary. — Fig. 5 is a graphical representation 
of the solution of the catenary. The theory is correct provided the load line 
has an infinite number of divisions — the more divisions the greater accuracy. 

First. — Let 1-6 (Fig. 5) represent the load line 
or weight of half cable (from tower to point of 

greatest deflection) or-——. Lay off Hf, from 6 to 0, 

equal to the horizontal component of tension in Eau/7/bn'um 
cable. Then will d^ be the angle which the direction Polygon 
of cable makes with the horizontal at top of tower; j>^, 

WiS s ^^^ 

and tan ^i==?rT7=7r~ • Divide the load line into 
2/1 2m 

any number of equal parts, the more the better for 
accuracy. Then from this force polygon the equi- 
librium polygon maybe drawn: Draw 1', equal in 
length to the length of cable weighing^ part, paral- 

lei to 1-0; draw 1' , equal in length to the length of P*— ' Directrix 

cable weighing one part, parallel to 2-0; similarly 

with lengths equal to 2', draw 3', parallel to 3-0; -^^S- o. 

4', parallel to 4-0; 5', parallel to 5-0; and lastly, draw 6', equal in length 

to V, parallel to 6-0. Then the curve l — E', shown imperfectly dotted, 

IS a catenary curve. 

Second. — Now instead of dividing the load line into five parts as above, 
if it were possible to divide it into an infinite ntmiber of parts, we should 
get from such a force polygon the true catenary curve l — E for the equili- 
brium polygon; and ii \ — E represents the half length (-^ ) of the cable 

between towers, the middle ordinate m will represent the length of cable 
whose weight Wim will be equal to the tension H, at E. 

Third, — From the construction of Fig. 5 it is evident that the stress in 
V is 1-0; in 2' is 2-0; in 3' is 3-0; etc. 

Fourth. — If the weight Wj of the cable is assumed as unity then l—E 
will equal 1-6 in length, and m will equal H. 

The Transformed Catenary. — This curve applies as well to the arch as 
to the suspension bridge (see Sec. 44, Arches). The general equation of the 
transformed catenary is 

nix X \ 

(9) 




y = Y Y '" +^~'") 



* By development. eU 1 +^ + ^l+^-+_g^ X 

similarly, also. .-^ = 1-^+2-^,-2^ + 2^^^^, 

See preceding page. 

t H = Wim, exactly, = -^ ( 1 -f- . 16yj , very nearly, = —j-, approx. 



754 iS.— SUSPENSION BRIDGES. 

Comparing equations (1) and (2) we note that any ordinate y of the trans- 
formed catenary = — times y of the true catenary, for constant values of icand 
m 

m. In fact the catenary is a special case of the trans- ^ 

formed catenary where a = m, just as the circle is a special /|B 

case of the ellipse where the semi axes a and b become / \^ 

equal to each other and are called r, the radius of circle. / 

In Fig. 6, let A B C D be the diagram of a catenary, 'M/ 

similar to one-half of Fig. 4. The curve A B is that due ^"h^yS 

to the weight Wi per lineal foot of the cable itself. Now ,,^^^ I 

if there is an additional weight v/ per horizontal lineal IT^*'**''^ \ 
foot imposed upon the cable the latter will assume ^ 

the exaggerated form AB\ the middle ordinate will f ^J 

become a, and the directrix, FE. Let Sx be the length O -^iHf^ 

of curve from A to any point p' , whose coordinates are x ij) if^ 

and y. Then will the load on 5x be proportional to the ^ — 2? i J 

area Ap'GF, in the same way that the load on the arc T ^ E 

Ap of the true catenary is proportional to the area ApkD. Fig. 6. 

PRACTICAL HINTS. 

Cables or Chains. — These may be composed of wire cables or of steel 
eye-bars, as follows: (1) For short spans, twisted wire ropes are generally 
preferred as they can be manufactured and handled conveniently. (2) For 
spans of moderate lengths, strands composed of parallel wires, laid together 
near the bridge site and then hoisted into position, possess an advantage in 
economy of material over twisted strands, which latter develop only about 
90% of the strength of straight wires. (3) For very long spans, the cables 
are made up of parallel wires as in the second case, but they are built in 
place as it would be impossible to raise them bodily on account of their 
great weight. (4) Steel eye-bars may be used instead of wire cables, as at 
first proposed* for the Manhattan suspension bridge over the East River, 
New York. This bridge to be composed of a central span of 1,470 ft., and 
two end spans of 725 ft. each. The four main chains to lie in vertical planes 
20 and 48 ft. distant from the axis of the bridge. The eye-bars to be of 
nickel steel, 3i to 3i% nickel; not over 0.05% sulphur; not over 0.06% 
phosphorus if made by acid O-H process, and not over 0.04% phosphorus 
if made by basic O-H process. The required ultimate strength was 85000 
lbs. per sq. in.; actual elastic limit, 48000; percentage of elongation in 18 ft., 
9; percentage of reduction at fracture, 40. Among the advantages claimed 
for the eye-bar cables are: They offer better connections for the vertical 
suspenders; they are better adapted to form integral parts of the stiffening 
trusses to equalize moving loads on the bridge; and they can be propor- 
tioned economically with varying cross-section to the exact stresses in 
various parts of the cables, whereas wire cables must have a uniform cross- 
section. (For revised plans and specifications, see pages 756, etc.) 

The parabolic curve may be used in making up preliminary estimates, 
and also for designing small suspension spans in general. See Fig. 3 and 
equations (1) to (8). 

Example. — What will be the tension in each cable at top of tower, at 
mid span, and at quarter span, for a clear span of 600 ft., and a center de- 
flection of 50 ft.; assuming total load at 8000 lbs. per lin. ft. of bridge, and 
supported by two cables? 

Solution. — From the preceding equations: 

11)1 _________ 

At towers, each cable, equation (7),^= i 'M'^^'^ 16(^2= 3,795,000 lbs. 

At mid span, each cable, equation (8), / = i . H = J . g^ = 3,600,000 lbs. 

At quarter span, each cable, equa's (6)(3),^=i.Hv'l + tan2 (?= 3,650,000 lbs. 

Towers and Backstays. — Provision for expansion and contraction of the 

cables may be made at the towers in several ways: 

(1) One of the most common methods is to sling the cables over the 
main pierS on Saddles resting on a nest of rollers as in Fig. 7. Note in this 

* Eye-bars not adopted; wire cable used in plans finally approved. 



CABLE. TOWER. ANCHORAGE. 



755 




Fig. 7. 



case that the load and the resultant reaction 

at pier must necessarily be vertical, as the 

rollers would eliminate any possibility of a 

horizontal component to the loading; also, 

that the main cable and the backstay make 

the same angle 6^ with the horizontal, and 

are stressed alike. Generally, any change in 

temperature or loading on main span will affect the angle of inclination 

of main cable at top of towers, and in order that such angle may equal the 

angle of backstay the saddle moves automatically to restore equilibrium. 

(2) The Truck, Fig. 8, gives the same move- 
ment to the cable as have the rollers a and b, 
whereas in Fig. 7 the cable has double the move- 
ment of the rollers. In Fig. 8, h represents the 
horizontal stress in main cable and in backstay, j^'^ 
as well as the stress in link joining rollers a an^i b\ 
consequently, stress in main cable = /t sec dx, and 
stress in backstay = /i sec 62- Vertical reaction at 
a = h tan 6^, and vertical reaction at b = h tan 62', 
showing that the vertical reaction is directly pro- 
portional to tang of angle of inclination of cable. 

(3) The Suspended Link (Fig. 9) suspended from 
top of tower at ,a, and the Toggle (Fig. 10) supported 
on tower at b, give the required lateral motion to 
cables, caused by temperature changes and moving 
loads. It is to be noted, however, in these cases that 
the direction of loading on towers and the resultant 
reaction are not confined to the vertical as in the 
two preceding cases, but take the direction of a line 
passing through pins a and b. The triangle of forces 
is shown graphically in each case. The towers should 
be designed to take care of these loads for any 
possible variation in direction from the vertical, as 
well as for the vertical loading itself. 

(4) The Hinged Tower, Fig. 11, is simply an elon- 
gated toggle. The cables may be fastened directly to 
top of tower, and the latter being hinged at the bottom 
will lean in the direction of the resultant load and re- 
action, or give direction to same. 

Anchorages. — ^The maximum stress in the backstay, 
at the point where the latter enters the^ ground, must 
of course be borne by the anchorage, with the proper 
factor of safety. The anchorage proper consists of the 
ends of the backstays imbedded in masonry having sufficient weight or 
resistance to overcome the stress. For this purpose the backstays may 
continue in a straight line into the masonry or they 
may curve downward as in Fig. 12. The main back- 
stays above the point a may be composed of wire 
cables, while the anchorage chain below a may be 
composed of eye-bars in order to provide better con- 
nections with the anchors. In some cases the anchor- 
age chains are imbedded in solid concrete masonry, 
which probably insures permanent protection to the Fig. 12. 

metal; in other cases it has been deemed advisable to allow the anchorage 
chains to pass through open galleries so as to be accessible for inspection. 
See plans of anchorage for Budapest suspension bridge in Eng. News, 

Aug. 24, 1905; also, Fig. 17, page 759. 

Cable Bcmci Cable Wrappings. — Fig. 13 shows the 
method of wrapping the cables for the 

CoffonDuck 




Fig. 10. 



Fig. 11. 





■CoffonDuck 
Wrappings 

Williamsburg Suspension 
Bridge, New York City, 



Fig. 13. 



756 



id.— SUSPENSION BRIDGES. 



DETAILS OF MANHATTAN SUSPENSION BRIDGE. 

Description. — Wire cable bridge of three spans: central span, 1470 ft.; 
two side spans, 725 ft. each. Four cables. 

Cables. — About 20^'' in dia.j each consisting of 37 strands and contain- 
ing 9472 parallel wires of 0192 in. dia. before galvanizing, and not more 
than 0.197 in. dia. after galvanizing. 



I 




31// 7 Jd^ua^ 



Live Loads. — (1) For the cables, trusses and towers: (a) a load of 8000 
lbs. per lin. ft. of bridge as "regular," or (b) 16000 lbs. per lin. ft. of bridge 
as "congested" traffic. (2) For the hangers, floor beams and floor system: 
(c) on each elevated track a load of 52 tons on four axles, 6'-10'-6', the motor 
ends of cars of the Interborough R. T. Co.; (d) on each street car track 
either a load of 26 tons on 2 axles 10 ft. apart or a load of 1800 lbs. per lin. ft. 



MANHATTAN BRIDGE DETAILS. 



757 



of track; (e) on any part of the roadway a load of 24 tons on two axles 10 
ft. apart and 5 ft. gage (assumed to occupy a width of 12 ft. and a length of 
30 ft.), and upon the remaining portion of the floor a load of 100 lbs. per sq. 
ft.; (f) on the footwalks a load of 100 lbs. per sq. ft. 

The wind pressure is assumed as a moving load acting in either direc- 
tion horizontally with 2000 lbs. per lin. ft., but due allowance is made by 
taking the lateral deflection into account. 

Specifications for Material. — The wii'e for cables, hand ropes and sus- 
penders shall have an ultimate strength of not less than 215,000 lbs. per sq. 
in. before galvanizing, and an elongation of not less than 2% in 12 ins. of 
observed length, the stretch to be measured while the specimen is in the 
testing machine.' The bright wire shall be capable of coiling cold around a 
rod 1J1 times its own diameter without sign of fracture.^ The serving wire 
shall be No. 9 gage before galvanizing. After galvanizing, the steel wire 
shall have an ultimate strength of not less than 200,000 lbs. per sq. in. of 
gross section. (Continued on next page.) 




Fig. 15. Cross Section of Bridge. 




r^i.o.^o'o"; * 



TiePfs.?n'^s ^ 
LaffiBarsSH 



S—B — y-TS S — B S- .1. o- 

° ° ° ? y 1 !• I 'I i 



Lait.BarsSVi' 



9 B"ii"i)°?|°|g"9''g°9"B°n°n 



L^^ 



*.....l|....... 



\ III // 






9'9^^%^y^»; 



Fig. 16. Details of Floor-Beam Connections. 



768 



id.— SUSPENSION BRIDGES. 



m 



The suspenders shall be IM in. in dia. and weigh not less than 5.1 
per lin. ft. They, shall be composed of six strands of 19 wires each, laid 
around an independent wire rope center consisting of 49 wires, left hand lay. 
The suspenders shall be, preferably of long lay, but the lay must not be 
long enough to cause trouble in keeping the core in its true position during 
any of the operations before the suspenders are in their final position and 
loaded with the superstructure. 

Required Physical Properties of Finished Material. 
(Manhattan Bridge.) 



Ultimate 
Material. Strength. 

Lbs. per 
sq. in. 
Carbon Steel. 

Shapes and universal mill plates 60-68,000 

Eyebars, pins and rollers 64-72,000 

Rivet rods 50-58,000 

High carbon steel for trusses 85-95,000 

Sheared plates 60-68.000 

Nickel Steel. 

Shapes and plates 85-95,000 

Rivet steel 70-80.000 

Steel Castings. 

Test pieces from annealed castings 65.000 

Allowable Maximum 



Minimum 
Elastic 

Limit. 
Lbs. per 

sq. in. 

33,000 1 
35.000 
30.000 > 
45,000 
33.000 J 



Minimum Minimum 

Elongat'n. Reducti'n. 

Per cent Per cent 

In 8 Ins. of area. 



r44 

1.500.000 40 

divided by { 50 

ultimate. 35 

144 



per ct. 



55,000 
45.000 



1 1.600.000 f 
J ultimate \ 



40 per ct. 



Material, and Parts of Structure. 



In 2 Ins. 
35,000 20% 

Unit Stresses. 

For dead load, 

For dead load, temperat're and 

temperature congested live 

and regul'r live load, or for dead 

load, or for load, regular 
dead load, tem- live load, tem- 
perature and perature and 

wind. wind. 

Lbs. per Sq. In. Lbs. per Sq. In. 



60.000 
30.000 



20,000 



73.000 



40.000 

*40.000-150i-5-r 

20,000 

35.000 



t25.000 



Wire: 

Main cables 

Suspenders 

Nickel Steel : 

Tension in stiffening trusses 

Compression in stiffening trusses 

Shear on rivets in stiffening trusses; field 

Bearing on rivets in stiffening trusses; field 

Structural Steel in Towers: 

Tension 

Compression *22.000-90 l^r *t27,000-100i-j-r 

Shear on shop rivets and bolts 13.000 16.000 

Bearing on shop rivets and bolts 25.000 30,000 

Structural Steel in Stiffening Trusses: 

Tension 20,000 24,000 

Compression *20,000-90 l^r *24,000-100 l^r 

Shear on shop rivets 13,000 

Bearing on shop rivets 25,000 

Structural Steel In Floor System of Roadway and 
Footways: 

Tension chords 

Shear on shop rivets, bolts and web-plate net section 
Bearing on shop rivets and bolts 

Structural Steel in Floor System for Railroad and 
Trolley Tracks : 

Tension chords 

Shear on shop rivets, bolts and web-plate net section 
Bearing on shop rivetsand bolts 

Structural Steel in Anchorages: 

Tension in eye-bars 

Bearing on diameter of pins 

Bending or outer fibre of pins 

Shear on pins 

High Carbon Steel: 

Tension In stiffeninc trusses 

Compression in stiffening trusses 

Note. — For Weights of Materials in Manhattan Bridge, see page 760. 



15.000 
10.000 
20.000 



10.000 

7.000 

14.000 

16,000 
22.000 
22,000 
12,000 



20.000 



35,000 
*35,000-135;-Hr 



*Where Z=length and r= least radius of gyration, both In Inches. 
tincluding secondary stresses. 



MANHATTAN BRIDGE SPECIFICATIONS. 



759 



\<-;f£,9^ 



K- -,0,09 







760 iZ.'^USPENSION BRIDGES. 

Approximate Weights op Materials in Manhattan Bridge. 

Anchorages. Towers. Cables. Main span. Side spans. Totals. 

Lbs. Lbs. Lbs. Lbs. Lbs. Lbs. 

Nickel steel 7,349,600 8,897,800 16,247,400 

Structural steel 1.335.600 21.333.800 30,200 10.602,600 10.447,200 43.749,400 

Wire 12,176,200 12.176.200 

Suspenders, etc 1.153,600 1.153,600 

Eye-bars 3.731,900 3.731.900 

Castings, steel 1,500 3,385.200 1.744.600 13.700 28.200 5,173,200 

Castings, Iron 18,500 189.100 54,500 7,600 24,600 294,300 

Pins, bolts, nuts, etc. 307,500 119,100 383,000 10,000 22.600 842,200 



Totals of steel .. . 5.395,000 25.027.200 15.542.000 17.983.500 19.420.400 83.368.200 

Concrete (cu. yds.) 930 930 

Bronze 400 12,000 2.100 4,200 18,700 

Zinc 25.200 25.200 

Lead 20,600 7,400 28,000 

Economic Considerations. — ^The approximate costs of materials, erected, 
in a suspension bridge of about 1200 to 1500-ft. spans designed for combined 
railway and highway traffic, are as follows: Steel in wire cables, 6 to 6.25 
cents per lb. (add about 4% to this for copper covering) ; riveted steel in 
center span, 4.25 to 4.5 cents per lb. (nickel steel, 6 to 6.25 cents) ; steel in 
towers, side spans and anchorage, 3.4 to 3.6 cents per lb. ; steel in viaduct 
spans, about 3 cents per lb.; anchorage masonry, $5.50 to $6.50 per cu. yd. 

Steel in caw^^V^wr spans, erected, about 4.5 cents per lb. (nickel steel, 
about 6 cents per lb.). 

EXCERPTS AND REFERENCES. 

Waterproof Wrapping for the Cables of the New East River Suspen- 
sion Bridge (By Wilhelm Hildenbrand. Eng. News, Nov. 13, 1902). — Illus- 
trated. Discussed by A. H. Sabin, in Eng. News, Nov. 20, 1902. 

The Williamsburg Bridge Across the East River at New York City 

(Eng. News, Nov. 17, 1903). — Illustrations of anchorages and details. 

A Rational Form of Stiffened Suspension Bridge (By Gustav Linden- 
thai. Trans. A. S. C. E., Vol. LV) .—Discussions by W. Hildenbrand, 
Joseph Mayer, R. S. Buck, W. W. Crehore, Theodore Cooper, C. C. Schnieder, 
Owsald Erlinghagen, H. W. Hodge, F. Schule, J. Melan, L. S. Moisseiff, 
A. Rieppel. 

The Monongahela River Suspension Bridge at Morgantown, W. Va. 

(By W. H. Boughton. Eng. News, April 18, 1907). — Illustration of saddle 
and top of tower. 

The Towers of the Manhattan Bridge Over the East River at New 
York City (Eng. News, April 16, 1908).— Illustrated. 

Report on the Manhattan Suspension Bridge at New York City (By Ralph 
Modjeski. Eng. News, Oct. 14, 1909). — Calculations, (a) Extract from 
specifications for superstructure, with table of unit stresses; (b) Derivation 
of formulas used in the calculation of stresses in the stiffening trusses, with 
moment and shear diagrams; (c) Method used in the calculation of stresses 
in the tower and cable, with formulas and diagrams; (d) Method used in the 
calculation of stresses in the lower floorbeams and lateral system, with formu- 
las and diagrams. Illustrated: Fig. 1 (not reproduced here) is a diagram of 
dead-loads and cable ordinates; showing (1) Panel load of susp. structure, 
(2) Total panel load, (3) Actual ordinates, in feet, (4) Ordinates to para- 
bolas, in feet. For discussions of calculations, see Eng. News, Mar. 3. 1910. 

Illustrations. 
Description. Eng. Rec. 

Machine for winding wire around main cables — Manhattan 

bridge July 31, '09 



44.— ARCHES. 

General Discussion. — An arch is a structure so designed that the loading, 
including the weight of the structure itself, produces a thrust at the abut- 
ments, in such a manner that the resultant horizontal reactions at those 
points tend to relieve, wholly or in part, the bending-moment effect on 
the span. 

The. Ideal Arch would naturally take the (inverted) form which the 
cables of a suspension bridge assume for the particular conditions of load- 
ing imposed; for then the horizontal reactions at the abutments will relieve 
the arch ring of any bending moment whatsoever, the stresses throughout 
being purely of axial compression. Either of the curves described in Sec. 43, 
Suspension Bridges, if inverted, will become ideal arches for the given load- 
ings. Such arches are called linear arches. The circular arch also is a linear 
arch when the resulting forces are radial, as in the case of a circular dam 
where the hydrostatic pressure is normal to the up-stream face. 

The Circular Arch as a Dam. — Let r, Fig. 1, be 
the radius in feet of the up-stream face of the dam; 
h, the depth in feet below the surface of the water 
to the level at which the pressure is to be considered ; 
p, the pressure in lbs. per sq. ft. at depth h; W, the ^ \ 

weight in lbs. of a cu. ft. of water. Then p = Wh; and ^ 

the tangential compressive stress at any point of the Fig. 1. 

circle for one foot vertical, and at depth h, is T = pr = Whr= 62.5 hr lbs. = R. 
If t is the thickness of the masonry in feet at depth h, the compressive stress 

T hr 

in lbs. per sq. ft. on the masonry is -- = 62.5 — . For overturning effect see 

Sec. 49, Dams. 




The Catenary. — Let Fig. 1 represent a vertical arch of uniform thick- 
ness t and supporting its own weight only; then in order to be a true linear 
arch its curvature should be that of a catenary instead of a circle. 

^ The Parabola.— Let Fig. 1 represent a vertical arch supporting, including 
weight of arch itself, a uniform horizontal load; then in order to be a true 
linear arch its curvature should be that of a parabola. 

The Transformed-Catenary Arch. — Let it be re- 
quired to design a masonry arch of 40 ft. span, 2 ft. 
in depth at the crown and 8 ft. at the springing line; 
and supporting a live load of 140 lbs. per sq. ft. ^^^ __ _ 
Assuming the weight w of masonry at 140 lbs. per cu. *^ * " '" 
ft., and the spandrel filling F to be solid at the same ^^^' ^' 

weight; and, further, reducing the live load to equivalent depth of masonry, 
namely, one foot, we have the outline diagram as shown in Fig. 2. The 
problem is to find the curve of the intrados (a transformed catenary) so 
that the line of resultant pressure shall trace the center line of the arch 
stones (practically) as shown by the middle dotted line. Note that the 
thinner the arch stones and the flatter the arch, the more nearly will this 
be true. 




m = 



Solution. — First, find the value of m in the equation* 
0.4342945 c 



com log 



(!-VS-') 



= 11. 34+ ; and substitute the values of a(= 2+1) 



* See also the equation of the catenary and transformed catenary in 
Sec. 43, Suspension Bridges. 



761 



762 U.— ARCHES. 

and m in the general equation *y= -^ i e^+ e ^7='7r( e^^^-i ) . Then: 

2 \ / 2 \ ein,/ 

For x= 0. 6. 10. 15. 20. 

y= 3.00 3.30 4.24 6.03 9.00 

Depth of masonry = 2.00 2.30 3.24 5.03 8.00 

The above calculation is very simple, requiring only a few minutes' 
_L 1 
time, remembering that log^°'=— log e, =0.4342945^11.34 = 0.03828 for 

fK 

this particular case; and log e^ = 0.03828 ic. Also note that e~'^is the 

X I X 

reciprocal of ^ m and hence log e~ m = 1 — log e m . 

The Horizontal Thrust H at any point of the arch for each foot in 
length (perpendicular to face) is if = i£;w2= 140X (11.34)2= 18 000 lbs.; and 
at crown of arch=9000 lbs. per sq. ft. = 62.5 lbs. per sq. in. 

The Vertical Shear Px, at any point distant x from the center of arch, is 



H 



Px= — \/y^ — a^ = wm \/y'^ — a^\ and at the abutment the shear is equal to 
m 

the vertical reaction P = — v'/i2_a2=^^ V/j2-a2 = 140X 11.34X V81-9 = 
m 

13471 lbs. — per lin. ft. (axial) of bridge. 

A 

The Area -»- of Half the catenary, between directrix and soffit, is 

A p 1 Q471 

:^ = w\//i2_a2=-=-i^ =96.22 sq. ft. 
2 w 140 

The area of face of masonry = 96.22- 20= 76.22 sq. ft. 

The depth of arch stones is assumed to be 24 inches, but with a uniform 

live load as above it may sa fely be m uch less. The tangential thrust at the 

springing line is equal to VH'^ + P^ = 22482 lbs. or only 78 lbs. per sq. in. 

The tangent of the angle which this thrust makes with the horizontal at 
1 p 

that point is tan ^i = —\//t2-a2 = — = 0.7484; therefore ^i = 36°-49', the 
m ri 

thrust being tangent to the soffit of the arch. At any point distant x from 

center of span, tan^ = — x/'y'^—a^. With a variable live load on the arch 
m 

the resultant line of pressure through the middle of the arch stones will 
change its form and deviate from a true central position. The amount of 

1 24 
deviation in the present instance cannot be more than-2-X-^= 4 inches, or 

one-half the "middle third," without producing tension in the masonry 
joints, which is not allowable. (See Masonry Arches, following.) 

The live load on the span is assimied at 140 lbs. per sq. ft. of floor, 
equivalent to a depth of one foot of masonry. Now it is plainly evident 
that the curve of the intrados of the arch (Fig. 2) will not be affected by 
any relative change of live load to dead load provided the ''loading contour" 
remains the same. Thus, we may increase the uniform live load to 280 lbs. 
per sq. ft. by decreasing the depth of masonry one foot, making the depth 
of arch stones 12 ins. instead of 24 inches. This, of course, would double 
the thrust on the arch stones per sq. in., making the horizontal thrust at 
crown 125 lbs. per sq. in., and the tangential thrust at springing 156 lbs. 
per sq. in. The line of resultant pressure would not change its form but 
would now conform more nearly with the (new) middle line of the arch stones. 
They will never exactly coincide unless the arch ring is reduced to a thin 
plate. 

The resultant line of pressure as previously determined will evidently 
be affected by changes in the loading contour: (1) By raising the loading 
contour, that is, increasing the uniform load per sq. ft.; (2) by lowering the 
loading contour, that is, decreasing the uniform load; (3) by considering 
the uniform load at middle of span only; (4) by considering the uniform 

* Log ^=0.4342945. 



TYPES, PARTS AND KINDS OF ARCHES. 



763 



load on one side of the span only; (5) by placing concentrated loads on the 
span either at center of arch or over the haunches on either side, it is 
these various conditions of loading, partly, at least, that render the trans- 
formed catenary a curve of theory and not of practice, m the masonry arch. 
(See Masonry Arches, following.) 




MASONRY ARCHES. 

Parts of the Arch (Fig. S).— Crown (c), the highest part of the 
arch. Sofflt, the inner or concave surface of the arch. Intrados, curve of 
intersection of soffit with a plane perpen- 
dicular to axis of arch. The kind of 
curve determines the kind of arch, 
whether segmental (circular), elliptical, 
etc. * Radius (oi segmented arch), radius 
of the intrados, with center at c. Axis, 
a line joining the centers c. Face or 
Head of arch, the end walls. Arch-sheet- 
ing, all of the arch masonry, cylindrical 
shell or voussoirs, not including the face 
voussoirs. Back of arch, the upper or 
convex surface of the arch-sheeting. 
Extrados, curve of intersection of back 
of arch with a plane perpendicular to axis; 
or, curve touching the outer extremities 
of the radial joints between the voussoirs. ^ ^S- o. 

Voussoirs (or arch-stones) , the stones radiating from the soffit, and composing 
the arch proper; the arch masonry. Ring Stones, the voussoirs or arch-stones 
which show at face of arch. Keystone, the middle ring stone. Springing line, 
the intersection of intrados with pier or abutment; the lower edge of the 
intrados. Skewhack, (a) the inclined surface or joint from which an arch 
springs; (h) the stone itself which supports the arch. Springer, the lowest 
voussoir or arch stone. Haunch, that part of the arch about midway 
between crown and skewback. Span, the perpendicular distance between 
springing lines. Rise, the height of intrados at crown above plane of 
springing lines. Abutment, the skewback and masonry supporting it. 
Spandrel walls, the walls which rise above the arch-stones or voussoirs. 
These walls stiffen the arch-sheeting and have considerable stiffness them- 
selves. Interior spandrel walls are placed under railway and street -car 
tracks, or wherever heavy, concentrated loads are liable to occur. Longi- 
tudinal walls, between the spandrel walls, give added stiffness, and are used 
for long spans. Spandrel, the space between spandrel walls; or the space 
between back of arch and roadway. Spandrel-filling, the filling which 
is placed in the spandril, composed of earth or masonry, or both. The 
spandrel filling may support the superimposed loads; or, they may be 
supported by girders resting on the walls above the arch. Backing, masonry 
carried outside the arch-sheeting and above the skewbacks. Siring Course^ 
a course of voussoirs extending from end to end of arch. Coursing Joint, 
a (continuous) joint between string courses, extending from end to end 
of arch. Heading Joint, a (broken) joint at right angle with axis of arch. 
King Course, the voussoirs between two adjacent heading joints. 

Kinds of Arches (from shape of intrados). — Circular arch, intrados is 
part of a circle; includes the two following. Semi-circular or " full-centered " 
arch, intrados is a semi-circle. Segmental arch, intrados is less than a semi- 
circle. This is not as strong as the preceding, but it is particularly adapted 
to long spans, having less tendency to " bulge " 
upward, or to become " depressed " at the haunches, 
when subjected to live concentrated loads. For 
window openings, make rise -r- span = vers 30°. 
Gothic or " pointed " arch (Fig. 4), intrados is two 
equal arcs with span as radii; sometimes defined as 
two equal arcs with radius greater than half -span. «-^-— ^/TOff-- 

Used particularly in Architecture, as is also the fol- 
lowing. Fig. 4. 




The " radius " of any arch is the radius of intrados at crown. 



764 



4A.^ARCHES. 





Fig. 7. 
or arches cross each other, 
around vertical axis through center of keystone. 



Tudor arch, modified Gothic with intrados compounded. The " four- 
centered " Tudor arch is shown in" Fig. 5. The 
radii are i and 1| of the span, respectively. Ellip- 
tical arch, intrados is part of an ellipse. The major 
axis may be either horizontal or vertical. (Any arch 
is said to be surbased when the rise is less than the 
half-span; and surmounted when the rise is greater 
than the half-span.) Oval or ** basket-handle " arch, 
intrados composed of arcs or circles approaching the 
elliptic arc in form. The "three-centered " oval is 
very common. Parabolic arch, intrados is parabolic. 
Catenarian arch, intrados is catenarian. 

Further Classification of Arches. — Right arch, one 

whose ends or faces are perpendicular to axis of 

arch. Skew arch^ an arch whose ends or faces are 

oblique to axis of arch. There are two kinds: 
First (Fig. 6), the skew arch proper, with 
smooth cylindrical soffit, spiral joints and un- 
broken axis; Second (Fig. 7), the oblique arch 
being made up of a number of short, right arches 
called ribs, which are offset transversely and 
successively in one direction. Flat arch, flat soffit 
with wedge shaped voussoirs; used for doors and 
windows. Vault, surface generated by the in- 
trados of an arch moving in a straight line on 
the springing lines. Cloistered vault, formed 
when two vaults or arches meet; as " vaulted 
ceiling." Groined vault, formed when two vaults 
Dom,e, formed by right section of arch revolving 
(Also, see Glossary.) 

Brick Arches. — The three principal methods of bonding brick arches 
are illustrated in Fig. 8, and described as follows: 

Rowlock bond (R). — All the bricks are laid as stretchers in concen- 
tric rings. The average thickness of R 

joint is less under this construction, ^.<<\''''ljj£njlr'^'"-^ x, 

and hence more bricks and less mor- .C<\\\ iVi ii/tt 7>A« 

tar are required. It is cheaper to 
lay, and is employed largely in tunnel 
work, sewers, etc., where architec- 
tural effect is not important. 

Header and Stretcher bond (H. ^ \ ^^2- °' 

& 5.). — Radial joints continuous, and bricks laid as headers and stretchers. 
The average thickness of joint is greater under this method, and hence 
more mortar and less (the least) number of bricks are required. It is 
more pleasing to the eye than the rowlock bond, and is used largely in 
the fronts of buildings. 

Block in Course bond (B. in C). — ^This bond aims to attain with brick 
the wedge-shaped voussoirs of the stone arch. ^ That is, the bricks are 
grouped in sections bounded by continuous radial joints. Adjacent sec- 
tions are of different bond, any good bond being permissible. Bonds with 
continuous radial joint are usually made narrower than those bonds where 
radial joints are broken. 

The Masonry Arch Is a Statically Indeterminate Structure, and its solu- 
tion is based on various assumptions more or less approximate. These will 
be considered in the material order of design. To begin with, we have to 
assume the finished arch and then examine it for stability (resistance to 
deformation), strength (resistance to crushing), economy, etc. 

Curve of Intrados. — ^The form of curve selected for the intrados will 
naturally affect the thickness of the arch ring and the economy of the 
whole structure. We have seen (Fig. 2) that the transformed catenary is the 
ideal curve of soffit where the loading contour is a horizontal line, and this 
curve would undoubtedly be used were the conditions of loading constant 
as shown. It will be noted that in the transformed catenary the curve is 
quite flat at the crown and gradually becomes sharper toward the springing. 
The ellipse is a curve that can be made to approximate the modified catenary, 





MASONRY ARCH— ARCH RING. 765 

and is more simple than the latter. It may be used in the form of the 
elliptic arc or of the semi-ellipse*. If the latter is used the major axis be- 
comes the span of the arch, and the semi-minor axis the rise. One of the 
largest elliptic arches in the world is the ("New") London bridge over the 
Thames, England, for street traffic. It is a granite arch of 152 ft. span, 
29.5 ft. rise, and with a crown radius of 162 ft. The thickness at crown is 
4.75 ft., and at springing, 9.00 ft. The basket-handle or many-centered arc is 
a modification of the ellipse, and has certain apparent advantages over 
the latter when applied to an arch of cut stone. To this class belongs 
the recently constructed stone arch bridge at Plauen, Saxony, with a 
clear span of 295.2 ft., the longest ever built. The total rise is 59 ft., and 
the thickness of arch stones at crown is 4.92 ft. It is 5-centered, with 
radii of 344.4 ft., 191.8 ft., and 101.7 ft. The circular form, using the arc 
of a circle, is the most frequently employed on account of its simplicity, in 
the drafting room, for the stone cutter, and in laying out the centers for 
erection in the field. The "segmental" f arch subtends a central angle at 
60° and is mainly employed in building construction. The semi-circular 
arch subtends a central angle of 180° and is particularly adapted to 
openings through high embankments where the rise of the arch is 
not limited, and where the haunches of the arch are called upon 
to resist inclined forces due to earth pressure. The largest semi- 
circular arch known to the writer is the Ballochmyle, over the Ayre, Scotland. 
It is a railway arch of 181 ft. span. The thickness of ring at crown is 4.5 ft., 
and at springing 6 ft. The parabola is sometimes used for fiat, concrete 
arches, but is not particularly economical for a masonry arch because of the 
distribution of the loads; that is, a parabola would call for a uniformly 
distributed load over the structure, whereas in a masonry arch this condi- 
tion cannot even be approximated. In conclusion, we may state that, 
generally speaking, the basket-handle is doubtless the best form to use for 
long spans, of cut-stone work — 5-centered for extremely long spans, and 
3-centered for medium long spans, say up to 200 feet. For concrete arches 
the elliptic arc may be used for any length of span; the semi-ellipse, for any 
length of span where the rise is considerable; also for short spans in general; 
the circular arc, for moderate and short spans, especially of cut-stone. 

The Parabolic Oval is a new arch curve designed by Chas. Worthington, 
M. Am. Soc. C. E., and is described very elaborately in Eng. News of April 
15, 1909, accompanied by numerous illustrations. 

Thickness of Arch Ring. — ^Theoretically, the arch stones should have 
greater thickness at the springing than at the crown, because of the greater 
tangential thrust. For practical reasons, however, the thickness of tha 
arch ring is usually made uniform throughout for spans up to 75 ft. For 
spans greater than 75 ft. the thickness 'at springing is usually from 25 to 
100 per cent greater than at crown J, depending upon length of span, rise, etc. 

Rules for assuming thickness of arch stones. — Having selected the most 
probable curve for the intrados the next step is to approximate as closely 
as possible the thickness of the arch ring at crown and at springing. No 




* The radius of curvature r for any point p (whose coordi- c 
nates are x and y) of the ellipse, whose semi-major axis is a,^ 
and semi-minor axis b (Fig. 9) , is 

(g^ 4-&%2)2^ Pqj. ^j^g semi-ellipse where b = the rise, and 
a^b^ 
2a the span, the radius of curvature at crown C is (for Fig. 9. 

^ , • a2 (half span) 2 j ^, ,. <. ^ , . . 

^=0, y = o),r = -r-= : ; and the radius of curvature at spnngmg 

c • /r A \ ^^ (rise)2 

5 IS (for 3^=0, x.= a), r = — 



a half span 

t The term "segmental" is here used in its restricted meaning* but in 
its broader sense any circular arch is a segmental arch. 

t A notable exception to this rule is the railway arch bridge of 4 spans 
over the Etherow river, England. The spans are each 100 ft.; rise, 25 ft.; 
and uniform thickness of ring, 4 ft. Such a depth at crown, however, is exces- 
sive. 



766 



iL— ARCHES. 



absolute, simple rules can be devised, but the writer presents the following 
as giving very close results for first-class concrete and cut stone work: 
Thickness ic at crown (all dimensions in feet) : 

For highway bridges te at crown =-. /O.Ol span (— : 1-3) -1-0.15 (1) 

For high H. W. em- ) / Tsoan \~ 

bankments, or >tc at crown =^/0.01 span (—. — +4) -f-0.20 (2) 

For railroad bridges ) \ ^ ^^^ ' 

^°'^^tSs:"'- }'' -t-°- =y0.01spa„(^f+5)+0.25 (3) 

Thickness is at springing (all dimensions in feet) : 
For all cases, ^s at springing = ^c [1-f 0.002 (span -f- 2 X rise)] (4) 



1. — Crown Thickness t^ (Feet) for Masonry Arches. 
[Calculated from Formulas (1), (2) and (3).] 





Ris 
Sp 


3-T- 

an 












Span, 


in Feet. 


















i 




2 


5 


10 


15 


25 


40 


60 


80 


100 


125 


150 


175 


200 


240 


280 


320 




1-10 


.10 


.66 


.96 


1.29 


1.55 


1.95 


2.43 


2.94 


3.37 


3.76 


4.18 


4.57 


4.92 


5.25 


5.74 


6.18 


6.60 


& 


1-8 


.125 


.62 


.89 


1.20 


1.43 


1.81 


2.25 


2.72 


3.12 


3.47 


3.86 


4.21 


4.54 


4.84 


5.29 


5.70 


6.08 


1-7 


.143 


.60 


.86 


1.15 


1.37 


1.73 


2.15 


2.60 


2.98 


3.31 


3.69 


4.02 


4.33 


4.62 


5.05 


5.44 


5.81 


11 


l-6f 


.15 


.59 


.85 


1.13 


1.35 


1.70 


2.12 


2.56 


2.93 


3.26 


3.63 


3.96 


4.26 


4.55 


4.97 


5.35 


5.71 


1-6 


.167 


.57 


.82 


1.10 


1.31 


1.65 


2.05 


2.47 


2.83 


3.15 


3.50 


3.82 


4.12 


4.39 


4.80 


5.17 


5.52 




1-5 


.20 


.55 


.78 


1.04 


1.25 


1.56 


1.94 


2.34 


2.68 


2.98 


3.31 


3.61 


3.89 


4.15 


4.53 


4.88 


5.21 


1-4 


.25 


.52 


.74 


.99 


1.17 


1.47 


1.82 


2.20 


2.52 


2.80 


3.11 


3.39 


3.65 


3.89 


4.25 


4.58 


4.88 


o^ 


1-3 


.333 


.50 


.70 


.92 


1.10 


1.37 


1.70 


2.05 


2.34 


2.60 


2.89 


3.15 


3.39 


3.61 


3.94 


4.25 


4.53 


P=4 


1-2* 


.40 


.48 


.67 


.89 


1.06 


1.32 


1.63 


1.97 


2.25 


2.50 


2.77 


3.02 


3.25 


3.47 


3.78 


4.07 


4.35 




1-2 


.50 


.47 


.65 


.86 


1.02 


1.27 


1.56 


1.88 


2.15 


2.39 


2.65 


2.89 


3.11 


3.31 


3.61 


3.89 


4.15 


2 ^-o 


1-10 


.10 


.73 


1.04 


1.38 


1.65 


2.07 


2.57 


3.10 


3.55 


3.94 


4.38 


4.78 


5.15 


5.49 


6.00 


6.46 


6.89 


1-8 


.125 


.69 


.97 


1.30 


1.54 


1.93 


2.39 


2.88 


3.30 


3.66 


4.07 


4.44 


4.78 


5.10 


5.57 


6.00 


6.40 


1-7 


.143 


.67 


.94 


1.25 


1.48 


1.86 


2. -30 


2.77 


3.17 


3.52 


3.91 


4.26 


4.59 


4.89 


5.34 


5.75 


6.13 




1-6^ 


.15 


.66 


.93 


1.23 


1.46 


1.83 


2.27 


2.73 


3.12 


3.47 


3.85 


4.20 


4.52 


4.82 


5.26 


5.66 


6.04 


1-6 


.167 


.65 


.91 


1.20 


1.42 


1.78 


2.20 


2.65 


3.03 


3.36 


3.74 


4.07 


4.38 


4.67 


5.10 


5.49 


5.86 


1-5 


.20 


.62 


.87 


1.15 


1.36 


1.70 


2.10 


2.52 


2.88 


3.20 


3.55 


3.87 


4.17 


4.44 


4.85 


5.22 


5.57 


orR. 

and 

ighwa 


1-4 


.25 


.60 


.83 


1.09 


1.30 


1.61 


1.99 


2.39 


2.73 


3.03 


3.36 


3.66 


3.94 


4.20 


4.58 


4.93 


5.26 


1-3 


.333 


.57 


.79 


1.04 


1.22 


1.52 


1.87 


2.25 


2.57 


2.85 


3.16 


3.44 


3.70 


3.94 


4.30 


4.63 


4.93 


1-2* 


.40 


.56 


.77 


1.01 


1.19 


1.47 


1.81 


2.17 


2.48 


2.75 


3.05 


3.32 


3.57 


3.81 


4.15 


4.47 


4.76 


^ W 


1-2 


.50 


.55 


.75 


.97 


1.15 


1.42 


1.75 


2.10 


2.39 


2.65 


2.94 


3.20 


3.44 


3.66 


3.99 


4.30 


4.58 




1-10 


.10 


.80 


1.12 


1.47 


1.75 


2.19 


2.70 


3.25 


3.71 


4.12 


4.58 


4.99 


5.37 


5.73 


6.25 


6.73 


7.18 




1-8 


.125 


.76 


1.06 


1.39 


1.65 


2.05 


2.53 


3.04 


3.47 


3.86 


4.28 


4.67 


5.02 


5.35 


5.84 


6.28 


6.70 


1-7 


.143 


.74 


1.02 


1.35 


1.59 


1.98 


2.44 


2.93 


3.35 


3.71 


4.12 


4.49 


4.83 


5.15 


5.62 


6.05 


6.45 


5i 


l-6t 


.15 


.73 


1.01 


1.33 


1.57 


1.96 


2.41 


2.90 


3.30 


3.67 


4.07 


4.43 


4.77 


5.08 


5.54 


5.97 


6.36 


1-6 


.167 


.72 


.99 


1.30 


1.53 


1.91 


2.35 


2.82 


3.22 


3.57 


3.96 


4.31 


4.64 


4.94 


5.39 


5.80 


6.18 


•Sfd 


1-5 


.20 


.70 


.96 


1.25 


1.47 


1.83 


2.25 


2.70 


3.08 


3.41 


3.79 


4.12 


4.43 


4.72 


5.15 


5.54 


5.91 


W_| 


1-4 


.25 


.67 


.92 


1.20 


1.41 


1.75 


5.15 


2.57 


2.93 


3.25 


3.60 


3.92 


4.22 


4.49 


4.90 


5.27 


5.62 


©a 


1-3 


.333 


.65 


.88 


1.14 


1.35 


1.66 


2.04 


2.44 


2.78 


3.08 


3.41 


3.71 


3.99 


4.25 


4.63 


4.98 


5.31 


1-2* 


.40 


.64 


.86 


1.12 


1.31 


1.62 


1.98 


2.37 


2.70 


2.99 


3.31 


3.60 


3.87 


4.12 


4.49 


4.83 


5.15 




1-2 


.50 


.62 


.84 


1.09 


1.27 


1.57 


1.92 


2.30 


2.62 


2.90 


3.21 


3.49 


3.75 


3.99 


4.35 


4.68 


4.98 



Remarks. — ^The above table is for solid arch rings. Where arch rings 
are ribbed the thickness or depth of ribs should be increased. For spans 
under 75 ft. an arch ring of uniform thickness may be used by increasing 
the crown thickness obtained from above table by from 5 to 20%. See 
Table 2. 

Tables 5 and 6, pages 774 to 781, contain various data from arches that 
have been built. 



MASONRY ARCH— THICKNESS OF RING, 



767 



2. — ^Table for Obtaining Thickness ts of Arch Stones at 

Springing. 
Values of [1 + 0.002 (span + 2 X rise)] in Equation (4). 
(Thickness at springing = thickness at crown X tabular values below.) 



Rise-J- 
Span. 


Span, in Feet. 


Rise^ 
Span. 


Span, in Feet. 


£ 


Q 


75 


100 


150 


200 


250 


300 


6 


d 


75 


100 


150 


200 


250 


300 


1-10 

1-8 
1-7 
l-6f 
1-6 


.10 
.125 
.143 
.15 

.167 


1.18 
1.19 
1.19 
1.20 
1.20 


1.24 
1.25 
1.26 
1.26 
1.27 


1.36 
1.38 
1.39 
1.39 
1.40 


1.48 
1.50 
1.51 
1.52 
1.53 


1.60 
1.63 
1.64 
1.65 
1.67 


1.72 
1.75 
1.77 

1.78 
1.80 


1-5 
1-4 
1-3 

1-2^ 
1-2 


.20 

.25 

.333 

,40 

.50 


1.21 
1.23 
1.25 
1.27 
1.30 


1.28 
1.30 
1.33 
1.36 
1.40 


1.42 
1.45 
1.50 
1.54 
1.60 


1.56 
1.60 
1.66 
1.72 
1.80 


1.70 
1.75 
1.83 
1.90 
2.00 


1.84 
1.90 
2.00 
2.08 
2.20 



Note from Tables 1 and 2 that for any given span the actual thickness 
of arch ring at springing remains nearly constant as the rise approaches the 
half span. 

Problem. — In designing a railroad masonry arch of 200 ft. span and 
40 ft. rise, what thickness of arch ring shall be assumed, tentatively, at 
crown? Also at springing? 

Solution. — At crown, 4.44 ft. (from Table 1). At springing, 4.44X1.56 
(from Table 2) = 6.93 ft. Hence use 4.5 and 7. respectively, for trial. 

Forces Acting on a Masonry Arch. — ^These may be classified as the 
"outer" and the "inner" forces, the same as in any other structure. The 
outer forces include (1) the moving loads due to trains, vehicles, etc.; (2) the 
mixed loads, due to weight of track, roadway, embankment, spandrel fill- 
ing, etc.; (3) the dead loads due to weight of arch ring itself; (4) the reac- 
tions. The inner forces comprise the stresses in the arch ring, usually 
calculated at points of imaginary joints. These stresses are examined for 
compression, shear, and possible tension. The compression is determined 
from the line of resultant pressure through the arch ring, which also reveals 
possible tension. The shear at any point of the arch is equal to the algebraic 
simi of the outer forces at the left of that point, taken in the direction of 
the cutting plane. Furthermore, we know that the algebraic sum of the 
outer forces, either of the whole structure or of that portion at the left of 
any given cutting plane, is equal to zero. The same is true of the inner 
forces; also of the outer and inner forces combined. Hence, an inner force 
in a complete structure may be determined by assuming it to be an outer 
force in maintaining equilibrium in a portion of the structure. 

Direction of Outer Forces. — Let Fig. 10 represent a masonry arch 
with arch ring drawn to scale. Let MM represent 
the mixed static loading above the arch ring due to 
spandril filling, roadway and track, and with depth 
reduced to equivalent masonry loading, so that d, in 
feet, at any point distant x from left abutment, will 
represent the intensity of the total static loading at ^ 
that point in terms of the weight of masonry per 
cubic foot. If, now, we add the moving load m 
for any particular case of loading', say on the right half 
duce the same to equivalent depth of masonry, we obtain the* loading con- 
tour for calculating the arch for that particular case of loading. 

Vertical Loading. — In fixing the loading contour as above we have 
assumed all loading to be vertical, which, although not quite correct, is the 
usual assiimption. Note that the line of resultant pressure (dotted) through 
the arCh ring approaches the upper edge on the more heavily loaded half, 
and the lower edge on the less heavily loaded half, indicating a tendency to 
cause tension in joints at t, which must be guarded against, as well as undue 
compression at c. The thrust T at crown will not be horizontal but will 
incline upward to the right because of the heavier loading on that side, 




Fig. 10. 
of span, and 



768 



U.— ARCHES. 




Fig. 11. 




Fig. 12. 



producing positive vertical shear V at crown. The horizontal component H of 

thrust T is, H = \/T^— y2 = horizontal component of Ri or of R2, also. 

Inclined Loading. — ^The loading at the crown may^ always be assumed 
as vertical; also at the haunches if the spandril filling is of rough masonry, 
or of earth or clay well tamped and kept thoroughly drained. But if the 
filling is of loose earth, sand or gravel, and especially if liable to become 
saturated with rain, from poor drainage, the material will have a tendency 
to slide along a low angle of repose, with resultant inclined loading at the 
haunches, as FFF, Fig. 10. Such inclined loading will produce a different 
line of resultant pressure in the arch ring than will vertical loading. 

Three=Hinged Masonry Arch. — (The following theories are approximate 
only.) The three hinges, a f k (Fig. 11), placed 
at crown and at springing, are necessarily points of 
zero moment and hence the line of resultant pres- 
sure must pass through those points. The method 
of procediure is as follows: Divide the arch ring into 
imaginary voussoirs with radial joints. From the 
upper extremities of these joints draw vertical lines 
up to the loading contour L. C Find the area and 
the center- of gravity of each of these figures, bounded ' 
by full lines, between the L. C. and the soffit of arch. 
Then the area multiplied by the weight of the masonry 
per cu. ft. will give the vertical loads Pi, P2, P3, etc., passing 
through the centers of gravity of the figures. The reactions 
i?i and i?2» as well as their horizontal and vertical compo- 
nents, may be found graphically (see Fig. 17, Sec. 16, page 
315) by treating each load separately or by using one or 
more resultant loads. With H, Vu V2, and the loads Pi, 
P2, etc., the force polygon (Fig. 12) maybe constructed, and 
the equilibrium polygon shown in Fig. 11 may be drawn; 
thus, draw Ri through a, parallel to Ri of the force poly- 
gon, to meet Pi produced; from this intersection draw 1' 
parallel to 1 to meet P2 produced, etc. If the work is done carefully 
the equilibrium polygon will pass through the hinges / and k. See also 
Fig. 14. 

The Line of Resistance or line of resultant pressure (not shown in Fig. 11) 

may be drawn by connecting the points a b c d—f k 

where the chords of the equilibrium polygon, as shown, in- 
tersect the masonry joints. But it must be remembered 
that if, in the equilibrium polygon, any point as p due to a 
load as P' falls outside the area from which the loading is 
derived, then the chords of the equilibrium polygon must 
be produced to meet the joint and give position to points 
on the line of resistance. Thus, in Fig. 13 the line of resistance 
will pass through ^c instead of through be. 

It is to be noted that the pressure at each of the joints may be obtained 
by scaling the rays of the force polygon, using the proper scale; thus, the 
pressure at c, intersected by 2' (Fig. 11), is obtained by scaling ray 2; d, by 
scaling ray 3; /, by scaling ray 5. The force polygon also gives the vertical 
shear V at crown. As the loading is all vertical the value of H (horizontal 
thrust) is constant throughout the arch. 

Three-hinged masonry arches are usually constructed of concrete or 
reinforced concrete. The hinges, at crown and springing, usually consist of 
steel shoes with pin bearings; sometimes lead sheets are used. 

The Criterion of Stability of a masonry arch is whether the true line of 
resistance for any case of assimied loading lies wholly within the middle 
third. If it does, the arch is stable for that loading. Now, while we can 
draw practically a true line of resistance for an arch with three hinges 
(Fig. 11), it is impossible to draw a true line of resistance for an ordinary 
arch with no hinges, and in order to determine its stability we have to make 
certain assumptions. 

Ordinary Masonry Arch — No Hinges. — Dr. Winklers' theory for the line 
of resistance of an arch is as follows: "For an arch of constant cross-sfection, 
that line of resistance is approximately the true one which lies nearest to the 
axis of the arch ring, as determined by the method of least squares." Hence, 
for all practical purposes, if a line of resistance can be drawn within any 
portion of the arch ring, say the middle third, the true line of resistance 




Fig. 13. 



MASONRY ARCH— CALCULATIONS, 



769 




Fig. 14. 




will lie within that portion.* Furthermore, in the clear language of Prof. 
Lanza'.f "An arch and its load being given, a line of resistance can always 
be made to pass through any twoX given points; hence, if any two points of 
a line of resistance are given, the line is determined." In practice, we 
assume two points within the portion of the arch ring, say the middle third, 
and draw a line of resistance passing through them. If this line of resist- 
ance lies wholly within the portion of the ring, the arch is stable; if not, 
two points are again selected (one of which may remain unchanged) and a 
new line of resistance is drawn, etc., until it is determined whether a line of 
resistance can be drawn within the given portion of the arch ring. If such 
a line cannot be drawn, the thickness of such ring is increased, or the 
curve of the arch is rnodified , or both. 

Symmetrical Loading. — Let Fig. 14 represent one half of a symmetrically 
loaded arch; Pi, P2, P3, P4 and P5, the loads on that 
half; and P, the resultant of those loads. It is re- 
quired to determine whether the arch is stable with 
reference to the middle third, which is bounded by 
dashes. Assume two points a and 6, the former at 
the lower edge, and the latter at the upper edge, of 
the middle third; and pass a line of resistance through 
these points. The thrust H Sit b must be horizontal 
as the arch is symmetrical in form and loading, and 
its value may be determined by taking moments 

about a; thus, Hh = Pd, or H =P-j-. Now draw the force 

polygon. Fig. 15, laying off the value of H thus obtained, 
horizontally, from the foot of the load line, to the point O; 
and draw the rays R, 1, 2, 3, etc. Now draw the equilibrium 
polygon shown dotted in Fig. 14: producing H to P5; then 
chord 4', parallel to ray 4, to P4; 5' parallel to 5; etc. The - ' ^ 
last chord is i?i, from Pi, passing through the point a. -nv- le 
Points on the line of resistance (not shown) will be at ^ ^S- 15. 
intersection of the chords of the equilibrium polygon, thus found, with the 
masonry joints; and by inspection we see that the line of resistance, if 
drawn, will pass outside the middle third of the first three joints to the left 
of b. This, however, does not prove that the arch is unstable, for, by 
another trial, we select bi instead of b, and readily obtain a line of resistance 
lying wholly within the middle third and passing through a bi. The new 
force polygon will have the pole Oi instead of O, and distant Hi horizontally 

from the foot of the load line, the value of Hi being equal to P7-. A mere 

hi 

glance at Fig. 14 will indicate that the true line of resistance, although it 

has not been determined definitely, will lie nearer the axis of the arch and 

pierce the springing and crown joints well within the middle third instead 

of at the edge as indicated by points a and bi. 

U nsym,m.etrical Loading. — If we analyze the preceding case of sym- 
metrical loading, critically, we cannot fail to see that we have really assumed 
the whole arch, of which Fig. 14 represents one-half, to be 3-hinged — one 
hinge at the crown and one at each springing; that we have taken the 
liberty of assuming these hinges to be at any part of the joint within the 
selected middle portion of the ring; and that, because of the symmetry, it 
has been necessary to consider one-half the arch only. 

The present case of unsymmetrical loading calls for but little special 
treatment when viewed from the above stand- 
point. Let Fig. 16 be the arch under consideration, 
with loads Pi to Pio derived from a loading contour 
(not shown in the Fig.). Assume three points within 
the given middle portion of the ring to act as hinges 
for an assumed 3-hinged arch, and calculate the arch^' 
as such by any method which will give a line of re- 
sistance passing through the three selected points, 
a.sab c. If such a line of resistance lies wholly within Fig. 16. 

* Dr. Winkler himself claims that this is not strictly true, but it is ac- 
cepted by many noted authorities. 

t See Applied Mechanics, by Gataeno Lanza, page 804. 
^ % Note that in a 3-hinged arch, three points are given because they are 
points of zero moment absolutely determ,ined. 




770 



U.— ARCHES. 




Fig. 17. 



the selected middle portion of the arch ring the arch is considered stable; 
if not, select three points again until, by repeated trial, it is determined 
whether or not the arch is stable, as explained for Fig. 14. Two methods will 
be shown for drawing the equilbritim polygon, from which the line of resist- 
ance is determined. 

First Method. — ^Time will be saved by a proper selection of points to be 
assumed for the hinges. In Fig. 16 where the heavier loading is on the right 
half of span the hinge c on that end is assumed at the inner edge of the 
middle portion; on the left end, carrying the lighter load, 
the hinge a is assumed at the outer edge of the middle 
portion; while at center of span the hinge b is assumed at 
the middle of the vertical joint. ^ Draw the equilibriimi 
polygon passing through those points, a b c, treating the 
arch as 3-hinged, as explained for Fig. 11. Examine for sta- 
bility. 

Second Method. — Select the hinges as above (Fig. 16). 
Lay off the load line in the force polygon. Fig. 17. Choose 
any pole as O, and draw the rays (dotted) of the force poly- 
gon. Construct the equilibrium polygon (dotted)in Fig. 16, be- 
ginning at c and ending at a'. Draw the trial closing line a' c. 
Draw OM (Fig. 17) parallel to aV to meet the load line at M. From M 

draw MOi parallel to the true closing line ca. Make the distance MOi = k-j- , 

in which fe = the horizontal distance to the load line. Then will point Oi be 
the true pole of the force polygon. Draw the rays (full) from Oi, and con- 
struct the equilibrium polygon (full) in Fig. 16, beginning at c. It will be 
found that the chords will pass through the hinges b and a. Exajnine for 
stability. If unstable select new points for hinges, and proceed as before. 

The "middle third" of the arch ring is usually selected as the "middle 
portion" within which to draw the line of resistance because, theoretically, 
when the line of resistance passes within the middle third there can be no 
tension in any part of the masonry, nor opening of any of the joints. If the 
arch ring is stiffened greatly by solid spandril walls the criterion of stability 
may not require that the line of resistance be confined within the middle 
third, but, say, within the middle half. 

The cases of loading for an arch are: (1) dead load only; (2) full load, 
including live load over whole span; (3) including live load over one-half 
of span; (4) including live load in any position; (5) including concentrated 
loads. It must not be presumed however that for all these cases of loading 
it is necessary to assume the hinges in any one fixed position. 



Centers for Arches. 

Definition. — An arch center is a framework for supporting an arch 
during construction, and hence of temporary character. It is so designed 
that it can be removed readily after the completion of the arch; and in the 
removal it has to be lowered somewhat in order to give sufficient clearance 
at the soffit of the arch, as the latter settles and becomes self-supporting. 

Parts of the Arch Center. — Fig. 18 shows two half -sections of arch 
centers with an end view of one of the frames. One half -section illustrates 




Fig. 18. 
the unbraced rib which supports the sheeting or lagging on which the arch 
atones are laid. The other half-section is a trussed frame, suitable for longer 



CENTERS FOR MASONRY ARCHES, 771 

spans. When trussed, the rib is called the ftac^ piece. The truss shown in Fig. 
18 is the Kingpost, the inclined traces transmitting the stresses from the 
haunches directly to the vertical post, and then down the inclined struts 
to ends of span. Various forms of trusses are used for the frames, depend- 
ing upon length of span, rise, etc. The frames are placed perpendicular to 
the axis, spaced in parallel planes, and supported by wedges which are 
" struck " when the center is removed. Instead of the wedges, jacks or 
sand cylinders can be used for this purpose. 

Loads. — ^The loads which an arch center has to support are: 
Live loads, including (1) The percentage of weight of placed masonry 
supported, in different sections of the arch; (2) Same with regard to un- 
placed masonry and other materials, machines, derricks, men, etc.; (3) 
Impact due to handling material during construction of arch. 

Dead loads, comprising weight of center itself, including sheeting, 
frames, trestling, etc. 

Calculation of live loads. — ^Three cases will be considered, as noted above: 

(1) Masonry in place. — In Fig. 19 let the ^^^ C 
shaded portion show the voussoir V or sec- ''v^ 
tion of masonry whose weight is W. Then: ^"^ 

(a) If we assume V as merely resting ^^K^ ^ 
without friction on the joint / and on the \^ 
back of the arch center, we have, letting /^^ 
P equal the angle of inclination of the nor- / y^ 
mal pressure Wn with the horizontal, / / 

Wn^WsinP (1) / / 

and the vertical and horizontal components /ws' 

of Wn are W, = W sin2 /?, and Wh = W sin p I L 

cos 0, respectively. Wt( = W cos /?) is the ' — ^ ""^"""^Tq 

tangential thrust. It is worthy of note that ^^^- ^^' 

the normal pressure Wn of equation (1) would equal W for any voussoir 
at the crown c, and zero for any voussoir at the hor. springing 5. Authori- 
ties differ, however, as to the value of the equation for sections between 
these two points, on account of the unknown effects of friction, which 
latter would tend to reduce the value of Wn. 

(b) If we assume the voussoir V as hinged at o, then, by taking moments, 

W.h==W/.d, or Wr! = w\ ...(2) 

a 

Now W/ of equation (2) will equal W„ of equation (1) only when the 
voussoir V is extremely thin, practically a plate, and when its intrados is a 
straight line instead of a curve as shown in Fig. 19. Otherwise its value 
will be less than Wn of equation (1). 

(c) If we assume friction at the joint / we have, calling a the angle 
which the joint makes with the horizontal, and Q the angle of repose of the 
masonry. 

Normal pressure W„" = Wn — W tan Q cos a, nearly (3) 

or Wn" = Wn' - W tan 6 cos a, nearly (4) 

in which tan 6 may vary from 0.50 to 0.65; that is, the angle of repose 
may vary between, say, 27° to 33°. The problem becomes more difficult 
if we attempt to include the friction of the soffit of V on the back of the 
arch center. 

(2) Unplaced masonry, machines, etc. — ^These would tend to increase 
the stresses and must be allowed for liberally, especially over the haunches 
where the stresses are particularly indeterminate. 

(3) Impact. — ^The amount of impact would depend largely on the size 
of the stones used; it can be taken into account in a general way, if small, 
by the " factor of safety " adopted. 

Practical formula for live loads. — Let W = weight of section or voussoir 
V (Fig. 19), Wn = normal pressure due to W; and ^9 = angle of Wn with the 
horizontal. Then 

Wn = W sin3 y9 (5) 



712 



^,— ARCHES. 



. — ^Values op Wn for Successive Values op ^ 
From ^=90° (at crown) to /5=0°. Equation (5), 





Normal 




Normal 




Normal 




Normal 


Angle 


Pressure 


Angle 


Pressure 


Angle 


Pressure 


Angle 


Pressure 


/?. 


Wn. 


/?. 


Wn. 


/?. 


Wn. 


/?. 


W^. 


90° 


1.000 W 


70° 


0.830 W 


50° 


0.450 W 


30° 


0.125 W 


88° 


.998 " 


68° 


.797 " 


48° 


.410 " 


28° 


.103 " 


86° 


.993 " 


66° 


.762 " 


46° 


.372 " 


26° 


.084 " 


84° 


.984 " 


64° 


.726 " 


44° 


.335 " 


24° 


.067 *• 


82° 


.970 " 


62° 


.688 " 


42° 


.300 " 


22° 


.053 " 


80° 


.955 " 


60° 


.650 " 


40° 


.266 " 


20° 


.040 " 


78° 


.936 " 


58° 


.610 " 


38° 


.233 " 


15° 


.017 " 


76° 


.914 " 


. 56° 


.570 •' 


36° 


.203 " 


10° 


.005 " 


74° 


.888 " 


54° 


.530 " 


34° 


.175 " 


5° 


.001 " 


72° 


.860 •• 


52° 


.489 " 


32°° 


.149 " 


0° 


.000 " 



Use of Table: Example. — A section of arch masonry weighing 20 tons 
is supported radially by a brace of the arch center acting normal to the in- 
trados and making an angle y?=50° with the horizontal. Find the com- 
pressive stress on the brace due to this load? 

Solution: T7n for /?= 50° is . 45 PF= 9 tons. Ans. 

Types of arch centers. — Fig. 20 illustrates the most simple form of 
centering, that for a fiat -soffit arch. The illustration is that of a Flat 
Arch of the recessed type. They are built up to 8 or 10 ft. in span. 




Wf\ 


ff°*Tf^*»5fc 


f^:- 


1 [■ :| ^^ 


\ 1 ^ 


k 


^ u 



Fig. 20. 



Fig. 21. 



The caps supporting the lagging rest on longitudinal stringers, which 
are in turn supported by posts. Note the economy of material in lagging 
by increasing the depth of strips at the expense of width. The strength 
is proportional to width and to (depth) 2. 

Fig. 21 is the standard segmental arch used over doors and windows. 
The "back" of the center consists of templates of thick boards cleated 
together. The lagging is spaced closer together if the arch is of brick. 
Note that the points o a a form an equilateral triangle in both Figs. 20 
and 21. 

Fig. 22 shows the make-up of braced or unbraced wooden rib as illus- 
trated in Fig. 18. The segmental pieces are 
sawed from plank and (nailed or) bolted together 
into two or more leaves according to the strength --'^'* • •-'•*^-* 

desired. The ribs should, preferably, be braced, 
although unbraced ribs have been used up to t,. «« 

50 or 60 ft. spans. Fig. 22. 

Fig. 23 represents the leaves of the rib laid flat-wise instead of ver- 
tical as in the preceding case. This type is the same as that used in 
chords of ordinary wooden bowstring truss 
bridges. 

Iron or steel ribs are sometimes used instead 
of wooden ribs in cases where great strength 
and stiffness are required. I-beams are the 
best for this purpose. Fig. 23. 




CENTERING P'OR MASONRY ARCHES. 



773 



Fig. 24 illustrates a supported trussed frame, mainly of the Warren 
type. The left half-section is shown supported by vertical posts. The 
right half-section is shown supported by inclined posts, which system is 




Fig. 24. 



Fig. 25. 



sometimes adopted for economy, as in the case of swift currents or ice. 
The same type may also be used without interior supports. In fact the 
end supports may also be omitted by allowing the ends of frames to rest 
on projections or in (artificial) recesses of the abutments. 

Fig. 25 is a skeleton outline of center used in the erection of 60-ft. arch 
of Washington bridge, "^ New York City. The sand cylinders s, are used 
instead of wedges or jacks for lowering the center after the arch is completed. 
They consist of 12-in. plate-iron cylinders filled with sand on which rests 
the plunger, which forms a support to the centering above. By manipu- 
lating a plug at the bottom of the cylinder the outflow of sand is regulated 
and consequently the lowering of the center. In using sand cylinders 
under centers, care must be used to keep the sand perfectly dry. They 
have met with varying success. 

In Table 6, next to last column, reference is made to the files of Eng. 
News, by date, of descriptions of some recently constructed arches. Many 
of these descriptions embody the designs of the centers quite as much as 
of the masonry. 

Striking the center. — ^This consists in lowering the center for removal 
after the masonry arch is completed, either by striking the wedges, operating 
the jacks or manipulating the sand-boxes to relieve the pressure at the soffit. 

The time when this should be done will depend upon the length of span; 
size and kind of voussoirs, whether brick or stone; character of bond; char- 
acter and thickness of joint; quality of mortar, etc. ^ For instance, there 
could be little objection to striking the center almost immediately of a first- 
class stone arch bridge of moderate span with large voussoirs, well bonded, 
with close-fitting joints. On the other hand, it would be unwise to do so 
in the case of a brick arch of long span, small rise, thick joints, and bricks 
poorly bonded, as considerable unhealthy deformation might take place. 
Again, in the case of arches in building walls, considerable time should 
elapse before the centers are removed, at least until the bricklaying above 
the arch is beyond its sphere of influence in causing additional weight or 
pressure. Three or four months is the outer limit for any case, as that gives 
ample time for hardening of the mortar. The presence of the centering 
need in no way interfere with the traffic over the structure. 

Camber of center. — In laying out and erecting the center of an arch, it 
is sometimes advisable to give the frames a slight^ camber, amounting, at 
the crown, to about 5 or i per cent, of the radius of the intrados. Of 
course the object of this camber is to provide for all settlement so that the 
intrados of the finished masonry structure shall be the true curve as de- 
signed, and on which the calculations of the structure are based. 

The total camber to be provided for will equal (1) the deflection of 
frame due to its own dead weight; (2) the deflection due to weight of masonry 
of completed arch; (3) settlement when center is removed; (4) further 
settlement due to superimposed loads on arch. 

If the center is rigidly constructed, the arch stones well laid with close 
joints, and the arch itself not of the pronounced " flat " type, very little, 
if any, camber is required. 

* Wm. R. Hutton, chief engineer. 



774 U.^ARCHES. 

Tables of Arches. 

Table 5, following, is a list of some notable arches tliat have been built, 
with principal dimensions and important remarks. 

Table 6 comprises some typical modern arches. Reference is given to 
the files of Engineering News, where fuller descriptions are given. 

5. — Some Notable Arches that Have Been Built. 

Note. — Ris = rise; Rad = radius at crown; Cr = thickness of ring at 
crown; 5^ = thickness at springing; all dimensions in feet and decimals. 
Rein.-conc.= reinioTced concrete; 3-c^«. = 3-centered, etc. 

No. Span. (Dimensions.) Name and "Kind." [Engineer and Remarks. 

Location. Date.] 

1* 295.27 (Ris=60±, Rad=344.48, Cr=4.92, Sp=11.15). Plauen, 
Saxony. "Stone; 3-cen." [1905.] Flattest stone arch ever 
built, and longest span. Solid arch (without hinges). Arched 
abutments rise from solid rock foundation so that static 
(elastic) arch proper has span of 213.2 and rise of 21.2 ft. 
Live loads: (1) A 15-ton vehicle 9 .84 ft. bet. axles and a unif. 
load of 114.6 lbs. per sq. ft.; (2) 3 steel rollers total weight 
25.16 tons and a unif. load of 114.6 lbs. per sq. ft. Calc'd 
pressure, including temp., was 981.4 lbs. per sq. in. on top 
edges of joints at crown, and 745 . 1 lbs. on lower edges of 
joints. Max. pres. on foundations 355.6 lbs. per sq. in., the 
foundation rock having crushing strength of 23760 lbs. per sq. 
in. Mortar for foundations, 1 Portland cement to 4 sand; 
for main and secondary arch, 1 Port. cem. to 3 sand. 
Large spandrel openings. 

2* 275.6 (Ris=100±, Rad= , Cr= , Sp= ). Luxemburg. 
" Stone, -cen." [1901.] 

3 251. (Ris = 88, Rad = 133, Cr=4). Trezzo, over Adda, Italy. 

"Granite; circ." [Built, 1380; destroyed intentionally, 1420 
±.] 

4* 233. (Ris=70.25, Rad=118.75, Cr=5.5, Sp = 9.5). Walnut 
Lane, Phila. "Concrete, 3-centered." [Webster, 1907.] 
Twin arch rings. Spandrel arches; and arch approaches. 
Actual deflection on removing centers was re in. at crown and 
1-32 in. at quarter points, the calculated deflection at crown 
being f", the difference being accounted for partly to rise in 
temp. 

5 220. (Ris = 57, Rad = 134, Cr = 4.2+4, Sp=6-!-15). Cabin John, 

Washington, D. C, aqueduct. " Stone, circ." [Meigs, 1853- 
9.] The voussoirs of the arch ring are of Quincy granite 
making depth at crown 4.2 ft., and thickness at springing 
6 ft. The spandrel filling of sandstone is laid partly with 
radial joints so as to increase effective depth of ring to 8 . 2 ft. 
at crown and 21 ft. at springing. The arch carries a 20-ft. 
roadway and a 9-ft. conduit. 

6 213. rRis=59.6, Cr= 6.9, Sp= 10.2). J aremcze, Anstna.. "Circ." 

[1892.] Spandrel openings. Railroad. 
7* 211.5 (Ris = 87.6, Rad = 89.25, Cr=4.33, Sp = 6.5). Kempten, 
Bavaria. " Concrete; basket -handle." [1907.] 4-track rail- 
road. Twin arch rings. 

8* 210. (Ris=52.5). Gutach Riv., Bavaria. "Stone." [1905?] 

Spandrel walls. Railroad. 
9* 209.9 (Cr=3.4). Bogenhausen, over Isar R., Bavaria. "Stone." 

[1902.] 3-hinged arch, 21.4 ft. rise. Highway. 
10 200. (Ris=42, Rad = 140, Cr=4.6, Sp=7). Grosvenor, over 

Dee. Chester, Eng. " Circular." [Hartley, 1833.] 



* Described also in Table 6. 



MASONRY ARCHES ERECTED. 775 

5. — Some Notable Arches that Have Been Built. — Continued. 

No. Span. (Dimensions.) Name and " Kind." [Engineer and Remarks. 

Location. Date.] 

11 196.8 (Ris=52.8, Cr=5.6, Sp=13.8). Gour Noir, France. 
" Circular." [Draux, 1888.] Railroad. 

12* 187.5 (Ris=32.2). Lautrach, over Iller, Bavaria. "Concrete." 

^ [1907.] 
13* 187. (Ris = 55.8, Cr=6.56, Sp = 9.18). Schwaenderholz, on N. 

& D. Ry. " Stone." Railroad. 

14 181. (Ris=90.5, Rad = 90, Cr=4.5, Sp = 6). Ballochmyle, Ayre, 

Scotland. " Circular." [Miller.] Railroad. 

15 164. (Ris=16.4, Cr=3.3, Sp=3.6). Munderkingen, over Dan- 

ube. " Circular." [Bois, 1893.] Hinged arch. Railroad. 

16* 164. (Ris=15.75, Cr=1.77, Sp = 2.98). Chatiellerault, France. 

" Reinforced concrete." [1902-.] Four arched ribs. Hen- 
nebique system. Highway. 

17 159. (Ris = 28, Cr = 4.5. Sp=6). Wheeling, W, Ya. "Circular." 

[Hogue and White, 1893.] 

18 152. (Ris=37, Rad=162, Cr=4.9, Sp=10). London, over 

Thames, Eng. "Ellip." [Rennie, 1831.] 5 spans. 

19 150. (Ris=35, Rad = 98. Cr=4.5). Gloucester, over Severn, 

Eng. "Ellip." [Telford.] 

20 150. (Ris=27, Cr=3.8, Sp = 4.5). Elyria, Ohio». "Sandstone; 

circ." [Kinney, 1886.] Highway. 

21 148. (Ris=18, Rad=160, Cr=4.92). Turin, Italy. "Circu- 

lar." [Mosca.] 

22 144. (Ris=19.3, Cr=4.2). Pwm^j;, over Thames, Eng. [Bazal- 

zette, 1886.] 4 other spans. 
23* 143.87 (Ris=19.03, Cr=4.10, Sp = 4.78). Orleans, over the 
Loire, France. " Stone; catenoid." [Renardier, 1906.] 
Highway. 70 spans. 

24 141. (Ris-28, Rad = 103, Cr=4.92). Alma, Paris. "Small, 

cement rubble; ellip." [Darcel.] Railroad. 

25 140. (Ris=35, Rad = 88, Cr=1.5+1, Sp = 2.5 + ). Pont-y-Prydd, 

over Taff, So. Wales. " Rough rubble in lime mortar; 
circular." [Built by a stone mason in 1750, to replace a 
former bridge of the same general design which fell on 
striking centers; present bridge, however, has spandrel 
openings which former bridge did not.] 
26* 140. (Ris=30, Cr=5+2). ///. Cent. R. R., over Big Muddy. 

" Concrete; ellip." [1903.] 

27 131.2 (Ris=18.2, Cr=3, Sp = 3). Coulouvreniere,FTSince. -Circ." 

[Bois, 1896.] 

28 128.2 (Ris=32, Cr = 5.1). Neuilly, Seine, France. "Ellip." 

[Perronet, 1773.] Arch settled 2 ft. on removing center, 
and radius at crown increased from 160 to about 250 ft. 

29 128. (Ris = 24.2, Rad = 169, Cr=5.25, Sp=7.2). Maidenhead, 

Eng. " Brick in cement; Ellip," [Brunei, 1837.] Railroad. 

30* 125. (Ris=39, Rad = 67.75, Cr=5, Sp=7.67). Piney Branch, 

Wash., D. C. " Concrete; parabolic." [Douglas and 
Darwin, 1906.] Highway. 

31 124. (Ris=6.92, Rad-281, Cr=2.67, Sp=3.60). Experimental 

Arch at Souppes, France — See No. 32. " Cut granite in 
Portland cement; circular." [Vaudrcy, 1866.J Width, 
12 ft. Ratio span to rise, 17.83. Centers struck in 4 mos.; 
deflection 1 i ins. Tested without injury by distributed load 
of about 500 lbs. per sq. ft.; also by weight of about 5 tons 
falling 18 ins. on key. 



* Described also in Table 6- 



776 a.— ARCHES. 

5. — Some Notable Arches that Have Been Built. — Continued. 



No. Span. (Dimensions.) Name and "Kind." [Engineer and 

Location. Date.] 



Remarks. 



32 124 



33 120. 



34* 120. 



35 118 



116. 



37 116. 
38* 112. 



39 100. 



40 100. 



41 100. 



42 90. 



43 90. 



44 



84. 



45 83. 



46 
47 



81 
80 



48 80. 



49 80. 



60* 80. 



61 


79. 


62 


79. 


63* 


79. 


64 


78. 


65 


76. 



66 75. 



Ris = 6.92, Rad = 281, Cr=2.67, Sp=3.60). Bourbonnais, 

France. " Cut granite; circ." [Vaudrey.] Very bold; 

built after experiment at Souppes, preceding. Railroad. 
Ris=31, Rad=112, Cr=4.5, Sp = 8.) Waterloo, over 

Thames, London. " Granite; ellip." [Rennie, 1816.] 

8 other spans. 

Ris = 12, Cr=2.33). Jacaguas Riv., Porto Rico. "Rein- 
concrete." [Thacher, 1901.] Highway. Two other spans, 
each 100 ft. 

Ris=38, Rad = 65, Cr=3.6, Sp=3.6). Tongueland, over 

Dee, Eng. " Circular." [Telford, 1801.] Highway. 
Ris=21.2, Cr=3.5, Sp = 4). Cresheim, Fairmount Park, 

Phila. " Sandstone; circ." [Webster, 1893.] Sewer. 
Ris=14.8, Rad = 120, Cr=4). Napoleon, Paris. "Small 

rough rubble in cement; circ." [Couche.] Railroad. 
Ris=17.7, Cr=2.46, Sp = 2.79). Miltonburg, over the 

Main, Germany. " Concrete." [Fleischman, 1899.] 

Highway. 5 other spans. 
Ris = 25, Cr = 4, Sp = 4). Etherow Riv,, Eng. "Circular." 

[Haskoll.] Railroad. 3 other spans. 
Ris = 22, Cr=1.83, Sp=1.83.) Bishop-Aukland, Eng. 

" Circular." [1388.] Highway. 

Ris=15, Cr=3, Sp=7). Wellington, over Aire, Leeds, 

Eng. [Rennie, 1819.] 
Ris=30, Rad = 49, Cr=3). Dean, near Edinburgh, Scot. 

" Circular." [Telford.] Highway. 
Ris=15, Rad=75, Cr = 2.83). Licking Aqueduct, Ches. & 

O. Canal. " Circular." [Fisk.] 

Ris = 27.9, Cr=3, Sp = 4). Elkader, la. "Limestone; 
circ." [Tschirgi, 1888.] One other span. 

Ris = 1 1 . 75, Cr = 4 . 6). Over the Oise, France. " Circular." 
Railroad. 

Ris = 28, Cr=4.5). Trilport, France. "Ellip." Railroad. 
Ris = 40, Rad = 40, Cr=3, Sp=3.50). .Conemaugh Viaduct, 

Penn. R. R. " Sandstone in lime without sand; circ." 
Ris = 40, Rad = 40, Cr=2.66). Royal Border Viaduct, 

Eng. " Brick in cement; circ." 
Ris =16, Rad = 58, Cr=4.66). Posen Viaduct, Germany. 

" Brick in cement; circ." 
Ris =12, Rad = 88, Cr=1.33). Clifty Creek, Greensburg, 

Ind. "Rein. -con.; 5-cen." [Luten, 1906.] Highway. 
Ris = 26.3, Cr=4). Orleans, France. "Ellip." Railroad. 
Ris =13, Cr=3.5, Sp = 4.5). Hutcheson, Glasgow, Scot. 

[Stephenson.] 
Ris =11, Rad = 88, Cr=1.5, Sp = 2.5). Grand Rapids, 

Mich. " Rein. -con.; 3-cen." [Anderson.] Highwav. 
Ris=25, Rad = 43, Cr=3). Falls, P. & R. Ry. "Circ." 
Ris =38, Rad = 38, Cr=7.6, Sp = 14). Westminster, over 

Thames, London. "Circ." [Labelye, 1747.] 12-1-2 other 

spans. 
(Ris =15, Cr=2.4, Sp = 2.4). Albany St., New Brunswick, 

N. J. " Brick ring; circ." [1893.] Small skew; 6 other 

spans. 



* Described also in Table 6. 



57 


75 


58 


72 


59* 


70 


60 


70 


61 


70 


62 


70 


63 


66 


64 


65 


65 


64, 



MASONRY ARCHES ERECTED. 777 

5. — Some Notable Arches that Have Been Built. — Concluded. 

No. Span. (Dimensions.) Name and " Kind." [Engineer and Remarks. 

Location. Date.] 

(Ris=11.5, Cr=2.5, Sp=3). Allentown, Eng. " Circ." 
[Stephenson.] Highway. 

(Ris==16.5, Rad = 47, Cr=2.75, Sp = 2.75). Black Rock 
Tunnel Br., P. & R. Ry. " Circular." [Robinson.] 

(Ris=19.75, Cr=3, Sp = 3). Rockville Br., Penn. R. R. 
" Stone; circ." [Brown, 1901.] 47 other spans. 

(Ris=25, Cr=3.5, Sp=3.5). Swatara, P. & R. Ry. 
" Brick; circ." [Osbom.] 

(Ris =17.6, Rad = 44, Cr = 3) . Brent Viaduct, Eng. " Brick 

in cement; ellip." [Brunei, 1837.] Railroad. 7 other 

spans. 
(Ris=17.5, Cr=2 + .5). Wellesley, Limerick. "Ellip." 4 

other spans. 
(Ris =13.8, Rad = 47, Cr=2.5). Bow, over Lea, Eng. 

" Granite; ellip." [Walker. 1837.] Highway. 
(Ris =32.5, Cr=2.75, Sp = 2.75). Houghton Riv., Eng. 

" Circular." [Haskoll.] Railroad. 

(Ris=16.5, Rad =39.28, Cr=3, Sp=3). Watertown, Wis.; 
C. M. & St. P. R. R. " Stone; circular." [Loweth, 1903.1 
Double track; 3 other spans. 

66 60. (Ris =30, Cr=2.7, Sp = 2.7). Conemaugh Viaduct, Penn. 

R. R. " Stone; circ." [Brown, 1890.] One other span; 
3 tracks, on curve. 
(Ris =20, Rad =33, Cr=2.2). Bewdley, Eng. "Circular." 
[Telford.] Highway. 

(Ris=18, Rad=34, Cr=2.5). Chestnut St., Phila. "Brick 
in cement; circ." [Kneass.] 

(Ris=8.5, Rad = 42.33, Cr=1.75, Sp = 2.5). Sandy Hill, 
over Hudson. " Rein. -cone. ; circ." [Kasson, 1906.] 
Highway & Elec. Ry. 

(Ris = 29, Rad = 29, Cr=2.5, Sp = 2.5). Carrollton, near 
Baltimore. " Granite; circ." Railroad. 

(Ris=17, Rad = 33, Cr=1.5). Llanrwast, Wales. "Circ." 
[Jones, 1636.] Highway. 

(Ris = 28, Rad = 28, Cr = 3 . 1 7, Sp = 3 . 1 7) . Raritan Riv. ; P. 
R. R. " Stone; circ." [Bowles, 1903.] 

(Ris =9, Rad = 45, Cr=2.5). Monacacy Viaduct, Ches. & 
O. Canal. " Ellip." [Fisk.] 

(Ris=10.3, Cr=2.75). Sterling, over Fourth. "Circ." 
(Ris= 3 . 8, Cr = 3 . 16). Nemours, France. " Circ." [Perronet.j 
(Ris = 5.1, Cr=3). Ahhatoir St., Vsivis. "Circ." Railroad. 
(Ris = 1 5, Rad = 28 . 3, Cr = 2) . Avon Viaduct, Eng. ' ' Brick 
in cement; ellip." [Vignoles.] 

(Ris= 7, Cr= 2). Filbert St., Phila.; Penn. R. R. " Brick in 
lime mortar; circ." 

(Ris = 7, Rad = 47, Cr = 2 + . 67) . James Riv. Aqueduct, 
Va. " Circ." [Ellet.] 

(Ris= 3.83, Cr= 3.83). P^sm^s, France. "Circ." [Bertrand.] 
(Ris=6.1, Cr=3). Conturette, France. "Circ." 
(Ris=15, Rad = 21, Cr=2). Touoloway Culvert, under 
Ches. & O. Canal. " Rubble in cement; circ." [Fisk.] 

* Described also in Table 6, 



67 


60. 


68 


60. 


69 


60. 


70 


58. 


71 


68. 


72* 


56. 


73 


54. 


74 


53. 


75 


53. 


76 


53. 


77 


50. 


78 


50. 


79 


50. 


80 


45. 


81 


43. 


82 


40. 



778 



iL— ARCHES. 



6. — Some Typical — 
Note. — ^The next to the last column in table makes reference, by 



Name and 
Location. 



Mate- 
rial. 



Bridge 
Ft. 



One 
Arch 
Span. 



Rise. 



Rad. 

at 
Crown 



Curve 

of 
Intra- 
dos. 



Crown Sp'g'g 



Thickness at 



Foun- 
da- 
tions. 



Plauen Arch B., 
Plauen, Saxony. 
Luxemburg B. 
(Valley of the 
Pretrusse.) 
Walnut Lane B., 
Philadelphia. 



Kempton B., 
Bavaria (Across 
the lUer R.) 



Gutach RIv. B 
(On Neustadt & 
Donaeuschingen 
Ry). 

BogenhausenB., 
Bavaria (Across 
Isar R.). 
Lautrach B., 
Bavaria (Across 
the Iller R.). 
Schwaenderholz 
B. (On Neus- 
tadt & Donaeu- 
schingen Ry.). 
Chatellerault B., 
France. 

Orleans B., 
France (Across 
the Loire R.). 
Illinois Cent. R. 
R. B. (Across 
Big Muddy R.). 
Piney Branch 
B., Washington, 
D. C. 

Jacaguas R. Br., 
Porto Rico. 



Miltenburg B 
Germany (Over 
the River Main) 



Laibach B., 
Austria. 



Venice, Cal. 

Yorktown B., 
Indiana. 

Dayton B., Ohio 
(Across Miami 
R.). 



Stone 
Stone 

Cone. 



492, 



585. 



Cone. 

Stone 

Stone 
Cone. 
Stone. 

Rein- 
cone. 

Stone 
Cone. 
Cone. 



Rein- 
Cone. 



Cone. 



Reln.- 
Conc. 



R.-C. 

Reln.- 
Conc. 

Rein- 
Cone. 



442.8 

(bet. 

abuts.) 

1088.8 



272. 
404. 

733. 

Skew. 
200. 



295.27 
(213.2) 
275.6 



233, 



211'-6« 
(166) 



210. 

209.9 
187.5 
187. 

164. 



143.87 

(70 
spans) 
140. 



125. 



120. 
(3 
spans) 

112. 
(6 
spans) 



102.8 

96. 
95. 



60. ± 
(21.2) 
100. ± 



70' 



87.6 
(29) 



52.5 



344.48 



3-cen. 



4.92 



11.15 



118'- 



89.25 



3-cen. 



Basket 
-h'ndle 



5'-( 



4'-4'' 



9'-6'' 



6 '-6" 
(6.1) 



Cone. 
Cone. 

Cone. 



3.4 



32.2 
55.8 

15.75 

19.03 

30. 

39. 

12. 

17.7 

14.6 

13'- 10* 
11.1 



Cone. 



67' 



Cate- 
noid 

EUip. 
Parab. 



6.56 

1.77 
4.10 
5.-F-2 
5'-0'' 
2'-4'' 

2.46 



9.1 

2.98 
4.78 



Cone. 
Cone. 
Pile. 
Cone. 



Pile & 
Cone. 



79 



Ellip 



2'-2'' 



S'-O" 



588. 



132. 



3-cen. 



I'-S" 



Pile & 
Cone. 



Pile 
Cone. 

Cone. 



VARIOUS TYPES OF ARCHES ERECTED. 



779 



— Modern Arches. 
date, to files of Eng. News, where more complete descriptions may be found. 





Date 








o Kind 


of 




Engin- 


Ref. 




g of 


Erec- 


Width 


eers. 


Eng. 


Remarks. 


Traffic 


tion. 






News. 




1 


H.W. 


1905 


36.1 + 
19.68 




8-17-05 


The longest single span masonry arch in 
the world (1907). 


2 


H. W. 


1901 


52.48 




10-10-01 
2-27-02 
3-5-03 


Two separate parallel arches 19.68 ft 
apart in clear, with intervening space 
spanned with reinforced-concrete slabs. 


3 


H.W 


1907 


40+ 
20 


Webster 


1-31-07 
8-15-07 


Twin arch rings each 2 1 . 5 ft. wide by 9 . 5 
ft. thick at haunch, and 1 8 ft. wide by 
5.5 ft. thick at crown. Abuts, rein- 
forced, 1* sq. rods. Rubble concrete 
1 : 3 : 6 for abuts. ; concrete 1 : 2 : 5 for 
arch rings. 


4 


R. R. 
(4-trk) 


1907 


54. 
(max.) 




5-2-07 


Twin arches spaced 4 ins. Concrete: 
Large arch ring, 1 : 2^ : 5; abut piers 
1:3:6; small piers, 1:4:8; founda- 
tions, 1:5:9; backing for steel hinges. 
1:2:2; with computed stresses re- 
spectively 498. 370. 370. 270. and 938 
lbs. per sq. In. 


5 




1905? 






8-17-05 




6 


H.W. 


1902 


39.+ 
20. 





9-12-02 


3-hinged arch; rise between centers of 
hinges, 21.4 ft. Limestone arch ring 
with granite blocks at steel hinges. 


7 




1907 


13.8 





5-2-07 




8 


R. R. 





14.4 





8-17-05 
12-26-01 


Arch ring built In two thicknesses or 
layers, toothed. 


9 


H.W. 




26.24 




4-10-02 


Hennebique system. 4 arched ribs re- 
inforced with round iron rods. Live 
load, 82 lbs. per sq. ft. 


10 


H.W. 


1906 


44.13 


Renard- 
ler. 


3-28-07 


Ratio of span to rise =7.5. Special pro- 
vision for expansion. 


11 


R. R. 
H.W. 


1903 
1906 






11-12-03 
11-16-05 


Traffic maintained during erection. 


12 




Douglas 


Half bridge erected on one side of center 










and 


4-19-06 


line ; in future when other half is erected 










Darwin. 


1-20-07 


roadway will be 6 1 ft. 


13 


H.W. 


1901 


18. 
(20) 


Thacher. 


8-1-01 


Two end spans each 100 ft., rise 11.28 
ft. Bridge tested under light loads 
after 30 days, for deflection. Sand 
boxes. 


14 


H.W. 


1899 


14.4 
+ 9.8 


Fleisch- 
mann. 


2-25-01 


Arches, 23 ft. wide, 3-hInged, with nar- 
row strips of lead (5 to 6 ins.) 0. 78 in. 
thick. Max. pres. In masonry 360 lbs. 
per sq. in. Factor safety, 8. Con- 
crete in arches, 1:^:3. 


15 


H.W. 


1903 


42. 
(50) 


Melan. 


7-16-03 


3-hinged. 109. 4 ft. c.-c. pins. Live load 
96 lbs. per sq. ft., and 2 13-ton 
wagons. Concrete, arch ring, 1:3:3; 
top of foundations, 1:3-5. 


16 


H.W. 


1907 


16. 


Ehlers. 


8-29-07 


1 cement to 5 ocean gravel; light con- 
struction throughout. 


17 


H.W. 


1905?116. 


Luten. 


5-11-05 


3 ribs; middle rib 12 ft. wide, 2 side ribs. 














each 3 . 5 ft. wide. Settlement at crown i 














in. after striking centers, 30 days. 


18 


H.W. 


1904 


54. 
(clear) 




5-19-04 


Concrete for piers, 1:3:6; for abuts. 
1 : 3 : 7; for arch ring, 1:2:4. Melan 
system; 2-^x2^x5-16 angles. Live load 
150 lbs. per sq. ft. and 2 lines 40-ton 
clec. cars. 



780 ii,— ARCHES. 

6. — Some Typical — 
Note. — The next to the last column in table makes reference, by 



1 



Name and 
Location. 



Mate- 
rials. 



Bridge 
Ft 



One 
Arch 
Span. 



Rise. 



Rad. 

at 
Crown 



Curve 

of 
Intra- 
dos. 



Crown Spg'g. 



Thickness at 



Foun- 
da- 
tions. 



19 



20 



22 



23 



24 



25 



26 



27 



28 



29 



32 



Santa Ana Via 
duct; San Pe- 
dro, Los Ang. & 
St. Lake R. R. 
Belvldere B., 111. 
(Elec. Ry.) (A- 
cross Klshwau- 
kee R.). 



Cllfty Creek B., 

Greensburg, 

Ind. 



Grand Rapids B, 
Mich. (Across 
Grand R.). 
Krosno B., Gal- 
Icla, Austria. 



Rockvllle B., 
Penn., Penn. R 
R. 



Guayo River B., 
Porto Rico. 

C. M. & St. Paul 
R. R. Br., 
Watertown, 
Wis. 

Sandy Hill B., 
N. Y. (Across 
the Hudson R.) 

Decatur B., 111. 
(Wabash R. R.) 
(Across San- 
gammon R.) . 



Rarltan R. Br. 
(Penn. R.R.) 
New Brunswick 
N.J. 
Como Park B., 
St. Paul. Minn 
Standard Over 
head B's. (Lo- 
cal) (N.Y.N.H. 
& H.R.R.) 
L.S. & M.S.R. 
R. Arch B's. 
(Local Stand 
ard). 



Cone. 



Reln.- 
Conc. 



984. 



(8 
spans) 

81. 

(4 

spans) 



36.9 



10.5 



43.5 



83.36 



Clrc. 



Clrc. 



3'-6'' 



3. 



4'-6F 



Rein. 
Cone. 



Rein.- 
Conc. 

Reln.- 
Conc. 



Stone 



Rein.- 
Conc. 

Stone 



Reln.- 
Conc. 



Reln.- 
Conc. 



Stone 



Rein.- 
Conc. 
Reln.- 
Conc. 



Rein.- 
Conc. 



257. 
3820. 

270. 
360. 

1025. 



Skew 
(4 
spans) 



.500. 



80. 



79. 
(5 
spans) 

75. 



70. 

(48 

spans) 



70. 

(3 

spans) 

64. 

(4 

spans) 

60. 
(6 
spans) 

59. 



12. 



5-cen. 



3-cen. 



l'-4- 



I'-e" 

r-o- 



19.75 

7.50 
16'- 6' 

8'-6- 



Clrc. 



39'-3t' 



42'-4' 



Clrc. 



Clrc. 



l'-9'" 



3 '-9" 



56. 

(21 
spans) 

50. 

31. 



30. 



28. 

12'- 6* 
6'- 8" 



Clrc. 



25. 



Elllp. 



3-cen. 



3'-2'' 

O'-IO' 
l'-2"' 

2'-r 



2'- 6* 



Z'-G' 



V-l" 



2'-6'' 



6'- 5" 



Cone. 



Cone. 



Cone. 



Cone. 



Pile & 
Cone. 

Pile & 
Cone. 



Cone. 



Cone. 



Cone. 
(Piers) 



Cone. 



VARIOUS TYPES OF ARCHES ERECTED. 



781 



— Modern Arches — Concluded, 
date, to files of Eng. News, where more complete descriptions may be found. 



Kind 

of 
Traffic 



Date 

of 
Erec- 
tion. 



Width 



Engi- 
neers. 



Ref. 
Eng. 

News. 



Remarks. 



20 



R. R. 



Elec. 
(l-trk) 



1903 



1906 



17. 



14. 



Strauss. 



10-22-03 



8-30-06 



H. W. 



H. W. 



R. R. 



H. W. 



R. R. 

(2-trk) 



H. W. 
& elec 



R. R. 

(2-trk) 



R. R. 



H. W. 



R. R. 



.906 



16. 



1901 

1901 
1903 

1906 

1907 



51 



18. 
(20) 

30. 
(o to 0) 



3 2'- 4" 
(clear) 



34. 



1903 



57. 



17-2 
(15) 



1903 



Vari- 
able 

(Cul- 
verts) 



Luten. 



Ander- 
son. 



6-28-06 



12-1-04 



Brown 

Thacher 
Loweth. 

Kasson. 



Cunning- 
ham. 



Bowles. 



Wilson. 



12-12-01 

8-1-01 
3-26-03 

5-9-07 

3-21-07 



6-18-03 

4-6-05 
4-11-07 

7-14-04 



Concrete arch rings, 1:2:4*; founda- 
tions, piers and spandrel walls, 1:11. 



Two arched ribs 2.5 ft. wide spaced 
8'- 10" c.-c. Concrete, without tem- 
perature, 500 lbs. comp., 50 lbs. tens.; 
with temperature, 650 and 75 lbs. 
Steel 10,000 lbs. without temp.; 13,000 
lbs. with temp, variation. E of con- 
crete. 1 400 000; E or steel, 28 000 000. 
Factor of safety, 4, for 40-ton double- 
truck cars. Novel mode of erection, 
without centering. 

Live load 200 lbs. per sq, ft., and 20-toii 
roller. Mr. Luten gives general for- 
mulas for design of R.-C. bridges — 
curve of intrados, crown thickness, 
curve of extrados, abutment ties, arch 
rods — in Eng. News, June 28, 1906. 

Concrete for arch rings, 1:2:4; for 
piers, 1 : 31 : 7; for abuts., 1 : 2^ : 4. 
Thacher bars 1 i" dia. at intrados. 

Arch: 4 ribs 18 ins. thick. Concrete 420 
lbs. per sq. in.; wt. at 140 lbs. Live 
load 20 lbs. per sq. ft. and a continuous 
string of 1 6-ton road cars. 

Arch rings, spandrel walls and pier fac- 
ings, stone; filling and backing, con- 
crete. One half longitudinal section 
constructed at a time, re-using center- 
ing for other half. Longest stone-arch 
bridge in the world (1907). 

Two end spans each 70 ft., rise 7 ft. 
Sand boxes 10 ins. dia., 12 ins. high, 
with 6 to 8 ins. sand — not satisfactory. 

Traffic maintained during construction. 



Live loads; 12-f-lOO lbs. per sq. ft.; 
elec. cars, 60-ton; for floor plates, 12- 
ton truck on base 6x10 ft. Concrete, 
1:2:4 and 1:3:5. 

Span 58'-3i" meas. square. Concrete: 
For arches, slabs, carrying tracks, and 
footings for piers, 1 : 2^ : 5; for piers, 
spandrels, abuts., and lower slabs, 
1:3:6. Live loads 9000 lbs. per lin. 
ft. per track. Working stresses, 600 
lbs. for cone, in comp., and 12000 lbs. 
for steel in tens. 

Sandstone of excellent quality from 
Cambria and Clearfield Counties, Pa. 
(Conemaugh Stone Co.). 

Concrete, 1:2:4; mortar facing, 1 : 2. 

Old-rail reinforcement — economical. 



Concrete, 1 : 3 
Monier plan. 



6. Johnson steel rods — 



782 



Ai.— ARCHES. 



Concrete Culverts. 

Fig. 26 is a section of a small railroad culvert, 3 
ft. wide and 3 ft. high, which has been used consider- 
ably on the Penn. R. R. as a standard. The amount 
of concrete required is about 0.41 cu. yd. per lin. 
ft. of culvert. Thickness at crown is 8 inches. 






Fig. 26. 



STEEL AND COMBINATION ARCHES. 

A few hints will be given to illustrate the method employed in the 
calculation of some of the forms of steel arches most commonly used in 
practice. Steel arches differ from stone arches in that they are designed to 
resist bending, as well as compression and shear. In other words, the line 
of resultant pressure is not confined within any given limits but may pass 
anywhere outside the middle third or even outside the rib of the arch 
altogether. Hence the form of arch selected, and depth of rib or truss, are 
matters of economy, adaptability and appearance, rather than of mere 
gravitational stability of the respective parts. 

In the following discussion it is assumed that the forms of the structiires 
have been determined, the problems being to find the stresses in the various 
members or parts under certain given loadings. The first thing to do in 
each case is to find (one or more of) the reactions at the points of support; 
and when these are known the remaining solution reduces to the simple case 
of "finding the inner forces or stresses when the outer forces are given;" 
that is, to the case of any simple structure. 

The methods used in finding the reactions may be wholly analytical or 
partly graphical. The latter is chosen here for simplicity. The first step is 
to draw the arch and its "position line," sometimes called the "locus line." 

Arch with No Hinges. — ^This arch (Fig. 27) is seldom used in practice, the 

2-hinged and 3-hinged arches being mainly employed. 
The position line is laid off by means of an equation 
containing x and y. From any point p the direction 
of the reactions for any load P may be obtained by 
either one of several methods: (1) by drawing lines 
from any point as p tangent to the reaction curves; 
(2) by calculating the vertical ordinates yt and y2 
from the points a and b, to points on the line of di- 
rection of the reactions; (3) similarly, by calculating 
Xi and 3^1 ; (4) by calculating the right angle offsets from a and b; (5) by cal- 
culating the horizontal thrust H; etc. The force polygon, above, gives the 
value and direction of Ri and R2. The equation for giving these values 
vary with the form of arch, direction of loading, etc., and will not be given 
here. The temperature stresses have also to be considered.* 

Two=Hinged Parabolic Arched Rib. — Let the curve ab fPig. 28) be the 

neutral axis of the rib with end hinges at a and b. '* 

The "position line" may be determined from the 

6 4 c2 /t 
formula y = -ri r» for ^^Y value of x on either 




Fig. 27. 




side of the vertical axis; c being the half span, and s] 

h the rise. The reactions Ri and R2 are obtained for 

any force P at point p by drawing the triangle of Fig. 28. 

forces as above. These reactions must pass through or act at the hinges 

a and b. Hi and Vi are the horizontal and vertical components of Ri\ H2 

and V2 of i?2. The bending moment at any point m is equal to R2 multiplied 

by the offset d. The axial thrust due to R2 must also be considered in 

designing the flanges of the girder; and the shear, in designing the web. 

A table may be made showing the bending, compressive and shearing stresses 

at various points m along the girder due to loads P at successive positions, 

and summarized, being careful to use the proper + and — signs. 

The above method is exact only for shallow, solid ribs of constant cross- 
section; but it is used also for plate girders and open ribs even when the 
flange plates do not extend to the hinges. 



* See A Treatise on Arches by Malverd A. Howe; also Trusses and 
Arches, Part III Arches, by Charles E. Green. 



CULVERTS. STEEL AND COMBINATION ARCHES. 783 

The effect of temperature is to increase the stresses very considerably, 
and for this reason many engineers prefer the 3-hinged arch, not affected by 
temperature, and statically determinate like any simple structure. 

Figs. 29 and 30 illustrate a design of a 100-ft. span, 4-track railroad, 
2-hmged steel arch bridge: 




Figs. 29. Half -Elevation, Half -Section and Details. 




Fig. 30. End of Main Rib, Enlarged. 



Three=Hinged Arch. — Let Fig. 31 be a braced arch with hinges at crown 
and springing; and let it be required to find the stress in the diagonal cd 




Fig. 31. 

due to the panel load P at e. Draw the Position Line as shown, and from the 
point p, directly above e, lay off by scale the load P, verticallv. Then by 
the triangle of forces find Rx acting in the direction ap. To find the stress s 
in cd draw a cutting plane through the panel and find the origin O at the 



784 a.— ARCHES. 

intersection of the other two active members ce and ad. Taking moments 
about O, of the forces acting at the left of the cutting plane, we have, using 

the lever anns m and «, mS = nRu or S= —Ri (tension). Similarly, the 

m 

stress in any member may be obtained for any loading. 

COST OF REINFORCED CONCRETE ARCH BRIDGES. 

Highway Bridges. — Cost per square foot of floor for50-ft. spans, $2.00 to 
$2.50; 75-ft. spans, $2.50 to $3.50; 100-ft. spans, $3.50 to $4.50; 125-ft. spans, 
$4.50 to $6.00; 150-ft. spans, $6.00 to $10.00. 

Electric=RaiIway Bridges. — Cost per square foot of floor for 50-ft. spans, 
$3.50 to $4.00; 75-ft. spans, $4.00 to $4.50; 100-ft. spans, $4.50 to $5.00; 125-ft. 
spans, $5.00 to $6.50; 150-ft. spans, $7.00 to $10.00. 

Steam =RaiI way Bridges. — Cost per lineal foot, double track, for 50-ft. 
spans, $200 to $250; 75-ft. spans, $250 to $275; 100-ft. spans, $275 to $300; 
125-ft. spans. $300 to $325; 150-ft. spans, $325 to $350. 

EXCERPTS AND REFERENCES. 

(See, also. Tables of Masonry Arches, pages 774, etc.) 
The Design of Arch Culverts (By D. B. Luten. Eng. News, June 13, 
1901).— Illustrated. 

The Design of a Reinforced=Concrete Arch Bridge (By D. B. Luten. 
Eng. News, May 8, 1902). — Illustrated. 

Stresses in Masonry- and Concrete Arches (By D. B. Luten. Eng. 
News, June 12, 1902). 

Steel Arch Bridge, 450=Ft. Span, Over the Rio Grande on the Pacific 
Ry., Costa Rica (By Theodore Cooper and Gunwald Aus. Eng. News, 
Oct. 23, 1902). — Illustration of steel shoe. 

Design and Construction of a 50=Ft. Brick Arch Culvert (By W. J. 
Douglas. Eng. News, Dec. 25, 1902). — Illustrated. 

A Wooden=Braced Arched Highway Bridge (By A. Munster. Eng. 
News, Jan. 8, 1903). — Illustrated. 

A Reinforced=Concrete 3»Hinged Arch Bridge (Prof. J. Melan. Eng. 
News, July 16, 1903). — Illustrated. 

Stress Diagram of Concrete Arch (By H. W. Parkhurst. Eng. News, 
Nov. 12, 1903).— Illustrated. 

A Reinforced-Concrete Highway Bridge, With Cost Data (P. A. Court- 
right. Eng. News, May 12, 1904). 

Three=Hinged Steel Arch Trusses for a St. Louis Exhibition Building 
(Eng. News, Sept. 29, 1904). — Span, 172 ft. Actual and estimated weights 
are given. 

A New Graphical Method for Stresses in 3-Hinged Arches (By J. W. 
Balet. Eng. News, Oct. 20, 1904). 

The Connecticut Ave. Concrete Arch Bridge (By Geo. S. Morison. 
Eng. News, Jime 1, 1905). — Illustrated. 

Three-Hinged Steel Arch Bridge at Exeter, England (Eng. News, 
July 20, 1905). — Illustrated details, and detailed elevation of half -rib. 

Parabolic Reinforced=Concrete Arch Bridge (Trussed Concrete Steel 
Co. Eng. News, Nov. 15, 1906). — Illustrated; 75-ft. span. 

Arch Rib Bridge of Reinforced Concrete at Grand Rapids, Mich. 

(L. W. Anderson. Eng, News, Mar. 22, 1906).— City Bridge; illustrated; 
cost data. 

Three-Hinged Concrete Arch Bridge, Brookside Park, Cleveland (By 
H. F. Hackerdom. Eng. News, May 10, 1906). — Illustrated details of 
hinges. 

Graphical Method of Laying Out a 5-Centered Arch (By A. Swartz. 
Eng. News, May 10, 1906). 

Empirical Formulas for Reinforced Concrete Arches (By D. B. 
Luten. Eng. News, June 28, 1906). — Illustrated. 



MISCELLANEOUS DATA. 785 

Low=Cost Concrete Culverts (By W. H. Whorley. Eng. News, 
July 5, 1906).— Tables. 

Special Form of Arch Centering (By J. H. Milbum. Eng. News, 
Aug. 23, 1906). — Illustrated form for 50-ft. arch. 

Reinforced Concrete Arch Bridge Built in Reinforced = Concrete 
Forms Without Centering (Eng. News, Aug. 30, 1906).— Illustrated. 

Some 3=Hinged Concrete Arches in Germany (Eng. News, May 2, 
1907).— Illustrated. 

The Oakland Steel Arch Bridge Without Hinges, at Pittsburg (By 

Willis Whited. Eng. News, May 16, 1907).— Illustrated. 

Itemized Cost of Reinforced Concrete Arches (By G, P. Carver. 
Eng. News, Aug. 22, 1907). — ^Table and diagram of costs. Illustration of 
a 100-ft. arch. 

The Elastic Theory and a Faulty Arch (By H. S. Pritchard. Eng. 
News, Jan. 9, 1908). 

A Combination Arch=Cantilever Concrete and Steel Bridge in France 
(Eng. News, Mar. 26, 1908). — Illustrated. 

A 3-Hinged Masonry Arch with Metal Joints and Concrete Super- 
structure ("Annales des Ponts et Chaussees," Vol. 29, part 5, 1907; Eng. 
News, Sept. 10, 1908). — Illustrated. 

A New Arch Curve, the Parabolic Oval (By C. Worthington. Eng. 
News, April 15, 1909). — Formulas and illustrations. 

Subdividing An Arch Ring for Stress Analysis (By F. E. Tumeaure. 
Eng. News, April 22, 1909). 

A 259=Ft. Concrete Arch Bridge in Switzerland (Eng. News, Aug. 5, 1909). 
— Illustrated. Table of load and temperature stresses. Sand boxes used. 

Reinforced Concrete Arch Bridges, Spans 281 ft. and 120 ft. (Eng. News, 
Sept. 2, 1909). — Illustrated, with plans of floor system, abutment and high 
retaining wall. 

List of Masonry Arch Bridges over 175-Ft. Span. (Eng. News, Sept. 2. 
1909). — Stone and concrete, 

180=Ft. Stone Arch Bridge at Wiesen, Switzerland (Eng. News, Sept. 16, 
1909). — Eleven illustrations. 

Formulas for the Volume o* Material in Groined Arches (By Chas. B. 
Buerger. Eng. News, Oct. 7. 1909). — Illustrated. 

Novel 3=Hinged Steel Arch, In Greece (Eng. News, Nov. 4, 1909). — 
Railwayarch. 193. 5-ft. span. Illustrations: — Planand elevation of viaduct; 
Details of half arch rib; Details of hinges at crown and springing; Scheme 
of erection. 

Walnut Lane Bridge, Phila. (By G. S. Webster and H. H. Quimby. 
Trans. A. S. C. E., Vol. LXV., Dec, 1909).— Plans, including central con- 
crete arch of 233-ft. span. Total cost of bridge, including electrical conduits 
and lamp standards, bush-hammering, and all extra work, was $267,000, 
which gives a rate of $7.60 per sq. ft. of floor surface and $0,880 per cu. ft. of 
space — area of profile by width of bridge. 

Reinforced=Concrete Viaduct, Harrisburg, Pa. (Eng. News, Jan. 13, 1910). 
— ^The viaduct proper is 1841 ft. long, 78 ft. above ground at its highest point, 
and carries on 19 arches a 28-ft. roadway and two 8-ft. sidewalks. Full 
description, illustrated. 

Tests of Model Concrete Arches by the New York State Engineer's Office 

("Barge Canal Bulletin" for February, 1910; Eng. News, Mar. 10 and July 
7, 1910).— Illustrated. 

Failure of Reinforced-Concrete Arch Highway Bridge (Eng. News, Mar. 
17, 1910).— Illustrated. 75 and 90-ft. spans. 

The Analytical Calculation of a Concrete Arch (By Malverd A. Howe. 
Eng. News, May 12, 1910). — Illustrated. Discussions: Horizontal thrust; 
Bending moments at the supports; Vertical reactions; Dead load; Live load; 
Temperature; Effect of direct stress; Changes in dimensions. There are 
nine tables for use in making calculations. Article continued in Eng. 
News, June 2, 1910. 



786 U.^ARCHES. 

The Meadow St. Reinforced=Concrete Arch Bridge, Pittsburg, Pa. (By 

N. S. Sprague. Eng. News, Dec. 1, 1910). — Description with 8 illustrations. 
The length of the main arch span is 209 ft. with a rise of 46.14 ft. and con- 
sists of three arch ribs, the two outside ribs being uniformly 3 ft. 9 ins. in 
thickness and the central rib 5 ft. and all three ribs having a depth var5H[ng 
from 5 ft. at the crown to 6 ft. 2 ins. at the springing line. 

The New Charles River Bridge, Boston Elevated Railway (Eng. Rec, 
Dec. 17, 1910). — A reinforced-concrete arch bridge of five arches of 122-ft. 
4-in. clear span, four of 98-ft. 4-in. clear span, a steel lift bridge at the river 
lock and a special span of 125 ft. 4 ins. at the small boat lock. The special 
illustrations of the 98-ft. span, accompanying this descriptive article are: — 
Details of hinge, including arch reinforcement and plan of skewback; longi- 
tudinal section of half span, part plan and details; reinforcement around 
stringers and of fioorbeam. 

Illustrations of Recent Arch Spans. 

Description. Eng. News. 

Steel A-arch, 3-hinged, 180-ft. span, 51-ft. rise Apr. 21, '10 

Arch dam design for the site of Shoshone dam Time 9, '10 

Wooden arch centering for 144-ft. masonry arch, Norway July 7, '10 

3-hinged rein-cone, arch, 88-ft. span, Paris Sept. 15,' 10 

Eng. Rec. 

Details of centering trusses for cathedral stone arches Tan. 23, '09 

Table of stresses in 280-ft. span, concrete arch bridge Jan. 23, '09 

Rein.-conc. arch bridge, Grand River, L. S. & M. S. Ry Apr. 24, '09 

3-hinged centers for building 150-ft. concrete arches Apr. 24, '09 

A combination steel and concrete arch bridge, 250-ft. span May 22, '09 

Reinforcement and erection of concrete arch .June 12, '09 

139-ft. rein.-conc. arch, Edmondson Ave. bridge, Baltimore c... .June 19, '09 

364-ft. steel spandrel-braced arches and details June 26, '09 

Elastic rein.-conc. railway arch span, 97^ ft., rise 36f ft June 26, '09 

Engineer's and contractor's falsework, Edmondson Ave. bridge .Aug. 14, '09 

Short-span (34 to 44 ft.) rein.-conc. R. R. bridges Mar. 19, '10 

Determination of wind stresses in 3-hinged arches — Eusink May 21, '10 

Rein.-conc. R. R. viaduct, five 120-ft. and two 100-ft. spans... .July 16, '10 
Details three 87.5 ft. rein.-conc. arch highway spans, Los Angeles.Aug. 13, '10 
Details of plate girder and rein.-conc. arch (70-ft. span) R. R. 

bridge Aug. 20,'10 

Highway viaduct — 28 70-ft. rein. cone, arch spans Oct. 15, '10 

Proposed 864-ft. arch across the Kentucky River. Nov. 26,' 10 

149-ft. rein.-conc. arch for electric R. R., Maritime Alps Dec. 3, *10 

Centerings for 80-ft., 90-ft., 100-ft. stone arches, B. & O. viaduct. 

Brand. Cr Dec. 17. '10 



45.— TRESTLES. 

A Trestle is a bridge composed of a series of relatively short beam- or 
girder spans resting on "bents," which take the place of ordinary piers. 

Pile Trestles. — Where the bents are composed of piles it is called a pile 
trestle or pile bridge. Each bent may consist of any number of piles driven 
in line transversely with the axis of the bridge, the number required depend- 
ing upon the kind of bridge, the width, loading, height of trestle, kind of 
soil, lateral stiffness required, whether on tangent or curve, etc. Generally, 
the piles are driven vertically but where great stiffness is required the end 
piles of each bent are driven commonly on a batter. In fact, this latter 
practice is usual with some railroads even for low trestles on tangents. The 
cost of driving such batter piles is not excessive, as the gins of the driver 
may be arranged easily so as to swing laterally on a pivot like a pendulum. 
For a foot bridge, two or more piles are used to the bent; for a highway 
bridge, three or more; and for a railroad bridge, foiu* or more. 

The maximum load on a pile should be limited to about 40,000 lbs., 
and a load of 25,000 to 30,000 lbs. is preferable, usually. If driven in soft 
material a less load should be used. Again, the loading on pile may be 
limited to the allowable compression "across grain" of the wooden cap 
resting upon it. Piles should be peeled before driving unless they are driven 
in salt water. 

Wqoden caps are usually 12 x 14 ins. for railroad bridges; 12x12, lOx 
12, 10x10, etc., for highway bridges — ^with greater dimension vertical. 
The piles when driven are sawed off on a plane, either level or inclined (to 
give proper elevation to outer rail if on a curve), and the caps are drift - 
bolted to them. The usual size of drift bolt is f to | in. in diam., and 18 to 
22 ins. long. Holes are first bored with an auger about J in. smaller in dia. 
than the bolt. 

Diagonal bracing for each bent consists usually of two planks, one on 
either side of bent and crossing at the middle line, with upper ends bolted 
to caps near their outer ends, and with lower ends bolted to the outer piles 
— using f-in. screw bolts and cast washers. At intersection with the inner 
piles the braces are fastened to same with two wrought spikes with length 
at least twice the thickness of the planks. Similarly, horizontal "sash" 
braces may be fastened to the piles on both sides of the bent at foot of 
diagonals. For railroad trestles, 4x10 in. bracing is common; and for 
highway bridges, 3x10, 3x8, 2x10, 2x8, etc. High trestles are double 
braced. 

Standard Plans for Pile Trestles. 

Each railroad has its own standards, differing in certaitt essential details 
from those of other roads, such, for instance, as spacing of stringers and 
jack stringers, arrangement and size of guard rails, and the design of the 
floor in general. 

Fig. 1 is an elevation of bent of single track trestle of the Oregon Short 
Line R. R., showing ballast floor. 

Timber Trestles. — The term "frame trestles" is applied to those trestles 
constructed of framed timbers, usually sawed to "dimension," but some- 
times consisting of straight, round poles or piling timbers denuded of the 
bark. The latter construction has been used considerably in the Pacific- 
Northwest. Where such poles are used the posts are commonly in one 
length, even for very high trestles. 

Ordinarily, however, the bents of high trestles are constructed in 
sections, vertically: A single-deck trestle is one in which the bents are in 
one section; double-deck, in two sections; 3-deck, in three sections; etc. 
Fig. 2 is an illustration of a 3-deck trestle showing the right half resting on 
piling, and the left half resting on mud-sills, the latter being used in cases 
where piles cannot be driven or where the cost of driving is prohibitive. 
The mud -sills should be of cedar, about 8x12 ins. by 4 ft. long, and laid 
side by side on a well tamped foundation thoroughly drained and free from 

787 



788 



i5.— TRESTLES. 



wash. Instead of driving piles, "false" piles, set by hand, are sometimes 
employed. Concrete or rubble masonry piers are desirable where a suitable 
sub-foundation is presented cheaply. 



•^^...^\ZT^^"j^ ff-6-fQ 9-Q~ Cross 77g >- r^- ^Curb 6'x8'-'/7-2'' 

-* #Mll>^^^#^^I^^^I^ ^^Ws< i open Joints. 




W.D.Panels 
long plus I'-O'to be dressed to 
exact depth on upper edge. 

f Drift bolts I'-Qlong. 

I" Drift bolts l-IO'lorig„ 



Fig. 1. (See page 787.) 

Fig. 2 illustrates a more or less typical railroad trestle which will be 
described briefly: The main posts, 12x12 ins., have a continuous batter of 
3:12 and 1:12. They are "dapped" into the caps and sills about f in., drift- 
bolted at tops, and doweled (with 1-in. round iron) and toe-nailed (with 
wrought spikes) at feet. (Mortise and tenon joints are no longer used.) 
The "batter" posts, used below the top deck, are likewise framed, drift- 
bolted, doweled and toe-nailed. They give lateral stiffness as well as vertical 
support. The main caps and sills are 12 x 14 ins., while the inter- 
mediate caps and sills are 12x12 ins., extending 12 ins. or more 
beyond the outer edge of posts. Each section of the bent is sway- 
braced with 4 X 10 in. plank, screw -bottled at ends, and spiked at inter- 
sections. The bracing between the bents consists of 12 x 12 in. longitudinal 
girts G, framed into the intermediate caps and sills, with lap joints, and 
thoroughly drift -bottled to them; and the longitudinal diagonal braces L 
consist of 4 X 10 in. plank, screw -bottled at ends. The floor of the bridge 
rests on two lines of main stringers and the two lines of jack stringers, the 
latter being used as safety stringers in case of derailment. Main stringers 
are ordered in 2-span lengths, that is, for bents 15 ft. centers they are 
ordered 30 ft. long, laid two, three or four to each line with alternate and 
butt joints, dapped over the caps and drift-bolted thereto. They are 
separated about 1 inch laterally by cast spool separators through which 
the screw bolts are inserted (using cast washers), four to each joint over 
the caps. Between the lines of stringers a 4-in. plank of proper length is 
spiked to top of caps to preserve the required stringer spacing. The jack 
stringers are ordered one foot longer than the main stringers (or say 32 ft. 
long) so that the joints can be "halved" and drift-bolted to caps with one 



PILE TRESTLES. TIMBER TRESTLES. 



789 



bolt. The practical sizes of main stringers are 8x 14, 9x 16, lOx 18, using 
two or three (or more) stringers under each rail. The width of jack stringer 
should be not less than 6 ins., and it should not be less than about one-third 
the width of one line of main stringers. High trestles are provided usually 
with outer guard rails or "bull" rails, which have, in certain cases, prevented 
trains from plunging over the side of the bridge. These bull rails are prefer- 
ably about 12 ins. high, and 10 x 12 in. timbers are commonly used. They 
may be ordered in any convenient lengths, are halved like the jack stringers, 
and seciu*ed firmly by screw bolts passing vertically through bull rail, tie 
and jack stringer. In addition to the bull rails or "outer" guard rails, there 
should be "inner" guard rails. The inner guard rails are usually comrnon 
rails placed inside the main rails, but wooden guard rails, say 5x8 ins. 
(5 ins. vertical), dapped over the ties to preserve proper spacing of the 



LonfQiis 




Fig. 2. 

latter, are frequently used. They are secured to the ties by wood or lag 
screws (8 or 9 in.) with heads flush with top of guard and bearing on fiat 
washers. Sections of bents are from 20 to 24 ft. in height and uniform 
from top of bridge downward so that the various decks are on planes parallel 
with the track. Longitudinal diagonal bracing may generally be omitted 
between alternate pairs of bents. 

Many modifications may be made of Fig. 2. 
For instance, the inner main posts may be ver- 
tical; all the main posts may be continuous 
(each composed of one stick, or of two sticks 
bolted together with alternate joints) and the 
transverse diagonal bracing divided by hori- 
zontal sash braces spaced vertically about the 
depth of one deck, with longitudinal girts 
bolted at intersections of sash braces and posts; 
instead of having separate intermediate caps and 
sills, one piece may serve to act as the cap of 
the section below as well as the sill of *the sec- 
tion above; etc. 

Grass-hopper bents. Fig. 3, are often used with 
economy on side-hill work; either to save cost 
of excavation or to avoid encroachment on ad- Fig. 3. 




790 



i5.— TRESTLES. 



joining property. The writer has used as many as three broken sills to 
the bent with perfect safety and even without concrete backing to resist 
thrust. At a, the upper broken sill is dapped into the post one inch and 
sash braces are spiked and bolted on as shown in Fig. 3. Careful at- 
tention should be paid to drainage. 

The bents of wooden trestles are spaced usually from 12 to 16 ft. centers, 
the ordinary spacing being, perhaps, 15 ft., calling for 30-ft. stringers which 
are shipped conveniently on the average fiat car. Unlike steel trestles the 
spacing is constant or nearly so throughout the bridge, regardless of the 
variation in height of bents. Where long spans are introduced in a wooden 
trestle, as for spanning a creek, the supporting bents are usually doubled 
or tripled. 

On curves, the caps are inclined in order to give proper elevation to the 
outer rail. This method is preferable to using level caps, with shins under 
the stringers or on the ties, etc., as is sometimes done. The writer generally 
supplies the foreman of the framing crews with tables giving the increased 
lengths of posts on outside of curve and the decreased lengths on inside of 
curve for each bent, to provide for the proper elevation of the outer rail 
and a like depression of the inner rail, each being one-half of the required 
"elevation." 

Pile and Timber Trestles. — A pile and timber trestle is one in which 
the piling of the foundation is cut off high enough above the ground to 







Fig. 4. 

constitute the lower deck, and on which the timber trestle proper is erected. 
Sometimes, however, the term is applied to a frame or timber trestle with 
simply a pile foundation. 

Fig. 4 is a section of a pile and timber trestle of the A., T. & S. F. R. R. 



WOODEN STRINGERS. STEEL TRESTLES. 



791 



1. — Allowable Bending Moments in Ft.-Lbs. on Wooden Stringers. 
By Formula, M=kfbd^-i-12 
In which /== allowable outer fiber stress in lbs. per sq. in.; 
b = breadth of stringer, in ins. ; 
d = depth of stringer, in ins. 

[Values of M in Ft.-Lbs.] 



Size of 
Stringer. 
Inches. 


1 Stringer. 


2 Stringers. 


3 Stringers. 


4 Stringers. 


/=900. 


/=1000. 


/= 900. 


/=1000. 


/=900. 


/=1000. 


/= 900. 


/=1000. 


*lxl6 
6x16 
7x16 
8x16 
9x16 


3 200 
19 200 
22 400 
25 600 
28 800 


3 556 
21 333 
24 889 
28 444 
32 000 


6 400 
38 400 
44 800 
51 200 
57 600 


7 111 
42 667 
49 778 
56 889 
64 000 


9 600 
57 600 
67 200 
76 800 
86 400 


10 667 
64 000 
74 667 
85 333 
96 000 


12 800 

76 800 

89 600 

102 400 

115 200 


14 222 

85 333 

99 556 

113 778 

128 000 


10x16 

*lxl8 

6x18 

7x18 

8x18 


32 000 
4 050 
24 300 
28 350 
32 400 


35 556 
4 500 

27 000 
31 500 

36 000 


64 000 
8 100 
48 600 
56 700 
64 800 


71 111 
9 000 

54 000 
63 000 

72 000 


96 000 
12 150 
72 900 
85 050 

97 200 


106 667 
13 500 
81 000 
94 500 

108 000 


128 000 
16 200 
97 200 

113 400 

129 600 


142 222 
18 000 
108 000 
126 000 
144 000 


9x18 
10x18 
12x18 
♦1x20 

8x20 


36 450 
40 500 
48 600 
5 000 
40 000 


40 500 
45 000 
54 000 
5 556 
44 444 


72 900 
81 000 
97 200 
10 000 
80 000 


81 000 
90 000 
108 000 
11 111 
88 889 


109 350 
121 500 
145 800 
15 000 
120 000 


121 500 
135 000 
162 000 
16 667 
133 333 


145 800 
162 000 
194 400 
20 000 
160 000 


162 000 

180 000 

216 000 

22 222 

177 778 


9x20 
10x20 
12x20 


45 000 
50 000 
60 000 


50 000 
55 556 
66 667 


90 0«) 
100 000 
120 000 


100 000 
111 111 
133 333 


135 000 
150 000 
180 000 


150 000 
166 667 
200 000 


180 000 
200 000 
240 000 


200 000 
222 222 
266 667 



*Moments are directly proportional to width of stringer, and to (depth )2. 

Example. — Find the size of wooden stringers required for 15-ft. spans, 

assuming the engine load to give a maximum bending moment of 298500 

ft. -lbs. per track, and assuming the floor to weigh 450 lbs. per lin. ft. per 

track, (and the stringers 420 lbs. per lin. ft. of track). 

Solution.— Total bending moment = 29 8500 4- ^^^^^^^^^ = 322969 ft.-lbs. 

o 

per track = 161485 ft.-lbs. per rail. From the above table this calls for 4 — 
10x18 stringers under each rail, if the outer fiber stress is assumed at 900. 

Steel Trestles. — The proper and economical design of a steel trestle for 
a given locality calls for a thorough examination of the site. In addition to 
the profile and contour map, borings may be required to determine the 
character of the sub-foundation. The spans are longer than for wooden 
trestles, and are made to increase with the height of the structure so that 
the economic rule "The cost of piers should equal cost of spansf" will apply 
(practically) . 

The distinctive feature of steel trestles is that there are few posts to the 
bent; in fact, two posts are stffficient for single track railroad trestles. 
Again, in high trestles the bents are usually arranged in pairs, each pair 
being braced horizontally and diagonally, both longitudinally and trans- 
versely, and sometimes laterally, really forming a steel pier. 

As a first consideration in design it may be noted that, as far as the floor 
(comprising the spans) alone is concerned, that floor is the most economical 
when the spans are of equal length; that is, for any given number of supports 
those supports should be spaced equidistant throughout to give the least cost 
of spans. But taking the structure as a whole we have second that the cost of 
any pier, together with its foundation, should equal the cost of the portion 
of the spans which it supports (Fig. 5). Hence the profile, character of 
foundation, etc., have considerable influence in varying the lengths of the 
spans. This applies as well to bents as to piers. In testing the economy 
of any design let us assume any portion as D G to be fixed and equal 
to L = 4 -I- /s -h /e. Now any small increase in I5 will require an increase in 



t Approximately; see Sec. 37, Bridges, page 683. 



792 



i5.— TRESTLES. 



the amount of material m both the span and bracing between bents B5 
and Be, and a decrease in material in spans S4 and Se- Conversely, any 
decrease in h will produce the opposite effect. Hence, /^ should be such 
that for any small change in its length the increase in material on the one 







Fig. 5. 



hand will just equal the decrease on the other. But note that in the simi- 
lar adjustment of adjacent portions the above spans may require re-adjust- 
ment; and likewise, all other sections. We have also to take into consid- 
eration the best relative lengths of I2, Ip /e* etc. 
Tables or diagrams showing the weights of 
spans, bents, and bracing for various lengths 
and heights for the particular loading will greatly 
facilitate the above adjustments. 

The floor system is arranged similar to that 
of an ordinary steel bridge, that is, with floor 
beams and stringers, the former being supported 
directly by the bents. For short spans, I-beams 
may be used; for longer spans, plate girders. 
For very long spans both floor beams and 
stringers may be trussed, using deck girders for 
this purpose. 

The posts of the bents (Fig. 6.) are battered 
about 1:6 — considerably less than for wooden 
trestles. This narrowing of the space between 
posts tends to call for less metal in the transverse 
bracing; for more metal in the posts, from wind 
stresses; but for slightly less metal in the posts 
from direct loads. The best designs have stiff 
diagonal bracing instead of rods, and large mem- 
bers with few connections are preferable. The 
feet of columns should be anchored. The wind 
stresses may be calculated by treating the bent 
as a cantilever arm, assuming all the joints to 
be hinged; but if the framework is stiffened by 
heavy gusset plates, and riveted connections are 
used, it becomes statically indeterminate. 

Elevated Railroad Trestles. — For a full discussion of this subject the 
reader is referred to Paper No. 806, Trans. Am. Soc. C. E., June, 1897, by 
Mr. J. A. L. Waddell 

Reinforced Concrete Trestles. — The outline shown in Fig. 6, above, for 
steel trestles, is a good design for a high trestle bent of reinforced concrete. 
It can also be modified as follows: (a) By omitting the horizontal braces, 
using the diagonal bracing only; (b) by omitting the diagonal braces, 
using the horizontal bracing only; (c) by omitting all bracing. In case (b), 
however, the mushroom connections of horizontal braces^ to posts must be 
of the large gusset type, in order to introduce bending resistance at ends of 
braces; and similarly at the floor-beam connections. Case (c) should be 
used for short bents only. 

The posts and bracing should be calculated for live;, dead- and wind 
loads; for centrifugal force due to moving load if trestle is on a curve (see 
page 702); and for longitudinal thrust or momentum due to stopping of 
trains (page 702). 

For proportioning the members, see pages 585 and 609; also page 446 for 
beam and column formulas. 

References to numerous designs and details may be found on the follow- 
ing page. 




Fig. 6. 



MISCELLANEOUS DATA. 793 

COST OF RAILROAD TRESTLES. 

Timber Trestles. — (a) Single Track. Cost in dollars per Hn. ft. (approx.) 
= 8 +0.2// 4-0.002^2. Double Track. Cost in dollars perlin. ft. (approx.) == 
16 +0.4/Z + 0.003/Z2. In which H = height of trestle bent, in feet. 

Steel Trestles. — About three to five times the cost of timber trestles. 

Reinforced Concrete Trestles. — Low trestles cost about the same as 
small arch spans. See page 784. 

EXCERPTS AND REFERENCES. 

Wooden Trestles, Utah Central Ry. (Eng. News, Jan. 17, 1901).— 
Illustrated. 

Steel Trestle Viaduct, C. & N.=.W. Ry. (Eng. News April 22, 1901).— 
Illustrated. 

Railway Trestle Bents of Reinforced Concrete (By W. A. Allen. Eng. 
News., Mar. 12, 1903).— Illustrated. 

Reinforced^Concrete Trestlework Viaduct for a Spanish Mineral Ry. 

(Eng. News, May 17, 1906). 

Reinforced=Concrete Viaduct on the Richmond & Chesapeake Bay 

Ry. (Eng. News, Dec. 12, 1907).— Illustrated. 

A Traveler for Viaduct Erection (By L. L. Jewel. Eng. News, Oct. 8, 
1908). — Illustrated. 

The Bear=River Steel Viaduct, Cal. (Eng. News, Mar. 11, 1909). 
— Illustrated details. 

Table for Estimating Quantities in Timber Trestles (By Emile Low. 
Eng. News, April 22, 1909). 

Reinforced=Concrete Viaduct with Some Structural Steel Reinforcement 

(Eng. News, July 1, 1909). — Tower bents composed of two posts, 18 x 18 ins. 
square at the high bents and 15 x 15 ins. at the low, spaced 12 ft. c. to c. at 
the girders and extending on an outward batter of 1 : 6 to spread founda- 
tions on solid rock. Posts are concrete reinforced at the four comers by 
straight round rods, encircled every 12 ins. with ^-in. wire. Longitudinal 
and transverse braces are 9-in. and 12-in. I-beams encased in concrete. 
Illustrated. 

Reinforced=Concrete Trestle, Pasadena, Cal. (Eng. News, Nov. 18, 
1909). — Consists of six girder spans resting on five tower bents; whole length 
divided into panels of 17 ft. 3 ins., each bent being of that length, and each 
span divided into three of such spaces by the floor beams, making a span 
length for each girder of 51 ft. 9 ins. Towers consist of four columns- each; 
columns 18x18 ins. and reinforced v/ith eight l^-in. round steel bars ex- 
tending into the pier footings to within one foot of their base. Nearly all 
the columns are over 50 ft. long and have longitudinal and transverse 
struts framing into their third points to stiffen them. Struts vary in size 
from 10 x 18 ins. to 12 x 24 ins. and are reinforced with four |-in. or four 
1-in. twisted steel bars laced as a column. Illustrations include details of 
expansion joint. 

Important Illustrations of Trestles and Details. 

Description. Eng. News. 

A 5-mile railway trestle across Albemarle Sound Apr. 21, '10 

Willow Creek Viaduct, Des Chutes Railway Aug. 11,' 10 

Eng. Rec 

Stresses in typical tower of a railway trestle Jan. 9, *09 

Design, construction and cost of a rein. -cone, trestle Feb. 20, '09 

Details reinforcement, rein. -cone. R. R. trestle Apr. 3, '09 

The Sligo reinforced-concrete highway viaduct May 15, '09 

Typical tower bent, manuf'rs railway steel viaduct May 15, '09 

A substitute for drift bolts on wooden trestles Tune 5, '09 

Details of steel trestle, Norfolk & Western Ry . . . Mar. 19, '10 

Details 6-post and 4-post pile R. R. trestle, Albemarle Sound.. .Apr. 30, *10 
Standard solid floor railroad trestles Oct. 29, '10 



46.— ROOFS. 

Wind Pressure. — ^The problem of wind pressure which indirectly presents 
itself to the engineer consists in discovering the existing relation between 
the velocity of the wind and its pressure against any surface — right, oblique, 
plane, curved, etc. Then, knowing the maximum velocity of the wind in 
the particular locality in which a structure is to be erected, the probable 
pressure on its surface is deduced within a reasonable degree of accuracy 
by this ratio. The subject never has been treated satisfactorily on the 
grounds of pure theory, while the few practical experiments recorded seem 
to give results not entirely in accord with each other nor with any theory 
yet advanced. 

VelociHes Attained. — ^The velocity of a volume of air moving along 
the surface of the earth increases with its distance above the average surface, 
hence high structures, or those in exposed positions, should be designed to 
resist the greater wind pressures in any locality. Prof. Henry claims that 
on Mt. Washington, N. H., 150 miles per hour has been recorded. This 
probably exceeds by over 50% the maximimi velocity ever attained at the 
average surface of the ground in that State. The tornado which tore up 
a portion of the St. Louis Bridge floor is credited with but 120 miles per 
hour. A hurricane such as occasionally visits the Atlantic coast may attain 
a velocity of 60 miles per hour upward, depending upon exposure. From 
90 to 100 miles per hour is probably the maximum velocity ever attained 
in New York City in the most exposed positions. The Pacific coast is never 
visited by the violent hurricanes incident to other sections of the country. 

Direct Wind Pressure. — When the atmosphere is at rest it exerts a pres- 
sure at sea level of about 14.7 lbs. per square inch; while a cubic foot of dry 
air under one atmospheric pressure (7 60 millimeters of mercury) weighs about 
0.081 lb. per cubic foot. Wind is air in motion caused by a tendency to 
restore equilibrium in atmospheric pressure, at about the same level, by 
air rushing in to replace a heated and rising atmosphere in another locality. 
Hence, the directing force is a tension or tendency to partial vacuum "ahead" 
of the wind, as well as a compression from behind. It is well to bear this 
in mind, as explaining in part at least the uplifting or overturning power 
exerted on roofs, due to suction. This suctional power is not well known, 
and its effect should be considered more in future experiments and inves- 
tigations. Some attempts have been made to deduce a rule for pressure, 
based on the weight of the volume of air moving against the exposed surface, 
taking into account temperature, humidity and barometric pressure, but 
the results have been more or less unsatisfactory. 

Smeaton, 150 years ago, made some crude experiments on wind pressure 
in connection with the power of windmills, and constructed a table from 
the formula 

P= . 005 y2 ; _ _ (1) 

in which P = horizontal normal pressure in lbs. per sq. ft., 

and V = velocity of wind in miles per hour. 
This formula seems to agree fairly well with many experiments on small 
surfaces. Another formula, of the same form as Smeaton's, giving results 
20% less, is used considerably:* 

P = . 004 y2 (2) 

In both of the above the pressure is assumed to be proportional to the 
square of the velocity. Lieut. Crosby's experiments near Baltimore, Md. 
(Engineering, June 13, 1890) to determine the resistance of the air to fast 
moving trains seem to indicate that the pressure P is directly proportional 
to the velocity V, and not to V^, But this conclusion is generally discredited. 

The conclusion from Baker's experiments in connection with the con- 
struction of the Forth Bridge is that the pressures given by Smeaton 'b 
formula (1) are too great for high velocities. In the light, or perhaps better 

* This formula is now used by the U. S. Signal Service. 

794 



WIND PRESSURE. 



795 



"darkness," of modern experimental data, the pressure P may be assumed 
to lie somewhere between the values given by formulas (1) and (2) — near 
the former for low velocities, and near the latter for high velocities. These 
values are deduced in the following table; and two columns are added giving 
results from the experiments of Eiffel and Stanton. 

1. — Direct Normal Wind Pressures. 



Velocity V 


P= .005^2 


P= .004^2 


P = .003iF2 


P = .003 F2 




of Wind 


(1) 


(2) 


(sm. areas) 


(Ig. areas) 




in Miles 


Pressure P 


Pressure P 


Pressure P 


Pressure P 


Remarks. 


per Hour. 


Lbs. per 


Lbs. per 


Lbs. per 


Lbs. per 






Sq. Ft. 


Sq. Ft. 


Sq. Ft.* 


Sq. Ft.* 




10 


.50 


.40 


.33 


.30 




20 


2.00 


1.60 


1.33 


1.20 


Brisk wind. 


30 


4.50 


3.60 


3.00 


2.70 




40 


8.00 


6.40 


5.33 


4.80 


High wind. 


50 


12.50 


10.00 


8.33 


7.50 




60 


18.00 


14.40 


12.00 


10.80 


Violent storm. 


70 


24.50 


19.60 


16.33 


14.70 




80 


32.00 


25.60 


21.33 


19.20 


Htirricane. 


90 


40.50 


32.40 


27.00 


24.30 




100 


50.00 


40.00 


33.33 


30.00 


Violent hurri- 


110 


60.50 


48.40 


40.33 


36.30 


cane. 


120 


72.00 


57.60 


48.00 


43.20 


Tornado. 



Before considering the resolution of a direct wind pressure into its nor- 
mal, vertical and horizontal components, as practically applied to roofs and 
other structures, it will be well to emphasize the above hints regarding 
pressure and tension (suction) on any exposed body. If the surface of a 
thin flat sheet is exposed to the direct force of the wind, there will be tension 
or suction on the leeward face, due to parti^ vacuum, thus producing- 
apparently additional pressure on the windward face and as these forces 
act in the same direction, the resultant "wind pressure" is thereby in- 
creased. The tension may be reduced by placing a long tapering pro- 
jection on the leeward face of the plate, flush with the edge, to prevent the 
formation of air eddies. Further, ii the windward face is convex, 
the pressure also will decrease, while if it is concave the pressure will 
increase. The thickness of the plate within certain limits is also a factor, 
as well as the density and humidity of the atmosphere. 




a^JA 

Fig. 1; 

Normal and Component Wind Pressures. — Let P, Fig. 1, be the direct 
horizontal pressure per square foot on any vertical surface, and Pn the 
normal pressure on the same unit of inclined stu-face, sloping at angle A 
with the horizontal. Then — 

By Abstract theory, Pn = P sin2 A (3) 

By Hutton's experiments, Pn = P (sin AY-^ cos a-i (4) 

By Duchemin's formula, Pn = P -t . „ ^ (5) 

l + sm2A 
The following table gives values of Pn deduced from these three for- 
mulas, assuming P = 50, 40 and 30 lbs. Use Hutton or Duchemin. 

*From experiments by M. Eiffel and Dr. Stanton; on velocities of 40 to 
90 miles per hour. See, also, remarks under Railroad Bridges, Section 38, 
page 697. 



796 



iQ.— ROOFS. 



-Normal Wind Pressures Pn on Inclined Surfaces.* 
For horizontal pressures of 50, 40 and 30 lbs. 





Pitch 




50 Lbs. 






40 Lbs 






30 Lbs 




Angle, 
A. 


CO O 

-^ (U 




Q 


cc O 


^i 
" ? 

ffi 


O 

Q 


CO O 




.2 
Q 


5° 




0.4 


6.5 


8.61 


0.3 


5.2 


6.89 


0.2 


3.9 


5.17 


10° 
15° 




1.5 
3.3 


12.1 
17.8 


17.00 
24.15 


1.2 

2.7 


9.7 
14.2 


13.59 
19.32 


0.9 
2.0 


7.2 
10.7 


10.19 
14.49 


18°-26' 
20° 


1-6 


5.0 

5.8 


21.2 
23.0 


28.75 
30.30 


4.0 
4.7 


17.0 
18.4 


23.00 
24.24 


3.0 
3.5 


12.7 
13.8 


17.25 
18.18 


21°-48' 
25° 


1-5 


6.9 
8.9 


24.8 
28.3 


32.66 
35.96 


5.5 

7.1 


19.8 
22.6 


26.13 

28.77 


4.1 
5.4 


14.9 
17.0 


19.60 
21.58 


26°-34' 
30° 


1-4 


10.0 
12.5 


29.7 
33.1 


37.27 
40.00 


8.0 
10.0 


23.8 
26.5 


29.82 
32.00 


6.0 
7.5 


17.8 
19.9 


22.36 
24.00 


33°-41' 
35° 


1-3 


15.4 
16.5 


36.6 
37.6 


42.42 
43.15 


12.3 
13.2 


29.2 
30.1 


33.93 
34.52 


9.2 
9.9 


21.9 
22.6 


25.45 
25.89 


40° 
45°-00' 


1-2 


20.7 
25.0 


41.6 
45.0 


45.50 
47.16 


16.5 
20.0 


33.3 
36.0 


36.40 
37.73 


12.4 
15.0 


25.0 
27.0 


27.30 
28.30 


50° 
55° 




29.3 
33.6 


47.6 
49.3 


48.30 
49.01 


23.5 

26.8 


38.1 
39.4 


38.64 
39.21 


17.6 
20.1 


28.6 
29.6 


28.98 
29.41 


60° 

65° 


1-1 


37.5 
41.1 


50.0 
50.0 


49.68 
49.78 


30.0 

32.9 


40.0 
40.0 


39.74 
39.82 


22.7 
24.6 


30.0 
30.0 


29.81 
29.87 


70° 

75° 




44.2 
46.7 


50.0 
50.0 


49.89 
49.95 


35.3 
37.5 


40.0 
40.0 


39.91 
39.96 


26.5 
28.0 


30.0 
30.0 


29.93 
29.97 


80° 

85° 




48.5 
49.6 


50.0 
50.0 


50.00 
50.00 


38.8 
39.7 


40.0 
40.0 


40.00 
40.00 


29.1 
29.8 


30.0 
30.0 


30.00 
30.00 


90° 




50.0 


50.0 


50.00 


40.0 


40.0 


40.00 


30.0 


30.0 


30.00 



*The normal pressures given in the table are in lbs. per square foot of 
inclined surface of exposed roof — from horizontal winds producing pressures 
of 50, 40 and 30 lbs. per square foot on vertical surfaces. 

Example. — Assuming direct wind pressure to be 50 lbs. per square foot 
of vertical surface, and using Hutton's formula, find from the above table 
the normal pressure in lbs. per square foot on a roof with a pitch of one in 
two. 

Answer — 45 lbs. per square foot. 



WIND PRESSURE. SNOW LOADS. 



797 



In designing roofs and buildings it is convenient to use the nonnal 
pressure against the roof surface, and to know also its vertical and hori- 
zontal components. The subjoined table gives these values for the 5 stan- 
dard pitches* of roofs, and based on Hutton's and Duchemin's formulas at 
50, 40 and 30 lbs. direct wind pressure. See table 2, preceding. 

3. — Wind Pressures in Lbs. per Square Foot on Roofs. 







1 


50. 


40. 


30. 


Nor- 


Hori- 


Ver- 


Nor- 


Hori- 


Ver- 


Nor- 


Hori- 


Ver- 


fo 


P^ 


< 


mal. 


zont'l 


tical. 
20.1 


mal. 


zont'l 


tical. 


mal. 


zont'l 


tical. 




1-6 


18°-26' 


21.2 


6.7 


17.0 


5.4 


16.1 


12.7 


4.0 


12.0 


§ 


1-5 


21°-48' 


24.8 


9.2 


23.0 


19.8 


7.4 


18.4 


14.9 


5.5 


13.8 




1-4 


26°-34' 


29.7 


13.3 


26.5 


23.8 


10.6 


21.3 


17.8 


8.0 


15.9 


^ 

K 


1-3 


33°-41' 


36.6 


20.3 


30.5 


29.2 


16.2 


24.3 


21.9 


12.1 


18.2 


1-2 


45°-00' 


45.0 


31.8 


31.8 


36.0 


25.5 


25.5 


27.0 


19.1 


19.1 


d 


1-6 


18°-26' 


28.8 


9.1 


27.3 


23.0 


7.3 


21.8 


17.3 


5.5 


16.4 


6 


1-5 


21°-48' 


32.7 


12.1 


30.4 


26.1 


9.7 


24.2 


19.6 


7.3 


18.2 


1-4 


26°-34' 


37.3 


16.7 


33.4 


29.8 


13.3 


26.7 


22.4 


10.0 


20.0 


^ 


1-3 


33°-4r 


42.4 


23.5 


35.3 


33.9 


18.8 


28.2 


25.5 


14.1 


21.2 


Q 


1-2 


45°-00' 


47.2 


33.4 


33.4 


37.7 


26.7 


26.7 


28.3 


20.0 


20.0 



Probably the low direct pressure value of 30 lbs. per square foot, reduced 
by Hutton'sf (or Duchemin's) formula will be sufficient for general cases; 
the next higher value, 40 lbs., for particularly exposed positions; and the 
highest value, 50 lbs., for special cases as in the tornado belts. 

In open sheds the maximum direct pressure may be assumed as acting 
normal to the inside leeward surface, and the "lifting " force may be ob- 
tained by multiplying the total direct pressure by cos A, the angle of incli- 
nation of roof with the horizontal. 

On a cylinder the theoretical pressure is % that on "a corresponding 
plane diametrical section or rectangular plate; but Borda, by experiment, 
found it to be only 0.57. Likewise, he found the pressure on a sphere to 
be but 0.41 of that on a corresponding circular plate of the same diameter, 
while theory gives 3^. In general, the pressure on a concavity, measured 
by the diametrical plane surface, is greater than unity; while on a convexity 
it is less than unity. The intensity of pressure may be increased by the 
deflection of the wind from an adjacent structure. 







f* 


^^ 


^^ 








^ 


^Y^ 


-=rtsf5?" 


^^ ^ 


?-» 


xij^ 


/Q^ 



Fig. 2. — Roof "Pitches." 

Snow Loads. — ^The snow loads which may come on a roof will vary with 
the latitude of the place; its altitude above sea level; the general humidity 
of the atmosphere; the winter temperature; the location with respect to 
mountain ranges; the pitch of the roof (Fig. 2); the character of roofing. 
The writer is familiar with the character of the snow fall in nearly 
every State in the Union and in Canada, For the Pacific slope west 
of the Coast and Cascade mountain ranges, there is no need to provide for 
any snow -load, but up in these ranges and in Eastern Washington, Eastern 
Oregon, Northern California, and in all sections eastward to the Atlantic 
coast, provision must be made. The heaviest snow-falls in the United 



* The pitch of a roof is one-half the natural tangent of inclination with 
the horizontal. 

t The writer believes Hutton's formula, founded on practical experi- 
mental data, to be quite well established and fairly reliable. Duchemin's 
formula gives values about 25% higher for the ordinary 1-4 pitch. 



798 



46.— i?OOFS. 



States are in the Central Northwest, New England and the Rocky Mountain 
regions. The following diagram, giving the snow load per horizontal 
square foot of roofs for different localities and for standard roof pitches, 
will be found practically reliable. The latitude of the place is considered 
as increased one degree for each thousand feet in altitude above sea level. 




•(a) For Pacific Slop? 50 32 54- 56 

*Cb)BaI.of N.AmJO 32 34 36 38 40 42 44 46 48 

Latitude in Degrees + one Degree for each WOOff. 
in elevation above Sea Level. 
Fig. 3. 

Example, — ^To find the snow load in pounds per horizontal square foot 
at Denver for a pitch of 1 in 5? The latitude of Denver, 40, plus itjoo of 
its altitude in feet above sea level, 5, gives 45. Using line_ (b) of above 
diagram this is equivalent to a snow load of 28.2 lbs. per horizontal square 
foot for a pitch of 1-5. 

Investigations by S. de Perrot, in Switzerland, according to the ''Engi- 
neer'' (London), show that where a heavy fall of snow is followed by thawing 
and freezing and then more snow, in repeated cycles, the laminar mass of 
snow and ice will have a weight of 36 to 38 lbs. per cu. ft. ; and the thickness 
of the mass on the roof, from 24 to 32 ins., will produce a load of 70 to 100 
lbs. per sq. ft., about 2 to 4 times the weight ordinarily assumed in calcu- 
lations. 

Roof Coverings. — Materials for roof covering are selected for protection 
against rain, snow and other natural agencies. They should be light, 
durable, economical and more or less artistic; and their selection will be 
dependent on the character of the building, its location with respect to 
climate, the amount of acids and injurious gases in the atmosphere, and 
the pitch of the roof. Among the most commonly used materials for 
"pitch" roofs are shingles, slate and tile for residences; slate, gravel and 
tile for railway structures; corrugated steel for warehouses. For tem- 
porary structures, as exposition buildings, the patented roofings are generally 
used, and then relaid elsewhere. For "flat" roofs, tin, tar and gravel, 
asphalt, and other compositions are preferred. 

*West of the Coast and Cascade Mt. ranges, tin general. 



ROOF COVERINGS— SHINGLE, SLATE. 



799 



Shingle Roofing. — Shingles are made of white cedar, red cedar, spruce, 
pine, fir and cypress.* In the United States the life, in years, of cedar 
shingles will correspond to about the latitude of the place in degrees; pine 
will last about one-third to one-half as long as cedar, depending upon the 
climate. The following table is based on shingles 4" wide, and with an 
average thickness of 1/5". The weight of cedar is assumed at 36 lbs. and 
pine at 40 lbs. per cubic foot. The length of shingle is a little over 3 times 
the "weather." 

4. — Weight op Shingles 0.2 Inch Thick, Laid on Roofs. 
(Weight is proportional to thickness of shingles.) 









Shingles 


Weight per 




Weight 




Assumed 


Weather 


per 


Square of 100 


Nails 


of Nails 


Length. 


Width. 


or 


Square 


Sq. Ft. 


per 


per 


Ins. 


Ins. 


Gage. 


of 100 




Square. 


Square. 










Ins. 


Sq. Ft. 
Number. 


Cedar. 
Lbs. 


Pine. 
Lbs. 


Number. 


Lbs. 


14 




4 


900 


210 


233 


1800 


4.50 


15 




4i 


800 


200 


222 


1600 


4.00 


16 




5 


720 


192 


213 


1440 


3.60 


18 




5i 


655 


197 


218 


1310 


3.28 


20 




6 


600 


200 


222 


1200 


3.00 


22 




6i 


554 


203 


226 


1108 


2.77 


24 




7 


515 


206 


229 


1030 


2.58 



Shingles are nailed directly on shingle laths or on solid V sheathing 
covered with tarred paper, using two 1^'' nails to each shingle. The laths are 
from two inches wide upward, spaced a few inches apart, and nailed hori- 
zontally to the jack-rafters. 

Slate Roofing. — Slates are laid shingle fashion. The length of slate is 
usually 2 times the "weather" + 3 inches. The following table is based on 
the above. The number of slates per square of 100 sq. ft. = 14,400-h (width 
X weather). 

5. — ^Number of Slates per Square, Laid on Roof. 
(See below for weight per sq. ft. of slate unlaid.) 



^ 




Slates 






Slates 






Slates 




Wea- 


per 100 




Wea- 


per 100 




Wea- 


per 100 


Size. 


ther or 
Gauge. 


Sq. Ft. 


Size. 


ther or 
Gauge. 


Sq. Ft. 


Size. 


ther or 
Gauge. 


Sq. Ft. 


Ins. 


Ins. 


Nimi- 
ber. 


Ins. 


Ins. 


Num-. 
ber. 


Ins. 


Ins. 


Niim- 
ber. 


6x12 


41 


533 


8x16 


6i 


277 


10x20 


H 


170 


7 12 




457 


9 16 


u 


246 


12 20 


« 


141 


8 12 


<( 


400 


10 16 


(( 


222 


14 20 


M 


121 


9 12 


" 


356 


11 16 


(( 


202 


12x22 


9i 


126 


10 12 


<( 


320 


12 16 


u 


185 


14 22 


(( 


108 


7x14 


5i 


374 


9x18 


n 


213 


12x24 


10^ 


114 


8 14 




327 


10 18 


u 


192 


14 24 




98 


9 14 


" 


291 


12 18 


M 


160 


16 24 


« 


86 


10 14 


a 


262 


14 18 


« 


137 


14x26 


lU 


89 



Note. — Sizes range up to 24''x44''. 

At 174 lbs. per cubic foot the weight of one square foot of slate at various 
thicknesses is as follows: 

Thickness, in inches Y iY i" ¥ V T V 1" 

Weight, in pounds . 1.81 2.72 3.62 5.44 7.25 9.06 10.88 14.50 



* Good heart -cypress shingles are now almost unprocurable. 



800 



i6.— ROOFS. 



6. — Total Weight of Slate per Square of Roof, 
(Weather or gage as per Table 5.) 



Length 

of Slate. 

Ins. 


Weight in Lbs. per Thickness. 


V 


^" 


r 


r 


r 


r 


r 


V 


12 
14 
16 
18 
20 
22 
24 
26 


483 
461 
446 
435 
427 
420 
414 
410 


725 

692 
669 
652 
640 
630 
621 
615 


967 
923 
892 
870 
853 
840 
828 
820 


1450 
1384 
1338 
1305 
•1279 
1259 
1242 
1230 


1934 
1845 
1785 
1740 
1706 
1679 
1657 
1639 


2417 
2307 
2231 
2175 
2133 
2099 
2071 
2048 


2900 

2768 
2677 
2610 
2559 
2518 
2485 
2459 


3867 
3691 
3569 
3480 
3412 
3358 
3313 
3278 



Note. — -ra" slates are the most common. 

Slate may be nailed on wooden sheathing, laths, porous terra cotta, 
reinforced concrete sheathing, etc., with or without felt between. The 
nails may be of malleable iron, copper, zinc, or composition metal. Iron 
or steel nails should be tinned or galvanized. The slates are often fastened 
directly to the sub-purlins by copper wire. Slater's cement makes a good 
tight bond and is recommended for fiat pitch, which should not be less than 
1-4 for slate roof. 

Tile Roofing. — ^Tiles are made of terra cotta (baked clay), glass, and 
metal. 

Clay tiles come in various shapes and under different names, as plain or 
flat, groove-and-fillet, pan, Spanish, etc. The plain clay tiles, say 6i''xl0^'' 
x^" thick, will weigh from 15 to 18 lbs. per sq. ft. of roof when laid 51'' to 
weather. The weight of porous terra cotta roofing in lbs. per square foot 
= 4 (thickness in inches -1- 1). 

The M. W. Powell Co.'s* Specifications for a Tile Roof are: First cover 
the roof foundation with 6 thicknesses of No. 1 wool roofing felt, weighing 
not less than 15 lbs. (single thickness) per 100 sq. ft.; the felt to be laid 
smoothly and evenly, and well cemented together, not less than 9 ins. 
between each layer, with roofing cement. All joinings along the walls 
and around the openings to be made carefully. The roof then to be covered 
with actinolite cement, and vitrified tile to be laid on this surface, the joints 
of the tile to be made with marmolite cement. The tile to be 6''x9"xf''' 
thick. All walls and openings should be flashed with copper. The surface 
of the roof foundation should be perfectly smooth before the felt is laid. 

Glass tiles are used principally for skylights. 

Metallic tiles, of copper, zinc, iron, tin, etc., are made up in artistic 
forms and laid as shingles. 

Tin Roofing. — ^Tin plate proper consists of thin sheets of iron or steel 
coated with tin by dipping and rolling. When the molten tin is adulter- 
ated with lead the product is terne plate, which is much cheaper. For the 
base, "charcoal" plates are better than "coke." 

Roofing tin comes commonly in sheets of two sizes, f 14''x20'' and 20''x 
28''. As manufactured by the American Sheet .and Tin Plate Co. of Pitts- 
burg, there are 112 14x20 sheets, or 56 20x28 sheets, in a box. The sheets 
are graded as "Primes" and "Wasters." The Primes or perfect sheets 
are branded according to thickness or weight of iron body, as IC (i lb. per 
sq. ft.) and IX (f lb. per sq. ft.). The IC 20x28 plates manufactured 
by Merchant and Evans, of Philadelphia, weigh about 215 lbs. net per box 
of 112 sheets, and the IX 20x28 plates weigh about 270 lbs. net. The IC 
and IX plates are supposed to have the same thickness of coating. 



* Manufacturers of patent roofing materials, cements, etc., 204 Dearborn 
St., Chicago, 111. 

t Sheets may be obtained in sizes 10x14 and multiples thereof. 



TILE-, TIN-, CORRUGATED-STEEL ROOFING. 



801 



Fig. 4. 




The tin sheets are laid on the roof in two ways, viz., with ^a^seam (Fig. 
4) , or with standing seam (Fig. 5) . The flat seam is preferred for roofs of 
small pitch, using 14x20 

plates; the standing seam . 

for steep roofs, using 20x28 
plates, generally. When 
laid with flat seam, a box 
of 112 sheets 20x28, con- 
taining 435 sq. ft. of plate, 
will cover about 384 sq. 
ft. of roof, about 12% be- 
ing used in seams; and 
laid with standing seam it 
will cover about 370sq. ft., 
with a loss of about 15%. 
If 14x20 sheets are used, Fig. 5. 

the percentage of loss is slightly greater. Inversely, to cover 1000 sq. ft. 
of roof will require 583 sheets of 14x20 if laid with flat seam, or 303 sheets 
of 20x28 if laid with standing seam. 

For fastening flat-seam roofing, use 1" barbed and tinned roofing nails 
about 6" apart and well under edges of seams. In soldering use rosin (not 
acid) as a flux. In laying standing-seam roofing the sheets are locked and 
soldered together in long rolls from ridge to eaves. The standing seams 
are not soldered, but are locked together and held in place with tin cleats 
spaced 15 to 18 ins. apart, through which nails are driven. 

Sheet Steel Roofing. — Sheet steel, "black" or galvanized, and of Nos. 
26, 27 and 28 gauge, is used for roofing. It comes in sheets, or in rolls up 
to 50 ft. in length. The sheets are laid with horizontal flat seams, or by 
lapping, and with vertical standing seams, crimped, with tin fastenings. 

Corrugated Steel Roofing. — Corrugated steel, "black" or galvanized, 
and of Nos. 16, 18, 20, 22, 24, 26 and 27 gage is usually laid directly on the 
purlins. To these sheets are fastened clips of various kinds by means of 
clinch rivets or bolts. The clips may 
hook under some edge of the purlin or 
pass completely around it, as in Fig. 6. 
They are usually spaced 12 inches apart. 
Corrugated sheets are rolled from plain 
sheets 30'' wide and in lengths up to 10 
feet. The standard corrugation is about 
2^" wide and f" deep, which narrows the 
30'' sheet down to 27^. Allowing for 
side lap, the net *'weather" width is re- 
duced to about 24". Lengthwise, the 
sheets should preferably rest on three 
purlins to give continuous-girder strength, 
although the strength is calculated for a 
simple span. The horizontal or "end" 
lap should be from 4" to 8", the latter for the flatter pitches and where 
the use of slater's cement is desirable. The average end lap is say 6", 
hence for sheets 4-ft. long the "weather" area is but 70% of the original 
flat sheet; 5-ft. long, 72%; 6-ft. long, 73.3%; 7-ft. long, 74.3%; 8-ft. long. 
75%; 9-ft. long, 75.6%,; 10-ft. long, 76%. In other words the weight 
of corrugated sheets per square foot of roof, laid, is respectively 43, 
39, 36i, 35, 33i 32^ and 3U per cent, greater than that of the plain sheet 
metal. Galvanizing adds about one-third of a pound per square foot to 
flat metal, or, say, three-eighths of a pound per square foot to corrugated. 

The strength of corrugated steel roofing may be obtained from the 
formula, 

M=fb dt-T- 45 
In which M = bending moment or resisting moment, in /t.-lbs.\ 
t = thickness of metal, in ins.; 

/ = allowable stress in lbs. per sq. in. in the metal; 
d = depth of corrugations, in ins.; 
b = breadth in ins. of loaded sheet before corrugating. 

Tar-Gravel Roofing. — Gravel roofing is usually laid on wooden sheathing 
covered with roofing felt. The felt is laid in several thicknesses, with lap 




802 



iQ.^ROOFS. 



joints well smeared with tar. It is then covered with a good coating of 
hot tar over which is spread a layer of clean, fine-screened gravel. About 
IJ cu. ft. (9^ gallons) of tar and 4^ cu. ft. of gravel will cover 100 sq. ft. 
of roof. Gravel roofing, above the sheathing, may be assumed to weigh 
about 8 lbs. per square foot. 

Cement-Gravel Roofing. — ^This is similar to tar-gravel roofing, except 
that cement is used in place of tar. It will weigh from 8 to 10 lbs. upward 
per square foot, above the sheathing. In the case of tall oflice buildings, 
a very heavy cement pavement is often used. 

Asphalt-Gravel Roofing. — ^This is like tar-gravel roofing, but with 
asphalt used instead of tar. It will weigh about 8 lbs. per square foot 
above the sheathing. Prepared asphalt roofing is sold in rolls of about 110 
square feet, weighing about Ij lbs. per square ft., gravel surfaced, and less 
than 1 lb. per square ft., sand surfaced. 

. Slag Roofing. — ^This is laid like the various kinds of gravel roofing, but 
using slag instead of gravel— with tar, roofing cement or asphalt. It is 
laid on several thicknesses of felt and weighs 8 lbs. or more per square ft., 
above the sheathing. 

Patented Roofings. — Many patented roofings are shipped in rolls or 
sheets ready to be put in place. Among them may be mentioned the fol- 
lowing: Prepared asphalt roofing, which may have a sand or gravel surface; 
Asbestos roofing, composed of asbestos felt on a canvas base; Sparham 
roofing, a pulverized talcose lime rock mixed with coal-tar and pitch and 
applied hot, with a trowel; Rubberoid, a wool felt saturated with a parafiine 
preparation; Perfected granite roofing, a tarred paper with a sea-grit or 
quartz-pebble surface; Ferroinclave, a special corrugated steel plastered 
with Portland cement and sand about J or f of an inch above and below 
the corrugations. There are a large number of patented roofings on the 
market, some of them excellent. The engineer should satisfy himself as to 
their fire-proof qualities, durability, etc. 

Weight of Roofing Materials. — ^The following table gives the approxi- 
mate weight of roofing materials furnished by the manufacturers. 

7. — Approximate Weight of Roofing Material. 
(See, also, Section 27.) 



Material. 



Weight per 

Square Foot. 

Lbs. 



Above 

the 

Sheathing 

or Wood 

Boarding. 



Sheathing. 



Clay Tiles, 103^x6Mx^ in., 5H ins. to weather 

Copper, 16 oz. standing steam 

Corrugated galvanized iron No. 20 

Corrugated galvanized iron No. 26 

Felt and asphalt or coal-tar 

Glass, H in. thick 

Lath and plaster ceiling (ordinary) 

Lead, about 3^ in. thick 

Mackite, 1 in. thick, with plaster 

Neponset roofing, felt, 2 layers 

Shingles, 6x18 ins., H to weather 

Skylight of glass, i^ to 3^ in., including frame. 

Slag roof, 4-ply 

Slate, 3^ in. thick, 3 in; double lap 

Slate, -rs in. thick, 3 in. double lap 

Spanish Tiles, 16 oz. Copper 

Spanish Tiles, IC Tin 

Teme Plate, IC 

Terne Plate, IX 

Tiles (plain), 10ix6ixf ins., 51 ins. to weather 
Tiles (Spanish), 14^x10^ ins., 7i ins. to weather 

Hemlock sheathing, 1 in. thick 

Spruce sheathing, 1 in, thick 

White pine sheathing, 1 in. thick 

Yellow pine sheathing, 1 in. thick 



18 

1 

2 

IH 
6 to 8 
6 to 8 
10 

2 

[ to 10 

4 

4H 
6M 

1 

18 
83^ 
2 

2H 
23^ 
4 



ROOFING MATERIALS. TRUSSES. 



803 



Common Types of Roof Trusses, (Figs. 7). 

(See, also, Stress Diagrams of five types, Figs. 8). 

Note. — Letters and numbers on diagrams are names of members — not 

Bow's notation. 



ett A -. . . "I KJ. ^: /- 




(b) Double A (c) Complex A. 
oad 




(6) Single and Double Fink ' if) Single and Complex Fan (g) Compound Fan 

WindLoad- 
Bofh endsfixed^-'\ 

A 

(/) Triangular 




WZ/jdload" 
Fixed Side \^\^ Fcr pitch i;4 

J' 



(k) Howe a)Compoundrtnk-Warren 




(m) Praff 




Wind Load-. xoV^* _.— - 






ff)) Cambered Compound Fink (^Compound Pratt- Warren (p) ParaboUc 
R. 
Figs. 7. Truss Diagrams. 

Stress Diagrams of 5 Types for Different Kinds of Loading, (Figs. 8). 
(See, also, Diagrams of Trusses, Figs. 7.) 

Hybrid Type (d). — Dead load stress diagram of one-half truss. Note 
that stress in member 3 is obtained analytically by taking moments about 
the apex. This reduces the number of unknown stresses to two — c and d — 
at the joint. Compare this case with the double Fink type (e) which is 
plainly more economical. 

Double Fink Type (e). — With this case as with the Hybrid type (d) 
the ambiguity is relieved by finding the stress in member 3, analytically. 
We know, however, that in the stress diagram, / is "in line with" a, so 
that it is not necessary in practice to resort to analytics. 

Types (h), (;) and (o). — The stress diagrams (Figs. 8) for these types 
are self explanatory. 

Remarks. — ^There are several methods in use in calculating the stresses 
in roof trusses, based on different assumptions: 

(A). — For practical expediency short spans are often designed on a basis 
of 40 lbs. per square foot of roof, this to include wind, snow and dead loads. 
This method is unscientific, although safe in most cases. 

(B). — A load of 40 lbs. per square foot of roof is sometimes specified to 
cover wind and snow loads, the weight of roof covering and trusses to be 
added. This might do for a certain class of structures, in a certain locality, 
but even then there is no good reason for its adoption. 



804 



I— ROOFS. 



(C) . — ^The only correct 
method is to make the as- 
sumptions as near the true 
conditions as possible. Thus: 

For short spans with both 
ends fixed, the wind load 
and reactions will be con- 
sidered normal to one face 
of the roof as shown in truss 
and stress diagrams (h). 

For long spans where one 
end of truss is fixed and the 
other end is a roller or slid- 
ing end, it should be cal- 
culated for that condition; 
namely, for wind pressure 
on the fixed side, and, again, 
for wind pressure on the 
roller side. See truss- and 
stress diagrams (/) and (o). 

Generally, in all cases, it is 
good practice to design both 
halves of the truss symmet- 
rically, using the maximum 
stresses obtainable by con- 
sidering the wind pressure 
on either side of the roof, 
and either end of truss roller 
or fixed. 



Sfress in 3, analyticaUy 
X3i ^4-6^2=4.00 

For Pitch l-^-^-^i fi 



Stress mS-^OO 




^^Deadload 
id) Hybrid 




f^,-ik-5'H 



%(i Snow Load 
ie) Double Fink 



WindLoad" 
Fixed Side 




Figs. 8.— 

5 Stress 

Diagrams. 



WindLoad 
ff oiler Side. 



to) Compound Pratf'Warren 

8. — Unit Stresses in Pratt Roof Trusses for Unit Loads P. 
(See Figs. 9, Next Page.) 

[+= tension; —= compression. For character of stress see 1 to 4 pitch.] 





A 


B 


C 


D 


I 


10-Panel Pratt. 


8-Panel Pratt. 


6-Panel Pratt. 


4-Panel Pratt. 


•a 


^eo 


^-^ 


^ui 


^co 


^th 


^»o 


rCiCO 


^^' 


A^ 


^co 


A^ 


A^ 


•^ 


f^o 


^o 


^!o 


.H o 


^o 


r. o 


^o 


-Bo 


^o 


^o 


^o 


4^0 






















•rH -^J 








P^rH 


Ph^ 


PM^ 


f^HTH 


PkrH 


^,H 


^r^ 


PhtH 


^^ 


PW.H 


PMrH 


Ph^ 


1 


6.75 


+ 9.00 


11.25 


5.25 


+ 7.00 


8.75 


3.75 


+ 5.00 


6.25 


2.25 


T3.OO 


3.75 


2 


6.00 


+ 8.00 


10.00 


4.50 


+ 6.00 


7.50 


3.00 


+ 4.00 


5.00 


1.50 


+ 2.00 


2.50 


3 


5.25 


+ 7.00 


8.75 


3.75 


+ 5.00 


6.25 


2.25 


+ 3.00 


3.75 


2.70 


-3.35 


4.04 


4 


4.50 


+ 6.00 


7.50 


3.00 


+ 4.00 


5.00 


3.61 


-4.47 


5.39 


2.70 


-3.35 


4.04 


5 


3.75 


+ 5.00 


6.25 


4.51 


-5.59 


6.73 


4.51 


-5.59 


6.73 


1.00 


-1.00 


1.00 


6 


5.41 


-6.71 


8.08 


5.41 


-6.71 


8.08 


4.51 


-5.59 


6.73 


1.25 


+ 1.41 


1.60 


7 


6.31 


-7.83 


9.42 


6.31 


-7.83 


9.42 


1.00 


-1.00 


1.00 








8 


7.21 


-8.94 


10.77 


6.31 


-7.83 


9.42 


1.25 


+ 1.41 


1.60 








9 


8.11 


-10.06 


12.12 


1.00 


-1.00 


1.00 


1.50 


-1.50 


1.50 








10 


8.11 


-10.06 


12.12 


1.25 


+ 1.41 


1.60 


1.68 


+ 1.80 


1.95 








11 


1.00 


-1.00 


1.00 


1.50 


-1.50 


1.50 














12 


1.25 


+ 1.41 


1.60 


1.68 


+ 1.80 


1.95 














13 


1.50 


-1.50 


1.50 


2.00 


-2.00 


2.00 














14 


1.68 


+ 1.80 


1.95 


2.14 


+2.24 


2.36 














15 


2.00 


-2.00 


2.00 




















16 


2.14 


+ 2.24 


2.36 




















17 


2.50 


-2.50 


2.50 




















18 


2.61 


+ 2.69 


2.80 





















UNIT STRESSES IN ROOF TRUSSES, 



805 



Note.— Nos. A, B, C and 
D correspond with similar 
Nos. in Tables 8 and 9. 



P 

8- Panel Pratt 



P 

3^ 




A 10- Panel Pratt 
9.- 






4'PanelPratt 

^4 



C. 6- Panel Pratt 



Figs. 9. 

-Unit Deductions for One-Half Truss (Lean-to). 
Supplementary to Table 8. 

(No deductions for web members.) 
For unit stresses in one-half of each of the above trusses, considered as 
supported at points o and h, or a and c, make the following deductions: 
Bottom chord members. — Deduct from the unit stress in each bottom chord 
member, as shown in Table 8, preceding, the unit stress in the center 
panel member of truss (as shown in black face type). The remainders 
are the unit stresses to be used. 
Top chord m,emhers. — Deduction for each top chord member will be the 
bottom chord deduction multiplied by secant of angle of inclination of 
roof with the horizontal. For 1 to 3 pitch, secant = 1 .202; 1 to 4 pitch, 
sec. = 1.118; 1 to 5 pitch, sec. = 1.077. The following data in connec- 
tion with Table 8 will be found useful: 

A. B. C. D. 

no^,.of (Pitch 1-3: 1.202X3.75 = 4.51; X3.00 = 3.61; X2.25 = 2.70; Xl.50 = 1.80. 

ueauct J .. j_4. 1.118x5.00 = 5.59; X4.00 = 4.47; X3.00 = 3.35-, X2.00 = 2.24. 

lor ^ .. j_g. 1.077X6.25 = 6.73; X5.00 = 5.39; X3.75 = 4.04; X2.50 = 2.69. 

10.— Unit Stresses in Fan and Fink Roof Trusses for Unit Loads P. 

(See Figs. 10, next page.) 
[4- =tension; —= compression. For character of stress see 1 to 4 pitch.] 





E 


F 


G 


H 




Compound Fan. 


Compound Fink. 


Simple Fan. 


Simple Fink. 




■g" 


-s^ 


-g^ 


•g" 


-s^ 


^^ 


•g" 


-g^ 


^^ 


•g" 


•g-^ 


-g-i 


^ 


-M 


-M O 


+J o 


-M o 


•4^ O 


■^ 


■+^ o 


+^ o 


+J 


•4J O 


4j o 


+J o 


e:: 


K:^ 


s::^ 


p^;:: 


'^t: 


s:^ 


'q^:^ 


s:^ 


p::: 


p::: 


s::^ 


p:^ 


1 


8.25 


+ 11.00 


13.75 


5.25 


+ 7.00 


8.75 


3.75 


+ 5.00 


6.25 


2.25 


+ 3.00 


3.75 


2 


6.75 


+ 9.00 


11.25 


4.50 


+ 6.00 


7.50 


2.25 


+ 3.00 


3.75 


1.50 


+ 2.00 


2.50 


3 


4.50 


+ 6.00 


7.50 


3.00 


+ 4 00 


5.00 


3.39 


-4.70 


5.98 


2.15 


-2.91 


3.66 


4 


7.14 


-10.06 


12.95 


4.65 


-6.48 


8.31 


3.53 


-4.55 


5.58 


2.71 


-3.35 


4.04 


5 


7.28 


-9.93 


12.54 


5.20 


-6.93 


8.68 


4.50 


-5.59 


6.73 


0.83 


-0.89 


0.93 


6 


8.25 


-10.96 


13.69 


5.76 


-7.38 


9.05 


0.93 


-1.08 


1.21 


0.75 


+ 1.00 


1.25 


7 


8.80 


-11.40 


14.07 


6.31 


-7.83 


9.42 


0.93 


-1.08 


1.21 








8 


8.94 


-11.25 


13.66 


0.83 


-0.89 


0.93 


1.50 


+ 2.00 


2.50 








9 


9.91 


-12.30 


14.80 


0.75 


+ 1.00 


1.25 














10 


0.93 


-1.08 


1.21 


1.66 


-1.79 


1.86 














11 


0.93 


-1.08 


1.21 


1.50 


+ 2.00 


2.50 














12 


1.50 


+ 2.00 


2.50 


0.75 


+ 1.00 


1.25 














13 


2.50 


-2.69 


2.79 


0.83 


-0.89 


0.93 














14 


2.25 


+ 3.00 


3.75 


2.25 


+ 3.00 


3.75 














15 


1.50 


+ 2.00 


2.50 




















16 


0.93 


-1.08 


1.21 




















17 


0.93 


-1.00 


1.21 




















18 


3.75 


+ 5.08' 6.25 





















806 



i&,— ROOFS. 



Note. — Nos. E, F, G and H 
correspond with similar Nos. 
in Tables 10 and 11. 



P I jfh F. Compound nnk H-Smp/ef/nk 






1 k-^y^ 






-2- 

£. Compound Fan 




G. Simple Fan 

Figs. 10. 



11. — Unit Deductions for One-Half Truss (Lean-to). 

Supplementary to Table 10. 

(No deductions for web members.) 

Bottom chord members. — Deduct from the unit stress in each bottom 
chord member as shown in Table 10, preceding, the unit stress in the center 
panel member of truss (as shown in black face type). The remainders are 
the unit stresses to be used. 

Top chord members. — Deduction for each top chord member will be the 
bottom chord deduction multiplied by secant of angle of inclination of roof 
with the horizontal. Following are unit deductions for top chords; 

Pitch 'E F G H 

1-3 5.41 3.61 2.70 1.80 

1-4 6.71 4.47 3.35 2.24 

1-5 8.08 5.39 4.04 2.69 

DESIGN OF COMBINATION ROOF TRUSSES. 

Problem. — ^Trusses, 8 panels @ 9 ft. = 72 ft., spaced 14 ft. centers; 
pitch 1 to 4; height, 72^4=18 ft. Covering, slate laid on felt and 1-inch 
spruce sheathing. Loads'. — Spruce sheathing, 3 lbs. per ft. B. M.; slate and 
felt, 9 lbs. per sq. ft.; snow, 20 lbs. per horizontal sq. ft.; wind, 30 lbs. per 
sq. ft. against vertical surface. 

Solution. — For lack of space, hints only can be given. In this calcu- 
lation it is assumed that the wind load and snow 2oad do not act on the same 
face of roof at the same time; but may act separately on either side, or simul- 
taneously on opposite sides: For maximum stresses in sheathing, jack- 
rafters and purlins, assimie the wind load or snow load to act with the 
dead loads; for maximum stresses in the trusses, the wind load is assumed 
to act on one side of the roof and the snow load on the other at the same 
time, the wind load being con- 
sidered as acting (1) on the fixed 
side, and (2) on the roller side; 
also (3) the snow load may be 
considered as acting on the 
whole roof without any wind 
load. The trusses are then 
designed symmetrically, using 
the maximum dimensions of 
twin members in either half of 
truss. The solution in detail is 
as follows: 

Spacing of Jack-Rafters. — 
The horizontal sheathing is 
nailed to the vertically inclined 
jack -rafters, and these are laid 
directly on the horizontal pur- 
lins, which in turn rest on the top chord joists of the roof trusses. 




DESIGN OF COMBINATION ROOF TRUSS. 807 

The strength of the sheathing will determine, within certain limits, 
the spacing of the jack rafters, which spacing will be constant and some 
division of 14 ft., the spacing of the trusses. For the purpose of reference, 
the normal and tangential pressures on roof surface for pitch of 1 to 4 (slope 
6'' in 12") are given below: 

Pressure in Lbs. per Sq. Ft. of Roof. 
Vert. Normal. Tangential. 
Wind (Table 3, from Button's rule) 30 . 15.9 17.8 
Snow (Normal = 20 Xcos2 26°-34'; 

tang = 20 X cos a sin a) 17.9 16.0 8.0 

Slate (Normal = 9 X cos a; tangen- 

tial=9Xsin a) 9.0 8.0 4.0 

Sheathing (Normal = 3 X cos a; tan- 
gential = 3 X sin a) 3.0 2.7 1.3 

It is to be noted in the above table that the normal wind pressure 
(17.8 lbs.) is considered the "direct" pressure, and 15.9 lbs. its vertical 
"component." On the other hand, the vertical loads due to snow, slate and 
sheathing are the "direct" loads, and the normal and tangential forces are 
their "components," The snow load per square foot of roof=20Xcos 26°- 
34'. For the sheathing, the normal pressure due to wind is greater than that 
due to snow, but with the latter there is a tangential pressure of 8 lbs. to 
be considered, which is lacking in wind pressure. Assuming the sheathing 
to be 12 ins. wide, the following two^ alternative conditions obtain as showing 
the forces per lin. ft. acting on the sheathing: 





Fig. 12. — Wind and dead loads on Fig. 13. — Snow and dead loads on 

sheathing. sheathing. 

If / = the allowable outer fiber stress per square inch ( = 800 lbs. for spruce 
sheathing), and ic = length of span in ft., we have: 



. 5.3X^2X6^^2 



144X8X1 
/. 800 = 1 (28.5 + ^K 
Or i<;=6.07ft. 



13.3X^2X6 
^ 144X8X1 

.-. 800=1 (26. 7+i|2^);t2 

Or ^=6.19 ft. 

Hence the jack rafters might be spaced 6 ft. apart, but for practical reasons 
they will be spaced -V = 4'-8" apart, centers. 

Size of Jack-Rafiers (yellow pine). — The unsupported length = panel 
length X secant 26°-34' = 9X 1 . 118= 10 ft. The jack-rafter must resist 
bending due to the maximum normal forces on roof, which are 17.84-8.0 
+ 2 . 7= 28 . 5 lbs. per square ft. (wind acting — no snow load), +• the normal 
force due to the weight of the jack rafter itself. Neglecting the latter for 
the present we have, iy = total normal load = 28.5X 4f X 10, and bending 

moment Mb = 28. 5 X ^ X ^^-^^-^^^ = H & ^2 

in which f = allowable fiber stress per sq. in. = 1300 lbs., 
o = breadth of rafter in inches, 
d = depth of rafter in inches; 

, , ,, 6Mb 6X28.5 14 100X12 .^ , 

hence, 6^2 = __ = _____._.____ =92.1. 

For depth £f = 6, 6 = 2.56; hence use rafters size 3'' x 6" to allow liberally for 
weight of rafter itself. At 4 lbs. per ft. B. M., weight of jack rafter = 6 lbs. 
per lin. ft.; normal component = 5. 4 lbs.; tangential component = 2 . 7 lbs. 



808 



iQ.— ROOFS. 




Purlins (white pine). — The span of the purlins is 14 feet, the distance 
center to center of trusses. They are laid horizontally, resting on the top 
chord joints, and hence spaced 10 ft. apart, centers. There are two con- 
centrated loads on each purlin, namely, where the jack rafters rest, at 
points 4' -8'' apart (Fig. 14). Let P and P\ Fig. 6, 
represent respectively the normal and tangential 
components of each of these loads on the purlin. <-4'-8->jg- 4'- 8'-:j<- 4'-a-?j . 

Neglecting the weight of the purlin itself, the acting r . ._ Tf 

forces are: **^~ ^'^ ^ 

Wind acting. Snow acting. 

' ■ ' ' ' • Fig. 14. 

P P' P P' ^ . 

Wind 831 « ^ i^ 

Snow 747 373 

Slate 373 187 373 187 

Sheathing 126 63 126 63 Vs^-<iiiiiifiiK> 

Jack-rafters. .54 27 54 27 "'i^liFf' 

Total, lbs.. 1384 277 1300 650 Fig. 15.— Purlin. 

Assuming the allowable outer fiber stress /= 1200 lbs. per sq. in. at a and 6, 
and the condition "snow-acting" for maximum, we have: 

use purlins 7"x 10'', which allows liberally for weight of purlins themselves. 
At 3 lbs. per ft. B. M. these will weigh 17^ lbs. per lin. ft. 

Trusses. — ^The trusses are designed for the following Cases, considering 
only the members in the left hand half of the symmetrical truss'. 
Case I. — Dead load over all. 
Case II. — Snow load over all. 
Case III. — Snow load on right half only. 
Case IV. — Snow load on left half only. 
Case V. — Wind on left; left end "fixed." 
Case VI. — Wind on left; left end "roller." 
Case VII. — Wind on right; left end "fixed." 
Case VIII. — Wind on right; left end "roller." 

For maxim \im stresses — 
Combine I with II. 

I with VI or VII. 

I and III with V or VI. 

I and IV with VII or VIII. 

Assuming the weight of the truss at 60 lbs. per lin. ft., the loads per 
joint of truss, for calculating stresses, are as follows: 
Dead load per joint. — Slate, felt and sheathing, 12X14X10=1680 lbs.; 

jack-rafters, 3X6X10=180 lbs.; purlins, 14X17^ = 245 lbs.; truss 540 

lbs.; total, 2645 lbs. 
Snow load per joint (vertical).— 20 X 14 X 9 = 2520 lbs. 
Wind load per joint (normal).— 17.8 X 14 X 10= 2492 lbs. 

Table 12, following page, gives a simimary of the stresses. 

Remarks. — The preceding principles used in the design of jack-rafters, 
purlins, trusses, etc., of the combination truss, can readily be applied to 
the design of steel roofs. 

For weight of steel trusses and purlins, see pages 810 and 811. 



DESIGN OF COMBINATION ROOF TRUSS. 



809 



Size 
of 
Mem- 
bers. 


00 

x: : : 


So 
x: : : 


•4-> +-> ex 

% 1/1% w% I^ 

„ CO aco aco ^ 
° X m « ;3 Xo (u 


Maxi- 
mum 
Stress. 


iCiO^CO 


t^COC<li-l 


i-H t^ t^ Tjl 00 OS 


ir^t^fMco 

CO CO CO (M 

+ + + + 


COOilOr-H 

1 1 1 1 


CO <M 1>- »0 OS >0 

1+1+17 


Cases 
Combined 

for 
Maximum. 


III, V. 
Ill, V. 
Ill, V. 
Ill, V. 


IV, VII (or VIII). 
IV, VII (or VIII). 
IV, VII (or VIII). 
IV, VII (or VIII). 


III, V (or VI). 
Ill, V (or VI). 
Ill, V (or VI). 
Ill, V (or VI). 
Ill, V (or VI). 
Ill, V (or VI). 
Ill, V (or VI). 


HHh-ll— IHH 


t— II— II— ll-H 




i 

o 

'd 
c 


Left 
End 

Roller. 

VIIL 


+ 5.6 
+ 5.6 
+ 5.6 
+ 5.6 


CO CO CO CO 

CD CO CO CO 

1 1 1 1 


OOOOOOiM 
OOOOOOtJh 

+ 


Left 
. End 
Fixed. 
VIL 


T-It— iT-tl— 1 


CO CO CO CO 


000000<M 


r-li-lrHT-l 
+ + + + 


CO CO CO CO 

1 1 1 1 


OOOOOOrJ^ 

+ 


M-l 

o 


Left 

End 

Roller. 

VI. 


TtlTjHCOOS 


CO^OtPCO 


T-H Tt^ 00 (M 


Oi 05 CO CO 

++++ 


»OI>-OSth 

1117 


CO rH ^ <M »0 '^J* 
1 + 1 + 1 + 


Left 

End 

Fixed. 

V. 


00(MTf< 


CO»Crt<CO 


Ot-HTtiO00O(M 


rJ<T^T-IOO 
+ + + ^ 


1 1 17 


CO i-H -^ (M lO Tt< 
1 + 1 + 1 + 




On 

Left. 

IV. 


cOcOtHiO 


co»ooqTH 


(M 10 »0 CO 00 


rHTHT-l ■ 
+ + + ^ 


»ooo^-^ 
''11 


0(MrHCO(M^CO 
i + I + 1 + 


On 

Right. 

IIL 


oooo 


CD CD CO CO 


00000000 


++ + + 


1 1 1 1 


OOOOOOCO 

+ 


Over 

All. 

11. 


COCOr-llO 


<NrHOOI>. 


05 (M iO »0 CO »0 


i-H T-H 1— I 1-H 
+ + + + 


i-l-^C0 05 

T-l l-H T-l 1-H . 

1 1 1 1 


CO i-H CO (M Ttl t^ 

1 + 1 + 1 + 


Dead 
Load. 

I. 


»010 05(M 


OOOOt^t- 


oocot-coooos 


00 00 lO CO 

>— 1»— « t— 1 T— 1 

++ + + 


1 1 1 1 


CO t-H CO <M Ttl t^ 

1 + 1 + 1 + 




1 

a 


T-l(MCOTji 

•pjoqo 
uio:^:^og 


lOCOt^OO 

•pjoqo 
dox 


Oi T-( (M CO tM 10 

T— t 1—K T— t T— 1 T— 1 T—t 

•SJ9qUJ9I^ 







•52 




I- 






1-4 


II 


""'fn 








►^ 


11 








-g 


ol^ 


(i3 




.2 


A 
^ 




^ 




^ 


r! 


CO 






<u 




C M 


00 






ri 


-— N 


0' 


-<l 


A<z> 


c3 
G 
CO 


(1) 




teo 


.Q 
nJ 


C) 


1 


&U3 








^+ 


cq 
1>. 





g 




cd < — 1 




> 


r, 


11 


aJ ^ 


4J 
0) 


■§ 





•S 


t3 M 


C/2 


i 


^ 




■^a 


[8 




u 

a 


«1 


(/2 


^~^ 


•S 


tfj 


1 




T1 


^ 


CO* 






3 


.s 


rj r/T 






rt 




V3 «H 


.5? 




r! 


aj 


^^ 


p^ 




^ 


1^ 


V>6 






(I) 


rU 








^ 













< 




810 



46.— ROOFS, 



Details of Design.— Beiore leaving the subject of the design for com- 
bination roof truss, three points in the details of design will be considered, 
namely, (a) the center lower chord splice, (6) the end corbel, and (c) the 
center lower chord block. 

(a). — ^The writer can conceive of no case in practice where the tensile 
strength of a full main wooden member is used in proportioning that mem- 
ber in tension. But in the case of a splice, as shown in Fig. 17, the tensile 

strength of the net sec- ^ Oak Sp lice 

tions of main member : [ ' "^ 

and of splice have tobe^ YeUo\itPme 
considered, as well as ^ ^'*^' i 
the shearing values j 

along the grain, and the 
end fiber bearing values. 
Fig. 17 is the splice de- 
signed for center of low- 
er chord of roof truss 
shown in Fig. 16. 



iCj. 



Top 



- ^'i' \ - 



View 



2- , lY 



YellowPSne 
B'kb' 



l<--a-l3"-X-b-/7' ->U— - 17"— >^~-l3'-->l 



3Jde i V/etv 



Fig. 17.— Bottom Chord Splice. (Tension 26,600 lbs.) 

^ ^, . , . ,, Yellow pine. Oak. 

Data. — Sheanng along gram, lbs. per sq. in 100 130 

Bearing 1500 1500 

Tension 1500 1500 



For shear: Oak keys, a = 
For bearing: c = 

For tension: Oak keys, o = 



= 13''; Yellow pine, 6 = 

in call ir. 



26600 
2X8X100 



17' 



= IV\ Yellow pine, y = 2y'. 



26600 

2X8X130 
26600 

2X8X1500 
26600 

2X8X 1500 
(6.) — Fig. 18 shows the detail at end 
of truss. When a corbel is used it has 
to be long enough to give proper shear- 
ing surface at the right of each key, 
while the shear on the bottom chord 
itself is at the left of each key. Vari- 
ous devices are used to resist the 
thrust of the rafter. In addition to 
the \Y notch at its toe, there may 
be a bolt or strap a b used in connec- 
tion with a corbel; or a shorter bolt 
a p without the corbel ; or (still without 
the corbel) straps a p and p s may be 
joined by a common pin p. 

(c.) — Detail of lower chord block 
(cast iron) is shown in Fig. 19. Where 
the block is not used the lower chord is 
simply notched as shown in Fig. 20. 

Framing Table or Table of Squares. ^i^- ^^- ^^S^ 20. 

— For calculating lengths of roof-truss members, use Tables of Squares, 
Sec. 33, pages 643 to 664. (See problems on page 638.) For calculating 
Howe truss braces and blocks see page 635. 

Weight of Steel Construction in Roofs. — Where trusses, purlins, etc., 
are of steel the following formulas may be used in obtaining the approxi- 
mate weights prior to actual calculation: 
Weight in lbs. of metal in Trusses, ) _ 1 f^/'^i ^\ n\ 

per hor. sq. ft. of roof } ~ 5 V "^ "^ 10/ ^ ^ 

Weight in lbs. of metal in Trusses, ^ i ( With minimum ) 

Piurlins and Bracing, per hor. >• =7^ S< value of about >• (2) 

sq.ft. of roof ) ^^ iior5. ) 

In which S = span of trusses, in feet. 

Weight of Steel Trusses and Purlins. — The above formulas, (1) and (2), 
are, of course, only approximate. 

Table 13, following, gives more exact weights of trusses and of purlins 
for the specified horizontal load of 50 lbs. per square foot of building. 




WEIGHT OF STEEL TRUSSES AND PURLINS. 



811 



13. — Steel Roofs — Approximate Weight of Trusses and Purlins. 
(Based on Uniform Load of 50 Lbs. per sq. ft. of Building.) 



Span 


Distance Center to Center of Trusses, in Feet. 


m 
Feet. 


6 


8 


10 


12 


14 


16 


18 


20 


22 


24 



16 


1.61 


1 1.51 


18 


1.77 


1.66 


20 


1.92 


1.81 


22 


2.07 


1.95 


24 


2.21 


2.08 


26 


2.34 


2.21 


28 


2.46 


2.33 


30 


2.59 


2.45 


35 


2.87 


2.72 


40 


3.13 


2.97 


45 


3.36 


3.20 


60 


3.57 


3.41 


55 


3.77 


3.60 


60 


3.95 


3.78 


65 


4.11 


3.95 


70 


4.26 


4.10 


75 


4.41 


4.25 


80 


4.55 


4.38 


85 




4.51 


90 




4.62 


100 




4.84 


110 






120 







1.42 


1.34 


1.27 


1.57 


1.48 


1.40 


1.70 


1.61 


1.53 


1.84 


1.74 


1.65 


1.96 


1.86 


1.77 


2.09 


1.98 


1.89 


2.20 


2.10 


2.00 


2.32 


2.21 


2.10 


2.58 


2.46 


2.35 


2.83 


2.70 


2.59 


3.05 


2.92 


2.80 


3.25 


3.13 


3.00 


3.45 


3.30 


3.19 


3.63 


3.49 


3.36 


3.79 


3.65 


3.52 


3.95 


3.80 


3.67 


4.09 


3.95 


3.81 


4.22 


4.08 


3.95 


4.35 


4.21 


4.07 


4.47 


4.33 


4.19 


4.69 


4.55 


4.41 


4.88 


4.74 


4.61 


5.05 


4.92 


4.79 











1.27 
1.39 
1.50 








1.33 
1.44 






1.38 




1.61 


1.54 


1.48 


1.42 


1.72 


1.65 


1.58 


1.52 


1.82 


1.75 


1.68 


1.61 


1.92 


1.85 


1.77 


1.70 


2.16 


2.08 


2.00 


1.92 


2.38 


2.29 


2.21 


2.13 


2.59 


2.49 


2.40 


2.32 


2.78 


2.68 


2.59 


2.50 


2.96 


2.86 


2.76 


2.67 


3.13 


3.02 


2.92 


2.83 


3.28 


3.18 


3 08 


2.98 


3.43 


3.32 


3.22 


3.13 


3.57 


3.46 


3.36 


3.26 


3.70 


3.59 


3.49 


3.39 


3.83 


3.72 


3.61 


3.51 


3.95 


3.84 


3.73 


3.63 


4.17 


4.06 


3.95 


3.85 


4.37 


4.25 


4.14 


4.04 


4.55 


4.43 


4.33 


4.23 



Weight of Trusses in Lbs. per sq. ft. of Building. 

1.21 
1.33 
1.46 
1.58 
1.69 
1.80 
1.90 
2.01 
2.25 
2.48 
2.69 
2.88 
3.07 
3.24 
3.40 
3.55 
3.69 
3.82 
3.95 
4.07 
4.28 
4.48 
4.66 

Weight of Purlins in Lbs. per sq. ft. of Building. 
I 0.15 I 0.20 I 0.25 I 0.30 I 0.35 | 0.40 | 0.45 | 0.50 | 0.55 | 0.60 

Reference. — See, also, Sec. 47, Buildings. 

EXCERPTS AND REFERENCES. 
Concrete Platform and Umbrella Roof of Union Station, at Dayton, O. 

(Eng. News, Aug. 8, 1901).— Illustrated. 

Howe Truss Roofs for Transportation Building, at St. Louis Expo- 
sition (Eng. News, May 19, 1904). — Illustrated. Also cost data of the prin- 
cipal buildings. 

Timber Roof Trusses (By J. F. Jackson. Eng. News, June 2, 1904). — 
Illustrated. 

Method of Erecting the Roof Trusses of the 71st Regiment Armory, 
N. Y. City (By W. T. McCarthy. Eng. News, June 16, 1904).— Illustrated. 

Wind Stresses in Knee=Braced Mill Buildings (By W. H. Dunham. 
Eng. News, Oct. 6, 1904). — Graphically illustrated. Discussions: Eng. 
News, Nov. 10, 1904; Jan. 25, 1905. 

Reinforced=Concrete Slab Roof for a Small Warehouse (Eng. News, 
June 22 1905).— Illustrated. 

Modified Saw=Tooth Roof (By M. S. Ketchum. Eng. News, Nov. 23, 
1905).— Illustrated. 

Reinforced=Concrete Shingles for Roofing (Eng. News, Aug. 30, 1906). 
— Illustration of hand molding machine for concrete shingles. 

Saw=Tooth Roofs for Factories (By K. C. Richmond. Eng. News, 
Dec. 13, 1906). — Illustrated. 

Steel Dome for Emporium Building, San Francisco (Eng. News, 
May 14, 1908). — Illustrated. 

An Improved Method of Saw=Tooth Roof Construction (By S. M. 
Green. Eng. News, Sept. 3, 1908). — Illustrations of gutter construction 
and ventilator design for use on weave sheds. 

Illustrations. 

Description. Eng. Rec. 

Roof of the Standard Steel Car Co., Butler, Pa Nov. 26,' 10 




Fig. 1. 



47.— BUILDINGS. 

Plastering. — Plastering usually consists of three coats, viz., (1) the 
rough or "scratch" coat which is applied directly to the wood- or metal 
laths; (2) the "brown" coat which is floated either on the scratch coat (the 
latter having previously been scratched with a comb in order to roughen it 
so the brown coat will adhere better), or sometimes directly on the wall; 
and (3) the finishing or "skim" coat which is applied to the brown coat 
after it has been finely scratched or roughened. The skim coat may be 
either "stucco" or "hard finish" (gage stuff). 

(1.) — ^The scratch coat is composed of a mixture of slaked lime, clear 
river or pit sand (essentially free from salt) and cattle hair* 
(preferably goat or cow) . These are mixed in the proportion 
of one part lime paste to two parts sand, with IJ bushels of 
hair to each barrel of unslaked lime. Less hair is required 
for walls than for ceilings. A barrel of Rockland, Me., lime 
weighs 220 lbs. net, contains about 3i cubic feet, and will 
make about 2. 6 barrels or 9 cubic feet of paste. A barrel of 
200 lbs. will make about 8 cubic feet of paste. Approxi- 
mately, 9 cubic feet of lime paste, 18 cubic feet of sand and 
4 bushels of hair will cover 40 square yards about f' thick 
on wooden laths (Fig. 1), and about 30 square yards on metal 
laths. 

(2.) — ^The brown coat is sometimes leaner in cement than the scratch 
coat and contains usually but half the quantity of hair. It is generally 
l*' thick, sometimes |". Plaster prepared in sheets and shipped ready for 
nailing is a common substitute for the scratch and brown coats. 

(3.) — ^The skim coat, usually j\ contains no hair. Stucco is composed 
of one part pure lime and two parts clear sand of the purest kind, white 
preferred. Hard finish is made from any of the patent plasters on the 
market. They are composed principally of plaster of Paris or gypsum, 
which gives the hard finish, and are recommended for general use, being 
more satisfactory in many ways than the ordinary lime mixture. A mixture 
of 2^ cubic feet each of lime, plaster of Paris and white sand or marble dust 
will skim-coat about 100 square yards from re'' to J" thick. 

Lathing and plastering is commonly estimated to weigh about 10 lbs. per 
square foot. 

Lathing. — Laths may be of wood or metal. Wooden laths are usually 
ly wide, i" thick and 4 ft. long. They are made of pine, spruce or hem- 
lock. The straight-grained split lath is preferable to the sawed. A bundle 
of 100 laths (50 sq. ft., solid— 12^ ft. B. M.— 37| lbs.), 

spaced I inch, will cover 6.48 sq. yds.: equal to 15431athsper 100 sq. yds. 
I .. .. .. g 94 .. *. .. J44J .. 

i " " " , 7.41 '• '* •' 1350 '• 

About 10 lbs. of nails are required per 100 sq. yds. of lathing. From the 
above it is to be noted that lathing weighs about | lb. per square foot in 
place. 

Metal lathing, either wire or expanded metal (see Figs. 7 to 10), is now 
universally used in fire-proof buildings. The weight ranges from 2f to i\ 
lbs. per sq. yd., or ^ to ^ lbs. per sq. ft. Generally, the weight of the ex- 
panded metal per unit of area is about one-half or less than the weight of the 
original sheet, or, in other words, a sheet is expanded so as to cover twice 
or more its original area, depending upon the mesh. Thus, Diamond 
gage 24 covers 2.2; gage 26 covers 2.4; "A" gage 24 covers 1.9; "B" gage 
27 covers 2.06, etc. 10 lbs. of staples will fasten 100 sq. yds. of expanded 
metal lathing. 

Partitions are either permanent (fixed) or temporary (movable). In 
addition to the ordinary wooden partition, hollow tile and expanded metal 
are very largely used. 



* Wood fiber is often used instead of hair, for cheap work. 



812 



PLASTERING. LATHING. PARTITIONS. 



813 



Wooden Partitions. — Fig. 2 represents the average partition used in a 
frame dwelling. The studding is spaced 16 ins. on centers. It will be noted 
that those pieces marked with a cross ( X ) would be superfluous if the door 
opening were omitted: a, c, c would complete the studding, and b the 




Fig. 2.- 



-Stud Partition; dimension stuff either all 2"x 6" 
or all rx r. 



bridging, across the door opening. In the above illustration, about 26| 
lineal feet of extra scantling is required for each door opening, over that for 
a plain partition. A partition of this kind will weigh 3 or 4 lbs. per square 
foot, exclusive of laths and plaster. 

Hollow-Tile Partitions. — Solid terra cotta weighs from 120 to 125 lbs. 
per cubic foot, 122^ lbs. being a good average; but the hollow blocks for 
partitions will usually not exceed, two-thirds that amount, and may weigh 
somewhat less. The following types of blocks are used: 






Fig. 3.— Plain Hollow Fig. 4. — Webbed Block. 
Block. 

Figs. 3 and 4 are ordinary blocks 
such as are used in a partition with 
I-beam studding (see Fig. 6). The 
patent blocks illustrated in Fig. 5 are 
sustained laterally by means of hori- 
zontal metal strips or bands of steel 
between vertical studding. 

Other Partitions than the hollow 
tile which may be used in connection 
with I-beam studding (Fig. 6) are 
(a) plaster boards, which are laid in 
between the beams and flushed for 
plastering; (b) wire lathing strung 
between the beams flush with the 
flanges and fastened to them, and also 
supported intermediately with angles 
about 2-ft. centers; (c) expanded metal 
lathing. 



t 

I 



Fig. 5.— Patent Block 
for plastering. 



~T] — 1 — \ — 1 — r 

i! 


n 


"It: . 




V -^.= . 






i 



Elevation 



rTie ro ds 



il^PgJSI^ 



Plan 

Fig. 6. — Ordinary Hollow-tile wall 
with steel I-beam studs. 



814 



il.—BUILDINGS. 



These three classes of material may also be used with other skeleton 
designs, provided the principle of rigidity is maintained. Figs. 7 and 8 are 
examples of hollow and of solid partition construction by the expanded 
metal system, and Figs. 9 and 10 are cuts of the metal lath used. The 
lathing can be plastered with ordinary lime plaster, but cement plaster is 
better. 




Fig. 7. — Plan of Expanded Metal Hollow Partition. 
Fig. 8. — Plan of Expanded Metal Solid Partition. 




Fig. 9. — "Diamond' 
24 Gage. Sheets, 18 ins. x 
26 Gage. Sheets, 24 ins. x 



Lath (Expanded Metal). 

?6 ins. 20 sq. yds. per bundle. 

96 ins. 16 sq. yds. per bundle. 




24 Gage. 

27 Gage. 
Floors, Ceilings, etc. 



Fig. 10.— "A" Lath (Expanded Metal). 
Sheets, 18 ins. x 96 ins. 12 sq. yds. per bundle. 

"B" Lath. 
Sheets, 18 ins. x 96 ins. 20 sq. yds. per bundle. 



-The following loads are considered in designing 
the floors of buildings: 
(1.) Live loads, due to — 

(a) People; 

(b) Safes, merchandise, furniture, machinery, etc.; 

(c) Partitions which are subject to change of position. 
(2.) Dead loads, due to — 

(d) Flooring or tiling; 

(e) Fireproof arches between the beams; 
(/) Ceiling, under the floor; 

(g) Beams directly supporting the above; 

(h) Girders, supporting the beams and in turn directly supported by 
the colimins or walls. 



FLOORS, CEILINGS. LIVE LOADS, 



815 



Live Loads. — ^There is great diversity of opinion among engineers re- 
garding the live loads to be assumed for each class of buildings, and this is 
aggravated by the decided lack of harmony of the various city building 
codes. For instance, the requirements of the ten leading cities of the United 
States are shown in the following table: 



1. — Minimum Live Loads for Floors and Roofs- 
Cities. 



-Ten Leading 





Pounds per Square Foot. 


Structure, 




^1 

i 


.3 

II 


S 


pq 


02 

h 

'S 


1 


si 


2^ 




1 
1 


Dwellings — one or two fami- 
lies 


60 
60 


40 
50 


70 
70 


50 
50 


40 
70 


50 
50 


40 
40 


50 

50 

75 
75 

75 

110 

110 

75 


60 

60 

160 
50 

75 


60 
60 

'150 
70 


52 


Lodging houses, apartment 
houses, tenement houses, 
hotels, etc 


56 


Halls, dining-rooms, caies, 
offices, etc., in hotels and 
apartment-houses 


68 


Office buildings, first floor. . 

Office buildings, above first 

floor 


150 

75 


50 
50 


100 
100 


100 
100 


70 
70 


ISO 

75 


60 
60 


97 

75 


Halls and lobbies in office 
buildings 


110 


Public assembly rooms: 

churches, theaters, etc. . . 
Schools 


90 

75 


100 
75 


120 
120 


200 
60 


100 
100 


125 
100 


80 
50 

250 
100 

ioo* 


**75 


100 
100 


103 
83 


Machine shops, armories, 
drill-rooms, etc 


250 


Light manufacturing and 
retail stores and store- 
houses 


120 
150 


100 
100 


120 
150 


125 

250 
70 


120 
150 


100 
200 


110 
200 


125 
250 


150 
150 


117 


Heavy storehouses, ware- 
houses, livery stables, 
etc 


160 


Stairways 


85 


Sidewalks 


300 
50t 
30J 










300 








300 


Roofs, per square foot of 
superficial surface 


25 


30 


40K 




30 


25 


.... 


401 




Roofs, per square foot of 
• horizontal projection 


40 


30 





















* The lower supports to carry two-thirds of the total weight. 

t Pitch less than 20 degrees, 
jt Pitch more than 20 degrees. 
i For flat roofs. 

^ It has been found by actual test that forty selected men with an average 
weight of 163.2 lbs. may be packed into a floor area of 6-ft. square, when 
each man tries to occupy as little space as possible. This is equivalent to 
an occupied area of x% of a sq. ft. per man, an average load of 181.3 lbs. per 
sq. ft. of floor, and a total load of 6528 lbs. Perhaps the passenger elevator 
would instance a load approaching the above ideal more closely than would 
any other case in practice, but even there a load of 120 lbs. per sq. ft. makes 
a very compact and uncomfortable mass, and is probably very rarely ob- 
tained. For a "mixed" crowd covering a considerable area as in public 
halls or corridors, 100 lbs. per sq. ft. may be taken as the extreme loading, 
even when the ordinary crowding takes place. People in a crowd do not 
stand perfectly erect and allow themselves to be packed together as in the 
ideal case above cited. Occasionally during a panic the people in the center 
of a crowd may be packed so as to produce a concentrated loading nearly 
equivalent to the ideal load of 180 lbs. per sq. ft. over a limited area, but by 



816 



il.— BUILDINGS, 



no means over the whole floor space or even over any considerable portion 
of it. Hence in designing floors where people congregate it is good practice 
to adopt heavier loadings for small floor areas in a building than for large 
ones. In other words the live load per sq. ft. of floor area may be allowed 
to decrease, in tall bmldings, from the maximiim loading for floor beams 
and arches, down to the minimum for columns and foundations — main 
girders taking an intermediate position. 

Loads from Safes. — One of the most important factors to be considered 
in the design of floors for office buildings and offices in general is the effect 
due to the weight of safes. Hence the following table is inserted as com- 
prising the heaviest safes in use (the 900's and 800's) and also those com- 
monly used for upper floor offices (the 500's). Ntmibers "A", "B" and"C" 
are very heavy compared with their dimensions and are liable to be placed 
in any office. The 600's and 700's, omitted in this table, range in weight 
between the 500's and 800's. The 2500-lb. safe, No. 513, is the lightest one 
considered. For any desired calculation columns 6 and 7, showing the 
distance apart of the supporting wheels, may be used in 'connection with 
either column 8, or with column 2 in connection with 10, 11 or any other 
assumed weight of the contents of the safe. In general, the use of colimin 
8 is considered good practice. 

2. — ^Table of Weights and Dimensions op Heavy Safes. 
(Hall's Safe Company of Cincinnati, Ohio.) 







Outside Dimen- 


Wheel Base 






Wt. in Lbs. 






Weight 


sions in Inches. 


in Ft. 


Dead 


tl 


of 






of Safe 








(Approx.) 


Weight 


§ 


Contents at 




No. 


(Emp- 
ty.) 












on Each 
Wheel, 


o 'o 






ECInd of Safe. 














100 


25 
















+ 10%. 


SS 


Lbs. 


Lbs. 








H'ht 


W'th 


D'th 


W'th 


D'th 




Cu. 


per 
Cu. 


per 
Cu. 






Lbs. 












Lbs. 


Ft. 


Ft. 


Ft. 




(1) 


(2) 


(3) 


(4) 


(5) 


(6) 


(7) 


(8) 


(9) 


(10) 


(11) 


(12) 


921 


16080 


86.5 


68.5 


36 


4.7 


2.7 


4422 


29.2 


2920 


730 


Double- 


920 


12530 


77.5 


59.5 


35 


4.0 


2.6 


3445 


20.2 


2020 


505 


door flre- 


923 


11100 


79 


51 


34 


3.4 


2.5 


3052 


16.0 


1600 


400 


proof, lined 
with steel. 


918 


9650 


69.25 


51 


34 


3.4 


2.5 


2653 


13.2 


1320 


330 


922 


8000 


73.5 


45.5 


34 


2.8 


2.5 


2200 


12.2 


1220 


305 


■ and steel in- 


916 


7270 


61 


44 


34 


2.7 


2.5 


2000 


9.7 


970 


243 


side doors. 


915 


6200 


57 


40 


34 


2.5 


2.5 


1705 


7.6 


760 


190 


For B a n- 


914 


5380 


53 


38 


34 


2.4 


2.5 


1480 


6.3 


630 


158 


kers. 


821 


11000 


86.5 


68.5 


30 


4.7 


2.3 


3025 


28.4 


2840 


710 




820 


8900 


77.5 


59.5 


30 


4.0 


2.3 


2448 


21.2 


2120 


530 


Double- 
door, fire- 
proof, lined 
with steel ; 
no inside 
doors. For 
Jewelers. 


823 


7900 


79 


51 


30 


3.4 


2.3 


2173 


18.3 


1830 


458 


818 


6400 


69 


51 


30 


3.4 


2.3 


1760 


15.1 


1510 


378 


822 


6300 


73.5 


45.5 


30 


2.8 


2.3 


1733 


14.1 


1410 


353 


819 


5250 


69 


38 


30 


2.4 


2.3 


1443 


11.0 


1100 


275 


816 


4900 


61 


44 


30 


2.7 


2.3 


1348 


11.3 


1130 


283 


815 


4200 


56.5 


39.5 


30 


2.5 


2.3 


1155 


8.5 


850 


213 


817 


4025 


61 


34 


30 


2.0 


2.3 


1107 


8.0 


800 


200 


814 


3700 


53 


38 


30 


2.4 


2.3 


1018 


7.4 


740 


185 




521 


8000 


86.5 


68.5 


34 


4.7 


2.5 


2200 


38.2 


3820 


955 


■ 


520 


6000 


77.5 


59.5 


32 


4.0 


2.4 


1650 


27.0 


2700 


675 


Double- 


523 


5350 


79 


51 


32 


3.4 


2.4 


1471 


23.2 


2320 


580 


door, fire- 


518 


5150 


69 


51 


30 


3.4 


2.3 


1416 


17.0 


1700 


425 


proof, with 


522 


4700 


73.5 


45.5 


30 


2.8 


2.3 


1292 


15.9 


1590 


398 


inside 


516 


3650 


61 


44 


30 


2.7 


2.3 


1003 


12.7 


1270 


318 


doors; con- 


519 


3550 


69 


38 


30 


2.4 


2.3 


976 


12.2 


1220 


305 


taining 


517 


3200 


61 


34 


30 


2.0 


2.3 


880 


8.9 


890 


223 


Banker's 


515 


3150 


57 


40 


29 


2.5 


2.2 


866 


9.4 


940 


235 


steel chest. 
For gener- 
al office use. 


514 


2850 


53 


38 


29 


2.4 


2.2 


783 


7.8 


7 80 


195 


512 


2800 


55.5 


34.75 


29 


2.0 


2.2 


770 


7.2 


720 


180 


513 


2500 


49 


36 


28 


2.2 


2.1 


688 


6.0 


600 


150 




"C" 


5750 


52 


30 


26 


1.9 


1.9 


1582 


9.2 


920 


230 


1 Square 


"B" 


4500 


45 


28 


25 


1.8 


1.8 


1238 


6.4 


640 


160 


\ door. For 


"A" 


3500 


41 


25 


24 


1.7 


1.7 


962 


4.5 


450 


113 J Bankers. 



LOADS FROM SAFES, FLOOR CONSTRUCTION. 



817 



Fixed Vaults with their contents may be considered as static loads and 
must be specially considered, but Movable Safes are liable to be placed in 
almost any position on the floor. 

Problem 1. — On the first floor of a building find what single concentrated 
load will give the same bending moment as safe No. 921,. Table 2, to one of 
a system of parallel beams spaced 5 ft. centers and 16 ft. long? 

Solution. — het A, B and C, Fig. 11, re pre- _ 

sent the beams of which B is the one under 
consideration; D and E the supporting girders; 
and 1, 2, 3 and 4 the four wheels supporting 
the safe, of which c is the center. From the 
principle of the maximum floor-beam reaction, 
page 692, B will support the greatest load when 
it bisects the normal distance between the 
center of gravity c and the line 1-2; that 
is, with maximum resultant at r. Likewise, the 
maximum moment on B will obtain at m, equi- ^ 

distant with r from the line F-F which bisects Fig. 11. 

the beams. If each of the corner weights of the safe is 4422 lbs., we have, 
for resultant at r, on beam B. 

f = 8844/ ^^~^-^ ) = 12912 lbs.. 




and maximimi moment, at m, =-z (8—1.175)2. 

The moment due to a load P at the center of the beam ■■ 
equating with the above, 

yX 8 =^(6.825)2 



X 8; whence. 



or P=.7278r= 9400 lbs. 



Hence, in the above case it will be noted that a concentrated load of 9400 
lbs. applied at the center of the beam B will produce a bending moment 
equal to that of the safe weighing 17,688 lbs., placed rectangularly in the 
most critical position. Furthermore, this is equivalent to a running load 
of 1175 lbs. per lineal foot of beam, or to a distributed load of 235 lbs. 
per square foot of floor. If safe is placed diagonally on beam, the bend- 
ing moment will be increased about 5 per cent. 



Examples op Floor and Ceiling Construction. 




Fig. 12. — Floor Framing. — a is a double hanger or stirrup; 
6 is a patent hanger; c is a common mortise. 

The under flooring may be spruce or hemlock, |" thick and laid diago- 
nally in order to brace or stiffen the floor. On top of this and at right angle 
with the tail beams is laid the finished flooring which may be white pine 
i" thick or any other flooring material. The ceiling may consist of lathing 
and plastering below the beams in the usual manner. 



818 



AT.— BUILDINGS. 

rS ^C 




Fig. 13. — Centering for 
Brick Arch. 



Fig. 14. — 4-inch Brick Arch. — c is concrete; s, 
wooden screed on which is nailed the flooring /; t, 
the rod; zy, washer; a, angle iron to receive the 
thrust of the arch; /, steel beam. 





Fig. 15. — Hollow Bricks. Fig. 16. — Skew-backs. 

Instead of the solid bricks, hollow bricks are often used in order to 
reduce the dead load of the arch. Figs. 15 and 16 are examples of ordinary 
hollow bricks and skew-backs. They can be manufactured to any desired 
pattern. 



Fig. 17. — Corrugated Steel and 
Concrete Floor. 



Fig. 



18.— Steel Trough and 
Concrete Floor. 



Cinder Concrets 



^ 



n3mm^m^Mm''m'^\ 




End View, 



Fig. 



19.— Typical Terra- 
Cotta Floor. 



Figs. 20." 



-End-Construction Hollow- 
Tile Floor. 




Fig. 21. 

'Expanded Metal 



Fig. 22. 




Fig. 25. Fig. 26. 



Fig. 27. 



Fig. 28. 



Fig. 29. 



FLOOR CONSTRUCTION. CITY CODES. 819 

DIGEST OF THE NEW YORK CITY BUILDING CODE (1906). 

[Also District of Columbia, practically the same.] 
QUALITY OF MATERIALS. 

1. Lime mortar. — 1 part Ifme, not > 4 parts saiid; lime properly 
slaked before being mixed with the sand. 

2. Cement mortar. — 1 part cement, not > 3 parts sand; mixed before 
adding water. Portland cement = cement that, when tested neat, will resist 
tension of at least 120 lbs. per sq. in., after 1 day air setting; and 300 lbs., 
after 1 day in air and 6 days in water. Other cements, 60 lbs. and 120 lbs., 
respectively. 

3. Cement and lime mortar. — 1 part cement, 1 part lime, not > 3 parts 
of sand to each. 

4. Concrete. — At least 1 cement; 2 sand; 5 clean broken stone (2 in. 
ring), or 5 clean gravel. 

5. Wrought iron. — Ultimate strength 48,000 lbs. per sq. in.; elastic 
limit not < 24,000; elongation 20% in 8 ins. (small specimens). 

6. Steel.— Ult str, 54-64,000; elastic limit, not < 32,000; elongation, 
not < 0.20. Rivet steel, 50-58.000 ult str. 

7. Cast iron. — One-inch square bar, 54-in. span, shall support central 
load of 450 lbs. before breaking. Tensile strength, not < 16,000 lbs. per 
sq. in. (small specimens). 

EXCAVATIONS AND FOUNDATIONS. 

8. Bearing capacity of soil. — Where no tests are made allow for soft 
clay, 1 ton per sq. ft. ; ordinary clay and sand together, in layers, wet and 
springy, 2 tons; loam, clay or fine sand, firm and dry, 3 tons; very firm, 
coarse sand, stiff gravel or hard clay, 4 tons. 

9. Pressure under footings of foundations. — For warehouses and fac- 
tories, full dead and full live loads; for stores, light factories, churches, 
school houses, and places of public amusement or assembly, full dead and 
75 per cent of live loads; for office buildings, hotels, dwellings, apartment 
houses, tenement houses, lodging houses, and stables, full dead load and 
60% of live load. 

10. Foundations. — Piles 20 ft. or less in length, not < 5 ins. at 

small end and 10 ins. at butt. Piles over 20 ft., not < 12 ins. 

at butt. Max load per pile, 40,000 lbs. Use Engineering News 

2wh* 
formula when pile is not driven to refusal: P= ^ . Safe load for stone, 

brick or concrete piers in caissons: to rock, not > 15 tons per sq. ft.; to 
firm gravel or hard clay, 10 tons; in open caissons or sheet pile trenches, 
8 tons. 

WOODEN BEAMS. GIRDERS AND COLUMNS. 

11. Wood beams. — Minimum thickness, 3 ins. Every wood header or 
trimmer more than 4 ft. long shall be hung in stirrup irons. All wood floor 
and wood roof beams bridged with cross bridging spaced not > 8 ft. Safe 

uniform load in lbs. per lineal foot for long spans: hemlock, 70 -^ ; spruce 

bd^ bd^ bd^ 

and white pine, 90 -r- ; oak, 120 -y ; yellow pine, 140 -j-; in which b = 

breadth of beam in inches., (i = depth in ins., L = length in ft. For short 
spans the shear must be considered. 

12. Timber for trusses. — Working stresses in timber struts of pin -con- 
nected trusses shall not exceed 75% of the working stresses established in 
par. 27-30. 

FIREPROOF BUILDINGS. 

13. Fireproof buildings. — For buildings exceeding 12 stories or 150 ft. 
the floors shall be of stone, cement, rock asphalt, tiling, etc., or the sleepers 
and floors may be of wood treated by some approved process to render them 
fireproof. 



* See Sec. 50, Foundations, page 871, 



820 47,'-BUILDINGS. 

14. Fireproof floors. — Shall be constructed with wrought iron or steel 

floor beams calculated to deflect no more than ^-inch per foot of sjgian 
under total (live and dead) load; and tie rods shall be spaced not > 8 
times depLh of beam. (1) Brick arches, springing from the lower flange of 
the steel beams, shall be designed with a rise not < li ins. for each foot of 
span, and with a thickness not < 4 ins. for spans of 5 ft. or less, and not 
< 8 ins. for spans over 5 ft. They shall be composed of good hard brick or 
hollow brick of ordinary dimensions laid to a line on the centers, properly 
and solidly bonded, each longitudinal line of brick breaking joints with the 
adjoining lines in the same ring and with the ring under it when more than 
4-in. arch is used; cement mortar joints. (2) Hollow tile arches of hard- 
burned clay or porous terra-cotta shall have an effective depth not < If 
ins. per ft. of span, some allowance (not over 6 ins.) being made if soffit of 
arch is straight; if segmental, the depth of tile shall be not < 6 ins. if rise 
is not < Ij ins. times span in ft,; cement mortar joints. (3) Portland 
cernent concrete arches, segmental, with rise not < IJ ins. times span in ft. 
Thickness at crown not < 4 ins. Mixed as per par. 4. Arches shall be rein- 
forced and protected on under side with corrugated or sheet steel, steel ribs, 
or metal in other forms weighing not less than 1 lb. per sq. ft., and having 
no openings larger than 3 sq. ins. (4) Reinforced floors of solid or hollow 
bt;Lrned clay, stone, brick, or concrete slabs in flat or curved shapes, in 
combination with wire cloth, expanded metal, wire strands, or wrought 
iron or steel bars, may be used, but proper tests shall be made as provided 
in the Code. 

IRON AND STEEL CONSTRUCTION. 

15. Skeleton construction. — Where columns are used to support iron 
or steel girders carrying inclosure walls, the said colimins shall be of cast 
iron, wrought iron, or rolled steel, and on their exposed outer and inner sur- 
faces be constructed to resist fire by having a casing of brickwork not less 
than 8 ins. thick on the outer surfaces, nor less than 4 ins. thick on the inner 
surfaces, all bonded into the brickwork of the inclosure walls. Exposed 
sides of girders protected with 4 ins. of brickwork; outer edges of flanges, 
2 ins. 

16. Steel and wrought iron columns. — Minimum thickness of metal, { 
inch. Least lateral dimension ^V length of coliimn, except as allowed in 
par. 26. 

17. Cast iron columns. — Minimum diameter, 5 ins.; minim tmi thick- 
ness of metal, f in. Least lateral dimension zu length of column, except as 
allowed in par. 26. 

18. Steel and iron girders. — Stiff eners shall be used at intervals not 
exceeding 120 times thickness of web if the unsupported depth of the web 
plate exceeds 60 times its thickness. 

19. Rolled beams used as girders. — Beams in pairs to form girders 
shall be connected together by bolts and separators at intervals of not more 
than 5 ft. Beams 12 ins. or more in depth shall have 2 bolts to each sepa- 
rator. 

20. Cast iron lintels. — Maximum span, 16 ft.; minimum thickness of 
metal, f in. 

21. Painting of structural metal work. — After erection all work shall 
be painted at least one additional coat. All iron or steel used under water 
shall be inclosed with concrete. 

FLOOR LOADS. 

22. Floor loads. — Dead loads = weight of walls, floors, roofs, partitions 
and all permanent construction. Live loads = variable loads = all loads 
other than dead loads. Live loads per sq. ft. of floor shall be assimied as 
follows: 

For dwelling, apartment, tenement, hotel or lodging house. not< 60 lbs. 

For office purpose, first floor, - - - - - - " 150 

" " *' all floors above the first, - - - " 75 

" school or place of instruction, ----- " 75 

" stable and carriage house purposes, - - - - " 75 

" place of public assembly, - - - - - - " 90 

" ordinary stores, light manufacturing, light storage, - " 120 

" warehouse, factory, heavy stores, - - - - "150 

" roofs with pitch less than 20°, per area of roof, - " 50 

" " " " more " 20°, per horizontal area, - " 30 

" sidewalks, between the curb and area lines, - - " 300 



NEW YORK BUILDING CODE. 821 

Note. — For the purpose of determining the carrying capacity of columns 
of dwellings, office buildings, stores, stables and public buildings when over 
5 stories in height, a reduction of the live loads shall be permissible as fol- 
lows: For the roof and top floor the full live loads shall be used; for each 
succeeding lower floor it shall be permissible to reduce the live load by 5% 
until 50% of the live loads fixed by this section is reached, when such re- 
duced loads shall be used for all remaining floors. 

CALCULATIONS.— STRENGTH OF MATERIALS. 

23. Safe load for masonry work. — Safe load per sq. ft.: Brickwork in 
lime mortar, 16,000 lbs.; in lime and cement mortar mixed, 23,000 lbs.; in 
cement mortar, 30,000 lbs. Rubble-stone work in Portland cement, 20,000 
lbs.; in other cement, 16,000 lbs., in lime and cement mixed, 14,000 lbs.; 
in lime mortar, 10,000 lbs. Portland cement concrete, 30,000 lbs. Other 
cement concrete, 16,000 lbs. 

24. Weights of certain materials. — In computing weight of walls, one 
cu. ft. of brickwork shall be deemed to weigh 115 lbs.; sandstone, white 
marble, granite and other kinds of building stone, 170 lbs. per cubic foot. 

25. Factors of safety. — While working stresses are not prescribed 
allow 4 for metals, 6 for timber, 10 for natural or artificial stones and brick 
or stone masonry. 

26. Strength of columns. — Safe load in lbs. per sq. in.: Cast iron, 

7 • 7 7 

11,300-30— ; steel, 15,200-58—; wrought iron, 14,000-80—. Yellow 
r r r 

pine (long leaf), 1000— 18-r; white pine, Norway pine, spruce, 800 — IS-r ; 
a a 

oak, 900-17^; chestnut and hemlock.-g- (800- 15-jj ; locust.-r- (SOO- 15-j\ . 

Columns eccentrically loaded shall have such stresses computed, and the 
combined stresses shall not exceed the allowable working stresses. The 
eccentric load of a column shall be considered to be distributed equally over 
the entire area of that colimin at the next point below at which the column 
is securely braced laterally in the direction of the eccentricity. 

27. Compression (Direct). — Safe load in lbs. per sq. in.: Rolled steel, 
16,000; cast steel, 16,000; wrought iron, 12,000; cast iron (in short blocks), 
16,000; steel pins and rivets (bearing), 20,000; wrought iron pins and 
rivets (bearing), 15,000. 



With Across 

Grain. Grain. 

Locust, - - 1,200 1,000 

-> Hemlock, - 500 500 

Chestnut, - 500 1,000 

stone, 4 - - - 230 

'• 5 - - - 208 

" 4 - - - 125 

" 5 - - - 111 



With Across 

Grain. Grain. 
Oak, - - - 900 800 

Yellow pine, - 1,000 600 

White pine, spruce 800 400 

Concrete, Portland: cement, 1; sand, 2; 

1; " 2; 

" Rosendale, or equal: " 1; " 2; 

1; " 2; 

Rubble stonework in Portland cement mortar - _ - 140 

" " " Rosendale " " - - - _ m 

" lime and " " - - - - 97 

** " " lime mortar ------ 70 

Brickwork in Portland cement mortar: cement, 1; sand, 3 250 

" Rosendale,or equal " " " 1; " 3 208 
" " lime and cement " " 1; lime, 1; 

sand, 6 160 

" " lime mortar: lime, 1; sand, 4 - - - - 111 

Granites, ace. to test, 1,000 to 2,400. Greenwich stone - 1,200 

Limestones," " 700 "2,300. Greiss (New York City) 1,300 

Marbles, " " 600 " 1,200. Bluestone, North River, 2,000 

Sandstones, " " 400 " 1,600. Slate, - - 1,000 
Brick (Haverstraw, flatwise) 300. 



* /= length of column, in ins.; r = radius of gyration, in ins.; d = least 
dimension, in ins. 



822 ^1.— BUILDINGS. 

28. Tension (Direct). — Safe stress in lbs. per sq. in.: Rolled steel, 16,- 
000; cast steel, 16,000; wrought iron, 12,000; cast iron, 3,000; yellow 
pine, 1,200; white pine, spruce, 800; oak, 1,000; hemlock. 600. 

29. Shear. — Safe stress in lbs. per sq. in.: Cast iron, 3,000. Web plates: 
Steel, 9,000; wrought iron, 6,000. Shop rivets and pins: Steel, 10,000; 
wrought iron, 7,500. Filed rivets: Ste 
Field bolts: Steel, 7,000; wrought iron, 5, 





With 


Across 




Fiber. 


Fiber. 


Yellow pine 


70 


500 


White pine, 


40 


250 


Spruce, 


50 


320 


Oak, - - 


100 


600 



;1, 8,000; 
lOO 


wrought iron, 6,000. 


Locust, 

Hemlock, 

Chestnut, 


With Across 
Fiber, Fiber. 

- - 100 720 

- - 40 275 

- - ... 150 



30. Bending. — Safe extreme fiber stess in lbs. per sq. in.: Rolled beams. 
Steel, 16,000; wrought iron, 12,000. Rolled pins, rivets and bolts: Steel, 
20,000; wrought iron, 15,000. Riveted beams (not flange section): Steel, 
14,000; wrought iron, 12,000. Cast iron: Compression side, 16,000; tension 
side, 3,000. Yellow pine, 1,200; white pine, spruce, 800; oak, 1,000; locust, 
1,200; hemlock, 600; chestnut, 800. Granite, 180; Greenwich stone, 150; 
gneiss (New York City), 150; limestone, 150; slate, 400; marble, 120; 
sandstone, 100; bluestone (North River), 300; brick (common), 50; brick- 
work (in cement), 30. Concrete: (Portland), 1:2:4, 30 lbs.; 1:2:5, 20 lbs.; 
(Rosendale, or equal), 1:2:4, 16 lbs.; 1:2:5, 10 lbs. 

31. Wind pressure. — All structures exposed to wind shall be designed 
to resist a horizontal wind pressure of 30 lbs. for every sq. ft. of surface thus 
exposed, from the ground to the top of same, including roof, in any direction. 
In no case shall the overturning moment due to wind pressure exceed 75% 
of the moment of stability of the structure. In all structures exposed to 
wind, if the resisting moments of the ordinary materials of construction 
such as masonry, partitions, floors and connections are not sufficient to 
resist the moment of distortion due to wind pressure, taken in any direction 
on any part of the structure, additional bracing shall be introduced sufficient 
to make up the difference in the moments. In calculations for wind bracing, 
the working stresses set forth above may be increased by 50%. In buildings 
under 100 ft. high, where the height does not exceed four times the average 
width of the base, the wind pressure may be disregarded. 

CONCRETE-STEEL CONSTRUCTION. 

1. The term "concrete-steel" shall mean an approved concrete mixture 
reinforced by steel of any shape, the steel to take up the tensional stresses 
and assist in the resistance to shear. 

2. Concrete-steel construction will be approved only for buildings 
which are not required to be fireproof by the building code, unless satisfac- 
tory fire and water tests shall have been made under the supervision of this 
bureau. 

3. Complete drawings and specifications must be filed with the supt. of 
buildings, showing all details of the construction, the size and position 
of all reinforcing rods, stirrups, etc., and giving the composition of the 
concrete. 

4. Execution of work shall be under the control of a competent fore- 
man or superintendent. 

5. Concrete to be mixed in the proportions of 1 cement, 2 sand, and 4 
stone or gravel; or the proportions may be such that the resistance of the 
concrete to crushing shall not be less than 2000 lbs. per sq. in. after harden- 
ing for 28 days. The concrete is to be what is usually known as a "wet" 
mixture. 

6. Only high-grade Portland cements shall be permitted; and shall de- 
velop a tensile strength of at least 300 lbs. per sq. in. after 1 day in water; 
at least 500 lbs. per sq. in. after 1 day in air and 6 days in water; and at 
least 600 lbs. per sq. in. after 1 day in air and 27 days in water. 

7. The sand must be clean and sharp, free from loam or dirt, and not 
finer than the standard sample submitted. 

8. The stone shall be clean, broken trap rock, or gravel, of a size that 
will pass through a three-quarter inch ring. 

9. The steel shall have an ult. str. of 54-64,000 lbs. per sq. in.; an elastic 
limit of not less than 32,000 lbs. per sq. in.; and a min. elong. of not less 
than 20% in 8 ins. 



NEW YORK BUILDING CODE. 823 

10. In concrete-steel design the following stresses in lbs. per square 
inch shall not be exceeded : 

Extreme fiber stress on concrete in compression 500 (650*) 

Shearing stress in concrete 50 

Concrete in direct compression. 350 (450*) 

Tensile stress in steel 16,000 

Shearing stress in steel 10,000 

11. The adhesion of concrete to steel shall be assumed to be not greater 
than the shearing strength of the concrete. 

12. The ratio of modulus of elasticity of concrete and steel shall be 
taken as 1 to 12. ^ 

13. The following assumptions shall guide in the determination of the 
bending moments due to the external forces: (a) Beams and girders shall 
be considered as simply supported at the ends, no allowance being made for 
construction over supports, (b) Floor plates, when constructed continuous 
and when provided with reinforcement at top of plate over the supports, 
may be treated as continuous beams, the bending moment for uniformly 
distributed loads being taken at not less than WL -^10; the bending moment 
may be taken as 1^L-t-20 in the case of square floor plates which are rein- 
forced in both directions and supported on all sides. The floor plates to the 
extent of not more than ten times the width of any beam or girder may be 
taken as a part of that beam or girder in computing its moment of resistance. 

14.^ In calculating the moment of resistance it is assumed that the 
stress in any fiber is directly proportional to its distance from the neutral 
axis (taking into consideration the proportionate moduli of elasticity of 
steel and concrete) . The tensile strength of the concrete shall not be con- 
sidered. 

15. When the shearing stresses developed in any part of a concrete- 
steel construction exceed the safe working strength of concrete, a sufficient 
amount of steel shall be introduced in such a position that the deficiency 
in the resistance to shear is overcome. 

16. When the safe limit of adhesion between the concrete and steel is 
exceeded, some provision must be made for transmitting the strength of the 
steel to the concrete. 

17. Concrete-steel may be used for columns in which the ratio of length 
to least side or diameter does not exceed 12. The reinforcing rods must be 
tied together at intervals of not more than the least side or dia. of column. 

18. The contractor must be prepared to make load tests on any portion 
of a concrete-steel construction, within a reasonable time after erection, as 
often as may be required by the supt. of buildings. The tests must show 
that the construction will sustain a load of 3 times that for which it is de- 
signed without any sign of failure. 

EXTRACT FROM CHICAGO BUILDING ORDINANCE (1910). 

REINFORCED CONCRETE. 

Ratio of Moduli of Elasticity — Adhesion — Bond. — (a) The calculations 
for the strength of reinforced concrete shall be based on the assumed ulti- 
mate compressive strength U, in pounds per square inch, given in the table 
below for the mixture to be used, (b) The ratio of the modulus of elasticity 
of steel to that of concrete is designated by the letter R: 

1:1 :2 mix (1 cement, 1 sand, 2 broken stone, gravel or slag) U=2,900; R=10. 
1:13^:3 mix (1 cement, 13^ sand, 3 broken stone, gravel or slag) 17=2,400; R=12. 
1:2 :4 mix (1 cement, 2 sand, 4 broken stone, gravel or slag) [7=2,000; R=15. 
1:23^:5 mix (1 cement, 2^ sand, 5 broken stone, gravel or slag) t/=l,750; R=18. 
1:3 :7 mix (1 cement, 3 sand, 7 broken stone, gravel or slag) t7=l,500; R=20. 



* Mayor Gaynor's views expressed under date of July 21, 1911, regarding 
the proposed revised code. He also says: As near as I can make out from 
all that has been explained to me the cost of cinder concrete should not be 
increased. It is not necessary that the cinders should be screened. The 
thickness of the arches should not be increased from the minimum of 4H to 
a minimum of 6H- That is unnecessary. The weight of the reinforcing 
material therein should not be raised to a minimum of one pound per square 
foot. One-quarter of a pound per square foot may suffice. A precise and 
unvarying rule should be made to govern. 



824 



4t7.— BUILDINGS. 



Adhesion — Bond. — For a concrete of 1:2:4 mix, the allowable adhesion 
in lbs. per sq. in. of surface of embedment shall not exceed the following: 
On plain bars of structural steel, 70; on plain bars of high carbon steel, 50; 
on plain flat bars in which width to thickness is not >2 to 1, 50; on twisted 
bars when twisting is not < one complete twist in eight diameters, 100. 

EXTRACT FROM BUILDING LAWS AND ORDINANCES OF 
PHILADELPHIA (1907). 

Live Loads for Floors. — Lbs. per sq. ft.: Dwellings, tenement houses, 
apartment houses, hotels, hospitals and asylums, 70 lbs. ; office buildings, 
100 lbs.; places of public assembly, light manufacturing and retail stores, 
120 lbs.; storehouses, warehouses and manufactories, 150 lbs. and upward 
in proportion to the loads they have to carry. 

Roofs shall be constructed to bear a safe weight of 30 lbs. per superfi- 
cial foot. 

Ultimate Stresses in lbs. per sq. in.: 





Cast 
Iron. 


Wrt. 
Iron. 


Mild 
Steel. 


Medi- 
um 
Steel. 


Hem- 
lock. 


Spruce 


•Long 

Leaf 

Yellow 

Pine. 


Tension (direct) 




50,000 
50,000 


58,000 
58,000 


65,000 
65,000 


4,000 
2,100 

3,600 


5,000 
3,000 

4,400 


7,200 


Compression (direct) . . . . 
Bending— extreme fiber 
(tension) 


70,000 
15,000 


4,500 
6,400 


Shear 


30,000 


35,000 


40,000 




Shear — perp. to grain. . . 




2,500 
250 


3,000 
300 


4,500 


Shear — parallel with 
grain 










400 



Working Stresses* in lbs. per sq. in.: 





Cast 
Iron. 


Wrt. 
Iron. 


Mild 
Steel. 


Medi- 
um 
Steel, 


Hem- 
lock. 


Spruce 


Long 

Leaf 

Yellow 

Pine. 


Tension 




12,500 
12,500 


14,500 
14,500 


16,250 
16,250 


1,000 
350 

250 

900 


1,250 
500 

300 

1,100 


1,800 


Compression 


11,667 


750 


Compression — perp. to 
grain 


550 


Bending — extreme fiber 
(tension) 


3,750 








1,600 


Shear 


7,500 


8,750 


10,000 




Shear — perp. to grain. . . 




4161 
41! 


500 
50 


750 


Shear — parallel to grain . 










66f 



* For columns, the safe working loads p in lbs. per sq. in. may be re- 
duced by the following formulas: — 



Cast iron, 



P = 



11667 



Mild steel, p = 




Wrought iron, p = - 



12500 



Meditmi steel, p 



1 + - 



13500r2 



+ 



l-f- 



15000r2 
16250 



llOOOr^ 



Hemlock, ^=350 



1-7- ; Spruce, :^= 500 -5-r; Yellow pine,:^ = 750- 7.5-t • 
a a a 

In which / = length, r=- least radius of gyration, cf = least diameter, all in 
inches. The allowable reduction of live load on columns and girders shall 
be as follows: "For all tenement houses, hotels, apartment houses, hospitals 
and office buildings the live loads on columns^ girders and foundations may 
be estimated by the formula X=100— |VA, and for light manufacturing 

buildings by the formula X= 100— gVA, in which "X" equals the percentage 
of live load to be used, and"A"equals area carried by any girder, column or 
foundation. 



PHILADELPHIA BUILDING CODE, 825 

Allowable Pressures in lbs. per sq. ft. — Concrete, 15 tons; brickwork in 
lime mortar, 8 tons; brickwork in lime and cement rnortar, 12 tons; brick- 
work in cement mortar, 15 tons; stonework (rubble) in lime mortar, 5 tons; 
stonework (rubble) in lime and cement mortar, 8 tons; stonework (rubble) 
in cement mortar, 10 tons. 

Reinforced Concrete. — Reinforced concrete construction will be accepted 
for fireproof buildings of the first class, if designed as hereinafter pre- 
scribed; provided, that the aggregate for such concrete shall be clean broken 
hard stone, or clean graded gravel, together with clean siliceous sand or fine 
grained gravel ; should the concrete be used for flooring between rolled steel 
beams, clean furnace clinkers entirely free of combustible matter, or suita- 
ble seasoned furnace slag may be used; when stone is used with sand or 
gravel it must be of a size to pass through a one-inch ring, and 25% of the 
whole must not be more than one-half the maximum size; and provided 
further, that the minimum thickness of concrete surrounding the reinforc- 
ing members of reinforced concrete beams and girders shall be 2 ins. on the 
bottom and one-half inch on the sides of said beams and girders. The 
minimum thickness of concrete under slab rods shall be one inch. All rein- 
forcement in columns to have a minimum protection of 2 ins. of concrete. 

For walls, reinforced concrete may be used in place of brick and stone 
walls, in which case the thickness may be two-thirds of that required for 
brick walls. Concrete walls in such cases must be reinforced in both direc- 
tions in an approved manner. 

All steel reinforcement shall be of standard grade of structural steel or 
iron of either grade to meet the "Manufacturers' Standard Specifications," 
revised Feb. 3, 1903. 

Slabs, beams and girders shall be designed on the assumption of a load 
four times as great as the total load (ordinary dead load plus ordinary live 
load). The steel to take all the tensile stresses. The stress-strain curve of 
concrete in compression is a straight line. 

Ratio of moduli of elasticity of concrete to steel: Stone or gravel con- 
crete, 1 to 12; slag concrete, 1 to 15; cinder concrete, 1 to 30. 

Allowable unit transverse stress (lbs. per sq. in.) upon concrete in com- 
pression: Stone or gravel concrete, 600; slag concrete, 400; cinder concrete, 250. 

Allowable unit transverse stress (lbs. per sq. in.) in tension: Iron, 12,000; 
steel, 16,000. 

Allowable unit shearing stress (lbs. per sq. in.) upon concrete: Stone or 
gravel concrete, 75; slag concrete, 50; cinder concrete, 25. 

Allowable unit adhesive strength (lbs. per sq. in.) of concrete: Stone or 
gravel concrete, 50; slag concrete, 40; cinder concrete, 15. 

Allowable unit stresses (lbs. per sq. in.) upon concrete in direct com- 
pression in columns: Stone or gravel concrete, 500; slag concrete, 300; cinder 
concrete, 150. 

Allowable unit stress upon hoop columns composed of stone or gravel 
concrete shall not be over 1000 lbs. per sq. in., figuring the net area of the 
circle within the hooping. The percentage of longitudinal rods and the 
spacing of the hoops to be such as to permit the concrete to safely develop 
the above unit stress with a factor of safety of 4. 

Floor slabs, when constructed continuously, and when provided with 
reinforcement at top of slab over the supports, may be treated as continu- 
ous beams, the bending moment for uniformly distributed loads being taken 
at not less than WL -^ 10. In case of square floor slabs which are reinforced 
in both directions and supported on all sides, the bending moment may be 
taken at WL h- 20; provided, that in floor slabs in juxtaposition to the walls 
of the building the bending moment shall be considered as WL -^ 8, when 
reinforced in one direction, and if the floor slab is square and reinforced in 
both directions the bending moment shall be taken as WL -r- 16. 

In columns the longitudinal rods will not be considered as taking any 
direct compression. 

EXTRACT FROM THE BUILDING LAW OF BOSTON (1909). 

MATERIALS— ALLOWABLE FIBER STRESSES. 
[Lbs. per square inch.] 
Timber. — Extreme fiber (bending): White pine and spruce, 1000; white 
oak, 1000; yellow pine (long leaf), 1500. Shearing along grain: White pine, 
80; white oak, 150; yellow pine (long leaf), 100. Compression perpendicular 
to grain : White pine and spruce, 250; white oak, 600; yellow pine (long leaf), 
500. Modulus of elasticity : White pine, 750,000; spruce, 900,000; white oak, 



826 \7— BUILDINGS. 

850,000; yellow pine (long leaf), 1,300,000. Columns (centrally loaded and 

flat ends) : White pine and spruce, 630 for L-^D =0 to 10, 595 for L-^D =10 to 
15. 560for L-e-L> = 15to20, 525for L^D = 20 to 25, 490 for L^D=2b to 30; 
white oak, 810 for L^D = to 10, 765 for L^D = 10 to 15, 720 for Lh-D = 15 to 
20, 675 for L^D = 20 to 25, 630 for Lh-D = 25 to 30; yellow pine (long leaf), 
900forL4-Z)=0tol0, 850forLH-D = 10tol5, SOOfor Lh-Z) = 15 to 20, 750 for 
L^D = 20 to 25, 700 for L^D = 25 to 30. No column shall be used with greater 
value than L-r-D = 30. For eccentric loads, see Methods of Computation, p. 827. 

Wrought Iron and Steel (steel at 55-65,000). — Extreme fiber of rolled 
beams or shapes: Wrought iron, 12,000; steel, 16,000. Tension: Wrought 
iron, 12,000; steel, 16,000. Compression in flanges of built beams: Wrought 
iron, 12,000; steel, 16,000. Shearing (including pins and rivets, but not bolts): 
Wrought iron, 9,000; steel, 10,000. Shearing (bolts): 80% of preceding 
values. Direct bearing (including pins and rivets, but not bolts) : Wrought 
iron, 15,000; steel, 18,000. Direct bearing (bolts): 80% of preceding values. 
Bending on pins: Wrought iron, 18,000; steel, 22,500. Modulus of elas- 
ticity: Wrought iron, 27,000,000; steel, 29,000,000. For compression mem- 
bers, use the formula: [12,000 for iron or 16,000 for steel] divided by 
[ 1 -I- (L2 -j- 20000 r2) ], in which L = length and r = radius of gyration in inches. 
Compression flanges of beams shall be proportioned to resist lateral flexure; 
if the ratio of unsupported length of flange to width of flange does not ex- 
ceed 20, no allowance need be made; if the ratio is 70, the above specified 
allowable fiber stress shall be reduced by one-half; and proportionate for 
values between 20 and 70. 

Cast Iron. — Extreme fiber stress: Tension, 3,000; compression, 16,000. 
Columns (centrally loaded and unsupported laterally): Average stress, 11,000 
forL-^r = 10, 10,700 for LH-r= 20, 10,400 for L-^r = 30, 10,000 for L^r = 40. 
9,800 for L-^r = 50, 9,500 for L^r = 60, 9,200 for L^r = 70. L and r in inches. 
Cast iron shall not be used for columns in buildings of more than 75 feet in 
height, nor in cases where L-i-r exceeds 70. 

[Tons (2000 lbs.) per square foot.] 

Stone Work, in Compression. — First quality dressed beds and builds, 
laid solid in mortar of one part Portland cement to three parts sand, or one 
part natural cement to two parts sand. Granite, 60; marble and limestone, 
40; sandstone, 30. When poorer mortar is used the above stresses shall be 
lowered (as approved) . 

Brickwork, in Compression. — (1.) For first class hard-burned bricks, 
including piers in which the height does not exceed six times the least di- 
mension, laid in: (a) One part Portland cement, three parts sand, by vol- 
ume, dry, 20; (b) One part natural cement, two parts sand, by volume, dry, 
18; (c) One part natural cement, one part lime and six parts sand, by vol- 
ume, dry, 12; (d) Lime mortar, one part lime, six parts sand, by volume, 
dry, 8. (2.) For brick piers of hard-burned bricks, in which the height is 
from six to twelve times the least dimension: Mortar (a), 18; mortar (b), 15; 
mortar (c), 10; mortar (d), 7. (3.) For brickwork made of "light-hard" 
bricks, the stresses shall not exceed two-thirds of the stresses for like work 
of hard-burned bricks. 

CONCRETE AND REINFORCED CONCRETE. 

Cement shall conform to the specifications of the American Society 
for Testing Materials, as modified from time to time by that association. 

Concrete. — When the structural use of concrete is proposed, a specifica- 
tion, stating the quality and proportions of materials, and the methods of 
mixing the same, shall be submitted to the building commissioner, who may 
issue a permit at his discretion and under such further conditions, in addi- 
tion to those stated below, as he sees fit to impose. 

A. In first class Portland cement concrete, containing one part cement 
to not more than six parts mixed properly graded aggregate, except in piers 
or columns of which the height exceeds six times the least dimension, the 
compressive stress shall not exceed 30 tons of 2000 lbs. per sq. ft. 

B. In piers and columns of first class Portland cement concrete, con- 
taining one part cement to not more than five parts mixed properly graded 
aggregate, where the height of the pier or column is more than six times and 
does not exceed twelve times its least dimension, the compressive stress 
shall not exceed 25 tons of 2000 lbs. per sq. ft. 

By "aggregate" shall be understood all the materials in the concrete 
except the cement. Cinder concrete shall be used constructively only for 
floors, roofs and for filling. 



BOSTON BUILDING CODE, 827 

Rules for the commutation of reinforced concrete columns may be for- 
mulated from time to time by the building commissioner with the approval 
of the board of appeal. 

In reinforced concrete beams or slabs subjected to bending stresses, the 
entire tensile stress shall be assumed to be carried by the steel, which shall 
not be stressed above the limits allowed for this material. First class Port- 
land cement concrete in such beams or slabs, containing one pr.rt cement to 
not more than five parts mixed properly graded aggregate, may be stressed 
in compression to not more than 500 lbs. per sq. in. In case a richer con- 
crete is used, this stress may be increased with the approval of the commis- 
sioner to not more than 600 lbs. per sq. in. 

In reinforced concrete the maximum shearing force upon the concrete, 
when uncombined with compression upon the same plane shall not exceed 
60 lbs, per sq. in., unless the building commissioner with the consent of the 
board of appeal shall fix some other value. 

If the embedded steel has no mechanical bond with the concrete, its 
holding power shall not exceed the allowable shearing strength of the con- 
crete. 

Reinforced Concrete. — Reinforced concrete slabs, beams or girders, if 
rendered continuous over supports by being unbroken in section, shall be 
provided with proper metal reinforcement at the top over said supports and 
may be computed as continuous beams, as hereinafter described. 

The modulus of elasticity of the concrete, if not shown by direct tests, 
may for beams and slabs be taken as one-fifteenth that of steel, and for col- 
umns one-tenth that of steel. 

The reinforcing metal shall be covered by not less than three-fourths 
inch of concrete in slabs, and by not less than one and one-half inches of 
concrete in beams and columns. 

Methods of Computation. — Beams or girders of metal or reinforced con- 
crete shall be considered as simply supported at their ends, except when 
they extend with unbroken cross-section over the supports, in which case 
they may be considered as continuous. 

The span of a beam shall be considered as the distance from center to 
center of the bed plates or surfaces upon which it rests. If it is fastened to 
the side of a column, the span will be measured to the center of the column. 

In slabs, beams or girders continuous over supports, provision shall be 
made for a negative bending moment at such supports equal to four-fifths 
of the positive bending moment that would exist at the center of the span 
if the piece were simply supported; and the positive bending moment at the 
center of the span may be taken equal to the negative bending moment at 
the support. 

In the case of a slab of reinforced concrete with parallel ribs or girders 
beneath, the rib or girder may be considered to include a portion of the slab 
between the ribs, forming a T-beam. The width of the T-beam on top shall 
not exceed one-third of the span of the rib nor the distance from center to 
center of the ribs. 

Reinforced concrete columns shall be proportioned on the assumption 
that the concrete and the steel are shortened in length in the same propor- 
tion. The steel members shall be tied together at intervals sufficiently 
short to prevent buckling. 

If a column is loaded eccentrically or transversely, the maximum fiber 
stress, taking account of the direct compression, the bending which it 
causes, its eccentricity and the transverse load, shall not exceed the maxi- 
mum allowable stress in compression. 

If a tension piece is loaded eccentrically or transversely, the maximum 
fiber stress, taking account of the direct tension, its eccentricity and the 
transverse load, shall not exceed the .maximum allowable stress in tension. 

An eccentric load upon a column shall be considered to affect eccentric- 
ally only the length of column extending to the next point below at which 
the column is held securely in the direction of the eccentricity. 

If a piece is exposed to tension and compression at different times, it 
shall be proportioned to resist the maximum of each kind, but the unit 
stresses shall be less than those used for stress of one kind, depending upon 
the ratio and the relative frequence of the two maxima. 



828 



ir.SUILDINGS. 



EXTRACT FROM BUILDING LAWS OF CITY OF BUFFALO (1909). 

CONCRETE CONSTRUCTION. 

Concrete may be used in buildings of all classes when such construction 
is approved by the Deputy Building Commissioner. 

Regulations regarding the use of concrete in hollow blocks and in rein- 
forced steel construction: 

Height of buildings. — Buildings whose exterior walls are of hollow con- 
crete blocks may be erected not to exceed 3 stories in height, and the thick- 
ness of such walls shall be as given below for brick walls; provided, however, 
that the materials of construction are not strained beyond the safe limits: 





Buildings of Class I: Sale, stor- 
age or manufacture of merchan- 
dise, and public livery, boarding or 
sale stables. 


Buildings of Classes II, III, IV: 
all other buildings, as hotels, 
hospitals, office buildings, halls, 
theaters, etc. 




Basement. 


Story. 


Basement. 


Story. 


o 

CO 


Stone. 


Brick. 


Gro'nd 


2 


3 


Stone. 


Brick. 


Gro'nd 


2 


3 


1 

2 
3 


18" 
18" 
20" 


12" 
16" 
16" 


12" 
12" 
16" 


12" 
12" 


12" 


18" 
18" 
20" 


12" 
16" 
16" 


12" 
12" 
12" 


12" 
12" 


12" 



Buildings whose exterior walls are of reinforced concrete steel may be 
erected 3 stories in height, and the thickness of such walls shall be as given 
in the following table; provided, that the materials of construction are not 
strained beyond the safe limits: 

Stories. Basement. 1st Story. 

1 8" 6" 

2 10" 6" 

3 12" 8" 



2nd Story. 3rd Story. 



6" 



Concrete must be mixed in the proportions of 1 of Portland cement, 
2 of sand, and 5 of stone or gravel; or the proportions may be such that 
the resistance of the concrete to crushing shall not be less than 2,000 lbs. 
per sq. in. after hardening for 28 days, by approved test. The concrete 
used in reinforced concrete steel construction must be what is usually known 
as a "wet mix:ture." 

''Reinforced concrete steel" shall be understood to mean an approved 
concrete mixture reinforced by steel of any shape, so combined that the steel 
will take up the tensional stresses and assist in the resistance to shear. 

Concrete construction will be approved only for buildings which are not 
required to be fireproof by the building ordinances, unless fire and water 
tests shall have been made under the supervision and to the satisfaction of 
the Deputy Building Commissioner. Each company offering a system of 
concrete construction for fireproof buildings must submit such construction 
to a fire and water test. 

Inspection and tests. — ^The execution of concrete work shall be confided 
to workmen who shall be under the control of a competent foreman or 
superintendent, and persons erecting buildings of concrete shall provide for 
expert inspection of the cement and inerts and a daily record shall be kept 
of the tests, the temperature in which the concrete was worked, and all 
other conditions which may be of importance in the construction, and a 
certified copy of such record shall be filed twice each week, or oftener if 
required. 

Quality of materials. — Only high grade Portland cement shall be per- 
mitted in concrete construction. Such cement when tested, after 1 day in 
air and 6 days in water, shall develop a tensile strength of at least 500 lbs. 
per sq. in.; and after 1 day in air and 27 days in water shall develop a 
tensile strength of at least 600 lbs. per sq. in. Other tests as to fineness, 
constancy, voliune, etc., shall be made in accordance with the standard 



BUFFALO BUILDING CODE. 829 

method prescribed by the Committee of the Am. Soc. C. E., as may from 
time to time be directed. 

The sand to be used must be clean, sharp grit sand, free from loam or 
dirt. 

The stone used in the concrete must be clean broken stone or gravel, 
of a size that will pass through a f-in. ring. In case it is desired to use other 
materials or other kinds of stone, samples of same must be submitted for 
approval. 

Reinforced concrete steel must be so designed that the stresses shall not 
exceed the following limits: 

Extreme fiber stress on concrete in compression, 500 lbs. per sq. in. 

Shearing stress in concrete, 50 lbs. per sq. in. 

Concrete in direct compression, 350 lbs. per sq. in. 

Tensile stress in steel, 16,000 lbs. per sq. in. 

Shearing stress in steel, 10,000 lbs. per sq. in; 

Adhesion of concrete to steel, not greater than shearing strength of 
concrete. 

Modulus of elasticity of concrete to steel, 1 to 12. 

Bending moments. — The following assumption shall guide in the de- 
termination of the bending moments due to the external forces: Beams 
and girders shall be considered as simply supported at the ends, no allow- 
ance being made for continuous construction over the supports. Floor 
plates when constructed continuous and when provided with reinforcement 
at top of plate over the supports, may be treated as continuous beams, 
the bending moment for uniformly distributed loads being taken at not less 
than W^L-i-10; the bending moment may be taken WL-*-20 in the case of 
square floor plates which are reinforced in both directions and supported 
on all sides. The floor plate may be taken as part of the beam or girder in 
computing its moment of resistance to the extent of not more than 10 times 
the width of that beam or girder. 

Resisting moments. — ^The moment of resistance of any reinforced con- 
crete steel construction under transverse loads shall be determined by 
formulas based on the following assumptions: 

The bond between concrete and steel is sufflcient to make the two ma- 
terials act together as a homogeneous solid. 

The strain in any fiber is directly proportionate to the distance of that 
fiber from the neutral axis. 

The modulus of elasticity of the concrete remains constant within the 
limits of the working stresses. 

The tensile strength of the concrete shall not be considered. 

When the shearing stresses developed in any part of the construction 
exceeds the safe working strength of the concrete, a sufficient amount of 
steel shall be introduced in such a position that the deficiency in the resist- 
ance to shear is overcome. 

When the safe limit of adhesion between the concrete and steel is ex- 
ceeded, some provision must be made for transmitting the strength of the 
steel to the concrete. 

Columns. — Reinforced concrete steel may be used for columns in which 
the ratio of length to least side or diameter does not exceed 16. The rein- 
forcing rods must be tied together at intervals of not more than the least 
side or diameter of the column. 

Tests. — ^Tests must show that the construction will sustain a load of 
3 times that for which that portion of the building is designed, without any 
sign of failure. 

^Hollow concrete blocks used for outside walls and partitions shall not 
be loaded to more than 150 lbs. per sq. in. of available or effective section, 
and the hollow spaces shall not exceed 14 the area of the blocks when 
using the tables for thickness of walls. 

Untried methods of construction may first require preliminary trial 
tests. 

Frost. — ^The influence of frost must be excluded when concrete work is 
done. 



* For specifications for hollow concrete building blocks in the city of 
Philadelphia, see Sec. 25, Masonry, page 450. 



830 i7.— BUILDINGS. 

EXCERPTS AND REFERENCES. 

Reinforced=Concrete Work at the Atlanta Railway Terminal Sta- 
tion (Eng. News, April 12, 1906). — Illustrated details. 

A System of Reinforced=Concrete Construction Without Wooden 
Forms (Eng. News, July 12, 1906). — Illustrated. 

Practical Hints for Concrete Constructors (By W. J. Douglas. Eng. 
News, Dec. 20, 1906, and Jan. 24, 1907).— Illustrated. 

A Reinforced=Concrete Shop with Steel Roof Trusses and Crane- 
Girders (By W. F. Tubesing. Eng. News, Jan. 10, 1907). — Illustrated. 

The 48=Story Tower of the Metropolitan Life Building, N. Y. City 

(By Purdy & Henderson. Eng. News, Jan. 31, 1907). — Illustrated details 
of column shoes and column connections. 

Table Showing Proportions of Value in the Various Items of Con- 
struction of Fireproof Buildings (By F. J. T. Stewart. Eng. News, Feb. 7, 
1907) . — ^The table embraces various kinds of buildings in New York, Chicago, 
Boston, Baltimore and St. Louis, and gives the percentage of cost of about 
50 items of construction arranged under the following headings: Founda- 
tions, steel frame, mason work, equipment, trim and finish, and general 
expenses. The cost per cubic foot is also given. Cost of foundations varies 
from 2.3 to 24.5% of the total cost of building. 

A Reinforced=Concrete Mill Building With Separately=Molded Mem- 
bers (Eng. News, July 4, 1907). — Ten illustrations, showing details of con- 
struction. 

Stresses in Qas=Holder Girder Frames (By H. Stoffels. Eng. News, 
Aug. 15, 1907). — Illustrated diagrams of stresses in a single-lift gas holder, 
with three different arrangements of guide rollers. 

The Singer Building and the City Investment Building, of New York 
City (Eng. News, Dec. 5, 1907). — Sixteen illustrations, including: Typical 
floor plan, foundation plan, cast steel shoe for column base, elevation of 
cupola, diagram of wind bracing, details of wind bracing, typical column, 
column anchorage, of Singer tower; foundation plan, details of cast-steel 
bases of columns, foundation girders, floor plan, typical column sections, 
details of portal girders and wind bracing, of City Investment Building. 

A Reinforced=Concrete Building With Concrete Domes: Cincinnati 
Zoological Garden (Eng. News, Feb. 20, 1908). — Illustrated details of 

typical column, girder and dome construction. 

Adjustable and Portable Forms for Concrete Building Construction 

(By L. G. Hallberg. Eng. News, Mar. 5, 1908). — Illustrated details of post 
and form for girder, with adjustable and portable centering. 

Steel Construction for Long Span Floors in the Chicago Athletic 
Assn. Building (Eng. News, Mar. 19, 1908). — Illustrations of steel- framing 
for floor, and plan of steel floor girder 43-ft. long. 

Reinforced=Concrete Cantilever Girders in the Bogertown Building, 
Phila. (Eng. News, April 23, 1908). — Illustration of the side-wall-bearing 
cantilevers, and saw-tooth roof construction. 

The 10=Story Reinforced Concrete Hostetter Building, Pittsburg 

(Eng. News, May 14, 1908). — Illustrated details of column reinforcement 
and cast-iron base. 

A Reinforced=Concrete Cold Storage Building (By W. F. Tubesing. 
Eng. News, July 11, 1908). — Illustrated details of Wall and footings, tank, 
and reinforced-concrete covered bridge between buildings. 

The Reinforced=Concrete Court House at New Orleans (Eng. News, 
July 2, 1908). — Illustrated details of floor construction. 

A Steel Frame Grand Stand at Dallas, Tex. (By Howard Arthur. 
Eng, News, Aug. 20, 1908). — Illustrated: Side elevation, showing dimen- 
sions of members. 

Conservatory Buildings of Steel Construction in Garfield Park, 
Chicago (Eng. News, Aug. 27, 1908). — Illustrated: Stress sheet of arched 
ribs of Palm House ; details of steel ribs. 



MISCELLANEOUS DATA. 



831 



Special Structural Steel Work in the La Salle Hotel, Chicago (Eng. 
News, Dec. 3, 1908). — Illustrated: Diagrams and details of trtisses and 
columns. 

Cost of Concrete Construction as Applied to Buildings (Paper pre- 
sented at Nat'l Assn. of Cement Users, Cleveland, Jan. 11 to 16, 1909. 
Eng. News, Jan. 14, 1909).— Tables. 

Advance in Reinforced=Concrete Construction; An Argument for 
Multiple=Way Reinforcement in Floor=Slabs (By C. A. P. Turner. Eng. 
News, Feb. 18, 1909). 

The Operation of Passenger Elevators (By R. P. Bolton. Trans. 
A. S. C. E., Vol. LXIV). 

Cost of Concrete Construction as Applied to Buildings (By L. C. Wason. 
Eng. Rec, Jan. 16, 1909). — Numerous tables of cost. 

Cost of Concrete Construction as Applied to Buildings (By L. C. Wason. 
Paper, Nat'l Assn. Cement Users. Eng. Rec, Jan. 16. 1909). — Summary 
of cost on several buildings. 

Reinforced=Concrete Building for a Large Manufacturing Plant (Eng. 
News, Aug. 26, 1909). — Nine illustrations. A table of reinforcing rods is 
given, showing the shop data required when rods are ordered. Floor loads 
ranged from 100 to 700 lbs. per sq. ft. In many places there was heavy 
vibrating machinery. Unit stresses were decided upon for different con- 
ditions, and formulas made up as shown in the following table: — 

Table op Formulas for Unit Stresses. 





Fiber Stress: 
lbs. per sq. in. 


Per 

Cent 

of Steel. 


Resist- 
ing moment, 
in. -lbs. 




Concrete. 


Steel. 


General conditions 


700 
550 
500 
450 
650 


16000 
15000 
14000 
13000 
13000 


0.85 
0.75 
0.70 
0.62 
1.10 


120 W2 


Slightly vibrating loads 

Moderate vibrating loads 

Violent vibrating loads 

Tanks containing liquid 

Concrete not reinforced 


100 6^2 
90 W2 
70 M2 

120 6J2 
14 6^2 



Sand and Tar Base for Plank Flooring on Reinforced=Concrete Floor- 
Slabs (Eng. News, Nov. 18, 1909). — For second-story floor of factory. 
Instead of laying the plank flooring directly on sleepers embedded in the 
concrete floor slab (ordinary method), in this case (for fear that dry rot 
would soon destroy the sleepers and the flooring immediately next to the 
concrete), a layer of sand mixed with coal tar was filled between the ground 
of the first story and its flooring: First, a mixture of 50 gals, of coal-tar 
(Barrett's No. 5) to each cu. yd. of sand was spread \\ ins. thick on the floor 
slab and leveled while still warm and soft with a straight-edge. Second, on 
this layer there was then placed a layer of 2-in. plank, then l-m. rough pine 
boards, and finally a wearing surface of li-in. square edge maple; the differ- 
ent layers of planking being placed in different directions, compacting the 
sand-tar layer to about 1 in. in thickness. The floor is solid but resilient 
and comparatively noiseless. 

Reinforced= Concrete Car Barns; Separately Molded vs. Monolithic 
Structures (By M. D. Pratt. Eng. News, Dec. 9, 1909). — Illustrations: 
Plans and details, separately-molded construction. Tables: (1) Cost of 
separately-molded construction; (2) Comparative cost between separately- 
molded and cast-in-place construction. Conclusion: Separately-molded 
construction is cheaper. 

Standard Building Regulations for the Use of Reinforced Concrete 
(Nat'l Assn. Cement Users, Chicago convention amendment, 1910; Eng. 
Rec, Mar. 5, 1910). — Condensed digest: — 'Walls — 9. Rein. -cone, walls, 
one-third the thickness called for brick or stone walls, may be used. Curtain 
walls, not less than 4 ins. thick. 10. Cone, walls must be reinforced in both 
directions. Max. spacing of bars to be 18 in. centers, reinforcement in both 
faces of wall to be considered. Total reinforcement not less than \%- 
Cinder Concrete. — 19. Shall not be used for reinforced-concrete structures; 



832 i7.^BUILDINGS. 

it may be used for fireproofing. Assumptions in Design. — 43. The span 
length for beams and slabs shall be the dist. c.-c. of supports, but not to 
exceed the clear span plus the depth of beam or slab ; brackets shall not be 
considered as reducing the clear span. 44. Length of columns shall be the 
max. unsupported length. 45. Where slabs and beams are figured as simple 
beams the length shall be the clear dist . between supports excluding brackets. 
Loads. — 47. Weight of rein. -cone, to be taken as 150 lbs. per cu. ft. 49. The 
roof shall be figured to carry 30 lbs. live load per sq. ft. unless otherwise 
noted. 50. A reduction of live load coming to the column supporting the 
floor below the roof of 5% to be allowed and a further reduction of 5% of 
the live load of each story below until tue total reduction shall amount to 
50% of the live load of any floor, after which all loads shall be figured net 
to the foundations. These reductions shall not apply to storage warehouses. 
51. No reduction of loads shall be allowed for figuring floor slabs. 52. Nor 
none for figuring beams. 53. A reduction of 15% live load may be allowed 
in figuring the girders, except in buildings used for storage purposes. 54. 
In assuming the load coming to the columns all beams and girders shall be 
considered as carrying a net load consisting of 100% each of live load, 
subject to the above reductions. Bending Moments. — 55. Slabs. — ^The 
bending moment of slabs uniformly loaded and supported at two sides only 
shall be taken as wP-^S, where w=unit load and /=span. 56. Continuous 
Slabs. — For interior slabs overhanging two or more supports the bending 
moment shall be taken as wl'^-i-12. The reinforcement at the top of the slab 
over supports must equal that used at the center. 57. Slabs Reinforced in 
Both Directions. — Slabs reinforced in both directions and supported on 
four sides and fully reinforced over the supports (the reinforcement passing 
into the adjoining slabs) may be figured on the basis of bending moments 
equivalent to wl^-r-F for load in each direction. When span under consider- 
ation is not continuous, F=8; when continuous over one support, F=10; 
when continuous over both supports, F=12. The distribution of the loads 
to be determined by the formula: r=L^-r(L^—b^), in which r=proportion 
of load carried by the transverse reinforcement, L=span, 6=breadth of 
slab. 58. The slab area may be reduced by one-half as above figured, 
when the reinforcement is parallel to and not further from the supports 
than i of the shortest side. The reinforcement spanning the shortest 
direction shall be below the reinforcement spanning the longer direction, 
and shall not be further apart than 2^ times the thickness of the floor in- 
cluding the finish. 59. Simple Beams. — The bending moment of beams 
supported at the ends only shall be figured as of simple beams. 60. Partially 
Restrained Beams. — Beams supported at one end and continuous at the 
other to be figured partially restrained with a bending moment of -^o that 
of a simple beam. When the over-all vertical distance of the tension members 
is greater than i of the total depth of the beam the stresses in each member 
shall be computed in proportion to the distance from the neutral axis. 
Beams supporting rectangular slabs reinforced in both directions shall be 
assumed to take the following load: The beams on which the shortest sides 
of the slab rest shall take the load of that portion of the slab formed by the 
isosceles triangle having this side as its base and half this side as its height. 
The load from the remaining portion of the slab shall go to the beams on 
which the long side of the slab rests. 62. Continuous Beams.-^When 
beams or girders are continuous over two or more supports, the interior 
beams may be considered as partially restrained, and the bending moments 
at the center and support figured as f that of a simple beam, unless the 
concrete at the bottom of the beam at the support shall by this considera- 
tion receive excess compression. 63. T-Beams. — In beam and slab construc- 
tion, an effective metallic bond should be provided at the junction of the 
beam and slab. When the principal slab reinforcement is parallel to the 
girder, transverse reinforcement shall be used extending over the girder 
and well into the slab. 64. — Where adequate bond between slab and web 
of beam is provided, the slab may be considered as an integral part of the 
beam, but its effective width shall not exceed ^ on either side of the beam, 
nor be greater than 6 times the thickness of the slab on either side of 
the beam. Measurement from the edge of the web. 65. In the design of 
T-beams acting as continuous beams, due continuation should be given to 
the compressive stresses at the supports at the bottom of the beam. Ratio 
of Moduli. — 75. The ratio of moduli of elasticity of concrete to steel shall be 
considered as 1 to 15. 76. The allowable tensile stress in reinforcement to 
be 16,000 lbs. per sq. in. for medium steel and 20,000 lbs. per sq. in. for 
high elastic limit steel with adequate mechanical bond. 77. The compres- 



MISCELLANEOUS DATA, 833 

sive stress in the steel reinforcement to be 15 times the allowed compression 
in concrete in which the steel is embedded. Fireproofing. — 78. For main 
reinforcement in columns, 2 ins. of concrete; girders, 1^ ins.; floor slabs, 
lin. 

Approximate Cost of Mill Buildings (By C. T. Main. Paper before N. E. 
Cotton Manuf'rs' Assn., April, 1904, and revised to Jan., 1910; Eng. News. 
Jan. 27, 1910). — Diagrams and tables of costs. 

New Buildings of the Jones & Laughlin No. 14 Mill (Eng. Rec, May 14, 
1910). — Illustrated details: Runway girder hangers, shipping yard columns, 
bloom yard columns, wall columns, 57-ft. riveted roof truss, semi-truss of 
main roof, crane girders. 

Agreements for Building Contracts (By W. B. Bamford. Trans. A. S. C. 
E., Vol. LXVII., June, 1910). — Bibliography. Suggested "Agreement and 
Schedule of Conditions of Contract." 

Underpinning the Cambridge Building, N. Y. City (By T. K. Thomson. 
Trans. A. S. C. E., Vol. LXVII., June, 1910).— Illustrated. 

Some Formulas for Statically Indeterminate Members and Frames (By 
C. T. Mitchell. Eng. News, May 26, 1910). — Formulas and illustrations. 
Cases discussed: (a) Eccentrically-loaded column; base fixed, top hinged, 
(b) Combined crane-and-roof column, (c) Frame of two columns and cap 
beam, (d) General case of two-story framed bent, (e) Kneebraced bent. 

Structural Steel Details in the Ball Realty Building, New York (Eng. 
Rec, Aug. 13, 1910). — Details of foundation cantilever, designed to obviate 
the expense of making large excavations and underpinning an adjacent 
building; special details of pier shelves, suspended wall beam, depressed 
floor beam, depressed wall channel. 

Cutting Structural Steelwork with the Oxy=Acetylene Flame, in Dis- 
mantling a Building (Eng. Rec, Aug. 27, 1910). — ^The acetylene flame was 
used for cutting 12 columns, four 12-in. I-beams, eight 20-in. I-beams, 2 
plate girders 32 ins. deep, and for cutting holes in 8 underpinning piles and 
also for cutting them in two in the middle; the piles were regular 16-in. steel 
pipes filled with concrete and about 70 ft. long. The average speed of 
cutting with this flame is one 24 x J-in. plate in 1 min., but on heavy solid 
steel a speed as great as a 4^ x 4i-in. section in 1 min. has been attained. 
This work was done with a portable plant including three acetylene safety 
cylinders like those used for car lighting apparatus and other purposes. 
Each cylinder was 12 ins. in diameter, and 26 ins. long and contained 225 
cu. ft. of acetylene gas at atmospheric pressure, compressed to a pressure of 
150 lbs. per sq. in. 

Wind Loads on Mill Buildings (By Albert Smith. Paper before Western 
Soc of Engrs., Nov. 9, 1910; Eng. News, Nov. 24, 1910). — Mr. Smith ad- 
vocates: — (1) Placing the wind loads equally on the two walls, and inward 
and outward on the windward and leeward roofs respectively, as giving 
important changes of stress in members of the roof-truss, as giving less 
stress in the knee-braces and columns, and as permitting the rational design 
of the girts. (2) An accurate determination of the point of contraflexure 
for fixed-end columns. (3) Making the amount of the wind-pressure for 
a given building a matter to be decided in the light of the exposure of that 
building. 

A Test of a Flat Slab Floor in a Reinforced=>Concrete Building (By Arthur 
R. Lord. Paper before Annual Conv. of Natl. Cement Users, New York, 
Dec 13, 1910; Eng. News, Dec. 22, 1910). — Description of test, with illus- 
trations and diagrams. 

Important Illustrations of Buildings and Details. 

Description. Eng. News, 

Cantilever truss for carrying upper front of building Mar. 10, '10 

Rein. -cone, and plaster block floor construction Mar. 10, ' 10 

Greenhouses in Berlin Botan. Gardens, with roof Mar. 17, '10 

Narrow 12-story steel building, beam conn, with brackets . . . .May 26, '10 
A reinforced-concrete cold storage warehouse, 72 ft. x 100 ft . . . Nov. 3, '10 

Eng. Rec 
Cold storage warehouse insulation, rein. -cone construction... .Jan. 30, '09 

Structural details of the Philadelphia Opera House Jan. 30, '09 

Ten-story reinforced-concrete warehouse, Pittsburg Feb. 6, '09 



834 4^1.— BUILDINGS. 

Description. Eng. Rec. 

Alaska Commercial Bldg., San Francisco, eng'g features Feb. 6, '09 

Columns with connections for wind bracing and cantilevers Feb. 20, '09 

Methods of hanging shafting, etc., in rein. -cone, buildings Mar. 13, '09 

Reinforced-concrete church, with dome, in Los Angeles Mar. 20, '09 

A light steel pier shed, roof 53-ft. span Mar. 27, '09 

Construction of the Baxter Bldg. (rein. -cone), Portland, Me.. . .Apr. 10, '09 

Methods of hanging wires and shafting to cone, beams Apr. 24, '09 

Reinforced-concrete dome of the Porto Rico Capitol May 1, '09 

Types of hangers used for piping in a power-house May 15, '09 

The Keewatin rein. -cone, flour mill May 29, '09 

Principal roof trusses and banquet hall girders, La Salle Hotel . . .June 5, '09 
Steel details, floor, girders, columns, cornice of Trust Building . .July 3, '09 
Engine house: sep. -molded roof members; engine and drop pits, 

etc July 10, '09 

Cement shed, mixing building, measuring tank, concrete plant. . .July 17, '09 

Section of framework, Copper Queen smelter building July 31, '09 

Details steel framework of large coal storage shed Sept. 4, '09 

Plans of sedimentation basin, Goderich, Ontario Sept. 4, '09 

Plan of wood-framed machine shop Sept. 4, '09 

Details concrete and steel work. Met. Life Bldg., San Francisco. Sept. 11, 'Q9 

Details safety equipment, Singer Building elevators Sept. 11, '09 

A reinforced-concrete sawmill Sept. 11, '09 

Floor beam plans, typical columns. Bank Bldg., N. Y Oct. 2, '09 

Structural steel frames of open hearth bldgs., Gary, Ind Oct. 9, '09 

Rein. -cone, joists with hollow tile fillers. Tables Oct. 9, '09 

Rein. -cone. (Quincy market) cold storage warehouse, Boston Nov. 13, '09 

General plans of Balloon house, U. S. Signal Corps Dec. 4, '09 

Details of forms for concrete floor beams and slabs Dec. 11, '09 

Details, steel truss supporting column, Martinique Hotel Jan. 1, '10 

Rein. -cone, grand stand at Minn. State Fair Grounds Jan. 15, '10 

Approx. cost of mill buildings; diagrams, tables Jan. 29, '10 

Details, columns, etc., large steel frame rolling mill Jan. 29, '10 

Cross-section rein. -cone, warehouse; details machinery Mar. 12, '10 

Steel and architectural details, N. Y. Municipal Building Mar. 19, '10 

Cast -steel column-pedestals (4 to 6^ ft. sq.), N. Y. Munic. Bldg. .Mar. 19, '10 

Rein. -cone, girder beam, 53 ft. clear span Mar. 5, '10 

12-story rein. -cone, bldg., 48 ft. x 110 ft., without inter. columnsMar. 5, '10 

Steel freight sheds at Winnepeg, C. N. & G. T. P. Ry Apr. 16, '10 

Arrangement of reinforcing at elevator and stairway Apr. 16, '10 

Framework 22d reg't armory; 3-hinged arch; cantilever May 5, '10 

Rein. -cone, grandstand, Minn. State Fair race track June 4, '10 

Formula for determining the elevation of grandstand seats June 4, '10 

Concrete and tile floors with 2-way reinforcement June 16,' 10 

Steel and architectural details of Chicago station, C. & N. W. Ry.June 18,' 10 

Structural details of Columbia Theatre, San Francisco June 23,' 10 

Column sections, bases and splices, Curtis bldg., Phila July 9, '10 

Concrete building with steel columns in lower stories July 30, '10 

Saw-tooth-roof machine shop for Georgia Ry July 30, '10 

Heating and ventilation of Union Passenger Sta., Wash., D. C.Jan. 2, '09 

Reinforced-concrete construction in the Hartford Armory Jan. 9, '09 

54-ft. rein. -cone, arch for supporting warehouse floor Jan. 16, '09 

Wall insulation of a cooling room Aug. 6, ' 10 

Half of steel roof truss for the Doe Memorial Library Aug. 27,' 10 

Diagram from column formula: 15000 — 60(/^r) Sept. 3, '10 

Steel and rein. -cone, grand-stand, baseball park, Chicago Sept. 3, '10 

Structural steel details in the Wick Bldg., Youngstown Sept. 24,' 10 

Details of cantilever beam in rein. -cone, storage warehouse Oct. 15, '10 

Deep underpinning (12-story steel bldg.) through sand Oct. 22, '10 

Depositing concrete by gravity in a 7-story building Oct. 22, '10 

The Tacoma High School Stadium (L. D. Howell) Oct. 29, ' 10 

Structural details of the Curtis Bldg., Phila., Pa Nov. 5, ' 10 

Structural details in the Soldan High School, St. Louis Nov. 5, ' 10 

Metal wall forms for concrete houses Nov. 19,' 10 

A large concrete coal breaker and washery building Dec. 3, '10 

Underpinning the Manhasset Building, New York Dec. 10, '10 

Typical beam, columns and floor reinforcement, cone, bldg Dec, 10, *10 

Grandstand of reinforced concrete, Cleveland (O.) B. B. Club. . .Dec. 17, '10 




48.— RETAINING WALLS. 

The forces acting on a retaining wall, due to the pressure of the earth 
behind it, are not susceptible to exact determination. Several theories 
have been advanced from time to time, based on assumptions more or less 
at variance with practical conditions, and from these theories formulas have 
been deduced, but they are not relied upon with any degree of certainty. 
We depend rather upon the proportions of existing structures for our 
designs. Some data have been obtained from tests of models, but the con- 
ditions under which the tests were made are not considered sufficiently 
reliable upon which to base a general working formula. 

Theory of Earth Pressure (No. I). — Any theory of earth pressure 
should be founded on assumptions clearly and carefully made, and leaning 
rather on the side of safety than otherwise. It is believed that in the 
present discussion a clearer concep- 
tion may be had by taking a concrete 
example for an illustration. 

Let us consider an earth fill 20 
ft. high, level, and of indefinite ex- 
tent. Imagine this fill to be cut by 
the vertical plane ab; then will there 
be on either side of the plane a set 
of equal and symmetrical forces act- 
ing on the plane, in equilibrium. . 

What is the nature of these forces, Ground Line 

their intensities and directions, and -p. ^ 

their resultants^ -^^ig* ■»• 

Firstly, it is assimied that the earth fill is dry and granular, in fact 
sand, as that will probably produce about the greatest pressure*; also that 
there is no cohesion among the grains of sand and hence there can be no 
tension in any part of the mass. If now the fill to the right of the plane ab 
is removed, it is clearly evident that certain forces as p, whose resultant is 
P, may be applied on the right face of the plane to hold the fill to the left 
of it in place, and maintain equilibrium as before. In Fig. 1 these forces 
are represented as acting horizontally, but no assumption is being made 
at present as to the direction of the original forces on the plane ab due to 
the earth fill removed, nor to the direction of the existing forces acting to 
the left of the plane due to the earth fill in place. It is assumed merely that 
the p forces represent in intensity the horizontal components of these forces, 
in the direction of the former and opposite in direction to the latter. 

Let us next remove the plane ab, the forces acting to the right of it, and 
also that portion of the ground line to the right of a, if we can so stretch our 
imagination. Immediately, the sand will begin to slide over fan-like planes 
radiating from a: There will be a tendency for the whole triangular prism 
as abd to slide en masse on some plane ad, and the general movement 'of 
the earth will not cease until some plane as ac is exposed and all the material 
above is removed. It will further be found that this plane ac makes an 
angle ^=33°-41' (about) with the horizontal. This angle is called the 
angle of repose, angle of friction, f or natural slope for that material. It is 
based on the engineer's slope of 1^ horizontal to 1 vertical, which earth fill 
in general assumes, and is the slope at which the material will just barely 
remain at rest. For instance, if we consider the mass of earth abc restored 
above the plane ac and assume it for the moment to have sufiicient cohesion 
so that no sliding plane above the plane ac can develop through it, that is, 
no part of the mass abc can slide on the other part, then will the niass be 
at rest, as the tendency to slide on the plane ac will just equal the resistance 
due to friction. Clearly, then, considering the prism abc as a united mass, 
it is evident from the foregoing that no horizontal pressure would be exerted 

* The exception to this is clean, coarse, uncemented gravel; see page 838. 
t The coefficient of friction = tan ^. 

835 



836 ^.--RETAINING WALLS, 

by it, and hence no forces p, acting on the face ab, would be needed to keep 
it in place. As such it would exert a vertical force = Wt , a normal force 
Wy COS = Wn , and a tangential force on the plane ac equal to Wv sin = 
Wt =Wn tan <f>=iWn XI) -^l^ = i Wn. The tangent of the natural slope, 
or tan 0, equal to f for earthwork, is commonly termed the coefficient of 
friction because it requires a force a little greater than f of the normal 
pressure on the natiu-al slope, to move the mass. Hence the united prism 
abc is just stationary on the IJ to 1 slope dc, because the total friction, 

i Wn = Wt , the tangential force (1) 

Slope of Maximum Pressure. — Consider any triangular prism abd. Fig. !» 
resting on the sliding plane ad whose base is x, height h, and length perpen- 
dicular to the paper is one — all in feet. The weight of the material com- 
posing the fill is taken at 100 lbs. per cu. ft. Then, with the center of 
gravity at 5, we have: 

Vertical force or weight, Wy = — ^ — ^^^- ' 

Normal force (to plane ad), Wn = Wy cos a: = 



2 V;j2+^2 

Tangential force (on plane ad) , Wt = Wy sin oc = 



Total friction (on plane ad), /= f W„ 



2 Vh^ + x^ 
lOOhx X 



Vh^ + x^ 



and the resultant force R, parallel with the sliding plane ad, will equal the 
tangential force minus the total friction, or 

^_ lOOhx h_ lOOhx x ^2) 

2 \/PT^2 3 VW+x^ 

By placing the first differential coefficient equal to zero, and solving for 
maximum, 

dR (h^-hx^)i (50 h^-mhx) - (50 h^x-S3^ hx^)x (h^-{-x^)-^ _^ 
dx h^-\-x2 
whenw, (h^ + x^) (50 h^- 66f hx) = 50 h-^x^- 33| h ^. 
3/^3 
or, ^ + 2 /j2 ^ = — (General equation is sfi -\- 2 h^x = h^ cot <t)) (3) 

Solving this cubic equation there is obtained 

ic= .627 /j; whence oc = 57° -55'. 
Hence, the maximum pressure in a direction parallel with the slope, against 
the vertical plane ab, obtains when we consider that the sliding plane makes 
an angle oc = 57°-55' with the horizontal; and when h = 20, x= 12.54. Sub- 
stituting the value of x in equation (2), or the values of OC and x in the 
following: 

R-=50hxs\ti Oi-ZZ\hxQos oc (4) 

wehave, i?= 12540X .847-8360 X .531 = 6182 lbs. Hence, from the 
above analysis, the resultant pressure R, due to the earth sliding on the 
plane of maximum slope pressure, is 6182 lbs., which force is assumed to 
act parallel with the direction of the sliding plane, through the centerof 
gravity B of the mass, and intersecting the vertical plane ab at a point 
distant f h below the top of fill (Fig. 1). 

In a similar manner the total horizontal pressure H ( = R cos oc), against 
the wall, may be found; thus, 

TT E, ^ 50h^x^-3S^hx^ ,.. 

H = R cos oc = / (5) 

h^ + x^ 

Placing the first differential coefficient— i — equal to and solving, we have, 

ax 
«3 + 3 A^j. ^ 3^3 (General equation is x3 + 3 A^x = 2 h^ cot 0), and for maximum 
value of H we have x = 0.8178 h = 16.36 ft.; whence oc = 50°-43', and H = 16360 
(sin oc — I cos oc ) cos = 3645 lbs., applied horizontally at c. Fig. 1. Neglect- 
ing friction on slope, H = 8016 lbs., making the lateral pressure about 40% of the 
vertical pressure, which may be considered a maximum for any earthy material. 




PRACTICAL DEDVCTIONS. 837 

Assumptions Regarding Friction. — Reverting to Fig. 1, there are two 
planes where friction may be considered, namely, ad and ab. We have 
found (equation 4) that the possible friction on the plane ad! is 3Si hx cos 
oc = 4439 lbs. which, deducted from the tangential force, 50 hx sin CC (or 
10621 lbs.), gives 6182 lbs., the resultant. Hence, if the friction is neglected 
our resultant will be increased from 6182 to 10621 lbs., an increase of nearly 
72%. Should this friction, for safety, be wholly or partly neglected? 
Before arriving at any conclusion in regard to this, let us consider the 
possible friction on plane ab. li a b represents the back of the retaining 
wall, the amount of possible friction will depend 
upon the construction of the wall itself. 

Let Fig. 2 represent a simple frame construction 
to better analyze the acting forces. The resultant 
force R is resisted by the strut c e, and hence the 
stress in the latter is equal and opposite to R. The 
point c is the center of gravity of the distributed 
earth pressure acting on the facing, which is sup- 
ported by the stud ab, which facing is stiff enough to 
resist bending, and which is balanced on the pivot 
or point of support c. As long as the resultant Ground, Mne, a ^ e 

pressure acts parallel to the plane a d, as indicated, Fig. 2. 

there will be, apparently, unstable equilibrium; but should it make a less 
angle than OC with the horizontal, then there would be a tendency for the 
strut c e to revolve about e. In doing so it would tend to raise the facing 
a b, thereby causing friction of the latter against the earth fill. Indeed, 
as the forces now act, the resultant R can be transmitted to the strut c e 
only on the assumption that there is sufficient frictional force downward 
on the "fill" face of the wall to resist the vertical component of the stress 
in the strut c e. Let us examine this: Assuming the coefficient of friction 
of the earth on the face of the wall as equal to | of the normal pressure, 
there is obtained, 

Total friction =i R cos ex: • 

= 2189 lbs. for 2?= 6182; 
= 3760 lbs. for i?= 10621. 
This friction on the "fill" face of the wall will be opposed by the vertical 
component of the stress in the strut c e, equal to 
R sin OC 
= 5236 lbs. for i?= 6182; 
= 8996 lbs. for i?= 10621. 
Hence it is evident that the wall may have to be anchored, at a, an amount 
equal to 

Vertical component of i?— (total friction +wt. of wall) 
of else the strut c e will have to have a less inclination with the horizontal. 
Similarly, the preceding form of analysis may be repeated, but using 
the maximum value of H obtained from equation (5) with the angle of 
slope 50°-43'. 

Practical Deductions. — ^The uncertainty of some of the foregoing assump- 
tions makes the problem a difficult one to solve. However, we are reason- 
ably certain that the resultant pressure on the wall parallel with the sliding 
plane ad (Fig. 1) is somewhere between 6182 and 10621 lbs. Assuming 
the latter as correct by neglecting friction on the slope (which can obtain 
only when the wall begins to tip) we have that the resultant normal pressure 
P acting horizontally at c is 

P=i? cos a=10621X. 531 = 5640 lbs. 

which is equivalent to a horizontal pressure per square foot on the vertical 
wall a b, varying uniformly from zero at b, to 564 lbs. at a. This is equiva- 
lent to stating that the horizontal pressure per square foot against the 
retaining wall at any point below the top is equal to 28t% per cent, of the 
weight of a vertical column of earth, one square foot in section, extending 
from the top of the fill to that point: that is, the lateral pressure is 28i^(j per 
cent, of the vertical pressure. 

For Temporary Shoring, this maybe reduced to 25 per cent., or even 
less, provided of course that the "earth" is not saturated with water so as 
to be in a muddy condition. Up to a certain point of wetness, moist earth 
will produce less percentage of lateral pressure than dry, granular earth. 

For Permanent Structures, the lateral pressure should be assumed at 
about 30 per cent, of the vertical, and even up to 33^ per cent, in places 
subject to considerable jarring, as for retaining walls supporting railway 



838 



i8.— RETAINING WALLS. 



.^3V— 8~. 




6000 




6000 



embankments. Where the weight of a train or of a structure falls within 
the range of the slope, it can be reduced to an equivalent volume of earth. 

If the material is coarse gravel, the ratio of lateral to vertical pressure 
may reach as high as 40 per cent., for which provision should be made. 

Graphical Solution of Preceding Problem. — Fig. 3 is a graphical solution 
for a masonry retaining wall 20 ft. high to restrain a level earth fill of the 
same height — the problem which has engaged our 
attention throughout the discussion of the theory 
of earth pressure. The weight of the earth fill is 
assumed at 100 lbs. per cu. ft., and the lateral pres- 
sure, 30 per cent, of the vertical. Hence the re- 

30 X 20^ 
sultant lateral pressiire P = ^ = 6000 lbs., acting 

horizontally at a point one-third the height from the 
bottom. Selecting the trapezoidal type of wall, the 
center of gravity is found by assuming the top and 
bottom widths; laying off the bottom width on either 
side of the top, and the top width on either side of the 
bottom; and finding the point of intersection of the 
diagonal lines joining the outer points. The weight 
of one foot section of wall at 150 lbs. per cu. ft.= 
16500 lbs., acting vertically through the center of 
gravity. From the intersection of the two acting 
forces the triangle of forces is drawn showing the 
resultant to intersect the base within the middle third, 
which is good practice. 

Fig. 4 shows another design for the same in which 
the top width of wall is one ft. instead of three. In 
this case the resultant falls outside the middle third, 
and hence there is tension on the face ah. In addition 
to this tension, the factor against overturning at t is 
lessened, and it is not as desirable a type as that 
shown in Fig. 3. 

Top of Fill, Sloping. — Up to the present, we 
have considered the top of fill level, and flush with 
the top of the wall. We will now consider what 
modification of pressure will be effected in case the 
surface of the fill should slope either upward or down- 
ward from the back or, in fact, show any profile 
whatever. _ Fig. 5, in which ah is the "fill" face of 
the retaining wall, shows the slopes of maximum 
pressure for: (1) a level fill bd, with maximum pres- 
sure slope ad making an angle of 57°-55' with the 
horizontal; (2) an upward surface slope hu, with 
maxim iim pressiire slope au slightly less than ad\ 
and (3) a downward surface slope hi, with maximum 
pressure slope al slightly greater than ad. It is to be 
noted that the slopes of maximum pressure au and 
al very nearly coincide with ad, the slope of maxi- 
mum pressure which we found for a level fill, and 
it might also be stated here that this slope, ad, 
may be varied either way several degrees without Fig. 5. 

materially affecting the resultant pressure. Also, as hu and hi are the upper 
and lower limits at which earth fill will stand (H to 1), then all other possible 
surface slopes must be less than these, and in approaching a level their 
slopes of maximum pressure a u and a I will approach o c^ in direction. For 
all practical purposes a d may he assumed as the slope of maximum pressure 
for any surface slope. This assumption, containing a small percentage of 
error, greatly simplifies the method of calculation for general cases. For 
depressed surface slopes, as hi, the assumption is on the side of safety; 
while for surcharged walls with upward slopes, as hu, the assumption in- 
volves a slightly opposite effect, to counteract which it would be well to 
have the resultant of all the forces (earth and wall) cut the base of the 
retaining wall at least f of the width from the toe instead of the customary \. 

General Cases. — By observing the following methods a retaining walJ 
may be designed for any surface slope: (1) Draw the back of the wall, ah 
(Figs. 6, 7, and 8) ; the slope of greatest pressure, au or o/, making an angle 



4. 




RANKINE'S THEORY OF EARTH PRESSURE. 



839 



of 57°-55' with the horizontal (tan 57°-55' = 1.595, cot = 0.627); and the 
surface slope bu or bL (2) Find the area and the center of gravity of the 
earth fill area, abu or abl, above the slope of greatest pressure, au or al. 
Multiply this area by the weight of the fill per cubic foot (average 100 lbs.). 
This weight, F, of one lineal foot of fill is laid off to scale of lbs. vertically 
downward from the center of gravity e.g. of the mass. The resultant 
tangential pressure T, parallel with the slope of maximum pressure, au or 
o/, is F sin oc; and the resultant pressure P, normal to the face of the wall, 
is F sin oc sin /?. (For a vertical wall, F sin oc sin jS = F sin OC cos oc = 
0.45 F*). Figs. 6 and 8 show the graphical solution for P, in intensity, 
direction, and point of application, at c. (3) Find the point of applica- 
tion c = the center of gravity of the total lateral earth pressure P on the 
wall ab. This point, c, is at the intersection of the back of the wall (or 
its average slope) with a line drawn through the center of gravity, c. g., 
of the earth fill, parallel with the slope of maximum pressure, au or al. 
The direction of P is normal to the back of the wall ab. (4) Assume 
any section of wall, as trapezoidal or rectangular, with a base about i% 
the height; and find its area and center of gravity. Multiply this area 
by the weight of the masonry per cubic foot (average 150 lbs. for good 
stone masonry) . This weight W, of one lineal foot of wall is laid off to scale 
of lbs., vertically downward from a point o at the intersection of a vertical 
line passing through the center of gravity of the masonry, and the force P. 
That is, W and P are laid off in sequence and the resultant R is drawn. If 
R cuts the base of the wall within the middle third the design is sufficiently 
strong to resist overturning, with a proper factor of safety. If the wall is 
heavily surcharged, as in Figs. 6 and 7, it is better to have the resultant 
cut at least | of the base from the toe intsead of J. (5) One or two trial 
wall sections may be necessary before the desired result is attained. (6) 
Read the remarks under " Practical Deductions," page 837. The fore- 
going are equivalent to a lateral pressure equal to 28^xi per cent, of the ver- 
tical pressure — for average conditions. If greater or less security is desired, 
P may be increased or dimished proportionately, depending upon the 
permanency of the structure and the quality of the material composing the 
fill. An increase of P of say 20 to 40 per cent, would provide for a per- 
manent structure supporting a gravel fill or a fill liable to become quite 
saturated with water. For a temporary wall or shoring, P may in some 
cases be decreased from 5 to 10 per cent. This increase or decrease in P 
is made prior to the dimensioning of the wall so that, in any case, the re- 
sultant should fall within the middle third. (7) If any superimposed load, 
as a train load, comes on the fill, it should be reduced to an equivalent 
sectional area of earth, "and may be considered as such in finding the weight 
F and the center of gravity, e.g. 



-v 


"^ 


!_ 


1 


\ -f' 


\ 


^ 


\n 


\ 


N 

*.j; 




e~ 


11 


a 


& 



Fig. 6. 




Fig. 7. 




*5^.-v 



Fig. 8. 



Rankine's Theory of Earth Pressure (No. 2). — This theory also assumes 
the earthy material to be granular and without power of cohesion, therefore 
all stresses are compressive throughout the mass. 



*This agrees with the deduction, page 837, that the lateral pressure for 
any depth below the surface of a level fill is 2S^a per cent, of the vertical 
pressure, although it is not apparent at first glance. 



840 ^.—RETAINING WALLS. 

Notation. 
w = weight of earth in lbs. per cu. ft.; 
X =vert. depth in ft. below surface, of any (conjugate) plane parallel 

with the surface plane (Fig. 9) ; 
6 = angle of inclination of conjugate plane with the horizontal; 
^ = angle of repose of earth,* or angle of greatest obliquity; 
pc = intensity of vert. pres. in lbs. per sq. ft. on any conjugate plane at 

depth X below surface; 

p, = intensity of pres. in lbs. per sq. ft., in a direction parallel with the 
conjugate plane, against a vertical plane (as a retaining wall) at 
depth. X below surface. 

Formulas. 

(Siirface plane assumed to be of indefinite extent.) 

In general, pc =w x cos d (1) 

and pr may have any value between 




Fig. 9. 



_ „ cos ^— V^COS^ ^ — C0S2^ ^^ . 

p^ vw X COS e — ^ (2a) 

cos ^+\/cos2 ^ — COS2^ 



J _ , cos ^+\/cOs2 d — COS^(f> ,--. 

and p^ \w X cos 6 ==: (2b) 

cos 6—\/cos^ 6 — cos^<p 
If h is the height of the retaining wall, the maximum intensity of pressure, 
at the bottom, may be found by substituting h for x in the preceding equa- 
tions; and as the intensity of pressure at the top of the wall is zero, the 
average will be one-half the maximimi. Hence the total pressure on the 
wall will equal the average intensity multiplied by the height h, and this 
to be applied at a point ^h from top or surface, and in a direction parallel 
with the conjugate plane. 

If the surface plane is horizontal: 
Equation (1) reduces to po =w x (3) 

" (2a) •• - p^^^^\^^ (4a) 

1 + sm 

•■ (26) •■ " p,xwx\^^^ (46) 

1 — sm 

If the surface plane is inclined at the angle of repose so that 6=^, then 

ps ^^w X cos 6 = w x cos (5) 

For water, the angle of repose = 0, and equations (4a) and (46) reduce 
to the hydraulic equation 

p, =w X (6) 



* Usually assimied at_ 33°-41' for earth fill, equal to slope of 1^^ to 1, 
making coefficient of friction ( = tan ^) = f. 



STANDARD TYPE. 



841 



N. Y. C. & H. R. R. R. Standard Retaining Wall. 
(W. J. Wilgus, Chief Engineer.) 




D Concrete 
r-iTi 



GmmtlLlne 
IbantJijKofi to salt 
/oeaf conditions but must 
aat be less than "f-C 'deep ^ 
enless good rock Is found. 

Outside pile to be 
bartered out 1:6 in 
saff mat&iuL Number of Piles If required to be 

determined by loading and character Horizf^ttal Joki. 
(^underlying material. 



[J UJ Ul Ul 



Weep holes not more than 
l5'-0' apart with vertical 
•^ blind drain extending to 
1 io top of wall. 
^-.-i Oid rails lO'Cto C. unless good 
•^ rockls found. Whenepilesare 
^, not used B.of Rail to be 6' 
from Bolto m^yy//^///'^ c*^"" 



Stone 



Fig. 10. 



1.— 


Cubic Yards op Masonry 


IN Retaining Walls (Fig. 10). 




Cu. Yds. per Running Ft. 




Cu. Yds. per Running Ft. 


Height 






Height 
















A 




Body 


Foundat'n 


A 




Body 


Foundat'n 




Coping. 


Wall. 


4' Deep. 




Coping. 


Wall. 


4' deep. 


5'0" 


0.111 


0.671 


0.833 


18' 0" 


0.111 


3.858 


1.544 


6'0" 


<t 


0.858 


0.920 


19' O'' 




4.204 


1.556 


7'0" 


" 


1.048 


0.932 


20' 0" 


(( 


4.553 


1.568 


8'0" 


<t 


1.241 


0.944 


21' 0" 


•' 


4.946 


1.744 


9'0" 


" 


1.455 


1.032 


22' 0" 


(( 


5.341 


1.756 


10' 0" 


(t 


1.673 


1.044 


23' 0" 


'* 


5.740 


1.768 


11' 0" 


*• 


1.894 


1.056 


24' 0" 


(i 


6.183 


1.946 


12' 0" 


t( 


2.136 


1.143 


25' 0" 


*• 


6.629 


1.958 


13' 0" 


♦• 


2.381 


1.155 


26' 0" 


♦' 


7.078 


1.970 


14' 0" 


<( 


2.630 


1.167 


27' 0" 


(( 


7.571 


2.146 


15' 0" 


•' 


2.922 


1.343 


28' 0" 


*• 


8.068 


2.158 


16' 0" 


(t 


3.217 


1.355 


29' 0" 


i( 


8.567 


2.170 


17' 0" 




3.516 


1.367 


30' 0" 


<« 


9.110 


2.346 



EXCERPTS AND REFERENCES. 
Designs of Reinforced=Concrete Retaining=WalIs (By J. Lehman. 

Eng. News, Aug. 7, 1902).— Illustrated. 

Typical Cross=Section of Retaining^Wall, and Details of Expansion 
Joint (Eng. News, June 2, 1904). — Illustrated. 

Analysis and Design of a Reinforced=Concrete Retaining-Wall (By 

F. F. Sinks. Eng. News, Jan. 5, 1905). — Comparison of cost with plain 
concrete. Illustrated. Discussion of this article in Eng. News, Feb. 16, 
1905. 



842 



^.—RETAINING WALLS. 



High Reinforced=Concrete Retaining=Wall Construction at Seattle, 
Wash. (By C. F. Graff. Eng. News, Mar. 9, 1905).— Illustrated. 

Difficult Reinforced=Concrete Retaining =Wall Construction on the 
Great Northern R. R. (By C. E. Graff. Eng. News, May 3, 1906).— Illus- 
trated. 

The Stability of Sea Walls (By D. C. Berber. Eng. News, Aug. 23, 
1906).— Illustrated. 

Reinforced=Concrete Retaining=Wall Design (By E. P. Bone. Eng. 
News, April 25, 1907).— Illustrated. 

Comparative Sections of Thirty Retaining Walls and Some Notes on 
Retaining Wall Design (By F. H. Carter. Eng. News, July 28, 1910). — 
The following table is compiled from the dimensions given on the cross- 
sections: — 



Retaining Wall. 



N. Y., N. H. & H. R. R. (stone 

masonry) 

Penn.,N. Y. & L. I. R. R. (concrete) 

wet gr'd r 

Penn., N. Y. & L. I. R. R. (concrete) 
Boston subway (cone, granite faced) 
East Boston tunnel (cone, granite 

faced) 

Penn. Ave. subway, Phila. (stone 

masonry) 

Detroit tunnel (concrete) 



Borough of Bronx, N. Y. City ( , 



111. Cent. R. R., Chicago (concrete) . . 
B. & M. R. R. (1st class mas. or con- 

B. & A. r! R.' '(.*.* .' .' .* .") level* earth 
emb'k't 

Penn. R. R., standard (stone mason- 
ry) 

N. Y. C. &H. R. R. R. ( ) 

Sea wall, Lynn shore (concrete) 

Sea wall, Cradock Br. (1:3:6 con- 
crete) 

At spillway, Wachusett dam ( ) 

Mass, Highway Comm. (stone mason- 
ry) 

Board Water Supply, N. Y. (con- 

QJ-g'f^g^ .......' 

Board Water Supply, N. * Y. '(rubble 
mas.*) • 

Board Water Supply, N. Y. (cyclo- 
pean mas.) 

Sea wall, Charlestown (stone mason- 
ry) 

Subway wall, Boston Term. Sta. 
(concrete) 

Harbor wall, Charles river (cone, 
granite faced) 

Sea wall, Charles River (dry coursed 
rubble) 

Retaining walls on transverse roads 
in connection with Boulevard, 
New York City. — 3 examples are 
given 

Retaining walls designed for Cam- 
bridge Main Street Subway. — 3 
examples are given 



Height 
h. 



25' 9 " 

23' " 
18' " 
13' 6 " 

17' " 

28' 4 " 
28' U" 
33' 2 " 
21' " 

20' '^ 

20' " 

25' " 
28' " 
18' " 

19' 9 " 

26' 4 " 

13' 6 " 
20' " 
18' " 
44' 6 " 
24' " 
16' 6 " 
23' 4 " 
15' 6 " 



Top 

Width 
t. 



3' 0" 

3' 4" 

3' 4" 

3' 0" 



2' 6" 
3' 0" 
2' 4" 
1' 5" 

1' 6" 

2' 0" 

3' 0" 
3' 0" 
3' 0" 

2' 0" 

3' 7" 

2' 4" 
2' 3" 
3' 6" 
3' 0" 
2' 6" 
3' 0" 
4' 6'' 
4' O'' 



Bottom 

Width 

b. 



12' '^ 

15' W 
9' " 

8' " 

5' " 

12' 4 '^ 
13' 7 " 

12' ir 
9' 3r 

8' " 

9' " 

13' 8 " 

12' 31" 

9' " 

9' " 
13' 6 " 

7' " 

9' 5i" 

11' 3r 

24' or 

12' " 

8' 6 " 

15' '^ 

9' 6 ♦ 



Ratio 
b^h. 



0.46 

0.69 
0.50 
0.59 

0.30 

0.43 
0.48 
0.36 
0.44 

0.40 

0.45 

0.55 
0.44 
0.50 

0.45 
0.51 

0.52 

0.47 

0.63 

0.54 

0.50 

0.52 

0.64 

0.61 



MISCELLANEOUS DATA. 843 

The Bracing of Trenches and Tunnels, With Practical Formulas for 
Earth Pressures (By J. C. Meem. Trans. A. S. C. E.. Vol. LX). 

A Reinforced^Concrete Retaining=Wall Along the Bank of the Ohio 
River (By F. A. Bone. Eng. News, June 3, 1909). — Illustrated. 

The Design of Retaining Walls (By Comm. on Masonry of the Am. Ry. 
Eng'g and M. of W. Assn. Eng. Rec, Sept. 11, 1909). — Includes many 
types of structures in actual use, and contains 32 illustrated sections. 

Tables for Determination of Earth Pressure on Retaining Walls (By 
C. K. Mohler. Eng. News, Nov. 25, 1909).— (1) Rankine's method after 
Howe; (2) Sliding prism theory. 

The Cracking and Partial Failure of Abutments and Retaining Walls (By 
C. K. Mohler. Eng. News, Oct. 13, 1910). — Criticism of present methods 
of design and construction. 

Illustrations of Various Types of Retaining Walls. 

Description. Eng. News. 

Reinforced-concrete retaining walls, bridge approaches Nov. 25,' 09 

Eng. Rec. 

32-ft. reinforced-concrete retaining wall Apr. 3, *09 

Section, 31-ft. rein.-conc. retaining wall, C. B. & Q. R. R Aug. 21, '09 

Design of retaining walls for Steptoe smelter Feb. 19, '10 

Combined rein.-conc. fence and retaining wall Feb. 26, '10 

Rein.-conc. retaining wall and roadway bridge and walk Sept. 24,' 10 

Types of French railway retaining walls Nov. 12, '10 



49.— DAMS. 

Common Fixed Types. — A dam is a structure designed to hold back a 
large body of water in an impounding reservoir, at a higher level than 
would naturally obtain. It should first of all be safe, and the site should 
be selected with due regard to present and future needs for storage, econ- 
omy of construction, efficiency of head, outlet, wasteway, etc. With re- 
spect to their structural features, dams may be classified as follows: 

_A Gravity dam is a masonry dam designed to resist overturning by the 
action of gravity alone. It must also resist any tendency to slide horizon- 
tally (down-stream) on its base or on any plane above (or below) the base; 
there must be no tension in the masonry at any point as in the up-stream 
face where it is most likely to occur; and the compression in any part as 
the down-stream face, where it is maximum, must not exceed the safe, 
allowable intensity per square inch. 

An Arched dam is a horizontally curved (arched) type with ends rigidly 
braced against the side walls of the canyon — a typical site for this class of 
dam. The arch is designed to relieve partly — not wholly — gravity re- 
quirements although many engineers use the full gravity section that would 
be required for a straight danr, thereby employing the arch to give greater 
stability, or in other words, to increase the factor of safety. Although the 
stresses in an arched dam cannot be determined with accuracy, especially 
as it acts partly as a gravity dam, being "fixedly" supported throughout 
its whole length, we know that the arch effect relieves the tendency to over- 
turning, caused by the action of the water pressure on the up-stream face, 
and this being true, it is evident that the base of the dam can be narrowed 
materially. This possible reduction of section, however, diminishes as we 
approach the top of the dam where the combined gravity-, arch- and tem- 
perature stresses bear in increased ratio to the gravity stresses alone. 
Indeed, it has been claimed that, under certain conditions, the arched dam 
requires a greater section near the top than does the gravity dam, although 
the writer has never met with such conditions in practice. 

A Buttressed dam is a gravity dam with buttresses, or thickened sections, 
spaced at stated intervals for the more economical distribution of the material 
than one of uniform cross-section. It may be built of concrete, steel- 
concrete, or stone masonry. The sections at the buttresses will be larger, 
and the sections between the buttresses will be smaller, than that of an 
equally safe gravity dam of uniform cross-section. Care should be taken 
in both the design and construction to avoid any possibility of tension, 
undue shearing, or excessive compression in any part of the masonry. The 
facing between the buttresses may be supported by horizontal reinforced 
concrete beams as used in building construction, or by concrete or stone 
masonry arches. The facing should practicably be impervious to water. 

A Braced dam is a dam with braced vertical or sloping face. It differs 
from a buttressed dam in that the solid masonry buttresses are replaced 
by open bracing of reinforced concrete, steel or wood. The facing may be 
of the same character of material, but not connected with the braced legs 
in such a manner as to form a series of arches between them. A good angle 
for the face of a braced dam is a slope of 45 degrees with the horizontal, 
in which case the cost of facing will generally about equal the cost of bracing. 
But if the facing is very expensive and the bracing cheap, this angle should 
be increased. The cost of repairs for each should also be considered in 
determining the economic angle. 

A Cantilever dam is designed on the cantilever principle and may be 
braced or buttressed. The facing is supported by beams or trusses projec- 
ting beyond the upper limit of the bracing. It may be framed of steel and 
concrete, reinforced concrete, steel or wood, or a combination of these 
materials. It should be secured firmly to a natural bed-rock or to a con- 
crete anchorage. ^ 

A Crib dam is a skeleton, box-like structure weighted with filling and 
usually faced to prevent undue leakage. The skeleton structure may be 
composed of logs or of hewn or sawed timbers, framed and bolted into a crib, 

844 



COMMON TYPES. GRAVITY DAM, 



845 



filled with rock, slag, gravel or other suitable material, sunk to the bottom 
(which has been previously prepared) and preferably bolted thereto. Plank- 
ing makes a cheap facing. 

A Composite dam is one composed of two or more radically different 
kinds of material, composite but not structural in character. A good ex- 
ample of a composite dam is the common brush and rock dam, the brush 
to prevent excessive leakage and the rock to give stability. 

A Filled dam is one not strictly structural in character, containing 
little or no cohesion and hence not capable of being "overturned," but 
which, if flooded (overtopped) with water, will disintegrate and wash away. 
It may be composed of any durable material of specific gravity greater 
than unity, as clay, earth, sand, gravel or loose rock. If it is of the finer 
materials it is called an Earth dam; if of the coarser (rock), a Rock-fill dam. 

(a) An Earth dam should be composed of material or materials which 
will pack well and allow but small voids, if any, and which, under the action 
of water, will not wash, leaving large holes or pockets. "Cement" gravel 
is probably the best material as it contains the proper natural binder for 
a solid mass. If materials are mixed, such as earth, sand, clay, etc., they 
are better if mixed thoroughly and uniformly — if not, there are liable to 
be seams of strata allowing the water to percolate and wash the materials. 
The ratio of 3 horizontal to 1 vertical for the wet slope and 2 to 1 for the 
dry slope is ordinarily good practice and generally should not be exceeded. 

{h) A Rock-fill dam is composed of loose rock dumped — not carefully 
laid — in place, with proper up-stream and down-stream slopes. The up- 
stream slope may sometimes be as steep as ^ horizontal to 1 vertical if the 
rock facing is laid by hand, otherwise 1 to 1 is the usual practice. The dry- 
or down-stream slope is usually broken, being steeper, say 1 to 1, for the 
upper and li to 1 or H to 1 for the lower section. The facing may be of 
double planking, caulked and asphalted, and nailed to wooden stringers, 
say e^'xC'', imbedded in the face rock. The thickness of the planking will 
of course be greater for the greater head of water. 

STABILITY OF GRAVITY DAMS. 

A gravity dam to be safe must be built on the natural, hard bed-rock 
not liable to disintegrate, and capable of withstanding the maximum 
intensity of pressure at the toe of the dam, to resist overturning. The 
outlet, for drawing off the water for domestic or commercial uses, is usually 
by (cast iron) pipes extending through the masonry wall of the dam itself, 
but preferably by tunnel construction through the solid rock at one side 
of and apart from the dam. Likewise, the wasteway, for carrying off the 
flood waters, should preferably be at some other 
point than at the dam itself. In many cases, how- 
ever, there is no alternative but to let the surplus or 
waste water from the full reservoir flow over the en- 
tire crest of the dam, or through a specially provided 
wasteway in a certain limited part of the crest. 

When the dam acts as a waste-weir, i.e., with 
the water flowing over the crest, additional forces 
must be considered as tending to overturn the 
structure. There are (1) the additional head of 
water hi above the top of the dam; (2) the tension 
or suction at the rear face of the dam, due to a 
partial vacuum at a. Fig. 1, caused by the falling 
water taking up the particles of air between it 
and the rear face of the dam; and (3) the pressure 
of ice and logs against the upper face. The vacuum 
effect may be lessened materially by rounding off 
the rear upper comer of the dam, as shown in 
Fig. 2. Fig. 

Hydrostatic Pressure. — In the discussion of dams and the forces acting 
against them it is convenient to assume a section of the dam and of the col- 
umn of water, etc. acting, as one foot thick. The weight of a cubic 
foot of water is generally assumed at 62.5 lbs., involving an error of \ of 
one per cent, on the side of safety. Therefore, the pressure of one square 
foot of surface at a depth of h feet below the surface is 62.5 h. It will thus 
be seen that the intensity of pressure increases uniformly with the depth, 
and furthermore it is always normal (at right angle) to the surface acted 




S46 



49.— DAMS. 



upon, because, being a frictionless liquid, the pressure is equal in all direc- 
tions. To find the total pressure, then, on any plane surface one foot wide, 
whether inclined or not, it is necessary only to find the intensity of pressure 
per square foot at its "middle point" and multiply this result by the length 
of the plane. Moreover, as will be shown in the next article, under "Center 
of Pressure," the total pressure, instead of being assumed as a distributed 
force, may be assumed to act at the center of pressure, i.e., at the center 
of gravity of pressiire on any "rigid" surface, thereby greatly simplifying 
the calculations. 

The following are common examples of total pressure, the heavy line 
in each figure representing the edge of the plane, one foot wide, acted upon: 

(H and h are in feet; pressure is in lbs. against the plane ab). 



""i 


1 

1 
i 


c 


Fig. 


3. 



Viatersurface 




)N of er surface 




(1). Vertical plane ah, just touching surface of water. 

T ^ 1 62.5H „ 62.5H2 „ 

Total pressure = — - — • H= — ^ — lbs. 



Fig. 5. 
(Fig. 3.) 



If the surface of the plane makes an angle 6 with the vertical, multiply 
the above result by secant 6. 

(2). Vertical plane ah, submerged below the surface. (Fig. 4.) 

Total pressure = 62.5 (^y^) {H - h) = 62.5 (~y^) ^^^• 

(3). Inclined plane ah, submerged below the surface. (Fig. 5.) 
Total pressure =62.5 ( ^ j sec ^ lbs. 

This pressure will not be horizontal, but normal to the plane ab; ex- 
ample (2) illustrates the horizontal component of this pressure. 

The above formulas will be found useful in calculating the total pressure 
on any section of a dam, whether the face is vertical or sloping. The point 
of application of the resultant pressure will now be discussed. 

Center of Pressure. — If the total pressure against any surface as ab, in 
the preceding illustrations, is concentrated as a single resultant force, this 
force will act at the center of pressure, and through the center of gravity 
of the distributed force. In Example 1, above, the resultant P of the 

distributed force is — '-= lbs., acting horizontally through the center of 

gravity e.g. of the pressure triangle a & c, at a point p on 
the face of the dam, distant f H below the water sur- 
face (Fig. 6). Thus, p is the center of pressure, for the 
head H, on the plane ab, whether vertical or inclined. 
Likewise, in Example 2, the resultant pressure on the 

62 5 
plane at is P = —^(H^—h^) lbs., and the center of pres- 
sure, p, is f f H+ „ , J feet below the water surface (Fig. 

7). In Example 3, the same value, f f H4- „ , ) , holds 

true for the distance to the center of pressure below the 
surface, while the total pressure is of course greater — or 
62 5 

—^ (//2 — h^) times sec angle of inclination with the ver- 
tical. For reference, these values are tabulated as fol- 
lows: — Fig. 7. 




CENTER OF PRESSURE, CENTER OF GRAVITY. 



847 



1. — ^Table op Hydrostatic Pressures on Vertical and Inclined Sur- 
faces. 



Case 



Pressure 
Plane. 






Surface 
of Water 
Above 
Top of 
Plane. 



Surface 

of Water 

Above 

Bottom 

of Plane. 

H 



Resultant 

Pressure 

on Plane 

1-Ft. Wide. 



Distance 
Center of 
Pressure 
Below Sur- 
face of Water. 



(1) 

(la) 
(2) 

(3) 



Vertical 

Inclined 

Vertical; 

submerged. 
Inclined ; 

submerged. 



Ft. 



h 

h 



Ft. 

H 

H 

H 

H 



Lbs. 
31.25 H2 

31.25 H2 sec 6 

31.25 (H2- h^) 

31.25 (m-h^)\ 
sec d i 






Ft. 



H-\- 



H-hhJ 



h 






Fig. 8. 



Center of Gravity of Trapezoid. — 

Let it be required to find the center of 
gravity of the trapezoid whose top is T 
and bottom, B (Fig. 8). Lay off 6 and 
h\ =B; and t and t\ =T. The diag- 
onals joining the extreme points inter- 
sect at the center of gravity, o. The 
line r o also bisects T and B. Ana- 
lytically, ^=Y-g- \ b + t ) ' ^^^^^^^" 
lating a large gravity dam the section 

of masonry or of water pressure is frequently divided into trapezoidal 
slices, the vertical forces of the masonry being the weight of each trape- 
zoid acting through its center of gravity. The area of the trapezoid = 

D — 2 — ' hence the weight of a vertical section of masonry one foot thick, 
and the water pressure due to the corresponding pressure trapezoid, are: 
For masonry, W= mD — ^ — » when m = wt. of one cubic foot of masonry. 

T -i- 7? 
For water pressure, P = wD — ^ — > when «;=wt. of one cubic foot of water. 

For water pressure, Pn = wD — - — sec ^; when w=wt. of one cu. ft. of water. 
These are illustrated in Fig. 9. 




Fig. 9. 

The Triangular Dam. — The simplest form of dam is the triangular 
section, as abc. Fig. 10. Let ac be any assumed base distant h below the 
top and also distant H below the water surface. If w is the weight per 



848 



49.— DAMS. 



cubic foot of water and m is the weight per cubic 
foot of masonry, we have, from the preceding: 

w H^ 
The resultant horizontal pressure, P = 



The total weight of masonry, 



W = 



2 

mbh 



Now, if it is desired that the resultant, R, shall in- 
tersect the base, b, at the edge of the "middle third," 
as shown in Fig. 10, we have that 

"tt^ = -g- "^ "q" "^ ^' "^^®^^® ^y substitution, 

wH^ ^ mb h 
2 • IT" 




= ^;or 



It H — h, this reduces to 



-V- 



(1) 



(2) 



If w (water) = 62.5, and m (masonry) = 146,* 6 = 0.654 h (3) 

It is desirable, sometimes, to have the resultant cut the base | b (instead of 
J b) back from the toe, in which case we will have, b = 0.7 h (4) 



I 



1 



4-J 



±t4t::i 



(Comp] 



Fig. 



b — 
12. 



Pressure on Foundations. — If two stijf pencil erasers, 
Fig. 11, are pressed moderately together by equal and 
opposite forces W and W applied near the left end, a, 
it will be seen that they do not remain in contact 
throughout their entire length, but separate at the right 
end, e. Moreover, it will be found that the length of con- 
tact to the right of the applied force is double the con- 
tact length to the left, and therefore f of the total 
contact b. Similarly, we have in the case of the Fig. 11. 

dam a b c. Fig. 10, that when the reservoir is empty and consequently there 
is no water pressure on the face db, the resultant force is W, the weight of 
the masonry, acting as shown in Fig. 11. Hence, we may say that for a 
triangular dam with a vertical face ab the resultant 
weight when dam is empty will produce a pressure in- 
tensity on the base varying uniformly from zero at the 
lower toe, c, to a maximum {m h) at the upper heel, o, 
Fig. 12. The exact results given above may be de- 
duced by mathematical analysis, which, however, will 
be omitted here. 

Let us now consider the forces acting on the plane a c. Fig. 10, due to 
the water pressure P. Imagine the dam to be a vertical beam of length 
a b and "fixed" at the lower end, a c. Consider the depth of the beam, 6, 
and the width (vertical with the page), unity. The acting force, P, is 

— 2~» the lever arm is -5-, and hence the bending moment about the section 

wH^ H wW 
a c is — zr- • -r- = — -— . The resisting moment of the beam at the section 

ac \s^ \ f 62. Equating, / = — ;x- = the compressive stress at c and the ten- 
sile stress at a, considering the neutral axis midway between 

stress on the foundation or on any plane above ^ '^ 

In addition to this there is the horizontal Fig. 13. 



If the 



By combining Figs. 12 and 13 we are enabled to 
obtain the vertical component of the distributed 




same. 



* Specific gravity= 146-^62.5 = 2. 336; often assumed at 2\. 

t The resisting moment of a rectangular beam about a neutral axis 
passing through the center of section is Mr = i / X breadth of beam X (depth 
of beam)2; in which / = outer fiber stress. 



PRESSURE ON FOUNDATIONS. 849 

shear ( = P) which may be considered as distributed in intensity varying 
directly with the vertical pressure at any point. Hence the resultant 
intensities will be parallel with the resultant R, Fig. 10. 

Combining the stress due (1) to the weight of the masonry dam, and 
(2) to the water pressure on the face a c, we have, for the triangular dam. 

Vertical stress at a per sq. ft. == U — —^ mh (5) 

*' •' c" "" =/c= 0~^ (6) 

The result is tension if "plus," compression if "minus." 

The following example is solved by equations (5) and (6). Wt. of 
water, w, is assumed at 62.5; masonry, m, 146 lbs. per cubic foot; see also 
Fig. 10. 

Example. — What is the effect on the resultant pressure, R, Fig. 10, 
when the reservoir is being filled ? 

Solution. — (a) When the reservoir is empty we have from equation (5), 
/a = — 146 /r, that is, the intensity of compression per sq. ft. at a is equal 
to 146 X height of dam in feet. If the limiting pressure on masonry is 
assumed at 30,000 lbs. per sq. ft. it will be seen that the limiting height of 

on OQA 

the dam will [be, h = = 205^ ft., when the reservoir is empty, and that the 

pressure on the foundation uniformly decreases from 30,000 lbs. per sq. ft. 
at a to zero at c (see Fig. 12). (b) If, now, the reservoir is gradually filled, 
the horizontal pressure P increases with H^ and the distance to center of 

pressure above the base increases with — . The resultant pressure R will 

gradually swing to the right from the vertical position W and with increas- 
ing magnitude. It will be found also that the intensity of pressure at a 
will gradually decrease with the corresponding increase of pressure at c, 
while R is traveling across the middle third of the base. By the use of 
equations (5) and (6) the intensities of pressure /» and fc can readily be 
obtained by substituting the values of the unknown quantities, (c) Lastly, 
we will consider the reservoir as full, that is, the water is assumed to be at 
its flood height. In this case, H may be slightly less than, equal to, or greater 
than, h. The resultant R, Fig. 10, is now cutting the base at, say, the 
extreme right edge of the middle third, whence the compression at a is 
reduced to zero, while the compression at c has reached the maximum. 
Therefore it is evident from equation (5) that if /a =0, 

-^— =mh -. (7) 

from which, the value of any factor may be found by assuming values for 

— r-, which compare with equation (1). 



'^f 



If H = h, this reduces to equation (2), or h = h^\—. Assuming water {w) 

m 



-^V' 



at 62.5, and masonry (m) at 146, we have, 6 = 0.654 h, equation (3). If 
now this value of b is substituted in equation (3) and w=62.5, we have, 
when reservoir is full, 

, _ 62.5H3 62.5 k^ 

'" (0.654/^)2 0.4277/^2 ~ 14b /i, or 

the same result which was obtained for fa with the reservoir empty. It is 
to be noted, also, that the most economical type of dam approaches the 
triangular section with a vertical up-stream face. 

General Formulas for Pressure on Foundations. — For any type of gravity 
dam the following formulas may be applied with a reasonable degree of 
accuracy, or at least sufficiently so for all practical purposes; 

Notation. 
W = total weight in lbs. of "section-foot" of masonry above plane a c. 
b = width of base a c in feet, with center of base at origin, a. 



850 



i9.—DAMS. 



-Fa 



= distance in feet from origin o, to point of applica- 
tion of the resultant, R, on a c\ ^xii down- 
stream, — :r if up-stream, from o. 

= vertical intensity of stress in lbs. per sq. ft. at a. 

= vertical intensity of stress in lbs. per sq. ft. at c. 

= angle of inclination of resultant R with the 
vertical. 

= intensity of stress in lbs. per sq. ft., parallel 
with R, at a. 

= intensity of stress in lbs. per sq. ft. parallel 
with R, aX c. 

"^ Formulas. 




-Fig. 14. 



= -^0-f) 

= -^(-f) 

= -I^(l-f)sec., 
= -f (-f)sec., 



If the result 

is minus, the 
stress is com- 
pression', if 
plus, it is ten- 
sion. 



L 



Note that /a 



(7) 

(8) 

.....(9) 

(10) 

also that when 



and /c are vertical components of Fa and Fc 
+ ^ or —xis equal to ^, the resultant pressure cuts the base at the edge of 



the middle third; whence /a is respectively equal to or to — 
2W 



2W 
b ' 



and fc 



is respectively equal to r-or to 0. That is to say, that when the resul- 

o 

tant cuts the base within the limits of the middle third there will be no 

tension in any part of the masonry. 

Table 2, next page, is based on allowable tension in the masonry, but 

in actual practice tension is not allowable in the masonry of a gravity dam. 

Factor of Safety Against Overturning. — With the resultant pressure R 
cutting the base at the lower edge of the middle third, Fig. 10, when the 
reservoir is full and the water pressure P is maximum, it is to be noted that 
the factor of safety of the dam, against overturning, is 2, because if P is 
doubled, R will pass through the lower toe at c, whence the dam will be on 
the point of overturning. The nearer to the center of the middle third that 
R intersects the base, the greater is the factor of safety. If it intersects 
in the middle third, the factor is 2 or greater; if in the outer third, the factor 
is between 1 and 2; if at the outer toe, c, the factor is 1; if outsidf the outer 
toe, the factor is less than 1, and hence the dam will overturn. These 
principles apply not only to the triangular dam but to any practical type. 

Shear. — ^The horizontal shear on any plane a c 
is equal to the horizontal component of the total 
water pressure on a section-foot above that plane, 
or equal to P cos 6. If W = th.e total weight of the 
masonry resting on the joint ac, and / = the coeffi- 
cient of friction, then / {W + P sin <9)=the total 
resisting force due to friction. Hence, if the joint 
ac is assumed to be a horizontal plane with no ad-'^ 
hesion, but simply friction, between the two sur- 
faces in contact, we have, for equilibrium, 

P cos d z,f (W + P sin 6) (11) 

In ordinary practice, f may be assumed at f , hence 
P cos d must not exceed f (W + P sin 6). Equa- 
tion (11) may be used safely in designing because 




Fig. 15. 



/ (W + P sin 6) does not represent the entire resisting force. In fact, hori- 
zontal and unbroken joints are avoided in practical construction, and hence 
considerable shearing resistance is introduced, thereby increasing the factor 
of safety against sliding, materially. 



FACTOR OF SAFETY, SHEAR. 



851 



2. — Pressure on Foundations for Various Values op x. 
(See Fig. 14). ■ 



Resul- 
tant 
Pres- 
sure 
at 
— re or 

X. 



Intensity of Pressure on Foundations in Lbs. 
per Square Foot. 



Fa 



Fc 



Remarks. 



-.1676 
-.156 
-.106 
-.05 6 
±0 
-I-.05 6 
+ .106 
+ .1256 
+ .15 6 
+ .1676 
+ .206 
+ .256 
+ .306 
+ .356 
+ .406 
+ .456 
+ .506 



2W 



6 


1.91^ 


6 


1.6T7 


6 


1.3W 



6 

'b 

0.7PF 

6 
0.4 Ty 

6 
(i.2bW 

b 

6 
± 
0.2 T^ 



6 

Q.bW 

6 

0.8 W 
6 

I.IT^ 
6 

6 

1.7Ty 

6 
2.0T7 



± 

0.1 T^ 

6 

0.4^ 

6 
0.7T7 

6 

_W 
6 
1.3T1^ 



6 


1.6T^ 


6 


1.75W 


6 


1.9T^ 


6 


2W 


6 


2.2T7 


6 


2.5W 


6 


2.8T^ 


6 


3.1 T7 


6 


SAW 


6 


3.7Ty 


6 


4T^ 



2P7 



1.91^ 



■Sec/? 



1.6T7 



6 
1.3T^ 



•Sec/? 
Sec/? 



± 

0.1 TT 



(Resultant at 
up-stream edge 
of middle third. 



■Sec;5 



6 
PAW 

6 
0.7T^ 



SecjS 
Sec^ 
Sec/? 



W 
--^Sec^ 
o 



W 
-■^Sec/?. 
o 



0.7W ^ . 

r — Sec/? 

6 

r — Sec^ 

6 



r — Sec^ 

o 

± 

+ MEsec^ 



H r — SecyS 



H r — Sec/? 

6 

6 
1 4iy 



6 

H r — Sec^ 

o 



r — Sec/? 

6 

— r — Sec/? 


— r — Sec/? 
o 

o 

— ^"T — Sec/? 
o 

-^ — Sec^S 
o 

— r — Sec/? 
o 

-^-T — Seci5 
o 

— ^-r — Sec/? 

H^ Sec/3 
6 

4T^ 
-^^Sec/? 
o 



[Resultant at 
Reenter of base 
[or origin, o. 



Resultant 
pressure %b 
from down- 
stream toe of 
dam. 

[Resultant at 
) down-stream 
ledge of mid- 
Idle third. 



These values 
of X give ten- 
sion in up- 
stream face, 
which is not 
allowable in 
practice. 



Note. — Plus == tension; minus = compression. See notation, page 849. 



852 



49.— DAM5. 



Author's Type of Dam. — The author submits Fig. 16 as a type which 
may be used, ordinarily, up to 200 ft. in height, every part to be enlarged 
pr9portionately. The dimensions shown in the figure are based on a 
height of unity. If the proposed height of dam is 50 ft., multiply each di- 
mension by 50; if 100 ft., multiply by 100; if 200 ft., muhiply by 200; etc. 
In approaching the upper limit of height, care must be used to see that the 
maximum allowable pressures per square foot on the base and foundations 
are not exceeded. Table 2, preceding, may be consulted with regard to the 
pressure on foundations. If for any proposed height, say 225 ft., the 
pressure on the foundations would be excessive, the type may be propor- 
tioned for a height which will not produce excessive pressure, say for 200 
ft., and the side lines projected further downward, below the E' level, the 
necessary distance. Note that the up-stream face below the B' level is a 
series of chords touching an imaginary parabolic curve at their vertices, 
while the down-stream face is a straight line; also that above the B' level 
the up-stream face is vertical and the down-stream face is a parabola. 
The design is based on the masonry having a specific gravity of 23^, which 
is equivalent to HS.SS'^S lbs. per cubic foot, hence the engineer is cautioned 
not to use this type, unmodified, for masonry of less specific gravity. 

Calculations of Author's Type of Dam 
(Fig. 16). — The triangular type of dam, with 
an apex at 6, Fig. 10, is never adopted in 
practice. The practical type is never allowed 
to come to a point at the top, but has a cer- 
tain top width, say about one-tenth the height 
of the dam, giving mass to resist any forces 
acting at the surface such as those due to 
floating ice, logs, etc. This width also pro- 
vides usually for foot-path and roadway, 
either expressly or incidentally, and for a 
general promenade if the dam is high and 
in a picturesque site. Another departure 
from the triangular dam is the upstream 
face a b, which is battered more or less in- 
stead of being vertical. With these condi- 
tions imposed the problem becomes, by com- 
parison, a complicated one when we wish 
to design a type having practical lines, 
containing the least amount of masonry, and 
whose resultant lines of pressure must gen- 
erally coincide with the edges of the middle -^'■ " '" ^z — *■ — ^/jg ' " * .ziei 
third. In Fig. 10, the triangular type, line „. .„ a xi„ » 'r ^f 

E represents the line of resultant pressure ^V?- l^.— Author s Type ot 
when the reservoir is empty, and line F when Dam.— Unit Dimensions, 
full. They are both straight lines and are drawn frorn b to the base, cut- 
ting the latter in three equal parts. Likewise they will cut_ any parallel 
plane above the base in a similar manner, and hence the triangular type 
is an ideal one, easy to calculate. 

There are several methods in use in designing dams. The "cut-and- 
try" method is the one here presented, and Fig. 16 is the result of the 
^'second trial." It is practical, economical, safe, and pleasing to the eye; 
in fact, the outline was determined not a little by the general effect, keeping 
in mind certain well-known principles. 

Notation. 

h = height of dam in feet — assumed as 1; 

t = width of top of dam in feet — equal to 0.1 h; 
w = specific gravity of water — equal to 1 ; 

m = specific gravity of masonry — assumed as 23^; 

P = total spec. grav. water pressure on dam above any plane in question; 
W = total spec. grav. weight of masonry above any plane in question; 
d = depth of water above the plane in question; or depth of section con- 
sidered, from top of dam downward. 

Calculation. 
Fig. 16 represents the second and final design. (The first design 
is not shown, but had a greater width at the A' level and contained about 
7 per cent, more masonry.) It is to be noted that in the following 




TRIAL METHOD OF DESIGN. 



853 



calculations the horizontal position of the center of gravity of the masonry 
W above any plane in question is desired, but not the vertical position; 
likewise, the vertical position of the center of pressure of the water pressure P, 
above any plane, is required. The vertical component of P, acting down- 
ward on the sloping up-stream face, will be neglected. This is on the side 
of safety, as, if considered, it would throw the line of resultant pressure for 
"full reservoir" inward toward the center of the middle third. Calculations 
will be made of the forces above the respective planes A' , B\ C , D' and E\ 
taken in order. In other words, the structure is considered as cut suc-^- 
cessively by these planes and the problem consists in finding the point of 
applicant of the resultant force on each plane, (a) for reservoir empty, and 
{b) for reservoir full. Connecting all the (a) points gives the line of resul- 
tant pressure when empty, and connecting all the (6) points gives the line 
of resultant pressure when full. 

A' Level. — ^This plane is 0.2 /^ below top of dam. Considering one 
section-foot of dam, and the specific gravity of the 
masonry equal 2^, we have the acting forces as shown 
in Fig. 17, in which Wa and Wh act alone (above 
the plane) when the dam is empty, and the force P 
is added when the dam is full. Or, we may con- 
sider Wa + Wh = W acting vertically through the 
center of gravity of the total section above A\ cut- 
ting the base at e and being resisted, when empty, by 
the equal and opposite force Re. But when the res- 
ervoir is full, the added force P will make the resul- 
tant take the direction o f, cutting the base at /, and 
being resisted by the equal and opposite force R?. 
Note that the point o is the intersection of the vertical force W with the 
horizontal force P, and that, relatively, 
Wa=2| (.IOX.20) h^, acting vertically through center of rectangle: 

3 j h^ acting vertically ^% X .035 h from left face of para- 

bolic spandril. Hence, 

3 ) hK acting vertically .00Q3* h to right of 

Wa, that is, through the point o. 
p^^d^=^ (.2A)2 = .02 h\ acting horizontally .06f h above A\ Taking 

moments, we find the distance e / = .06f /j ™ = .0256 h. It is to be noted 

that both resultants intersect the A' base well within the limits of the 
middle third. 

J5' Level. — ^This plane is O.i h below top of dam and is calculated in 
the same manner as the preceding. Note that Wc includes Wa. and 
that Wd includes Wh. Then, relatively, 
Wc =2i (.10X.40) h\ 




W ^W.^W =2^{-^) 



h\ 




acting vertically through o\ .0293 /j to right of Wc. 
Hence, Ro acts .0793 h from left hand face or .0007/^ 
outside of the middle third. For a dam 100 ft. 
high this amounts to only \ inch in a total width on 
the B' level of 24 ft. — an amount insignificant. 
p = ^ci2=H-4/j)2 = . 08/^2, acting 0.13^/^ above B' 
level. Taking moments in the case of the full reser- 

p 
voir we find the distance ^ / =0.13|/t-7- =.0779/t. 

C Level. — In Fig. 19, Wc+Wd is the relative weight of the masonry 
above the B' level, while We is the relative weight of the trapezoidal sec- 
tion-foot between the B' and the C levels. The vertical arrows indicating 



Re ^ 
Fig. 18. 



.0063 = 



.0605 W. 



w. + w. 

center of gravity of two or more parallel forces 



See page 295 for determining the position of the 



854 



49.— DAMS. 



^- 



the weights, pass through the respective 
weight W ( = Wo + Wd-{- We) intersects 
the C plane at e, distant .1225 /j from 
the face of the dam; or .0053 h out- 
side the Hmit of the middle third, 
amounting to about 6f ins. on a total 
base of 38 ft. 4 ins. for a dam 100 ft. 
high. For full reservoir, we have, 

We ==2^X.0mh\ 

W =2lX.121h\ 

P =iJ2= 1 (.6/^)2= .18/^2, acting 

0.20 h above C' level. Taking moments 

in the case of the full reservoir we 

p 
find the distance ^f = . 2 ;jr^ = .1275/j. 



gravity. 



total 




(/-..i 



Re Rf 

This is on the assumption that P acts horizontally. Note, however, that 
below the B' level the pressure is really normal to a sloping face and if so 
considered the resultant, under full reservoir pressure, would cut the C 
plane a little to the left of /, thereby increasing the factor of safety. 

D' Level. — Briefly discussed, the relative weight of the new trapezoidal 
section-foot, between the C and the D' levels is 
Wt =2^ X. 0911/^2, 
W =2^ X.212f /j2, 

p =|cf2=^ (. 8/1)2 = .32/^2^ acting 0.2Q^h above D' level. Completing 
the calculation it is found that when the reservoir is empty the resultant 
cuts the D' plane .1770 h from the up-stream face, or .0008 h outside the 
limit of the middle third, amounting to 1 in. on a total base of 53 ft. 4 ins. 
for a dam 100 feet high. When the reservoir is full the resultant cuts the 
D plane .1720 h farther from the face, and within the middle third. 

E' Level. — For the section between D' and E' levels, 
Wg =2i X. 122^/^2, 
W =2\ X. 335/^2. 

p =^d'^= \}fi^ acting \ h above E' level. 

With the reservoir empty the resultant cuts the base .2382 h from the up- 
stream face, which is .0082 h within the limit of the middle third. When 
the reservoir is full the resultant cuts the E' plane .2132 h farther from the 
up-stream face. 

For general dimensions and quantities, see Tables 3 and 4, following. 

QUANTITIES IN MASONRY, ROCK-FILL AND EARTH DAMS. 

Tables 4, 5 and 6, following, will be found useful in estimating the relative 
quantities in masonry, rock-fill and earth dams, from profiles or contour maps 
of any particular site. Quantities are in cubic yards per 100-ft. section of 
dam. For other sectional lengths the yardage will of course be proportional. 
Thus, for 50-ft. lengths, divide by 2; for 25 -ft. lengths, divide by 4; etc. 

The scientific design of masonry dams, laid in cement mortar, is subject 
to more accurate determination than are those of rock-fill or earth. For 
the two latter classes we are guided mainly by the behavior of the types 
that have been constructed, and by carefully studying the causes of failure 
of those that have not stood the test. The actual cross-section of the dam 
itself is only one of the necessary elements of strength. The foundations 
should be selected and prepared carefully to receive the superstructure or 
dam proper; the outlet constructed with care to prevent leakage; and the 
wasteway ample to provide for flood discharges. 

A word in regard to the effect of profile on the resisting power of a dam. 
In our calculations we assume one section-foot of dam and one section-foot 
of water pressure acting against it — the resultant forces in each section- 
foot acting independently of any other section-foot. This theory would 
hold true in practice for a straight dam of constant 
height and of indefinite length, but for all ordinary C-i 
cases, as for instance Fig. 20, where the ground line 
is irregular, calling for different heights for each 
section, it is impossible for any section to act inde- 
pendently of any other. Thus, under full reservoir 
head, the high section A will tend to deflect down- 
Stream a greater amount than will one of less height, 



Top or Ciam 




Fig. 20.— 
Profile of Dam-Site. 



QUANTITIES IN MASONRY DAMS, 



855 



say, B, because in the iormer case the deflection will extend ^iown to the 
base a-a, while in the latter case it will extend only as low as bh. Hence, 
shearing stresses are set up between adjacent sections, tending to decrease 
the overturning moment for the higher sections, near the middle of the dam. 



3. — General Table op Dimensions and Quantities for Masonry Dams- 
Author's Type, Fig. 16. 
(See also Table 4.) 













Cu. Yds. 




Depth d 






Width of 




per 100 




of Section 


Width of 


Width of 


Base of 


Area in 


Lin. Ft. 




Consid- 


Top of 


Base of 


Section 


Sq. Ft. of 


of Dam 




ered In 


Dam in 


Section 


in terms 


Section 


(for sec- 


Remarks. 


terms of 


terms of 


in terms 


of depth 


of depth d 


tion depth 




total 


total 


of total 


of sec- 


in terms 


d), in 




heigbt. h. 


height, h. 


height, h. 


tion, d. 


OtJl^. 


terms of 
hi. 
(6) 




(1) 


(2) 


(3) 


(4) 


(5) 


(7) 


.02 h 


.10 h 


.10035 h 


5.017 d 


.00200 hi 


.0074 hi 




.04 " 


" 


.10140 " 


2.535 " 


.00402 " 


.0149 " 




.06 " 


tt 


.10315 " 


1.719 " 


.00607 " 


.0225 " 




.08 " 


" 


.10560 " 


1.320 " 


.00815 " 


.0302 " 




.10 " 


t€ 


.10875 " 


1.088 " 


.01029 •• 


.0381 " 




.12 " 


" 


.11260 " 


.938 " 


.01250 " 


.0463 " 




.14 " 


" 


.11715 " 


.837 " 


.01480 " 


.0548 " 




.16 " 


« 


.12240 " 


.765 " 


.01724 " 


.0639 " 




.18 " 


" 


.12835 " 


.713 " 


.01970 " 


.0730 " 




.20 " 


" 


.13500 " 


.675 " 


.02233 " 


.0827 " 




.22 " 


« 


.14235 " 


.647 " 


.02511 " 


.0930 " 




.24 " 


•• 


.15040 " 


.627 " 


.02803 " 


.1038 " 




.26 " 


" 


.15915 " 


.612 " 


.03113 " 


.1153 " 


The quantities In 


.28 " 


« 


.16860 " 


.602 " 


.03440 " 


.1274 " 


this table are ex- 


.30 " 


(C 


.17875 " 


.596 " 


.03788 " 


.1403 " 


pressed, generally. 


.32 " 


" 


.18960 " 


.593 " 


.04156 " 


.1539 " 


in terms of h, the 


.34 " 


" 


.20115 " 


.592 " 


.04546 " 


.1684 " 


height of the stan- 


.36 " 


<< 


.21340 " 


.593 " 


.04961 " 


.1837 " 


dard type, Fig. 16. 


.38 " 


" 


.22635 " 


.596 " 


.05400 " 


.2000 " 


If it is desired to 


.40 " 


" 


.24000 " 


.600 " 


.05867 " 


.2173 " 


express them in 


.42 'I 


" 


.25433 " 


.606 " 


.06361 " 


.2356 " 


terms of t, the 


.44 " 


" 


.26867 " 


.611 " 


.06884 " 


.2550 '\ 


width of top of 


.46 " 


" 


.28300 " 


.615 " 


.07436 " 


.2754 " 


dam, we have only 


.48 " 


" 


.29733 " 


.619 " 


.08016 " 


.2969 " 


to substitute the 


.50 " 


•• 


.31167 " 


.623 " 


.08625 " 


.3194 " 


values of h and hi 


.52 " 


" 


.32600 •' 


.627 " 


.09263 " 


.3431 " 


expressed in terms 


.54 " 


" 


.34033 " 


.630 " 


.09929 " 


.3677 " 


of t and t2. Thus. 


.56 " 


(( 


.35467 " 


.633 " 


.10624 " 


.3935 " 


h 


.58 " 


" 


.36900 " 


.636 " 


.11348 " 


.4203 " 


as. = - 

hi 
and.2=_ 


.60 " 


" 


.38333 " 


.639 " 


.12100 " 


.4481 " 


.62 " 


" 


.39833 " 


.642 •* 


.12882 " 


.4771 " 


.64 " 


" 


.41333 " 


.646 " 


.13693 " 


.5071 " 


.66 " 


" 


.42833 " 


.649 " 


.14535 " 


.5383 " 


we have, h=\U 


.68 " 


" 


.44333 " 


.652 " 


.15407 " 


.5706 " 


and 7^2=100^2. 


.70 " 


<( 


.45833 " 


.655 " 


.16308 " 


.6040 " 


We may therefore 


.72 " 


" 


.47333 " 


.657 " 


.17240 " 


.6385 " 


use with this table 


.74 " 


" 


.48833 " 


.660 " 


.18202 " 


.6741 " 


the top width 


.76 " 


<( 


.50333 " 


.662 " 


.19193 " 


.7109 " 


instead of the height 


.78 " 


" 


.51833 " 


.665 " 


.20215 " 


.7487 " 


h. 


.80 " 


" 


.53333 " 


.667 " 


.21267 " 


.7877 " 




.82 " 


<< 


.54900 " 


.670 " 


.22349 " 


.8277 " 




.84 " 


•• 


.56467 " 


.672 " 


.23463 " 


.8690 " 


See, also. Table 4. 


.86 " 


" 


.58033 " 


.675 " 


.24608 •• 


.9114 " 




.88 " 


" 


.59600 " 


.677 •• 


.25784 " 


.9550 " 




.90 " 


" 


.61167 " 


.680 " 


.26992 " 


.9997 " 




.92 " 


" 


.62733 " 


.682 " 


.28231 " 


1.0456 " 




.94 " 


<< 


.64300 " 


.684 '• 


.29501 " 


1.0926 " 




.96 " 


" 


.65867 " 


.686 " 


.30803 •• 


1.1409 " 


I 


.98 " 


" 


.67433 " 


.688 " 


.32135 " 


1.1902 " 




1.00 " 


« 


.69000 " 


.690 " 


.33500 " 


1.2407 " 





856 



49.— DilMS. 



4. — Specific Table op Quantities in Masonry Dams — ^Author's Type. 

Fig. 16. 

(See also Table 3.) 





Cubic Yards of Masonry per 100 Lineal Feet of Dam, for Section 


Depth d 

of 
Section in 




Of Depth d from Top (Fig. 21). 






Top= 


Top= 


Top = 


Top = 


Top = 


Top= 


Top = 


Top = 


Top= 


Top= 


Feet. 


20'. 


18'. 


16'. 


14'. 


12'. 


10'. 


8'. 


6'. 


4'. 


2'. 


d 


h= 


h= 


h= 


h= 


h= 


h= 


h== 


n= 


ft== 


fe= 


200'. 


180'. 


160'. 


140'. 


120'. 


100'. 


80'. 


60'. 


40'. 


20'. 


(1) 


(2) 


(3) 


(4) 


(5) 


(6) 


(7) 


(8) 


(9) 


(10) 


(11) 


4 


296 


267 


237 


208 


178 


149 


119 


90 


61 


33 


8 


596 


536 


478 


419 


360 


302 


244 


187 


132 


87 


12 


899 


810 


723 


636 


549 


463 


379 


298 


224 


179 


16 


1207 


1091 


976 


861 


748 


639 


529 


386 


348 


315 


20 


1524 


1381 


1239 


1099 


961 


827 


701 


588 


511 


496 


24 


1852 


1683 


1516 


1351 


1191 


1038 


898 


782 


717 


28 


2193 


1998 


1807 


1621 


1442 


1274 


1126 


1017 


966 


32 


2554 


2330 


2118 


1912 


1717 


1539 


1391 


1294 


1260 


36 


2919 


2680 


2448 


2227 


2020 


1837 


1696 


1613 


1600 


40 


3309 


3051 


2802 


2568 


2354 


2173 


2044 


1976 


1985 


44 


3720 


3445 


3183 


2939 


2722 


2550 


2435 


2384 




48 


4153 


3864 


3591 


3342 


3129 


2969 


2869 


2836 




52 


4612 


4311 


4031 


3781 


3577 


3431 


3345 


3333 




56 


5096 


4787 


4504 


4259 


4067 


3935 


3866 


3877 




60 


5612 


5296 


5014 


4778 


4600 


4481 


4431 


4467 




64 


6157 


5840 


5563 


5339 


5176 


5071 


5041 






68 


6735 


6420 


6153 


5943 


5793 


5706 


5696 






72 


7350 


7041 


6786 


6589 


6453 


6385 


6398 






76 


8000 


7702 


7461 


7278 


7157 


7109 


7146 






80 


8692 


8406 


8178 


8009 


7905 


7877 


7941 






84 


9424 


9152 


8938 


8784 


8698 


8690 








88 


10199 


9941 


9741 


9601 


9535 


9550 








92 


11016 


10772 


10586 


10463 


10417 


10456 








96 


11876 


11645 


11473 


11369 


11343 


11409 








100 


12778 


12561 


12404 


12320 


12314 


12407 








104 


13723 


13519 


13379 


13315 


13332 








108 


14710 


14520 


14399 


14354 


14396 








112 


15739 


15564 


15463 


15438 


15507 








116 


16812 


16653 


16572 


16567 


16664 








120 


17926 


17787 


17725 


17743 


17867 








124 


19084 


18965 


18922 


18965 




Top 

r-t-n 






128 
132 


20286 
21533 


20187 


20164 


20234 
21549 








21454 


21451 




T' 
1 


'y//////^ 


— ""■« 




136 


22825 


22765 


22785 


22911 




y/^Y/0^ 


, 




140 


24160 


24121 


24165 


24319 




I 
1 


^^^ 


S*"^ 




144 


25541 


25521 


25592 






I 
1 




V^ 




148 


26966 


26966 


27065 






1* 


v/y/yyyy/y 




152 


28434 


28458 


28585 






{/////////, 




156 


29948 


29996 


30151 








\ 




160 


31507 


31581 


31763 






1 

1 




\ 




164 


33110 


33212 








i 




\ 




168 


34760 


34889 








/ 


\ 




172 


36456 


36613 








1 / 


\ 




176 


38199 


38383 








^ / 


> 


\ 


180 


39988 


40200 
























Fig. 2 


1. 




184 

188 


41824 
43705 




Not 


e.— Fc 


)r absolute quantities, use the 


192 


45634 




second 


colur 


nn only; the 


last 


nine columns 


196 


47607 




are for 


possit 


le sections wh 


ichmt 


ist be verified 


200 


49630 




or moc 


iified b 


y trial calculj 


itions. 





TABLES OF QUANTITIES. 857 

5. — Table of Quantities in Rock-Fill Dams. 



Depth d 


Area of 


Cubic Yards of 


Masonry per 100 Lineal Feet of Dam, 


of 


Up- 


for Section of Depth d 


from Top (Fig. 22) 




Section 


stream 














from 
Top of 


Face, 
100 Ft 
































Dam, In 


Long 




















Feet. 


and of 


Top = 


Top = 


Top = 


Top = 


Top = 


Top = 


Top = 


Top = 


Top= 


d 


Depth d. 
Sq. Ft. 


24'. 


22'. 


20'. 


18'. 


16'. 


14'. 


12'. 


10'. 


8'. 


(1) 


(2) 


(3) 


(4) 


(5) 


(6) 


(7) 


(8) 


(9) 


(10) 


(11) 


4 


447 


400 


370 


341 


311 


281 


252 


222 


193 


163 


8 


894 


889 


830 


770 


711 


652 


593 


533 


474 


415 


12 


1342 


1467 


1378 


1289 


1200 


1111 


1022 


933 


844 


756 


16 


1789 


2133 


2015 


1896 


1778 


1659 


1541 


1422 


1304 


1185 


20 


2236 


2889 


2741 


2593 


2444 


2296 


2148 


2000 


1852 


1704 


24 


2683 


3733 


3556 


3378 


3200 


3022 


2844 


2667 


2489 


2311 


28 


3130 


4667 


4459 


4252 


4044 


3837 


3630 


3422 


3215 


3007 


32 


3578 


5689 


5452 


5215 


4978 


4741 


4504 


4267 


4030 


3793 


36 


4025 


6800 


6533 


6267 


6000 


5733 


5467 


5200 


4933 


4667 


40 


4472 


8000 


7704 


7407 


7111 


6815 


6519 


6222 


5926 


5630 


44 


4919 


9289 


8963 


8637 


8311 


7985 


7659 


7333 


7007 




48 


5366 


10667 


10311 


9956 


9600 


9244 


8889 


8533 


8178 




52 


5814 


12133 


11748 


11363 


10978 


10593 


10207 


9822 


9437 




56 


6261 


13689 


13274 


12859 


12444 


12030 


11615 


11200 


10785 




60 


6708 


15333 


14889 


14444 


14000 


13556 


13111 


12667 


12222 




64 


7155 


17067 


16593 


16119 


15644 


15170 


14696 


14222 






68 


7602 


18889 


18385 


17881 


17378 


16874 


16370 


15867 






72 


8050 


20800 


20267 


19733 


19200 


18667 


18133 


17600 






76 


8497 


22800 


22237 


21674 


21111 


20548 


19985 


19422 






80 


8944 


24889 


24296 


23704 


23111 


22519 


21926 


21333 






84 


9391 


27074 


26452 


25830 


25207 


24585 


23963 








88 


9838 


29363 


28711 


28059 


27407 


26756 


26104 








92 


10286 


31756 


31074 


30393 


29711 


29030 


28348 








96 


10733 


34252 


33541 


32830 


32119 


31407 


30696 








100 


11180 


36852 


36111 


35370 


34630 


33889 


33148 








104 


11627 


39556 


38785 


38015 


37244 


36474 








108 


12074 


42363 


41563 


40763 


39963 


39163 








112 


12522 


45274 


44444 


43615 


42785 


41956 








116 


12969 


48289 


47430 


46570 


45711 


44852 








120 


13416 


51407 


50519 


49630 


48741 


47852 








124 


13863 


54630 


53711 


52793 


51874 










128 


14310 


57956 


57007 


56059 


55111 










132 


14758 


61385 


60407 


59430 


58452 










136 


15205 


64919 


63911 


62904 


61896 




% 






140 


15652 


68556 


67519 


66481 


65444 






144 


16099 


72296 


71230 


70163 


t 


'fm% 


'""T" 




148 


16546 


76141 


75044 


73948 


1 <• 


■^ 


1 ! 


152 


16994 


80089 


78963 


77837 


1 V 


MM 


J "o 


156 


17441 


84141 


82985 


81830 


r 
1 


160 


17888 


82896 


87111 


85926 




.6Q-:^.J30., 


■^ 


\.i 


164 


18336 


92556 


91341 






5j?:__[?i.__./5 


0' ^ 


vi 


168 


18783 


96919 


95674 




!i 


^ 


172 


19230 


101385 


100111 




V/k 


10' !^ 


230' 


N 


176 


19677 


105956 


104652 












180 


20125 


110630 


109296 






Fig. 22 






184 


20572 


115407 














188 


21019 


120289 














192 


21466 


125274 






Note 


—Area of up-stream face Is 


196 


21913 


130363 






given 1 


n order to estin 


late the 


; quan- 


200 


22361 


135556 






tity of 


material in th 


e facing 


'• 



858 



i9,—DAMS. 



6. — ^Table op Quantities in Earth Dams. 
Fig. 23. 





Area of 












Up- 


Cubic Yards of Earth per 100 Lineal Feet of Dam. for 


Depth d 


stream 




Section of Depth d, from Top (Fig. 23). 


of Sec- 


Face, 










tion 


100-ft. 










from Top 


Long 






























of Dam, 


and of 


Top = 


Top = 


Top = 


Top = 


Top== 


Top = 


Top = 


Top= 


Top= 


In Feet. 


Depth d. 
Sq. Ft. 


28' 


26' 


24' 


22' 


20' 


18' 


16' 


14' 


12' 


(1) 


(2) 


(3) 


(4) 


(5) 


(6) 


(7) 


(8) 


(9) 


(10) 


(11) 


4 


1077 


563 


533 


504 


474 


444 


415 


385 


356 


326 


8 


2154 


1422 


1363 


1304 


1244 


1185 


1126 


1067 


1007 


948 


12 


3231 


2578 


2489 


2400 


2311 


2222 


2133 


2044 


1956 


1867 


16 


4308 


4030 


3911 


3793 


3674 


3556 


3437 


3318 


3200 


3081 


20 


5385 


5778 


5630 


5481 


5333 


5185 


5037 


4889 


4741 


4593 


24 


6462 


7822 


7644 


7467 


7289 


7111 


6933 


6756 


6578 


6400 


28 


7539 


10163 


9956 


9748 


9541 


9333 


9126 


8919 


8711 


8504 


32 


8616 


12800 


12563 


12326 


12089 


11852 


11615 


11378 


11141 


10904 


36 


9693 


15733 


15467 


15200 


14933 


14667 


14400 


14133 


13867 


13600 


40 


11770 


19111 


18815 


18519 


18222 


17926 


17630 


17333 


17037 


16741 


44 


12847 


22785 


22459 


22133 


21807 


21481 


21156 


20830 


20504 


20178 


48 


13924 


26756 


26400 


26044 


25689 


25333 


24978 


24622 


24267 


23911 


52 


15001 


31111 


30726 


30341 


29956 


29570 


29185 


28800 


28415 


28030 


56 


16078 


35763 


35348 


34933 


34519 


34104 


33689 


33274 


32859 


32444 


60 


17155 


40711 


40267 


39822 


39378 


38933 


38489 


38044 


37600 


37156 


64 


18233 


45956 


45481 


45007 


44533 


44059 


43585 


43111 


42637 


42163 


68 


19310 


51496 


50993 


50489 


49985 


49481 


48978 


48474 


47970 


47467 


72 


20387 


57333 


56800 


56267 


55733 


55200 


54667 


54133 


53600 


53067 


76 


21464 


63467 


62904 


62341 


61778 


61215 


60652 


60089 


59526 


58963 


80 


22541 


69896 


69304 


68711 


68119 


67526 


66933 


66341 


65748 


65156 


84 


23618 


76622 


76000 


75378 


74756 


74133 


73511 


72889 


72267 


71644 


88 


24695 


83763 


83111 


82459 


81807 


81156 


80504 


79852 


79200 


78544 


92 


25772 


91200 


90519 


89837 


89156 


88474 


87793 


87111 


86430 


85744 


96 


26849 


98933 


98222 


97511 


96800 


96089 


95378 


94667 


93956 


93244 


100 


27926 


106963 


106222 


105481 


104741 


104000 


103259 


102519 


101778 


101037 


104 


29003 


115289 


114519 


113748 


112978 


112207 


111437 


110667 


109896 


109126 


108 


30080 


123911 


123111 


122311 


121511 


120711 


119911 


119111 


118311 


117511 


112 


31157 


132830 


132000 


131170 


130341 


129511 


128681 


127852 


127022 


126193 


116 


32234 


142193 


141333 


140474 


139615 


138756 


137896 


137037 


136178 


135319 


120 


33311 


151852 


150963 


150074 


149185 


148296 


147407 


146519 


145630 


144741 


124 


34388 


161807 


160889 


159970 


159052 


158133 


157215 


156296 


155378 


154459 


128 


35465 


172059 


171111 


170163 


169215 


168267 


167319 


166370 


165422 


164474 


132 


36542 


182607 


181630 


180652 


179674 


178696 


177719 


176741 


175763 


174785 


136 


37619 


193452 


192444 


191437 


190430 


189422 


188415 


187407 


186400 


185391 




Up-Str$am Face 



Fig. 23. 

Note.— Area of up-stream face Is given In order to estimate the quantity of 
material in the facing. 



QUANTITIES IN EARTH DAMS, 



859 



EXCERPTS AND REFERENCES. 

The Periyar Dam and Irrigation Works, So. India (Eng. News, 
Oct. 24, 1901). — Illustrated, with section of dam. Article contains a table 
showing the principal dimensions of Periyar dam as compared with those 
of other masonry dams over 150 ft. in total height, as follows: 



Name. 



New Croton, N. Y. 

Periyar, India 

Furens, France . . . . 

Villar, Spain 

San Mateo, Cal . . . . 

Puentes, Spain 

Tache, France 

Ban, France 

Gillippe, Belgium . . 



Depth 

of 
Water, 

Ft. 



150, 
155, 
164, 
162, 



153 



137 
147 



Ht. 

Above 

Base, 

Ft. 



219. 

176. 

170.6 

170.33 

170. 

164.24 

161.43 

156.82 

154. 



Top 

Width 

Ft. 



18. 
12. 

9.91 
14.75 
20. 
35.73 



16.40 
49.22 



Base 

Width 

Ft. 



200. 

138.5 

161. 

154.5 

176. 

144.29 



126.98 
216.5 



Area 
Profile 

Sq.Ft. 



28400 
10772 
10712 
11596 
16660 
16349 



6780 
18708 



Top 
Length 

Ft. 



2240 
1200 
328 
546 
700 
925 



771. 



Mate- 
rial. 



Stone. 

Cone. 

Stone. 

Stone. 

Cone. 

Stone. 

Stone. 

Stone. 

Stone. 



A Proposed New Type of Masonry Dam (By Geo. L. Dillman. Trans. 
A. S. C. E., Vol. XLIX). 

Comparative Cross=Sections of the High Earth Dams of the World 

(Eng. News, Nov. 28, 1901). — Illustrations of 17 dams, giving sections, 
slopes, nature of core wall, etc., including: The earth portion of New Croton 
Dam (ht. 200 ft., masonry core wall, water slope 2:1); San Leandro Dam, 
Cal. (ht. 110 ft., puddle wall of stiff blue clay, water slope 3:1, down- 
stream slope 2.4:1); Turner's Embankment, Liverpool (ht. 105 ft.; clay 
puddle core wall extending down 90 ft. into sand, earth, stones, gravel and 
shale; water face 3:1 with 15" pitching); Titicus Dam, N. Y. (ht. 100 ft., 
masonry core wall, down -stream face 2.5:1, water face 2.4:1 with 12" 
broken stone and 18" paving. 

Lake Cheesman Dam and Reservoir (By C. L. Harrison and S. H. 
Woodard. Trans. A. S. C. E., Vol. LIII). — See page 131 of Transactions 
for comparative sections of the following dams: New Croton, Lake Chees- 
man, Chatrain, Periyar, Furens, Villar, Ban, and Sweetwater dams — in the 
order of their height, beginning with the highest. 

Steel Dam With Concrete Foundations, at Redridge, Mich. (Eng. 
News, Aug. 15, 1901). — Illustrated. 

A Rational Formula for the Length of Waterways (By J. P. Frizell. 
Eng. News, Aug. 8, 1901). — For discussions, see Eng. News, Sept. 5 and 
Oct. 10, 1901. 

The Tabeaud High Earth Dam, Cal. (By B. Bassell. Eng. News, 
July 10, 1902). — Illustrated, with very complete description of method of 
construction — rolling, etc. 

Recent Practice in Hydraulic=Fill Dam Construction (By J. D. 

Schuyler. Trans. A. S. C. E.. Vols. LVIII and LX). 

Classified Review of Dam and Reservoir Failures in the U. S. (By 

W. R. Hill. Eng. News, June 19, 1902).— Ten listed under "Insufficient 
Spillways and Overtopping;" 10 under "Failures Due to Water Leaking 
Along Pipes Laid Through Embankments;" 6 under "Failures of Reservoir 
Bottoms;" 5 under "Embankments and Walls Undermined When Built 
Upon Porous or Yielding Material;" 6 under "Poor Workmanship or 
Faulty Design;" 4 under "Ice Pressure;" 8 under "Miscellaneous or Un- 
known Causes." In addition to the above, 48 other failures of dams and 
reservoirs are referred to, the causes of failure being unrecorded. 

Macadam as a Core for Dams and Reservoir Embankments (Eng. 
News, June 25, 1903). 

Diagrams for Hollow Reinforced-Concrete Dams (By H. W. Foster. 
Eng. News, Dec. 24, 1903). — Illustrated. 

The Stability of Surcharged Masonry Dams (By E. Sherman Gould. 
Eng. News, Mar. 3, 1904). 



860 i9.—DAMS. 

Rubble Concrete Dam for the Atlanta Water & Electric Power Co. 

(Eng. News, July 7, 1904). — Illustrated section of rollway portion of dam 
with graphical determinations of resultant pressures. "The advantages of 
rubble concrete for many kinds of masonry work, and particularly for 
massive structures like masonry dams, are gradually being recognized by 
engineers. In many cases where large yardage of masonry is required the 
use of rubble concrete will effect a saving in time and cost over rubble 
masonry work or fine concrete work. As an illustration, rubble concrete 
composed of 40% large stone and 60% of 1:2J:5 concrete requires 7% of 
the volume to be of cement, while rubble masonry composed of 65% large 
stone and 35% of 1:2^ mortar requires 10% of the volume to be of cement." 
Investigation of Stresses* in High Masonry Dams of Short Spans (By 
G. Y. Wisner and E. T. Wheeler. Eng. News, Aug. 10, 1905).— Relates to 
the proposed curved Pathfinder dam (cross-section: height 210', top width 
10', bottom width 94', up-stream batter 0.15, down-stream batter 0.25), 
with diagrams giving results of calculations. 

Earth Dams with Concrete Core Walls (By Clemens Herschel. Eng. 
News, Sept. 7, 1905). 

Computation of Height of Backwater Above Dams (Eng. News, 
Nov. 1, 1906; also Nov. 29, 1906, with tables). 

Large Reinforced=Concrete Dam at Ellsworth, Me. (Eng. News, 
May 23, 1907). — Illustrated section; 64 ft. high. 

Large Electrically Operated Gates for the Roosevelt Dam, Ariz. (By 
F. W. Hanna. Eng. News, Mar. 30, 1907). — Illustrated. 

Electrically Operated Service Gates for the Shoshone and Pathfinder 
Dams (By F. W. Hanna. Eng. News, Jan. 2, 1908). — Illustrated. 

Reinforced=Concrete Diaphrams for Earth Dams (By B. M. Hall. 
Eng. News, Feb. 6, 1908). — Illustrated. 

Combination Dam and Bridge of Reinforced Concrete (Eng. News, 
April 9, 1908).— Illustrated. 

The Design of Buttressed Dams of Reinforced Concrete (By R. C. 
Beardsley, Eng. News, April 23, 1908). — Illustrated. 

Progress on the Roosevelt Dam ; with Cost Data (By C. W. Smith. 
Eng. News, Sept. 10, 1908). 

Movable Dams and Lock at the Power Plant on the Chicago Drain- 
age Canal (Eng. News, Nov. 12, 1908). — Illustrated. 

Cast=Iron Sluice Gates for the Fens Gate Chamber, Charles River 
Basin, Boston, Mass. (By W. H. Sears. Eng. News, Feb. 25, 1909). — Illus- 
trated. 

Investigations of the Saturization of Earth Dams (Eng. Rec, Aug. 21, 
1909). — Results of experiments by Desmond Fitzgerald: 1. Clay banks 
are more completely saturated than well-drained banks. 2. Clay banks are 
more slowly saturated and part more slowly with their water of saturation. 
3. In high banks it is unsafe to have nothing but clayey material, a down- 
stream section of well-drained material being essential. 

Partial Failure Through Undermining of the Zuni Dam, New Mexico 

(Eng. News, Dec. 2, 1909).— Combined hydraulic earth fill, 60,120 cu. yds., 
up-stream section; and rock fill, 40,160 cu. yds., down-stream section. 
Up-stream: slope 3:1, rock rip-rap 18 ins. deep on gravel 12 ins. deep. 
Down -stream: slope li : 1. Illustrated. The spillway, south abutment 
and extreme south end of dam were undermined by the passage of water 
underneath a cap of lava rock which flanked the dam and extended beneath 
the spillway. 

The Eastwood Multiple=Arch Dam (Eng. Rec, Jan. 15, 1910).— Abstract 
of article by John S. Eastwood, in the "Journal of Electricity, Power and 
Gas," Oct. 30, 1909. The Hum.e-Bennett dam consists of 12 arches, each 
50-ft. span, resting on 13 buttresses, the end walls of the last buttress at 
each end being extended into the opposite bank as a core wall, as they are 
above the normal water line and have no water load. The elevation of the 
water line is 5,300, that of the crest of the middle six arches is 5,303, and 
the remainder of the crest to the ends is 5,304 ft. above mean tide level. 
This allows a 330-ft. crest for any freshet that may occur when the spillway 
flashboards were accidently left in their openings. The entire structure 



MISCELLANEOUS DATA. 861 

rests on sound bedrock. Mr. Eastwood found that to give the required 
stability with the greatest economy it was desirable to build the top 16 ft. 
of the dam with vertical arches, and all arches up to 20 ft. high at the spring 
line were built vertical. All arches higher than 20 ft. are carried vertical 
to within 16 ft, of the top at the crown line, and then slope to the founda- 
tions at an angle of 32 deg. The arch ring thickness is increased as required 
for the water pressure. All of the vertical part of the arch wall is 18 in. 
thick; the wall increases in thickness from this point at the rate of 1 ft. to 
each 24 ft. vertical, or a little more than required for water load. The 
buttresses are all 2 ft. thick at the top and project 8 ft. from the inside 
spring line to the down-stream end, all comers being clipped. The batter 
of the down -stream end is 5 in. to 1 ft., and of the sides 1 in 24 to the base 
on each side. Each buttress is finished on its down -stream end with wing 
buttresses or counterforts. The 12 spillway openings are located in the 
three middle arches of the dam, the openings being 5 x 8 ft. each, to be closed 
to any desired height by means of flashboards. The structure was rein- 
forced throughout when necessary by means of railroad iron scrap and old 
logging cable. . . . The buttress forms consist of 2 x 4 in. studding 
set about 20 in. apart on centers and spliced where not long enough to reach 
the top; a framework was first built and lined with 12-in. lumber of 10 and 
12-in. widths, lightly nailed to the inside of the studs, the studs being braced 
to the trestle. The shapes of the counterfort forms were such that they 
braced themselves when once boarded up. The arch forms were built up 
from the bottom, using 2 x 4-in. studding and ^ x 6 in. stuff nailed on double. 
. Crushed granite was used for the coarse aggregate, the crusher- 
run being mixed with sand from an adjacent pit; the mix was approx. 1:2:4. 
The forms were not removed for at least a week after the concrete was laid, 
and the walls were kept wet by a night watchman and a day crew. A 
union of the different day's work was made by scarifying the surface of the 
old work and washing off with a hose ; dry cement was then sprinkled over 
this surface and concreting begun, a few batches of concrete with excess of 
mortar being laid in contact, the work being carried up as nearly level as 
possible. The junctions of the walls were made on the center of buttresses, 
the reinforcement being left protruding to tie them together. . . . The 
water face and the parts of the down-stream face were plastered, the water 
face with two coats of 1 to 1 2- cement plaster and selected sand and a wash 
coat of neat cement on the bases of the middle arches. A base seal of 
mortar was placed along the line of contact with the rock. . . . The 
structure contains 2,207 cu. yds. of concrete and was built in 114 days. 
All parts of dam are in compression, max. stress 187.5 lbs. per sq. in. 
(safety factor 16) being at bases of arch rings. Max. stress in shear, 50 lbs. 
per sq. in. Rates of overturning, 1 : 3.6. . . . Cost of dam, in- 
cluding plastering, about $21 per cu. yd. or^ $46,000 for the structure; 
cement costing a little over $5.00 per bbl., delivered. 

Arched Masonry Dam at Las Vegas, N. M. (By C. W. Sherman. Eng. 
News, Oct. 27, 1910). — Description and illustrations. Also contains a 
table of 23 curved masonry dams, giving max. height, base thickness, top 
thickness, max. stress in arch, radius of up-stream face, top length, charac- 
ter of rock, date of building. 

Movable Dams on the New York State Barge Canal (Eng. News, Dec. 8, 
1910). — Description, with 14 illustrations, of the type of dam known as the 
bridge dam with the Boule gates. 

Illustrations of Various Types of Dams. 

Description. Eng. News. 

Cone, rubble-faced dam 47' high June 13, 1901. 

Rock-fill dam 100' high with steel core Jan. 2, '02. 

Assuan dam (66' high) and reservoir on the River Nile Aug. 14, '02. 

Wigwam masonry dam 70' high, Waterbury, Conn. May 7, '03. 

Spier Falls masonry dams 50' and 152' high June 18, '03. 

Hollow rein. -cone, dam 11' high (Ambursen type) Nov. 5, '03. 

Arched concrete dam 110' high, Barossa, So. Australia April 7, '04. 

Arched Masonry dam 230' high. Lake Cheesman, Colo. May 12, '04. 

Timber dam 30' high on the Penobscot River Sept. 1, '04. 

Roosevelt masonry dam 260' high, Arizona Jan. 12, '05. 

Rolling steel dam at Sweinfurt, Bavaria Jan. 19, '05. 

Two small concrete dams — one on pile foundations Feb. 9, '05. 



862 



49.— DAM5. 



Description. 

Hollow rein. -cone, dam 25' high, at Schuylerville 

Debris Barrier No. 1, Yuba River, Cal. 

Arched masonry dam 80' high, Cheyenne, Wyo. 

Gatun dam 270' to bed rock (Panama Canal) 

Pile foundation for movable dam 

Movable dam and lock of the Rice I. & I. Assn, La. 

Structural steel dams (F. H. Bainbridge) 

Crib dam 20' high with sheet steel piling 

New Croton dam, with balanced gate valve (Wegmann) 

The Mercedes curved masonry dam 1 30' high 

Collapsible steel dam crest, Bear River, Utah 

*Hauser Lake steel dam, Missouri R., Mont. 

Lock Gates of the Charles River dam 

Butterfly dam on Chicago Drainage Canal 

Revolving segmental sluice-gates for Sterling dam 

Plan and cross-section of Shoshone dam 

Failure of concrete dam, 10 ft. high, at Danville, N. Y. 

Designs for rebuilding the Austin dam, Texas 

Curved masonry dam.s in New South Wales 

Construction of Cataract Dam, Sidney, N. S. W. 

Reinforced buttressed dam, Ottawa, Can. 

Dike, mosquito extermination work, Welfleet, Mass. 

Design and constr. of movable dam and lock, Lockport 

The La Prele, hollow reinforced-conc. dam, 130 ft. high 

Buttressed masonry dam reinforced with steel I-beams 

Diamond drill borings for a dam, Clackamas R., Ore. 

Experiments with rubber models of dams, to study stresses 
Rein. -cone, dam, buttressed, 16' 6" high, 11' 6" base 
Section of the Arrowhead hydraulic fill dam 
Olive Bridge dam (cyclopean masonry) ; earth dike 
Diagram, principal stresses and planes in the m.asonry dam 
Cross-section of hydraulic fill dam. South Carolina 
Cross-section Kensico dam, Catskill water supply 
Cross-section rock -fill crib dam, power develop., Mont. 
Construction plant for the Holter dam, Montana 



Eng 

April 

June 

June 

July 

July 

Sept. 

Sept. 

Nov. 

Oct. 

Nov. 

Oct. 

Nov. 

July 

July 

Aug. 

Dec. 

Jan. 

Apr. 

May 

June 

Jime 

Aug. 

Oct. 

Nov, 

Nov. 

Dec. 



, News, 

27, "05. 
15, '05. 
29, '05. 

27, '05. 
27. '05. 

28, '05. 
28, '05. 

3, '05. 

4. '06. 

1, '06. 

3, '07. 
14, '07. 

9, '08. 
22, '09. 

5, '09. 

9. '09. 
13, '10. 
14, '10. 
19, '10. 



23. 
30, 
11, 
6, 
10, 
24, 



'10. 
'10. 
'10. 
'10. 
'10. 
10. 



22, '10. 
Eng. Rec. 
Mar. 6, '09. 
Mar. 27, '09. 
Apr. 3, '09. 
Sept. 4, '09. 
Oct. 2, '09. 
Oct. 2, '09. 
Dec. 25, '09. 
Mar. 12, '10. 
Oct. 29. '10. 



*See Eng. News of April 30, 1908, for description of failure. 



50.— FOUNDATIONS. 

General Discussion. — Equally as important as the superstructure, is 
the substructure or foundation. This may be defined as that part of the 
structure designed to carry and distribute safely the forces from the super- 
structure to the foundation hed,^ and vice versa, with resulting stability and 
without injury or undue stress in any part. Good foundation work in both 
design and execution is less dependent on theory than structural work in 
general. It has to draw mainly on practical experience in the particular 
class of work at hand, being largely tempered of course with good judg- 
ment and certain well-known, established principles. 

In the design of any structure we must know, and keep clearly in mind, 
the "acting forces," hence it is necessary in designing foundations to know: 
(1) the nature of the superimposed loads, (2) the nature of the foundation 
loads, and (3) the nature of the reactions, at the foundation bed. The first 
named may be calculated with a reasonable degree of accuracy, assuming 
the foundation and foundation bed to be relatively rigid; the second is 
susceptible of equally accurate determination; while the deduced stresses 
in the foundations, and the resultant reactions at the foundation bed, are 
usually proportioned to the loads above. If now for any reason unequal 
settlement takes place, new stresses will be introduced which may be con- 
siderably at variance with the calculated stresses in many of the parts, 
and which may work serious injury to the structure as a whole. It there- 
fore devolves upon the engineer to so proportion the foundation and founda- 
tion bed that not only will each part be sufficiently strong to support the 
loads, but also that they will he relatively strong and rigid, or at least sufficiently 
so to prevent unequal settlement. 

Foundation Bed. — In general, the character of material desirable for 
foundation bed will take rank in about the following order: 

(a.) Solid rock (granite, limestone, sandstone) ; 

(b). Hardpan or cemented gravel (composed of gravel, sand, and lime 

or clay binder) ; 
(c.) Gravel and sand (compact and dry, as in gravel pit) ; 
(d.) Indurated clay (hard, dry and well drained); 



(e.) Sand (dry); 

(f.) Gravel; 

(g.) Boulders and gravel ; 

(h.) Clay and sand; 

(i.) Common clay; 

(j). Sand and loam; 

(k.) Loam. 



This classification, or any other for that 
matter, may be considered to be more or 
less elastic, depending (1) upon the amount 
of moisture present; (2) upon the lateral 
confinement of the material; (3) upon the 
underlying strata; (4) upon the nature of 
[ the superimposed loads. 
The various classes of material for foundation bed- will be discussed in 
detail as follows: 

(a.) Solid Rock. — Granite, when tested in small cubes exhibits a safe- 
average, so-called crushing strength of, say, 12000 lbs. per square inch; 
gneiss, basalt and trap, not less than that amount; limestone (and marble), 
8000 lbs. ; sandstone, 5000 lbs. Some of the best specimens, however, which 
have been tested give, respectively, twice the above value for granite, or 
24000 lbs.; three times the above value for limestone, or 24000 lbs.; and 
three times the above value for sandstone, or 15000 lbs. With these large 
values for small specimens, where in reality the blocks are ruptured by 
"shearing stresses," it would be problematical to assign even approximate 
values for the crushing strength of well-constructed masonry foundations 
en-masse, in which the "crumbling resistance" would increase faster than the 
area of the resisting surface. And, furthermore, it would be mere conjecture 
to place an upper limit to the resisting power of rock in situ and forming 
the foundation bed. A good, solid rock bed is the best that can be had, 
and will safely support any structure which man is liable to build upon it. 

In preparing the foundation bed, it should be leveled or "stepped" so 
that its plane or planes shall be normal to the acting forces. All perishable 
or weathered material should be removed and the exposed surface of the 

863 



864 50.— FOUNDATIONS. 

rock should not be so smooth as to unsafely decrease resistance to sliding 
of the foundation above. Especially should this be looked after in the case 
of a submerged pier or of a dam, where the lateral pressure, due to the 
current of the stream, or to ice, logs, or hydrostatic pressure, would be 
considerable. 

A bed-rock foundation is not always easily obtainable on account of its 
depth below the ground surface or, in marine work, below the water surface. 
It is therefore often more economical to choose for a foundation bed a 
poorer character of material not so deep, thereby saving considerable in 
excavation but requiring, generally, a more expensive foundation "footing" 
(see Figs. 1, 2 and 3). Hence it is that often after the excavation for the 
foundation is started the plans are changed when it has become evident 
that a stratum has been reached that is "good enough" for all practical 
purposes; or, on the other hand, when the expected stratum as revealed by 
the borings does not meet the requirements, and another stratum at greater 
depth is decided upon. (Excavation processes are described farther on.) 

1. — Actual Bearing Pressures on Bed Rock. 

Foundations of New Croton Dam. — Calculated pressures limited to 1 5 tons per sq. ft. 

at base of dam on rock surface, the resultant pressure being kept within the 

middle third of section. 
Manhattan Life Building, New York City. — Pressure at base of caissons on bed rock, 

10.8 tons per sq. ft. 
American Surety Building. — Pressure at base of caissons, 7i tons per sq. ft. 
GUlender Building. — Pressure at base of caissons on bed rock, 12 tons per sq. ft. 

(b.) Hard-Pan. — Next to solid rock there is no better material for 
foundation bed than cemented gravel or hard-pan. When well cemented, 
and in thick and extensive beds, it is capable of sustaining safely without 
injury and with comparatively slight settlement, a quiescent loading of 
10 tons per square foot (= 138.8^8 lbs. per square inch). 

(c.) Gravel and Sand. — Large, thick beds of well-compacted gravel and 
sand, free from wash by the action of water, can generally be counted on 
to sustain about 8 tons per square feet of quiescent loading, or equal to lll.l'^l 
lbs. per square inch. For such a loading, however, the foundation bed 
should beat least 10 or 12 feet below the surface of the ground, and the 
under strata must be firm. 

{d.) Indurated Clay. — By the term "indurated" we mean the hard 
variety, usually containing in sufficient quantities such binding materials 
as carbonate of lime, silicates of aluminum and magnesium, iron oxides, etc. 
When in the natural bed, at considerable depth, and free from the softening 
action of water, this material may be loaded safely to 6 tons per square 
foot (equivalent to 83. S^'^S lbs. per square inch). If it contains a proper 
intermixture of gravel and sand it approaches hard-pan, and the bearing 
power is increased. 

{e.) Dry Sand. — Under the most favorable conditions, that is, where 
the material is confined as in a deep trench and not allowed to spread out 
from the pressure above, and also where it is reasonably dry and free from 
wash, sand will easily support a load of 4 tons per square foot ( = 55.5^5 lbs. 
per square inch). For large buildings, however, where unequal settlement 
to any considerable extent would be liable to produce cracks in the masonry 
walls, the maximum limit of allowable pressure should be fixed at 3 tons per 
square foot. 2\ tons or even 2 tons is quite common practice under ordi- 
nary conditions of wetness, but where there is no wash from running water. 
For the above conditions the underlying strata must of course be firm. 
Where the amount of settlement is of minor importance there is really no 
practical limit to the bearing power of sand, hence the wide range of values 
assumed. In some instances 10 tons per square foot has been exceeded. 

2. — Actual Bearing Pressures on Sand. 
Washington Monument, Washington, D. C. — Pressure at base resting on sand bed 2 ft. 

thick, about 1 1 tons per sq. ft. ; and with wind pressure added, about 1 4 tons. 
Piers of Brooklyn Suspension Bridge. — At base of piers, 44 ft. below bed of river, 

pressure on layer of sand 2 ft. thick resting on bed rock, 5^ tons per sq ft. 
St. Paul Building, New York City. — Continuous grillage over entire area. Pressure 

on compact sand, 3. 2 tons per sq. ft. 
World Building, New York City. — Inverted arches over continuous concrete footings. 

Pressure on dense, fine sand, 4.7 tons per sq. ft. 
Spreckles Building, San Francisco. — Continuous grillage. Pressure on dense, wet 

sand, 2i tons per sq. ft. 



FOUNDATION BED— BEARING PRESSURES. 865 

(/.) Gravel; (g.) Boulders and Gravel. — Some very important structures 
are found on these classes of material, both above and under water. If 
properly prepared and protected from lateral bulging and wash such a 
foundation may be loaded with 4 tons per square foot ( = 55.5'^5 lbs. per 
square inch). If, further, it is compacted with dry or moist sand not subject 
to wash, the bearing power will be increased, say to 6 or 8 tons, depending 
upon the binding quality of the material and on local conditions; if a 
moderate quantity of clay is intermixed with the sand and the resultant 
matrix is sufficient to fill the voids of the gravel the higher value (8 tons) 
may be used with safety. Such a material if thoroughly hardened would 
form hard-pan. 

3. — Actual Bearing Pressures on Gravel. 
Piers of Cincinnati Suspemion Bridge. — At base of piers, 12 ft. below low water, 
pressure on gravel bed (neglecting skin friction of piers), 4 tons per sq. ft. 

(h.) Clay and Sand\^ (i.) Common Clay. — Ordinary soft clay or clay and 
sand, when wet, is subject to considerable displacement if heavily loaded; 
and under ordinary conditions it is best not to allow more than 1 ton per 
square foot (=13.8"^8 lbs. per square inch) for important buildings where 
much settlement would be harmful. But this bearing power may be greatly 
increased by proper drainage or by a close (lateral) confinement of the 
material, so that well-drained clay or such clay containing some sand may 
be loaded with 2 tons per square foot. A naturally dry common clay bed of 
considerable thickness and extent will stand three tons with allowable 
settlement; and this may be increased to 4 tons for a mixture of sand and 
clay where the former predominates and the latter is sufficient for a binder. 
Hard, firm clay unmixed with sand will also stand 4 tons and if mixed as 
a binder with good, coarse sand the bearing power will be greatly augmented. 
Fine clay and sand when thoroughly saturated with water forms quick- 
sand. 

4. — Actual Pressures on Clay and Sand. 

Capital Building. Albany. N. Y. — Clay with some sand. Pressure allowed, 2 tons 

per sq. ft. 
Congressional Library, Washington. D. C. — Yellow clay mixed with sand. Pressure 

allowed, 2^ tons per sq. ft. 

(/ ) Sand and Loam; (k.) Loam. — Loam is too spongy and compressible 
to be relied upon, from an engineering standpoint, to support any structure, 
and hence the excavation is always carried through it to some firmer 
foundation bed beneath. It is a mixture of clay, sand, vegetable mould, 
and decayed animal matter. But when mixed with a large proportion of 
sand it is less objectionable especially when well tamped. Railway trestle 
bents which rest on mud sills supported by this material, are frequently 
subject to considerable settlement. 

Practical Tests of Soils. — ^Where the bearing power of any particular 
soil is in question it may be tested by loading a vertical timber of given 
cross-section (the larger the better) and resting on the soil at the bottom 
of a pit, with a weight equal to or greater than the proposed intensity of 
loading. Allowance will be made, however, for the results obtained as the 
settlement due to loading such a limited area would be excessive. 

Selection of Site for Building. — Generally, the site for the erection of a 
structure is determined from necessity rather than from choice. That is to 
say, conditions other than those of local character govern and fix the 
location. But there are many instances where other sites than those 
selected would have been chosen from pure economy of cost of foundation 
had direction been given to its consideration. 

Examination of Soil. — The simplest and most satisfactory method of 
determining the character of the soil is by digging pits. This should be 
done generally before making estimates and letting contracts, on all im- 
portant work requiring excavation in general, and including foundation 
work at no great depth. For the former we may cite railroad cuts, irriga- 
tion canals, and waterways in general; while for the latter, may be men- 
tioned dam sites, and sites for bridge piers, buildings, etc. The pits need 
be only large enough for a man to find room to excavate. Where great depth 
is required borings must be made. 



866 50.— FOUND A TIONS. 

Borings in Soil. — For soft earth and clay, up to about 100 feet in depth, 
a common wood auger with levers, turned by hand, may be used. It may 
have a diameter of 1^ to 2 J inches and be worked on the end of a sectional 
iron rod easily made by any blacksmith. It is usually started by one-man 
power, but after being sunk a short distance two or more men may be re- 
quired. During the process of boring, samples are brought up and recorded 
together with their distances below the surface. 

For the harder strata and at greater depth, artesian well boring tools 
are employed. (See Eng. News, Vol. XXI, page 324, for illustrations.) 

Estimating Loads on Foundations. — ^These loads include: 
(1.) Forces due to the weight of the structure itself, so distributed as 
to conform with the reactions, whether vertical or inclined. This is not 
always an easy matter, but the principle should be kept clearly in mind. 

(2.) The live loads or that percentage of them liable to act in unison to 
produce maximum stresses in any part of the foundation. The term "live 
loads" will here be considered to embrace those loads for which the structiire 
was principally designed, as for instance people, furniture and merchandise 
for buildings, water pressure for dams, etc. 

(3.) The wind loads (also other natural forces of an accessory character 
such as snow, pressure of ice, etc.), considering possible tension in the 
foundations as well as compression; also shearing. 

(4.) Temperature stresses, due to changes in temperature affecting the 
length of certain principal members confined between parts of the founda- 
tion, as for instance in the two-hinged steel arch. 

(5.) Impact loads due to moving machinery, as steam engines, steam 
hammers, dynamos, etc. 

For Buildings. — There are two main classes of foundations in use for 
buildings, namely, continuous foundations, and independent piers. The 
latter are generally employed where a suitable, natural foundation bed 
exists only at great depth, in which case the continuous foundation would 
be more expensive. As more or less settlement always occurs after a build- 
ing is erected, due almost wholly to the constant pressure of the dead load, it is 
best to proportion the area of all foundation footings to the dead load only, 
using a "reduced" and uniform working pressure for same, so that, selecting 
that pier which we will call the "Index" pier or part of foundation which 
sustains the least ratio of dead load to total load, we have, 

Reduced working pressure Dead load stress at footing of column 

Allowable maximum pressure Maximum stress at footing of column "^ ^ 

Under the subject of Buildings, page 822, it will be noted that for high 
buildings the maximum stress at footing of column is reduced by diminish- 
ing the live load for each floor below the top one a certain percentage, as it 
is not likely that all floors "in column" will be loaded fully at the same time. 

It may be stated here that vault shafts running up through buildings 
should be carried on a separate foundation from that of the main building. 
Engine and boiler foundations should also be independent. 

Care should be taken that the line of resultant pressure at the founda- 
tion footing should come well within the middle third if the foundation 
bed is solid rock, and it should be practically central for any bed which is 
soft or springy. Eccentricity of loading may unduly increase the intensity 
of pressure, and also cause tipping, cracking, and perhaps rupture, of the 
masonry wall. 

For City Building Codes, see Table 5, following page. 

For Dams. — To save the reader the time and trouble of looking into 
the matter it will be stated that for a gravity dam with reservoir empty, 
it will not be necessary to consider any additional stresses in the masonry 
or on the foundation bed, due to wind pressure against the down -stream 
face. Assuming a dam 200 ft. high with a base of 69 ft., and considering a 
wind pressure of 50 lbs. per sq. ft. acting horizontally (with no vertical 
component), then under the most unfavorable conditions the intensity of 
stress at either toe would not exceed 9 lbs. per square inch. In calculating 
the resultant stress due to any pressure at the top of the dam, as for instance 
logs or ice, it is necessary only to consider the dam as a beam fixed at one 
end with length equal to the height of the dam and with depth equal to its 
base, and assuming the neutral axis of the section to bisect the base. 



LOADS ON FOUNDATIONS. 



m 



6. — Bearing Power op Soils for Buildings. 

(Extracts from various City Codes.) 

[Loads are in tons of 2000 lbs. per sq. ft.] 

New York (1906). — Where no test is made, different soils, excluding mud, at bottom 
of footings, shall be deemed safe to sustain the following loads per sq. ft. : Soft 
clay, 1 ton; ordinary clay and sand together, in layers, wet and springy, 2 tons; 
loam, clay or fine sand, firm and dry, 3 tons; very firm, coarse sand, stiff 
gravel or hard clay, 4 tons, or as otherwise determined by the Commissioner of 
Buildings. 

Chicago (1907). — If the soil is a layer of pure clay at least 15 ft. thick, without ad- 
mixture of any foreign substance excepting gravel, the load shall not exceed 
If tons per sq. ft. ; pure clay in layers at least 1 5 ft. thick, dry and thoroughly 
compressed, 2 J tons; dry sand, at least 15 ft. thick, and without admixture of 
clay, loam or other foreign substance, 2 tons; clay and sand mixed, li tons. 

Philadelphia (1907). — Foundations of other materials than piles shall be so propor- 
tioned that the loads upon the soil shall not exceed the limits for the different 
kinds of soil than herein given, to wit: Sand and loose gravel, 3^ tons per sq. ft. ; 
dry, hard clay, 3i tons; cemented gravel, 6 tons. 

Cleveland (1907). — Good, sound, natural earth shall not be loaded to more than the 
following: Gravel and coarse sand well cemented, or rock or hard shale unex- 
posed to the action of air, frost or water, 8 tons per sq. ft ; dry, hard clay or fine 
sand, compact and well cemented, 4 tons; moderately dry clay or clean, dry 
sand, 2 tons; soft, wet sand, 1 ton; quicksand or alluvial soils, i ton; the sand 
underlying the City of Cleveland above the lake level, commonly called "quick- 
sand," when drained of its ground water without puddling or disturbing the 
foundation may be loaded, 3 tons. 

San Francisco (1910). — Soft clay, 1 ton per sq. ft.; sand and clay mixed, 2 tons; 
firm, dry clay, 3 tons; hard clay, 4 tons; loam or fine, dry sand, 3 tons; com- 
pact sand, 4 tons; coarse gravel, 6 tons; shale rock, 10 tons; hard rock, 20 tons. 

Buffalo (1909). — In no case shall the soil under any building be loaded with a weight 
greater than 3^ tons per sq. ft. If the soil is composed of other materials than 
hard clay or gravel then the area of the foundation shall be extended as directed 
until the pressure is reduced to a safe limit. 

District of Columbia (1906). — (Practically the same as N. Y. City Code.) Soft clay, 
1 ton per sq. ft.; ordinary clay and sand together, in layers, wet, 2 tons; loam, 
clay, or fine sand, firm and dry, 3 tons; very firm, coarse sand, stiff gravel, or 
hard clay, 4 tons, or as otherwise determined by the Inspector of Buildings. 

For Machines, Dynamos, etc. — The weight of the machine as a whole is 
not alone to be considered. A more massive foundation may be required 
for a light machine than for a heavy machine with the same weight of 
moving parts. The foundation should be on a natural, hard bed, but 
where this is not obtainable the foundation itself should be massive enough 
to absorb the shock or impact of the machine. Manufacturers usually 
prefer to install their own machines and prepare the foundations, and, 
where possible, it is best for them to do so on account of their practical 
knowledge of the requirements of the case. 

Types of Foundation Footings: — 




Foundation Bed 



III 



i}wiii/iiiiiiiii'ii'iiiiiii'ii"iii"ii///ii'i>'' 

Foundation Bed 



Fig. 1. — Least dimensions for ordi- 
nary concrete footing under 
brick or rubble wall or pier. 
Where more bearing area is re- 
quired on foundation bed the 
offset and thickness are increased 
or the sides of the footing sloped, 
as in Fig. 2. 



Fig. 2. — Concrete footing with or 
without I-beams, for heavily 
loaded walls, yielding founda- 
tions, or both. One or more of 
the beams is often imbedded in 
the concrete to give stiffness and 
absorb shock due to heavy ma- 
chinery, as in power houses, etc. 



868 



50.— FOUNDATIONS. 



Problem. — Let it be required to de- 
sign a footing of I-beam construction 
under a steel column, carrying at bottom 
of footing a maximum load of 748,000 
lbs., and on a soil capable of sustaining 
3 tons (6000 lbs.) per square foot. 

Solution. — By using formula (1), 
page 866, we find that the reduced 
working pressure on foundation is, say, 
5200 lbs. per square foot, hence the 
total area of the footing should be 
748.000 H- 5200= 143.8 square feet, or say 
a square base of length B = 12 ft. Now, 
tentatively, we may assume that A = 

|- = 3ft.;a=|- = 3ft.;6=-^=4i ft.- 
c=|- = Uft. 



^S 



n i 1 M I i I r 

^LLVATIOH 



8 




l< b — 

u — .. 



-A— 1- — b -^-H 
B— — --— -H 

Plan 

Fig. 3. — I-beam Footing for Inde- 
pendent (isolated) Piers. Beams 
to be imbedded in concrete. 



Then for the bottom tier 

of beams each cantilever arm a will sus- 
tain a total upward pressure of 5200 X a 
XB = 5200X3X12= 187200 lbs., pro- 
ducing a bending moment of 1 87200 X 

-|-=280,800ft.-lbs. Using a fiber stress 

of 16000 lbs. per square inch for steel 

I-beams the above moment calls for 

either 12 ^ 21 lb. beams, 9 10'' 25 lb. 

beams, or 6 12'' 31 lb. beams*. The 

12" beams are the more economical, 

while the 10" beams give closer and 

better spacing and will be adopted. 

Similarly, for the middle tier of beams, 

each cantilever arm h will sustain a total upward pressure (neglecting 

weight of lower tier of beams) of 5200X^X5 = 280800 lbs., producing 

a bending moment of 631800 ft. lbs., and calling for 5 20" 65 lb. beams. 

Lastly, for the upper tier, each cantilever arm c will (be assumed to) 

support one-quarter the total load or 187,200 lbs., producing a bending 

moment of 140,400 ft. -lbs., and calling for 5 10" 25 lb. beams. 

(Dther proportions may be assumed for the areas of the tiers of beams 
if it is thought that economy may result. 

Coffer=Dams.— A coffer-dam is a fixed enclosure built in situ around a 
proposed foundation for the purpose of shutting out the water during con- 
struction of the latter. Such a dam may be formed by building an earth 
embankment around the site; by constructing a water-tight casing of any 
material, as iron or wood; by sinking cribs; by driving sheet piling; or by 
a combination of these methods. In any event there will be more or less 
leakage, and either hand or machine pumps will have to be installed to keep 
the site dry. 

By Earth Embankment. — A tight dam may be made with gravel and 
clay, but other soils may be used if there is a matrix sufficient to fill the 
voids. This construction is used where the water is shallow and quiet 
or where there is but little current. The width of dam at top may vary 
from 3 ft. upward, with side slopes of 2 or 2^ to 1. For a depth greater 
than 5 ft. other methods are generally used. Sometimes sand in bags is 
employed to good advantage, especially where there is some current. 

By Water-Tight Casing. — The casing may be in the form of a large 
wooden crib with single or double shell. If of single shell, the seams are 
calked tight, while if the shell is double the intervening space or chamber is 
packed with clay puddle, to prevent leakage. The 'size of the shell timbers 
depends on the depth of water and also on the interior bracing. For coffer- 
dams in ordinary bridge foundation work, 12"xl2" timbers are frequently 
used, dove-tailed or rather halved at the joined ends or corners. The 
bracing may extend entirely across the crib from shell to shell, and be re- 
placed by shorter struts abutting against the masonry as the latter inter- 



* See Tables of Properties of I-beams, pages 554 and 555. 



COFFERDAMS. SHEET PILING. 



869 



cepts it in being projected upward. In designing the shell, remember that 
the hydrostatic pressure in lbs. per sq. ft. = ^2.bH, in which H is the depth 
in ft. below the water surface. (See Dams, page 846.) Thus for a horizontal 
shell timber of length, say, 6 ft. between interior bracing, and whose center 
is 30 ft. below the surface, we have, if d equals thickness of shell. 

^ ,. , w/2 62.5X30X6X6X12 

Bendmg moment = -5- = 5 — 

o o 

Resisting moment = ^ f b d^; in which /= 1000, and 6= 12. 
_ ^. „ 62.5X30X6X6X12X6 _ ^^ , „. 

Equatmg, d^ = 8X1000X12 = 50.63; or, say d=8 ms. 

The design of such a crib is quite simple under favorable circumstances, 
but in a swift stream with an uneven bed, part of which may be rock, it 
offers many difficulties. The bottom of the crib in this case should conform 
closely with the rock bottom. For a silt bed the bottom of crib should be 
sharpened to penetrate the silt. Piles may be driven to hold the crib in 
place. This leads to another construction, namely, that of driving one or 
more rows of piles and sheathing them, thus obtaining the same result in 
another way. The piles should be well diagonal-braced where necessary to 
resist bending, remembering that where not so braced the resisting moment 
of the section in inch-lbs. is = 0.0982 diameter^X allowable outer fiber 
stress / of pile per square inch — say /= 1000. 

By Sinking Cribs. — A large crib coffer-dam is sometimes formed by 
sinking several small cribs in line around the site, and planking and calking 
them on the outside. The cribs are made rectangular, of squared timbers 
framed in the ordinary manner, and containing cells or chambers, somewhat 
above the level of the bottom, to be weighted with stone so as to sink into 
the river bed. Guide piles are often driven for this work. In sinking cribs 
where there is current two or more clusters of piles may be driven up stream, 
or auxiliary cribs may be sunk to which are attached adjustable steel cables 
for "dropping" the cribs into position. 

By Driving Sheet Piling. — ^This is one of the most common methods in 
use, both in wet ground and in shallow water; and where it can be employed 
effectively, is simple and cheap. Some of the principal forms of wooden 
sheet piling are the following: 



Fig. 4. 



-Large Square 
Piles. 



Fig. 5. 



-Plain Single 
Sheeting. 



Eir 



1 r 



rzi 



Fig. 6. — Double Sheeting. 



IX 



rr 



i^ > M 



Fig. 7. — Triple Sheeting. 



Fig. 8.— Matched Piles. 



[Z 



U 



Fig. 9. — V-Matched Piles. 



"3 



E 






S 



Fig. 10. — Tongue-and-Groove Piles. 



Fig. 11.— Built Matched 
Pile. 



\ i! ii \ 



Fig. 12.— Built V-Matched 
Pile. 



\ 11 Ii \ 

\_L il ^ 

Fig. 13.— Built W- 
Matched Pile. 



n: 



d 



□L 



n 



C3 



0: 



-EL 



£3 



Fig. 14. 



Fig. 15. 



Fig. 16. 



Figs. 14, 15, 16.— Details of Wakefield Sheet Piling. 



870 



m.— FOUNDATIONS, 



Other forms are in use but they are no improvement over the above. 
The lower ends of the piles are usually cut on the slant so that in driving 
they -will crowd against those in place and close the joints 
(Fig. 17). They are also sharpened so as to "broom" if 
there is solid rock bottom. A good method to secure 
tight joints is to drive from both ends of the line toward 
the center and then drive home a good tight-fitting "key" 
pile. Sheet piles are usually driven between guides or hori- 
zontal waling pieces which may be supported by a row of 
ordinary round piles, or sawed piles, previously driven. After 
the sheeting is driven it is keyed tightly or wedged in the 
wales. Fig. 17.' 

Steel Sheet Piling comes in many forms, some of which are composed 
of standard rolled sections riveted together, while others are special rolled 
sections not requiring any riveting. 




Fig. 18. 
Fig. 18 illustrates an interlocking channel-bar piling caisson driven 
in place and ready for excavating. The piling is that of the Friestedt* 
type. This piling in place will weigh from 23 to 67.5 lbs. per sq. ft. Piling 
which will weigh when interlocked about 41 lbs. per sq. ft. will have a 
moment of inertia of 69.21 and a radius of gyration of 1.6. 




Fig. 19. 

Fig. 19 illustrated the plain rolled section as manufactured by the 
United States Steel Piling Co. of Chicago. The actual cost of this 
piling depends somewhat upon the specifications. If ordered cut to lengths 
it can be furnished on cars at approximately S42.0Oper net ton. Where cor- 
ner pieces are required there is a net extra of approximately $10.00 per 
net ton for the comers only, and there is also a net extra of $2.00 per ton 
for punching each piece with pulling holes where purchasers prefer shop 
punching, prior to shipment. The above prices are for August, 1906. 



* Manufactured by the Friestedt Interlocking Channel Bar Co., of 
Chicago. 



SHEET PILING. PILE FOUNDATIONS. 871 

Leakage in coffer-dam construction may often be stopped by the 
use of clay puddle, with perhaps an intermixture of straw and gravel if the 
leak is at the foundation bed. A more expensive but effective method, and 
which may be applied to deep rock fissures, is by grouting. Sometimes 
canvas or tarpaulin coated with asphalt and weighted to the bottom will 
answer. If fact the use of canvas can generally be recommended in various 
ways as cheap and convenient. The joints of the upper-work shotild be 
calked, and the seams may be asphalted or greased if desired. 

After the coffer-dam is constructed the water is pumped out and the 
material is excavated to a firm foundation bed. 

Pile Foundations. — In soft, marshy ground it is often best to use a pile 
foundation, and for permanent construction the piles should be cut off at 
least two feet below the level of so-called continual saturation, or low 
water mark. Many of our tall office buildings, up to 12 stories or more in 
height, rest on piles, as do also many of our large masonry bridge piers. 
The term "pile" unless specially named otherwise will be assumed to be of 
timber. 

Supporting Power of Piles. — Some engineers have claimed that piles 
should be driven with the large end downward; and this would be logical 
were it true that the direct bearing of the lower end were all the supporting 
power that the pile received.* But it must be remembered that with the butt 
end upward and the pile tapering downward it receives considerable vertical 
"support" along the sides in addition to the largely increased "friction." 
In addition to its being, the most practical way there is another point in 
favor, of driving the small end downward, namely, that generally the soil 
increases in compactness with the depth below the surface and hence tends 
to give greater lateral stiffness at the lower end where it is required. This 
is very important where a pile is driven to a hard stratum, through com- 
paratively soft materials, thereby acting as a column subject to bending. 
Usually such a column may be considered as "fixed" at the lower end.f 
For very long piling so supported and driven through loose, soft material 
the loading should not exceed ^ to J the allowable load on a short column 
of the same sectional area. In any case the loading should be limited by 
the following formula J which takes into consideration the resistance at the 
last blow only (see Table 6, following page) : 

t^=7^ a) 

where P = safe|| load in tons, on pile; 

«; = weight of hammer, in tons; 

/t = height of free fall of hammer, in feet; 

5 = penetration of pile at last blow, in inches. 
It is good practice to drive a pile ordinarily until the penetration at the 
last blow is less than ^ inch, using of course a moderately heavy hammer; 
but care must be taken that the pile itself is not being injured by "brooming," 
either at the bottom, top, or any other part. Many piles are seriously 
injured by overdriving. An experienced man can tell, generally, when a 
pile has been "driven" by the action of the pile and hammer, and the 
impact. 

Pile Drivers. '^ — ^The main essentials of a pile driver are (1) a hammer 
(or ram), (2) a derrick with vertical (or inclined) leads with guides for the 
hammer to work in, and (3) power to operate the hammer, so that in its 
descent it will strike the head of the pile and ram the latter into the ground. 
We will not consider such accessory devices as the explosion of gunpowder 
or dynamite on the head of the pile as the hammer descends, because the 
economy and practical utility of such methods have not been demonstrated. 

* Under certain conditions, such as to prevent subsequent uplifting, 
piles may be driven butt end downward , as when the sub-stratum is quick-sand . 

t Even so, it is supported more or less throughout its length by the 
lateral resistance of the material through which it is driven unless it be 
water. Theoretically, a pile may assume all the conditions of strength 
ranging from a short strut to that of a long column with round ends, as 
when it is driven through silt to a solid rock bed. 

^ w h 

t Wellington's formula for drop-hammer. For steam-hammer, ^=~rQT* 

1 1 Factor of safety of 6. 

t For description of Pile-pulling machines, see Eng. News, April 16, 1903. 



872 



50 —FOUNDATIONS. 



6. — Safe Bearing Power op Piles, in Tons op 2000 Lbs. 

Driven by Drop Hammer. 

[By Wellington's Formula (1) preceding page: P = 2 wh^{s+l).] 



Penetra- 
tion 
at Last 




Weight of Hammer in 


Tons, for Drop of 20 Feet.* 






















Blow. 


^ ton 


f ton 


1 ton 


Utons 


1^ tons 


1 1 tons 


2 tons 


2\ tons 


2itons 


Inches. 


(1000 


(1500 


(2000 


(2500 


(3000 


(3500 


(4000 


(4500 


(5000 




lbs.) 


lbs.) 


lbs.) 


lbs.) 


lbs.) 


lbs.) 


lbs.) 


lbs.) 


lbs.) 


Vs 


17.78 


26.67 


35.56 


44.44 


53.33 


62.62 


71.11 


80.00 


88.89 


H 


16.00 


24.00 


32.00 


40.00 


48.00 


56.00 


64.00 


72.00 


80.00 


H 


14.55 


21.82 


29.09 


36.36 


43.64 


50.91 


58.18 


65.45 


72.73 




13.33 


20.00 


26.67 


33.33 


40.00 


46.67 


53.33 


60.00 


66.67 


5^ 


12.31 


18.46 


24.62 


30.77 


36.92 


43.08 


49.23 


55.38 


61.54 


M 


11.43 


17.14 


22.86 


28.57 


34.29 


40.00 


45.71 


51.43 


57.14 


% 


10.67 


16.00 


21.33 


26.67 


32.00 


37.33 


42.67 


48.00 


53.33 


1 


10.00 


15.00 


20.00 


25.00 


30.00 


35.00 


40.00 


45.00 


50.00 


1^ 


8.89 


13.33 


17.78 


22.22 


26.67 


31.11 


35.56 


40.00 


44.44 


1^ 


8.00 


12.00 


16.00 


20.00 


24.00 


28.00 


32.00 


36.00 


40.00 


m 


7.27 


10.91 


14.55 


18.18 


21.82 


25.45 


29.09 


32.73 


36.36 


2 


6.67 


10.00 


13.33 


16.67 


20.00 


23.33 


26.67 


30.00 


33.33 


2M 


6.15 


9.23 


12.31 


15.38 


18.46 


21.54 


24.62 


27.69 


30.77 


23^ 


5.71 


8.57 


11.43 


14.29 


17.14 


21.00 


22.86 


25.71 


28.57 


2^ 


5.33 


8.00 


10.67 


13.33 


16.00 


18.67 


21.33 


24.00 


26.67 


3 


5.00 


7.50 


10.00 


12.50 


15.00 


17.50 


20.00 


22.50 


25.00 


3^ 


4.44 


6.67 


8.89 


11.11 


13.33 


15.56 


17.78 


20.00 


22.22 


4 


4.00 


6.00 


8.00 


10.00 


12.00 


14.00 


16.00 


18.00 


20.00 


5 


3.33 


5.00 


6.67 


8.33 


10.00 


11.67 


13.33 


15.00 


16.67 


6 


2.86 


4.29 


5.71 


7.14 


8.57 


10.00 


11.43 


12.86 


14.29 



The drop-hammer. — ^This may consist of a heavy 
block of oak when some hastily improvised machine is 
desired, but the cast-iron ram as shown in Fig. 20 is the 
type quite universally employed. In the illustration 
the hammer h is engaged by the nippers n, and is 
released when the latter are drawn up against the wedge 
w fastened to the guides g, at the top of the derrick. 
The hammer may also be released at any height below 
the top by pulling a tripping rope attached to the nip- 
pers. The hammer line or hoisting rope r runs over a 10'' 
to 18" sheave (2 sheaves are fixed at the top — one for 
the hammer line and the other for the pile line) and 
can be operated either by horse power or by hoisting 
engine. These hammers weigh from 1200 lbs. upward, 
2000 to 2500 lbs. being a very satisfactory medium 
weight. Greater speed can be obtained by the use of 
a hoisting engine with a friction clutch or friction drum, 
so that the rope and hammer may be released by the 
engine driver at any moment. In such a case the ham- 
mer rope is fastened permanently to the hammer which 
is level on top, instead of being depressed as in Fig. 
20. It is to be noted that the hammer should be 
heavier for this method as it has to "overhaul" the 
hammer rope in its descent. Such hammers weighing 
efficient. 




SECTION A-A 

Figs. 20. 
Drop-Hammer 
with Nippers. 
3500 lbs. are very 



The steam-hammer. — Fig. 21 is an illustration of an improved type of 
steam-hammerf with a "gravity" action. The total machine, which may 

* For any other drop the bearing is proportional. Thus, for 15 ft. drop, 
multiply by ^; for 25 ft. drop, multiply by ^U\ etc. 

t The Warrington steam hammer as manufactured by the Vulcan Iron 
Works, Chicago. The first steam hammer for pile driving was applied by 
James Nasmyth in 1845. 



PILE DRIVING, 



878 



Fig. 21. 

Steam Pile- 

Driver. 



weigh as much as 5 tons, is suspended from the top and 
between the leads or gins of the derrick, like a common 
drop-hammer; but in this case it is allowed simply to rest 
on the pile to be driven. The ram h, whose weight is about 
half that of the total machine, slides vertically on four circu- 
lar guides and is connected to a piston rod operated from the 
cylinder c. Steam is led into the lower part of the cylinder 
through a flexible tube, thus raising the ram which is then 
allowed to drop by its own weight. The amount of "drop" 
may be lessened if the cylinder is double acting. 

The derrick. — ^The ordinary pile driver derrick is a simple 
affair consisting of two upright leaders with guides for di- 
recting the hammer (Fig. 22), and supported by a frame- 
work (for fore-and-aft and lateral bracing) resting on a plat- 
form or horizontal frame. The frame may rest on a scow, 
on a car, or on rollers. If on a scow, it is usually fixed; on 
a car, it is allowed to swing laterally around a vertical pivot ; 
on rollers, it is given forward and lateral movement by using 
two sets of wooden rollers, one above the other, at right 
angles to each other. A tilting driver for driving batter piles 
may be constructed by allowing the leaders to swing on a 
horizontal pivot attached to the A-frame of the derrick near 
the top ; or by pivoting the whole derrick frame on V-bol- 
sters which will allow lateral tipping. Ratchet devices may 
be used for tilting the leads or the derrick frame, and hold- 
ing them in position while driving the piles. The follow- 
ing dimensions may be taken as mere "hints" for design 
for scow and heavy land derricks: For a 60-ft. derrick, 
leads 8''x 10''; guides 4"x4'' sheathed with 4''x|" iron; spread- 
ers (lateral) 8''xl0'' for height of 55 ft. and total base of 18 
ft.; ladder strings (fore and aft) 6''xl2'' for height of 60 
ft. and base of 18 ft.; horizontal bracing 6''x8" to 8''xl0" 
(the latter supporting platforms); diagonal bracing 4''xl2'"; 
platform frame 12''xl2" and 12''xl4"; caps (for supporting sheaves) 
8''x20". For a height less than 60 feet the dimensions may be proportioned 
about ^s the square root of the height; thus, for a derrick 30 ft. high multiply 
by \/i or 7o. Less amount of bracing is of course needed for a low derrick 
than for a high one and where it is not over 20 or 25 ft. the diagonal bracing 
is not required. For a steel frame, proportion the members for equal 
strength and stiffness to the above. A car-derrick may be made to hinge 
at the foot of the ladder strings and lie flat when not in use. If a light, 
portable land driver is required the above proportions may be reduced. 

The power. — ^This may be either man, horse, steam, gas, hydraulic, pneu- 
matic, or electric. The first named is now only employed in driving sheet 
piling, and to a limited extent. The horse is used frequently in outlying 
districts where it would be expensive to ship a hoisting engine for the 
limited amount of work to be done. Steam is mostly employed, and usually 
by the hoisting engine or by operating the steam ham- 
mer (see page 872). Hydraulic pressure may be used 
for driving piles for foundation work in submarine tun- 
nels where a steady pressure, without shock, is abso- 
lutely necessary. For a large amount of work requir- 
ing several drivers, electric power, delivered from a 
central plant, has proven economical. Fig. 22 is a light, 
portable steam pile driver for height up to 50 feet. 
More diagonal bracing may be used if deemed advisa- 
ble but should not be added, to increase the portable 
weight, unless the work is heavy and demands it. 
Pile driving is often assisted by the water jet. 

The water jet. — If the soil is sandy great assistance 
to driving piles may be rendered by attaching the end of 
a small pipe to the foot of a pile and playing a stream 
of water into the sand as the pile descends. If the ma- 
terial is pure sand, driving often becomes unnecessary, 
the pile settling readily under the weight of the hammer. Fig. 22. 

In some cases a driver is not used at all, the pile being Light, Portable 
forced down by bringing some other weight to bear Land Driver. 




874 ^.—FOUNDATIONS. 

upon it, as by block and tackle or by lever. Softer wood for piles can be 
used in such cases than would be required for ordinary driving. 

Pile shoes. — ^These may be used in hard driving to prevent brooming of 
pile. The shoe may be of cast iron or steel, usually fitted to the point of 
pile after the latter has been sharpened or shaped to receive it. Cones 
made of sheet steel are sometimes used. Provision should be made, by lugs 
or straps, for spiking or bolting the shoes to the piles. Shoes should have 
either a point or an edge, for penetration. o 

Pile Rings. — For top of pile. 

Common Pile Foundations. — ^There is danger in driving 
piles too close. Instances are frequent when piles already 
driven have been weakened considerably by subsequent 
too-close driving. 2' -9" centers is about as close as ordinary 
piles should be driven, and 3'-(K' is much better; but 
local conditions sometimes demand a minimum spacing 
of 2'-Q" or even 2', especially if the piles are small. 

Piles should be driven to firm foundation, and it is 
sometimes necessary to drive them in "tandem" or one 
above the other to reach a suitable supporting soil. Fig._ 23 
shows the dowel connection used for splicing same. Piles 
spliced in this manner have been driven considerably Fig. 23. 
over 100 ft. in depth. Dowel Splice. 

Cutting off piles. — Grades for cutting off piles are given by, say, driving 
a tack in the side of the pile either at the desired level or at certain established 
distance below it. The cut-off is made with a cross-cut saw resting on two 
short horizontal guide sticks nailed on opposite sides of pile, with top 
edges at grade. For cutting off-piles under water there are several methods: 
(1) The above method may be employed sometimes during absolute low 
water, by sawing off two or three feet below the surface, provided the cut- 
off is near the river bottom; (2) under the same conditions as above it may 
be advisable to cut off all the piles just above the surface and then nail 
horizontal guide strips on which to suspend a "gage" saw operated by hand 
and which will cut off the piles at the true level a certain distance below 
the guides; (3) a circular saw fixed horizontally at the lower end of a 
vertical shaft operated from a scow, may be used; (4) if the water is too 
rough or deep and greater care is required it is best to cut the piles off 
above water as in method number (2) and construct a level platform on 
which to operate the circular saw, say 4 ft. diameter, from a movable 
machine (a pile driver is sometimes rigged up for this purpose) mounted on 
rollers. 

Foundations on piles are of many kinds and are discussed under Caissons, 
and what follows. 

''Dead-Men." — ^These are short piles sometimes "planted" through soft 
material to a firm bed and packed around the sides with gravel or sand, 
wet and well tamped. They should be well braced laterally. 

Iron Piles. — ^These are cast iron, wrought iron and steel, being best 
(most durable) in the order named. Cast iron piles may sometimes be used 
to advantage in hard-driving soil where a wooden pile would broom and 
where the ground becomes alternately wet and dry. Steel shapes, as I-beams, 
may be used in wharf construction (fresh water) where considerable lateral 
thrust has to be resisted. They should be coated with asphalt before 
driving. 

Screw Piles.'*' — A screw pile consists of a cast iron shoe, shaped some- 
thing like a cartridge, surrounded by about 1^ turns of spiral disk thread 
5 feet more or less in diameter and mounted on the lower end of a vertical 
shaft which, when turned, screws the pile into the ground. The shaft is 
usually of steel and about to the diameter of the screw. Screw piles are 
particularly adapted to soft soil, but are seldom used in this country except- 
ing for wharf work, anchoring beacons, and light -house construction. Their 
use in bridge foundations has been extremely limited. The circular area 
of the screw presents resistance against an uplifting as well as downward 
force. 

Disk Piles. i — A circular "disk" of cast iron fastened to the lower end 
of an iron shaft and sunk by the water jet (see page 873) is the usual form 



* See Eng. News, Oct. 15, 1903. 

t See Trans. A. S. C. E.. Vol. VIII, page 227-37; Coney Island Pier. 



KINDS OF PILES. 



875 



of disk pile. The thickness of the cast disk may be, say, about one-third its 
diameter, the latter being proportioned to the required bearing area, and 
usually about 2 ft. for an ordinary shore-sand foundation. The diameter 
of the shaft is proportioned to the superimposed loading and may be equal 
to the thickness of the disk, more or less. Screw piles are more common 
than disk piles. 

Sand Piles. — Sand piles are formed by driving ordinary piles, pulling 
them, and immediately filling the hole with wet sand, well compacted in 
thin layers. Such piles if well constructed will usually have a greater 
bearing power than the wooden piles withdrawn, provided, of course, the 
ground is firm enough to give lateral support. 

Concrete Piles. — A concrete pile may be constructed in a similar manner 
to the sand pile, just described — differing from a concrete pier in that it 
requires lateral support from the soil, which the pier does not. But the 
usual form of concrete pile, in place, is either (1) a concrete core enclosed 
in an iron or steel shell, or (2) a steel reinforcement encased in concrete 
(similar to the ordinary concrete-steel column). In either case the piles 
may be driven with the drop- or the steam hammer; or they may be sunk 
(in sand) with the water jet, by a small metal tube centrally located in, 
and terminating at the point of, the pile; or the water jet may be used 
merely to assist the driving. Under this heading we will include only 
those piles encased in a metal shell, and then follow with a discussion of 
reinforced-concrete piles, under a separate heading. 




waferpipe. 
Telescopa shell in aboti 
d'lengfhs. 



P 

Spreader of Joints 
as<jb. 
Jef^ Dogg^S^Cast disk or shoe. 

Fig. 24. 



^ Water- Jet Concrete Piles.*- — ^Fig. 24 illustrates 
a simple method of constructing a concrete pile 
in situ and of sinking same by the water jet. 
Note that the telescopic shell (similar to a collapsi- 
ble pocket drinking cup) is assembled at the sur- 
face where the pile is to be sunk; the shoe, with 
central nozzle, is attached to the water pipe, 
and water, under pressure of 40 lbs. more or less, 
is forced through the nozzle into the sand; as 
fast as the telescopic sections sink they are filled 
with concrete, and the "spreaders" are introduced at the joints to prevent ' 
collapse. The process is simple and effective. The piles can be sunk to 
any practicable depth in sand, and they form a safe and permanent founda- 
tion. 

Metal-Shell Concrete Piles.'*' — ^The finished pile consists of a metal "shell" 
of say No. 20 sheet iron, driven into the soil and then filled with concrete. 
The shell is driven by inserting into it a tightly fitting iron "core" which 
may be considered as the "temporary pile" and which is driven in the 
usual manner by the drop or the steam hammer, and with 
the use of a follower to protect the head of the core from 
injury. After driving, the metal core is collapsed (made to 
occupy less diameter) and withdrawn, leaving the metal shell 
in place, which latter is now filled with concrete. The metal 
core consists essentially of three main parts, split longitudinally, 
the two outer "halves" being split by the th;rd, acting as a 
key or wedge between them. When the key is lowered the 
two sides are spread apart as in Fig. 25. When raised, the 
core collapses ready to be withdrawn. The diameter of the 
core may be 6 ins. at the point and, say. Id to 20 ins. at the 
butt. 

Reinforced-Concrete Piles. — ^These may be built in various 
sectional forms, as circular, (modified) triangular, (modified) 
square, rectangular (for sheet piling), etc. Figs. 26, 27, 28 and Fig. 25. 
29 show simple sections. 

Sometimes four more rods are placed intermediately between those shown 
in Fig. 29, making 8 reinforced rods instead of 4. For rectangular section, 
6 rods are often used — 3 on each side. 

Instead of rods or expanded metal, I-beams are sometimes used, 
especially for sheet piling where lateral stiffness is required. The I-beams 
may be in pairs, connected at intervals by riveted diaphragms. 



* Invented and patented by A. A. Raymond, Chicago, 111. 



876 



60.— FO UNDA TIONS. 



At the point of the pile the reinforcement rods are brought together, 
and may be banded with wire, or welded. At the top, the rods are imbedded 
in concrete; but after driving, the concrete may be picked away and the rods 
gxposed in order to bind with the new-laid concrete for foundations. 







Fig. 26. Fig. 27. Fig. 28. Fig. 29. 

Fig. 26. — Circular, tapering pile, reinforced with f"° and 1^° steel rods, 

united at bottom of pile. 
Fig. 27. — Cylindrical pile, reinforced with expanded metal. 
Fig. 28. — (Modified) triangular pile, reinforced with 1"° steel rods, and 

Y'° wire ties. 
Fig. 29. — "Square" pile, reinforced with 4 — 1"° steel rods with various 

arrangements of l"° wire ties. The central hole is for water jet. 

Concrete piles are made in moulds. They should be kept sufficiently 
Wet while curing, and away from the sun. In driving, a follower should be 
used to protect the concrete, at the head of the pile, from injury. 




Fig. 30. 

Open Caissons. — An "open" caisson is a water-tight box without a top, 
in which a masonry pier is built and sunk on to a prepared foundation. 
The sides are detachable from the bottom (see Fig. 30), and hence may be 
removed after the pier is sunk in place, and used in the construction of 
succeeding piers. Guide piles are usually driven to assist in sinking the 
caisson. 

The following rules will be found useful in proportioning the thickness 
of planking on the sides of the caisson, and subject to hydrostatic pressure 
(taking into consideration the deflection as well as the strength): Hard 
woods: For hydrostatic head of 36 ft. make thickness of planking in inches 
equal to the unsupported span in feet; for 4^ ft. head make the thickness 
one-half of the above; for intermediate heads make thickness directly pro- 
portional between the two. Soft woods: For hydrostatic head of 30 ft. 
make thickness of planking in inches equal to the unsupported span in feet; 
for 3| ft. head make the thickness one-half of the above; for intermediate 
heads make thickness directly proportional between the two. 



OPEN CAISSONS. PIERS. 877 

In proportioning the uprights, note that for deep caisson work they 
may be braced by horizontal struts from side to side or from the sides to 
the built masonry pier. 

For the construction of concrete piers in open caissons, the wooden forms 
are built inside and separate from the sides of the caisson — leaving a space 
between, all around. 

For large or deep piers the floor of the caisson may be of two or three 
courses of 12-inch timber; and the sides may be planked with two thick- 
nesses of planking (the inner course being laid diagonally), or they may be 
composed of crib timbers dovetailed or halved together and well drift- 
bolted. Guide piles are sometimes driven in pairs instead of in single line. 

Crib Piers. — A crib pier is simply a wooden crib (see page 869) or box 
constructed of logs or of squared timbers framed and bolted together, and 
sunk to a natural or a prepared foundation bed, by filling the chambers 
with gravel, rock, etc. A crib differs from a caisson in that the latter is 
supposed to be water-tight. The crib may or may not have a bottom. 
If it is sunk to a natural bed the bottom is omitted (which is the usual 
form) and the sides are projected downward a few feet below the inner 
bracing and chamber floors so as to cut into the bed and get a good stable 
bearing. The bottom of the sides of the crib is sometimes extended by 
blocking, to conform with the natural bed, especially if the latter is solid 
rock, in which case the crib is often bolted thereto. 

The crib may be of rectangular form (with vertical or inclined sides) or 
it may have a V-end up-stream, and perhaps down-stream also. The up- 
stream end should be protected with angle iron or steel rail, against floating 
ice and logs. In framing the sides, the timbers (say 12"xl2'0 may rest 
squarely on each other or be separated a few inches; and framed into the 
cross timbers, say every four feet apart. Interior longitudinal timbers are 
also framed in, about the same distance apart, forming vertical chambers 
about 4-ft. square. The bottoms of these chambers are planked, to hold 
the filling. All the timbers should be securely drift -bolted together. If the 
crib is designed to support a heavy superimposed load the chambers should 
be made smaller by increasing the number of longitudinal and transverse 
timbers. 

The crib is usually sunk directly on the bottom, if hard; but if soft, it 
is sometimes sunk on a prepared pile foundation, and the bottom protected 
from scour by a deposit of rip-rap. If the crib projects above low water it 
becomes merely a temporary structure as the timbers are exposed to rot. 
Crib piers are specially permanent when entirely submerged below low 
water mark, and when not subject to the attacks of the teredo (see page 360). 
As such, a crib may act as a sub-foundation on which may be constructed a 
concrete or stone masonry pier. 

Pile Piers. — A pile pier is a pile sub-foundation projected upward to 
support the superstructure direct. It is at best a temporary affair and 
lacks the lateral stability of masonry. Pile piers and abutments are used 
largely in every new country where timber abounds, for supporting railway 
and highway bridge spans, on account of low first cost and rapidity of 
construction. As Bridge Engineer of a western railroad the writer con- 
structed many of the bridges from end of track, as the latter was laid, 
causing little delay in the tracklaying, and utilizing the construction trains 
for the transportation of material to the bridge sites. 

A simple form of bridge pier consists of two or. more rows of piles (5 or 
more in a row), each row capped with a 12'''xl4" cap dapped ^ inch and 
drift -bolted with l''°x20'' drift bolts. On top of these caps, cross-caps are 
drift -bolted to receive the pedestals of the span. The inside rows of piles 
may be projected up-stream in the form of a V. The rows of piles should 
be sway -braced with 4''xl2'' planking, both longitudinally and trans- 
versely; secured at the ends, to piles and caps, with screw bolts; and 
double spiked at intersections with piles. The outsides of piers may be 
sheathed all around with 3-inch or 4-inch planking, laid longitudinally with 
butt joints. The nose of the pier may be driven with batter piles and 
sheathed with iron as a protection against ice and logs. Rip-rap may be 
deposited around the piles, to prevent scour; also pile clusters may be 
driven up-stream, as ice breakers or fenders. 

Tubular Piers. — Under this, heading the writer desires to bring to the 
attention of the reader a variety of piers which are tubular in form, com- 



878 50.— FOUND A TIONS. 

posed of various materials, sunk by various processes, and occupying 
various positions when in place. When we say "tubular in form" we mean 
that they are composed of a tubular "shell" which is lowered into position 
and filled with, say, concrete or some other equally serviceable material. 

Shape. — The circular section is the one most commonly used and 
hence if the section is uniform throughout, the pier is cylindrical in form; 
if the section decreases uniformly upward it is conical. Often two cylinders 
of different diameters (the smaller one above) are joined by a conical frus- 
tum. They are usually placed in pairs for supporting bridge trusses, being 
braced to each other by a web plate, or by horizontal and diagonal bracing. 
The center pier of a drawbridge rnay be one large circular cylinder or may 
consist of, say, 6 or 8 separate cylinders surrounding a central one of larger 
diameter, all braced rigidly together. When used singly for center piers of 
draw spans the oval or elliptical section is sometimes preferred. 

Materials. — ^Timber staves, say from 6" to 12'' thick and planed with 
sides radial have been constructed in the circular or "barrel" form similar 
to the sides of a water tank or a section of stave pipe. The bottom is pro- 
vided with an iron shoe if it is to be sunk into the soil. Instances are on 
record where a masonry shell, as of brick, has been used as a tubular pier 
and sunk into the ground to considerable depths. But the most commonly 
used materials are the metals — cast iron, wrought iron and steel. Cast-iron 
shells with metal 1-in. thick, more or less, and with flanges inside for bolting 
the sections together, present a smooth exterior surface for sinking, and are 
very durable. Wrought iron and steel are generally preferred. The metal 
is from i-in. to f-in. thick and the riveted tubes are composed of sections 
from 5 to 6 ft. in length. Both butt- and lap joints are used. The cylinders 
are stiffened at the top with an outside circular angle-iron and the top 
covered, when finished, with a plate. There is no reason why a reinforced- 
concrete shell should not prove economical under certain favorable con- 
ditions. 

Sinking in Place. — ^The method of placing or sinking the tubes, or their 
method of support after being placed, often determines the name of the 
pier. For instance, a metal cylinder encasing a cluster of piles is called a 
Gushing cylinder pier, "Gushing pier," or Gushing pile, named after the 
inventor of the system. If the same cylinder (or usually two cylinders) is 
set on a timber platform resting on piles, it is called a "platform pier," 
platform cylinder pier, or simply a cylinder pier. If it is sunk in the ground 
it may receive the name of cylinder pier, or "tubular pier." If sunk by the 
pneumatic process (compressed air) it is called a "pneumatic cylinder," 
pneumatic tube, or pneumatic pile. Sinking the cylinder in place, in the 
foundation bed, may be accomplished either by: (1) Weighting the tube 
with pig iron and excavating inside, whence it descends by overcoming the 
frictional resistance* on the sides; (2) Dredging, when in water, and loading 
as before; (3) Pneumatic Process, for great depth under water, excavating 
in a "working chamber" under compressed air, as with the pneumatic 
caisson. 

Gushing Piers. — Fig. 31 illustrates two styles of casing and two styles of 
piling ordinarily used for Gushing piers. The wrought -iron or steel cylinders 
(say j' to I" metal) are usually placed after the piles are driven and bolted 
together, but sometimes one of more bottom sections (5 to 6 ft. lengths each) 
are set up and riveted together in place and then the piles are driven inside 
of them. Gare must be used not to bulge the sides in driving. After the 
cylinders are set and connected with bracing (at least at the bottom) they 
are filled with concrete, deposited in layers and thoroughly tamped around 
the piles. Soft silt, logs, boulders, etc, should be removed (previously) 
from the bottom and the cylinders should rest on as firm a foundation bed 
as practicable. The horizontal bracing between the cylinders may be com- 
posed of two channels with pin connections to angles or bent plates riveted 
to the cylinders. They may be connected top and bottom, with tie plates 
or lattice bars, or be reinforced with plates. Two 10^' channels are common 
for small piers. Adjustable rods, singly or in pairs, are often used for the 
diagonal bracing, but stiff members are preferred. Howe truss bracing 



* The frictional resistance for cylinder piers may vary from about 
300 lbs. per sq. ft. in mud, up to as much as say 1500 lbs. per sq. ft. in 
gravel. 



CVSHING PIERS. CYLINDER PIERS, 



879 



has been used considerably in the West where timber is plentiful; that is, 
the diagonal braces are of timber, say 10" to 12", tied with horizontal rods, 
li"° and upward. The web brac- 
ing is usually sheathed on both 
sides with 2-in. planking, projecting 
a little beyond high and low water. 
One of the best forms of bracing is 
a web plate connected with the 
cylinders by vertical angles riveted 
thereto. The web is stiffened at in- 
tervals by horizontal angle stiffen- 
ers. Much of this description, espe- 
cially that relating to the bracing, 
will apply to Platform Piers, which 
will next be described. Plenty of 
riprap should be used around the 
piers to prevent scour. Ends of 
piles should be cut off at different 
elevations. 



Platform Cylinder Piers. — Plat- 
form piers are constructed by driv- 
ing two or more rows of piles, cap- 
ping them below low water mark 
with 12" timbers laid longitudinally 
with the pier, and drift -bolted to 
the piles; and then laying a plat- 
form of 6" to 12" timbers trans- 
versely and spiked to the caps. 
The cylinders are then placed on 
this platform and securely bolted 




fUN 

Fig. 31. 

thereto through holes in the bottom flange angles. Sometimes the 
a, a, (Fig. 32), are allowed to project upward through the platform 
the cylinders, thus forming a com- 
bined Platform and Gushing pier. The 
cylinders are filled with concrete and 
braced similar to the "Gushing piers" 
just described. For heavy masonry 
piers the platform consists of a "gril- 
lage" of timbers (say 12''xl2'0 usually 
three solid courses or more, laid at 
right angle, drift-bolted together and 
sunk on the piles. On this grillage the 
masonry is laid. 



piles 
into 




tvsl IvJ LvJ 



Elevation 



Pneumatic Cylinder Piers. — Instead 
of supporting cylinder piers by the 
Gushing process or on platforms, as 
per methods just described, they are 
often sunk to a firm foundation bed 
by excavating the material from the 
inside and below the bottom, as they descend, and 
heavily weighting them. If the tubes are sunk 
under water the material may be removed often 
by dredging, using an orange-peel bucket for this 
purpose. Where this can be done it will be found 
very economical. But for deep foundation work 
the pneumatic* process is generally used. 

Pneumatic Process. — In Fig. 33, W is the com- 
pressed-air working chamber, connected with the 
outer air by the air lock. When the workmen or 
materials pass through the lock into or out of the 
chamber, the doors a and h are worked like the 
gates of a canal lock, because the compressed air 



r 


UMj 


b" 


- 


-- 


-In 


) 


n 


" 


': 


m 




X (a) 1 


^_ ^-^ M 


3 N°) I 


Nfe^^JV' 




- 


... 


( 


=c- 


- 


-- 




^\K?L,i 



Plan 
Fig. 32. 



*By the pneumatic process is meant the plenum 
or com-pressed-air process. The vacuum, process is 
now seldom used. 




x^'ig. 33. 



880 



50.— FOUNDATIONS. 



in W is at a higher pressure than the atmospheric pressure. The compressed 
air, sand and water pumps are on the scow. The material excavated is 
raised by a windlass. When the air lock has been sunk to water level, 
a new section of cylinder is inserted and the air lock placed above it. 
Guide piles may be driven to guide the descending cylinders. 



Pneumatic Foundations. — ^The main essentials for the prosecution of 
deep foundation work iinder water, for large bridge piers, are: 

(1) The pneumatic caisson (an inverted, air-tight, "open" caisson), which 

forms the working chamber, and supports the masonry pier; 

(2) The crib, a cob-like (sometimes solid grillage) construction of timbers 

above the pneumatic caisson, really forming a part of it, and on 
which the masonry pier is built ; 

(3) The coffer-dam (sometimes omitted), built on top of the crib so that 

masonry can be laid dry even when below the water level; 

(4) The pneumatic tubes, consisting of air shafts, air locks, etc.; 

(5) The machinery scow, containing boilers, air compressors, engines, and 

dynamos for lighting; 

(6) The excavating tools, as picks, shovels, windlass for hoisting, etc. 

(7) The sand lift, for forcing out the sand; 

(8) The mud pump, for pumping out the mud. 

Fig. 34 illustrates the first four essentials, and Fig. 33 shows the arrange- 
ment of main shaft and air lock, enlarged. Note that the air lock occupies 
(Fig. 34) about a central position in the shaft, high enough to be out of 
danger from flooding, and low enough to be economical. After the pier is 
sunk to bedrock the shafts, as well as the working chamber, are filled solid 
with concrete. 

The main essentials will be discussed briefly in detail, as follows:* 



■Aux. Shaft 




The Pneumatic Caisson. — ^The caisson 
may be separate from the crib, as shown in 
Fig. 34, in which case the roof is supported 
by longitudinal and transverse trusses extend- 
ing through the working chamber (but not 
shown in the illustration). But the more 
modem method is the combined crib and 
caisson, by which means the crib becomes a 
part (sometimes the whole) of the truss sys- 
tem, to transfer the central weight, over the 

chamber, on to the cutting edges of the Working Chamber^ 

caisson, during the process of sinking. The p- o/i c r ' 

combined crib and caisson is typically repre- •^^^- ^^- separate Caisson, 
sented in Fig. 35, which is a general plan and longitudinal section of the 
coffer-dam and caisson for South Pier of Brooklyn tower foundation of 
the new East River Bridge (Williamsburg) New York City. Fig. 36 shows 
a vertical section of same, transverse to bridge axis. Figs. 37 to 40 
show details of caisson for North Pier, and Figs 41 to 43 show details of 
coffer-dam for either pier. 

The working chamber (Fig. 35) was 7 ft. high and divided by bulkheads. 
All seams were calked with two strands of oakum; the chamber was then 
lined with 3-in. plank, the joints (and spikes to fasten them) being treated 
as above, and then painted with white lead, making it air-tight. Fig. 40 
shows the roof plan of caisson. 

^ The Crib. — Where a crib is used above the caisson proper, it may be a 
solid grillage, or it may be divided into separate vertical compartments to 
be filled separately with concrete as the caisson is sunk; or the compart- 
ments may be staggered (offset) vertically or open so the concrete filling will 
form one monolithic mass. The accompanying illustrations (Figs. 36 to 
39) show an open crib of the latter type. 



* The illustrations of Williamsburg bridge foundations are adapted from 
the official working drawings. 



PNEUMATIC FOUNDATIONS. 



881 




rr 



Section pgrgllel to Bridqe Axis 
..ji^.^ 



i SeclionA-A l_.. Section C-C ,r:r-^. 



1 iLiLJ.. 




Section BB 



SecYionB'D 



Fig. 35. — Combined Crib and Caisson. 
(See pages 880 and 882.) 



882 



50.— FO UNDA TIONS. 



The Coffer-Dam. — ^The coffer-dam as applied to the pneumatic founda- 
tion, is really an open caisson (see page 876) resting on top of the crib; 
or if the crib is omitted, it rests directly on the pneumatic caisson. But 
sometimes the coffer-dam itself is omitted, as when the masonry is built 
directly on the crib or on the pneumatic caisson, and its top kept above high 
water as the pier sinks. If this is done, no coffer-dam is needed to shut out 
the water. But it is not always safe, advisable, or possible (economically) 
to construct the masonry with the rate of progress corresponding with the 
sinking of the pier, and hence the coffer-dam is employed. It is usually 
erected in sections, one above the other. Note in Fig. 35 the substi- 
tuted bracing against the masonry as the latter is built upward. Figs. 41. 
42 and 43 show the bracing in detail. 



(.i.S 




l^//M^Am 



Section tranbversetoBndqe.Axis 

Fig. 36. — Combined Crib and Caisson. (See page 880.) 
The Freezing Process. — Soft, flowing mud and quicksand (especially) 
are the most difficult materials to be encountered in sinking foundations 
and shafts, excavating for wells, or driving tunnels; and it has naturally 
occurred to a few inventors to devise means for freezing this material, 
somewhat beyond the area to be excavated (leaving frozen walls for tem- 
porary lateral support) , so the actual excavation can be made by ordinary 
methods, as with the pick and shovel. 

Any practical system consists essentially in driving a number of tubes 
into the ground around the proposed excavation, and a little beyond its 
outer limits. These are called the "freezing tubes" and may be, say, from 
4" to 10" in diameter. They are closed at the lower end and should pene- 
trate the full depth of the soft material. Inside of these tubes are fitted 
the "circulating pipes," about \" to \Y diam., with lower ends open, and 
extending practically to the bottom of the tubes. The circulating pipes are 
connected at their tops by a "circulating ring" into which the freezing 
liquid is pumped ; descending through the pipes into the large tubes where 
by absorbing the heat from the surrounding soil, the latter is frozen; then 
emerging from the tops of the tubes into the "collector rings" and reservoir. 
From the reservoir it passes to the refrigerating machine, and then to the 
pump, thus completing the cycle. Certain modifications of this process 
have been introduced, but the process in any form is seldom used. The 
circulating liquid may be a solution of calcium chloride or brine. Ammonia 
machines have been used with success, with magnesium chloride as the 
circulating medium. 



THE FREEZING PROCESS. 



883 



M/?!/Bz 



r**y)'Drift$Gftonecfri'ver)ihe<Kft 
>5f/vf:afae(y/nfer secfion 




Scmy&S. 

r'iS" , 

C5.head\ 

a^ya/Iy 

Z'aparf\ 



mw^^^^mm^M m^^^^ mmmm 










^^d"5feel/?(^5.enc/5up5e^^2"'3fee/fiod5'er7a^^^ 
eiL-6'5'M (£f/9'5"3fee//foc/5*-^-i^zr/8'/r5reeM5 

enc/si/p5ef,5pacecfa5on7r-cfn5^'er5e$ecf/OfT 
--Section parallel to Bridg e Axis^ 



x/f /ntarmo^inTa ^ Cjo^"^^^ Dovetail-" -Joiot in horiz - 



Oak 



^iim/0'/2 "i'*^""*^i^'f^ . - uorner uoveTou- -joint m nonz - 
^^' .O.v^ln.!^ubin. ^r^^a^'' 




i -Pl<?r) ^ Chambef ''*"'"' '"'' 
Fig. 37. — Open Crib Caisson. (See page 880.) 



884 



50.— FO UNDA TIONS. 




Section ond Elevationtransverseto Bridge Axis ^ shpwing Bulkhead 




Section showing Chamber-Struts 




Elevations and Sedbnof Bui khead '' ^^ .^|^ 

...^ ^^sS,. .^_ Hi 

Plon Of Bulkhead 
Fig. 38.— Details of Caisson. (See page 880.) 



PNEUMATIC CAISSONS. 



885 




— Notes on Oetglls ^ 

(RQIJiOnnBolfsfobermcvf 

fromonfiharybarlSbewifhoutffeixi 
incf^yifhb/untpcintrwghlymadi' 
encfmerp^wundedformoyeal! 
''j/vr/J edges. A/t Drift Bolfi to be 
'c/rivenjn ^8 'Wiam. holes, 
5cf&y8off5fobenn)n.dh\&t 
'n ftghflyfiffingho/e5,w/ff)6'£ffcm^ 
Ca5fHif5hers.C.5.orplain 05 shown 
B"5feelMstohayc{2''cfmi 
CasfPVashers. 
TMM'^flTimberlnmfffs, 



^ohes/zedfo uniform cf/'mens/m 
•hefons SJZ/na, 



7/na, 



Course, ancf/n Inner 5heaff)/h^ 
Course, fotepufon wifh TdS.tyef 



maferiaf. 

A//oufer5eam5fff/2'f2^/foC 

M/a//7jmbers.(2/7cf also a/f joints 

(^ Chamber l/hin^f?anf(s-fy he 

thoroughly caof/ec/m/hOafiu/Tr 

'andsefyeafwiffyP/fc/?. 



Scmvi 




U5ua/i]) 



'^ctipn showing ieaaggtVM ^Sfi^^ 
Fig. 39. — Details of Caisson. (See page 

.»._-«-.. — - — j9 .- — --- — — ..I)} 

— ■ — "BEg^ESSBFi/ 




.r'45 



A//Dnf/So//s ed:exceptA'hefeof/7erm5er}o/ed 
AIIScreyvBolb 45 "- ^ - ^ 



ieif?-.. 




f?oofTfmber.in every joints ^^ 3'^/2'Cai{/m 

^^'in every d'^ /Too fjmfxn 
/nmryjoinr 

Fig. 40. — Roof Plan of Caisson. (See page 880.) 



886 



PVei^ehemmW 



50.— FO UNDA TIONS. 




SectioDjS:/? parallel to Bridge Axie> 






deamBoanf-^ 



'39'- 



^.|j^ .Zironsfcr 

J) irgn^ tightening "Cotfer- 

j^ -Dam at Corners • 

. — Notes on Oetaib— 

ii^^^^'AI/fforWaf/VmbersfobesJzaf 
on45/<^e5;Jc/'nf5 iocomeaf/erm/fPosts 
mc/eoc/] Coi/rsefabreak-phfs with Course 



Centre 



t/ne 



enct fasfemc/iv/y/i'ijDr/ff Bo/f: 

_ Cro5S-5trut5 to/nferxcf/nsa/mp/ar?e,one 
tinebe//yconf/nuouiff]wu;/hinferjecf/onjf!e 
ct/ier//r?ebe//pf/gM/y'j!)uffec/'£?^a'n5h/7e . 
f/r5fanc/'f/5fKd'/og'eft?eronfcpmcfAof' 
fomw///?J'>/P'-^'p/anks,proper/y3p/^ai ^^^^^ 

Jo/'afsofCrossS/rufsfo come rear //jfersect/oos arcf/o^- 1 Plnn nf Ton nf rrrffpr-nnm 

//orJo/ntsoftpseofeac/isecf/onofCof^r-Damfotje 
toi'frecfJb/SeamBaar(/5Cff/(ycaoMyrk;'dt5eryv?^my/?Wd^^ 

Fig. 41.— Details of CofEerdam. (See pages 880, 882.) 



— ipr 



i 




3T" 



P 



COFFERDAMS. 



887 



/^" /^ocfs, nofvpsef- 



/f.^yv\ 



^A/7/^P/hef7oonn^-T&G. ^i^i^o' 




6ectioa4i4anci Elevafjbn tran^ver^e 'to Bridge Axis 






a 


^ 


'B/oc/r/nq 
Coffer^- 








^ 






Jii . 






3 












^ 






■•' 






SS ' ' 












1 



for_ 
Da/hPoff. 



1.1. ni- 
IJ3 — tjr 



y iW ^^J HJ 

Part Plon-of Top of Caia^on 

Fig. 42.— Details of Cofferdam. (See pages 880, 882.) 



888 



m.— FOUNDATIONS. 



if ra 




Detail of Ironsfor Anchor 
jng CofferDam to Caisson 

0/?e Pa/'rcppoiite each fine 
of/i^'Vnterior/rvnRod'i „ 




Detail of j 
foot of. Wall PosB^irans- 
verse side of Coffer-Dam. 
long/fucf/na/3/i/e s/i77j7ar 



Fig. 43. — Details of Cofferdam. (See pages 880, 882.) 



Masonry Piers.* — In sectional plan, masonry piers are designed largely 
to accommodate the superimposed loadings. Thus, for swing bridges the 
center pier would naturally be circular, octagonal, hexagonal or square — 
the circle being the most economical and the square the least. If the draw 
is center-bearing, the pier may be of solid masonry; if rim-bearing, it may 
consist of a circular shell to support the rim or track, and perhaps a central 
pier or core; the rim and core may be joined by steel struts, or concrete 
webs, radiating from the latter; or the radiating struts may be used with 
the central core omitted. But there are other considerations which affect, 
more or less, the shape of the center pier, namely, the sides of the draw 
openings for the passage of water craft should be straight, or in continuous 
line with the up-stream and down -stream arms of the draw rest ; where the 
water way is limited and the stream is swift, the pier should be designed 
to offer the least resistance to the flow, to ice and to logs; and lastly, for 
structural reasons, the sectional plan should be simple. 



Fig. 44. 



< 



> CJ 



Fig. 45. 



3 



Fig. 46. 



C 



■\ 



9^ 



Fig. 47. 



Fig. 48. 



Fig. 49. 



For ordinary river piers supporting the ends of spans, we naturally 
choose the rectangular section, or one of its modified forms. The modifica- 
tions are based on such sections as will offer but moderate resistance to the 
flow of the stream and at the same time economize in masonry. Figs. 44 
to 49 show various sectional plans of piers, from the rectangular to the 
double diamond. The right-hand ends are "up-stream." Note that Fig. 46 
shows two types of ends, namely, the semi-circular and the 45° pointed; 
also that the down-stream ends of Figs. 45, 46 and 47 maybe either square, 
or symmetrical with the up-stream ends. Figs. 48 and 49 illustrate the 
saving in masonry over Figs. 46 and 45, respectively, by comparing the full 



* For masonry abutments, see Sec. 25, Masonry, pages 436 and 437. 



MASONRY PIERS, 



889 



lines with the dotted (between the ends of the latter); and although not 
altogether pleasing in appearance they may be (and have been) used as 
concrete piers in replacing some existing cylinder piers, where the other 
forms above would have overloaded the pile foundation. Where concrete 
piers are built hollow, or cobbed (with cross walls) , they may generally be 
reinforced with steel rods. 

Summarizing in general, the rectangular type, Fig. 44, is suitable for a 
land pier; Fig. 46, with semicircular ends, for a land or shore pier; and 
Fig. 47, with circular arcs (a, b and c being equidistant), for a channel pier 
in the swiftest current. By combining 47 with 46 (semicircular ends), using 
the former type below high water and the latter type above, there is obtained 
a pier at once efficient, economical and graceful (see Fig. 50), and suitable 
for our deepest rivers and swiftest currents. 




Fig. 50. — Practical Type for High, River Piers. 

Contents of Piers by Prismoidal Formula. — Where the sides of the piers 
are battered in straight lines the average sectional area multiplied by the 
vertical height will give the cubic contents. The average sectional area is 
equal to 

\ (top area 4- 4 times middle area + bottom area) . 
For an ordinary masonry pier of rectangular 
cross-section. Fig. 51, let 

/= length of top of pier, under coping, in ft.; 

iy = width of top of pier, under coping, in ft.; 

/j = height of pier (between coping and footing), in 
ft.; 

6= batter of masonry (horizontal -^ vertical). 

h 




Fig. 51. 



Then, cubic contents in ieet=-^[wl^r i{w+bh){l+bh) + {w+2bh){l-\-2bh)'\. .{I) 



= h[wl +bh{l+w)+ ib%^ 
If the batter is 1 in 24, we have, since b = ^. 



Cubic contents in feet 



=hVi 



^n\wl+^-^±^ + 



h^ 



(2) 



(3) 



4X6 6X8X9J 

in yards = (above, divided by 3X9) (4) 

For small piers, the batter is usually 1 in 12, or | in 12; for large piers, 



1 in 24. 



890 50.— FOUNDATIONS. 

EXCERPTS AND REFERENCES. 

Foundations for the New Singer Building, New York City (By T. K. 

Thomson. Trans. A. S. C. E., Vol. LXIII). 

Pressures on Foundation Footings for the Walls of Buildings (Eng. 
News, Jan. 31, 1901). — Formulas by Chas. E. Greene and Frank T. Daniels. 

Experience With Foundations in Boston (By J. R. Worcester. Eng. 
News, Feb. 5, 1903). — Contains a formula for the bearing power of piles: 
S= (W — p)-^f: where 5 = area in sq. ft. of pile in contact with the earth, 
W = load in lbs. on pile. p = a, factor for bearing of pile ( = 5000 to 6000 lbs. 
for sand and gravel, and for silt),/ = a factor for friction of soil ( = 100 to 
300 lbs. per sq. ft. in soft material, 300 to 500 lbs, per sq. ft. in mixed 
material, 400 to 600 lbs. per sq. ft. in sand and gravel) ; p and / having been 
determined by experiment. 

A Novel Tilting Pile Driver (By J. H. Baer. Eng. News, Sept. 3, 1903) 
— Drawing and dimensions of driver. 

Concerning the Holding Power of Anchor Bolts (Eng. News, Jan. 5, 
1905). — References where data can be obtained. 

A Novel Water Jet for Driving Piles (By S. A. Jubb. Eng. News, 
May 4, 1905). — Illustrated. 

Design and Construction of High Bridge Piers of Reinforced Concrete 

(By W. M. Torrence. Eng. News, May 25, 1905).— Illustrated. 

Spread=Foundation of Reinforced=Concrete for a Six=Story Building 
(Eng. News, July 20, 1905). — Illustrated. 

Construction of Cofferdams (By T. P. Roberts. Paper, Engrs. Soc. 
West. Pa., May 23, 1905; Eng. News, Aug. 10, 1905). 

New Concrete Covering for Timber Piles in Teredo=Infested Waters 
(Eng. News, Jan. 4, 1906). — A pipe armor; illustrated. 

The Design of High Abutments (By W. M. Torrence. Eng. News, 
Jan. 11, 1906). — Illustrated; with quantities and costs. 

A Form for Applying Concrete Armoring to Timber Piles (Eng. News, 
May 24, 1906).— Illustrated. 

A Method of Manufacturing Reinforced=Concrete Piles by Rolling 

(A. C. Chenoweth. Eng. News, July 26, 1906). — Illustrated. "The cost of 
a pile 61 ft. long and 13 ins. in dia. is about $60. It is reinforced to carry 
its own weight in handling. A pile 30 ft. long could be made and driven 
for $1 per ft., so the price of a 60-ft. or 100-ft. pile would be no guide for 
estimating the cost of shorter length." 

Allowable Pressures on Deep Foundations (By E. L. Corthell. Eng. 
News, Dec. 20, 1906). 

Telescoping Leads for Pile Drivers (By H. P. Shoemaker. Eng. 
News, Nov. 14, 1907). — Illustrated. 

Cost of Small Concrete Piers (By J. H. Ryckman. Eng. Rec, Jan. 23, 
1909). 

Reinforced=Concrete Caissons: Their Development and Use for Break- 
waters, Piers and Revetments (By W. V. Judson. Eng. News, July 8. 
1909). — Illustrated: Fig. 2 shows method of computing stresses in steel in 
caisson (for breakwater) shown in Fig. 1. (Figs. 1 and 2 are not reproduced 
here.) 

Steel Sheeting and Sheet=Piling (By L. R. Gifford. Trans. A. S. C. E., 
Vol. LXIV., Sept., 1909). — Illustrations of various types, with discussions. 

The Design, Manufacture, Driving and Cost of Reinforced-^Concrete 
Piles (Eng. Rec, Mar. 27, 1909). — Two papers presented before the Boston 
Soc. of C. E., Sept. 16, 1908. 

Caisson Disease and Its Prevention (By Henry Japp. Trans. A. S. C. E., 
Vol. LXV., Dec, 1909). — Illustrations: Medical air-lock used in the East 
River tunnels of the Penna. Tunnel and Terminal R. R.; automatic constant- 
rate decompression valve; automatic constant-rate decompression and 
ventilating valve; air-lock with middle decompressing chamber. 

The Sixth Street Viaduct, Kansas City (By E. E. Howard. Trans. A. S. 
C. E., Vol. LXV., Dec, 1909). — Illustrations: Details of concrete pier No. 
2, Kaw River; general details of steel shoes of Kaw River bridge. 



MISCELLANEOUS DATA. 891 

Concrete Piles (By H. J. Cole. Trans. A. S. C. E., Vol. LXV., Dec, 
1909). — Illustrated. Cost comparisons. 

Laying Concrete Under Water on the Detroit River Tunnel (By Olaf 
Hoff. Paper, Nat'l Assn. of Cement Users, Feb. 21-26, 1910; Eng. News, 
Mar. 17, 1910). — Illustrations: section of tunnel; details of tremie scow. 

Substructure of Cantilever Bridge over the Ohio R. at Beaver, Pa., P. & 
L. E. R. R. (Eng. News, May 5, 1910). — Side elevation of bridge and detail 
plans of piers. 

Long=Time Tests on Concrete in Sea Water (By A. Poulson. Report, 
Fifth Congress of the International Assn. for Testing Materials, Copen- 
hagen, 1909; Eng. News, July 7, 1910). — Description of methods of making 
tests. Tests not completed. 

Chicago Rules for Measuring Excavation and Concrete Work (Eng. 
News, Nov. 3, 1910). — Rules formulated by a joint committee composed of 
15 members, five each from the Western Society of Engineers, the Chicago 
Architects' Business Association, and the Masons' and Contractors' Asso- 
ciation of Chicago; and were adopted by the associations in June, 1910. 
They are quite comprehensive and lengthy. 

Illustrations of Various Types of Foundations: — 

Description. Eng. News. 

Ties for securing concrete forms July 10, 1902. 

Design of concrete piers with metal shells (Cooper) Nov. 6, '02. 

Foundation piers for the Quebec Cantilever bridge Jan. 29, '03. 

Concrete abutment and parapet wall for a skew R. R. bridge Sept. 24, '03. 

Large concrete pier with a sheet pile and canvas coif erdam May 30, '05. 

A circular wooden caisson for pivot pier June 7, '06. 

Reinforced concrete caisson for dock-wall construction Jan. 17, '07. 

A concrete bridge abutment of T-section Feb. 14, '07. 

Sinking a shaft with steel and concrete lining, soft ground Feb. 21, '07. 

Concrete foundations in shifting ground May 21. '08. 

Reinforced-concrete caissons for breakwater at Algoma, Wis. Oct. 5, '08. 

Pile-and-cylinder founciations for a R. R. drawbridge June 10, '09. 

Pneumatic caissons (30' x 640, B. & O. R. R. bridge Nov. 18, '09. 

Shield method of sinking building footings, Chicago Nov. 18, '09. 

Concrete pile with cone, shell and removable steel core Dec. 16, '09. 

Foundations for McKinley Bridge across Mississippi River Jan. 6, '10. 

Concrete casings for protecting piles against teredo Jan. 13, '10. 

Saw, with guide bracket, for under-water piles June 16, '10. 

Depositing concrete under water, slope walls, Lachine canal Aug. 11, '10. 

Eng. Rec. 

Loading platform for foundation bearing tests July 10, '09. 

Forms for reinforced-concrete piles Dec. 11, '09. 

Pier 3 of the Boston terminal of the B. & A. R. R. Dec. 25, '09. 

Rein. -cone, piles for Evansville filter foundation ' Feb. 19, '10. 

Underpinning of 300-ton column on quicksand May 14, '10. 

Rein.-conc. abutments for double track R. R. bridge July 23, '10. 
165-ft. rein.-conc. R. R. bridge pier, Willamette Riv., PortlandAug. 6, '10. 

The north caisson of the Quebec bridge Oct. 1, '10. 

Foundations of St. Louis Municipal Bridge structure Oct. 15, '10. 

Plant for foundations for Municipal Building, New York Nov. 5, '10. 

Details of concrete shaft lining and working chamber Nov. 12, '10. 

River pier and caisson for Beaver Bridge, Pittsburg Nov. 19, '10. 

Placing a floating concrete crib for a lighthouse Nov. 19, '10. 

Sheet pile cofferdam for laying 48-in. gas main, New York City Dec. 3, '10. 

Caissons and method of sinking, Bankers Trust Co.'s. Bldg. Dec. 10, '10. 



51.— WHARVES, PIERS AND DOCKS. 

(References: Foundations, Sec. 50; Breakwaters, Sec. 52.) 

Definitions. — A wharf is essentially a platform structure projecting 
outward from the shore, and alongside of which water-crafts may be moored 
for the exchange of freight or passengers. The term "quay" is applied dis- 
tinctly to a wharf which skirts the shore, and runs about parallel with, and 
extends to no great distance beyond, the shore line. A pier, on the other 
hand, is a wharf which projects outward from the shore a considerable 
distance; is supported usually on piles or piers; and hence has "open 
waterways" beneath the platform. 

A Dock is an artificial receiving basin for water-craft — for loading, un- 
loading, repairing, etc. It need not necessarily be "closed." A common 
form of dock is the "open" basin between adjacent wharves or piers; thus, 
we speak of "docking" a vessel, or bringing her up alongside one of the 
wharves or piers. Where the rise and fall of the tide is excessive and would 
interfere with loading and unloading, "closed" docks may be used, as those 
at London and Liverpool. These are provided with gates which are opened 
only at full tide. It is but a step from the common "closed" dock to the 
Graving Dock.* 

Foundations. — One of the most important features to be considered in 
the construction of wharves and piers is the foundation. The kind of 
foundation most advisable to use will depend upon: (1) the uses for which 
the structure is designed; (2) the nature and character of the soil and 
foundation bed, and depth of same; (3) the currents, tide limits, and depths 
of water; (4) the restrictions by the Government, State (and City) as called 
for by the established — 

Pierhead and Bulkhead Lines. — Piers with open waterways may be 
constructed to the pierhead lines, while wharves of solid construction may 
be built only to the bulkhead lines. Where two sets of lines exist, one set 
established by the Government and the other set by the State, the set 
nearest the shore is supposed to govern. The U. S. Engineers have author- 
ity to interpret the true position of established Government harbor lines, 
and no encroachment is allowed beyond without the approval of the Sec'y 
of War. Such approval may often be obtained to meet certain exigencies 
in local conditions. 

Construction Methods. — Common methods of wharf construction inside 
of bulkhead lines are: (a) By sinking timber cribs filled with stone, around 
the sides (inside) of the wharf area, and then filling in behind with earth, 
perhaps by dredging from the outside, (b) By constructing wharf walls of 
stone or concrete masonry, instead of sinking cribs, and filling in behind 
them. Care should be taken to have these walls rest on a good sub-founda- 
tion or foundation bed, that is, either natural or artificially prepared, as 
they are really retaining walls of the most treacherous kind: the earth 
backing is saturated and has a flat angle of repose, and when the wharf is 
loaded the overturning force may be increased greatly; moreover, the 
resistance of the wall itself to overturning is decreased considerably when 

* A Graving Dock (commonly called a Dry -dock) is an (excavated) 
basin into which a vessel can be floated, the gates closed, the water forced 
out, and the hull exposed for inspection, repairs, cleaning, painting, etc. 
(See Paper No. 1016, Trans. A. S. C. E., June, 1906, entitled "A New Graving 
Dock at Nagasaki, Japan.") A Floating Dock (commonly called a Floating 
Dry-dock) is what its name implies and need not be defined. (See Paper 
No. 1042, Trans. A. S. C. E., June, 1907, entitled "The Naval Floating Dock 
— Its Advantages, Design and Construction," by Leonard M. Cox. Mr. 
Cox defines the "Marine Railway" and the "Lift Dock" as additional 
forms of Repair Docks.) 

892 



CONSTRUCTION METHODS. 893 

immersed in water. It is a good plan to back such walls with stone spalls 
or slag, before filling, (c) By any of the methods used in the construction 
of open piers, which will now be explained. 

Piers are usually of timber or concrete — at least the floors — supported 
on piling. The latter may be timber-, screw-, disk-, concrete- or cylinder-. 
Timber Piles are most commonly used. They are driven in rows (some of 
the piles are frequently driven slanting to give lateral stability) , and capped 
with say 12"xl2" timbers thoroughly drift -bolted to piles. The floor may 
be composed of one or more thicknesses of 3'' or 4" planking laid on say 
4''xl4'' stringers resting on the caps and drift -bolted or toe-nailed thereto. 
A guard from 5''x8'' to 10"xl2" surrounds the edge of the pier. Snubbing 
piles are driven where required near edge of pier, and allowed to project 
above the floor level. Fender piles, singly or in cluster, are driven usually 
at the corners of piers to receive the shock of vessels making landing. The 
piles of the pier are thoroughly sway -braced with S^'xlO'' to 4'''xl2'' planking. 
Where the soil is sandy, screw piles or disk piles are sometimes employed. 
The long iron pier at Coney Island is supported on disk piles with disks 24" 
dia and 9" thick, the wrought-iron shafts (tubes) being 8f" outside diameter. 
Such piles are calculated to support 5 tons or more per sq. ft. of disk area. 
They are sunk with the water jet. Where timber piles are subject to the 
attacks of the toredo nevalis* (see page 360) the expense of repairs is con- 
siderable, hence concrete piles and cylinder piles are often employed. 

Ferry Slips and Bridge Aprons. — A ferry slip is a dock for a ferry boat; 
the bridge or apron being an adjustable roadway, for rise and fall of tides, 
to and from the ferry in the slip. Probably the largest ferry boats in the 
world are those plying between San Francisco and Oakland, Cal. Some of 
the slips in New York City are designed for boats 250 ft. upward in length. 
The tremendous force with which these boats sometimes strike the slips in 
landing, especially during foggy weather, requires the latter to be constructed 
in the strongest manner compatible with the requisite elasticity, and im- 
provements are constantly being made in their design. Local conditions 
must be studied for each particular case, as those which fit one locality 
may not fit another. 

The throat or shore end of the slip is usually made to conform closely 
with the shape of the ferry boat for say i to ^ its length, and from thence 
it flares outward more or less to the mouth. Sometimes one wing only is 
extended beyond the middle in order to facilitate landing, under certain 
directions of wind and tide. 

The following illustrations, were prepared by the Dept. of Docks and 
Ferries, New York City, Chas. W. Staniford, Engineer-in-Chief, and were 
furnished the writer through the courtesy of Mr. S. W. Hoag, Jr., of the 
Engineering Department. 

39th Street Ferry, Manhattan: 

(a) Plans of Crib with Dolphin : 
Fig. 1. — General Plan of Outer End of West Pier with Dolphin. 
Figs. 2 to 7. — General Details of Crib. 

(b) Plans of Ferry Dolphin: 
Fig. 8. — Part Plan of Ferry Dolphin. 
Figs. 9 to 13. — General Detail of Ferry Dolphin. 

(c) Plans of Ferry Bridge. 
Fig. 14. — Part Plan and Cross-Section of Bridge. 
Figs. 15 to 17. — General Elevation and Details of Bridge. 



* The piling along the Seattle (Wash.) water front is often rendered 
unsafe by this sea-worm in two or three years' time, being almost completely 
honey-combed. 



894 



51,— WHARVES, PIERS AND DOCKS, 




FERRY CRIB AND DETAILS. 



895 




896 



51.— WHARVES, PIERS AND DOCKS. 



>i(;K'^i 




Fig. 8.— Part Plan of Ferry Dolphin. 
(For Details, see Figs. 9 to 13, on following page.) 



FERRY DOLPHIN AND DETAILS. 



897 



0' 5' 

i I I 1 1 I 1 ! ,1, 



3 Turns 




\ \ \ I 




Fig. 9.— Section A-B. 



Cr/b SOV— 



Fig. 10.— Section C-D. 



/C, 



--"--~-'>l 
^ 



k- lo'o"- 



1FE7F 









'^---^■^^ 






^F? g"'"^^^^^3^3#^^ 



i Unl , l i i i _|L^ , !i !i :^ 



m -i.--- g"--ir4.:^j^^^^I^^it^^ 






■Fish PJorf-es „ '' 



Wk/S' 



Fig. 11. — Part Side Elevation. 



0' 5' 10* 




Fig. 12.— Section E-F. 



Fig. 13. 



898 



51.— WHARVES, PIERS AND DOCKS, 



•Sot 6ui>jOC>y J.O uoi+OAai3 



irHT 



TITT 



rf^TT- 



TTZ 



IB 



-"^ " - . ■■ ^ ^1 



S 




7,^?/,^i" ^^-9^ ^a^-:yi:y/ j/iPOJ/>;,^/ H 

•;,^,^f-v r- -.""T>ll 







(For Elevation of Bridge and Details, see Figs. 15, 16 and 17, 

on following page.) 



FERRY BRIDGE AND DETAILS, 



899 



J3qqn^. 



5 ^ 



tmJh 






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IS 
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(For Plan of Bridge, see Fig. 14, on preceding page.) 



900 51.— WHARVES, PIERS AND DOCKS, 

EXCERPTS AND REFERENCES. 

The U. S. Steel Floating Dry=Dock for Cavite, Philippine Islands (By 

J. S. Schultz. Eng. News, Dec. 10, 1903). — Illustrated. 

Novel Steel Pier Construction at Lome, Africa ("Le Genie Civil" of 
July 15, 1905; Eng. News, May 24, 1906). — Illustrated structural details. 

The Terminal Station and Ferry House of D., L. & W. R. R., at 
Hoboken (By C. C. Hxirlbut. Eng. News, Sept. 20, 1906).— Sixteen illus- 
trations. 

Steamship Terminal With Concrete Pile Piers at Brunswick, Qa., 
A. & B. Ry. (Eng. News, Dec. 20, 1906).— Illustrated. 

Dock Walls at the Port of Koenigsberg, Prussia (Eng. News, Aug. 22, 
1907). — Illustrations showing methods of driving piles, and bracing. 

New Piers for Transatlantic Steamships, Chelsea Improvement, N. Y. 
City (Eng. News, Jan. 14, 1909). — ^Twelve illustrations and double-page 
insert. 

Light Reinforced=Concrete Wharf Construction, Madras Harbor, India 
(Eng. News, Nov. 11, 1909). — Construction comprises reinforced-concrete 
piles (reinforced with 1-in. rods) driven 8 ft. apart to a depth of 8 ft. below 
bottom and tied back to an anchor by means of 30-lb. old rails completely 
encased in concrete. Back of these piles are sunk reinforced-concrete slabs 
to a point below river bottom, the slabs acting as a retaining wall to hold 
back the earth. The tops of the piles are joined together by an arch con- 
struction topped by a coping of old rails. A 1 : 2 : 4 concrete was used for 
all the work. Illustrated. 

Reinforced Concrete Wharf of the United Fruit Company, Bocas del- 
Toro, Panama (By T. H. Barnes. Trans. A. S. C. E., Vol. LXVI., Mar., 
1910). — Illustrated, with cost data. 

Illustrations Useful for Reference: — 

Description. Eng. News. 

Standard car-ferry transfer bridge Dec. 19, 1901. 

Large ore dock of the C. & N.-W. Ry., Escanaba, Mich. July 30, '03. 

New graving dock at Kobe, Japan Sept, 24, '03. 

Concrete dry-dock at Kiel, Germany Dec. 3, '03. 

Details of Reinforced-concrete crib-work wharf May 26, '04. 

Details of standard pile pier, Dept. Docks and Ferries, N. Y. May 18, '05. 

Elevation and plan of fireproof wharf, Tampico, Mex. June 8, *05. 

Solid pier construction in Baltimore harbor July 19, '06. 

Plan and details of reinforced-concrete pier, Atlantic City July 26, '06. 

Reinforced-concrete retaining wall and quay April 20, '09. 

Design of Concrete Naval Dry -Dock, Pearl Harbor, Hawaii Dec. 9. '09. 

Rein. -cone, piers at U. S. Naval Station, Philippine Islands Dec. 15, '10. 

Eng. Rec. 

Walls, girders, ties, anchorage, etc., Baltimore piers May 8, '09. 

Details of ferry platforms and bridges, C. R. R. of N. J. May 22, '09. 

Sewells Point coal pier, Virginia Ry. Feb. 5, '10. 

Sea-Wall bulkheads in connection with streets and buildings Feb. 26, '10. 

Cross-section of deck of rein.-conc. deck of ore dock Nov. 12, '10. 



52.— BREAKWATERS. 



General Discussion. — This subject forms one of the most important in 
River and Karbor Improvements. The term "breakwater" is quite distinct 
from the so-called "reaction (curved) breakwater," which latter belongs 
rather under the head of Jetties (page 905) and might be termed a "break- 
water" jetty or "lee" jetty. 

A Breakwater is an arm-like construction projected in the water and 
designed to form an artificial harbor for sea-going crafts. The foundation 
is stone, rip-rap, gravel, etc., either deposited loose or sunk in timber cribs. 
Where the timber cribs are used they are, for permanent construction, 
projected upward only to within about 2 ft. of low water, thus forming a 
substructure on which the superstructure is built. The superstructure* is 
best constructed of stone- or concrete blocks weighing from 1 to 10 or 15 
tons; or of concrete deposited en masse. If the bottom is soft or silty, a 
trench should first be dredged on the line of the breakwater, rernoving all 
soft material liable to cause trouble by excessive settlement. This method 
will generally be more satisfactory than the one sometimes employed of 
using a gravel core in the breakwater and trusting permanent settlement 
to take place as the breakwater is built up. Although the latter method 
has proved successful in some instances, there are records of utter failure 
during the first heavy storm after completion of the work. 

For an excellent discussion of breakwater construction, see Paper No. 
971, Trans. A. S. C. E., June, 1904, entitled "The Breakwater at Buffalo, 
N. Y.," by Emile Low. See also Paper No. 15 of Transactions, Vol. IV — 
Part A, page 324, entitled "The Delaware, Sandy Bay and San Pedro 
Breakwaters," by C. H. McKinstry. (^Considerable valuable data on break- 
water construction is contained in Vol. VIII — Part 4, Annual Report (1904) 
of Chief of Engineers, War Department (U. S.). The following illustrations 
are from these sources: 



*/?-t^ 



-iT^i^ 




Fig. 1.— Buffalo (N. Y.) Breakwater. 
Types of Breakwaters. — Fig. 1 is a plan of minimum cross-section for 
replacing the decayed timber superstructure of a portion of the old Buffalo 
breakwater, with a new superstructure of concrete and stone "shell con- 
struction," The maximum cross-section is 32 ft. on the harbor side and 
22 ft. on the lake side, making a total width of 78 ft. 







Fig. 2. — Sandy Bay (Mass.) Breakwater. 
Fig. 2 is a plan of the Sandy Bay (Mass.) breakwater, showing the section 
adopted in 1902. 

* The use of timber cribs projecting above high water is becoming more 
and more obsolete. But see Fig. 4. 

901 



902 



52.—BREA KWA TERS, 



Fig. 3 is a cross-section of the Delaware Bay outer breakwater, con- 
structed, 1897-1901. The tqtal length of superstructure is 7960 ft., costing 






Sea Slope 
Elev.O 




Sand and Mud 



Cross Section 



Sand and Mud 



Fig. 3. — Delaware Bay Outer Breakwater. 

$39.43 per lin. ft.; total length of substructure, 8040 ft., costing SI 78. 17 
per lin. ft. 



d'P/ahk 



MCAN Lake Levo.. 




Fig. 4. — Oswego (N. Y.) Outer Breakwater. 
Fig. 4 is a cross-section of the Oswego (N. Y.) outer breakwater (timber 
crib), constructed, 1884-1900. 

Averages Per Lineal Foot for Figs. 3 and 4. 



Materials. 



Volumes 
Cu.Yds. 



Cross-section above mean low water. , 
Cross-section below mean low water. . 
Total cross-section above sea bottom 

Rubble stone 

Capping stone 

Cost of superstructure 

Cost of substructure 

Total cost 

(Approximate voids: rubble, 39%; 
capping stone, 10%.) 



20.37 

108.30 

128.67 

120.14 

8.53 



Air Wt. 
Tons. 



33.31 
150.51 
183.82 
167.00 

16.82 



Section. 
Sq. Ft. 



550 

2924 

3474 

93.37% 

6.63% 



Cost. 



$197.69 

19.91 

39.43 

178.17 

217.60 



Cross-section above mean lake level. . 
Cross-section below mean lake level. . 
Total cross-section above lake bottom 

Timber 

Stone 

Cost of old structure 

Cost of foundation 

Cost of new superstructure 

Total cost 2970 ft 

Total cost 570 ft 



7.5 
29.81 
37.31 

7.44 
28.20 



7.4 
29.66 
37.06 

1.02 
32.71 



202.25 
805.00 
1007.25 
21.0% 
79.0% 



$120.04 

18.87 

25.18 

145.18 

164.05 



Notable Breakwaters. — ^The following table was compiled in 1903, under 
the direction of Maj. Theo. A. Bingham, Corps of Eng'rs, U. S. A. 



STATISTICS OF BREAKWATERS, 



903 



T 



n 

t 


ong sea slope. 

ong sea slope; does not include cost of 
quarrying stone; done by convict labor, 
uilt of slag; price paid for removal of slag 
covered cost of construction. 
sa slope long'l ; great saving of material ; 
concrete, blocks 13 cu. yds. each, 
oncrete blocks, 13 cu. yds.= 20 tons each. 


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[ chalk, firm, was leveled off. 
f rubble mound. 

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904 52.— BREAKWATERS. 

EXCERPTS AND REFERENCES. 

The Sea Wall of La Punta, Havana (By W. M. Black. Eng. News, 
Nov. 14, 1901). — Illustrated: Fig. 1 shows section through steps of seawall; 
Fig. 3 shows section of concrete toe with projecting stones to check run of 
waves. The article gives the composition of the various concretes used. 

The Materials for the Concrete of the Buffalo Breakwater (By Emile 
Low. Eng. News, Sept. 11, 1902). — The gravel and sand were obtained 
from the bed of the Niagara River by means of a so-called "sand-sucker," 
described as follows: The vessel consists of a wooden hull 132 ft. long, 
30.2 ft. beam and 7.2 ft. depth. The propelling machinery consists of a 
double non-condensing engine with a steam cylinder 14'' dia. and 16'' stroke. 
The boiler is allowed to carry 30 lbs. steam pressure. At the bow is located 
a centrifugal pump, driven by two direct-connected engines with cylinders 
9" dia. and 9" stroke. The suction pipe is 12" dia., and is also a discharge 
pipe. Located on the deck of the scow is a large wooden box, 85' long, 
24' 9" wide and 3' 10" deep, divided into two compartments by wooden 
bulkheads 4" thick. The capacity of the box is 325 cu. yds. The water 
charged with gravel and sand is pumped from the river bed (generally 12' 
deep) and flows into the flume, with screens of J" wire spaced y apart in 
frames ll"x 24". Five tables are given showing various properties of the 
aggregates, and data regarding the manufactured concrete blocks. 

Wave Action in Relation to Engineering Structures (By D.D. Gaillard. 
Professional Paper No. 31 of the Corps of Engrs., U. S. A.; Eng. News, 
Feb. 23, 1905). — Paper deals with: Definitions and theory; Height and 
Length of Waves; Reduction in Height of Waves on Passing into a Closed 
Harbor; Velocity of Waves; Per Cent of Wave Abo ve Water Level ; Depth 
in which Waves Break; Dynamometer Tests of Wave Force; Comparison 
of Theoretical Wave Force; etc. Tables and formulas are given. 

Reinforced=Concrete Caissons for Breakwaters (By W. V. Judson. Eng. 
News, July 8, 1909). — Illustrated: plan of caisson and method of computing 
stresses. Estimated cost of Algoma breakwater, if built of stone-filled 
wooden cribs, placed on pile foundation and capped with a standard con- 
crete superstructure (the cheapest permanent form) was $105.18 per lin. 
ft. Estimate for caisson breakwater was $103.74. Actual cost of caisson 
breakwater was $75. 67 -f $2. 62 per lin. ft. 

Illustrations of Breakwaters, for Reference: — 

Description. Eng. News. 

Plans of crib breakwater at Welland Canal entrance May 15, 1902. 

Plans of concrete breakwater at Buffalo (T. W. Symons) May 29, '02. 

Section and views of Cleveland breakwater Mar. 23, '07. 

Concrete breakwater at Harbor Beach, Mich. Mar. 28, '07. 

Sections of breakwaters in fishery harbors of Scotland; Cost Dec. 22, '10. 



53.— JETTIES. 



General Discussion. — At the mouths of navigable rivers where there is 
cross-current, whether the river empties into another or into the sea, sand 
bars are liable to form, shoaling the water and obstructing navigation. 
The cost of dredging deep-water channels through these bars and main- 
taining them is sometimes enormous, hence jetties are often constructed to 
reduce this annual expense. 

Jetties are structures designed to change the shape and velocity of the 
moving volume of water so that it will do the work, in part at least, of 
deepening the channel. The volume is contracted laterally and deepened 
vertically, and the velocity is increased, thereby cutting out the channel 
and carrying the material further on, some of it out to sea. 

"Twin" jetties are formed by two jetty arms converging. One is called 
the "windward" jetty and the other the "lee" jetty. The windward jetty 
is placed on the side of the channel on which the sand-drift predominates; 
the "lee" jetty, on the other side. As twin jetties are very expensive it is 
customary on large projects to build one jetty first, watch the effect of 
scouring of channel, shifting of sand bars, etc., and then with this addi- 
tional data plan the second arm. In selecting the single jetty, sometimes 
the windward- and sometimes the lee- jetty is chosen. The argument in 
favor of the latter is that the channel is practically confined from further 
movement "leeward," as the sand drifts from the "windward"* and is swept 
on or carried away by the current; whereas with the single windward 
jetty the channel may_ fluctuate in position instead of remaining permanent 
and deep. Each particular case requires a special solution. 

The "reaction breakwater," so-called, is a single S-shaped jetty de- 
signed to increase the scour by creating a centrifugal force to the current, 
thereby narrowing the volume laterally and increasing the velocity, similar 
to the natural winding cuttings in river beds. The effect of this type of jetty 
has not fully been demonstrated, and will be watched with much interest. 

Jetty Construction. — Most of the jetties now built are of rock fill, that 
is, rubble stone dumped from scows or trains. The trains are run out on 
temporary pile trestles constructed along the line of the jetty. The material 
is paid for by the yard or ton. Brush and rock are sometimes used but the 
use of brush has practically given way to rock alone, especially on large 
work, on account of the action of the teredo and liability to unequal settlement. 

Small jetties and bank protections of streams are often constructed by 
driving two rows of piles, facing them on the inside with planking, bolting 
and bracing the rows together, and filling the space between with brush 
and rock. The brush is often tied together in bundles or fascines (see Fig. 1). 
The bottom layer is placed on the bed 

of the stream transversely to the di- /if'J?" >j 

rection of the jetty, and on these K""' 'llZlI SSZ^^Illl^l^,^^^^^ 
are placed other fascines laying longi- ^^'^^^^;=^ ^^^'i^^^_^ fe— ' '^^ 

tudinally between the piles. Between ■ <^ == i~^.a E g = fJ 1 wrf^zi — u* -^ 

the fascines and the bottom of the 

side planking, t boxes filled with -p- i i? • 

rock are sunk between the piles; and ^'^^- •^* rascme. 

the balance of the space above, between the planking, is filled with rock, 

forming a pile crib. This type is often used for the protection of bridge 

abutments, and frequently as real jetties in deepening the channel. 

EXCERPTS AND REFERENCES. 

The Improvement of the Entrance to Cumberland Sound, Georgia 
and Florida (By J. H. Bacon. Eng. News, May 12, 1903). — Shows general 
plans of the jetty-work improvement; table of official record of contract 
work; and table of sand movement at Cumberland Sound. 

Complicated Reinforced=Concrete Jetty=Head, Thames Haven, Eng- 
land (Eng. News, April 22, 1909).— Illustrated. 

* The terms "windward" and "lee" relate to the sand drift and not to 
the direction of the wind, although they are frequently in the same direction. 

t The side planking consists of plank spiked or bolted to the inside of 
piles and laid as far below low water as practicable. 

905 



^ 



54.— EARTHWORK.* 

Uncertain Cost. — In any engineering work the chance of an accurate 
estimate of the cost diminishes as earthwork becomes the important item; 
for in the purchase of materials and suppHes a fairly correct estimate may 
be had in advance, but where the labor factor enters largely in a direct way 
great uncertainty exists. Especially is this true where the character of 
work to be performed, as in grubbing, clearing, and "earthwork," is proble- 
matical, the quality of labor uncertain, and the rate of wages unstable. 

Before making an estimate the engineer should consider: (1) the kind 
and quality of material to be handled, by digging test pits and by boring; 
(2) the most approved method of doing the work; (3) the availability of 
good contractors, superintendents, and foremen; (4) the quality and price 
of labor. These will be considered in the order mentioned. 

Kind and Quality of Material. — No contract should be let or work 
started before full information as to the character of the material has been 
obtained. The cost of pits and borings is merely incidental. Not only are 
they a practical guarantee of the correctness of the estimates but they give 
timely information necessary for the pruchase and proper disposition on 
the work of the necessary tools and machinery. The writer has seen steam 
shovels delivered where nothing short of dynamite could be used, and the 
expense of such changes and incidental delays is enormous. For trench 
work, test pits will generally determine in advance whether shoring plank 
will be required, so they can be ordered in time. One of the first things to 
look for is the material "hard-pan," "cemented gravel" or "glacial drift," 
as it is variously termed. When not classified it is a source of contention. 
Lying beneath the soil, unseen, it is often from 3 to 5 times as expensive 
to remove as is the material above, especially in trench work where it can- 
not be loosened economically or handled with the steam shovel. It comes 
between the earth and the loose rock or the sand-rock classifications and 
should be mentioned in specifications. Stiff clay is sometimes as hard to 
work as many of the hard-pans, and some idea should be had of the propor- 
tionate amount of this material. If it is large the estimate of cost of "earth" 
should be raised accordingly. Loam is about the easiest material to shovel 
by hand, while sand and gravel are more difficult. ^ Loose and solid rock 
are considered under the next subject heading (Section 55). Sandrock is a 
partially formed sandstone. It may be in various degrees of hardness, and 
should receive a separate classification. 

Approved Methods of Handling. — Space will permit only of the briefest 
mention of the methods employed, simply to recall them to the attention 
of the reader. 

Clearing and Grubbing. — After the trees are cut down the stumps are 
usually blown out with giant powder (No. 2). An effective method, how- 
ever, which is sometimes employed, is to snatch them out with a donkey 
engine. This has been done economically on railway right-of-way in the 
State of Washington, where the stumpage is thick. Quite a common 
method is to twist them out by using horses at the end of a horizontal 
lever chained rigidly to the stump; but only small stumps can be removed 
in this manner. The method of burning stumps is a slow process and not 
generally resorted to on engineering work. It takes many years for stumps 
to rot. The brush-hook is very serviceable in cutting brush, especially 
small willows and alders. Grubbing the roots of trees is best accomplished 
with the grubbing-hoe or mattock. Strange to say, estimates for clearing 
and grubbing are almost universally too low. The price is usually per 
acre, or sometimes per station on railroad work. In reservoir work for 
domestic water supply the grubbing must be done most carefully in order 
that the top soil may be removed unobstructed. Under railroad fills the 



* For earthwork Tables, see Sec. 59, Railroads, page 1017, etc. 

906 



METHODS OF HANDLING. fl07 

stumps are often left standing, unless the depth of fill is small. For trench 
work, as for pipe-laying, it is best that the grubbing and excavation con- 
tracts go together — to the same party. 

Loosening Earth. — The team with driver, and plow with holder, are the 
most economical outfit generally; but four or six horses, with two men at 
the plow, are often used on railroad work especially if the material is hard. 
The proverbial "six-mule-team" requirement often denotes the upper limit 
of common earth classification. Hard-pan may be plowed generally with 
such an outfit but not always economically and it may be cheaper to use 
powder (coarse, black). For narrow trench work and for shallow bank ex- 
cavation in such material, the common pick is used. 

Loading and Conveying.— ¥ or widening and deepening large railroad 
cuts, the steam shovel stands pre-eminent. The size of the shovel (capacity 
of bucket, dipper, or scoop) should be gaged to the depth of cut or rather to 
the amount of material which can be loaded on the cars from one position, 
as "moving up" is expensive. A shallow cut would call for a pony shovel, 
and if very shallow, hand shoveling might be cheaper. A cu.-yd. bucket is a 
good size for an average cut. In considering the use of any (steam) plant 
the interest on cost, the depreciation, the operating expenses, and the 
repairs must be proportioned to the total yardage, hence the greater the 
yardage the more expensive may be the outfit, provided of course that it 
is sufficiently economical while in operation. Errors are often made in 
installing too complete a plant for the work; and also in the other direction. 
The material is usually loaded in trains of self-dumping cars operated by 
locomotive. If the delivery track to the steam shovel is on a proper grade, 
say about 2%, the cars may be operated by gravity in making-up the 
loaded trains. If fiat cars are used they may be unloaded cheaply with a 
plow drawn the full length of train, or sluiced by hydraulic jet. 

In taking out long sand cuts, where the material can be hand shoveled 
without loosening, a good outfit consists of small dump-cars (side-dumping) 
operated in train on a narrow gage track. There is usually more or less of 
a down grade from cut to fill, and this gravity power may be assisted by 
horses which draw the empty cars back. 

Four-wheeled carts with removable slat bottoms, and two-wheeled 
dump carts are used in excavating cellars where the material is removed from 
the site, but they have practically been crowded out by the wheel and drag 
scraper in ordinary railroad grading. The drag scraper is specially efficient 
for side-wasting from cuts and for short leads generally; the wheel scraper, for 
longer leads. The Fresno scraper is popular in the West and is considered 
one of the most efficient scrapers on the market. Wheelbarrows are used 
in narrow, muddy places. 

Numerous conveyor systems have been employed for side-waste from 
large, deep cuttings. The plane rubber belt and the movable-bucket types 
have not yet demonstrated their practicability. The combined hoist and 
conveyor* are generally employed in deep sewer excavations, large buckets 
being used. Steel cableways are sometimes used in mining and river-bottom 
excavation for dams and foundations; also drag buckets. 




Fig. 1. 

The excavation of the Chicago Drainage Canal developed types of the 
movable crane and movable bridge conveyors unique in their magnitude. 
They were simply steel structures for conveying loaded cars from the canal 
trench out over the spoil banks for dumping. The first named was a canti- 
lever, the second a simple span, placed transversely to the canal and resting 
on tracks parallel with the canal (see Fig. 1). 

* The Carson trenching machine is of this type. See "Cable Hoist- 
Conveyors" for general use on large work; published by The Trenton Iron 
Co. 



908 5i.— EARTHWORK, 

There are many graders, trench machines and excavators on the market, 
many of which are greatly overrated in efficiency and capacity. Some of 
them work well in loose soils but are useless in the harder materials liable 
to be encountered. Caution should be exercised in their selection. 

Under favorable conditions where there is plenty of water, excavation 
and fill by hydraulic method is sometimes the cheapest. The favorable 
conditions are a gravity supply of water obtained with little expense, the 
right kind of material as sand or fine gravel, sufficient grade from cut to 
fill, and a short conveying distance. The material is washed from the natural 
bank by concentrating a stream from a nozzle, or giant, and is carried in 
a sluice box under a constant stream of running water. The required 
grade of the sluice may be as great as 8 or 10 per cent for very coarse ma- 
terial. Earth dams constructed in this manner are plentiful in California. 
Part of the water front at Tacoma, Wash., has been filled by washing down 
the steep bank and sluicing the material into the bay. If there is no gravity 
supply and pumping is required the chances of the hydraulic method being 
the cheapest are greatly lessened. 

One other method which may be touched upon in this connection is 
by hydraulic dredging. By the use of a rotary cutter and a suction pump 
the fine material from the bottom is sucked up and forced into a sheet-iron 
pipe leading to the shore. Filling operations along our water fronts bear 
witness to the efficiency of this method. Material is frequently delivered 
one-half mile from the dredge, and sometimes to a much greater distance. 

Superintendence. — Some years ago a certain engineer known to the 
writer entered into a term contract as Chief Engineer with a contracting 
firm at a stated salary and a certain percentage of the profits. One of the 
first contracts secured was for the construction of headworks and a 30-mile 
pipe line for water works in one of our prominent western cities. The 
amount of the contract was over $900,000.00. The estimate was based on 
prevailing prices, the prices for material being guaranteed in most cases. 
A profit of 25% was added to all estimated costs, including materials fur- 
nished. The Treasurer* volunteered to go up and superintend the work, 
as it would be a nice outing for him. They had other contracts on hand 
but this was the largest, and the profit on the steel pipe material alone, 
delivered, was about $50,000.00. The wood-stave pipe construction was 
under an experienced man. The Treasurer was a fair book-keeper. When 
.the work was about half completed the City Engineer became desperate 
and called at the main office. "Fully a hundred thousand dollars had been 
wasted on the line. Men all over the line working without sufficient tools; 
one gang of 12 with only one shovel between them, and no pick. Material 
from the trench being wasted improperly, that had to be used again for 
back-fill." Upon investigation it was found that these were facts; that the 
office force was up to its ears in book-keeping; that there was no active 
superintendent or line boss; that not one of the foremen had been furnished 
with a profile of the line or had been instructed where to make his spoil 
banks; and after the pipe had been laid, extra borrow pits had to be opened 
for back-fill! It proved to be an expensive "outing," and the loss fell most 
heavily upon the Treasurer himself who was the principal owner. Here is 
a case where a good superintendent could have saved the company ten 
times his salary. 

Good foremen keep track of how much each man is doing, know what 
he ought to do, and whether he is doing a day's work. It is a mistake to 
drive men who are resting occasionally, as the one who "keeps moving" 
may not be doing half as much work. Poor foremen can wipe out the profits 
in a contract even although they may appear to be "hustlers." 

Labor. — The common labor problem is a difficult one to solve. Generally, 
"white" labor is the most profitable but it is hard to secure, exclusively, 
on large work. Italian and Chinese come next in order. A method com- 
monly employed with some contractors is to select one or two good men in 
each gang, secretly pay them from 25 to 50 cents more per day and let 
them "set the pace." As a further incentive a small bonus is sometimes 
offered to the other men if a certain amount of work is accomplished. In 



* There was considerable rivalry between the Treasurer and the Secre- 
tary as to who should "boss" the job, which was acknowledged to have been 
the best one ever landed on the Pacific Coast up to that time. 



SUPERINTENDJENCE, SHRINKAGE. 909 

estimating the cost of earthwork the labor cost is the main item. It may 
be set down as a fixed rule that, with the same class of men the cost of 
work increases more rapidly than the increase in wage, because under 
increased pay demanded by "good times" the men will actually do less 
work per day; they will not work as steadily; and there will be more 
changes in the working force, preventing thorough organization. Thus, if 
a piece of work cost A dollars with labor at $1 per day, it might cost 2 A 
dollars with the same labor at $1.75 per day, the extra cost, iA, being 
enough to reduce materially the contractor's profit. Hence, estimates based 
on different labor costs should be used with caution. 

Earth Embankment. — Dry earth embankment for railroad and other 
similar work is universally carried up with slope of If horizontal to 1 verti- 
cal? earth dams and reservoir embankments, much flatter, as 2 to 1, 2^ to 1, 
or 3 to 1. The latter are built in horizontal layers and usually rolled, 
while the former are built largely by side and end dumping. To illustrate 
a principle, the best reservoir embankments are composed of fine gravel 
with enough clay to form a good binder, being puddled by sheep or cattle 
tramping over it during construction, and sprinkled with a moderate 
amount of water. "Spike" rollers are an imitation of the above. Grooved 
rollers are better than plane rollers. 

Shrinkage of Earth, — Earth in place swells when loosened, but under 
the action of pressure and moisture it tends to recede to its original volume 
and, in some cases, to lesser volume than it previously occupied. The 
latter is often illustrated in laying water pipe: A trench is dug, pipe is laid, 
and the back-filling is done with thorough wetting and tamping. Often, 
material has to be borrowed to fill the trench, notwithstanding that the pipe 
itself has displaced a small portion of it. 

The subject of shrinkage has received much attention from various 
writers, and has provoked considerable discussion, especially during^ the 
past fifty years. The variance in the difi^erent conclusions drawn is of 
course due mainly to the lack of definite and careful experiments and ex- 
perimental data, and partly to incorrect analysis. The special problems 
relating to railway embankments are : 

(1) What quantities in "cut" (unmeasured) shall be assumed to be 

equivalent to the volume of the material after being placed in 
"fill" (and measured therein for final estimate) ? 

(2) What shall be the tentative cross-section of a fill so that when prac- 

tically final settlement has obtained, the lines of the fill or em- 
bankment shall be those of another cross-section, predetermined? 

In discussing these questions a great many factors have to be considered, 
some of which are the following: 

What are Earths? Earths are mainly metallic oxides which are prac- 
tically insoluble in water, as alumina, glucina, yttria, thorina, etc. — the 
alkaline earths, as baryta, lime, magnesia, strontia, etc., being somewhat 
soluble in water. Chemically speaking, earths are dry, inodorous, un- 
inflammable substances, fusible only at extremely high temperatures. 

What are Soils? Soils are earths, composed of the broken and weathered 
rock of the earth's crust, mixed with varying proportions of organic matter 
called humus, either decomposed or in a state of decomposition. Sedentary 
soils are those soils which rest upon the parent rock from which they were 
formed. Transported soils are those soils which have been removed and 
deposited elsewhere by the action of water, glaciers, or wind; if by wind, 
they are usually termed Aeolian soils. 

The Density of Soil in its natural bed depends upon its composition, 
formation, and method of deposition; also upon the action of the vegetable 
and animal life in the soil, in connection with the action of the percolating 
waters and of frost. Thus, heavy soils, as hard clay and cemented gravel, 
deposited by water or glacial action, under pressure, will be very dense, 
while light soils, as volcanic ash, deposited dry by the wind, will naturally 
have less density. But these light soils may become dense through the 
water and glacial action above described; while the heavy surface soils, so 
formed, may in turn become light and porous through the action of the 
ground water — dissolving certain of the chemical elements and giving them 
up to the plants or carrying them away, or carrying away (perhaps into 
the deeper soils), the small (or even large) suspended particles themselves; 



910 H,— EARTHWORK. 

and the voids thus created being further increased by the action of frost, 
tending to swell the soil. Hence it is seen that soils may have different 
densities, even if of the same composition, when lying in their natural beds. 

The Effect of Water on excavated soils is to settle them, mechanically, 
and to make them more compact * with the notable exception of clay, which 
swells when moistened and shrinks again when dried. Thus, with most 
soils the water will deposit the finely suspended particles of matter into the 
voids of the coarser material, decrease the friction, and settle it into a 
denser mass; while with clay the admixture of water produces a colloidal 
state, causing the mass to expand. 

The Effect of Temperature on soil- or earth embankment is mainly the 
effect of drying out the water, or of producing frost action; the direct ex- 
pansion or contraction due to temperature, being extremely slight, and 
negligible. 

The Effect of Vertical Compression or Jarring on an earth embankment, 
is a downward tendency to settlement and a lateral tendency to expansion. 
This statement as to lateral expansion does not of course take into consider- 
ation any side-surface slides of the mass, or wash of slopes from rain. 

The Effect of Stirring or Puddling is to make the soil or earth denser by 
decreasing the percentage of voids. The following experiment was made 
with coarse sand, which had all passed through a i-in. sieve (16 meshes to 
the inch) : A box of one cu. ft. capacity was filled with the sand, then jarred 
to settle it, and again filled to the top. Water was then poured in, filling 
all the voids in the sand, the amount of water used being 0.342 cu. ft., thus 
measuring the voids in the sand as 34.2%. The wet sand was then thorough- 
ly stirred, and settled in the box to 82.5% of its original volume, the voids 
in the stirred sand being therefore 20.2% of the final volume. 

In the following discussion the term earth will be considered to include 
soil. 

Swellage (w) of Earth takes place, usually, when it is first dug. This is 
due to the loosening of the material, thus increasing the voids. The more 
thoroughly it is loosened the greater will be the swellage. But it can im- 
mediately be Compressed or compacted to its original volume, or be made 
to Shrink below its original volume, if water and sufficient pressure are 
applied. The Percentage of Swellage is the percentage of increase in volume 
of the loose material excavated, based on the original volume of the material 
in situ. 

Compression (k) of Earth takes place, more or less, in forming any 
embankment; that is, the loose material is compacted somewhat, even by 
its own weight, while being placed. • A low embankment formed by shovel 
work, or by dumping from a cableway, or from a train supported on a 
trestle, especially during dry weather, would naturally show little compres- 
sion during the short period of time required in its construction; while the 
material forming a high embankment might be compressed appreciably 
under the same conditions of construction, owing to the increased weight of 
the bank and the element of time, both of which are important factors. 
If the track were supported directly on the embankment itself, or the 
material delivered in wheelbarrows, or especially if carts or scrapers were 
used, the compression would be increased; while if tlje material were spread 
in thin layers and rolled with a heavy roller, the compression would be 
much greater. The Percentage of Compression is the percentage of re- 
duction of volume in placing the material in the embankment, based on the 
Volume of the loose material after being excavated. 

Contraction (c) of Earth in embankment continues, perhaps, indefinitely ; 
but the rate of contraction, under constant conditions, decreases with time. 
That is to say, under the same conditions the rate of contraction decreases 
as the material in the embankment becomes more dense. The rate of 
contraction, however, may be increased by pressure, by jarring or shaking, 
and usually by moisture or sprinkling. Thus we have as factors tending to 
contract the material in the embankment: The pressure of the embankment 
itself, the pressure and jarring due to moving trains, and the sprinkling due 
to rainfall; added to this, is the wash of the slopes which, for our immediate 
purpose, will be included under the present heading. The Percentage of 
Contraction in Embankment is the percentage of decrease in volume at any 
stated time after the work is completed, based on the volume of the com- 
pacted materials as placed in embankment. 

Shrinkage (s) of Earth is also measured in per cent, and is the ratio of the 
loss in volume of earth (measured) in embankment at any specified time, 



SHRINKAGE DATA. 



9U 



based on the original volume which the same material occupied before being 
excavated (that is, in situ). 

Shrinkage may be obtained approximately by the following formula: 

Notation (from preceding discussion): 

w = Swellage % = ratio of increase in volume of loose material excavated, to 
volume of same material before being excavated; 

k = Compression •%=ratio of decrease in volume of placed material in 
embankment, to volume of same material loose after being ex- 
cavated ; 

c = Contraction % = ratio of decrease in volume of material in embankment 
(at any specified time), to volume of same material when placed in 
embankment ; 

s = Shrinkage % = ratio of decrease in volume of material in embankment 
(at any specified time), to volume of same material before being 
excavated. 

Formulas: 

Then, Shrinkage s^l-(l+w)(l-k)(l-c) (1) 

And, when measured immediately embankment is completed, 

s=l-(l-hw)(l-k) (2) 

Many wrong conclusions arise in current discussions on shrinkage of 
earth, partly from the erroneous assumption of some writers that the 
shrinkage is based on the volume of earth in embankment or fill, instead of 
on the original volume in excavation. 

Tables 1, 2 and 3, following, give the approximate values of (1 +«£/), 
(1—^), (1— c) and 5, in Formula (1). They are intended merely as a 
guide, covering average and extreme ranges, and in no way absolute, as 
will become apparent from the greatly varying conditions of construction 
previously discussed. 



1. — Approximate Values op (l+w) to be Used in Formula (1). 

(w = Swellage.) 



Class. 



Material. 

Blasted rock, large masses, Irregular 

Broken rock, smaller masses, as for riprapplng 

Crushed trap, granite and harder rocks, run of crusher. 
Crushed limestone, sandstone and softer rocks, run of 

crusher 

Quartz rock crushed line — loose 

Limestone crushed to fine grains — loose 

Glacial drift, cemented gravel, clay-and-gravel, extra 

dense 

Cemented gravel, hard — well loosened 

Cemented gravel, muck, compact hard-pan, av 

Clay-and-gravel, ordinary — well loosened 

Clay-sand-gravel mixture, average 

Sand-and-gravel, compact — well loosened 

Loam, sandy loam, average 

Sand or gravel, ordinary 



Voids. 



(14-tt?). 



.47 


1.80 


.43 


1.75 


.41 


1.70 


.39 


1.65 


.37 


1.60 


.35 


1.55 


♦(.33) 


1.50 


*(.31) 


1.45 


♦(.29) 


1.40 


♦f.26) 


1.35 


*(.23) 


1.30 


*(.20) 


1.25 


*(.17) 


1.20 


*(.13) 


1.15 



^Additional voids due to loosening of material. 



912 5i,— EARTHWORK, 



-Approximate Values op (1—^) to be Used in Formula (1). 
{k = Compression.) 



Class, 



Material, and Method of Placing in Embankment. 



O-k). 



10 



11 



12 



13 



Blasted rock, large masses 

Broken rock as for riprap : (a) Carelessly dumped 

(b) More carefully placed 

Crushed trap, granite and the harder rocks: 

(a) Loosely placed 

(b) Thoroughly shaken in transportation. . 

(c) Thoroughly rolled 

Crushed limestone, sandstone and the softer rocks: 

(a) Loosely placed 

(b) Thoroughly shaken in transportation . . 

(c) Thoroughly rolled 

Quartz rock crushed to sand, loose 

Limestone crushed to fine grains, loose 

Glacial drift, cemented gravel, clay-and-gravel, extremely dense: 

(a) Cableways or wheelbarrows used, dry weather, low embankment. 

(b) Carts or scrapers used, some rain, medium embankment 

(c) Material thoroughly sprinkled and rolled, high embankment . . 
Cemented gravel, or clay-and-sand, very hard, well loosened: 

(a) Cableways or wheelbarrows used, dry weather, low embankment 

(b) Carts or scrapers used, some rain, medium embankment 

(c) Material thoroughly sprinkled and rolled, high embankment . . . 
Cemented gravel, muck, and compact hard-pan, average:. 

(a) Cableways or wheelbarrows used, dry weather, low embankment. 

(b) Carts or scrapers used, some rain, medium embankment 

(c) Material thoroughly sprinkled and rolled, high embankment . . . 
Clay-and-gravel, ordinary, well loosened: 

(a) Cableways or wheelbarrows used, dry weather, low embankment 

(b) Carts or scrapers used, some rain, medium embankment 

(c) Material thoroughly sprinkled and rolled, high embankment . . . 
Clay-sand-gravel mixture, average: 

(a) Cableways or wheelbarrows used, dry weather, low embankment 

(b) Carts or scrapers used, some rain, medium embankment 

(c) Material thoroughly sprinkled and rolled, high embankment . . 
Sand-and-gravel, compact: 

(a) Cableways or wheelbarrows used, dry weather, low embankment 

(b) Carts or scrapers used, some rain, medium embankment 

(c) Material thoroughly sprinkled and rolled, high embankment . . . 
Loam, sandy loam, average: 

(a) Cableways or wheelbarrows used, dry weather, low embankment. 

(b) Carts or scrapers used, some rain, medium embankment 

(c) Material thoroughly sprinkled and rolled, high embankment . 
Sand or gravel, ordinary: 

(a) Cableways or wheelbarrows used, dry weather, low embankment. 

(b) Carts or scrapers used, some rain, medium embankment 

(c) Material thoroughly sprinkled and rolled, high embankment . . 



1.00 

1.00 

.80 

1.00 
93 

.77 

1.00 
.91 

.74 



.80 
70 
60 

.80 
.71 
,60 

,80 

,72 
,60 

,80 
,73 
,62 

.80 
.75 
.65 

.85 
.80 
.67 

.90 
.80 
.60 

.95 
.90 
.70 



SHRINKAGE DATA, 



913 



-Approximate Values op (1-c) to be Used in Formula (1). 
(c = Contraction.) 





Material, and Method of Placing. 


One Year After 

Embankment is 

Completed. 


After Final 
Settlement of 
Embankment. 




Weather— 


Weather — 


O 


1 


4J 


fl^ 


4^ 
1 


ft 


4i 




4^ 


7 
8 
r 

10 

11 

12 
13 
14 


Glac. drift, cem.-grav., etc., ex. den: 

(a) Cableways, dry, low emb'km't. 

(b) Carts, scrapers, rain, med. embk 

(c) Material rolled wet, high embk . 
Cem. grav., clay-and-sand, very hard 

(a) Cableways, dry, low emb'km't . 

(b) Carts, scrapers, rain, med. embk 

(c) Material rolled wet, high embk . 
Cem. grav., muck, hard-pan, average: 

(a) Cableways, dry, low emb'km't . 

(b) Carts, scrapers, rain, med. embk 

(c) Material rolled wet, high embk . 
Clay-and-grav., ord., well loosened: 

(a) Cableways, dry, low emb'km't 

(b) Carts, scrapers, rain, med. embk 

(c) Material rolled wet, high embk . 
Clay-sand-gravel mixture, average : 

(a) Cableways, dry, low emb'km't . 

(b) Carts, scrapers, rain, med. embk 

(c) Material rolled wet, high embk . 
Sand-and-gravel, compact: 

(a) Cableways, dry, low emb'km't . 

(b) Carts, scrapers, rain, med. embk 

(c) Material rolled wet, high embk . 
Loam, sandy loam, average: 

(a) Cableways, dry, low emb'km't. 

(b) Carts, scrapers, rain, med. embk 

(c) Material rolled wet, high embk . 
Sand or gravel, ordinary: 

(a) Cableways, dry, low emb'km't . 

(b) Carts, scrapers, rain, med. embk 

(c) Material rolled wet, high embk . 


(1-c) 

.92 

.95 

1.00 

.92 

.95 

1.00 

.92 

.95 

1.00 

.92 
.95 
.99 

.92 
.96 
.99 

.93 

.96 

1.00 

.92 
.95 
.99 

.96 

.98 

1.00 


(1-C) 

.84 

.90 

1.00 

.84 

.90 

1.00 

.85 

.90 

1.00 

.86 
.91 
.99 

.87 

.91 

99 

.88 

.92 

1.00 

.84 
.90 
.98 

.92 
.96 
.99 


*s 

-.10 
.00 
.10 

-.07 
.02 
.13 

-.03 
.04 
.16 

.01 

.06 
.17 

.04 
,06 
.16 

.02 
.04 
.16 

.01 
.09 
.29 

-.05 

-.02 

.19 


-.01 
.05 
.10 

.03 
.07 
.13 

.05 
.09 
.16 

.07 
.10 
.17 

.10 
.11 
.16 

.07 
.08 
.16 

.09 
.14 
.29 

.00 
.00 
.20 


(1-c) 

.90 
.94 
.99 

.90 
.94 
.99 

.90 
.94 
.99 

- .90 
.94 
.99 

.91 
.94 
.99 

.92 

.95 
.99 

.90 
.94 
.98 

.95 
.97 
.99 


(1-C) 

.80 
.88 
.99 

.80 
.88 
.99 

.80 
.88 
.99 

.84 
.89 
.99 

.84 
.88 
.99 

.85 
.89 
.99 

.80 
.87 
.97 

.90 

.95 
.98 


*s 

-.08 
.01 
.11 

- 04 

.03 

14 

-.01 
.05 
.17 

.03 

.07 
.18 

.05 
.08 
.17 

.04 
.05 
.17 

.03 

.10 

30 

-.03 

-.01 

.20 


*s 

.04 
.08 
.11 

.07 
.09 
.14 

.10 
.11 
.17 

.09 

.12 
.18 

.13 
.14 

.17 

.10 
.11 
.17 

.14 
.17 
.30 

.02 
.01 
.21 



Experiments on the Shrinkage of Earth are scattered, and most of them 
are unsatisfactory because the data are not complete. In all experiments 
on shrinkage the prismoidal formula should be used in the calculations, 
otherwise errors amounting to from 3 to 10 per cent would be quite usual in 
some cases, and enough to destroy the value of the results obtained. 

If it becomes necessary to base an estimate on the quantity in embank- 
ment, whereas the contract price was based on the quantity in excavation, 
experiments should be made on the shrinkage of the material. f The usual 
method of making these experim^ents is to dig a rectangular-shaped pit, or 
trench, and then place the excavated material back in the pit or trench 
under conditions similar to those in making the embankment. The shrink- 
age per cent will then be the difference between the original volume of the 
pit and the final volume of the placed material, divided by the original 
volume of the pit. 

Elwood Morris, about two-thirds of a century ago, made some measure- 
ments on a large scale, on the shrinkage of earth. "The embankments were 
formed in layers by carts and scrapers, and one winter intervened between 



*Values of 5 are actual shrinkage averages from original volume in 
excavation. Where the minus sign appears before these values, it indicates 
a minus shrinkage, or actual increase in volume by that amount. 

tMany contracts on railroad work stipulate that when material is 
measured in embankment a certain percentage, say 10%, shall be added 
for shrinkage. 



914 



5i.— EARTHWORK, 



the commencement and completion of the work." The results of the 
measurements, involving nearly 44,000 cubic yards, are: Shrinkage of 
yellow clayey soil, 9.25 to 10.15 per cent; shrinkage of light sandy soil, 12.93 
per cent ; mean average shrinkage, 10.3 per cent. Based on these experiments 
some authors give the "shrinkage of earth" as 9 to 13 per cent, while others 
have widened the range to 8 to 15 percent. Most railroads use about 10 
per cent as an average working basis. 

The Am. Ry. Eng. & M, of W. Assn. Committee Report for 1907 recom- 
mends shrinkage allowance for both height and width in new banks, 20 
replies favoring this, while two favored allowance for vertical shrinkage 
only, and two for horizontal shrinkage only. The following shrinkage 
values were recommended: 

Suggested by 
Members. 

Black dirt, trestle filling 7 to 30% 

Black dirt, raising under traffic 4 to 20% 

Clay, trestle filling 5 to 30% 

Clay, raising under traffic 2 to 20% 

Sand, trestle filling 3 to 15% 

Sand, raising under traffic 2 to 15% 

In building the Tabeaud Dam, 1900-1902, near Jackson, Cal., tests of 
the earth material used showed the following average weights per cu. ft.: 

Dust dry soil (angle of repose 36°) 84.0 lbs. 

Soil fully saturated, 52% moisture (angle of repose 23°) 101 . 7 lbs. 

Natural bank soil, 19% moisture (angle of repose 44°) 116.5 lbs. 

Delivered from wagons, moist and loose (showing swellage of 52%) 76 . 6 lbs. 
Loose dirt from dam, shaken down and measure struck (swellage 

45.6%) ... 80.0 

Material in dam, 38% gravel and grit, thoroughly sprinkled and 

rolled (showing shrinkage of 12 . 4%) 133.0 

Experiments made by Mr. D. C. Henny, on earth material for the Cold 
Springs Dam, Umatilla Irrigation Project, Oregon, show the following results: 



Recommended 

by Committee. 

15% 

5% 

10% 

5% 

6% 

5% 



lbs. 
lbs. 





Specific Gravity. 


Percentage Voids. 


an 
p 


u 


* Sample. 


Constit- 
uents. 


Mass. 




Dry. 


Wet 
Rammed. 




Loose. 


Compact. 


hi 


A. Surface soil 

B. Fine subsoil 

C. Gravel 


2.52 
2.65 
2.90 
2.66 
2.93 

2.64 
2.83 
2.90 
2.87 
2.83 
2.91 
2.84 


1.41 
1.55 
1.91 
0.94 
1.57 

1.75 
1.76 
1.91 
1.95 
2.01 
2.04 
1.88 


59 
54 
42 

74 
55 

50 
47 
42 
41 

47 
47 
41 


49 
43 
37 
68 
49 

39 
41 
33 
35 
43 
40 
35 


44 
45 
34 
65 
46 

38 
39 
34 
32 
29 
30 
34 


3.9 
4.0 
9.2 
5.5 
1.5 

3.0 


.023 

025 

.559 


D. Volcanic ash . . . 

E. Coarse subsoil 

Mixtures: 
B C 

75% 25%.... 
67% 33%.... 
50% 50%.... 
33% 67%.... 
25% 75%.... 
20% 80%.... 
15% 85% 


.010 
.079 

.066 


14.7 
20.0 
21.0 
76.7 
18.0 


.076 
.076 
.086 
.091 
.102 



* Sample A is the 12-in. surface soil in the bottom lands, being darker 
in color than the deeper-lying soil, but containing a sensibly greater quantity 
of organic matter, as roots and vegetable fibers. Sample B is taken from 
1 to 4 ft. below the surface, being slightly lighter in color than A, and with 
less vegetable matter. Sample C is coarse sand gravel and from the steep 
side-hill, the mass of the material being a coarse sand or fine gravel with a 
considerable proportion of gravel that would be retained on a one-inch 
mesh; also occasional large cobbles, and a scantiness of fine sand. Sample 
D is volcanic ash, almost pure white in color; when in place it appears to be 
slightly indurated. Sample E is a coarse soil, coarser in appearance than 
A and B but otherwise greatly resembling them; it lies from 1 to 4 ft. below 
the surface. Samples A, B and E all contain a considerable proportion of 
volcanic ash and vegetable matter, and are fairly representative of soils in 
that section of the West. tMassachusetts State Board of Health standard. 



SHRINKAGE DATA, 



915 



Experiments made by Mr. Emery Sudler. on soils for use in construc- 
tion of earth dam for water-works reservoir, Baltimore, Md,, show the 
following results: 



Properties or Condition of Material. 



Yellow to Brown 
Clay, from layer 
2-ft. to 4-ft. thick. 



Soft, rotten 
Serpentine Rock, 
below the clay. 



Weight in lbs. 
per cubic 
foot when 
moist 



In place 

Loose 

Compressed (in 4-in. layer in 6- 
in. test pipe, at about 150 

lbs. per sq. in.) 

Volume in cubic feet correspond- "1 Loose 

ing to one cubic foot in place . . J Compressed 
Absorption by weight 




108. 
74.5 



129.3 
1.45 
.84 
.007 



This table shows that when loosened the clay swelled 50% and the 
rotten rock swelled 45%; after being compressed the loose clay showed a 
compression of 40.7% and the loose rotten rock a compression of 42.4%; 
while the ultimate shrinkage of the material in place when compressed, 
was: clay, 11.9% and rotten rock, 16.5%. 

Shrinkage in Volume = Vertical Shrinkage. — If each particle in an earth 
embankment " shrinks " vertically (not laterally) how will the settlement of 
the top of an embankment compare with the " shrinkage " in volume? 



— - t -— -^ 




Fig. 2. 

Solution.— Let the full lines, Pig. 2, represent the original fill, and the 
dotted lines the top and slopes after vertical "shrinkage." Then will d -v- h 
represent the ratio of vertical " shrinkage," as well as " shrinkage " in volume. 
For 



Final volume 



=(k 






h-d 



1- 



Original volume ^" ' 2 '" 2 h ^ h 

Earthwork is always measured in excavation where possible; but it 
often happens that a contractor will start in on a borrow-pit before it is 
cross-sectioned. For this and other reasons it is sometimes necessary or 
advisable to measure up the embankments as a basis for payment. Con- 
siderable judgment must therefore be exercised in the question of "shrink- 
age." • 

Performance of Work. — ^The following items are selected and digested 
from the columns of Engineering-Contracting,'*' and give what may be con- 
sidered fair averages of good work under the conditions named. For 
detailed information see original articles. References are made to the files 
of Eng'r-Contr. in the following manner; thus, E.-C, (date, page). 

Sewer Trench in stiff clay, wet and soft in spots, but tough digging; at West 
Allis (near Milwaukee), Wis.; by "Buckeye Traction Ditcher," a machine with 
buckets on the periphery of a large wheel, operated by steam, costing about $4600.00, 
and manufactured by the Van Buren, Heck & Marvin Co., of Findlay, O. Estimated 
performance Gess than actual) 900 lin. ft. of trench 2 ft. wide by 7 ft. deep in 10 hrs. 
at cost of 5 cts. per cu. yd., including all items of pay roll, fuel, interest and depre- 
ciation.— ^.-C. Jan., 1906. p. 7. 



* Published weekly by the Myron C. Clark Publishing Co., Chicago, III. 



916 H.— EARTHWORK. 

New York Subway. Earth excavation in a typical section of about h mile, 
Brooklyn Extension: Excavation proper (labor 1.60, materials and plant 0.32, 
power 0.02, dump charges at 60 cts. per load 0.25), $2.19 per cu. yd.; bracing and 
sheeting (labor 0.78, materials and plant 0.37), $1.15 per cu. yd.; pumping and 
draining (labor 0.01, materials and plant 0.01, power 0.01), $0.03 per cu. yd.; 
bridges and barricades (labor 0.10, materials and plant 0.14), $0.24 per cu. yd.; 
backfilling Oabor), $0.01 per cu. yd. Grand total. $3.62 per cu. yd. — E.-C, Feb., 
1906, p. 30. 

Panama Canal. Cost per cu. yd. of mixed excavation (96052 c. y. hard rock, 
254252 c. y. soft rock, 3 91340 c. y. earth), for a total of 741644 cu. yds., between 
July 1, 1904, and June 30, 1905: July, 65.4 c., Aug., 50.6 c. Sept., 57.3 c., Oct. 54.1c., 
Nov. 50.1 c, Dec. 52.8 c, Jan. 47.8 c, Feb. 46.5 c. Mar. 43.3 c, Apr. 52.5 c. May 
83.8 c, June 102.7 c. Excavators and steam shovels were used. Roughly, the 
above costs include up-keep, depreciation, etc., of plant. — E.-C, Feb., 1906, p. 44. 

Intercepting Sewers, Chicago. (1) Excavating with Ennis type derrick, equipped 
with a 1 cu. yd. Haywood orange-peel bucket, an 8i" by 10" double-drum hoisting 
engine and 60-ft. boom. Cost, exclusive of wear and tear of machinery, 6.6 cts per 
cu. yd. (2) Other methods were used also. — E.-C, Apr., 1906, p. y6. 

R. R. Excavation with Elevating Grader. 7 examples, by D. J. Hauer. Ma- 
terial, average earth when dry. Machine built by the Nat'l Drill and Mfg. Co. of 
New York City. Wagons, drop bottom patent dump, of 2-yds. capacity. (1) R. R. 
cut, 400 ft. long, 45 ft. wide on top. Cost Goading 0.130. handling 0.111, dumping 
0.041, water boy 0.001, foreman 0.012), 29.5 cts, per cu. yd. Av. output. 206 cu. 
yds. per day; lead, 400 ft. (2) R. R. cut about 1400 ft. long, 20 ft. wide and 2 ft. 
deep. Cost (loading 0.067. handling 0.078, dumping 0.011, water boy 0.002, foreman 
0.007), 16.5 cts per cu. yd. Av. capacity, 3 80 cu. yds. per day; lead 1000 ft. (3) R. 
R. cut, 30 ft. roadbed for double track. Cost Ooad. 0.046, haul. 0.072, dump. 0.016. 
water 0.001, foreman 0.005), 14 cts. per cu. yd. Max. output at top of cut, 510 cu. 
yds. per day; av. output, 300 cu. yds; lead 600 ft. (4) R. R. cut for double track. 
Cost (load. 0.108, haul. 0.149, dump. 0.019, water 0.003, foreman 0.010), 28.9 cts. 
per cu. yd. Av. output, 284 cu. yds. per day; lead, 700 ft. (5) R. R. cut. Cost 
a. 0.061, h. 0.077, d. 0.018. w. 0.002, f. 0.006), 16.4 cts. per cu. yd. Av. output, 417 
cu. yds. per day; lead, 500 ft. (6) R. R. borrow-pit. Cost 0- 0.098, h. 0.094, d. 
0.049, w. 0.003. f. 0.009), 25.3 cts. per cu. yd. Av. output, 260 cu. yds. per day; 
could have been increased if more wagons had been used. Lead, 500 ft. (7) R. R. 
borrow-pit. Cost (1.0.153, h. 0.260, d. 0.050, w. 0.002, f. 0.015). 48 cts per cu. yd. 
Av. output, 167 cu. yds. per day; material hauled 1700 ft., management inferior. — 
E.-C, Apr.. 1906, p. 102. 

Steam Shovel Work on Ann Arbor R. R. In 1895. Cost figures cover loading, 
hauling and placing under track, but make no allowance for rental of plant, loco- 
motive or cars, nor for depreciation of plant. Labor, $1.1 5. Cost per cu. yd.: Sand, 
7.22 to 13.88 cts.; sand (very light face), 17.24 cts.; sand (all work lowering by 
hand charged against this cut). 25.44 cts.; sand Oight face). 13.25 cts.; quicksand. 
13.98 and 15.95 cts.; gravel, 8.93 and 14.37 cts.; gravel Gong haul). 19.81 cts.; 
clay, 9.60 to 14.01 cts.; clay (hard pan), 17.65 cts.; sand and gravel. 6.48 and 8.58 
cts.; sand and gravel (very light face), 17.31 cts.; sand and clay. 10.49 cts. — E.-C, 
May 30, 1906. p. 151. 

Sewer Tunnel at Cleveland, Using Hydraulic Shield. Sewer, 13^ ft. dia., built 
of four rings of No. 1 shale brick laid in Portland cem. mortar. Shield, 16i ft. dia.. 
4 ft. long, of I" metal, and weight about 16 tons; upper half provided with follower 
7 ft. long, of f " steel, bolted to shield. Shield pushed forward by 1 2 hydraulic jacks. 
5" dia. and 26" long. Water led to jacks by pipes with swinging joint; av. pressure 
about 700 lbs. per sq. in., but pump could develop 6000 lbs. Material, hard, dry 
quicksand, sometimes mixed with clay. Cost of tunnel per lin. ft.: 8 c. y. excav., 
underground labor, at 73 cts. = $5.44; 8 c. y. excav., surface labor, at 48 cts.= 
$3.82, 2.62 c. y. brickwork, underground labor, at $1.12= $2.95; 2.62 c. y. brick- 
work, surface labor, at 73 cts.= $1.91; 1100 bricks @$9 per M=$9.90; 2.1 bbls. 
cement (1:3 mortar), at $1.70= $3.57; 1 c. y. sand at $1=$1.00; plant, 50% of first 
cost, distributed over 1625 lin. ft. = $5.00; lumber=$1.05; shafts or manholes= 
$1.00. Total $3 5.64.— ^.-C. July 25, 1906, p. 22. 

Clearing and Grubbing Land; and Blasting Stumps. Area, 9 acres; trees. 
6 ins. to 3 ft. in dia., with average about 20 Ins., consisting of oak, hickory, chestnut, 
etc.; number of trees cut was over 1100, and number of stumps blasted was 1212. 
Trees under 6" dia. classed as brush, and stumps were grubbed with mattocks. 
For blasting stumps the following were used: 1 churn drill. 1 large auger, and 
1 bucket, costing in all about $80. Total cost per acre was as follows, with 
Italian labor at $1.25: Chopping $18.84, grubbing and clearing $15.53. making 
cord wood $10.14, blasting $73.73, grubbing after blasting $35.26, grinding axes 
$0.65. tools S9.00; total cost per acre $163.2 5.— J5;.-C., Feb. 27, 1907, p. 93. 

Wash Drill Borings. Deep Waterways Survey. Great Lakes to Atlantic Tide 
Waters. 1897-1900. The process consisted In alternately "driving casing" and 
"drilling" until "bottom" was reached. Where obstructions were encountered that 
could not be passed by drilling, they were removed by pulling the drill rod and 
lifting the casing 3 or 4 ft. and then firing a stick or two of dynamite at the bottom 
of the hole, (a) On the Tonawanda-Olcott and La Salle-Lewiston Routes, compris- 



PERFORMANCE OF WORK. 



917 



Ing 404 holes bored to an aggregate depth of 9624 ft. The cost of borings (including 
total cost of plant) was 68.63 cts. per lin. ft., the material being sand, gravel, clay, 
and hardpan. (b) On the Western Division of the Oswego-Mohawk Route — from 
Oswego to Rome — 7 50 holes were bored to an aggregate depth of 33711 ft., and at 
an average cost of 70,07 cts. per lin. ft. For the Oswego river and harbor work, the 
machines were mounted on small flatboats with open wells at the center; and the 
work on Oneida Lake was done through the ice. (c) On the Eastern Division of the 
Oswego-Mohawk Route there were made 290 soundings by hand with a steel rod, 
and 1562 actual borings, together amounting to 55521 ft., aggregate depth, at a 
cost of 54.19 cts. per lin. ft. The borings varied from a few feet to 190 ft. in depth. 
Four types of machines were used, viz.: 1 Pierce well-boring machine, 1 Sullivan 
wash drill, and 2 home-made affairs. The material encountered was. sand, 20706 
ft.; clay, 9880 ft.; earth, 7611 ft.; sand and clay, 3176 ft.; gravel, 2815 ft.; sand 
and gravel, 2728 ft.; sand, clay and gravel, 1843 ft.; quicksand, 1529 ft.; clay and 
shale, 902 ft.; sand, loam and mud, 900 ft., clay and gravel, 760 ft.; rock, 626 ft.; 
mud, 417 ft.; miscellaneous, 1628 ft. (d) On the Champlain Route from Ogdens- 
burg to Lake St. Francis there were 148 sand borings totaling 7052 ft., and 151 
water borlDgs totaling 2123 ft. The cost of the 9175 ft. of borings was 84.18 cts: 
per lin. ft. (e) On the Hudson River Division of the Champlain Route 57 991 lin. 
ft. of borings cost 12.35 cts. per lin. ft. (f) On the Hudson River Survey, Hudson 
to Troy, the borings were made with an outfit mounted on a catamaran and on 
scows. Silt, clay, coarse and fine sand, gravel, and boulders were penetrated. A 
2Hn. casing and "B drill rods," with X-bits were used. In all, 1385 borings were 
made aggregating 28965 ft. in depth and costing 25.07 cts. per lin. ft. — E.-C, 
Mar. 27, 1907. p. 132. 

Diamond Drill Borings, Deep Waterways Survey, Great Lakes to Atlantic Tide 
Waters, 1897-1900: 



Route. 



No. of holes 

Depth in feet 

Standpipe, feet 

Rock drilled, feet 

Cost of boring, per lin, ft 

Rental of drilling outfit. 

Carbon 

Labor 

Teamster 

Teaming, extra 

Superintendence 

Repairs 

Coal (and wood) 

Lumber 

Core boxes 

Freight and Express — 

Traveling expenses 

Sundries 



(1) 
Tonawan- 
da-Olcott. 



1022, 
118. 
904. 



(2) 
La Salle- 
Lewiston. 



4 

5 

$2,494 
0.769 
0.423 
0,542 
0.124 
0.077 
0.321 
0.042 
0.039 
0.030 
0.015 
0.042 
0.042 
0.028 



5 

574.4 
101.0 
473.4 



(3) 

Oswego, 

W'n Div'n. 



7 
521.6 
241.7 
279.9 
$3,738 
1.150 
0.139 
1.016 
0.345 
0.177 
0,479 
0.101 
0.066 

6!6i6 
0.143 
0.058 
0.054 



(4) 
Champlain; 
N'n DIv n 



4 
342,5 
90.7 
251.8 
$4,366 
1.099 
0.490 
1.024 
0.368 
0.190 
0.547 
0.072 
0.073 



0.020 
0.225 
0.201 
0.057 



—E.-C, Mar. 13, 1907, p. 108. 
R. R. Grading with Wheeled Scrapers. Five examples: 

Ex. 1. Ex. 2. Ex. 3. Ex. 4. Ex. 5. Ay. % 
Material. Sandy Good Wet Fine Loam- 
loam, clay. clay. sand. clay- 
sand. 

Lead. In ft 260. 300. 400. 500. 700. 432. 

Foreman ($3.00).... $.017 $.019 $.026 $.024 $.020 $.021 6.0 

Scrapers ($4,75) 138 .158 .216 .222 .210 .189 51.6 

Plowing ($9,20) 052 .057 .080 .073 .053 .063 17.2 

Snatching ($6.00)... .034 .037 .052 .050 .030 .040 10,9 

Loaders ($1.60) 018 .020 .028 .026 .020 .022 6.0 

Dumping (men $1.50) .008 .016 .039 .027 .033 .025 6.8 

Water boy ($1.00). .006 .004 .009 .008 .001 ,006 1.5 

Total cost per cu.yd. $.273 $.311 $.450 $.430 $.367 $.366 (100.0) 

—E.-C, Sept. 25, 1907, p. 184. 

Trenching and Backfilling for Pipe Sewer, CentervIIle, Iowa. Data from which 
table was compiled was furnished by Mr. M. A. Hall, engineer In charge, from dally 
reports by Inspectors, and Mr. Hall suggests adding 10% for possible omissions la 



918 



5i.— EARTHWORK, 



labor roll. There were 53 Jobs in all, and these are numbered In the following table 
by "side and top" method. The following information is furnished for each job: 
p= dia. of pipe, in ins. ; t= size of trench, width by depth; d= cost in cents per cubic 
yard for digging trench, &=cost in cents per cubic yard for back-filling. Labor at 
1 8 cts. per hour. 

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EARTHWORK CLASSIFICATION. 919 

Two Hydraulic Fill Dams. (1) Croton dam on the Muskegon River, Mich.; a 
few hundred feet below the junction of the Big and Little Muskegon. Sluiced em- 
bankment, 250 ft. long, 55 ft. high and 20 ft. wide on top; containing 104000 cu. 
yds. of sand and gravel. Maximum distance sluiced, 800 ft.; grades generally 8 to 
9%. Slopes of finished embankment, 2 to 1 and 3 to 1. Two Ramsey and two 
Gould 6-in. rotary fire pumps were used, operated by electric power. Total cost 
including plant, and superintendence (^ cent), 7.3 cents per cu. yd. (2) Lyons dam. 
Grand River valley. Embankment 150 ft. long and containing 23000 cu. yds. of 
sand and gravel was sluiced into place. Total cost including plant minus salvage, 
and superintendence (3 cents), 36.01 cts. per cu. yd. The high cost was due to 
expensive trestles to carry the troughs, and from trouble with Ice. — E.~C., Nov. 13, 
1907, p. 276. 

Panama: Steam Shovel Excavation. (1) Earth. Excavation by steam shovel 
10.81 cts., transportation 19.42 cts., tracks 8.30 cts., dumps 15.48 cts., general ex- 
pense 1.93 cts., arbitrary to cover cost of plant 12.00 cts.; grant total 67.94 cts. per 
cu. yd. (2) Rock. Add 25.57 cts, to above for blasting, making grand total of 
93.51 cts. per cu. yd. (3) Dredging. Cost of dredging 1,213,597 cu. yds. of soft ma- 
terial, place measurement, by one dredge was about lOi cts. per cu. yd. — E.-C, 
Jan. 8, 1908, p. 32. 

Earthwork Classification. — ^The object of classification is to fix a basis 
for estimating the cost of work by unit prices which will be fair for any 
kind of material encountered and for any reasonable changes in the original 
plans. First of all, the engineer who draws up the specifications should 
have clearly in mind the kinds of material he is classifying, not merely by 
name, which is often misleading, but by a thorough practical knowledge 
that can be demonstrated to intending bidders. The writer, as contracting 
engineer, has been on work where test pits were dug showing material 
which the engineer who drew up the specifications could not classify him- 
self! The impression received, however, was that it would be classified as 
"cemented gravel," but when the work was completed it was paid for as 
"common earth" excavation, although blasting was the cheapest method 
of loosening it. In fairness to the engineer it should be stated that he 
finally allowed the powder bill, which was quite an item; but it did not 
cover the extra expense for that material over our unit price for "common 
earth." 

In view of the above, the writer believes that all material which cannot 
be described accurately in specifications should be exposed to view in test 
pits as samples of classification, and referred to as such, together with 
descriptions of same as accurate as possible. 

Common Classifications. — ^The following is from the very valuable paper 
by W. F. Dennis, entitled "Uniformity of Requirement and Clearness of 
Specification in Agreements for the Graduation of Railroads," published in 
Trans. A. S. C. E., June, 1907 (p. 321). 

Solid Rock. (Abstracts from Specifications.) 
A^. Y., N. H. & H. — All rock or stone containing 1 cu. yd. or more. All other 
material earth. Erie. — Rock in masses exceeding 1 cu. yd., which cannot be re- 
moved without blasting. P. R. R. — [Same as Erie.] B. & O. — Rock in solid beds or 
masses, which may best be removed by blasting. Ches. & O. — Rock in ledges and 
detached masses exceeding ^ cu. yd., which may best be removed by blasting. 
Nor-f. & W. — Rock in masses which may best be removed by blasting. Southern. — 
Rock in masses of more than 1 cu. yd,, which may best be removed by blasting. 
"Big Tom." — Stone in solid masses or ledges. C, B. & Q. — Stratified rock weighing 
more than 140 lbs. per cu. ft., which can only be removed by blasting. Chi. & Alt. — 
All stratified rock and rock occurring in masses which can only be removed by blast- 
ing must ring under hammer. Great Nor. — Rock In place, in removing which 

It is necessary to resort to drilling and blasting. A., T. & S. F. — Rock In solid beds 

or masses in its original or stratified position other material which can be 

removed without continuous drilling and blasting, but which Is as difficult 

as solid rock or limestone. III. Cent. — Rock in solid beds or masses in its original 
position which may best be removed by blasting. Everything else classi- 
fied as common excavation. N. P. — All rock In masses that cannot be removed 
without drilling and blasting. Mo. P. — Rock In solid beds or masses. In Its original 
position, which can only be removed by continuous blasting. 

Loose Rock. (Abstracts from Specifications.) 
Erie. — Slate, shale or other rock which can properly be removed by steam 
shovel, pick or bar, without blasting, although blasting may be resorted to on favor- 
able occasions to facilitate the work detached masses, 3 cu. ft. to 1 cu. yd. 

P. R. R. — Stone and detached rock lying in separate and continuous masses con- 
taining not over 1 cu. yd. ; also all slate or other rock that can be quarried without 



920 64 —EARTHWORK, 

blasting, although blasting may be resorted to occasionally. B. & O. — Slate, coal, 
shale, soft friable sandstone and soapstone, detached masses 3 cu. ft. to 1 cu. yd. 
Ches. & O. — Shale, slate, ochre, which can be removed with pick and bar, and is 
soft and loose enough to be removed without blasting, although blasting may be 
resorted to occasionally. Detached masses 3 cu. ft. to 1 cu. yd. Nor1. & W. — Shale 
soapstone, and other rock which can be removed by pick and bar, and is soft and 
loose enough to be removed without blasting, although blasting may be resorted to 
occassional! y. Detached masses 1 cu. ft. to 1 cu. yd. Southern. — [Same as Norf. & 
W.) "Big Four." — Shale, coal, slate, soft sandstone, soapstone, conglomerate 
stratified limestone in layers less than 6 in. Detached masses 3 cu. ft. to 1 cu. yd. 
C. B. & Q. — Stratified rock which can be removed by pick and bar and weighing 
more than 140 lbs. per cu. ft. Detached masses 3 cu. ft. to 1 cu. yd. Chi. & Alt. — 

Stratified rock which can be removed by pick and bar and masses between 

3 cu. ft. and 1 cu. yd. Great Nor. — Slate and other rock, and loose enough to be re- 
moved without blasting, although blasting may be resorted to occasionally. Detached 

masses 3 cu. ft. to 1 cu. yd. A., T. <& S. F. — Hard shale or soapstone in 

original or stratified position, boulders in gravel, cemented gravel, hardpan and 

other material requiring use of pick and bar or which cannot be plowed with 

10-ln. plow and 6-horse team. III. Cent. — (No loose rock. Everything but solid rock 
classed as common excavation.) N. P. — Slate, soft sandstone, or other rock than 

can be removed without blasting. Detached rock between 1 cu. ft. and 1 cu.yd. 

Mo. P. — All rock which requires for its removal steam shovel or pick and bar. 

without blasting, although blasting may be resorted to at the option of the 
contractor. Detached masses 1 to 1 8 cu. ft. 

EXCERPTS AND REFERENCES. 

Some References to Earthwork and Especially to Shrinkage. — 

(1). Notes on Earthwork. By Geo. J. Specht. Tech. Soc. Pac Coast 
Transactions, May, 1885. A collection and digest of data on shrinkage up 
to that time. (2). Shrinkage of Earthwork. By P. J. Flynn. Tech. Soc. 
Pac. Coast Transactions, read June 5, 1885. Refers to experiments on 
shrinkage, made in India, and gives tables of shrinkage. Also gives nu- 
merous references and data. (3) . Shrinkage — Growth. J. C. Nagle. ' 'A Field 
Manual for Railroad Engineers," 1887. (4). Shrinkage of Earthwork. 
W. M. Patton. "Civil Engineering." Patton says: Sand shrinks about 10%; 
sand and gravel, 8%; earth and loam, 10 to 12%; gravelly clay, 8 to 10%; 
puddled "clay and soil, 20 to 25%; rock excavation produces a larger mass 
by from 25_% in cases of small fragments and 60 or 70% when in blocks 
carelessly piled up. (5). Shrinkage of Macadam Under Rolling. See Eng. 
News of Feb. 11, 1904. See, also, Eng. News, Jan. 14 ,1904, under Macadam 
Road Construction Along Charles River. 

A Novel Method of Constructing High Embankments in Switzerland 
(Eng. News, Aug. 21, 1902). — By use of temporary suspension bridge; 
illustrated. 

The Cost of Hydraulic Excavation for Embankments and for Placer 
Mining (Eng. News, Nov. 27, 1902). — Cost data on several works, also 
references to other data. 

The Buckeye Trench Digging Machine (Eng, News, Aug. 6, 1903). — 
Illustrated. 

Discussions on Clearing and Grubbing (Eng. News, Jan., 1904.) 

Time Required to Load Wagons with a Steam Shovel (By J. S. Ely. 
Eng. News, July 14, 1904).— Table. 

A Cross=Cut Excavating Machine for Drainage Ditches (Eng. News, 
Sept. 7, 1905). — Illustrated. 

Machine for Spreading and Leveling Material (Eng. News, Jan. 4, 
1906).— Illustrated. 

Cost of Steam Shovel Work by Railway Force and by Contract (By 
T. C. Sesser. Bulletin 81, Nov., of Am. Ry. Eng. & M. of W. Assn.; Eng. 
News, Jan. 17, 1907). — Work by company force, 18.7 cts. per cu.yd.; 
contract, 26.0 cts.; saving, 7.3 cts. 

Excavating Machines on the N. Y. State Barge Canal (Eng. News, 
June 6, 1907).— Illustrated. 

Gravel Spreader Used on the Colorado River Levee Construction 
(By H. T. Cory. Eng. News, July 11, 1907). — Illustrated. "Cost of work 
done by the machine was about one-tenth of a cent per yard of material 
spread. Machine cost $300. Its operation required a locomotive and four 



MISCELLANEOUS DATA. 921 

An Unloading: Machine for Dumping Cars in Building Embank- 
ments on the Western Pacific Ry. (Eng. News, Aug. 22,1907). — Illustrated; 
consists of a circular loop at end of top of embankment, for cars to run 
around and dump. 

Hydraulic Construction of Large Embankments on the Chi., Mil. & 
Puget Sound Ry. (Eng. News, May. 27, 1909). — ^Twelve illustrations of con- 
struction work. 

Machine for Excavating Trenches and Foundations in Frozen 
Ground (Eng. News, June 17, 1909.) — Used in Winnipeg, Canada. "In 
that city the frost penetrates to a depth of 5 to 6 J ft. Previous to the in- 
troduction of this machine, all earth excavation during 5 or 6 months of 
the winter had been done by hand picking or by blasting the frozen earth 
with black powder, the former costing about $1.35 per cu. yd., and the 
latter about $1.25 per cu. yd. down to 93 cents per cu. yd. By the machine, 
the cost has been reduced to from 11 to 30 cents per cu. yd. The ("Drop- 
Chisel") machine is illustrated. 

Excavation Methods, Fourth Avenue Subway, Brooklyn (Eng. Rec, 
Dec. 3, 1910). — Described, and illustrated. 



55.— ROCK EXCAVATION. 

Subject Divisions.— Under "Quarrying" (see page 419) are discussed 

the methods of excavating "dimension" and other stone for subsequent use 
in masonry construction. The present subject deals with the most approved 
methods of excavating open rock cuts and trenches for railways, highways, 
canals, sewers, etc. For subaqueous excavation, including blasting under 
water, see "Dredging," page 927, and for tunnel work see "Tunneling" 
page 933. 

Open Rock Cuts. — ^The cheapest form of open cut is the side-hill cut 

(Fig. 1), in which c is the cut,/ the fill, and w the 
waste. A common method of operation is to begin 
drilling with holes a at the bottom of the cut, 
using moderate blasts and working back in steps 
to b. But where the quantity^ in cut c greatly 
exceeds the fill /, it is often advisable to begin at 
b, using deep holes and large charges of explo- 
sives. By this means a more effective use of the 
powder can be had in not only loosening the rock 
but in wasting a large proportion of it at the same 
time. Solid rock is supposed to stand about "ver- Fig. 1. 

tical," hence the drill holes b as shown. If the material is seamy and con- 
tains more or less loose rock the latter may be taken out to slope as re- 
quired, but it is not necessary to take out the whole cut in the same slope.* 

Drilling. — ^The most common form of rock cut is the "thorough" cut 
as shown in Fig. 2. The rock is excavated in 
benches, a, b and Cy but not so regularly as 
shown in the Fig. unless channeling machines 
are used as was the case on the Chicago 
Canal. The rise or lift may be assumed as 
about equal to the tread, but this depends on 
the size of blocks that can be handled eco- 
nomically, the depth of economical drilling, 
the quality and character of stone and seams, 
and whether horizontal holes at the bottom 
of rise are drilled to assist in blasting. 





Section 



Fig. 2. 

The spacing of the holes may 
be assumed about equal to their depth, if in single row and the rock 
is not too hard. For trap and granite the spacing should be closer, ordi- 
narily. The rise may vary from a few feet up to 12 or even 20 ft., the 
deeper holes requiring the larger drills, say 3 to 3| ins., and operated 
by machine. Chum drills up to 3 ins. in dia. (If" bar) may be used for 
vertical holes in soft rock, where machine drills are not available. Each 
drill is operated by two or more men who raise it, tiim it 'roiind a little 
and let it plunge back into the hole. Sometimes the drill is loaded to 
give more weight and hence become more effective. For two-hand ham- 
mer drilling (one man holding and turning drill, and two men striking), 
the hole is usually started with a li to IJ-in. bit and using 10-lb. hammers. 
The dia. of bit is decreased with depth of hole, to prevent binding, and 
the limiting depth is about 8-ft. Octagonal bars from |-in. to 1-in. are used. 
For one-hand drilling (hammer 4^ lbs.), the drills are usually 1 in to \\ in. 
with octagonal bars f-in to |-in. in dia. The minimum diameter of hole 
(at bottom) for use of dynamite is about f in. The hand drill is often 
called a jumper. Rotary drills, rotating drills, or augers are names given 
to (hand or machine) drills which bore solid holes, or annular rings around 
solid cores. They are generally used in the softer rocks where heavy blasts 
are required. Such drills can, however, penetrate almost any material 
liable to be encountered, instances being recorded in which borings have 
been made through imbedded steel rails used for foundations. Core drills 

* The writer knows of an instance where a whole rock cut was taken out 
to slope of i to 1, where 1% of it could have been vertical, or at least \ to 1. 



922 



OPEN ROCK CUTS. 923 

ere particularly designed for well-borings, both water and oil; and for 
ordinary borings to determine the geological formation at different depths, 
samples of which are brought up in the form of cylindrical cores. They 
may be either steel (hardened) or diamond (black) -pointed, for effective 
cutting. 

Trenching in Rock. — ^Trench Excavation, for water pipes, sewers, etc., 
is more expensive than open-cut excavation, but less expensive than tunnel 
excavation. The increase in cost over open-cut work is due (1) to the greater 
amount of difficulty of drilling, per cubic yard removed; (2) to the limiting 
charges of explosives, thereby reducing the duty of drilling, powder and 
labor; (3) to the large proportion of excavation required outside of "neat 
lines" oi trench, without added compensation; (4) to the great amount of 
trimming up of sides and bottom of trench preparatory to laying the pipe. 
These are discussed as follows: 

(1.) Owing to the large amount of drilling required, machine drills should 
be installed where the preliminary cost, reduced to rental, is warranted by 
the total yardage. Comparing cost of operation alone with that of hand 
drilling, a saving of 40 to 60 cents per cubic yard is often effected. The 
direct -acting steam drill is perhaps the most commonly used for this work, 
but often compressed air may be cheaper. Gasoline compressors are some- 
times cheaper than steam compressors especially on small jobs. Electric 
motors also should be considered where electric power is available. 

(2.) Larger blasts may be fired if ballasted with logs, timbers or blasting 
mat, where the flying rocks are liable to cause damage, as in cities. Mr, 
H. P. Gillette in his "Rock Excavation," describes a blasting mat woven 
with old hemp rope, H in dia. or larger. The mat is woven like a cane- 
seated chair, the first set of strands being looped over 1-in. gas pipes. It is 
weighted with heavy timbers, thus effectively checking the progress of the 
smaller fragments of rocks. The narrow outlines of the trench, however, 
limit the duty of the charge. 

(3.) In estimating the cubic yards in trench excavation it is well to 
assume the trench 24 ins. wider than the external diameter of pipe or 
Bewer to be laid, and 6 ins below grade. Bell holes have to be dug at the 
pipe joints and there is more or less trimming to be done along the sides 
and bottom^ so that this seems no more than a just compensation. Engi- 
neers sometimes let contracts based on the lineal feet of pipe laid irrespective 
oft depth of trench or quality of material to be excavated. This throws the 
burden of examination of the soil, etc., upon the contractor (who usually 
makes a guess) , and is not recommended. If for any reason the line of the 
trench is changed after such a contract has been let, complications invari- 
ably arise between engineer and contractor. 

(4.) Reference has just been made to trimming up the bottom and 
sides of the trench to receive the pipe for laying. There is another item 
which must not be overlooked, namely, lowering the grade after the trench 
has been finished. Instances are of record where contractors have been 
obliged to lower the grade from a few inches to a few feet at a price based 
on the original bid per cubic yard, whereas such work sometimes costs 
4 or 5 times the amount received, the specifications admitting of no redress. 
In line with the above is a trench mostly in earth excavation but with a 
small amount of rock at the bottom. In such a case the lack of proper 
borings may prove very expensive to the contractor. 

Chicago Drainage Canal. — ^The material was classified as glacial drift 
(26,000,000 cu. yds.) and solid rock (12,300,000 cu. yds). The glacial drift 
comprised tough, indurated clay and cemented gravel or hard pan, inter- 
mixed with boulders and capped with prairie soil. The solid rock was 
limestone, increasing in hardness with the depth. All the material except- 
ing the top soil, required blasting, the hardpan standing vertically like 
solid rock. The best method of loosening the hardpan was by dynamite 
(40%) discharged at the ends of small tunnels. Fig. 3 shows sections 
of the canal in the two classified materials named. 

The rock section is 160 ft. wide at base, and 36 ft. deep. The drills 
were operated mostly by compressed air at 80 to 90 lbs. pressure per sq. in., 
from a central compressor, through 8 or 10-in. mains with 6 to 8-in. branches. 
Three drills were commonly supplied with 2-in. feed pipes 175 to 230 ft. 
long, feeding into 3i x 6i in. cylinders. Bits of the X pattern were found 
to be the best. The rock was taken out in 3 12-ft. Hfts (Fig. 2); and 40% 



924 



55.— ROCK EXCAVATION. 



dynamite proved to be the best generally. The sticks were H x 6 ins., 
weighing f lb., and 10 to 25 sticks were charged in a hole. The price per lb., 
including caps and fuse, was about 12 cents and about 1 lb. of dynamite 
was used per cu. yd. of rock. 




Chicago Main Drainage Channel. 

Fig. 3. 

Channeling machines of the Sullivan and Ingersoll types were used in 
order to secure smooth side walls to the canal. The average machine 
weighed 11000 lbs., was operated on a track 30 ft. long, and struck 250 
blows per min. The width of the channel cut by the bit was 2| ins. at the 
top, decreasing J in. for each 2 ft. of depth. The speed of channeling was 130 
to 200 sq. ft. per 10 hrs. on the upper lift (where the rock was softer) and 
about half as much on the two lower lifts. The lifts were 12 ft. each. The 
cost of the channeling varied from 8 to 25 (say 17) cents per sq. ft., or from 
3 to 7 (say 6) cents per cu. yd. excavated. 

Steam shovels of the Bucyrus type were employed to a limited extent 
in loading rocks on cars. The cars were operated by incline and hoist 
methods. But these were generally more expensive than the cantilever, 
cableway and derrick methods of conveyance as shown in the two following 
tables. 

1. — Cost in Cents per Cu. Yd. (Solid). 



S s 1 '§§ -a & .o a ^ 

Brown Cantilever 3.9 4.1 8.0 3.2 1.0 3.6 14.6 0.0 38*.3 

Lidgerwood Cableway. . 3.7 3.8 8.4 2.7 1.0 3.6 15.6 0.0 38.8 
Hullett-McMyler Derrick 3.9 4.0 7.4 2.5 1.8 5.3 18.3 0.0 43.2 

Hullett Conveyor 4.1 3.7 8.5 3.8 1.2 6.2 21.4 0.0 48.9 

Car Hoist, No. 1 3.7 3.9 9.1 2.7 0.8 3.124.8 5.1 53.1 

Car Hoist, No. 2 3.9 3.6 8.9 3.2 0.9 1.2 22.9 2.3 47.1 

Car Hoist, No. 3 4.0 5.0 10.7 3.1 1.2 1.2 26.4 4.8 56.5 

Note. — Shop repairs, drill sharpening and plant rental not included. 

2. — Output in Cubic Yards by Conveyors.* 




Section. 


Cu. Yds. 


Cu. Yds. 
per 10 Hrs. 


Cu. Yds. 
per Man in 

Pit 
per 10 Hrs. 


Brown Cantilever 


10 
8 
7 
7 
9 
8 

10 
9 

14 

14 


443,750 
600,725 
180,406 
109,397 
178,839 
131,674 

60,341 
308,531 
324,880 

63,700 


478 
397 
217 
235 
335 
285 
269 
463 
282 
153 


10.45 


Lidgerwood Cableway 

Hullett-McMyler Derrick. . . . 
Hullett Cantilever 


10.25 

8.52 
9.91 


Hullett Conveyor. . . . 


6.85 


Car Hoist, No. 1 


6.96 


Car Hoist, No. 2 


6.98 


Car Hoist, No. 3 


6.82 


Double Boom Derrick 

St. Paul Derrick 


8.22 
8.22 



* Compiled by Mr. W. G. Potter, 
hour. 



Common labor received 15 cents per 



CHICAGO CANAL. PERFORMANCE OF WORK. 925 

Performance of Work. — A few examples* are given to illustrate methods 
and cost of various classes of rock excavation, as follows: 

A Lobnitz Rock Breaker for excavating rock under water, designed by F. W. 
Allen, Eng'r of Empire Eng. Corporation. f used in excavating (about 260000 cu. 
yds. of) rock about 18 to 25 ft. under water, in improvement of Black Rock Harbor. 
Buffalo, N. Y. Contract price, $1.85 per cu. yd. The rock is broken by a cutter 
consisting of a steel cylinder 28 ins. dia., about 25 ft. long and weighing 46000 lbs., 
fitted at th# bottom with a hardened steel cutting point. The weight is hoisted 
and let fall upon the rock bottom from a height of six ft. or more. — XEng. News, 
Feb. 28. 1907. p. 236. 

Drilling and Blasting Soft Shale Rock Under Water, at Ashtabula Harbor, Ohio. 
Drill boats 85 ft. long by 30 ft. wide, held in position by four spuds, one at each 
corner. Equipped with Ingersoll-Sargent steam drils on trucks for horizontal 
movement, and operated vertically by hydraulic lifts. Drill Boat "A" worked days 
only. Cost of drilling and blasting 4744 cu. yds. (place measure) during month of 
May, 1907, was 28.5 cts. per cu. yd., the average depth of holes being 6.15 ft., spaced 
about 6.7 ft. apart. Of this amount, the labor cost of drilling and blasting was 6.7 
cts., and cost of explosives was 6.7 cts. Drill Boat "B" worked both day and night 
shifts. Cost for the day shift was 23.6 cts. per cu. yd., while for the night shift it 
amounted to only 20.4 cts. per cu. yd., there being less breeze during the night 
The total average amount of explosive (45%-dynamite) used was f lb. per lin. ft. 
of holes, and costing 1 5 cts. per lb. The blasted material was excavated with dipper 
dredges.— ^nfir.-CoTO^r., Aug. 7. 1907. p. 83. 

Excavating Granite in Open Cuts on the Grand Trunk Pacific R. R. — Data 
furnished by Geo. C. McFarlane, eng'r and contr. Steam drills used in large cuts, 
and hand drills in smaller cuts. 1" steel to make drills gaged to If. used for entire 
depth of hole. Gang of three men drilled per day average of 2 9 ft. in dark hornblende, 
20 ft. in red granite, and 18 to 19 ft. in trap and diabase rock. Rock-cuts 20 ft. wide 
at bottom; side slopes 1:4 or as material will stand. On most open cut work it has 
been the custom to drill the blast hole on or near the center line of the cut, so as to 
get the charge of explosive at the center of the mass to be moved. A 1 2-ft. hole is 
given a 12-ft. burden, while a 25-ft. hole is given from 20 to 25-ft. burden, but Mr. 
McFarlane used for the 20-ft. roadbed two holes 10-ft. on each side of center line, 
the burden not to exceed 1 5 ft. After drilling, the bottom of the holes are chambered 
to receive the charge by springing with dynamite. In a bottom bench where a heavy 
lift is required, no more than a ft. is chambered. In upper benches it is permissible 
to chamber 2 or 3 ft. of hole. The first springs are held down by water tamping. 
In springing, water tamping is nearly as good as sand, except when charges of over 
30 sticks are used. With heavy charges the water tamping permits the dynamite 
to shake up the hole and make it ragged. Sand tamping has to be drawn after each 
shot. If the sand is free from clayey material it can be drawn quickly by spudding 
it with a churn drill and pumping up the cuttings. Sometimes it can be blown out 
with the steam blowpipe. With water tamping, a cap and fuse are used, unless the 
hole is ragged, when the spring is made with electric exploders. The fuse, usually 

12 ins. long, after being split is held under water for 5 or 6 seconds to kill any fire 
hanging in the tape, and then dropped into the hole. Unless the drop fuse were 
dipped in water, it might ignite dynamite adhering to the sides of the hole, causing 
a premature explosion. After each water-tamped spring, the hole is blown out with 
steam, or pumped out with a sludge pump. The springing opens up the rock jointing 
and indicates very closely where the burden of the shot will cleave from the solid, 
and the successive springing charges indicate the ratio of enlargement of the pocket. 
A few items of cost are as follows: (a) Fully f of the rock cuts are sublet to gangs 
of station men at from $1.15 to $1.35 per cu. yd. They are given free cars and rails; 
and are charged $4.50 per week for board, $5.50 per day for team and driver, $11.50 
per box for 60% dynamite, and $2.75 per keg for black powder, (b) Cost of block 
holing a red granite boulder 10 x 12 x 9 ft., containing 36 cu. yds., was (48" drill 
hole $2.50, 3 lbs. 60% dynamite $0.66, reblasting 2 fragments $1.42) $4.50or 12^ 
cts. per cu. yd. (c) Cost of block holing red granite boulder 1 6 x 14 at top and 

13 X 10 at bottom, 15 ft. deep, containing 92 cu. yds., was (96" machine drill hole 
$7.25, 16 lbs. 60% dynamite $3.52, reblasting 2 fragments $1.45) $12.24 or 13i cts. 
per cu. yd. (d) CJost of blockholing black granite boulder triangular in shape, 4 x 5 x 
5-5 ft. high, containing 1.1 cu. yds., was (7" hole 50 cts., J lbs. 60% dynamite 8 cts.) 
58 cts. or 52i cts. per cu. yd. — Erug -Contr., Nov. 27. 1907. p. 301. 



* Reference to articles giving more complete description are given at 
the end of each example. 

tNew York City. J See also Eng.-Contr., Dec. 18, 1907, p. 343, for 
illustrations of cutter and detailed information. 



926 55.— ROCK EXCAVATION. " " '^' 

EXCERPTS AND REFERENCES. 

The Well Driller for Drilling Blasting Holes (Eng. News, June 23, 
1904). — On the excavation work for the Wabash R. R., in Ohio, in drilling 
brown sandstone the holes were put down to a depth of 24 ft. with a 3'' bit, 
and the drill averaged two such holes per day of 10 hours. The cost of 
labor, fuel and water was about 12| cents per ft. of hole drilled. In the 
blue standstone, which is softer, an average of 60 ft. per day -yas drilled. 
Formerly, with chum drills by hand, the cost in brown sandstone has been 
38 cents per ft. of hole, the holes being 20 to 30 ft. deep; the steam drills, 
up to depths of 20 ft., and in sandstone no harder, reducing this cost but 
very little. The well-drill holes being large (3'0 are never "sprung" more 
than three times in the sandstone, whereas the steam-drill holes (If'O must 
be sprung 4 or 5 times. 

Methods of Subaqueous Rock Excavation, Buffalo Harbor, N. Y. 

(Eng. News, July 6, 1905). 

Rock Excavation by Mechanical Power Instead of Explosives (Eng. 
News, June 25, 1908). — Editorial on the Lobnitz rock breaker and similar 
machines. 

Illustrations of Machines, Tools, etc. 

Pescription. Eng. News. 

Austrian drill boat with screw-operated spuds May 26, '10. 

Eng. Rec. 
Excavating submerged rock with a drill boat, N. Y., N. H. & 
H. R. R. Jan. 8, '10. 



56.— DREDGING. 

General Discussion.— In estimating the cost of dredging in any locality 
much will depend upon the amount of dredging to be done, as well as the 
kind of material, method of disposal, depth and roughness of water, climatic 
conditions as affecting the number of working days per year, and othe? 
considerations. The cost of dredging may vary from 3 to 4 cents up to about 
$1 per cu. yd. for earthy material; and for solid rock where subaqueous 
drilling and blasting are required the total expense of removal may amount 
to $3 or more per cubic yard. 

There are three methods of measuring dredged material, namely, 
(1) in place, by prior and by subsequent surveys or soundings; (2) in scows, 
by capacity or by displacement ; and ( 3) in fill. The first method is generally 
unsatisfactory especially in shallow cuts where any error in average depth 
would amount to considerable in percentage; and again, the excavated 
channel is almost sure to become partly filled again with current drift, 
which amount would represent an error in yardage against the contractor. 
But such measurements are very useful as a check in many cases. The 
second method, in scows, is usually adopted and is probably the best where 
the main object is "dredging" and not "filling" or reclamation. In this 
connection it should not be forgotten that dredged earth will swell from 
8 to 40 per cent (average say 25) and the method of measurement should 
be stated in the specifications and contract. Scows should be self-dumping 
so that the cost of delivermg or wasting will be that_ of towage mainly. 
The third method of measurement is in fill where the object is reclamation. 
But scow measurements are also generally kept as a check or even as the 
main basis for payment. The writer was employed in an engineering 
capacity* in connection with the filling of what is now known as a part of 
Back Bay Park, Boston, being the trapezium bounded by Beacon St., West 
Chester Park (now Massachusetts Ave.), the Boston & Albany R. R., and 
Brookline Ave. Much of the area was under water. It was cross-sectioned 
in 20-ft. squares and soundings from rafts were taken at range and station- 
line intersections, in connection with gage (tide) readings at stated intervals 
of time. In taking the soundings (2 or 3 men at a pole) there were usually 
3 readings, namely, top of mud, bottom of soft mud, and "stiff mud" 
readings. The filling consisted of dredged material from the Charles River, 
floated in on scows as long as the depth of water would permit, then the 
top filling was brought in on B. & A. R. R. cars from gravel banks at 
Weston, 20 miles distant. The character of the mud bottom precluded the 
possibility of estimating the quantity of fill accurately by the method of 
cross-sectioning and sounding, and payments were made by scow and 
gravel-pit measurements. It was estimated, however, that the amount of 
fill required below the top of the mud was equal to from 20 to 40% of the 
stiff and soft mud respectively. In hydraulic dredging for reclamation it 
is customary to estimate "in fill" only, and it should be stated clearly in 
the contract. 

Dredges. — Any machine that will excavate material under water is 
called a dredge. Perhaps the simplest form of dredge is the drag scraper 
made to ply back and forth over the bottom by the use of cables. There 
are two forward cables to give the bucket the correct incline for scooping 
up the material, and one back cable to steady the bucket and haul it back. 
Dredges are usually mounted on scows or floating hulks and may be Operated 
by steam or electricity. The latter is seldom used however excepting 
where the dredging is centralized, of great magnitude, and practically con- 
tinuous, in which case there may be a central power station on shore, feeding 
one or more dredges. 

Dredges are usually classified according to the excavating machine, of 
which there are four principal types, namely, (1) the dipper dredge, (2) the 



* With Fuller & Whitney, Boston, Mass. 

927 



928 



SQ.^DREDGING. 




grapple dredge, (3) the bucket elevator dredge, (4) the hydraulic dredge. 
Each of these is particularly adapted to certain kinds of dredging and there 
are many modifications. 

The Dipper dredge (Fig. 1) is really 
a long-handled dipper or shovel which 
is filled by pressing the scoop down 
into the mud while being swung radi- 
ally outward. The material is dumped 
through the bottom of the bucket, 
which is movable (usually hinged). 
This is the best all-around type of 
dtedge for general use. The buckets 
average 1 to 2 cu. yds. , but may be of 
any size up to 6, 10 or even 15 cu. yds. 

capacity. They are capable of working ^ 

in very soft and very hard material and ^^^^__ 

are generally useful around wharves, ^^f^f^f^ 

and for channel and canal excavation. 

30 ft. of water is about the limiting ^ Fig. 1. 

depth for good work. Dipper Dredge. 

The Clam-shell bucket (Fig. 2) is suspended from a derrick boom on a 
scow and is a form of grapple dredge. The bucket is lowered with jaws 
open, and sinks into the mud by its own weight. 
The jaws are then closed around the material and 
the bucket is raised and dumped. The capacity 
of a clam-shell bucket is about the same as that 
of a dipper, but it is especially adapted to deep 
dredging. 

The Orange-peel bucket (Fig. 
3) is another form of grapple, 
and may have 3 or more seg- 
ments which are open during 
its descent and, after pressing 
into the mud, are closed. In 
addition to its regular work in 
dredging, it is frequently em- 
ployed in excavating inside of 
cylinders which are being sunk 
for foundations. 

The (Bucket) Elevator 
Dredge or bucket ladder dredge 
consists of an endless ladder 
chain to which buckets are at- 
tached. These buckets scoop 
up the material from the bot- 
tom, are elevated to the top of 
the ladder, and the material is dumped in a chute or on a belt conveyor 
which deposits it as required. They are particularly adapted to "skim" 
dredging in soft material for canal work, etc. They leave a smooth bot- 
tom. The buckets may have a capacity of say 0.1 to 1.5 cu. yd. each, and 
a speed of 30 to 40 ft. per min. 

For discussion of sub-aqueous electric power cable, see Eng. News, May 
7, 1903, and Aug. 13. 1903. 

The Hydraulic Dredge* is used in the improvement of waterways and 
in the reclamation of low lands. It works best in fine material which 
remains suspended in a swift-moving volume of water. The principal 
features. of the hydraulic dredge are the rotary cutter or stirrer to loosen 
the material and keep it in suspension, and the centrifugal pump to pump 
the suspended material up into the delivery pipe and to its destination. 
If the material is light and loose the rotary cutter is sometimes replaced by 
the water-jet, but ordinarily the cutter is better, and must be used for 
compacted materials. Silt, mud, sand and clay are handled easily, but sharp 
sand wears out the pumps quickly. High-speed pumps will handle coarse gravel, 
coral and small boulders. The delivery pipe is of sheet iron or steel (sometimes 
wood-stave pipe), with an average diameter of say 10 to 20 ins., and may be 




Fig. 2. 
Clam-shell Bucket. 



Fig. 3. 
Orange-peel Bucket. 



* For valuable data on Dredges, see Eng, News of July 28, 1904. 



TYPES OF DREDGES. SUBMARINE BLASTING, 



929 



2000 to 6000 ft. in length. Between the dredge and the shore it is supported on 
pontoons, either single or double. 

Dredging "with" and "against" the current of a stream are subjects for 
discussion, and depend somewhat on local conditions. 

Blasting Under Water is required in removing ledge rock and large 
boulders. The following is from Gillette's "Rock Excavation," from data 
furnished by C. Y. Dixon, U. S. Ass't Engr.: 

At the mouth of the Detroit River It is usually necessary to drill and blast the 
material before it may be excavated. The drill boats in use (1903) are from 60 to 
80 ft. long and from 2 5 to 30 ft. wide, and are held in position by four "spuds," one 
at each corner. They are equipped with two or more IngersoU steam drills supported 
on vertical frames having trucks to permit of the drills being moved horizontally 
along the edge of the boat. The drills are raised and lowered during the operation 
by hydraulic lifts. The boiler furnishes the steam for operating the drills, the 
pumps used in connection with the hydraulic lifts, the forge the electric light plant 
and other machinery with which the ordinary drill boat is equipped. The holes, 
about 2^ ins. in dia., are made at the corners of 5-ft. squares to a depth of about 
3 ft, below the required depth, at the rate of about 5 ft. per hour per drill. One pound 
of 60% dynamite is used per lin. ft. of drill hole. The holes are charged by inserting 
sticks of dynamite with the exploder and battery wires attached to the bottom of a 
long pipe, the battery wires leading out through a slit in the side of the pipe. This pipe 
is lowered into the drill hole, the dynamite shoved down with a long ram-rod, and 
the pipe withdrawn, a wire spring clamped to the dynamite stick preventing its com- 
ing out of the hole. The wires are then attached to the battery and the dynamite 
exploded. During this operation the drill boat is not moved, nor does the work of 
operating the other drills cease except at the time of firing. On two occasions, 
however, the charge of dynamite came out of the hole, and was exploded directly 
underneath the boat, causing it to sink almost immediately. This was due to careless- 
ness, as by pulling the wires up taut they will indicate whether the charge is in place. 
One day's supply is kept at the drill boat, stored in a small scow trailing down- 
stream at a safe distance. Dipper dredges are used in excavating the material. 
They vary In length from 80 to 135 ft. and" in width from 30 to 40 ft., and are held 
In position by three spuds (36 ins. square), two at the bow and one at the stern. 
After the entire width of the area to be improved has been worked over by the 
dredge, cut by cut, the derrick scow follows after to remove such loose pieces of 
rock as may have been left projecting above the required depth. The derrick scows 
are usually from 80 to 100 ft. long and from 20 to 25 ft. wide and they are equipped 
with an ordinary hoisting engine and derrick capable of lifting from 12 to 18 tons, 
and a complete diving outfit. When lifting the boulders, the derrick scow is pinned 
up and supported on two spuds, each about 1 ft. square. The material to be moved 
Is found by means of an iron bar. about 30 ft. long, suspended from the side of the 
scow to the required depth. Any obstruction struck by this bar, as the scow is swept 
over the improved area. Is removed by the derrick by means of a chain, which Is 
placed in position by a diver. 

The following summary of the above will be of interest. It should be noted that 
the material at Ballard's Reef was limestone bedrock, clay hardpan (50%), and 
boulders, with two ft. of loose material overlying the bedrock; at Lime Kiln Crossing 
the material was mainly limestone bedrock, with no overlying material; at Am- 
herstburg Reach the material was limestone bedrock, clay and boulders, with gener- 
ally one to two ft. of loose material above. 

Dredging Detroit River, 1900-1903. 



Ballard's 


Lime Kiln 


Amherstb'g 


Reef. 


Crossing. 


Reach. 


74.143 


101,072 


98.332 


135,548 


121,707 


209.821 


153,097 


168,633 




225,000 


61,000 


225,000 


1 ft. 


.5 ft. 


1.3 ft. 


1.8 ft. 


6 ft. 


2.8 ft. 


21 ft. 


18 ft. 


20.07 ft. 


7.248 


3,945 


9,021 


3,386 


1,490 


2,890 


10.634 


5,435 


12.911 


34 


17.4 


39 


$3,000 


$3,200 


$3,200 


$102,000 


$55,720 


$124,800 


$1.38 


$0.55 


$1.27 


$0.7 5 


$0.46 


$0.60 


19.0 


43.0 


23.3 


12.8 


22.4 


16.2 


150 


250 


200 



Cu. yds. above 23 ft. depth (place measure)... 

Cu. yds., total (place measure) 

Cu. yds., total (scow measure) 

Area dredged, sq. yds 

Average depth dredged, pay material 

Average depth dredged, total exc 

Average depth of water over material 

Dredge hours, worked 

Dredge hours, delayed 

Dredge hours, total 

Dredge months (12-hr. days) 

Cost per month 

Total cost 

Cost per cu. yd. (place measure), pay material 

Cost per cu. yd. (place measure), total exc 

Av. cu. yds. per hr., working time , 

Av. cu. yds. per hr., total time 

Maximum cu. yds., per hr., soft material . . ._ 



930 



m.— DREDGING, 
Drilling. 



Ballard's 
Reef. 



Lime Kiln 
Crossing. 



Amherstburg 
Reach. 



[Worked 

Drill hours \ Delayed 

iTotal 

Number of holes drilled 

Number of feet drilled 

Ft. per hr., actual work 

Ft. per cu. yd., pay material 

Ft. per cu. yd., total exc 

Distance between holes 

Average depth of holes 

Average depth of pay material 

Percentage of drilling below pay depth 

No. of lbs. of 60% dynamite 

Lbs. per cu. yd., pay material 

Lbs. per cu. yd., total excav 

Total cost of drilling 

Cost per cu. yd., pay material 

Cost per cu. yd., total excav 

Cost per lin. ft. drilled 

Cost per drill hour 



24.442 

982 

25.424 

30,023 

191.850 

7.9 

2.6 

1.4 

5 ft. 

6.2 ft. 

1.0 ft. 

84.0 % 

110,305 

0.5 

0.8 

$59,235 

$0.80 

$0.44 

$0.31 

$2.25 



37,746 
1.278 
39.024 
29,236 
240.591 
6.4 
2.4 
1.9 
5 ft. 
8.2 ft. 
5. ft. 
37.5 % 
222.396 
2.2 
1.8 
$105,245 
$1.04 
$0,865 
$0.44 
$2.69 



38.441 

38;44i 

35.432 

181.421 

4.7 

1.8 

0.9 

5 ft. 

5.1 ft. 

1.3 ft. 

75.0% 

263.672 

2.7 

1.2 

$96,470 

$0.98 

$0.46 

$0.53 

$2.51 



Note. — 75% dynamite often produces the best results. 

Derrick Scows. 



Cost. 



Cost per sq. yd. of area Improved 

Cost per cu. yd. of material removed by 
diver 



Ballard's 


Lime KUn 


Amherstburg 


Reef. 


Crossing. 


Reach. 


at $9.70 


13.0 mos. 


16.0 mos. 


11.2 mos. 


at $9.70 


at $9.70 


$10,865 


$12,610 


$15,520 


Tug service 


Tug service 


Tug service 


included in 


included in 


included in 


dredging. 


dredging. 


dredging. 


$0.0475 


$0.22 


$0.07 


$5.73 


$2.84 


$2.04 



Summary of Cost. 



Ballard'8 
Reef. 



Lime Kiln 
Crossing. 



Amherstburg 
Reach. 



Dredging 

Drilling 

Derrick Scows 

Totals 

Cost per cu yd. of pay material 

Cost per cu. yd. of total excavation, 



$102,000 
59,235 
10,865 

$172,100 

$2.32 
$L27 



$55,720 

105,245 

12,610 

$173,575 

$1,718 
$1,425 



$124,800 
96.470 
15.520 

$236,790 

$2.41 
$1.04 



Gold Dredging. — This has become a very profitable industry in many 
of the streams of. California. Mr. W. P. Hammon, who has had a large 
experience in this class of work, writes the author from Marysville, Cal., 
under date of Sept. 12. 1906, as follows: 

In handling gravel in our work we use elevator dredges entirely, the hydraulic 
dredge not being so efficient or economical; in fact, in most places along the river 
the latter is altogether impracticable for the reason that where heavy cobbles or 
boulders are frequent, in the operation of a hydraulic dredge, they are inclined to 
nest at the intake of the suction pipe thereby preventing gold from moving with the 
flow of water. 

We have operated elevator dredges with steam and electric current both, and 
from our experience consider that electric power Is at least 50 per cent more eco- 
nomical, to say nothing of the great convenience electricity has over steam. This 
estimate Is based on power at li cents per kilowatt-hour, and wood at $4.00 per 
cord aboard ship. 



GOLD DREDGING. PERFORMANCE OF WORK, 931 

Performance of Work. — ^The following examples are given mainly for 
reference to the articles themselves, in which complete details and illustra- 
tions are given; the abstracted matter being sufficiently full to present to 
the reader some idea of the nature of the article, and to bring out some 
principles in the design and execution of this class of work. 

Powerful Dipper Dredge "Majestic' with Edward Cable-Storage Drum. De- 
signed by and built for Edward Bros., of Sault Ste. Marie, Mich., sub-contractors on 
the improvement of the West Neebish Channel in the St. Mary's River, Mich. 
Steel hull 116x40 ft., and 13 ft. deep, with two steel trusses 24 ft. deep extending 
the entire length. Steel boom 60 ft. long, stepped into a steel casting at the bow 
and supported by four cables 2^ ins. dia.; boom swung by two 2-in. cables on a 
24-ft. turntable. Dipper handle 54 ft. long and weighing 15 tons; with 5-ft. dipper 
It will excavate to a depth giving 30 ft. of water; operated by horizontal engine 
of 250 H. P., capable of exerting, through the gearing, a pull of 200 tons upon the 
dipper cable, and of dredging solid beds of soft limestone rock without blasting. 
The bow spuds are 43 ins. square by 52 ft. long, each composed of four sticks dressed 
to 21i ins., fitted into cast steel shoe at bottom and bolted together for 8 ft. from 
the bottom with 24 bolts \\ ins. in dia. At the top, the sticks are held together by 
an Iron band f x 10". For the intermediate space there is no bolting, the timbers 
being free so as to give a certain flexibility or spring. The spud foot is 10x12 ft. 
composed of timbers 1 6 ins. square. The dredge can be lifted about 2 ft. above Its 
floating position, by means of a 2-in cable running over the top of each spud, thus 
forcing the spuds from 2 to 6 ft. into the mud. The capacity of the dredge is about 
40 to 50 cu. yds. per hour in rock alone, and about 10 times that amount when 
working in soft material and using an 8-yd. dipper. The Jackson & Church Co., of 
Saginaw, Mich., controls the patents for the Edward cable-storage drum, which 
provides for a cable about 1000 ft. long to be used, and when this breaks, only about 
80 or 90 ft. are lost, the working length being replaced by paying out sufficient cable 
from the storage part of the drum. — Eng. News, Feb. 7, 1907, p. 145; and Oct. 10, 
1907, p. 373. 

Supporting the Discharge Pipe of a Dredge. Device used in dredging the harbor 
at Gary, Ind. The 24-in. pipe line was suspended from pontoons made of pairs of 
24-in. cylinders built of boiler plate, water-tight, and from 10 to 80 ft. in length. 
Each pair of cylinders is fastened together by timbers under which the discharge 
pipe is swung, the latter being laid with flexible joints at intervals, for the necessary 
bends. Material is delivered from 1000 to 3000 ft. from dredge. — Eng.-Contr., Sept. 
11, 1907. p. 160, 

Hydraulic Dredge "Oneida," used on the New York State Barge Canal. De- 
scribed by Mr. Emile Low. Designed by Mr. Lindon W. Bates, Pres. the Empire 
Engineering Corporation, contractors. Length over all, 97 ft.; extreme beam, 17.5 
ft. ; molded depth to deck, 10 ft. ; light draft, 5. 5 ft. Can be taken through locks. 
Two suction pipes, on sides, extending from centrifugal pump aft to cutter forward; 
Inside dia., 19J Ins., No 9 B. W. G. metal, connected forward to two Bates curved 
telescopic joints. Two movable suction pipes or suction ladders are built-up pipes, 
each having an area of 291 sq ins., at the lower ends of which are attached the 
cutter heads 5 ft. 6 ins. dia. and 3 ft 8 Ins. high The cutters and dredge are illus- 
trated in Eng. News. When working, pontoons are attached to side of dredge to 
give stability. During Oct., 1906, 38173 cu. yds. were dredged at cost of 50.5 cts, 
per cu. yd.; during Nov., 1906, (two weeks spent in repairs), 144 882 cu. yds. were 
dredged at cost of 30.8 cts. per cu. yd. — Eng. News, Dec. 5, 1907, p. 619. 

Hydraulic Dredge "Francis T. Simmons," used in Lincoln Park Reclamation 
from Lake Michigan. Dredge designed by Mr. A. W. Robinson, and built by Atlantic 
Equipment Co., Ill Broadway, New York City. Hull, 148 ft. long, 38 ft. wide, and 
10 ft. 6 Ins. deep. Main pump has 30-in. suction and discharge, and main engines 
are of the triple expansion marine type of 1200 I. H. P There are two double- 
ended marine boilers 1 1 ft. 6 Ins. x 1 8 ft. long with eight corrugated furnaces. The 
suction pipe is carried by a very strong steel frame, and is fitted with a powerful 
cutter for digging the clay. A capacity of 3000 cu. yds, per hour In clay Is said to 
have been reached. — Eng.-Contr., Dec. 25, 1907, p. 356. 

EXCERPTS AND REFERENCES. 

Methods of Measuring Dredged Material in the Channel Improve- 
ment of the Delaware and Schuylkill Rivers (C. H, Ott. Proc. Engrs. Club 
of Phila,, April number; Eng. News, July 3, 1902). — Illustrations of sound- 
ing raft, submerged sweep used with rowboat, and sweep and derrick used 
with diver. Description and methods used. 

An Improved Centrifugal Dredging Pump (By M. M. Patrick. Eng, 
News. July 9, 1903). — Illustrated. 

The Electrical Equipment of a Gold Dredge (By R. L. Montague. 
Eng. News, July 16, 1903). — Illustrated, 



1 



932 56.— DREDGING. 

Cutters for Hydraulic Dredges Working in Hard Material (Eng. News, 

June 15, 1905). — Used at Alameda, Cal. Illustrated. 

The Work of a Ladder Dredge and Belt Conveyor System on the Fox 
River, Wisconsin (By L. M. Mann. Eng. News, Oct. 25, 1906). — Illustrated. 

Dredging Operations at Warroad, Lake of the Woods, Minn., by U. S. 

Gov't (By Emile Low. Eng. News, Nov. 29, 1906). — Cost data. 

The Fruhling System of Suction Dredging (By John Reid. Eng. 
News, Mar. 5, 1908), — Table of comparison of this type with American 
hydraulic hopper dredges. 

Large Elevator Dredge for Work in Boston Harbor (Eng. News, Jan. 27, 
1910). — Capacity of each bucket is Ij cu. yds. and the bucket chain is 
driven ordinarily at a rate of 14 buckets per min., so that the dredging 
capacity with full buckets would be 1,100 cu. yds. per hour. This has been 
exceeded by more than 30% for shorter periods in actual use. For driving 
the bucket chain, a double tandem compound steeple engine is provided, 
with cylinders 12x16 ins. by 18-in. stroke. This is entirely separate 
from the engines for driving the vessel's propeller shaft. The ladder 
frame is of steel and is of sufficient length to work at a depth of 51 ft. 
After a year's service, the buckets with their pins and bushings are re- 
ported in good condition and the general loss of time and cost of repairs is 
said to compare favorably with that of a dipper dredge on the same class 
of work. The material encountered at different parts of the channel in- 
cluded hardpan, clay, stone and gravel. The ordinary yardage for a single 
day's work was about 8,000, and under favorable conditions as much as 
10,800 cu. yds. 

Working Costs of Gold Dredging in California (By Charles Janin and 
W. B. Winston. Mining and vScientific Press, July, 1910). — The total oper- 
ating expense per cu. yd. varies from 9.23 cts., for difficult digging, down to 
2.30 cts. for easy digging (fine gravel). 

Illustrations. 

Description. Eng. News. 

Light-draft stem-wheel suction dredge for Niger river June 16, '10. 

Eng. Rec. 
Gold dredging and rock crushing in California July 16, '10, 



57— TUNNELING. 

Definitions. — In a broad sense, a tunnel is an artificial underground 
passage; and tunneling treats of the methods of construction, embracing 
the processes of excavation, timbering and lining. Either or both of the 
last named may sometimes be omitted as when the excavation is through 
hard, solid rock. Tunnels are usually horizontal or nearly so; but they 
may be inclined in any direction: a vertical tunnel is called a shaft. For 
subsequent enlargement, a small tunnel projected ahead of the full section 
is called a heading if in upper part of section,* and a drift if at bottom of 
section. A loop heading is a small auxiliary tunnel loop from the main 
section. In blasting, the quantity of rock which should be loosened by a 
correct charge of powder is called the burden. Muck is a term applied 
to the excavated material (including broken rock) from a tunnel. 

Kinds of Tunnels. — Railway-, highway-, subway-, aqueduct-, mining-, 

drainage-, sewer-, ventilation-, sub-aqueous-. 

Methods of Tunneling. — ^There are many methods in use — some of them 
distinctly national or provincial — dependent upon the class of material to 
be excavated, the machinery available, the position of the tunnel, and local 
circumstances: 

Open-cut, dry tunneling — for subways, etc. 
Through, dry tunneling (including the deep subaqueous-). 
In solid rock, by headings or by drifts. 

In common earth, by driving full section at once; by driving either 
the upper or the lower half first; by leaving a middle core sub- 
pillar to be taken out last. 
Through, wet tunneling — not subaqueous. 

In quickstand, by shield or by freezing process. 
Subaqueous tunneling — not deep. 

By shield — ^with or without compressed air. 
Surface-bottom tunneling — ^by dredging, and — 
Constructing coffer dams. 
Sinking caissons. 
Aqueous "tunneling." (Proposed for English Channel but not adopted.) 
Tubes suspended, supported, or sub-floated in water. 

Open=Cut Tunneling. — Much of our modem subway construction is 
done, in part at least, by the open-cut method, if the required cutting is not 
very deep. The sections to be built are opened up in convenient lengths of 
say 12 ft. or more, and the street traffic is supported on wooden framework 
during the processes of excavating and lining the tunnel. The construction 
is somewhat analogous to that of a big sewer, but much more intricate. 
The sides of the cut may be shored up by sheet piling, say 2\ ins. thick, 
driven behind longitudinal girts braced apart by transverse struts of 10 x 
12 in. timbers with single or double wedges at the ends (see Fig. 1). 

Tunnel portals also niay be constructed in open- 
cut as well as by the shield method. In Europe the r^ 
usual practice is to construct the portals first, while % 
in the United States they are finished after the main M 

tunnel is completed. -W 

Open cuts were formerly' very rare at depths *|v 

greater than 65 ft., for any considerable distance, but ^% 
recently the practice is quite common to run them 
up to about 80 ft. in depth. Fig. 1. — Shoring. 

*'Heading=" vs. "Drift=" Method.— The first named is the distinctive 
feature of American tunneling. A wedge of rock is first blasted out on the 
center at the top of the section, by drilling two rows of holes with the drills 

* Sometimes the term "heading" is used in a broad sense as applying to 
any advance tunnel smaller than the full section. 

933 



Shoring 



2. 



934 



57.— TUNNELING, 



converging toward each other at their points. This "center cut" is then 
widened laterally into a "heading" from which the bench below is easily- 
worked by vertical drilling supplemented by horizontal drilling at the sides 
of the section. Sometimes a central core is left to support the centering 
temporarily. There are various modifications of the above. They differ 
essentially however from the prevailing European method of first driving a 
"drift" at the bottom of the section and working upward. In either method, 
where possible, the work progresses in benches so two or more drilling gangs 
may be employed at the same time. Headings and drifts are often advanced 
a few hundred feet beyond the main, full section — sometimes a thousand 
feet of more. 

Drilling and Blasting. — Hand drilling, and even steam drilling, is being 
supplanted by drills operated with compressed air or electricity. Where 
the necessary water power is available hydraulic drills are used, and some- 
times show great economy over any other power; at the same time each 
drill is supplied with a jet of water for cleaning out the hole and laying the 
dust in the tunnel. Compressed air has the advantage of incidentally 
ventilating the tunnel. Electricity becomes economical as the depth of 
tunnel increases from the face or portal. 

Drills may be mounted on tripods, on cars, on columns (one or two 
drills to each column) which brace against the top and bottom of the tunnel 
and have a wide range of action, on bars which brace against the sides of 
the tunnel, etc. The size of an air drill is denoted by -the inner diameter 
of the cylinder. 

Dynamite of high grade, say 75% down to 60 or 50, is used in center- 
cut blasting in hard rock, as granite, trap, basalt, syenite, gneiss, etc., 
while for soft or seamy rock and for trimming up on the sides of the tunnel 
a lower grade of dynamite, say 40%, is better. 

The "Radialaxes" channeling machine, manufactured by the Ingersoll- 
Rand Co., has been used to some extent, and its general efficiency will be 
watched with interest. 

Timbering. — In hard, solid rock, timbering is not required and the 
finished tunnel is often left rough without lining. But in loose, seamy 
rocks and in the common soils, the sides and roofs have to be supported. 
Round timbers are generally used for timbering because they are cheaper, 
although sawed timbers are more easily framed and handled. In very 
loose material, lagging or sheeting is driven longitudinally of the tunnel 
behind transverse, segmental girts, or it may be driven transversely behind 
longitudinal girts. The girts are supported by timber props or struts, size 
about 12x12 ins. if sawed. Another method is to omit the lagging and lay 
the longitudinal girts close together, supported by segmental ribs of timbers. 
If these segmental ribs are placed close together even the girts may be 
omitted. Figs. 2 and 3 show simple types of rafter timbering in small 
tunnels in which the lower space is left open. 




Fig. 2. 



Fig. 3. 



Lining. — ^The lining may be of timber, brick, stone masonry, concrete, 
etc. In subaqueous work the lining is frequently of cast iron, steel or rein- 
forced concrete. The thickness of plain concrete or brick lining in the upper 
or arched section will of course depend largely upon the kind of material 
through which the tunnel is driven. In loose material of considerable depth 
the thickness of the concrete lining may be assumed at not less than 2 ft. for 
a single track railroad tunnel, and somewhat more for a double track 
tunnel. This thickness increases gradually in the lower bench walls, 



TIMBERING AND LINING. 



935 



below the middle of the section, to about 150% neat thickness at the foot- 
ings. Fig. 4 is a section of the *Gallitzin tunnel through the Alle- 
ghenies on the Pennsylvania R. R j -..^c i -r 

The foundation footings are sometimes very expensive and difficult, it 
through caverns, seamy rock or soft material If the last named is en- 




Fig. 4. 

countered the side walls are frequently continued below the track in the 
form of inverts or inverted arches, of concrete; and "weep holes" should 
be left in the lower part of the walls to drain off the surplus of outside 
water, which if allowed to accumulate might cause undue pressure on the 
lining by softening the sustained material. 

Alinement and Grade. — ^The following remarks apply particularly to 
railroad tunnels: 

A straight alinement conduces to economy in both length and sectional 
area of tunnel; and in cost of excavation, timbering ai;d lining, per unit 
quantity of material as well as in total. It also minimizes cost of operation, 
and reduces risk of accident. Curves may be introduced when, other con- 
siderations outweigh the above, or from necessity. 

Grades are absolutely essential in long tunnels, for drainage. As tunnels 
are located generally at the highest points of the lines the grade summits are 
fixed usually near their centers, so that in driving the tunnels up-grade 
from each end they are drained naturally. If it is assured, however, that 
the prevailing heavy freight traffic will be in one direction it might be better 
to place the whole tunnel on a down-grade in that direction, thus reducing 
the smoke nuisance (aiding ventilation) and the chance of trains being 
stalled. For drainage, a grade of about 2 ft. per mile is the minimum — 
10 ft. per mile is better. 

Ventilation. — ^The greatest difficulties met with in the ventilation of 
ordinary railroad tunnels are during their construction. It is estimated 
that each workman fouls about 25 cubic feet of air per minute; and a horse 



See Eng. News, Dec. 6, 1906. 



936 57.— TUNNELING. 

5 or 6 times as much. If we add to this about 10 cubic feet per man for 
fouHng by dust and gases from explosions, we have a basis for estimating 
the minimum number of cubic feet of air which must be supplied at the head- 
ing. This must be increased when men are working in different parts of the 
tunnel and also when locomotives are employed to handle the cars. The 
purity of the atmosphere can be aided materially by the use of compressed 
air drills and also by the use of the water jet in the drill holes and by water 
sprinkling elsewhere to lay the dust. The carbonic acid gas exhaled in 
breathing may be augmented by "fire damp" or carburetted hydrogen gas 
which lies pocketed in rocks in the coal measures, sometimes encountered 
by railway and other tunnels. 

Vertical shafts in tunnel construction are sometimes sunk to increase 
the number of headings and push the work, but the cost of sinking them 
and the subsequent cost of handling tunnel excavation through them is 
very great. These shafts are left open for ventilation after the tunnel is 
completed. It is a serious question however whether they are of much value 
for this purpose. The best ventilator is an express train running down- 
grade through the tunnel and emitting little or no (black) smoke, in which 
case the tunnel shaft is not only of no aid but a positive hindrance to clear- 
ing the tunnel. A central shaft is sometimes beneficial if located in about 
the middle of a very long tunnel which point also happens to be the summit 
of two ascending grades; but usually where the whole tunnel is on one 
grade the shaft is rather a drawback. Artificial ventilation is accomplished 
usually by forcing air through pipes to the center of the tunnel, or by 
suction at the ends (through closed doors), or both. With the advent of 
electricity in the operation of trains the difficulty disappears. 

"Shield" Method. — This method is employed in boring subways and 
subaqueous tunnels. The shield consists of a cylindrical steel shell with a 
cutting edge pressed against the head of the tunnel to be excavated. A 
detachable hood is provided, when boring through gravel and sand, to 
replace the upper and side portions of the cutting edge. As the excavation 
progresses, the whole shield is pressed forward by hydraulic jacks or shoving 
rams; and in order to preserve true alinement in the finished tunnel it is 
necessary to steer the shield by exerting more pressure on some of the jacks, 
thereby giving the shield a certain "lead" or angular direction, either later- 
ally or vertically, or both. It is sometimes necessary also to give the shield 
a constant lead in one direction in order to produce a true alinement when 
passing through certain classes of material. For subaqueous tunneling, 
where compressed air is used, the shield is a very complicated machine, and 
the progress of the work hinges largely on its proper design.* 

"Dredging" Method. — ^This method was employed by McMullen & 
McBean.t sub-contractors, in the construction of the west half of the river 
portion of the Harlem River Tunnel, New York City, for the Rapid Transit 
Railroad. Mr. McBean describes the work as follows: 

First a channel was dredged across the river bottom to within a few feet of the 
lull depth of excavation required to build the tunnel. In this channel, foundation 
piles and a row of specially prepared heavy timber sheeting, along each side and 
across the ends, were driven and cut off to a true plane about 25 ft. below the surface 
of the water. This sheeting forms the sides and ends of a pneumatic working chamber. 
For the roof of this chamber a platform of timber 40 ins. in thickness and extending 
the full width and length of the tunnel section, was built and sunk and rested on 
the cut off sheeting which formed the sides and ends as above described. Simultane- 
ously with pumping the water from under this roof, compressed air was forced into 
the chamber under a pressure corresponding to the pressure of the water above the 
roof. Inside this chamber the west half of the tunnel was built and then the timber 
roof was removed. 

"Caisson" Method. — ^This consists in building and sinking caissons to 
the required level below the surface of the river bed and later connecting 
them together so that they form sections of the completed tunnel. See Eng. 
News of Feb. 15, 1906, and April 11, 1907, for illustrated description of 



* See articles on "The Construction of the Pennsylvania R. R. Tunnels 
Under the Hudson River at New York City," by James Forgie, in Eng. News 
of Dec. 13, 1906, and Feb. 28, 1907. 

t Mr. D. D. McBean is credited with the design of this method of tun- 
neling. 



METHODS OF TUNNELING. 937 

this method, proposed by Mr. L. Chagnand, a Paris contractor, and applied 
in the construction of a tunnel crossing the Seine on Line 4 of the Metro- 
politan railway of Paris. It may be noted here that in connecting the 
caisson sections in place, the "Freezing" method is proposed. The first 
caisson sunk measured 54 ft. wide, 41 ft. high, 118 ft. long, and weighed 
617.000 lbs. without the cast iron lining. See also Le Genie Civil, Dec. 2, 
1905. 

Performance of Work. — References are given here to recent tunnel con- 
struction under the most approved methods; particular pains being taken 
to bring to the attention of the young engineer the various classes of work 
in tunneling. 

Cost of Building Subways in New York City. — Nearly four pages devoted to 
detailed costs of about i-mile section of Brooklyn extension. Cost, including plant, 
about $810,000. Some of the unit costs are: Eartli excavation, $3.62 per cu. yd.; 
rock excavation, $5.71 per cu. yd.; concrete for foundations, $4.61 per cu. yd.; 
concrete for arches, $7.69 per cu. yd.; steel work, about 3i cts. per lb. in place; 
brick backing, about $24.59 per cu. yd. — Eng.-Contr., Feb., 1906, p. 29. 

Cost of a Small Rock Tunnel, 9 ft. diam. and about 3000 ft. long. About 2.46 
cu. yds. of excavation, in hard trap rock, per lin. ft. of tunnel. Total cost, including 
plant depreciation, repairs, etc., $9.36 per cu. yd. or $23.02 per lin. ft. 75% dyna- 
mite was used. Data by F. L. Pruyn. — Eng.-Contr., May 16, 1906. 

The Catskill Aqueduct of the Additional Water Supply System of New York 
City. — Over three pages of descriptive matter, with illustrations of standard types 
of aqueduct: in dry loose earth, on embankment, for grade tunnel (timbered and 
untimbered rock), and for Hunter's Brook siphon (single and double ring sections). — 
Eng.-Contr., Dec. 19, 1906. 

Chicago (Lawrence Ave.) Intercepting Sewer. — Length of land portion. Lake 
Michigan to Chicago River, 8220 ft., through clay. Internal dia. of lining, 16 ft.; 
external dia. of shield, 20'-l''; brick lining in 4 rings, backed with solid timbering 
8" thick. Contract price, $79.50 per lin. ft. Article contains about 2^ pages of 
descriptive matter relating to methods and cost, with illustrations of draw knives 
and shield.— ^wfir.-Conir., Feb. 6, 1907. 

Progress of Work in Lowering the Tunnels under the Chicago River. — ^About 
two pages of descriptive matter, illustrated. Tunnels at Van Buren, Washington 
and La Salle Sts. to be lowered to meet Federal Government requirement for 26-ft. 
depth Instead of 17-ft. depth heretofore maintained. — Eng. News, Sept. 13, 1906, 
and May 7. 1907. 

Pipe Subways in British Cities and in Paris. — Over four pages, with numerous 
Illustrations. — Eng. News, Mar. 14, 1907 

Blaw Collapsible Steel Centering for Concrete Conduit Work. — Illustrated. 
System used for sewer work. — Eng.-Contr., Feb. 13, 1907. 

Method of Relining the Hodges Pass Tunnel, Oregon Short Line Ry. — Two 
pages with illustrations. — Eng.-Contr., Feb. 20, 1907. 

Highway Tunnel under Thames River, at London, Eng. — Four pages with 
copious illustrations including plans of shield — Eng. News, Dec. 19, 1907. 

Methods of drilling and Mucking in a Rock Tunnel. — Abstract of article in May, 
1907, issue of Mine and Quarry,* by W. P. J. Dinsmore. An interesting article of 
over two pages, illustrated, treating of the plan of work, arrangement of drill holes, 
handling the muck, ventilation, etc. — Eng.-Contr., Jan. 1, 1908. 

Some Notable Tunnels. — ^The following data have been compiled from 
various sources: 

Simplon Tunnel. — Twin single-track R. R. tunnels Nos. 1 and 2, 55.76 ft. apart 
centers. Through Alps between Baffl. Switzerland, and Iselle, Italy, on line of Jura- 
Slmplon Ry. Length 19729 meters= 64728 ft. = 12.26 miles. Tunnel No. 1 con- 
structed (1898-1905) with clean width of 14.75 to 16.4 ft., and height above rail, 
18 ft.; net sectional area, 250 sq. ft.; cost $180 per lin. ft.; drift method; thickness 
of arch ring, 1 4 to 24 inches. Tunnel No. 2 was simultaneously put through as a head- 
ing to provide for ventilation and drainage; to be enlarged later. 

Saint Gothard Tunnel. — (1872-82.) Double-track R. R. Between Goschenen 
and Alrolo, Switzerland. Length 48887 ft. = 9.26 miles. Clear width 26.24 ft.; clear 
height 19.68 ft. Cost $246 per lin. ft. 

Mont Cenis Tunnel. — ( 1 862-7 1.) Double track R. R. Between Fourneau, France, 
and Bardonn^che, Italy. Length 42157 ft.= 8 miles. Clear width 26.24 ft.; clear 
height 20.6 ft. Cost $380 per lin. ft. 

Ariberg Tunnel. — (1880-84.) Double-track R. R. Austria-Switzerland. Width 
26.40 ft.; height 25 ft. Cost $208 per lin. ft. 

Hoosac Tunnel. — (1854-76.) Double-track; Troy and Greenfield R. R., Mass. 
Length 25081 ft. = 4.75 miles. Clear width 26 ft.; clear height 21 ft. Cost about 
$400 per lin. ft. 



* Published by the Sullivan Machinery Co., Chicago, 111. 



938 h7.— TUNNELING, 

Cascade Tunnel. — (1897-1900.) Single track, through Cascade Mts., on Great 
Northern Ry. Length 13813 ft. = 2.62 miles. Clear width 16 ft.; clear height above 
base of rail. 21.5 ft. Concrete lining, 2 to 3i ft. thick, replacing temporary timber 
lining. 

Stampede Tunnel. — (1886-88.) Single track, through Cascade Mts., on Northern 
Pacific R. R. Length 9850 ft.= 1.87 miles. Contract price $118 per lin. ft., without 
masonry lining. 

Busk Tunnel. — (1890-93.) Single track, through Rocky Mts., on Colorado Mid- 
land R. R. Length 93 95 ft. = 1.78 miles. Clear width 15 ft.; clear height 21 ft. 

Musconetcong Tunnel. — (1872-7 5.) Double-track R. R., on Easton and AmtH:)y 
(L. V. R. R.). Length 4879 ft. ; clear width 26 ft. ; clear height 21 ft. [H. S. Drinker, 
author of "Tunneling," was a resident engineer on this work.] 

EXCERPTS AND REFERENCES. 

A Proposed Ventilating System for the Park Ave. Tunnel, N. Y. City 

(By A. H. Gary. Eng. News, Aug. 8, 1901). — Illustrated. 

Subaqueous Tunnel Siphons of the Mass. Pipe Line Gas Co. (W. W. 

Cummings. Jl. Assn. of Eng. Sec, June, 1901; Eng. News, Oct. 3, 1901). — 
Illustrated details. 

Difficult Work in Repairing a Swiss Railway Tunnel (Eng. News, 
Sept. 25, 1902). — Shows, by illustration, method of centering for caving 
roof. 

Freezing Process for Building the River Tunnels of the P. R. R. at 

N. Y. City (By Charles Sooysmith, Inventor. Eng. News, Dec. 4, 1902). — 
Illustrated. 

Methods of Work Adopted in Constructing the Chicago Telephone 
Tunnels (Eng. News, Feb. 19, 1903). — Illustrated. 

Tunnel at Michel Creek Loop, Crow's Nest Pass Line, Can. Pac. Ry. 

(By C. R. Coutlee. Eng. News, April 2, 1903).— Illustrated. 

Construction of the Simplon Tunnel (Eng. News, Aug. 13, 20, 27 
1903). — Complete description of the tunnel and its construction. Illus- 
trated. 

The Pennsylvania R. R. Tunnel Under the North River, N. Y. City 
(Eng. News, Oct. 15, 1903). — Complete detailed description of method of 
construction. Illustrated. 

The East River Division of the Penn. R. R. Tunnel, at N. Y. City 
(Eng. News, Oct. 29, 1903).— Illustrated. 

The Cost of Concrete Tunnel Lining and of Tunnel Excavation (By 
Geo. W. Lee. Eng. News, Dec. 17, 1903). — Illustrated sections with cost 
data. 

Remarkable Progress of the Hudson River Tunnel for the N. Y. & 
N. J. R. R. Co. (Eng. News, Nov. 10, 1904). — Illustrations of shield. 

Waterproofing the P. R. R. Tunnels at New York (Eng. News, 
June 29, 1905). — Specifications. 

Concrete Stringers and Tracks in Mine Shafts (Eng. News, Jan. 25, 
1906). — Illustrated. 

The Ventilation of Tunnels (By C. S. Churchill. Trans. A. S. C. E., 
Vol. LVII). — Includes the ventilation of subways. 

Tunnel Lining Work in the Far West (Eng. News, Dec, 6, 1906).— 
Descriptive and illustrated article on lining and relining; masonry and 
timber. 

The Construction of the P. R. R. Tunnels Under the Hudson River 
at N. Y. City (By James Forgie. Eng. News, Dec. 13, 20, 1907). — Complete 
Description of the tunnel and its construction. Plans of shield. Tables 
VI and VII (Eng. News, Feb. 28): Comparative Statement Giving Partic- 
ulars of Some of the Principal Tunnels (Railroad and Other Than Railroad) 
Constructed Under Waterways. Very comprehensive, 

Alpine Railway Tunnel Data (Eng. News, Dec. 5, 1907) —Table, 

giving: Particulars of the Four Principal Alpine Railway Tunnels. 

Records in Rock Tunneling (Eng. News, April 2, 1908). — Shows 
progress figures — greatest advance made in any one month, in feet, for a 
single heading. See, also, Eng. News, Nov. 19, 1908. 



MISCELLANEO US DATA. 9 39 

Some Detail Tunnel Costs in Tunnel No. 7, Los Angeles Aqueduct (By 

C H. Richards. Eng. News, Nov. 18, 1909).— Finished cross-section of 
rock tunnel, with 8-in. concrete lining, 8 ft. 6 ins. wide and 10 ft. 6| ins. high 
at center. Over-breakage was 17%, making 6| cu. yds. broken material 
per ft. of tunnel. Total cost of tunnel, timbered and ready for lining, 
$16,432 per lin. ft. Drilling. — Done by No. 7 Leyner drill, water being 
forced through the hollow steel. Drill used approximately 66 cu ft. free 
air per min. at 83 lbs. pressure, drilling holes to 10 ft. in depth. During 
al5-dayperiod 150holeswere drilled, aggregating 1203.3 ft. in length; average 
depth of hole being 8 ft. Average speed of drilling was 21.74 ft. hole per 
hour of actual drilling time, or 15.84 ft. per hour if lost time is included: 
which means that the average 8-ft. hole was drilled in 22 mins. Average 
cost of drilling was 3.6 cts. per ft. of hole. Explosives. — There were used 
650 lbs. of U-in. 40% gelatine; 250 lbs. 1-in. 40% gelatine; and 150 lbs. 
1-in. 60% gelatine; a total of 1,050 lbs., or 11.6 lbs. per lin. ft of tunnel, or 
3.3 lbs. percu. yd. place measurement. Cost (del.) was $140.87. Adding 
cost of fuse (2,700 ft., $13.85). caps (306=$2.49), and tamping stick ($0.48), 
makes the total cost $157.69, which is about $0.50 per cu. yd. place measure- 
ment and $0.27 per cu. yd. loose. Tables of detailed costs are not reproduced 
here. 

The Use of Steel for Mine Timbering (By R. B. Wood worth. Paper, 
"Mine Timbering in Steel," before W. Va. Coal Mining Inst., Dec. 7, 1909; 
Eng. News, Feb. 24, 1910). — Illustrated. 

The New York Tunnel Extension of the Penn. R. R. (Trans. A. S. C. E.. 
Vol. LXVIII., Sept., 1910).— Following papers presented:— 
Paper No. 1150. General Discussion. By C. W. Raymond. 

'• 1151. The North River Division. By C. M. Jacobs. 
" 1152. The East River Division. By Alfred Noble. 
" " 1153. Meadow Div. «,nd Harrison Transfer Yards. By E. 
B. Temple. 
" 1154. The Bergen Hill Tunnels. By F. Lavis. 
•' 1155. The North River Tunnels. By B. H. M. Hewett and 

W. L. Brown. 
" 1156. The Terminal Station-West. By B. F. Cresson, Jr. 
*• 1157. The Site of the Terminal Station. By G. C. Clarke. 
" 1158. The Cross-Town Tunnels. By J. H. Brace and F. 

Mason. 
•• 1159. The East River Tunnels. By Brace, Mason and 
Woodward. 

Illustrated Cross=-Sections of Some Tunnels: — 

Description. Eng. News. 
Lining of Musconetcong tunnel during traffic Nov. 7, 1901. 

Typical cross-sections Albina tunnel, Switzerland Dec. 19, '01, 

St 'd tunnel sections for Colo. Springs & Cripple Cr. Dist. Ry. May 1, '02. 

Subway construction, Long Island R. R. May 22, '02. 

Cross-section of Pryor Gap tunnel, Mont., B. & M. R. R. July 3, '02. 

Cross-sections of telephone tunnels in Chicago J^ly 17, '02. 

Subway construction in N. Y. City Sept. 18, '02. 

Subway construction in N. Y. City Feb. 12, '03. 

Example of wide arch soft ground tunneling June 4, '03. 

Concrete arch subway construction in open cut June 11, '03. 

Lining of Gallitzih tunnel, P. R. R., with concrete Sept. 24, '03. 
Subway construction under Harlem River, N. Y. City Oct. 1 and 8, '03. 

London underground railway tunnel in soft ground Oct. 29, '03. 

Subway construction in N. Y. City Nov. 19, '03. 

Example of wide arch rock tunnel at Genoa, Italy Nov. 26, *03. 

Cross-section of Winston tunnel, Chi. Gt. Western Ry. Aug. 4, '04. 

Cross-section of P. R. R. tunnel under Capitol Hill, Wash. D.C. Sept. 14, '05. 

Water works tunnels. Salt Lake City Dec. 14, '05. 

Proposed method of tunnel construction by sinking caissons Feb. 15, '06. 

Proposed tunnel under Detroit River for Mich. Cent. R. R. Feb. 15, '06. 

Subways in London in connection with street improvement July 12, '06. 

The tunnel work of the P. R. R. under East River, N. Y. City July 12, '06. 

Concrete blocks for tunnel lining. Mex. Cent. Ry. July 26,' 06. 

Cross-sections of pipe subways in London Mar. 14, '07. 

New water supply tunnels at Chicago Nov. 14, *07. 



940 57.— TUNNELING. 

Description. Eng. News. 

Cross-section of Cascade tunnel, for electric system Nov. 18, '09. 

Lining and grouting tunnels in water-bearing material Nov. 25, *09. 

Railway tunnels; cross-sec, grades, lining, drainage June 2, '10. 

Reconstruction Washington St. Tunnel under Chicago river July 21, '10. 

Typical sections of timnels, Catskill aqueduct Oct. 20, '10. 

Eng. Rec. 

Concrete -lined tunnel, 8' x 9', Los Angeles aqueduct July 3, '09. 

Sections of the Washington St. timnel, Chicago Dec. 11, '09. 

The Bergen Hill 4-track tunnel, Erie R. R. Dec. 18, '09. 

Sections of Main St. Subway, Cambridge, Mass. Jan. 1, '10. 

Relining railroad tunnel with cast-iron segments Jan. 1, '10. 

Terryville tunnel, double track, N. Y., N. H. & H. R. R. Feb. 26, '10. 
Construction of tunnel for Great Western Power Plant * July 16, '10. 

Section of tramway and pipe subway in Kingsway, London Dec. 10, '10. 



58.— SURVEYING, MAPPING AND 
LEVELING. 

Care of Instruments. — The correct use of surveying instruments takes 
into consideration their lack of precision. No instrument, however carefully 
it may be adjusted in the shop or field, will remain permanently accurate 
with ordinary field use. New instruments hold their adjustments better 
than old ones because the parts of precision are less worn. With good care 
in handling, the life of an instrument can be doubled without decreasing 
its duty. For the Level, there is but one main object of adjustment, namely, 
to preserve a horizontal plane of sight around a vertical axis. For the plain 
Transit it is necessary, in addition to the above "horizontal plane" adjust- 
ments, to preserve a vertical plane of sight passing through the point of the 
suspended plumbbob. Note that when the instruments are out of adjust- 
ment these "planes" of sight are warped into "cones" of sight, with the 
apex of the cone at the center of the telescope; hence the adjustments con- 
sist in changing "cones" into "planes" of sight. 

To Adjust the Level.* — In running a line of levels, if the back-sights and 
fore-sights are of equal length the results of the leveling will be accurate 
even if the instrument is out of adjustment, as shown in Fig. L Thus, the 
elevation of each of the turning points at 
R will be accurately determined, but the 
"height of instrument" in each case will 
be wrong because the lines of sight are 
not level. For general leveling, however, 
the instrument should be adjusted so that Fig. 1. 

all sights will be level. 

First, the cross-hairs, or line of collimation. — "Set up" the level firmly 
(although it need not be accurately leveled) with one diagonal pair of leveling 
screws / in line with the telescope. Open the clips c c. Fig. 2, so the tele- 



I t 




Fig. 2.— The Level. 

scope will be free to revolve in the Ys. Adjust the eye-glass e by the milled 
head e' , so the cross hairs appear distinctly. Sight the telescope T on a 
distant point (at the intersection of horizontal and vertical lines) about 
250 to 300 feet away, as this is the usual length of sight for accurate leveling. 
Move the object glass o by the milled head o' so that the object is seen 
distinctly without "parallax" (i. e., without the cross-hairs "dancing" or 
appearing to move away from a straight line of sight). With the bubble 
tube below the telescope, as shown in the illustration, move the line of 

* See also "peg-adjustment" described in the fourth adjustment of the 
transit, p. 944. 



941 



942 



58,— SURVEYING, MAPPING AND LEVELING, 



sight, by means of the leveling screws / and the tangent screw t, so the 
intersection of the cross-hairs will cover the distant point. Now, theoretic- 
ally, if the telescope is revolved in the Ys by revolving the bubble tube 
completely around it, the cross-hairs are in adjustment if their intersection 
continues to cover the distant point. That is, the line of sight is along the 
central axis of the telescope and pierces the center of the object glass. 
When either cross-hair leaves the distant object it is out of adjustment. 
The cross-hairs are stretched across a circular, movable ring or diaphragm, ■ 
called the reticule, held in position by four adjusting screws 5. To move the 
horizontal cross-hair upward, loosen the top screw and tighten the bottom 
one. (See also Fig. 6.) The reverse operation will lower it. Similarly, the 
vertical cross-hair can be moved laterally by the two side screws. These 




Fig. 3. — ^Level Telescope. 

screws are turned with capstan or adjusting pins. To adjust the horizontal 
cross-hair bring it on to the distant object, with the bubble tube below the 
telescope; revolve the telescope half round, in the Ys, bringing the bubble 
tube above; correct one-half the variation from the distant object, by operat- 
ing the top and bottom screws s. Repeat until accurately adjusted. The 
vertical cross-hair can be adjusted in the same manner. 

Second, the bubble tube. — With the clips c open as before, level the 
instrument and lightly clamp the telescope over one set of leveling screws /, 
by means of the tangent clamp. Now level the bubble B accurately, gently 
lift the telescope out of the Ys and set it back reversed, end for end. If the 
bubble is still level it is in adjustment. If it moves either way from a 
central position it is out of adjustment. Correct one-half the variation from 
the central position with the leveling screws and the balance by operating 
the capstan nuts n at the right-hand end of the tube as shown in Fig. 2, 
thus rising one end of the tube to a level position. Repeat (after leveling 
the instrument again with the leveling screws /) until the bubble remains 
central in the tube upon reversing the telescope in the Ys. 

Third, the Ys. — In the two previous adjustments, we have fixed the 
line of sight centrally through the telescope, and adjusted the bubble tube 
parallel with it. It now remains to adjust the Ys, in which the telescope 
rests, to the same level, so that when the instrument is leveled up the 
vertical center pin will be truly vertical, and the line of sight truly horizontal 
in any direction. To do this, level up the instrument carefully, and start 
with the telescope over one set of the leveling screws /. With the telescope 
clamped firmly in the Ys, swing half round on the vertical center pin so the 
telescope will rest over the same screws but reversed in direction. Correct 
one-half the variation of the bubble, from the central position, with the 
leveling screws /, and the balance by means of the large capstan nuts iV, 
operated with large adjusting pins, and similar to the nuts n described in 
the second adjustment. (Level up and) repeat, with the same screws, 
until adjustment is effected. The adjustment may be repeated over the 
other set of screws in order to test the workmanship of the vertical center 
pin and bearings, but it is rarely done. 

To Adjust the Transit. — ^Even with a transit considerably out of adjust- 
ment, leveling can be done by using equi-distant back- and fore-sights; 
straight lines can be run by half-revolving the alidade (commonly called 



LEVEL AND TRANSIT ADJUSTMENTS. 



943 



"turning half around") between reversals of the telescope; and angles can 
be measured by the method of continuous readings (called "dQubling") 
and, if the "vertical" cross-hair is not perpendicular, by proper inclination 
of the telescope. But the only safe and proper way is to keep the instrument 
in adjustment. There are four or five principal adjustments, as follows: 
First, the plate-bubbles. — ^The two bubbles b b are usually set at right 
angle with one another, one on a pair of standards and the other on the 
upper horizontal plate. When the bubbles are adjusted and are central in the 
tubes, the central axis or center pin of the transit is vertical. To adjust 




Fig. 4.-— The Transit. 

either of the bubbles, revolve the alidade so the bubble will be parallel 
with one set of diagonal leveling screws / /; level the instrument; revolve 
the alidade 180° and correct ow^-/ta// the variation of the bubble from a 
central position, by the leveling screws /, and the other half by raising 
or lowering one end of the tube, using the adjusting pin in the capstan 
headed screw or nut at one end. Repeat until the bubble remains central in 
the tube when the alidade is revolved. The other bubble is adjusted 
likewise. 




Fig. 5.— Transit Telescope. 

Second, the standards. — In order that the telescope will revolve about 
its axis in a true vertical circle, the standards 5 must support the axis of the 
telescope in a perfectly horizontal position. In some instruments there is 
provision for raising or lowering one end of the axis in the standard, while 
in others there is no adjustment possible without returning the instrument 
to the shop. To test the standards, level the instrument, sight on some 
high point with the instrument clamped against lateral movement, revolve 





944 58.— SURVEYING, MAPPING AND LEVELING. 

the telescope vertically downward and fix a low point on the line of sight. 
Now revolve the alidade 180°, sight again on the high point and get another 
low poin£ beside the first one. A point midway be- 
tween the two low points will lie in a vertical plane 
passing through the high point. The adjustment 
can then be made on the standard, using these two 
fixed points. 

Third, the vertical hair or line of collimation. 
— By the two preceding adjustments we have 
secured a vertical axis for the alidade, and a hori- 
zontal axis for the telescope. It now remains to ad- 
just the vertical cross-hair in the telescope so that 
in revolving through a vertical circle the Hne of 
sight will describe a vertical plane instead of a Fig. 6. 

"cone," or in other words, so that a straight line may be produced by 
"reversing the instrument" (telescope). Set up the instrument on fairly 
level ground. Sight on a fixed point a, Fig, 7, reverse the telescope ver- 
tically, and set a point b on line in the opposite direction from a; revolve 
the alidade 180°, set the instrument again on a, 
reverse the telescope, and set a point c on line 
opposite the point b. Correct the vertical cross- 
hair by moving it laterally so the line of sight 
strikes d (cd = i cb). This is done by operating the 
side capstan-headed screws c which move (laterally) p,. _ 

the diaphragm ring orreticule to which thecross-hairs -^^^S* '• 

are attached. Repeat the whole operation until, on reversing, the first and 
second points (b and c) coincide at e, on the straight line produced. If the 
"vertical" cross-hair is not truly perpendicular (which can be tested by 
sighting at a plumb-string) it should be made so during this adjustment. 
This is done by loosening two adjacent screws of the reticule and tapping 
gently against their heads in the direction required. 

Fourth, the telescope bubble. — This is generally accomplished by what 
is called the "peg-adjustment," sometimes also adopted in the adjustment 
of the engineer's level. Two leveling pegs, a 
and b, are driven nearly level and about 250 ft. 
apart (Fig. 8). The instrument is set up at A, 
about a foot from the rod at a, and rod readings 
are taken at a and b, these readings being respec- ^ 
tively At, and A\,. Similarly, with the instru- " ^ 
ment at B, readings Bb and J5a are taken. Then, pjg g^ 

from the nature of the problem, 

A^-B^ = Ab-Bb±2c .'...(1) 

in which c = the inclinatiorr of the line of sight from the horizontal, between 
the two rods. Now with the instrument at 5, fix a point on the rod at a, 
at elevation J5a ±^ above the peg; sight the telescope on this point, using 
the horizontal cross-hair; and bring the telescope bubble to a level in the 
tube by raising or lowering one end of the latter, with the adjusting screws. 
Equation (1), in practice, will determine whether c is -I- or — . Repeat the 
above for a new and more refined adjustment. 

Fifth, the vertical arc or circle. — If the transit has an attachment for 
reading vertical angles, it is desirable that the angle shall read zero when 
the telescope is horizontal. To adjust the vertical arc or circle, level up the 
instrument, and then level the telescope by the attached bubble. Adjust 
the vernier (attached to the standards) so the zero mark will coincide with 
the zero of the vertical circle or arc. If the vernier is not adjustable, record 
the "index error" to be used with all vertical angles meastured. 

The Solar Attachment. — ^This is a small instrument attached to the 
upper part of the transit telescope, for determining the meridian, latitude, 
time, etc. It is used largely in government land surveying where the 
section and township-lines are supposed to nm in the direction of the 
principal points of the compass. 

Fig. 9 illustrates the solar apparatus manufacttired by the Messrs. 
Gurley, of Troy, New York. 

The circles shown represent those supposed to be drawn upon the concave 
surface of the heavens. When the telescope Is set horizontal by Its spirit level, the 
hour angle will be In the plane of the horizon, the polar axis will point to the zenith. 



THE SOLAR ATTACHMENT. 



945 



and the zeros of the vertical arc and its vernier will coincide. If we Incline 
the telescope, directed north as shown in the cut, the polar axis will d.escend from 
the direction of the zenith. The angle through which it moves, being laid off on the 
vertical arc, will be the co-latitude (90°— latitude) of the place where the instrument 




Fig. 9. — ^Transit with Solar Attachment. 

Isused, the latitude itself being found by subtracting this angle from 90°. When the 
sum is above (north of) or below (south of) the equator, its declination, or angular 
distance above or below, as given in the Ephemeris or Nautical Almanac, can be set 
off upon the declination arc, and its image brought into position as before. In order 
to do this, however, it is necessary not only that the latitude and declination are 
set off correctly upon their respective arcs, but also that the Instrument is moved 
In azimuth until the polar axis points to the pole of the heavens, I. e., is placed in the 
plane of the meridian. Thus the position of the sun's image will indicate not only 
the latitude of the place, the declination of the sun for the given hour, and the 
apparent time, but It will also determine the meridian, or true north and south line 
passing through the place where the observation is made. Allowance for declina- 
tion: By referring to the Ephemeris, and setting off on the arc the declination of 
the sun for the given day and hour, we are still able to determine Its position with 
the same certainty as if It remained on the equator. When the sun's declination is 
south (from Sept. 22 to Mar. 20) the arc is turned downward, toward the plates of 
the transit; during the remainder of the year the arc is turned upward. — From 
GuTley's Manual, page 71, etc. 



946 



58.— SURVEYING, MAPPING AND LEVELING. 



The Solar Instrument* manufactured by Keuffel & Esser, New York 
(see Fig. 10), is perhaps one of the neatest and most efficient on the 
market. Attached to the transit it may be used for (A) determining the 
astronomical meridian, (B) the latitude, and (C) the time; it serves also as 
a vertical sighting telescope in mining engineering. 




Fig. 10. — Solar Instrument. 

It consists of a small telescope with prism to eyepiece, mounted In a Y-shaped 
standard which revolves upon a vertical axis attachable to ttie telescope of the 
transit. This small telescope, called the solar telescope, is capable of rotation in 
altitude and azimuth, slow motion being imparted to it in either sense by means of 
tangent screws. The vertical axis, called the polar axis, can be inclined to correspond 
with the axis of the earth's rotation by inclining the transit telescope, to which it is 
attached, the vertical circle giving the inclination. A level which surmounts the 
solar telescope is provided with two pointers, so placed that when the shadow of one 
of them falls upon the other, the sun will be in the field of view. 

(A). For determining the meridian, (a). Incline the 
transit telescope until the angle of declination f corrected 
for refraction, is indicated by the vertical limb or arc, de- 
pressing the telescope if the sun's declination is north, and 
elevating it If it is south. See Fig. 11. 

(&). Bring the solar telescope into the vertical plane 
of the transit telescope, (without disturbing the position of 
the latter) and also to a horizontal position by means of its 
level. The two telescopes will now enclose an angle equal 
to the amount of the declination. See Fig. 12. 

(c). Without disturbing the relative positions of the 
two telescopes, elevate the transit telescope (and with it the 
solar) until the amount of the co-latitude is indicated by 
the vernier of the vertical limb. See Fig. 13. 

(d). Revolve the two telescopes together upon their 
respective vertical axes until the image of the sun is brought 
into the field of the solar telescope; when the sun is accu- 
rately bisected the transit telescope will be in the meridian 
and the compass needle will indicate the amount of its de- 
clination at the place of observation. It will of course con- 
siderably facilitate this last operation if, before commenc- 
ing to revolve the two telescopes, the transit one is 
approximately pointed toward the south by means of the 
transit compass needle. 

(B). For ascertaining the latitude. Direct the transit telescope towards the 
south, incline it to an amount equal to the sun's meridian declination uncorrected 
for refraction, depressing the telescope if the declination is north and elevating it if 
It is south. Now bring the solar telescope into the vertical plane of the transit 
telescope and to a perfectly horizontal position by means of its level, then clamp it. 
A few minutes before noon (the moment of the sun's culmination) bring the sun's 
image between the two horizontal wires of the solar telescope by moving only the 
transit telescove in altitude and azimuth. By means of the tangent screws of the 
transit, keep the sun, as it continues to rise and travel southwards, in this position 
relatively to the cross-hairs of the solar telescope. When it has ceased to rise, take 
the reading of the vertical arc of the transit, deduct from it the refraction due to 
this altitude, and the remainder Is the co-latitude, which deducted from 90° gives 
the latitude. The position of the two telescopes Is identical with that shown In 
Fig. 13. 

(CO. For observing the time. Having brought the two telescopes Into their 
final positions for determining the meridian, that is the transit one in the meridian 
and the solar telescope bisecting the sun, revolve them both upon their horizontal 
axis, without disturbing the vertical axis, until they are both perfectly level. The 




* Invented by Saegmuller, 1881. 

t See American Ephemeris or Nautical Almanac. 



SOLAR OBSERVATIONS, 



947 



angle formed by their respective lines of sight, which can be determined by sight- 
Jng with the two telescopes upon any clearly defined distant object, and taking the 
difference of the respective readings of the transit horizontal limb, is the hour angle. 
This is then reduced to time before or after apparent noon: One degree of arc = 4 
minutes of time and 1 minute of arc= 4 seconds of time. The time obtained by 
such an observation is reliable to a few seconds.^ 

To Adjust the Solar Attachment. — Attach the instrument to a transit 
in perfect adjustment, or adjust the transit as explained, page 942. In 
the Saegmuller solar (Fig. 10) there are two adjustments as follows: 

First, the polar or vertical axis. — Level the transit and also its tele- 
scope. Now revolve the solar telescope horizontally about its polar axis 
and note whether the bubble in the small tube attached above it moves 
from a central position. If so, correct one-half the variation by the screw v^ 
Fig. 10, and the other half by adjusting the bubble tube to a level position, 
either raising or lowering one end by the capstan-headed screws. Repeat, 
until the bubble remains central when the solar telescope is revolved. 

Second, the horizontal hair. — ^The horizontal lines of sight of the transit 
telescope and the solar telescope must be parallel with each other. To 
accomplish this the reticule or diaphragm of the latter is moved upward or 
downward by means of upper and lower capstan-headed screws s. Fig. 10, 
as explained in the adjustment of the level. 

Solar Observation with Transit Alone. — Mr. A. W. French gives the 
following method* for determining the meridian f from the sun by direct 
observation with the transit and without the use of solar attachment; 

"i. Observe the sun directly, by the aid of colored glass, and bring his image 
tangent to the horizontal and vertical wires; read the vertical circle and the hori- 
zontal plates. Suppose that in the first pointing the image was approaching the 
wires; then bring it into the opposite quarter of the field of view, where the image 
recedes from the wires; bring the wires tangent and read as before. The mean of 
th5se readings will give the apparent altitude and the plate reading for the sun's 
eerier. If the transit has a full vertical circle, the telescope should be reversed 
b-it-ween the two pointings, to eliminate all errors of adjustment. If the transit has 
OD'y a vertical arc, no reversal can be made, and great care must be taken that the 
p5p*e levels and standards are In good adjustment and the Index error accurately 
determined. 

"2. Read plates when pointing at any convenient mark, thus finding the angle 
between the sun's center and the mark. 

"3. Computation of the P Z S triangle enables us to find the angle between the 
sun and the north ; then, addition or subtraction of the angle between the sun and 
tfce mark (as the mark is north or south of the sun) gives the angle made by the line 
(station to mark), and the meridian. 

"The accompanying form of notes and reductions needs but a few remarks. 

Observation. 





Horizontal Circle Readings. 


Vertical Circle 
Readings. 




Telescope. 


On Mark. 


On Sun. 


Date and Time. 


Direct Sf 

Reversed TO 


240° 41' 00" 
60° 41' 00" 


282° 51' 30" 
102° 43' 00" 


30° 57' 00" 
31° 09' 00" 


Sept. 26, 1896, 2.30 p. m.. 
standard time, 75th merld. 


Averages. . . 


240° 41' 00" 


282° 47' 15" 


31° 03' 00" 





* See Eng. News, May 20, 1897. 

t The meridian is the great circle whose plane passes through the zenith 
and the poles. The azimuth of a body is the horizontal arc measured on the 
horizon between the north point and the vertical circle passing through the 
body; thus, the azimuth of the sun or of a star in determining the meridian. 
The declination of a body is its angular distance from the equator measured 
on an hour circle; thus the declination of the sun is south in winter, north 
in summer, and zero when it crosses the equator on March 20 and Sept. 20. 
The altitude of a body is its distance from the horizon measured on a vertical 
circle. The zenith distance is 90° minus the altitude. "Co" — before declina- 
tion, latitude, etc., means the complement of (90° — ) these quantities. 



948 



^.—SURVEYING, MAPPING AND LEVELING, 



Computation. 
Declination at Greenwich noon= 7 a. m. standard time 75th 

meridian = 1° 32' 33* south 

Hourly change= 58.5". Change for 7i hours= 58.5 X 7i = 7' 39* south 



Declination at 2 p. m - 

Average vertical angle by observation 
Correction for refraction - - - 



1° 40' 12" south 



31° 03' 00* 
1' 40* 



True altitude 



Latitude of Thayer School =43 
Station about 1 mile south = 



= 31' 
42' 



' 01' 
10* 



20* 



r 00" 



Latitude of station 



= 43° 41' 10" 



Cos hP 



ZS= SJ' 



sin i 5 X sin ih S — co-decl.) 



sin co-alt. X sin co-lat. 
where S= co-decl. + co-alt+ co-lat. 
co-decl. = 91° 40' 12" 

co-alt. = 58° 58' 40" 

co-lat. = 46° 18' 50" 



5=196° 57' 42* 



|/S= 98° 28' 51* 

co-decl. = 91° 40' 12* 

§5-co-decl.= 6° 48' 39" 

log sin 98° 28' 51" = 

" • 6° 48' 39" = 

a. C. " " 58° 58' 40" = 

a. C. " " 46° 18' 50" = 



9.995225 
9.074052 
0.067035 
0.140781 



19.277093 



tlj 






o 



CD C 



logcos^PZS = 9.638546 
^ PZS= 64° 12' 40* 
PZ5=128° 25' 20" 
Azimuth of sun from north = 128° 25' 20* 
Angle between sun and mark = 42° 06' 15* 

Angle north — station — mark =170° 31' 35* 
"If a single observation is made, the altitude must be changed by the semi- 

1 fi' 
diameter (16') and the horizontal angle by. not 16', but cQg pf ^it. ' The dihedral 

angle, whose edge Is the vertical line through the instrument subtended by the 
semi-diameter, varies with the altitude of the sun, from 16' for alt.= 0°, to 46' for 
alt.= 67° on June 21st. in this latitude (43° 42')." 



Meridian from the North Star. — Polaris is a distant "fixed" star called 
the north star. It lies about in line with the two "pointers" of the "big 
dipper" (the two stars farthest from the "handle") and facing its open top. 
It is the brightest star of a cluster of three forming one end of "Urser Minor" 
and can be recognized easily.* The earth's axis if produced will never pierce 
the north star, but wabbles around it in a circle. This makes the star 
appear to move in a circle around the true north point (Fig. 14), and hence 
we consider the pole P as fixed, and the star to move around it. The 



* The accompanying 
diagram will aid in finding 
Polaris at any time of the 
year: Hold the diagram, 
while facing the north, in 
such a position that the 
month, during which the 
observation is made, will *-Nov. 
point vertically upward. 
Polaris crosses the merid- 
ian on April 10th of each 
year at about noon. 



\ 



f 



f Cassiopeia's 
4 Chair 

V-- 



/ 



v<: 



y 



pointf 



Polaris t^ 

Pole Star ^J _._^/ May -* 



UjftleDippi 
\ in Ursa Mil 



*Cin Ursa Minor 
Nt UnieBear 



sP 



/ 



Big'-Dipper 
VOrsa Major 
GreafBear 



V-e 



■\ 



TO DETERMINE THE MERIDIAN. 



949 



polar distance, or radial distance from pole to star varies from a little 
more than 1° to a little less than 2° angular measure, varying from day 
to day, from year to year, and also with the latitude of 
the place of observation. When the star appears at E or W 
it is at its eastern or western elongation, respectively; when 
at U or L it is at its upper or lower culmination. Now if we 
know the polar distance of the star when it is at its eastern 
or western elongation, and also the latitude of the place of 
observation, it is a very simple matter to find the azimuth 
of the star or the horizontal angle from the star to the pole, 
and this is what the surveyor desires in getting the true meri- 
dian : 

_. r A • .1 i? o^ Sine of Polar Distance ,^v 

Sme of Azimuth of Star=p^ t-t— — 7 — \ 7-^r —r- — • • • -(l) 

Cos. of lat. of place of observation 

Table 1, below, is a table of polar distances from latitude 0, the equator, 
on Jan. 1 of each 3rd year, 1906-1930; and may be used in connection with 
the above formula (1) for any latitude, and for any time (year) by interpo- 
lation. 

• 

1. — Polar Distances of Polaris for Lat. 0, the Equator. 




Time, 
Jan. 1 

Polar 
Dlst. . 



Log sin 
of p. d. . 



Diff. of 
Lost sines 



1906.0 



t If 

1 11 41 



8.31910 



1909.0 



o / » 

1 10 45 



31341 



1912.0 



r IT 

1 9 49 



8.30765 



1915.0 



O t If 

8 53 



8.30181 



1918.0 



1 7 58 



8.29594 



921.0 



1 7 2 



28999 



1924.0 



o / » 

6 7 



8.28401 



1927.0 



I n 

1 5 12 



!. 27794 



1930.0 



1 4 16 



8.27169 



-.00569 -.00584 -.00595 —.00607 

-.00576 -.00587 -.00598 -.00625 



To find the polar 
distance {p. d.) of 
Polaris on July 1, 
1914, for latitude iV 
( + )38°-30'? Use 
Table 1 and formula 
(1). 



Example of Use of Table 1. 

Log sin p. d. for Jan. 1, 1915 

Add i diff (1915-1912) for July. 



= 8.30181 

1. 1914....= 0.00097 



Log sin p. d. for July 1, 1914 = 8 . 30278 

Log cos 38°- 30' (latitude) = 9.89354 



.-. ^. £i. = l°-28'-13'' Log sin = 8.40924 

The polar dist. at elongation of star is called the Azimuth at Elongation. 

For latitudes + 25° to + 72°, the direct Azimuths are given in Table 2, 
on following page. 

The Observation of Polaris for Azimuth will now be described. The 
transit should be in adjustment and be provided with a reflector for illu- 
minating the cross-hairs. This may consist of a circular piece of bright tin 
about the diameter of the object glass, and with a hole in it about half its 
diameter. This annular ring may be fastened to a wire, bent around the 




Fig. 15. 
object-end of the telescope as shown in Fig. 15. It should project outward 
at an angle of about 45° and the under side may be painted white so as to 
reflect the light from a dark lantern (held beneath it) into the telescope. 
A fore-sight or illuminated target should be placed several hundred feet 
away from the transit, and about due north to serve in fixing a base line 
for the azimuth reading to the star. The target may consist of a vertical 
slit (say about iV of an inch in width) in a box with a light behind it. 
(Continued on page 951.) 



950 



58.— SURVEYING, MAPPING AND LEVELING. 



2. — Azimuth op Polaris at Elongation Jan. 1. 
For Various Years and Latitudes. 



Lat. 


1900.0 


1905.0 


1906.0 


1907.0 


1908.0 


1909.0 


1910.0 


Yearly 
Decrease. 


o 


/ 


o / 


o / 


o / 


o / 


o / 


o / 


o / 


+ 25 


1 21.2 


1 19.4 


1 19.1 


1 18.7 


1 18.4 


1 18.1 


1 17.7 


0.35 


26 


21.8 


20.1 


19 8 


19.4 


19.1 


18.7 


18.4 


0.35 


27 


22.5 


20.8 


20.5 


20.1 


19.8 


19.4 


19.1 


0.35 


28 


23.3 


21.6 


21.3 


20.9 


20.5 


20.1 


19.8 


0.35 


29 


24.1 


22.4 


22.1 


21.7 


21.3 


20.9 


20.5 


0.36 


30 


1 24.9 


1 23.1 


1 22.8 


1 22.4 


1 22.1 


1 21.7 


1 21.3 


0.36 


31 


25.8 


24.0 


23.6 


23.2 


22.9 


22.5 


22.2 


0.36 


32 


26.7 


24.9 


24.5 


24.1 


23.8 


23.4 


23.1 


0.37 


33 


27.7 


25.9 


25.5 


25.1 


24.7 


24.3 


24.0 


Oi.37 


34 


28.7 


26.9 


26.5 


26.1 


25.7 


25.3 


25.0 


0.38 


35 


1 29.8 


1 27.9 


1 27.5 


1 27.1 


1 26.8 


1 26.4 


1 26.0 


0.38 ^• 


36 


30.9 


29.0 


28.6 


28.2 


27.9 


27.5 


27.1 


0.39 § 


37 


32.1 


30.1 


29.7 


29.3 


29.0 


28.6 


28.2 


0.39 2 


38 


33.3 


31.4 


31.0 


30.6 


30.2 


29.8 


29.4 


0.40 ^ 
0.40 '« 


39 


34.7 


32.7 


32.3 


31.8 


31.4 


31.0 


30.6 


40 


1 36.0 


1 34.0 


1 33.6 


1 33.2 


1 32.8 


1 32.4 


1 32.0 


0.41 ■§ 


41 


37.5 


35.4 


35.0 


34.6 


34.2 


33.8 


33.4 


0.41 ^ 


42 


39.0 


36.9 


36.5 


36.0 


35.6 


35.2 


34.8 


0.42 4, 


43 


40.6 


38.5 


38.1 


37.6 


37.2 


36.8 


36.3 


0.43 5 


44 


42.3 


40.1 


39.7 


39.2 


38.8 


38.4 


37.9 


0.44 1 


45 


1 44.0 


1 41.8 


1 41.4 


1 40.9 


1 40.5 


1 40.1 


1 39.6 


0.44 S 


46 


45.9 


43.7 


43.2 


42.7 


42.3 


41.9 


41.4 


0.45 -^ 


47 


47.9 


45.6 


45.1 


44.6 


44.2 


43.7 


43.3 


0.46 « 
0.47 S 


48 


49.9 


47.7 


47.2 


46.7 


46.3 


45.8 


45.3 


49 


52.1 


49.8 


49.3 


48.8 


48.4 


47.9 


47.4 


0.48 ■;; 

a 

0.49 i 


50 


1 54.4 


1 52.0 


1 51.5 


1 51.0 


1 50.6 


1 50.1 


1 49.6 


51 


56.9 


54.4 


54.0 


53.5 


53.0 


52.5 


52.0 


0.49 2 


52 


59.5 


57.0 


56.4 


55.9 


55.4 


54.9 


54.4 


0.51 ^ 


53 


2 02.2 


59.6 


59.1 


58.6 


58.1 


57.6 


57.1 


0.51 >? 


54 


05.1 


2 02.5 


2 02.0 


2 01.5 


2 00.9 


2 00.4 


1 59.9 


0.52 ^ 

0.55 1 
0.56 S 
0-57 ^ 


55 


2 08.3 


2 05.6 


2 05.0 


2 04.4 


2 03.9 


2 03.4 


2 02.8 


56 


11.6 


08.8 


08.2 


07.7 


07.1 


06.6 


06.0 


57 


15.1 


12.2 


11.7 


11.1 


10.5 


10.0 


09.4 


58 


18.8 


15.9 


15.3 


14.7 


14.2 


13.6 


13.0 


0.58 S 


59 


22.8 


19.8 


19.2 


18.6 


18.0 


17.4 


16.8 


0.60 5 

3 

0.62 a 


60 


2 27.1 


2 24.0 


2 23.4 


2 22.8 


2 22.1 


2 21.5 


2 20.9 


61 


31.7 


28.5 


27.9 


27.2 


26.6 


25.9 


25.3 


0.64 2 


62 


36.7 


33.4 


32.7 


32.1 


31.4 


30.8 


30.1 


0.66 Z 


63 


42.1 


38.6 


38.0 


37.3 


36.6 


35.9 


35.2 


0.69 S 
0.70 ^ 


64 


47.8 


44.3 


43.6 


42.9 


42.2 


41.5 


40.8 


65 


2 54.1 


2 50.4 


2 49.7 


2 49.0 


2 48.3 


2 47.5 


2 46.8 


0.73 


66 


3 00.9 


57.1 


56.3 


55.6 


54.8 


54.1 


53.3 


0.76 


67 


08.3 


3 04.4 


3 03.6 


3 02.8 


3 02.0 


3 01.2 


3 00.4 


0.79 


68 


16.4 


12.3 


11.5 


10.7 


09.8 


09.0 


08.2 


0.82 


69 


25.3 


21.0 


20.1 


19.3 


18.4 


17.6 


16.7 


0.86 


70 


3 35.2 


3 30.6 


3 29.7 


3 28.8 


3 27.9 


3 27.0 


3 26.1 


0.91 


71 


46.1 


41.3 


40.3 


39.4 


38.4 


37.5 


36.5 


0.96 


72 


58.2 


53.2 


52.1 


51.1 


50.1 


49.1 


48.1 


1.01 



Ex. — Elongation of Polaris for Lat N( + )3S°-3(y during June-July, 
1914, or 4i years after Jan. 1, 1910, is 1°- SO' minus ih times 0°-0'.40=» 
l<>-28'.2. 



POLARIS— OBSERVATIONS. 951 

(a) Observation op Polaris at Elongation. 
Just before the time of elongation of Polaris, bring the telescope into 
position, cutting the star with vertical wire; and with the horizontal angle 
plate maintained at zero on the vernier, follow the star by operating the 
lower-plate slow-motion screw. When the star ceases to move laterally it 
has practically reached its elongation. Now measure the horizontal angle 
to the illuminated target, previously set (p. 949). By working rapidly, the 
angle may be doubled as a check or for greater accuracy*. Better still, for 
correcting any error of adjustment of the transit, it may be well to re- 
volve the alidade 180° for the second reading, with the telescope reversed. 
The mean of these two readings will be the correct one to use, and this 
angle will be the horizontal angle between Polaris at elongation and the 
established base line fixed by the target. The angle between this base line and 
the true meridian can now be calculated by taking into consideration the 
azimuth of Polaris at its elongation, for the particular time and latitude, 
from Tables 1 or 2, and 3t; thus, it will either be equal to the sum or the diff 
of the measured angle and the azimuth. The operator must have clearly 
in mind whether the elongation is east or westf and in which direction 
the measured angle is taken, in order to lay off the meridian in the right 
direction. A pencil sketch should be drawn showing the angles. 

(6) Observation op Polaris at any Hour Angle. 

Referring to Fig. 14, the hour angle of Polaris (or of any star) is the time 
which has elapsed since its upper culmination, at U. The time occupied by 
Polaris in making a complete cycle from U to U, is 23/j 56.1 m, and in 
making a half cycle, from U to L or from L to U, is assumed to be one-half 
this or 11/t 58m. Therefore, if we know the time of upper culmination we 
have but to subtract or add 11/t 58m to find the preceding or the following 
time of its lower culmination. To an observer at the equator J (lat. 0), the 
successive hour angles of Polaris are: dh Om at U, bh 59w at W, \\h 58.1m 
at L, nh hi. Im at E, and 23/t 56. lm=Oh Om at U again. Now for any point 
in N. lat., the hour angle at western elongation will be diminished, while at 
eastern elongation it will be increased; the amount of decrease or increase 
varying with the lat. of the observer. Thus, in 40° N. latitude the hour angle 
at western elongation is 5h 55m, and at eastern elongation is ISh 01m, 
varying about 4m in each case from the above as observed at the equator. 

It will be seen from the above that the azimuth of the star varies with 
the star's hour angle and the latitude\\ of the place of observation, as well as 
with the time of observation. 

There are two kinds of tim£ which must be kept clearly in mind, viz.. 
Civil (clock) Time and Astronomical Time. An astronomical day begins at 

*The variation (deduction) of azimuth angle for time of observation 
just prior to or after elongation is very small, being less than 0°— 01' for 
30 min. in time. The following table shows the relation for lat. 45° A^. and 
for the year 1910, the azimuth at elongation being about 1°— 40'= lOO'. 



Diff. in Time. 


Diff. in Azimuth. 


Diff. In Time. 


Diff. in Asimuth. 


O/t Im 


0° 0.'001= 0."06 


0/t 25m 


0° 0.'594= 35."6 


5 


0. 024= 1. 4 


30 


0. 856= 51. 4 


10 


0. 095= 5. 8 


32. 4 


1. = 60. 


15 


0. 214=12. 8 


35 


1. 164= 69. 8 


20 


0. 381 = 22. 9 


45. 9 


2. =120. 



t Froni Table 3 we can find whether the elongation is east or west, as 
explained in the foot-note accompanying that table. 

X The difference in time between the upper culmination and the adja- 
cent elongation diminishes with increase in latitude; while the azimuth at 
elongation increases with increase in latitude. 

1 1 The variation for longitude is very slight, and for points in the U. S. 
it is negligible for ordinary surveying Table 3 gives the local mean (astro- 
nomical) time of the upper culmination of Polaris, computed for longitude 
108° (7h. 12m.) west of Greenwich. The meridian passes through the 
western parts (near the edges) of Colorado and New Mexico, and is con- 
sidered fairly "central" for U. S. territory. (The eastern edge of Maine is 
about 67° west, the western edge of Oregon about 125° west, and the western 
edge of Alaska about 168° west.) The error is of course greatest at the 
culminations (at which time observation should be avoided when possible 
to do so) and least at the elongations. 



952 58.— SURVEYING, MAPPING AND LEVELING. 

noon of a civil day, and each is 24 hrs. in length. Thus, the A. M of a 
civil day corresponds to the last half of the preceding astronomical day, 
each lapping by 12 hrs.* 

In making an observation on Polaris for azimuth, it is necessary to know: 
(1) the latitude of the place of observation, which may be obtained from a 
map; (2) the mean local time of observation, which can be calculated from 
the particular "standard time"t used in that locality when the longitude of 
the place of observation is known; (3) the horizontal measured angle from 
Polaris to the previously established base line, explained under (a), pre- 
ceding, together with the mean local time of observation; (4) the hour 
angle of Polaris at time of observation, to be calculated from Parts I and II 
of Table 3; (5) the azimuth of Polaris from the hour* angle, date, and lati- 
tude of place of observation, to be obtained from Table 4. (6) Find the sum 
or difference of the azimuth and the measured angle, explained under (a) 
preceding, and lay it off in the right direction from the established base line. 

Practical Example of Use of Tables 3 and 4. 
Place of observation, lat. 41° N., long, 100° W.; Mountain time, 8 30 p.m., 
Nov. 8, 1911; measured angle from Polaris eastward to base line, 2° — 40'. Find the 
azimuth of Polaris, and the angle of base line with true meridian? Then we have — 

h m 

Standard time of observation (merid. 105° W.) 8 30. 

For merld. 100° W., add 5x 4m 20.0 

Mean local time of obs., 1911, Nov. 8. . . , 8 50. 

Equivalent time to Nov. 7 (add 24 h.) 32 50. 

h m 

Astrom. time U. C Polaris, Nov. U (Table 3. Part I) 10 48. 

Reduction to Nov. 7 (Table 3, Part 11)11 Subtract 23.6 10 24.4 

Hour angle of Polaris at observation 22 25.6 

Subtract from 23 56 . 1 

Time argument for Table 41f 1 30.5 

Azimuth of Polaris, at observation (lat. 41°).. 0° 36'. 5 E. 

Measured angle eastward from Polaris to base line 2° 40' E. 

Angle of base line eastward from true meridian 3° 1 6' . 5 E. 

Therefore lay off angle 3° 16'. 5 westward from base line to true meridian. 



* Civil time P» M. = astronomical time with the P. M. omitted; thus, 
5.30 P. M. = 5h 30m of the same date. Civil time A. M., + \2 hrs. = astrono- 
mical time of the preceding date; thus, 5.30^4.. M. June 2= 17h 30m June 1. 
Astronomical time under 12h = civil time P. M of the same date* thus 5h 30m 
= 5.30 P.M. of same date. Astronomical time over 12h, with 12 hrs. de- 
ducted from it = civil time A.M. of the following date; thus, 17h 30m 
June 1 = 5.30 A. M. June 2. 

t Standard railway time for longitude west from Greenwich: Inter- 
colonial time, for 60° west, Eastern time, for 75° west; Central time, for 
90° west; Mountain time, for 105° west; Pacific time for 120° west. 15° of 
longitude =1 hr. of time (1°=4 m.); 15' of long. = 1 min. of time {V = 
4 sec); 15" of long. = 1 sec. of time (r'=0.6f sec). Ex.—Th.e place of 
observation is, say, 108° west, and the observer has Mountain time; hence 
the mean local time is 3X4 m. = 12 m. slower than his watch. For 102° 
west it would be 12 m. faster t\i2in his watch. West of standard time meri- 
dian means deduct', east means add. 

% This is the nearest date preceding, in the table. Values are given in 
Part I for the 1st and 15th of each month; and in Part II the reduction is 
given for succeeding dates. 

1 1 Caution. — Be sure to use the same "Diff. for 1 day" in Part II as 
obtained from Part I, and opposite the day of the month Subtract', don't 
add. 

tSee ''Exr below Table 3. 



UPPER CULMINATION OF POLARIS. 



953 



3. — Local Mean (Astronomical) Time of the Upper Culmination of 

Polaris, computed for Longitude 108° (7h. 12m.) west of Greenwich. 
[The time on line with any date in Part I is the hours and minutes elapsed 
(common watch time) since the preceding noon.] 

Part I. 





































Diffj for 


Date. 


1901. 


1902. 


1903. 


1904. 


1905. 


1906. 


1907. 


1908. 


1 day. 




h. 


m. 


h. 


m. 


h. 


m. 


h. 


m. 


h. 


m. 


h. 


m. 


h. 


m. 


h. 


m. 


m. 


Jan. 1 


6 


39.5 


6 


41.0 


6 


42.4 


6 


43.9 


6 


41.4 


6 


42.8 


6 


44.3 


6 


45.7 


3.95 


15 


5 


44.2 


5 


45.7 


5 


47.1 


5 


48.6 


5 


46.1 


5 


47.5 


5 


49.0 


5 


50.4 


3.95 


Feb. 1 


4 


37.1 


4 


38.6 


4 


40.0 


4 


41.5 


4 


39.0 


4 


40.4 


4 


41.9 


4 


43 3 


3.95 


15 


3 


41.9 


3 


43.4 


3 


44.8 


3 


46.3 


3 


43.8 


3 


45.2 


3 


46.7 


3 


48.1 


3.95 


Mar. 1 


2 


46.6 


2 


48.1 


2 


49.5 


2 


47.0 


2 


48.5 


2 


49.9 


2 


51.4 


2 


48.9 


3.94 


15 


1 


51.5 


1 


53.0 


1 


54.4 


1 


51.9 


1 


53.4 


1 


54.8 


1 


56.3 


1 


53.8 


3.94 


April 1 





44.6 





46.1 





47.5 





45.0 





46.5 





47.9 





49.4 





46.8 


3.94 


15 


23 


45.6 


23 


47.1 


23 


48.5 


23 


46.0 


23 


47.5 


23 


48.9 


23 


50.4 


23 


47.8 


3.93 


May 1 


22 


42.8 


22 


44.3 


22 


45.7 


22 


43.2 


22 


44.7 


22 


46.1 


22 


47.6 


22 


45.1 


3.93 


15 


21 


47.8 


21 


49.3 


21 


50.7 


21 


48.2 


21 


49.7 


21 


51. 1 


21 


52.6 


21 


50.1 


3.92 


June 1 


20 


41.2 


20 


42.7 


20 


44.1 


20 


41.6 


20 


43.1 


20 


44.5 


20 


46.0 


20 


43.5 


3.92 


15 


19 


46.4 


19 


47.9 


19 


49.3 


19 


46.8 


19 


48.3 


19 


49.7 


19 


51.2 


19 


48.7 


3.91 


July 1 


18 


43. £ 


18 


45.3 


18 


46.7 


18 


44.2 


18 


45.7 


18 


47.1 


18 


48.6 


18 


46.1 


3.91 


15 


17 


49.0 


17 


50.1 


17 


51.9 


17 


49.4 


17 


50.9 


17 


52.3 


17 


53.8 


17 


51.3 


3.92 


Aug. 1 


16 


42.4 


16 


43.9 


16 


45.3 


16 


42.8 


16 


44.3 


16 


45.7 


16 


47.2 


16 


44.7 


3.92 


15 


15 


47.6 


15 


49.1 


15 


50.5 


15 


48.0 


15 


49.5 


15 


50.9 


15 


52.4 


15 


49.9 


3.92 


Sept. 1 


14 


41.0 


14 


42.5 


14 


43.9 


14 


41.4 


14 


42.9 


14 


44.3 


14 


45.8 


14 


43.3 


3.92 


15 


13 


46.1 


13 


47.6 


13 


49.0 


13 


46.5 


13 


48.0 


13 


49.4 


13 


50.9 


13 


48.4 


3.93 


Oct. 1 


12 


43.3 


12 


44.8 


12 


46.2 


12 


43.7 


12 


45.2 


12 


46.6 


12 


48.1 


12 


45.6 


3.93 


15 


11 


48.3 


11 


49.8 


11 


51.2 


11 


48.7 


11 


50.2 


11 


51.6 


11 


53.1 


11 


50.6 


3.93 


Nov. 1 


10 


41.4 


10 


42.9 


10 


44.3 


10 


41.8 


10 


43.3 


10 


44.7 


10 


46.2 


10 


43.7 


3.93 


16 


9 


46.4 


9 


47.9 


9 


49.3 


9 


46.8 


9 


48.3 


9 


49.7 


9 


51.2 


9 


48.7 


3.94 


Dec. 1 


8 


43.3 


8 


44.8 


8 


46.2 


8 


43.7 


8 


45.2 


8 


46.6 


8 


48.1 


8 


45.6 


3.94 


15 


7 


48.1 


7 


49.6 


7 


51.0 


7 


48.5 


7 


50.0 


7 


51.4 


7 


52.9 


7 


50.4 


3 95 





Part I.— Continued. 








Part II. 




















Difl. 


Reduction of tabular times to intermedi- 


Date. 


1909. 


1910. 


19n. 


. for 


ate dates. 














1 day. 


Subtract ttie reduction when computing 
from a preceding, or add it when 


















h, m. 


h. 


m. 


h. 


m. 


m. 


working from a following date. 


Jan. 1 
15 


6 43.2 
5 47.9 


t 


44.7 
49.4 


6 
5 


46.1 
50.8 


3.95 
3.95 






Reduction. Arg.— "Diff. for 


No. 


Feb. 1 


4 40.8 


4 


42.3 


4 


43.7 


3.95 


Day of 


1 day." 


of 


Ig 


3 45.6 


3 


47.1 


3 


48.5 


3.95 


the 




days 












Mar. 1 


2 50.3 


2 


51.8 


2 


53.2 


3.94 


month 


m. 


m. 


m. 


m. 


m. 


elap- 


16 


1 55.2 


1 


56.7 


1 


58.1 


3.94 




3.91. 


3.92. 


3.93. 


3.94. 


3.95. 


sed. 


April 1 


48.3 





49.8 





51.2 


3.94 






























15 


23 49.3 


23 


50.8 


23 


52.2 


3.93 




m. 


m. 


m. 


m. 


m. 




May 1 


22 46.5 


22 


48.0 


22 


49.4 


3.93 


2orl6 


3.9 


3.9 


3.9 


3.9 


3.9 


1 


15 


21 51.5 


21 


53.0 


21 


54.4 


3.92 


3orl7 


7.8 


7.8 


7.9 


•7.9 


7.9 


2 


June 1 


20 44.9 


20 


46.4 


20 


47.8 


3.92 


4orl8 


11.7 


11.8 


11.8 


11.8 


11.8 


3 


15 


19 50.1 


19 


51.6 


19 


53.0 


3 91 


5orl9 


15.6 


15.7 


15.7 


15.8 


15.8 


4 


July 1 


18 47.5 


18 


49.0 


18 


50.4 


3.91 


6or20 


19.5 


19.6 


19.6 


19.7 


19.7 


5 


15 


17 52.7 


17 


54.2 


17 


55.6 


3.92 


7or21 


23.5 


23.5 


23.6 


23.6 


23.7 


6 


Aug. 1 


16 46.1 


16 


47.6 


16 


49.0 


3.92 


8or22 


27.4 


27.4 


27.5 


27.6 


27.6 


7 


15 


15 51.3 


15 


52.8 


15 


54.2 


3.92 


9or23 


31.3 


31.4 


31.4 


31.5 


31.6 


8 


Sept 1 


14 44.7 


14 


46.2 


14 


47.6 


3.92 


10or2 4 


35.2 


35.3 


35.4 


35.5 


35.5 


9 


15 


13 49.8 


13 


51.3 


13 


52.7 


3.93 


llor2 5 


39.1 


39.2 


39.3 


39.4 


39.5 


10 


Oct. 1 


12 47.0 


12 


48.5 


12 


49.9 


3.93 


12or26 


43.0 


43.1 


43.2 


43.3 


43.4 


11 


15 


11 52. C 


11 


53.5 


11 


54.9 


3.93 


13or27 


46.9 


47.0 


47.2 


47.3 


47.4 


12 


Nov. 1 


10 45.1 


10 


46.6 


10 


48.0 


3.93 


14or28 


50.8 


51.0 


51.1 


51.2 


51.3 


13 


15 


9 50.1 


9 


51.6 


9 


53.0 


3.94 


29 


54.7 


54.9 


55.0 


55.2 


55.3 


14 


Dec. 1 


8 47.0 


8 


48.5 


8 


49.9 


3.94 


30 


58.6 


58.8 


58.9 


59.1 


59.2 


15 


15 


7 51.8 


7 


53.3 


7 


54.7 


3.95 


31 


62.6 


62.7 


62.9 


63.0 


63.2 


16 



Ex. — To find the time ofupper culmination of Polaris on Nov. 8, 1911. for longi- 
tude 108° west of Greenwich? 

Solution. — From Part I of above table we have for Nov. 1, 1911, lOh. 48.0m., and 
the diff. (deduction) for one day is 3.93 m. Now under this diff. In Part II, and in line 
with the day of the month. 8, in the left-hand column (or in line with the number of 
days elapsed since Nov. 1, namely 7, in the right-hand column) we obtain the total 
deductionof 27.5m., making the astronomical time of upper culmination lOh. 48.0m. 
minMs 27.5m. = 10h. 20.5m., or 10.20^ P.M., local civil time. Note that In the practical 
example preceding, the local time of observation was 8.50 P.M. or Ih. 30.5m. earlier, 
and this ih. 30.5m. becomes the time argument in Table 4 for that example. 



954 



5S.-^SURVEY1NG, MAPPING AND LEVELING. 



4. — Azimuth of Polaris 
[The hour angles are expressed In mean solar time. The occurrence of a period after 



Star and Azimuth. 
W. of N. when hour angle is less than 

11'' SS"". 
E. of N. when hour angle is greater than 

111* 58™. 
Time argument, the star's hour angle (or 
23'» 56"> .1 mirms the star's hour angle), 
for the year- 



Polaris above the Pole. 
To determine the true meridian, the azi- 
muth will be laid off to the east when 
the hour angle is less than 1 \^ 58", and 
to the 2^es< when f7reaZer than lli> 58™. 





i 


r^ 


i 


1 


S 

a 


1 


^'125 


1 





M 


Azimuths for Latitude — 


1 


0\ 


o* 


0^ 


a 


a 














1 

















C 


•^ 


■^ 


^ 


"" 


** 


■"^ 


"■ 


"^ 


"■ 


^ 


** 


30 


32 


34 


36 


38140 


42 


44 


46 


48 


50 


h. 


m. 


m. 


m. 


m. 


m. 


m. 


m. 


m. 


m. 


m. 


m. 


/ 


' 


/ 


/ 


/ 


' 


' 


/ 


/ 


1 


/ 









































































4. 


4. 


4. 


5 


5 


5 


5 


5 


5 


5 


5 


2 


2 


2 


2 


2 


2 


2 


2 


2 


2. 


2. 




9. 


9. 


9. 


9. 


9. 


9. 


9. 


9. 


10 


10 


10 


3. 


3. 


3. 


4 


4 


4 


4 


4. 


4. 


4. 


5 




14 


14 


14 


14. 


14. 


14. 


14. 


14. 


14. 


14. 


15 


5. 


5. 


5. 


5. 


6 


6 


6 


6. 


6. 


7 


7 




19 


19 


19 


19 


19 


19 


19. 


19. 


19. 


19. 


19. 


7 


7 


7. 


'7. 


8 


8 


8. 


8. 


9 


9 


9. 




23. 


23. 


23. 


24 


24 


24 


24 


24. 


24. 


24. 


24. 


9 


9 


9 


9. 


9. 


10 


10. 


10. 


11 


11. 


12 




28. 


28. 


28. 


28. 


29 


29 


29 


29 


29. 


29. 


29. 


10. 


11 


11 


11. 


11. 


12 


12. 


13 


13. 


14 


14. 




33 


33 


33. 


33. 


33. 


34 


34 


34 


34. 


34. 


34. 


12. 


12. 


13 


13 


13. 


14 


14. 


15 


15. 


16 


17 




38 


38 


38 


38. 


38. 


38. 


39 


39 


39 


39. 


39. 


14 


14. 


14. 


15 


15. 


16 


16. 


17 


18 


18. 


19 




42. 


42. 


43 


43 


43. 


43. 


44 


44 


44 


44. 


44. 


16 


16 


16. 


17 


17. 


18 


18. 


19. 


20 


21 


21. 




47. 


47. 


48 


48 


48 


48. 


48. 


49 


49 


49. 


49. 


17. 


18 


18. 


19 


19. 


20 


20. 


21. 


22. 


23 


24 




52 


52. 


52. 


53 


53 


53. 


53. 


54 


54 


54. 


54. 


19. 


20 


20. 


21 


21. 


22 


22. 


23. 


24. 


25. 


26. 




57 


57. 


57. 


58 


58 


58. 


58. 


59 


59 


59. 


T 


21. 


21. 


22 


22. 


23. 


24 


25 


26 


27 


28 


29 


-r 


T 


— 


— 


— 


-T 


— 


— 


— 


4. 


— 


5 


23 


23. 


24 


24. 


25 


26 


27 


28 


29 


30 


31. 




7 


7 


7. 


7. 


8 


8. 


8. 


9 


9. 


9. 


10 


25 


25. 


26 


26. 


27 


28 


29 


30 


31. 


32. 


34 




12 


12 


12. 


13 


13 


13. 


14 


14 


14. 


15 


15 


27 


27 


27. 


28. 


29 


30 


31 


32 


33. 


35 


36. 




17 


17 


17. 


18 


18 


18. 


19 


19. 


19. 


20 


20. 


28. 


29 


29. 


30 


31 


32 


33 


34. 


36 


37. 


39 




22 


22 


22. 


23 


23. 


23. 


24 


24. 


25 


25. 


26 


30. 


31 


31. 


32 


33 


34 


35 


36. 


38 


39. 


41. 




27 


27 


27. 


28 


28. 


29 


29. 


29. 


30 


30. 


31 


32 


32. 


33 


34 


35 


36 


37 


38. 


40 


42 


43. 




32 


32. 


33 


33. 


33. 


34 


34. 


35 


35. 


36 


36. 


33. 


34. 


35 


36 


37 


38 


39. 


41 


42. 


44 


46 




37 


37. 


38 


38. 


39 


39. 


40 


40. 


41 


41. 


42 


35. 


36 


37 


38 


39 


40 


41. 


43 


44. 


46. 


48. 




42. 


43 


43. 


44 


44. 


45 


45. 


46 


46. 


47 


47. 


37 


38 


38. 


39. 


40. 


42 


43. 


45 


46. 


48. 


50. 




47. 


48 


48. 


49 


50 


50. 


51 


51. 


52 


52. 


53 


39 


39. 


40. 


41. 


42. 


44 


45. 


47 


49 


51 


53 




53 


53. 


54 


54. 


55 


55. 


56. 


57 


57. 


58 


58. 


40. 


41. 


42. 


43. 


44. 


46 


47. 


49 


51 


53 


55 




58. 


59 


59. 





0. 


1. 


2 


2. 


3 


4 


4. 


42. 


43 


44 


45. 


46. 


48 


49. 


51. 


53. 


55. 


57. 


"2- 


4 


4. 


5 


6 


6. 


7 


7. 


8. 


9 


9. 


10. 


44 


45 


46 


47 


48. 


50 


51. 


53. 


55. 


57. 


60 




9 


10 


10. 


11. 


12 


12. 


13. 


14 


15 


15. 


16. 


46 


47 


48 


49 


50. 


52 


53. 


55. 


57. 


-6F 


62. 




15 


16 


16. 


17 


18 


18. 


19. 


20 


21 


21. 


22. 


47. 


48. 


49. 


51 


52. 


54 


56 


57. 


"eT 


62 


64. 




21 


21. 


22. 


23 


24 


24. 


25. 


26 


27 


28 


28. 


49. 


50. 


51. 


53 


54. 


56 


58 


"W 


62 


64. 


67 




27 


27. 


28. 


29 


30 


30. 


31. 


32. 


33 


34 


35 


51 


52 


53. 


55 


56. 


58 


"60" 


62 


64. 


66. 


69. 




33 


33. 


34. 


35 


36 


37 


38 


38. 


39. 


40. 


41. 


53 


54 


55. 


56. 


58. 


— 


62 


64 


66. 


69 


72 




39 


40 


40. 


41. 


42. 


43. 


44. 


45 


46 


47 


48 


54. 


56 


57 


58. 


— 


62 


64 


66 


68. 


71. 


74. 




45 


46 


47 


48 


49 


50 


51 


52 


53 


54 


55 


56. 


57. 


59 


— 


62 


64 


66 


68. 


71 


73. 


76. 




51. 


52. 


53. 


54. 


55. 


56. 


57. 


58. 


59. 


1 


— 


58 


59. 


— 


62. 


64 


66 


68 


70. 


73 


76 


79 




58. 


59. 


0. 


1. 


2. 


3. 


4. 


6 


7 


8 


9 


60 


61. 


63 


64. 


66 


68 


70 


72. 


75 


78 


81. 


"T" 


— 


6. 


7. 


8. 


10 


11 


12 


13 


14. 


15. 


17 


61. 


63 


64. 


66 


68 


70 


72 


74. 


77. 


80. 


84 




12. 


13. 


14. 


16 


17 


18. 


19. 


21 


22 


23. 


25 


63. 


65 


66. 


68 


70 


72 


74. 


77 


79. 


82. 


86 




19. 


21 


22 


23. 


25 


26 


27. 


29 


30. 


31. 


33 


65 


66. 


68. 


70 


72 


74 


76. 


79 


82 


85 


88. 




27. 


28. 


30 


31. 


33 


34. 


35. 


37 


38. 


40. 


42 


67 


68. 


70 


72 


74 


76 


78. 


81 


84 


87 


91 




35. 


37 


38. 


39. 


41 


43 


44. 


46 


47. 


49. 


51 


69 


70. 


72 


74 


76 


78 


80. 


83 


86 


89. 


93. 




44 


45. 


47 


48. 


50 


52 


53. 


55 


57 


59 


— 


W. 


72 


74 


75. 


77. 


80 


82. 


85. 


88. 


91. 


95. 




53 


54. 


56 


58 


59. 


"T 


— 


— 


— 


— 


11. 


72. 


74 


76 


77. 


79. 


82 


84. 


87. 


90. 


94 


98 


"T 


— 


4. 


— 


— 


— 


12. 


14. 


16. 


19 


21 


23. 


74 


76 


77. 


79. 


81. 


84 


86. 


89. 


92. 


96 


100. 




13 


15 


17. 


19. 


22 


24 


26. 


29 


32 


34. 


37. 


-76 


77. 


79. 


81. 


83. 


86 


88. 


91. 


95 


98. 


103 




24. 


27 


29. 


32 


34. 


37. 


40. 


43. 


46. 


50 


53. 


77. 


79. 


81. 


83. 


85. 


88 


90. 


94 


97 


101 


105 




38 


40. 


43. 


46. 


50 


53. 


57. 


— 


— 


11 


16. 


79. 


81. 


83 


85 


87. 


90 


93 


96 


99. 


103 


107. 




54 


57. 


1. 


5 


— 


16 


IL 


32 


42. 




~~" 


81 


83 


85 


87 


89. 


92 


95 


98 


101. 


105. 


116 


T 


— 


— 


29. 


40. 
















83 


85 


87 


89 


91. 


94 


97 


100 


103. 


107. 


112 



Note. — From the "Ex." below Table 3, preceding page, we find that the upper cul- 
mination of Polaris occurs at 10.20^ P. M. on Nov. 8, 191 4 ; and that the time of observation 
Is fixed at 8.50 P. M., mean local time.or Ih. 30.5m. earlier. Hence the position of Polaris 
atthe time of observation Is just to the right of C7, Fig. 14, p.949. Moreover, the star's hour 
angle at observation is 23h. 56.1m. minus Ih. 30 5m.=:22h. 25.6m., which corresponds 
with that given In the practical example preceding. Have graphically in mind the posi- 
tion of the star, and its apparent motion, and the calculations become simple. 



AZIMUTH OF POLARIS, 



955 



FOR USE OF Land Surveyors. 

minutes of time or of an hour angle Indicates that Its value Is 0"^. 5 greater than printed.) 







Star and Azimuth 


. 












Polaris below the Pole. 






W, of N. when hour angle is less than 


To determine the true meridian. 


the azi- 


in SS-". 


muth will be laid off to the east when 


E. of N. when hour angle is greater than 


the hour angle is less than 1 1^ 58'», and 
to the west when greater than 11 58™. 


Time argument, the star's hour angle (or 










23h 56^.1 minus the star's hour angle). 










for the year — 










1 





i 


i 


i 


i 


«o 


1 





1 


© 


.... 


Azimuths for Latitude — 


1 


0^ 


a 


Ov 



































C 




^* 


*" 


"* 


^ j mm 


"* 


"* 


"* 


"^ 


** 


30 


32 


34 


36 


38 


40 


42 


44 


46 


48 


50 


h. 


m. 


m. 


m. 


m. 
















' 


' 


/ 


/ 


' 


/ 


1 


/ 


' 


' 


t 


6 


34 


28 


20. 


9. 


m. 


m. 


"mT 


mT 


~ 






83 


85 


87 


89 


91. 


94 


97 


100 


103. 


107. 


112 




56 


52. 


48. 


45 


40. 


34 


27 


18 


8 


~ 


m. 


81. 


83 


85 


87 


89. 


92 


95 


98 


101. 


105. 


109. 


"7" 


"12: 


— 


— 


4 


-0: 


56. 


52. 


48. 


44 


39 


34 


79. 


81. 


83 


85 


87. 


90 


93 


96 


99. 


103 


107 




26 


23. 


21 


18. 


16 


— 


— 


T 


4 


"0: 


57 


78 


79. 


81. 


83. 


85. 


88 


90. 


93. 


97 


100. 


104. 




37. 


35. 


33. 


31. 


29 


26. 


24 


21. 


19 


16 


13 


76 


77. 


79. 


81. 


83. 


86 


88. 


91. 


95 


98. 


102 




48. 


46. 


44. 


42. 


40. 


38. 


36. 


34. 


32 


29. 


27 


74. 


76 


77. 


79. 


81. 


84 


86. 


89. 


92. 


96 


100 




58 


56. 


55 


53 


51. 


49. 


47. 


45. 


43. 


41. 


39. 


72. 


74 


76 


77. 


79. 


82 


84. 


87. 


90. 


94 


97. 


-r 


-T 


"T 


4 


— 


1 


59. 


57. 


-5i 


54 


52. 


50. 


71 


72. 


74 


76 


77. 


80 


82. 


85. 


88. 


91. 


95 




16 


14. 


13 


11. 


10 


— 


— 


T 


"T 


T 


— 


69 


70. 


72 


74 


75. 


78 


80. 


83 


86 


89. 


92. 




24 


23 


21. 


20 


18. 


17. 


16 


14. 


13 


11. 


10 


67. 


68. 


70. 


72 


74 


76 


78. 


81 


84 


87 


90 




32 


31 


29. 


28 


27 


25. 


24. 


23 


21. 


20. 


19 


65. 


67 


68. 


70 


72 


74 


76. 


79 


81. 


84. 


88 




39. 


38. 


37 


36 


35 


33. 


32. 


31 


30 


28. 


27. 


64 


65 


66. 


68 


70 


72 


74. 


77 


79. 


82. 


85. 




47 


46 


44. 


43. 


42. 


41. 


40 


39 


38 


36. 


35. 


62 


63. 


64. 


66. 


68 


70 


72 


74. 


77. 


80 


83 




54 
0. 


53 

59. 


52 
58. 


50. 

57. 


49. 


48. 
55. 


47. 
54. 


46. 
53. 


45. 
52. 


44. 
51. 


43 
50. 


60. 


61. 


63 
61 


64. 
62. 


66 
64 


68 
66 


70 
68 


72. 
70. 


75 
73 


78 

75. 


80. 


"9" 


58. 


59. 


78 






a^BV 


••B^ 






^BtS 
















^— • 




















7. 


6. 


5. 


4. 


3. 


2. 


1. 


1 


T 


59 


58 


57 


58 


59 


60. 


62. 


64 


66 


68 


70. 


73 


76 




13. 


13 


12 


11 


10. 


9. 


8. 


7. 


6. 


5. 


5 


55 


56 


57. 


— 


60. 


62 


64 


66 


68. 


71 


73. 




20 


19 


18. 


17. 


17 


16 


15 


14. 


13. 


12. 


11. 


53. 


54. 


55. 


57 


58. 


60 


62 


64 


66 


68. 


71 




26 


25. 


24. 


24 


23 


22. 


21. 


20. 


20 


19 


18 


51. 


52. 


53. 


55 


56. 


— 


_60^ 


62 


64 


66. 


68. 




32 


31. 


31 


30 


29. 


28. 


28 


27 


26. 


25. 


24. 


49. 


50. 


52 


53 


54. 


56 


58^ 


59. 


62 


64 


66. 




38 


37. 


37 


36 


35. 


35 


34 


33. 


32. 


32 


31 


48 


49 


50 


51 


52. 


54 


55. 


57. 


■59: 


61. 


64 




44 


43. 


43 


42 


41. 


41 


40 


39. 


38. 


38 


37. 


46 


47 


48 


49. 


50. 


52 


53. 


55. 


57. 


59. 


61. 




50 


49 


48. 


48 


47. 


47 


46 


45. 


45 


44 


43. 


44. 


45. 


46. 


47. 


48. 


50 


51. 


53. 


55 


57 


"59" 




55. 


55 


54. 


54 


53 


52. 


52 


51. 


51 


50 


49. 


42. 


43. 


44. 


45. 


46. 


48 


49. 


51 


53 


55 


57 


^^i^ 














































10 


1 


0. 





59. 


59 


58. 


57. 


57 


56. 


56 


55. 


41 


41. 


42. 


43. 


44. 


46 


47. 


49 


50. 


52. 


54. 




6. 


6 


5. 


5 


4. 


4 


3. 


3 


2. 


2 


1. 


39 


40 


40. 


41. 


43 


44 


45. 


47 


48. 


50 


52 




12 


11. 


11 


10. 


10 


9. 


9 


8. 


8 


7. 


7 


37. 


38 


39 


40 


41 


42 


43. 


45 


46. 


48 


49 




17. 


17 


16. 


16 


15. 


15 


14. 


14 


13. 


13 


12. 


35. 


36 


37 


38 


39 


40 


41 


42. 


44 


45. 


47. 




23 


22. 


22 


21. 


21 


20. 


20 


19. 


19. 


19 


18. 


34 


34. 


35 


36 


37 


38 


39 


40. 


42 


43. 


45 




28 


27. 


27. 


27 


26. 


26 


25. 


25 


25 


24. 


24 


32 


32. 


33. 


34 


35 


36 


37 


38. 


39. 


41 


42. 




33. 


33 


32. 


32 


32 


31. 


31 


30. 


30 


30 


29. 


30 


31 


31. 


32 


33 


34 


35 


36 


37. 


39 


40 




38. 


38 


38 


37. 


37 


36. 


36. 


36 


35. 


35 


35 


28. 


29 


29. 


30. 


31 


32 


33 


34 


35. 


36. 


38 




43. 


43. 


43 


42. 


42. 


42 


41. 


41. 


41 


40. 


40 


26. 


27 


28 


28. 


29 


30 


31 


32 


33 


34 


35. 




49 


48. 


48 


48 


47. 


47. 


47 


46. 


46. 


46 


45. 


25 


25. 


26 


26. 


27 


28 


29 


30 


31 


32 


33 




54 


53. 


53 


53 


52. 


52. 


52 


52 


51. 


51 


51 


23 


23. 


24 


24. 


25. 


26 


27 


27. 


28. 


29. 


31 




59 


58. 


58. 


58 


58 


57. 


57. 


57 


57 


56. 


56 


21. 


22 


22 


22. 


23 


24 


25 


25. 


26. 


27. 


28 


— 


— 


— 


3. 


— 


— 


■^ 


— 


— 


— 


1. 


1. 


19. 


20 


20. 


21 


21. 


22 


22. 


23. 


24 


25 


26 




9 


8. 


8. 


8. 


8 


8 


7. 


7. 


7 


7 


6. 


18 


18 


18. 


19 


19. 


20 


20. 


21. 


22 


23 


23. 




14 


13. 


13. 


13. 


13 


13 


12. 


12. 


12. 


12 


12 


16 


16. 


16. 


17 


17. 


18 


18. 


19 


20 


20. 


21. 




19 


18. 


18. 


18. 


18 


18 


18 


17. 


17. 


17. 


17 


14 


14. 


15 


15 


15. 


16 


16. 


17 


17. 


18 


19 




24 


23. 


23. 


23. 


23 


23 


23 


22. 


22. 


22. 


22 


12. 


12. 


13 


13. 


13. 


14 


14. 


15 


15. 


16 


16. 




28. 


28. 


28. 


28. 


28. 


28 


28 


28 


27. 


27. 


27. 


10. 


11 


11 


11. 


11. 


12 


12. 


12. 


13 


13. 


14 




33. 


33. 


33. 


33. 


33 


33 


33 


33 


33 


32. 


32. 


9 


9 


9 


9. 


9. 


10 


10. 


10. 


11 


11. 


12 




38. 


38. 


38. 


38. 


38 


38 


38 


38 


38 


38 


37. 


7 


7 


7. 


7. 


8 


8 


8 


8. 


.9 


9 


9. 




43. 


43. 


43. 


43. 


43 


43 


43 


43 


43 


43 


43 


5. 


5. 


5. 


5. 


6 


6 


6 


6. 


6. 


7 


7 




48. 


48. 


48 


48 


48 


48 


48 


48 


48 


48 


48 


3. 


3. 


3. 


4 


4 


4 


4 


4 


4. 


4. 


4. 




53 


53 


53 


53 


53 


53 


53 


53 


53 


53 


53 


2 


2 


2 


2 


2 


2 


2 


2 


2 


2. 


2. 




58 


58 


58 


58 


58 


58 158 


58 


58 


58 


58 











01 01 


















Note. — The time of lower culmination of Polaris occurs llh. 54m. before or after the 
time of upper culmination. The time of eastern and of western elongation occurs 
approximately 5h. 55m. before and after the time of upper culmination. Hence the 
time of elongation may be obtained from the above table. See also Fig. 14, p. 949, and 
Table 2, p. 950. tor data in this connection. 



955a 



58.— SURVEYING, MAPPING AND LEVELING. 



4a. — Polaris, 1912, for Meridian of Greenwich. 
Civil Date and Clock Time. 



Date 


Upper Cul- 


Elongation, 


Decli- 


Date 


Upper Cul- 


Elongation, 


Decll- 


1912. 


mination. 


Lat. 40°. 


n'tion. 


1912. 


mination. 


Lat. 40°. 


n'tion. 








O / 

+88 50 








O 1 

+88 50 


Jan. 


hm 


hm 


n 


Mar. 


hm 


hm 


It 


1 


6 47.4 P.M. 


W.E. 46.4 A.M. 


30.8 


1 


2 50.5 P.M. 


W.E. 8 45.6 P.M. 


27.7 


2 


6 43.5 


42.5 


31.0 


2 


2 46.6 


8 41.7 


27.5 


3 


6 39.5 


38.6 


31.1 


3 


2 42.6 


8 37.7 


27.2 


4 


6 35.6 


34.6 


31.3 


4 


2 38.7 


8 33.8 


27.0 


5 


6 31.6 


30.7 


31.5 


5 


2 34.7 


8 29.8 


26.7 


6 


6 27.7 


26.7 


31.6 


6 


2 30.8 


8 25.9 


26.4 


7 


6 23.7 


22.8 


31.7 


7 


2 26.9 


8 22.0 


26.1 


8 


6 19.8 


18.8 


31.8 


8 


2 22.9 


8 18.0 


25.8 


9 


6 15.8 


14.9 


31.9 


9 


2 19.0 


8 14.1 


25.6 


10 


6 11.9 


10.9 


31.9 


10 


2 15.0 


8 10.1 


25.3 


11 


6 7.9 


7.0 


31.9 


11 


2 11.1 


8 6.2 


25.1 


12 


6 4.0 


[ 3.0 A.M. 
I 11 59.1 P.M. 


J32.0 


12 
13 


2 7.2 
2 3.2 


8 2.3 
7 58.3 


24.9 
24.6 


13 


6 0.0 


11 55.1 


32.0 


14 


1 59.3 


7 54.4 


24.4 


14 


5 56.1 


11 51.2 


32.0 


15 


1 55.3 


7 50.4 


24.1 


15 


5 52.1 


11 47.2 


32.1 


16 


1 51.4 


7 46.5 


23.9 


16 


5 48.2 


11 43.3 


32.2 


17 


1 47.5 


7 42.6 


23.6 


17 


5 44.2 


11 39.3 


32.2 


18 


1 43.5 


7 38.6 


23.3 


18 


5 40.3 


11 35.4 


32.3 


19 


1 39.6 


7 34.7 


22.9 


19 


5 36.3 


11 31.4 


32.3 


20 


1 35.6 


7 30.7 


22.6 


20 


5 32.4 


11 27.5 


32.4 


21 


1 31.7* 


7 26.8 


22.3 


21 


5 28.4 


11 23.5 


32.4 


22 


1 27.8 


7 22.9 


22.0 


22 


5 24.5 


11 19.6 


32.4 


23 


1 23.8 


7 18.9 


21.7 


23 


5 20.5 


11 15.6 


32.3 


24 


I 19.9 


7 15.0 


21.4 


24 


5 16.6 


- 11 11.7 


32.3 


25 


1 16.0 


7 11.1 


21.1 


25 


5 12.6 


11 7.7 


32.2 


26 


1 12.0 


7 7.1 


20.9 


26 


5 8.7 


11 3.8 


32.2 


27 


1 8.1 


7 3.2 


20.7 


27 


5 4.7 


10 59.8 


32.1 


28 


1 4.1 


6 59.2 


20.4 


28 


5 0.8 


10 55.9 


32.1 


29 


1 0.2 


6 55.3 


20.1 


29 


4 56.8 


10 51.9 


32.0 


30 


56.3 


6 51.4 


19.8 


30 


4 52.9 


10 48.0 


32.0 


31 


52.3 


6 47.4 


19.5 


31 


4 48.9 


10 44.0 


32.0 










Feb. 

1 


4 45.0 P.M. 


W.E. 10 40.1 P.M. 


32.0 


Apr. 
1 


48.4 P.M. 


W.E. 6 43.5 P.M. 


19.1 


2 


4 41.0 


10 36.1 


32.0 


2 


44.5 


6 39.6 


18.8 


3 


4 37.1 


10 32.2 


32.0 


3 


40.5 


6 35.6 


18.5 


4 


4 33.1 


10 28.2 


31.9 


4 


36.6 


6 31.7 


18.1 


5 


4 29.2 


10 24.3 


31.8 


5 


32.7 


6 27.8 


17.8 


6 


4 25.2 


10 20.3 


31.6 


6 


28.7 


6 23.8 


17.5 


7 


4 21.3 


10 16.4 


31.5 


7 


24.8 


6 19.9 


17.2 


8 


4 17.3 


10 12.4 


31.3 


8 


20.9 


6 16.0 


17.0 


9 


4 13.4 


10 8.5 


31.2 


9 


16.9 


6 12.0 


16.7 


10 


4 9.4 


10 4.5 


31.1 


10 


13.0 


6 8.1 


16.4 


11 


4 5.5 


10 0.6 


31.0 


11 


9.1 


6 4.2 


16.1 


12 


4 1.5 


9 56.6 


30.8 


12 


5.1. 


6 0.2 


15.8 


13 


3 57.6 


9 52.7 


30.7 


13 


1.2 P.M. 


W.E. 5 56.3 P.M. 


15.5 


14 


3 53.7 


9*48.8 


30.6 


14 


11 57.3 A.M. 


E.E. 6 2.2 A.M. 


15.2 


15 


3 49.7 


9 44.8 


30.5 


15 


11 53.4 


5 58.3 


14.9 


16 


3 45.8 


9 40.9 


30.4 


16 


11 49.4 


5 54.3 


14.5 


17 


3 41.8 


9 36.9 


30.2 


17 


11 45.5 


5 50.4 


14.2 


18 


3 37.9 


9 33.0 


30.0 


18 


11 41.6 


5 46.5 


13.9 


19 


3 33.9 


9 29.0 


29.9 


19 


11 37.6 


5 42.5 


13.6 


20 


3 30.0 


9 25.1 


29.7 


20 


11 33.7 


5 38.6 


13.3 


21 


3 26.0 


9 21.1 


29.4 


21 


11 29.8 


5 34.7 


13.0 


22 


3 22.1 


9 17.2 


29.2 


22 


11 25.9 


5 30.8 


12.8 


23 


3 18.1 


9 13.2 


29.0 


23 


11 21.9 


5 26.8 


12.5 


24 


3 14.2 


9 9.3 


28.8 


24 


11 18.0 


5 22.9 


12.3 


25 


3 10.2 


9 5.3 


28.6 


25 


11 14.1 


5 19.0 


12.0 


26 


3 6.3 


9 1.4 


28.4 


26 


11 10.2 


5 15.1 


11.8 


27 


3 2.4 


8 57.5 


28.2 


27 


11 6.2 


5 11.1 


11.5 


28 


2 58.4 


8 53.5 


28.1 


28 


11 2.3 


5 7.2 


11.1 


29 


2 54.5 


8 49.6 


27.9 


29 


10 58.4 


5 3.3 


10.8 


30 


2 50.5 


8 45.6 


27.7 


30 


10 54.4 


4 59.3 


10.5 










31 


10 50.5 


4 55.4 


10.2 



TABLES OF POLARIS, 1912, 



965b 



4a. — Polaris, 1912, for Meridian op Greenwich. — Continued. 
Civil Date and Clock Time. 



Date 


Upper Cul- 


Elongation, 


DecII- 


Date 


Upper Cul- 


Elongation, 


DecII- 


1912. 


minatioD. 


Lat. 40°. 


n'tion. 


1912. 


mination. 


Lat. 40°. 


n'tion. 








o / 

+ 88 50 








O / 

4-88 50 


May 


h m 


hm 


n 


July 


h m 


h m 


tf 


1 


10 50.5 A.M. 


E.E. 4 55.4 A.M. 


10.2 


1 


6 51.5 A.M. 


E.E. 56.4 A.M. 


1.6 


2 


10 46.6 


4 51.5 


9.9 


2 


6 47.6 


52.5 


1.7 


3 


10 42.7 


4 47.6 


9.7 


3 


6 43.7 


48.6 


1.7 


4 


10 38.8 


4 43.7 


9.4 


4 


6 39.8 


44.7 


1.7 


5 


10 34.8 


4 39.7 


9.2 


5 


6 35.9 


40.8 


1.7 


6 


10 30.9 


4 35.8 


8.9 


6 


6 32.0 


36.9 


1.7 


7 


10 27.0 


4 31.9 


8.7 


7 


6 28.0 


32.9 


1.7 


8 


10 23.1 


4 28.0 


8.5 


8 


6 24.1 


29.0 


1.7 


9 


10 19.1 


4 24.0 


8.3 


9 


6 20.2 


25.1 


1.7 


10 


10 15.2 


4 20.1 


8.0 


10 


6 16.3 


21.2 


1.8 


11 


10 11.3 


4 16.2 


7.8 


11 


6 12.4 


17.3 


1.9 


12 


10 7.4 


4 12.3 


7.5 


12 


6 8.5 


13.4 


2.0 


13 


10 3.4 


4 8.3 


7.2 


13 


6 4.6 


9.5 


2.1 


14 


9 59.5 


4 4.4 


6.9 


14 


6 0.7 


5.6 


2.2 


15 
16 


9 55.6 
9 51.7 


4 0.5 
3 56.6 


6.7 
6.5 


15 


5 56.7 


{ 1.6 A.M. 
11 57.7P.M. 


1 2.3 


17 


9 47.8 


3 52.7 


6.3 


16 


5 52.8 


11 53.8 


2.5 


18 


9 43.9 


3 48.8 


6.1 


17 


5 48.9 


11 49.9 


2.6 


19 


9 39.9 


3 44.8 


5.9 


18 


5 45.0 


11 46.0 


2.7 


20 


9 36.0 


3 40.9 


5.8 


19 


5 41.1 


11 42.1 


2.7 


21 


9 32.1 


3 37.0 . 


5.6 


20 


5 37.2 


11 38.2 


2.8 


22 


9 28.2 


3 33.1 


5.5 


21 


5 33.3 


11 34.2 


2.9 


23 


9 24.3 


3 29.2 


5.3 


22 


5 29.3 


11 30.3 


3.0 


24 


9 20.3 


3 25.2 


5.1 


23 


5 25.4 


11 26.4 


3.1 


25 


9 16.4 


3 21.3 


4.8 


24 


5 21.5 


11 22.5 


3.2 


26 


9 12.5 


3 17.4 


4.6 


25 


5 17.6 


11 18.6 


3.4 


27 


9 8.6 


3 13.5 


4.4 


26 


5 13.7 


11 14.7 


3.6 


28 


9 4.7 


3 9.6 


4.2 


27 


5 9.8 


11 10.8 


3.7 


29 


9 0.7 


3 5.6 


4.0 


28 


5 5.9 


11 6.8 


3.9 


30 


8 56.8 


3 1.7 


3.9 


29 


5 1.9 


11 2.9 


4.1 


31 


8 52.9 


2 57.8 


3.7 


30 


4 58.0 


10 59.0 


4.3 










31 


4 54.1 


10 55.1 


4.4 


Jutie 
1 


8 49.0 A.M. 


E.E. 2 53.9 A.M. 


3.6 


Aug. 
1 


4 50.2 A.M. 


E.E.IO 51.2 P.M. 


4.6 


2 


8 45.1 


2 50.0 


3.5 


2 


4 46.3 


10 47.3 


4.7 


3 


8 41.2 


2 46.1 


3.4 


3 


4 42.4 


10 43.4 


4.9 


4 


8 37.2 


2 42.1 


3.3 


4 


4 38.5 


10 39.4 


5.0 


5 


8 33.3 


2 38.2 


3.2 


5 


4 34.5 


10 35.5 


5.2 


6 


8 29.4 


2 34.3 


3.0 


6 


4 30.6 


10 31.6 


5.4 


7 


8 25.5 


2 30.4 


2.9 


7 


4 26.7 


10 27.7 


5.6 


8 


8 21.6 


2 26.5 


2.8 


8 


4 22.8 


10 23.8 


5.8 


9 


8 17.7 


2 22.6 


2.6 


9 


4 18.9 


10 19.9 


6.0 


10 


8 13.7 


2 18.6 


2.5 


10 


4 15.0 


10 16.0 


6.3 


11 


8 9.8 


■ 2 14.7 


2.3 


11 


4 11.1 


10 12.0 


6.6 


12 


8 5.9 


2 10.8 


2.2 


12 


4 7.1 


10 8.1 


6.8 


13 


8 2.0 


2 6.9 


2.1 


13 


4 3.2 


10 4.2 


7.1 


14 


7 58.1 


2 3.0 


2.1 


14 


3 59.3 


10 0.3 


7.3 


15 


7 54.2 


1 59.1 


2.0 


15 


3 55.4 


9 56.4 


7.5 


16 


7 50.3 


1 55.2 


2.0 


16 


3 51.5 


9 52.5 


7.7 


17 


7 46.3 


1 51.2 


2.0 


17 


3 47.6 


9 48.5 


7.9 


18 


7 42.4 


1 47.3 


2.0 


18 


3 43.6 


9 44.6 


8.1 


19 


7 38.5 


1 43.4 


1.9 


19 


3 39.7 


9 40.7 


8.4 


20 


7 34.6 


1 39.5 


1.9 


20 


3 35.8 


9 36.8 


8.6 


21 


7 30.7 


1 35.6 


1.8 


21 


3 31.9 


9 32.9 


8.9 


22 


7 26.8 


1 31.7 


1.7 


22 


3 28.0 


9 29.0 


9.2 


23 


7 22.8 


1 27.7 


1.7 


23 


3 24.1 


9 25.0 


9.5 


24 


7 18.9 


1 23.8 


1.6 


24 


3 20.1 


9 21.1 


9.8 


25 


7 15.0 


1 19.9 


1.5 


25 


3 16.2 


9 17.2 


10.1 


26 


7 11.1 


1 16.0 


1.5 


26 


3 12.3 


9 13.3 


10.4 


27 


7 7.2 


1 12.1 


1.5 


27 


3 8.4 


9 9.4 


10.7 


28 


7 3.3 


1 8.2 


1.5 


28 


3 4.5 


9 5.4 


11.0 


29 


6 59.4 


1 4.3 


1.6 


29 


3 0.5 


9 1.5 


11.2 


30 


6 55.5 


1 0.4 


1,6 


30 


2 56.6 


8 57.6 


11.5 


31 


6 51.5 


56.4 


1.6 


31 


2 52.7 


8 53.7 


11.7 










32 


2 48.8 


8 49.8 


12.0 



S55c 



58,--SURVEYING, MAPPING AND LEVELING. 



4a. — Polaris, 1912, for Meridian op Greenwich. — Concluded. 
Civil Date and Clock Time. 



Date 


Upper Cul- 


Elongation, 


Decli- 


Date 


Upper Cul- 


Elongation, 


Decll- 


1912. 


miDation. 


Lat. 40°. 


n'tion. 


1912. 


mination. 


Lat. 40^ 


n'tion. 








, 

+ 88 50 








o > 

+88 50 


Sep. 


h m 


hm 


If 


Nov. 


hm 


hm 


* 


1 


2 48.8 A.M. 


E.E. 8 49.8P.M. 


12.0 


1 


10 45.4P.M. 


W.E. 4 44.4 A.M. 


34.9 


2 


2 44.9 


8 45.9 


12.3 


2 


10 41.4 


4 40.5 


35.3 


3 


2 41.0 


8 41.9 


12.6 


3 


10 37.5 


4 36.5 


35.6 


4 


2 37.0 


8 38.0 


12.9 


4 


10 33.6 


4 32.6 


36.0 


5 


2 33.1 


8 34.1 


13.2 


5 


10 29.6 


4 28.7 


36.3 


6 


2 29.2 


8 30.2 


13.6 


6 


10 25.7 


4 24.7 


36.7 


7 


2 25.3 


8 26.3 


14.0 


7 


10 21.8 


4 20.8 


37.0 


8 


2 21.4 


8 22.3 


14.4 


8 


10 17.8 


4 16.9 


37.3 


9 


2 17.4 


8 18.4 


14.7 


9 


10 13.9 


4 12.9 


37.7 


10 


2 13.5 


8 14.5 


15.1 


10 


10 9.9 


4 9.0 


38.1 


11 


2 9.6 


8 10.6 


15.4 


11 


10 6.0 


4 5.0 


38.5 


12 


2 5.7 


8 6.6 


15.7 


12 


10 2.1 


4 1.1 


38.9 


13 


2 1.7 


8 2.7 


16.0 


13 


9 58.1 


3 57.2 


39.3 


14 


1 57.8 


7 58.8 


16.3 


14 


9 54.2 


3 53.2 


39.6 


15 


1 53.9 


7 54.9 


16.6 


15 


9 50.3 


3 49.3 


40.0 


16 


1 50.0 


7 51.0 


16.9 


16 


9 46.3 


3 45.4 


40.4 


17 


1 46.1 


7 47.0 


17.3 


17 


9 42.4 


3 41.4 


40.7 


IS 


1 42.1 


7 43.1 


17.7 


18 


9 38.4 


3 37.5 


41.0 


19 


1 38.2 


7 39.2 


18.0 


19 


9 34.5 


3 33.5 


41.3 


20 


1 34.3 


7 35.3 


18.4 


20 


9 30.6 


3 29.6 


41.6 


21 


1 30.4 


7 31.3 


18.8 


21 


9 26.6 


3 25.7 


41.9 


22 


1 26.4 


7 27.4 


19.2 


22 


9 22.7 


3 21.7 


42.2 


23 


1 22.5 


7 23.5 


19.6 


23 


9 18.7 


3 17.8 


42.5 


24 


1 18.6 


7 19.6 


19.9 


24 


9 14.8 


3 13.8 


42.9 


25 


1 14.7 


7 15.6 


20.3 


25 


9 10.9 


3 9.9 


43.2 


Z6 


1 10.7 


7 11.7 


20.6 


26 


9 6.9 


3 6.0 


43.6 


27 


1 6.8 


7 7.8 


20.9 


27 


9 3.0 


3 2.0 


43.9 


28 


1 2.9 


7 3.9 


21.3 


28 


8 59.0 


2 58.1 


44.3 


29 


59.0 


7 0.0 


21.6 


29 


8 55.1 


2 54.1 


44.6 


30 


55.1 


6 56.0 


22.0 


30 


8 51.2 


2 50.2 


44.9 


Oct. 








Dec. 








1 


51.1 A.M. 


E.E. 6 52.1P.M. 


22.3 


1 


8 47.2P.M. 


W.E. 2 46.3 A.M. 


45.2 


2 


47.2 


6 48.2 


22.7 


2 


8 43.3 


2 42.3 


45.5 


3 


43.3 


6 44.3 


23.2 


3 


8 39.3 


2 38.4 


45.7 


4 


39.4 


6 40.3 


23.6 


4 


8 35.4 


2 34.4 


46.0 


5 


35.4 


6 36.4 


24.0 


5 


8 31.4 


2 30.5 


46.2 


6 


31.5 


6 32.5 


24.4 


6 


8 27.5 


2 26.5 


46.5 


7 


27.6 


6 28.5 


24.8 


7 


8 23.5 


2 22.6 


46.7 


8 


23.6 


6 24.6 


25.2 


8 


8 19.6 


2 18.6 


47.0 


9 


19.7 


6 20.7 


25.6 


9 


8 15.7 


2 14.7 


47.3 


10 


15.8 


6 16.7 


25.9 


10 


8 11.7 


2 10.8 


47.6 


11 


11.8 


6 12.8 


26.3 


11 


8 7.8 


2 6.8 


47.9 


12 


7.9 


6 8.9 , 


26.6 


12 


8 3.8 


2 2.9 


48.2 


13 


4.0 


E.E. 6 5.0 P. k. 


27.0 


13 


7 59.9 


1 58.9 


48.4 


14 


0.1 A.M. 
11 56.2P.M. 


|W.E.5 55.2 A.M. 
'' 5 51.3 


f 27.4 
1 27.8 


14 

15 


7 55.9 
7 52.0 


1 55.0 
1 51.0 


48.7 
48.9 


15. 


11 52.2 


'28.2 


16 


7 48.0 


1 47.1 


49.1 


16 


11 48.3 


5 47.3 


28.6 


17 


7 44.1 


1 43.1 


49.3 


17 


11 44.4 


5 43.4 


29.1 


18 


7 40.1 


1 39.2 


49.5 


18 


11 40.4 


5 39.5 


29.5 


19 


7 36.2 


1 35.2 


49.6 


19 


11 36.5 


5 35.5 


29.9 


20 


7 32.3 


1 31.3 


49.8 


20 


11 32.6 


5 31.6 


30.2 


21 


7 28.3 


1 27.4 


50.0 


21 


11 28.6 


5 27.7 


30.6 


22 


7 24.4 


1 23.4 


50.2 


22 


11 24.7 


5 23.7 


31.0 


23 


7 20.4 


1 19.5 


50.4 


23 


11 20.8 


5 19.8 


31.3 


24 


7 16.5 


1 15.5 


50.7 


24 


11 16.8 


5 15.9 


31.7 


25 


7 12.5 


1 11.6 


50.9 


25 


11 12.9 


5 11.9 


32.0 


26 


7 8.6 


1 7.6 


51.1 


26 


11 9.0 


5 8.0 


32.4 


27 


7 4.6 


1 3.7 


51.3 


27 


11 5.0 


5 4.1 


32.8 


28 


7 0.7 


59.7 


51.5 


28 


11 1.1 


5 0.1 


33.2 


29 


6 56.7 


55.8 


51.6 


29 


10 57.2 


4 56.2 


33.6 


30 


6 52.8 


51.8 


51.7 


30 


10 53.2 


4 52.3 


34.0 


31 


6 48.8 


47.9 


51.3 


31 


10 49.3 


4 48.3 


34.4 


32 


6 44.9 


43.9 


51.9 


32 


10 45.4 


4 44.4 


34.9 











TABLES OF POLARIS, 1912. 



955d 



4b. — Polaris, 1912, for Various Latitudes. 
(To accompany Tables 4a and 4c.) 





Corrections to the 
Times of Elonga- 
tions for Different 
Latitudes. 


AZIMUTHS OF POLARIS AT ELONGATION, 1912. 


UUtude. 


Decl. + 88»60' 




0" 


10" 


20" 


30" 


40" 


50" 





W.E. mE.E. 


o / // 


o / // 


9 t II 


/ // 


t It 


1 H 


25 


+1.7- 


1 17 14 


117 3 


1 16 52 


1 16 41 


1 16 30 


1 16 19 


26 


1.6 


1 17 53 


1 17 42 


1 17 31 


1 17 20 


1 17 8 


1 16 57 


27 


1.5 


1 18 34 


1 18 23 


1 18 11 


1 18 


1 17 49 


1 17 38 


28 


1.4 


1 19 17 


1 19 6 


1 18 54 


1 18 43 


1 18 32 


1 18 20 


29 


1.3 


1 20 2 


1 19 51 


1 19 39 


1 19 28 


1 19 16 


1 19 5 


30 


1.2 


1 20 50 


1 20 38 


1 20 27 


1 20 15 


1 20 4 


1 19 52 


31 


1.1 


1 21 40 


1 21 28 


1 21 17 


1 21 5 


1 20 53 


1 20 42 


32 


1.0 


1 22 33 


1 22 21 


1 22 9 


1 21 57 


1 21 46 


1 21 34 


33 


0.9 


123 28 


1 23 16 


1 23 4 


i 22 52 


122 40 


1 22 28 


34 


0.7 


124 26 


1 24 14 


1 24-2 


123 50 


1 23 38 


1 2326 


35 


0.6 


125 27 


1 25 15 


1 25 3 


1 24 51 


1 24 39 


1 24 26 


36 


0.5 


126 32 


1 26 19 


1 26 7 


1 25 55 


1 25 42 


1 25 30 


37 


0.4 


127 39 


1 27 27 


1 27 14 


1 27 2 


1 26 49 


1 26 37 


38 


0.2 


128 50 


1 28 37 


1 28 25 


1 28 12 


1 27 59 


1 27 ,47 


39 


+0.1- 


130 5 


1 29 52 


1 29 39 


1 29 26 


129 13 


129 


40 


0.0 


1 31 23 


1 31 10 


1 30 57 


1 30 44 


1 30 31 


130 18 


41 


-0.1+ 


132 45 


1 32 32 


1 32 19 


1 32 6 


1 31 62 


13139 


42 


0.3 


1 34 12 


1 33 58 


1 33 45 


1 33 32 


1 33 18 


133 5 


13 


0.4 


135 43 


1 35 30 


1 35 16 


1 35 2 


1 34 48 


1 34 35 


44 


0.6 


137 19 


137 5 


1 36 51 


1 36 37 


1 36 23 


136 9 


15 


0.8 


1 39 


1 38 46 


1 38 32 


1 38 18 


1 38 4 


1 37 50 


46 


1.0 


1 40 47 


1 40 32 


1 40 18 


1 40 3 


' 1 39 49 


1 39 35 


47 


1.2 


1 42 39 


1 42 24 


1 42 10 


1 41 55 


I 41 40 


1 41 26 


48 


i.4 


1 44 37 


1 44 22 


1 44 7 


1 43 52 


.1 43 38 


1 43 23 


49 


1.5 


1 46 42 


1 46 27 


1 46 12 


1 45 57 


1 45 42 


1 45 26 


SO 


1.7 


1 48 55 


1 48 39 


1 48 24 


1 48 8 


1 47 52 


1 47 37 


51 


1.9 


1 51 15 


1 50 59 


1 50 43 


1 50 27 


1 50 11 


149 55 


62 


2.1 


1 53 43 


1 53 26 


1 53 10 


1 52 54 


1 52 38 


1 5222 


53 


2.3 


1 56 20 


1 56 3 


1 55 47 


1 55 30 


1 55 13 


1 54 57 


54 


2.6 


1 59 6 


1 58 49 


1 58 32 


1 58 15 


1 57 58 


1 57 41 


55 


2.8 


2 2 3 


2 1 46 


2 1 29 


2 1 11 


2 54 


2 36 


56 


io 


2 5 12 


2 4 54 


.2 4 36 


2 4 18 


2 4 


2 3 42 


57 


3.3 


2 8 33 


2 8 14 


2 7 56 


2 7 38 


2 7 19 


2 7 1 


58 


3.6 


2 12 7 


2 11 48 


2 11 29 


2 11 10 


2 10 52 


2 10 33 


59 


3.9 


2 15 56 


2 15 37 


2 15 17 


2 14 58 


2 14 39, 


2 14 19 


60 


4.2 


220 2 


2 19 42 


2 19 22 


2 19 2 


2 18 42 • 


2 18 22 


61 


4.5 


2 24 25 


2 24 4 


2 23 44 


2 23 23 


2 23 3 


2 22 42 


62 


4.9 


2 29 8 


2 28 47 


2 28 26 


. 2 28 4 


2 27 43 


2 27 22 


63 


5.2 


2 34 14 


2 33 52 


2 33 30 


2 33 8 


2 32 46 


2 32 24 


64 


5.6 


2 39 44 


2 39 21 


2 38 58 


2 38 35 


2.38 12 


2 37 50 


65 


6.0 


2 45 41 


2 45 18 


2 44 54 


2 44 30 


2 44 6 


2 43 43 


66 


6:4 


2 52 10 


2 51 45 


2 51 20 


, 2 50 56- 


2 50 31 


250 7 


67 


6.9 


2 59 13 


258 48 


258 22 


2 57 56 


2 57 31 


2 57 5 


68 


7.5 


3 6 56 


3 6 30 


3 6 3'^ 


3 5 36 


3 5 10 


3 443 


69 


8.1 


3 15 25 


3 14 57 


3 14 29' 


3 14 1 


3 13 34 


3 13 6 


70 


-8.7+ 


3 24 46 


3 24 17 


3 23 48 


3 23 19 


3 22 49 


3 22 20 



965e 



58.— SURVEYING, MAPPING AND LEVELING. 



4c, — Polaris, 1912, General Table. 
(To accompany Tables 4a and 4b.) 



MEAN TIME HOTTR ANGLES, 1912. j 


AZIMUTHS OF POLARIS, 1912. 




Decl. + 88' 5(y 


Lititude. 


i 

o 

W 


0" 


10" 


20" 


20" 


40" 


50" 


30" 


32" 


34" 


36° 


38" 


40" 


42" 


44" 


46' 


48' 


60» 




m 


m 


m 


m 


m 


m 


, 


/ 


/ 


/ 


/ 


/ 


/ 


/ 


/ 


/ 


, 





0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 o.ol 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 




4.9 


4.9 


4.9 


5.0 


6.0 


5.0 


1.8 


1.8 


1.8 


1.9 


1.9 


2.0 


2.1 


2.1 


2.2 


2.3 


2.4- 




9.8 


9.8 


9.8 


9.9 


9.9 


10.0 


3.5 


3.6 


3.7 


3.8 


3.9 


4.0 


4,1 


4.3 


4,4 


4.6 


4.8' 




14.8 


14.8 


14.8 


14.9 


14.9 


15 


5.3 


5.4 


5.5 


5.7 


5.8 


6.0 


6.2 


6.4 


6.6 


6.9 


7.2 




19.7 


19.7 


19-8 


19.8 


19.9 


19.9 


7.0 


7.2 


7.4 


7.6 


7.8 


8.0 


8.3 


8.6 


8.9 


9,2 


9.6 




24.6 


24.7 


24.7 


24.8 


24.9 


24.9 


8.8 


9.0 


9.2 


9.4 


9.7 


10.0 


10.3 


10.7 


11.1 


11.5 


12.0 




29.6 


29.7 


29.7 


29.8 


29.9 


29.9 


10.6 


10.8 


11.1 


11.3 


11.7 


12.0 


12.4 


12.8 


13.3 


13.8 


14.4 




34.6 


34.7 


34.8 


34.8 


34.9 


35.0 


12.3 


12.6 


12.9 


13.2 


13.6 


14.0 


14.5 


15.0 


15.5 


16.1 


16.8 




■39.6 


30.7 


39.8 


39.8 


39.9 


40.0 


14.1 


14.4 


14.7 


15,1 


15.5 


16.0 


16.5 


17.1 


17.7 


18,4 


19.2 




44.6 


44.7 


44.8 


44.9 


45.0 


45.1 


15.8 


16.2 


16.6 


17.0 


17.5 


18.0 


18.6 


19.2 


19.9 


20.7 


2L6 




49.6 


49.7 


49.8 


50.0 


50.1 


50.2 


17.6 


18.0 


18.4 


18.9 


19.4 


20.0 


20. d 21.41 


22.2 


23.0 


24.0 




54.7 
59.7 


54.8 
59.8 


54.9 


55.0 


55.2 


55.3 


19.4 
2L1 
22.9 


19.8 
2L6 
23.4 


20.3 
22.1 
24.0 


20.8 
22.7 
24.6 


21.4 
23.3 
25.3 


22.0 
24.0 
26.0 


22.7 
24.8 
26.8 


23.5 
25.6 
27.8 


24.4 
26.6 
28.8 


25.3 
27.6 
29,9 


26.4 




0.0 
5.1 


0.2 
5.3 


0.3 
5.5 


0.4 

5.6 


28.8 


1 


4.8 


5.0 


31.2 




10.0 


10.2 


10.3 


10.5 


10.7 


10.8 


24.6 


25.2 


25.8 


26.5 


27,2 


28.0 


28.9 


29.9 


31.0 


32,2 


33.6 




.15.2 


15.4 


15.6 


15.8 


15.9 


16.1 


26.4 


27.0 


27.6 


28 4 


29.1 


30.0 


31.0 


32.0 


33.2 


34.5 


36,0 




20.4 


20.6 


20.8 


21.0 


21.2 


21.4 


28.2 


28.8 


29.5 


30.2 


31.1 


32.0 


33.0 


34.2 


35.4 


36.8 


38.4 




25.7 


25.9 


26.1 


26.4 


26.6 


26.8 


29.9 


30.6 


31.3 


32.1 


33,0 


34.0 


35.1 


36.3 


37.6 


39.1 


40,8 




31.0 


31.2 


31.5 


31.7 


31.9 


.32.2 


31.7 


32.4 


33.2 


34.0 


35.0 


36.0 


37.2 


38.4 


39.8 


41.4 


43,2 




•36.4 


36.6 


36.9 


37.2 


37.4 


37.6 


33.4 


34.2 


35.0 


35.9 


36,9 


38.0 


39.2 


40.6 


42.1 


43,7 


45:6 




41.8 


42.1 


42.3 


42.6 


42.9 


43 1 


35.2 


36.0 


36.8 


37.8 


38.8 


40.0 


41.3 


42,7 


44,3 


46,0 


48,0 




47.4 


47,7 


48.0 


48.2 


48.5 


48.8 


37.0 


37.8 


38.7 


39.7 


40.8 


42.0 


43.3 


44.8 


46.5 


48,3 


50.4 




53.0 


53.3 


53.6 


53.9 


64.2 


54.5 


38.7 


39.6 


40.5 


41.6 


42.7 


44.0 


45.4 


47.0 


48.7 


60.6 


52,8 




58.6 


53.9 


59.3 


59.6 


J!:! 


0.3 
6.1 


40.5 
42.3 


4L4 
43.2 


42.4 
44.2 


43.5 
45.4 


44.7 
46.6 


46.0 
48.0 


47,5 
49.5 


49.1 
51.2 


50,9 
63,1 


52.9 
55.2 


55.2 


2 


4.4 


4.7 


6.1 


5.4 


5.7 


57,5 




10.2 


10.6 


10.9; 


• 11.2 


11.6 


12.0 


44.0 


45.0 


46.1 


47.2 


48.6 


60.0 


51.6 


53.4 


65,3 


57,5 


69.9 




16.2 


16. e 


17.0 


17.4 


17.7 


18.1 


45.8 


46.8 


47.9 


49.1 


60.5 


62.0 


53.7 


55.5 


57.5 


69,8 


62.3 




■22.3 


22.7 


23.1 


23.5 


23.9 


24.3 


47.6 


48.6 


49.8 


51.0 


52.4 


64.0 


65.7 


67.6 


59.7 


"ell 


64.7 




28.5 
34.7 


28.9 
35.2 


29.4 
35.6 


29.8 
36.1 


30.2 
36.6 


30.7 
37.0 


49:3 
61.1 


50.4 
52.2 


6L6 
63.5 


62.9 
54,8 


54.4 
56.3 


56.0 
58,0 


57.8 
59.8 


59.7 


61.9 
64.1 


64.4 
66.7 


67,1 




61.9 


69.5 


























^^M^ 














41.1 


41.6 


42.1 


42.6 


43.0 


43.5 


52.8 


64.0 


55.3 


66,7 


58.3 


60,0 


61.9 


64.0 


66.4 


69.0 


71.9 




47.8 


48.3 


48.8 


49.4 


49.9 


50.4 


5^.6 


65.8 


5.. 


58.6 


60.2 


62,0 


64.0 


66.1 


68.6 


71.3 


74.3 




54.6 


55.1 


55.7 


66.2 


56.8 


57.3 


56.4 
58.2 
59.9 


57.6 


59.0 
60.9 
62.7 


60,5 
62.4 
64.3 


62.2 
64.1 
66.1 


64.0 
66.0 
68.0 


66.0 
68,1 
70.1 


68.3 
70.4 
72.5 


70.8 
73.0 
75.2 


73,6 
75.8 
78.1 


76.6 


"^ 


L7 

8.9 


2.3 
.9:5 


2.9 
10.1 


3.4 
10.8 


4.0 
11.4 


12.0 


79.0 




.61.3 


81.4 




16.4 


17.1 


17.7 


18.4 


19.0 


19.7 


61.7 


63.1 


64.6 


66.2 


68.0 


70.0 


72.2 


74.6 


77.4 


80.4 


83.8 




24.4 


25.1 


25.8 


26.5 


27.3 


28.0 


63.5 


64.9 


66.4 


68.1 


69.9 


72.0 


74.3 


76.8 


79,6 


82.7 


86.1 




32.7 


33.5 


34.2 


35.0 


35.8 


36.6 


65.3 


66.7 


68.3 


70.0 


71.9 


74.0 


76.3 


78.9 


8L8 


85.0 


88.5 




41.4 


42.2 


43.1 


43.9 


44.8 


45.7 


67.0 


68.5 


70.1 


71.9 


73.8 


76.0 


78.4 


81.0 


84,0 


87.2 


90.9 




50.6 


61.5 


52.4 


53.3 


54.3 


55.2 


68.8 
70.6 


70.3 
72.1 


72.0 
73.8 


73,8 
75.7 


75.8 
77.7 


78,0 
80,0 


80.5 
82.5 


83.2 
85.3 


86.2 
88.4 


89.5 
91.8 


93.3 


4 


0.3 


1.3 


2.3 


3; 3 


4.3 


5.4 


95.6 




11.1 


12.2 


13.4 


14.5 


15.7 


16.8 


72.4 


73.9 


75.7 


77.6 


79.7 


82.0 


84. e 


87.4 


90.6 


94.1 


98.0 




23.0 


24.3 


25.6 


26.9 


28.3 


29.6 


74.2 


75.8 


77.5 


79,5 


81.6 


84.0 


86. e 


89.5 


92.7 


96.3 


100.3 




36.5 
52.8 


38.0 
54.8 


39.6 
66.9 


41.3 
69.1 


43.0 


44.7 


76.0 
77.8 
78.7 


77.6 
79.4 
80.4 


79.4 
81.3 
82.2 


8L4 
83.3 
84,3 


83.6 
85.5 
86.5 


86.0 
88,0 
89.0 


88.7 
90.7 
91.7 


91.6 
93.7 
94,8 


94.9 
97.1 
98.2 


98.5 
100.8 
101.9 


102.6 




1.3 
13.7 


16.6 


105.0 


"^ 


"Tl 


5.6 


8.2 


10.9 


106.1 




8.9 


11.8 


14.9 


18.2 


'21.7 


25.3 


79.1 


80.8 


82.7 


84.7 


87.0 


89.5 


92.2 


95.3 


98.7 


102.5 


106,7 




15.3 


18.6 


22.2 


26.1 


30.9 


36.8 


79.6 


81.3' 8.3.1 


85.2 


87.5 


90.0 


92- E 


95.9 


99.3 


103.1 


107,3 




23.3 


27.5 


32.5 


38.9 


61.6 




80.0 


81.7 83.6 


85.7 


88.0 


90.5 


93.3 


96.4 


99.8 


103.6 


107.9 




34.2 


41.4 

• 


' 









80.5 


82.2 84.1 


86.2 


88.5 


91.0 


93. S 


96.9 


100.4 


104.2 


108.5 



TABLES OF POLARIS, 1912, 



955f 



4c. — Polaris, 1912, General Table.— Concluded. 
(To accompany Tables 4a and 4b.) 



MKAN TIME EOTm ANGLES, 1912. 


AZIMUTHS OF POLARIS, 1912. 


i 


Decl.+88»50' 




Latitude. 


n 


0" 


10" 


20" 


30" 


40" 


SO" 


30« 


32* 


34° 


36« 


38' 


40« 


42* 


44" 


46- 


48- 


50» 


c 


m 


m 


m 


zn 


m 

59.6 


m 



/ 
80.5 
80.1 
79.6 


/ 
82.2 
81.8 
81.3 


84.1 
83.6 
83.2 


/ 
86.2 
85.7 
85.2 


88.6 
88.0 
87.5 


9L0 
90.5 
90.0 


93.8 
93.3 
92.8 


/ 
96.9 
96.4 
95.8 


/ 
100.3 
99.8 
99.2 


• ■ 


# 


6 


16.9 
26.8 
34.8 


8.8 
22.7 
31.5 


104.1108.4 




17.7 
28.0 


11.3 
24.] 


103.6107.8 




19; 3 


13.4 


103.0107.2 




4L3 


38.6 


35.8 


32.8 


29.2 


25.4 


79.2 


80.9 


82.7 


84.8 


87.0 


89.5 


92.2 


95.3 


98.7 


102. 4 106. 6 




47.3 


44.9 


42.4 


39.8 


36.9 


33.8 


78.7 


80.4 


82.3 


84.3 


86.5 


89.0 


91.7 


94.8 


98.1 


101.8106.0 


7 


57.7 


55.7 


53.6 


5L4 


49.1 


46.8 


77.9 
76.1 


79.5 

77.7 


81.3 
79.5 


83.3 
8L5 


85.6 
83.6 


88.0 
86.0 


90.7 

88.6 


93.7 
91.5 


97.0 
94.8 


100.7104.8 

1 




14.2 


12.6 


11.0 


9.4 


7.7 


6.0 


98. 4 102. 3 




28.0 


26.7 


25.4 


24.1 


22.7 


2L3 


74.4 


76.0 


77.7 


79.6 


81.7 


84.0 


86.6 


89.4 


92.5 


96.0 


99.9 




40.0 


38.9 


.37.7 


36.6 


35.4 


34.2 


72.7 


74.2 


75.9 


77.7 


79.8 


82.0 


84.5 


87.2 


90.3 


93.7 


97.5 




50.9 


49.9 
59.8 


48.9 
58.9 


47.9 
57.9 


46.8 
57.0 


45.7 
56.1 


70.9 
69.2 
67.4 


72.4 
70.6 
68.8 


74.0 
72.2 
70.3 


75.8 
73.9 
72.0 


77.8 
75.9 
73.9 


80.0 
78.0 
76.0 


82.4 
80.4 
78.3 


85.1 
83.0 
80.8 


88.1 
85.9 
83.6 


9L4 
89.1 

86.8 


96.0 


8 


0.7 
10.1 


92.7 




9.3 


8.4 


7.6 


6.7 


5.8 


90.3 




19.0 


18.3 


17.5 


16.7 


15.9 


15.1 


65.6 


67.0 


68.5 


70.2 


72.0 


74.0 


76.2 


78.7 


8L4 


84.5 


87.9 




27.4 


26.7 


26.0 


25.3 


24.6 


23.9 


63.9 


65.2 


66.6 


68.3 


70.0 


72.0 


74.2 


76.6 


79.2 


82.2 


85.5 




35.6 


34.9 


34.3 


33.6 


33.0 


32.3 


02.1 


03.4 


64.8 


66.4 


68.1 


70.0 


72.1 


74.4 


77.0 


79.9 


83.1 




43.2 


42.6 


42.0 


4L4 


10.8 


40.2 


60.3 


6L6 


63.0 


64.5 


66.1 


68.0 


70.0 


72.3 


74.8 


77.6 


8a7 




50.7 


50.1 


49.5 


49.0 


48.4 


47.8 


68.6 


59.8 


6L1 


62.6 


64.2 


66.0 


08.0 


70.2 


72.6 


75.3 


78.3 




57.9 


57.4 


56.8 


56.3 


5.5.8 


55.2 


56.8 
55.0 


68.0 
56.2 


69.3 
57.4 


60.7 

— 


62.3 
60.3 


64.0 
62.0 


65.9 
63.8 


65.9 


70.4 
68.2 


73.0 
70.7 


7&9 


~ 


5.0 


4.5 


4.0 


3.5 


3.0 


2.5 


73.5 




11.8 


n.3 


10.8 


10.4 


9.9 


9.4 


53.3 


54.4 


55.6 


56.9 


~ 


60.0 


6L8 


63.8 


66.0 


68.4 


71.1 




f8.4 


a8.o 


17.5 


17.1 


16.6 


16.2 


53.5 


52.6 


53.7 


55.0 


56.4 


58.0 


~ 


61.6 


63.8 


66.1 


68.8 




24.9 


24.5 


24.0 


23.0 


23.2 


22.7 


49.7 


50.7 


51.9 


53.1 


54.5 


56.0 


57.7 


q 


61.6 


63.8 


66.4 




3L2 


30.8 


30.4 


30.0 


29.6 


29.2 


47.9 


48.9 


50.0 


5L2 


52.5 


54.0 


55.6 


57.4 


~ 


61.6 


64.0 




37.5 


37.1 


36.7 


36.4 


36.0 


35.6 


46.2 


47.1 


48.2 


49.3 


50.6 


52.0 


53.5 


55.3 


57.2 


69.3 


61.6 




43. C 


43.2 


42.9 


42.5 


42:1 


41.8 


44.4 


4&.3 


46.3 


47.4 


48.6 


50.0 


51.5 


53.1 


55.0 


57.0 


59.3 




49.6 


49.3 


48.9 


4&6 


48.2 


47.9 


42.6 


43.5 


44.5 


46.5 


46.7 


48.0 


49.4 


61.0 


52.8 


54.7 


6a9 




65.5 


65.2 


54.9 


516 


54.2 


53.9 
59.8 


40.9 
39.1 


4L7 
39.9 


42.6 
40.8 


43.6 
4L7 


44.8 
42.8 


46.0 
44.0 


47.4 
45.3 


48.9 
46.8 


50.6 
48.4 


52.4 
50.1 


64.5, 


10 


L4 


1.1 


o.s 


0.5 


P.1 


52.1; 




7.1 


G.8 


6.5 


6.3 


6.0 


5.7 


37.3 


38.1 


38.9 


39.8 


40.9 


42.0 


43.2 


44. jD 


46.2 


47.9 


49i7 




12.8 


12.5 


12.3 


12.0 


11.7 


11.6 


35.5 


36.3 


.37.1 


38.0 


38.9 


40.0 


41.2 


42.5 


44.0 


45.6 


47.4 




18.4 


18.2 


17.9 


17.7 


17.4 


17.2 


338 


34.4 


35.2 


36.1 


37.0 


38.0 


39.1 


40.4 


4L8 


43.3 


4&0 




24.0 


23.8 


23.5 


23.3 


23.0 


22.8 


32.0 


32.6 


33.4 


34.2 


35.0 


36.0 


37.1 


38-2 


39.6 


41.0 


42.6 




29.5 


29.3 


29.1 


28.8 


28.6 


28.3 


30.2 


30.8 


3L5 


32.3 


33.1 


34.0 35.0 


36.1 


37.4 


38.7 


40.3 




34.9 


34.7 


34.6 


34.3 


34.1 


33.9 


28,4 


29.0 


29.7 


30.4 


3L1 


32. 0; 32.91 .34.0 


35.2 


36.4 


37.9 




40.3 


40.1 


39.0 


39.8 


39.6 


39.4 


26,7 


27.2 


27.8 


28.6 


29.2 


30.0, 30.9 


31.9 


33.0 


34.2 


3&5 




45.7 


45.5 


45.3 


46.2 


45.0 


44.8 


24.9 


25.4 


2.'>.9 


26.6 


27.2 


28.0; 28.8 


29.7 


30.8 


31.9 


33.1 




51.0 


50.8 


50.7 


60.6 


50.3 


60.2 


23.1 


23.6 


24.1 


24.7 


25.3 


26.0 26.8 


27.6 


28.6 


29.6 


30.8 




56.3 


56.2 


56.0 


65.9 


65.7 


55.6 


21.3 
19.5 


2L8 
19.9 


22.2 
20.4 


22.8 
20.9 


23.3 
21.4 


24.0 24.7 
22.0 22.6 


25.5 
23.4 


26.4 
24.2 


27.3 
•25.0 


28.4 


11 


1.5 


1.4 


1.2 


1.1 


1.0 


0.8 


26.0 




6.7 


6.6 


6.5 


6.4 


6.2 


6.1 


17,8 


18.1 


18.5 


19.0 


19.4 


20.0 20.6 


21.2 


22.0 


22.8 


23.7 




11.9 


11.8 


11.7 


11.6 


11.5 


11.4 


16.0 


16.3 


16.7 


17.1 


17.5 


18.0, 18.5 


19.1 


19.8 


20.5 


2L3 




17.1 


17.0 


16.9 


16.9 


16.8 


16.7 


14.2 


14.5 


14.8 


16.2 


16.6 


16. Oi 16.6 


17.0 


17.6 


18.2 


18.9 




22.3 


22.2 


22.1 


22.1 


22.0 


21.9 


12.4 


12.7 


13.0 


13.3 


13.6 


14.0 14.4 


14.9 


15.4 


16.9 


16.6 




27.4 


27.3 


27.3 


27.2 


27.1 


27.1 


10.7 


10.9 


ILl 


11.4 


11.7 


12.01 12.3 


12.7 


13.2 


13.7 


14.2 




32.6 


32.5 


32.6 


32.4 


32.3 


32.3 


8.9 


9.1 


9.3 


9.6 


97 


10.0 10.3 


10.6 


11.0 


11.4 


11.8 




37.7 


37.7 


37.6 


37.0 


37.5 


37.6 


7.1 


7.3 


7.4 


7.6 


7.8 


8.0 8.2 


8.5 


8.8 


9.1 


9.5 




42.8 


42.8 


42.7 


42.7 


42.7 


42.6 


5.3 


5.4 


5.6 


6.7 


6.8 


6.0 6.2 


6.4 


6.6 


6.8 


7.1 




47.9 


47.9 


47.8 


47.8 


47.8 


.47.7 


3.6 


3.6 


3.7 


3.8 


3.9 


4.0 4.1 


4.2 


4.4 


4.6 


4.7 




.•>3.0 


53.0 


63.0 


53.0 


52.9 


62.9 


1.8 


1.8 


L9 


1.9 


1.9 


2.0 2.1 


2.1 


2.2 


2.3 


2.4, 




58.0 


68.0 


68 


58.0 


58.0 


58.0 


0.0 


0.0 


0.0 


0.0 


0.0 


0.0 0.0 


0.0 


0.0 


0.0 


0.0 



956 



-SURVEYING, MAPPING AND LEVELING. 



Tapes. — ^The best form of tape for general surveying is a 100-ft. steel 
ribbon with graduations to feet, tenths, and hundredths. The zero end of 
the tape should be the end of the steel ribbon itself, and it may be provided 
with a small handle or a large detachable handle, the latter for continuous 
line measurements. The 100-ft. end should also be provided with detach- 
able handle so it can be wound in a box. 

Every city surveyor should keep, in his office, a standard tape tested by 
the U. S. Coast and Geodetic Dept. at Washington, and certified to as cor- 
rect at a certain standard temperature, say 60° F., and for a certain pull, say 
10 to 15 lbs., when uniformly supported. This should be used as a test tape 
only, and never for field work. The advantage of such a standard over a 
fixed base (as on a pavement) is that the temperature* of the tapes need not 
be taken during the test. Nor is it necessary to use the spring balance, as 
the tapes can be brought to practically the same tension without it. It is 
necessary of course to use the same pull for the field measurements and 
this is one of the tricks of chaining, f The "misuse" of the thermometer in 
the field is often a source of "error." That is to say, the temperature of the 
thermometer will never indicate the temperature of the tape, exactly. 
Both will be affected differently by the sun's rays, surrounding air, wind 
and temperature of ground. The best chaining is done on cloudy days and 
in still air. 

Temperature corrections should be added to a measured length between 
two fixed points in the field when the temperature of the tape is above the 
standard temperature', subtracted when below. Reverse the above when 
laying out a fixed distance, as setting one point from another. 

5. — ^Temperature Corrections in Feet per 100 Feet. 
Note. — Use signs as per table for measurements between fixed objects. 
Reverse signs in table when laying out fixed distances. 

(From the author's "Railway Right -of- Way Surveying." J) 







100-foot Tape Standard at following Temperatures. 






40° 


45° 


50° 


55° 


60° 


65° 


70° 


75° 


80° 


85° 






0° 

5° 

10° 


-.027 
- .023 
-.02 


-.03 

-.027 

-.023 


-.033 
-.03 
- .027 


— .037 
-.033 
-.03 


-.04 

-.037 

-.033 


- .043 
-.04 

- .037 


-.047 
- .043 
-.04 


-.05 

-.047 

-.043 


- .053 

-.05 

-.047 


-.057 
-.053 
-.05 


0° 

5° 

10° 




15° 
20° 
25° 


-.017 
-.013 
-.01 


-.02 
-.017 
- .013 


- .023 

-.02 

-.017 


- .027 
-.023 
-.02 


-.03 
-.027 
- .023 


-.033 
-.03 
- .027 


-.037 
-.033 
-.03 


-.04 
-.037 
- .033 


-.043 

-.04 

-.037 


— .047 
-.043 
-.04 


. 15° 
20° 
25° 




30° 
35° 
40° 


-.007 
-.003 


-.01 
-.007 
- .003 


-.013 
-.01 
- .007 


-.017 

- .013 

- .01 


-.02 
-.017 
- .013 


-.023 
-.02 
- .017 


-.027 
-.023 
- .02 


-.03 
-.027 
- .023 


- .033 
-.03 

- .027 


- .037 
-.033 
-.03 


30° 
35° 
40° 




45° 
50° 
55° 


+ .003 
+ .007 
+ .01 


+ !663 
+ .007 


-.003 
+ !663 


-.007 
— .003 


-.01 
-.007 
- .003 


- .013 

- .01 

- .007 


-.017 
-.013 
- .01 


-.02 
-.017 
- .013 


-.023 

-.02 

-.017 


-.027 
-.023 
-.02 


45° 
50° 
55° 




60° 
65° 
70° 


+ .013 
+ .017 
+ .02 


+ .01 
+ .013 
+ .017 


+ .007 
+ .01 
+ .013 


+ .003 
+ .007 
+ .01 


+ !663 
+ .007 


- .003 
+ !603 


-.007 
- .003 


-.01 
-.007 
- .003 


-.013 

-.01 

-.007 


-.017 
-.013 
- .01 


€0° 
65° 
70° 




75° 
80° 
85° 


+ .023 
+ .027 
+ .03 


+ .02 
+ .023 
+ .027 


+ .017 
+ .02 
+ .023 


+ .013 
+ .017 
+ .02 


+ .01 
+ .013 
+ .017 


+ .007 
+ .01 
+ .013 


+ .003 
+ .007 
+ .01 


+ .'663 
+ .007 


-.003 
+ !663 


-.007 
-.003 


75° 
80° 
85° 




90° 

95° 

100° 


+ .033 
+ .037 
+ .04 


+ .03 
+ .033 
+ .037 


+ .027 
+ .03 
+ .033 


+ .023 
+ .027 
+ .03 


+ .02 
+ .023 
+ .027 


+ .017 
+ .02 
+ .023 


+ .013 
+ .017 
+ .02 


+ .01 
+ .013 
+ .017 


+ .007 
+ .01 
+ .013 


+ .003 
+ .007 
+ .01 


90° 

950 

100° 



* The tapes should lie unwound in the same atmosphere for some little 
time before making the test. 

t On very accurate city work it is often desirable to let the chainmen 
use the spring balance until they get accustomed to the proper tension, 
after which it may be discarded generally, or used only occasionally to keep 
them in tune. % Published by McGraw-Hill Book Company, New York. 



TAPES AND THEIR USE. 957 

The Sag and the Stretch of a Tape supported at two or three points, 
Fig. 16, positions 2 and 3, may be made to equalize each other, by apply- 
ing the proper tension or pull at the ends, so that the horizontal measured 
length will remain the same, and equal also to the length of the tape when 
supported throughout as in position 1. Practical advantage is taken of the 
above fact in measuring with a single- or double-sagged tape. The pull Pj, for 
position 1, is of course fixed when the tape is standardized, while P2andP3, 
for positions 2 and 3, are obtained with the spring balance and by plumbing 
over the fixed points a and b If it is desired , however, to maintain a constant 
tension = Pi for all three positions of the tape (and this is generally the 






^ 



' .-Length of Fietd Tape at 

EKistinfj Temperature^ 

Fig. 16. 

better method) then the horizontal distance error must be determined and 
recorded for positions 2 and 3. These "distance errors" will become the 
"index" errors or "sag" errors of the tape, and are to be used in addition 
to the "temperature" corrections. Care must be used in maintaining the 
standardized tension while measuring, as a difference of 6 lbs. in pull, when 
the tape is uniformly supported, will affect an ordinary 100-ft. steel tape 0.01 
of a foot. The exact amount depends upon the size (area) of the ribbon, and 
the modulus of elasticity of the steel. It can best be determined with the 
particular tape, using the spring balance and watching the amount of 
stretch. The apparent stretch of a sagged tape is greater than when lying 
flat. Tapes differ very much in area, varying from about .0015 to .008 sq. 
inch. 

The Engineer's Chain (100 1-ft. links) is recommended for rough work 
in a mountainous or lava country ^ but it is often used where the steel (ribbon) 
tape would prove more expeditious. 

The Surveyor's or Gunter's Chain (100 7.92-in. links), 66 ft. long, is 
used mainly in the West in connection with government land surveys. 
It is also useful in retracing old land surveys made before the engineer's 
100-ft. chain came into use. 



958 



58.— SURVEYING. MAPPING AND LEVELING. 



6. — ^Table op Feet and Chains (Gunter's or Surveyor's). 

(1 chain = 100 links; 1 link = 7.92 ins.) 

(a) Chains to Feet (Exact). 



Ch'ns. 


Feet. 


Chains. 


Feet. 


Chains. 


Feet. 


Chains. 


Feet. 


Chains. 


Feet. 


.01 


.66 


.21 


13.86 


.41 


27.06 


.61 


40.26 


.81 


53.46 


.02 


1.32 


.22 


14.52 


.42 


27.72 


.62 


40.92 


.82 


54.12 


.03 


1.98 


.23 


15.18 


.43 


28.38 


.63 


41.58 


.83 


54.78 


.04 


2.64 


.24 


15.84 


.44 


29.04 


.64 


42.24 


.84 


55.44 


.05 


3.30 


.25 


16.50 


.45 


29.70 


.65 


42.90 


.85 


56.10 


.06 


3.96 


.26 


17.16 


.46 


30.36 


.66 


43.56 


.86 


56 76 


.07 


4.62 


.27 


17.82 


.47 


31.02 


.67 


44.22 


.87 


57.42 


.08 


5.28 


.28 


18.48 


.48 


31.68 


.68 


44.88 


.88 


58.08 


.09 


5.94 


.29 


19.14 


.49 


32.34 


.69 


45.54 


.89 


58.74 


.10 


6.60 


.30 


19.80 


.50 ^ 


33.00 


.70 


46.20 


.90 


59.40 


.11 


7.26 


.31 


20.46 


.51 


33.66 


.71 


46.86 


.91 


60.06 


.12 


7.92 


.32 


21.12 


.52 


34.32 


.72 


47.52 


.92 


60.72 


.13 


8.56 


.33 


21.78 


.53 


34.98 


.73 


48.18 


.93 


61.38 


.14 


9.24 


.34 


22.44 


.54 


35.64 


.74 


48.84 


.94 


62.04 


.15 


9.90 


.35 


23.10 


.55 


36.30 


.75 


49.50 


.95 


62.70 


.16 


10.56 


.36 


23.76 


.56 


36.96 


.76 


50.16 


.96 


' 63.36 


.17 


11.22 


.37 


24.42 


.57 


37.62 


.77 


50.82 


.97 


64.02 


.18 


11.88 


.38 


25.08 


.58. 


38.28 


.78 


51.84 


.98 


64.68 


.19 


12.54 


.39 


25.74 


.59 


38.94 


.79 


52.14 


.99 


65.34 


.20 


13.20 


.40 


26.40 


.60 


39.60 


.80 


52.80 


1.00 


66.00 









(b) Feet to Chains. 








Feet. 


Chains. 


Feet. 


Chains. 


Feet. 


Chains. 


Feet. 


Chains. 


Feet. 


Chains. 


1 
2 


.015^15 
.030^30 


3 
4 


.045^45 
.060^^60 


5 
6 


.075^75 
.090^90 


7 
8 


.106^06 
.121^21 


9 
10 


.136^36 
.151^51 



Note. — ^The inverted caret indicates repeating decimal; thus, 1 ft. 
0.015 15 15 15 chain. 



Ex.— Reduce 18 c/t. 45 l.[to feet? 

Solution: 

.18 ch= 11.88 ft. .-.18 ch= 1188.00ft. 

.45 ch = 29.70 ft. .'.45/ = 29.70ft. 



Ans. 18ch451 = 1217.70ft. 

Or, mult. 18.45 by 66. 



Ex.— Reduce 482.73 ft. to ch. and /? 

Solution: 

400ft. = 6.0606ch. . 70ft. = . 0106 ch. 

80ft. = 1.2121ch. .03ft. = .0005 ch. 

2ft. = 0.0303ch. 

Ans. 7ch. 31.411. 
Or, divide 482.73 by 6 and 11. 



FEET AND CHAINS. PLATTING ANGLES, 



959 



Methods of Platting Angles. — ^There are three principal methods of 
platting angles: 

1st. — By the Protractor. The cheapest, simplest and perhaps most 
satisfactory for general use is the paper protractor, a complete circle 14 ins. 
in dia. and graduated to \ degrees. Steel and German silver protractors 
with movable arms and verniers are convenient but expensive, costing from 
$10.00 to $40.00 and upward, while the large paper protractor may be 
procured for about 30 cts. 

2nd. — By Tangents. This consists of laying oflE ^.^^ 

the angles by the use of the Table of Natural Tangents ^^^ %^ 

(see Table 3, p. 144.) Thus, in Fig. 17, let it be J^ Base.woo'^ 

required to lay off the angle a=28°— 16' from the 
base line. Lay off a length of Base, to any scale, -^^S- 17. 

equal to some multiple of 10, say 1000; then the length of Altitude, at 
right angle to the Base, will equal 1000 mult, by tang a:= lOOOX 0.5377 = 
637.7 to same scale. Draw the Hypothenuse, which now makes the required 
angle a with the Base. This method is desirable for small angles. For 
large angles, and for very long lines, the method of Chords is often preferred. 

3rd. — By Chords. Assume a length of Base as some multiple of 10, say 
1000, to any scale. With a radius of 1000 describe the arc a b, Fig. 18. 
From the following Table of Chords to Radius 1 find 
the value corresponding to the angle a, assumed here 
as 28°- 16'. This value, 0.4884, mult, by 1000, the 
assumed Base, =488.4 the desired Chord, or radius 
of arc, with center at a, to intersect at b. Draw Ob; 
then aOb is the desired angle a. Note that the 
Chord = Base (i.e., radius of circle), mult, by twice 
sin |cx, when a does not exceed 90°. pig i^ 

7. — Chords to Radius 1. 




M 


0° 


1° 


2° 


3° 


4° 


5° 


6° 


70 


8° 


90 


10° 


M. 


"o' 


.0000 


.0175 


.0349 


.0524 


.0698 


.0872 


.1047 


.1221 


.1395 


.1569 


.1743 


0' 


2 


.000*6 


.0180 


.0355 


.0529 


.0704 


.0878 


.1053 


.1227 


.1401 


.1575 


.1749 


2 


4 


.0012 


.0186 


.0361 


.0535 


.0710 


. 0884 


.1058 


.1233 


.1407 


.1581 


.1755 


4 


5 


.0015 


.0189 


.0364 


.0538 


.0713 


.0887 


.1061 


.1235 


.1410 


.1584 


.1758 


5 


6 


.0017 


.0^2 


.0366 


.0541 


.0715 


.0890 


.1064 


.1238 


.1413 


.1587 


.1761 


6 


8 


.0023 


.0198 


.0372 


.0547 


.0721 


.0896 


.1070 


.1244 


.1418 


.1592 


.1766 


8 


10 


.0029 


.0204 


.0378 


. 0553 


.0727 


.0901 


.1076 


.1250 


.1424 


.1598 


.1772 


10 


12 


.0035 


.0209 


.0384 


.0558 


.0733 


.0907 


.1082 


.1256 


.1430 


.1604 


.1778 


12 


14 


.0041 


.0215 


.0390 


.0564 


.0739 


.0913 


.1087 


.1262 


.1436 


.1610 


.1784 


14 


15 


.0044 


.0218 


.0393 


.0567 


.0742 


.0916- 


.1090. 


.1265 


.1439 


.1613 


.1787 


15 


16 


.0047 


.0221 


.0396 


.0570 


.0745 


.0919 


.1093 


.1267 


.1442 


.1616 


.1789 


16 


18 


.0052 


.0227 


.0401 


.0576 


.0750 


.0925 


.1099 


.1273 


.1447 


.1621 


.1795 


18 


20 


.0058 


.0233 


.0407 


.0582 


.0756 


.0931 


.1105 


.1279 


.1453 


.1627 


.1801 


20 


22 


.0064 


.0239 


.0413 


.0588 


.0762 


.0936 


.1111 


.1285 


.1459 


.1633 


.1807 


22 


24 


.0070 


.0244 


.0419 


.0593 


.0768 


.0942 


.1116 


.1291 


.1465 


.1639 


.1813 


24 


25 


.0073 


.0247 


.0422 


.0596 


.0771 


.0945 


.1119 


.1294 


.1468 


.1642 


.1816 


25 


26 


.0076 


.0250 


.0425 


.0599 


.0774 


.0948 


.1122 


.1296 


.1471 


.1645 


.1818 


26 


28 


.0081 


.0256 


.0430 


.0605 


.0779 


.0954 


.1128 


.1302 


.1476 


.1650 


.1824 


28 


30 


.0087 


.0262 


.0436 


.0611 


.0785 


.0960 


.1134 


.1308 


.1482 


.1656 


.1830 


30 


32 


.0093 


.0268 


.0442 


.0617 


.0791 


.0965 


.1140 


.1314 


.1488 


.1662 


.1836 


32 


34 


.0099 


.0273 


.0448 


.0622 


.0797 


.0971 


.1145 


.1320 


.1494 


.1668 


.1842 


34 


35 


.0102 


.0276 


.0451 


.0625 


.0800 


.0974 


.1148 


.1323 


.1497 


.1671 


.1845 


35 


36 


.0105 


.0279 


.0454 


.0628 


.0803 


.0977 


.1151 


.1325 


.1500 


.1674 


.1847 


36 


38 


.0111 


.0285 


.0460 


.0634 


.0808 


.0983 


.1157 


.1331 


.1505 


.1679 


.1853 


38 


40 


.0116 


.0291 


.0465 


.0640 


.0814 


.0989 


.1163 


.1337 


.1511 


.1685 


.1859 


40 


42 


.0122 


.0297 


.0471 


.0646 


.0820 


.0994 


.1169 


.1343 


.1517 


.1691 


.1865 


42 


44 


.0128 


.0303 


.0477 


.0651 


.0826 


.1000 


.1175 


.1349 


.1523 


.1697 


.1871 


44 


45 


.0131 


.0305 


.0480 


.0654 


.0829 


.1003 


.1177 


.1352 


.1526 


.1700 


.1873 


45 


46 


.0134 


.0308 


.0483 


.0657 


.0832 


.1006 


.1180 


.1355 


.1529 


.1703 


.1876 


46 


48 


.0140 


.0314 


.0489 


.0663 


.0838 


.1012 


.1186 


.1360 


.1534 


.1708 


.1882 


48 


50 


.0145 


.0320 


.0494 


.0669 


.0843 


.1018 


.1193 


.1366 


,1540 


.1714 


.1888 


50 


52 


.0151 


.0326 


.0500 


.0675 


.0849 


.1023 


.1198 


.1372 


.1546 


.1720 


.1894 


52 


54 


.0157 


.0332 


.0506 


.0681 


.0855 


.1029 


.1204 


.1378 


.1552 


.1726 


.1900 


54 


55 


.0160 


.0334 


.0509 


.0683 


.0858 


.1032 


.1206 


.1381 


.1555 


.1729 


.1902 


55 


56 


.0163 


.0337 


.0512 


.0686 


.0861 


.1035 


.1209 


.1384 


.1558 


.1732 


.1905 


56 


58 


.0169 


.0343 


.0518 


.0692 


.0867 


.1041 


.1215 


.1389 


.1563 


.1737 


.1911 


58 


60 


.0175 


.0349 


.0524 


.0698 


.0872 


.1047 


.1221 


.1395 


.1569 


.1743 


.1917 


60 



960 



56.— SURVEYING, MAPPING AND LEVELING. 



o c^'^iOfocoo c«q "<j< in to oo o eq ««j< lo «o oo o n -«*< »o co oo o n •>* no «o oo o ca ""i* >« «o oo o 

^H ,-1 T-< ^-H 1— 1 ii cj cj e<i yj c~j eg 00 cc co cc oo co ■<» -»»«•«» t» ■^ -^ >o io>o>oio>oto 

«o CQ oo o CO C5 Tj< o to 05 »-< t^ cj 00 '^ c» a» ITS ^H to ffsi i^ t^ CO OS -^ o CO to ^H t^ ca 00 »-4 ■«* o> lo 

t>. ooGOOJCios^ 1— I'rt^-icge^aeo eo"<f-<j*"<}<mto toi>.t^c^oooo oics^^^n— < e^caeocoeo'^ 

:z! ri'::;'::;'::;'"'^ e^c>cicvic<ie^cq e>qcvje>qcQe^e>q cqc<)(Nic<ie>acg eacoeOcococo cooocoeccoco 

io laiatoioiom vrs inirj »o io m u^iotammm ira irj in> lo irs >« toioioioioto ^aiaioioioia 

oo CO oj e>q ■»«» o to r-it^oeooo-^ o m oo •>-< to c<ci oocotooiino to -^ -««* t- ec oo ■>d< o cquo ^ to 

o ^-HC<ie»qcofo •Tf<->*<mioirtto t^t^b-oooooj Oiooo^-^pa c^ioococo'^-^ »otototot-t>. 

O O O O C5 O O <000<000 OOOOOC5 Ci -h _ ^ ^ ,-< ,-h ^ ^ ^ ^ ^ _i ^ ^ ^ ,^ ^ 

m io »£5 »o »o ko »o kA lo m lo lo no >o uis lo lo m lo in in m iis lo lo lo m m »o mio »o »« »o io lo m 

» ^oco»n,-Ht>- CM 00 •— I •<# 03 in •^ to Ci e^ t^ cci oi-^t-oto^ t^ oo m oo •><*< o> in^eotoegoo 

00 ■^ininiototo c^c-ooooooos c>oo-— <-^(m caooco-^-ifvn mtototot^t>. ooosojoi^o 

oo Qooooooooooo oooooooooooo a:! oi Oi en <s><yi oioiojososo oiososcioios ojtJioiOiOO 

•^ ■•^•^•^-er-^Tj* T»<-^T}<-^'»^Tjt Tjf-^-^Tjiiti-^jt ■^•rf<'<*i-^-««<'^ Tt<-^->^'»*->^'«j< 1^'^-^'^mta 

o> inoootooqb- co oo ^ ■* o m -—i t^ o c<] od '4< a> »o oo — < to eg ooootoosino to e<i '* t~- co oo 

to t-oooooooo5 oO'-HT^cMcq ecco-^-^f-^m mtotoc^r^oo oooiOso>C)^H ^-le^eQC^aeooo 

to tOtOeOtOtOtO t>. tr^ t^ t- t^ t- t^ t^ t- t^ t^ I>. t^ t^ t~ t~ t^ t>. t~ C^ t^ t^ 00 00 OOOOOOOOOOOO 

■^ -Tjf Tj< Ti« -^ -i^ ■»»( -^ ■rj< Tf< ■>»< -<i< -«i< ^ ,^ ^ ^ ^ ^ .^^^^,^^ -^ -^ ^ ^ ^ ^ .^.^^^^^ 

05 lOOootONt^ CO oi »-i ->* o ?o -^ i>- o oo 00 -"i* oiooo-Hb-ea oo -^ to os m ^ to e<i m oo co a> 

OS 0-— ^^^^^egevi ooco-«tiT*<ioin totoc^t^t^oo osojosoo-h •^^ege^aegoo-^ ■^irsinmtoto 

•>* mxnmmmm mmmmiom mmmmioio miciotototo totototototo totototocoto 

OS -rf o CO to e^ tr^ oo OS -H 5j< o to »-< t^ o CO oo -«»< oinoo-Ht-oa oo -^ to os >« ^h toe>ainoocoo> 

eg oo-^'^-«*<mio totot-t^oooo O5os<oot:>'-^ ocicge^cooo-^ •>*im mjo to c^ t» oo oo oo os 

CO oo 00 CO CO CO oo oocooooococo oo co -<*<-«*< -«r -* ■<*< >s< -<»<-<»< -rt< •»*< -^ .^ ^ .^ ^ ^ •<)< -^ rf '>«< -^ii) 

■«»< ■<»< -^ '^ 1* -<*< M< ^^^^-^^ -,»< -^ ^ Tt< -<»< '(t* -.»< ".J* ■<»< 1^ r»< ^ ^ -^ I*! -* -^ tH -"U* '<i»< ■*■»!< •»«« 

00 '<«< o CO m 1-H t^ cvj 00 »^ -Tj< OS in »-i to os c<i oo "*• osmoo^tocq oocotiosino <oc4mt>-coos 

in tot>-t^t^oooo 050>o<oo-H cMcgc^qcooo-^ ■<*iinmtotot^ t^oooooooso 0'-H.-<i-<c>aesi 

^H ^^^^^^ i-H -H e^a c^ c^ c^a c^a e^ eg eg oa cm cq e^a «^a eg cm e^a eg eg evi eva cm co coeoeocococo 

■^ ■^ -^ "^ -^ -^ "<*< •.«< -^ -,f< -^ -^ -^ ,^ .^ .^ ^ ^ -^ ^ .^ ^ ^ ^ ,^ ^ -l!j< ^ -,}« lj< ^ .^ ^ ^ .^ ^ .^ 

t^ CO OS cvj ■>* o to eq t^ o oo OS ■«*< o to os »^ t^ oo os -^ t^ cs to »-^ t- co to oo -^ o in -h •»*< t* co oo 

oo oso500»-«-H c<iogooooco-«*i inmintotob- t^ooooososo c>»-H^<--Hegoo oo-*'<<»<'<*inin 

OS OSOSC50<00 OOOOOO O O C3 O to O OOOOOt-H ,-( ^ _ ^ ^ ^ ^H ^ — H »-< »- 

CO oo 00 -»J< "^ -»«< -Tt* -^ -^ -«*<■«»< -<J< -»»* ^ ^ ,^ ^ ,^ ^ .^ ^ ^ .^ ^ ^ .^ ^ ^ ^ ^ ^ 1^^^^^ 

to Moooooosin o to OS oa 00 oo osinoootoe<i oocotoosino to oa in t^ oo os inoootoc^it^ 

•»-i Saegooooco-"^ ininintotoi>. t>.ooooososo ^-H»-<'-He<aoo co-^-^-'finin tot^t^t^oooo 

oo oooooooooooo oooooooooooo OOOOOOOOOOOS OS050st3SasOS OSOSOSOSOSOS OSCSOSOSOSOS 

oo coeoeocococo oocooooococo eocococoooco eo co eo co oo oo eo oo co oo oo eo cococoeoeoeo 

in c>toosoaoooo osmoootocg oocotoosino toe^amoocoos ino eo'to oaoo eocjse^amoto 

•«»< ininintotot^ t^ooooososo ^»-<^H^He>aco co-^-*Ttiinin toc^t^t^oooo ososcs^^-ii-t 

to tOtOtOtOtOtO tOtOtOtOtOi>. t^t-t^t-t^t— t~t-~b^t^t>.t^ t^t~t^t^t>.t>. t~^-000000Q0 

CO oo CO CO CO oo CO CO CO 00 eo co oo co co oo co co co CO CO CO oo CO oo CO CO oo CO CO CO CO oo oo oo oo CO 

CO OS -^ t>. ^ to eg t~- oo to os in o to esi in t^ co os in o co to oa t^ oo oi eg in ^ to e*a oo o oo os in 

t>, t^ooooososo o-H-^'-Hoaoo coTf<Tti-^inin.toi>.t~t:^oooo ososoOi-^'.-h csiesieoooco-^ 

■^ •^ -^t* ■^ •>!*< -T»< in m in in in in in mininininm m m in m in in inmtotototo totototototo 

eo coooeoeoeoco coeoeocococo coeoeocococo eocococoooco eoeooocococo cococoeoeoeo 

»-i t*e>ainoo-*o in-H-^t~cooo T»«oooto»-<t^ oooo-H-^t<oto ^-Mt^oooos"^ otoosegb-co 

^ ^»-<T-^.-Hegoo co-^'^'<timm toi>.t^t^oooo ososoo»-<i— i evie^aeoooco'^ minintotot^ 

CO eocococoooco eocococoooco coeoeocococo co co ■<»< -^^ ■* -<j" .,*< >^ <,»« ■,»<•,}< ^ ^ .^ ^ ^ ^ ^ 

eo cococoeoeoeo coeoeocococo coeocooococo eoeooocococo cococoeoeoeo coeocotococo 

OS •^ocotoegi>. cooscginoto e>acoocoosin ■-^toose^aoo'**' osinoo— <i>.esi ooTt<t^osin 

eg eo-^T»<Ttiinin totot~-t~oooo ososoocs— < e^ae»aoaco^o■^ -^mmtotot^ t— ooooooos_ 

^ ^HT-i.-H»-^T-i^H _ii-i-H-^— <^H 1-H•^c^aegesle^a c>aoac^aoac^^c<a eooacgoaoae>a e<ie>aegegegeo 

CO coeoeocococo coooeoeoeoco eoeococococo cococoeoeoeo cococoeoeoeo eoeoeococoeo 

to eg 00 1-1 CO OS in ^-i to os eg oo -^ os in oo -h t^ eg oo •**< t^ o in t-^ b- co to oo ■^ o to-H^t»<oo» 

in to to t- t- t^ 00 OS OS OS o o '^ ^ cm eg oo oo -"i* -^tj* in m to to i>. t~- oo oo oo os o o «-h — < »-< eg cm 

OS OSOSOSOSOSOS OS OS OS O C3 O O CD O O O O O O O O <=> O OOOOO-H «^^^^^ 

eg eg co co eg co eo co co oa co co co co co co co oo co co co eo co co eo co oo co oo co co co co eo co eo co 

eo osinoo-Ht^eo oo -<t< t^ o in ^ t^ oo m oo ■«*< o to —•"<»« t^ co os -^t* o eo to eo t^ cooscginoto 

oo ooososoo^ ,-1 CO eo CO oo -^ -^i^ in in in to t^ t^ oo oo oo os os o »-« ^ »-i eg eg eo oo -<»<•<«« in in 

t-- t«. tr^ t^ oo 00 00 oooooooooooo oooooooooooo oooooooooooo OSOSOSOSOSOS OSOSOSOSOSOS 

CO eocoeocoegco eo eo eg eg eg eo eo eo eg eo co eo cococoeoeoeo eo co eo eo eo eo eg eg eg co co eo 

■m^ tocoinoo-^os in.-H'xjtt-cooo -^ocom— <t>. eoos'r-i"<tioto cot^ocoosin ^tooscococo 

^ ^eoeoeocoeo -<»< in in in to to t- oo oo oo os os o o »-< -h eg eg oo oo -<*< ^ ■««< in to to to t^ t^ oo 

to totototototo totototototo totototototo t^t--t^t>.t»t^ t^t^t^b^b^t^ t^b-t-b-t^t>. 

eo cococoeoeoeo eoeococococo co eo eg eg co eg eo eo eo eg eo eg eoeoeococoeo co eo eg co eo co 

t^ cooscoinono eo 00 —I •* crs in »- i>. o eo oo ■<*» o to oo -h t^ oo os -^r< t^ o to co £-eo<05>2 

CO iij^-^SinSto t^t^ooooooos oo.-'^— <eo oo co co '<*< -^ m in to to t^ t- oo oo<5os<50»^ 

12 2, 2; is -tfJ "S "S ^ -<»< -^ -*-<»< -<}* inininininin in in in in in in in in in m in m inminintoto 

eo ^eoeoeococo eo eo eo eo eg eo eoeoeococoeo co co eo eg eo eo co co co co co eg coeoeoeocoeo 

_u <-aema^^4t^eo osinb-.otoeo ooootoosin-n t^oamoo-^o in»-<ii<t~ooo> -^ocotoeot^ 

S SSSooooS osoo-H^og eocococo-^in in to to to tr- oo oo os os os o o ^ co eg eg co oo 

eo eoeoeoegeoS eo eo oo oo fo co eoeococococo coeoeocococo co co oo oo ^ '^ S IS S :?: S S 

CO coeoeoeocoeo eoeoeoegeoco cocgegegeoco cocoeoeococg egegeacoeoeo cococoeoeoeo 

^ toeoinoo"*os lo ^ ->*< t- co oo -<»< o oo to co t^ oo os co in o to eooo-n-^osin •-; ?2 S 52 9S r2! 

g i§i§S^ ^^2^22^:^ t2:2S2^^ 2222Sg J^j:S?3?5?^S3 S?5Si5SS 

CO cocgeocococo coeoeocococo coeocgeoeocg eocoegeocoeo egcocoeoeoeo eo eo^eo eo eo eo 

h, cooo-H-^oto cob-ocoosin i-itoojeooo-<t< omoo — t^co oo5't--otoeo J::J^JP^J5^ 

i^ SSeoM-^-S inKStotot^ ooooooososo -:;-:-:; co eg co co ;2^ ;2^ in jn to «o pr Jr pr S S 

2 222^222 222222 22222S SSSSSS SSSSSS SSSS.SS 



b c^^iotoooo ca;2:i222S ?5^SSI5^ f^^^^^^ 5?5i5^5S SSIgSSS 



CHORDS FOR PLATTING ANGLES, 



961 



N-^»r5«oooo c>q-*»ft«oooo N-^iotoooo cq-^irstoooo ca-^iotoooo N"«mjb«coo 



in Ln «o CO (o b« c» ooooosoioo »-h ,-( t-h cm ca co co ■<»<■<»< -^ to »o to to t- 1>. t- oo oooaoioo^-i 
'^ rtt'Tti^tt'^'^^ ^t*'^^t)'<t<mm lommtototn mkOiOiOtdto tomiotototo tnmmcococo 

00 COOOOOOOOOOO OOOOOOOOOOOO OOOOOUOOOOOO OOOOOOCOOOOO OOCXJOOOJQOQO 0OQOC»0O0OQO 



T»« OS -"I** i>. o to o eo »-i -^ CO ^ t^ cMt^ocoooco oo -<*' <o Oi -^ o mocoto-^co i-< i— os e>j t>. esi 

o> os^^^H<-ievi CM CO CO eo •^ -^ lo to <o «o co t>. t^ c» oo oo os ^ o »-i i-i »-h cq «m co co co ■«*< Tt< to 

M cci CO CO CO 00 CO eococoeoeoco cocoeoeoeoco co eo co co eo "^jt -^ tx -u* t}« -rj* -»}< ■<*< -xi* ■•*< -^j^ ■<»' ■«»< 

00 OOOOOOOOOOOO OOOOOOOOOOOO OOlOOOOOOOOO OOOOOOOOOOOO OOOOOOOOOOOO OOOOOOOOOOOO 



5? 



m o <o 00 -^ to cd t» eq in 00 CO oo if os c<i -^ o »o ^cooo^h«op<i t- «^ to oo eo oo co os »-< "!t< os to 

i>. oooooooiOiO C5 i-H ^ ^-H ca e<i co co -.^ m^ to to co co <a3 t^ t^ oo ooosososoo ^-h ,-i oq e<i e^a eo 

0» OS 05 05 OS 0> ^ ^Cd^^O^ C> O O C> C3 O O O O O O «3 O O O ^ r-i »-< ,-H »-< ^^ ^^ -rt T-H 

C>. t- t- C- t^ t^ OO OOOOOOOOOOOO OOOOOOOOOOOO OOOOOOOOOOOO OOOOCOOOOOOO OOOOOOOOOOOO 



to o to 00 -H <o ^H t^cqtoi^eooo co os r-t -^ o to o?oooi-i«oN b- e<i to 00 CO oo rH os e<i •^ o to 

1^ e^ e<i c<i CO CO T(« ^^tototococo t^t^ooooosos c>oo^^»-h«vi e^ co co co -^ ■<*< to to to «o t>. t^ 

00 OOOOOOOOOOOO OOOOOOOOOOOO OOOOOOOOOOOO OS OS OS OS OS OS OS OS OS OS OS OS OS OS OS OS OS OS 

t>. t^t^t^t^t^t- t-t^t^t^t^t* t>.l:^C~t^t~t>- t^t^-t^t^-t^t- t^t>.C^C— C^t^ t^C^t-t>.t>.t^ 



•^ OS ■»*< t^ ^ to »^ «o "H -^ t~ e^ !>. CO 00 i-t CO OS •»!< oioooo«o.-< «o c^a -^ t~ cq oo eo os ^-« rt< os ^ 

to to CO CO t>. t^ 00 OOOSOSOSOO »-< ,-1 C^ CCJ C<1 CO ■««< -xf Tf to to CO CO t^ I>- t^ 00 00 oso>ooo<-i 

CO cocoeoeoeoco CO CO CO CO t^ t>. t^ l>. c^ t^ t^ t^ t~ t^ t^ t~ t^ !>• t^ t>. t^ t>- 1>. t>. t^ t>. OO 00 oo oo 

t» t» t^ t^ t^ t^ t>. t>- t^ t»- t^ t» t>. t* t- t» t^ t^ t- t^ t- t~ t^ !>• t^ t~ t>- !>• t- t- !>• t^ C— t^ !>• I>» t~ 



c>q oocoeooo-xfos •^ocoto.-Hco »-ii>.ose<iooco oo-^fcoosoo io»-(CocO'-<t>. eMoooeooo-xf 

O* OS O ^ O »-H »-l e^ CO CO CO rf Tt< to to to CO CO t- t~ 00 OO OO OS C3 CD —H ^H ^H CM cq CO CO -^ ■^ "<S 

■««4 Tftotototoio lOiotoiototo tototototoio totototoirjco cocococococo cocoeoeoeoco 

t>i t^ t^ !>• !>. C~ t>. I>- t>- t^ t>. t^ t- t~ t^ t>. !>. t~ t>. t~ C^ t» t» t^ C>- t» t» «>• C^ t^ t>. t- t>. t-- C^ C^ t^ 



O »0-H-<j<coc<it>. caoo^eoos-^ otooooco-n t^e^tot-cooo eooscq-<j<<=)to »-icoos-Ht>.cq 

CO co-^-^-^toto cocot>.t^t^oo osososoo^ T-(c>ae^ae^coco •«*<-^totocoeo t-c^c^ooooos 

CO eococoeoeoco cocoeoeoeoco eo eo co •^ •^ •^ ■^ ■^ ^^ -^ "^ "^ i}< •»*( t*< -^ ^:j< ■«ti •^ ■><*<•<!*<•<*< ^i 

t^ t-t^t"t^C~-t>» t»t>.t~t~t«t>. t^t>-t^t>>t>.t^ «>.|>-t^t>»t^t>. fc-.t-t-.t~l>.!>. t-.t^t-.t~t-t~ 



cooo^'ifosio o to 00 1-H CO CM t~oqtooocoos -^oe^ato^co »-i t^ os c<i oo eo oo-^coostoo 

t»t-.ooaooooft ^^CS"— iT-icq cqeocoeoTf-^ iocoeocot-.t>- ooooooosos^ ^»-<>— i»-(cqeo 

i-H >— I T-i »H i-H T-i cq CM e^ e^ e<i pq c<i c>q e<i esi cm oq e<i c<» e^ cq e^ eq e»q e^ esi cm oq co eococoeoeoco 

t- t- 1~ t» t~ t« t~t-.t-.t-.t~t- t~ t-. c~ t~ t- c- t~ t- t~ t~ t-. t- t~t-.t-t-.t-t~ t- c~ t~ t~ c~ t~ 



oiooooeoi-i t~ e^q to oo CO OS '^oscqtooco i-H i:~ OS e>q oo CO oo-*t~osioo CO »M .»j» CO e«q t- 

1-l«•4^HC<le^qco eo-'f-^Tfioto cocot^t~oooo osososoOt-^ ■.-ie^«^csieo'«»< ^fiototococo 

O^O^OO OCDOOCDO OCDOOOO O <=> C3 r-^ ^m ,^ ^^^^^^-4 .^ ,^ ^ ^ ^ ^ 

t~ t~ t~ t~ t- t~ t~ t~ t~ t- t- t~ t~ t- t- t~ t~ t~ t~ t~ t~ t~ t~ t~ t~ t~ t~ t~ t~ t~ t~ t~ t~ t~ t~ t- 



o CO -^ -^ t- e>q oo co os »-4 -5t< o to ■■-.< co os ^ t~ cq oo eo co os >«t< o to *-< eo co ■•^ e~ eo oo o eo os ■^j* 

rji -^ftototococo t~t~oooooso> c>o<=>-H— tcqie^eoeocorj«io toco<ocot>-t~ ooooososctso 

00 OO 00 00 JO OO OO OO 00 00 OO OO JO OS OS OS OS OS OS OS OS OS OS OS OS <» OS OS OS OS OS OS OS OS OS OS c> 

CO CO CO CO CO eo CO CO CO co co CO CO co CO co co co co co co co co co co co co co co co co co co eo co co t— 



CO c<i t~ o eo 00 ■«** ostot~oto^-H CO e^ to t~ eo 00 ■<*< os c>q to o co ^-i t- os e>q oo eo os -«j« t~ os to o 

t— ooooososos^ ^f-HT-ic<ievieo co'»*<'Tf'>*<ioto cocot~t~oooo stsososoo-th THe^3e^^e^Jco 

CO CO to CO CO CO fc.- t~t~t-t~t^t~ t-t~t-t~t~c~ t~ t- t~ t^ t~ t~ t~t~t^oooooo oooooooooooo 

CO CO CO CO CO CO CO <o CO CO co co CO CO CO eo co co eo co eo co co co co co eo eo co co co <o co eo co eo co 



-H t~eMtooocoo> "I** o eo to -H CO cvi t- o co oo -.s* O5iooooco»-i t^ c^a to oo co os ■<*< o evj to -h co 

rH TH e>a CM e>q CO CO ^f to to to co co t~ t~ oo oo oo os os o o — ^ '-^ c<i cvj co eo eo •«** •»!< to eo co co i>. t- 

to lotokotototo lotototototo OO tototototo tococDcococo cocoeoeoeoco cocoeoeoeoco 

CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO eo co co eo co co co co to co co to co co co co co eo co co co co 



CO e^t-oeooc^ ostot-oeorH t~ e>q to oo eo Js •»j<oeoto^eo e^t~oeooo-^ ostoooocoT- 

-^ »otocoeo«ot~ t~ooooososo o »-H -H »-( es] e^a eo •<*<-*' -f to to coeot~t~t~oo ooososoo— < 

eo eoeoeoeoeoeo coeoeococo-^ .»*«-<ij<'^Tti-<tiTtt •«if-<f-«f-Ti'-^-ct» .T»<-»*i-»*"Tfi-^Tt« -<»«Tti-rt<iotoio 

CO CO CO CO CO CO CO CO CO co co co co co co co co eo co co co co eo co co co co co to to to to to co co co co 



o CO —I -* t~ 53 CO rf OS cq to o CO T-i t~ OS c<i 00 OO osif t~oto^ CO eva to t~ eo 00 •<*« o e<i lo »- co 

00 ooososessoo i-h ^ cm cq co eo •»»< t»< •<}< to to eo eo t- t~ oo oo os osooo^»-( e^a eo eo eo 'tt" "<»< 

r^ ,_( .^H »-i 1— t e^a e<i e^a cq e^a e^a co e>q e>q co eo e^a cm eva eo eo eo co evj eo eoeoeoeoeoeo eoeoeoeoeoeo 

CO CO eo CO eo CO eo CO co co co co eo co co eo eo eo co co eo co co eo co co co eo co co co co eo eo eo co co 



^ 52Sni'^^'''' !::? 55 "^ '^ "^ '^ oj ■<*< t- o to »h coeotot~eoos -^ <D eo to »-< eo eo t~ o co os ■»*< 

31 iS i2 5S ^ E:: £;: 22 55 92 ^ ''^ <=> o-n^eoeoco co -^ -<*<-«*< to to co i>- 1~ t~ oo oo crsosoooi-i 

52 P2'5°2°292?° ooooooooooos cjsososososos osososososos oscrsosososos ososocsoo 

to tOlOtOlOtOtO tOtOtOtOtOtO tOtOtOtOlOtO tOtOtOtiitOtO tototoiototo totoeoeococo 



52 95SJ^SJ2:i !S5S?2^^^ -^foeoto^t- eoooT-neoosto ocooo^t~c<i oo-^fcoottoo 

S lSrSS?fofS;« S!J2H5"^*®'0 t~ooooooosos cs o »-< ^ i-i eo eo eo ro ><»<•*»< to tocoeocot-oo 

J2 fSfSiScSfSJS J2M2'^'""^"' lOtototototo eoeoeoeoeoeo eoeoeoeoeoeo eoeoeoeoeoeo 

to totototototo iOtototototo totototototo totototototo totototototo totrstototoio 



i2 SSSSr^S? 2°3^^2y2:=J coeotoooeoos ■^oeoeo^t~ co oo --< -^ os to o co cd eo t- co 

2J J2t5i-;_^Er t~ooQOC7sosO o --i r-i ,-i eo eo eo ■*-«*' ""^i to to coeot~t~t~oo os os os o cs .-c 

52 r2TS^**'^52 CO CO CO CO CO -<J< -^ -»t< -<*< r»< -^ -^ -*». -XT -xf -^ ■<»< -* -i»< -^ -tt< -^ -<»< -<t< ->»< -"f -*i to to to 

to to to to tfj irj to totototototo totototototo totototototo totototototo totototototo 



962 



58,— SURVEYING, MAPPING AND LEVELING. 



o cq -* »f5 «o 00 o cq ■* in ?o 00 o c<i ■«»« in «o oo o ea'^»o«oooo w<*<»nec90 ca'*»o«oooo 
^H ■_< .^.^ .^^ ^^ ^M e<i CM esj esi o3 gq ev; co co CO co eo -;>< ■^y Tf Tf t}« •«»< lo vo to tn tn >n to 

c<i «Oi-HCQ»i-iir3 e>ioc^C3-«TC5 -"ifcrs-^cococo occMir3t^csit>. t-ics05i-i«jC3 ioOcMiocrs-'S' 

t>. t>i00ocooc30> ooc3»-i»Hi-H c^acsicococoTf •^s^o-omtoco c>.t>.c>.ajooos o*ooc=>c3'^ 

•>!»* Tjtr»<T*<'«j<Tt<'«^ lo la in la la in louoiomiOiO iriiOOiOioo lOiomwmio »n«o<o!ioi;o«o 

,^ .,-|.,-<.^^,^.,^,^ T-l-.-<'>^^Hl-^T-l 5I-I-,— I^M't-H^-ll-iH ,_(^^T-l>-<l-^»i^ ,-<^<,^,H^HyH l-<»-<^Nl-<'^-<->^ 

00 C000Occit^e<I t- oq -^ CO '^ «0 i-iCOOOOmO lis Ci C>q •«»» 05 tJ* OiOO«0000000 cqt«oc<it«e<i 

e>q CO do •^ ■^ •^i< m lo «o CO to c^ i>. ooooooo^oo ^^»-ivhi— icq cvj co co co •^ ^< m uo eo co co t>> 

CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO ^ ^^ ^i* "^ ^fi "^ ^f* ^* ^^ ^^ ^tf* ^ 1-H ^1 ^J< ^< ^3* ^}* "^ 

T}4 Oi •<}< CO 00 CO 00 coooocqt^cq r>. cq •* co — < co i-<cooO'->mo lo cd cq irj os t}< o> •<# co o> co oo 

00 GO <3» 0:> Ci C3 C=> ^^ -H C^q cq cq CO CO n< tj< •rt» ITS »0 CO CO CO t>. t^ 00 00 0> 05 05 o» O O <-h -h .^ e>q cq 

»_, ^^^^e^cNq e<i c<i CQ cq cq cq c^q cq cq e^j CNq cq cq cq cq evq cq evq c>q cq c^q cq ccj co eocococococo 

»-* ».H i-H 1-H i-l 1— I ^H T-i ,-( ,-1 ,-< ^H •^H ,— 1 ,-( i-H T-l 1— I tH »-1 1—1 tH »-i i—( 1-H t-i i-^ i-h vH tH i-I »h i— I i— I »-( »-H ^ 

yj ,.; ^' ^' ^' ^' ^ ^^^^^^ ^^^^^^ ^^^^^^ _; ^ ^ ^ ^ ^ ^' ^ ^ ^' ^' ^ 

o» »*< 00 »^ CO 00 CO 00 cs in 00 cq t^ e>q t^ ci cq t^ ^h co —< co co i-< co omoooino in 05 c»a rt< o» •<}< 

CO ^f* •»}< m in m CO co i>. ^~ t— 00 00 cscscioo-H »-i cq cq eg co co rt« -^ tj* in m co cocot^t^t^oo 

O 000000 000000 OOOi-tT-i^ ^^rtrt„^ ^^^^^^ ^rt^^„^ 

_| ,_| _4 _| _| _| _| _| ^ ^ _| _| _| _| ,-1 iF^ _l _< tH ,-1 ,-1 t-4 t-( ,-h tH ,-1 ^h 1-4 »-( i-( ^H _i t-i 1-1 1-4 t-i i-H 

tH ,-i^.^l-J^^ ^'^^-J^^ ^^'^'^'^^ ,-^^^,-^^.4 l-J^',.-<l-<^^ ^^,i^T^^T-J 

CO 00 CO m t>. gq b» c<j t- 05 c^i t- cq co .^ •<:t< co — co 1-icoooomo inocqinorjt o» •«!*« t, o» T*t os 

o> OS ^ c> o — ^ 1— I cq ca cq CO CO -^ ^j< in in in co co t^ t-- tr^ 00 oc oi os o o o ■— < ^h i^cQcqcqcoco 

00 00 OS OS OS OS OS OS OS OS OS OS OS OS OS OS OS OS OS OS OS OS OS OS OS OS ^ O O O O O O O O O O 

O 000000 000000 OOOOOCD 000000 o '-I -^ »-• —I 1-1 ^«-«»^^»^i-i 

cc ^H CO oo ^^ CO -^H m o CO m o in csint>.oino •«*< os cq -^ os -rj* os Tt< co os co 00 cooo-hcoooco 

■Tj< in «n in CO CO t>. tr~oooocoosos ooO'— i»-Hcq cq c<i co co 00 ■>!»< -"ti in in o co co t^ t^ od 00 00 os 

t>. !>. t- t^ t^ !>• t>. t^ t^ t>. t>. t^ t>. 00 00 00 00 CO 00 00 00 00 00 Ou 00 00 00 OO CO OO 00 QO OO OO 00 00 00 

O OOOOOO C50000 oooooo oooooo oooooo oooooo 



s 



3 



OO CO 00 »^ CO 00 CO OO CO in 00 CO OO coooocqt^cq ^^ cq in t» cq t^ cqt^oscqt^^^ co »-< •^ o ^-h co 

OS ^^-^-H,-ic«q «sj CO CO CO -"f ■># inmcococot^ t-- oo oo oo os os o o o ^ -h cq cq co co eo -^ ■<*< 

in eocococococo eocococococo eocococococo eocococococo t^ t^ t- t^ c^ t^ t- t^ t>> ^~ t^ t^ 

o oooooo oooooo oooooo oooooo oooooo oooooo 



^ 



t 



o inocqinoin oini>.oino •«!»< os cq -^t* os ■^ os •«* t>. os -<*< os •«* os -^ -^i* os ^jt os ■rj^ co os co oo 

in CO CO CO c^ t^ OO oo OO OS OS o o o '-h -h ■^^ cq cq co co ro -^ •>*" m m co co co i>. t^ooooooosos 

■1 Tj* Tt< -"^ Tfi Tj< ■«!ti-^->:j<'rj<-«*<in ininininmin mininininin ininminujin inininininm 

looooo oooooo oooooo oooooo oooooo oooooo 



s; 



»-< co^HCOco^Hco 1-1 CO OO T^ CO »H coi-Hcocooin oinoooino inocoinoin omoooino 

^ ^ »-i 1-1 1-1 cq cq cococo'^-^m in CO CO CO t^ t-- ooooocoscso o -h i-i i-i e^q cm co co co -^ -^ m 

CO eocococococo eocococococo eocococococo eo co co co eo tJ< •«»< t*4 ■«a< -^ti tJ* -^ Tt< tJ< rt< tj* ■^ ■^ 

o ^oo^oo oooooo C300000 oooooo oooooo oooooo 



1-1 CO -^ CO CO »-f CO i-< CO 00 -- CO -H CO -H CO CO 1-1 CO ^ CO 00 T-< CO -w CO 1^ CO CO —I CO »^ CO 00 —I CO -^ 

in m CO CO CO t^ t>. ooooooososo C3 i-i i-i -^ cq cq co co oo -^ ">9< in in co co co i>. i>. ooooooososcs 

^ ^ _! ^ ,^ ^ _ _ ^ ^ .^ .^ e^a cq cq cq evj ca cq cq cq ca e^q cq cq evj cq cq cq cq e^5 cq cq evq cvq cq co 

o oooooo oooooo oooooo oooooo oooooo oooooo 



I 



in o CO m o in ciinooomcs mocoinoin c> co oo »-i co i— i co -h co co i— i co i— i co oo i-< co »- 

c> »^ —( ••^ c^a c* CO CO CO rj< «Ti .n in o <o o t- t^ ooooooososo o i-* i-i th cq e^q co co eo -^ tj< in 

OOC300C5 oooooo OOOOOO O O O O O -H >-(,-( _4 ^H -^ ^ 1-1 ^ _ ,-( ^ T-< 

oooooo oooooo <0000<=><=> 00C3000 oooooo OOOOOO 



00 ■<*< OS 1-1 -^ OS •rt< OS nt* CO OS ■^ OS ^ OS cq -^ OS -xj* os i^f t^ os in o inocqinoin oint^omo 

Tf m in CO CO CO t^ t^ooooooosos ^ ^ i— 1 1-< i— i cq evi co co co •rf in in co co co t>. t^ ooooooososo 

00 OOOOOOOOOOOO oo OO 00 00 00 00 0S0S0S050S0S C?s OS OS OS OS OS OSOSOSOSOSOS OSOSOSOSOSO 

OS OS OS OS OS OS OS OS OS os os os os os os os os os os os os os os os os os os os os os os os os os os os ^ 



»-4 CO OS — H b- cq t>. cq "f t^ cq t^ e>q t- o ca t^ c^q oocoinoocooo coooocoooco oococooocooo 

oc>^i— '1— iCQ cqcococo'*'!}* m in co co co t>. .t- oo oo oo os os ooi— ii-i^nesi cqeococoTt"T»< 

t~ t^ t^ t~ t^ !>. t~ t^ t^ t^ t>. b- t^ t^ t^ !>• tr- t- t- t- b- t>i l>- t>. OO oo 00 00 CO 00 OOOOOOOOOOOO 

OSOSOSOSOSOS OSOSOSOSOSOS osososososos osososososos osososososos osososososos 



CO 00 CO CO OS ""ii* OS ■Tjt OS 1— I Tjt OS ^ OS 'i^ t^ ^ in o lnoc^alnoln oinco— hco»h co^^coco^^co 

■<s< •^ininincoco t^t^ooooooos osoo— <ir- e^a ca co co co ■<*< ■<*< m in in co co t>. t~ oo oo oo os os 

in inininininin ininininmin incococococo eocococococo eocococococo eocococococo 

OS OS OS OS OS OS OS OS OS OS OS os os os os os os os os os os os os os os os os os os os os os os os os os os 



OS inocQinoin omooocoi-i co i-< •«*< co — co ca t- os ca t- cq b^ cq in «>. eq oo coooocoooco 

00 OS ^ o ^ -H 1— I cq cq cvj CO CO ■^ ^ij* m in in co co t^ t^ t^ oo oo as osooo-hi-i e>q cq co co co ^j" 

CO co"*-^"<t<TfTf< M<'*''^'^Tf'!}< •Tf'TfTf-^Tr-'*' T*<«tj<xt<Tti-<*'-^ ■^ininininin inininminin 

OS osososososos osososososos osososososos osososososos osososososos osososososos 



m oinoo^co'— < cO'^'^cO'-Ht>. c^at>.o>cs3t>.cq t^coinoocooo cooo^^coos'^ osTfc--os-^05 

CO •^•^Tjtininco coi>-t^t^oooo ososcsoo-h i-icqcacacoco ■Tf<-xtiinininco cot>.t--t^oooo 

cq cq «^ esi cq e^a e^a cq ca e^a cq e^ e>q e^ ca cq co co co eocococococo eocococococo eocococococo 

OS osososososos osososososos osososososos osososososos osososososos osososososos 



o inocoin^Hco i-i co os i-i «o cq t^ cq in t~ e>q t^ coooocoooco oo t}< co os •»)< os ^jtoscqinom 

OO oo OS OS OS C5 o T-i 1-1 »-< cq e^ CO co is* i* Tti in in co co t- t^ t^- oo ooosososo^ i— > t— cq e^ co co 

^ C3000^Hi-i »-4 T-c ^ 1-1 -H ^ ,_^ ^ ^ ^ ^ ^ ^^^^^^ _ „ ^ ^ esj eva e^ cq cq e^a cq e^ 

OS osososososos osososososos osososososos osososososos osososososos osososososos 



•^ os^*tt^^ino inocoeo^^eo ^-i co os cq t^ cq t>. cq in oo co oo co oo — h Tt< os ■^ os -xt* b- os in o 

cq cq CO CO ■<»* -^ in in co co co t^ t>. oo oo oo cjs os o o -h ^ i-i ca ca co co ■«*< -^ -^ in incocoeot^oo 

OS osososososos osososososos os os os os os co c> ^ ^ o c> o C3 ^ ^ ^ o o oooooo 

00 OOOOOOOOOOOO OOOOOOOOOOOO oo 00 00 00 00 OS osososososos osososososos osososososos 



t^ CO 00 o CO 00 '<*< OS -Tj* b- OS ij< o in o CO in -h CO i— coosi^t^cq t^ cq in b<. co oo co oo i^ ■*»< os t*< 

eb I?- t^ oo 00 00 OS osoooi-icq cq co co co -^ •«*< in in in co co t>. t~ oo oo oo os os o O i-i i— i-< cq 

b- t-i>.t^t-t^t>. t^oooooooooo OOOOOOOOOOOO OOOOOOOOOOOO OOOOOOOOOOOO osososososos 

00 OOOOOOOOOOOO OOOOOOOOOOOO OOOOOOOOOOOO OOOOOOOOOOOO OOOOOOOOOOOO OOOOOOOOOOOO 



o m 1-H CO CO — < CO cqtr^ocqb^co oo co co oo "^J* os -!j< os cq in ^ in ^eooo»Hco»H b-tM'^r^cqt* 

1-1 1-1 cq cq cq CO CO -»j< i*< in in in co eot^t^t^oooo ososoo^i-i e<i e<i cq co co -^ -^j* m m in eo co 

CO eocococococo eocococococo eocococococo co co t^ t^ t^ r>. t^ b- t>. t^ t>. t>. t» t— t>. t^ t~ t» 

00 OOOOOOOOOOOO OOOOOOOOOOOO OOOOOOOOOOOO OOOOOOOOOOOO OOOOOOOOOOOO OOOOOOOOOOOO 



CHORDS FOR PLATTING ANGLES. 



963 



00 e<i «o 05 T-H ID OS CO !>. CT> 1-1 ITS o ■<* oo o M «o o -^ OS »w CO b- 1-H lo OS T-4 cio t^ Cvi «o o e<i ■^ oo e<i 
^ CM e^a M CO CO 00 -^ -^^ -^ \a) »n <o co «3 t- t- t- oo ooooosososo oo— <««-iesi e^a co co co eo -"m 

C3 OOOOOO C300C500 OOC3C300 OOOOOi-H ,-(,-(,-i^,-(,m ,-(,-i^r-<^T-i 



CO t^CQ-^COO-^ OO CQ m t>- ^ UO OS CO lO t^ CCJ «0 O -^ CO 00 e<l t- 1-1 m t^ OS CO b» C>3«0000-^00 

OS osooo^i-t ^-ie>qcq^cicoco co-««<-*ti<»oio scyseoeot-t^. ooooooooosos ocso— ii-h— < 

OO 00 OS OS OS Od OS OS OS OS OS OS OS OS OS OS OS OS OS OS OS OS OS OS OS OS OS OS OS OS OS o o o ^ o ^ 

CO COeoCOCOCOeO COCOCOCOCOCO COCOOOCOCOCO COCOCOCOCOOO COCOCOCOCOCO 'tt* Tti -<}< -5H tH I*' 



t>. i-ieoooO"*oo CM b- OS 1-1 lo OS cooooc<icoo ■^os^coe^th ift o e>a •^ oo c<i «o-hco»oo4co 

CO i>- t- t^ 00 00 00 osososoo^ »-< ^ c^a e^ai e^ CO co co '<*< -<»<-<*< m mcocococot^ t^ oo oo oo oo os 

t>i t^ t^ t~ t~ !>. t- Ir^ t^ t>. 00 00 00 OO 00 00 OO 00 GO 00 00 00 00 00 00 000000000000 00 OO 00 00 00 00 

CO COCOCOCOCOCO COCOCOCOCOCO COCOCOCOCOCO COCOCOCOCOCO COCOCOCOCOCO COCOCOCOCOCO 



o M* 00 i-H CO t^ i-i ID o cq M* 00 cq t-- r-^ co m os -ij* oo cq ■«*< co ^ lo os co i« oo pci co o ■<»< b- os eo t» 

rj« "^ -^ la \a m to cot—t^tr^t^oo ooososososo ^i— i»Mi-<e^3evi c<icococO'^'*' loiriirsmcoco 

CD COCOCOCOCOCO COCOCOCOCOCO COCOCOCOCOI>. t-t-.t>.l--t>.t^ l>.E^E^t--.b^t^ t-t^t^t^b^tr^ 

CO COCOCOCOCOCO COCOCOCOCOCO COCOCOCOCOCO COCOCOCOCOCO COCOCOCOCOCO COCOCOCOCOCO 



ca cooeoioosco ooc<i-^cooio oscoiftt^c^co o lo t>. os co r>. c<icoooort<o> cot^os^-icoo 

^^ —I e^ CQ N c<j CO CO -*-<»< ->i< lO lO in CO CO CO t^ t- oooocoooosos o o o >— ^ i-i e>q cq c<i co co -^ 

Id lC5k£0»OU0lDlO jrjlOlOlOiOlft lOlOiOlOlOlO ^OlOl^^lOlOlO totocotocOCO COCOCOCOCOCO 

CO COCOCOCOCOCO COCOCOCOCOCO COCOCOCOCOCO CO CO »D CO CO CO COCOCOCOCOCO COCOCOCOCOCO 



CO t^-MCOcoo-^ OS CO ifi t- "-I CO o -^ c^ C3S CO !>. cacooooifsos cot^oeacoo »o os »^ co oo c<i 

OO 00 OS OS OS O ^ C> ^^ ^H .— I Cd Cq CO CO CO CO •* Tf< lO lO lO CO CO CO C^ t^ OO OO OO OS OSOSOOCS-h 

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CO COCOCOCOCOCO COCOCOCOCOCO COCOCOCOCOCO COCOCOCOCOCO COCOCOCOCOCO CO CO CO 03 CO CO 



f 



"f 



s 



c>q t>--Hco»ftO-^ e75coiot>.cqco oiot^oscooo ccicooot-hioos Tt<oooc<it^»H mocM^ooeo 

\a locococot^t- b-ooooooosos oooo^-^t-w evievie^ieocooo 'sj^-^ioifimco cot-t-^b^t^oo 

ca eg e^3 eg cq e<i cq c^ e^3 eg (^3 cci cq co co co co co co cocococococo co co co co in co co co eo co co eo 

CO COCOCOCOCOCO COCOCOCOCOCO COCOCOCOCOCO COCOCOCOCOCO COCOCOCOCOCO COCOCOCOCOCO 



^H CO O e<I -<*< OS CO t~ e^ '•J* CO »-< »0 OS -^ co OO co l>- »-< «0 OO O »»< OS COOOOCQCO»-t »0 OS — < ■»*« 00 CM 

eg eg CO CO CO CO -"^ •>i*< ko imo co co «o t^ t^ t^ oo oo osososocao »-h »-< ca eg cq oo co co -^ -^ m^ m 

_( »_i,^._H^H»-^'rH _i-^^H»-i— !,-( _<^H»-H^H.^Hi— I 1— i»-i— «ege^qeM pqcgcgeqcNicg cg«^ac^ege<ieg 

OO COCOCOCOCOCO COCOCOCOCOCO COCOCOCOCOCO COCOCOCOCOCO COCOOOCOCOCO COCOCOCOCOCO 



OS CO OO o e<i t^ •^-4 m o e^a -rH OS CO oo e<i ^f co r-4 m o -^ co oo co i>. e<jcoooomos •rj4 oo o cq t«- »^ 

OO OSCJSOOO^ -H cvq evq cq eg CO CO ■<*' -tf "^ l« lO CO CO CO CO t^ l>. 00 OO 00 OS crs OS O O -H — < ^ CQ 

OS ososoooo o o C3 o o o oocsooc) oocDcooo oocDooo ,-. _i ^ .^ ^ l;^ 

eg eq eg CO CO CO CO cocococococo cocococococo cocococococo cocococococo cocococococo 



CO oir)t^os-«*coo egt-osr^coo loos^-^cocq t>.^-^coom osrt<«oooe>qb^ i-ncoooomos 

m cocococot^t^ ooooooososo C)Or-i— h»-i(ni e^cococo-<:*<-^ ^jtioirsiococo t^t^r>.cooooo 

00 00 00 00 00 00 00 00 00 00 00 00 OS OS OS OS OS C35 OS OS CJS OS OS OS OS OS OS OS OS OS OS OS OS OS OS OS OS 

cq eg eg eg eg eg eg e^ eg eg eg eg oq eg eg cq e^q «^a eg cq eg eg e^q esi eg eg eg eg eg cq eg eg cm eg eg eg eg 



eg CO — H CO uo o Tt« oo co »n t- eg co i-h m co o -^ os cooooegt^-n eoocqiftosco oo eg >A r» «-< co 

eg e<i CO eo CO ■Tf ■Tf< ■rti lo m lo co co t~ t>- 1>. oo oo oo ososo^C3»-h »-< eg e^a eg eg co co •rf ^f rf m uo 

!>. t- t— t^ t- t^ t^ t- t>. t^ t- t- t^ t~ t^ C- t— t^ J>i t~ t^ OO OO 00 00 OO 00 OO 00 00 00 00 00 00 00 00 00 

eg eg eg eg eg «>a eg eg eg e>q eg eg cq eg eg eg eg eg oq eg eg eg eg eg eg eg eg evj eg eg eg pq eg eg eg eg eg 



CO »-i »o 00 o ■<*< OS •«*< 00 o CO t~ eg co— (COiootj^ oscoxcooegb- »h co oo o lo o» -^ooocot^eg 

OO osososooo 1— c '-( eg cq CM CO co -rf -rf -* lo m lo co co co i>. t>. oo oo oo cjs c?s os o ^ i— i »-i i— i eg 

lo lo »o lo CO CO CO cocococococo cocococococo cocococococo cocococococo t- t^ t>. t- tr^ t^ 

c>q eg eg eg eg eg CM cq eg sm eg eg eg eg eg eg cq eg eg eg eg eg eg eg oa eg eg cm eg eg eg eg eg eg eg eg cvj 



eo 00 eg in tr^ eg CO ^-ninooo-^os ^t* oo o co t>. eg co —<•<*< co o lO os ■«*< co oo co oo cgt^os»-<coo 

•-H »-Hegcgegeoeo TttTfT^immm cocot^r^t^oo ooososcssoo o»-<'— ii— legeg cococoT}<rf<io 

CO cocococococo cocococococo CO CO CO CO CO- CO eOCOCOCO-<9<-rJ< rt<-^"<*<'<I<"^*<-Tt< T*<Tt4'r»<Tj<-.^Tj< 

eg eg eg eg eg eg eg eg eg eg eg eg eg eg eg eg eg eg eg eg eg eg eg eg eg eg eg eg eg eg eg eg eg eg eg eg eg 



\n o •"4* t~ OS -<*< OO cooooegb-^H co-^como-* os -^ co oo co b<. eg t>. os -^ co o lo os eg Tt< os co 

i>» ooooooooosos oO"-^— <-Heg egcococo-^-"!** -<tiio»oio,coco t^t^t^ooooos ososooo-h 

,^ _^^^^^ eg eg eg eg eg eg eg eg eg eg eg eg eg eg eg eg eg eg eg eg eg eg eg eg eg eg eo co co co 

eg egegcgegegeg egegegegegeg egegegegegig cgcgegegegeg cgegegegegeg cgegegcgcgeg 

CO i-icoooomo ■^ OS i-H CO 00 CO t>. eg "^ t-» ^ co »-<»ot^o-*04 •«*< oo »-i co oo eg t- — i •^ co »-h in 

CO ■^ T}4 •^ in in CO CO CO t^ b^ t^ 00 ooosososo^ t-i i— 1 1— i eg eg eg eo co -^ ■^ ■«*< in in co co co t^ t>> 

O ©0C>00<3 ^OOCSCSO OOOO-Hi— I ,-h_i^h^ht-h-H »-!,—(— )»^t-(^h ,_i^^,_i,h-hi— i 

eg egegegegegeg egegegegegeg cgegegegegeg cgcgegegegeg egegegegegeg cgegegegegeg 

CO 1— icooooino •>* OS -^ •^ 00 CO ooegint^egco i-ncooo^mo ■»*< os »-< t*i oo eo ooegint^cgco 

OS ooo-^-^cg eg eg CO eo CO -^ •»}< in in m co co t^ i>- t^ oo oo o» oso5000«^ — i eg eg eg eo co 

OO OSOSCSOSOSOS OSOSOIOSOSOS OSOSOSOSOSOS OSOSOSOSOSOS OSOS^O^O oooooo 

^ _ _ ^ _i rt ,_! Ti ^ T-iT-H rH 1-1 ,-( ,-< ,-( ^ ^ ,-H _( -H »-H ^ .-H — < 1-1 1-1 eg eg eg eg egegegegegeg 

CO oinb-oinos ■^os-icoooeo t^eg-'S^f^-^co r^coooomo -^ a> ■r-i -.t^ co co ooegint>.egco 

in CO CO CO f- t^ i>- ooooosososo o .-h »^ »-< eg eg co co eo -rt< ^^t* in in in co co co c^ t* oo oo oo os os 

t>. t^ t- t^ t~ t^ t^ t^ b- t^ t~ t^ OO 00 OO 00 OO OO OO C» OO 00 00 00 00 OO 00 OO 00 OO OO CO 00 00 00 OO 00 

-.*< OS ■«4< CO 00 CO 00 egt^oegt--i co »-^ co co O in o -^ r^ os i>*< os coooocot^eg b»eg-^co»ico 

—4 — <egegcgcoco ■<t<-»*<inininco cot--t>.t^oooo ojosctsosoo '-i»-<egegcgco co-^-^-^finin 

CO cocococococo cocococococo cocococococo CO CO CO CO «>■ t>. t~ t-- t>. t^ r-~ t^ t~ t:^ t» t^ t~ t^ 

lO eg-^incoooo 

icg «g eg cq eg eg CO 



964 



l-^SURVEYING, MAPPING AND LEVELING, 



Farm Surveying. — Let it be required to make a survey of a farm of 
about 150 acres, locating the roads, fences, buildings, 
determining the acreage, and making the map. As |"T 
the farm is, say, about 30 miles from the city the sur- 
veyor decides that he will prepare to spend one day in 
the field, only. 

The Equipment consists of 1 transit, 2 100-ft. 
tapes, 2 flag poles (Fig. 19), 12 pins (Fig. 20) with red 
flannel tied at their tops to prevent losing them, 1 axe, 
1 hatchet, 1 transit plumbbob (Fig, 21), 1 plumbbob 
(Fig. 22) for each of the men, 1 steel frost pin (Fig. 
23) if the ground is frozen, and the stakes (Fig. 24), ^ ^ J^^^'Js 





Fig. 21. Fig. 22. Fig. 23. 



Fig. 24, 



V 
Fig. 25. 



Fig. 19. Fig. 20. 




hubs (Fig. 25), tacks, etc. (The stakes may perhaps be procured on the 
ground, but it is often cheaper to take them from the office.) Four 
men, say, besides the chief of party, comprise the working outfit. 

The Traverse of the farm is represented (Fig. 26) by the broken instru- 
ment-line A B C D E A, which closely follows the fence lines (not shown). 
In running around the farm with this transverse, lettered or numbered 
stakes are set on the instrument lines opposite all bends in the fence lines; 
these stakes are located by base line measure- 
ments, and from them the offset distances are 
measured to the fences, thus completely tying- 
in the farm boundary. Any building as H may 
be located by running a spur instrument line 
from some point on the traverse, as B. The trav- 
erse itself is determined by the lengths of the 
measured base lines, A B, BC, CD, etc., and 
by the measured angles at A, B, C, etc. Inaccu- 
racies in measurement, both in base line distances 
and in angles, will usually creep into the work ~v- 

and hence the traverse will seldom close, that is, -^^S- •^6. 

it will have to be adjusted. Absolute errors, however, can be eliminated 
by measuring the base lines twice and by "repeating" all angles, and 
these should be examined carefully before leaving the field. 

The Adjustment of the Traverse may be made in several ways, but 
the simplest and most practical is as follows: Let A, B, C, etc., be the 
interior angles, measured in the field, at the respective comers of the tra- 
verse (Fig. 26). The sum of these angles should equal 540° (=180°X 
number of sides— 360°) if there is perfect accuracy in the field work. If the 
angles add up to within a few minutes of 540° this variation may be propor- 
tioned among all the angles so their sum will equal 540°. But greater 
weight should be given to those measured 
angles with the (longest and) clearest fore- j-- 
sights, as A, especially if the angle "doubled" ' 
accurately when measured in the field; and 
less weight should be given to, say, C and D,^ J» 
especially if the angles did not "double" or ^ >? 
"repeat" properly. Having adjusted the angles f^ ^ 
(Fig. 27) so their sum is 540°, assume one "'' 
side of the traverse as a base line, say A B, 
and cut the exterior of the traverse into right 
angle triangles as shown by the dotted lines; 
calculate the angles a, Ci, C2 and e, from the Fig. 27. 



£2//.39. 




FARM SURVEYING, 



965 



adjusted interior angles A, B, etc.; solve the sides of the triangles, using 
the measured base lines of the traverse, in each case, as the hypothenuse 
in connection with the calculated angles a, Ci, etc. 

The final test is now made by adding together the horizontal distances 
at the top of the traverse, which we will call W, and comparing with the 
sum of the horizontal distance at the bottom, which we will call E; like- 
wise, the sum of the right-hand vertical distances iV should "equal" the left- 
hand vertical distances S. If there has been no radical error in the field 
work, they should check fairly well; thus. Fig. 27, starting from A, We have, 

( + )E ( + )N (-)W (-)S 

+ 2977.69 +1187.06 - 458.60 -2387.76 

+ 694.70 + 917.47 -2211.39 

+ 283.61 -1001.88 



+ 3672.39 
-3671.87 



+ 2388.14 
-2387.76 



-3671.87 



-2387.76 



+ . 52 and + . 38 = closing errors. (See Fig. 28.) 

To "close" the traverse, construct a closing diagram as per Fig. 29, 
laying off the base line measurements A B, B C, etc., consecutively and 
with sufficient accuracy, from Fig. 27. At the right-hand end, at A, lay off 
the closing errors, 0.38, 0.52 and the hypothenuse or closing line 0.64, 




-o-oo-oa 297764 



Fig. 28. — Adjusted Traverse. 

vertically above the base line A A, and draw rays to A, intersecting verticals 
erected from B, C, D and E. The scaled distances from the base line to 
their intersections will be the closing errors to be used at these points, B. 
C, D and E, in Fig. 28, the closing line at each point being parallel to the 
closing line shown dotted at A, and the errors at any point being propor- 
tional to the total distance on the base line from the initial point A, in the 




Fig. 29. — Closing Diagram. 

direction of the lettering. Fig. 28 need not be drawn to scale, the adjusted 
traverse being shown by full lines and the unadjusted traverse (greatly 
exaggerated by using an enlarged scale for the closing errors) being shown by 
dash-and-dot lines. Now construct external triangles around the adjusted 
traverse lines, calculate their sides (applying difference in errors obtained 
from Fig. 28 to the calculated sides in Fig. 27) and their angles. From 
these angles, calculate, by addition and subtraction, the new interior an- 
gles for the adjusted traverse. All calculations should be made in regular 
office calculation books (unruled pages), numbered and indexed. Cross- 
references are always desirable for field books, calculation books and office 
plans. 



d66 5S.— SURVEYING, MAPPlN(^ AND . LEVEUNG. 

The Office Plan is made on heavy manila detail paper. The adjusted 
traverse is shown in red ink. Angle points on the instrument lines are indi- 
cated by being enclosed in a small triangle; and ordinary hubs, at other 
points, by a small circle. The property lines are in black (India) ink, as are 
also the outlines of buildings. Lanes may be indicated by parallel dotted 
black lines, and streets by full shaded lines. Fences may be shown by fine 
black lines (unless they mark the property boundary) and should be lettered 
''Fence '^ "Stone wall," etc.; or they may be indicated by broken lines as 
for instance alternate dashes and dots. All corrected measurements and 
angles are given on the office plan as they appear in the field book or calcu- 
lation book, and cross reference is made to each other. The lettering is 
slanting, and is made by single strokes of the pen, including names of streets, 
buildings, etc. All information should be recorded, including the acreage 
of the property. This is usually obtained by calculating the area of the 
traverse and making the necessary additions and subtractions when the 
boundary line of the property is respectively exterior or interior 'to the 
traverse lines. Field measurements are taken in such a way as to simplify office 
calculations. Both magnetic and true north are shown. The title is prefer- 
ably in the lower right-hand corner and should include the name of the 
property (owner), location, date of survey and by whom, reference to field 
book, date of plan, and scale. The author has found it convenient to cut 
out a small 45° triangle from the lower right-hand corner of the plan so that 
when the plans are lying in the case drawer, the thumb can be inserted and 
the number of the plan sought can be seen readily. The plan numbers are 
adjacent to the comers so cut. 

The Finished Map which is furnished to the owner of the property is on 
mounted white paper with rough surface. Paragon paper or its equal is 
recommended. The transfer is made with the steel or needle point from 
the office plan. All instrument lines are omitted, but the property lines are 
shown with distances and angles just as though the instrument lines had 
actually traced them and they were the real traverse. In other words, the 
property line traverse should contain sufficient data so it can be plotted to 
"close," and also be described in deed. All lines should be in India ink. The 
slant-block lettering, plain or fancy, is easy to make, neat, and clean. The 
upright Roman lettering for street names is perhaps more desirable than 
the block lettering, but requires greater care in proportioning and execution. 
The title should be in taste with the general map, neat and compact. It 
should always include the scale, date, and surveyor's name. It is a mistake 
to color a map too highly. The property boundary may be shaded with a 
diluted carmine, and the streets tinted with burnt sienna. Buildings may 
be tinted with the proper colors* to represent wood, brick, 
stone, etc. The north point should be neat and artistic but not 
coarse. It should be placed in a position to "balance" the 
map. The border may consist of a heavily leaded line be- 
tween two iiner lines; or one of the latter may be omitted. 
The corners are generally made as in Fig. 30, but may be 
curved to various patterns. pjg^ 30, 

City-Lot Surveying. — ^The functions of the surveyor are almost judicial 
in character. No statutes are framed or can be framed to meet all cases of 
conflicting deed lines. These conflicts arise from inaccurate surveys in the 
past when land was cheap, and also from improper wording of deeds of 
conveyance. The inaccurate surveys were due in part to the use of chains 
which were longer or shorter than the present standard. This is the cause 
of much of the "surplus" and "deficiency" existing in many of our city 
blocks, amounting in some cases to several inches per hundred feet. Jersey 
City, N. J., is a notable example of surplus, some of the blocks being 4 ins. 
per 100 ft. too long. The sensible way is to distribute this surplus propor- 
tionately or, in other words, to use the same length of chain by which the blocks 
were laid out originally. But this cannot be held to in all cases because 






* Technical or conventional colors may be purchased in liquid form 
(25 cts. per bottle) in the following colors: 1 cast iron, 2 wrought iron, 3 
steel, 4 copper, 5 brass, 6 machinery, 7 leather, 8 light wood, 9 dark wood, 
10 brick, 11 stone, 12 brown stone, 13 Prussian blue, 14 gamboge, 15 yellow 
ochre, 16 vermillion, 17 burnt sienna, 18 carmine. The same may be pur- 
chased in water colors ( 1 cts. per half pan) with the exception that Chinese 
white is substituted for burnt sienna (17). 



istice to no one. It is well to re- I ^K ^ B,ock /2 « <J^ 1 

making out deeds of original T^ ^bj e s 4 3 a /^ k\i' 

its, the description of each lot J ^p U^J .,U. I.I.LI 45 (_[] 

•c^A +n +Vic» CQmo cfr«3^f lino >l ~ " 



c/ri^ Lors. government land. m 

some of the lines have been established by mutual consent, expressed or 
implied. Perhaps one or more buildings have been erected and have absorbed 
nearly if not quite all the surplus in the block. If on the other hand there 
is a deficiency of total measurement, we have a more serious problem to 
confront. Those having the prior deeds are apt to oppose any proportionate 
distribution of the "deficiency," throwing it entirely in the last lot con- 
veyed. Existing buildings have much to do with the solution of these 
questions. The purpose of the surveyor is to 
distribute any benefits or losses _as_ equally as 
possible, with injustice to no one. 
member that in " 
conveyance of lots, 

should be referred to the same street line. ^. . . 

Thus, in Fig. 31, the west line of 1st St. may > ^^^- ^-y 

be selected as the initial base of all lots in block "" • ~~* ^ 

12, and the width of lot 8 should read "25 ft. • Fig. 31. 

more or less" to the east line of 2nd St. 

Street lines in small cities should be fixed by stone monuments set on 
center lines at the street intersections. As the cities grow in size these 
monuments are bound to be disturbed, but they have served their purpose 
in convenience for ready use. Offsets to buildings may now be used or the 
monument points transferred to the manholes* which have supplanted the 
stone monuments. The point selected on the building should be such as 
can readily be described, as above or helow the water table; the corner of a 
building is the best, as being definite. Where sewers, water works and 
street car lines have been introduced more rapidly than substantial build- 
ings have been erected, it is quite customary to transfer the stone monu- 
ments from the centers to the corners of the streets, say on 2-ft. to 10-ft. 
offset lines. 

Government Land Surveying. — ^The following is a digest of General 
Land Office "Circular on Restoration of Lost and Obliterated Comers and 
Subdivision of Sections," Revision of June 1, 1909. 

An "obliterated" corner is one where no visible evidence remains of the 
work of the original surveyor in establishing it; but it is not a "lost" comer 
if its location has been preserved beyond all question by acts of land- 
owners, and by the memory of those who knew and recollect the true situs 
of the original monument. 

Synopsis of Acts op Congress. 

May 20, 1785. Prescribing mode of survey for the "Western Territory, said 
territory to be divided into "townships of six miles square, by running lines due N 
and S, and others crossing them at right angles," as near as may be. Further pro- 
vided that the first line running N and 5 should he on the Ohio river, at a point due 
A^ from the western terminus of a line run as the south boundary of the State of 
Penn., and the first line running E and W should begin at the same point and extend 
through the whole territory. In these initial surveys only the exterior lines of the 
townships were surveyed, but the plats were marked by subdivisions into sections 
1 mile square, numbered from 1 to 36, commencing with No. 1 In the southeast corner 
of the township, and running from >S to iV in each tier to No. 36 in the northwest 
corner of the township; mile corners were established on the township lines. The 
region embraces what is known as the "Seven Ranges" in Ohio. 

May 18, 1796. "Territory northwest of the River Ohio, and above the mouth 
of the Kentucky River." Section 2 provided for dividing lands "by N and S lines 
run according to the true meridian, and by others crossing them at right angles, so 
as to form townships of 6 miles square," etc. Also that "one-half of said townships, 
taken them alternately, should be subdivided into sections containing, as nearly as 
may be. 640 acres each, by running through the same each way parallel lines at the 
end of every two miles; and by marking a corner on each of said lines at the end of 
every mile." Also that "the sections shall be numbered, respectively, beginning 
with No. 1 in the northeast section, and preceding west and east alternately through 
the township, with progressive numbers till the 36th is completed. t" 

May 10, 1800'. Amendatory to the foregoing. "Townships west of the Musking- 
um, which are directed to be sold in quarter townships, to be subdivided Into half 
sections of 320 acres each, as nearly as may be, by running parallel lines through the 
same from E to W, and from S to N, at a distance of one mile from each other, and 



* Four marks with a cold chisel on the fixed iron rim and at quadrant 
points will determine the true center. 

t This method of numbering sections is still In use. 



968 58.— SURVEYING, MAPPING AND LEVELING. 

marking corners, at the distance of each half mile on the lines running from E to 
W, and at the distance of each mile on those running from S to N. And the Interior 
lines of townships intersected by the Muskingum, and of all townships lying east of 
that river, which have not heretofore been actually subdivided into sections, shall 
also be run and marked . And In all cases where the exterior lines of the town- 
ship thus to be subdivided into sections or half-sections, shall exceed or shall not extend 
six miles, the excess or deficiency shall be specially noted, and added to or deducted 
from the western or northern ranges of sections or half-sections In such townships, 
according as the error may be in running the lines from E to W or from S to N." 

June 1, 1796. Act "regulating the grants of land appropriated for military 
services, etc., provided for dividing the "U. S. Military Tract," in Ohio, Into town- 
ships 5 miles square, each* to be subdivided in quarter townships containing 4000 
acres. 

March l, 1800. Amendatory of the foregoing act. Section 6 enacted that the 
Secretary of the Treasury was authorized to subdivide the quarter townships into 
lots of 100 acres, bounded as nearly as practicable by parallel lines 160 perches In 
length by 100 perches In width. [1 perch = 1 rod= 16.5 ft. = i chain.] These subdi- 
visions into lots were made upon the plats In the office of the Secretary of the Treasury 
and did not agree with the actual survey made later, many fractional lots being 
entirely crowded out. This fact may explain some of the difficulties met with In 
the district thus subdivided. 

February 11, 1805. This act directs the subdivision of land Into quarter sections, 
and provides that all corners marked in the field shall be established as the proper 
corners of the sections or quarter sections which they were intended to designate, 
and that corners of half and quarter sections not marked shall be placed as nearly as 
possible "equidistant from those two corners which stand on the same line." Also 
that "the boundary line actually run and marked [in the field] shall be established as 
the proper boundary lines of the sections, or subdivisions, for which they were in- 
tended, and the length of such lines as returned by either of the surveyors aforesaid 
shall be held and considered as the true length thereof and the boundary lines which 
shall not have been actually run and marked as aforesaid shall be ascertained by 
running straight lines from the established corners to the opposite corresponding 
corners, but in those portions of the fractional townships where no such opposite or 
corresponding corners have been or can be fixed, the said boundary lines shall be 
ascertained by running from the established corners due A^ and S, or E and W lines, 
as the case may be, to the water course, Indian boundary line, or other external 
boundary of such fractional township." 

April 24, 1820. This act provides for the sale of public lands in half-quarter 
sections, and requires that "In every case of the division of a quarter section the 
line for the division thereof shall run N and S, and fractional sections, con- 
taining 160 acres and upward, shall in like manner, as nearly as practicable, be 
subdivided Into half-quarter sections, under such rules and regulations as may be 
prescribed by the Secretary of the Treasury; but fractional sections containing less 
than 160 acres shall not be divided." 

May 24, 1824. This act provides "that whenever. In the opinion of the President 
of the U. S., a departure from the ordinary mode of surveying land on any river, 
lake, bayou or water course would promote the public interest, he may direct the 
surveyor-general in whose district such land is situated, and where the change Is 
intended to be made, under such rules and regulations as the President may pre- 
scribe, to cause the lands thus situated to be surveyed in tracts of two acres in width, 
fronting on any river, bayou, lake, or water course, and running back the depth 
of forty acres." 

April 5, 1 83 2. This act directed the subdivision of the public lands Into quarter- 
quarter sections; that In every case of the division of a half-quarter section the 
dividing line should run E and W, and that fractional sections should be subdivided, 
under rules and regulations prescribed by the Secretary of the Treasury. Under 
this provision the Secretary directed that fractional sections containing less than 
160 acres, or the residuary portion of a fractional section, after the subdivision into 
as many quarter-quarter sections as it is susceptible of, may be subdivided into lots, 
each containing the quantity of a quarter-quarter section as nearly as practicable, 
by so laying down the line of subdivision that they shall be 20 chains wide, which 
distances are to be marked on the plat of subdivision, as are also the areas of the 
quarter-quarters and residuary fractions. 

These two last acts provided that the corners and contents of half-quarter and 
quarter-quarter sections should be ascertained as nearly as possible In the manner 
and on the principles prescribed In the act of February 11, 1805. 

General Rules from the Foregoing Acts. 

1st. Boundaries established and returned by the duly appointed Government 
surveyors, when approved by the surveyor general and accepted by the government, 
are unchangeable. 

2nd. Original township, section, and quarter-section corners established by the 
Government surveyors must stand as the true corners which they were intended to 
represent, whether the corners be in place or not. 



GOVERNMENT LAND SURVEYING. 969 

3rd. That quarter-quarter corners not established by the Government surveyors 
shall be placed on a straight line joining the section and quarter-section corners and 
midway between them, except on the last half-mile of section lines closing on the 
north and west boundaries of the township, or on other lines between fractional 
sections. 

4th. That all subdlvisional lines of a section running between corners established 
in the original survey of a township must be straight lines, running from the proper 
corner in one section line to its opposite corresponding corner in the opposite section 
line. 

5th. That In a fractional section where no opposite corresponding corner has 
been or can be established, any required subdivision line of such section must be run 
from the proper original corner in the boundary line due E and W, or N and S, as 
the case may be, to the water course, Indian reservation, or other boundary of such 
section, with due parallelism to sectional lines. 

Rem. Extinct corners of the Government surveys must be restored to their 
original locations, whenever it is possible to do so; and hence resort should always 
first be had to the marks of the survey in the field. The locus of the missing corner 
should first be identified on the ground by the aid of the mound, pits, line trees, 
bearing trees, etc., described in the original survey, as their indentification affords the 
best means of relocating the missing corner in its original position. Next to this, 
clear and convincing testimony of citizens as to the locality it originally occupied 
should be taken, if such can be obtained. In any case, whether the locus of the 
corner is fixed by the one means of the other, such locus should always be tested and 
confirmed by measurements to known corners. 

Exceptional cases may be noted as follows: (a) When new measurements are 
made on a single line to determine the position thereon for a restored lost corner 
(or a quarter corner on a section line), or when new measurements are made between 
original corners on two lines for the purpose of fixing by their intersection the posi- 
tion of a restored missing corner (as a corner common to 4 sections or townships), 
proportionate measurements, using the original field notes, must be used, recogniz- 
ing distributed errors in the original chaining. (&) When the relocated corner cannot 
be made to harmonize with the field notes In all directions it sometimes becomes the 
task of the surveyor to place it according to the requirements of one line and against 
the calls of another line. For instance, if the line between sections 30 and 31, reported 
78 chains In length [2 chains short,] would draw the missing corner on range line 
1 chain eastward out of range with the other exterior corners, the presumption 
would be strong that the range line had been run straight and the length of the sec- 
tion line wrongly reported, because experience shows that west random lines are 
regarded as less important than range lines and more liable to error, (c) Again, 
where a corner on a standard parallel has been obliterated, it is proper to assume 
that it was placed in line with other corners, and if an anomolous length of line reported 
between sections 3 and 4 would throw the closing corner into the northern town- 
ship, a surveyor would properly assume that the older survey of the standard line Is 
to control the length of the later and minor line. The marks or corners found on 
such a line closing to a standard parallel fix its location, but its length should be limited 
by its actual intersection, at which point the lost closing corner may be placed. 
id) The strict rule of the law that "all corners marked in the field shall be established 
as the corners which they were intended to designate," and the further rule that 
"the length of lines returned by the surveyors shall be held and considered as the 
true length thereof," are found In some cases to be impossible of fulfillment In all 
directions at once, and discretion must be applied by the surveyor, (e) In a case of an 
erroneous but existing closing corner, which was set some distance out of the true 
State boundary of Missouri and Kansas, it was held by this office that a surveyor 
subdividing the fractional section should preserve the boundary as a straight line, 
and should not regard said closing corner as a proper corner of the adjacent fractional 
lots. The said corner was considered as fixing the position of the line between two 
fractional sections, but that its length extended to a new corner to be set on the new 
boundary line. (/) The principle of the preponderance of one line over another of 
less importance has been recognized in the rule for restoring a section corner common 
to two townships. The new corner should be placed on the township line; and 
measurements to check its position by distances to corners within the townships are 
useful to confirm it if found to agree well, but should not cause it to be placed off the 
line if found not to agree, if the general condition of the boundary supports the pre- 
sumption that it was properly alined. 

To Restore Lost or Obliterated Corners. 

1. To restore corners on base lines and standard parallels. — By proportionate 
measurements on the line, using original field notes and joining the nearest identified 
original standard corners on opposite sides of missing corner or corners, (a) "Standard 
Connors" means standard township, section, quarter-section, and meander corners; 
also such closing corners as are used in the original survey to determine the position 
of a standard parallel, or established during the survey of the same (&) A lost or 
obliterated closing corner from which a standard parallel has been initiated or to 
which it has been directed will be reestablished In Its original place by proportionate 



970 58.— SURVEYING, MAPPING AND LEVELING. 

measurement from the comers used In the original survey to determine its position; 
measurements from corners on the opposite side of the parallel will not control, 
(c) A missing closing corner originally established during the survey of a standard 
parallel as a corner from which to project surveys south will be restored to its original 
position by considering it a standard corner and treating it accordingly, (d) There- 
fore, from the preceding, using proportionate measurements, we have: "As the 
original field-note distance between the selected known corners is to the new measure 
of said distance, so is the original field-note length of any part of the line to the 
required new measure thereof, (e) As existing original corners must not be disturbed, 
discrepancies between the new and the original field-note measurements of the line 
joining the selected original corners will not affect measurements beyond said corners. 
Proportionate measurements are to be used between them. (/) After having checked 
each new location by measurement to the nearest known corners, new corners will 
be established permanently and new bearings and measurements taken to prominent 
objects, and recorded for future reference. 

2. Restoration of township corners common to four townships — Two cases: 1st. 
Where the position of the original corner has been made to depend upon measure- 
ments on two lines at right angles to each other: A line will first be run connecting 
the nearest identified original corners on the meridional township lines, north and 
south of the missing corner, and a temporary corner will be placed at the proper 
proportionate distance, thus determining the corner in a north and south direction 
only. Next, the nearest original corners on the latitudinal township lines will be 
connected ^nd a point thereon determined In a similar manner, near the intersection 
with the meridional line just run. The Intersection of these two lines will define the 
position for establishing the true corner. 2nd. — Where the original corner has been 
located by measurements on one line only; for example, as a guide meridian: Resto- 
ration of corner is effected by proportionate measurements on said line, as previously 
explained. 

3. Reestablishment of corners common to two townships. — The two nearest known 
corners on the township line (the same not being a base or a correction line) to be 
corrected as in case No. 1, by a right line, and the missing corner established by 
proportionate distance, and to be "checked" upon by measurements laterally to 
nearest known section or quarter-section corners. 

4. Reestablishment of closing corners. — Measure from the quarter-section, section, 
or township corner east or west, as the case may be, to the next preceding or suc- 
ceeding corner In the order of original establishment, and reestablish the missing 
corner by proportionate measurement. 

5. Reestablishment of interior section corners. — Same manner as corners common 
to four townships. When a number of corners are missing on all sides of the one 
sought to be reestablished, the entire distance must be measured between the nearest 
existing recognized corners both N and *S, and E and W, in accordance with the 
rule laid down, and the new corner reestablished by proportionate measurement. 

6. Reestablishment of quarter-section corners on township boundaries. — Only one 
set of quarter-section corners are actually marked In the field on township lines, 
and they are established when the township exteriors are run. When double section 
corners are found, the quarter-section corners are considered generally as standing 
midway between the corners of their respective sections, and when required to be 
established or reestablished, they should generally be so placed. 

7. Reestablishment of quarter-section corners on closing section lines between 
fractional sections.— Must be reestablished according to the original measurement of 
40 chains from the last Interior section corner, or rather that distance corrected by 
proportional measurement of original field notes and the new measurement on 
closing line. 

8. Reestablishment of interior quarter-section corners. — The missing quarter- 
corner (In the later surveys) must be established equidistant between the section 
corners marking the line, according to the field notes of the original survey. 

9. Where double corners were originally established, one of which is standing, to 
reestablish the other. — It being remembered that the corners established when the 
exterior township lines were run, belong to the sections in the townships north and 
west of those lines, the surveyor must first determine beyond a doubt to which sections 
the existing corner belongs. This may be done by testing the corners and distances 
to witness trees or other objects noted in the original field notes of the survey, and 
by remeasuring distances to known corners. Having determined to which township 
the existing corner belongs, the missing corner may be reestablished In line north or 
south of the existing, as the case may be, at the distance stated in the field notes 
of the original survey, by proportionate measurement, and tested by retracement 
to the opposite corresponding corner of the section to which the missing corner 
belongs. 

Subdivision of Sections. 

1. Subdivision of sections into quarter sections. — Run straight lines from the 
established quarter-section corners, U. S. surveys, to the opposite corresponding 
corners. The point of Intersection of these lines will be the legal center of the section. 
(a) Upon the lines closing on the north and west boundaries of a township, the 



GOVERNMENT LAND SURVEYING. 971 

quarter-section corners are established by the U. S. deputy surveyors, but In sub- 
dividing such sections said quarter-corners should be so placed as to suit the calcu- 
lations of the areas of the quarter sections adjoining the township boundaries as 
expressed upon the official plat, adopting proportionate measurements where the 
new measurements of the north and west boundaries of the section differ from the 
original measurements. 

2. Subdivision of fractional sections. — Where opposite corresponding corners have 
not been or cannot be fixed, the subdivision lines should be ascertained by running 
from the established corners due N, S, E or W, as the case may be. to the water course, 
Indian boundary line, or other boundary of such fractional section, (a) The law 
presumes the section lines surveyed and marked in the field by the U. S. deputy sur- 
veyors to be due A^ and S or E and W lines, but in actual experience this is not always 
the case. Hence, In order to carry out the spirit of the law. it will be necessary in 
running the subdivlsional lines through fractional sections to adopt mean courses 
where the section lines are not due lines, or to run the subdivision line parallel to 
the E, S, W, or N boundary of the section, as conditions may require, where there 
Is no opposite sectional line. 

3. Subdivision of quarter sections into quarter-quarters. — Preliminary to the sub- 
division of quarter sections, the quarter-quarter corners will be established at points 
midway between the section and quarter-section corners, and between quarter cor- 
ners and the center of the section, except on the last half mile of the lines closing on 
the north or west boundaries of a township, where they should be placed at 20 chains, 
proportionate measurement, to the north or west of the quarter-section corner, 
(a) The quarter-quarter section corners having been established as directed 
above, the subdivision lines of the quarter section will be run straight between 
opposite corresponding quarter-quarter section corners on the quarter-section 
boundaries. The Intersection of the lines thus run will determine the place for the 
corner common to the lour quarter-quarter sections. 

4. Subdivision of fractional quarter sections. — The subdivision lines of fractlolnal 
quarter sections will be run from properly established quarter-quarter section corners 
(par. 3) due N, S, E or W, to the lake, water course, or reservation which renders 
such tracts fractional, or parallel to the east, south, west, or north boundary of the 
quarter section, as conditions may require. (See par. 2 (a).) 

5. Proportionate measurement. — By "proportionate measurement" Is meant a 
measurement having the same ratio to that recorded In the original field notes as 
the length of chain used In the new measurement has to the length of chain used In 
the original survey, assuming that the original and new measurements have been 
correctly made. For example: the length of the line from the quarter-section corner 
on the west side of sec. 2, T. 24 N., R. 14 E, Wisconsin, to the north line of the town- 
ship, by the United States deputy surveyor's chain, was reported as 45.40 chains, 
and by the county surveyor's measure is reported as 42.90 chains; then the distance 
which the quarter-quarter section corner should be located north of the quarter- 
section corner would be determined as follows As 45,40 chains, the Government 
measure of the whole distance, is to 42.90 chains, the county surveyor's measure of 
the same distance, so Is 20.00 chains, original measurement, to 18.90 chains by the 
county surveyor's measure, showing that by proportionate measurement In this case 
the quarter-quarter section corner should be set at 18.90 chains north of the quarter- 
section corner. Instead of 20.00 chains north of such corner, as represented on the 
official plat. In this manner the discrepancies between original and new measure- 
ments are equitably distributed. 

Useful Tables in Public Lands Surveys. 
(From the Manual of 1902.) 

The system of rectangular surveying, authorized by law May 20, 1785 
(see p. 967), was first employed in the survey of U. S. public lands in 
the State of Ohio. 

The boundary line between the States of Penn. and Ohio, known as 
"Ellicott's line," in longitude 80° 32' 20" west from Greenwich, is the 
meridian to which the first surveys are referred. The townships east of 
the Scioto R., in Ohio, are numbered from south to north, commencing 
with No. 1 on the Ohio River, while the ranges are numbered from east to 
west, beginning with No. 1 on the east boundary of the State, except in the 
tract designated "U. S. Military Land," in which the townships and ranges 
are numbered, respectively, from the south and east boundaries of said tract. 

Since 1875, numbered and locally-named principal meridians and base 
lines have been established as shown by Table 8, following. 



972 



5S.— SURVEYING, MAPPING AND LEVELING, 



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PUBLIC LAND SURVEYS— TABLES. 



973 



9. — ^Azimuths of the Secant, and Offsets, in Feet, to the Parallel. 

Arguments: latitude in left-hand column, and distance from starting 

point at top of table. 

(For example of use of table, see Fig. 32.) 









Azimuths and Offsets at — 






Deflect 'n 

Angle 
and nat. 
tan. to 
Rad.66ft 




miles. 


i mile. 


1 mile. 


U miles. 


2 miles. 


2i miles. 


3 miles. 


30 


89° 58'. 5 
1.93 N. 


89° 58' 7 
0.87 N. 


89° 59'.0 
0.00 


89° 59'. 2 
0.67 S. 


89° 59'.5 
1.15 S. 


89° 59'.7 
1.44 S. 


90°(E.orW.) 
1.54 S. 


3' 00". 2 
0.69 ins. 


31 


89° 58'. 4 
2.01 N. 


89° 58'.6 
0.91 N. 


89° 58'.9 
0.00 


89° 59' 2 
0.70 S. 


89° 59'.5 
1.20 S. 


89» 59'.7 
1.50 S. 


90°(E.orW.) 
1.60 S. 


3' 07". 4 
0.72 ins. 


32 


89° 58'.4 
2.09 N. 


89° 58'.6 
0.94 N. 


89° 58'. 9 
0.00 


89° 59'.2 
0.73 S. 


89« 59'.5 
1.25 S. 


89'* 59'.7 
1.56 S. 


90°(E. orW.) 
1.67 S. 


3' 15".0 
0.75 ins. 


Z3 


89° 58'.3 
2.17 N. 


89° 58'. 5 
0.97 N. 


89° 58'.8 
0.00 


89° 59'.1 
0.76 S. 


89° 54' 4 
1.30 S. 


89° 59' 7 
1.62 S. 


90°(E. orW.) 
1.73 S. 


3' 22".6 
0.78 ins. 


34 


89° 58' .2 
2.25 N. 


89° 58'.5 
1.01 N. 


89° 58'.8 
0.00 


89° 59'.1 
0.79 S. 


89° 59'.4 
1.35 S. 


89° 59'.7 
1.69 S. 


90°(E. orW.) 
1.80 S. 


3' 30",4 
0.81 ins. 


35 


89° 58'.2 
2.33 N. 


89° 58'.5 
1.05 N. 


89® 58'. 8 
0.00 


89° 59'.1 
0.82 S. 


89° 59'.4 
1.40 S. 


89" 59'.7 
1.75 S. 


90° (E. orW.) 
1.87 S. 


3' 38". 4 
0.84 ins. 


36 


89° 58'.1 
2.42 N. 


89° 58'. 4 
1.09 N. 


89° 58'.7 
0.00 


89° 59'. 
0.85 S. 


89" 59'.4 
1.46 S. 


89« 59' 7 
1.82 S. 


90°(E.orW.) 
1.94 S. 


3' 46".4 
0.87 ins. 


37 


89° 58'.0 
2.51 N. 


89° 58'.3 
1.13 N. 


89° 58'.6 
0.00 


89° 58-.9 
0.88 S. 


89* 59' 3 
1.51 S. 


89° 59' 7 
1.89 S. 


90°(E.orW.) 
2.01 S. 


3' 55". 
0.90 ins. 


38 


89° 58'. 
2.61 N. 


89° 58'. 3 
1.17 N. 


89° 58'.6 
0.00 


89° 58'. 9 
0.91 S. 


89'* 5 9 '.3 
1.56 S. 


89° 59'.7 
1.95 S. 


90°(E.orW.) 
2.08. S. 


4 03". 6 
0.93 ins. 


39 


89° 57'. 9 
2.70 N. 


89° 58'.2 
1.21 N. 


89° 58'. 6 
0.00 


89° 58' 9 
0.94 S. 


89° 59'.3 
1.62 S. 


89° 59'.7 
2.02 S. 


90°(E.orW.) 
2.16 S. 


4' 12". 6 
0.97 ins. 


40 


89° 57'.8 
2.79 N. 


89° 58'. 1 
1.25 N. 


89° 58'. 5 
0.00 


89° 58'. 9 
0.98 S. 


89° 59'.3 
1.68 S. 


89° 59'.7 
2.10 S. 


90°(E.orW.) 
2.24 S. 


4'. 21". 6 
1.00 ins. 


41 


89° 57'. 7 
2.89 N. 


89° 58'. 
1.30 N. 


89° 58'.4 
0.00 


89° 58' 8 
1.02 S. 


89° 59'.2 
1.74 S. 


89° 59'.6 
2.17 S. 


90°(E.orW.) 
2.32 S. 


4' 31". 2 
1.04 ins. 


42 


89° 57'.7 
3.00 N. 


89° 58'.0 
1.35 N. 


89° 58'. 4 
0.00 


89° 58'.8 
1.05 S. 


89° 59'.2 
1.80 S. 


89° 59'.6 
2.25 S. 


90° (E. or W ) 
2.40 S. 


4' 40". 8 
1.08 ins 


43 


89° 57'.6 
3.11 N. 


89° 58'.0 
1.40 N. 


89° 58'.4 
0.00 


89° 58'.8 
1.08 S. 


89° 59'.2 
1.86 S. 


89° 59'.6 
2.33 S. 


90°(E. orW.) 
2.48 S. 


4' 50". 8 
1.12 ins. 


44 


89° 57'.5 
3.22 N. 


89° 57'. 9 
1.45 N. 


89° 58'. 3 
0.00 


89° 58'.7 
1.12 S. 


89° 59'.2 
1.93 S. 


89° 59'.6 
2.41 S. 


90°(E.orW.) 
2.57 S. 


5' 01".0 
1.16 ins. 


45 


89° 57'.4 
3.33 N. 


89° 57'.8 
1.50 N. 


89° 58'.3 
0.00 


89° 58'.7 
1.16 S. 


89° 59'.1 
2.00 S. 


89° 59'.5 
2.49 S. 


90°(E.orW.) 
2.66 S. 


5' 11".8 
1.20 ins. 


46 


89° 57^3 

3.44 N. 


89° 57'.7 
1.55 N. 


89° 58'.2 
0.00 


89° 58'.6 
1.21 S. 


89° 59'. 1 
2.07 S. 


"89° 59'. 5 
2.59 S. 


90°(E.orW.) 
2.76 S. 


5'. 2 2". 8 
1.24 ins. 


47 


89° 57'.2 
3.57 N. 


89° 57'.6 
1.61 N. 


89° 58'. 1 
0.00 


89° 58'. 6 
1.25 S. 


89° 59'. 1 
2.14 S. 


89° 59'.5 
2.67 S. 


90° (E. or W.) 
2.86 S. 


5' 34".2 
1.28 ins. 


48 


89° 57'.1 
3.70 N. 


89° 57'. 5 
1.66 N. 


89° 58'.0 
0.00 


89° 58'.5 
1.30 S. 


89° 59'.0 
2,22 S. 


89° 59'.5 
2.78 S. 


90° (E. or W.) 
2,96 S. 


5' 46". 2 
1.33 ins. 


49 


89° 57'.0 
3.82 N. 


89° 57'. 5 
1.72 N. 


89° 58'. 
0.00 


89° 58'. 5 
1.34 S. 


89° 59'.0 
2.30 S. 


89° 59'.5 
2.87 S. 


90°(E. orW.) 
3.06 S. 


5' 58". 6 
1.38 ins. 


50 


89° 56'.9 
3.96 N. 


89° 57'. 4 
1.78 N. 


89° 57'. 9 
0.00 


89° 58'.4 
1.39 S. 


89° 59'.0 
2.38 S. 


89° 59'.5 
2.97 S. 


90° (E.orW.) 
3.17 S. 


6' 11". 4 
1.43 ins. 




'^•'3N.,R.2IE. 



/ %y^*rd Standard Parallel Nor+b,^^ - 



N.89h'E. Secant Line S.SSh'E. 



-* — 
Zm. 




Regular 



Qf-fse+s pjg 32.— Example. 



974 



f^.— SURVEYING, MAPPING AND LEVELING. 



10. — Azimuths of the Tangent to the Parallel. 

The azimuth is the smaller angle the tangent makes with the true meridian 

and always measured from the north and towards 

the tangential points. 



Lati- 
tude. 


1 mile. 


2 miles. 


3 miles. 


4 miles. 


5 miles. 


6 miles. 





o 


/ 


» 


o 


/ 


/r 


o 


/ 


It 


o 


/ 


/r 


o 


/ 


» 


o 


/ • 


30 
31 
32 


89 
89 
89 


59 
59 
59 


30.0 
28.8 
27.5 


89 
89 
89 


58 
58 
58 


59.9 
57.5 
55.0 


89 

89 
89 


58 
58 
58 


29.9 
26.3 
22.5 


89 
89 
89 


57 
57 
57 


59.9 
55.0 
50.0 


89 57 
89 57 
89 57 


29.9 
23.8 
17.5 


89 
89 
89 


56 59.8 
56 52.5 
56 45.0 


33 
34 
35 


89 
89 
89 


59 
59 
59 


26.2 
24.9 
23.6 


89 
89 
89 


58 
58 
58 


52.5 
49.9 
47.2 


89 
89 
89 


58 
58 
58 


18.7 
14.8 
10.8 


89 
89 
89 


57 
57 
57 


44.9 
39.7 
34.4 


89 
89 
89 


57 
57 
56 


11.2 
04.6 
58.0 


89 

89 
89 


56 37.4 
56 29.6 
56 21.6 


36 

37 

38 


89 
89 
89 


59 
59 
59 


22.2 
20.8 
19.4 


89 
89 
89 


58 
58 
58 


44.4 
41.6 
38.8 


89 
89 
89 


58 
58 
57 


06.8 
02.5 
58.2 


89 
89 
89 


57 
57 
57 


28.9 
23.3 
17.5 


89 
89 
89 


56 
56 
56 


51.1 
44.1 
36.9 


89 
89 
89 


56 13.4 
56 05.0 
55 56.3 


39 
40 
41 


89 
89 
89 


59 
59 
59 


17.9 
16.4 
14.8 


89 

89 
89 


58 

58 
58 


35.8 
32.8 
29.6 


89 

89 
89 


57 

57 
57 


53.7 
49.2 
44.4 


89 
89 
89 


57 
57 
56 


11.6 
05.5 
59.3 


89 
89 
89 


56 
56 
56 


29.6 
21.9 
14.1 


89 
89 
89 


55 47.5 
55 38.3 
55 28.S 


42 
43 
44 


89 
89 
89 


59 
59 
59 


13.2 
11.5 
09.8 


89 
89 
89 


58 
58 
58 


26.4 
23.1 
19.6 


89 
89 
89 


57 
57 
57 


39.6 
34.6 
29.5 


89 
89 
89 


56 
56 
56 


52.8 
46.2 
39.3 


89 
89 
89 


56 
55 
55 


06.0 
57.7 
49.1 


89 
89 
89 


55 19.2 
55 09.2 
54 58.9 


45 
46 
47 


89 
89 
89 


59 
59 
59 


08.0 
06.2 
04.3 


89 
89 
89 


58 
58 
58 


16.1 
12.4 
08.6 


89 
89 
89 


57 
57 
57 


24.1 
18.6 
12.9 


89 
89 
89 


56 
56 
56 


32.1 
24.8 
17.1 


89 
89 
89 


55 
55 
55 


40.2 
31.0 
21.4 


89 
89 
89 


54 48.2 
54 37.2 
54 25.7 


48 
49 
50 


89 
89 
89 


59 
59 
58 


02.3 
00.2 
58.1 


89 
89 
89 


58 
56 
57 


04.6 
00.5 
56.2 


89 
89 
89 


57 
57 
56 


06.9 
00.7 
54.3 


89 
89 
89 


56 
56 
55 


09.2 
00.9 
52.6 


89 
89 
89 


55 
55 
54 


11.5' 
01.2 
50.5 


89 
89 
89 


54 13.8 
54 01.4 
53 48.5 



Lati- 
tude. 


7 miles. 


8 miles. 


9 miles. 


10 miles. 


11 


m 


les. 


12 miles. 


o 


o 


/ 


n 


o 


/ 


It 


o 


» 


ti 


o 


/ 


n 





/ 


H 


o 


, m 


30 
31 
32 


89 
89 
89 


56 
56 
56 


29.8 
21.3 
12.5 


89 
89 
89 


55 
55 
55 


59.8 
50.0 
40.0 


89 
89 
89 


55 
55 
55 


29.8 
18.8 
07.6 


89 
89 
89 


54 
54 
54 


59.7 
47.6 
35.1 


89 
89 
89 


54 
54 
54 


29.7 
16.3 
02.6 


89 
89 
89 


53 59.7 
53 45.1 
53 30.1 


33 
34 
35 


89 

89 
89 


56 
55 
55 


03.6 
54.5 
45.2 


89 
89 
89 


55 
55 
55 


29.9 
19.4 
08.8 


89 
89 
89 


54 
54 
54 


56.1 
44.4 
32.3 


89 
89 
89 


54 
54 
53 


22.3 
09.3 
55.9 


89 
89 
89 


53 
53 
53 


48.5 
34.2 
19.5 


89 
89 
89 


53 14.8 
52 59.1 
52 43.1 


36 
37 

38 


89 
89 
89 


55 
55 
55 


35.6 
25.8 
15.7 


89 
89 
89 


54 
54 
54 


57.8 
46.6 
35.1 


89 
89 
89 


54 
54 
53 


20.0 
07.4 
54.5 


89 
89 
89 


53 
53 
53 


42.3 

28.2 
13.9 


89 
89 
89 


53 
52 
52 


04.5 
49.1 
33.2 


89 
89 
89 


52 26.7 
52 09.9 
51 52.6 


39 
40 
41 


89 
89 
89 


55 

54 
54 


05.4 
54.7 
43.7 


89 
89 
89 


54 
54 
53 


23.3 
11.1 
58.5 


89 
89 
89 


53 
53 
53 


41.2 
27.5 
13.4 


89 
89 
89 


52 

52 
52 


59.1 
43.8 
28.2 


89 
89 
89 


52 
52 
51 


17.0 
00.2 
43.0 


89 
89 
89 


51 34.9 
51 16.6 
50 57.8 


42 
43 
44 


89 
89 
89 


54 
54 
54 


32.4 
20.8 
08.7 


89 
89 
89 


53 
53 
53 


45.6 
32.3 
18.5 


89 
89 
89 


52 
52 
52 


58.8 
43.8 
28.4 


89 
89 
89 


52 
51 
51 


12.0 
55.4 
38.2 


89 
89 
89 


51 
51 
50 


25.2 
06.9 
48.0 


89 
89 
89 


50 38.4 
50 18.5 
49 57.8 


45 
46 
47 


89 
89 
89 


53 
53 
53 


56.3 
43.4 
30.0 


89 
89 
89 


53 
52 
52 


04.3 
49.5 
34.3 


89 
89 
89 


52 
51 
51 


12.3 

55.7 
38.6 


89 
89 
89 


51 
51 
50 


20.4 
01.9 
42.9 


89 
89 
89 


50 
50 
49 


28.4 
08.1 
47.2 


89 
89 
89 


49 36.4 
49 14.3 
48 51.4 


48 
49 
50 


89 
89 
89 


53 
53 
52 


16.1 
01.7 
46.6 


89 
89 
89 


52 
52 
51 


18.4 
01.9 
44.7 


89 

89 
89 


51 
51 
50 


20.7 
02.1 
42.8 


89 
89 
89 


50 
50 
49 


23.0 
02.4 
40.9 


89 
89 
89 


49 
49 
48 


25.3 
02.6 
39.0 


89 
89 
89 


48 27.6 
48 02.8 
47 37.1 



Note. — For example of use of table, see i'^ig 33^, next page. 



PUBLIC LAND SURVEYS— TABLES. 



975 



11, — Offsets, in Chains, from Tangent to Parallel. 
[Chains.] 















Miles 














Lat- 
























itude. 


























Deg. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


11 


12 


30 


0.006 


0.023 


0.053 


0.09 


0.14 


0.21 


0.29 


0.37 


0.47 


0.58 


0.71 


0.84 


31 


0.006 


0.024 


0.055 


0.10 


0.15 


0.22 


0.30 


0.39 


0.49 


0.60 


0.74 


0.88 


32 


0.006 


0.025 


0.057 


0.10 


0.16 


0.23 


0.31 


0.40 


0.51 


0.63 


0.76 


0.91 


33 


0.007 


0.026 


0.059 


0.10 


0.16 


0.24 


0.32 


0.42 


0.53 


0.65 


0.79 


0.95 


34 


0.007 


0.027 


0.061 


0.11 


0.17 


0.25 


0.33 


0.43 


0.55 


0.68 


0.82 


0.98 


35 


0.007 


0.028 


0.064 


0.11 


0.18 


0.25 


0.35 


0.45 


0.57 


0.70 


0.86 


1.02 


36 


0.007 


0.029 


0.066 


0.12 


0.18 


0.26 


0.36 


0.47 


0.59 


0.73 


0,89 


1.06 


37 


0.008 


0.031 


0.068 


0.12 


0.19 


0.27 


0.37 


0.48 


0.61 


0.75 


0.91 


1.10 


38 


0.008 


0.032 


0.071 


0.13 


0.20 


0.28 


0.38 


0.50 


0.64 


0.78 


0.95 


1.14 


39 


0.008 


0.033 


0.074 


0.13 


0.20 


0.29 


0.40 


0.52 


0.66 


0.81 


0.99 


1.18 


40 


0.008 


0.034 


0.076 


0.13 


0.21 


0.30 


0.41 


0.54 


0.68 


0.84 


1.02 


1 22 


41 


0.009 


0.035 


0.079 


0.14 


0.22 


0.32 


0.43 


0.56 


0.70 


0.87 


1.06 


1.26 


42 


0.009 


0.036 


0.082 


0.14 


0.23 


0.33 


0.44 


0.58 


0.73 


0.90 


1.09 


1.31 


43 


0.009 


0.038 


0.085 


0.15 


0.24 


0.34 


0.46 


0.60 


0.75 


0.93 


1.14 


1.35 


44 


0.010 


0.039 


0.088 


0.16 


0.24 


0.35 


0.48 


0.62 


0.79 


0.97 


1.18 


1.40 


45 


0.010 


0.040 


0.091 


0.16 


0.25 


0.36 


0.49 


0.64 


0.81 


1.00 


1.22 


1.45 


46 


0.010 


0.042 


0.094 


0.17 


0.26 


0.37 


0.51 


0.66 


0.84 


1.04 


1.26 


1.50 


47 


0.011 


0.044 


0.097 


0.17 


0.27 


0.39 


0.53 


0.68 


0.87 


1.07 


1.31 


1.56 


48 


0.011 


0.045 


0.101 


0.18 


0.28 


0.40 


0.55 


0.71 


0.91 


1.12 


1.35 


1.61 


49 


0.012 


0.046 


0.104 


0.19 


0.29 


0.42 


0.57 


0.74 


0.93 


1.16 


1.40 


1.67 


50 


0.012 


0.048 


0.108 


0.19 


0.30 


0.43 


0.59 


0.77 


0.97 


1.20 


1.45 


1.73 



Note. — For use of above table, see example below (Fig. 33). Note 
that in the table the offsets are in chains, and not in feet. 



»i: 



^. 



V5^ 



^.^' 



^t 




T.\3N.,K.'^^^ 



•s^ 



^\J:fj&'i.^^ 



r>cV^ 






I*' <eiO> ,^»'T o\ c>J, rr'. 









:-^.f»>; 









c; NO ui '^ 

CJ «3 -: «^ 



us 



V^^ 



Vf)> 



Fig. 33. — Example of use of Tables 10 and 11, for 
latitude 45 deg. 34.5 m. N. 



976 



&&.— SURVEYING, MAPPING AND LEVELING. 



12. — Correction op Randoms. 
Links, and Minutes of Arc, showing departure in running 80.00 chs, at any 
course from 1 to 60 minutes (or difference in latitude for 90° minus 
angle) . 



An- 


De- 


An- 


De- 


An- 


De- 


An- 


De- 


An- 


De- 


An- 


De- 
part- 
ure 


gle. 


parture 


gle. 


parture 


gle. 


parture 


gle. 


parture 


gle. 


parture 


gle. 


MIn. 


Links. 


Min. 


Links. 


Min. 


Links. 


Min. 


Links. 


Min. 


Links. 


Min. 


Links. 


1 


2* 


11 


25§ 


21 


49 


31 


72* 


41 


95f 


51 


119 


2 


4f 


12 


28 


22 


mi 


32 


74f 


42 


98 


52 


121* 


3 


7 


13 


30i 


23 


531 


33 


77 


43 


100^ 


53 


123f 


4 


H 


14 


32f 


24 


56 


34 


79* 


44 


102- 


54 


126 


5 


llf 


15 


35 


25 


581 


35 


81f 


45 


105 


55 


128J 


6 


14 


16 


37^ 


26 


60t 


36 


. 84 


46 


107* 


56 


130§ 


7 


16i 


17 


39f 


27 


63 


37 


86* 


47 


109f 


57 


133 


8 


18f 


18 


42 


28 


65i 


38 


88f 


48 


112 


58 


135* 


9 


21 


19 


m 


29 


671 


39 


91 


49 


114* 


59 


137f 


10 


231 


20 


m 


30 


70 


40 


93* 


50 


116f 


60 


140 



Remarks. — ^Table 12, showing the departure or falling at 80 chains 
distance, for any number of minutes up to 60, can also be used in finding 
the minutes of correction of a random course corresponding to the number 
of links of falling. For distance less than 1 mile, the links of falling must be 
proportionately increased; for example, if the falling at 70 chains is 28 
links, the correction of the course will be 14 minutes for 32 links. For 
township exteriors and other long lines, the number of links of falling must 
be divided by the number of miles to bring the calculation to the basis of 
the table. 

Table 12 may be used to determine the return from the random course, 
also, by keeping in mind clearly just what is being done. 



PUBLIC LAND SURVEYS— TABLES. 



977 



13. CONVERGENCY OP MERIDIANS SIX MILES LONG AND SIX MILES 

APART, AND OTHER RELEVANT DATA, TO LATITUDE 70° NORTH. 





Convergency. 


Difference of longitude 


Difference 


of latitude 








per range. 


for— 


Latitude. 














On the 














parallel. 


Angle. 


In arc. 


In time. 


1 mile in arc. 


1 Tp. in arc. 


o 


Links. 


/ „ 


/ If 


Seconds. 






30 


41.9 


3 


6 0.36 


24.02 






31 


43.6 


3 7 


6 4.02 


24.27 






32 


45.4 


3 15 


6 7.93 


24.53 


0'871 


5'.225 


33 


47.2 


3 23 


6 12.00 


24.80 






34 


49.1 


3 30 


6 16.31 


25.09 






35 


50.9 


3 38 


6 20.95 


25.40 






36 


52.7 


3 46 


6 25.60 


25.71 






37 


54.7 


3 55 


6 30.59 


26.04 


0'.870 


5'. 221 


38 


56.8 


4 4 


6 35.81 


26.39 






39 


58.8 


4 13 


6 41.34 


26.76 






40 


60.9 


4 22 


6 47.13 


27.14 






41 


63.1 


4 31 


6 53.22 


27.55 






42 


65.4 


4 41 


6 59.62 


27.97 


0'.869 


5'.217 


43 


67.7 


4 51 


7 6.27 


28.42 






44 


70.1 


5 1 


7 13.44 


28.90 






45 


72.6 


5 12 


7 20.93 


29.39 






4G 


75.2 


5 23 


7 28.81 


29.92 






4 7 


77.8 


5 34 


7 37.10 


30.47 


0'.869 


5'.212 


48 


80.6 


5 46 


7 45.79 


31.05 






49 


83.5 


5 59 


7 55.12 


31.67 






50 


86.4 


6 12 


8 4.83 


32.32 






51 


89.6 


6 25 


8 15.17 


33.01 






52 


92.8 


6 39 


8 26.13 


33.74 


0'.868 


5'. 2 07 


53 


96.2 


6 54 


8 37.75 


34.52 






54 


99.8 


7 9 


8 50.07 


35.34 






55 


103.5 


7 25 


9 3.18 


36.22 






56 


107.5 


7 42 


9 17.12 


37.14 






57 


111.6 


8 


9 31.97 


38.13 


• 0'.867 


5'.202 


58 


116.0 


8 19 


9 47.83 


39.19 






59 


120.6 


8 38 


10 4.78 


40.32 






60 


125.5 


8 59 


10 22.94 


41.52 






61 


130.8 


9 22 


10 42.42 


42.83 






62 


136.3 


9 46 


11 3.38 


44.22 


0'.866 


5M98 


63 


142.2 


10 11 


11 25.97 


45.73 






64 


148.6 


10 38 


11 50.37 


47.36 


J 




65 


155.0 


11 8 


12 16.82 


49.12 






66 


162.8 


11 39 


12 45.55 


51.04 






67 


170.7 


12 13 


13 16.88 


53.12 


0'.866 


5'. 195 


68 


179.3 


12 51 


13 51.15 


55.41 






69 


188.7 


13 31 


14 28.77 


57.92 






70 


199.1 


14 15 


15 10.26 


60.68 


0'.866 


5'. 193 



Remarks on Table 13. 

The second column of Table 13 contains the convergency of two 
meridians six miles long and six miles apart, measured on a parallel of 
latitude. When the parallel of latitude passing through the south end of 
such meridians, and forming the south boundary of the township of which 
the meridians form the meridional boundaries, is coincident with a tabular 
latitude given in the first column, the required convergency will be ob- 
tained directly from the second column (see Fig. 34) ; while for other than 
the tabular latitudes, it will be obtained by simple proportion (Fig. 35). 
The third column contains the angle of convergency. (abc. Figs. 34 and 35.) 

For the purpose of computing convergency within the boundaries of a 
regular township, said boundaries may be regarded as straight lines and the 
township a plane figure, generally a trapezoid; the convergency of any 
rectangular part thereof, bounded by meridional and latitudinal section 
jines, will be determined as follows: Multiply the convergency for the 
township, determined as above directed, by the length of the tract in miles 



978 



5S.— SURVEYING, MAPPING AND LEVELING. 



and decimals of a mile, divided by 6, and the product by the width of the 
tract divided by 6; the resulting product will be the convergency required. 
(See Fig. 34.) 

To obtain the convergency of the meridional boundaries of any tract 
bounded by section lines, or other lines of legal subdivision, within a town- 
ship, proceed as follows: Divide the tract into the least possible number of 
rectangular parts and compute the convergency for each tract; then, take 
the sum of the convergencies thus determined. (See example, Fig. 36). The 
convergency of two meridians lii equal length, in the same latitude, is pro 
portional to their distance apart; e. g., the convergency of two meridians 
6 miles long, separated by 5 ranges, latitude 38°, is 56.8 Iks. X 5 = 2.84 
chains. 

Convergency of meridians in the same latitudes, and not exceeding 24 
miles in length, may be computed by an approximate proportion, which 
combines the advantages of convenience with an accuracy sufficient for the 
ordinary wants of the land surveyor; the proportion is this: The cosines 
of the latitudes are to each other as the lengths of the intercepted parallels. 
The following example illustrates the use of this rule: 

The distance between the Principal Meridian and first range line west, 
in latitude 42° 39' 07", is 6 miles; what is the convergency of the two range 
lines at the Base Line, the meridional distance being 24 miles? 

cos 42° 39'07'' : cos 43° :: 480.00 chs : 477.31 chs., which proportion may 
be worked with natural cosines, or more expeditiously by logarithms, as 



follows: 



a. c. log cos 
log cos 
log 

log 



42° 39' 
43° 
480.00 

477.30 



07" 



0.133427 
9.864127 
2.681241 

2.678795 



The difference 2.70 chs. is the convergency required. 



ac = Convergency on the Parallel. 
abc = Angle of Convergency. 

Table 13; opposite Latitude 44°, will be found 70.1 
links, the convergency. 

North Boundary = 480.00 -0.70 = 479. 30 chs. 

For Convergency of the meridians sh and /g, we 
have: 

70.1X1X1=17.15 Links, as in the text. 




Required the Convergency for a Township in Lat. 

38° 29' N. 
From Table 13: 

Convergency in Latitude 38°, = 56.8 Links; 
39°, = 58.8 " 



Difference = 2.0 



b £ast48a00chs. - , 
LatM'Nortk 

Fig. 34. 
Ea$f 478.28c/fs: 



Also; 29' = 0°.48; 

then, 0°. 48X2. 0=0. 96 links; 

ac= 56.8 + 0.96=57.76 links, the convergency required. 

North boundary = 478.86- 0.58*= 478.28 chains. 

* Taken to nearest whole link. 
Tabular Convergency, is 80.6 links. 

Convergency for the tract abcdefgh: S06/As 
Conv. ofA; 80.6X^X1 = 22.39 links; 

" " B; 80.6XgXi = 13.42 " 

" " C; 80.6XiX§= 8.96 " 



b. East 478.86 cha. 



ll 



Fig« 36. 



Latitude 
36-89'Norfh. 



Convergency Required =44.77 



Also; 

Conv. for E. tract is 
.. S.W. " " 

Total convergency 



17.92 " 
17.91 " 

80.60 links, for Township. 



a' ' ■ ■ b 




A 




h 


L 




- 


B 


. 


■ 


-d-J 


. 




C 




, 


f , i 





Latitude 
46' North. 



Fig. 36. 



LENGTH OF A DEGREE OF LATITUDE. 



979 



14. — ^-Length op a Degree- of Latitude Computed to Minutes. 



t 


29° 


30° 


31° 


32° 


33° 


34° 


35° 


36° 


37° 


38° 


i 


t 


Chains 


Chains 


Chains 


Chains 


Chains 


Chains 


Chains 


Chains 


Chains 


Chains 


t 





5509.15 


5509.97 


5510.82 


5511.67 


5512.55 


5513.44 


5514.34 


5515.25 


5516.18 


5517.11 





1 


09.16 


09.99 


10.83 


11.69 


12.56 


13.45 


14.35 


15.27 


16.19 


17.13 


1 


2 


09.17 


10.00 


10.84 


11.70 


12.58 


13.47 


14.37 


15.28 


16.21 


17.14 


2 


3 


09.19 


10.01 


10.86 


11.72 


12.59 


13.48 


14.38 


15.30 


16.22 


17.16 


3 


4 


09.20 


10.03 


10.87 


11.73 


12.61 


13.50 


14.40 


15.31 


16.24 


17.17 


4 


5 


09.21 


10.04 


10.89 


11.75 


12.62 


13.51 


14.42 


15.33 


16.25 


17.19 


5 


6 


09.23 


10.06 


10.90 


11.76 


12.64 


13 53 


14.43 


15.34 


16.27 


17.20 


6 


7 


09.24 


10.07 


10.91 


11.78 


12.65 


13.54 


14.45 


15.36 


16.28 


17.22 


7 


8 


09.25 


10.08 


10.93 


11.79 


12.67 


13.56 


14.46 


15.38 


16.30 


17.23 


8 


9 


09.27 


10.10 


10.94 


11.81 


12.68 


13.57 


14.48 


15.39 


16.32 


17.25 


9 


10 


09.28 


10.11 


10.96 


11.82 


12.70 


13.59 


14.49 


15.41 


16.33 


17.27 


10 


11 


09.30 


10.13 


10.97 


11.83 


12.71 


13.60 


14.51 


15.42 


16.35 


17.28 


11 


12 


09.31 


10.14 


10.99 


11.85 


12.73 


13.62 


14.52 


15.44 


16.36 


17.30 


12 


13 


09.32 


10.15 


11.00 


11.86 


12.74 


13.63 


14.54 


15.45 


16.38 


17.31 


13 


14 


09.34 


10.17 


11.01 


11.88 


12.76 


13.65 


14.55 


15.47 


16.39 


17.33 


14 


15 


09.35 


10.18 


11.03 


11.89 


12.77 


13.66 


14.57 


15.48 


16.41 


17.34 


15 


16 


09.36 


10.19 


11.04 


11.91 


12.79 


13.68 


14.58 


15.50 


16.42 


17.36 


16 


17 


09.38 


10.21 


11.06 


11.92 


12.80 


13.69 


14.60 


15.51 


16.44 


17.38 


17 


18 


09.39 


10.22 


11.07 


11.94 


12.81 


13.71 


14.61 


15.53 


16.46 


17.39 


18 


19 


09.41 


10.24 


11.09 


11.95 


12.83 


13.72 


14.63 


15.54 


16.47 


17.41 


19 


20 


09.42 


10,25 


11.10 


11.96 


12.84 


13.74 


14.64 


15.56 


16.49 


17.42 


20 


21 


09.43 


10.26 


11.11 


11.98 


12.86 


13.75 


14.66 


15.57 


16.50 


17.44 


21 


22 


09.45 


10.28 


11.13 


11.99 


12.87 


13.77 


14.67 


15.59 


16.52 


17.45 


22 


23 


09.46 


10.29 


11.14 


12.01 


12.89 


13.78 


14.69 


15.61 


16.53 


17.47 


23 


24 


09.47 


10.31 


11.16 


12.02 


12.90 


13.80 


14.70 


15.62 


16.55 


17.49 


24 


25 


09.49 


10.32 


11.17 


12.04 


12.92 


13.81 


14.72 


15.64 


16.56 


17.50 


25 


26 


09.50 


10.33 


11.19 


12.05 


12.93 


13.83 


14.73 


15.65 


16.58 


17.52 


26 


27 


09.51 


10.35 


11.20 


12.07 


12.95 


13.84 


14.75 


15.67 


16.60 


17.53 


27 


28 


09.53 


10.36 


11.21 


12.08 


12.96 


13.86 


14.76 


15.68 


16.61 


17.55 


28 


29 


09.54 


10.38 


11.23 


12.10 


12.98 


13.87 


14.78 


15.70 


16.63 


17.56 


29 


30 


09.56 


10.39 


11.24 


12.11 


12.99 


13.89 


14.79 


15.71 


16.64 


17.58 


30 


31 


09.57 


10.41 


11.26 


12.12 


13.01 


13.90 


14.81 


15.73 


16.66 


17.60 


31 


32 


09.58 


10.42 


11.27 


12.14 


13.02 


13.92 


14.82 


15.74 


16.67 


17.61 


32 


33 


09.60 


10.44 


11.29 


12.15 


13.04 


13.93 


14.84 


15.76 


16.69 


17.63 


33 


34 


09.61 


10.45 


11.30 


12.17 


13.05 


13.95 


14.86 


15.77 


16.70 


17.64 


34 


35 


09.63 


10.46 


11.31 


12.18 


13.07 


13.96 


14.87 


15.79 


16.72 


17.66 


35 


36 


09.64 


10.48 


11.33 


12.20 


13.08 


13.98 


14.89 


15.81 


16.74 


17.67 


36 


37 


09.65 


10.49 


11.34 


12.21 


13.10 


13.99 


14.90 


15.82 


16.75 


17.69 


37 


38 


09.67 


10.50 


11.36 


12.22 


13.11 


14.01 


14.92 


15.84 


16.77 


17.71 


38 


39 


09.68 


10.52 


11.37 


12.24 


13.13 


14.02 


14.93 


15.85 


16.78 


17.72 


39 


40 


09.69 


10.53 


11.39 


12.26 


13.14 


14.04 


14.95 


15.87 


16.80 


17.74 


40 


41 


09.71 


10.55 


11.40 


12.27 


13.16 


14.05 


14.96 


15.88 


16.81 


17.75 


41 


42 


09.72 


10.56 


11.42 


12.29 


13.17 


14.07 


14.98 


15.90 


16.83 


17.77 


42 


43 


09.74 


10.57 


11.43 


12.30 


13.18 


14.08 


14.99 


15.91 


16.84 


17.78 


43 


44 


09.75 


10.59 


11.44 


12.31 


13.20 


14.10 


15.01 


15.93 


16.86 


17.80 


44 


45 


09.76 


10.60 


11.46 


12.33 


13.21 


, 14.11 


15.02 


15.94 


16.88 


17.82 


45 


46 


09.78 


10.62 


11.47 


12.34 


13.23 


14.13 


15.04 


15.96 


16.89 


18.83 


46 


47 


09.79 


10.63 


11.49 


12.36 


13.24 


14.14 


15.05 


15.98 


16.91 


17.85 


47 


48 


09.80 


10.65 


11.50 


12.37 


13.26 


14.16 


15.07 


15.99 


16.92 


17.86 


48 


49 


09.82 


10.66 


11.52 


12.39 


13.27 


14.17 


15.08 


16.01 


16.94 


17.88 


49 


50 


09.83 


10.67 


11.53 


12.40 


13.29 


14.19 


15.10 


16.02 


16 95 


17.89 


50 


51 


09.85 


10.69 


11.54 


12.42 


13.30 


14.20 


15.11 


16.04 


16.97 


17.91 


51 


52 


09.86 


10.70 


11.56 


12.43 


13.32 


14.22 


15.13 


16.05 


16.98 


17.93 


52 


53 


09.87 


10.72 


11.57 


12.45 


13.33 


14.23 


15.15 


16.07 


17.00 


17.94 


53 


54 


09.89 


10.73 


11.59 


12.46 


13.35 


14.25 


15.16 


16.08 


17.02 


17.96 


54 


55 


09.90 


10.74 


11.60 


12.48 


13.36 


14.26 


15.18 


16.10 


17.03 


17.97 


55 


56 


09.92 


10.76 


11.62 


12.49 


13.38 


14.28 


15.19 


16.11 


17.05 


17.99 


56 


57 


09.93 


10.77 


11.63 


12.51 


13 39 


14.29 


15.21 


16.13 


17.06 


18.00 


57 


58 


09.94 


10.79 


11.65 


12.52 


13.41 


14.31 


15.22 


16.15 


17.08 


18.02 


58 


59 


09.96 


10.80 


11.66 


12.53 


13.42 


14.32 


15.24 


16.16 


17.09 


18.04 


59 


60 


5509.97 


5510.82 


5511.67 


5512.55 


5513.44 


5514.34 


5515.25 


5516.18 


5517.11 


5518.05 


60 



980 58.— SURVEYING, MAPPING AND LEVELING. 

14. — Length of a Degree of Latitude. — Concluded. 



+5 


39° 


40° 


41° 


42° 


43° 


44° 


45° 


46° 


47° 


48° 


_i 


/ 


Chains 


Chains 


Chains 


Chains 


Chains 


Chains 


Chnins 


Chains 


Chains 


Chains 







5518.05 


5519.00 


5519.96 


5520.92 


5521.88 


5522 85 


5523 81 


5524.78 


5525.75 


5526 72 





1 


18.07 


19.02 


19.97 


20.93 


21.90 


22.86 


23.83 


24. 80 


25.77 


26.73 


1 


2 


18.08 


19.03 


19.99 


20.95 


21 91 


22.88 


23.85 


24.82 


25.78 


26 75 


2 


3 


18.10 


19.05 


20.00 


20.96 


21.93 


22.89 


23.86 


24.83 


25.80 


26.76 


3 


4 


18.11 


19.06 


20.02 


20.98 


21.94 


22.91 


23.88 


24.85 


25.82 


26 78 


4 


5 


18.13 


19.08 


20.04 


21.00 


21.96 


22 93 


23.90 


24.86 


25 83 


26 80 


5 


6 


18.15 


19.10 


20.05 


21.01 


21 98 


22.94 


23.91 


24 88 


25.85 


26 81 


6 


7 


18.16 


19.11 


20.07 


21.03 


21.99 


22.96 


23.93 


24.90 


25.86 


26.83 


7 


8 


18.18 


19.13 


20.08 


21.04 


22.01 


22.98 


23.94 


24.91 


25 88 


26.84 


8 


9 


18.19 


19.14 


20.10 


21.06 


22.02 


22.99 


23.96 


24.93 


25.90 


26.86 


9 


10 


18.21 


19.16 


20.12 


21.08 


22.04 


23.01 


23.98 


24.94 


25.91 


26.88 


10 


11 


18.22 


19.18 


20.13 


21.09 


22.06 


23.02 


23.99 


24.96 


25 93 


26.89 


11 


12 


18.24 


19.19 


20.15 


21.11 


22.07 


23.04 


24.01 


24.98 


25 94 


26 91 


12 


13 


18.26 


19.21 


20.16 


21.12 


22 09 


23.06 


24.02 


24.99 


25.96 


26.92 


13 


14 


18.27 


19.22 


20.18 


21.14 


22.11 


23.07 


'24.04 


25.01 


25.98 


26.94 


14 


15 


18.29 


19.24 


20.20 


21.16 


22.12 


23.09 


24.06 


25.03 


25.99 


26 96 


15 


16 


18.30 


19.25 


20.21 


21.17 


22.14 


23 10 


24.07 


25.04 


26.01 


26 97 


16 


17 


18.32 


19.27 


20.23 


21 19 


22.15 


23.12 


24.09 


25.06 


26.02 


26.99 


17 


18 


18.34 


19.29 


20.24 


21.20 


22.17 


23.14 


24.11 


25.07 


26 04 


27.00 


18 


19 


18.35 


19.30 


20.26 


21.22 


22.19 


23 15 


24.12 


25.09 


26.06 


27.02 


19 


20 


18.37 


19.32 


20.28 


21.24 


22.20 


23.17 


24.14 


25.11 


26.07 


27.04 


20 


21 


18.38 


19.33 


20.29 


21.25 


22 22 


23 19 


24.15 


25.12 


26.09 


27 05 


21 


22 


18.40 


19.35 


20.31 


21.27 


22.23 


23.20 


24.17 


25.14 


26.10 


27 07 


22 


23 


18.41 


19.37 


20.32 


21.29 


22.25 


23.22 


24.19 


25.15 


26.12 


27.09 


23 


24 


18.43 


19.38 


20.34 


21.30 


22.27 


23.23 


24.20 


25.17 


26 14 


27.10 


24 


25 


18.45 


19.40 


20.36 


21.32 


22.28 


23.25 


24.22 


25.19 


26.15 


27.12 


25 


26 


18.46 


19.41 


20.37 


21.33 


22.30 


23.27 


24.23 


25 20 


26.17 


27.13 


26 


27 


18.48 


19.43 


20.39 


21.35 


22.31 


23.28 


24.25 


25.22 


26 19 


27.15 


27 


28 


18.49 


19.45 


20.40 


21.36 


22.33 


23.30 


24.27 


25.23 


26 20 


27.17 


28 


29 


18.51 


19.46 


20.42 


21.38 


22.35 


23.31 


24.28 


25.25 


26.22 


27.18 


29 


30 


18.53 


19.48 


20.44 


21.40 


22.36 


23.33 


24.30 


25.27 


26 23 


27.20 


30 


31 


18.54 


19.49 


20.45 


21.41 


22.38 


23.35 


24.32 


25.28 


26.25 


27.21 


31 


32 


18.56 


19.51 


20.47 


21.43 


22.40 


23.36 


24.33 


25.30 


26.27 


27.23 


32 


33 


18.57 


19.53 


20.48 


21.45 


22.41 


23.38 


24.35 


25.32 


26.28 


27.25 


33 


34 


18.59 


19.54 


20.50 


21.46 


22.43 


23.40 


24.36 


25.33 


26.30 


27.26 


34 


35 


18.60 


19.56 


20.52 


21.48 


22.44 


23.41 


24.38 


25.35 


26 31 


27.28 


35 


36 


18.62 


19.57 


20.53 


21.49 


22.46 


23.43 


24 40 


25.36 


26 33 


27.29 


36 


37 


18.64 


19.59 


20.55 


21.51 


22.48 


23.44 


24.41 


25.38 


26.35 


27.31 


37 


38 


18.65 


19.60 


20.56 


21.53 


22.49 


23.46 


24 43 


25.40 


26 36 


27.33 


38 


39 


18.67 


19.62 


20.58 


21.54 


22.51 


23.48 


24.44 


25.41 


26^38 


27.34 


39 


40 


18.68 


19.64 


20.60 


21.56 


22.52 


23.49 


24.46 


25.43 


26.39 


27.36 


40 


41 


18.70 


19.65 


20.61 


21.57 


22.54 


23.51 


24.48 


25.44 


26 41 


27.37 


41 


42 


18.72 


19.67 


20.63 


21.59 


22.56 


23.52 


24.49 


25.46 


26.43 


27.39 


42 


43 


18.73 


19.68 


20.64 


21.61 


22.57 


23.54 


24.51 


25.48 


26.44 


27.41 


43 


44 


18.75 


19.70 


20.66 


21.62 


22.59 


23.56 


24.52 


25.49 


26 46 


27.42 


44 


45 


18.76 


19.72 


20.68 


21.64 


22.6a 


23.57 


24.54 


25 51 


26 47 


27.44 


45 


46 


18.78 


19.73 


20.69 


21.65 


22.62 


23.59 


24.56 


25.52 


26.49 


27.45 


46 


47 


18.79 


19.75 


20.71 


21.67 


22.64 


23.60 


24.57 


25.54 


26.51 


27.47 


47 


48 


18.81 


19.76 


20.72 


21.69 


22.65 


23.62 


24.59 


25 56 


26.52 


27.49 


48 


49 


18.83 


19.78 


20.74 


21.70 


22.67 


23.64 


24.61 


25.57 


26.54 


27.50 


49 


50 


18.84 


19.80 


20.76 


21.72 


22 69 


23.65 


24.62 


25.59 


26.56 


27.52 


50 


51 


18.86 


19.81 


20.77 


21.74 


22.70 


23.67 


24.64 


25.61 


26.57 


47.53 


51 


52 


18.87 


19 83 


20.79 


21.75 


22.72 


25^69 


24 65 


25 62 


26.59 


27.55 


52 


53 


18.89 


19.84 


20.80 


21.77 


22.73 


23 70 


24-67 


25 64 


26.60 


27.57 


53 


54 


18.91 


19.86 


20.82 


21.78 


22.75 


23.72 


24.69 


25-65 


26.62 


27.58 


54 


55 


18.92 


19.88 


20.84 


21.80 


22.77 


23.73 


24 70 


25 67 


26.64 


27.60 


55 


56 


18.94 


19.89 


20.85 


21.82 


22.78 


23.75 


24.72 


25 69 


25.65 


27.61 


56 


57 


18.95 


19 91 


20.87 


21.83 


22.80 


23.77 


24.73 


25 70 


26-67 


27.63 


57 


58 


18.97 


19.92 


20.88 


21.85 


22.81 


23.78 


24.75 


25-72 


26 68 


27 65 


58 


59 


18.98 


19.94 


20.90 


21.86 


22.83 


23.80 


24.77 


25.73 


26.70 


27 66 


59 


60 


5519.00 


5519.96 


5520. 92 


5521.88 


5522.85 


5523.81 


5524.78 


5525.75 


5526.72 


5527.68 


60 



LENGTH OF A DEGREE OF LONGITUDE. 



981 



15. — -Length op a Degree of Longitude Computed to Minutes. 



t 


29° 


30° 


31° 


32° 


33» 


34° 


35° 


36° 


37° 


38° 


1 


/ 


Chains 


Chains 


Chains 


Chains 


Chains 


Chains 


Chains 


Chains 


Chains 


Chains 


/ 





4843.17 


4795.82 


4747.01 


4696.75 


4645.06 


4591.96 


4537.45 


4481.56 


4424.29 


4365.68 





1 


42.40 


95.02 


46.19 


95.90 


44.19 


91.06 


36.53 


80.61 


23.33 


64.69 


1 


2 


41.62 


94.22 


45.36 


95.05 


43.32 


90.16 


35.61 


79.67 


22.36 


63.70 


2 


3 


40.84 


93.42 


44. 53 


94.20 


42.44 


89.26 


34.69 


78.73 


21.40 


62.72 


3 


4 


40.06 


92.61 


43.71 


93.35 


41.57 


88.37 


33.77 


77.78 


20.43 


61.73 


4 


5 


39.28 


91.81 


42.88 


92.50 


40.69 


87.47 


32.84 


76.84 


19.46 


60.74 


5 


6 


38.50 


91.01 


42.05 


91.^5 


39.82 


86.57 


31.92 


75.89 


18.49 


59.75 


6 


7 


37.72 


90.20 


41.22 


90.80 


38.94 


85.67 


31.00 


74.95 


17.53 


58.76 


7 


8 


36.94 


89.40 


40.39 


89.94 


38.06 


84.77 


30.08 


74.00 


16.56 


57.77 


8 


9 


36.16 


88.59 


39.56 


89.09 


37.19 


83.87 


29.15 


73.05 


15.59 


56.77 


9 


10 


35.38 


87.79 


38.73 


88.24 


36.31 


82.97 


28.23 


72.11 


14.62 


55.78 


10 


11 


34.60 


86.98 


37.90 


87.38 


35.43 


82.07 


27.30 


71.16 


13.65 


54.79 


11 


12 


33.82 


86.18 


37.07 


86.53 


34.55 


81.17 


26.38 


70.21 


12.68 


53.80 


12 


13 


33.04 


85.37 


36.24 


85.67 


33.68 


80.26 


25.46 


69.26 


11.71 


72.81 


13 


14 


32.26 


84.56 


35.41 


84.82 


32.80 


79.36 


24.53 


68.32 


10.74 


51.81 


14 


15 


31.47 


83.76 


34.58 


83.96 


31.92 


78.46 


23.60 


67.37 


09.77 


50.82 


15 


16 


30.69 


82.95 


33.75 


83.11 


31.04 


77.56 


22.68 


66.42 


08.80 


49.83 


16 


17 


29.91 


82.14 


32.92 


82.25 


30.16 


76.65 


21.75 


65.47 


07.82 


48.83 


17 


18 


29.12 


81.33 


32.08 


81.40 


29.28 


75.75 


20.83 


64.52 


06.85 


47.84 


18 


19 


28.34 


80.52 


31.25 


80.54 


28.40 


74.85 


19.90 


63.57 


05.88 


46.84 


19 


20 


27.55 


79.71 


30.42 


79.68 


27.52 


73.94 


18.97 


62.62 


04.91 


45.85 


20 


21 


26.77 


78.90 


29.58 


78.82 


26.64 


73.04 


18.04 


61.67 


03.93 


44.85 


21 


22 


25.98 


78.09 


28.75 


77.97 


25.75 


72.13 


17.11 


60.72 


02.96 


43.85 


22 


23 


25.20 


77.28 


27.92 


77.11 


24.87 


71.23 


16.19 


59.77 


01.98 


42.86 


23 


24 


24.41 


76.47 


27.08 


76.25 


23.99 


70.32 


15.26 


58.81 


01.01 


41.86 


24 


25 


23.62 


75.66 


26.25 


75.39 


23.11 


69.41 


14.33 


57.86 


4400.04 


40.86 


25 


26 


22.83 


74.85 


25.41 


74.53 


22.22 


68.51 


13.40 


56.91 


4399.06 


39.87 


26 


27 


22.05 


74.04 


24.57 


73.67 


21.34 


67.60 


12.47 


55.96 


98.08 


38.87 


27 


28 


21.26 


73.22 


23.74 


72.81 


20.45 


66.69 


11.54 


55.00 


97.11 


37.87 


28 


29 


20.47 


72.41 


22.90 


71.95 


19.57 


65.78 


10 61 


54.05 


96.13 


36.87 


29 


30 


19.68 


71.60 


22.06 


71.09 


18.69 


64.88 


09.67 


53.09 


95.16 


35.87 


30 


31 


18.89 


70.78 


21.22 


70.22 


17.80 


63.97 


08.74 


52.14 


94.18 


34.87 


31 


32 


18.10 


69.97 


20.39 


69.36 


16.91 


63.06 


07.81 


51.19 


93.20 


33.87 


32 


33 


17.31 


69.16 


19.55 


68.50 


16.03 


62.15 


06.88 


50.23 


92.22 


32.87 


Zi 


34 


16.52 


68.34 


18.71 


67.64 


15.14 


61.24 


05.94 


49.27 


91.25 


31.87 


34 


35 


15.73 


67.53 


17.87 


66.77 


14.26 


60.33 


05.01 


48.32 


90.27 


30.87 


35 


36 


14.94 


66.71 


17.03 


65.91 


13.37 


59.42 


04.08 


47.36 


89.29 


29.87 


36 


37 


14.15 


65.89 


16.19 


65.05 


12.48 


58.51 


03.14 


46.41 


88.31 


28.87 


37 


38 


13.35 


65.08 


15.35 


64.18 


11.59 


57.60 


02.21 


45.45 


87.33 


27.87 


38 


39 


12.56 


64.26 


14.51 


63.32 


10.70 


56.68 


01.28 


44.49 


86.35 


26.87 


39 


40 


11.77 


63.44 


13.67 


62.45 


09.81 


55.77 


4500.34 


43.53 


85.37 


25.86 


40 


41 


10.98 


62.52 


12.82 


61.59 


08.93 


54.86 


4499.40 


42.57 


84.39 


24.86 


41 


42 


10.18 


61.81 


11.98 


60.72 


08.04 


53.95 


98.47 


41.62 


83.41 


23.86 


42 


43 


09.39 


60.99 


11.14 


59.85 


07.15 


53.03 


97.53 


40.66 


82.42 


22.85 


43 


44 


08.59 


60.17 


10.30 


58.99 


06.26 


52.12 


96.59 


39.70 


81.44 


21.85 


44 


45 


07.80 


59.35 


09.45 


58.12 


05.36 


51.21 


95.66 


38.74 


80.46 


20.85 


45 


46 


07.00 


58.53 


08.61 


57.25 


04.47 


50.29 


94.72 


37.78 


79.48 


19.84 


46 


47 


06.21 


57.71 


07.76 


56.38 


03.58 


49.38 


93.78 


36.82 


78.49 


18.84 


47 


48 


05.41 


56.89 


06.92 


55.51 


02.69 


48.46 


92.84 


35.86 


77.51 


17.83 


48 


49 


04.61 


56.07 


06.07 


54.65 


01.80 


47.55 


91.91 


34.89 


76.53 


16.82 


49 


50 


03.82 


55.25 


05.23 


53.78 


00.90 


46.63 


90.97 


33.93 


75.54 


15.82 


SO 


51 


03.02 


54.43 


04.38 


52.91 


4600.01 


45.71 


90.03 


32.97 


74.56 


14.81 


51 


52 


02.22 


53.60 


03.54 


52.04 


4599.12 


44.80 


89.09 


32.01 


73.57 


13.80 


52 


53 


01.42 


52.78 


02.69 


51.17 


98.22 


43.88 


88.15 


31.04 


72.59 


12.80 


53 


54 


4800.62 


51.96 


01.84 


50.30 


97.33 


42.96 


87.21 


30.08 


71.60 


11.79 


54 


55 


4799.82 


51.13 


01.00 


49.42 


96.44 


42.04 


86.27 


29.12 


70.62 


10.78 


55 


56 


99.02 


50.31 


4700.15 


48.55 


95.54 


41.13 


85.32 


28.15 


69.63 


09.77 


56 


57 


98.22 


49.49 


4699.30 


47.68 


94.64 


40.21 


84.38 


27.19 


68.64 


08.76 


57 


58 


97.42 


48.66 


98.45 


46.81 


93.75 


39.29 


83.44 


26.22 


67.66 


07.75 


58 


59 


96.62 


47.84 


97.60 


45.94 


92.85 


38.37 


82.50 


25.26 


66.67 


06.74 


59 


60 


4795.82^ 


1747.01 


4696.75 


4645.06 


4591.96 


4537.45 


4481.56 


4424.29 


4365.68 


4305.73 


60 



982 



58.— SURVEYING, MAPPING AND LEVELING. 



15. — ^Length of a Degree of Longitude. — Concluded. 





39*^ 


40° 


41° 


42° 


43° 


44° 


45° 


46° 


47° 


48° 


i 


/ 


Chains 


Chains 


Chains 


Chains 


Chains 


Chains 


Chains 


Chains 


Chains 


Chains 


f 





4305.73 


4244.47 


4181.91 


4118.06 


4052.96 


3986.62 


3919.05 


3850.28 


3780.33 


3709.22 





1 


04.72 


43.44 


80.85 


16.99 


51.87 


85.50 


17.91 


49.12 


79.15 


08.03 


1 


2 


03.71 


42.41 


79.80 


15.91 


50.77 


84.38 


16.78 


47.97 


77.98 


06.83 


2 


3 


02.70 


41.37 


78.75 


14.84 


49.67 


83.27 


15.64 


46.81 


76.80 


05.63 


3 


4 


01.69 


40.34 


77.69 


13.76 


48.58 


82.15 


14.50 


45.65 


75.63 


04.44 


4 


5 


4300.68 


39.31 


76.64 


12.69 


47.48 


81.03 


13.36 


44.50 


74.45 


03.24 


5 


6 


4299.67 


38.27 


75.58 


11.61 


46.38 


79.91 


13.23 


43.34 


73.27 


02.05 


6 


7 


98.65 


37.24 


74.52 


10.53 


45.28 


78.79 


11.09 


42.18 


72.00 


3700.85 


7 


8 


97.64 


36.20 


73.47 


09.46 


44.19 


77.68 


09.95 


41.02 


70.92 


3699.65 


8 


9 


96.63 


35.17 


72.41 


08.38 


43.09 


76.56 


08.81 


39.86 


69.74 


98.46 


9 


10 


95.61 


34.13 


71.36 


07.30 


41.99 


75.44 


07.67 


38.70 


68.56 


97.26 


10 


11 


94.60 


33.10 


70.30 


06.22 


40.89 


74.32 


06.53 


37.54 


67.38 


96.06 


11 


12 


93.59 


32.06 


69.24 


05.14 


39.79 


73.20 


05.39 


36.38 


66.20 


94.86 


12 


13 


92.57 


31.02 


68.18 


04.07 


38.69 


72.08 


04.25 


35.22 


65.02 


93.66 


13 


14 


91.56 


29.99 


67.12 


02.99 


37.59 


70.96 


03.11 


34.06 


63.84 


92.46 


14 


15 


90.54 


28.95 


66.07 


01.91 


36.49 


69.S4 


01.97 


32.90 


62.66 


91.26 


15 


16 


89.52 


27.91 


65.01 


4100.83 


35.39 


68.72 


3900.83 


31.74 


61.48 


90.60 


16 


17 


88.51 


26.87 


63.95 


4099.75 


34.29 


67.59 


3899.69 


30.58 


60.30 


88.86 


17 


18 


87.49 


26.84 


62.89 


98.67 


33.19 


66.47 


98.54 


29.42 


59.12 


87.66 


18 


19 


86.48 


24.80 


61.83 


97.58 


32.09 


65.35 


97.40 


28.26 


57.94 


86.46 


19 


20 


85.46 


23.76 


60.77 


96.50 


30.98 


64.23 


96.26 


27.09 


56.76 


85.26 


20 


21 


84.44 


22.72 


59.71 


95.42 


29.88 


63.11 


95.12 


25.93 


55.57 


84.06 


21 


22 


83.42 


21.68 


58.65 


94.34 


28.78 


61.98 


93.97 


24.77 


54.39 


82.86 


22 


23 


82.40 


20.64 


57.58 


93.26 


27.67 


60.86 


92.83 


23.60 


53.21 


81.66 


23 


24 


81.39 


19.60 


56.52 


92.17 


26.57 


59.73 


91.68 


22.44 


52.02 


80.46 


24 


25 


80.37 


18.56 


55.46 


91.09 


25.47 


58.61 


90.54 


21.28 


50.84 


79.25 


25 


26 


79.35 


17.52 


54.40 


90.01 


24.36 


57.49 


89.40 


20.11 


49.66 


78.05 


26 


27 


78.33 


16.48 


53.44 


88.92 


23.26 


56.36 


88.25 


18.95 


48.47 


76.85 


27 


28 


77.31 


15.43 


52.27 


87.84 


22.15 


55.24 


87.11 


17.78 


47.29 


75.64 


28 


29 


76.29 


14.39 


51.21 


86.75 


21.05 


54.11 


85.96 


16.62 


46.10 


74.44 


29 


30 


75.27 


13.35 


50.14 


85.67 


19.94 


52.98 


84.81 


15.45 


44.92 


73.24 


30 


31 


74.24 


12.31 


49.08 


84.58 


18.84 


51.86 


83.67 


14.29 


43.73 


72.03 


31 


32 


73.22 


11.26 


48.02 


83.50 


17.73 


50.73 


82.52 


13.12 


42.55 


70.83 


32 


33 


72.20 


10.22 


46.95 


82.41 


16.62 


49.60 


81.37 


11.95 


41.30 


69.62 


33 


34 


71.18 


09.18 


45.89 


81.33 


15.52 


48.48 


80.23 


10.79 


40.18 


68.42 


34 


35 


70.16 


08.13 


44.82 


80.24 


14.41 


47.35 


79.08 


09.62 


38.99 


67.21 


35 


36 


69.13 


07.09 


43.75 


79.15 


13.30 


46.22 


77.93 


08.45 


37.80 


66.01 


36 


37 


68.11 


06.04 


42.69 


78.07 


12.19 


45.09 


76.78 


07.28 


36.62 


64.80 


37 


38 


67.09 


05.00 


41.62 


76.98 


11.09 


43.96 


75.63 


06.11 


35.43 


63.59 


38 


39 


66.06 


03.95 


40.55 


75.89 


09.98 


42.83 


74.48 


04.95 


34.24 


62.39 


39 


40 


65.04 


02.90 


39.49 


74.80 


08.87 


41.71 


73.34 


03.78 


33.05 


61.18 


40 


41 


64.01 


01.86 


38.42 


73.71 


07.76 


40.58 


72.19 


02.61 


31.86 


59.97 


41 


42 


62.99 


4200.81 


37.35 


72.62 


06.65 


39.45 


71.04 


01.44 


30.67 


58.76 


42 


43 


61.96 


4199.76 


36.28 


71.53 


05.54 


38.32 


69.89 


3800.27 


29.48 


57.56 


43 


44 


60.93 


98.72 


35.21 


70.44 


04.43 


37.18 


68.74 


3799.10 


28.30 


56.35 


44 


45 


59.91 


97.67 


34.14 


69.35 


03.32 


36.05 


67.58 


97.93 


27.11 


55.14 


45 


46 


58.88 


96.62 


33.08 


68.26 


02.21 


34.92 


66.43 


96.76 


25.92 


53.93 


46 


47 


57.85 


95.57 


32.01 


67.17 


4001.10 


33.79 


65.28 


95.59 


24.73 


52.72 


47 


V48 


56.83 


94.52 


30.93 


66.08 


3999.98 


32.66 


64.13 


94.41 


23.53 


51.51 


48 


49 


55.80 


93.47 


29.86 


64.99 


98.87 


31.53 


62.98 


93.24 


22.34 


50.30 


49 


50 


54.77 


92.42 


28.79 


63.90 


97.76 


30.39 


61.82 


92.07 


21.15 


49.09 


50 


51 


53.74 


91.37 


27.72 


62.81 


96.65 


29.26 


60.67 


90.90 


19.96 


47.88 


51 


52 


52.71 


90.32 


26.65 


61.71 


95.53 


28.13 


59.52 


89.72 


18.77 


46.67 


52 


53 


51.68 


89.27 


25.58 


60.62 


94.42 


26.99 


58.36 


88.55 


17.58 


45.46 


53 


54 


50.66 


88.22 


24.51 


59.53 


93.31 


25.86 


57.21 


87.38 


16.38 


44.25 


54 


55 


49.63 


87.17 


23.43 


58.43 


92.19 


24.73 


56.06 


86.20 


15.19 


43.03 


55 


56 


48.59 


86.12 


22.36 


57.34 


91.08 


23.59 


54.90 


85.03 


14.00 


41.82 


56 


57 


47.56 


85.07 


21.29 


56.25 


89.96 


22.46 


53.75 


83.86 


12.80 


40.61 


57 


58 


46.53 


84.02 


20.21 


55.15 


88.85 


21.32 


52.59 


82.68 


11.61 


39.40 


58 


59 


45.50 


82.96 


19.14 


54.06 


87.73 


20.19 


51.44 


81.51 


10.41 


38.18 


59 


60 


4244.47 


4181.91 


4118.06 


4052.96 


3986.62 


3919.05 


3850.28 


3780.33 


3709.22 


3636,97 


60 



STADIA SURVEYING. 



The Stadia. — ^Transits may be ordered with stadia, usually without 
extra expense. The stadia is a ring or diaphragm inserted between the eye- 
piece and the center of the telescope, and across which are stretched two 
horizontal wires, called "stadia wires," so spaced that they will subtend a 
distance on a leveling rod equal to 0.01* the distance of the rod from a 
certain point P (where the visual rays cross), a few inches ( = the focal 
length /) in front of the objective. The distance from the center of the 
instrument (intersection of plumb-line with telescope) to the point P for 
any instrument is called the "constant" for that instrument, f This con- 
stant, or its coordinate values if the line of sight is not horizontal, must be 
added to the values reduced from stadia readings. 

The Stadia Reduction Table, following, was calculated by Mr. Arthur 
Winslow for use in the Pennsylvania Geological Survey, and may be used 
in finding the horizontal distance and the elevation of an object located from 
the transit by stadia. 

Example op Use of Table. 

Assume "constant" for instrument == 1.00; 

Stadia reading on rod = 4.26 ft. (rod always vertical); 

(The instrument is set up over a hub whose elevation is known, and 
sighted to a point on rod whose reading, from the middle cross-wire, is 
equal to height of instrument above the hub ; and in this position the stadia 
reading and the vertical angle are taken.) 

Vertical angle = 12°— 19'. Then from Table, by interpolation: 
Horizontal dist. of object from inst. = 95.45X4.26+0.98 = 407.60 ft. 
Vertical dist. of object from hub = 20. 84 X 4.26+ 0.22= 89.00 ft. 

Form of Field Notes. (Left-hand Page.) 
(Right-hand page is for notes and sketches.) 



Dist. 
(Stadia.) 

Instrume 
426 ft. 



Hor. Ang. % 

nt at Sta 84 

318°- 30' 



Vert. Ang. 

(Elev. = 287. 

12°- 19' 



Cor. Dist. 

5). Zero (ho 

407.6 



Diff . in 
Elev. 
r. ang.) on 
89.0 



83. 



Elev. 
376.5 



* Any ratio may be used when ordering; but 0.01 is usual and con- 
venient. The diaphragm may be ordered with the stadia wires either "fixed" 
or "adjustable." 

t Each instrument has its particular "constant," whose value is fur- 
nished by the maker, and may vary from 0.75 to 1.25 ft. The constant is 
equal to .the distance from center of instrument, or axis, to the objective, 
when the instrument is focussed on a distant object, plus the focal distance /, 
or distance from the plane of the cross-wires to the objective. Hence, the 
constant for any instrument may readily be found by focussing to some 
distant object and measuring from the objective to the cross-wires and to 
the center of the axis, and adding these measurements together. This sum 
gives the distance from center of axis to the point P, noted above. 

t Measured right-hand from back-sight [Sta. 83] up to 360**. 



984 



m.— SURVEYING, MAPPING AND LEVELING. 



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II II II 



STADIA REDUCTION TABLE. 



985 



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b-. OS CO 


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CD ^-^ 

II II II 

« v» « 



986 



^.^SURVEYING, MAPPING AND LEVELING. 



k 




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^ ^^^^^ ^^^^^ ^^^^^ ^^^^^ ^^^^^ ^^^^^ 


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an 


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in to 00 
to 000 


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II II 11 

tt v> u 



LEVELING— CURVATURE AND REFRACTION. 987 

Leveling Correction for Earth's Curvature and Refraction. — A level 
"surface" around the earth is a spheroid. Now this spheroid is of the 
exact size and shape of the "mean" earth when every point of its surface is 
at zero elevation, corresponding generally with mean sea level. This is the 
datum spheroidal "plane" used in leveling. Any "plane" above said datum, 
i. e., at a higher elevation, if extended around the globe, will form a spher- 
oidal level surface whose perpendicular distance from the datum plane will 
be constant at points of equal latitude; but will decrease gradually from 
the equator toward the poles. Hence, a meridian line of levels if run at a 
considerable height above the datum plane would be subject to correction; 
but the error is generally so small compared with other errors that it is 
disregarded in practical leveling. 

Ordinary sources of error in leveling are eliminated by taking equal 
back-sights and fore-sights and by using other pre- 
cautions. Those due to curvature of the earth and 
refraction of light rays passing through the atmos- 
phere must be corrected, where a single sight is taken 
on a distant object from a fixed position of the level. 
Let t (Fig. 37) be the point of sight of the telescope 
and let it be required to find the elevation of the 
point p upon which rests a leveling rod hp. Let the Fig. 37. 

point I on the rod be at the same elevation as t; then Ip, the desired height, 
is the height of instrument above p, and the radius of the curve tl is the 
radiiis of the earth. If no atmosphere were present to cause refraction, the 
line of sight would be the horizontal line th, but on account of refraction 
the real line of sight takes the curve tr whose radius is about 7 times the 
radius of the earth; hence hi =7 times hr, or rl^y^hl (nearly). Now as tr 
and tl are short curves of very large radii we may consider them as para- 
bolas and intersecting the leveling rod at angles of 90°. If D represents 
the horizontal distance from instrument to rod, we have, from the nature of 
the problem, that hr, rl and hi are each proportional to D^. Moreover, 
while the correction for curvature adds to the apparent elevation of the 
distant object, the correction for refraction subtracts from this by about ^ 
part. The combined (difference) correction for curvature and refraction is 
additive, as per the following table. 




988 



6&.—SURVEYING, MAPPING AND LEVELING. 



17. — Correction for Earth's Curvature, and Refraction. 
(Add "Curvature and Refraction" to apparent Elevation of object.) 





Curva- 
ture 


Dis- 


Correct 


ion in Feet for — 


Dis- 


Correction in Feet for — 


Dis- 














tance. 


and 


tance 






Curva- 


tance. 






Curva- 


Feet. 


Refrac- 


Miles. 


Curva- 


Refrac- 


■tureand 


Miles. 


Curva- 


Refrac- 


ture and 




tion. 




ture. 


tion. 


Refrac- 
tion. 




ture. 


tion. 


Refrac- 
tion. 


300 


.002 


1 


0.7 


0.1 


0.6 


34 


771.0 


108.0 


663.3 


400 


.003 


2 


2.7 


0.4 


2.3 


35 


817.4 


114.4 


703.0 


500 


.005 


3 


6.0 


0.8 


5.2 










600 


.007 


4 


10.7 


1.5 


9.2 


36 


864.8 


121.1 


743.7 


700 


.010 


5 


16.7 


2.3 


14.4 


37 
38 


913.5 
963.5 


127.9 
134.9 


785.6 
828.6 


800 


.013 


6 


24.0 


3.4 


20.6 


39 


1014.9 


142.1 


872.8 


900 


.017 


7 


32.7 


4.6 


28.1 


40 


1067.6 


149.5 


918.1 


1000 


.020 


8 


42.7 


6.0 


36.7 










1100 


.025 


9 


54.0 


7.6 


46.4 


41 


1121.7 


157.0 


964.7 


1200 


.030 


10 


66.7 


9.3 


57.4 


42 
43 


1177.0 
1233.7 


164.8 
172.7 


1012.2 
1061.0 


1300 


.035 


11 


80.7 


11.3 


69.4 


44 


1291.8 


180.8 


1111.0 


1400 


.040 


12 


96.1 


13.4 


82.7 


45 


1351.2 


189.2 


1162.0 


1500 


.046 


13 


112.8 


15.8 


97.0 










1600 


.052 


14 


130.8 


18.3 


112.5 


46 


1411.9 


197.7 


1412.2 


1700 


.059 


15 


150.1 


21.0 


129.1 


47 
48 


1474.0 
1537.3 


206.3 
215.2 


1267.7 
1322.1 


1800 


.066 


16 


170.8 


23.9 


146.9 


49 


1602.0 


224.3 


1377.7 


1900 


.074 


17 


192 8 


27.0 


165.8 


50 


1668.1 


233.5 


1434.6 


2000 


.082 


18 


216.2 


30.3 


185.9 










2200 


.099 


19 


240.9 


33.7 


207.2 


51 


1735.5 


243.0 


1492.5 


2400 


.118 


20 


266.9 


37.4 


229.5 


52 
53 


1804.2 
1874.3 


252.6 
262.4 


1551.6 
1611.9 


2600 


.139 


21 


294.3 


41.2 


253.1 


54 


1945.7 


272.4 


1673.3 


2800 


.161 


22 


322.9 


45.2 


277.7 


55 


2018.4 


282.6 


1735.8 


3000 


.184 


23 


353.0 


49.4 


303.6 










3200 


.210 


24 


384.3 


53.8 


330.5 


56 


2092.5 


292.9 


1799.6 


3400 


.237 


25 


417.0 


58.4 


358.6 


57 
58 


2167.9 
2244.6 


303.5 
314.2 


1864.4 
1930.4 


3600 


.266 


26 


451.1 


63.1 


388.0 


59 


2322.7 


325.2 


1997.5 


3800 


.296 


27 


486.4 


68.1 


418.3 


60 


2402.1 


336.3 


2065.8 


4000 


.328 


28 


523.1 


73.2 


449.9 










4200 


.362 


29 


561.2 


78.6 


482.6 


61 


2482.8 


347.6 


2135.2 


4400 


.397 


30 


600.5 


84.1 


516.4 


62 
63 


2564.9 
2648.3 


359.1 
370.8 


2205.8 
2277.5 


4600 


.434 


31 


641.2 


89.8 


551.4 


64 


2733.0 


382.6 


2350.4 


4800 


.472 


32 


683.3 


95.7 


587.6 


65 


2819.1 


394.7 


2424.4 


5000 


.512 


33 


726.6 


101.7 


624.9 


66 


2906.5 


406.9 


2499.6 



Ex. — ^The rod reading on an object distant 3200 ft. from the level is 
5.00 ft.; and the height of instrument (H. /.) is 300.00 ft. Find the eleva- 
tion of the object. 

Ans. — ^The apparent elevation is 295.00 ft.; and the true elevation is 
295.21 ft. 



LEVELING— TABLES, 



989 



18. — Allowable Errors in Leveling. 



Distance. 


Distance, 


Allowable 


Distance. 


Distance, 


Allowable 


Miles. 


Feet. 


Error.* 


Miles. 


Feet. 


Error.* 


.1 


528 


.005 


.0189 


100 


.002'2 


.125 


660 


.006 


.04 


200 


.003 


.25 


1320 


.008 


.06 


300 


.004 


.333 


1760 


.009 


.08 


400 


.004 


.5 


2640 


.011 


.09 


500 


.005 


.625 


3300 


,013 


.11 


600 


.005 


.75 


3960 


.014 


.13 


700 


.006 


1 


5280 


.016 


.15 


800 


.006 


1.25 


6600 


.018 


.17 


900 


.007 


1.5 


7920 


.020 


.19 


1000 


.007 


1.75 


9240 


.021 


.23 


1200 


.008 


2 


10560 


.023 


.27 


1400 


.008 


2.5 


13200 


.025 


.30 


1600 


.009 


3 


15840 


.028 


.38 


2000 


.010 


4 


21120 


.032 


.47 


2500 


.011 


5 


26400 


.036 


.57 


3000 


.012 


6 


31680 


.039 


.66 


3500 


.013 


8 


42240 


.045 


.76 


4000 


.014 


10 


52800 


.051 


.85 


4500 


.015 


12 


63360 


.055 


.95 


5000 


.016 


15 


79200 


.062 


1.14 


6000 


.017 


20 


105600 


.072 


1.33 


7000 


.018 


25 


132000 


.080 


1.52 


8000 


.020 


30 


158400 


.088 


1.70 


9000 


.021 


40 


211200 


.101 


1.89 


10000 


.022 


50 


264000 


.113 


2.08 


11000 


.023 


60 


316800 


.124 


2.27 


12000 


.024 


70 


369600 


.134 


2.46 


13000 


.025 


80 


422400 


.143 


2.65 


14000 


.026 


90 


475200 


.152 


2.84 


15000 


.027 


100 


528000 


.160 


3.03 


16000 


.028 


125 


660000 


.179 


3.22 


17000 


.029 


150 


792000 


.196 


3.41 


18000 


.030 


175 


924000 


.212 


3.79 


20000 


.031 


200 


1056000 


.226 


4.17 


22000 


.033 


250 


1320000 


.253 


4.73 


25000 


.035 


300 


1584000 


.277 


5.68 


30000 


.038 


350 


1848000 


.299 


6.63 


35000 


.041 


400 


2112000 


.320 


7.58 


40000 


.044 


450 


2376000 


.339 


8.52 


45000 


.047 


500 


2640000 


.358 


9.47 


50000 


.049 



* Error (in feet) leveling "up and back" must not exceed 
.016\/distance in miles (one way). 



.990 



58.--SURVEYING, MAPPING AND LEVELING, 



EXCERPTS AND REFERENCES. 

The Plane-Table for Small Topographical Surveys (By W. P. Bullock. 
Eng. News, May 29, 1902). — Sketch illustrating use. 

A City Engineer's Card Index of Plans and Notes (By A. H. Pratt. 
Eng. News, April 16, 1903). — Cards illustrated. 

Some Remarkable Records in Taking Soundings Through Ice on 
Lake Superior (By G. A. Taylor. Eng. News, Mar. 24, 1904). — The cost of 
the soundings was 3 cents each for field work alone. 

Boring Sounding Holes Through Ice (Eng. News, May 26, 1904). — 
Illustrated details of ice auger. 

A Rapid Method of Taking Soundings in Shallow Water (By A. E. 
Collins. Eng. News, June 16, 1904.) — Device illustrated. 

Electrical Devices for Deep Borehole Surveying (By H. F. Marriott. 
Eng. News, July 27, 1905). — Includes 19 illustrations. 

Methods of Rod=Holding in Stadia Surveying and Description of a 
New Stadia Slide Rule (By A. L. Bell. Eng. News, Nov. 9, 1905). — Stadia 
formulas ; illustrated . 

Wash Drill Borings: (I) On New York State Barge Canal: (2) On 
the Deep Waterways Surveys ; (3) For the Rapid Transit Commission, N, Y. 
City (Eng. News, Jan. 17, 1907). — Methods and cost data. 

Cost of Earth Auger Borings on the N. Y. State Barge Canal (By 
Emile Low. Eng. News, Mar. 21, 1907). 

Testing Steel Tapes at the National Bureau of Standards (By H. T. 
Wade. Eng. News, Aug. 13, 1908). — Illustrated. 

North=Points for Maps (By A. W. Bedell. Eng. News, Oct. 7, 1909).— 
Over 40 different designs of north points illustrated. 

Description of Four Stadia Surveys and Their Cost (By A. W. Tidd. 
Eng. News, Oct. 21, 1909). — Illustrations: — Typical portion of topographical 
map; Page of stadia notes; Portion of page of traverse notes: Stadia rod; 
Diagram for reading differences of elevation from stadia notes; Diagram for 
reducing stadia readings to distance; Protractor for plotting stadia notes; 
Device for interpolating contours. 

Summary of Costs. 





Detail 
topography. 


Topography for 
5-ft. contours. 




Deens 
Bridge. 


The Hem. 
locks site. 


Lower End 
of basin. 


West 
Branch. 


Area, in acres 


213 
2392 
17990 
5 
27 
7h 
S171 50 
$181.10 
29 
11 
84 

$0.81 

$0.85 


51 
1010 
8600 


1310 

3215 

52350 


200 


Number of Shots 

Length of traverse, in feet 

Number of traverses 


781 

10520 

3 


Number of courses 


25 

8 

$150.40 

$190.40 

6 

20 

168 

$2.94 

$3.74 




62 

16 
$434.40 
$514.40 

82 
2.4 

40 

$0.33 
$0.39 


16 


Days work, Sat.= full day 

Total cost, excl. transportation . 

Total cost, incl. transportation. . 

Acres surveyed per day 

Shots per acre 

Feet of traverse, per acre 

Cost per acre, excl. transporta- 
tion 

Cost per acre, incl. transporta- 
tion 


3 
$77 80 
$107.80 
67 

3.9 
52 

$0.39 

$0.54 







Illustrations. 

Description. 
Concrete boundary monuments, with costs 



Eng. Reo. 
Jan. 1. '10. 



59.— RAILROADS. 

A.— GENERAL DISCUSSION. 

Existing Mileage, etc. — There are (1907) about 212,500 miles of steam 
railroads in operation in the United States, or one mile to each 400 inhabi- 
tants. The total assets of these roads are estimated at $15,500,000,000 or 
$73,000 per mile or $180 per inhabitant; the gross annual earnings, $2,000,- 
000,000 or $9400 per mile or $23 per inhabitant; and the annual operating 
expenses, $1,340,000,000 or $6300 per mile or $16 per inhabitant. There 
are about 7i miles of line per 100 sq. miles of land, equivalent to 13.8 sq. 
miles per mile of line. 

Both the earning capacity and operating expenses of a road are depen- 
dent to a greater or less extent upon its location: the former upon the 
general route selected, and the latter upon the detailed location adopted 
by the engineer. This last factor embraces the two important subjects of 
Gradients and Curvature, and the problems which confront the locating 
engineer are: What shall be their maximum values on a particular line 
or on a particular Division ? and what price shall be paid for their reduction, 
per cent of grade and per degree of curvature? These questions, especially 
the former, also appear constantly during the operation of the road, in the 
ordinary improvements of the line. 

Economic Principles. — Private capital very seldom seeks investment 
unless it "promises" to be a paying one from the start, and railroad building 
offers no exception to this rule. In the design and construction of works of 
public utility from public funds, provision is made for future increase of popula- 
tion, on the grounds of economy. For instance, the estimated population 
20 years hence is generally assumed as a basis for determining the capac- 
ities of reservoirs and sizes of distributing mains for domestic water supply 
systems installed in cities and towns. The reason for this is that they are 
works of quite permanent character and cannot be "enlarged" without 
great expense. Hence we have an illustration of a plant over-efficient for 
a number of years after construction, but economically so because the 
cost of gradually increasing the capacity of the plant would have exceeded 
the interest on the surplus capital invested in the beginning. On the 
other hand, let us take the case of a growing industry such as a manufac- 
turing plant. Provision is here made (wisely) for future expansion of 
business. The site is selected where additional land may be purchased 
when required. Additional buildings are erected also when needed. There 
is but little surplus or idle capital invested in the plant dependent on the 
growth of the business, and furthermore, when additions are made, ma- 
chinery of the latest pattern is installed. Thus the plant is efficient and 
economical at all times. 

Railroad Projection cannot be governed wholly by either of the above 
extreme cases. The first would require too great an outlay at the start 
which would not be warranted by the early receipts of the road, and the 
second would make subsequent improvement too expensive. The New 
York West Shore and Buffalo Railroad is an example of a brand new road 
made to order for future generations, while the Pennsylvania System is 
built up of short lines which served at one time local interests. The first 
named now caters to second-class traffic, while the second has grown strong 
by virtue of its natural location and expansion. On the other hand there 
are roads which have been too much localized. They can never become 
"through" lines unless they reduce their gradients Sind curvature, and the 
"through freight" traffic is usually the best source of revenue. Exceptions 
may be made, however, to some small roads like the Columbia Southern 
and the Central of New Jersey, where the prevailing freight traffic (coal, in 
the latter case) is in one direction — that of the descending grades. 

Gradients, in a line, may be established either (1) from necessity, as 
when the two terminal points to be joined together are at different eleva- 

991 



992 5^.— RAILROADS. 

tions above sea level; (2) to reduce distance, as in passing over a long range 
of hills instead of around it; (3) to reduce the cost of construction incident 
to deep cuttings, high embankments and long tunnels. 

The "ruling" grade on a location is the maximum grade allowable on 
any part of the line, and is sometimes called the "limiting" grade. For 
mountainous sections it is generally fixed at about 2%, more or less, and 
arbitrarily adhered to* by the locating engineers. Sometimes, however, the 
ruling grade is changed after location is begun. If a low mountain pass is 
discovered it may be decreased; if certain unforeseen difficulties are en- 
countered in the topography of the country it may be increased. It is 
always best to have the heavy grades bunched together continuously if 
possible and not scattered throughout the line. By this arrangement they 
may be taken from the class of "limiting" grade, for single engine trains, 
and placed in the class of "pusher" grade, where assistant engines or pushers 
can be operated economically at one point. In this way the lirniting grade 
proper, on the line, may be said to be lowered, which is decidedly advan- 
tageous to the reduced operating expenses of the road. 

The Traction Force of a locomotive is the train resistance which it can 
overcome. It cannot exceed the "adhesion" of the driving wheels to the 
rails; the "adhesion" should not exceed the "cylinder" power of the engine; 
the "cylinder" power should not exceed the "boiler" power. We will 
assume then, that the engine is properly designed: that the cylinder power 
is a little in excess of the adhesion, and that the boiler power is just suffi- 
cient to cause slipping of drivers, when using sand. Then tractive force 
equals adhesion. The adhesion or tractive force may be assumed, for our 
present purpose, at if the total weight on the drivers, that is, this force can 
be exerted horizontally in moving the train. 

The train resistance on a level track comprises rolling friction proper, 
journal friction, reduced effect of traction due to curvature, air resistance, 
etc. We will assume it to be about 8 lbs. % per short ton or V250 the total 
weight of train — engine, tender and cars. Hence, using the same unit of 
weight throughout, 

ry, ^. r Total wt. on drivers Total wt. of train ^ . . ^ .,v 

Traction force == -. = -^f. =Train resistance (1) 

4 lb\j 

Total wt. of train = 62.5 X total wt. on engine drivers (2) 

Gross car loads = 62.5 X total wt. on drivers — wt. of engine and tender. . . (3) 

Net car loads = 62.5 X total wt. on drivers — wt. of engine, tender and 

empty cars (4) 

The effect of an ascending grade on train resistance is calculated easily. 
The level-grade tractive force is simply increased by the total weight of 
trainXthe rate (or %) of the grade\\. Hence, for any ordinary grade and 
using the same unit of weight throughout, we have, from (1), 

rry J.. r Total wt. on drivers o^^i^r^^/i. ^ r 

Tractive force == -. = Total wt. of tram (250 + rate of 

4 
grade) • (5) 

rp ^ , ^ . ^ . Total wt. on drivers ,„. 

Total wt. of tram = „,„ . .^ — ; 7 r- (6) 

.0164- 4 X rate of grade 

n/r J. / frf\ £ J Total wt. on drivers -^. .-. 

Max. rate (or %) of grade = - .^ ^ ^ . — 7 — ft — : .004 (7) 

4 X total wt. of tram 



* Compensation amounting to .04 (or .05) % of grade per degree of 
curve is introduced in order to equalize the tractive resistance, 
t May vary from ^ to i; recent experiments give 0.23 to 0.235. 
% May vary from 4 to 10; 7 to 8 lbs. is usually assumed. 
II This is a slight approximation. The rate (or %) of grade is equal to 

-r or tangent a (Fig. 1), the angle of inclination which 
grade line makes with the horizontal, whereas the 
multiplier should he -r or sine a. For such slight in- 
clinations as railroad grades the error is inappreciable. 
It is on the side of safety. 




RULING GRADE. ENGINE TRACTION. 993 

Problem 1. — An engine with tender weighs 138 tons. The weight on 
the drivers is 80 tons. 

(a). Find total weight of train for a level haul? 
(b). Find total weight of train for haul up a 1% grade? 
(c). Find the ruling grade for a total train weight of 83 3 J tons? 

Solution.— For (a) use (2), (b) use (6), (c) use (7); then, 

(a). Total weight of train = 62.5 X 80= 5000 tons. 

80 
(6). Total weight of train = ^^^^ ^^ =1429 tons. 

80 
(c) . Ruling (maximum) grade = -rg— — .004 = .02, or 2% grade. 

Note. — As the engine and tender weigh 138 tons this amount must be 
deducted from the total weight of train to get the gross-car-loads. To find 
the net freight the weight of empty cars must also be deducted. The 
weight of cattle-, box- and coal cars is about 3*5* their normal capacity; 
gondola- and platform cars, ^. Hence if the above train of 833 tons is a 
cattle train the net freight would = ? (833— 138) = 496-1- tons, or about to 
the total weight of the train for a 2% grade. If we take into consideration 
the weight of the caboose and also that the cars are not all loaded to their 
normal capacity, the ratio of net freight is still further reduced. Moreover, 
the ratio of weight on the drivers, approximately, — 

To total weight of engine and tender, is 1 : 1.725; 

To total weight of train 1 : 10.417; 

To weight of train minus engine and tender 1 : 8.692; 

To net freight (cattle) only, not including wt. of cars 1 : 6.200. 

The following table is calculated from equation (6) and gives the total 
weight of train (including engine and tender) which may be hauled up 
various grades, asstiming the weight on the drivers as unity. 



* That is, weight empty = f weight loaded. 



994 



59,--RAILROADS, 



-Ratio op Total Weight op Train, T, to Weight on Drivers, D, 
FOR Various Grades. 
(Weight on Drivers, D, Assumed as Unity.) 
Calculated from Formula 6, preceding. 



Rate of Grade. 


Total 

Wt. 

of Train. 


Rate of Grade. 


Total 

Wt. 

of Train . 


Rate of Grade. 


Total 
Wt. 














of Train, 


Per 100. 


Ft. per 
Mile. 


T 
D 


Per 100. 


Ft. per 
Mile. 


T 
D 


PerlOO. 


Ft. per 
Mile. 


T 
D 


Level 


Level 


62.50 


1.00 


52.800 


17.86 


2.00 


105.600 


10.42 


0.02 


1.056 


59.52 


1.02 


53.856 


17.61 


2.02 


106.656 


10.33 


0.04 


2.112 


56.82 


1.04 


54.912 


17.36 


2.04 


107.712 


10.25 


0.06 


3.168 


54.35 


1.06 


55.968 


17.12 


2.06 


108.768 


10.16 


0.08 


4.224 


52.08 


1.08 


57.024 


16.89 


2.08 


109.824 


10.08 


0.10 


5.280 


50.00 


1.10 


58.080 


16.67 


2.10 


110.880 


10.00 


0.12 


6.336 


48.08 


1.12 


59.136 


16.45 


2.12 


111.936 


9.92 


0.14 


7.392 


46.30 


1.14 


60.192 


16.23 


2.14 


112.992 


9.84 


0.16 


8.448 


44.64 


1.16 


61.248 


16.03 


2.16 


114.048 


9.77 


0.18 


9.504 


43.10 


1.18 


62.304 


15.82 


2.18 


115.104 


9.69 


0.20 


10.560 


41.67 


1.20 


63.360 


15.63 


2.20 


116.160 


9.62 


0.22 


11.616 


40.32 


1.22 


64.416 


15.43 


2.22 


117.216 


9.54 


0.24 


12.672 


39.06 


1.24 


65.472 


15.24 


2.24 


118.272 


9.47 


0.26 


13.728 


37.88 


1.26 


66.528 


15.06 


2.26 


119.328 


9.40 


0.28 


14.784 


36.76 


1.28 


67.578 


14.88 


2.28 


120.384 


9.33 


0.30 


15.840 


35.71 


1.30 


68.640 


14.71 


2.30 


121.440 


9.26 


0.32 


16.896 


34.72 


1.32 


69.696 


14.53 


2.32 


122.496 


9.19 


0.34 


17.952 


33.78 


1.34 


70.752 


14.37 


2.34 


123.552 


9.12 


0.36 


19.008 


32.89 


1.36 


71.808 


14.20 


2.36 


124.608 


9.06 


0.38 


20.064 


32.05 


1.38 


72.864 


14.04 


2.38 


125.664 


8.99 


0.40 


21.120 


31.25 


1.40 


73.920 


13.89 


2.40 


126.720 


8.93 


0.42 


22.176 


30.49 


1.42 


74.976 


13.74 


2.42 


127.776 


8.87 


0.44 


23.232 


29.76 


1.44 


76.032 


13.59 


2.44 


128.832 


8.80 


0.46 


24.288 


29.07 


1.46 


77.088 


13.44 


2.46 


129.888 


8.74 


0.48 


25.344 


28.41 


1.48 


78.144 


13.30 


2.48 


130.944 


8.68 


0.50 


26.400 


27.78 


1.50 


79.200 


13.16 


2.50 


132.000 


8.62 


0.52 


27.456 


27.18 


1.52 


80.256 


13.02 


2.52 


133.056 


8.56 


0.54 


28.512 


26.60 


1.54 


81.312 


12.89 


2.54 


134.112 


8.50 


0.56 


29.568 


26.04 


1.56 


82.368 


12.76 


2.56 


135.168 


8.45 


0.58 


30.624 


25.51 


1.58 


83.424 


12.63 


2.58 


136.224 


8.39 


0.60 


31.680 


25.00 


1.60 


84.480 


12.50 


2.60 


137.280 


8.33 


0.62 


32.736 


24.51 


1.62 


85.536 


12.38 


2.62 


138.336 


8.28 


0.64 


33.792 


24.04 


1.64 


86.592 


12.25 


2.64 


139.392 


8.22 


0.66 


34.848 


23.58 


1.66 


87.648 


12.14 


2.66 


140.448 


8.17 


0.68 


35.904 


23.15 


1.68 


88.704 


12.02 


2.68 


141.504 


8.12 


0.70 


36.960 


22.73 


1.70 


89.760 


11.90 


2.70 


142.560 


8.06 


0.72 


38 016 


22.32 


1.72 


90.816 


11.79 


2.72 


143.616 


8.01 


0.74 


39.072 


21.93 


1.74 


91.872 


11.68 


2.74 


144.672 


7.96 


0.76 


40.128 


21.55 


1.76 


92.928 


11.57 


2.76 


145.728 


7.91 


0.78 


41.184 


21.19 


1.78 


93.984 


11.47 


2.78 


146.784 


7.86 


0.80 


42.240 


20.83 


1.80 


95.040 


11.36 


2.80 


147.840 


7.81 


0.82 


43.296 


20.49 


1.82 


96.096 


11.26 


2.82 


148.896 


7.76 


0.84 


44.352 


20.16 


1.84 


97.152 


11.16 


2.84 


149.952 


7.71 


0.86 


45.408 


19.84 


1.86 


98.208 


11.06 


2.86 


151.008 


7.67 


88 


46.464 


19.53 


1.88 


99.264 


10.96 


2.88 


152.064 


7.62 


0.90 


47.520 


19.23 


1.90 


100.320 


10.87 


2.90 


153.120 


7.58 


0.92 


48.576 


18.94 


1.92 


101.376 


10.78 


2.92 


154.176 


7.53 


0.94 


49.632 


18.66 


1.94 


102.432 


10.68 


2.94 


155.232 


7.49 


0.96 


50.688 


18.38 


1.96 


103.488 


10.59 


2.96 


156.288 


7.44 


0.98 


51.744 


18.12 


1.98 


104.544 


10.50 


2.98 


157.344 


7.40 


1.00 


52.800 


17.86 


2.00 


105.600 


10.42 


3.00 


158.400 


7.35 



Note. — ^To find the gross weight of train behind the tender'. Multiply the 
weight of the driving wheels by the figures under "Total wt. of Train," for 
the particular grade, and deduct weight of engine and tender. 

For the net freight, multiply this result by f, approximately; but see 
page 993. 



TRACTION ON GRADES. GRADE REDUCTION. 995 

The Allowable Expense for Grade Reduction will now be considered. 
A few hints only can be given and these based on data of very general 
character. Let Fig. 2 represent a modem freight train, in which 



feo 



-r-_A - /7252r~ T kl^^^l^^rp; 



loooojo 



O OP 



Fig. 2. 



D = weight on engine drivers (consolidation type) ; 
L = weight of locomotive and tender =-= 1.725 D\ 

T = weight of train (multiplying values^ in preceding Table by Dj ; 
C = weight of loaded cars, which (see Table 1) = ( ^ - 1.725 j D; 

F = weight of net freight hauled, which = ? (d "" l-'^25j D, approx. 

We will assume that trains are made up and hauled over a Division of 
100 miles, without regard to the nature of the traffic on the balance of the 
road; that there are 1,000,000 tons of freight annually, hauled by 138-ton 
engines with 80 tons on drivers; that the cost per train mile is $1.00; 
and that no pusher engines are used. 



-100 Mile Divis'm 



Fig. 3. 

Ques. — What will be the ruling grade*, on the Division, provided a 
saving of $200,000 can be effected in cost of construction for each 0.1% 
grade above a level grade; the interest value of money being at the rate of 
5 per cent ? 

Ans. — ^We can calculate readily the cost of hauling this freight by 
determining, for various grades, from the preceding discussion: (1) The 

net freight F = f (-^- 1.725 j D, hauled per train; (2) The number of 

trains per year required to haul the 1,000,000 tons; (3) The total cost of 
haul, at $1.00 per train mile. Column (4), in the following table, gives the 
increased cost of the annual haul for any grade over that for a grade 0.1 per 
cent less. Column (5) shows the annual interest on the $200,000 at 5 per cent, 
which equates nearly with $9910 in column (4), opposite a 2% grade, which 
is therefore the required ruling grade. Any other rate of interest than 5 per 
cent would equate differently and give a different ruling grade. Of course 
other considerations naturally affect the problem to a greater or less extent. 

Remarks. — Problems of this character are, in their nature, extremely 
concrete, and therefore cannot be truly represented by merely abstract for- 
mulas, which serve only as guides. For instance, the probable increase in 
future traffic (either immediate or remote) should be taken into account. 
If the future traffic is almost certain to be immediately and largely increased, 
the grade reduction should be proportionately great. Another fact to be 
borne in mind is, that the cost of grade reduction after track is laid is much 
greater than before, and especially so when under heavy traffic. But, as a 
compensating effect, the road is better able to stand this extra expense at 
such a time, both as to available cash and traffic economy. The "Lake 
Shore" road has exp'^nded hundreds of thousands of dollars in reducing 
grades by a small fraction of one per cent for a distance of a few miles, and 
while this work was going on the writer counted on one Sunday, 27 sections 
of a "single freight train." 

* Of course the grade must be assumed to be long so the momentum of 
the train cannot be counted on as affecting the problem. 



996 



.—RAILROADS. 



2. — Cost of Haul on Various Grades. 
And determination of Ruling Grade. 
(See preceding discussion.) 









Total Cost 


Increased 










No. Of 


of Hauling 


Cost of 


Interest 




Grade 


Net Freight 


Trains * 


1,000,000 


Haul 


on 




per 


per Train. 


per Year 


Tons at $1 


over that 


$200,000 


Remarks. 


100. 


Tons. 


Required. 


per 
Train Mile 


for grade 
0.1 lower. 


at 5%. 






(1) 


(2) 


(3) 


(4) 


(5) 




Level 


3472.9 


287.9 


$28,790 









0.1 


2758.6 


362.5 


36.250 


$7,460 






0.2 


2282.6 


438.1 


43.810 


7,560 




ggfei 


0.3 


1942.0 


514.9 


51.490 


7,680 




S f^^ 


0.4 


1687.1 


592.7 


59,270 


7,780 




'^'2+3 


0.5 


1488.9 


671.7 


67,170 


7,900 






0.6 


1330.0 


751.8 


75,190 


8.020 




•§^-Sg 


0.7 


1200.3 


833.1 


83,310 


8.120 




|>;.^5 


0.8 


1091.7 


916.0 


91.600 


8.290 




w§ .-g 


0.9 
1.0 


1000.3 
922.0 


999.7 
. 1084.6 


99.970 
108.460 


8,370 
8.490 






1.1 


854.0 


1171.0 


117,100 


8.640 




^'S |S 


1.2 


794.6 


1258.5 


125,850 


8.750 




ii|zl 


1.3 


742.0 


1347.7 


134,770 


8,920 




1.4 


695.1 


1438.6 


143.860 


9.090 






1.5 


653.4 


1530.4 


153,040 


9,180 




iSfsg 


1.6 


615.7 


1624.1 


162.410 


9.370 




5 .gflS 


1.7 


581.4 


1719.9 


171.990 


9.580 




r=ll 


1.8 


550.6 


1816.3 


181.630 


9.640 




1.9 


522.6 


1913.6 


191.360 


9.730 






2.0 


496.9 


2012.7 


201,270 


9.910 


$10,000 


Ruling Grade. 


2.1 


472.9 


2114.8 


211,480 


10.210 






2.2 


451.7 


2218.0 


221,800 


10.320 






2.3 


430.6 


2322.5 


232.250 


10.450 




» OC CO 


2.4 


411.7 


2428.9 


242.890 


10.640 




... ^ ^ 

1^1 


2.5 


394.0 


2538.1 


253.810 


10.920 




IS- 


2.6 


377.4 


2649.5 


264.950 


11.140 




2.7 


362.0 


2762.4 


276.240 


11.290 




I'^^s^ 


2.8 


347.7 


2875.9 


287.590 


11.350 




g-d "^c^ 


2.9 


334.4 


2991.5 


299.150 


11,560 




5|g> 


3.0 


321.4 


3111.1 


311,110 


11.960 




wo e8 




GRADIENT AND CURVATURE ECONOMICS. 997 

Curvature,* with increased Distance, in a line may arise from four primary 
wonsiderations, namely, (1) to increase the revenue of the road by passing 
through towns not on an air line between terminal points; (2) to reduce cost 
of construction, as by avoiding deep cuts, high fills, long tunnels, expensive 
bridge crossings, etc.; (3) to reduce cost of operation, as by avoiding steep 
grades; and (4) to reduce cost of maintenance, as by choosing a line with a 
permanent roadbed, cheaply maintained, instead of a "structural" line 
involving much expense in repairs and renewals. 

(1). In a new country, thinly settled and without competing roads, 
any departure from an economically located "air" line, to tap a lateral 
region, is justifiable when the "richness" of that region is about proportional 
to the additional cost of reaching it, assuming that the cost per mile of such 
changed line as well as its future growth of business will be, say, proportional 
to that of the whole line. If there is any question on this latter point it 
may better be tapped by a spur. The population of a region is in no wise 
the only safe criterion on which to base probable business. A cattle-raising 
country sparsely populated is very deceptive in this respect. 

(2). We have just considered a case (1) where increased curvature 
(and distance) on a line would be justified by an increase in revenue about 
proportional to the increased cost of the 
line. Let ABC, Fig. 4, be such a line 
giving increased revenue over the direct 
line A aC. We will consider now whether 
some change in location from A aC would 
be justified by reduced cost of construction, Fig. 4. 

of such amount that the annual interest on same would be equal to the in- 
creased net revenue received in case ( 1) ? In the first case the line ABC is con- 
sidered more expensive to construct than the line A aC, while in the present 
case the longer line is cheaper. As a matter of fact there is no relation 
between the two cases, although at first glance there appears to be. In the 
former case we are increasing our business say proportionately to our in- 
vestment by running the line through B. In the latter case we are not 
willing to increase the length of our line to pass through B, but rather 
through some point B' (where there is no business) so that the annual in- 
crease in the operating expenses on the line A B'C over that of A aC shall not 
exceed the interest on the decreased cost of construction. The actual cost of the 
line between A and C is another matter. This should be fixed within certain 
limits, however, having due regard to the amount and quality of traffic 
between 0-0', the terminal centers-of -gravity of haul; and also to the 
quality of the improvements made on those parts of the line A O and C O' . 
If the bulk of traffic is "through" traffic, O and O' may be considered practi- 
cally to" be at the terminal points of the line. 

(3). Let us now suppose that instead of a deep cut or tunnel on the 
line AaC (Fig. 4) as in the preceding case (2), we are confronted with a 
long ridge which will have to be surmounted with heavy grades if the "air" 
line is to be maintained. Here we may resort to the expedient of selecting 
some route 3.s AhC, carrying with it reduced grades and increased curva- 
ture and distance. In fixing the new route we now apply the opposite rule 
to that of case (2): The line should pass through some point b so that the 
annual decrease in the operating expenses on the line Ah C over that of A aC 
should exceed the interest on the increased cost of construction. 

(4). The cost of maintenance should generally be considered as a part 
of the "operating" expenses in the two preceding cases, (2) and (3). But 
the term may have a greater significance apart from such association. Two 
lines having equally objectionable grades may be constructed, (a) following 
the natural contour of the country, and (b) shortening the distance and 
cutting out curvature by the use of bridges and trestles. The difference in 
cost of maintenance and renewals in favor of (a) may outweigh all other 
considerations against it. The increased curvature would have to be con- 
siderable to become a serious matter, as the cost of maintaining curved 
track is but slightly more than that for maintaining straight track. The 
life of ties in curved track is shortened from 3 to 5% annually per degree of 
curvature. The excess cost per train mile is inappreciable for a slightly 
increased length of line (not at all proportional to the length), and it is 
also but slightly affected by the introduction of moderately flat ciirves. 
Steep grades are especially to be avoided. 

* Curvature is used, generally, in this discussion, in its broadest sense as 
including distance. 



998 



'RAILROADS, 



Location of the Line. — ^This comprises three main operations, as follows: 

B. Reconnoissance, or general field inspection. 

C. Preliminary Survey, with instruments. 

D. Location, or final determination of the line. 

Topographical maps of many sections of the country may be had from 
the Government and from the several States. Those of the Geological Sur- 
veys are especially valuable in fixing the general route of the line in the 
reconnoissance and in the subsequent detailed surveys. 



B.— THE RECONNOISSANCE. 

This is the Field Examination necessary in fixing the "critical" points 
on the line, prior to the preliminary survey, in order to reduce the expense 
of the latter. It picks out the low mountain passes, the tunnel locations, 
and the favorable river- and other crossings. If well conducted it may 
dictate also, within close limits, the ruling grade and maximum curvature. 
The principal instruments used are the aneroid barometer, thermometer, 
pedometer (if on foot), cyclometer ( if by wheel), odometer (if by wagon). 
These three latter are used for measuring the distance traveled. A hand 
level will be found useful, also a pocket compass, if detailed information is 
necessary in any particular locality. The thermometer is used in connec- 
tion with the aneroid barometer. The transit with stadia is often used 
advantageously at this early stage. (See Stadia Reduction Table, page 984.) 

The Aneroid Barometer is useful in determining the altitude of any 
point above sea level, or the relative difference in altitude between two or 
more points. It consists of a small, 
circular, air-tight box, in vacuo, with 
one side sensitive to the pressure of the 
atmosphere outside. The heavier the 
atmospheric pressure (the lower the 
altitude) the more it is pressed inward, 
and this movement is multiplied and 
transmitted to a recording index, like 
the hand of a clock, which denotes the 
pressure in inches (on the inner circle) 
corresponding to the mercurial column 
of an ordinary barometer. It is to be 
noted that outside the mercurial scale 
there is also a direct reading scale, to 
hundreds of feet, giving the direct alti- 
tude approximately. All aneroids now 
sold by the best makers are "compen- 
sated" for change or difference. in tem- 
perature (see Fig. 5) ; but this does not 
mean necessarily that they are absolutely 
exact, but merely that they are nearly so and may be used "singly" with some 
degree of accuracy. When, however, extreme accuracy is required in get- 
ting the difference in elevation between two stations — one called the upper 
station and the other the lower station — two compensated aneroids are 
used, one at each station. Readings are taken at the same time and — 

The difference in elevation = (ii? — /?) ( 77^7^7^ | (1) 




ru Tx/QOO + r + A 
= ^^-^H 1000 ) 



1000 

in which H = reading in feet on elevation scale at upper station; 

/j = reading in feet on elevation scale at lower station; 
r = temperature in degrees Fahrenheit at upper station; 

^ = temperature in degrees Fahrenheit at lower station. 
If the temperature at both stations is 50°F. it is seen that H — h represents 
the true difference in elevation, and there is no correction for temperature. 
Of course the aneroids will have to be compared by taking observations 
together at some station and the difference or index error noted for subse- 
quent observations. The author has seen it stated by some of our 
prominent writers that the smaller aneroids of If to 2^ inches diameter give 
as accurate results as the larger ones. The author's observations are to the 
contrary and are confirmed by the following from Messrs. Keuffel & Esser, 
New York City: "All our aneroids are compensated ind their readings do 
not require any further corrections than the one referred to [formula (1) 



RECONNOISSANCE SURVEY, BAROMETER. 



999 



preceding]. It is our judgment that the small pocket aneroids are less accu- 
rate than the larger ones; not only are the dials of the larger instruments more 
closely graduated and permit of finer reading, but the larger instruments 
are also more sensitive on account of the larger size of their vacuum boxes. 
Furthermore, the instrumental error arising from the elastic reaction of the 
counter spring and rocker is greatly reduced." 

The Mercurial Barometer is seldom used in railroad reconnoissance, 
having been supplanted by the aneroid, previously described. The 
aneroid is much more convenient to carry; there is no correction for lati- 
tude nor for variation in gravity due to altitude above the earth. For 
extremely accurate work, however, in establishing absolute elevations above 
sea level, the mercurial barometer is used. The following Tables are from 
Appendix 10, Report U. S. Coast and Geodetic Survey for 1881, 



3. — Barometric Elevations for Temperature 50** F. 
(For use with Mercurial Barometer.) 
Note. — For temperatures other than 50°, see Correction Table, No. 4. 
[Elevation in Feet above sea level.] 





Height 
























of 


.0 


.1 


.2 


.3 


.4 


.5 


,6 


.7 


.8 


.9 


Rem. 


Barom. 
























Ins. 
























11 


27336 


27090 


26846 


26604 


26364 


26126 


25890 


25656 


25424 


25194 




12 


24966 


24740 


24516 


24294 


24073 


23854 


23637 


23421 


23207 


22995 




13 


22785 


22576 


22368 


22162 


21958 


21757 


21557 


21358 


21160 


20962 


^ • 


U 


20765 


20570 


20377 


20186 


19997 


19809 


19623 


19437 


19252 


19068 


•«i3 


15 


18886 


18705 


18525 


18346 


18168 


17992 


17817 


17643 


17470 


17298 


B 


16 


17127 


16958 


16789 


16621 


16454 


16288 


16124 


15961 


15798 


15636 


1 


17 


15476 


15316 


15157 


14999 


14842 


14686 


14531 


14377 


14223 


14070 


18 


13918 


13767 


13617 


13468 


13319 


13172 


13025 


12879 


12733 


12589 


60 


19 


12445 


12302 


12160 


12018 


11877 


11737 


11598 


11459 


11321 


11184 


2c3 


20 


11047 


10911 


10776 


10642 


10508 


10375 


10242 


10110 


9979 


9848 




21 


9718 


9589 


9460 


9332 


9204 


9077 


8951 


8825 


8700 


8575 


II 


22 


8451 


8327 


8204 


8082 


7960 


7838 


7717 


7597 


7477 


7358 


23 


7239 


7121 


7004 


6887 


6770 


6654 


6538 


6423 


6308 


6194 


a« 


24 


6080 


5967 


5854 


5741 


5629 


5518 


5407 


5296 


5186 


5077 


f 


25 


4968 


4859 


4751 


4643 


4535 


4428 


4321 


4215 


4109 


4004 


26 


3899 


3794 


3690 


3586 


3483 


3380 


3277 


3175 


3073 


2972 


1, 


27 


2871 


2770 


2670 


2570 


2470 


2371 


2272 


2173 


2075 


1977 


28 


1880 


1783 


1686 


1589 


1493 


1397 


1302 


1207 


1112 


1018 


a 


29 


924 


830 


736 


643 


550 


458 


366 


274 


182 


91 


30 


000 


—91 


—181 


—271 


—361 


—451 


—540 


—629 


—717 


—805 





Ex. 1. — The mean temp, of two stations whose diff. of elev. is desired 
is 50° F. The barom. reading at upper station is 24.62 ins.; and at lower 
station, 28.165. Find the difference in elevation of the two stations. 

Solution. — Using proportional differences — 

Elev. upper station (24.62 ins.) = 5385 ft. 
Elev. lower station (28.165 ins.) = 1720 ft. 



Ans. — Diff. in elevation.. . . = 3665 ft. 



1000 



59.— RAILROADS. 



table. 



4. — Barometric Correction Table for Temperature. 
(To be used in connection with Table 3, preceding.) 
Note. — Mult, value obtained from Table 3, by the Coefficients in this 



[Coefficients.] 



Mean 


Coef. 


Diff. 


Mean 


Coef. 


Diff. 


Mean 


Coef. 


DlfC. 


Temp. 


per Deg. 


Temp. 


per Deg. 


Temp. 


per Deg. 


0° 


.8975 


.00219 


30° 


.9620 


.00214 


60° 


1.0262 


.00210 


10° 


.9194 


.00214 


40° 


.9834 


.00215 


70° 


1.0472 


.00205 


20° 


.9408 


.00212 


50° 


1.0049 


.00213 


80° 


1.0677 


.00202 


30° 


.9620 




60° 


1.0262 




90° 


1.0879 





Ex. 2. — Now for any other mean temp, of the two stations than 60" F., 
say 65° F., we find the coef. for mean temp, in Table 4, and mult, this into 
the result obtained from Table 3 for 50° F. Thus, for 65° F., the coef. is 
1.02624-. 00210X5= 1.0367. Solving Ex. 1, for a mean temp, of 65° F., we 
have 3665X1.0367= 3800 ft. = diff. in elev. of the two stations. 



C— THE PRELIMINARY SURVEY. 

The Organization for the preliminary survey, as ordinarily conducted, 
comprises full parties for transit-, level-, and topographic work. This 
survey consists in developing a broken line on the ground, that can be used 
later as a base in projecting the final location. The preliminary line should 
be nearly identical, practically, with the final location so that the latter will 
be "fully covered" by the topography. In cases where it is evident that 
any part of the location will be radically different from the preliminary line 
as run, a new preliminary line should be run immediately covering that 
portion, and the old line "abandoned" and marked so in the note books. 
Equated stationing should be used instead of introducing the terms "long 
station" or "short station." 

The Locating Engineer will select, usually, some critical point where 
the line has to come, in both position and elevation, for a starting point of 
the survey. Where a mountain range has to be crossed it is customary to 
begin at the summit and work down, but this does not always hold. If 
there is no pass low enough, and a tunnel is imperative, it is often necessary 
to make a quick topographical survey "along" the main range, running 
transverse lines from the main summit line at points where it is thought the 
length of tunnel will be the least, at the desired 'elevation. This will not 
necessarily be at the "pass" and may be a considerable distance from it. 
(In connection with tunnel location, the geological formation should be 
studied and also the possibility of one or more shafts to facilitate construc- 
tion.) 

The Transitman should be careful to hold closely to any grade line 
which may have been decided upon from the reconnoissance data. For this 
purpose the transit should be provided with a Gradienter Screw by which 
the telescope may be inclined to the required grade. It consists of a clamp 
and slow-motion screw, so that one complete revolution of the latter raises 
or lowers the line of sight of the telescope 1 foot vertically in a horizontal 
distance of 100 feet. The edge of the head is divided into 100 parts for 
minute readings, and the number of complete turns of the screw are indi- 
cated by a graduated bar. Stakes are set every 100-ft., or station, and 
hubs at every transit point. In a true preliminary line no curves are run, 
but it is sometimes convenient to fit in a curve around a hillside to facilitate 
the work of topography and later location. The accuracy with which pre- 
liminary lines are run depends somewhat upon the circumstances of the 
case. The policy of the Southern Pacific R. R. Co. is to run very accurate 
preliminary surveys so that the subsequent location can be calculated to a 
nicety in the office. Magnetic bearings should be taken at every position of 
the transit, as a check on the angles. 



PRELIMINARY SURVEY. GRADE TABLES. 



1001 



5. — Grades in Feet per 100-Ft. and Feet per Mile. 
Part I. [Feet per Mile. (Exact.)] 



Ft. per 

100 ft. 



.5 
.6 
.7 
.8 
.9 
1.0 

l!2 
1.3 
1.4 

1.5 
1.6 
1.7 
1.8 
1.9 
2.0 
2.1 
2.2 
2.3 
2.4 

2.5 
2.6 
2.7 
2.8 
2.9 
3.0 
3.1 
3.2 
3.3 
3.4 

3.5 
3.6 
3.7 
3.8 
3.9 



00 



5.280 
10.560 
15.840 
21.120 

26.400 
31.680 
36.960 
42.240 
47.520 

52.800 
58.080 
63.360 
68.640 
73.920 

79.200 
84.480 
89.760 
95.040 
100.320 

105.600 
110.880 
116.160 
121.440 
126.720 
132.000 
137.280 
142.560 
147.840 
153.120 
158.400 
163.680 
168.960 
174.240 
179.520 

184.800 
190.080 
195.360 
200.640 
205.920 



.01 



.528 
5.808 
11.088 
16.368 
21.648 
26.928 
32.208 
37.488 
42.768 
48.048 

53.328 
58.608 
63.888 
69.168 
74.448 
79.728 
85.008 
90.288 
95.568 
100.848 

106.128 
111.408 
116.688 
121.968 
127.248 
132.528 
137.808 
143.088 
148.368 
153.648 

158.928 
164.208 
169.488 
174.768 
180.048 
185.328 
190.608 
195.888 
201.168 
206.448 



,02 



1.056 
6.336 
11.616 
16.896 
22.176 
27.456 
32.736 
38.016 
43.296 
48.576 



856 
136 
416 
696 
976 

256 
536 
816 
096 
376 

656 
936 
216 
496 
776 



53. 
59. 
64. 
69. 
74. 

80. 
85. 
90. 
96. 
101. 

106. 

111. 

117. 

122. 

127. 

133.056 

138.336 

143.616 

148.896 

154.176 

159.456 

164.736 

170.016 

175.296 

180.576 

185.856 

191.136 

196.416 

201.696 

206.976 



03 



1.584 

6.864 

12.144 

17.424 

22.704 

27.984 
33.264 
38.544 
43.824 
49.104 



54. 
59. 
64. 
70. 
75. 
80. 
86. 
91. 
96. 
101. 

107. 

112. 

117. 

123. 

128. 

133.584 

138.864 

144.144 

149.424 

154.704 

159.984 
165.264 
170.544 
175.824 
181.104 

186.384 
191.664 
196.944 
202.224 
207.504 



,04 



2.112 

7.392 

12.672 

17.952 

23.232 

28.512 
33.792 
39.072 
44.352 
49.632 
54.912 
60.192 
65.472 
70.752 
76.032 



81. 
86. 
91. 
97. 
102. 

107. 
112. 
118. 
123. 
128. 



134.112 
139.392 
144.672 
149.952 
155.232 
160.512 
165.792 
171.072 
176.352 
181.632 
186.912 
192.192 
197.472 
202.752 
208.032 



,05 



2.640 
7.920 
13.200 
18.480 
23.760 
29.040 
34.320 
39.600 
44.880 
50.160 

55.440 
60.720 
66.000 
71.280 
76.560 



81. 
87. 
92. 
97. 
102. 

108. 
113. 
118. 
124. 
129. 



840 
120 
400 
680 
960 

240 
520 
800 
080 
360 



134.640 
139.920 
145.200 
150.480 
155.760 

161.040 
166.320 
171.600 
176.880 
182.160 
187.440 
192.720 
108. 000 
203.280 
208.560 



.06 



3.168 

8.448 

13.728 

19.008 

24.288 



29. 
34. 
40. 
45. 
50. 

55. 
61. 
66. 
71. 
77. 
82. 
87. 
92. 
98. 
103. 

108. 
114. 
119. 
124. 
129.888 

135.168 
140.448 
145.728 
151.008 
156.288 
161.568 
166.848 
172.128 
177.408 
182.688 
187.968 
193.248 
198.528 
203.808 
209.088 



568 
848 
128 
408 
688 

968 
248 
528 
808 
088 

368 
648 
928 
208 
488 
768 
048 
328 
608 



.07 



3.696 

8.976 

14.256 

19.536 

24.816 

30.096 

35.376 

40.656 

45.936 

51.216 

56.496 

61.776 

67.056 

72.336 

77.616 

82.896 

88.176 

93.456 

98.736 

104.016 

109.296 

114.576 

119.856 

125.136 

130.416 

135.696 
140.976 
146.256 
151.536 
156.816 
162.096 
167.376 
172.656 
177.936 
183.216 
188.496 
193.776 
199.056 
204.336 
209.616 



.08 



4.224 
9.504 
14.784 
20.064 
25.344 
30.624 
35.904 
41.184 
46.464 
51.744 



024 
304 
584 
864 
144 

424 
704 
984 
264 
544 

824 
104 
384 
664 
944 



57. 
62. 
67. 
72. 
78. 
83. 
88. 
93. 
99. 
104. 

109. 
115. 
120. 
125. 
130. 

136.224 
141.504 
146.784 
152.064 
157.344 
162.624 
167.904 
173.184 
178.464 
183.744 
189.024 
194.304 
199.584 
204.864 
210.144 



09 



4.752 
10.032 
15.312 
20.592 
25.872 
31.152 
36.432 
41.712 
46.992 
52.272 



57. 
62. 
68. 
73. 
78. 

83. 

89. 

94. 

99. 
105. 
110. 
115. 
120. 
126. 
131. 
136.752 
142.032 
147.312 
152.592 
157.872 
163.152 
168.432 
173.712 
178.992 
184.272 
189.552 
194.832 
200.112 
205.392 
210.672 



P P. 



■ 528 
.053 
.106 
.158 
.211 
.264 
.317 
.370 
.422 
.475 



Ex. 1. — Rate of grade =1.937 ft. 
per 100 ft. Then ft. per mile 



Ex. 2. — By inverse operation, 
the rate of grade may be obtained 
"101.904+0.37=102.274. when ft. per mile is given. 

Part II. [Feet per Mile. (Exact.)] 



Ft. per 

100 ft. 


.0 


.1 


.2 


.3 


.4 


.5 


.6 


.7 


.8 


.9 


P.P. 


4 


211.20 
264.00 


216.48 
269.28 


221.76 
274.56 


227.04 
279.84 


232.32 
285.12 


237.60 
290.40 


242.88 
295.68 


248.16 
300. 96 


253.44 
306.24 


258.72 
311.52 


5.28 


5 


1 .528 
21.056 


6 


316.80 


322.08 


327.36 


332.64 


337.92 


343.20 


348.48 


353.76 


359.04 


364.32 


3 1.584 


7 


369.60 


374.88 


380.16 


385.44 


390.72 


396.00 


401.28 


406.65 


411.84 


417.12 


12.112 


8 


422.40 


427.68 


432.96 


438.24 


443.52 


448.80 


454.08 


459.36 


464.64 


469.92 


52.640 


9 


475.20 


480.48 


485.76 


491.04 


496.32 


501.60 


506.88 


512.16 


517.44 


522.72 


5 3.168 
7 3.696 


10 


528. 00 


533.28 


538.56 


543.84 


549.12 


554.40 


559.68 


564.96 


570.24 


575.52 


3 4.224 
H.752 



Ex. 3. — By direct operation: 
Rate of grade = 4.25 per 100 ft. 
Then ft. per mile = 224.40. 



Ex. 4. — By inverse operation: 
Ft. per mile = 250. Then rate of 
grade per 100 ft. = 4. 735. 



6. — Number of 100-Ft. Stations Corresponding to Distance in Miles. 
(Use above Table, No. 5, with revised headings.) 



Ex. a. — By direct operation: 
2.25 miles = 118.8 stations; 22.5 
miles = 1188 stations. 



Ex. b. — By inverse operation: 
500.00 stations =9. 4 + .06''96 miles; 
5000 sta. = 94.6''96 miles. 



1002 



h— RAILROADS. 



7. — Grade Angles Corresp'd'g to Rates of Grade in Feet per 100-Ft. 
Part I. [Grade Angle.] 



Ft. per 
100 ft. 


.00 


.01 


.02 


.03 


.04 


.05 


.06 


.07 


.08 


.09 




o / # 


O / /r 


O / If 


/ If 


O t If 


O t If 


f If 


O f If 


1 It . 


O f ff 


.0 





21 


41 


1 02 


1 23 


1 43 


2 04 


2 24 


2 45 


3 06 


.1 


3 26 


3 47 


4 08 


4 28 


4 49 


5 09 


5 30 


5 51 


6 11 


6 32 


.2 


6 53 


7 13 


7 34 


7 54 


8 15 


8 36 


8 56 


9 17 


9 38 


9 58 


.3 


10 19 


10 39 


11 00 


11 21 


11 41 


12 02 


12 23 


12 43 


13 04 


13 24 


.4 


13 45 


14 06 


14 26 


14 47 


15 08 


15 28 


15 49 


16 09 


16 30 


16 51 


.5 


17 11 


17 32 


17 53 


18 13 


18 34 


18 54 


19 15 


19 36 


19 56 


20 17 


.6 


20 38 


20 58 


21 19 


21 39 


22 00 


22 21 


22 41 


23 02 


23 23 


2d 43 


.7 


24 04 


24 24 


24 45 


25 06 


25 26 


25 47 


26 08 


26 28 


26 49 


27 09 


.8 


27 30 


27 51 


28 11 


28 32 


28 53 


29 13 


29 34 


29 54 


30 15 


30 36 


.9 


30 57 


31 17 


31 38 


31 58 


32 19 


32 39 


33 00 


33 21 


33 41 


34 02 


1.0 


34 23 


34 43 


35 04 


35 24 


35 45 


36 05 


3.6 26 


36 47 


37 08 


37 28 


1.1 


37 49 


38 09 


38 30 


38 51 


39 11 


39 32 


39 53 


40 13 


40 34 


40 54 


1.2 


41 15 


41 35 


41 56 


42 17 


42 38 


42 58 


43 19 


43 39 


44 00 


44 21 


1.3 


44 41 


45 02 


45 23 


45 43 


46 04 


46 24 


46 45 


47 06 


47 26 


47 47 


1.4 


48 08 


48 28 


48 49 


49 09 


49 30 


49 51 


50 11 


50 32 


50 52 


51 13 


1.5 


51 34 


51 54 


52 15 


52 36 


52 56 


53 17 


53 37 


53 58 


54 19 


fA 39 


1.6 


55 00 


55 21 


55 41 


56 02 


56 22 


56 43 


57 04 


57 24 


57 45 


58 06 


1.7 


58 26 


58 47 


59 07 


59 28 


59 49 


1 00 09 


1 00 30 


1 00 51 


1 01 11 


1 01 32 


1.8 


1 01 52 


1 02 13 


1 02 34 


1 02 54 


1 03 15 


103 35 


1 03 56 


1 04 17 


1 04 37 


1 04 58 


1.9 


1 05 19 


1 05 39 


1 06 00 


1 06 20 


1 06 41 


1 07 02 


1 07 22 


1 07 43 


1 08 04 


1 08 24 


2.0 


1 08 45 


1 09 05 


1 09 26 


1 09 47 


1 10 07 


1 10 28 


1 10 48 


1 11 09 


1 11 30 


1 11 50 


2.1 


1 12 11 


1 12 32 


1 12 52 


1 13 13 


1 13 33 


1 13 54 


1 14 15 


1 14 35 


1 14 56 


1 15 16 


2.2 


1 15 37 


1 15 58 


1 16 18 


1 16 39 


1 17 00 


1 17 20 


1 17 41 


1 18 01 


1 18 22 


1 18 43 


2.3 


1 19 03 


1 19 24 


1 19 44 


1 20 05 


1 20 26 


1 20 46 


1 21 07 


1 21 28 


1 21 48 


1 22 09 


2.4 


1 22 29 


1 22 50 


1 23 11 


1 23 31 


1 23 52 


1 24 12 


1 24 33 


1 24 54 


1 25 14 


1 25 35 


2.5 


1 25 56 


1 26 16 


1 26 37 


1 26 57 


1 27 18 


1 27 39 


1 27 59 


1 28 20 


1 28 40 


1 29 01 


2.6 


1 29 22 


1 29 42 


1 30 03 


1 30 24 


1 30 44 


1 31 05 


1 31 25 


1 31 46 


] 32 07 


1 32 27 


2.7 


1 32 48 


1 33 08 


1 33 29 


1 33 50 


1 34 10 


1 34 31 


1 34 51 


1 35 12 


1 35 33 


1 35 53 


2.8 


1 36 14 


1 36 35 


1 36 55 


1 37 16 


1 37 36 


1 37 57 


1 38 18 


1 38 38 


1 38 59 


1 39 19 


2.9 


1 39 40 


1 40 01 


1 40 21 


1 40 42 


1 41 02 


1 41 23 


1 41 44 


1 42 04 


1 42 25 


1 42 45 


3.0 


1 43 06 


1 43 27 


1 43 47 


1 44 08 


1 44 29 


1 44 49 


1 45 10 


1 45 30 


1 45 51 


1 46 12 


3.1 


1 46 32 


1 46 53 


1 47 13 


1 47 34 


1 47 55 


1 48 15 


1 48 36 


1 48 56 


1 49 17 


1 49 38 


3.2 


1 49 58 


1 50 19 


1 50 39 


1 51 00 


1 51 21 


1 51 41 


1 52 02 


1 52 22 


1 52 43 


1 53 04 


3.3 


1 53 24 


1 53 45 


1 54 05 


1 54 26 


1 54 47 


1 55 07 


1 55 28 


1 55 48 


1 56 09 


1 56 30 


3.4 


1 56 50 


1 57 11 


1 57 32 


1 57 52 


1 58 13 


1 58 33 


1 58 54 


1 59 15 


1 59 35 


1 59 56 


3.5 


2 00 16 


2 00 37 


2 00 58 


2 01 18 


2 01 39 


2 01 59 


2 02 20 


2 02 41 


2 03 01 


2 03 22 


3.6 


2 03 42 


2 04 03 


2 04 24 


2 04 44 


2 OS 05 


2 05 25 


2 05 46 


2 06 07 


2 06 27 


2 06 48 


3.7 


2 07 08 


2 07 29 


2 07 50 


2 08 10 


2 08 31 


2 08 51 


2 09 12 


2 09 33 


2 09 53 


2 10 14 


3.8 


2 10 34 


2 10 55 


2 11 16 


2 11 36 


2 11 57 


2 12 17 


2 12 38 


2 12 58 


2 13 19 


2 13 40 


3.9 


2 14 00 


2 14 21 


2 14 41 


2 15 02 


2 15 23 


2 15 43 


2 16 04 


2 16 24 


2 16 45 


2 17 06 



P. p. 



Ex. 1. — Rate of grade =1.937 ft. 
per 100 ft. Then grade angle = 
1°06'20''+15"=1°06'35'^ 



Ex. 2. — By inverse operation, 
the rate of grade may be obtained 
when grade angle is given. 









Part II. 


[Grade Angle.] 










Ft. per 

100 ft. 


.0 


.1 


.2 


.3 


.4 


.5 


.6 


.7 


.8 


.9 


P.P. 


4 

5 
6 
7 
8 
9 

10 


f n 

2 17 26 

2 51 45 

3 26 01 

4 00 15 

4 34 26 

5 08 34 

5 42 38 


Q 1 n 

2 20 52 

2 55 10 

3 29 27 

4 03 40 

4 37 51 

5 11 59 

5 46 02 


O 1 If 

2 24 18 

2 58 36 

3 32 52 

4 07 06 

4 41 16 

5 15 23 

5 49 26 


/ If 

2 27 44 

3 02 09 

3 36 18 

4 10 31 

4 44 41 

5 18 48 

5 52 50 


O f H 

2 31 10 

3 05 27 

3 39 43 

4 13 56 

4 48 06 

5 22 12 

5 56 15 


O 1 If 

2 34 36 

3 08 53 

3 43 08 

4 17 21 

4 51 30 

5 25 37 

5 59 39 


O / n 

2 38 01 

3 12 19 

3 46 34 

4 20 46 

4 54 55 

5 29 01 

6 03 03 


O f . 

2 41 27 

3 15 44 

3 49 59 

4 24 11 

4 58 20 

5 32 25 

6 06 27 


O t ff 

2 44 53 

3 19 10 

3 53 24 

4 27 36 

5 01 45 

5 35 50 

6 09 51 


o » w 

2 48 19 

3 22 36 

3 56 50 

4 31 01 

5 05 09 

5 39 14 

6 13 14 


3 25 
10 21 

2 41 

3 1 02 

4 1 22 
51 43 

6 2 03 

7 2 24 

8 2 44 

9 3 05 



Ex. 3. — By direct operation: 
Rate of grade =4.25 per 100 ft. 
Then grade angle = 2« 26' 01". 



Ex. 4.^By inverse operation: 
Grade angle = 2° 42' 39''. Then rate 
of grade per 100 ft. = 4. 735. 
Notes. 
Grades may be run with the transit by the use of the gradienter attach- 
ment. See also Table No. 8, following. 



GRADE ANGLES AND RATES OF GRADES. 



1003 



8. — Grades in Feet per Mile Reduced to Feet per 100-Ft. 
[Grade in Feet per 100 Ft.] 



Ft. per 


0. 


.. 


2. 


3. 


4. 


5. 


6. 


7. 


8. 


9. 


P.P. 


MUe. 
























0—. 




.01894 


.03788 


.05682 


.07576 


.09470 


.11364 


.13258 


.15152 


.17045 








1—. 


.18939 


.20833 


, 22727 


.24621 


.26515 


.28409 


.30303 


.32197 


.34091 


.35985 


1894 


2—. 


.37879 


.39773 


.41667 


.43561 


.45455 


.47348 


.49242 


.51136 


.53030 


.54924 


1 


189 


3-. 


.56818 


.58712 


.60606 


.62500 


.64394 


.66288 


.68182 


.70076 


.71970 


.73864 


?, 


379 


4—. 


.75758 


.77652 


.79545 


.81439 


.83333 


.85227 


.87121 


.89015 


.90909 


.92803 


3 


568 
758 
947 


.5—. 


.94697 


.96591 


.98485 


1.00379 


1.02273 


1.04167 


1.06061 


1.07955 


1.09848 


1.11742 


4 
5 


6—. 


1.13636 


1.15530 


1.17424 


1.19318 


1.21212 


1.23106 


1.25000 


1.26894 


1.28788 


1.30682 


6 


1136 


7—. 


1.32576 


1.34470 


1.36364 


1.38258 


1.40152 


1.42045 


1.43939 


1.45833 


1.47727 


1.49621 


7 


1326 


8—. 


1.51515 


1.53409 


1.55303 


1.57197 


1.59091 


1.60985 


1.62879 


1.64773 


1.66667 


1.68561 


8 


1 51 5 


9—. 


1.70455 


1.72348 


1.74242 


1.76136 


1.78030 


1.79924 


1.81818 


1.83712 


1.85606 


1.87500 


9 


1705 


10—. 


1.89394 


1.91288 


1.93182 


1.95076 


1.96970 


1.98864 


2.00758 


2.02652 


2.04545 


2.06439 




11—. 


2.08333 


2.10227 


2.12121 


2.14015 


2.15909 


2.17803 


2.19697 


2.21591 


2.23485 


2.25379 


1893 


12—. 


2.27273 


2.29167 


2.31061 


2.32955 


2.34848 


2.36742 


^.38636 


2.40530 


2.42424 


2.44318 


1 


189 


13—. 


2.46212 


2.48106 


2.50000 


2.51894 


2.53788 


2.55682 


2.57576 


2.59470 


2.61364 


2.63258 


2 


379 


14—. 


2.65152 


2.67045 


2.68939 


2.70833 


2.72727 


2.74621 


2.76515 


2.78409 


2.80303 


2.82197 


3 

4 
5 


568 
757 
947 


15—. 


2.84091 


2.85985 


2.87879 


2.89773 


2.91667 


2.93561 


2.95455 


2.97348 


2.99242 


3.01136 


16—. 


3.03030 


3.04924 


3.06818 


3.08712 


3.10606 


3.12500 


3.14394 


3.16288 


3.18182 


3.20076 


6 


1136 


17-. 


3.21970 


3 23864 


3.25758 


3.27652 


3.29545 


3.31439 


3.33333 


3.35227 


3.37121 


3.39015 


7 


1325 


18—. 


3 40909 


3.42803 


3.44697 


3.46591 


3.48485 


3.50379 


3.52273 


3.54167 


3.56061 


3.57955 


8 


1514 


19—. 


3 59848 


3.61742 


3.63636 


3.65530 


3.67424 


3.69318 


3.71212 


3.73106 


3.75000 


3.76894 


9 


1704 


20—. 


3.78788 


3.80682 


3.82576 


3.84470 


3.86364 


3.88258 


3.90152 


3.92045 


3.93939 3.95833 





Ex. 1.— The grade of a road is 95.3 ft. per mile- (1.79924 + .00568) ft. 
per 100 ft. 

O.-'-Grade Angles Corresponding to Grades in Feet per Mile. 

[Grade Angle.] 



Ft. per 
Mile. 



0— . 
1— . 
2—. 
3—. 



5—. 
6—. 
7—. 
8—. 



10-. 
11—. 
12—. 
13—. 
14—. 

15—. 
16—. 
17—. 
18—. 
19—. 

20—. 



0. 



6 31 

13 01 

19 32 

26 03 

32 33 
39 04 
45 34 
52 05 
i58 36 

1 05 06 

1 11 37 

1 18 07 

1 24 38 

1 31 08 

1 37 38 
1 44 09 
1 50 39 

1 57 

2 03 39 

2 10 09 



1. 



39 

7 10 

13 40 

20 11 

26 42 

33 12 
39 43 
46 13 
52 44 
59 15 

1 05 45 

1 12 16 

1 18 46 

1 25 17 

1 31 47 

1 38 17 
1 44 48 
1 51 18 

1 57 48 

2 04 18 

2 10 48 



1 18 

7 49 

14 19 

20 50 

27 21 

33 51 
40 22 
46 53 
63 23 
59 54 

06 24 
12 55 
19 25 
25 56 
32 26 

38 56 
45 27 
51 57 
58 27 
04 57 



2 11 27 



1 57 

8 28 

14 

21 29 

28 00 

34 30 
41 01 
47 32 
54 02 
1 00 33 

1 07 03 

1 13 34 

1 20 04 

1 26 35 

1 33 05 

1 39 35 
1 46 06 
1 52 36 

1 59 06 

2 05 36 

2 12 06 



4. 



2 36 

9 07 
15 38 
22 08 
28 39 

35 09 
41 40 
48 11 
54 41 
1 01 12 



07 42 
14 13 
20 43 
27 14 
33 44 

40 14 
46 45 
53 15 
59 45 
06 15 



2 12 45 



3 15 

9 46 

16 17 

22 47 

29 18 

35 49 
42 19 
48 50 
55 20 
1 01 51 

1 08 21 

1 li 52 

1 21 22 

1 27 53 

1 34 23 

1 40 53 
1 47 24 

1 53 54 

2 00 24 
2 06 54 

2 13 24 



3 54 
10 25 
16 56 
23 26 
29 57 

36 28 
42 58 
49 29 
55 59 
1 02 30 

1 09 00 

1 15 31 

1 22 01 

1 28 32 

1 35 02 

1 41 32 
1 48 03 

1 54 33 

2 01 03 
2 07 33 

2 14 03 



4 33 
11 04 
17 35 
24 05 
30 36 

37 07 
43 37 
50 08 
56 38 
08 09 



1 09 39 

1 16 10 

1 22 40 

1 29 11 

1 35 41 

1 42 11 
1 48 42 

1 55 12 

2 01 42 
2 12 

2 14 42 



5 13 
11 43 
18 14 
24 44 
31 15 

37 46 
44 16 
50 47 
57 17 
1 03 48 

1 10 18 

1 16 49 

1 23 19 

1 29 50 

1 36 20 

1 42 50 
1 49 21 

1 55 51 

2 02 21 
2 08 51 

2 15 21 



9. 



5 52 
12 22 
18 53 
25 24 
31 54 

38 25 
44 55 
51 26 
57 57 
1 04 27 

1 10 58 

I 17 28 

1 23 58 

1 30 29 

1 36 59 

1 43 30 
I 50 00 

1 56 30 

2 03 00 
2 09 30 

2 16 00 



p 


P. 


39" 


1 


4 


2 


8 


3 


12 


4 


16 


5 


20 


6 


23 


7 


27 


8 


31 


9 


35 


40" 


1 


4 


2 


8 


3 


12 


4 


16 


5 


20 


6 


24 


7 


28 


8 


32 


9 


36 



Ex. 1. 

i° or 51" 



—The grade of a road, is 95.3 ft. per mile; hence, grade angles 
+ 12''. 



1004 59.— RAILROADS. 

The Levelman starts usually from an established bench mark (B. M.), 
frequently at an assumed elevation, as near as possible to the correct eleva- 
tion above sea level. He follows closely behind the transit man, taking 
elevations on the ground at every station and also at intermediate points 
where the profile demands. He should keep the transit at the correct eleva- 
tion at every transit point and sometimes give levels ahead of the transit. 
He establishes temporary bench marks on every transit hub, and permanent 
bench marks, say, every half mile. They should be described accurately in 
the note book by exact stationing on the line and by distance to right or 
left of same. Check levels should be run every 10 miles, more or less. If 
work is slack, the levelman can often aid the topographer over certain 
stretches by taking side levels. Two rodmen can often be used to great 
advantage. The combined target- and self-reading rod is the best — the 
former for turning points, and the latter for ordinary ground elevations. 

The Topographer gets the ground elevations at stations on the center 
line from the levelman, either the night before or during the day, at intervals. 
His duties are to take such notes that contour lines can be platted accu- 
rately on the maps. It is perhaps needless to say that in some cases topog- 
raphy should be taken extremely accurate, and especially so if the loca- 
tion lines are determined primarily in the office. Perhaps the most common 
method of taking topography is with the hand level, for elevation, and by 
pacing, for distance. Often the (cloth) tape is used. There are two methods 
of keeping the notes: (1) To give the relative elevation at right angle to 
the center line at each station, and the distance out, to each break in the 

+ 43 
ground, as ' ; and (2) to "sketch in" the contour lines directly in the 

field, giving the distances out from center line. The clinometer is very 
useful in moderately sloping country. The transitman may often help the 
topographer on very steep hillsides by taking vertical slope angles at right 
angle to the line. 

The Mapping consists in platting the instrument line from the transit 
notes, and the topography or contour lines from the topography notes. 
The former may be platted with a protractor or by tangent- or chord 
deflection,* from each previous tangent line; or a base (say north and 
south) may be established on the map from which each tangent line is laid off 
by calculated angle. Sometimes the lines are platted from calculated latitudes 
and departures of the angle points or points of intersection (P. Z.'s) of the 
tangents. This latter method has the advantage of accuracy. The magnetic 
bearings are a check on the calculated bearings of each tangent and errors 
of reading angles in the field, both in amount and direction. 

D.— THE LOCATION SURVEY. 

The Location is the Objective, or the desired end sought; the other 
surveys are simply means to this end. The reconnoissance may sometimes 
be reduced to a mere inspection of the country in a most casual manner. 
The preliminary survey may often consist in running a few compass lines 
to determine the general route, or even these may be omitted. But the 
location survey consists in the final establishment on the ground of the line 
as it is to be built: running in the tangents and joining them with the 
proper curves. Of course there are generally minor changes in the line 
during its construction, but these subsequent changes might be considered 
part of the location proper. 

The Profile and Grades are subjects for constant study; the latter even 
after the road is constructed and in operation. The grade line ( = base of 
cut or top of fill, and is called sub-grade during construction) is usually 
adjusted to the profile by the use of a fine thread stretched over the latter 
in various positions, studying at the same time the "equalization of cuts 
and fills." By this phrase we do not mean necessarily that the quantities 
in the cuts should equal those in the fills, although this might hold true for 
certain saw-toothed profiles or profiles with short haul from cut to fill; and 
especially where the material in the cuts, instead of borrowed material, 
would naturally be used in the fills. If the material in cuts is rock, and 
borrowed material may be had more cheaply for the fills, the grade line 
would naturally be raised to reduce the cost of cutting. Generally, it may 
be stated that the grade line is so adjusted that the quantities in cut are 

* See Table of Chords, page 959. 



LOCATION SURVEY. CURVES. 1005 

somewhat less than those in fill. In this connection see earthwork tables * 
page 1017, etc; and also notes on shrinkage, page 909. 

» Railroad Curves. — Curves are introduced in both the grade and aline- 
ment of roads to give easy change of direction to moving trains. The kind 
of curve used for either purpose is chosen by reason of its convenience in 
laying out on the ground. Hence we use, generally, parabolic vertical 
curves for connecting grade lines, and circular horizontal curves for con- 
necting tangents. 

Vertical Parabolic Curves are easily calculated and laid out. The principle 
of the parabola is that the o-^set from a fixed base is proportional to the 
square of the distance in a uniform direction from a fixed point. In Fig. 6, 




Fig. 6. — Parabolic. Vertical Curve. 

let V be the vertex or summit of two grade lines passing through A and B, 
respectively; and let it be required to put in a parabolic curve AvB of any 
length, say 800 feet Then A and B, the limiting points of the curve, will 
be 4 stations either side of the vertex. Calculate the elevations of these two 
points. Produce AV to T and calculate the maximum offset TB. Then 

TTi 

X square of distance in stations from A to any other station is equal to 

8X8 

the vertical offset at that station from the line A VT to the parabolic curve. 

TTi T Ti 

Thus, Vv = — /T-^ X 4 X 4 = —t- . The level and rod are used for vertical off- 

oX o * 

sets (elevations), and the chain for horizontal distances. Valley curves are 
treated in a similar manner to summit curves. 

Horizontal Circular Curves may be simple, compound or reversed. To 
these may be added the infinitely compounded curves — the parabola, and 
the cubic parabola; and also the many forms of spiral and easement curves 
which are approximations to the latter. 

Simple Curves are arcs of circles, and are denoted either by the radius 
of the circle or by the "degree of curve," generally the latter. The degree 
of curve may be defined as the angle subtended at the center of the circle 
by a "100-ft. station" on the arc, whether the chord lengths are 100-ft. 
(used for flat curves as of 5° or less), 50-ft. (used for sharp curves from 5° 
to 10°), or 25-ft. (used for curves above 10° and for turnouts and switches). 
Thus, it is seen that the degree of curve is comparable with the radius only 
when the length of chord by which the curve is laid out, is given; hence, 
For 100-ft. chords, the sine of \ the degree of curve = 50 ft. -^radius in ft. 
" 50-ft. " " " A " ** " =25 ft.-H " 

*• 25-ft. " " " i " " " =12ift.-^ " 

from which the degree of curve may be obtained by multiplying the corres- 
ponding angles by 2, 4 and 8, respectively. Throughout the following 
discussion 100-ft. chords will be assumed unless otherwise noted. 

To lay out a simple curve of D degrees from a point, P. C. (called point 
of curve or beginning of curve), situated on the tangent line Ti, Fig. 
7: Set up the transit at the P. C., and turn off, by addition, from the 

tangent produced, the successive angles -x- for each 100-ft. station. The 



*These tables may be copied on profile paper in the form of cubic yards 
per 100-ft. station and arranged so that the quantities can be read off 
directly by matching the zero of the table to the grade line of the profile, 
and reading the quantity opposite the ground line. The tables are to feet 
in height, or to feet and half feet, and the heights are to the same vertical 
scale as the profile. Caution: Note that the quantities must be increased 
when the ground slope is not level. 



1006 



-RAILROADS. 



angle -^ is called the deflection angle per station and is always equal to 

i the central angle per station, or i the degree of curve, D. From this 
it will be seen that the total deflection angle to any point on the curve, from 
a tangent to the curve at instrument point, is equal to ^ the total central 




Fig. 7. — Circular Horizontal Curve, 
angle subtended by the two points. Thus, the deflection angle to point 2 



is 2X-^\ to point 3, is 3 X y 



and to the P. T., is {^+x) X y, in which x is 



any decimal part of a 100-ft. station. If, in Fig. 7, the degree of curve 
p=30°-30', and x=&0 ft., we have that the total deflection to the end of 

D 3°-30' 

curve or point of tangent P.T. = {Z + x) X-^=2>.QX — ^ — = 6°-18'; and that 

the total central angle = 12°36'. Moreover, the central angle subtended by 
the total length of curve, from the P. C. to the P. T., is e-qual to the angle 
of intersection I at the point of intersection P. I. of the adjoining tangents 
Ti and T2 produced. The portion of a tangent produced which lies between 
the P. /. and the P. C. or P. T. is variously termed the "semi-tangent," 
"tangent distance" or "vertex distance." The length of a curve is the dis- 
tance between the P. C. and P. T., measured in chord lengths, and not the 
true length of the arc. Thus, in Fig. 7, if the last chord x is to of a 100-ft. 
station, the total length of curve is 360 feet. If the degree of curve is 



3°-30', the radius = 50- 



sin ^-f^ = 1637.28 ft. 



section / is 12°- 3 6', the semi-tangent 



and if the angle of inter- 



1 2**- ?fi' 
radius X tan ^7^^ = 180.76 ft. 



The last is simply the solution of a right-angle triangle with hypothenuse 



joining o and P. I., and with angle at the base (at o) equal to -^. 



The 



temal" is the shortest distance from the P. I. to the curve, and is measured 
along the hypothenuse just mentioned. Clearly, it is the length of the 
hypothenuse minus the radius, and in the present instance is equal to 

1»2°- 36' 
radius X exsec* = 9.95 ft. In fitting a curve in between two adjoin- 
ing tangents of a preliminary survey, the external distance is often of 
primary importance in arbitrarily fixing the position (and degree) of the 
curve. When the degree of the curve is determined the semi-tangents are 
calculated and laid off from the P. /., and the curve run in as explained 
above. 



* Exsecant = secant — 1. 



CIRCULAR CURVES, 



1007 



10. — Radii of English Curves, in Feet.* (Chords 100 Feet.) 

Note. — See Foot-note for using this table for Metric curves. 

[Radii in Feet.] 



Min- 
utes. 


Degree of Curve. 


0° 


1° 


2° 


3° 


4° 


5° 


6° 


70 


0' 


Infinite 


5729.65 


2854.93 


1910.08 


1432.69 


1146.28 


955.366 


819.020 


1 


343775. 


5635.72 


2841.26 


1899.53 


1426.74 


1142.47 


952.722 


817.077 


2 


171887, 


5544.83 


2817.97 


1889.09 


1420.85 


1138.69 


950.093 


815.144 


3 


114592. 


5456.82 


2795.06 


1878.77 


1415.01 


1134.94 


947.478 


813.238 


4 


85943.7 


5371.56 


2772.53 


1868.56 


1409.21 


1131.21 


944.877 


811.303 


5 


68754.9 


5288.92 


2750.35 


1858.47 


1403.46 


1127.50 


942.291 


809. 39f 


6 


57295.8 


5208.79 


2728.52 


1848.48 


1397.76 


1123.82 


939.719 


807.499 


7 


49110.7 


5131.05 


2707.04 


1838.59 


1392.10 


1120.16 


937.161 


805.611 


8 


42971.8 


505^.59 


2685.89 


1828.82 


1386.49 


1116.52 


934.616 


803.731 


9 


38197.2 


4982.33 


2665.08 


1819.14 


1380.92 


1112.91 


932.086 


801.860 


10 


34377.5 


4911.15 


2644.58 


1809.57 


1375.40 


1109.33 


929.569 


799.997 


11 


31252.3 


4841.98 


2624.39 


1800.10 


1369.92 


1105.76 


927.066 


798.144 


12 


28647.8 


4774.74 


2604.51 


1790.73 


1364.49 


1102.22 


924.576 


796.299 


13 


26444.2 


4709.33 


2584.93 


1781.45 


1359.10 


1098.70 


922.100 


794.462 


14 


24555.4 


4645.69 


2565.65 


1772.27 


1353.75 


1095.20 


919.637 


792.634 


15 


22918.3 


4583.75 


254.6.64 


1763.18 


1348.45 


1091.73 


917.187 


790.814 


16 


21485.9 


4523.44 


2527.92 


1754.19 


1343.15 


1088.28 


914.750 


789.003 


17 


20222.1 


4464.70 


2509.47 


1745.26 


1337.65 


1084.85 


912.326 


787.210 


18 


19098.6 


4407.46 


2491.29 


1736.48 


1332.77 


1081.44 


909.915 


785.405 


19 


18093.4 


4351.67 


2473.37 


1727.75 


1327.63 


1078.05 


907.517 


783.618 


20 


17188.8 


4297.28 


2455.70 


1719.12 


1322.53 


1074.68 


905.131 


781.840 


21 


16370.2 


4244.23 


2438.29 


1710.56 


1317.46 


1071.34 


902.758 


780.069 


22 


15626.1 


4192.47 


2421.12 


1702.10 


1312.43 


1068.01 


900.397 


778.307 


23 


14946.7 


4141.96 


2404.19 


1693.72 


1307.45 


1064.71 


898.048 


776.552 


24 


14323.6 


4092.66 


2387.50 


1685.42 


1302.50 


1061.43 


895.712 


774.806 


25 


13751.0 


4044.51 


2371.04 


1677.20 


1297.58 


1058.16 


893.388 


773.067 


26 


13222.1 


3997.49 


2354.80 


1669.06 


1292.71 


1054.92 


891.076 


771.336 


27 


12732.4 


3951.54 


2338.78 


1661.00 


1287.87 


1051.70 


888.776 


769.613 


28 


12277.7 


3906.64 


2322.98 


1653.01 


1283.07 


1048.48 


886.488 


767.897 


29 


11854.3 


3862.74 


2307.39 


1645.11 


1278.30 


1045.31 


884.211 


766.190 


30 


11459.2 


3819.83 


2292.01 


1637.28 


1273.57 


1042.14 


881.946 


764.489 


31 


11089.6 


3777.85 


2276.84 


1629.52 


1268.87 


1039.00 


879.693 


762.797 


32 


10743.0 


3736.79 


2261.86 


1621.84 


1264.21 


1035.87 


877.451 


761.112 


33 


10417.5 


3696.61 


2247.08 


1614.22 


1259.58 


1032.76 


,875.221 


759.434 


34 


10111.1 


3657.29 


2232.49 


1606.68 


1254.98 


1029.67 


873.002 


757.764 


35 


9822.18 


3618.80 


2218.09 


1599.21 


1250.42 


1026.60 


870.795 


756.101 


36 


9549.34 


3581.10 


2203.87 


1591.81 


1245.89 


1023.55 


868.598 


754.445 


37 


9291.29 


3544.19 


2189.84 


1584.48 


1241.40 


1020.51 


866.412 


752.796 


38 


9046.75 


3508.02 


2175.98 


1577.21 


1236.94 


1017.49 


864.238 


751.155 


39 


8814.78 


3472.59 


2162.30 


1570.01 


1232.51 


1014.50 


862.075 


749.521 


40 


8594.42 


3437.87 


2148.79 


1562.88 


1228.11 


1011.51 


859.922 


747.894 


41 


8384.80 


3403.83 


2135.44 


1555.81 


1223.74 


1008.55 


857.780 


746.274 


42 


8185.16 


3370.46 


2122.26 


1548.80 


1219.40 


1005.60 


855.648 


744.661 


43 


7994.81 


3337.74 


2109.24 


1541.86 


1215.30 


1002.67 


853.527 


743.055 


44 


7813.11 


3305.65 


2096.39 


1534.98 


1210.82 


999.762 


851.417 


741.456 


45 


7639.49 


3274.17 


2083.68 


1528.16 


1206.57 


996.867 


849.317 


739.864 


46 


7473.42 


3243.29 


2071.13 


1521.40 


1202.36 


993.988 


847.228 


738.279 


47 


7314.41 


3212.98 


2058.73 


1514.70 


1198.17 


991.126 


845.148 


736.701 


48 


7162.03 


3183.23 


2046.48 


1508.06 


1194.01 


988.280 


843.080 


735.129 


49 


7015.87 


3154.03 


2034.37 


1501.48 


1189.88 


985.451 


841.021 


733.564 


50 


6875.55 


3125.36 


2022.41 


1494.95 


1185.78 


982.638 


838.972 


732.005 


51 


6740.74 


3097.20 


2010.59 


1488.48 


1181.71 


979.840 


836.933 


730.454 


52 


6611.12 


3069.55 


1998.90 


1482.07 


1177.66 


977.060 


834.904 


728.909 


53 


6486.38 


3042.39 


1987.35 


1475.71 


1173.65 


974.294 


832.885 


727.370 


54 


6366.26 


3015.71 


1975.93 


1469.41 


1169.66 


971.544 


830.876 


725.838 


55 


6250.51 


2989.48 


1964.64 


1463.16 


1165.70 


968.810 


828.876 


724.312 


56 


6138.90 


2963.71 


1953.48 


1456.96 


1161.76 


966.091 


826.886 


722.793 


57 


6031.20 


2938.39 


1942.44 


1450.81 


1157.85 


963.387 


824.905 


721.280 


58 


5927.22 


2913.49 


1931.53 


1444.72 


1153.97 


960.698 


822.934 


719.774 


59 


5826.76 


2889.01 


1920.75 


1438.68 


1150.11 


958.025 


820.973 


718.273 


60 


5729.65 


2864.93 


1910.08 


1432.69 


1146.28 


955.366 


819.020 


716.779 



* Table 10, above, may be used for Metric curves: Radii of curves in 
meters = values in above table mult, by r^o chord in meters. Ex. — For 

1637 28 
chord = 20 meters, and degree of curve = 3° 30': Radius = ^ — meters. 



1008 



.—RAILROADS. 



10. — Radii of English Curves, in Feet. — Concluded. 
(Chords 100 Ft.) 
Note. — See Foot-note preceding page, for use of this table for Metric curves. 
[Radii in Feet.] 





Degree of Curve.' 




Degree of Curve. 


Degree of 













Curve. 


i 


8° 


90 


3 


10° 


12° 


14° 


16° 


18° 


0' 


716 779 


637.275 


0' 


573.686 


478.339 


410.275 


359.265 


319.623 


20° 0' 


287.939 


1 


715.291 


636.099 


2 


571.784 


477.018 


409.306 


358.523 


319.037 


10 


285.583 


2 


713.810 


634.928 


4 


569.896 


475.705 


408.341 


357.784 


318.453 


20 


283.267 


3 


712.335 


633.761 


6 


568.020 


474.400 


407.380 


357.048 


317.871 


30 


280. 988 


4 


710.865 


632.599 


8 


566.156 


473.102 


406.424 


356.315 


317.292 


40 


278.746 


5 


709.402 


631.440 


10 


564.305 


471.810 


405.473 


355.585 


3U6.715 


50 


276.541 


6 


707.945 


630.286 


12 


562.466 


470.526 


404.526 


354.859 


316.139 


21° 0' 


274.370 


7 


706.493 


629.136 


14 


560.638 


469.249 


403.583 


354.135 


315.566 


10 


272.234 


8 


705.048 


627.991 


16 


558.823 


467.978 


402.645 


353.414 


314.993 


20 


270.132 


9 


703.609 


626.849 


18 


557.019 


466.715 


401.712 


352.696 


314.426 


30 


268.062 


10 


702.175 


625.712 


20 


555.227 


465.459 


400.782 


351.981 


313.860 


40 


266.024 


11 


700.748 


624.579 


22 


553,447 


464.209 


399.857 


351.269 


313.295 


50 


264.018 


12 


699.326 


623.450 


24 


551.678 


462.966 


398.937 


350.560 


312.732 


22° 0' 


262.042 


13 


697.910 


622.325 


26 


549.920 


461.729 


398.020 


349.854 


312.172 


10 


260.098 


14 


696.499 


621.203 


28 


548.174 


460.500 


397.108 


349.150 


311.613 


20 


258.180 


15 


695.095 


620.087 


30 


546.438 


459.276 


396.200 


348.450 


311.056 


30 


256.292 


16 


693.696 


618.974 


32 


544.714 


458.060 


395.296 


347.752 


310.502 


40 


254.431 


17 


692.302 


617.865 


34 


543.001 


456.850 


394.396 


347.057 


309.949 


50 


252.599 


18 


690.914 


616.760 


36 


541.298 


455.646 


393.501 


346.365 


309.399 


23° 0' 


250.793 


19 


689.532 


615.660 


38 


539.606 


454.449 


392.609 


345.676 


308.850 


10 


249.013 


20 


688.156 


614.563 


40 


537.924 


453.259 


391.722 


344.990 


308.303 


20 


247.258 


21 


686.785 


613.470 


42 


536.253 


452.073 


390.838 


344.306 


307.759 


30 


245.529 


22 


685.419 


612.380 


44 


534.593 


450.894 


389.959 


343.625 


307.216 


40 


243.825 


23 


684.059 


611.295 


46 


532.943 


449.722 


389.084 


342.947 


306.675 


50 


242.144 


24 


682.704 


610.214 


48 


531.303 


448.556 


388.212 


342.274 


306.136 


24° 0' 


240.487 


25 


681.354 


609.136 


50 


529.673 


447.395 


387.345 


341.598 


305.599 


10 


238.853 


26 


680.010 


608.062 


52 


528.053 


446.241 


386.481 


340.928 


305.064 


20 


237.241 


27 


678.671 


606.992 


54 


526.443 


445.093 


385.621 


340.260 


304.531 


30 


235.652 


28 


677.338 


605.926 


56 


524.843 


443.951 


384.765 


339.595 


304.000 


40 


234.084 


29 


676.008 


604.864 


58 


523.252 


442.814 


383.913 


338.933 


303.470 


50 


232.537 


30 


674.686 


603.805 




11° 


13° 


15° 


17° 


19° 


25° 0' 


231.011 


31 


673.369 


602.750 


0' 


521.671 


441.684 


383.065 


338.273 


302.943 


10 


229.506 


32 


672.056 


601.698 


2 


520.100 


440.559 


382.220 


337.616 


302.417 


20 


228.020 


33 


670.748 


600.651 


4 


518.539 


439.440 


381.380 


336.962 


301.893 


30 


226.555 


34 


669.446 


599.607 


6 


516.986 


438.326 


380.543 


336.310 


301.371 


40 


225.108 


35 


668.148 


598.567 


8 


515.443 


437.219 


379.709 


335.660 


300.851 


50 


223.680 


36 


666.856 


597.530 


10 


513.909 


436.117 


378.880 


335.013 


300.333 


26° 0' 


222.271 


37 


665.568 


596.497 


12 


512.385 


435.020 


378.054 


334.369 


299.816 


10 


220.879 


38 


664.286 


595.467 


14 


510.869 


433.929 


377.231 


333.727 


299.302 


20 


219.506 


39 


663.008 


594.441 


16 


509.363 


432.844 


376.412 


333.088 


298.789 


30 


218.150 


40 


661.736 


593.419 


18 


507.865 


431.764 


375.597 


332.451 


298.278 


40 


216.811 


41 


660.468 


592.400 


20 


506.376 


430.690 


374.786 


331.816 


297.768 


50 


215.489 


42 


659.205 


591.384 


22 


504.896 


429.620 


373.977 


331.184 


297.260 


27° 0' 


214.183 


43 


657. 947 


590.372 


24 


503.425 


428.557 


373.173 


330.555 


296.755 


10 


212.893 


44 


656.694 


589.364 


26 


501.962 


427.498 


372.372 


329.928 


296.250 


20 


211.620 


45 


655.446 


588.359 


28 


500.507 


426.445 


371.574 


329.303 


295.748 


30 


210.362 


46 


654.202 


587.357 


30 


499.061 


425.396 


370.780 


328.689 


295.247 


40 


209.119 


47 


652.963 


586.359 


32 


497.624 


424.354 


369.989 


328.061 


294.748 


50 


207.891 


48 


651.729 


585.364 


34 


496.195 


423.316 


369.202 


327.443 


294.251 


28° 0' 


206.678 


49 


650.499 


584.373 


36 


494.774 


422.283 


368.418 


326.828 


293.756 


10 


205.480 


50 


649.274 


583.385 


38 


493.361 


421.256 


367.637 


326.215 


293.262 


20 


204.296 


51 


648.054 


582.400 


40 


491.956 


420.233 


366.859 


325.604 


292.770 


30 


203.125 


52 


646.838 


581.419 


42 


490.559 


419.215 


366.085 


324.996 


292.279 


40 


201.969 


53 


645.627 


580.441 


44 


489.171 


418.203 


365.315 


324.390 


291.790 


50 


200.826 


54 


644.420 


579.466 


46 


487.790 


417.195 


364.547 


323.786 


291.303 


29° 0' 


199.696 


55 


643.218 


578.494 


48 


486.417 


416.192 


363.783 


323.184 


290.818 


10 


198.580 


56 


642.021 


577.526 


50 


485.051 


415.194 


363.022 


322.585 


290.334 


20 


197.476 


57 


640.828 


576.561 


52 


483.694 


414.201 


362.264 


321.989 


289.851 


30 


196.385 


58 


639.639 


575.599 


54 


482.344 


413.212 


361.510 


321.394 


289.371 


40 


195.306 


59 


638.455 


574.641 


56 


481.001 


412.229 


360.758 


320.801 


288.892 


50 


194.240 


60 


637.275 


573.686 


58 


479.666 


411.250 


360.010 


320.211 


288.414 


30° 0' 


193.185 








60 


478.339 


410.275 


359.265 


319.623 


287.939 







CIRCULAR CURVE TABLES, 



1009 



11. — *Semi-Tangents and Externals to a 1° English Curve, in Feet.I 

(Chords 100 Feet.) 
Note. — See Foot-note for using this table for Metric curves. 



Ex 
terna.l. 



.00 

.22 

.87 

1.96 

3.49 

5.46 

7.86 

10.71 

13.99 

17.72 

21.89 

26.50 

31.56 

37.07 

43.03 

49.44 

56.31 

63.63 

71.42 

79.67 

88.39 

97.58 

107.24 

117.38 

128.00 

139.11 

150.71 

162.81 

175.41 

188 51 

202.12 

216.25 

230.90 

246.08 

261.80 

278.05 

294.86 

312.22 

330.15 

348.64 

367.72 

387.38 

407.64 

428.50 

449.98 

472.08 

494.82 

518.20 

542.23 

566.94 

592.32 



Semi-Tangent. 



Dis- 
tance 

S-T 



0.00 
50.00 
100.01 
150.04 
200.08 
250.16 
300.28 
350.44 
400.66 
450.93 
501.28 
551.70 
602.21 
652.81 
703.51 
754.32 
805.25 
856.30 
907.49 
958.81 
1010.3 
1061.9 
1113.7 
1165.7 
1217.9 
1270.2 
1322.8 
1375.6 
1428.6 
1481.8 
1535.3 
1589.0 
1643.0 
1697.2 
1751.7 
1806.6 
1861.7 
1917.1 
1972.9 
2029.0 
2085.4 
2142.2 
2199 
2257 
2314 
2373 
2432 
2491 
2551 
2611 
2671.8 



Dlfl. 

for 

0° 10' 

d_ 
6 



8.333 
8.335 
8.338 
8.340 
8.347 
8.353 
8.360 
8.370 
8.378 
8.392 
8.403 
8.418 
8.433 
8.450 
8.468 
8.488 
8.508 
8.532 
8.553 
8.582 
8.600 
8.633 
8.667 
8.700 
8.717 
8,767 
8.800 
8.833 
8.867 
8.917 
8.950 
9.000 
9.033 
9.083 
9.150 
9.183 
9.233 
9.300 
9.350 
9.400 
9.467 
9.533 
9.600 
9.650 
9.733 
9.800 
9.867 
9.950 
10.033 
10.100 






d ft o 
O 

Add 



,000 
.001 
.001 
.002 
.003 
.003 
.004 
.005 
.005 
.006 
.006 
.007 
.008 
.008 
.009 
.010 
.010 
.010 
.011 
.012 
.013 
.014 
.014 
.015 
.016 
.017 
.017 
.018 
.019 
.019 
.020 
.021 
.021 
.022 
.023 
.024 
.024 
.025 
.026 
.026 
.027 
.028 
.028 
.029 
.030 
.031 
.031 
.032 
.033 
.034 
.034 






50° 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

61 

62 

63 

64 

65 

66 

67 

68 

69 

70 

71 

72 

73 

74 

75 

76 

77 

78 

79 

80 

81 

82 

83 

84 

85 

86 

87 



90 
91 
92 
93 
94 
95 
96 
97 
98 
99 
100 



Ex- 
ternal. 



E 



592.32 
618.39 
645.17 
672.66 
700.89 
729.85 
759.58 
790.08 
821.37 
853.46 
886.38 
920.14 
954.75 
990.24 
1026.6 
1063.9 
1102.2 
1141.4 
1181.6 
1222.7 
1265.0 
1308.2 
1352.6 
1398,0 
1444.6 
1492.4 
1541.4 
1591.6 
1643.0 
1695.8 
1749,9 
1805.3 
1862.2 
1920.5 
1980.4 
2041.7 
2104.7 
2169.2 
2235.5 
2303.5 
2373.3 
2444.9 
2518.5 
2594.0 
2671.6 
2751.3 
2833.2 
2917.3 
3003.8 
3092.7 
3184.1 



Semi-Tangent. 



Dis- 
tance. 

S-T 



2671.8 
2732.9 
2794.5 
2856.7 
2919.4 
2982.7 
3046.5 
3110.9 
3176.0 
3241.7 
3308.0 
3375.0 
3442.7 
3511.1 
3580.3 
3650.2 
3720.9 
3792.4 
3864.7 
3937.9 
4011.9 
4086.9 
4162.8 



4239.7 
4317.6 
4396.5 
4476.5 
4557.6 
4639.8 
4723.2 
4807,7 
4893.6 
4980.7 
5069.2 
5159.0 
5250.3 
5343.0 
5437.2 
5533.1 
5630.5 
5729.7 
5830.5 
5933.2 
6037.8 
6144.3 
6252.8 
6363.4 
6476.2 
6591.2 
6708.6 
6828.3 



Diff. 

for 

0° 10' 

d_ 
6 



O ft o 

Add 



10.183 
10.267 
10.367 
10.450 
10.550 
10.633 
10.733 
10.850 
10.950 
11.050 
11.167 
11.283 
11.400 
11.533 
11.650 
11.783 
11.917 
12.050 
12.200 
12.333 
12.500 
12.650 
12.817 
12.983 
13,150 
13,333 
13.517 
13.700 
13.900 
14.083 
14.317 
14.517 
14.750 
14.967 
15.217 
15.450 
15.700 
15.983 
16.233 
16.533 
16.800 
17.117 
17.433 
17.750 
18.083 
18.433 
18.800 
19.167 
19.567 
19.950 



.034 
.035 
.036 
.036 
.037 
.038 
.039 
.040 
.040 
.041 
.042 
.043 
.044 
.044 
.045 
.046 
.047 
.048 
.049 
.050 
.051 
.052 
,053 
.054 
,055 
.056 
,057 
.058 
.059 
.060 
.061 
.062 
.063 
.064 
.065 
.066 
.067 
.068 
.070 
.071 
.072 
.073 
.075 
.076 
,078 
,079 
.080 
.082 
.083 
.085 
.086 



Examples of Use 
of Table. 



Note. — Where ex- 
ternals are required 
accurately, add the 
following correc- 
tion to result ob- 
tained from table: 

(a) For /= 0° to 50°, 
Cor. = + .000003/2 2). 

(b) For 7= 100°, 
Cor.= -f .00004 /2D. 

In which / = inter- 
section angle, and 
Z)=degree of curve, 
both in degrees. 



fl CO "~ 
jj o 

tJo 

<o II 

« o5 
o > o 

■^ CO 

w 

6q 



IS 

II 'O 



o QJO 

II P. I 



00 o'^ 



1 ^11 

go 



a^ 



4.3 ■ 

sf 

X3 O g 

Ho 



'. 03 



11 lb ' 

^H fe t4 b tH 

o o o ^ o 



X 



3 fl 



00 .M c^ 

6. So 



o 



These results, to 
hundredths, are 
closer than will or- 
dinarily be meas- 
ured in the field. 
Note that the cor- 
rections are very 
small and can gen- 
erally be disre- 
garded when / and 
D are small. 



* Semi-tangents and externals are (almost exactly) inversely propor- 
tional to the degree of curve D, for the same intersection angle /. 

t Table 11, above, may be used for Metric curves: Semi-tangents and 

externals in meters = values in above table mult, by t^o chord in meters. 

Ex. — For chord = 20 meters, and intersection angle = 12°; then semi-tang = 

602.21 ^ ^ . 1 31.56 . . ,o 

■ — = — meters, and external -= — ■_ — meters, for a 1° curve. 





lOiO 



l^RAILROADS. 



12. — ^Minutes and Seconds Reduced to Decimals op a 
Degree or Hour.* 

(Either Angular Measure or Time Measure.) 
Note.— -See Foot-note regarding use of this table. 
[Decimals of a Degree or Hour.] 



•o o 



p.p. 

.00139 
00028 
00056 
00083 

coin 



d 


Seconds. 


i 


0" 


5" 


10* 


15" 


20" 


25" 


30" 


35" 


40" 


45" 


50" 


55* 


c 


.00000 


.00139 


00278 


00417 


.00556 


.00694 


.00833 


.00972 


.01111 


.01250 


.01389 


01528 


1 


.01667 


.01806 


.01944 


02083 


.02222 


.02361 


.02500 


.02639 


.02778 


.02917 


.03055 


03194 


2 


.03333 


.03472 


03611 


03750 


.03889 


.04028 


.04167 


.04306 


.04444 


.045831.047221 


04861 


3 


.05000 


.05139 


05278 


.05417 


.05556 


.05694 


.05833 


.05972 


.06111 


.06250 


.06389 


06528 


4 


.06667 


.06806 


.06944 


.07083 


.07222 


.07361 


.07500 


.07639 


.07778 


.07917 


. 08056 


.08194 


5 


.08333 


.08472 


.08611 


.08750 


.08889 


.09028 


.09167 


09306 


.09444 


.09583 


.09722 


09861 


6 


.10000 


.10139 


.10278 


.10417 


.10556 


.10694 


.10833 


.10972 


.11111 


.11250 


.11389 


11528 


7 


.11667 


.11806 


11944 


.12083 


.12222 


.12361 


.12500 


.12639 


.12778 


.12917 


.13056 


.13194 


8 


.13333 


.13472 


.13611 


.13750 


.13889 


.14028 


.14167 


.14306 


.14444 


.14583 


.14722 


.14861 


9 


.15000 


.15139 


.15278 


15417 


.15556 


.15694 


.15833 


.15972 


.16111 


.16250 


.16389 


.16528 


10 


.16667 


.16806 


.16944 


17083 


.17222 


.17361 


.17500 


.17639 


.17778 


.17917 


.18056 


.18194 


11 


.18333 


.18472 


.18611 


.18750 


.18889 


.19028 


.19167 


.19306 


.19444 


.19583 


.19722 


.19861 


12 


.20000 


.20139 


.20278 


.20417 


.20556 


,20694 


.20833 


.20972 


.21111 


.21250 


.21389 


.21528 


13 


.21667 


.21806 


.21944 


.22083 


.22222 


.22361 


.22500 


.22639 


.22778 


.22917 


.23056 


.23194 


14 


.23333 


.23472 


.23611 


.23750 


.23889 


.24028 


.24167 


.24306 


.24444 


.24583 


.24722 


.24861 


15 


. 25000 


.25139 


.25278 


.25417 


.25556 


.25694 


.25833 


.25972 


.26111 


.26250 


.26389 


.26528 


16 


.26667 


.26806 


.26944 


.27083 


.27222 


.27361 


.27500 


.27639 


.27778 


.27917 


.28056 


.28194 


17 


.28333 .28472 | 


.28611 


.28750 


.28889 


.29028 


.29167 


.29306 


.29444 


.29583 


.29722 


.29861 


18 


.30000 


.30139 


.30278 


.30417 


.30556 


.30694 


.30833 


.30972 


.31111 


.31250 


.31389 


.31528 


19 


.31667 


31806 


,31944 


.32083 


.32222 


.32361 


.32500 


.32639 


.32778 


.32917 


.38056 


.33194 


20 


.33333 


.33472 


.33611 


.33750 


.33889 


.34028 


.34167 


.34306 


.34444 


.34583 


.34722 


.34861 


21 


.35000 


.35139 


.35278 


.35417 


.33556 


.35694 


.35833 


.35972 


.36111 


.36250 


.36389 


.36528 


22 


.36667 


.36806 


.36944 


.37083 


.37222 


.37361 


.37500 


.37639 


.37778 


.37917 


.38056 


.38194 


23 


.38333 


.38472 


.38611 


.38750 


.38889 


.39028 


.39167 


.39306 


.39444 


.39583 


.39722 


.39861 


24 


.40000 


.40139 


.40278 


.40417 


.40556 


.40694 


.40833 


.40972 


.41111 


.41250 


.41389 


.41528 


25 


.41667 


.41806 


.41944 


.42083 


.42222 


.42361 


.42500 


.42639 


.42778 


.42917 


.43056 


.43194 


26 


.43333 


.43472 


.43611 


.43750 


.43889 


.44028 


.44167 


.44306 


.44444 


.44583 


.44722 


.44861 


27 


.45000 


.45139 


.45278 


.45417 


.45556 


.45694 


.45833 


.45972 


.46111 .46250 


.46389 


.46528 


28 


.46667 


.46806 


.46944 


.47083 


.47222 


.47361 


.47500 


.47639 


.47778 


.47917 


.48056 


.48194 


29 


.48333 


.48472 


.48611 


.48750 


.48889 


.49028 


.49167 


.49306 


.49444 


.49583 


.49722 


.49861 


30 


'. 50000 


.50139 


.50278 


.50417 


.50556 


.50694 


. 50833 


. 50972 


.51111 


.51250 


.51389 


.51528 


31 


.51667 


.51806 


.51944 


.52083 


.52222 .52361 


.52500 


.52639 


,52778 


.52917 


.53056 


.53194 


32 


.53333 


.53472 


.53611 


.53750 


.53889 


. 54028 


.54167 


.54306 


.54444 


. 54583 


.54722 


.54861 


33 


. 55000 


.55139 


.55278 


.55417 


. 55556 


.55694 


.55833 


.55972 


.56111 


.56250 


.56389 


.56528 


34 


.56667 


.56806 


.56944 


.57083 


.57222 


.57361 


.57500 


.57639 


.57778.57917 


.58056 


.58194 


35 


. 58333 


.58472 


.58611 


.58750 


. 58889 


59028 


.59167 


.59306 


.59444 .59583 


.59722 


.59861 


36 


.60000 


.60139 


.60278 


.60417 


.60556 


.60694 


.60833 


.60972 


.61111 


.61250 


.61389 


.61528 


37 


.61667 


.61806 


.61944 


.62083 


.62222 


.62361 


.62500 


.62639 


.62778 


.62917 


.63056 


.63194 


38 


.63333 


.63472 


.63611 


63750 


.63889 


.64028 


.64167 


.64306 


.64444 


.64583 


.64722 


.64861 


39 


.65000 


.65139 


.65278 


.65417 


.65556 


.65694 


.65833 


.65972 


.66111 


.66250 


.66389 


.66528 


40 


.66667 


.66808 


.66944 


.67083 


.67222 


.67361 


.67500 


.67639 


.67778 


.67917 


.68056 


.68194 


41 


.68333 


.68472 


.68611 


.68750 


.68889 


.69028 


.69167 


.69306 


.69444 


.69583 


.69722 


.69861 


42 


.70000 


.70139 


.70278 


•70417 


.70556 


.70694 


.70833 


.70972 


.71111 


.71250 


71389 


.71528 


43 


.71667 


.71806 


.71944 


.72083 


.72222 


.72361 


.72500 


.72639 


.72778 


.72917 


.73056 


.73194 


44 


.73333 


.73472 


.73611 


•73750 


.73889 


.74028 


.74167 


.74306 


.74444 


.74583 


.74722 


.74861 


45 


.75000 


.75139 


.75278 


.75417 


.75556 


.75694 


.75833 


.75972 


.76111 


.76250 


.76389 


.76528 


46 


.76667 


.76806 


.76944 


.77083 


.77222 


.77361 


.77500 


.77639 


.77778 


.77917 


.78056 


.78194 


47 


.78333 


.78472 


.78611 


.78750 


.78889 


.79028 


.79167 


.79306 


.79444 


.79583 


.79722 


.79861 


48 


.80000 


.80139 


.80278 


.80417 


.80556 


.70694 


.80833 


.80972 


.81111 


.81250 


.81389 


.81528 


49 


.81667 


81806 


.81944 


.82083 


.82222 


.82361 


.82500 


.82639 


.82778 


.82917 


.83056 


.83194 


50 


.83333 


.83472 


.83611 


.83750 


.83889 


.84028 


.84167 


.84306 


.84444 


.84583 


.84722 


.84868 


51 


.85000 


.85139 


.85278 


.85417 


.85556 


.85694 


.85833 


.85972 


.86111 


.86250 


.86389 


.86528 


52 


.86667 


.86806 


.86944 


.87083 


.87222 


.87361 


.87500 


.87639 


.87778 


.87917 


.88056 


.88194 


53 


.88333 


.88472 


.88611 


.88750 


.88889 


.89028 


.89167 


.89306 


.89444 


.89583 


.89722 


.89861 


64 


. 90000 


.90139 


.90278 


.90417 


. 90556 


. 90694 


.90833 


.90972 


.91111 


.9125C 


.91389 


.91528 


55 


.91667 


.91806 


.91944 


.92083 


.92222 


.92361 


.92500 


.92639 


.9277? 


.92917 


. 93056 


.93194 


56 


.93333 


.93472 


.93611 


.93750 


.93889 


.94028 


.94167 


.94306 


.94444 


.94583 


.94722 


.94861 


57 


.95000 


.95139 


.95278 


.95417 


.95556 


.95694 


.95833 


.95972 


.96111 


.9625C 


.96389 


.96528 


58 


. 96667 


.96806 


.96944 


.97083 


.97222 


..97361 


.97500 


.97639 


.97778 


.97917 


.98056 


.98194 


59 


.98333 


.98472 i. 98611 


.98750 -.98889 1.99028 


.99167 


.99306 


.99444). 99583'. 99722 


.99861 



♦Ex. In Angular measure: 20° 35' 17" = 20° 35' 15" + 2" = 20.58806 deg. 
Ex. In Time measure; 20b . 35m.. 17 s = 20h.. 35 m., i5 s + 2 s = 20.58806 hrs. 



MINUTES TO DEGREES. CURVE PROBLEMS. 



1011 



The Various Problems in Simple Curves may all be solved without the 
use of formulas. It is necessary only to draw a sketch of the conditions of 
the problem for any particular case, set down the known data governing the 
case, and solve trigonometrically for the unknown. The position of the 
center of the circle or circular arc is the key to all solutions. A few hints 
will be given to illustrate: ^ 

To move a curve (shown in full lines, Fig. 8) >>'^^''^^^^Ss^ T 

joining two tangents {Tx and T2), so it will end ^^^'\'^"^^^^%^ 

(dotted lines) in a parallel tangent (^3): The dis- \y/\ V ,y^^^^ 

tance moved ( = m)=the perpendicular distance ^ \ ?V "^" 

between the tangents {d) -^ sine of intersection angle \M^ 

(/). of 

To change the radius r (full) to r' (dotted) so that Fig. 8. 

curve ending in tangent T2 shall end in parallel tan- ^,«s=::: — ^^^ 

gent Tz (Fig. 9) : The end of the new curve lies on y^-- — -^■>=3^^^> 

the long chord as shown. Then n—d-^sin -77, and ^ ^^'\ / ^^S^ 

r— /= Y"^^^^Y' ^^^^®^°^® ^'~^~ (^"^2 sin2— j , in 

which cf = perpendicular distance between 7^2 and T3. Or, r' = r — d-^ (1 — 
cos /) = r — cJ -7- vers /. The reverse holds true when radius is increased. 

To join a curve (c)' and a point (P) by a tan- ^ r \' 

gent {T 2)'. The tangent Ti and the curve c fit ><'''''^5l 

the contour line around the hill H, and it is 'S'^\ H /%^*"*vv.J 
proposed to throw off a tangent T2, from some ^ '^v— i' • ^^^^ 
point on the curve, so it will pass through the ^^IVijgo' ^g^^s^^ 

point P, ahead. One practical method of doing "3 t~-^^ 

this in the field is to assume the position -p. ^^ 

of the P. T.y turn the angle for the tangent ^^^' "* 

ahead, measure the distance and offset to P, and from this data calculate 
the true position of the P. T., and "move ahead" or "back up" accordingly. 
If, however, the position of the point P has been located, the position of the 
P. T. can be fixed accurately by calculation. Reduce the location of P 
(no matter how located) to the distance d from the center of the circle at d>, 
and calculate the angle a. The distance r is the radius of the curve. This 
gives a calculated tie from the P. C. to P. Now 

x-^-^y'^ = r'^..,. (1) 

and Tan^ = - = — — ; oty'^ + x^^dx (2) 

y d — x 

Equating (1) and (2). dx = r'^, °^^==T ^^^ 

From which y, tan /?( = — ) ,/ (= a— 90°+i9), etc., can be found. 

Compound Curves are made up of two or more curves of different radii, 
directly joining each other, and curving in the same direction. They are 
not difficult of analysis. They key to the solution of any problem lies in 
putting the data in the right shape and work- ^ ^ 

ing from the centers of the curves. Fig. 11 ^^\ ^-i^.^ 

needs no explanation other than that the f^' ^^If/'^'u 

P. C. C. is the point of compound curve, and .j^-'^V'-^. ^I /^/ ^.^> V 

the total angle^ of intersection between tan- ""^/"^^^""'^^ ^-l ^"""■^^'' 

gents Ti and T2 is the sum of the separate in- ^ y^ a:j ,;iCv > 
tersection angles 7 and i. Note also that the J^^'CX ' ^'''^* ^** 

length of tangent to the P. C. C. is equal to ^ \ l*/V 

the S-T of the large curve + the s-t of the ^k j/ 

small curve; hence the total vertex distances, \ ['0| 

y and V, from the P. C. and P. T., respec- \\«! 

tively, bear a relation to the rest of the data. \ I 

The following table illustrates a few simple lO 

problems and solutions which may be had „. * 

from Fig. 11. The known quantities are indi- l^ig. 11. 

cated by letters; and the answers required, by blank spaces in each line. 
The formula for solution in each case is at the right. Problems with other 
data may be reduced to these forms before solving. 



1012 



.—RAILROADS, 



13. — Solutions op Compound Curve Problems. 
(See Fig. 11). 
Note, — Capital letters refer to the curve of larger radius; small letters 
to curve of smaller radius. 



No. 


Angles of 
Intersection. 


Radii. 


Vertex 
Dlstanc's 


Solution. 




/+i 


I 


i 


R 


r 


V 


V 


(Descriptions refer to Fig. 1 1.) 


1 


I+i 

I+i 
I+i 


I 

I 


i 
ri 


R 
R 

. . R 


r 

r 

r 
r 
r 






/Solve for S-T, s-t, small triangle, V, v. 


2 
3 


V 
V 
V 

V 


V 
V 

V 
V 


\(I+i). 

{R-r= [R vers (/+ i)-V sin (I+i)] -i- vers i; 

]and v=R sin iI+i)—(R—r) sin i—V cos 

[(I+i). 

fVers i=[R vers (I+i)-V sin (/+i)I-5- 

i (R—r) ; and v=R sin (,1+i)— (R—r) sin i— 


4 






. . R 


IF cos (I+i). 

fTan ^I=[R vers (I+i)-V sin (I+i)]-r- 

\[R sin (I+i)—V cos (I+i) — v]; and R—r 


5 


I 


ri 


. . R 


[ = [R sin (7+ i)-V cos (7+ i) - ?;] h- sin i. 

(R-r=[v sin (I+i)-r vers (7+ i) ] ^ vers 7 ; 

^and y=(72-r) sin I+r sin (7+i)-v cos 

[(I+i). 

fVers 7=[v sin (74-i)-r vers (7+^)l-^ 

U72- r) ; and F = (72- r) sin 7+ r sin (7+ i) 

[—v cos (7+i). 

fTan ^I=[v sin (7+i)-r vers (I+i)]-i- 

^[F+2;cos(7+i)-rsin (74-i); and72-r== 

[[V+v cos (74- i)-r sin (74-i)]^sin 7. 


7 

















Note. — For Nos. 1, 2 and 5, either / or i may be given. 

_ Reversed Curves should be avoided, especially for main line traffic. 
This may be done by "separating" the two simple curves on either side 
of the point of reversed curve (P. R. C.) and joining them by a short 



cr-:^- 




/ 


T, * 


^ 


oo 




-4 


¥ 




Fig. 12. 

tangent. The curves may be separated by moving them back from the 
P. C. and P. T., or by making them "sharper." To find the relation exist- 
ing between the known and unknown data in Fig. 12: 
Let d = th.e length of the line joining the P. C. with the P. T.; 

a = the angle which the line d makes with Ti ; 

^ = the angle which the line d makes with T2. 
Then sin A = ^ (cos a 4- cos /?), from which the angle A, is obtained; 
I=a+90°-A\ 
t = /?+90°-A=7-a4-/?; 

and R=(d sin A) -i- (sin 7 + sin -j) (1) 

In Fig. 13 the distance d is measured between 
points on two tangents Tt and T2', and the angles/ and 
i, which the line d makes with these tangents, are 
known. It is required to fit in a reverse curve of com- 
mon radius R: 




tan h 1 + tan \ i 



Fig. 13. 



COMPOUND, REVERSED AND EASEMENT CURVES, 1013 




The Cubic Parabola is the principal form of parabolic curve used, and 
is fundamental to most so-called spiral- or easement curves. The equation 
of the cubic parabola is y = nx^, in which x is the abscissa, and y the ordi- 
nate 'to any pointy, Fig. 14. The constant « is a decimal, and may have 
any value: small for a flat curve, and 
large for a sharp curve. If the cubic para- 
bola is laid off by deflection angles from 
the point o on the tangent T, then the 
deflection angle* to point 1 is d; to point 
2, id; to point 3, 9d; to point 4, 16d; to 
point 5, 25d\ etc. That is, while the ordi- 
nate y to any point p is proportional to 
x^, the deflection angle is proportionate 
to x^. It is to be noted also that x is 
measured along the tangent produced and not along the curve. The effect 
of this is, of course, to gradually increase the lengths of the "stations" from 
point o: as 4-5>3-4>2-3> 1-2 >0-l. For this reason the cubic parabola 
has never been popular, but has given way to the spiral curve. Both of 
them are very flat at the ends and grow rapidly sharper toward the center 
of the curve. 

The Spiral Curve is a modification of the 
cubic parabola. It is based on chords of equal 
length, with the curve compounded at the end 
of each chord. The chords may be of any 
length from 10 ft. to 50 ft., while 30 ft. is 
quite usual. The degree of curve of the first 
arc, subtended by the chord 0-1, is the com- 
mon difference for the degree of curve of 
successive arcs. Thus, if the central angle 
of the curve 0-1 is A, then 0-1 will incline 
at an angle of ^A with the tangent T; 1-2, at an angle of \AXi = 2A\ 
2-3, MX9=4M; 3-4, MX16=8A; 4-5, M X 25= 12^; etc. This can 
be proved by making the central angles A, 2A, SA, iA, 5A, respectively. 
The offsets from tangent T to points 1, 2, 3, etc. are calculated successively, 
knowing the length of chords and their angles of inclination with T; as are 
also the horizontal distances between the offset lines, as xu X2, etc. Then, 




Fig. 15. 



the tangent of the deflection angle from o to any point on the curve is 



yt 




Simple Easement Curves are sometimes approximated from simple 
curves, as follows: Run out the regular simple curve from the tangent T 
as shown dotted. Lay off the required curve in- 
side, by measuring the offset distance d from 
the outer curve. Lay off the point P. S. of the 
spiral starting from the tangent, and also the 
end of the spiral at the P. C. C. where it joins 
the new curve, equidistant from the point c, which 
bisects the offset distance d at the P. C. The 
offset distance may average from 2 to 4 ft.; and 
the distance from P. S. to P. C. may vary from 
50 ft. to 100 ft. The writer can recommend these curves for fast-train 
service. Note that the inner curve may be "run in" directly with the 
instrument, the outer curve being omitted. 

E.— RIGHT OF WAY. 
Filing of Location. — After the final location is adopted it is filed with the 
Secretary of State for the particular State through which the line is pro- 
jected. The following is a typical description of location: Beginning at a 
stake on the shore of Huron Bay, near high-water mark and westerly 
about 325 feet from the westerly corner of the old stone fort in the town of 
Stamford, county of Huntoon, and State of Ohio; thence north 84° 30' 
West, four thousand three hundred ten (4310) feet to a stake in the county 
road in front of Judge Pritchard's farm-house; thence by a curve to the 

* Approximately; but almost exact for flat curves. To be strictly exact, 
the ^'natural tangent of the deflection angle" should be substituted for "de- 
flection angle." 

t The "Railroad Spiral," by W. H. Searles, contains tables for laying 
out any spiral in the field, with instrument at any point on the spiral. 



1014 



59.— RAILROADS, 



right, with a radius of nineteen hundred ten (1910) feet, a distance of three 
hundred fifty (350) feet to a stake; thence by a tangent to said curve 
North 74° West, 12652 feet to a stake etc. 

Purchase and Condemnation. — The Real -Estate Agent of the road is 
provided with maps of the located line showing land lines, owners' names, 
and widths of right-of-way desired through the various parcels of land. 
The usual width is 100 feet, but this should be exceeded in the case of heavy 
cuts and fills. The following table. No. 14, will be useful for reference in 
this connection. Where land cannot be purchased at a reasonable figure it 
may be condemned and the condemnation price so fixed is called an 
"award." 

Oftentimes the award may include a parcel of land of considerable size, 
not required strictly for right-of-way purposes. In such cases it is a great 
mistake for the R. R. Co. to dispose of any of this land without first con- 
sidering whether it is liable to be needed for a passing siding, or for a spur 
track to some prospective manufacturing plant. The writer can instance 
many cases in which supposed surplus property has been disposed of at a 
low figure and repurchased for an amount four or five times as great. 

14. — ^Table for Finding Width of Right-of-Way for Cuts and 

Fills. 

Note. — Total width between slope stakes = width of base of roadway -j- 
sum of horizontal distances for slopes (from table) . 

[Horizontal Distance in Feet for One Slope.] 



•^ . 
























ss 








Side Slope. 












|H 


itol. 


itol. 


ito 1. 


1 to 1. 


ltd. 


litol. 


litol. 


Iftol. 


2 to 1. 


2itol. 


2itol. 


4 


.8 


1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


8 


1.6 


2 


4 


6 


8 


10 


12 


14 


16 


18 


20 


12 


2.4 


3 


6 


9 


12 


15 


18 


21 


24 


27 


30 


16 


3.2 


4 


8 


12 


16 


20 


24 


28 


32 


36 


40 


20 


4.0 


5 


10 


15 


20 


25 


30 


35 


40 


45 


50 


24 


4.8 


6 


12 


18 


24 


30 


36 


42 


48 


54 


60 


28 


5.6 


7 


14 


21 


28 


35 


42 


49 


56 


63 


70 


32 


6.4 


8 


16 


24 


32 


40 


48 


56 


64 


72 


80 


36 


7.2 


9 


18 


27 


36 


45 


54 


63 


72 


81 


90 


40 


8.0 


10 


20 


30 


40 


50 


60 


70 


80 


90 


100 


44 


8.8 


11 


22 


33 


44 


55 


66 


77 


88 


99 


110 


48 


9.6 


12 


24 


36 


48 


60 


72 


84 


96 


108 


120 


52 


10.4 


13 


26 


39 


52 


65 


78 


91 


104 


117 


130 


56 


11.2 


14 


28 


42 


56 


70 


84 


98 


112 


126 


140 


60 


12.0 


15 


30 


45 


60 


75 


90 


105 


120 


135 


150 


64 


12.8 


16 


32 


48 


64 


80 


96 


112 


128 


144 


160 


68 


13.6 


17 


34 


51 


68 


85 


102 


119 


136 


153 


170 


72 


14.4 


18 


36 


54 


72 


90 


108 


126 


144 


162 


180 


76 


15.2 


19 


38 


57 


76 


95 


114 


133 


152 


171 


190 


80 


16.0 


20 


40 


60 


80 


100 


120 


140 


160 


180 


200 


84 


16.8 


21 


42 


63 


84 


105 


126 


147 


168 


189 


210 


88 


17.6 


22 


44 


66 


88 


110 


132 


154 


176 


198 


220 


92 


18.4 


23 


46 


69 


92 


115 


138 


161 


184 


207 


230 


96 


19.2 


24 


48 


72 


96 


120 


144 


168 


192 


216 


240 


100 


20.0 


25 


50 


75 


100 


125 


150 


175 


200 


225 


250 


104 


20.8 


26 


52 


78 


104 


130 


156 


182 


208 


234 


260 


108 


21.6 


27 


54 


81 


108 


135 


162 


189 


216 


243 


270 


112 


22.4 


28 


56 


84 


112 


140 


168 


196 


224 


252 


280 


116 


23.2 


29 


58 


87 


116 


145 


174 


203 


232 


261 


290 


120 


24.0 


30 


60 


90 


120 


150 


180 


210 


240 


270 


300 



Ex. — Width of base of roadway =28 ft.; height. 48 ft. (ground slope 
about level;) side slopes, li to 1. Then width between slope stakes=28 + 
60+60=148 ft. (Retaining walls are often used to narrow the required 
right-of-way.) 



RIGHT-OF-WAY TABLES, 



1015 



15. — Acres per 100-Ft. Station for Various Widths in Feet. 
Note. — Supply space occupied by dash in left-hand column by the unit 
figure at top of columns. 

[Acres per 100-Ft. Station.] 



P. P 



^5 


0. 


1. 


2. 


3. 


4. 


5. 


6. 


7. 


8. 


9. 


0— . 




. 00230 


.00459 


.00689 


.00918 


.01148 


.01377 


.01607 


.01837 


.02066 




1— . 


.02296 


.02525 


.02755 


.02984 


.03214 


.03444 


.03673 


.03903 


.04132 


. 04362 


2—. 


.04591 


.04821 


.05051 


.05280 


.05510 


.05739 


.05969 


.06198 


.06428 


.06657 


3—, 


.06887 


.07117 


.07346 


.07576 


.07805 


.08035 


.08264 


.08494 


.08724 


.08953 


4—. 


.09183 


.09412 


.09642 


.09871 


.10101 


.10331 


. 10560 


.10790 


.11019 


.11249 


5—. 


.11478 


.11708 


.11938 


.12167 


.12397 


.12626 


.12856 


.13085 


.13315 


.13545 


6~. 


.13774 


.14004 


.14233 


.14463 


.14692 


.14922 


.15152 


.15381 


.15611 


.15840 


7—. 


.16070 


.16299 


.16529 


.16758 


.16988 


.17218 


.17447 


.17677 


.17906 


.18136 


8—. 


.18365 


.18595 


.18825 


.19054 


.19284 


.19513 


.19743 


.19972 


.20202 


.20432 


9—. 


.20661 


.20891 


.21120 


.21350 


.21579 


.21809 


.22039 


.22268 


.22498 


.22727 


— . 


.22957 


.23186 


.23416 


.23646 


.23875 


.24105 


.24334 


.24564 


.24793 


.25023 



For 

Tenths 

.00230 



. 00023 
. 00046 
.00069 
. 00092 
.00115 
.00138 
.00161 
.00184 
.00207 



Ex.— Width =90. 75 ft. Then we have 0.20661 + 0.00161 + 0.00011 = 
0.20833 acre per 100-ft. station. 

16. — Acres per Mile for Various Widths in Feet. 
(See also Table 17, following.) 
Note. — Supply space occupied by dash in left-hand column by the unit 
figure at top of column. 

[Acres per Mile.] 



5^ 
























2P^ 


0. 


1. 


2. 


3. 


4. 


5. 


6. 


7. 


8. 


9. 


P. P 


0—. 




.121 


.242 


.364 


.485 


.606 


.727 


.848 


;970 


l.,091 


For 




1—. 


1.212 


1.333 


1.455 


1.576 


1.697 


1.818 


1.939 


2.061 


2.182 


2.303 


Tenths 


2—. 


2.424 


2.545 


2.667 


2.788 


2.909 


3.030 


3.152 


3.273 


3.394 


3.515 


.121 


3—. 


3.636 


3.758 


3.879 


4.000 


4.121 


4.242 


4.364 


4.485 


4.606 


4.727 


1 


.012 


4—. 


4.848 


4.970 


5.091 


5.212 


5.333 


5.455 


5.576 


5.697 


5.818 


5.939 


2 


.024 


5—. 


6.061 


6.182 


6.303 


6.424 


6.545 


6.667 


6.788 


6.909 


7.030 


7.152 


6 

4 


.036 
.048 


6—. 


7.273 


7.394 


7.515 


7.636 


7.758 


7.879 


8.000 


8.121 


8.242 


8.364 


5 


.061 


7—. 


8.485 


8.606 


8.727 


8.848 


8.970 


9.091 


9.212 


9.333 


9.455 


9.576 


6 


073 


8—. 


9.697 


9.818 


9.9^9 
11.152 


10.061 


10.182 


10.303 


10.424 


10.545 


10.667 


10.788 


7 


.085 


9—. 


10.909 


11.030 


11.273 


11.394 


11.515 


11.636 


11.758 


11.879 


12. poo 


8 


.097 


10—. 


12.121 


12.242 


12.364 


12.485 


12.606 


12.727 


12.848 


12.970 


13.091 


13.212 


y 


.109 



Ex.— Width = 90. 75 ft. Then we have 10.909+0.085 + 0.006=11.000 
acres per mile. 

17. — Acres for 1 Foot in Width and for Various Distances in 

Miles. 
(Use above Table No. 16, changing the headings as below.) 
Note. — For strip 100 ft. wide, mult, values in table by 100. 
[Acres for 1 Foot in Width.] 



Dis- 




■ 




















tance. 


0. 


1. 


2. 


3. 


4. 


5. 


6. 


7. 


8. 


9. 


P.P. 


Miles. 






























Body of Table same as Table 16. 









Ex. — Strip 1 ft. wide and 90.75 miles long. Then we have 10.909 + 
0.085+0.006= 11.000 acres. For strip 100 ft. wide we have 1100 acres. 



1016 59.— RAILROADS. 

F.— CONSTRUCTION. 

Earthwork Calculations. — The preliminary and location siirvey maps 
and profiles should give all the information necessary to estimate the cost 
of constructing the line. Borings should be made or test-pits dug occasion- 
ally in order to classify the material in excavation. Several methods are 
used in making preliminary estimates of earthwork, and the three following 
are worthy of note: 

(1). — ^The cross-section at each station, and at intermediate points if 
necessary, may be plotted on cross-section paper from the profile and 




Fig. 17. 

topography notes, as shown in Fig. 17. The distances D, d, H, h (and p) 
are then scaled and used in the following formula for area of cross-section 
in "cut," and similarly in "fill:" 

Area in sq. ft. = ^ [p{D + d) + 10 {H+h)'\ (1) 

where 10 ft. is J the width of roadbed in excavation. For quantities in fill, 
10 would be replaced by 7, if the roadbed is to be 14 ft. wide. The cubical 
contents in feet of the solid figure of which the above cross-section is con- 
sidered as the "average area" is found by multiplying this area by ^ the sum 
of the distances to the adjacent cross-sections on either side. The summa- 
tion of the contents in cubic feet is reduced to cubic yards, as the cost price 
is in that denomination. (Use planimeterif preferred.) 

(2) . — If the topography has been taken with a clinometer and the ground 
line is straight, as in Fig. 18, instead of broken as 
shown in Fig. 17, there are three methods in use for 
determining areas: 

(a). — Plat the ground line G by means of the 
profile height h and the angle of inclination a; scale 
G and H. Then, for the etched portion or cut: 
ATea.= ^GH — C\ in which C is the area of triangle 
below roadbed base, to V. In the figure, the con- 
stant C is 100 sq. ft. It varies with the side slopes 
and width of roadbed. (Use planimeter if preferred.) 

(b). — Prepare tables of areas for various heights h, and various ground 
slopes a; and for the standard roadbed and side slopes. For loose rock in 
excavation the side slope may be say i : 1; in embankment, 1 : 1. Earth 
1 : 1 for cut, and 1^:1 for fill. The calculation of the table may be based 
on the following formula: 

Area required, cut or fill = K^ + ^)^ cos2a[cot(./34- a)4-cot(;9- a)]-C (2) 

In which Area required = area of the etched portion in Fig. 18, 
/j = center height of cut or fill, 
7; = vertex distance below roadbed, 
a = angle of ground slope with the horizontal, 
^ = angle of side slopes with the horizontal, 
C' = area of triangle (ht. = 'y) below roadbed. 

When /?=45° the formula is somewhat simplified. It is to be noted* that 
{h-{-v) may be assumed' as unity for various ground and side slopes, and 
afterward expanded by squares, finally deducting the value of the constant 
area C. The ground slope must not intercept the roadbed. 

(c). — ^Tables may be prepared giving correction areas for slopes, to be 
added to tables for "level sections" (see Case 3), instead of tables of 
actual areas as previously described. From Fig. 18 it will be seen that 
sloping ground always gives a plus correction, as the small triangle below 
the elevated side of the ground line G is greater than the one above the 
depressed side. The correction tables may be for areas of cross-section, or 




CONSTRUCTION. EARTHWORK. 1017 

for cubic yards or cubic feet per station of 100 ft., 50 ft., etc. See Table 22, 
page 1030; also Table 27, page 1039. 

(3), — Tables of "level sections" may be used in preliminary estimates 
where the ground is fairly "level;" or in connection with the previously 
described correction tables, where the transverse ground line is sloping. 
These tables may be calculated from the following formula, modified from 
formula (2), page 1016, by making a = 0, and using the previous notation: 

^=Area required = (/j + z;) 2 cot 0-C (3) 

Or, giving C ( = v'^ cot P) its value in terms of v and /?, 

A = (h^+2hv) cot (3) 

When the side slopes are 1:1, /? = 45°, and 

A = (h-hv)^'-C=h^-h2hv (4) 

When the side slopes are IJ : 1, 

A = i (h + vy-C = i(h^-{-2 hv); (5) 

and similarly for any other side slope. 

These tables may be copied on profile paper in the form of cubic yards 
per 100-ft. station, and arranged so that the quantities can be read off 
directly by matching the zero of the table to the grade line of the profile, 
and reading the quantity opposite the ground line. The profile tables are 
usually to feet in height; or to feet and half feet. 



List op Earthwork Tables, Following. 
(And also tables relevant thereto.) 

Table No. Description. Page. 

18. Multiplication table, up to 60X60 1018-1019 

18a. Multiplication table, up to 89X35 1020 

19. End areas reduced to cu. yds. per station (Equiv. 1-10) 1021 

20. End areas (1-27000) reduced to cu. yds. per station 1022-1027 

21. Method of calculating earthwork tables 1028-1029 

22. Formulas for calculating ground-slope quantities 1030-1033 

23. Level sections; Heights, 0-60 ft. ; Roadway, 14 ft. ; Slopes. IH to 1 1034 

24. " " " 60-120" " 14" " li^tol 1035 

25. •• " '• 0-60 " " 16 " " Itol 1036 

26. •• *• " 0-60" " 16" " 13^tol 1037 

27. •* " •• 0-60 " " 18 " " Itol 1038 

with corrections for ground slopes 1039 

28. Level sections; Heights, 60-100 ft. ; Roadway, 18 ft. ; Slopes, 1 to 1 1040 

with corrections for ground -slopes 1041 

29. Formulas for extending tables of level sections to any side slopes 1042 
29a. Factors for extending tables of level sections to other widths 

and slopes 1042 

30. Level sections ; Heights, 0-60 ft. ; Roadway, 18 ft. ; Slopes, IH to 1 1043 

31. " " •* 0-60 " " 20 " " Mtol 1044 

32. " *' •• 0-60 " " 20 " " Ktol 1045 

33. '• " •• 0-60 " •• 20 " " Itol 1046 

34. *• *• •• 0-60 " •• 22 " " Itol 1047 

35. •' •• " 0-60" " 24" " IJ^tol 1048 
.36. •• •• •• 0-60" " 26" " li^tol 1049 

37. " •• *• 0-60 " " 28 " " Itol 1050 

38. " " •• 0-60" " 28" " IJ^tol 1051 

39. " " " 0-60 " " 30 " " Itol 1052 

40. " " " 60-100" " 14-30" " Itol 1053 

41. " " " 60-100" •• 14-30 •• " IHtol 1054 

42. Prismoidal correction table 1057-1058 



1018 



RAILROADS. 



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(I, b w 

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s 



MULTIPLICATION TABLE, 



1019 





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CO 






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OS 
C<q 




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oin 05<MinoO'-H^b-ococo05cainoo^-*t-ococoa5cainoO'^-<*<t^ 
05 ca c<] C5 t- -* (M OS CO ^ ^ 00 in CO o t- in eg a> t^ ■* i-H 00 CO CO <D 00 in ca 
»-H <M t^ c^ CO CO CO in in in in -<*• -«4< -«*< -:t< CO CO CO cq <M c<i CQ i-H T-H »-H i-H 


S^ 




CO 


■*o<o «o o "* 00 ca CO o -* 00 (M CO o -=j* 00 (^a CO o '^ oo «vj CO o "* oo cq CO 
cq CO 05 t^ in c^ 0:1 1-- -^ e^a <35 CO -* ^ 02 CO CO ^ 00 CO CO o 00 in CO o t>. in e<i 
cq CM c^a CO CO CO in in in in -«*< TjH T}< "rj< CO CO CO CO ca cq e^ c^a 1-1 r-< ^ r-i 


^ 




^ 


00 in ca 05 mo in m in mo incD in in ino in ino ino m 
in oi CO CO cq t- m cci t^ in csj CD t>. m c^a c- m cq t^ in cq c» t>. m CO 
ca eg CO CO <o co m m m m -^ -<a< •"*<■* co co co eo cq eo cq cq i-h i-< 1-1 r-i 


in 

CCl 




00 
CO 


cqoooco-rt< CO ca 00 -:!< CO cq 00 ^ CO cq 00 'i* CO ca 00 -* CO e<i QO -* 
C35cocoo-^ t-mc^aooomcocDoococo— <ooco-^^a3co-*cKioic>.->4<c^ 
ca 00 CO ■<*< "* in in m in rh -^ -«*< ->*< CO CO CO CO ca e^ ca e^a i-H »-< .-H T-H 


cq 




^ 


CO m -r}< CO ca T-H 01 co co t^ ■<* ^-h 00 m ca cft co co t^ -<*< -h 00 in ca 05 co co 


CO 




o 


illliii ii55isiiiissi§gss2»s3s 


cq 
CQ 




■^ 


Ssssiiii 5iiSiisisissi§2Ss»sss 


cq 




cci 


^gSliiiSS iiiS|||g|S|§ssg|ss§g 




ca 




CO 

■'1' 


iissssggis iSiiiissiisgsH— »"s 


OS 




4J 


^ 


iiiiiiiiiii siiaisss§s3sissss2 


00 


in 


in in CO CO c^ t- 00 00 00 05 CJ5 CD ca ca cq ca cq cq t-h ,-h ^ ,-1 ,-h i-n 


t— 




«3 


isi§Hiiii§is iiii2Si2i3-°s??3s 


CO 


^ 
^ 


^ 


isiSiiigSiiSsi i2iliS2ls°-SSg2 


m 




05 


CO 


iiiliiiiiiliiii ssi^sss^-sssss 


-* 


t 



5 




CO 


^ 


o 


mmmmmimm ^b^^^^^-^^^ 


cq 


K 

^ 


in 


im%mrr^Mimmi sss.^.^.... 


- 




»n 


iiiiiiiiiiiiiiiiiii i^^^^^-^^ 







s 




Oi 








00 




g 




t- 




S 


iiiiiiiiiiiiiiiiiiiiiii ^^^^^" 


co 




in 


iiiiiiiiiiiiiiiiiiiiiiii -^^^^^ 


m 




s 


iiiiiiiiiiiiiiiiiiiiiiiii-^^"^ 


■<i< 




g 


cvacvacqe<ievic<icae^acqcsae^ac^acqcvacv3cacvjcococococococococo 


CO 




o 


i§§iiiiiiis§iiii§i§siiieiii '" 

cvIe^ac^ac^acqc^ae^3(^ae^ae^aca«^acqe<l«^acqcococococococococococo 


N 






^JSJ^?S^S55:5S§SS:3:i5^^^5gSSS:SSSteSSS 


- 





fo t-- CO 1-1 ooo 
«^co ca m 



CO 






-OH 



>;^ W-O O.M 



Jmm "^ 



XX 



00'^ OJ (U "? o> 
i^l^O^ So 

mi, K, o 3o ^ 






^ m^-i 



■00"* 

XX 

cr.;^; 






1020 



59.— RAILROADS. 






evic»5'*i«<ot>.ooo»0'-4CsicO'^»o«ot»ooa40^e>cico^>r5eot^ooo*o«-Hesieo'<j«»o 



c» t^ «o lo •<* CO eva 



ic>aM-rt<»r3;c)c^ooooa50i-ic^coTH -- 



rr '^^ ■"^ "-^ ^*^ "■" ^^ *•'•' ""^ '"' *— ' ^^ ^^-' ■•'^ ^^ ^^ ^^ C*0 C<J 1-H d Ci 00 t^ iO aO 

Oit~coi£2Tt<ci5e^a.-iOosoo«oio-r)«eoca'-iooiOOt^iCi->:nev5w>-i 
ooa50i-(c^coTinn;c>?oi:^ooo50TMesico-n<->s<ir5toc>.oooiO^ 



coM*c<)00o<c-<*cKic300to-^evic50o<o-^e^c>c»to-^csjooo«o-^csiooo!x>-^esic3 
t>.50W-«t<c>a^00500<x>)0-^ev3Mc»0500t^eo-<«<co<^q^C)oot^tC)io-»*<c<i'--icr>Oioo 

^esiOr5->»»l050C^C^OOCiO-HCNJO:iT»»rJ<kO;Ot^<»050i-icq<MCO'^10'X>C^00050>0 

"* ^ <-< — —, ^ T-Hcv] CM e^a e^eo 



ooinicqo>«oc20t~-*r^ookoeM(3icococ3t^-<«<^ooiOMostocoot^-»t<'-HOO»o 



o o <=> ^ ^ ^ ^ CM e^a eva 

«>qc30'^ococKioo'^o«oo3oo-^C)<xie>qoo-^o«oe^aoo-^C5«5oqoo-^o 
t>.if5Tt"eO'-^c>oot^tO'^co»-ic>05i>-coTt<coeMc>a3i>.coioa3eMooiOO 



-^- ~ CO W 00-<ifO 
OiOO«0»OCOC<l»-l 



omoicioioounoiooiooiocsiocDiocDiooiooinjomoinomoinioio 
t>.»o■^e^al-H05<XDcolOcopq<oo5^-cox^'co-H<oool^-in■^«^3l— iC50Q?c)iiococ<ac>ojt^ 



O O O ^^ ■^H .^^ ^-H Tl CM c<i . 

ooevicoOTtiooeMtoO'^coe>q«oc>-«tioocqcoo-<+iooevi«30-^ooevj':o' 
«o»ocoNC>oot^ir5-*cviC305t^<x>-»t<e<i'— iOiOoco-«*<ec»HC>oocoLracoi 
i-He<ico-^iniira<:ot-ooo500'-<e^c3-r}<»oir5«3t>.ooo50T-i^cqoo'>»<: 



cooiM»nioor^'<tit~oeo«oojCMmoo 



C) •"*! 00 esi «o O 

**^ -^T* fcXJ T— < t^J V^^ v,4,^ -^T* VJ ^— ( V.^ WU ^i^ lij i^J C^ ^^ 00 l>» XO ^< 

e^TO-^kOirstoc^ooosOT-i,— iCQOO'^iototot^oooj 



oor^.^t~oeo«oojCMmooT-i-rf<t^ococooie>a)ooo^'«*<t^oe<5«oo5e<iio 
OlOO«^->*cc^C5t^co■^e^'-c05t-«o-rJ^<^3C5cy5t^xO'*^(^aoo5t^mec^ao 
■^iOcot-ooosoiO'-icoco'^'^'iocoi^.oocsoOi— (ev>coTj<'^»r50t^ooo5 



Tt<«ooo«M'*'coooOM-^ix>ooo«si-*cooocieM'»ti«ooo^e>aTti«ooooca'»<coooo 

cc>-^e>a»H05t>.irtcocQOOotO'^coi-<a>t^io-^c^JC>oo«oioeOT-<03t>-coTt<e>ciooot^ 

1— ieMcr5-^-«tiin)«£>t>-ooo50>o»— le^acoeo'^iOcot-ooooosOr-icMNCC-^mtot^t^oo 

o o O »-< >-< -^ ^ C'a e^ e>a 



e<^co•^»o<ol>•oooiO'-^e<^c'^^J•lO«or>.ooo5C^»-^e«c^co^lo^o^~oooi^1-«e>^^co■Tf^lf^ 

<OT»<e>q^ooeo-^cci'^o>t^»oec»^03t^iococq<oooco'^evi<3oo<»Tt'C'O^H03t>.xofo 

i-ie<ico"^"<f^«©t^cooooO'-<e^eMCo-^io«3t>-i>-ooo5C>»-H,— iesiccrti»Qio«ot^oo 

o o o -^ ^ r^ '-I g^ yi e^a 



0O0O0O00OOOO0OOO0OOOO00O00OOO0OC30O 

«o■^caooo«o■^<^aooo«£>■^e^^ooo^c>•^c^ooo«£>■^e^acDoo«^■^^le^ac50occ>Tt^cclO 
i-<Nco-*-^io«£ji>.oooooiC3T-icaevaco-^io<»<»t^ooa500i-ic^eC'^-*i<kocot^oo 
o o (Z) -r^ y-i -^ -H ^ eg e^a e>a 



oot^«o»A■^coe<^^oo»oot*«o»n•<*eoe>c^»-<oo>oo^««om•^c^5ev^T-^oo500^-to»D 

u:50o^Ha>t^m^o»-^05C£>■^eg<:=)OOlio•^c^^<oool^)Co^H05l^^lr5eo■'-^05t-■^e^aoooco 

T-^e^ae»^eo-^m^ot>.t^ooa>0'-l^c<^a:l-«^»olo^clt>.ooooo50l-HC>ae^^o3-<Ji»o«^«^^^ 

o o o ^ ^ ^i r-i e~a e*a e^ 



o 



«o-^e>QOooco-^e<iOQO«o-^e<JOOo<orj<e>aooo«OTjie<iooo«o-«*<c>cioooeo-<!}<e«ao 

»rseO'-Ha5«o-«fe>cioooioeo«-^04t>»'*'c<ic30o«oco»-<05t^iopa^ooco-^'— loit^meo 
^-He<|Coe»o■^»rt«o^^t^oooiOO'^e^eo•^•^»0«OC*t^ooo>0»-lT-le^^co•TJ^•»<'kf5«Oc^ 



•^^oo»ocq05«ooooc^'^'^oo»o«^o:>«occot^'^^^oome>qo»«ocoot^Tj<-^oo!0 
\ocooooeoeo^o>t>i-«9*MOt^»ocooooi:i5->*i— <05t~-«»<c^ac3t>.ioeO'-Hooco-<9<T-HOj 
»-He>Qooc'3-^m<ocot~ooc500'-He<icoc3'<fio«o«ot>.ooaiOO^^c<ie»5ei5Tri»fs;o«o 



evioo-«j<o«5c>acOTt<o<oegoo-^o<£>c<ioo 
r-^e^ae»^co■«rlo«o«ot>. 



<oegoo-^o<£>c<ioo-«*'cr)coe<ioo'<*<ocoe<aoo 

CO'-tOOtO"*'— i05CO'^e^05t^-TfC>qOt>.ir3Cq 

ooo>osOT-ie^c>q<:c-<*<»o»o«3t^ooo50>kr5T-< 
o ^ -H ^ ^ ^ e>ci 



^. O to N OO ■'J* O 
^ .. .>. O 00 lO 00 o 00 CO 

oiOkOi-MevaMCO-^mioeo 
■r-i -H e>a CM e*a 



omo^o»oomoiooioo»«o>o©»f50iooiftOiooioo»Domou50io 
^oeK^o^^lOccl^^^»f3c^a^^^"^es^ot»»oe<^C)t>.^r5CQOl^«loe^aot^lr3e^aot-lf5e<J 

»-ie»acoco'*»n«ocot~oooio»Oi-He>acacoTj»in>iocot>.oooooiOr-(»-<cacor»(-^fiir5cc 
o o o ^ 1^ ^-< ^ ^ oj N e<i 



OOe<I<00•«J^OOCCI«00'<*'00«^J«0 0■«t<OOe^3COO•«J<ODe<^eOO'^J^OOe>CICOO•<**OOe^acOO 

•^Most-'^^^osco-^'— ioocoe»3'^ooioe»3C50oif5CMO«?^me>qa>t^-^ccioico'^^-cr) 
»— le^aoaoo-^iomcot^oooooiO'^'-tCQCOTr-^mcot^t^oooiOiOi— cpqe^jps-^mio 



OiCO 

I N CO 



N-H Oico eo • 



.occeoo>«<i»o 00 »-<•«»< 

scoot--*e<305co'*^ 

Tttiomcot^ooooasoot^esicc 



t-ooocooieMiooo^•>!»^t^^eocooie^alf5 

oocococ)^~lOeMo:>^^•«*^-HOltoooooolO 

oo-^kococot^ooooaiOi— iT-<eqc5-^'^m 

-^ ^ '^ CM evj cNa 



ooo 

00 CO 

eg CO 



^ oocoeo o t- 



oooesj'<j<cooooe<i'*«ooooe<> 
'<*' eva 05 
CO t- c^ 



I 00 lo e^a CJ5 CO -^ 
oioor-He^icdCC-^jo 



^tooooeg 
OOlO CM <r> t:^ 
lOCO t>. OO 00 



TH CO 00 O M M< I 

"^ -H OO CO CO o 

a>C3<=) ^ e^a CO ! 



i^j<m«ot»ooo»0'-HC<ico'!t<i««ot*ooCT> 
ooioe»a05cocO'-iooiOcgo5cococ>t^'<*< 

eq CO 



•»*•-»*< O CO t^ t^ 00 < 



CM CO Tti VOCO 
CO CO O t^ ■>*' 

«cgcqcO"*"^»ocoi>.t^oo 



t^ooo 
— I 00 »n 

Oi Oi ^ 



coo t^ 
i-H eg eg 



CO-* »o 
"rf ^ 00 
CO •^ Tt< 



' O o< 
00 »o < 
CM CO- 

o 



>ooo 
I coo b- 

)CO t- t- 

o 



ooooo< 
Tf -^^ 00 m eg ( 

00 Oi o^ C> »-H 1 



>oo 

I O t^ 

1 -^r T»< 



ooooo 

■»ti — ( ooioeg 
irscoco t^ 00 



ooo 

a> CO CO 

00 05 o 



ooo 

o t^ ■«*< 

-H ,-ieg 



ooo 

— OOlO 

coco rj* 



, CO »jC5 " 

eg CO 



CO o t^ -* —< 00 in 



^O 05( 

ego5»f5eMOicocoot^T}<»- 

CO CO b^ 00 00 Oi O »-l 1-4 < 



i00t»co»f5"«!fieoeg»-i 
cgi 



T-4 00 meg 05 
»r5 ifi CO t^ i>i 



eo eg — H 
CO CO o 

OOC5 o 



O <3> 00 

t-coo 

o ^ eg 



t~co in 
b- -<*< '^ 
eg CO -^ 



ie*ao( 
eg CO • 



levj ^ ( 

1 -H 00- 
>COCOt>.OOOOCROO 



I CO -itf* eg o 00 CO ■<i< 
»-< 00 in eg 00 in e^a 



ico^ eg ^00 

OS CO coo CO 

■*j< xo CO t>. t~ 



CO Th eg 

CO o t^ 

00 05 05 



OOO CO 

•^ o t^ 

O^ ,-1 



•<f ego 
-<*< — 00 
eg coco 



-<*<»-<oom«MO><oeoOb» 

I O t^ CO 
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I 00lft< 

CO CO ( 
eg CO ■ 



■«s<^ooioego}coeo< 

C>b-eoOt>.COC3t>.- 



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iooo50o^egcgeo-<r 



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t- "«»< O b- ■"»< 

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OS CO CO 

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eg eg CO 



00 -^i* o CO e^ 00 

05 CO CO OS CO eg 
'-I eg CO CO -^ »n 



Tj<oeoegoo->*'Ocoecjoo'<»< 
oicoegoikocgoiioegooio 
m CO b» 



t^ooo5050T-(r-ieg 



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egco-^mcot^oooso-^i 



MULTIPLICATION TABLE, AREAS TO CU. YDS. 1021 



19. — Unit and Decimal Areas of Cross-Section Reduced to 
Cu. Yds. per 100-Ft. Station. 

(See Table 20, following, for general use.) 

Note. — ^The values in the table are cu. yds. per 100-ft. station corres- 
ponding to the unit area in the first column when used in the proper decimal 
place or denomination as per headings of colum-ns. 

[Cu. Yds. per 100-Ft. Station.] 





MU. 


H 


T 


Thous. 


h 


t 


Units 


1 Decimals 






(7) 


(6) 


(5) 


(4) 


(3) 


(2) 


(1) (.1)(.2) 


S3 


o 


, 


o 


o , 


o 


o 


o . o o 




1 


3 703 704 


370 370 


37 037 


3 703.7 


370.4 


37.04 


3.70 


0.37 


.04 


1 


2 


7 407 407 


740 741 


74 074 


7 407.4 


740.7 


74.07 


7.41 


0.74 


.07 


2 


3 


11 111 HI 


1 111 111 


111 111 


11 111. 


1 111.1 


111.11 


11.11 


1.11 


.11 


3 




(7) 


(6) 


(5) 


(4) 


(3) 


(2) 


(1) 


(.1) 


(.2) 




4 


14 814 815 


1 481 481 


148 148 


14 815. 


1 481.5 


148.15 


14.81 


1.48 


.15 


4 


5 


18 518 519 


1 851 852 


185 185 


18 519. 


1 851.9 


185.19 


18.52 


1.85 


.19 


5 


6 


22 222 222 


2 222 222 


222 222 


22 222. 


2 222.2 


222.22 


22.22 


2.22 


.22 


6 




(7) 


(6) 


(5) 


(4) 


(3) 


(2) 


(1) 


(.1) 


(.2) 




7 


25 925 926 


2 592 593 


259 259 


25 926. 


2 592.6 


259.26 


25.93 


2.59 


.26 


7 


8 


29 629 630 


2 962 963 


296 296 


29 630. 


2 963.0 


296.30 


29.63 


2.96 


.30 


8 


9 


33 333 333 


3 333 333 


333 333 


33 333. 


3 333.3 


333.33 


33.33 


3.33 


.33 


9 



Ex. — ^The average sectional area of 
one station (100 ft.) of the Culebra cut 
was 40600 sq. ft. Find the cu. yds. in 
the station? 



Solution. — From above Tabic 
4 (5) = 148 148 
6 (3) = 2 222 

150 370 cu. yds. Ans. 



1022 



5^.— RAILROADS, 



20. — Cubic Yards in 100-Ft. Station, for — 
(Column headings are Units of niimbers in first column.) 

[Cu. Yds., from formula g .] 



Area. 
Sq. Ft. 



0. 



1. 



4. 



5. 



6. 



7. 



P.P. 



1-. 
2-. 
3-. 

4-. 

5-. 
6-. 

7-. 
8-. 
9-. 

I 10-. 

i 11-. 

■3 12-. 

w 13-. 

^ 14-, 

o 15-. 

: 16-. 

^ 17-. 
o 18-. 
^ 19-. 

* 20-. 
^21-. 
. 22-. 
7 23-. 
w 24-. 

§ 25-. 
■O 26-. 

5 28-. 
>> 29-. 

"S 30-. 

•3.31-- 

3 32-. 

g 33-. 

o 34-. 
S 

ft 35-. 

to 36-. 

o 37-. 

5 38-. 
>.39-. 

"S 

ft 40-. 

43-. 
44-. 

45-. 
46-. 
47-. 
48-. 
49-. 



0.0 

37.0 

74.1 

111.1 

148.1 

185.2 

222.2 

259.3 

296 

333.3 

370.4 

407. 

444. 

481. 

518. 

555.6 
592.6 
629.6 
666.7 
703.7 

740.7 

777.8 

814 

851.9 

888.9 

925. 

963.0 
1000.0 
1037.0 
1074.1 

1111.1 
1148.1 
1185.2 
1222.2 
1259.3 

1296.3 
1333.3 
1370.4 
1407.4 
1444.4 

1481.5 
1518.5 
1555.6 
1592.6 
1629.6 

1666.7 
1703.7 
1740.7 
1777.8 
1814.8 



3.7 

40.7 

77.8 

114.8 

151.9 

188.9 

225 

263.0 

300.0 

337.0 

374.1 

411.1 

448.1 

485 

522.2 

559.3 
596.3 



633. 
670. 
707. 



744 

781 

818 

855.6 

892.6 

929.6 
966.7 
1003.7 
1040.7 
1077.8 

1114.8 
1151.9 
1188.9 
1225.9 
1263.0 

1300.0 
1337.0 
1374.1 
1411.1 
1448.1 

1485.2 
1522.2 
1559.3 
1596.3 
1633.3 

1670.4 
1707.4 
1744.4 
1781.5 
1818.5 



7.4 

44.4 

81.5 

118.5 

155.6 

192 

229.6 

266.7 

303 

340.7 

377.8 

414.8 

451.9 

488 

525.9 

563.0 
600.0 
637.0 
674.1 
711.1 

748.1 
785.2 
822.2 
859.3 
896.3 

933.3 

970.4 

1007.4 

1044.4 

1081.5 

1118.5 
1155.6 
1192.6 
1229.6 
1266.7 

1303.7 
1340.7 
1377. 
1414 8 
1451.9 

1488.9 
1525.9 
1563.0 
1600.0 
1637.0 

1674.1 
1711.1 
1748.1 
1785.2 
1822.2 



11.1 

48.1 

85.2 

122.2 

159.3 

196 

233.3 

270.4 

307.4 

344 

381.5 
418.5 
455.6 
492.6 
529.6 

566.7 

603.7 

640.7 

677 

714.8 

751.9 

788.9 

825 

863.0 

900.0 

937.0 
974.1 

1011.1 

1048 

1085.2 

1122.2 

1159.3 

1196.3 

1233. 

1270.4 

1307 

1344.4 

1381 

1418 

1455.6 

1492 

1529.6 

1566 

1603 

1640.7 

1677.8 
1714.7 
1751.9 
1788.9 
1825.9 



14.8 

51.9 

88.9 

125.9 

163.0 

200.0 

237.0 

274 

311.1 

348 

385.2 
422.2 
459.3 
496.3 
533. 

570.4 

607.4 

644.4 

681 

718 



755.6 
792.6 
829.6 
866.7 
903.7 

940.7 
977.8 

1014.8 

1051 

1088 



1125.9 
1163.0 
1200.0 
1237.0 
1274.1 

1311.1 

1348 
1385.2 
1422 
1459 

1496 

1533 

1570 

1607.4 

1644.4 

1681.5 
1718.5 
1755.6 
1792.6 
1829.6 



18.5 
55.6 
92.6 
129.6 
166.7 

203.7 

240 

277 

314.8 

351.9 

388.9 
425.9 
463.0 
500.0 
537.0 

574.1 
611.1 
648.1 
685.2 
722.2 

759.3 

796 

833 

870 

907 



944 

981.5 
1018.5 
1055 
1092 



1129.6 

1166.7 

1203.7 

1240 

1277.8 

1314.8 

1351 

1388.9 

1425 

1463.0 

1500.0 

1537 

1574 

1611.1 

1648.1 

1685 

1722.2 

1759 

1796 

1833.3 



22.2 
59.3 
96.3 
133.3 
170.4 

207.4 
244.4 
281.5 
318.5 
355.6 

392.6 

429 
466 
503.7 
540 



577 

614.8 

651. 

688. 

725. 

763.0 
800.0 
837.0 
874.1 
911.1 

948.1 

985.2 

1022.2 

1059.3 

1096.3 

1133.3 
1170.4 
1207.4 
1244.4 
1281.5 

1318.5 
1355.6 
1392.6 
1429.6 
1466.7 

1503.7 

1540.7 

1577.8 

1614 

1651.9 

1688 

1725.9 

1763.0 

1800.0 

1837.0 



25.9 

63.0 

100.0 

137.0 

174.1 

211.1 

248.1 
285.2 
322.2 
359.3 

396.3 

433.3 

470 

507 

544 

581.5 
618.5 
655.6 
692.6 
729.6 



766 
803 
840 
877 
914.8 



951.9 

988.9 

1025.9 

1063.0 

1100.0 

1137.0 
1174.1 
1211.1 
1248.1 
1285.2 

1322.2 
1359.3 
1396.3 
1433.3 
1470.4 

1507.4 
1544.4 
1581.5 
1618.5 
1655.6 

1692.6 
1729.6 
1766.7 
1803.7 
1840.7 



29.6 

66.7 

103.7 

140.7 

177.8 

214.8 
251.9 
288.9 
325.9 
363.0 

400.0 

437.0 

474 

511.1 

548.1 

585 

622.2 

659. 

696. 

733. 



770.4 
807.4 
844.4 
881.5 
918.5 



955.6 
992.6 
1029.6 
1066.7 
1103.7 

1140.7 
1177.8 
1214.8 
1251.9 
1288.9 

1325.9 
1363.0 
1400.0 
1437.0 
1474.1 

1511.1 
1548.1 
1585.2 
1622.2 
1659.3 

1696 

1733.3 

1770.4 

1807.4 

1844.4 



33.3 

70.4 

107.4 

144.4 

181.5 

218.5 
255. e 
292.6 
329.6 
366 

403.7 

440.7 

477. 

514.8 

551.9 

588.9 
625.9 
663.0 
700.0 
737.0 

774.1 
811.1 
848.1 
885.2 
922.2 

959 

996.3 
1033 
1070.4 
1107.4 

1144.4 
1181.5 
1218.5 
1255.6 
1292.6 



1329.6 

1366 

1403 

1440 

1477.8 



1514.8 
1551.9 
1588.9 
1625.9 
1663.0 



1700.0 
1737.0 
1774.1 
1811.1 
1848.1 



3.7 
.4 
.7 
1.1 
1.5 
1.9 
2.2 
2.6 
3.0 
3.3 

3.8 
.4 
.8 
1.1 
1.5 
1.9 
2.3 
2.7 
3.0 
3.4 



50-. 1851.9 1855.6 1859.3 1863.0 1866.7 1870.4 1874.1 1877.8 1881.5 1885.2 

Ex. — Average area of cross-section 321. =1188.9 

of station is 321.4 sq. ft. Find yardage (P. P. col.) .4= 1.5 

for station? ^ns. 1190.4 cu. yds. 



EARTHWORK—AREAS TO CU. YDS. PER STA, 



1023 



— Given Areas in Sq. Ft. op Cross-Section. 
(Move decimal point to right or left for both Areas and Cu. Yds.) 

[Cu. Yds., from formula » ^ .] 



Area. 
























Sq. Ft. 


0. 


1. 


2. 


3. 


4. 


5. 


6. 


7. 


8. 


9. 


P.P. 


50-. 


1851.9 


1855.6 


1859.3 


1863.0 


1866.7 


1870.4 


1874.1 


1877.8 


1881.5 


1885.2 




51-. 


1888.9 


1892.6 


1896.3 


1900.0 


1903.7 


1907.4 


1911.1 


1914.8 


1918.5 


1922.2 




52-. 


1925.9 


1929.6 


1933.3 


1937.0 


1940.7 


1944.4 


1948.1 


1951.9 


1955.6 


1959.3 




53-. 


1963.0 


1966.7 


1970.4 


1974.1 


1977.8 


1981.5 


1985.2 


1988.9 


1992.6 


1996.3 




54-. 


2000.0 


2003.7 


2007.4 


2011.1 


2014.8 


2018.5 


2022.2 


2025.9 


2029.6 


2033.3 




55-. 


2037.0 


2040.7 


2044.4 


2048.1 


2051.9 


2055.6 


2059.3 


2063.0 


2066.7 


2070.4 




.56-. 


2074.1 


2077.8 


2081.5 


2085.2 


2088.9 


2092.6 


2096.3 


2100.0 


2103.7 


2107.4 




57-. 


2111.1 


2114.8 


2118.5 


2122.2 


2125.9 


2129.6 


2133.3 


2137.0 


2140.7 


2144.4 




58-. 


2148.1 


2151.9 


2155.6 


2159.3 


2163.0 


2166.7 


2170.4 


2174.1 


2177.8 


2181.5 




59-. 


2185.2 


2188.9 


2192.6 


2196.3 


2200.0 


2203.7 


2207.4 


2211.1 


2214.8 


2218.5 




fl 60-. 
1 61-. 
1 62-. 


2222.2 


2225.9 


2229.6 


2233.3 


2237.0 


2240.7 


2244.4 


2248.1 


2251.9 


2255.6 




2259.3 


2263.0 


2266.7 


2270.4 


2274.1 


2277.8 


2281.5 


2285.2 


2288.9 


2292.6 




2296.3 


2300.0 


2303.7 


2307.4 


2311.1 


2314.8 


2318.5 


2322.2 


2325.9 


2329.6 




« 63-. 


2333.3 


2337.0 


2340.7 


2344.4 


2348.1 


2351.9 


2355.6 


2359.3 


2363.0 


2366.7 




•S 64-. 


2370.4 


2374.1 


2377.8 


2381.5 


2385.2 


2388.9 


2392,6 


2396.3 


2400.0 


2403.7 




o 65-. 


2407.4 


2411.1 


2414.8 


2418.5 


2422.2 


2425.9 


2429.6 


2433.3 


2437.0 


2440.7 




^ 66-. 


2444.4 


2448.1 


2451.9 


2455.6 


2459.3 


2463.0 


2466.7 


2470.4 


2474.1 


2477.8 


3.7 


tS 67-. 


2481.5 


2485.2 


2488.9 


2492.6 


2496.3 


2500.0 


2503.7 


2507.4 


2511.1 


2514.8 


1 


.4 


Qj 68-. 


2518.5 


2522.2 


2525.9 


2529.6 


2533.3 


2537.0 


2540.7 


2544.4 


2548.1 


2551.9 


2 


.7 


S 69-. 


2555.6 


2559.3 


2563.0 


2566.7 


2570.4 


2574.1 


2577.8 


2581.5 


2585.2 


2588.9 


3 
4 
5 


1.1 
1.5 
1.9 


!'70-. 


2592.6 


2596.3 


2600.0 


2603.7 


2607.4 


2611.1 


2614.8 


2618.5 


2622.2 


2625.9 


^ 71-. 


2629. 6 


2633.3 


2637.0 


2640.7 


2QUA 


2648.1 


2651.9 


2655.6 


2659.3 


2663.0 


6 


2.2 


. 7Z-. 


2666.7 


2670.4 


2674.1 


2677.8 


2681.5 


2685.2 


2688.9 


2692.6 


2696.3 


2700.0 


7 


2.6 


7 73-. 


2703.7 


2707.4 


2711.1 


2714.8 


2718.5 


2722.2 


2725.9 


2729.6 


2733.3 


2737.0 


8 


3.0 


^74-. 


2740.7 


2744.4 


2748.1 


2751.9 


2755.6 


2759.3 


2763.0 


2766.7 


2770.4 


2774.1 


9l 3.3 


"i 75-. 


2777.8 


2781.5 


2785.2 


2788.9 


2792.6 


2796.3 


2800.0 


2803.7 


2807.4 


2811.1 




•o 76-. 


2814.8 


2818.5 


2822.2 


2825.9 


2829.6 


2833.3 


2837.0 


2840.7 


2844.4 


2848.1 


3.8 


2 77-. 


2851.9 


2855.6 


2859,3 


2863.0 


2866.7 


2870.4 


2874.1 


2877.8 


2881.5 


2885.2 


1 


.4 


5 78-. 


2888.9 


2892.6 


2896.3 


2900.0 


2903.7 


2907.4 


2911.1 


2914.8 


2918.5 


2922.2 


2 


.8 


>. 79-. 


2925.9 


2929.6 


2933.3 


2937.0 


2940.7 


2944.4 


2948.1 


2951.9 


2955.6 


2959.3 


3 
^5 


1.1 
1.5 
1.9 


•a 80-. 


2963.0 


2966.7 


2970.4 


2974.1 


2977.8 


2981.5 


2985.2 


2988.9 


2992.6 


2996.3 


H 81-. 

g 82-. 


3000.0 


3003.7 


3007.4 


3011.1 


3014.8 


3018.5 


3022.2 


3025.9 


3029.6 


3033.3 


6 


2.3 


3037.0 


3040.7 


3044.4 


3048.1 


3051.9 


3055.6 


3059.3 


3063.0 


3066.7 


3070.4 


7 


2.7 


§ 83-. 


3074.1 


3077.8 


3081.5 


3085.2 


3088.9 


3092.61 3096.3 


3100.0 


3103.7 


3107.4 


8 


3.0 


o 84-. 


3111.1 


3114.8 


3118.5 


3122.2 


3125.9 


3129.6 


3133.3 


3137.0 


3140.7 


3144.4 


9 


3.4 


§ 85-. 


3148.1 


3151.9 


3155.6 


3159.3 


3163.0 


3166.7 


3170.4 


3174.1 


3177.8 


3181.5 




& 86-. 


3185.2 


3188.9 


3192.6 


3196.3 


3200.0 


3203.7 


3207.4 


3211.1 


3214.8 


3218.5 




o 87-. 


3222.2 


3225.9 


3229.6 


3233.3 


3237.0 


3240.7 


3244.4 


3248.1 


3251.9 


3255.6 




5 88-. 


3259.3 


3263.0 


3266.7 


3270.4 


3274.1 


3277.8 


3281.5 


3285.2 


3288.9 


3292.6 




^89- 


3296.3 


3300.0 


3303.7 


3307.4 


3311.1 


3314.8 


3318.5 


3322.2 


3325.9 


3329.6 




S 90-. 


3333.3 


3337.0 


3340.7 


3344.4 


3348.1 


3351.9 


3355.6 


3359.3 


3363.0 


3366.7 




g 91-. 


3370.4 


3374.1 


3377 8 


'3381.5 


3385.2 


3388.9 


3392.6 


3396.3 


3400.0 


3403.7 




"^ 92-. 


3407.4 


3411.1 


3414.8 


3418.5 


3422.2 


3425.9 


3429.6 


3433.3 


3437.0 


3440.7 




93-. 


3444.4 


3448.1 


3451.9 


3455.6 


3459.3 


3463.0 


3466.7 


3470.4 


3474.1 


3477.8 




94-. 


3481.5 


3485.2 


3488.9 


3492.6 


3496.3 


3500.0 


3503.7 


3507.4 


3511.1 


3514.8 




95-. 


3518.5 


3522.2 


3525.9 


3529.6 


3533.3 


3537.0 


3540.7 


3544.4 


3548.1 


3551.9 




96-. 


3555.6 


3559.3 


3563.0 


3566.7 


3570.4 


3574.1 


3577.8 


3581.5 


3585.2 


3588.9 




97-. 


3592.6 


3596.3 


3600.0 


3603.7 


3607.4 


3611.1 


3614.8 3618.5 


3622.2 


3625.9 




98-. 


3629.6 


3633.3 


3637.0 


3640.7 


3644.4 


3648.1 


3651.9 


3655.6 


3659.3 


3663.0 




99^. 


3666.7 


3670.4 


3674.1 


3677.8 


3681.5 


3685.2 


3688.9 


3692.6 


3696.3 


3700.0 




100-. 


3703.7 


3707.4 


3711.1 


3714.8 


3718.5 


3722.2 


3725.9 


3729.6 


3733.3 


3737.0 





Ex. — Average area of cross- 
section is 3214 sq. ft. Find yard- 
age for station ? 



Yardage for 321.4 sq. ft. = 1190.4 cu. yds. 
•' 3214 " " =11904 " " 
Ans. 



1024 



.—RAILROADS. 



20. — Cubic Yards in 100-Ft. Stations, for — 
(Column headings are Units of numbers in first column.) 



[Cu. Yds., from formula 



lOOA 
3X9 



Area. 
Sq. Ft. 



0. 


1. 


2. 


3. 


4. 


5. 


6. 


7. 


8. 


9. ] 


3703.7 


3707.4 


3711.1 


3714.8 


3718.5 


3722.2 


3725.9 


3729.6 


3733.3 


3737.0 


3740.7 


3744.4 


3748.1 


3751.9 


3755.6 


3759.3 


3763.0 


3766.7 


3770.4 


3774.1 


3777.8 


3781.5 


3785.2 


3788.9 


3792.6 


3796.3 


3800.0 


3803.7 


3807.4 


3811.1 


3814.8 


3818.5 


3822.2 


3825.9 


3829.6 


3833.3 


3837.0 


3840.7 


3844.4 


3848. 1 


3851.9 


3855.6 


3859.3 


3863.0 


3866.7 


3870.4 


3874.1 


3877.8 


3881.5 


3885.2 


3888.9 


3892.6 


3896.3 


3900.0 


3903.7 


3907.4 


3911.1 


3914.8 


3918.5 


3922.2 


3925.9 


3929.6 


3933.3 


3937.0 


3940.7 


3944.4 


3948.1 


3951.9 


3955.6 


3959.3 


3963.0 


3966.7 


3970.4 


3974.1 


3977.8 


3981.5 


3985.2 


3988.9 


3992.6 


3996.3 


4000.0 


4003.7 


4007.4 


4011.1 


4014.8 


4018.5 


4022.2 


4025.9 


4029.6 


4033.3 


4037.0 


4040.7 


4044.4 


4048.1 


4051.9 


4055.6 


4059.3 


4063.0 


4066.7 


4070.4 


4074.1 


4077.8 


4081.5 


4085.2 


4088.9 


4092.6 


4096.3 


4100.0 


4103.7 


4107.4 


4111.1 


4114.8 


4118.5 


4122.2 


4125.9 


4129.6 


4133.3 


4137.0 


4140.7 


4144.4 


4148. 1 


4151.9 


4155.6 


4159.3 


4163.0 


4166.7 


4170.4 


4174.1 


4177.8 


4181.5 


4185.2 


4188.9 


4192.6 


4196.3 


4200.0 


4203.7 


4207.4 


4211.1 


4214.8 


4218.5 


4222.2 


4225.9 


4229.6 


4233.3 


4237.0 


4240.7 


4244.4 


4248.1 


4251.9 


4255.6 


4259.3 


4263.0 


4266.7 


4270.4 


4274.1 


4277.8 


4281.5 


4285.2 


4288.9 


4292.6 


4296.3 


4300.0 


4303.7 


4307.4 


4311.1 


4314.8 


4318.5 


4322.2 


4325.9 


4329.6 


4333.3 


4337.0 


4340.7 


4344.4 


4348.1 


4351.9 


4355.6 


4359.3 


4363.0 


4366.7 1 


4370.4 


4374.1 


4377.8 


4381.5 


4385.2 


4388.9 


4392.6 


4396.3 


4400.0 


4403.7 2 


4407.4 


4411.1 


4414.8 


4418.5 


4422.2 


4425.9 


4429.6 


4433.3 


4437.0 


4440.7 3 

4477.8 5 


4444.4 


4448 1 


4451.9 


4455.6 


4459.3 


4463.0 


4466.7 


4470.4 


4474. 1 


4481.5 


4485.2 


4488.9 


4492.6 


4496.3 


4500.0 


4503.7 


4507.4 


4511.1 


4514.86 


4518.5 


4522.2 


4525.9 


4529.6 


4533.3 


4537.0 


4540.7 


4544.4 


4548.1 


4551.9 7 


4555.6 


4559.3 


4563.0 


4566.7 


4570.4 


4574.1 


4577.8 


4581.5 


4585.2 


4588.9 8 


4592.6 


4596.3 


4600.0 


4603.7 


4607.4 


4611.1 


4614.8 


4618.5 


4622.2 


4625.9 9 


4629.6 


4633.3 


4637.0 


4640.7 


4644.4 


4648.1 


4651.9 


4655.6 


4659.3 


4663.0 


4666.7 


4670.4 


4674.1 


4677.8 


4681.5 


4685.2 


4688.9 


4692.6 


4696.3 


4700.0 


4703.7 


4707.4 


4711.1 


4714.8 


•4718.5 


4722.2 


4725.9 


4729.6 


4733.3 


4737.01 


4740.7 


4744.4 


4748.1 


4751.9 


4755.6 


4759.3 


4763.0 


4766.7 


4770.4 


4774.12 


4777.8 


4781.5 


4785.2 


4788.9 


4792.6 


4796.3 


4800.0 


4803.7 


4807.4 


4811.13 

4 

4848.15 


4814.8 


4818.5 


4822.2 


4825.9 


4829.6 


4833.3 


4837.0 


4840.7 


4844.4 


4851.9 


4855.6 


4859.3 


4863.0 


4866.7 


4870.4 


4874.1 


4877.8 


4881.5 


4885.2 6 


4888.9 


4892.6 


4896.3 


4900.0 


4903.7 


4907.4 


4911.1 


4914.8 


4918.5 


4922.2 7 


4925.9 


4929.6 


4933.3 


4937.0 


4940.7 


4944.4 


4948.1 


4951.9 


4955.6 


4959.3 8 


4963.0 


4966.7 


4970.4 


4974.1 


4977.8 


4981.5 


4985.2 


4988.9 


4992.6 


4996.3 9 


5000.0 


5003.7 


5007.4 


5011.1 


5014.8 


5018.5 


5022.2 


5025.9 


5029.6 


5033.3 


5037.0 


5040.7 


5044.4 


5048. 1 


5051.9 


5055.6 


5059.3 


5063.0 


5066.7 


5070.4 


5074.1 


5077.8 


5081.5 


5085.2 


5088.9 


5092.6 


5096.3 


5100.0 


5103.7 


5107.4 


5111.1 


5114.8 


5118.5 


5122.2 


5125.9 


5129.6 


5133.3 


5137.0 


5140.7 


5144.4 


5148.1 


51.51.9 


5155.6 


5159.3 


5163.0 


5166.7 


5170.4 


5174.1 


5177.8 


5181.5 


5185.2 


5188.9 


5192.6 


5196.3 


5200.0 


5203.7 


5207.4 


5211.1 


5214.8 


5218.5 


5222.2 


5225.9 


5229.6 


5233.3 


5237.0 


5240.7 


5244.4 


5248.1 


5251.9 


5255.6 


5259.3 


5263.0 


5266.7 


5270.4 


5274.1 


5277.8 


5281.5 


5285.2 


5288.9 


5292.6 


5296.3 


5300.0 


5303.7 


5307.4 


5311.1 


5314.8 


5318.5 


5322.2 


5325.9 


5329.6 


5333.3 


5337.0 


5340.7 


5344.4 


5348.1 


5351.9 


5355.6 


5359.3 


5363.0 


5366.7 


5370.4 


5374.1 


5377.8 


5381.5 


5385.2 


5388.9 


5392.6 


5396.3 


5400. 


5403.7 


5407.4 


5411.1 


5414.8 


5418.5 


5422.2 


5425.9 


5429.6 


5433.3 


5437.0 


5440.7 


5444.4 


5448.1 


5451.9 


5455.6 


5459.3 


5463.0 


5466.7 


5470.4 


5474.1 


5477.8 


5481.5 


5485.2 


5488.9 


5492.6 


5496.3 


5500.0 


5503.7 


5507.4 


5511.1 


5514.8 


5518.5 


5522.2 


5525.9 


5529.6 


5533.3 


5537.0 


5540.7 


5544.4 


5548. 1 


5551.9 


5555.6 


5559.3 


5563.0 


5566.7 


5570.4 


5574.1 


5577.8 


5581.5 


5585.2 


5588.9 



P.P. 



100-. 
101-. 
102-. 
103-. 
104-. 

105-. 
106-. 
107-. 
108-. 
109-. 



I 111-. 

w 113-. 
- 114-. 

6 115-. 
^ 116-. 
ti 117-. 

118- 
S 119-. 
bo 

"^ 120-. 

^ 121-. 

. 122- . 

7 123-. 
w 124- . 

1 125-. 
•O 126-. 
o 127-. 
S 128-. 
>. 129-. 

£i 

•d 130- . 
•2 131-. 
§* 132-. 
§ 133-. 
o 134-. 

U 135- . 
S- 136- . 



137-. 
138-. 
13 9-. 

140- . 
141-. 
142- . 
143-. 
144- . 

145-. 
146-. 
147-. 
148- . 
149-. 

150-. 



3.7 
.4 
.7 
1.1 
1.5 
1.9 
2.2 
2.6 
3.0 
3.3 

3.8 
.4 
.8 
1.1 
1.5 
1.9 
2.3 
2.7 
3.0 
3.4 



Ex. — Average area of cross- 
section is 1295.3 sq. ft. Find 
yardage for station ? 



(P. P. Col.) 



1295. =4796.3 

.3= y. 

A.ns. 4797.4 cu. yds. 



EARTHWORK— AREAS TO CU. YDS. PER STA. 



1025 



— Given Areas in Sq. Ft. of Cross-Section. — Continued. 
(Move decimal points to right or left for both Areas and Cu. Yds.) 

fCu. Yds., from formula „ ^ .] 



0. 



1. 



2. 



3. 



5. 



7. 



9. 



P.P. 



5555.6 
5592.6 
5629.6 
5666.7 
5703.7 

5740.7 
5777.8 
5814.8 
5851.9 
5888 

5925.9 
5963.0 
6000.0 
6037.0 
6074.1 

6111 

6148.1 

6185.2 

6222.2 

6259. 

6296.3 
6333.3 
6370.4 
6407.4 
6444.4 

6481.5 
6518.5 
6555.6 
6592.6 
6629.6 



6703.7 
6740.7 
6777.8 
6814.8 

6851.9 
6888.9 
6925.9 
6963.0 
7000.0 

7037.0 
7074.1 
7111.1 
7148.1 
7185.2 

7222.2 
7259.3 
7296.3 
7333.3 
7370.4 

7407.4 



5559.3 
5596.3 
5633.3 
5670.4 
5707.4 

5744.4 
5781.5 
5818.5 
5855.6 
5892.6 

5929.6 
5966.7 
6003.7 
6040.7 
6077.8 

6114.8 

6151.9 

6188.9 

6225 

6263.0 

6300.0 

6337 

6371 

6411.1 

6448.1 

6485.2 

6522.2 

6559 

6596 

6633 

6670.4 

6707 

6744.4 

6781.5 

6818.5 

6855.6 
6892.6 
6929.6 
6966.7 
7003.7 

7040.7 
7077.8 
7114.8 
7151.9 
7188.9 

7225.9 
7263.0 
7300.0 
7337.0 
7371 



5563.0 
5600.0 
5637.0 
5674.1 
5711.1 

5748. 1 
5785.2 
5822.2 
5859.3 
5896.3 

5933.3 
5970.4 
6007.4 
6044.4 
6081 

6118.5 

6155.6 

6192 

6229.6 

6266 

6303.7 
6340.7 
6377.8 
6414.8 
6451.9 

6488. 

6525. 

6563.0 

6600 

6637.0 

6674.1 
6711.1 
6748.1 
6785.2 
6822.2 

6859.3 
6896.3 
6933.3 
6970.4 
7007.4 

7044.4 
7081.5 
7118.5 
7155.6 
7192.6 



7229.6 
7266.7 
7303.7 
7,340.7 
7377.8 



5566.7 
5603.7 
5640.7 
5677.8 
5714.8 

5751.9 
5788.9 
5825.9 
5863.0 
5900.0 

5937.0 
5974.1 
6011.1 
6048.1 
6085.2 

6122.2 

6159. 

8196. 

6233. 

6270. 

6307.4 
6344.4 
6381.5 
6418.5 
6455.6 

6492.6 
6529.6 
6566.7 
6603.7 
6640.7 

6677.8 

6714.8 

6751.9 

6788 

6825.9 

6863.0 
6900.0 
6937.0 
6974.1 
7011.1 

7048.1 
7085.2 
7122.2 
7159.3 
7196 

7233.3 

7270. 

7307. 

7344. 

7381. 



5570.4 
5607.4 
5644.4 
5681.5 
5718.5 

5755.6 
5792.6 
5829.6 
5866.7 
5903.7 

5940.7 

5977.8 

6014.8 

6051 

6088 

6125.9 
6163.0 
6200.0 
6237.0 
6274.1 

6311.1 
6348.1 
6385.2 
6422.2 
6459.3 

6496.3 
6533.3 
6570.4 
6607.4 
6644.4 

6681.5 
6718.5 
6755.6 
6792.6 
6829.6 

6866.7 
6903.7 
6940.7 
6977.8 
7014.8 

7051.9 

7088 

7125.9 

7163.0 

7200.0 

7237.0 

7274.1 

7311. 

7348. 

7385.2 



5574.1 
5611.1 
5648.1 
5685.2 
5722.2 

5759.3 
5796.3 
5833.3 
5870.4 
5907.4 

5944.4 
5981.5 
6018.5 
6055.6 
6092.6 

6129.6 
6166.7 
6203.7 
6240.7 
6277.8 

6314.8 

6351 

6388 

6425.9 

6463 



6500.0 
6537.0 
6574.1 
6611.1 
6648. 1 

6685.2 
6722.2 
6759.3 
6796.3 
6833.3 



6870 

6907 

6944.4 

6981.5 

7018.5 

7055.6 
7092.6 
7129.6 
7166.7 
7203.7 

7240.7 
7277 8 
7314.8 
7351.9 
7388.9 



7411.117414.8 7418.517422.2 7425.9 7429.6 7433.317437.0 7440.7 



5577.8 
5614.8 
5651.9 
5688.9 
572§.9 

5763.0 

5800.0 
5837.0 
5874.1 
5911.1 

5948.1 

5985.2 

6022 

6059.3 

6096.3 

6133.3 

6170.4 

6207. 

6244. 

6281. 

6318. 

6355. 

6392.6 

6429 

6466.7 

6503.7 
6540.7 
6577.8 
6614.8 
6651. 

6688.9 
6725.9 
6763.0 
6800.0 
6837.0 

6874.1 
6911.1 
6948.1 
6985.2 
7022.2 

7059.3 

7096.3 

7133 

7170.4 

7207.4 

7244 
7281 
7318 5 
7355.6 
7392.6 



5581.5 
5618.5 
5655.6 
5692.6 
5729.6 

6766.7 
5803.7 
5840.7 
5877.8 
5914.8 

5951.9 

5988 

6025 

6063.0 

6100.0 

6137.0 
6174.1 
6211.1 
6248.1 
6285.2 

6322.2 
6359.3 
6396.3 
6433.3 
6470.4 

6507.4 
6544.4 
6581.5 
6618.5 
6655.6 

6692.6 
6729.6 
6766.7 
6803.7 
6840.7 

6877.8 

6914.8 

6951.9 

6988 

7025 

7063.0 

7100 

7137.0 

7174.1 

7211 



7248. 



7285 
7322.2 
7359.3 
7396.3 



5585.2 
5622.2 
5659.3 
5696.3 
5733.3 

5770.4 
5807.4 
5844.4 
5881.5 
5918.5 

5955.6 
5992.6 
6029.6 
6066.7 
6103.7 

6140.7 
6177.8 
6214.8 
6251.9 
6288.9 



6325.9 
6363.0 
6400.0 
6437.0 
6474. 

6511.1 

6548.1 

6585 

6622 

6659 

6696.3 
6733.3 
6770.4 
6807.4 
6844.4 

6881.5 

6918.5 

6955.6 

6992 

7029.6 

7066.7 
7103.7 
7140.7 
7177.8 
7214.8 

7251.9 
7288.9 
7325.9 
7363.0 
7400.0 



5588.9 
5625.9 
5663.0 
5700.0 
5737.0 

5774. 
5811.1 
5848. 1 
5885.2 
5922.2 

5959.3 
5996.3 
6033.3 
6070.4 
6107 

6144 
6181 
6218 
6255. 6|2 
6292.6 3 

4 
6329.6 
6366.7 
6403.7 
6440.7 
6477.8 9 

6514.8 
6551.9 



6588.9 


.4 


6625.9 2 


J .8 


6663.0: 


\ 1.1 




I 1.5 


6700.0. 


5 1.9 


6737.0 f 


5 2.3 


6774.1 ' 


J 2.7 


6811. U 


3 3.0 


6848. 1< 


) 3.4 



6885.2 

6922.2 

6959 

6996 

7033.3 



7070 

7107 

7144.4 

7181.5 

7218.5 

7255.6 

7292.6 

7329.6 

7366 

7403 



3J 
.4 
.7 
1.1 
1.5 
1.9 
2.2 
2.6 
3.0 
3.3 



Ex. — Average area of cross- 
section is 1845.8 sq. ft. Find 
yardage for station? 



(P. P, 



1845. =6833.3 
Col.) .8= 3.0 



Ans. 6836.3 cu. yds. 



1026 



m.— RAILROADS, 



20. — Cubic Yards in 100-Ft. Station, for — 
(Column headings are Units of numbers in first column.) 

[Cu. Yds., from formula .] 

oxy 



Area. 
























Sq. Ft. 


0. 


1. 


2. 


3. 


4. 


5. 


6. 


7. 


8. 


9. I 


^ p. 


200-. 


7407.4 


7411.1 


7414.8 


7418.5 


7422.2 


7425.9 


7429.6 


7433.3 


7437.0 


7440.7 




201-. 


7444.4 


7448.1 


7451.9 


7455.6 


7459.3 


7463.0 


7466.7 


7470.4 


7474. li 7477.81 




202-. 


7481.5 


7485.2 


7488.9 


7492.6 


7496.3 


7500.0 


7503.7 


7507.4 


7511,1 


7514.8 




203-. 


7518 5 


7522.2 


7525.9 


7529.6 


7533.3 


7537.0 


7540.7 


7544.4 


7548. 1 


7551.9 




204-. 


7555.6 


7559.3 


7563.0 


7566.7 


7570.4 


7574. 1 


7577.8 


7581.5 


7585.2 


7588.9 




205-. 


7592.6 


7596.3 


7600.0 


7603.7 


7607.4 


7611.1 


7614.8 


7618.5 


7622.2 


7525.9 




206-. 


7629.6 


7633.3 


7637.0 


7640.7 


7644.4 


7648.1 


765L9 


7655.6 


7659.3 


7663.0 




207-, 


7666.7 


7670.4 


7674,1 


7677.8 


7681.5 


7685.2 


7688,9 


7692.6 


7696.3 


7700.0 




208-. 


7703.7 


7707.4 


7711.1 


7714.8 


7718.5 


7722.2 


7725.9 


7729.6 


7733.3 


7737.0 




209-. 


7740.7 


7744.4 


7748.1 


7751.9 


7755.6 


7759.3 


7763.0 


7766.7 


7770.4 


7774.1 




i 210-. 


7777.8 


7781.5 


7785.2 


7788.9 


7792.6 


7796.3 


7800.0 


7803.7 


7807.4 


7811.1 




1 211-. 


7814.8 


7818.5 


7822.2 


7825.9 


7829.6 


7833.3 


7837.0 


7840.7 


7844.4 


7848. 1 




■3 212-. 
§ 213-. 


7851.9 


7855.6 


7859.3 


7863.0 


7866.7 


7870.4 


7874.1 


7877.8 


7881.5 


7885.2 




7888.9 


7892.6 


7896.3 


7900.0 


7903.7 


7907.4 


7911.1 


7914.8 


7918.5 


7922.2 




^ 214-. 


7925.9 


7929.6 


7933.3 


7937.0 


7940.7 


7944.4 


7948.1 


7951.9 


7955.6 


7959.3 




&215-. 


7963.0 


7966.7 


7970.4 


7974.1 


7977.8 


7981.5 


7985.2 


7988.9 


7992.6 


7996.3 




^ 216-. 


8000.0 


8003.7 


8007.4 


8011.1 


8014.8 


8018.5 


8022.2 


8025.9 


•8029.6 


8033.3 


3.7 


^ 217-. 


8037.0 


8040.7 


8044.4 


8048. 1 


8051.9 


8055.6 


8059.3 


8063.0 


8066.7 


8070.4 


1 .4 


© 218-. 


8074.1 


8077.8 


8081.5 


8085.2 


8088.9 


8092.6 


8096.3 


8100.0 


8103.7 


8107.4 


2 .7 


S '''" 


8111.1 


8114.8 


8118.5 


8122.2 


8125.9 


8129.6 


8133.3. 


8137.0 


8140.7 


8144.4 


31.1 


•^ 220-. 


8148.1 


8151.9 


8155.6 


8159.3 


8163.0 


8166.7 


8170.4 


8174.1 


8177.8 


8181.5 


41.5 
51.9 


^ 22 1-. 
. 222-. 


8185.2 


8188.9 


8192 6 


8196.3 


8200.0 


8203.7 


8207.4 


8211.1 


8214.8 


8218.5 


62.2 


8222.2 


8225.9 


8229.6 


8233.3 


8237.0 


8240.7 


8244.4 


8248.1 


8251.9 


8255.6 


72.6 


7 223-. 


8259.3 


8263.0 


8266.7 


8270.4 


8274.1 


8277.8 


8281.5 


8285.2 


8288.9 


8292.6 


83 


t 224-. 


8296.3 


8300.0 


8303.7 


8307.4 


8311.1 


8314.8 


8318.5 


8322.2 


8325.9 


8329.6 


93.3 


1 225-. 


8333.3 


8337.0 


8340.7 


8344.4 


8348.1 


8351.9 


8355.6 


8359.3 


8363.0 


8366.7 




•O 226-. 


8370.4 


8371.4 


8377.8 


8381.5 


8385.2 


8388.9 


8392.6 


8396.3 


8400.0 


8403.7 


3.8 


« 227-. 


8407.4 


8411.1 


8414.8 


8418.5 


8422.2 


8425.9 


8429.6 


8433.3 


8437.0 


8440.7 


1 .4 


:S 228-. 


8444.4 


8448.1 


8451.9 


8455.6 


8459.3 


8463.0 


8466.7 


8470.4 


8474.1 


8477.8 


2 .8 


>» 229-. 

.Q 

-o 230-. 


8481.5 


8485.2 


8488.9 


8492.6 


8496.3 


8500.0 


8503.7 


8507.4 


8511.1 


8514.8 


31.1 
41.5 
51.9 


8518.5 


8522.2 


8525.9 


8529.6 


8533.3 


8537.0 


8540.7 


8544.4 


8548.1 


8551.9 


•1 231-. 
§ 232-. 


8555.6 


8559.3 


8563.0 


8566.7 


8570.4 


8574.1 


8577.8 


8581.5 


8585.2 


8588.9 


62.3 


8592.6 


8596.3 


8600.0 


8603.7 


8607.4 


8611.1 


8614.8 


8618.5 


8622.2 


8625.9 


72.7 


8 233-. 


8629.6 


8633.3 


8637.0 


8640.7 


8644.4 


8648.1 


8651.9 


8655.6 


8659.3 


8663.0 


83.0 


o 234-. 


8666.7 


8670.4 


8674.1 


8677.8 


8681.5 


8685.2 


8688.9 


8692.6 


8696.3 


8700.0 


913.4 


1 235-. 
& 236-. 


8703.7 


8707.4 


8711.1 


8714.8 


8718.5 


8722.2 


8725.9 


8729.6 


8733.3 


8737.0 




8740.7 


8744.4 


8748.1 


8751.9 


8755.6 


8759.3 


8763.0 


8766.7 


8770.4 


8774.1 




2 237-. 


8777.8 


8781.5 


8785.2 


8788.9 


8792.6 


8796.3 


8800.0 


8803.7 


8807.4 


8811.1 




5 238-. 


8814.8 


8818.5 


8822.2 


8825.9 


8829.6 


8833.3 


8837.0 


8840.7 


8844.4 


8848. 1 




>, 239-. 
<-< 


8851.9 


8855.6 


8859.3 


8863.0 


8866.7 


8870.4 


8874.1 


8877.8 


8881.5 


8885.2 




ft 240-. 


8888.9 


8892.6 


889G.3 


8900.0 


8903.7 


8907.4 


8911.1 


8914.8 


8918.5 


8922.2 




g 241-. 


8925.9 


8929.6 


8933.3 


8937.0 


8940.7 


8944.4 


8948.1 


8951.9 


8955.6 


8959.3 




242-. 


8963. C 


8966.7 


8970.4 


8974.1 


8977.8 


8981.5 


8985.2 


8988.9 


8992.6 


8996.3 




243-. 


9000. C 


9003.7 


9007.4 


9011.1 


9014.8 


9018.5 


9022.2 


9025.9 


9029.6 


9033.3 




244-. 


9037. C 


9040.7 


9044.4 


9048.1 


9051.9 


9055.6 


9059.2 


9063. C 


9066.; 


9070.4 




245-. 


9074.1 


9077.8 


9181. E 


9085.2 


9088. S 


9092. C 


9096. c 


9100. C 


9103.; 


9107.4 




246-. 


9111.1 


9114. e 


9118. E 


9122.2 


9125. £ 


9129.6 


9133. c 


9137. C 


9140.; 


9144.4 




247-. 


9148.1 


9151. S 


9155.f 


9159. S 


9163. ( 


9166.-/ 


9170.^ 


[ 9174.1 


9177. J 


9181.5 




248- 


9185.2 


9188. i 


9192. ( 


9196.C 


1 9200. ( 


9203.7 


9207.^ 


I 9211.1 


9214. i 


9218.5 




249-. 


9222.2 


9225. J 


9229. ( 


5 9233.: 


\ 9237. ( 


9240.7 


9244. '^ 


[ 9248.] 


9251.1 


9255.6 




250-. 


9259. 3I 9263. ( 


9266.' 


J 9270.^ 


[ 9274. 


I 9277. { 


9281.! 


) 9285.2 


I 9288. J 


9292.6 




Ex.— Average area of cross- 2436. =9022.2 


sectior 


1 is 2 

._ r 


i36.2 


sq. ft. 


Find 




(P.I 


\ Col.: 


. 


2= _ 


.7 





yardage for station? 



Ans. 9022.9 cu. yds 



EARTHWORK— AREAS TO CU. YDS. PER STA, 



1027 



— Given Areas in Sq, Ft. of Cross-Section. — Concluded. 
(Move decimal point to right or left for both Areas and Cu. Yds.) 

[Cu. Yds., from formula » „ .] 



Area. 
























Sq. Ft. 


0. . 


1. 


2. 


3. 


4. 


5. 


6. 


7. 


8. 


9. 


P.P. 


. 250- 


.9259.3 


9263.0 


9266.7 


9270.4 


9274.1 


9277.8 


9281.5 


9285.2 


9288.9 


9292.6 




7 251- 


.9296.3 


9300.0 


9303.7 


9307.4 


9311.1 


9314.8 


9318.5 


9322.2 


9325.9 


9329.6 


3.7 


w 252- 


.9333.3 


9337.0 


9340.7 


9344.4 


9348.1 


9351.9 


9355.6 


9359.3 


9363.0 


9366.7 


1 


.4 


^ 253- 


.9370.4 


9371.4 


9377.8 


9381.5 


9385.2 


9388.9 


9392.6 


9396.3 


9400.0 


9403.7 


2 


.7 


rt .254- 


,9407.4 


9411.1 


9414.8 


9418.5 


9422.2 


9425.9 


9429.6 


9433.3 


9437.0 


9440.7 


3 


1.1 


^|255- 


.9444.4 


9448. 1 


9451.9 


9455.6 


9459.3 


9463,0 


9466.7 


9470.4 


9474.1 


9477.8 


4 
5 


1.5 
1.9 


.9481.5 


9485.2 


9488.9 


9492.6 


9496.3 


9500.0 


9503.7 


9507.4 


9511.1 


9514.8 


6 


2.2 


>'S257- 


.9518.5 


9522.2 


9525.9 


9529.6 


9533.3 


9537.0 


9540.7 


9544.4 


9548.1 


9551.9 


7 


2.6 


So258- 
1 (2.259- 

g 4^260- 


.9555.6 


9559.3 


9563.0 


9566.7 


9570.4 


9574.1 


9577.8 


9581.5 


9585.2 


9588.9 


8 


3.0 


.9592.6 


9596.3 


9600.0 


9603.7 


9607.4 


9611.1 


9614.8 


9618.5 


9622.2 


9625.9 


9 


3.3 


.9629.6 


9633.3 


9637.0 


9640.7 


9644.4 


9648.1 


9651.9 


9655.6 


9659.3 


9663.0 




g«^261- 


.9666.7 


9670.4 


9674.1 


9677.8 


9681.5 


9685.2 


9688.9 


9692.6 


9696.3 


9700.0 


3.8 


oj 2262- 


.9703.7 


9707.4 


9.711.1 


9714.8 


9718.5 


9722.2 


9725.9 


9729.6 


9733.3 


9737.0 


1 


.4 


^ S)263- 
g<«264- 

|'^265- 


.9740.7 


9744.4 


9748.1 


9751.9 


9755.6 


9759.3 


9763.0 


9766.7 


9770.4 


9774.1 


2 


.8 


.9777.8 


9781.5 


9785.2 


9788.9 


9792.6 


9796.3 


9800.0 


9803.7 


9807.4 


9811.1 


3 

4 
5 


1.1 
1.5 
1.9 


.9814.8 


9818.5 


9822.2 


9825.9 


9829.6 


9833.3 


9837.0 


9840.7 


9844.4 


9848.1 


5>. 266- 


.9851.9 


9855.6 


9859.3 


9863.0 


9866.7 


9870.4 


9874.1 


9877.8 


9881.5 


9885.2 


6 


2.3 


t 267- 


.9888.9 


9892.6 


9896.3 


9900.0 


9903.7 


9907.4 


9911.1 


9914.8 


9918.5 


9922.2 


7 


2.7 


£. 268- 


.9925.9 


9929.6 


9933.3 


9937.0 


9940.7 


9944.4 


9948.1 


9951.9 


9955.6 


9959.3 


8 


3.0 


g 269- 


.9963.0 


9966.7 


9970.4 


9974.1 


9977.8 


9981.5 


9985.2 


9988.9 


9992.6 


9996.3 


9 


3.4 



Ex. — Average area of cross- 
section is 2583.7 sq. ft. Find 
yardage for station ? 



2583. =9566.7 
(P. P. Col.) .7= 2.6 



Ans. 9569.3 cu. yds. 



Remarks on above Table 20. 



Table 20 gives the number of cubic yards in 100-ft. station corresponding 
to areas in sq. ft. of cross-section up to 2700 sq. ft., equivalent to 10000 cu. 
yds. per 100-ft. station for the area named. The main table advances by 
unit areas, the units being at tops of columns; the Prop. Parts colimins 
being used for interpolation for tenths of sq. ft. Thus, for an average area 
of cross-section of 2622 sq. ft. the yardage per station equals 9711.1 cu. yds. 
If the area is increased by 0.4 sq. ft. the yardage would be increased by 1.5 
cu. yds., giving 9712.6 cu. yds for an area of 2622.4 sq. ft. By manipulating 
the decimal point to the right of left simultaneously for both areas and cu. 
yds., the rang^ of the table may be extended. Thus, for an area of 26224 
sq. ft. we have 97126 cu. yds. For areas up to 270 sq. ft. it will be found 
convenient to move the decimal point one place to the left, considering the 
column headings as tenths of sq. ft. Thus, for an area of 262.2 sq. ft. we 
have 971.1 cu. yds., obtained by moving the decimal point for the yardage 
one place to left also. For areas above 270 and up to 2700 sq. ft., the decimal 
system in the table should be preserved. 

If the average area of cross-section is for less than a 100-ft. station, say 
for 50 ft., the yardage would of course be proportional, i. e., i for 50 ft., 
i for 20 ft., etc. 



1028 b^.-^RAILROADS. 

21. — Method op Calculating Earthwork Tables for Level 

Sections. 

(From a page of the writer's calculation of Table 26, page 1037.) 

Isi. — Select foolscap paper, horizontally ruled. 

Snd. — Draw vertical lines, in pencil, to represent the decimal points of 
numbers in columns 1, 2, 3, 4, 5, etc.; thus insuring the figures being "in 
column" and saving considerable time in making decimal points. 

3rd.— het w/= width of roadway, in ft. (w=16 in the present case); 
5 = the side slopes of excavation or embankment (thus, in the present case, 
using slopes of IJ to 1, 5=1.5); ^ = height of level cutting, in ft.; and 
j2 = quantity in cu. yds. per 100-ft station for a level cut or fill groimd line. 

Then Q = (2sh+2w)^ . ^^; or, =(2sh-{-2w)h--~; or, =(sh-i-w)h-~. 

Sometimes one and sometimes another of the above equations will be found 
most convenient, depending upon s and h. 

4th. — Calculate Q for h = 0.1 as for line c, colimin 1*: 

100 
O=(2Xl.5X0.1 + 2Xl6)0.lX-~r7:= 5.981 cu. yds. Similarly. 

X y 

100 
for ;>= 0.2, C? = (2X1.5X0.2 + 2X16)0.1X^^ = 12.074 cu. yds. 

u X y 

.'.Diff. for 0.1ft. from ;j= 0.1 to ;? = 0.2 ft. = 6.093 cu. yds. (See opp. line 1.) 
This number, 6.093, forms a primary base for calculating all the Differences 
(see numbers in italics) in the following table ; and these Differences increase 
by a constant increment which will next be explained. 

5^^.— Find the successive Differences, lines 1, 3, 5, 7, etc. (in Italics), by 
calculating the constant incremental increase i for the particular slope s 
(in the present case 5=1.5) and for the successive increase in height d (in 
the present case d = 0.1 ft.); and adding this increment successively, using 
the first Difference, 6.093, as a base. Now the value of i in cu. yds. for any 

25 

slope =" oTTq cu. yds.; therefore for slope of 1^ to 1, as in the present case, 

t = ^ = 0.111'^l. Hence we have for Diff erences in column 1 : 6.093+0.111'^l 
= 6.204 (line 3V, 6.204+0.111^1 = 6.315 (line 5); 6.3154-0.111^1 = 6.426 
(line 7) ; etc. Now glancing down the column we notice a similarity of the 
three groups of numbers, the number (Difference) in each group being a 
unit larger than the corresponding one in the group above. Hence it is 
essential that these Differences be arranged in groups, each successive group 
of Differences being set down from the preceding group by mental calculation. 
Note in this connection also that the Differences in successive columns are 
increased by 3.000 from those in the same line in the preceding column. 
Thus, 6.093, 9.093, 12.093, etc., indefinitely. Now, as to the number of 
Differences in each group, before repetition occurs: In the present case this 
number is 9 because i = g; for a slope of 1 to 1, •t = 2T, therefore if 27 Differ- 
ences are arranged in each column they will increase by 2.000 in line hori- 
zontally, when the slope is 1 to 1. Hence, the number of Differences in 
each column should be arranged for the particular slope. For a slope of 
1^ to 1, as in the present case, any number of groups of 9 will be convenient. 

6th. — By successive addition, the quantities for each successive height h 
may now be obtained. Thus, for /j= 0.3 ft., (Q= 18.278 cu. yds.; for h= lit., 
(2=64.815 cu. yds.; for ;j=9.7 ft., Q= 1097.538 cu. yds., etc. 

Remarks. — ^The value of 5 at foot of each column is the stim of the Differ- 
ences (in italics) in that column. Only one column need be added (these 
values of 5 are simply for checking the regular additions in each column) 
as the successive sums will increase by, in the present case, 27X3=81. 
Note method of checking each column by lines a, b and c. 



* See table on opposite page. 



EARTHWORK TABLE CALCULATIONS. 



1029 



The following is the tabular arrangement, explained p. 1028, for calcu- 
iating the quantities Q in cu. yds. in Table 26, with ground line level, road- 
way w= 16ft., slopes s= 13^ to 1. The numbers in italics, opposite the odd 
line Nos., are Differences for successive tenths of feet in height, while the 
quantities in cu. yds., corresponding to the height in feet., are given oppo- 
site the even line Nos. 

[Ex. — For height of 3.2 ft. there are 246.519 cu. yds. per station.] 



No 


O ^ . Col. 1 . 4^- 


. Col. 2. ^ . 


Col. 3. ^ 


. Col. 4. J 


. Col 5. x; . 


Col. 6 


a- 




S 
S 


€203 


50G ^t 


284 


500 -a^ 365 


500 f,€ 446 500 -£§ 


527 


500 


b- 


S 5 


981 -3;^ 

— W 


209 


481 %^ 493 


981 ^^ 859 


''' gS 


1305 


981 








W 




HH 




w 






c- 


.1= 5 


981 


209 


481 


493 


981 


859 


481 


1305 


981 


1833 


481 


1 


6 


0P5 


9 


093 


12 


093 


15 


093 


18 


093 


21 


093 


2 


.2= 12 


074 


218 


574 


506 


074 


874 


574 11.= 


= 1324 


074 


1854 


574 


3 


6 


204 


P 


^04 


12 


^04 


15 


^04 


18 


^04 


21 


204 


4 


18 


278 3.= 


= 227 


778 


518 


278 


889 


778 


1342 


278 


1875 


778 


5 


6 


5i5 


9 


315 


12 


315 


25 


525 


25 


315 


21 


315 


6 


24 


593 


237 


093 


530 


593 8.5 = 


= 905 


093 


1360 


593 


1897 


093 


7 


6 


426 


P 


4;^5 


12 


426 


15 


426 


18 


426 


21 


426 


8 


.5= 31 


019 


246 


519 


543 


019 


920 


519 


1379 


019 14.= 


= 1918 


519 


9 


6 


538 


5> 


538 


12 


538 


25 


538 


25 


538 


21 


538 


10 


37 


557 


256 


057 6.= 


= 555 


557 


936 


057 


1397 


557 


1940 


057 


11 


6 


545 


9 


648 


12 


648 


15 


545 


25 


545 


21 


648 


12 


44 


205 


265 


705 


568 


205 


951 


705 11.5 = 


= 1416 


205 


1961 


705 


13 


6 


759 


5 


759 


12 


759 


25 


759 


25 


759 


21 


759 


14 


50 


964 3.5 = 


= 275 


464 


580 


964 


967 


464 


1434 


964 


1983 


464 


15 


6 


570 


9 


570 


12 


570 


25 


570 


25 


870 


21 


870 


16 


57 


834 


285 


334 


593 


834 9.= 


= 983 


334 


1453 


834 


2005 


334 


17 


6 


981 


5 


981 


12 


P52 


15 


552 


25 


981 


21 


981 


18 


1.= 64 


815 


295 


315 


606 


815 


999 


315 


1472 


81514.5 = 


= 2027 


315 


19 


7 


093 


io 


093 


13 


093 


25 


093 


19 


093 


22 


093 


20 


71 


908 


305 


408 6.5 = 


= 619 


908 


1015 


408 


1491 


908 


2049 


408 


21 


7 


^04 


10 


204 


13 


204 


16 


:g04 


19 


204 


22 


204 


22 


79 


112 


315 


612 


633 


112 


1031 


612 12.= 


= 1511 


112 


2071 


612 


23 


7 


315 


20 


315 


13 


315 


25 


315 


19 


315 


22 


315 


24 


86 


427 4.= 


= 325 


927 


646 


427 


1047 


927 


1530 


427 


2093 


927 


25 


7 


4^<? 


10 


4^5 


13 


4^5 


16 


4^5 


17 


426 


22 


426 


26 


93 


853 


336 


353 


659 


853 9.5 = 


= 1064 


353 


1549 


853 


2116 


353 


27 


7 


537 


10 


537 


13 


557 


25 


537 


19 


537 


22 


537 


28 


1.5=101 


390 


346 


890 


673 


390 


1080 


890 


1569 


390 15.= 


= 2138 


890 


29 


7 


545 


20 


545 


13 


648 


25 


545 


25 


545 


22 


648 


30 


109 


038 


357 


538 7.= 


= 687 


038 


1097 


538 


1589 


038 


2161 


538 


31 


7 


769 


20 


759 


13 


759 


16 


755> 


19 


759 


22 


759 


32 


116 


797 


368 


297 


700 


1^1 


1114 


297 12.5 = 


= 1608 


797 


2184 


297 


33 


7 


570 


20 


870 


13 


870 


25 


870 


25 


870 


22 


870 


34 


124 


667 4.5 = 


= 379 


167 


714 


667 


1131 


167 


1628 


667 


2207 


167 


35 


7 


981 


10 


981 


25 


^52 


16 


552 


19 


981 


22 


981 


36 


132 


648 


390 


148 


728 


648 10.= 


= 1148 


148 


1648 


648 


2230 


148 


37 


8 


093 


11 


0P5 


14 


093 


27 


093 


;20 


093 


23 


093 


38 


2. = 140 


741 


401 


241 


742 


741 


1165 


241 


1668 


74115.5 = 


= 2253 


241 


39 


8 


;g04 


22 


;g04 


^4 


204 


27 


^04 


^0 


204 


23 


;?04 


40 


148 


945 


412 


445 7.5 = 


= 756 


945 


1182 


445 


1688 


945 


22 V 6 


445 


41 


8 


315 


11 


525 


i4 


525 


17 


525 


20 


315 


23 


315 


42 


157 


260 


423 


760 


771 


260 


1199 


760 13.= 


= 1709 


260 


2299 


760 


43 


8 


4-^5 


22 


4^5 


14 


426 


27 


426 


;gO 


426 


23 


4^5 


44 


165 


686 5.= 


= 435 


186 


785 


686 


1217 


186 


1729 


686 


2323 


186 


45 


8 


537 


11 


537 


^4 


537 


17 


537 


20 


537 


23 


537 


46 


174 


223 


446 


723 


800 


223 10.5 = 


= 1234 


723 


1750 


223 


2346 


723 


47 


8 


545 


22 


545 


14 


648 


27 


648 


;gO 


648 


23 


545 


48 


2.5=182 


871 


458 


371 


814 


871 


1252 


371 


1770 


871 16.= 


= 2370 


371 


49 


8 


759 


22 


759 


^4 


759 


27 


755 


;gO 


759 


23 


755 


50 


191 


630 


470 


130 8.= 


= 829 


630 


1270 


130 


1791 


630 


2394 


130 


51 


8 


870 


11 


870 


^4 


570 


17 


570 


20 


870 


23 


570 


52 


200 


500 


482 


000 


844 


500 


1288 


000 13.5 = 


= 1812 


500 


2418 


000 


53 


8 


P5i 


22 


981 


14 


552 


27 


981 


^0 


981 


23 


552 


54 


209 


4815.5 = 


= 493 


981 


859 


481 


1305 


981 


1833 


481 


2441 


981 


S (Ch'k.)203 


500f81 


=284 


500-1-81= 


= 365 


500+81 


= 446 


500 + 81 


= 527 


500 + 81 


= 608 


500 



1030 



.—RAILROADS. 




a 
o 

o 



o= 



a= 



a 

I 

o 
a 

o- 

U CO 






CO O 



-go 



o 



D* O 

a :3 






O 



^ O 



O 

CO ;3 

^=5:1105 "=^105 "^lOS 

O I v/ O Ux ^ V 

o I A o A o A 

T-l'CQ T-l CO T-4 CO 



-5- 


« 


§ 


c 


•+-> 


rt 


1 


+-> 




CJ 


Q2. 


Qa. 


-(J 


§ 


^— ' 


+j 


^ 


^ 


a 


H<s 




+ 



o 
o 

I 

o 
o 



►^ <M 



a 


a 


ni 


a 


-i-^ 


4^ 


H 


H 


+ 


1 


'^ 


►Si 



« 


<5a. 






1 




-♦J 


+ 

►Si 



b q Q 



^,OS ^|0» ^|Oi 

thJco f-tjeo i-Hico 



a 


(i 


« 


G 




c 


rt 


« 


OJ 


+-> 


crt 


-*-> 




+j 










S 




S 


(M 


1 


<N 


+ 


CO 


+ 






« 




« 




a 




s 


4-> 


^ 


(M 


1 


<N 


+ 


■^ 


1 


►^ 




►^ 


Tt< 




««*< 







a 


a 



-♦-> 


+ 


-t-> 





1 


►Si 


+ 


►Si 



b Q q 



:5t 



EARTHWORK^-GROUND'SLOPE QUANTITIES. 



1031 



^ OS "^^ OS '^ 

o A o X o 

1-H CO T-H CO m^ 



Oi 



X§ X§ X 
|co t-h|co 1-iIcq 



o K/ o 
0|X o 



x§ 

CO tH 



^lOS jdlOS ^|Oi 

§ X§ X§ 

«d CO *-4 CO ^^ 



(|co thI 



§x 



CO 



"^ 1 05 "^ I OS ^ I OS 

o X o I X o X 

rHJCO 1-llcQ ,-h|C0 



(U> 
WcO 
S^ 
^^ 

r *** 

a II 

In rt 
. o 

»^ 2 
o ^1 

. w 

^7 

a II 
;gMW 

^7 






« 


« 


w 


§ 


§ 




-I-' 












rtl 




g 


1 


B 


+ 


00 


+ 


s 




►^ 



« 


« 


C 


« ^ 


01 


a ^ 


+-> 


rt -^ 


a 


+^ <N 






§ 




«|eo 


1 c^« 


+ 


i + 


-^ 


CO ^ 







« 




« 


C 


?J 


jn 


rt 




Gj 


+J 


^ 


+-I 


d 


ni 


(M 


g 




g 


HN 




Hc^ 


+ 


C<l 


+ 


^ 




s 



«i 


« « 


a 


r^ P! 


S 


5 «J 


-M 


CO +J 




■^ e^, 




50 --"^ 


^ 


^ ^ 


-<* 


1 ^ 


o 


<io 


4- 


<^« + 


►s» 


rH ^ 







►Si (M 



►^ «M 



-«il<N 



^l< 



« 




« 




« 

a 


c 

^ 


§ 


4J 


?J 


+ 


1 


I 


►^ 


C^ 


-J^i 


(M 




(>< 



« 




a 


C 


« 


c 


crt 




01 


-t-J 


a 


+-> 


^ 


05 


§ 


WIM 


1 


eilco 


+ 


r^n 


I 


►^ 


'"^ 


^ 


r4n 

1— 1 







« 




« 


C 


« 


C 


rt 


OJ 


+-> 


C! 


+J 


g 


Gj 


^ 


H« 


1 


He* 


+ 
-Si 


tH 


1 



« 


« 


c 


c 


oJ 


« OJ 


4-> 


_ +J 


g 


y s 


"* 


T^ 


o 


1 o 


+ 


Si 


00 


00 


o 


o 





« 


g 


s ^ 


+ 


■2 + 


►^ 


1 ^5i 




<N 



g 


« s 


«!« 




+ 


►Si 


1 .Si 




H« 



8 






g 


> 


^ 


03 -*^ 


o 


"V ® 


+ 

►Si 


^i 



o o 






II :i 
+^ oJ 

^ o 

* '^ 
O 



^ 



:^ 






:5t 



1032 



69.— RAILROADS, 






c/2 






p -r-l t-t 

! Si 53 



i X 



"^ 


oi -^ 


o 


V ol 
X o 


o 


r-i 


CO T-|l 



*=" V ® 

o X o 



§ X 

th'cO 



^ Oi "^ OS -^ 

o|X o X o 



< o 
o 



a 


w 


c 


V. C! 


aJ 


« rt 






^ 


r. ^ 


H" 


7h« 


+ 


J.4- 


s 


S 



+ J. 

xp. ^ 



« 


« 


c 


:^ 5=^ 


ni 


y CJ 


+j 


n r 




c3 5^ 


?^ 


^ rf 


wit> 


^c^ 


+ 


i + 


K? 


«!■* ^ 







a 




« 


c 


« 


C 


rt 




rt 


4-' 


a 


+-) 


C4 


rt 










?J 


1 


?3 


H* 


»o 


H* 


+ 


o 


+ 


^ 




w 



►Si (M 



f1 ^ 



win I 
►5i 



« 




8 










1 




+ 

►Si 


■*t- 


1 
-Ai 


•*t- 




«<«• 



« 


a 


Ci 


« a 


«J 


C! «« 


+-> 


e3 


T ^ 


H^ 


1 H^ 


+ 


:^ 1 


►Si 










w 




S3 


g 


S 


^)!J> 


eft- 


+ 


+-> 


+ 


►Si 


1 


►JS 





« 


s 


§ s 


H^ 


-M ,4* 


+ 


1 + 


►Si 


"^►^ 




o 



"73 • 



ctJ 

o 



b Q q 



::i Q d 



b Q Q 



rt G a, 
^ o 



::^ 






EARTHWORK— GROUND-SLOPE QUANTITIES, 



1033 




OS "^ 05 

X§ X 
CO 1-i'co 



« 


a 


c 


« c 


aJ 


S^ 


+-> 










c»|« 


-*i« «)« 


•rtl 


1 "* 


4- 


«l + 


►^ 


►Si 










y 


« 




1 "** 




+ 


J« 1 

►Si 


N|« 


e«|03 





« 


e*<5 




"^ 


+ 


►Si 


1 -Si 




«!« 



^ 8 



b Q Ci 



;i^ 



^ 






"a 



OS "^{Oi ^ 
P\ o K^ o 

CO th'cO tH 



« 


« 


c 


« C 


d 


a rt 


-)j 


nj +^ 


N 


4J O 






OS 


O' .OS 


+ 


1 + 


S 


"^ s 



►«i|c<i 



y 


« 


« 


C 




P! 


rt 


S 


rt 


-H 


nJ 


-*-> 


OS 


+-• 


OS 


+ 


1 


1 
►5i 



-(J o 





8 


OS 




+ 


►Si 


1 -Si 




y^ 






<L)cO 

OT "• 
(DO 

ft II 



ft c^ «> 
o tJ >• 

W Vh O 



V^ 4J ft 

. II rt 

-'-Hi 
in 



'S II PI 
:5 O'd 

TOO) 



PlrHlnOO 

Clji-KM 
4j CO 

o o>. 

:-^ 

°«+, <U 

o ftH 
^ ° ^ 

rt o w 

<u »- 3 



o 



O r* -^ 



1034 



m.— RAILROADS, 



23. — Level Sections (Earthwork); Height., 0-60 Ft. 
Base of Roadway, 14 Ft. Side Slopes, 13^ to 1. 
Note. — ^The last two columns enable us to use any other base than 14 ft. 
Ex. — Given height, 34.5 ft.; roadway 12 ft. Then we have, 8401.4- 
(251.85+ 3.70) = 8145.8 cu. yds. (For Ht. >60 ft., see Tables 24, 41.) 

ECu. Yds. per 100-Ft. Station.] 



1 






















Width 






Ht. 


.0 


.1 


.2 


.3 


.4 


.5 


.6 


.7 


.8 


.9 


of 2 Ft. 






Ft. 






















Cu.Yds 




w 







5.2 


10.6 


16.1 


21.6 


27.3 


33.1 


39.0 


45.0 


51.2 






•2 . 


1 


'*57!4 


63.8 


70.2 


76.8 


83.5 


90.3 


97.2 


104.2 


111.3 


118.6 


" 7".4i 




2 


125.9 


133.4 


141.0 


148.6 


156.4 


164.4 


172.4 


180.5 


188.7 


197.1 


14.81 




tH 


3 


205.6 


214.1 


222.8 


231.6 


240.5 


249.5 


258.7 


267.9 


277.3 


286.7 


22.22 






4 


296.3 


306.0 


315.8 


325.7 


335.7 


345.8 


356.1 


366.4 


376.9 


387.5 


29.63 




5 


398.1 


408.9 


419.9 


430.9 


442.0 


453.2 


464.6 


476.1 


487.6 


499.3 


37.04 




doi 


6 


511.1 


523.0 


535.0 


547.2 


559.4 


571.8 


584.2 


596.8 


609.5 


622.3 


44.44 


^ 




7 


635.2 


648.2 


661.3 


674.6 


687.9 


701.4 


714.9 


728.6 


742.4 


756.3 


51.85 






8 


770.3 


784.5 


798.7 


813.1 


827.5 


842.1 


856.8 


871.6 


886.5 


901.5 


59.26 


9 


916.7 


931.9 


947.3 


962.7 


978.3 


994.0 


1009.8 


1025.7 


1041.7 


1057.8 


66.67 


w 


10 


1074.1 


1090.4 


1106.9 


1123.5 


1140.1 


1156.9 


1173.9 


1190.9 


1208.0 


1225.2 


74.07 


d 


flJ^d 


11 


1242.6 


1260.1 


1277.6 


1295.3 


1313.1 


1331.0 


1349.0 


1367.2 


1385.4 


1403.8 


81.48 


43 


-53 


12 


1422.2 


1440.8 


1459.5 


1478.3 


1497.2 


1516.2 


1535.3 


1554.6 


1573.9 


1593.4 


88.89 






13 


1613.0 


1632.6 


1652.4 


1672.4 


1692.4 


1712.5 


1732.7 


1753.1 


1773.6 


1794.1 


96.30 


fe 


14 


1814.8 


1835.6 


1856.5 


1877.5 


1898.7 


1919.9 


1941.3 


1962.7 


1984.3 


2006.0 


103.70 


•l-l 


OXM 


15 


2027.8 


2049.7 


2071.7 


2093.8 


2076.1 


2138.4 


2160.9 


2183.5 


2206.1 


2228.9 


111.11 


o 


16 


2251.8 


2274.9 


2298.0 


2321.2 


2344.6 


2368.1 


2391.6 


2415.3 


2439.1 


2463.0 


118.52 


'p 


17 


2487.0 


2511.2 


2535.4 


2559.8 


2584.2 


2608.8 


2633.5 


2658.3 


2683.2 


2708.2 


125.93 


■*" 


18 


2733.3 


2758.6 


2783.9 


2809.4 


2835.0 


2860.6 


2886.4 


2912.4 


2938.4 


2964.5 


133.33 


Wl 


11 


19 


2990.7 


3017.1 


3043.6 


3070.1 


3096.8 


3123.6 


3150.5 


3177.5 


3204.7 


3231.9 


140.74 


20 


3259.3 


3286.7 


3314.3 


3342.0 


3369.8 


3397.7 


3425.7 


3453.8 


3482.1 


3510.4 


148.15 


a 


21 


3538.9 


3567.5 


3596.1 


3624.9 


3653.8 


3682.9 


3712.0 


3741.2 


3770.6 


3800.0 


155.56 


2 


^1 


22 


3829.6 


3859.3 


3889. 1 


3919.0 


3949.0 


3979.1 


4009.4 


4039.7 


4070.2 


4100.8 


162.96 


< 


23 


4131.5 


4162.3 


4193.2 


4224.2 


4255.3 


4286.6 


4317.9 


4349.4 


4381.0 


4412.6 


170.37 


=3^ 


24 


4444.4 


4476.4 


4508.4 


4540.5 


4572.7 


4605.1 


4637.6 


4670.1 


4702.8 


4735.6 


177.78 




25 


4768.5 


4801.5 


4834.7 


4867.9 


4901.3 


4934.7 


4968.3 


5002.0 


5035.8 


5069.7 


185.19 


P.P. 


26 


5103.7 


5137.8 


5172.1 


5206.4 


5240.9 


5275.5 


5310.1 


5344.9 


5379.8 


5414.9 


192.59 


7.41 


Eg 


27 


5450.0 


5485.2 


5520.6 


5556.1 


5591.6 


5627.3 


5663.1 


5699.0 


5735.0 


5771.2 


200.00 


1 .74 


28 


5807.4 


5843.8 


5880.2 


5916.8 


5953.5 


5990.3 


6027.2 


6064.2 


6101.3 


6138.6 


207.41 


2 1.48 


3§ 


29 


6175.9 


6213.4 


6251.0 


6288.6 


6326.4 


6364.4 


6402.4 


6440.5 


6478.7 


6517.1 


214.81 


3 2^22 


•S'' 


30 


6555.6 


6594.1 


6632.8 


6671.6 


6710.5 


6749.5 


6788.7 


6827.9 


6867.3 


6906.7 


222.22 


4 2.96 


ss 


31 


6946.3 


6986.0 


7025.8 


7065.7 


7105.7 


7145.8 


7186.1 


7226.4 


7266.9 


7307.5 


229.63 


5 3.70 


2 -r 


32 


7348.1 


7388.9 


7429.9 


7470.9 


7512.0 


7553.2 


7594.6 


7636.1 


7677.6 


7719.3 


237.04 


6 4.44 


3g 


33 


7761.1 


7803.0 


7845.0 


7887.2 


7929.4 


7971.8 


8014.2 


8056.8 


8099.5 


8142.3 


244.44 


7 5^19 


■"S 


34 


8185.2 


8228.2 


8271.3 


8314.6 


8357.9 


8401.4 


8445.0 


8488.6 


8532.4 


8576.3 


251.85 


8 5.93 


c-° 


35 


8620.4 


8664.5 


8708.7 


8753.1 


8797.6 


8842.1 


8886.8 


8931.6 


8976.5 


9021.5 


259.26 


9 6! 67 




36 


9066.7 


9111.9 


9157.3 


9202.7 


9248.3 


9294.0 


9339.8 


9385.7 


9431.7 


9477.8 


266.67 




37 


9524.1 


9570.4 


9616.9 


9663.5 


9710.1 


9756.9 


9803.9 


9850.9 


9898.0 


9945.2 


274.07 


§ 


:2J 


38 


9992.6 


10040 


10088 


10135 


10183 


10231 


10279 


10327 


10375 


10424 


281.48 


>Hi: 


39 


10472 


10521 


10569 


10618 


10667 


10716 


10765 


10815 


10864 


10913 


288.89 


3i 


40 


10963 


11013 


11062 


11112 


11162 


11212 


11263 


11313 


11364 


11414 


296.30 


a 


41 


11465 


11516 


11567 


11618 


11669 


11720 


11771 


11823 


11874 


11926 


303.70 


B 


42 


11978 


12029 


12081 


12134 


12186 


12238 


12291 


12343 


12396 


12449 


311.11 


3 


43 


12502 


12555 


12608 


12661 


12715 


12768 


12822 


12875 


12929 


12983 


318.52 


8 


d 4J 


44 


13037 


13091 


13145 


13200 


13254 


13309 


13363 


13418 


13473 


13528 


325.93 


bO 


sl 


45 


13583 


13639 


13694 


13749 


13805 


13861 


13916 


13972 


14028 


14084 


333.33 


a 


46 


14141 


14197 


14254 


14310 


14367 


14424 


14480 


14537 


14595 


14652 


340.74 


1 


^-^ 


47 


14709 


14767 


14824 


14882 


14940 


14998 


15056 


15114 


15172 


15230 


348.15 


CO u 


48 


15289 


15347 


15406 


15465 


15524 


15583 


15642 


15701 


15761 


15820 


355.56 


6^ 


49 


15880 


15939 


15999 


16059 


16119 


16179 


16239 


16300 


16360 


16421 


362.96 


p« 


50 


16481 


16542 


16603 


16664 


16725 


16787 


16848 


16909 


16971 


17033 


370.37 


5 




51 


17094 


17156 


17218 


17280 


17343 


17405 


17467 


17530 


17593 


17656 


377.78 




52 


17719 


17782 


17845 


17908 


17971 


18035 


18098 


18162 


18226 


18290 


385.19 


53 


18354 


18418 


18482 


18546 


18611 


18675 


18740 


18805 


18870 


18935 


392.59 


s 


-a o 


54 


19000 


19065 


19131 


19196 


19262 


19327 


19393 


19459 


19525 


19591 


400.00 


fj 


55 


19657 


19724 


19790 


19857 


19923 


19990 


20057 


20124 


20191 


20259 


407.41 




56 


20326 


20393 


20461 


20529 


20596 


20664 


20732 


20800 


20869 


20937 


414 81 




l^bO 


57 


21005 


21074 


21143 


21212 


21280 


21349 


21419 


21488 


21557 


21627 


422.22 




c 


58 


21696 


21766 


21836 


21906 


21976 


22046 


22116 


22186 


22257 


22327 


429.63 




:3 


59 


22398 


22469 


22540 


22611 


22682 


22753 


22825 


22896 


22968 


23039 


437.04 




60 


23111 


23183 


23255 


23327 


23399 


23472 


23544 


23617 [23689 1 


23762 


444.44 





EARTHWORK TABLES— LEVEL SECTIONS. 



1035 



24. — ^Level Sections (Earthwork); Height, 60-120 Ft. 

Base op Roadway, 14 Ft. Side Slopes, 13^ to I. 

Note. — ^The last two columns enable us to use any other base than 14 ft.: 
Ex. — Given height, 94.5 ft.; roadway 15 ft. Then we have, 54513 + 
i (696.30+ 3.70) = 54863 cu. yds. (See also Table 41.) 

[Cu. Yds. per 100-Ft. Station.] 



























Width 






Ht. 


.0 


.1 


.2 


.3 


.4 


.5 


.6 


.7 


.8 


.9 


of 2 Ft. 






Ft. 






















Cu.Yds 




m 


60 


23111 


23183 


23255 


23327 


23399 


23472 


23544 


23617 


23689 


23762 


444.44 




:3 

5 


61 


23835 


23908 


23981 


24055 


24128 


24201 


24275 


24349 


24422 


24496 


451.85 




62 


24570 


24645 


24719 


24793 


24868 


24942 


25017 


25092 


25167 


25242 


459.26 




.^ 


63 


25317 


25392 


25467 


25543 


25618 


25694 


25770 


25846 


25922 


25998 


466.67 




u 
o 

•+J 


64 


26074 


26150 


26227 


26303 


26380 


26457 


26534 


26611 


26688 


26765 


474.07 




65 


26843 


26920 


26998 


27075 


27153 


27231 


27309 


27387 


27465 


27544 


481.48 




^^ 


66 


27622 


27701 


27779 


27858 


27937 


28016 


28095 


28175 


28254 


28333 


488.89 


tS 


'V.o 


67 


28413 


28493 


28572 


28652 


28732 


28812 


28893 


28973 


29054 


29134 


496.30 


■S) 


Ceo 


68 


29215 


29296 


29377 


29458 


29539 


29620 


29701 


29783 


29864 


29946 


503.70 


•s 


69 


30028 


30110 


30192 


30274 


30356 


30438 


30521 


30603 


30686 


30769 


511.11 


w 


CO M 


70 


30852 


30935 


31018 


31101 


31185 


31268 


31352 


31435 


31519 


31603 


518.52 


d 


<o^ 


71 


31687 


31771 


31855 


31940 


32024 


32109 


32193 


32278 


32363 


32448 


525.93 




S^ 


72 


32533 


32619 


32704 


32789 


32875 


32961 


33046 


33132 


33218 


33305 


533.33 


« 


"^ w 


73 


33391 


33477 


33564 


33650 


33737 


33824 


33911 


33998 


34085 


34172 


540.74 


£ 


^1 


74 


34259 


34347 


34434 


34522 


34610 


34698 


34786 


34874 


34962 


35050 


548.15 


O 


75 


35139 


35227 


35316 


35405 


35494 


35583 


35672 


35761 


35851 


35940 


555.56 


76 


36030 


36119 


36209 


36299 


36389 


36479 


36569 


36660 


36750 


36841 


562.96 


a 




77 


36931 


37022 


37113 


37204 


37295 


37387 


37478 


37569 


37661 


37753 


570.37 


+3 


78 


37844 


37936 


38028 


38121 


38213 


38305 


38398 


38490 


38583 


38676 


577.78 




So 


79 


38769 


38862 


38955 


39048 


39141 


39235 


39328 


39422 


39516 


39610 


585.19 


80 


39704 


39798 


39892 


39986 


40081 


40175 


40270 


40365 


40460 


40555 


592.59 


a 


81 


40650 


40745 


40841 


40936 


41032 


41127 


41223 


41319 


41415 


41511 


600.00 


•d 


^^ 


82 


41607 


41704 


41800 


41897 


41993 


42090 


42187 


42284 


42381 


42479 


607.41 


1 




83 


42576 


42673 


42771 


42869 


42966 


43064 


43162 


43260 


43359 


43457 


614.81 


< 


84 


43556 


43654 


43753 


43852 


43951 


44050 


44149 


44248 


44347 


44447 


622.22 




85 


44546 


44646 


44746 


44846 


44946 


45046 


45146 


45246 


45347 


45447 


629.63 


P.P. 


>r. 


86 


45548 


45649 


45750 


45851 


45952 


46053 


46155 


46256 


46358 


46459 


637.04 


7^41 




87 


46561 


46663 


46765 


46867 


46969 


47072 


47174 


47277 


47379 


47482 


644.44 


1 T/l 


88 


47585 


47688 


47791 


47895 


47998 


48101 


48205 


48309 


48412 


48516 


651.85 


1 

2 

3 

4 

5 

6 

7 

8 
q 


. 1 1 

1.48 
2.22 
2.96 
3.70 
4.44 
5.19 
5.93 

c an 


Ji ^ 


89 


48620 


48725 


48829 


48933 


49038 


49142 


49247 


49352 


49457 


49562 


659.26 


23 


90 


49667 


49772 


49877 


49983 


50088 


50194 


50300 


50406 


50512 


50618 


666.67 


^o 


91 


50724 


50830 


50937 


51044 


51150 


51257 


51364 


51471 


51578 


51685 


674.07 


wco 

:5oo 


92 


51793 


51900 


52008 


52115 


52223 


52331 


52439 


52547 


52655 


52764 


681.48 


93 


52872 


52981 


53089 


53198 


53307 


53416 


53525 


53635 


53744 


53853 


688.89 


94 


53963 


54073 


54182 


54292 


54402 


54513 


54623 


54733 


54844 


54954 


696.30 


r^ <U 


95 


55065 


55176 


55287 


55398 


55509 


55620 


55731 


55843 


55954 


56066 


703.70 


•-^ 


96 


56178 


56290 


56402 


56514 


56626 


56738 


56851 


56963 


57076 


57189 


711.11 


y u. ui 




97 


57302 


57415 


57528 


57641 


57755 


57868 


57982 


58095 


58209 


58323 


718.52 


, 


98 


58437 


58551 


58665 


58780 


58894 


59009 


59123 


59238 


59353 


59468 


725.93 


o 


>hi 


99 


59583 


59699 


59814 


59929 


60045 


60161 


60276 


60392 


60508 


60624 


733.33 


5J 


100 


60741 


60857 


60974 


61090 


61207 


61324 


61441 


61558 


61675 


61792 


740.74 


fl 


101 


61909 


62027 


62144 


62262 


62380 


62498 


62616 


62734 


62852 


62970 


748.15 


B 


'drH 
CJOO 


102 


63089 


63207 


63326 


63445 


63564 


63683 


63802 


63921 


64041 


64160 


755.56 





103 


64280 


64399 


64519 


64639 


64759 


64879 


64999 


65120 


65240 


65361 


762.96 


o 
o 


fli VM 


104 


65481 


65602 


65723 


65844 


65965 


66087 


66208 


66329 


66451 


66573 


770.37 


<u° 


105 


66694 


66816 


66938 


67060 


67183 


67305 


67428 


67750 


67673 


67796 


777.78 




106 


67919 


68042 


68165 


68288 


68411 


68535 


68658 


68782 


68906 


69030 


785.19 




107 


69154 


69278 


69402 


69526 


69651 


69775 


69900 


70025 


70150 


70275 


792.59 


5 


C0-J5 


108 


70400 


70525 


70651 


70776 


70902 


71027 


71153 


71279 


71405 


71531 


800.00 


2 


o-^ 


109 


71657 


71784 


71910 


72037 


72163 


72290 


72417 


72544 


72671 


72799 


807.41 


A 


rtS 


110 


72926 


73053 


73181 


?3309 


73436 


73564 


73692 


73821 


73949 


74077 


814.81 


5 


pq^ 


111 


74206 


74334 


74463 


74592 


74721 


74850 


74979 


75108 


75237 


75367 


822.22 




15 '^ 


112 


75496 


75626 


75756 


75886 


76016 


76146 


76276 


76406 


76537 


76667 


829.63 


c4 w, 

-M 4-> 


113 


76798 


76929 


77060 


77191 


77322 


77453 


77585 


77716 


77848 


77979 


837.04 


g 


114 


78111 


78243 


78375 


78507 


78639 


78772 


78904 


79037 


79169 


79302 


844.44 


p 


^?^ 


115 


79435 


79568 


79701 


79835 


79968 


80101 


80235 


80369 


80502 


80636 


851.85 






116 


80770 


80905 


81039 


81173 


81308 


81442 


81577 


81712 


81847 


81982 


859.26 




117 


82117 


82252 


82387 


82523 


82658 


82794 


82930 


83066 


83202 


83338 


866.67 




118 


83474 


83610 


83747 


83883 


84020 


84157 


84294 


84431 


84568 


84705 


874.07 




^ 


119 


84843 


84980 


85118 


85255 


85393 


85531 


85669 


85807 


85945 


86084 


881.48 






120 


86222 


86361 


86499 


86638 


86777 


86916 


87055 


87195 


87334 


87437 


888.89 







1036 



,— RAILROADS, 



25. — ^Level Sections (Earthwork); Height, 0-60 Ft. 
Base of Roadway, 16 Ft. Side Slopes, 1 to 1. 
Note. — ^The last two columns enable us to use any other base than 14 ft.: 
Ex. — Given height, 20.3 ft.; roadway 14 ft. Then we have, 2729.2- 
(148.15+2.22) = 2578.8cu. yds. (For Ht. >60 ft., see Tables 28. 40.) 

[Cu. Yds. per 100-Ft. Station.] 



























Width 






Ht. 


.0 


.1 


.2 


.3 


.4 


.5 


.6 


.7 


.8 


.9 


of 2 Ft. 






Ft. 






















Cu.Yds 




^ 







6.0 


12.0 


18.1 


24.3 


30.6 


36.9 


43.3 


49 8 


56 3 








1 


"63!b 


69.7 


76.4 


83.3 


90.2 


97.2 


104.3 


111.4 


118.7 


126 


"*7!4i 




2 


133.3 


140.8 


148.3 


155.9 


163.6 


171.3 


179.1 


187.0 


195.0 


203.0 


14.81 




. ^ • 


3 


211.1 


219.3 


227.6 


235.9 


244.3 


252.8 


261.3 


270.0 


278.7 


287.4 


22.22 






4 


296.3 


305.2 


314.2 


323.3 


332.4 


341.7 


351.0 


360.3 


369.8 


379.3 


29.63 




5 


388.9 


398.6 


408.3 


418.1 


428.0 


438.0 


448.0 


458.1 


468.3 


478.6 


37.04 






6 


488.9 


499.3 


509.8 


520.3 


531.0 


541.7 


552.4 


563.3 


574.2 


585.2 


44.44 


s 


7 


596.3 


607.4 


618.7 


630.0 


641.3 


652.8 


664.3 


675.9 


687.6 


699.3 


51.85 


•a 




8 


711.1 


723.0 


735.0 


747.0 


759.1 


771.3 


783.6 


795.9 


808.3 


820.8 


59.26 




9 


833.3 


846.0 


858.7 


871.4 


884.3 


897.2 


910.2 


923.3 


936.4 


949.7 


66.67 


M 


10 


963.0 


976.3 


898.8 


1003.3 


1016.9 


1030.6 


1044.3 


1058.1 


1072.0 


1086.0 


74.07 


a 


11 


1100.0 


1114.1 


1128.3 


1142.6 


1156.9 


1171.3 


1185.8 


1200.3 


1215.0 


1229.7 


81.48 


4A 




12 


1244.4 


1259.3 


1274.2 


1289.2 


1304.3 


1319.4 


1334.7 


1350.0 


1365.3 


1380.8 


88.89 


S 


13 


1396.3 


1411.9 


1427.6 


1443.3 


1459.1 


1475.0 


1491.0 


1507.0 


1523.1 


1539.3 


96.30 


^ 


14 


1555.6 


1571.9 


1588.3 


1604.8 


1621.3 


1638.0 


1654.7 


1671.4 


1688.3 


1705.2 


103.70 


^ 


ll 


15 


1722.2 


1739.3 


1756.4 


1773.7 


1791.0 


1808.3 


1825.8 


1843.3 


1860.9 


1878.6 


111.11 


1 


16 


1896.3 


1914.1 


1932.0 


1950.0 


1968.0 


1986.1 


2004.3 


2022.6 


2040.9 


2059.3 


118.52 


|o 


17 


2077.8 


2096.3 


2115.0 


2133.7 


2152.4 


2171.3 


2190.2 


2209.2 


2228.3 


2247.4 


125.93 




18 


2266.7 


2286.0 


2305.3 


2324.8 


2344.3 


2363.9 


2383.6 


2403.3 


2423.1 


2443.0 


133.33 


H 


3S 
Si 


19 


2463.0 


2483.0 


2503.1 


2523.3 


2543.6 


2563.9 


2584.3 


2604.8 


2625.3 


2646.0 


140.74 


b 


20 


2666.7 


2687.4 


2708.3 


2729.2 


2750.2 


2771.3 


2792.4 


2813.7 


2835.0 


2856.3 


148.15 


2 


21 


2877.8 


2899.3 


2920.9 


2942.6 


2964.3 


2986.1 


3008.0 


3030.0 


3052.0 


3074.1 


156.56 


2 


9. u 


22 


3096.3 


3118.6 


3140.9 


3163.3 


3185.8 


3208.3 


3231.0 


3253.7 


3276.4 


3299.3 


162.96 


^ 


23 


3322.2 


3345.2 


3368.3 


3391.4 


3414.7 


3438.0 


3461.3 


3484.8 


3508.3 


3531.9 


170.37 


c3 w 


24 


3555.6 


3579.3 


3603.1 


3627.0 


3651.0 


3675.0 


3699.1 


3723.3 


3747.6 


3771.9 


177.78 




^'P 


25 


3796.3 


3820.8 


3845.3 


3870.0 


3894.7 


3919.4 


3944.3 


3969.2 


3994.2 


4019.3 


185.19 


P. P. 




26 

27 


4044.4 
4300.0 


4069.7 
4326.0 


4095.0 
4352.0 


4120.3 
4378. 1 


4145.8 
4404.3 


4171.3 
4430.6 


4196.9 
4456.9 


4222.6 
4483.3 


4248.3 
4509.8 


4274. 1 
4536.3 


192.59 
200.00 




7.41 




1 


.74 


28 


4563.0 


4589.7 


4616.4 


4643.3 


4670.2 


4697.2 


4724.3 


4751.4 


4778.7 


4806.0 


207.41 


2 


1 48 


J^^ 


29 


4833.3 


4860.8 


4888.3 


4915.9 


4943.6 


4971.3 


4999.1 


5027.0 


5055.0 


5083.0 


214.81 


3 


2.22 


o«2 

|S3 


30 


5111.1 


5139.3 


5167.6 


5195.9 


5224.3 


5252.8 


5281.3 


5310.0 


5338.7 


5367.4 


222.22 


4 


2.96 


31 


5396.3 


5425.2 


5454.2 


5483.3 


5512.4 


5541.7 


5571.0 


5600.3 


5629.8 


5659.3 


229.63 


6 
6 


3.70 


to (U 


32 


5688.9 


5718.6 


5748.3 


5778.1 


5808.0 


5838.0 


5868.0 


5898.1 


5928.3 


5958.6 


237.04 


4.44 


*^ ^ 


33 


5988.9 


6019.3 


6049.8 


6080.3 


6111.0 


6141.7 


6172.4 


6203.3 


6234.2 


6265.2 


244.44 


7 


5.19 




34 


6296.3 


6327.4 


6358.7 


6390.0 


6421.3 


6452.8 


6484.3 


6515.9 


6547.6 


6579.3 


251.85 


8 


5.93 


35 


6611.1 


6643.0 


6675.0 


6707.0 


6738.1 


6771.3 


6803.6 


6835.9 


6868.3 


6900.8 


259.26 


g 


ft fi7 


36 


6933.3 


6966.0 


6998.7 


7031.4 


7064.3 


7097.2 


7130.2 


7163.3 


7196.4 


7229.7 


266.67 




37 


7263.0 


7296.3 


7329.8 


7363.3 


7396.9 


7430.6 


7464.3 


7498.1 


7532.0 


7566.0 


274.07 


■2 




38 


7600.0 


7634.1 


7668.3 


7702.6 


7736.9 


7771.3 


7805.8 


7840.3 


7875.0 


7909.7 


281.48 


39 


7944.4 


7979.3 


8014.2 


8049.2 


8084.3 


8119.4 


8154.7 


8190.0 


8225.3 


8260 8 


288.89 


p 
o 




40 


8296.3 


8331.9 


8367.6 


8403.3 


8439.1 


8475.0 


8511.0 


8547.0 


8583.1 


8619.3 


296.30 


ca 


5^ 


41 


8655.6 


8691.9 


8728.3 


8764.8 


8801.3 


8838.0 


8874.7 


8911.4 


8948.3 


8985.2 


303.70 


g 


»a«« 


42 


9022.2 


9059.3 


9096.4 


9133.7 


9171.0 


9208.3 


9245.8 


9283.3 


9320.9 


9358.6 


311.11 


!3 


S° 


43 


9396.3 


9434.1 


9472.0 


9510.0 


9548.0 


9586.1 


9624.3 


9662.6 


9700.9 


9739.3 


318.52 


O 


6 bo 


44 


9777.8 


9816.3 


9855.0 


9893.7 


9932.4 


9971.3 


10010 


10049 


10088 


10127 


325.93 


45 


10167 


10206 


10245 


10285 


10324 


10364 


10404 


10443 


10483 


10523 


333.33 


be 

n 




46 


10563 


10603 


10643 


10683 


10724 


10764 


10804 


10845 


10885 


10926 


340.74 




47 


10967 


11007 


11048 


11089 


11130 


11171 


11212 


11254 


11295 


11336 


348.15 


s 


48 


11378 


11419 


11461 


11503 


11544 


11586 


11628 


11670 


11712 


11754 


355.56 


OJ 


49 


11796 


11839 


11881 


11923 


11966 


12008 


12051 


12094 


12136 


12179 


362.96 


A 




50 


12222 


12265 


12308 


12351 


12395 


12438 


12481 


12^25 


12568 


12612 


370.37 


5 
¥ 


W^ 


51 


12656 


12699 


12743 


12787 


12831 


12875 


12919 


12963 


13008 


13052 


377.78 


^o 


52 


13096 


13141 


13185 


13230 


13275 


13319 


13364 


13409 


13454 


13499 


385.19 


,£3 O 


53 


13544 


13590 


13635 


13680 


13726 


13771 


13817 


13863 


13908 


13954 


392.59 


^ 


54 


14000 


14046 


14092 


14138 


14184 


14231 


14277 


14323 


14370 


14416 


400. 00 


P 




55 


14463 


14510 


14556 


14603 


14650 


14697 


14744 


14791 


14839 


14886 


407.41 




■^T 


56 


14933 


14981 


15028 


15076 


15124 


15171 


15219 


15267 


15315 


15363 


414.81 






57 


15411 


15459 


15508 


15556 


15604 


15653 


15701 


15750 


15799 


15847 


422.22 




58 


15896 


15945 


15994 


16043 


16092 


16142 


16191 


16240 


16290 


16339 


429.63 




^ 


59 


16389 


16439 


16488 


16538 


16588 


16638 


16688 


16738 


16788 


16839 


437.04 






60 


16889 


16939 


16990 


17040 


17091 


17142 


17192 


17243 


17294 


17345 


444.44 







EARTHWORK TABLES— LEVEL SECTIONS. 



1037 



26. — Level Sections (Earthwork) ; Height, 0-60 Ft. 
Base of Roadway, 16 Ft. Side Slopes, 1^ to 1. 
Note. — ^The last two columns enable us to use any other base than 16 ft.: 
Ex. — Given height, 39.7 ft.; roadway 14 ft. Then we have, 11109 — 
(288.89+5.19) = 10815 cu. yds. (For Ht. >60 ft., see Tables 24, 41.) 

[Cu. Yds. per 100-Ft. Station.] 



























Width f 




Ht. 


.0 


.1 


.2 


.3 


.4 


.5 


.6 


.7 


.8 


.9 


of 2 Ft.l 




Ft. 






















Cu.Yds 




tn 







6.0 


12.1 


18.3 


24.6 


31.0 


37.6 


44.2 


51.0 


57.8 






2-H- 


1 


"64!8 


71.9 


79.1 


86.4 


93.9 


101.4 


109.0 


116.8 


124.7 


132.6 


"*7!4i 




■^ o 

-t-> 


2 


140.7 


148.9 


157.3 


165.7 


174.2 


182.9 


191.6 


200. £ 


209.5 


218.6 


14.81 




3 


227.8 


237.1 


246.5 


256.1 


265.7 


275.5 


285.3 


295.3 


305.4 


315.6 


22.22 




Ig 


4 


325.9 


336.4 


346.9 


357.5 


368.3 


379.2 


390.1 


401.2 


412.4 


423.8 


29.63 




5 


435.2 


446.7 


458.4 


470.1 


482.0 


494.0 


506.1 


518.3 


530.6 


543.0 


37.04 




(U o 


6 


555.6 


568.2 


581.0 


593.8 


606.8 


619.9 


633.1 


646.4 


659.9 


673.4 


44.44 


■a 


7 


687.0 


700.8 


714.7 


728.6 


742.7 


756.9 


771.3 


785.7 


800.2 


814.9 


51.85 


8 


829.6 


844.5 


859.5 


874.6 


889.8 


905.1 


920.5 


936.1 


951.7 


967.5 


59.26 


1 


a« 


9 


983.3 


999.3 


1015.4 


1031.6 


1047.9 


1064.4 


1080.9 


1097.5 


1114.3 


1131.2 


66.67 


:l 


10 


1148.1 


1165.2 


1182.4 


1199.8 


1217.2 


1234.7 


1252.4 


1270.1 


1288.0 


1306.0 


74.07 


d 


11 


1324.1 


1342.3 


1360.6 


1379.0 


1397.6 


1416.2 


1435.0 


1453.8 


1472.8 


1491.9 


81.48 


+» 


<u aj 


12 


1511.1 


1530.4 


1549.9 


1569.4 


1589.0 


1608.8 


1628.7 


1648.6 


1668.7 


1688.9 


88.89 


S 


•5^ 


13 


1709.3 


1729.7 


1750.2 


1770.9 


1791.6 


1812.5 


1833.5 


1854.6 


1875.8 


1897.1 


96.30 


ft* 




14 


1918.5 


1940.1 


1961.7 


1983.5 


2005.3 


2027.3 


2049.4 


2071.6 


2093.9 


2116.4 


103.70 


^ 


^^ 


15 


2138.9 


2161.5 


2184.3 


2207.2 


2230.1 


2253.2 


2276.4 


2299.8 


2323.2 


2346.7 


111.11 




TJcm 


16 


2370.4 


2394.1 


2418.0 


2442.0 


2466.1 


2490.3 


2514.6 


2539.0 


2563.6 


2588.2 


118.52 


S o 


17 


2613.0 


2637.8 


2662.8 


2687.9 


2713.1 


2738.4 


2763.9 


2789.4 


2815.0 


2840.8 


125.93 


s 


Sw 


18 


2866.7 


2892.6 


2918.7 


2944.9 


2971.3 


2997.7 


3024.2 


3050.9 


3077.6 


3104.5 


133.33 


^ 


s ^ 


19 


3131.5 


3158.6 


3185.8 


3213.1 


3240.5 


3268.1 


3295.7 


3323.5 


3351.3 


3379.3 


140.74 


fH 


3*^ 


20 


3407.4 


3435.6 


3463.9 


3492.4 


3520.9 


3549.5 


3578.3 


3607.2 


3636.1 


3665.2 


148.15 


o 




21 


3694.4 


3723.8 


3753.2 


3782.7 


3812.4 


3842.1 


3872.0 


3902.0 


3932.1 


3962.3 


156.56 


-d 


22 


3992.6 


4023.0 


4053.6 


4084.2 


4115.0 


4145.8 


4176.8 


4207.9 


4239.1 


4270.4 


162.96 


< 


23 


4301.9 


4333.4 


4365.0 


4396.8 


4428.7 


4460.6 


4492.7 


4524.9 


4557.3 


4589.7 


170.37 


"■g. 


24 


4622.2 


4654.9 


4687.6 


4720.5 


4753.5 


4786.6 


4819.8 


4853. 1 


4886.5 


4920. 1 


177.78 




•3^ 


25 


4953.7 


4987.5 


5021.3 


5055.3 


5p89.4 


5123.6 


5157.9 


5192.4 


5226.9 


5261.5 


185.19 


P. P. 


&s 


26 
27 


5296.3 
5650.0 


5331.2 


5366.1 
5722.1 


5401.2 


5436.4 


5471.8 


5507.2 


5542.7 


5578.4 


5614.1 
5977.8 


192.59 


7.41 


5686.0 


5758.3 


5794.6 


5831.0 


5867.6 


5904.2 


5941.0 


200.00 


1 .74 

2 1.48 

3 2.22 

4 2.96 

5 3.70 

6 4.44 

7 5.19 

8 5.93 

9 6.67 


62 


28 


6014.8 


6051.9 


6089.1 


6126.4 


6163.9 


6201.4 


6239.0 


6276.8 


6314.7 


6352.6 


207.41 


29 


6390.7 


6428.9 


6467.3 


6505.7 


6544.2 


6582.9 


6621.6 


6660. 5 


6699.5 


6738.6 


214.81 


(Uo 


30 


6777.8 


6817.1 


6856.5 


6896.1 


6935.7 


6975.5 


7015.3 


7055.3 


7095.4 


7135.6 


222.22 


^^« 


31 


7175.9 


7216.4 


7256.9 


7297.5 


7338.3 


7379.2 


7420.1 


7461.2 


7502.4 


7543.8 


229.63 




32 


7585.2 


7626.7 


7668.4 


7710.1 


7752.0 


7794.0 


7836.1 


7878.3 


7920.6 


7963.0 


237.04 


33 


8005.6 


8048.2 


8091.0 


8133.8 


8176.8 


8219.9 


8263.1 


8306.4 


8349.9 


8393.4 


244.44 


13^ 


34 


8437.0 


8480.8 


8524.7 


8568.6 


8612.7 


8656.9 


8701.3 


8745.7 


8790.2 


8834.9 


251.85 


■^ o 


35 


8879.6 


8924.5 


8969.5 


9014.6 


9059.8 


9105.1 


9150.5 


9196.1 


9241.7 


9287.5 


259.26 


d^ 


36 


9333.3 


9379.3 


9425.4 


9471.6 


9517.9 


9564.4 


9610.9 


9657.5 


9704.3 


9751.2 


266.67 


•!h . 


37 


9798. 1 


9845.2 


9892.4 


9939.8 


9987.2 


10035 


10082 


10130 


10178 


10226 


274.07 


. 


^^ 


38 


10274 


10322 


10371 


10419 


10468 


10516 


10565 


10614 


10663 


10712 


281.48 


•a 


2- 


39 


10761 


10810 


10860 


10909 


10959 


11009 


11059 


11109 


11159 


11209 


288.89 


.o^ 


40 


11259 


11310 


11360 


11411 


11462 


11513 


11563 


11615 


11666 


11717 


296.30 


d 

d 


d« 


41 


11769 


11820 


11872 


11923 


11975 


12027 


12079 


12132 


12184 


12236 


303.70 




42 


12289 


12342 


12394 


12447 


12500 


12553 


12606 


12660 


12713 


12767 


311.11 


43 


12820 


12874 


12928 


12982 


13036 


13090 


13145 


13199 


13254 


13308 


318.52 


o 


§^ 


44 


13363 


13418 


13473 


13528 


13583 


13638 


13694 


13749 


13805 


13861 


325.93 


u 


«r^ 


45 


13917 


13973 


14029 


14085 


14141 


14198 


14254 


14311 


14368 


14425 


333.33 


rt* 


^ u 


46 


14481 


14539 


14596 


14653 


14711 


14768 


14826 


14883 


14941 


14999 


340.74 


S 


47 


15057 


15116 


15174 


15232 


15291 


15350 


15408 


15467 


15526 


15585 


348.15 


<i> 


C/D.O 


48 


15644 


15704 


15763 


15823 


15882 


15942 


16002 


16062 


16122 


16182 


355.56 


0^ 


M-t 


49 


16243 


16303 


16364 


16424 


16485 


16546 


16607 


16668 


16729 


16790 


362.96 


p, 


^":^ 


50 


16852 


16913 


16975 


17037 


17099 


17161 


17223 


17285 


17347 


17410 


370.37 


A 


CQ, 


51 


17472 


17535 


17598 


17661 


17723 


17787 


17850 


17913 


17977 


18040 


377.78 


s 


jj o 


52 


18104 


18167 


18231 


18295 


18359 


18424 


18488 


18552 


18617 


18682 


385.19 


*^ 




53 


18746 


18811 


18876 


18941 


19006 


19072 


19137 


19203 


19268 


19334 


392.59 


s 


^ o 


54 


19400 


19466 


19532 


19598 


19665 


19731 


19798 


19864 


19931 


19998 


400.00 


S 


"^s 


55 


20065 


20132 


20199 


20266 


20334 


20401 


20469 


20537 


20605 


20673 


407.41 




•i^ 


56 


20741 


20809 


20877 


20946 


21014 


21083 


21152 


22221 


21289 


21359 


414.81 




^g) 


57 


21428 


21497 


21567 


21636 


21706 


21775 


21845 


21915 


21985 


22056 


422.22 






58 


22126 


22196 


22267 


22338 


22408 


22479 


22550 


22621 


22692 


22764 


429.63 




1 


59 


22835 


22907 


22978 


23050 


23122 


23194 


23266 


23338 


23411 


23483 


437.04 




60 1 


23556 


23628 


23701 


23774 J 


23847 


23920 


23993 


24066 


24140 


24213 


444. 44 





1038 



).— RAILROADS. 



27. — Level Sections (Earthwork) ; Height, 0-60 Ft. Also — 

Base of Roadway, 18 Ft. Side Slopes, 1 to 1. 

Note. — ^The last two columns enable us to use any other base than 18 ft.: 
Ex. — Given height, 14.8 ft.; roadway 19 ft. Then we have, 1797.9 + 
^(103.70+ 5.93) = 1852.7 cu. yds. (For Ht. >60 ft., see Tables 28, 40.) 

[Cu. Yds. per 100-Ft. Station.] 

























Width 






Ht. 


.0 


.1 


.2 


.3 


.4 


.5 


.6 


.7 


.8 


.9 


of 2 Ft. 






Ft. 






















Cu.Yds 




xa 







6.7 


13.5 


20.3 


27.3 


34.3 


41.3 


48.5 


55.7 


63.0 






S_: 


1 


"70!4 


77.8 


85.3 


92.9 


100.6 


108.3 


116.1 


124.0 


132.0 


140.0 


***7!4i 


^ 


^^ 


2 


148.1 


156.3 


164.6 


172.9 


181.3 


189.8 


198.4 


207.0 


215.7 


224.5 


14.81 




o 


3 


233.3 


242.3 


251.3 


260.3 


269.5 


278.7 


288.0 


297.4 


306.8 


316.3 


22.22 




g 


4 


325.9 


335.6 


345.3 


355.1 


365.0 


375.0 


385.0 


395.1 


405.3 


415.6 


29.63 




5 


425.9 


436.3 


446.8 


457.4 


468.0 


478.7 


489.5 


500.3 


511.3 


522.3 


37.04 






6 


533.3 


544.5 


555.7 


567.0 


578.4 


589.8 


601.3 


612.9 


624.6 


636.3 


44.44 


4.3 

.a 


7 


648.1 


660.0 


672.0 


684.0 


696.1 


708.3 


720.6 


732.9 


745.3 


757.8 


51.85 


^ 


ri o 


8 


770.4 


783.0 


795.7 


808.5 


821.3 


834.3 


847.3 


860.3 


873.5 


886.7 


59.26 


& 


2 c/3 


9 


900.0 


913.4 


926.8 


940.3 


953.9 


967.6 


981.3 


995.1 


1009.0 


1023.0 


66.67 




10 


1037.0 


1051.1 


1065.3 


1079.6 


1093.9 


1108.3 


1122.8 


1137.4 


1152.0 


1166 7 


74.07 


B 


11 


1181.5 


1196.3 


1211.3 


1226.3 


1241.3 


1256.5 


1271.7 


1287.0 


1302.4 


1317.8 


81.48 


t 


^^ 


12 


1333.3 


1348.9 


1364.6 


1380.3 


1396.1 


1412.0 


1428.0 


1444.0 


1460.1 


1476.3 


88.89 


s 


•^^ 


13 


1492.6 


1508.9 


1525.3 


1541.8 


1558.4 


1575.0 


1591.7 


1608.5 


1625.3 


1642.3 


96.30 


^^ 


14 


1659.3 


1676.3 


1693.5 


1710.7 


1728.0 


1745.4 


1762.8 


1780.3 


1797.9 


1815.6 


103.70 


o 


-^S^ 


15 


1833.3 


1851.1 


1869.0 


1887.0 


1905.0 


1923.1 


1941.3 


1959.6 


1977.9 


1996.3 


111.11 


1 


?.^ 


16 


2014.8 


2033.4 


2052.0 


2070.7 


2089.5 


2108.3 


2127.3 


2146.3 


2165.3 


2184.5 


118.52 


;S° 


17 


2203.7 


2223.0 


2242.4 


2261.8 


2281.3 


2300.9 


2320.6 


2340.3 


2360.1 


2380.0 


125.93 




a^ 


18 


2400.0 


2420.0 


2440.0 


2460.2 


2480.5 


2500.8 


2521.2 


2541.7 


2562 3 


2582.9 


133.33 


H 


'■+J aJ 


19 


2603.6 


2624.4 


2645.2 


2666.2 


2687.2 


2708.2 


2729.5 


2750.7 


2772.0 


2793.4 


140.74 


1 


'^t 


20 


2814.8 


2836.3 


2857.9 


2879.6 


2901.3 


2923 1 


2945.0 


2967.0 


2989.0 


3011.1 


148.15 




as 


21 


3033.3 


3055.6 


3077.9 


3100.3 


3122.8 


3145.4 


3168.0 


3190.7 


3213.5 


3236.3 


155.56 




<45 


22 


3259.3 


3282.3 


3305.3 


3328.5 


3351.7 


3375.0 


3398.4 


3421.8 


3445.3 


3468.9 


162.96 


< 


yQ (A 


23 


3492.6 


3516.3 


3540.1 


3564.0 


3588.0 


3612.0 


3636.1 


3660.3 


3684.6 


3708.9 


170.37 




CO . 


24 


3733.3 


3757.8 


3782.4 


3807.0 


3831.7 


3856.5 


3881.3 


3906.3 


3931.3 


3956.3 


177.78 




25 


3981.5 


4006.7 


4032.0 


4057.4 


4082.8 


4108.3 


4133.9 


4159.6 


4185.3 


4211.1 


185.19 


P. P. 


>>?^ 


26 


4237.0 


4263.0 


4289.0 


4315.1 


4341.3 


4367.6 


4393.9 


4420.3 


4446.8 


4473.4 


192.59 


7.41 


a§ 


27 


4500.0 


4526.7 


4553.5 


4580.3 


4607.3 


4634.3 


4661.3 


4688.5 


4715.7 


4743.0 


200.00 


1 .74 


28 


4770.4 


4797.8 


4825.3 


4852.9 


4880.6 


4908.3 


4936.1 


4964.0 


4992.0 


5020.0 


207.41 


2 1.48 




29 


5048.1 


5076.3 


5104.6 


5132.9 


5161.3 


5189.8 


5218.4 


5247.0 


5275.7 


5304.5 


214.81 


3 2.22 


30 


5333.3 5362.3 


5391.3 


5420.3 


5449.5 


5478.7 


5508.0 


5.537.4 


5566.8 


5596.3 


222.22 


4 2.96 


31 


5625.9 


5655.6 


5685.3 


5615.1 


5645.0 


5675.0 


5605.0 


5635.1 


5665.3 


5695.6 


229.63 


5 3.70 


^o 


32 


5725.9 


5956.3 


5986.8 


6017.4 


6048.0 


6078.7 


6109.5 


6140.3 


6171.3 


6202.3 


237.04 


6 4.44 


w ci 


33 


6233.3 


6264.5 


6295.7 


6327.0 


6358.4 


6389.8 


6421.3 


6452.9 


6484.6 


6516.3 


244.44 


.7 5.19 


'j^^ 


34 


6548.1 


6580.0 


6612.0 


6644.0 


6676.1 


6708.3 


6740 6 


6772.9 


6805.3 


6837.8 


251.85 


8 5.93 


T^ 


35 


6870.4 


6903.0 


6935.7 


6968.5 


7001.3 


7034.3 


7067.3 


7100.3 


7133.5 


7166.7 


259.26 


9 6.67 


36 


7200.0 


7233.4 


7266.8 


7300.3 


7333.9 


7367.6 


7401.3 


7435.1 


7469.0 


7503.0 


266.67 




. ■ 


37 


7537.0 


7571.1 


7605.3 


7639.6 


7673.9 


7708.3 


7742.8 


7777.4 


7812.0 


7846.7 


274.07 


fA 


^- 


38 


7881.5 


7916.3 


7951.3 


7986.3 


8021.3 


8056.5 


8091.7 


8127.0 


8162.4 


8197.8 


281.48 


■a 


gcc 


39 


8233.3 


8268.9 


8304.6 


8340.3 


8376.1 


8412.0 


8448.0 


8484.0 


8520.1 


8556.3 


288. 89 


o 


.t>. 


40 


8592.6 


8628.9 


8665.3 


8701.8 


8738.4 


8775.0 


8811.7 


8848.5 


8885.3 


8922.3 


296.30 


c 




41 


8959.3 


8996.3 


9033.5 


9070.7 


9108.0 


9145.4 


9182.8 


9220.3 


9257.9 


9295.6 


303.70 


B 


42 


9333.3 


9371.1 


9409.0 


9447.0 


9485.0 


9523.1 


9561.3 


9599.6 


9637.9 


9676.3 


311.11 


3 


^- 


43 


9714.8 


9753.4 


9792.0 


9830.7 


9869.5 


9908.3 


9947.3 


9986.3 


10025 


10064 


318.52 


I 


^-a 


44 


10104 


10143 


10182 


10222 


10261 


10301 


10341 


10380 


10420 


10460 


325.93 


be 




45 


10500 


10540 


10580 


10620 


10661 


10701 


10741 


10782 


10822 


10863 


333.33 


C 


a^ 


46 


10904 


10944 


10985 


11026 


11067 


11108 


11149 


11191 


11232 


11273 


340.74 


"O 


^M 


47 


11315 


11356 


11398 


11440 


11481 


11523 


11565 


11607 


11649 


11691 


348.15 


u 


"^^ 


48 


11733 


11776 


11818 


11860 


11903 


11945 


11988 


12031 


12073 


12116 


355.56 


P. 


Ci T-H 


49 


12159 


12202 


12245 


12288 


12332 


12375 


12418 


12462 


12505 


12549 


362.96 


50 


12593 


12636 


12680 


12724 


12768 


12812 


12856 


12900 


12945 


12989 


370.37 


•B 


pq.4-. 


51 


13033 


13078 


13122 


13167 


13212 


13256 


13301 


13346 


13391 


13436 


377.78 


¥ 


^o 


52 


13481 


13527 


13572 


13617 


13663 


13708 


13754 


13800 


13845 


13891 


385.19 


<u 




53 


13937 


13983 


14029 


14075 


14121 


14168 


14214 


14260 


14307 


14353 


392.59 


s 


54 


14400 


14447 


14493 


14540 


14587 


14634 


14681 


14728 


14776 


14823 


400.00 


5i) nj 


55 


14870 


14918 


14965 


15013 


15061 


15108 


15156 


15204 


15252 


15300 


407.41 




^„ 


56 


15348 


15396 


15445 


15493 


15541 


15590 


15638 


15687 


15736 


15784 


414.81 




'55 


57 


15833 


15882 


15931 


15980 


16029 


16079 


16128 


17177 


16227 


16276 


422.22 




58 


16326 


16376 


16425 


16475 


16525 


16575 


16625 


17675 


16725 


16776 


429.63 




;:< 


59 


16826 


16876 


16927 


16977 


17028 


17079 


17129 


17180 


17231 


17282 


437.04 






60 


17333 


17384 


17436 


17487 117538 


17590 


17641 


17693 


17745 


17796 


444.44 



EARTHWORK TABLES, WITH GROUND SLOPES. 



1039 



— Corrections for Ground Slopes Not Level. 

Base of Roadway, 18 Ft, Side Slopes, 1 to I. 

(See Explanation and Formulas in Table 22.) 

Note. — a = angle of ground slope (G. 5.) to right of left of center line; 

and *'G. S. = l:10," "(J. 5. = 2:10," etc., means tan a. Up or ( + ) slopes from 

the center indicate additive corrections, and down or ( — ) slopes subtract- 

ive, from quantities in table on opposite page for level sections. 

[Additive (+) and subtractive (— ) Corrections in Cu. Yds, per 100-Ft. Sta.] 





*l4i 


G.S 


.= 1:10. 


G.S 


. = 2:10. 


G.S 


. = 3:10. 


G.S 


. = 4:10. 


G.S 


. = 5:10. 




5£ 
g*i 

OK 


a 


= 5°.7 


a = 


= 11°.3 


a = 


= 16°.7 


a = 


= 21°.8 


a = 


= 26°.6 




+ Up 


— Down 


+ Up 


— Down 


+ Up 


— Down 


+ Up 


— Down 


+ Up 


— Down 




1 


21 


17 


46 


26 


79 


29 


123 


31 


185 


31 




2 


25 


20 


56 


37 


96 


49 


149 


56 


224 


59 


3 


30 


24 


67 


44 


114 


62 


178 


75 


267 


83 


OS 
C4 


4 


34 


28 


78 


52 


134 


72 


208 


89 


313 


104 




6 


40 


33 


91 


60 


156 


84 


242 


104 


363 


121 


11 


6 


46 


38 


104 


69 


179 


96 


278 


119 


417 


139 


o 


7 


53 


43 


119 


79 


203 


109 


316 


135 


474 


158 


s 


8 


59 


49 


134 


89 


229 


124 


357 


153 


535 


178 


c* 


9 


67 


55 


150 


100 


257 


138 


400 


171 


600 


200 


1 


10 


74 


61 


167 


111 


287 


154 


446 


191 
212 


669 


223 


C3S 
00 


11 


82 


67 


185 


123 


317 


171 


494 


741 


247 


CO 


12 


91 


73 


204 


136 


350 


188 


544 


233 


817 


272 


^-' 


13 


100 


81 


224 


149 


384 


207 


597 


256 


896 


299 


'^ «^ 


14 


109 


89 


245 


163 


420 


226 


653 


280 


980 


327 


^^ 


15 


119 


97 


267 


178 


457 


246 


711 


305 


1067 


356 


16 


129 


105 


289 


193 


496 


267 


771 


331 


1157 


386 


g'=^ 


17 


139 


114 


313 


209 


537 


289 


834 


358 


1252 


417 


J- 


18 


150 


123 


338 


225 


579 


312 


900 


386 


1350 


450 


H-S 


19 


161 


132 


363 


242 


622 


335 


968 


415 


1452 


484 


^ a 


20 


173 


142 


389 


260 


667 


359 


1038 


445 


1557 


519 


21 


185 


152 


417 


278 


714 


385 


nil 


476 


1667 


556 


^ o 


22 


198 


162 


445 


297 


763 


411 


1186 


508 


1780 


593 




23 


211 


172 


474 


316 


813 


438 


1264 


542 


1896 


632 




24 


224 


183 


504 


336 


864 


465 


1344 


576 


2017 


672 


§1 


25 


238 


195 


535 


357 


917 


494 


1427 


612 


2141 


714 


26 


252 


206 


567 


378 


971 


524 


1512 


648 


2269 


756 


27 


267 


218 


600 


400 


1029 


554 


1600 


686 


2400 


800 


-^i 


28 


282 


230 


634 


423 


1087 


585 


1690 


724 


2535 


845 


u w 

^1 


29 


297 


243 


669 


446 


1146 


617 


1783 


764 


2674 


891 


30 


313 


256 


704 


469 


1207 


650 


1878 


805 


2817 


939 


g£ 


31 


329 


269 


741 


494 


1270 


684 


1975 


847 


2963 


988 


o^ 


32 


346 


283 


778 


519 


1334 


718 


2075 


889 


3113 


1038 


IT 


33 


363 


297 


817 


544 


1400 


754 


2178 


933 


3267 


1089 


.2 <^ 


34 


380 


311 


856 


571 


1467 


790 


2283 


978 


3424 


1141 


35 


398 


326 


896 


598 


1537 


827 


2390 


1024 


3585 


1195 


^? 


36 


417 


341 


938 


625 


1607 


865 


2500 


1071 


3750 


1250 


S" 


37 


435 


356 


980 


653 


1679 


904 


2612 


1120 


3919 


1306 


[26 

O 


38 


455 


372 


1023 


682 


1753 


944 


2727 


1169 


4091 


1364 


39 


475 


388 


1067 


711 


1829 


985 


2844 


1219 


4267 


1422 




40 


494 


404 


1112 


741 


1906 


1026 


2964 


1270 


4446 


1482 


41 


514 


421 


1157 


772 


1984 


1068 


3086 


1323 


4630 


1543 


42 


535 


438 


1204 


803 


2064 


1112 


3211 


1376 


4817 


1606 


fn-S 


43 


556 


455 


1252 


835 


2146 


1156 


3338 


1431 


5007 


1669 


.5Prt 


44 


578 


473 


1300 


867 


2229 


1200 


3468 


1486 


5201 


1734 


2^ 


45 


600 


491 


1350 


900 


2314 


1246 


3600 


1543 


5400 


1800 


t^° 


46 


622 


509 


1400 


934 


2401 


1293 


3734 


1601 


5602 


1867 


O w 


47 


645 


528 


1452 


968 


2489 


1340 


3871 


1659 


5807 


1936 


^V 


48 


669 


547 


1504 


1003 


2579 


1388 


4011 


1719 


6017 


2006 


g>^ 


49 


692 


566 


1557 


1038 


2670 


1438 


4153 


1780 


6230 


2077 


<M o 


50 


716 


586 


1612 


1074 


2763 


1488 


4297 


1842 


6446 


2149 


51 


741 


606 


1667 


nil 


2857 


1538 


4444 


1905 


6667 


2222 


lioo 


52 


766 


626 


1723 


1148 


2953 


1590 


2593 


1969 


6891 


2297 


^^ 


53 


791 


647 


1780 


1186 


3051 


1643 


4746 


2034 


7119 


2373 


54 


817 


668 


1838 


1225 


3150 


1696 


4900 


2100 


7350 


2450 


S5?5 


55 


843 


690 


1896 


1264 


3251 


1750 


5057 


2167 


7585 


2528 


T 


56 


869 


711 


1956 


1304 


3353 


1806 


5216 


2235 


7824 


2608 


'. o 


57 


896 


733 


2017 


1344 


3457 


1862 


5378 


2305 


8067 


2689 


58 


924 


756 


2078 


1385 


3563 


1918 


5542 


2375 


8313 


2771 




59 


951 


778 


2141 


1427 


3670 


1976 


5708 


2447 


8563 


2854 


60 


980 


802 


2204 


1469 


3779 


2035 


5878 


2519 


8817 


2939 



1040 



69.— RAILROADS. 



28. — Level Sections (Earthwork); Height, 60-100 Ft. Also — 

Base of Roadway, 18 Ft. Side Slopes, 1 to 1. 
Note. — The last two columns enable us to use any other base than 18 ft.: 
Ex.— Given height, 88.5 ft.; roadway 17 ft. Then we have, 34908- 
i(651.85+ 3.70) = 34580 cu. yds. (See also Table 40.) 
[Cu. Yds. per 100-Ft. Station.] 

























Width 




Ht. 


.0 


.1 


.2 


.3 


A 


.6 


.6 


.7 


.8 


.9 


of 2 Ft. 




Ft. 
















• 






Cu.Yds 




60 


17333 


17384 


17436 


17487 


17538 


17590 


17641 


17693 


17745 


17796 


444.44 




61 


17848 


17900 


17952 


18004 


18056 


18108 


18161 


18213 


18265 


18318 


451.85 




62 


18370 


18423 


18476 


18528 


18581 


18634 


18687 


18740 


18793 


18847 


459.26 




63 


18900 


18953 


19007 


19060 


19114 


19168 


19221 


19275 


19329 


19383 


466.67 




64 


19437 


19491 


19545 


19600 


19654 


19708 


19763 


19817 


19872 


19927 


474.07 


Is 


65 


19981 


20036 


20091 


20146 


20201 


20256 


20312 


20367 


20422 


20478 


481.48 


66 


20533 


20589 


20645 


20700 


20756 


20812 


20868 


20924 


20980 


21036 


488.89 


67 


21093 


21149 


21205 


21262 


21318 


21375 


21432 


21488 


21545 


21602 


496.30 


68 


21659 


21716 


21773 


21831 


21888 


21945 


22003 


22060 


22118 


22176 


503.70 




69 


22233 


22291 


22349 


22407 


22465 


22523 


22581 


22640 


22697 


22756 


511.11 


70 


22815 


22873 


22932 


22991 


23049 


23108 


23167 


23226 


23285 


23344 


518.52 


aa 


71 


23404 


23463 


23522 


23582 


23641 


23701 


23761 


23820 


23880 


23940 


525.93 


•^-s 


72 


24000 


24060 


24120 


24180 


24241 


24301 


24361 


24422 


24482 


24543 


533.33 


•O o 

<& 


73 


24604 


24664 


24725 


24786 


24847 


24908 


24969 


25031 


25092 


25153 


540.74 


74 


25215 


25276 


25338 


25400 


25461 


25523 


25585 


25647 


25709 


25771 


548.15 




75 


25833 


25896 


25958 


26020 


26083 


26145 


26208 


26271 


26333 


26396 


555.56 


p. p. 


76 


26459 


26522 


26585 


26648 


26712 


26775 


26838 


26902 


26965 


27029 


562.96 


7.41 


77 


27093 


27156 


27220 


27284 


27348 


27412 


27476 


27540 


27605 


27669 


570.37 


1 .74 


76 


27733 


27798 


27862 


27927 


27992 


28056 


28121 


28186 


28251 


28316 


577.78 


2 1.48 


79 


28381 


28447 


28512 


28577 


28643 


28708 


28774 


28840 


28905 


28971 


585.19 


3 2.22 


80 


29037 


29103 


29169 


29235 


29301 


29368 


29434 


29500 


29567 


29633 


592.59 


4 2.96 


81 


29700 


29767 


29833 


29900 


29967 


30034 


30101 


30168 


30236 


30303 


600.00 


5 3.70 


82 


30370 


30438 


30505 


30573 


30641 


30708 


30776 


30844 


30912 


30980 


607.41 


6 4.44 


83 


31048 


31116 


31185 


31253 


31321 


31390 


31458 


31527 


31596 


31664 


614.81 


7 5.19 


84 


31733 


31802 


31871 


31940 


32010 


32079 


32148 


32217 


32287 


32356 


622.22 


8 5.93 


85 


32426 


32496 


32565 


32635 


32705 


32775 


32845 


32915 


32985 


33056 


629.63 


9 6.67 


86 


33126 


33196 


33267 


33337 


33408 


33479 


33549 


33620 


33691 


33762 


637.04 




87 


33833 


33904 


33976 


34047 


34118 


34190 


34261 


34333 


34405 


34476 


644.44 


bp 


88 


34548 


34620 


34692 


34764 


34836 


34908 


34981 


35053 


35125 


35198 


651.85 


a 


89 


35270 


35343 


35416 


35488 


35561 


35634 


35707 


35780 


35853 


35927 


659.26 


"S 


90 


36000 


36073 


36147 


36220 


36294 


36368 


36441 


36515 


36589 


36663 


666.67 


it 


91 


36737 


36811 


36885 


36960 


37034 


37108 


37183 


37257 


37332 


37407 


674.07 


92 


37481 


37556 


37631 


35006 


37781 


37856 


37932 


38007 


38082 


38158 


681.48 


93 


38233 


38309 


38385 


38460 


38536 


38612 


38688 


38764 


38840 


38916 


688.89 


94 


38993 


39069 


39145 


39222 


39298 


39375 


39452 


39528 


39605 


39682 


696.30 


^s 


95 


39759 


39836 


39913 


39991 


40068 


40145 


40223 


40300 


40378 


40456 


703.70 


a> S3 


96 


40533 


40611 


40689 


40767 


40845 


40923 


41001 


41080 


41158 


41236 


711.11 


§1 


97 


41315 


41393 


41472 


41551 


41629 


41708 


41787 


41866 


41945 


42024 


718.52 


98 


42104 


42183 


42262 


42342 


42421 


42501 


42581 


42660 


42740 


42820 


725.93 




99 


42900 


42980 


43060 


43140 


43221 


43301 


43381 


43462 


43542 


43623 


733.33 




100 


43704 


43784 


43865 


43946 


44027 


44108 


44189 


44271 


44352 


44433 


740.74 





Note that Base, Slope, and Cu. yds. in above table may all be multiplied 
by the same factor; thus, using factor of H for height of 66.3 ft., we have 
13800 cu. yds. for base of 12 ft. and slopes H to 1. 

' Examples of Use of Table 28. 



Ex. 1. — Find the number of cu. yds. in 
a 100-ft. station: roadway 18 ft., excav. 
slopes 1 to 1, ground slope 3 in 10 straight 
across, and center height 63 ft? 



Ex. 2. — Same, but slope "up" 4 in 10 
to left of center, and "down" 1 in 10 to 
right? 



Solution: 

Level cutting 18900 cu.yds. 

Up slope, add . . . . 4114 " 

23014 " •• 
Down slope, sub.. 2215 " " 

Ans 20799 '* " 

Solution: +18900 cu.yds. 

+ 6400 " " 
- 873 " " 

ian5....+ 24427 '* " 



EARTHWORK TABLES, WITH GROUND SLOPES. 



1041 



— Corrections for Ground Slopes Not Level. 

Base op Roadway, 18 Ft. Side Slopes, 1 to 1. 

(See Explanation and Formulas in Table 22.) 

Note. — a = angle of ground slope (G. S.) to right or left of center line; 
and "6^. S. = 1:10," "6^. 5. = 2:10," etc., means tan a. Up or ( + ) slopes 
from the center indicate additive corrections, and down or ( — ) slopes sub- 
tractive, from quantities in table on opposite page for level sections. 
[Additive ( + ) and subtractive ( — ) Corrections in Cu. Yds. per 100-Ft. Sta.] 



Center 


G. S 


. = 1:10. 


G.S 


.=2:10. 


G.S 


. = 3:10. 


G.S 


.=4:10. 


G.S 


. = 5:10. 


Height. 
Ft. 


a 


= 5°.7 


a = 


= ll°.3 


a = 


= 16°.7 


a= 


= 21°.8 


a = 


= 26°.6 


4- Up 


— Down 


+ Up 


— Down 


+ Up 


— Down 


+ Up 


— Down 


+ Up 


— Down 


60 


980 


802 


2204 


1469 


3779 


2035 


5878 


2519 


8817 


2939 


61 


1008 


825 


2269 


1512 


3889 


2094 


6049 


2593 


9074 


3025 


62 


1037 


848 


2334 


1556 


4001 


2154 


6223 


2667 


9335 


3112 


63 


1067 


873 


2400 


1800 


4114 


2215 


6400 


2743 


9600 


3200 


64 


1097 


897 


2467 


1645 


4229 


2277 


6579 


2820 


9869 


3290 


65 


1127 


922 


2535 


1690 


4346 


2340 


6760 


2897 


10141 


3380 


66 


1157 


947 


2604 


lV36 


4464 


2404 


6944 


2976 


10417 


3472 


67 


1188 


972 


2674 


1783 


4584 


2468 


7131 


3056 


10696 


3565 


68 


1220 


998 


2745 


1830 


4706 


2534 


7320 


3137 


10980 


3660 


69 


1252 


1024 


2817 


1878 


4829 


2600 


7511 


3219 


11267 


3756 


70 


1284 


1051 


2889 


1926 


4953 


2667 


7705 


3302 


11557 


3852 


71 


1317 


1077 


2963 


1975 


5079 


2735 


7901 


3386 


11852 


3951 


72 


1350 


1105 


3038 


2025 


5207 


2804 


8100 


3471 


12150 


4050 


73 


1384 


1132 


3113 


2075 


5337 


2874 


8301 


3558 


12452 


4151 


74 


1417 


1160 


3189 


2126 


5467 


2944 


8505 


3645 


12757 


4252 


75 


1452 


1188 


3267 


2178 


5600 


3015 


8711 


3733 


13067 


4356 


76 


1487 


1216 


3345 


2230 


5734 


3088 


8920 


3823 


13380 


4460 


77 


1522 


1245 


3424 


2283 


5870 


3161 


9131 


3913 


13696 


4565 


78 


1557 


1274 


3504 


2336 


6007 


3235 


9344 


4005 


14017 


4672 


79 


1593 


1304 


3585 


2390 


6146 


3309 


9560 


4097 


14341 


4780 


80 


1630 


1334 


3667 


2445 


6287 


3385 


9779 


4191 


14669 


4890 


81 


1667 


1364 


3750 


2500 


6429 


3462 


10000 


4286 


15000 


5000 


82 


1704 


1394 


3834 


2556 


6572 


3539 


10223 


4381 


15335 


5112 


83 


1742 


1425 


3919 


2612 


6717 


3617 


10449 


4478 


15674 


5225 


84 


1780 


1456 


4004 


2669 


6863 


3696 


10678 


4576 


16017 


5339 


85 


1818 


1488 


4091 


2727 


7013 


3776 


10909 


4675 


16363 


5454 


86 


1857 


1519 


4178 


2785 


7163 


3857 


11142 


4775 


16713 


5571 


87 


1896 


1552 


4267 


2844 


7314 


3938 


11378 


4876 


17067 


5689 


88 


1936 


1584 


4356 


2904 


7467 


4021 


11616 


4978 


17424 


5808 


89 


1976 


1617 


4446 


2964 


7622 


4104 


11857 


5081 


17785 


5928 


90 


2017 


1650 


4538 


3025 


7779 


4188 


12100 


5186 


18150 


6050 


91 


2059 


1684 


4630 


3086 


7937 


4274 


12346 


5291 


18519 


6173 


92 


2099 


1717 


4723 


3148 


8096 


4359 


12594 


5397 


18891 


6297 


93 


2141 


1752 


4817 


3211 


8257 


4446 


12844 


5505 


19267 


6422 


94 


2183 


1786 


4912 


3274 


8420 


4534 


13098 


5613 


19647 


6549 


95 


2226 


1821 


5007 


3338 


8584 


4622 


13353 


5723 


20030 


6677 


96 


2269 


1856 


5104 


3403 


8750 


4712 


13611 


5833 


20417 


6806 


97 


2312 


1892 


5202 


3468 


8917 


4802 


13872 


5945 


20807 


6936 


98 


2356 


1927 


5300 


3534 


9087 


4893 


14135 


6058 


21202 


7067 


99 


2400 


1964 


5400 


3600 


9257 


4985 


14400 


6171 


21600 


7200 


100 


2445 


2000 


5500 


3667 


9429 


5077 


14668 


6286 


22002 


7334 



Ex. — G. 5. = +2 in 10 to left of center, and — 3 in 10 to right; center 
height, 80 ft. Then add 3667 cu. yds. to, and subtract 3385 cu. yds. from, 
29037 cu. yds. obtained from table of level sections on opposite page. 

Interpolations for Table 28. 

(a). — When center height is in feet and tenths, interpolate for tenths 
directly from table. 

(b). — When the ground slope is intermediate between those slopes 
selected to head the columns, direct interpolation will be close enough for 
ordinary purposes for preliminary estimates, etc. Otherwise, use the exact 
formulas m Table 22. 

(c). — When the roadway, «/, is greater or less than 18 ft., multiply 

values in table by ( , ^ j , 



1042 



-RAILROADS. 



29. — Formulas for Extending Tables of Level Sections to any 

Side Slopes. 



Side 
Slopes 



Htol 
^to 1 

H to 1 

1 to 1 

IM to 1 
13^ to 1 
IH to 1 

2 to 1 



For Use with Tables with Slopes 

IH to 1. 
+ means add to ) the quanti- 
— means sub. from j ties in table. 



100/^2 10 

■ 3X9 * 8 
100/^2 

■ 3X9 
100/^2 

4X9 
100/^2 

6X9 
100/^2 

12X9 



c.y. 



3X9 ■■ " J 




For Use with Tables with Slopes 
Itol. 

+ means add to ) the quanti- 
— means stib. from J ties in table. 



100 ;j2 

4X9 


c.:j;. 


= ^^.m^i^cy. 


100/^2 

6X9 


<i 


^--^m •■ 


100 /j2 
12X9 


*' 


= -^[M^^] '• 


100/^2 

12X9 


c.y. 


= -^t«-l " 


100 /j2 

6X9 


•• 


= + 3X9 [^^^3 " 


100/^2 
4X9 


•• 


= +-^>.. •■ 


100/^2 

3X9 


" 


-i^C-1" 



* Note. — Quantities in brackets [] are + or — areas in sq.ft. due to 
change in cross-section by change in side slope, and correspond with areas 
[A] in Table 20. Hence, the cu. yds. to be added or subtracted may be 
obtained from [A] by the use of Table 20. 



29a. — Factors (F) for Extending Tables (T) of Level Sections to 

Given Side Slopes (First Column) and Widths of Roadway {R). 

(Tables 24, 28, 40 and 41 are not included here.) 



Side 
Slopes. 


R 


T 


F 


R 
6 


T 
39 


F 

V5 


R 
8 


T 
32 


F 
0.4 


16 


T 
31 


F 
0.8 


R 


T 


F 


7? 


T 


F 


Vs to 1 


4 


33 


V5 










M tol 


3 


30 


Vfi 


4 


25 


H 


5 


33 


% 


7 


37 


3^ 


10 


32 


y? 


20 


31 


1 


^tol{ 


6 


30 


H 


8 


25 


V?, 


9 


27 


H 


10 


33 




11 


34 


H 


14 


37 


^ 


15 


39 


Vo 


20 


32 


1 


40 


31 


2 




















5€tol{ 


7 


23 


Vo 


8 


26 


V? 


9 


30 


v>, 


12 


35 


V? 


13 


36 


H 


14 


37 


v? 


15 


33 


H 


21 


37 


% 


30 


22 


m 


60 


31 


3 














1 tol| 


12 


30 


K 


16 


25 


1 


18 


27 


1 


20 


33 


1 


22 


34 


1 


28 


37 


1 


30 


39 


1 


40 


32 


2 


80 


31 


4 




















IM tol 


15 


30 


Vfi 


20 


25 


l¥ 


25 


33 


m 




















1^ to 1 { 


14 


23 


1 


16 


26 


1 


18 


30 


1 


24 


35 


1 


26 


36 


1 


27 


27 


m 


28 


38 


1 


30 


33 


m 


33 


34 


m, 


45 


39 


w. 


60 


32 


3 


120 


31 


6 


m to 1 


21 


30 


1V« 


28 


25 


\% 


35 


33 


m 




















2 tol 


24 


30 


IK 


32 


25 


2 


36 


27 


2 


40 


33 


2 


44 


34 


2 


60 


39 


2 


2i^tol 


21 


23 


IV^ 


24 


26 


IH 

2H 


27 


30 


IH 


36 


25 


214 


45 


33|2^ 








21^ tol 


30 


30 


1% 


40 


25 


45 


27 


2^2 


50 


33 


23^ 


55 


342>^ 








2Mtol 


33 


30 


IVr, 


44 


25 


m 


55 


33 


2^ 




















3tol{ 


28 
66 


23 
34 


2 
3 


32 


26 


3 


35 


30 


2 


48 


35 


2 


55 


2; 


3 


60 


33 


3 



ort 

ment. R may be increased or decreased by consulting the last two columns 
of the tables referred to. 

Ex. — To find the number of cu yds. in a 100-ft. station for a canal 120 
ft. wide at bottom and with side slopes 13^ to 1: Consult Table 31 and mul- 
tiply the cu. yds. corresponding to any given height by the factor 6. 



EARTHWORK TABLES—LEVEL SECTIONS. 



1043 



30. — Level Sections (Earthwork) ; Height, 0-60 Ft. 
Base op Roadway, 18 Ft. Side Slopes, I^to 1. 
Note. — ^The last two columns enable us to use any other base than 18 ft.: 
Ex. — Given height, 46.4 ft.; roadway 20 ft. Then we have, 15054 + 
(340.74 + 2.96) = 15398 cu. yds. (For Ht. >60 ft., see Tables 28, 40.) 

[Cu. Yds. per 100-Ft. Station.] 



























Width 






Ht. 


.0 


.1 


.2 


.3 


.4 


.5 


.6 


.7 


.8 


.9 


of 2 Ft. 






Ft. 






















Cu.Yds 




w" 







6.7 


13.6 


20.5 


27.6 


34.7 


42.0 


49.4 


56.9 


64.5 






3 . 


1 


*""72!2 


80.1 


88.0 


96.1 


104.2 


112.5 


120.9 


129.4 


138.0 


146.7 


' '7!4i 




2 


155.5 


164.5 


173.5 


182.7 


191.9 


201.3 


210.8 


220.4 


230.1 


240.0 


14.81 




O 


3 


249.9 


260.0 


270.1 


280.4 


290.8 


301.3 


311.9 


322.6 


333.4 


344.5 


22.22 




u^ 


4 


355.5 


366.7 


378.0 


389.4 


400.9 


412.5 


424.2 


436.0 


448.0 


460.0 


29.63 




2-^ 


5 


472.2 


484.5 


496.9 


509.4 


522.0 


534.7 


547.6 


560.5 


573.6 


586.7 


37.04 




rt« 


6 


600.0 


613.4 


626.9 


640.5 


654.2 


668.1 


682.0 


696.1 


710.2 


724.5 


44.44 


i 


'^K 


7 


738.9 


753.4 


768.0 


782.7 


797.6 


812.5 


827.6 


842.7 


858.0 


873.4 


51.85 


<D O 


8 


888.9 


904.5 


920.2 


936.1 


952.0 


968.1 


984.2 


1000.5 


1016.9 


1033.4 


59.26 


a> 


S w 


9 


1050.0 


1066.7 


1083.6 


1100.5 


1117.6 


1134.7 


1152.0 


1169.4 


1186.9 


1204.5 


66.67 


WT3 


10 


1222.2 


1240.1 


1258.0 


1276.1 


1294 2 


1312.5 


1330.9 


1349.4 


1368.0 


1386.7 


74.07 


d 


ol 


11 


1405.6 


1424.5 


1443.6 


1462.7 


1482.0 


1501.4 


1520.9 


1540.5 


1560.2 


1580.1 


81.48 


43 


^^ 


12 


1600.0 


1620.1 


1640.2 


1660.5 


1680.9 


1701.4 


1722.0 


1742.7 


1763.6 


1784.5 


88.89 


« 


4J 


13 


1805.6 


1826.7 


1848.0 


1869.4 


1890.9 


1912.5 


1934.2 


1956.1 


1978.0 


2000.1 


96.30 


fe 


>»4; 


14 


2022.2 


2044.5 


2066.9 


2089.4 


2112.0 


2134.7 


2157.6 


2180.5 


2203.6 


2226.7 


103.70 


o 


-^(N 


15 


2250.0 


2273.4 


2296.9 


2320.5 


2344.2 


2368.1 


2392.0 


2416.1 


2440.2 


2465.5 


111.11 


09 


'S^ 


16 


2488.9 


2513.4 


2538.0 


2562.7 


2587.6 


2612.5 


2637.6 


2662.7 


2688.0 


2713.4 


118.52 


5 




17 


2738.9 


2764.5 


2790.2 


2816.1 


2842.0 


2868.1 


2894.2 


2920.5 


2946.9 


2973.4 


125.93 


S 


18 


3000.0 


3026.7 


3053.6 


3080.5 


3107.6 


3134.7 


3162.0 


3189.4 


3216.9 


3244.5 


133.33 


H 


S^ 


19 


3272.2 


3300.1 


3328.0 


3356.1 


3384.2 


3412.5 


3440.9 


3469.4 


3498.0 


3526.7 


140.74 


b 


p 


20 


3555 6 


3584.5 


3613.6 


3642.7 


3672.0 


3701.4 


3730.9 


3760.5 


3790.2 


3820.1 


148.15 


a 


21 


3850.0 


3880.1 


3910.2 


3940.5 


3970.9 


4001.4 


4032.0 


4062.7 


4093.6 


4124.5 


155.56 


2 


.si 


22 


4155.6 


4186.7 


4218.0 


4249.4 


4280.9 


4312.5 


4344.2 


4376.1 


4408.0 


4440.1 


162.96 


< 


23 


4472.2 


4504.5 


4536.9 


4569.4 


4602.0 


4634.7 


4667.6 


4700.5 


4733.6 


4766.7 


170.37 


^ tfl 


24 


4800.0 


4833.4 


4866.9 


4900.5 


4934.2 


4968.1 


5002.0 


5036.1 


5070.2 


5104.5 


177.78 




cStJ 


25 


5138.9 


5173.4 


5208.0 


5242.7 


5277.6 


5312.5 


5347.6 


5382.7 


5418.0 


5453.4 


185.19 


p.p. 


as 


26 
27 


5488.9 
5850.0 


5524.5 
5886.7 


5560.2 
5923.6 


5596.1 
5960.5 


5632.0 
5997.6 


5668.1 
6034.7 


5704.2 
6072.0 


5740.5 
6109.4 


5776.9 
6146.9 


5813.4 
6184.5 


192.59 
200.00 


7.41 


1 .74 


28 


6222.2 


6260.1 


6298.0 


6336.1 


6374.2 


6412.5 


6450.9 


6489.4 


6528.0 


6566.7 


207.41 


2 1.48 


0)0 


29 


6605.6 


6644.5 


6683.6 


6722.7 


6762.0 


6801.4 


6840.9 


6880.5 


6920.2 


6960.1 


214.81 


3 2.22 


30 


7000.0 


7040.1 


7080.2 


7120.5 


7160.9 


7201.4 


7242.0 


7282.7 


7323.6 


7364.5 


222.22 


4 2.96 


31 


7405.6 


7446.7 


7488.0 


7529.4 


7570.9 


7612.5 


7654.2 


7696.1 


7738.0 


7780.1 


229.63 


5 3.70 


^jCM 


32 


7822.2 


7864.5 


7906.9 


7949.4 


7992.0 


8034.7 


8077.6 


8120.5 


8163.6 


8206.7 


237.04 


6 4.44 


.23 ^ 


33 


8250.0 


8293.4 


8336.9 


8380.5 


8424.2 


8468.1 


8512.0 


8556.1 


8600.2 


8644.5 


244.44 


7 5.19 


l| 


34 


8688.9 


8733.4 


8778.0 


8822.7 


8867.6 


8912 5 


8957.6 


9002.7 


9048.0 


9093.4 


251.85 


8 5.93 


35 


9138.9 


9184.5 


9230.2 


9276.1 


9322.0 


9368.1 


9414.2 


9460.5 


9506.9 


9553.4 


259.26 


9 6.67 


C <u 


36 


9600.0 


9646.7 


9693.6 


9740.5 


9787.6 


9834.7 


9882.0 


9929.4 


9976.9 


10025 


266.67 




.5^ 


37 


10072 


10120 


10168 


10216 


10264 


10313 


10361 


10409 


10458 


10507 


274.07 


>> 


M 


38 


10556 


10605 


10654 


10703 


10752 


10801 


10851 


10901 


10950 


11000 


281.48 


-a 


g4i 


39 


11050 


11100 


11150 


11200 


11251 


11301 


11352 


11403 


11454 


11505 


288.89 


o 


40 


11556 


11607 


11658 


11709 


11761 


11813 


11864 


11916 


11968 


12020 


296.30 


d 


^^ 


41 


12072 


12125 


12177 


12229 


12282 


12335 


12388 


12441 


12494 


12547 


303.70 


B 


^^ 


42 


12600 


12653 


12707 


12761 


12814 


12868 


12922 


12976 


13030 


13085 


311.11 


d 

1 


^^ 


43 


13139 


13193 


13248 


13303 


13358 


13413 


13468 


13523 


13578 


13633 


318.52 




44 


13689 


13745 


13800 


13856 


13912 


13968 


14024 


14081 


14137 


14193 


325.93 


bO 


45 


14250 


14307 


14364 


14421 


14478 


14535 


14592 


14649 


14707 


14765 


333.33 


d 


46 


14822 


14880 


14938 


14996 


15054 


15113 


15171 


15229 


15288 


15347 


340.74 


"S 


47 


15406 


15465 


15524 


15583 


15642 


15701 


15761 


15821 


15880 


15940 


348 15 


p. 


^ Vh 


48 


16000 


16060 


16120 


16181 


16241 


16301 


16362 


16423 


16484 


16545 


355.56 


-J 


49 


16606 


16667 


16728 


16789 


16851 


16913 


16974 


17036 


17098 


17160 


362.96 




50 


17222 


17285 


17347 


17409 


17472 


17535 


17598 


17661 


17724 


17787 


370.37 


§ 


51 


17850 


17913 


17977 


18041 


18104 


18168 


18232 


18296 


18360 


18425 


377.78 


^ 


52 


18489 


18553 


18618 


18683 


18748 


18813 


18878 


18943 


19008 


19073 


385.19 


0) 




53 


19139 


19205 


19270 


19336 


19402 


19468 


19534 


19601 


19667 


19733 


392.59 


s 


So 


54 


19800 


19867 


19934 


20000 


20068 


20135 


20202 


20269 


20337 


20405 


400.00 


x>^ 


55 


20472 


20540 


20608 


20676 


20744 


20813 


20881 


20949 


21018 


21087 


407.41 






56 


21156 


21225 


21294 


21363 


21432 


21501 


21571 


21641 


21710 


21780 


414.81 




=1. 


57 


21850 


21920 


21990 


22061 


22131 


22201 


22272 


22343 


22414 


22485 


422.22 




58 


22556 


22G27 


22698 


22769 


22841 


22913 


22984 


23056 


23128 


23200 


429.63 




t/3 


59 


23272 


23345 


23417 


23489 


23562 


23635 


23708 


23781 


23854 


23927 


437.04 




Pf 


60 


24000 


24073 


24147 


24221 


24294 


24368 


24442 


24516 


24590 


24665 


444.44 





1044 



m.'-RAILROADS. 



31. — Level Sections (Earthwork); Height. 0-60 Ft. 

Base op Roadway, 20 Ft. Side Slopes, 3^ to I. 

Note. — The last two columns enable us to use any other base than 20 ft.: 
Ex. — Given height, 43.9 ft.; roadway 18ft. Then we have, 5036.3— 
(318.52+6.67) = 4711.1cu. yds. 

[Cu. Yds. per 100-Ft. Station ] 



Ht. 

Ft. 



.0 



.2 



.6 



,8 



Width 
of 2 Ft. 
Cu.Yds 



-a; 
ft 
o 



CO 



■So 



<D' 






r-H O 



c3 >> 



-s^^ 






^^ 

r o 

;::;'+-''o 
g O CO 

pqo 5 

*^ ^ 
4J o cr 

c3 ^ (U 
^. <^ 52 



75.0 
151.8 
230.6 
311 
393.5 
477.8 
563.9 
651.8 
741.7 
833.3 
926.9 
022.2 
1119.4 
1218.5 
1319.4 
1422.2 
1526.9 
1633.3 
1741.7 
1851.9 
1963.9 
2077.8 
2193.5 
231L1 
2430.6 
2551.9 
2675.0 
2800.0 
2926.9 
3055.6 
3186.1 
3318.5 
3452.8 
3588.9 
3726.9 
3866.7 
4008.3 
4151.9 
4297.2 
4444.4 
4593.5 
4744.4 
4897.2 
5051.9 
5208.3 
5366.7 
5526.9 
5688.9 
5852.8 
6018.5 
6186.1 
6355.6 
6526.9 
6700.0 

55 6875.0 

56 7051.9 

57 7230.6 

58 7411. 

59 7593. 

60 7777. 



7 

82 
159 
238 
319 
401 
486 
572 
660 
750 
842 
936 
1031 
1129 
1228 
1329 
1432 
1537 
1644 
1752 
1863 
1975 
2089 
2205 
2323 
2442 
2564 
2687 
2812 
2939 
3068 
3199 
3331 
3466 
3602 
3740. 
3880. 
4022. 
4166. 
4311. 
4459. 
4608. 
4759. 
4912. 
5067. 
5224. 
5382. 
5543. 
5705. 
5869. 
6035. 
6203. 
6372. 
6544. 
6717. 
6892. 
7069. 
7248. 
7429. 
7611. 
7796. 



14 
90 
167, 
246, 
327, 
410 
494, 
581, 
669. 
759, 
851. 
945, 

1041. 

1139, 

1238. 

1339. 

1443. 

1548. 

1654. 

1763. 

1874. 

1986. 

2100. 

2216. 

2334. 

2454. 

2576. 

2699. 

2825. 

2952. 

3081. 

3212. 

3345. 

3479. 

3616. 

3754. 

3894. 

4036. 

4180. 

4326. 

4474. 

4623. 

4774. 

4928. 

5083. 

5239. 

5398. 

5559. 

5721. 

5885. 

6051. 

6219. 

6389. 

6561. 

6734. 

6910. 

7087. 

7266. 

7447. 

7630. 
814, 



22. 
97. 

175. 

254. 

335. 

418. 

503. 

590. 

678. 

769. 

861. 

955. 
1051. 
1149. 
1248. 
1350. 
1453. 
1558. 
1665. 
1774. 
1885. 
1997. 
2112. 
2228. 
2346. 
2466. 
2588. 
2712. 
2837. 
2965. 
3094. 
3225. 
3358. 
3493. 
3630. 
3768. 
3809. 
4051. 
4195. 
4341. 
4489. 
4638. 
4790. 
4943. 
5098. 
5255. 
5414. 
5575. 
5737. 
5902. 
6068. 
6236. 
6406. 
6578. 
6752. 
6927. 
7105. 
7284. 
7465. 
7648. 
7833. 



29 
105, 

183. 
262. 
343. 
427, 
512, 
598 
687 
778 
870 
964 

1060, 

1158, 

1258, 

1360, 

1463, 

1569, 

1676, 

1785. 

1896, 

2009, 

2123. 

2240. 

2358. 

2478. 

2600. 

2724. 

2850. 

2978. 

3107. 

3238. 

3372. 

3507. 

3643. 

3782. 

3923. 

4065. 

4209. 

4355. 

4503. 

4653. 

4805. 

4958. 

5114. 

5271. 

5430. 

5591. 

5754. 

5918. 

6085. 

6253. 

6423. 

6595. 

6769. 

6945. 

7123. 

7302. 

7483. 

7667. 

7852, 



37.3 
113.2 
191.0 

270.6 

352.1 

435.4 

520.6 

607 

696.5 

787.3 

879.9 

974.3 

1070.6 

1168.8 

1268.8 

1370.6 

1474.3 

1579.9 

1687.3 

1796.5 

1907.6 

2020.6 

2135.4 

2252.1 

2370.6 

2491.0 

2613.2 

2737.3 

2863.2 

2991.0 

3120.6 

3252.1 

3385.4 

3520.6 

3657.6 

3796.5 

3937.3 

4079.9 

4224.3 

4370.6 

4518.8 



4820.6 

4974.3 

5129.9 

5287.3 

5446.5 

5607.6 

5770.6 

5935.4 

6102.1 

6270.6 

96441.0 

6613.2 

6787.3 

5 6963.2 

7141.0 

7320.6 

9 7502.1 

07685.4 

7870.6 



44 
120 
198 
278 
360 
443 
529 
616 
705 
796 
889 
983 

1080 

1178, 

1278, 

1380, 

1484, 

1590, 

1698, 

1807, 

1918, 

2032, 

2147, 

2263, 

2382, 

2503, 

2625, 

2749, 

2875. 

3003. 

3133. 

3265. 

3398. 

3534. 

3671. 

3810. 

3951. 

4094. 

4238. 

4385. 

4533. 

4683. 

4835. 

4989. 

5145. 

5303. 

5462. 

5623. 

5787. 

5952. 

6118. 

6287. 

6458. 

6630. 

6804. 

6980. 

7158. 

7338. 

7520. 

7703. 

7789. 



52. 

128. 

206. 

286. 

368. 

452. 

537. 

625. 

714. 

805. 

898. 

993. 
1090. 
1188. 
1289. 
1391. 
1495. 
1601. 
1709. 
1818. 
1930. 
2043. 
2158. 
2275. 
2394. 
2515. 
2637. 
2762. 
2888. 
3016. 
3146. 
3278. 
3412. 
3547. 
3685. 
3824. 
3965. 
4108. 
4253. 
4400. 
4548. 
4699. 
4851. 
5005. 
5161. 
5319. 
5478. 
5640. 
5803. 
5968. 
6135. 
6304. 
6475. 
6647. 
6822. 
6998. 
7176. 
7356. 
7538. 
7722. 
7807. 



59. 

136. 

214. 

294. 

376. 

460. 

546. 

634. 

723. 

814. 

908. 
1003. 
1099. 
1198. 
1299. 
1401. 
1505. 
1611. 
1719. 
1829. 
1941. 
2054. 
2170. 
2287. 
2406. 
2527. 
2650. 
2774. 
2901. 
3029. 
3159. 
3291. 
3425. 
3561. 
3699. 
3838. 
3979. 
4123. 
4268. 
4414. 
4563. 
4714. 
4866. 
5020. 
5176. 
5334. 
5494. 
5656. 
5819. 
5985. 
6152. 
6321. 
6492. 
6665. 
6839. 
7016. 
7194. 
7374. 
7556. 
7740. 
7826. 



67.4 
144.1 
222 
303.0 
385.2 
469 
555.2 
643.0 
732.6 
824. 
917.4 
1012.6 
1109. 
1208.5 
1309 
1411.9 
1516.3 
1622.6 
1730.8 
1840.8 
1952.6 
2066.3 
2181.9 
2299.3 
2418.5 
2539.6 
2662.6 
2787.4 
2914.1 
3042.6 
3173.0 
3305.2 
3439.3 
3575.2 
3713.0 
3852.6 
3994.1 
4137.4 
4282.6 
4429.6 
4578.5 
4729.3 
4881 
5036.3 
5192.6 
5350.8 
5510.8 
5672.6 
5836.3 
6001.9 
6169.3 
6338.5 
6509.6 
6682.6 
6857.4 
7034.1 
7212.6 
7393.0 
7575.2 
7759.3 
7845.2 



7.41 

14.81 

22.22 

29.63 

37.04 

44.44 

51.85 

59.26 

66.67 

74.07 

81.48 

88.89 

96.30 

103.70 

111.11 

118.52 

125.93 

133.33 

140.74 

148.15 

156.56 

162.96 

170.37 

177.78 

185.19 

192.59 

200.00 

207.41 

214.81 

222.22 

229.63 

237.04 

244.44 

251.85 

259.26 

266.67 

274.07 

281.48 

288.89 

296.30 

303.70 

311.11 

318.52 

325.93 

333.33 

340.74 

348.15 

355.56 

362.96 

370.37 

377.78 

385.19 

392.59 

400.00 

407.41 

414.81 

422.22 

429.63 

437.04 

444.44 



P. P. 

7.41 
.74 
1.48 
2.22 
2.96 
3.70 
4.44 
5.19 
5.93 
6.67 

-a 

o 
a 
B 



EARTHWORK TABLES— LEVEL SECTIONS. 



1045 



32. — Level Sections (Earthwork); Height, 0-60 Ft. 

Base of Roadway, 20 Ft. Side Slopes 3^ to 1. 

Note. — ^The last two columns enable us to use any other base than 20 ft.: 
Ex. — Given height, 18.1 ft.; roadway 22 ft. Then we have 1947.4 + 
(133.33 + 0.74) = 2081.5 cu. yds. 

[Cu. Yds. per 100-Ft. Station.] 



























Width 






Ht. 


.0 


.1 


.2 


.3 


.4 


.5 


.6 


.7 


.8 


.9 


of 2 Ft. 






Ft. 






















Cii.Yds 




w" 







7.4 


14.9 


22.4 


29.9 


37.5 


45.1 


52.8 


60.4 


68.2 






5 


1 


'*"75!9 


83.7 


91.6 


99.4 


107.3 


115.3 


123.3 


131.3 


139.3 


147.4 


'"7!4i 




S: 


2 


155.6 


163.7 


171.9 


180.2 


188.4 


196.8 


205.1 


213.5 


221.9 


230.4 


14.81 




^^rH 


3 


238.9 


247.4 


256.0 


264.6 


273.3 


281.9 


290.7 


299.4 


308.2 


317.1 


22.22 




^"-S 


4 


325.9 


334.8 


343.8 


352.8 


361.8 


370.8 


379.9 


389.1 


398.2 


407.4 


29.63 




S' 


5 


416.7 


425.9 


435.3 


444.6 


454.0 


463.4 


472.9 


482.4 


491.9 


501.5 


37.04 




6 


511.1 


520.8 


530.4 


540.2 


549.9 


559.7 


569.6 


579.4 


589.3 


599.3 


44.44 


*i 




7 


609.3 


619.3 


629.3 


639.4 


649.6 


659.7 


669.9 


680.2 


690.4 


700.8 


51.85 




8 


711.1 


721.5 


731.9 


742.4 


752.9 


763.4 


774.0 


784.6 


795.3 


805.9 


59.26 


9 


816.7 


827.4 


838.2 


849.1 


859.9 


870.8 


881.8 


892.8 


903.8 


914 8 


66.67 


w 


10 


925.9 


937.1 


948.2 


959.4 


970.7 


981.9 


993.3 


1004.6 


1016.0 


1027.4 


74.07 


d 


^^ 


11 


1038.9 


1050.4 


1061.9 


1073.5 


1085.1 


1096.8 


1108.4 


1120.2 


1131.9 


1143.7 


81.48 




-ss 


12 


1155.6 


1167.4 


1179.3 


1191.3 


1203.3 


1215.3 


1227.3 


1239.4 


1251.6 


1263.7 


88.89 


<v 


>» • 


13 


1275.9 


1288.2 


1300.4 


1312.8 


1325.1 


1337.5 


1349.9 


1362.4 


1374.9 


1387.4 


96.30 


s 


Si^ 


14 


1400.0 


1412.6 


1425.3 


1437.9 


1450.-7 


1463.4 


1476.2 


1489.1 


1501.9 


1514.8 


103.70 


<M 


T3<=> 


15 


1527.8 


1540.8 


1553.8 


1566.8 


1579.9 


1593.1 


1606.2 


1619.4 


1632.7 


1645.9 


111.11 


o 


di CO 


16 


1659.3 


1672.6 


1686.0 


1699.4 


1712.9 


1726.4 


1739.9 


1753.5 


1767.1 


1780.8 


118.52 


3 


T-^tt-l 

.ao 
^3 rt 


17 


1794.4 


1808.2 


1821.9 


1835.7 


1849.6 


1863.4 


1877.3 


1891.3 


1905.3 


1919.3 


125.93 


^ 


18 


1933.3 


1947.4 


1961.6 


1975.7 


1989.9 


2004.2 


2018.4 


2032.8 


2047.1 


2061.5 


133.33 


e 


19 


2075.9 


2090.4 


2104.9 


2119.4 


2134.0 


2148.6 


2163.3 


2177.9 


2192.7 


2207.4 


140.74 




20 


2222.2 


2237.1 


2251.9 


2266.8 


2281.8 


2296.8 


2311.8 


2326.8 


2341.9 


2357.1 


148.15 


a 


21 


2372.2 


2387.4 


2402.7 


2417.9 


2433.3 


2448.6 


2464.0 


2479.4 


2494.9 


2510.4 


156.56 


"d 


^1 


22 


2525.9 


2541.5 


2557.1 


2572.8 


2588.4 


2604.2 


2619.9 


2635.7 


2651.6 


2667.4 


162.96 


•d 


23 


2683.3 


2699.3 


2715.3 


2731.3 


2747.3 


2763.4 


2779.6 


2795.7 


2811.9 


2828.2 


170.37 


< 


^i 


24 


2844.4 


2860.8 


2877.1 


2893.5 


2909.9 


2926.4 


2942.9 


2959.4 


2976.0 


2992.6 


177.78 




25 


3009.3 


3025.9 


3042.7 


3059.4 


3076.2 


3093.1 


3109.9 


3126.8 


3143.8 


3160.8 


185.19 


P. P. 




26 


3177.8 


3194.8 


3211.9 


3229.1 


3246.2 


3263.4 


3280.7 


3297.9 


3315.3 


3332.6 


192.59 


7.41 


27 


3350.0 


3367.4 


3384.9 


3402.4 


3419.9 


3437.5 


3455.1 


3472.8 


3490.4 


3508.2 


200.00 


1 .74 


28 


3525.9 


3543.7 


3561.6 


3579.4 


3597.3 


3615.3 


3633.3 


3651.3 


3669.3 


3687.4 


207.41 


2 1.48 


29 


3705.6 


3723.7 


3741.9 


3760.2 


3778.4 


3796.8 


3815.1 


3833.5 


3851.9 


3870.4 


214.81 


3 2.22 


•§5 


30 


3888.9 


3907.4 


3926.0 


3944.6 


3963.3 


3981.9 


4000.7 


4019.4 


4038.2 


4057.1 


222.22 


4 2.96 


31 


4075.9 


4094.8 


4113.8 


4132.8 


4151.8 


4170.8 


4189.9 


4209.1 


4228.2 


4247.4 


229.63 


5 3.70 


2i 


32 


4266.7 


4285.9 


4305.3 


4324.6 


4344.0 


4363.4 


4382.9 


4402.4 


4421.9 


4441.5 


237.04 


6 4.44 


33 


4461.1 


4480.8 


4500.4 


4520.2 


4539.9 


4559.7 


4579.6 


4599.4 


4619.3 


4639.3 


244.44 


7 5.19 


34 


4659.3 


4679.3 


4699.3 


4719.4 


4739.6 


4759.7 


4779.9 


4800.2 


4820.4 


4840.8 


251.85 


8 5.93 


35 


4861.1 


4881.5 


4901.9 


4222.4 


4942.9 


4963.4 


4984.0 


5004.6 


5025.3 


5045.9 


259.26 


9 6.67 




36 


5066.7 


5087.4 


5108.2 


5129.1 


5149.9 


5170.8 


5191.8 


5212.8 


5233.8 


5254.8 


266.67 




37 


5275.9 


5297.1 


5318.2 


5339.4 


5360.7 


5381.9 


5403.3 


5424.6 


5446.0 


5467.4 


274.07 


>. 


38 


5488.9 


5510.4 


5531.9 


5553.5 


5575.1 


5596.8 


5618.4 


5640.2 


5661.9 


5683.7 


281.48 


•a 


39 


5705.6 


5727.4 


5749.3 


5771.3 


5793.3 


5815.3 


5837.3 


5859.4 


5881.6 


5903.7 


288.89 


o 


d'"*' 


40 


5925.9 


5948.2 


5970.4 


5992.8 


6015.1 


6037.5 


6059.9 


6082.4 


6104.9 


6127.4 


296.30 


1 


0"5 


41 


6150.0 


6172.6 


6195.3 


6217.9 


6240.7 


6263.4 


6286.2 


6309.1 


6331.9 


6354.8 


303.70 


^'^ 


42 


6377.8 


6400.8 


6423.8 


6446.8 


6469.9 


6493.1 


6516.2 


6539.4 


6562.7 


6585.9 


311.11 


^° 


43 


6609.3 


6632.6 


6656.0 


6679.4 


6702.9 


6726.4 


6749.9 


6773.5 


6797.1 


6820.8 


318.52 




44 


6844.4 


6868.2 


6891.9 


6915.7 


6939.6 


6963.4 


6987.3 


7011.3 


7035.3 


7059.3 


325.93 


bO 


45 


7083.3 


7107.4 


7131.6 


7155.7 


7179.9 


7204.2 


7228.4 


7252.8 


7277.1 


7301.5 


333.33 


d 


a-S 


46 


7325.9 


7350.4 


7374.9 


7399.4 


7424.0 


7448.6 


7473.3 


7497.9 


7522.7 


7547.4 


340.74 


« 


^-^ 


47 


7572.2 


7597.1 


7621.9 


7646.8 


7671.8 


7696.8 


7721.8 


7746.8 


7771.9 


7797.1 


348.15 


S 


l^VH 


48 


7822.2 


7847.4 


7872.7 


7897.9 


7923.3 


7948.6 


7974.0 


7999.4 


8024.9 


8050.4 


355.56 


S. ■ 


► o 

0) *+-• 


49 


8075.9 


8101.5 


8127.1 


8152.8 


8178.4 


8204.2 


8229.9 


8255.7 


8281.6 


8307.4 


362.96 




50 


8333.3 


8359.3 


8385.3 


8411.3 


8437.3 


8463.4 


8489.6 


8515.7 


8541.9 


8568.2 


370.37 


5 


51 


8594.4 


8620.8 


8647.1 


8673.5 


8699.9 


8726.4 


8752.9 


8779.4 


8806.0 


8832.6 


377.78 


■^ 


M-l 


52 


8859.3 


8885.9 


8912.7 


8939.4 


8966.2 


8993.1 


9019.9 


9046.8 


9073.8 


9100.8 


385.19 


o 


^2 


53 


9127.8 


9154.8 


9181.9 


9209.1 


9236.2 


9263.4 


9290.7 


9317.9 


9345.3 


9372.6 


392.59 


§ 


-So 


54 


9400.0 


9427.4 


9554.9 


9482.4 


9509.9 


9537.5 


9565.1 


9592.8 


9620.4 


9648.2 


400.00 


R. O 


55 


9675.9 


9703.7 


9731.6 


9759.4 


9787.3 


9815.3 


9843.3 


9871.3 


9899.3 


9827.4 


407.41 




1^ 


56 


9955.6 


9983.7 


10012 


10040 


10068 


10097 


10125 


10154 


10182 


10210 


414.81 




57 


10239 


10267 


10296 


10325 


10353 


10382 


10411 


10439 


10468 


10497 


422.22 




58 


10526 


10555 


10584 


10613 


10642 


10671 


10700 


10729 


10758 


10787 


429.63 




•J2 


59 


10817 


10846 


10875 


10905 


10934 


10963 


10993 


11022 


11052 


11082 


437.04 




S 


60 


11111 


11141 


11170 


11200 


11230 


11260 


11290 


11319 


11349 


11379 


444.44 





1046 



59.— RAILROADS, 



33. — Level Sections (Earthwork) ; Height, 0-60 Ft. 
Base of Roadway, 20 Ft. Side Slopes 1 to 1. 
Note. — The last two columns enable us to use any other base than 20 ft.: 
Ex. — Given height, 54.7 ft.; roadway 21 ft. Then we have, 15134 + 
i (400.00+ 5.19) = 15337 cu. yds. (For Ht. >60 ft., see Tables 28, 40.) 

[Cu. Yds. per 100-Ft. Station.] 



























Width 






Ht. 


.0 


.1 


.2 


.3 


.4 


.5 


.6 


.7 


.8 


.9 


of 2 Ft. 






Ft. 






















Cu.Yds 




tn 







7.4 


15.0 


22.6 


30.2 


38.0 


45.8 


53.7 


61.6 


69.7 






^ 


1 


**77!8 


86.0 


94.2 


102.6 


111.0 


119.4 


128.0 


136.6 


145.3 


154.1 


'**7!4i 




Sr^ 


2 


163.0 


171.9 


180.9 


190.0 


199.1 


208.3 


217.6 


227.0 


236.4 


246.0 


14.81 




.-O 


3 


255.6 


265.2 


275.0 


284.8 


294.7 


304.6 


314.7 


324.8 


335.0 


345.2 


22.22 




4 


355.6 


366.0 


376.4 


387.0 


397.6 


408.3 


419.1 


430.0 


440.9 


451.9 


29.63 




5:^ 


5 


463.0 


474.1 


485.3 


496.6 


508.0 


519.4 


531.0 


542.6 


554.2 


566.0 


37.04 




6 


577.8 


589.7 


601.6 


613.7 


625.8 


638.0 


650.2 


662.6 


675.0 


687.4 


44.44 


■¥^ 


^w 


7 


700.0 


712.6 


725.3 


738.1 


751.0 


763.9 


776.9 


790.0 


803.1 


816.3 


51.85 


•a 




8 


829.6 


843.0 


856.4 


870.0 


883.6 


897.2 


911.0 


924.8 


938.7 


952.6 


59.26 


s 


9 


966.7 


980.8 


995.0 


1009.2 


1023.6 


1038.0 


1052.4 


1067.0 


1081.6 


1096.3 


66.67 


w 


Uu^ 


10 


1111.1 


1126.0 


1140.9 


1155.9 


1171.0 


1186.1 


1201.3 


1216.6 


1232.0 


1247.4 


74.07 


fl 


^.^ 


11 


1263.0 


1278.6 


1294.2 


1310.0 


1325.8 


1341.7 


1357.6 


1373.7 


1389.8 


1406.0 


81.48 


■f9 




12 


1422.2 


1438.6 


1455.0 


1471.4 


1488.0 


1504.6 


1521.3 


1538.1 


1555.0 


1571.9 


88.89 


s 


13 


1588.9 


1606.0 


1623.1 


1640.3 


1657.6 


1675.0 


1692.4 


1710.0 


1727.6 


1745.2 


96.30 


£ 


J^d 


14 


1763.0 


1780.8 


1798.7 


1816.6 


1834.7 


1852.8 


1871.0 


1889.2 


1907.6 


1926.0 


103.70 


O 


•^S 


15 


1944.4 


1963.0 


1981.6 


2000.3 


2019.1 


2038.0 


2056.9 


2075.9 


2095.0 


2114.1 


111.11 




16 


2133.3 


2152.6 


2172.0 


2191.4 


2211.0 


2230.6 


2250.2 


2270.0 


2289.8 


2309.7 


118.52 


.a 




17 


2329.6 


2349.7 


2369.8 


2390.0 


2410.2 


2430.6 


2451.0 


2471.4 


2492.0 


2512.6 


125.93 


a 


ao 


18 


2533.3 


2554.1 


2575.0 


2595.9 


2616.9 


2638.0 


2659.1 


2680.3 


2701.6 


2723.0 


133.33 


fH 


S ^ 


19 


2744.4 


2766.0 


2787.6 


2809.2 


2831.0 


2852.8 


2874.7 


2896.6 


2918.7 


2940.8 


140.74 


t-i 


^ ^ 


20 


2963.0 


2985.2 


3007.6 


3030.0 


3052.4 


3075.0 


3197.6 


3120.3 


3143.1 


3166.0 


148.15 


o 


B^ 


21 


3188.9 


3211.9 


3235.0 


3258.1 


3281.3 


3304.6 


3328.0 


3351.4 


3375.0 


3398.6 


155.56 


-d 


^^ 


22 


3422.2 


3446.0 


3469.8 


3493.7 


3517.6 


3541.7 


3565.8 


3590.0 


3614.2 


3638.6 


162.96 


5 


23 


3663.0 


3687.4 


3712.0 


3736.6 


3761.3 


3786.1 


3811.0 


3835.9 


3860.9 


3886.0 


170.37 


r-. W 


24 


3911.1 


3936.3 


3961.6 


3987.0 


4012.4 


4038.0 


4063.6 


4089.2 


4115.0 


4140.8 


177.78 




'rt'P 


25 


4166.7 


4192.6 


4218.7 


4244.8 


4271.0 


4297.2 


4323.6 


4350.0 


4376.4 


4403.0 


185.19 


P.P. 




26 


4429.6 


4456.3 


4483.1 


4510.0 


4536.9 


4563.9 


4591.0 


4618.1 


4645.3 


4672.6 


192.59 


7.41 


27 


4700.0 


4727.4 


4755.0 


4782.6 


4810.2 


4838.0 


4865.8 


4893.7 


4921.6 


4949.7 


200.00 


1 .74 


B"" 


28 


4977.8 


5006.0 


5034.2 


5062.6 


5091.0 


5119.4 


5148.0 


5176.6 


5205.3 


5234.1 


207.41 


2 1.48 




29 


5263.0 


5291.9 


5320.9 


5350.0 


5379.1 


5408.3 


5437.6 


5467.0 


5496.4 


5526.0 


214.81 


3 2.22 


30 


5555.6 


5585.2 


5615.0 


5644.8 


5674.7 


5704.6 


5734.7 


5764.8 


5795.0 


5825.2 


222.22 


4 2.96 


31 


5855.6 


5886.0 


5916.4 


5947.0 


5977.6 


6008. 3 


6039.1 


6070.0 


6100 9 


6131.9 


229.63 


5 3.70 


^ oT 


32 


6163.0 


6194.1 


6225.3 


6256.6 


6288.0 


6319.4 


6351.0 


6382.6 


6414.2 


6446.0 


237.04 


6 4.44 


.^s 


33 


6477.8 


6509.7 


6541.6 


6573.7 


6605.8 


6638.0 


6670.2 


6702.6 


6735.0 


6767.4 


244.44 


7 5.19 


-5^ 


34 


6800.0 


6832.6 


6865.3 


6898.1 


6931.0 


6963.9 


6996.9 


7030.0 


7063.1 


7096.3 


251.85 


8 5.93 




35 


7129.6 


7163.0 


7196.4 


7230.0 


7263.6 


7297.2 


7331.0 


7364.8 


7398.7 


7432.6 


259.26 


9 6.67 


36 


7466.7 


7500.8 


7535.0 


7569.2 


7603.6 


7638.0 


7672.4 


7707.0 


7741.6 


7776.3 


266.67 




37 


7811.1 


7846.0 


7880.9 


7915.9 


7951.0 


7986.1 


8021.3 


8056.6 


8092 


8127.4 


274.07 


(A 


J^^ 


38 


8163.0 


8198.6 


8234.2 


8270.0 


8305.8 


8341.7 


8377.6 


8413.7 


8449.8 


8486.0 


281.48 


"3 


CO 


39 


8522.2 


8558.6 


8595.0 


8631.4 


8668.0 


8704.6 


8741.3 


8778.1 


8815.0 


8851.9 


288.89 


o 


40 


8888.9 


8926.0 


8963.1 


9000.3 


9037.6 


9075.0 


9112.4 


9150.0 


9187.6 


9225.2 


296.30 


s 


^^ 


41 


9263.0 


9300.8 


9338.7 


9376.6 


9414.7 


9452.8 


949L0 


9529.2 


9567.6 


9606.0 


303.70 


£ 


o^ 


42 


9644.4 


9683.0 


9721.6 


9760.3 


9799.1 


9838.0 


9876.9 


9915.9 


9955.0 


9994.1 


311.11 


^ 


^3+^ 


43 


10033 


10073 


10112 


10151 


10191 


10231 


10270 


10310 


10350 


10390 


318.52 


8 


o u 


44 


10430 


10470 


10510 


10550 


10590 


10631 


10671 


10711 


10752 


10793 


325.93 


bo 


45 


10833 


10874 


10915 


10956 


10997 


11038 


11079 


11120 


11162 


11203 


333.33 


B 


46 


11244 


11286 


11328 


11369 


11411 


11453 


11495 


11537 


11579 


11621 


340.74 


•5 


47 


11663 


11705 


11748 


11790 


11832 


11875 


11918 


11960 


12003 


12046 


348.15 


o 


c^^ 


48 


12089 


12132 


12175 


12218 


12261 


12305 


12348 


12391 


12435 


12479 


355.56 


^ 


/iTXtH 


49 


12522 


12566 


12610 


12654 


12698 


12742 


12786 


12830 


12874 


12919 


362.96 


A 


^S 


50 


12963 


13007 


13052 


13097 


13141 


13186 


13231 


13276 


13321 


133C6 


370.37 


Si 

4.9 


2^ 


51 


13411 


13456 


13502 


13547 


13592 


13638 


13684 


13729 


13775 


13821 


377.78 


1 


52 


13867 


13913 


13959 


14005 


14051 


14097 


14144 


14190 


14236 


14283 


385.19 


■^v. 


53 


14330 


14376 


14423 


14470 


14517 


14564 


14611 


14658 


14705 


14753 


392.59 


-^5 


54 


14800 


14847 


14895 


14943 


14990 


15038 


15086 


15134 


15182 


15230 


400.00 


"S^ 


55 


15278 


15326 


15374 


15423 


15471 


15519 


15568 


15617 


15665 


15714 


407,41 






56 


15763 


15812 


15861 


15910 


15959 


16008 


16058 


16107 


16156 


16206 


414.81 




;^ bO 


57 


16256 


16305 


16355 


16405 


16455 


16505 


16555 


16605 


16655 


16705 


422.22 




^ C 


58 


16756 


16806 


16856 


16907 


16958 


17008 


17059 


17110 


17161 


17212 


429.63 




1 


59 


17263 


17314 


17365 


17417 


17468 


17519 


17571 


17623 


17674 


17726 


437.04 




60 


17778 


17830 


17882 


17934 


17986 118038 


18090 


18143 


18195 


18247 


444 44 





EARTHWORK TABLES— LEVEL SECTIONS. 



1047 



34. — Level Sections (Earthwork) ; Height, 0-60 Ft. 
Base op Roadway, 22 Ft. Side Slopes, 1 to 1. 
Note. — ^The last two columns enable us to use any other base than 22 ft.: 
Ex. — Given height, 17.2 ft.; roadway 24 ft. Then we have, 2497.2 + 
(126.93+1.48) = 2624.6cu. yds. (For Ht. >60 ft., see Tables 28, 40.) 

[Cu. Yds. per 100-Ft. Station.] 



























Width 






Ht. 


.0 


.1 


.2 


.3 


.4 


.5 


.6 


.7 


.8 


.9 


of 2 Ft. 






Ft. 






















Cu.Yds 




.!• 







8.2 


16.4 


24.8 


33.2 


41.7 


50.2 


58.9 


67.6 


76.3 






1 


*'85!2 


94.1 


103.1 


112.2 


121 3 


130.6 


139.9 


149.2 


158.7 


168.2 


'"7!4i 




Jo'-* 


2 


177.8 


187.4 


197.2 


207.0 


216.9 


226.9 


236.9 


247.0 


257.2 


267.4 


14.81 




3 


277.8 


288.2 


298.7 


309.2 


319.9 


330.6 


341.3 


352.2 


363. 1; 374.1 


22.22 




:3\^ 


4 


385.2 


396.3 


407.6 


418.9 


430.2 


441.7 


453.2 


464.8 


476.4 


488.2 


29.63 




^c^. 


5 


500.0 


511.9 


523.9 


535.9 


548.0 


560.2 


572.4 


584.8 


597.2 


609.7 


37.04 




4J Cfl 


6 


622.2 


634.9 


647.6 


660.3 


673.2 


686.1 


699.1 


712.2 


725.3 


738.6 


44.44 


^ 


o o 


7 


751.9 


765.2 


778.7 


792.2 


805.8 


819.4 


833.2 


847.0 


860.9 


874.9 


51.85 




8 


888 9 


903.0 


917.2 


931.4 


945.8 


960.2 


974.7 


989.2 


1003.9 


1018.6 


59.26 




9 


1033.3 


1048.2 


1063.1 


1078.1 


1093.2 


1108.3 


1123.6 


1138.9 


1154.2 


1169.7 


66.67 


« 


10 


1185.2 


1200.8 


1216.4 


1232.2 


1248.0 


1263.9 


1279.9 


1295.9 


1312.0 


1328.2 


74.07 







11 


1344.4 


1360.8 


1377.2 


1393.7 


1410.2 


1426.9 


1443.6 


1460.3 


1477.2 


1494 1 


81.48 




12 


1511.1 


1528.2 


1545.3 


1562.6 


1579.9 


1597.2 


1614.7 


1632.2 


1649.8 


1667.4 


88^89 


? 


13 


1685.2 


1703.0 


1720.9 


1738.9 


1756.9 


1775.0 


1793.2 


1811.4 


1829.8 


1848.2 


96.30 


£ 




14 


1866.7 


1885.2 


1903.9 


1922.6 


1941.3 


1960.2 


1979.1 


1998.1 


2017.2 


2036.3 


103.70 




15 


2055 6 


2074.9 


2094.2 


2113.7 


2133.2 


2152.8 


2172.,4 


2192.2 


2212.0 


2231.9 


111.11 





>%ll 


16 


2251.9 


2271.9 


2292.0 


2312.2 


2332.4 


2352.8 


2373.2 


2393.7 


2414.2 


2434.9 


118.52 


M 


;QW 


.17 


2455.6 


2476.3 


2497.2 


2518.1 


2539.1 


2560.2 


2581.3 


2602.6 


2623.9 


2645.2 


125.93 


d 


'S^ 


18 


2666.7 


2688.2 


2709 8 


2731.4 


2753.2 


2775.0 


2796.9 


2818.9 


2840.9 


2863.0 


133.33 


S 


.^^ 

w 


19 


2885.2 


2907.4 


2929.8 


2952.2 


2974.7 


2997.2 


3019.9 


3042.6 


3065.3 


3088.2 


140.74 


20 


3111.1 


3134.1 


3157.2 


3180.3 


3203.6 


3226.9 


3250.2 


3273.7 


3297.2 


3320.8 


148.15 





21 


3344.4 


3368.2 


3392.0 


3415.9 


3439.9 


3463.9 


3488.0 


3512.2 


3536.4 


3560.8 


155.56 


1^ 


h 


22 


3585.2 


3609.7 


3634.2 


3658.9 


3683.6 


3708.3 


3733.2 


3758.1 


3783.1 


3808.2 


162.96 


"5 


23 


3833.3 


3858 6 


3883.9 


3909.2 


3934.7 


3960.2 


3985.8 


4011.4 


4037.2 


4063.0 


170.37 


< 




24 


4088.9 


411^.9 


4140.9 


4167.0 


4193.2 


4219.4 


4245.8 


4272.2 


4298.7 


4325.2 


177.78 




25 


4351.9 


4378.6 


4405.3 


4432.2 


4459.1 


4486.1 


4513.2 


4540.3 


4567.6 


4594.9 


185.19 


P.P. 


26 


4622 2 


4649.7 


4677.2 


4704.8 


4732.4 


4760.2 


4788.0 


4815.9 


4843.9 


4871.9 


192.59 


7.41 


to 


27 


4900.0 


4928.2 


4956.4 


4984.8 


5013.2 


5041.7 


5070.2 


5098.9 


5127.6 


5156.3 


200.00 


1 .74 


28 


5185.2 


5214.1 


5243.1 


5272.2 


5301.3 


5330.6 


5359.9 


5389.2 


5418.7 


5448.2 


207.41 


2 1.48 


B^ 


29 


5477.8 


5507.4 


5537.2 


5567.0 


5596.9 


5626.9 


5656.9 


5687.0 


5717.2 


5747.4 


214.81 


3 2.22 


cX 


30 


5777.8 


5808.2 


5838.7 


5869.2 


5899.9 


5930.6 


5961.3 


5992.2 


6023.1 


6054.1 


222.22 


4 2.96 


O "^ 


31 


6085.2 


6116.3 


6147.6 


6178.9 


6210.2 


6241.7 


6273.2 


6304.8 


6336.4 


6368.2 


229.63 


5 3.70 


'2 "^ 


32 


6400 


6431.9 


6463.9 


6495.9 


6528.0 


6560.2 


6592.4 


6624.8 


6657.2 


6689.7 


237.04 


6 4.44 


TO o 
O 


33 


6722.2 


6754.9 


6787.6 


6820.3 


6853.2 


6886.1 


6919.1 


6952.2 


6985.3 


7018.6 


244.44 


7 5.19 


34 


7051.9 


7085.2 


7118.7 


7152.2 


7185.8 


7219.4 


7253.2 


7287.0 


7320.9 


7354.9 


251.85 


8 5.93 


30 


35 


7388.9 


7423.0 


7457.2 


7491.4 


7525.8 


7560.2 


7594.7 


7629.2 


7663.9 


7698.6 


259.26 


9 6.67 


^c; 


36 


7733.3 


7768.2 


7803.1 


7838.1 


7873.2 


7908.3 


7943.6 


7978.9 


8014.2 


8049.7 


266.67 




•s| 


37 


8085.2 


8120.8 


8156.4 


8192.2 


8228.0 


8263.9 


8299.9 


8335.9 


8372.0 


8408.2 


274.07 


>» 


38 


8444.4 


8480.8 


8517.2 


8553.7 


8590.2 


8626.9 


8663.6 


8700.3 


8737.2 


8774.1 


281.48 


•a 


^^ 


39 


8811.1 


8848.2 


8885.3 


8922.6 


8959.9 


8997.2 


9034.7 


9072.2 


9109.8 


9147.4 


288.89 





SI 


40 


9185.2 


9223.0 


9260.9 


9298.9 


9336.9 


9375.0 


9413.2 


9451.4 


9489.8 


9528.2 


296.30 






1^ '^ 


41 


9566.7 


9605.2 


9643.9 


9682.6 


9721.3 


9760.2 


9799.1 


9838.1 


9877.2 


9916.3 


303.70 


Ai 


42 


9955.6 


9994.9 


10034 


10074 


10113 


10153 


10192 


10232 


10272 


10312 


311.11 


QM-I 


43 


10352 


10392 


10432 


10472 


10512 


10553 


10593 


10634 


10674 


10715 


318.52 


^0^ 


44 


10756 


10796 


10837 


10878 


10919 


10960 


11001 


11043 


11084 


11125 


325.93 


to 


CO 


45 


11167 


11208 


11250 


11291 


11333 


11375 


11417 


11459 


11501 


11543 


333.33 


a 


/i?M-l 


46 


11585 


11627 


11670 


11712 


11755 


11797 


11840 


11883 


11925 


11968 


340.74 


1 


a^ 


47 


12011 


12054 


12097 


12140 


12184 


12227 


12270 


12314 


12357 


12401 


348.15 




0+^ 


48 


12444 


12488 


12532 


12576 


12620 


12664 


12708 


12752 


12796 


12841 


355.56 


A 


c^'g 


49 


12885 


12930 


12974 


13019 


13064 


13108 


13153 


13198 


13243 


13288 


362.96 


6% 


50 


13333 


13379 


13424 


13469 


13515 


13560 


13606 


13651 


13697 


13743 


370.37 


5 


JgrCi 


51 


13789 


13835 


13881 


13927 


13973 


14019 


14066 


14112 


14159 


14205 


377.78 


"^ 


PQ bj 


52 


14252 


14299 


14345 


14392 


14439 


14486 


14533 


14580 


14628 


14675 


385.19 


Qi 


^^ 


53 


14722 


14770 


14817 


14865 


14912 


14960 


15008 


15056 


15104 


15152 


392.59 


U 


|«^ 


54 


15200 


15248 


15296 


15345 


15393 


15442 


15490 


15539 


15588 


15636 


400.00 


55 


15685 


15734 


15783 


15832 


15881 


15931 


15980 


16029 


16079 


16128 


407.41 




^•^ 


56 


16178 


16227 


16277 


16327 


16377 


16427 


16477 


16527 


16577 


16627 


414.81 




^° 


57 


16678 


16728 


16779 


16829 


16880 


16931 


16981 


17032 


17083 


17134 


422.22 




^ 


58 


17185 


17236 


17288 


17339 


17390 


17442 


17493 


17545 


17596 


17648 


429.63 


^ 


4-^ 



59 


17700 


17752 


17804 


17856 


17908 


17960 


18012 


18065 


18117 


18170 


437.04 




^ 


60 


18222 


18275 


18328 


18380 


18433 


18486 


18539 


18592 


18645 


18699 


444.44 





1048 



SQ.-^RAILROADS. 



35. — Level Sections (Earthwork) ; Height, 0-60 Ft. 
Base of Roadway, 24 Ft. Side Slopes, IJ^ to 1. 
Note. — ^The last two columns enable us to use any other base than 24 ft.: 
Ex. — Given height, 43.1ft.; roadway 22 ft. Then we have, 14151 - 
(318.52+ 0.74) = 13832 cu. yds. (For Ht. > 60 ft., see Tables 24, 41.) 

[Cu. Yds. per 100-Ft. Station.] 



























Width 






Ht. 


.0 


.1 


.2 


.3 


.4 


.5 


.6 


.7 


.8 


.9 


of 2 Ft. 






Ft. 






















Cu.Yds 




in 







8.9 


18.0 


27.2 


36.4 


45.8 


55.3 


64.9 


74.7 


84.5 






rJ 


1 


"94!4 


104.5 


114.7 


124.9 


135.3 


145.8 


156.4 


167.2 


178.0 


188.9 


*"7!4i 






2 


200.0 


211.2 


222.4 


233.8 


245.3 


256.9 


268.6 


280.5 


292.4 


304.4 


14.81 






3 


316.6 


238.9 


341.2 


353.7 


366.3 


379.0 


391.9 


404.8 


417.8 


431.0 


22.22 




...1— t 

Is 


4 


444.4 


457.8 


471.3 


484.9 


498.6 


512.4 


526.4 


540.4 


554.6 


568.8 


29.63 




5 


583.3 


597.8 


612.4 


627.1 


642.0 


656.9 


671.9 


687.1 


702.3 


717.7 


37.04 




^^S 


6 


733.3 


748.9 


764.7 


780.5 


796.4 


812.5 


828.7 


844.9 


861.3 


877.8 


44.44 


4^ 


7 


894.4 


911.2 


928.0 


944.9 


962.0 


979.2 


996.4 


1013.8 


1031.3 


1048.9 


51.85 


■g. 


a; xn 
§ ^ 

TO O 


8 


1066.7 


1084.5 


1102.4 


1120.5 


1138.7 


1156.9 


1175.3 


1193.8 


1212.4 


1231.2 


59.26 


•53 


9 


1250.0 


1268.9 


1288.0 


1307.2 


1326.4 


1345.8 


1365.3 


1384.9 


1404.7 


1424.5 


66.67 


W 


10 


1444.4 


1464.5 


1484.7 


1504.9 


1525.3 


1545.8 


1566.4 


1587.2 


1608.0 


1628.9 


74.07 


a 


"-§ 


11 


1650.0 


1671.2 


1692.4 


1713.8 


1735.3 


1756.9 


1778.7 


1800.5 


18?2.4 


1844.5 


81.48 


4^ 


ft)»ri 


12 


1866.7 


1888.9 


1911.3 


1933.8 


1956.4 


1979.2 


2002.0 


2024.9 


2048.0 


2071.2 


88.89 


1 


^1 


13 


2094.4 


2117.8 


2141.3 


2164.9 


2188.7 


2212.5 


2236.4 


2260.5 


2284.7 


2308.9 


96.30 




14 


2333.3 


2357.8 


2382.4 


2407.2 


2432.0 


2456.9 


2482.0 


2507.2 


2532.4 


2557.8 


103.70 




15 


2583.3 


2608.9 


2634.7 


2660.5 


2686.4 


2712.5 


2738.7 


2764.9 


2791.3 


2817.8 


111.11 


o 


16 


2844.4 


2871.2 


2898.0 


2924.9 


2952.0 


2979.2 


3006.4 


3033.8 


3061.3 


3088.9 


118.52 


s 


"^CSI 


17 


3116.7 


3144.5 


3172.4 


3200.5 


3228.7 


3256.9 


3285.3 


3313.8 


3342.4 


3371.2 


125.93 


4^ 


18 


3400.0 


3428.9 


3458.0 


3487.2 


3516.4 


3545.8 


3575.3 


3604.9 


3634.7 


3664.5 


133.33 


ao 


19 


3694.5 


3724.5 


3754.7 


3784.9 


3815.3 


3845.8 


3876.4 


3907.2 


3938.0 


3968.9 


140.74 


H 


2 ^ 


20 


4000.0 


4031.2 


4062.4 


4093.8 


4125.3 


4156.9 


4188.7 


4220.5 


4252.4 


4284.5 


148.15 


S 


g^ 


21 


4316.7 


4348.9 


4381.3 


4413.8 


4446.4 


4479.2 


4512.0 


4544.9 


4578.0 


4611.2 


155.56 


«M 


S-Q 


22 


4644.4 


4677.8 


4711.3 


4744.9 


4778.7 


4812.5 


4846.4 


4880.5 


4914.7 


4948.9 


162.96 


"d 


^1 


23 


4983.3 


5017.8 


5052.4 


5087.2 


5122.0 


5156.9 


5192.0 


5227.2 


5262.4 


5297.8 


170.37 


< 


24 


5333.3 


5368.9 


5404.7 


5440.5 


5476.4 


5512.5 


5548.7 


5584.9 


5621.3 


5657.8 


177.78 




7^ w 


25 


5694.4 


5731.2 


5768.0 


5804.9 


5842.0 


5879.2 


5916.4 


5953.8 


5991.3 


6028.9 


185.19 


P. P. 




26 


6066.7 


6104.5 


6142.4 


6180.5 


6218.7 


6256.9 


6295.3 


6333.8 


6372.4 


6411.2 


192.59 


7.41 


27 


6450.0 


6488.9 


6528.0 


6567.2 


6606.4 


6645.8 


6685.3 


6724.9 


6764.7 


6804.5 


200.00 


1 .74 

2 1.48 

3 2.22 

4 2.96 

5 3.70 

6 4.44 

7 5.19 

8 5.93 

9 6.67 


6 o 


28 


6844.4 


6884.5 


6924.7 


6964.9 


7005.3 


7045.8 


7086.4 


7127.2 


7168.0 


7208.9 


207.41 


29 


7250.0 


7291.2 


7332.4 


7373.8 


7415.3 


7456.9 


7498.7 


7540.5 


7582.4 


7624.5 


214.81 




30 


7666.7 


7708.9 


7751.3 


7793.8 


7836.4 


7879.2 


7922.0 


7964.9 


8008.0 


8051.2 


222.22 


31 


8094.4 


8137.8 


8181.3 


8224.9 


8268.7 


8312.5 


8356.4 


8400.5 


8444.7 


8488.9 


229.63 


05 OJ 


32 


8533.3 


8577.8 


8622.4 


8667.2 


8712.0 


8756.9 


8802.0 


8847.2 


8892.4 


8937.8 


237.04 


si 


33 


8983.3 


9028.9 


9074.7 


9120.5 


9166.4 


9212.5 


9258.7 


9304.9 


9351.3 


9397.8 


244.44 


34 


9444.4 


9491.2 


9538.0 


9584.9 


9632.0 


9679.2 


9726.4 


9773.8 


9821.3 


9868.9 


251.85 


35 


9916.7 


9964.5 


10012 


10061 


10109 


10157 


10205 


10254 


10302 


10351 


259.26 


d <a 


36 


10400 


10449 


10498 


10547 


10596 


10646 


10695 


10745 


10795 


10845 


266.67 


.5^ 


37 


10894 


10945 


10995 


11045 


11095 


11146 


11196 


11247 


11298 


11349 


274.07 


. 


w - 


38 


11400 


11451 


11502 


11554 


11605 


11657 


11709 


11761 


11812 


11865 


281.48 


>> 


Tl+i 


39 


11917 


11969 


12021 


12074 


12126 


12179 


12232 


12285 


12338 


12391 


288.89 


o 


>H^ 


40 


12444 


12498 


12551 


12605 


12659 


12713 


12766 


12821 


12875 


12929 


296.30 


f3 


. 00 


41 


12983 


13038 


13092 


13147 


13202 


13257 


13312 


13367 


13422 


13478 


303.70 


a 


CJ^ 


42 


13533 


13589 


13645 


13701 


13756 


13813 


13869 


13925 


13981 


14038 


311.11 


5 




43 


14094 


14151 


14208 


14265 


14322 


14379 


14436 


14494 


14551 


14609 


318.52 


O 


44 


14667 


14725 


14782 


14840 


14899 


14957 


15015 


15074 


15132 


15191 


325.93 


t> 


^-a 


45 


15250 


15309 


15368 


15427 


15486 


15546 


15605 


15665 


15725 


15785 


333.33 


B 


(u 53 


46 


15844 


15905 


15965 


16025 


16085 


16146 


16206 


16267 


16328 


16389 


340.74 




^-S 


47 


16450 


16511 


16572 


16634 


16695 


16757 


16819 


16881 


16942 


17005 


348.15 


o 


?§^ 


48 


17067 


17129 


17191 


17254 


17316 


17379 


17442 


17505 


17568 


17631 


355.56 


<o 


^^^ 


49 


17694 


17758 


17821 


17885 


17949 


18013 


18076 


18141 


18205 


18269 


362.96 


A 


6 o 


50 


18333 


18398 


18462 


18527 


18592 


18657 


18722 


18787 


18852 


18918 


370.37 


.d 




51 


18983 


19049 


19115 


19181 


19246 


19313 


19379 


19445 


19511 


19578 


377.78 


4J 


«^ 


52 


19644 


19711 


19778 


19845 


19912 


19979 


20046 


20114 


20181 


20249 


385.19 


"^ 


.pO 


53 


20317 


20385 


20452 


20521 


20589 


20657 


20725 


20794 


20862 


20931 


392.59 


^ 


^1 


54 


21000 


21069 


21138 


21207 


21276 


21346 


21415 


21485 


21555 


21625 


400.00 


P 


55 


21694 


21765 


21835 


21905 


21975 


22046 


22116 


22187 


22258 


22329 


407.41 




^^ 


56 


22400 


22471 


22542 


22614 


22685 


22757 


22829 


22901 


22972 


23045 


414.81 




'Tr, 


57 


23117 


23189 


23261 


23334 


23406 


23479 


23552 


23625 


23698 


23771 


422.22 




58 


23844 


23918 


23991 


24065 


24139 


24213 


24286 


24361 


24435 


24509 


429.63 




59 


24583 


24658 


24732 


24807 


24882 


24957 


25032 


25107 


25182 


25258 


437.04 




CO 


60 


25333 


25409 


25485 


25561 


25636 


25713 


25789 


25865 


25941 


26018 


444.44 





EARTHWORK TABLES— LEVEL SECTIONS. 



1049 



36. — ^Level Sections (Earthwork); Height, 0-60 Ft. 
Base of Roadway, 26 Ft. Side Slopes 13^ to 1. 
Note. — ^The last two columns enable us to use any other base than 26 ft.: 
Ex. — Given height, 22.2 ft.; roadway 27 ft. Then we have, 4875.8 + 
J (162,96+ 1.48) = 4958.0 cu. yds. (For Ht. >60 ft., see Tables 24, 41.) 

[Cu. Yds. per 100-Ft. Station.] 



























Width 






Ht. 


.0 


.1 


.2 


.3 


.4 


.5 


.6 


.7 


.8 


.9 


of 2 Ft. 






Ft. 






















Cu.Yds 




M 







9.7 


19.5 


29.4 


39.4 


49.5 


59.8 


70.1 


80.6 


91.2 








1 


'ioiio 


112.6 


123.6 


134.6 


145.7 


156.9 


168.3 


179.8 


191.3 


203.0 


*"7!4i 




2 


214.8 


226.7 


238.7 


250.9 


263.1 


275.5 


287.9 


300.5 


313.2 


326.0 


14.81 




« 'tH 


3 


338.9 


351.9 


365.0 


378.3 


391.6 


405.1 


418.7 


432.4 


446.1 


460.1 


22.22 




So 


4 


474.1 


488.2 


502.4 


516.8 


531.3 


545.8 


560.5 


575.3 


590.2 


605.2 


29.63 




5 


620.4 


635.6 


651.0 


666.4 


682.0 


697.7 


713.5 


729.4 


745.4 


761.5 


37.04 




^^ 


6 


777.8 


794.1 


810.6 


827.2 


843.9 


860.6 


877.6 


894.6 


911.7 


928.9 


44.44 


4i 


7 


946.3 


963.8 


981.3 


999.0 


1016.8 


1034.7 


1052.7 


1070.9 


1089.1 


1107.5 


51.85 


■g 


Is 

rt o 


8 


1125.9 


1144.5 


1163.2 


1182.0 


1200.9 


1219.9 


1239.0 


1258.3 


1277.6 


1297.1 


59.26 


*53 


9 


1316.7 


1336.4 


1356.1 


1376.1 


1396.1 


1416.2 


1436.4 


1456.8 


1477.3 


1497.8 


66.67 


w 


10 


1518.5 


1539.3 


1560.2 


1581.2 


1602.4 


1623.6 


1645.0 


1666.4 


1688.0 


1709.7 


74.07 


d 


11 


1731.5 


1753.4 


1775.4 


1797.5 


1819.8 


1842.1 


1864.6 


1887.2 


1909.9 


1932.6 


81.48 


43 


12 


1955.6 


1978.6 


2001.7 


2024.9 


2048.3 


2071.8 


2095.3 


2119.0 


2142.8 


2166.7 


88.89 


S 


i: 


13 


2190.7 


2214.9 


2239.1 


2263.5 


2287.9 


2312.5 


2337.2 


2362.0 


2386.9 


2411.9 


96.30 


P^ 


14 


2437.0 


2462.3 


2487.6 


2513.1 


2538.7 


2564.4 


2590.1 


2616.1 


2642.1 


2668.2 


103.70 


o 


1:2 


15 


2694.4 


2720.8 


2747.3 


2773.8 


2800.5 


2827.3 


2854.2 


2881.2 


2908.4 


2935.6 


111.11 




16 


2963.0 


2990.4 


3018.0 


3045.7 


3073.5 


3101.4 


3129.4 


3157.5 


3185.8 


3214.1 


118.52 


.a 


1^ 


17 


3242.6 


3271.2 


3299.9 


3328.6 


3357.6 


3386.6 


3415.7 


3444.9 


3474.3 


3503.8 


125.93 


? 


18 


3533.3 


3563.0 


3592.8 


3622.7 


3652.7 


3682.9 


3713.1 


3743.5 


3773.9 


3804.5 


133.33 


H 


38i 
6^ 


19 


3835.2 


3866.0 


3896.9 


3927.9 


3959.0 


3990.3 


4021.6 


4053.1 


4084.7 


4116.4 


140.74 


t-i 


20 


4148.1 


4180.1 


4212.1 


4244.2 


4276.4 


4308.8 


4341.3 


4373.8 


4406.5 


4439.3 


145.15 


a 


21 


4472.2 


4505.2 


4538.4 


4571.6 


4605.0 


4638.4 


4672.0 


4705.7 


4739.5 


4773.4 


155.56 


•d 


^^ 


22 


4807.4 


4841.5 


4875.8 


4910.1 


4944.6 


4979.2 


5013.9 


5048.6 


5083.6 


5118.6 


162.96 


-d 
< 


^ o 


23 


5153.7 


5188.9 


5224.3 


5259.8 


5295.3 


5331.0 


5366.8 


5402.7 


5438.7 


5474.9 


170.37 




24 


5511.1 


5547.5 


5583.9 


5620.5 


5657.2 


5694.0 


5730.9 


5767.9 


5805.0 


5842.3 


177.78 




^4 


25 


5879.6 


5917 1 


5954.7 


5992.4 


6030.1 


6068.1 


6106.1 


6144.2 


6182.4 


6220.8 


185.19 


P. P. 


iS 


26 
27 


6259.3 
6650.0 


6297.8 
6689.7 


6336.5 
6729.5 


6375.3 
6769.4 


6414.2 
6809.4 


6453.2 
6849.5 


6492.4 
6889.8 


6531.6 
6930.1 


6571.0 
6970.6 


6610.4 
7011.2 


192.59 
200.00 


7.41 


eg 


1 .74 


28 


7051.9 


7092.6 


7133.6 


7174.6 


7215.7 


7256.9 


7298.3 


7339.8 


7381.3 


7423.0 


207.41 


2 1.48 


VrH 


29 


7464.8 


7506.7 


7548.7 


7590.9 


7633.1 


7675.5 


7717.9 


7760.5 


7803.2 


7846.0 


214.81 


3 2.22 


S°2 


30 


7888.9 


7931.9 


7975.0 


8018.3 


8061.6 


8105.1 


8148.7 


8192.4 


8236.1 


8280.1 


222.22 


4 2.96 


1- 


31 


8324.1 


8368.2 


8412.4 


8456.8 


8501.3 


8545.8 


8590.5 


8635.3 


8680.2 


8725.2 


229.63 


5 3.70 




32 


8770.4 


8815.6 


8861.0 


8906.4 


8952.0 


8997.7 


9043.5 


9089.4 


9135.4 


9181.5 


237.04 


6 4.44 


.52 <u 

5S 


33 


9227.8 


9274.1 


9320.6 


9367.2 


9413.9 


9460.6 


9507.6 


9554.6 


9601.7 


9648.9 


244.44 


7 5.19 


34 


9696.3 


9743.8 


9791.3 


9839.0 


9886.8 


9934.7 


9982.7 


10031 


10079 


10127 


251.85 


8 5.93 


■=1 


35 


10176 


10225 


10273 


10322 


10371 


10420 


10469 


10518 


10568 


10617 


259.26 


9 6.67 


36 


10667 


10716 


10766 


10816 


10866 


10916 


10966 


11017 


11067 


11118 


266.67 




37 


11169 


11219 


11270 


11321 


11372 


11424 


11475 


11526 


11578 


11630 


274.07 


>, 


^ T 


38 


11681 


11733 


11785 


11838 


11890 


11942 


11995 


12047 


12100 


12153 


281.48 


13 


>^^ 


39 


12206 


12259 


12312 


12365 


12418 


12472 


12525 


12579 


12633 


12687 


288.89 


o 


fjeo 


40 


12741 


12795 


12849 


12903 


12958 


13013 


13067 


13122 


13177 


13232 


296.30 


c 


41 


13287 


13342 


13398 


13453 


13509 


13564 


13620 


13676 


13732 


13788 


303.70 


a 


Xi^ 


42 


13844 


13901 


13957 


14014 


14071 


14127 


14184 


14241 


14298 


14356 


311.11 




1*^ 


43 


14413 


14470 


14528 


14586 


14643 


14701 


14759 


14818 


14876 


14934 


318.52 


O 


^° 


44 


14993 


15051 


15110 


15169 


15228 


15287 


15346 


15405 


15464 


15524 


325.93 


be 


o-E 


45 


15583 


15643 


15703 


15763 


15823 


15883 


15943 


16003 


16064 


16125 


333. 33 


13 


^•§^ 


46 


16185 


16246 


16307 


16368 


16429 


16490 


16552 


16613 


16675 


16736 


340.74 


i 


-g^ 


47 


16798 


16860 


16922 


16984 


17046 


17109 


17171 


17234 


17297 


17359 


348.15 




^t 


48 


17422 


17485 


17548 


17612 


17675 


17738 


17802 


17866 


17929 


17993 


355.56 


1 


<u«2 


49 


18057 


18122 


18186 


18250 


18315 


18379 


18444 


18519 


18574 


18639 


362.96 




50 


18704 


18769 


18834 


18900 


18965 


19031 


19097 


19163 


19229 


19295 


370.37 


5 


51 


19361 


19427 


19494 


19561 


19627 


19694 


19761 


19828 


19895 


19962 


377.78 




52 


20030 


20097 


20165 


20232 


20300 


20368 


20436 


20504 


20572 


20641 


385.19 


■^u 


53 


20709 


20778 


20847 


20915 


20984 


21053 


21122 


21192 


21261 


21330 


392.59 


-55 


54 


21400 


21470 


21539 


21609 


21679 


21750 


21820 


21890 


21961 


22031 


400.00 


r^ 


55 


22102 


22173 


22244 


32315 


22386 


22457 


22528 


22600 


22671 


22743 


407.41 




1-^ 


56 


22815 


22887 


22959 


23031 


23103 


23175 


23248 


23321 


23393 


23466 


414.81 






57 


23539 


23612 


23685 


23758 


23832 


23905 


23979 


24052 


24126 


24200 


422.22 




^.S 


58 


24274 


24348 


24422 


24497 


24571 


24646 


24721 


24795 


24870 


24945 


429.63 




i 59 


25020 


25096 


25171 


25246 


25322 


25398 


25473 


25549 


25625 


25702 


437.04 




^ 60 25778 I 


25854 


25931 


26007 


26084 


26161 


26238 


26315 26392 j 


26469 


444.44 





1050 



.—RAILROADS. 



37. — ^Level Sections (Earthwork) ; Height, 0-60 Ft. 
Base op Roadway, 28 Ft. Side Slopes, 1 to 1. 
Note. — ^The last two coliimns enable us to use any other base than 28 ft. 
Ex. — Given height, 57.5 ft.; roadway 26 ft. Then we have, 18208- 
(422.22+ 3.70) = 17782 cu. yds. (For Ht. >60 ft., see Tables 28, 40.) 

[Cu. Yds. per 100-Ft. Station.] 



Ht. 
Ft. 



.0 



.1 



.2 



.4 



.5 



.7 



Width 
of 2 Ft. 
Cu.Yd8 



0-t-> 



v% 



+3.03 





16 

17 


ao 

•^ to 


18 
19 


;3 OS 


?,{) 


a^ 


21 


^^ 


22 


^*2 


23 


'^ w 


24 


la^ 


25 




26 


27 


28 


»— 1 o 


29 


^^ 


8(1 


(dec 
4J 


31 




32 


33 
34 


C •u 


35 


•2^ 


36 


(0 . 


37 


:^+^ 


38 


pH«*-l 


39 


a:s 


40 
41 


*o«*-i 


42 


a 


43 


cd^ 


44 


cT-S 


45 


P 


46 

47 


^^^H 


48 


a;«i^ 


49 




50 
51 


+j o 


52 


3i. 


53 




54 
55 




56 


;?;bo 


57 


c 


58 


52 


59 


;3 


60 



107.4 
222.2 
344 
474.1 
611.1 
755.6 
907 
1066.7 
1233.3 
1407. 
1588. 
1777. 
1974. 
2177.8 
2388.9 
2607.4 
2833.3 
3066.7 
3307.4 
3555.6 
3811.1 
•1074.1 
4344.4 
4622.2 
4907.4 
5200.0 
5500.0 
5807.4 
6122.2 
6444.4 
6774.1 
7111.1 
7455.6 
7807.4 
8166.7 
8533.3 
8907.4 
9288.9 
9677.8 
10074 
10478 
10889 
11307 
11733 
12167 
12607 
13056 
13511 
13974 
14444 
14922 
15407 
15900 
16400 
16907 
17422 
17944 
18474 
19011 
19556 



10.4 
118.6 
234.1 
357.1 

487.4 
625.2 
770.4 
923.0 
1083.0 
1250.4 
1425.2 
1607 
1797.1 
1994.1 
2198.6 
2410.4 
2629 
2856.3 
3090.4 
3331 
3580.8 
3837.1 
4100.8 
4371.9 
4650.4 
4936.3 
5229.7 
5530.4 
5838.6 
6154.1 
6477.1 
6807.4 
7145.2 
7490.4 
7843.0 
8203.0 
8570.4 
8945.2 
9327.4 
9717.1 
10114 
10519 
10930 
11350 
11776 
12210 
12652 
3101 
13587 
14021 
14492 
14970 
15456 
15950 
16450 
16959 
17474 
17997 
18527 
19065 
19610 



20.9 

129.8 

246.1 

369.8 

500.9 

639.4 

785.4 

938.7 

1099.4 

1267.6 

1443.1 

1626.1 

1816 

2014.2 

2219.4 

2432 

2652.0 

2879.4 

3114.2 

3356.4 

3606.1 

3863 

4127 

4399.4 

4678.7 

4965.3 

5259.4 

5560.9 

5869 



6186.1 

6509.8 

6840.9 

7179.4 

7525.3 

7878.7 

8239.4 

8607.6 

8983.1 

9366.1 

9756.4 

10154 

10559 

10972 

11392 

11819 

12254 

12696 

13146 

13603 

14068 

14539 

15019 

15505 

15999 

16501 

17010 

17526 

18050 

18581 

19119 

19665 



31.4 

141.1 

258.1 

382.6 

514.4 

653.7 

800.4 

954.5 

1115.9 

1284. 

1461.1 

1644.8 

1835 

2034.4 

2240.3 

2453 

2674.4 

2902.6 

3138.1 

3381.1 

3631.4 

3889.2 

4154.4 

4427.0 

4707.0 

4994.4 

5289.2 

5591.4 

5901.1 

1 



6218 

6542 

6874.4 

7213.7 

7560.3 

7914.4 

8275.9 

8644.8 

9021.1 

9404.8 

9795.9 

10194 

10600 

11014 

11434 

11863 

12298 

12741 

13191 

13649 

14114 

14587 

15067 

15554 

16049 

16551 

17061 

17578 

18103 

18634 

19174 

19720 



42.1 
152.4 
270.2 
395.4 
528.0 
668.0 
815,5 
970.3 
1132.4 
1302.1 
1479.1 
1663.6 
1855.4 
2054.7 
2261.3 
2475.4 
2696.9 
2925.8 
3162.1 
3405.8 
3656.9 
3915.4 
4181.3 
4454.7 
4735.4 
5023 
5319.1 
5622 
5932.4 
6250.2 
6575.4 
6908.0 
7248.0 
7595 
7950.2 
8212.4 
8682.1 
9059.1 
9443.6 
9835.4 
10235 
10641 
11055 
11477 
11906 
12342 
12786 
13237 
13695 
14161 
14635 
15115 
15604 
16099 
16602 
17112 
17630 
18155 
18688 
19228 
19775 



[9.4 



52.8 
163.9 
282.4 
408.3 
541.7 
682.4 
830.6 
986.1 

1149.1 

1319.4 

1497.2 

1682.4 

1875.0 

2075 

2282 

2497 

271 

2949.1 

3186 

3430 

3682.4 

3941.7 

4208. 

4482. 

4763. 

5052. 

5349. 

5652.8 

5963.9 

6282 

6608 

6941 

7282 

7630 

7986 

8349.1 

8719.4 

9097.2 

9482.4 

9875.0 

10275 

10682 

11097 

11519 

11949 

12386 

12831 

13282 

13742 

14208 

14682 

15164 

15653 

16149 

16653 

17164 

17682 

18208 

18742 

19282 

19831 



63.6 
175.4 
294.7 
421.3 
555.4 
696.9 
845.8 
1002.1 
1165.8 
1336.9 
1515.4 
1701.3 
1894 7 
2095.4 
2303. G 
2519.1 
2742.1 
2972.4 
3210.2 
3455 
3708.0 
3968.0 
4235 
4510 
4792 
5082.1 
5379 
5683.6 
5995.4 
6314. 
6641. 
6975.4 
7316.9 
7665.8 
8022.1 
8385.8 
8756.9 
9135.4 
9521.3 
9914.7 
10315 
10724 
11139 
11562 
11992 
12430 
12875 
13328 
13788 
14255 
14730 
15212 
15702 
16199 
16704 
17215 
17735 
18261 
18795 
19337 
19886 



74.4 
187.0 
307.0 
434.4 
569.2 
711.4 
861.1 
1018.1 
1182.6 
1354.4 
1533.7 
1720.3 
1914.4 
2115.9 
2324.8 
2541.1 
2764.8 
2995 
3234.4 
3480 
3733.7 
3994 
4262.6 
4538.1 
4821. 
5111.4 
5409 
5714.4 
6127.0 
6347.0 
6674.4 



7009 

7351.4 

7701 

8058 

8422.6 

8794.4 

9173.7 

9560.3 

9954.4 

10356 

10765 

11181 

11605 

12036 

12474 

12920 

13374 

13834 

14303 

14778 

15261 

15751 

16249 

16754 

17267 

17787 

18314 

18849 

19391 

19941 



85 

198.7 
319.4 
447.6 
583.1 
726.1 
876.5 
1034.2 
1199.4 
1372.0 
1552.0 
1739.4 
1934.2 
2136.4 
2346.1 
2563.1 
2787.6 
3019 
3258 
3505 
3759.4 
4020.9 
4289.8 
4566 
4849.8 
5140.9 
5439 
5745.3 
6158 
6379.4 



6707.6 
7043.1 
7386.1 
7736.4 
'4.2 
8459.4 
8832.0 
9212.0 
9599.4 
9994.2 
10396 
10806 
11223 
11648 
12079 
12519 
12965 
13419 
13881 
14350 
14826 
15310 
15801 
16299 
16805 
17319 
17839 
18368 
18903 
19446 
19996 



96 
210.4 
331.9 
460.8 
597.1 
740.8 
891.9 
1050.4 
1216 3 
1389.7 
1570.4 
1758.6 
1954.1 
2157.1 
2367.4 
2585.2 
2810.4 
3043.0 
3283.0 
3530.4 
3785.2 
4047.4 
4317.1 
4594.1 
4878.6 
5170.4 
5469 
5776.3 
6190.4 
6411.9 
6740.8 
7077.1 
7420.8 
7771.9 
8130. 
8496.3 
8869.7 
9250.4 
9638.6 
10034 
10437 
10847 
11265 
11690 
12123 
12563 
13010 
13465 
1'8927 
14397 
14874 
15359 
15850 
16350 
16856 
17370 
17892 
18421 
18957 
195G1 
20052 



7.41 

14.81 

22.22 

29.63 

37.04 

44.44 

51.85 

59.26 

66.67 

74.07 

81.48 

88.89 

96.30 

103.70 

111.11 

118.52 

125.93 

133.33 

140.74 

148.15 

155.56 

162.96 

170.37 

177.78 

185.19 

192.59 

200.00 

207.41 

214.81 

222.22 

229.63 

237.04 

244.44 

251.85 

259.26 

266.67 

274.07 

281.48 

288.89 

296.30 

303.70 

311.11 

318.52 

325.93 

333.33 

340.74 

348.15 

355.56 

362.96 

370.37 

377.78 

385.19 

392.59 

400.00 

407.41 

414.81 

422.22 

429.63 

437.04 

444.44 



P. P. 

7.41 
.74 
1.48 
2.22 
2.96 
3.70 
4.44 
5.19 
5.93 
6.67 



EARTHWORK TABLES— LEVEL SECTIONS. 



1051 



38. — Level Sections (Earthwork); Height., 0-60 Ft. 
Base op Roadway, 28 Ft. Side Slopes, l}^ to 1. 
Note. — ^The last two columns enable us to use any other base than 28 ft.: 
Ex.— Given height, 33.6 ft.; roadway 30 ft. Then we have, 9756.4 + 
(244.44+4.44) = 10005.3 cu. yds. (For Ht. >60ft., see Tables 24, 41.) 

[Cu. Yds. per 100-Ft. Station.] 



























Width 






Ht. 


.0 


.1 


.2 


.3 


.4 


.5 


.6 


.7 


.8 


.9 


of 2 Ft. 






Ft. 






















Cu.Yds 




52" 







10.4 


21.0 


31.6 


42.4 


53.2 


64.2 


75.3 


86.5 


97.9 






S . 

■4JtH 


1 


*i69!3 


120 8 


132.5 


144.3 


156.1 


168.1 


180.2 


192.4 


204.8 


217.2 


"■7!4i 




2 


229.6 


242.3 


255.0 


267.9 


280.9 


294.0 


307.2 


320.5 


334.0 


347.5 


14.81 




•" o 


3 


361.2 


374.9 


388.8 


402.8 


416.9 


431.1 


445.4 


459.9 


474.4 


489.1 


22.22 




1^ 


4 


503.7 


518.6 


533.6 


548.6 


563.9 


579.3 


594.7 


610.2 


625.6 


641.6 


29.63 




5 


657.5 


673.4 


689.5 


705.7 


722.1 


738.5 


755.0 


771.7 


788.4 


805.3 


37.04 




C4 in 


6 


822.2 


839.3 


856.5 


873.8 


891.2 


908.8 


926.4 


944.2 


962.0 


980.0 


44.44 


^ 


'*"! 


7 


998.1 


1016.4 


1034.7 


1053.1 


1071.6 


1090.3 


1109.0 


1127.9 


1146.9 


1166.0 


51.85 


•a 




8 


1185.2 


1204.5 


1223.9 


1243.5 


1263.1 


1282.9 


1302.7 


1322.7 


1342.8 


1363.0 


59.26 


•s 


9 


1383.3 


1403.8 


1424.3 


1444.9 


1465.7 


1486.6 


1507.6 


1528.6 


1549.8 


1571.2 


66.67 


w 


10 


1592.6 


1614.1 


1635.8 


1657.5 


1679.4 


1701.4 


1723.5 


1745.7 


1768.0 


1790.4 


74.07 


d 


11 


1813.0 


1835.6 


1858.4 


1881.2 


1904.2 


1927.3 


1950.5 


1973.8 


1997.3 


2020.8 


81.48 


4^ 


^rt 


12 


2044.4 


2068.2 


2092.1 


2116.1 


2040 1 


2164.3 


2188.7 


2213.1 


2237.6 


2262.3 


88.89 


« 


^i 


13 


2287.0 


2311.9 


2336.9 


2362.0 


2387.2 


2412.5 


2437.9 


2463.5 


2489.1 


2514.9 


96.30 


£ 


14 


2540.7 


2566.7 


2592.8 


2619.0 


2645.3 


2671.8 


2698.3 


2724.9 


2751.7 


2778.6 


103.70 


o 

to 




15 


2805.6 


2832.6 


2859.9 


2887.2 


2914.6 


2942.1 


2969.8 


2997.5 


3025.4 


3053.4 


111.11 


16 


3081.5 


3109.7 


3138.0 


3166.4 


3195.0 


3223.6 


3252.4 


3281.2 


3310.2 


3339.3 


118.52 


1 


p. 


17 


3368.5 


3397.8 


3427.3 


3456.8 


3486.4 


3516.2 


3546.1 


3576.1 


3606.1 


3636.4 


125.93 


?. 


18 


3666.7 


3697.1 


3727.6 


3758.3 


3789.0 


3819.9 


3850.9 


3882.0 


3913.2 


3944.5 


133.33 




19 


3975.9 


4007.5 


4039.1 


4070.9 


4102.7 


4134.7 


4166.8 


4199.0 


4231.3 


4263.8 


140.74 


20 


4296.3 


4328.9 


4361.7 


4394.6 


4427.6 


4460.6 


4493.9 


4527.2 


4560.6 


4594.1 


148.15 


a 


21 


4627.8 


4661.5 


4695.4 


4729.4 


4763.5 


4797.7 


4832.0 


4866.4 


4901.0 


4935.6 


155.56 


•d 


^^ 


22 


4970.4 


5005.2 


5040.2 


5075.3 


5110.5 


5145.8 


5181.3 


5216.8 


5252.4 


5288.2 


162.96 


5 


S« 


23 


5324.1 


5360.1 


5390.1 


5432.4 


5468.7 


5505.1 


5541.6 


5578.3 


5615.0 


5651.9 


170.37 


rt-ra 


24 


5688.9 


5726.0 


5763.2 


5800.5 


5837.9 


5875.5 


5913.1 


5950.9 


5988.7 


6026.7 


177.78 






25 


6064.8 


6103.0 


6141.3 


6179.8 


6218.3 


6256.9 


6295.7 


6334.6 


6373.6 


6412.6 


185.19 


P. P. 


26 
27 


6451.9 
6850.0 


6491.2 
6890.4 


6530.G 
6931.0 


6570. 1 
6971.6 


6609.8 
7012.4 


6649.5 
7053.2 


6689.4 
7094.2 


6729.4 
7135.3 


6769.5 
7176.5 


6809.7 
7217.8 


192.59 
200.00 


7.41 


1 .74 




28 


7259.3 


7300.8 


7342.4 


7384.2 


7426.1 


7463.1 


7510.1 


7552.4 


7594.7 


7637.1 


207.41 


2 1.48 


29 


7679.6 


7722.3 


7765.0 


7807.9 


7850.9 


7894.0 


7937.2 


7980.5 


8023.9 


8067.5 


214.81 


3 2.22 


30 


8111.1 


8154.9 


8198.7 


8242.7 
8688.^ 


8286.8 


8331.0 


8375.3 


8419.8 


8464.3 


8508.9 


222.22 


4 2.96 


^T-( 


31 


8553.7 


8598.6 


8643.6 


8733.9 


8779.2 


8824.6 


8870. 1 


8915.8 


8961.5 


229.63 


5 3.70 


W (U 


32 


9007.4 


9053.4 


9099.5 


9145.7 


9192.0 


9238.4 


9285.0 


9331.6 


9378.4 


9425.2 


237.04 


6 4.44 


^ 


33 


9472.2 


9519.3 


9566.5 


9613.8 


9661.3 


9708.8 


9756.4 


9804.2 


9852.1 


9900.1 


244.44 


7 5.19 


34 


9948.1 


9996.4 


10045 


10093 


10142 


10190 


10239 


10288 


10337 


10386 


251.85 


8 5.93 


.s « 


35 


10435 


10484 


10534 


10583 


10630 


10683 


10732 


10782 


10832 


10882 


259.26 


9 6.67 


^^ 


36 


10933 


10983 


11034 


11084 


11135 


11186 


11237 


11288 


11339 


11391 


266.67 






37 


11443 


11494 


11546 


11598 


11649 


11701 


11753 


11806 


11858 


11910 


274.07 


>% 


>t<tJ 


38 


11963 


12016 


12068 


12121 


12174 


12227 


12281 


12334 


12387 


12441 


281.48 


"3 


. ^4 


39 


12494 


12548 


12602 


12656 


12710 


12764 


12819 


12873 


12928 


12982 


288.89 


o 


^2 


40 


13037 


13092 


13147 


13202 


13257 


13312 


13368 


13423 


13479 


13535 


296.30 


d 


oS 


41 


13591 


13647 


13703 


13759 


13815 


13872 


13928 


13985 


14042 


14099 


303.70 


a 


»o^ 


42 


14156 


14213 


14270 


14327 


14385 


14442 


14500 


14558 


14615 


14673 


311.11 


d 


S 


43 


14731 


14790 


14848 


14906 


14965 


15024 


15082 


15141 


15200 


15259 


318.52 


8 




44 


15318 


15378 


15437 


15497 


15556 


15616 


15676 


15736 


15796 


15856 


325.93 


bo 


oTm 


45 


15917 


15977 


16038 


16098 


16159 


16220 


16281 


16342 


16403 


16465 


333.33 


d 




46 


16526 


16587 


16649 


16711 


16773 


16835 


16897 


16959 


17021 


17084 


340.74 


o 


47 


17146 


17209 


17272 


17335 


17398 


17461 


17524 


17587 


17651 


17714 


348.15 


A 


C/3 iH 

o 


48 


17778 


17842 


17905 


17969 


18033 


18098 


18162 


18226 


18291 


18356 


355.56 


oT- 


49 


18420 


18485 


18550 


18615 


18680 


18746 


18811 


18877 


18942 


19008 


362.96 


d):?: 


50 


19074 


19140 


19206 


19272 


19339 


19405 


19472 


19538 


19605 


19672 


370.37 


,d 

4^ 


W^ 


51 


19739 


19806 


19873 


19940 


20008 


20075 


20143 


20211 


20279 


20347 


377.78 


■^ 


r— 1 »^_, 


52 


20415 


20483 


20551 


20620 


20688 


20757 


20826 


20894 


20963 


21032 


385.19 


a> 


c« u. 


53 


21102 


21171 


21241 


21310 


21380 


21450 


21519 


21589 


21659 


21730 


392.59 


S 


-5° 


54 


21800 


21870 


21941 


22012 


22082 


22153 


22224 


22295 


22366 


22438 


400.00 




55 


22509 


22581 


22652 


22724 


22796 


22868 


22940 


23012 


23085 


23157 


407.41 




56 


23230 


23302 


23375 


23448 


23521 


23594 


23667 


23741 


23814 


23888 


414.81 




hi: 


57 


23961 


24035 


24109 


24183 


24257 


24331 


24405 


24480 


24554 


24629 


422.22 




58 


24704 


24779 


24854 


24929 


25004 


25079 


25155 


25230 


25306 


25381 


429.63 




•J^ 


59 


25457 


25533 


25609 


25686 


25762 


25838 


25915 


25992 


26068 


26145 


437.04 




::} 


60 


26222 


26299 


26376 


26454 


26531 


26609 


26686 


26764 


26842 


26920 


444.44 





1052 



59.^RAILROADS. 



39. — ^Level Sections (Earthwork); Height, 0-60 Ft. 

Base of Roadway, 30 Ft. Side Slopes, 1 to 1. 

Note. — ^The last two columns enable us to use any other base than 30 ft.: 
Ex. — Given height, 41.1 ft.; roadway 32 ft. Then we have, 10823+ 
(303.70+ 0.74) = 11127 cu. yds. (For Ht. >60ft., see Tables 28, 40.) 

[Cu. Yds. per 100-Ft. Station.] 



























Width 






Ht. 


.0 


.1 


.2 


.3 


.4 


.5 


.6 


.7 


.8 


.9 


of 2 Ft. 






Ft. 






















Cu.Yds 




?2 







11.1 


22.4 


33.7 


45.0 


56.5 


68.0 


79.6 


91.3 


103.0 






4-> 


1 


"iiils 


126.7 


138.7 


150.7 


162.8 


175.0 


187.3 


199.6 


212.0 


224.5 


'*'7!4i 




2 


237.0 


249.7 


262.4 


275.1 


288.0 


300.9 


313.9 


327.0 


340.1 


353.4 


14.81 






3 


366.7 


380.0 


393.5 


407.0 


420.6 


434.3 


448.0 


461.8 


475.7 


489.7 


22.22 




k""" 


4 


503.7 


517,8 


532.0 


546.3 


560.6 


575.0 


589.5 


604.0 


618.7 


633.4 


29.63 




So 


5 


648.1 


663.0 


677.9 


692.9 


708.0 


723.1 


738.4 


753.7 


769.0 


784.5 


37.04 




^:^ 


6 


800.0 


815.6 


831.3 


847.0 


862.8 


878.7 


894.7 


910.7 


926.8 


943.0 


44.44 


+5 


7 


95&.3 


975.6 


992.0 


1008.5 


1025.0 


1041.7 


1058.4 


1075.1 


1092.0 


1108.9 


51.85 


■E 


^o 


8 


1125.9 


1143.0 


1160.1 


1177.4 


1194.7 


1212.0 


1229.5 


1247.0 


1264.6 


1282.3 


59.26 


tUD 

1 


Ba 


9 


1300.0 


1317.8 


1335.7 


1353.7 


1371.7 


1389.8 


1408.0 


1426.3 


1444.6 


1463.0 


66.67 


^^ 


10 


1481.5 


1500.0 


1518.7 


1537.4 


1556.1 


1575.0 


1593.9 


1612.9 


1632.0 


1651.1 


74.07 




11 


1670.4 


1689.7 


1709.0 


1728.5 


1748.0 


1767.6 


1787.3 


1807.0 


1826.8 


1846.7 


81.48 


^1 


12 


1866.7 


1886.7 


1906.8 


1927.0 


1947.3 


1967.6 


1988.0 


2008.5 


2029.0 


2049.7 


88.89 


i 


13 


2070.4 


2091.1 


2112.0 


2132.9 


2153.9 


2175.0 


2196.1 


2217.4 


2238.7 


2260.0 


96.30 


pC4 


^d 


14 


2281.5 


2303.0 


2324.6 


2346.3 


2368.0 


2389.8 


2411.7 


2433.7 


2455.7 


2477.8 


103.70 


15 


2500.0 


2522.3 


2544.6 


2567.0 


2589.5 


2612.0 


2634.7 


2657.4 


2680.1 


2703.0 


111.11 


O 


12 


16 


2725.9 


2748.9 


2772.0 


2795.1 


2818.4 


2841.7 


2865.0 


2888.5 


2912.0 


2935.6 


118.52 


3 


17 


2959.3 


2983.0 


3006.8 


3030.7 


3054.7 


3078.7 


3102.8 


3127.0 


3151.3 


3175.6 


125.93 


4^ 

d 


!&*o 


18 


3200.0 


3224.5 


3249.0 


3273.7 


3298.4 


3323.1 


3348.0 


3372.9 


3397.9 


3423.0 


133.33 


5 ^ 


19 


3448.1 


3473.4 


3498.7 


3524.0 


3549.5 


3575.0 


3600.6 


3626.3 


3652.0 


3677.8 


140.74 


H 


ll 


20 


3703.7 


3729.7 


3755.7 


3781.8 


3808.0 


3834.3 


3860.6 


3887.0 


3913.5 


3940.0 


148.15 


g 


21 


3966.7 


3993.4 


4020.1 


4047.0 


4073.9 


4100.9 


4128.0 


4155.1 


4182.4 


4209.7 


156.56 


*.* 


^ ^ 


22 


4237.0 


4264.5 


4292.0 


4319.6 


4347.3 


4375.0 


4402.8 


4430.7 


4458.7 


4486.7 


162.96 




£- 


23 


4514.8 


4543.0 


4571.3 


4599.6 


4628.0 


4656.5 


4685.0 


4713.7 


4742.4 


4771.1 


170.37 


< 


'c3 M 


24 


4800.0 


4828.9 


4857.9 


4887.0 


4916.1 


4945.4 


4974.7 


5004.0 


5033.5 


5063.0 


177.78 




y^"^ 


25 


5092.6 


5122.3 


5152.0 


5181.8 


5211.7 


5241.7 


5271.7 


5301.8 


5332.0 


5362.3 


185.19 


P. P. 


S*^ 


26 


5392.6 


5423.0 


5453.5 


5484.0 


5514.7 


5545.4 


5576.1 


5607.0 


5637.9 


5668.9 


192.59 


7.41 


Sg 


27 


5700.0 


5731.1 


5762.4 


5793.7 


5825.0 


5856.5 


5888.0 


5919.6 


5951.3 


5983.0 


200.00 


1 


.74 


28 


6014.8 


6046.7 


6078.7 


6110.7 


6142.8 


6175.0 


6207.3 


6239.6 


6272.0 


6304.5 


207.41 


2 


1.48 


^o 


29 


6337.0 


6369.7 


6402.4 


6435.1 


6468.0 


6500.9 


6533.9 


6567.0 


6600.1 


6633.4 


214.81 


3 


2.22 




30 


6666.7 


6700.0 


6733.5 


6767.0 


6800.6 


6834.3 


6868.0 


6901.8 


6935.7 


6969.7 


222.22 


4 


2.96 


31 


7003.7 


7037.8 


7072.0 


7106.3 


7140.6 


7175.0 


7209.5 


7244.0 


7278.7 


7313.4 


229.63 


5 


3.70 


■^i 


32 


7348.1 


7383.0 


7417.9 


7452.9 


7488.0 


7523.1 


7558.4 


7593.7 


7629.0 


7664.5 


237.04 


6 


4.44 


33 


7700.0 


7735.6 


7771.3 


7807.0 


7842.8 


7878.7 


7914.7 


7950.7 


7986.8 


8023.0 


244.44 


7 


5.19 


:- 


34 


8059.3 


8095.6 


8132.0 


8168.5 


8205.0 


8241.7 


8278.4 


8315.1 


8352.0 


8388.9 


251.85 


8 


5.93 


35 


8425.9 


8463.0 


8500.1 


8537.4 


8574.7 


8612.0 


8649.5 


8687.0 


8724.6 


8762.3 


259.26 


9 


6.67 




36 


8800.0 


8837.8 


8875.7 


8913.7 


8951.7 


8989.8 


9028.0 


9066.3 


9104.6 


9143.0 


266.67 




37 


9181.5 


9220.0 


9258.7 


9297.4 


9336.1 


9375.0 


9413.9 


9452.9 


9492.0 


9531.1 


274.07 


>» 


38 


9570.4 


9609.7 


9649.0 


9688.5 


9728.0 


9767.6 


9807.3 


9847.0 


9886.8 


9926.7 


281.48 


•a 


39 


9966.7 


10007 


10047 


10087 


10127 


10168 


10208 


10248 


10289 


10330 


288.89 


o 


do 


40 


10370 


10411 


10452 


10493 


10534 


10575 


10616 


10657 


10699 


10740 


296.30 


d 


Oeo 


41 


10781 


10823 


10865 


10906 


10948 


10990 


11032 


11074 


11116 


11158 


303.70 


B 


'Om-i 


42 


11200 


11242 


11285 


11327 


11369 


11412 


11455 


11497 


11540 


11583 


311.11 


1 


43 


11626 


11669 


11712 


11755 


11798 


11842 


11885 


11928 


11972 


12016 


318.52 




44 


12059 


12103 


12147 


12191 


12235 


12279 


12323 


12367 


12411 


12456 


325.93 


bfi 


o.^ 


45 


12500 


12544 


12589 


12634 


12678 


12723 


12768 


12813 


12858 


12903 


333.33 


o 


a <u 


46 


12948 


12993 


13039 


13084 


13129 


13175 


13221 


13266 


13312 


13358 


340.74 


•5 


0.^:5 


47 


13404 


13450 


13496 


13542 


13588 


13634 


13681 


13727 


13773 


13820 


348.15 


u 


U:^ 


48 


13867 


13913 


13960 


14007 


14054 


14101 


14148 


14195 


14242 


14290 


355.56 




(U"*^ 


49 


14337 


14384 


14432 


14480 


14527 


14575 


14623 


14671 


14719 


14767 


362.96 




50 


14815 


14863 


14911 


14960 


15008 


15056 


15105 


15154 


15202 


15251 


370.37 


5 


51 


15300 


15349 


15398 


15447 


15496 


15545 


15595 


15644 


15693 


15743 


377.78 


■^ 




52 


15793 


15842 


15892 


15942 


15992 


16042 


16092 


16142 


16192 


16242 


385.19 


1 


53 


16293 


16343 


16393 


16444 


16495 


16545 


16596 


16647 


16698 


16749 


392.59 


,C3 O 

4J +J 


54 


16800 


16851 


16902 


16954 


17005 


17056 


17108 


17160 


17211 


17263 


400.00 


«b ^ 


55 


17315 


17367 


17419 


17471 


17523 


17575 


17627 


17680 


17732 


17784 


407.41 




1^ 


56 


17837 


17890 


17942 


17995 


18048 


18101 


18154 


18207 


18260 


18313 


414.81 




:§bo 


57 


18367 


18420 


18473 


18527 


18581 


18634 


18688 


18742 


18796 


18850 


422.22 




.s 


58 


18904 


18958 


19012 


19066 


19121 


19175 


19229 


19284 


19339 


19393 


429.63 




*55 


59 


19448 


19503 


19558 


19613 


19668 


19723 


19778 


19834 


19889 


19944 


437.04 




P 


60 


20000 


20056 


20111 


20167 


20223 


20279 


20335 


20391 


20447 


20503 


444.44 





EARTHWORK TABLES— LEVEL SECTIONS. 



1053 



40. — Level Sections (Earthwork); Height, 60-100 Ft. 
Bases of Roadway, 14-30 Ft. Side Slopes, I to 1. 
Note. — ^The last two columns enable us to use any other base than given: 
Ex. — Given height. 71.5 ft.; roadway 32 ft. Then we have, 26879+530 
27409 cu. yds. (See also Table 28.) 

[Cu. Yds- per 100-Ft.. Station.] 





Ht. 

Ft. 


Width of Roadway In Feet. 


Width 
of 2 Ft. 


Width 

of 10 Ft. 


























14 


16 


18 


20 


22 


24 


26 


28 


30 


Cu. Yds 


Cu.Yds 


V 


60 


16444 


16889 


17333 


17778 


18222 


18667 


19111 


19556 


20000 


444.44 


2222.2 


& 


.5 


16694 


17142 


17590 


18038 


18486 


18934 


19382 


19831 


20279 


448.15 


2240.7 


61 


16944 


17396 


17848 


18300 


18752 


19204 


19656 


20107 


20559 


451.85 


2259.3 


% 


.5 


17197 


17653 


18108 


18564 


19019 


19475 


19931 


20386 


20842 


455.56 


2277.8 


62 


17452 


17911 


18370 


18830 


19289 


19748 


20207 


20667 


21126 


459.26 


2296.3 


o 


.5 


17708 


18171 


18634 


19097 


19560 


20023 


20486 


20949 


21412 


462.96 


2314.8 


s 


63 


17967 


18433 


18900 


19367 


19833 


20300 


20767 


21233 


21700 


466.67 


2333.3 




.5 


18227 


18697 


19168 


19638 


20108 


20579 


21049 


21519 


21990 


470.37 


2351.9 


o 


64 


18489 


18963 


19437 


19911 


20385 


20859 


21333 


21807 


22281 


474.07 


2370.4 


.5 


18753 


19230 


19708 


20186 


20664 


21142 


21619 


22097 


22575 


477.78 


2388.9 


M 


65 


19019 


19500 


19981 


20463 


20944 


21426 


21907 


22389 


22870 


481.48 


2407.4 


<o 


.5 


19286 


19771 


20257 


20742 


21227 


21712 


22197 


22682 


23168 


485.19 


2425.9 


^ 


66 


19556 


20044 


20533 


21022 


21511 


22000 


22489 


22978 


23467 


488.89 


2444.4 




.5 


19827 


20319 


20812 


21305 


21797 


22290 


22782 


23275 


23768 


492.59 


2463.0 


>» 

^ 


67 


20100 


20596 


21093 


21589 


22085 


22582 


23078 


23574 


24070 


496.30 


2481.5 


.5 


20375 


20875 


21375 


21875 


22375 


22875 


23375 


23875 


24375 


500.00 


2500.0 




68 


20652 


21156 


21659 


22163 


22667 


23170 


23674 


24178 


24681 


503.70 


2518.5 


.5 


20931 


21438 


21945 


22453 


22960 


23468 


23975 


24482 


24990 


507.41 


2537.0 


5 


69 


21211 


21722 


22233 


22744 


23256 


23767 


24278 


24789 


25300 


511.11 


2555.6 


'•4-3 

T— I 


.5 


2J494 


22008 


22523 


23038 


23553 


24068 


24582 


25097 


25612 


514.81 


2574.1 




70 


21778 


22296 


22815 


23333 


23852 


24370 


24889 


25407 


25926 


518.52 


2592.6 


.5 


22064 


22586 


23108 


23631 


24153 


24675 


25197 


25719 


26242 


522.22 


2611.1 


<u 


71 


22352 


22878 


23404 


23930 


24456 


24982 


25507 


26033 


26559 


525.93 


2629.6 


,Q 


.5 


22642 


23171 


23701 


24231 


24760 


25290 


25819 


26349 


26879 


529.63 


2648.1 


:z! 


72 


22933 


23467 


24000 


24533 


25067 


25600 


26133 


26667 


27200 


533.33 


2666.7 


gJ 


.5 


23227 


23764 


24301 


24838 


25375 


25912 


26449 


26986 


27523 


537. 04 


2685.2 


b 


73. 


23522 


24063 


24604 


25144 


25685 


26226 


26767 


27307 


27848 


540.74 


2703.7 


1 


.5 


23819 


24364 


24908 


25453 


25997 


26542 


27086 


27631 


28175 


544.44 


2722.2 


74 


24119 


24667 


25215 


25763 


26311 


26859 


27407 


27956 


28504 


548. 15 


2740.7 


o 


.5 


24419 


24971 


25523 


26075 


26627 


27179 


27731 


28282 


28834 


551.85 


2759.3 




75 


24722 


25278 


25833 


26389 


26944 


27500 


28056 


28611 


29167 


555.56 


2777.8 


1 


.5 


25027 


25586 


26145 


26705 


27264 


27823 


28382 


28942 


29501 


559.26 


2796.3 




76 


25333 


25896 


26459 


27022 


27585 


28148 


28711 


29274 


29837 


562.96 


2814.8 




.5 


25642 


26208 


26775 


27342 


27908 


28475 


29042 


29608 


30175 


566.67 


2833.3 


77 


25952 


26522 


27093 


27663 


28233 


28804 


29374 


29944 


30515 


570.37 


2851.9 


r* 


.5 


26264 


26838 


27412 


27986 


28560 


29134 


29708 


30282 


30856 


574.07 


2870.4 


.a 


78 


26578 


27156 


27733 


28311 


28889 


29467 


30044 


30622 


31200 


577.78 


2888.9 


CO 


.5 


26894 


27475 


28056 


28638 


29219 


29801 


30382 


30964 


31545 


581.48 


2907.4 


TJ 


79 


27211 


27796 


28381 


28967 


29552 


30137 


30722 


31307 


31893 


585.19 


2925.9 


>< 


.5 


27531 


28119 


28708 


29297 


29886 


30475 


31064 


31653 


32242 


588.89 


2944.4 


3 


80 


27852 


28444 


29037 


29630 


30222 


30815 


31407 


32000 


32593 


592.59 


2963.0 


o 


81 


28500 


29100 


29700 


30300 


30900 


31500 


32100 


32700 


33300 


600.00 


3000.0 




82 


29156 


29763 


30370 


30978 


31585 


32193 


32800 


33407 


34015 


607.41 


3037.0 


83 


29819 


30433 


31048 


31663 


32278 


32893 


33507 


34122 


34737 


614.81 


3074.1 


rt q; 


84 


30489 


31111 


31733 


32356 


32978 


33600 


34222 


34844 


35467 


622.22 


3111.1 


a;5 


85 


31167 


31796 


32426 


33056 


33685 


34315 


34944 


35574 


36204 


629.63 


3148.1 


a^ 


86 


31852 


32489 


33126 


33763 


34400 


35037 


35674 


36311 


36948 


637.04 


3185.2 


^M 


87 


32544 


33189 


33833 


34478 


35122 


35767 


36411 


37056 


37700 


644.44 


3222.2 


C/2 G 


88 


33244 


33896 


34548 


35200 


35852 


36504 


37156 


37807 


38459 


651.85 


3259.3 


cJ?, 


89 


33952 


34611 


35270 


35930 


36589 


37248 


37907 


38567 


39226 


659 26 


3296.3 




90 


34667 


35333 


36000 


36667 


37333 


38000 


38667 


39333 


40000 


666.67 


3333.3 


91 


35389 


36063 


36737 


37411 


38085 


38759 


39433 


40107 


40781 


674.07 


3370.4 


T^a 


92 


36119 


36800 


37481 


38163 


38844 


39526 


40207 


40889 


41570 


681.48 


3407.4 


^0 


93 


36856 


37544 


38233 


38922 


39611 


40300 


40989 


41678 


42367 


688.89 


3444.4 


^+^ 


94 


37600 


38296 


38993 


39689 


40385 


41082 


41778 


42474 


43170 


696.30 


3481.5 


to 


95 


38352 


39056 


39758 


40462 


41166 


41869 


42573 


43277 


43981 


703.70 


3518.5 


96 


39111 


39822 


40533 


41244 


41956 


42667 


43378 


.44089 


44800 


711.11 


3555.6 


1? 


97 


39878 


40596 


41315 


42033 


42752 


43470 


44189 


44907 


45626 


718.52 


3592.6 


98 


40652 


41378 


42104 


42830 


43556 


44282 


45007 


45733 


46459 


725.93 


3629.6 


73 


99 


41433 


42167 


42900 


43633 


44367 


45100 


45833 


46567 


47300 


733.33 


3666.7 


w 


100 


42222 


42963 


43704 


44444 


45185 


45926 


46667 


47407 


48148 


740.74 


3703.7 



1054 



I— RAILROADS. 



41. — ^Level Sections (Earthwork); Height, 60-100 Ft. 
Bases of Roadway, 14-30 Ft. Side Slopes, IH to 1. 
Note. — The last two columns enable us to use any other base than given: 
Ex.— Given height, 68.5 ft.; roadway 32 ft. Then we have, 33679+507 
= 34186 cu. yds. (See also Table 24.) 

[Cu. Yds. per 100-Ft. Station.] 





Ht. 


Width of Roadway In Feet. 


Width 


Width 




Ft. 






of 2 Ft. 


oflOFt. 


























14 


16 


18 


20 


22 


24 


26 


28 


30 


Cu.Yds 


Cu.Yds 


<0 


60 


23111 


23556 


24000 


24444 


24889 


25333 


25778 


26222 


26667 


444.44 


2222.2 


^ 


.5 


23472 


23920 


24368 


24816 


25264 


25713 


26161 


26609 


27057 


448.15 


2240.7 


61 


23835 


24287 


24739 


25191 


25643 


26094 


26546 


26998 


27450 


451.85 


2259.3 




.5 


24201 


24657 


25113 


25568 


26024 


26479 


26935 


27390 


27846 


455.56 


2277.8 


s 


62 


24570 


25030 


25489 


25948 


26407 


26867 


27326 


27785 


28244 


459.26 


2296.3 


.5 


24942 


25405 


25868 


26331 


26794 


27257 


27720 


28183 


28646 


462.96 


2314.8 


% 


63 


25317 


25783 


26250 


26717 


27183 


27650 


28117 


28583 


29050 


466.67 


2333.3 


uZ 


.5 


25694 


26164 


26635 


27105 


27575 


28046 


28516 


28987 


29457 


470.37 


2351.9 


a 


64 


26074 


26548 


27022 


27496 


27970 


28444 


28918 


29393 


29867 


474.07 


2370.4 


.5 


26457 


26935 


27413 


27890 


28368 


28846 


29324 


29801 


30279 


477.78 


2388.9 


65 


26843 


27324 


27806 


28287 


28769 


29250 


29731 


30213 


30694 


481.48 


2407.4 


s 


.5 


27231 


27716 


28201 


28687 


29172 


29657 


30142 


30627 


31113 


485.19 


2425.9* 


4J 


66 


27622 


28111 


28600 


29089 


29578 


30067 


30556 


31044 


31533 


488.89 


2444.4 




.5 


28016 


28509 


29001 


29494 


29987 


30479 


30972 


31464 


31957 


492.59 


2463.0 


^ 


67 


28413 


28909 


29406 


29902 


30398 


30894 


31391 


31887 


32383 


496.30 


2481.5 


.5 


28813 


29313 


29813 


30313 


30813 


31313 


31813 


32313 


32813 


500.00 


2500.0 




68 


29215 


29719 


30222 


30726 


31230 


31733 


32237 


32741 


33244 


503.70 


2518.5 


• rH 


.5 


29620 


30127 


30635 


31142 


31650 


32157 


32664 


33172 


33679 


507.41 


2537.0 


.^ 


69 


30028 


30539 


31050 


31561 


32072 


32583 


33094 


33606 


34117 


511.11 


2555.6 




.5 


30438 


30953 


31468 


31983 


32498 


33012 


33527 


34042 


34557 


514.81 


2574.1 


6 


70 


30852 


31370 


31889 


32407 


32926 


33444 


33963 


34481 


35000 


518.52 


2592.6 


.5 


31268 


31790 


32313 


32835 


33357 


33879 


34401 


34924 


35446 


522.22 


2611.1 


^ 


71 


31687 


32213 


32739 


33265 


33791 


34317 


34843 


35369 


35894 


525.93 


2629.6 


tQ 


.5 


32109 


32638 


33168 


33698 


34227 


34757 


35287 


35816 


36346 


529.63 


2648.1 


r-< 


72 


32533 


33067 


33600 


34133 


34667 


35200 


35733 


36267 


36800 


533.33 


2666.7 


li 


.5 


32961 


33498 


34035 


34572 


35109 


35646 


36183 


36720 


37257 


537.04 


2685.2 


>, 


73 


33391 


33930 


34471 


35012 


35553 


36093 


36634 


37175 


37717 


540.74 


2703.7 


rt 


.5 


33824 


34368 


34912 


35457 


36001 


36546 


37090 


37635 


38179 


544.44 


2722.2 


6 


74 


34259 


34807 


35356 


35904 


36452 


37000 


37548 


38096 


38644 


548.15 


2740.7 


Q) 


.5 


34698 


35250 


35801 


36353 


36905 


37457 


38009 


38561 


39113 


551.85 


2759.3 


3 


75 


35139 


35694 


36250 


36806 


37361 


37917 


38472 


39028 


39583 


555.56 


2777.8 


rt 


.5 


35583 


36142 


36701 


37261 


37820 


38379 


38938 


39498 


40057 


559.26 


2796.3 


-M 


76 


36030 


36593 


37156 


37719 


38281 


38844 


39407 


39970 


40533 


562.96 


2814.8 


.^ 


.5 


36479 


37046 


37613 


38179 


38746 


39313 


39879 


40446 


41013 


566.67 


2833.3 




77 


36931 


37502 


38072 


38643 


39213 


39783 


40354 


40924 


41494 


570.37 


2851.9 


.S 


.5 


37387 


37961 


38535 


39109 


39683 


40257 


40831 


41405 


41979 


574.07 


2870.4 


78 


37844 


38422 


39000 


39578 


40156 


40733 


41311 


41889 


42467 


577.78 


2888.9 


w 


.5 


38305 


38887 


39468 


40050 


40631 


41212 


41794 


42375 


42957 


581.48 


2907.4 


'V 


79 


38769 


39354 


39939 


40524 


41109 


41694 


42280 


42865 


43450 


585.19 


2925.9 


>^ 


.5 


39235 


39824 


40412 


41001 


41590 


42179 


42768 


43357 


43946 


588.89 


2944.4 


6 
o 


80 


39704 


40296 


40889 


41481 


42074 


42667 


43259 


43852 


44444 


592.59 


2963.0 


81 


40650 


41250 


41850 


42450 


43050 


43650 


44250 


44850 


45450 


600.00 


3000.0 


82 


41607 


42215 


42822 


43430 


44037 


44644 


45252 


45859 


46467 


607.41 


3037.0 




83 


42576 


43191 


43806 


44420 


45035 


45650 


46265 


46880 


47494 


614.81 


3074.1 


^.^ 


84 


43556 


44178 


44800 


45422 


46044 


46667 


47289 


47911 


48533 


622.22 


3111.1 


(u'-S 


85 


44546 


45176 


45806 


46435 


47065 


47694 


48324 


48954 


49583 


629.63 


3148.1 


a^ 


86 


45548 


46185 


46822 


47459 


48096 


48733 


49370 


50007 


50644 


637.04 


3185.2 


Stc 


87 


46561 


47206 


47850 


48494 


49139 


49783 


50428 


51072 


51717 


644.44 


3222.2 


^.5 


88 


47585 


48237 


48889 


49541 


50193 


50844 


51496 


52148 


52800 


651.85 


3259.3 


aT'S 


89 


48620 


49280 


49939 


50598 


51257 


51917 


52576 


53235 


53894 


659.26 


3296.3 




90 


49667 


50333 


51000 


51667 


52333 


53000 


53667 


54333 


55000 


666.67 


3333.3 


91 


50724 


51398 


52072 


52746 


53420 


54094 


54768 


55443 


56117 


674.07 


3370.4 


92 


51793 


52474 


53156 


53837 


54519 


55200 


55881 


56563 


57244 


681.48 


3407.4 


ct3 O 


93 


52872 


53561 


54250 


54939 


55628 


56317 


57006 


57694 


58383 


688.89 


3444.4 


,a -^ 


94 


53963 


54659 


55356 


56052 


56748 


57444 


58141 


58837 


59533 


696.30 


3481.5 


o o 


95 


55065 


55769 


56472 


57176 


57880 


58583 


59287 


59991 


60695 


703.70 


3518.6 


96 


56178 


56889 


57600 


58311 


59022 


59733 


60444 


61156 


61867 


711.11 


3555.6 


97 


57302 


58020 


58739 


59457 


60176 


60894 


61613 


62331 


63050 


718.52 


3592.6 


98 


58437 


59163 


59889 


60615 


61341 


62067 


62793 


63519 


64245 


725.93 


3629.6 


TD 


99 


59583 


60317 


61050 


61783 


62517 


63250 


63983 


64717 


65450 


733.33 


3666.7 


'w 


100 


60741 


61481 


62222 


62963 


63704 


64444 


65185 


65926 


66667 


740.74 


3703.7 



SLOPE-STAKING. PRISMOIDAL FORMULA. 



1055 



£iff¥.3i_ 



7'-*Grade(5ubgni(k) 




Slope-Staking is one of the first operations after the location has been 
adopted and filed with the proper authorities to secure condemnation rights. 
It consists in setting slope-stakes at 
points where the side slopes of cut and fill 
intersect the ground line; and embraces 
also, in its widest sense, intermediate 
"cross-sectioning" as shown in Fig. 19. 
The illustration shows the "grade rod" 
method, which is considered the best. 
The "grade rod" is the difference in ele- 
vation between the height of instru- Fig. 19. 
ment and the top of fill or bottom of cut, for the station at which 
the slope stakes are to be set ; and is used directly with ground-rod readings. 
Thus, for the left-hand slope stake, 10.24- 2.3= 12. 5 = distance helow top of 
fill, and is marked —12.5. The "distance out" (from center line stake) 

12 5 

corresponding to — 12.5 is 12. 5X 1^+ 7.0=25.8. Hence ^- ' is the posi- 

tion of the slope stake. Sometimes two or three trials have to be made to 
select the point which will give the proper relation between elevation, and 
distance from center. 

The field notes are kept as shown at the bottom of sketch. Fig. 19; 
they are on the right-hand page of the note book, under Left - Center - Right. 
On the left-hand page are Station, Grade Elevation, -{-S., H. L, —S., B. M. and 
Grade Rod, if the complete records of bench marks and turning points are 
kept.* Sometimes, however, the records of the turnings are kept on loose 
sheets and thrown away and the balance of the above notes, including the 
cross-section notes, are all on the left-hand page, with an added column 
for Ground Elevation; the ^ right-hand page being reserved for office 
calculations of quantities in excavation (including solid rock, loose 
rock and earth) and embankment. The first named method is the best as 
there is a complete record of all field operations. The office copy may be 
in the form last mentioned ; and it is best to copy the field notes every day 
for office record. 

Earthwork Computation from cross-section notes as in Fig. 19, is 
made by cutting the figure up into triangles, rectangles and trapezoids, 
that can be calculated directly from the field notes. Thus, the area of the 

right-hand half of the figure = i|^X (26.2- 7.0) + 7.0 X-^^^^^^^^^. In this 

case note that the intermediate cross-section is taken 7.0 ft. out from 
center, one-half width of roadbed, a wise thing to do where practicable 
(provided an intermediate elevation is necessary) as it simplifies office 
calculation. On the left-hand side this was not done and we have for area 

of left-half of figure: ll.OX ^iM+i^^^ +14.8;x (^^^^y^^) - 12.5X 

'25.8- 7.0> 



.(^^^T 



The deduction is for the triangle outside the slope at T, as 

it was included in the previous quantities. Fig. 17, page 1016, presents the 
simplest form of sketch for computation and this will obtain when only the 
center cut or fill is given in addition to the slope-stake notes. Thus, area 

a + h==^{D-^d),andareac-\-d==^{H+h). 

Such a section is called a "three-level" 
section. 

Tlie Prism oidal Formula and Pris=» 
moidal Correction Formula are used for 
the accurate determination of the vol- 
umes of prismoids, where the method 
of "end areas" will not suffice. The fol- 
lowing discussion is based on the "three- 
level" section, Fig. 20. 




Fig. 20. 



"^^ Station = the number of the station, as 1095+50; Grade Elevation^ 
the elevation of sub-grade, i. e., bottom of cut or top of fill; +5 = back- 
sight, or the rod reading on the bench mark (B. M.) to determine the height 
of instrument (H. /.); — S = fore-sight, or the rod reading to determine the 
elevation of turning point (T. P.) or B. M., from the H. /. 



1056 



I— RAILROADS. 



Let E = area of the end section jE, 



w= " " middle " m, 
L = perp dist between E and e, 
H, Hr, Hi = respective elevations above roadbed at £, 
h, h,, h = " " " " e, 

Dt, D[ = respective distances out from center at £, 

dry d\ = respective distances out from center at ^, 
d = di + d\, 
W = one-half width of roadbed at £, 
w= " " " e. 

Se = solidity or volume by end areas, 
Vp = volume by prismoidal formula, 
Cp = prismoidal correction volume = ± (Se — Fp ) . 

Then, £=y(D.-f- A) + j(Hr+HO, 

^=Y(c?r + c^i)4--2-(^ + ^i). and since W=w, 

H + h W 

^ = £lp?(D + J)+i| (^Hr + m + hr + h). 

By end areas in cu. ft., 

5 =k(^E + e)=^iHD + hd + W{Hr + Hx + hr + h)^ ..(1) 

By prismoidal formula, in. cu. ft., 

V^=^{E+^m-\-e) = ^[HD + hd+ZW{Hr + H, + hr + h) + 

(H-hh)(D + d)-] (2) 

By prismoidal correction, in cu. ft., 

Cp = (5e- Fp) = -^(HD-hhd-hD-Hd) = ^{H-h)(D-'d) (3) 

By prismoidal correction, in cu. yds. for 100-ft. station, 
C, = i^^(H-h) iD-d)=^^ iH-h)(D-d) 

= 0.'308642(£?-/j) (D-d) (4) 

The correction Cp is to be subtracted if (H — h) {D — d) is positive (usual) . 
The correcton Cp is to be added if (H — h) (D — d) is negative (rare). 

The Prismoidal Correction table on pages 1057-8 is made up from the 
following equivalents, which may be used direct, if desired: 



(H-h)(D-d). 
(Feet.) 


Pris. Cor. Cp 
for 100-Ft. Sta. 
(Cubic Yards.) 


(H-h)(D-d). 
(Feet.) 


Pris. Cor. Cp 
for 100-Ft. Sta. 
(Cubic Yards.) 


1 
2 
3 
4 
5 


.308 642 

.617 284 

.925 926 

1.234 568 

1.543 210 


6 
7 
8 
9 
10 


1.851 852 
2.160 494 
2.469 136 
2.777 778 
3.086 420 



Example. — ^The following cross-sections were taken at stations 1 and 2, 
roadbed 20 ft. wide, and side slopes 1 on 1: 



Sta. 2. 



Sta. 1. 



LEFT. 
-h6.4 



16 

1-8. 



18.7' 



CENTER. 



4.2 



4-5.5 




14.3" 



REMARKS. 



h = L2: ff= 16.4+ 13.1=29 5 



H = 5.5; !)= 18.7+ 14.3=33.0 



Find the quantity of earth to be removed from Station 1 — 2? 

Solution. — By end areas, Se = 491 .11 Cu. Yds. 

Prismoidal correction for (H — h) (D — d), + 4:. 55 = 1 .40 " 

Therefore, by prismoidal formula, quantity = 489 . 71 '* Ans. 



PRISMOIDAL FORMULA AND CORRECTION, 



1057 



42. — -Prismoidal Corrections* 

[Cu. Yds. per 100-Ft. Station.] 



r_ 100 .rr 

•■ 12X27^ 



h) (D-d)l 



iH-h) 

X 
(D-d) 








Tenths 


, 






.0 


.1 


.2 


.3 


.4 


.5 


.6 


.7 


.8 


.9 


0. 




.0309 


.0617 


.0926 


.1235 


.1543 


.1852 


.2160 


.2469 


.2778 


1. 


'!3b86* 


.3395 


.3704 


.4012 


.4321 


.4629 


.4938 


.5247 


.5556 


.5864 


2. 


.6173 


.6481 


.6790 


.7099 


.7407 


.7716 


.8024 


.8333 


.8642 


.8950 


3. 


.9259 


.9568 


.9877 


1.019 


1.049 


1.080 


1.111 


1.142 


1.173 


1.204 


4. 


1.235 


1.265 


1.296 


1.327 


1.358 


1.389 


1.420 


1.451 


1.481 


1.512 


5. 


1.543 


1.574 


1.605 


1.636 


1.666 


1.697 


1.728 


1.759 


1.790 


1.821 


6. 


1.852 


1.883 


1.914 


1.944 


1.975 


2.006 


2.037 


2.068 


2.099 


2.130 


7. 


2.160 


2.191 


2.222 


2.253 


2.284 


2.315 


2.346 


2.377 


2.408 


2.438 


8. 


2.469 


2.500 


2.531 


2.562 


2.593 


2.624 


2.654 


2.685 


2.716 


2.747 


9. 


2.778 


2.809 


2.840 


2.870 


2.901 


2.932 


2.963 


2.994 


3.025 


3.056 


10. 


3.086 


3.117 


3.148 


3.179 


3.210 


3.241 


3.272 


3.303 


3 333 


3.364 




3.395 


3.426 


3.457 


3.488 


3.518 


3.549 


3.580 


3.611 


3.642 


3.673 


12! 


3.704 


3.735 


3.765 


3.796 


3.827 


3.858 


3.889 


3.920 


3.951 


3.981 


13. 


4.012 


4.043 


4.074 


4.105 


4.136 


4.167 


4.197 


4.228 


4.259 


4.290 


14. 


4.321 


4.352 


4.383 


4.413 


4.444 


4.475 


4.516 


4.537 


4.568 


4 599 


15. 


4.629 


4.660 


4.691 


4.722 


4.753 


4.784 


4.815 


4.846 


4.877 


4.907 


16. 


4.938 


4.969 


5.000 


5.031 


5.062 


5.093 


5.123 


5.154 


5.185 


5.216 


17. 


5.247 


5.278 


5.308 


5.339 


5.370 


5.401 


5.432 


5.463 


5.494 


5.524 


18. 


5.556 


5.586 


5.617 


5.648 


5.679 


5.710 


5.740 


5.772 


5.802 


5.833 


i ''' 


5.864 


5.895 


5.926 


5.957 


5.988 


6.019 


6.049 


6.080 


6.111 


6.142 


^ 20. 


6.173 


6.204 


6.235 


6.265 


6.296 


6.327 


6.358 


6.389 


6.420 


6.451 


.S 21. 


6.481 


6.512 


6.543 


6.574 


6.605 


6.636 


6.667 


6.698 


6.728 


6.759 


Z 22. 


6.790 


6.821 


6.852 


6.883 


6.914 


6.944 


6.975 


7.006 


7.037 


7.068 


V^: 


7.099 


7.130 


7.160 


7.191 


7.222 


7.253 


7.284 


7.315 


7.346 


7.377 


7.407 


7.438 


7.469 


7.500 


7.531 


7.562 


7.593 


7.623 


7.654 


7.685 


-g 25. 


7.716 


7.747 


7.778 


7.809 


7.840 


7.870 


7.901 


7. 932 


7.965 


7.994 


«J 26. 


8.024 


8.056 


8.086 


8.117 


8.148 


8.179 


8.210 


8.241 


8.274 


8.302 


Q 27. 


8.333 


8.364 


8.395 


8.426 


8.457 


8.488 


8.519 


8.549 


8.582 


8.611 


- 28. 
^. 29. 

7 30. 


8.642 


8.673 


8.704 


8.735 


8.765 


8.796 


8.827 


8.858 


8.891 


8.920 


8.950 


8.981 


9.012 


9.043 


9.074 


9.105 


9.136 


9.167 


9.200 


9.228 


9.259 


9.290 


9.321 


9.352 


9.383 


9.414 


9.444 


9.475 


9.508 


9.537 


J; 31. 


9.568 


9.599 


9.630 


9.660 


9.691 


9.722 


9.753 


9.784 


9.817 


9.846 


^ 32. 


9.877 


9.907 


9.938 


9.969 


10.00 


10.03 


10.06 


10.09 


10.12 


10.15 


33. 


10.19 


10.22 


10.25 


10.28 


10.31 


10 34 


10.37 


10.40 


10.43 


10.46 


^ 34. 


10.49 


10.52 


10.56 


10.59 


10.62 


10.65 


10.68 


10.71 


10.74 


10.77 


35. 


10.80 


10.83 


10.86 


10.90 


10.93 


10.96 


10.99 


11.02 


11.05 


11.08 


36. 


11.11 


11.14 


11.17 


11.20 


11.23 


11.27 


11.30 


11.33 


11.36 


11.39 


37. 


11.42 


11.45 


11.48 


11.51 


11.54 


11.57 


11.60 


11.64 


11.67 


11.70 


38. 


11.73 


11.76 


11.79 


11.82 


11.85 


11.88 


11.91 


11.94 


11.98 


12.01 


39. 


12.04 


12.07 


12.10 


12.13 


12.16 


12.19 


12.22 


12.25 


12.28 


12.31 


40. 


12.35 


12.38 


12.41 


12.44 


12.47 


12.50 


12.53 


12.56 


12.59 


12.62 


41. 


12.65 


12.69 


12.72 


12.75 


12.78 


12.81 


12.84 


12.87 


12.90 


12.93 


42. 


12.96 


12.99 


13.02 


13.06 


13.09 


13.12 


13.15 


13.18 


13.21 


13.24 


43. 


13.27 


13.30 


13.33 


13.36 


13.40 


13.43 


13.46 


13.49 


13.52 


13.55 


44. 


13.58 


13.61 


13.64 


13.67 


13.70 


13.73 


13.77 


13.80 


13.83 


13.86 


45. 


13.89 


13.92 


13.95 


13.98 


14.01 


14.04 


14.07 


14.10 


14.14 


14.17 


46. 


14.20 


14.23 


14.26 


14.29 


14.32 


14.35 


14.38 


14.41 


14.44 


14.48 


47. 


14.51 


14.54 


14.57 


14.60 


14.63 


14.66 


14.69 


14.72 


14.75 


14.78 


48. 


14.81 


14.85 


14.88 


14.91 


14.94 


14.97 


15.00 


15.03 


15.06 


15.09 


49. 


15.12 


15.15 


15.19 


15.22 


15.25 


15.28 


15.31 


15.34 


15.37 


15.40 


BO. 


15.43 


15.46 


15.49 


15.52 


15.56 


15.59 


15.62 


15.65 


15.68 


15.71 



* Correction to be subtracted when (H — h) (D 
** " added " " 



d) is positive. (Usual.) 
" negative. (Rare.) 



1058 



59.— RAILROADS. 



100 

42. — Prismoidal Corrections* [= (H — h) (D — d)]. — Concluded. 

1^ X ^ 1 

[Cu. Yds. per 100-Ft. Station.] 



X 
iD-d) 








Tenths. 






.0 


.1 


.2 


.3 


.4 


.5 


.6 


.7 


.8 


.9 


50. 


15.43 


15.46 


15.49 


15.52 


15.56 


15.59 


15.62 


15.65 


15 68 


15.71 


51. 


15.74 


15.77 


15.80 


15.83 


15.86 


15.90 


15.93 


15.96 


15 99 


16.02 


52. 


16.05 


16.08 


16.11 


16.14 


16.17 


16.20 


16.23 


16.27 


16.30 


16.33 


53. 


16.36 


16.39 


16.42 


16.45 


16.48 


16.51 


16.54 


16.57 


16 60 


16.64 


54. 


16.67 


16.70 


16.73 


16.76 


16.79 


16.82 


16.85 


16.88 


16.91 


16.94 


55. 


16.98 


17.01 


17.04 


17.07 


17.10 


17.13 


17.16 


17.19 


17,22 


17.25 


56. 


17.28 


17.31 


17 35 


17.38 


17 41 


17.44 


17.47 


17.50 


17.53 


17.56 


57. 


17.59 


17.62 


17.65 


17.69 


17.72 


17.75 


17.78 


17.81 


17.84 


17.87 


58. 


17.90 


17.93 


17.96 


17.99 


18.02 


18.06 


18.09 


18.12 


18.15 


18.18 


59. 


18.21 


18.24 


18 27 


18.30 


18.33 


18.36 


18.40 


18.43 


18.46 


18.49 


60. 


18.52 


18.55 


18.58 


18.61 


18.64 


18.67 


18.70 


18.73 


18.77 


18.80 


61. 


18.83 


18.86 


18.89 


18.92 


18.95 


18.98 


19.01 


19.04 


19.07 


19.11 


62. 


19.14 


19.17 


19.20 


19.23 


19.26 


19.29 


19.32 


19.35 


19.38 


19.41 


63. 


19.44 


19.48 


19.51 


19.54 


19.57 


19.60 


19.63 


19.66 


19.69 


19.72 


64. 


19.75 


19.78 


19.81 


19.85 


19.88 


19.91 


19.94 


19.97 


20.00 


20.03 


65. 


20.06 


20.09 


20.12 


20.15 


20 19 


20.22 


20.25 


20.28 


20.31 


20.34 


66. 


20.37 


20.40 


20.43 


20.46 


20.49 


20.52 


20.56 


20.59 


20.62 


20.65 


67. 


20.68 


20.71 


20.74 


20.77 


20.80 


20 83 


20.86 


20.90 


20.93 


20.96 


. 68. 


20.99 


21.02 


21.05 


21.08 


21.11 


21.14 


21.17 


21.20 


21.23 


21.27 ' 


t 69. 


21.30 


21.33 


21.36 


21.39 


21.42 


21.45 


21.48 


21.51 


21.54 


21.57 


c2 

2 ??: 


21.61 


21.64 


21.67 


21.70 


21.73 


21.76 


21.79 


21.82 


21.85 


21.88 


21.91 


21.94 


21.98 


22.01 


22.04 


22.07 


22.10 


22.13 


22.16 


22.19 


t 72. 


22.22 


22.25 


22.28 


22.31 


22.35 


22.38 


22.41 


22.44 


22.47 


22.50 


cJ 73. 


22.53 


22.56 


22.59 


22.62 


22.65 


22.69 


22.72 


22.75 


22.78 


22.81 


-Q 74. 
§ 76- 


22.84 


22.87 


22.90 


22.95 


22.96 


22.99 


23.02 


23.06 


23.09 


23.12 


23.15 


23.18 


23.21 


23.24 


23.27 


23.30 


23.33 


23.36 


23.40 


23.43 


23.46 


23.49 


23.52 


23.55 


23.58 


23.61 


23.64 


23.67 


23.70 


23.73 


23.77 


23.80 


23.83 


23.86 


23.89 


23.92 


23.95 


23.98 


24.01 


24.04 


►St 78. 


24.07 


24.10 


24.14 


24.17 


24.20 


24.23 


24.26 


24.29 


24.32 


24.35 


80. 


24 38 


24.41 


24.44 


24.48 


24.51 


24.54 


24.57 


24.60 


24.63 


24.66 


24.69 


24.72 


24.75 


24.78 


24.81 


24.85 


24.88 


24.91 


24.94 


24.97 


6 81. 


25.00 


25 03 


25.06 


25.09 


25.12 


25.15 


25.19 


25.22 


25.25 


25.28 


^ 82. 
& 83. 
'^ 84. 


25.31 


25.34 


25.37 


25.40 


25.43 


25.46 


25.49 


25.52 


25.56 


25.59 


25.62 


25.65 


25.68 


25.71 


25.74 


25.77 


25.80 


25.83 


25.86 


25.90 


25.93 


25.96 


25.99 


26.02 


26.05 


26.08 


26.11 


26.14 


26.17 


26.20 


• 85. 


26.23 


26.27 


26.30 


26.33 


26.36 


26.39 


26.42 


26.45 


26.48 


26 51 


86. 


26.54 


26.57 


26.60 


26.64 


26.67 


26.70 


26.73 


26.76 


26.79 


26.82 


87. 


26.85 


26.88 


26 91 


26 94 


26.98 


27.01 


27.04 


27.07 


27.10 


27.13 


88. 


27.16 


27.19 


27.22 


27.25 


27.28 


27.31 


27.35 


27.38 


27.41 


27.44 


89. 


27.47 


27.50 


27.53 


27.56 


27.59 


27.62 


27.65 


27.69 


27.72 


27.75 


90. 


27.78 


27.81 


27.84 


27.87 


27.90 


27.93 


27.96 


27.99 


28.02 


28.06 


91. 


28.09 


28.12 


28.15 


28.18 


28.21 


28.24 


28.27 


28.30 


28.33 


28.36 


92. 


28.40 


28.43 


28.46 


28.49 


28.52 


28.55 


28.58 


28.61 


28.64 


28.67 


93. 


28.70 


28.73 


28.77 


28.80 


28.83 


28.86 


28.89 


28.92 


28.95 


28.98 


94. 


29.01 


29.04 


29.07 


29.10 


29.14 


29.17 


29.20 


29.23 


29.26 


29. 2C 


95. 


29.32 


29.35 


29.38 


29.41 


29.44 


29.48 


29.51 


29.54 


29.57 


29.60 


96. 


29.63 


29.66 


29.69 


29.72 


29.75 


29.78 


29.81 


29.85 


29 88 


29.91 


97. 


29.94 


29.97 


30.00 


30.03 


30.06 


30.09 


30.12 


30.15 


30.19 


30.22 


98. 


30.25 


30.28 


30.31 


30 34 


30.37 


30.40 


30.43 


30.46 


30.49 


30.52 


99. 


30.56 


30.59 


30.62 


30.65 


30.68 


30.71 


30.74 


30.77 


30.80 


30.83 


100. 


30.86 


30.90 


30.93 


30.96 


30.99 


31.02 


31.05 


30.08 


31.11 


31.14 



* Correction to bt subtracted when {H ■ 
" added 



h) (D — d) is positive. (Usua/.) 
" negative. (Rare.) 




EARTHWORK COMPUTATION. HAUL. ROADBED. 1059 

Correction for Curvature, in earthwork computation, is very often 
neglected. Let A, Fig. 21, be the total area of the cross-section at any 
station on a curve; ^a the horizonta! distance 
from the center of the section to the center 
of gravity of A*; R the radius of the curve. 
Then the correction for curvature may be em- 
bodied by using a new area, Ac=A ( 1 ± -^ ) , 
and maintaining the distance between stations 
as measured on the center line; ^to be added Fig. 21. 

if Ra, the radius to the center of gravity of A, is greater than R, and sub- 
tracted if Ra < R. This is based on the theory that the volume of a solid 
of revolution is equal to the area revolved multiplied by the length of the 
path traced by its center of gravity. From this, we have. Volume for one sta- 

tion = AX100 ^1 ±^') ; but this is clearly equal to 100 Ac =100A (l ±^) . In 

one case the length of station is maintained while the area of the section 
is considered to be increased or decreased; in the other case the reverse is 
assumed. 

"Haul" is a term applied to the average "lead" or horizontal distance 
between the centers of gravity of the same material "in place" and "in fill." 
In Fig. 22, let G. L. be the grade line, H the haul, 
F. H. the free haul, and O. H. the overhaul or paid J^^^ 

haul. Then, overhaul = haul — free haul. The free g -xt^^ ^^^^^_J;:r 
haul may be 500 ft. more or less, according to the 'J>^ J {J^^H 

specifications and contract. The centers of gravity l^"irH-:J J 

are determined after the division lines d are fixed, pj„ 22 

from the estimated quantities, in much the same 

way as the center of gravity of A, Fig. 21, was determined. "Shrink- 
age" is a refinement which may be considered if the quantities are large. 
"Waste" and "borrow" will affect haul, and notes should be made (on the 
profile) of the manner in which all material on the work has been handled. 

Roadbed. — The standard roadbed cross-sections should be adopted 
before the grade line is established, in order to equalize cuts and fills, where 



Fig. 23. 

necessary, during construction. Each road has its own standard or 
rather standards, for the cross-section varies with the amount of traffic, 
height of embankment, whether on tangent or curve, main or branch 
line, etc. Fig. 23 shows about the minimum width of roadway that should 
be considered, namely, 14 ft. and 18 ft. at sub-grade for embankment and 
excavation, respectively. These should be increased rather than diminished, 
especially if on the main line, with high embankment, on a long curve, 
under heavy traffic. The width of roadway in embankment on existing 
roads varies from 12 ft. (minimum for cheap roads) to 20 ft. Allowance 
must be made for shrinkage of embankment as the width of roadbed at sub- 
grade will narrow three times the amount of vertical shrinkage, when side 
slopes are 1^ to 1. For double track lines the roadbed for single track is 
increased by the distance between track centers, say 13 ft. The size of 
ditches in cuts will depend on the length of cut, amount of drainage, and 
slope of ditch. Drain tile should be placed below the frost line. 

* A = Oi + 02 + aa -f- 04; and di, = [a2((ii— ^2) + ozidx -f- d-^) + a^{dx ■\- d^^lA —dx. 



1060 



59— RAILROADS. 



Rails and Fastenings. — Rails are rolled usually in 33-ft. lengths, al- 
though longer rails have lately been introduced on main-line track. Shorter 
lengths, varying by two feet, or even by one foot, are furnished for pre- 
serving "opposite" or "broken" joints in track laid on curves, and for 
switch "leads," etc. Rails are designated by the "weight per yard," as 
60-lb., 70-lb., 80-lb., etc. For instance, a 30-ft. 80-lb rail will weigh 800 lbs. 
In order to decrease the number of existing standard rail sections and the 
consequent expense in rolling, as well as to improve the type and facilitate 
quick delivery from the mills, the Am. Soc. of C. E., in 1893, adopted a 
standard type called the "American Society Standard," shown in Fig. 24 
and in Table 43, following. 

Splices for rail joints have developed from the primitive "chair" 
which simply gave vertical support to the rail joint placed directly over the 
tie, to the "fish plate" (invented in the early 40's) which gave vertical 
stiffness to the joint (see Fig. 27, below), and later to the "angle bar" 
which gives lateral as well as vertical stiffness to the joint. Many improved 
forms of angle bars have appeared on the market from time to time, but 
only those of uniform cross-section, that can be rolled continuously, have 
become standard. Figs. 24, 26 and 28 show standard types. 

Bolts for fastening the splice bars to the rail are often provided with 
lock-nuts to prevent the bolts from working loose, or at least from working 
loose too rapidly. One of the simplest and perhaps most common forms is 
a steel ring of square or hexagonal section cut beveled with sharp comers 
and forming a complete spiral curve of one turn. This is placed on the bolt 
and when the nut is screwed on, the spiral form of the lock-nut is compressed 
to a nearly circular form, and the sharp points pressing against the steel on 
either side prevent the nut and bolt from loosening. Weights and dimen- 
sions of splice bolts are given in Table 43. 

-— C 



Table 43, giving standard 
dimensions of rails, splice 
plates and bolts, will be 
found on the two following 
pages. The accompanying 
illustrations, Figs. 24, 26 and 
27, are also a part of this 
table, as per references given 
therein. Fig. 24 is the Am. 
Soc. C. E. Standard rail sec- 
tion (see pages 1061-2). Figs. 
26 and 27 are Penn. Steel Co. 
standards (see page 1062). 
Figs. 25 and 28, with tables 
of dimensions, will be found 
on pages 1061 and 1062, re- 
spectively. 



i'for35'fo 110* > i I 




Fig. 26. 



RAILS, SPLICES AND BOLTS. 



1061 



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1062 



m.— RAILROADS. 









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RAILS AND FASTENINGS. MIDDLE ORDINATES, 1063 

For Single Track, the number' of 30-ft. rails required per mile o£ track 

= ^^^^^^^ = 352; of 33-ft. rails, 320; of 50-ft. rails, 211.2. The number of s/ior/ 

tons (of 2000 lbs.) per mile of single track= 1.76Xwt. in lbs. of single rail 
per yd.; thus, there are required 176 short tons of 100-lb. rails per mile of 
single track. The number of long tons (of 2240 lbs.) per mile of single track 
= V-Xwt. in lbs. of single rail per yard.; thus, there are required 157^ long 
tons of 100-lb. rails per mile of single track. For 90-lb. rails, mult. 176 and 
157^, respectively, by ^^\ for 80-lb. rails, mult, by x%; etc. 

44. — ^Weight of Rails per Yard Reduced to Tons per Mile op 

Single Track. 

(Short tons at 2000 lbs.; long tons at 2240 lbs.) 



Wt. 


Short 


Long 


Wt. 


Short 


Long 


Wt. 


Short 


Long 


per 


Tons 


Tons 


per 


Tons 


Tons 


per 


Tons 


Tons 


Yard. 


per 


per 


Yard. 


per 


per 


Yard. 


per 


per 


Lbs. 


Mile. 


Mile. 


Lbs. 


Mile. 


Mile. 


Lbs. 


Mile. 


Mile. 


8 


14.08 


12V7 


56 


98.56 


88 


76 


133.76 


1193/7 


12 


21.12 


I8V7 


57 


100.32 


89V7 


78 


137.28 


1222/7 


16 


28.16 


251/7 


60 


105.60 


942/7 


80 


140.80 


1255/7 


25 


44.00 


39V7 


62 


109.12 


973/7 


85 


149.60 


133V7 


30 


62.80 


471/7 


64 


112.64 


IOOV7 


90 


158.40 


I4IV7 


35 


61.60 


55 


65 


114.40 


1021/7 


95 


167.20 


1492/7 


40 


70.40 


626/7 


68 


119.68 


10 66/7 


100 


176.00 


1571/7 


45 


79.20 


70V7 


70 


123.20 


110 


105 


184.80 


165 


60 


88.00 


78V7 


.72 


126.72 


1131/7 


110 


193.60 


1726/7 


52 


91.52 


8IV7 


75 


132.00 


117V7 


120 


211.20 


188V7 



Note. — Values in above table are exact. Fractions of long tons may be 
reduced to decimals of long tons and to pound equivalents as follows: 
^ = 0.14286 l.t. = 320 lbs.; f = 0.28571 l.t. = 640 lbs.; f = 0.42857 l.t. = 960 
lbs.; f= 0.57143 1. 1.= 1280 lbs.; f = 0.71429 l.t. = 1600 lbs.; f = 0.85714 l.t.= 
1920 lbs. 



Middle Ordinates for "bending" (curving) rails, to be laid on curves, 
may be obtained from the following formulas: 



.001L2Z) 
9X5 

.004L2D 
3X5 



(nearly exact) (1) 



(nearly exact) 



(2) 



L2 
M' = -^-^ (practically exact) ; or, M' = 

3L2 
ilf = -r^- (practically exact) ; or, M" = 

in which M' = middle ordinate to the curved rail, in feet; 

M" = middle ordinate to the curved rail, in inches; 

L = length of rail in feet ; 

R = radius of curve in feet ; 

D = degree of curve in degrees and decimals. 
Note that in the above formulas either the radius or the degree of curve 
may be used, both being exact for the fiat curves, say up to 4° or 5°. For a 
30-ft. rail and 10° curve, the "radius" formula gives M' = 0.196 (exact), 
while the "degree of curve" formula gives M'=0.200 (2% large); same 
rail for 20° curve gives M' = 0. 391 and M' = 0.400, respectively, instead of 
0.392, the correct value. For shorter rails the errors decrease, hence it 
is seen that either formula may be used for all practical purposes. 



1064 



.—RAILROADS. 



45. — ^Middle Ordinates in Inc^hes for Curving Rails. 

(Degrees of curve and radii are for 100-ft. chords.) 

For Ordinates in Feet, see Table 46. 

Note. — Ordinates are practically proportional to the square of length of 

rail. Thus, for 60-ft. rail, mult, value for 30, by 4; for 45-ft. rail, mult value 

for 30, by 2>^; for 8-ft. rail, divide value for 16, by 4; etc. 

[Middle Ordinates, in Inches.] 



o 








Length of Rail, or Arc 


. in 


Feet. 






t^ 


Radius. 
Feet. 


















































100 


50 


33 


30 


28 


26 


24 


22 


20 


18 


16 


14 


12 


10 


0.5 


11459.2 


lA 


11 


¥ 


i 


in 


5=4 


f 


5\ 


5^. 


3^2 


52 


bV 


5^ 


5^4 


1. 


5729.7 


2il 






15 

54 


"IQ 


^% 


^? 


32 


5\ 


T6 


1 

32 


5^2 


8^ 


1.5 


3819.8 


311 


II 




T6 


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1^ 


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2864.9 


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32 


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2.5 


2292.0 


611 


11 


M 


54 


If 


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55 


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3. 


1910.1 


711 


Iff 


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& 


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4. 


1432.7 


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211 


Hi 


li 




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si 


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11 


#5 


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4.5 


1273.6 


nil 


2il 


1.% 


lA 


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II 




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5. 


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111 


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54 


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6. 


955.4 


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311 


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49 
54 






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■•■54 


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10. 


573.7 


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611 


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2in 


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64 


ll 


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15. 


383.1 


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Hi 


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Se% 


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H^j 


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16. 


359.3 


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10t6 


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J-64 


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13^2 


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32 


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nil 


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3t^ 


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2in 




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19. 


302.9 


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12f 


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411 


311 


311 


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ill 


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84 


20. 


287.9 


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511 


411 


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21. 


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HI 


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f 


24. 


240.5 


62b«j 


1511 


611 


5f 


411 


4i 


3^1 


33^. 


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2S 


lei 


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25. 


231.0 


641i 


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7i^ 


511 


55^ 


411 


311 


33^. 


2^1 


26^5 


H4 


HI 


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26. 


222.3 


6711 


mi 


73^ 


65^ 


511 


411 


31 


33^1. 


2n 


211 


Iff 


Hi 


H^ 


n 


27. 


214.2 


69ii 


I7ii 


71 


63^ 


511 


41 


4g^4 


3^ 


211 


23^2 




HI 


U\ 


n 


28. 


206.7 


72i| 


18i 


7il- 


m 


511 


4H 


Hi 


3H 


2U 


211 


■•■e' 


lA 


U\ 


f 


29. 


199.7 


7411 


181 


Shk 


611 


5ii 


53^ 


m 


311 


3 


2i^ 


HI 


H^ 


U\ 


30. 


193.2 


77ii 


19f 


811 


611 


65^ 


5i 


4y 


31 


Sii 


2ii 


25^. 


H5 


US 


ti 



Note. — For reduction of inches and fractions to decimals of a foot, see 
Table 10, page 223. No particular refinement is necessary in curving rails: 
ordinates to the nearest ^" are close enough, usually. 

The quarter ordinates are practically three-fourths the middle ordinate. 



ORDINATES FOR CURVING RAILS. 



1065 



46. — ^Middle Ordinates in Feet for Curving Rails. 
(Degrees of curve and radii are for 100-ft. chords.) 
For Ordinates in Inches, see Table 45. 
Note. — Ordinates are practically proportional to the square of length of 
rail. Thus, for 60-ft. rail, mult, value for 30, by 4; for 45-ft. rail, mult, 
value for 30, by 2^; for 8-ft. rail, divide value for 16, by 4, etc. 
[Middle Ordinates, in Feet.] 





Rad- 
ius. 
Feet. 






Length of Rail, or Arc 


, in Feet. 






H> 
























si 


100 


50 


33 


30 


28 


26 


24 


22 


20 


18 


16 


14 


12 


10 


0.5 


11459.2 


.109 


.027 


.012 


.010 


.008 


.006 


.005 


.004 


.004 


.003 


.002 


.002 


.001 


.001 


1. 


5729.7 


.217 


.054 


.024 


.020 


.016 


.013 


.011 


.009 


.008 


.006 


.005 


.004 


.003 


.002 


1.5 


3819.8 


.327 


.082 


.036 


.029 


.026 


.021 


.018 


.016 


.013 


.010 


.008 


.006 


.004 


.003 


2. 


2864.9 


.436 


.109 


.047 


.038 


.034 


.029 


.025 


.021 


.017 


.014 


.011 


.008 


.006 


.004 


2.5 


2292.0 


.546 


.136 


.059 


.049 


.043 


.037 


.031 


.027 


.022 


.018 


.014 


.010 


.007 


.005 


3. 


1910.1 


.655 


.164 


.071 


.058 


.051 


.044 


.037 


.031 


.026 


.022 


.017 


.012 


.009 


.006 


3.5 


1637.3 


.763 


.191 


.083 


.070 


.061 


.052 


.043 


.037 


.031 


.025 


.020 


.015 


.011 


.008 


4. 


1432.7 


.872 


.218 


.095 


.079 


.069 


.060 


.050 


.042 


.035 


.029 


.023 


.018 


.013 


.009 


4.5 


1273.6 


.982 


.245 


.107 


.088 


.077 


.067 


.056 


.047 


.039 


.032 


.026 


.020 


.015 


.010 


5. 


1146.3 


1.090 


.273 


.119 


.099 


.086 


.074 


.063 


.053 


.044 


.035 


.029 


.022 


.016 


.011 


5.6 


1042.1 


1.199 


.300 


.131 


108 


.094 


.082 


.070 


.059 


.048 


.039 


.032 


.024 


.018 


.012 


6. 


955.4 


1.308 


.327 


.142 


.117 


.102 


.088 


.076 


.064 


.052 


.042 


.034 


.026 


.019 


.013 


6.5 


881.9 


1.417 


.354 


.154 


.128 


112 


.097 


.082 


.069 


.057 


.046 


.037 


.028 


.021 


.014 


7. 


819.0 


1.525 


.381 


.166 


.137 


.120 


.104 


.088 


.074 


.061 


.049 


.039 


.030 


.022 


.015 


7.5 


764.5 


1.634 


.408 


.178 


.146 


.127 


.111 


.094 


.079 


.065 


.053 


.042 


.032 


.024 


.016 


8. 


716.8 


1.743 


.436 


.190 


.158 


.137 


.119 


.100 


.085 


.070 


.056 


.045 


.034 


.025 


.017 


8.5 


674.7 


1.851 


.463 


.202 


166 


.145 


.126 


.106 


.090 


.074 


.060 


.048 


.036 


.027 


.018 


9. 


637.3 


1.961 


.490 


.214 


.175 


.153 


.133 


.112 


.095 


.078 


.063 


.050 


.038 


.029 


.019 


9.5 


603.8 


2.069 


517 


.225 


.187 


.163 


.141 


.119 


.101 


.083 


.067 


.054 


.042 


.031 


.021 


10. 


573.7 


2.178 


.545 


.237 


.196 


.171 


.148 


.125 


.106 


.087 


.071 


.057 


.045 


.032 


.022 


11. 


521.7 


2.394 


.598 


.261 


.216 


.188 


.163 


.139 


.117. 


.096 


.078 


.063 


.049 


.036 


.024 


12. 


478.3 


2.611 


.653 


.284 


.236 


.206 


.179 


.151 


.128 


.105 


.085 


.069 


.053 


.039 


.026 


13. 


441.7 


2.828 


.707 


.308 


.254 


.222 


.192 


.163 


.138 


.113 


.092 


.075 


.057 


.042 


.028 


14. 


410.3 


3.043 


.761 


.332 


.275 


.239 


.207 


.175 


.148 


.122 


.099 


.080 


.061 


.045 


.030 


15. 


383.1 


3.258 


.816 


.356 


.295 


.257 


.223 


.188 


.159 


.131 


.106 


.085 


.065 


.049 


.033 


16. 


359.3 


3.474 


.870 


.379 


.313 


.273 


.236 


.200 


.170 


.139 


.113 


.091 


.070 


.052 


.035 


17. 


338.3 


3.688 


.924 


.403 


.333 


.290 


.252 


.213 


.180 


.148 


.120 


.096 


.074 


.055 


.037 


18. 


319.6 


3.903 


.978 


.426 


.351 


.306 


.265 


.225 


.190 


.156 


.127 


.102 


.078 


.058 


.039 


19. 


302.9 


4.117 


1.031 


.450 


.371 


.324 


.280 


.238 


.201 


.165 


.134 


.108 


.082 


.061 


.044 


20. 


287.9 


4.330 


1.085 


.473 


.392 


.341 


.296 


.250 


.212 


.174 


.141 


.114 


.087 


.066 


.041 


21. 


274.4 


4.543 


1.138 


.496 


.410 


.357 


.309 


.262 


.222 


.182 


.148 


.120 


.091 


.069 


.046 


22. 


262.0 


4.756 


1.192 


.520 


.430 


.375 


.325 


.275 


.233 


.191 


.155 


.126 


.096 


.072 


.048 


23 


250. 8 


4.968 


1.245 


.543 


.450 


.390 


.338 


.287 


.243 


.199 


.162 


.131 


.100 


.075 


.050 


24. 


240.5 


5.178 


1.298 


.566 


.469 


.408 


.354 


.299 


.253 


.208 


.169 


.137 


.104 


.078 


.052 


25. 


231.0 


5.390 


1.351 


.589 


.486 


.424 


.367 


.311 


.263 


.216 


.176 


.142 


.108 


.081 


.054 


26. 


222.3 


5.600 


1.404 


.612 


.506 


.441 


.382 


.323 


.274 


.225 


.183 


.148 


.112 


.084 


.056 


27. 


214.2 


5.810 


1.457 


.635 


.524 


.457 


.396 


.335 


.284 


.233 


.190 


.153 


.116 


.087 


.058 


28. 


206.7 


6.019 


1.510 


.658 


.545 


.475 


.411 


.348 


.294 


.242 


.197 


.158 


.120 


.090 


.060 


29. 


199.7 


6.226 


1.563 


.681 


.564 


491 


.424 


.361 


.303 


.250 


.203 


.163 


.124 


.093 


.062 


30. 


193 2 


6.434 


1.615 


.704 


.582 


.507 


.438 


.373 


.313 


.259 


.210 


.168 


.128 


.096 


.064 



Note. — For reduction of decimals of a foot to inches and fractions, see 
Table 10, page 223. No particular refinement is necessary in curving rails: 
ordinates to the nearest M" are close enough, usually. 

The quarter ordinates are practically three-fourths the middle ordinate. 



1060 



\— RAILROADS, 



47. — Chord Lengths op Curved Rails. 

(Degrees of curve and radii are for 100-ft. chords.) 

[Chord Lengths, in Feet.] 





Radius. 
Feet. 


Length of Rail, or Arc, in Feet. 


Po 


100 


50 


33 


30 


26 


22 


18 


14 


10 


2 
4 
6 

8 
10 
12 

14 
16 
18 

20 
22 
24 

26 
28 
30 


2864.9 

1432.7 

955.4 

716.8 
573.7 
478.3 

410.3 
359.3 
319.6 

287.9 
262.0 
240.5 

222.3 
206.7 
193.2 


99.995 
99.980 
99.954 

99.919 
99.871 
99.818 

99.753 
99.677 
99.593 

99.498 
99.394 
99.281 

99.157 
99.027 
98.887 


49.999 
49.997 
49.994 

49.990 
49.983 
49.977 

49.969 
49.960 
49.949 

49.937 
49.924 
49.910 

49.894 
49.878 
49.860 


33.000 
32.999 
32.998 

32.997 
32.996 
32.994 

32.991 
32.988 
32.985 

32.982 
32.978 
32.974 

32.970 
32.965 
32.960 


30.000 
30.000 
29.999 

29.998 
29.997 
29.996 

29.993 
29.991 
29.989 

^9.986 
29.984 
29.981 

29.978 
29.974 
29.970 


26.000 
26.000 
26.000 

25.999 
25.998 
25.997 

25.996 
25.994 
25.993 

25.991 
25.989 
25.987 

25.985 
25.983 
25.980 


22.000 
22.000 
22.000 

22.000 
21.999 
21.998 

21.997 
21.997 
21.996 

21.995 
21.994 
21.993 

21.992 
21.990 
21.988 


18.000 
18.000 
18.000 

18.000 
18.000 
17.999 

17.999 
17.999 
17.998 

17.997 
17.996 
17.996 

17.995 
17.994 
17.993 


14.000 
14.000 
14.000 

14.000 
14.000 
14.000 

13.999 
13.999 
13.999 

13.999 
13.998 
13.998 

13.998 
13.997 
13.997 


IC.OOO 
10.000 
10 000 

10.000 
10.000 
10 000 

10.000 
10.000 
10.000 

9.999 
9.999 
9.999 

9.999 
9.999 
9.999 



Note. — For reduction of decimals of a foot to inches and fractions, 
Table 10, page 223. 



see 



To Find the Degree of Curvature op Laid Track. 
(See Formulas 1 and 2, and Notation, page 1063.) 

On maintenance work it is often necessary to find the degree of curva- 
ture of laid track, on a curve which has been^ more or less shifted and de- 
ranged by the trackmen; and then to run in a regular or a spiral curve 
which will best fit the existing track: so as to require the least possible 
shifting of the latter. 

From the approximate formulas (1) and (2), page 1063, it will be seen 
that there are direct ratios between the degree of ciirve (Z)), the curved 
length of rail (L), and the middle ordinate (M). If D is in degrees, and L 
and M in feet, we have, by transposition, 

^ 45000 M . . ^ . ,„, 

D= Yi — (approximate) (3) 

For a 100-ft. length of rail, L=100, hence, 

D—4i.5 M (approximate) (4) 

=4.58 M (nearly exact) (5) 

For a 30-ft. length of rail, L=30, hence, 

D=50 M (approximate) (6) 

=51 M (nearly exact) (7) 

For a 21' 3" length of rail, 

D=the middle ordinate in hundredths of a foot (8) 



CURVED RAILS--CHORDS, ORDI NATES, ETC. 



1067 



48. — Middle Ordinates for 20- and 30-Ft. Chords and 30-Ft. Arcs. 

Also, Chord Lengths of 30-Ft. Arcs. 

Note. — Radii aregiven in feet and tenths ; other dimensions In feet (') and inches ("). 



o^ 


Rad. 


Middle Ordinates for 


Ch.of 


Rad 


Middle Ordinates for 


Ch.of 


^c^ 
Q^ 


Ft. 




30' Arc. 


Ft. 




30' Arc. 


20'Ch. 


30' Ch. 


30'Arc 


20' Ch. 


30' Ch. 


30' Arc 





Infinite 


O'O '' 


O' " 


0' " 


30' " 


90 


0' 6W 


1' 33^" 


1' 24f" 


29' 10^" 


1 


5729.7 


0' y 


(f i'' 


u 
o 


30' " 


89 


0' 61" 


1' 34" 

y 33^" 


1' 34" 


29' 103^" 


2 


2864.9 


O' 1^'' 


0' A'' 


»+5 


30' " 


88 


0' QW' 


1' 33^" 


29'104 " 


3 


1910.1 


0' A'' 


0' W 


^ 


30' " 


87 


0' QW 


1' 3f" 


1' 34 " 


29' 104" 


4 


1432.7 


0' ^'' 


0' H'' 


0) 

a 


30' " 


86 


0' 7 " 


1' 311'' 


1' 3f" 


29' 103^" 


5 


1146.3 


0' h'' 


0' lA'' 


30' " 


85 


0' 7i^" 


1' 4 " 


1' 3i|// 


29'104" 


6 


955.4 


0' 1'' 


0' If' 


W 


30' " 


84 


0' 7A" 


1' 43^" 


1' 4 " 


29'103^" 


7 


819.0 


0' f '' 


0' If" 


^ 


30' " 


83 


0' 7i" 
0' 7^" 


1' 4f " 
1' 43^" 


1' 44" 
1' 43^" 


29'103^" 


8 


716.8 


0' w 


o' ir' 


+-> 


30' " 


82 


29'10 " 


9 


637.3 


(y w 


0' 2^" 


^ 


30' " 


81 


0' 73^" 


1' 4H" 


1' 4f " 


29' 941" 


10 


573.7 


o'W 


O' 2A" 


ll 


29'llif'' 


80 


0' 7i" 


1' 5 " 


1' 441" 


29' 94" 


11 


621.7 


O'U '' 


0' 2^" 


o 


29'llM" 


79 


0' 7f " 


1' 54" 


1' 53^" 


29' 94 " 


12 


478.3 


O'U'' 


0' 2W' 




29'llH'' 


78 


0' 71" 


1' 5r6" 


1' 54" 


29' 941" 


13 


441.7 


O'lf" 


(y s^" 


29'llif" 


77 


0' 7W 


y 544" 


1' 54" 


29' 9f " 


14 


410.3 


O'W 


(y Si" 


?i2 


29'IW 


76 


0' 71f" 


1' 541" 


1; 544;; 


29' 944" 


15 


383.1 


O'W 


0' 3i" 




29'lli" 


75 


0' 83^" 


1' 6A" 




29' 91" 


16 


359.3 


O'W 


0' 31" 


29'lli" 


74 


0' 84" 


1' 6^" 


y 6^" 


29' 93^" 


17 


338.3 


O'lf '' 


0' 4 " 


|o 


29'IU" 


73 


0' 8i" 


1' 644" 


1' g^// 


29' 94" 


18 


319.6 


o'lr' 


0' 4i " 


'Td -t-' 


29'IU" 
29'IU" 


72 


0' 8^" 


1' 6M" 


J' 644" 


29' 9f " 
29' 93^" 


19 


302.9 


0'2 '' 


r W 




71 


0' 8-" 


1' 73^" 


1' gi|.// 


20 


287.9 


0'2J^'' 


0' 4H'' 


1-1 ^ 

4JM 


29'IW 


70 


0' 8f " 


1' 74" 


1' 73^" 


29' 94" 


21 


274.4 


0'23^'' 


0' 4^" 


A^ 


29'llH'' 


69 


0' 8f " 
0' 8|" 


1' 741" 


V 74" 


29' 93^" 


22 


262.0 


0'2A'' 


O' 5i" 


g.s 


29' Hit'' 


68 


1' 84" 


1' 7f" 


29' 93^" 


23 


250.8 


0'2f '' 
0'2i '' 


0' 51" 


^« 


29' nil" 


67 


0' 9 " 


1' 81" 


1' 83^" 


29' 9 " 


24 


240.5 


O' 5f " 


3§ 


29'llf " 


66 


0' 9i" 


1' 81" 


1' 8f " 


29' 84" 


25 


231.0 


0'2| " 


0' 5W' 




29'llf " 


65 


0' 9i^" 


1' 9 " 


1' gii// 


29' 841" 


/^ 


225 


(y2W' 


0' 6 " 


u 
'd 


29'll4" 


64 


0' 93^" 


1' 9f " 


1' 9 " 


29' 844" 




220 


0'2f '' 


0' 6i" 


29'llii" 


63 


0' 9i^" 


1' 9f" 


1' 93^" 


29' 8f " 


6 


215 


0'2i|'' 


0' 6i" 


.0 


29'llli" 


62 


0' 9f " 


1'104" 


1' 944" 


29' 84" 


12; 


210 


0'2i '' 


O' 6^" 





2 9' 11-4" 


61 


0' 9f " 


1'104" 


I'lO " 


29' 8f " 


<0 


205 


0^211'' 


0' 6^" 


d 


2 9' 11^" 


60 


O'lOji^" 


1'104" 


I'lOf " 


29' 84 " 


3 


200 


0'3 '' 


0' 6f" 


^ 


29'llTi" 


59 


O'lOi" 


1'114" 


I'lOf " 


29' 84" 


^ 


195 


0'3ii^" 


0' QW 




29'11-i" 


58 


O'lO^" 


1'1144" 


l'll4" 


29' 8 " 


190 


0'3t^" 


(y 7i" 


CO 


29'11^" 


57 


O'lOf " 


2' 04" 


I'lW 


29' 74" 




185 


0'3i'' 


0' 7i^" 


^ 


29'llf" 
29'11^" 


56 


O'lOH" 


2' 0^" 


I'lllf" 


29' 744" 


C/3 


180 


0'3^'' 


O' 7i" 


tS 


55 


O'll " 


2' 1 " 


2' Of" 


29' 7 A" 




175 


0'3A'' 


0' 71i" 


to 


29'11^" 


54 


O'lW 


2' 14" 


2' 044" 


29' 7f " 




170 


0'3^'' 


0' 7W 


11 


29' 11^" 


53 


O'lW 


2' 2 " 


2' 1^" 


29' 73^" 




165 


0'3f '' 


(y 81" 


a 


29'lli" 


52 


O'llf " 


2' 24" 


2' 141" 


29' 7 " 




160 


0'3f '' 


0' 8A" 


^ 


29'IW 


51 


O'lli" 


2' 33^" 


2' 24 " 


29' 641" 




155 


0'3|'' 


0' 8ii" 





29' 11 A" 


50 


1' Oi" 


2' 3f" 


2' 244" 


29' 6f " 




150 


0'4 '' 


0' 9 " 


0^ 9 " 


29'llt" 


49 


1' Of" 


2' 44" 


2' 33^" 


29' 6f" 




145 


0'4i '' 


O' 91" 


9A" 


29'11|" 


48 


1' Of" 


2' 44" 


2' 34" 


29' 63^" 




140 


OM,^'' 


0' 9ii" 


0' 9f" 


29' IW' 


47 


1' Off'' 


2' 54" 


2' 44" 


29' 544" 




135 


0'4i^'' 


O'lO,^" 


O'lO " 


29'IU" 
29' IW' 


46 


y w' 


O' fi 3 // 


2' 53^" 


29' 5f " 




130 


0'4f '' 


O'lO^" 
O'lOl" 


O'lOf " 


45 


1' U" 


2' 64'" 


2' 5f" 


29' 5f " 




125 


O'^W' 


O'lOH" 


29'IU" 


44 


1' IW 


2' 7f " 


2' 6f" 


29' 53^" 




120 


0'5 '' 


O'lW 


O'lU'' 


29'IW' 


43 


1' 24" 


2' 8A" 


2' 7^" 


29' 4f " 




115 


0'5i '' 


O'llH'' 


O'llf '' 


29'11 " 


42 


1' 2i" 
1' 24" 


2' 94" 


2' 744" 


29' 44" 




110 


0'5^'' 


1' OA" 


1' Oi" 


29'lOi " 


41 


2'104" 


2' 8T^e" 


29' 4 " 




105 


0'5f '' 


r OH'' 


1' OH" 


29'10^" 


40 


1' 3i" 


2'11 " 


2' 9f" 


29' 3f" 




100 


0^6 '' 


1' W 


1' li" 


29'lOf " 


39 


1' 3f" 


3' " 


2' 10^" 


29' 3^" 




99 


0^6^'' 


V If" 


1; If- 


29'lOf " 


38 


1' 43^" 


3' 1 " 


2'IW 


29' 2f " 




98 


0'6i " 


r li" 


I'll" 


29'lOf " 
29'10^" 


37 


1' 44" 


3' 24" 


3' " 


29' 24" 
29' 144" 




97 


0'6^'' 


V 2 " 


1' ir' 


36 


1' 5 " 


3' 33^" 


3' 044" 




96 


0'6i '' 


r 21" 


1' 2^" 


29'10-5%" 


35 


1' 54" 


3' 44" 


3' 2 " 


29' 13^" 




95 


0'6A'' 


V 2i" 


1' 23^" 


29'10^ " 


34 


1' 63^" 


3' 54" 
3' 73^" 


3 '33^" 


29' Oi^'' 




94 


0'6f '' 


1' 2^" 


1' 2^" 


29'lOi " 


33 


1' 6f " 


3' 4A" 


28'llf " 




93 


0'6i '' 


1' 2f" 


r 2i" 


29'10i^" 


32 


1' 74" 


3' 8H" 


3' 53^" 


28' 1044" 




92 


0'6^" 


r 2f" 


V 2f" 


29'lOt" 


31 


1' 74" 


3' 103^-" 


3' 644" 


28'104" 


91 » 


0'6f ''1' 2|")1' 2ii'"29'10|'" 


30 1' 8^"i4' Oi "'3' 8^"i28' 9iV 



1068 



5Q.— RAILROADS. 



Track Spikes, or railroad spikes as they are commonly called, are square 
in cross-section, with a flared hook head and wedge point. Spikes of the 
same size vary considerably in weight, but the following table shows an 
average, and assumes 10560 spikes per mile of single track. 



49. — Track Spikes. 



Size Meas. 


Average 


Average 


Quantity of Spikes per 


Rails Used. 


Under 


Wt. 


Number 


Mile of Single Track. 


Weight per 


. Head. 


of 100 


per Keg 


Ties 2 feet c. toe. 


Yard. 




Spikes. 


of 


4 spikes per tie. 




Inches. 


Pounds. 


200 Pounds 






Pounds. 


Pounds. 


Kegs. 


5ixf 


66.67 


300 


7040 


35.20 


75 to 100 


5ixA 


53.33 


375 


5630 


28.16 


45 " 75 


5 x-^ 


50.00 


400 


5280 


26.40 


40 " 56 


5 X- 


44.44 


450 


4690 


23.47 


35 " 40 


4Jx^ 


37.74 


530 


3990 


19.93 


30 '♦ 35 


4 x^ 


33.33 


600 


3520 


17.60 


25 " 35 


4Jx^ 


29.41 


680 


3110 


15.53 


20 " 30 


4 x^ 


27.78 


720 


2930 


14.67 


20 •' 30 


S^x^ 


22.22 


900 


2350 


11.73 


16 " 25 


4 xf 


20.00 


1000 


2110 


10.56 


16 " 25 


3ixf 


16.81 


1190 


1770 


8.87 


16 '• 20 


3 x| 


16.13 


1240 


1700 


8.52 


16 " 20 


2ixf 


14.93 


1340 


1580 


7.88 


8 " 16 



Note. — ^There are 4 spikes per tie. The above table assumes the ties to 
be spaced 24'' centers. For other spacing allow as follows: 

For main line, 16 ties to 30-ft. rail, increase values in cols. 4 and 5 by xV« 
.. .. .. J7 .. 33_f^^ .. .. .. .. .. .. ^^ 

" " " 18 •• 33-ft. " " " " " '• tV- 

" sidings. 14 " 30-ft. " decrease " " " *• ^\. 



Rail Joints are almost universally "square." That is, the rails are cut 
off square as they come from the rolls. Such a joint is shown at 5 in Fig. 29. 



m 



M 



3 



Fig. 29. — Square, Miter and Lap Joints. 

In the same Fig. the "miter" joint is shown at M, and the "lap" joint at L. 
The miter- or beveled joint is made by sawing off the rails on a bevel of, 
say, 45° to 60° or 65°. The advantage claimed was that longer rails could be 
used with reduced effect of "hammering" on rail ends by trains, as the open 
joints caused by temperature contraction of rail would be robbed of their 
ill effects if beveled. Steel rails will vary about .0008 of their length under 
a change of temperature of 120° F.; hence 30-ft. rails if laid with closed 
joints at + 100°F. will have between \ and te in. joints when the tempera- 
ture falls to 20° below zero. The miter joint has never come into general 
use. It was formerly used quite extensively on the Lehigh Valley R. R., 
and may now be seen there, but it has generally been replaced with the 
square rail joint. The lap- or scarf joint, L, claims the advantage of the 
miter joint above mentioned; in addition, it is free from danger of any 
projecting point catching a wheel flange, especially if the track is curved. 
If the lap is long the joint is stiffened vertically, a decided advantage. It is 
seldom if ever employed in the United States. 



TRACK SPIKES. RAIL JOINTS, CROSS TIES. 



1069 



"Shims" are pieces of wood or iron inserted at rail joints when track is 
laid, in order to give the proper spacing of joint. The thickness of the 
shim depends upon the temperature. A good rule for thickness of shim 
is the following: 

S = . 00128 L(T-t) 
in which 5 = thickness of shim, in I6ths of an indh\ 
L = length of rail, in feet; 

T = "hottest" temp, to be expected in that locality, in degs. F. 
^ = prevailing temp, when track is laid, in degs. F. 

The Suspended Joint (i. e., where the joint comes between two sup- 
porting ties) has practically superseded the "supported" joint which rests 
directly on a single tie with the usual spacing. But better still, by far, is the 
use of three ties closely spaced at each joint with angle- or splice bars long 
enough (say 3^^ to 4 ft.) to get direct support from the two outside ties. 
Such an arrangement may properly be called a 3-tie "supported" joint. 

Alternate, Staggered or Broken Joints are usually preferred on main 
line to the "opposite" or "even" joints. The advantages of "broken" joints 
are: (1) The intensity of shock due to passing trains is reduced about 50% 
although the number of shocks is doubled; (2) the track can be kept in 
better surface and line on tangents; (3) on curves, one line of rails stiffens 
the other at the joints and aids in preserving uniform curvature even if the 
rails were not curved properly prior to laying. The advantage of "opposite" 
joints is purely one of cost in tracklaying, and hence for second-class yards, 
or for slow train service generally, they may do. 

Cross Ties. — Ordinary track ties are usually 8 or 8i ft. long; bridge ties 
for single track, 12 ft.; and switch ties, up to 15 ft. or more in length. 

Tables 50 and 51, respectively, give the number of cubic feet and feet 
board measure in ties of various dimensions. 

Tables 52 and 53 are bills of switch ties for No. 6 and No. 8 frog, respec- 
tively. To find the bill of material for any other frog: Draw the switch to 
scale, and scale off the lengths of ties. (For turnouts and switches, see 
page 1075.) 



50. — Cubic Feet in Wooden Ties of Various Dimensions. 
[Cubic Feet.] 



Lgth. 


Sectional Dimension, in Inches. 


tl 


6x8 


Gx9 


6x10 


7x8 


7x9 


7x10 


8x10 


8x12 


9x10 


9x 12 


1 


.028 


.031 


.035 


.032 


.036 


.042 


.046 


.056 


.052 


.063 


2 


.056 


.063 


.069 


.065 


.073 


.081 


.093 


.111 


.104 


.125 


3 


.083 


.094 


.104 


.097 


.109 


.122 


.139 


.167 


.156 


.188 


6 


.167 


.188 


.208 


.194 


.219 


.243 


.278 


.333 


.313 


.375 


9 


.250 


.281 


.313 


.292 


.328 


.365 


.417 


.500 


.469 


.563 


1 


.333 


.375 


.417 


.389 


.438 


.486 


.556 


.667 


.625 


.750 


2 


.667 


.750 


.833 


.778 


.875 


.972 


1.111 


1.333 


1.250 


1.500 


3 


1.000 


1.125 


1.250 


1.167 


1.313 


1.458 


1.667 


2.000 


1.875 


2.250 


6 


2.000 


2.250 


2.500 


2.333 


2.625 


2.917 


3.333 


4.000 


3.750 


4.500 


8 


2.667 


3.000 


3.333 


3.111 


3.500 


3.889 


4.444 


5.333 


5.000 


6.000 


8 6 


2.833 


3.188 


3.542 


3.306 


3.719 


4.132 


4.722 


5.667 


5.313 


6.375 


9 


3.000 


3.375 


3.750 


3.500 


3.938 


4.375 


5.000 


6.000 


5.625 


6.750 


10 


3.333 


3.750 


4.167 


3.889 


4.375 


4.861 


5.556 


6.667 


6.250 


7.500 


11 


3.667 


4.125 


4.583 


4.278 


4.813 


5.347 


6.111 


7.333 


6.875 


8.250 


12 


4.000 


4.500 


5.000 


4.667 


5.250 


6.833 


6.667 


8.000 


7.500 


9.000 


13 


4.333 


4.875 


5.417 


5.056 


5.688 


6.319 


7.222 


8.667 


8.125 


9.750 


14 


4.667 


5.250 


5.833 


5.444 


6.125 


6.806 


7.778 


9.333 


8.750 


10.500 


15 


5.000 


5.625 


6.250 


5.833 


6.563 


7.292 


8.333 


10.000 


9.375 


11.250 


16 


5.333 


6.000 


6.667 


6.222 


7.000 


7.778 


8.889 


10.667 


10.000 


12.000 


17 


5.667 


6.375 


7.083 


6.611 


7.438 


8.264 


9.444 


11.333 


10.625 


12.750 


18 


6.000 


6.750 


7.500 


7.000 


7.875 


8.750 


10.000 


12.000 


11.250 


13.500 



Ex.— A tie 7''x9"xl0'6" will contain (4.375+ .219 = )4.594 cu. ft.; and 
1000 such ties at 48 lbs. per cu. ft. will weigh 220,500 lbs. 



1070 



h^.— RAILROADS. 



51. — Feet Board Measure (B. M.) in Wooden Ties op Various 

Dimensions. 

(See also Table 4, Section 20.) 

[Ft. B. M.] 



Lgth. 


Sectional Dimension, in Inches. 




6x8 


6x9 


6x10 


7x8 


7x9 


7x10 


8x10 


8x 12 


9x10 


9 X 12 


1 


.33 


.38 


.42 


.39 


.44 


.49 


.56 


.67 


.63 


.75 


2 


.67 


.75 


.83 


.78 


.88 


.97 


1.11 


1.33 


1.25 


1.50 


3 


1.00 


1.13 


1.25 


1.17 


1.31 


1.46 


1.67 


2.00 


1.88 


2.25 


6 


2.00 


2.25 


2.50 


2.33 


2.63 


2.92 


3.33 


4.00 


3.75 


4.50 


9 


3.00 


3.38 


3.75 


3.50 


3.94 


4.38 


5.00 


6.00 


5.63 


6.75 


1 


4.00 


4.50 


5.00 


4.67 


5.25 


5.83 


6.67 


8.00 


7.50 


9.00 


2 


8.00 


9.00 


10.00 


9.33 


10.50 


11.67 


13.33 


16.00 


15.00 


18.00 


3 


12.00 


13.50 


15.00 


14.00 


15.75 


17.50 


20.00 


24.00 


22.50 


27.00 


6 


24.00 


27.00 


30.00 


28.00 


31.50 


35.00 


40.00 


48.00 


45.00 


54.00 


8 


32.00 


36.00 


40.00 


37.33 


42.00 


46.67 


53.33 


64.00 


60.00 


72.00 


8 6 


34.00 


38.25 


42.50 


39.67 


44.63 


49.58 


56.67 


68.00 


63.75 


76.50 


9 


36.00 


40.50 


45.00 


42.00 


47.25 


52.50 


60.00 


72.00 


67.50 


81.00 


10 


40.00 


45.00 


50.00 


46.67 


52.50 


58.33 


66.67 


80.00 


75.00 


90.00 


11 


44.00 


49.50 


55.00 


51.33 


57.75 


64.17 


73 33 


88.00 


82.50 


99.00 


12 


48.00 


54.00 


60.00 


56.00 


63.00 


70.00 


80.00 


96.00 


90.00 


108.00 


13 


52.00 


58.50 


65.00 


60.67 


68.25 


75.83 


86.67 


104.00 


97.50 


117.00 


14 


56.00 


63.00 


70.00 


65.33 


73.50 


81.67 


93.33 


112.00 


105.00 


126.00 


15 


60.00 


67.50 


75.00 


70.00 


78.75 


87.50 


100.00 


120.00 


112.50 


135.00 


16 


64.00 


72.00 


80.00 


74.67 


84.00 


93.33 


106.67 


128.00 


120.00 


144.00 


17 


68.00 


76.50 


85.00 


79.33 


89.25 


99.17 


113.33 


136.00 


127.50 


153.00 


18 


72.00 


81.00 


90.00 


84.00 


94.50 


105.00 


120.00 


144.00 


135,00 


162.00 



Ex.— A tie 8"xl0"xl2'3" will contain (80.000+ 1.667 = ) 81.667 ft. B. M.; 
and 1000 such ties will contain 81667 ft. B. M. 



52. — Bill of Switch Ties for No. 6 Frog. 
(8"xl0" is a good size; lengths are given to nearest 3".) 



1.2 


Length 


1^ 


Length 




Length 


i"5 


Length 




Length 




Length 


2 
6 
2 
3 


8' 3" 
8' 6" 
8' 9" 
9' 0" 


1 
2 
2 

1 


9' 3" 

9' 6" 

9' 9" 

10' 0" 


1 
1 
1 
1 


10' 3" 
10' 6" 
10' 9" 
11' 0" 


1 

1 
1 
1 


11' 3" 
11' 6" 
11' 9" 
12' 0" 


1 
1 

1 
1 


12' 6" 
12' 9" 
13' 0" 
13' 3" 


1 
3 

1 


13' 6" 
14' 0" 
14' 3" 



Total lin. ft. in above bill, using exact lengths = 380 lin. ft. 
Total lin. ft. in above bill, using 1-ft. lengths = 393 lin. ft. 



53. — Bill of Switch Ties for No. 8 Frog. 
(8"xl0" is a good size; lengths are given to nearest 3".) 



it 

12; p^ 


Lgth. 


6cr 

^p^ 


Lgtlh. 




Lgth. 


^p^ 


Lgth. 


^P^ 


Lgth. 


Ba 
^ ^ 

^P^ 


Lgth. 




Lgth. 


2 
3 
3 
3 


8' 0" 
8' 3" 
8' 6" 
8' 9" 


3 
2 
2 
1 


9' 0" 
9' 3" 
9' 6" 
9' 9" 


2 
1 

2 

1 


10' 0" 
10' 3" 
10' 6" 
10' 9" 


1 
1 

2 

1 


11' 0" 
11' 3" 
11' 6" 
11' 9" 


1 

2 
1 
1 


12' 0" 
12' 3" 
12' 6" 
12' 9" 


1 
1 
2 
3 


13' 0" 
13' 3" 
13' 6" 
14' 0" 


1 

2 


14' 3" 
14' 6* 



Total lin. ft. in above bill, using exact lengths = 484 lin. ft. 
Total lin. ft. in above bill, using 1-ft. lengths = 505 lin. ft. 



CROSS TIES. TIE PLATES. RAIL BRACES. 1071 

"Winter cut" ties are the best, and hewed ties are better than sawed. 
Planing improves sawed ties in shedding water and preventing decay. 
Creosoting lengthens the life of ties, and the creosote oil in ties so preserved 
has a beneficial effect on spikes in preventing rust. 

Wooden ties are the only kind used to any great extent in the United States 
at the present time. The old stone* tie has been abandoned. When the 
wooden tie is supplanted it will probably be by the steel tie, the steel- 
concrete tie, or the steel-paper tie; and then only gradually and on roads of 
the first class. 

Oak ties are the best, and white oak is the best variety as it holds the 
spikes better. The other varieties frequently used are bur oak, post oak, 
chestnut oak and red oak. Together they comprise at least one-half of the 
total number of ties in use. The average life of the best oak tie is about 
eight years, varying inversely with the humidity of the atmosphere, and 
depending upon the amount of traffic, position in the track (whether on 
curves or tangent), etc. Chestnut ties will last about as long as oak but do 
not hold a spike as well. Cedar ties are the longest lived, but they do not 
wear well under heavy traffic unless tie-plates are used, in which case they 
will last about twice as long as oak. Red cedar is not now readily obtainable, 
white cedar being much more plentiful. Next to oak, the various varieties 
of pine furnish most of the ties now being used: Yellow pine, in the South 
and East; loblolly pine in the Southwest; California mountain pine, Oregon 
pine (Douglas fir or Washington pine) , on the Pacific Coast ; Michigan pine 
in the Northwest ; etc. In the New England States, hemlock and spruce are 
much used, although inferior to the Southern yellow pine. Black walnut is 
used in the middle West, and redwood in California. The life of redwood is 
about up to that of cedar if tie-plates are used, the wood being very soft. 

Tie Plates are used on wooden ties, particularly those of soft wood, to 
prevent the rails from cutting into them; otherwise the life of such ties 
would be measured by the amount of heavy traffic passing over them, 
rather than by their resistance to the action of the weather, etc. The life 
of soft ties, then, may be increased: (1) by creosoting, to resist the action 
of the weather, as already explained; (2) by the use of tie plates, to resist 
abrasion. 

The usual construction of tie plates consists simply of flat, plain or 
ribbed plates say 6"x 8'', with bottom lugs or flanges say f" deep, which are 
driven into the tie. The plate is provided with 2, 3 or 4 square holes gaged 
to the rail flange, for spiking. In the earlier patterns there were lugs above 
the plate on one or both sides of the rail flange, but in more recent designs 
these are omitted. Figs. 30 to 33 show types of the Servis, Walhauper, Fox 
and Diamond tie plates; but plain plates (without ribs) are the best. 




^^^ Igr^^i — \r 



Fig. 30. Fig. 31. Fig. 32. Fig. 33. 

« 

The thickness of metal may vary from ^" to |". The cost of tie plates 
is merely nominal, varying from 5 to 15 cents each. They are used generally 
at certain points instead of universally, as at rail joints, where there^ is 
much hammering; on heavy grades where sand is much used ; on expensive 
bridge ties and switch ties; on curves; and in places generally where the 
renewal of ties would be unusually expensive, as at stations and crossings, 
and in tunnels. 

Rail Braces are designed to resist the outward lateral thrust of the rail 
due to passing trains; hence they are placed on the outside of rail, pressed 
firmly against it, and spiked solidly to the ties. The most primitive form of 
rail brace, and one which may be seen in almost any yard, at switches, is 



*Stone ties were tried in the early days on the Boston & Worcester R. R. 
(now part of the B. & A. R. R. system) but were abandoned on account of 
the cost and also because they did not furnish a sufficiently elastic 
track base. 



1072 



59.— RAILROADS. 




a bent fish-plate with one end pressing against the web of the rail, just undei 
the head, and the other end spiked to the tie. Many pat- 
terns are made of cast iron or cast steel. Fig. 34 shows the 
Atkins brace of forged steel; and Fig. 35 shows the Edwards 
brace, being a combination rail brace and tie plate. Rail 
braces are used on curves and at switches, and in general 
where double spiking is insufficient. 

Steel Ties are much used in Europe and somewhat in the Fig. 34. 
United States, but are at present avoided here on accoiint 
of the expense as compared with wooden ties. Those in 
Europe are generally of the trough type. Figs. 36, 37 and 
38 show a special I-beam section, after the original design 
of Mr. C. Buhrer, roadmaster of the L. S. & M. S. Ry., and 
manufactured by the Carnegie Steel Co., for the Bessemer & ^^S- ^^• 
Lake Erie Ry. The tie is 8' 6'' long and w^eighs 19.36 lbs. per ft. Section 
A - B of Figs. 37 shows the depression lug 6'' from ends of tie to prevent 




.tfr. 



47/^—- 



•"44. 



:::q--^- 2/|'— -^ 



-oH 



¥ 



— T 






n^ 



->|<-9i'->}<" — /tf|— •> 



Plan.. 




Figs. 36. 



!<•••• — -^i 




Section A-B. 

Figs. 37. 



Joint Clip. 



Pfciin 'Clip. 

Figs. 38. 



lateral movement in the ballast. Figs. 38 show details of rail fastenings, 
including those at joint where angle splice bars are used.* 

Concrete-Steel Ties may be said to be in the experimental stage. Those 
interested in these types may find designs of the fCampbell tie and the 
tPercival tie in Eng. News, Oct. 5, 1905, page 349. 



* For full description, see Eng. News, Aug. 24, 1905, page 202. 
t Mr. R. B. Campbell, Gen. Man., the Elgin, Joliet & Eastern Ry. 
X Mr. H. E. Percival, Galveston, Tex. 



TIES— STEEL, CONCRETE. BALLAST. TRACK GAGE. 1073 

Ballast may be broken stone, gravel, cinders, sand or dirt. The first 
named is by far the best and should be of about the size that will pass 
through a 2^ in. ring. Most roads using a great amount of broken stone have 
their own quarries and rock-crushing plants instead of breaking the stone 
by hand. Portable stone crushers are also used. The advantages of broken 
stone and gravel over the finer materials are: (1) good drainage, (2) firm 
bearing and solid ballast packing for the ties, (3) absence of frost, (4) lack 
of retention of moisture to rot the ties, (5) freedom from dust, unpleasant 
to passengers and injurious to the wearing parts of the rolling stock. 

For estimating the amount of ballast per mile of track, a sketch should 
be drawn of the ballast cross-section desired (see Fig. 23, page 1059) and 
from this deduct the cubic contents of the ties from Table 50, page 1069. 

Thus from the figure, we have, 

Cu. ft., gross, of ballast per lin. ft. of single track = 9.5 

Deduct for tie (6''x 8"- 8' spaced 24'' centers) by table = 1 . 333 



Cu. ft., net, of ballast per lin. ft == 8 . 167 

Cu. yds. net of ballast per mile= 8.167X ~|- = 1597. 

Hence 1600 cu. yds. of ballast per mile of single track is the very least 
that can be assumed. This is for a depth of 12 ins. and a top width of 8 ft. 
Generally the ballast is much deeper and wider, say 18" below top of tie, 
and 10 to 12 ft. wide. 

Gage of Track and Wheels. — ^The minimum "Standard" Gage* of track 
in the United States and Canada (also in England and most European 
countries) is 4' ^Y - On roads where this minimum gage is used for straight 
track, the gage is widened for track on curves, say about ^^" per each 
degree of curvature, as per the following table. 

54. — Increase in Gage for Various Degrees op Curvature. 
(Based on about ^^" per degree of curve.) 



Deg. of 
Curve . 


1° 


2° 


3° 


4° 


5° 


6° 


7° 


8° 


9° 


10° 


11° 


12° 


13° 


14° 


Incr. in 
Gage. 


^" 


#/ 


r 


^'' 


r 


r 


A" 


r 


^'' 


¥' 


r 


A" 


r 


r 



On some roads in the U. S., mostly in the South, the "Standard" Gage 
is 4' 9". This is true also of some main-line freight tracks on the P. R. R. 
system. In such cases the gage is seldom widened for ordinary curves. 
There is also another "Standard" Gage employed by a very few roads, 
viz., 4' 8l'\ which may be considered to be a compromise between the two 
gages above mentioned. 

Fig. 39 shows the Master Car Builders' (M. C. B.) standard wheel (and 
track) gage, which has become universally standard. Note that the side 








Fig. 39. 

play in the wheels is f" for a 4' 8^'' gage, and that for a 4' 9" gage it would 
amount to I''. 

* Gage of track is the distance between "inside heads," or "gage sides," 
of rails. 



1074 



59.— RAILROADS. 



*rhe following data regarding gages will be found useful in connection 
with the calculation of turnouts, switches, crossovers, crossings, etc. 

55. — Various Gages and Half Gages op Track, in Feet and Meters 
WITH Logarithmic Values. 



Gage 


Gage 




Gage 




g 


g 




g 




g. 


g. 


Log 


g. 


Log. 


2- 


2 • 


Log. 


2 • 


Log. 


Ft. Ins. 


Ft. 




Meters. 




Ft. Ins. 


Ft. 




Meters. 




2 


2. 


.3010300 


.6096 


9.7850458 


1 


1. 


.0000000 


.3048 


9.4840158 


2 3 


2.25 


.3521825 


.6858 


9.8361983 


1 1^ 


1.125 


.0511525 


.3429 


9.5351683 


2 6 


2.5 


.3979400 


.7620 


9.8819558 


1 3 


1.25 


.0969100 


.3810 


9.5809258 


2 9 


2.75 


.4393327 


.8382 


9.9233485 


1 4i 


1.375 


.1383027 


.4191 


9.6223185 


3 


3. 


.4771213 


.9144 


9.9611371 


1 6 


1.5 


.1760913 


.4572 


9.6601071 


3 3 


3.25 


.5118834 


.9906 


9.9958992 


1 n 


1.625 


.2108534 


.4953 


9.6948692 


3 6 


3.5 


.5440680 


1.0668 


0.0280838 


1 9 


1.75 


.2430380 


.5334 


9.7270538 


3 9 


3.75 


.5740313 


1.1430 


0.0580471 


1 m 


1.875 


.2730013 


.5715 


9.7570171 


4 


4. 


.6020600 


1.2192 


0.0860758 


2 


2. 


.3010300 


.6096 


9.7850458 


4 3 


4.25 


.6283889 


1.2954 


0.1124047 


2 1^ 


2.125 


.3273589 


.6477 


9.8113747 


4 6 


4.5 


.6532125 


1.3716 


0.1372283 


2 3 


2.25 


.3521825 


.6858 


9.8361983 


4 8i 


4.7083 


.6728672 


1.4351 


0.1568830 


2 4i 


2.3542 


.3718372 


.7176 


9.8558530 


4 8i 


4.7292 


.6747847 


1.4415 


0.1588005 


2 41 


2.3646 


.3737547 


.7207 


9.8577705 


4 9 


4.75 


.6766936 


1.4478 


0.1607094 


2 4^ 


2.375 


.3756636 


.7239 


9.8596794 


5 


5. 


.6989700 


1.5240 


0.1829858 


2 6 


2.5 


.3979400 


.7620 


9.8819558 


5 3 


5.25 


.7201593 


1.6002 


0.2041751 


2 n 


2.625 


.4191293 


.8001 


9.9031451 


5 6 


5.5 


.7403627 


1.6764 


0.2243785 


2 9 


2.75 


.4393327 


.8382 


9.9233485 


5 9 


5.75 


.7596678 


1.7526 


0.2436836 


2 10^ 


2.875 


.4586378 


.8763 


9.9426536 


6 


6. 


.7781513 


1.8288 


0.2621671 


3 


3. 


.4771213 


.9144 


9.9611371 



Various Track Gages are used in different countries as follows: In the 
United States, Canada, England and most European countries, the standard 
gage is 4' 8^''= L435 meters. This is true of Austria, Switzerland, Germany, 
France, Hungary, Italy, ^Sweden, and Balkan. In Russia, the standard 
gage is 1.524 meters with the exception of the Warschau-Wien and Wars- 
chau-Bromberg which have a standard of 1.435 meters. In Spain, the 
standard is 1.676 meters, and this gage also prevails mostly in the East 
Indies, Argentine and Chile. It corresponds to the old English gage of 
5' 6''. The gage of the Great Western R. R., in England, was changed from 
7 ft. (nn connection with 4' 81") wholly to standard gage m 1890. In Ireland, 
the standard gage is 5' S'\ Narrow gages are used in many countries. 
In Norway, Cape Colonies, South Australia, Japan and Java, a gage of 3' 6" 
is used largely. In Brazil, Algiers, Greece and Corsica, 1 meter is common. 
Gages of 1 meter, 0.75m. and 0.60m. are used to some extent in Germany 
for narrow gage extensions, etc. Many roads in Switzerland are of Im. 
gage. The gage of the Festiniog-Bahn, in Wales, is only 0.591 meter. 

The "Best" Standard Gage, for universal inter-trafhc, has gradually 
sifted down to that of about 4' 8^'. Wide gages of 5 to 6 ft., and narrow 
gages of 3 to 3^ ft., have been changed to the above standard almost uni- 
versally throughout North America, and the chances are that the same 
standardization will prevail ultimately in South America also. Pertinent 
to this question, the writer has lately been of the opinion that, with the 
enormous locomotives now being built, and with the limited head room, a 
wider gage, say 5' S" to 5' 6'' would have been a better standard to adopt 
than the present one. From the standpoint of the locomotive manufac- 
turer, the following letter, under date of September 8, 1906, from the 
Baldwin Locomotive Works, Burnham, Williams & Co., of Philadelphia, in 
reply to a query from the writer, is interesting: 

Your favor of September 4th was duly received. 

The capacity of any railway, as an instrument for transportation of goods, is 
proportionate to its gage. In fact, the practicable weight of locomotives and the 
practicable capacity of cars, increase directly in proportion to increased width of 
gage. In designing some of the heaviest locomotives now required for freight traffic. 
It would be a great comfort if the gage were wider than 4' 8^". As a broad proposi- 
tion, however, we do not think that even the congestion of traffic upon the principal 
trunk lines has yet reached a point limited by the size of locomotives, nor that 
wider gage than 4' 8^" has generally become necessary. 



TRACK GAGES. TURNOUTS, SWITCHES, FROGS, 1075 



Turnouts and Switches. — A turnout (for switching trains from one track 
to another) consists essentially of a switch, a frog, and the connecting 
"lead" rails. 

Switches may be classed under three main heads, namely, the stub- 
switch, the split-switch, and the Wharton switch. Where there is a double 
turnout from the same track, the diverting switch is called a three-throw 
switch. Fig. 40 shows a double "slip" switch, very useful in switching-yards 




Fig. 40. 

for economizing room, and particularly adapted to sharp crossings, with 
interchangeable traffic. For instance, a train may pass from either track 
on one side of the crossing to either track on the other, by operating the 
switches s. In the Fig., the switches are set for "crossing traffic" on tracks 
Aa and Bb. It is to be noted that the outside curved rails c are continuous 
throughout. The letters F denote position of frogs; and P, the points of 
switches. A single slip switch has but one curved rail c with its correspond- 
ing gage rail, instead of two as shown in Fig. 40. 

Frogs are devices for allowing the flanges of wheels to pass unobstructed 
along one rail crossing another rail. There are three distinct classes, namely, 
stiff or rigid frogs, spring or spring rail frogs, and movable-point frogs. 

Stiff frogs may be made up of rail sections, or of solid steel castings. 
Fig. 41 illustrates the shape of a stiff rail frog, the heads of rails only being 




l§j Moufh 



Fig. 41. 

shown. LW and RW are left- and right-wings respectively; and MP and 
SP are main- and side-points. Note that the "point of frog" is at the 
intersection of the outside lines of MP and SP and a little beyond the blunt 
point of tongue. All frogs are designated by numbers. Thus, if we let L 
equal the length from point of frog to heel, and W equal the width at heel, 
then 



The "frog number" n^^.^ 



(1) 



Frogs range from numbers 4 to 24, but the usual limits are 6 and 12, while 
8 and 9 are perhaps the most common. We will show further on how the 
jrog nuTYiber determines practically the _ radius or degree of the turnout 
curve, and it will be seen also what bearing the hind of switch and length of 
frog have on the problem, for any particular gage of track. 

Solid cast manganese steel frogs will greatly outwear the ordinary rail 
frog, say about 9 to 1, while they cost about 4 or 5 times as much. They 
may be had "one-sided" or "double sided" depending on whether the heavy 
traffic is mainly on one track, or is "balanced." There is a saving in cost of 
about 25% in favor of the former. 

Spring Rail Frogs are usually placed on the main line because there is 
little or no shock in riding over them, as there is over the rigid frogs. Fig. 
42 is a P. & R. Railway standard, showing the right wing RW pressed 
tightly against the tongue by the spring, thus forming practically a con- 

* As the ratio L -^ W is constant for any given frog, its number may 
best be determined by measuring the distance U, from the theoretical point 
of frog, to a point where I^' = say 4"; then « = L'-t-4. 



1076 



59.--RAILROADS. 



56. — Properties of Frog Angles <^ — 

With Logarithmic Values. 
Note. — Logarithmic values are exact for the given frog numbers; the 
frog angles are to the nearest second. (Angles to the nearest minute 
are close enough, usually.) 

For values of cosec and cot ^, see Table 64. 

Part I. — Properties of Frog Angles, <^. 



Frog 


Frog 


Nat 


Log 


Nat 


Log 


Nat 


Log 


Nat 


Log 

Sec 


No. 
n. 


Angle 
4>. 


Sin flS 


Sin <f> 


Cos (f> 


Cos <f> 


Tan 4> 


Tan 4> 


Sec 


4 


U^IS'OO' 


.246154 


9.3912067 


.969231 


9.9864272 


.253968 


9.4047794 


1.03175 




m 


12 40 49 


.219512 


9.3414586 


.975610 


9.9892761 


.225000 


9.3521826 


1.02500 




5 


11 25 16 


.198020 


9.2967086 


.980198 


9.9913138 


.202020 


9.3053948 


1.02020 




5^ 


10 23 20 


.180328 


9.2560628 


.983607 


9.9928214 


.183333 


9.2632415 


1.01667 




6 


9 31 38 


.165517 


9.2188433 


.986207 


9.9939680 


.167832 


9.2248753 


1.01399 


-f^ 


6^ 


8 47 51 


.152941 


9.1845244 


.988235 


9.9948604 


.154762 


9.1896640 


1.01190 


S 


7 


8 10 16 


.142132 


9.1526919 


.989848 


9.9955684 


.143590 


9.1571235 


1.01026 


o . 


m. 


7 37 41 


.132743 


9.1230128 


.991150 


9.9961396 


.133929 


9.1268732 


1.00893 


bc^ 


8 


7 09 10 


.124514 


9.0952169 


.992218 


9.9966071 


.125490 


9.0986099 


1.00784 


« lO 


m 


6 43 59 


.117241 


9.0690810 


.993103 


9.9969944 


.118056 


9.0720865 


1.00694 


1^ 


9 


6 21 35 


.110769 


9.0444192 


.993846 


9.9973192 


.111455 


9.0471000 


1.00619 


/-vO 


9H 


6 01 32 


.104972 


9.0210750 


.994475 


9.9975939 


.105556 


9.0234811 


1.00556 


r- II 


10 


5 43 29 


.099751 


8.9989157 


.995012 


9.9978285 


.100251 


9.0010871 


1.00501 


oj' 


10^ 


5 27 09 


.095023 


8.9778270 


.995475 


9.9980304 


.095455 


8.9797966 


1.00455 


«o 


11 


5 12 18 


.090722 


8.9577109 


.995876 


9.9982054 


.091097 


8.9595055 


1.00414 




11^ 


4 58 45 


.086792 


8.9384821 


.996226 


9.9983580 


.087121 


8.9401239 


1.00379 


-0-'"' 


12 


4 46 19 


.083189 


8.9200655 


.996534 


9.9984920 


.083478 


8.9215734 


1.00348 


CQ.^ 


12^ 


4 34 52 


.079872 


8.9023957 


.996805 


9.9986103 


.080128 


8.9037854 


1.00321 


8^ 


13 


4 24 19 


.076809 


8.8854147 


.997046 


9.9987151 


.077037 


8.8866996 


1.00296 


5PS 


U 


4 05 27 


.071338 


8.8533184 


.997452 


9.9988921 


.071520 


8.8544263 


1.00255 


O M 


15 


3 49 06 


.066593 


8.8234264 


.997780 


9.9990349 


.066741 


8.8243915 


1.00222 


16 


3 34 47 


.062439 


8.7954561 


.998049 


9.9991517 


.062561 


8.7963043 


1.00196 


Il7; 


17 


3 22 10 


.058773 


8.7691755 


.998271 


9.9992486 


.058874 


8.7699269 


1.00173 


■s-S 


18 


3 10 56 


.055513 


8.7443926 


.998458 


9.9993298 


.055598 


8.7450628 


1.00154 


IS 


19 


3 00 54 


.052595 


8.7209457 


.998616 


9.9993985 


.052668 


8.7215473 


1.00139 


20 


2 51 51 


.049969 


8.6986987 


.998751 


9.9994571 


.050031 


8.6992415 


1.00125 


1 


21 


2 43 40 


.047592 


8.6775346 


.998867 


9.9995076 


.047646 


8.6780270 


1.00113 


22 


2 36 14 


.045431 


8.6573530 


.998967 


9.9995513 


.045478 


8.6578017 


1.00103 




23 


2 29 27 


.043458 


8.6380670 


.999055 


9.9995895 


.043499 


8.6384776 


1.00095 




24 


2 23 13 


.041649 


8.6196003 


.999132 


9.9996230 


.041685 


8.6199773 


1.00087 





Vers ^ = sin ^.tan "«" = 2 sin^ — . 



tinuous main line rail. When a train takes the SP rail, the RW is crowded 
open by the wheel flange, but springs back after the train has passed. 
Spring-rail frogs are "rights" and "lefts" and hence care must be used in 
ordering them. Other types are the Vaughan, Wood, Ajax, Eureka, etc. 
Double spring rail frogs are seldom used. 




Fig. 42. — Spring Rail Frog. 

Movable-Point Frogs are shown in Fig. 43, as manufactured by Wm. 
Wharton, Jr., & Co., Philadelphia. Note that this type might well be used 
in Fig. 40 at the central frogs F. The cut is self-explanatory. The movable 
points also may be of manganese steel if required. 



FROGS. FROG ANGLES— PROPERTIES OF. 



1077 



-AND Properties of 3^ Frog Angles 



With Logarithmic Values. 
Note. — Logarithmic values are exact for the given frog numbers; the 
frog angles are to the nearest second. (Angles to the nearest minute 
are close enough, usually.) 

Part II. — Properties of ^ Frog Angles, -»-. 



Fros 


<l> 


Nat 


Log 


Nat 


Log 


Nat 


Log 


Nat 


Log 
Sec 


No. 
n. 


2* 


Sin-|- 


6 
Sm-|- 


cos 4 


cos 4 


Tanil 


Tan| 


Sec 4 




2 


-• 


4 


7007/30" 


.124035 


9.0935433 


.992278 


9.9966333 


.125 


9.0969100 


1.00778 






4U 


6 20 25 


.110432 


9.0430931 


.993884 


9.9973356 


.111111 


9.0457575 


1.00615 






5 


5 42 38 


.099504 


8.9978393 


.995037 


9.9978393 


.1 


9.0000000 


1.00499 


■6-l«^ 


5^1 5 11 40 


.090536 


8.9568201 


.995893 


9.9982128 


.090909 


8.9586073 


1.00412 






6 


4 45 49 


.083045 


8.9193160 


.996546 


9.9984973 


.083333 


8.9208188 


1.00347 







fi^ 


4 23 55 


.076696 


8.8847755 


.997055 


9.9987189 


.076923 


8.8860566 


1.00295 






7 


4 05 08 


.071247 


8.8527670 


.997459 


9.9988950 


.071429 


8.8538720 


1.00255 






7^ 


3 48 51 


.066519 


8.8229457 


.997785 


9.9990370 


.066667 


8.8239087 


1.00222 


1 


•>• 


8 


3 34 35 


.062378 


8.7950334 


.998053 


9.9991534 


.0625 


8.7958800 


1.00195 


1 


CO 


8^ 


3 21 59 


.058722 


8.7688011 


.998274 


9.9992499 


.058824 


8.7695511 


1.00173 






9 


3 10 47 


.055470 


8.7440584 


.998460 


9.9993308 


.055556 


8.7447275 


1.00154 


'^ 





9^ 


3 00 46 


.052559 


8.7206457 


.998618 


9.9993993 


.052632 


8.7212464 


1.00138 








10 


2 51 45 


.049938 


8.6984278 


.998752 


9.9994578 


.05 


8.6989700 


1.00125 




II 


io;< 


2 43 35 


.047565 


8.6772889 


.998868 


9.9995081 


.047619 


8.6777807 


1.00113 


li 


^ 


11 


2 36 09 


.045408 


8.6571292 


.998969 


9.9995518 


.045455 


8.6575773 


1.00103 




ii;^ 


2 29 22 


.043437 


8.6378622 


.999056 


9.9995899 


.043478 


8.6382722 


1.00094 


'^l"^^ 


]?< 


2 23 09 


.041631 


8.6194122 


.999133 


9.9996234 


.041667 


8.6197888 


1.00087 








]?M 


2 17 26 


.039968 


8.6017129 


.999201 


9.9996528 


.04 


8.6020600 


1.00080 


^ 


» 


18 


2 12 09 


.038433 


8.5847057 


.999261 


9.9996791 


.038462 


8.5850267 


1.00074 


bS) 


^ 


14 


2 02 43 


.035692 


8.5525652 


.999363 


9.9997232 


.035714 


8.552842C 


1.00064 


7^ 


M 


15 


1 54 33 


.033315 


8.5226375 


.999445 


9.9997589 


.033333 


8.5228787 


1.00056 







16 


1 47 24 


.031235 


8.4946380 


.999512 


9.9997880 


.03125 


8.4948500 


1.00049 


II 


. 


17 


1 41 05 


.029399 


8.4683333 


.999568 


9.9998122 


.029412 


8.4685211 


1.00043 


-0-1^ 


3 


18 


1 35 28 


.027767 


8.4435301 


.999614 


9.9998325 


.027778 


8.4436975 


1.00039 


ti 


19 


1 30 27 


.026307 


8.4200661 


.999654 


9.9998497 


.026316 


8.4202164 


1.00035 


CD 


20 


1 25 56 


.024992 


8.3978043 


.999688 


9.9998643 


.025 


8.3979400 


1.00031 


m 




21 


1 21 50 


.023803 


8.3766276 


.999717 


9.9998769 


.023810 


8.3767507 


1.00028 


be 





22 


1 18 07 


.022721 


8.3564351 


.999742 


9.9998878 


.022727 


8.3565473 


1.00026 


3 




23 


1 14 43 


.021734 


8.3371396 


.999764 


9.9998974 


.021739 


8.3372422 


1.00024 






24 


1 11 37 


.020829 


8.3186645 


.999783 


9.9999058 


.020833 


8.3187588 


1.00022 







Vers4 = sin 4 -tan 4 = 2 sin 2 4- 
2 2 4 4 




Fig. 43. 



ScctWN B-D 



1078 



59,— RAILROADS. 



SecTtoK A-A 



> Crossing Frogs are usully rigid, and made up of rail sections with perhaps 
cast steel frog junctions. There is a form of movable frog consisting of 
short pieces of rail on miniature turntables that can be turned in any 
desired direction, thus making a continuous rail of either track. 

Stub Switches. — ^The stub switch is the cheapest and most primitive 
form. Its use is now confined to second-class yards and spur connections 
with sidings, having disappeared entirely from main yards. It should never 
be used for main line connection. Fig. 44 illustrates the essential features. 
The head blocks H B are at 
the junction of the switch 
rails 5" with the main lead 
(rail) M L and turnout lead 
(rail) TL, which "lead" to 
the toe of frog; also with 
the continuous main line rail 
M and turnout rail T. The §4 
"throw" of the switch is 
clearly the distance between 
the gage sides of rails M and 
TL or ML and T, at the head 
blocks, and is usually 5, 51 
(or 6) ins.* The switch rails 
are tied together with a front 
rod and three or more back 
rods. The length of switch 
rails is governed by the frog 
number or by the degree of 
turnout curve. The follow- 
ing table was calculated by 
the author in 1887 while he 

was Resident Engineer of a v dA 

western road and was used by ■^^- **• 

the foremen in laying out all switches, including the terminal yards at 
Toledo. The sharpest frog ^as No. 9, and the table was used for both stub 
and split switches; but for frog numbers higher than 9 the table does not 
apply strictly to the latter. The virtue of the table lies in the offset dis- 
tances between the gage sides of ML and TL rails, Fig. 44, as given in the 
last seven columns but one. These offsets are measured at points 10-, 20-, 
30-ft., etc. from theoretical point of frog (Fig. 41). By this means the 
position of the TL rail is fixed quickly, and from it the T rail is gaged. The 
efficiency of the work per gang was increased from two switches in three 
days to one switch per day, in broken stone ballast. 




4 





57.- 


—Table for Laying 


OUT 


Switches. 


Gage 4' 


m". 


Throw 5". 






d 


Turnout 








Offset distances In ft. from gage 




o 


g 


Curve. 


Theo- 


Stub 


Split 


side of main lead to gage side of 


r^ 


^ 




reti- 
cal 
Lead. 


Switch, 
t 


Switch. 

t 


turnout lead at following dis- 
tances from point of frog. 


^ 


< 


De- 
gree. 


Ra- 
dius. 


& 


Rail 


Lead 


Rail 


Lead 


10ft. 


20ft. 


30ft. 


40ft. 


50ft. 


60ft. 


70ft. 


1 


n 


o / 


o / 


Ft. 


Ft. 


Ft. 


Ft. 


Ft. 


Ft. 


Ft. 


Ft. 


Ft. 


Ft. 


Ft. 


Ft. 


Ft. 


n 


6 


9 32 


16 58 


339.0 


56.5 


16.9 


39.6 


15 


54.6 


1.51 


2.73 


3.65 










6 


7 


8 10 


12 26 


461.4 


65.9 


19.6 


46.3 


15 


61.3 


1.31 


2.30 


3.13 


3.74 








7 


8 


7 09 


9 31 


602.7 


75.3 


22.5 


52.9 


15 


67.9 


1.16 


2.16 


2.99 


3.65 


4.14 






8 


9 


6 22 


7 31 


762.8 


84.8 


25.3 


59.5 


15 


74.5 


1.04 


1.96 


2.74 


3.39 


3.92 






9 


10 


5 43 


6 05 


941.7 


94.2 


28.0 


66.1 


15 


81.1 


0.95 


1.78 


2.52 


3.15 


3.66 


4.08 




10 


11 


5 12 


5 02 


1139.4 


103.6 


30.8 


72.7 


15 


87.7 


0.86 


1.64 


2.33 


2.93 


3.44 


3.87 


4.21 


11 


12 


4 46 


4 14 


1356.0 


113.0 


33.6 


79.4 


15 


94.4 


0.80 


1.52 


2.17 


2.76 


3.26 


3.68 


4.03 


12 



* Distance between rail heads should be about 3 ins.; and the throw is 
equal to this distance plus width of railhead. 

t "Rail" means switch rail; "lead" means distance from point of frog 
to toe of stub switch (if. B.), or to point of split switch. 



CROSSING FROGS. STUB SWITCHES. 



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STUB SWITCHES. TURNOUT CURVES, 



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^^lO^C^^' ^ ^' ^ ^ ^ ^ 



1082 



I— RAILROADS. ' 




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TURNOUTS FOR ANY GAGE, TURNOUT CURVES. 1083 

Turnout Curves for Stub Switches are simple curves. In staking them 
out on the ground it is necessary only to set two stakes, one marked P. F., 







Fig. 46. 

opposite the point of frog, and the other marked P. 5., opposite the point of 
switch. They should be set on the frog side of the track to indicate the 
direction of the turnout. The P. F. is usually located about half a frog 
length from a rail joint to save one cut of rail. No refinement is necessary 
in the calculation of turnouts, and the following formulas may be used 
where the main track is straight, as in Fig. 46: 

Let = frog angle = central angle of turnout curve; 

n = frog number (see Fig. 41); 

/= theoretical lead to point of main frog; 

/' = theoretical lead to point of crotch frog; 

g = gage of track; 

^ = radius of turnout curve; 

5 = switch length, P. S. to H. 5.; 
w = middle ordinate of turnout curve; 

<j = quarter ordinates of turnout curve. 

^ = throw of switch. 
All distances in feet. 

Then, tan— =j; 






4> I 



n=icot|- =— =Vi?H-2g; 
i? = g-^vers <^ — Y =2g«2. 

= Z-^sin ^ — ^; 

/=2g» = g cot^= [r+ -|) sin <^; 

s=2n\/gl =V2^^; 
t = s^^2R; 
m = g-^i (nearly); 
Q = Hg-^^ = -i6g (nearly). 
All the above formulas may be used with perfect safety for field work. 
Other formulas may be deduced from these by transposing or equating. 



1084 ' m.— RAILROADS. 

Double Turnouts ( = three-throw turnouts) require two main-Hne frogs 
and one crotch frog. If the two turnouts are opposite and of equal radii, 
then, Fig. 46, 0', the angle of the crotch frog, is equal to double the central 

angle y. But (^+f-) vers |- = -|-, iience vers y = |-*- (^ + |-) » ^^o^ 

which 0' is obtained. 

Further, if n' = the number of the crotch frog, 

and I' = th.e crotch frog (point) distance from the P. S., we have, 
i? = 4 n'^g (nearly) = 2 n^g; 

n' = n-^VY (nearly) = 0.7071 n (nearly); 
r = 0.707/ (nearly). 
Hence n' and V are respectively equal to n and I multiplied by 0.707 
(nearly). See Table 58. 

Curved Main Track. — ^The above remarks apply to turnouts from 
straight main-line track, in connection with the stub switch. It is to be 
remembered that nearly all the formulas used in practical track work are 
approximate and close enough to the exact values, which may be obtained 
by trigonometric calculation. We will show now how the foregoing formulas 
may be applied to turnouts from curved main-line track. In order to illus- 
trate we will consider Fig. 46 to be warped so the main line is curved instead 
of straight: 
Let r° = degree of curve of turnout from straight track; 

i° = degree of curve of turnout from curved main line; 

M° = degree of curve of main line after ciu-ving. 
Then t°=T° + M° when main line is curved toward turnout; 

t°=T° — M° when main line is curved away from turnout. 

For instance, a No. 9 frog calls for a 7° 30' curve from a straight track; 
it calls for 10° SO' curve from the inside of a 3° main-line ciu-ve; and for a 
4° 30' curve from the outside of a 3° main-line curve. Hence, when the 
two curves are in the same direction we have to find their difference, and 
when in opposite directions we have to find their sum, to get the degree of 
curve t° in Table 58, preceding, and the desired frog number, n, correspond- 
ing thereto. 

Split Switches. — In our consideration of the stub switch, which is really 
for second-class track work, we have assumed all frogs in the turnout 
curves to be curved, whereas they are straight, and should be so considered 
in first-class track work where the split switch is used, although formerly 
this refinement was not considered necessary. This introduces a short 
tangent in the turnout at the point of frog. Moreover, split switch points 
are straight and it is now customary to take this fact into consideration 
in calculating the turnout curve, which really extends only from the heel of 
switch (P. C.) to the toe of frog (P. T.), joining the two short tangents 
above mentioned, instead of from point of switch to point of frog as assumed 
in Table 57, preceding. , 

The following is the practice of the Weir Frog Co., of Cincinnati, Ohio: 



TURNOUTS FROM CURVES. SPLIT SWITCHES. 



1085 



61. — ^Turnouts for Split Switches and Spring Frogs. Gage 4' 8^' 
(Curve is Tangent to Switch Angle at Heel of Switch and to Frog 
Angle at Toe of Frog.) 

^-'s-"rur-z£^i^f(^r)ce from RS.fo Pr- 1 ^ 




Fig. 47. 
(a). — Switch and Frog Variable. 















Distance 






Frog 
No. 
n. 


Frog 

Angle. 

0. 


Radius 

of 
Curve. 

R. 


Degree 

of 
Curve. 


Switch 

Angle. 
a. 


Switch 

Length. 

s. 


from Point 

of Switch to 

Actual 

Point of 


Lgth 

of 
Frog. 


Length 

of Main 

Point of 

Frog. 












Frog. 






4 


14° 15' 


125.868' 


46° 49' 


3° 20' 29" 


7 


6" 


33' 9 " 


6'0" 


3' 9 " 


4^ 


12° 41' 


164.569' 


35° 22' 


" 




' 


36' 7 " 


6' 0" 


3' 9 " 


5 


11° 25' 


202,054' 


28° 39' 


2° 30' 27" 


10 


0" 


43' 7M" 


6' 0" 


V 10 " 


53^ 


10° 23' 


244.318' 


23° 37' 


" 




' 


46' 3M" 


6' 6" 


3' 103^" 


6 


9° 32' 


289.453' 


19° 53' 


1° 40' 16" 


15 


0" 


57' ly^" 


6' 8" 


4' 3 " 


6^ 


8° 48' 


343.249' 


16° 45' 


" 




' 


60' 3 " 


7' 0" 


4' 6 " 


7 


8° 10' 


403,942' 


14° 13' 


" 




♦ 


63' 4 " 


7' 0" 


4' 63^" 


l^i 


7° 38' 


468.794' 


12° 15' 


' 




' 


66' 3%" 


7' 6" 


5' " 


8 


7° 09' 


535.773' 


10° 42' 


" 




' 


69' 03^" 


8' 0" 


5' 3 " 


9 


6° 22' 


678.440' 


8° 27' 


" 




' 


74' 2 " 


10' 0" 


6' 43^" 


10 


5° 44' 


855.803' 


6° 42' 


" 




• 


79' 5M" 


11' 0" 


V XM" 


11 


5° 12' 


1048.987' 


5° 28' 


" 




• 


84' OM" 


12' 0" 


V 8 " 


12 


4° 46' 


1259.507' 


4° 33' 






88' 6>^" 


14' 0" 


8' 9 " 









(b).— 


15 Ft. Switches. 








6 


9° 32' 


231.21 


24° 56' 


1° 40' 16" 


15' 0" 


53' 9 " 


15' 0" 


8' 0" 


7 


8° 10' 


335.50 


17° 08' 


" 


" 


60' 1 " 


'• 


" 


8 


7° 09' 


461.08 


12° 27' 


" 


" 


64' 1%" 


•' 


" 


9 


6° 22' 


609.62 


9° 25' 


"' 


" 


71' 10%" 


" 


" 


10 


5° 44' 


783.02 


7° 19' 






7 7' 4%" 













(c).- 


18 Ft. Switches. 








6 


9° 32' 


228.97 


25° 15' 


1° 23' 34" 


18' 0" 


57' 6f^" 


15' 0" 


8' 0" 


7 


8° 10' 


331.04 


17° 22' 


" 


•♦ 


64' 23^" 


" 




8 


7° 09' 


453.04 


12° 40' 


♦* 


" 


70' 7 " 


•• 


" 


9 


6° 22' 


596.05 


9° 37' 


'« 


" 


76' 8^" 


•• 


•t 


10 


5° 44' 


761.28 


7° 32' 






82' 7 " 







1086 



59.— RAILROADS, 



62. — ^Three-Throw Turnouts. Split Switches. 
Gage 4' 8H"l Throws 5". 

L I*- — ~ ^1- Frog distance L 







Fig. 48. 









Distance from 




Length of 


Main Line 


Crotch Frog 


Crotch Frog 


Point of Switch to 


Length of 


Actual Main 


Frog No. 


Number. 


Angle. 


Actual Point of 
Crotch Frog. 


Crotch Frog. 


Point of 
Crotch Frog. 


4 


2.76 


20° 34' 


29' 93^" 


5' 0" 


3' 3 " 


4^ 


3.12 


18° 12' 


31' 8 " 


5' 0" 


3' 3 " 


5 


3.51 


16° 14' 


37' 5^^" 


5' 6" 


3' 6 " 


5^ 


3.83 


14° 53' 


39' 2 " 


5' 6" 


3' 6 " 


6 


4.21 


13° 34' 


41' 0%" 


6' 0" 


3' 9 ' 


6^ 


4.56 


12° 32' 


42' gM" 


6' 0" 


3' 9 " 


7 


4.91 


11° 37' 


44' 5%" 


6' 0" 


3' 10 " 


7^ 


5.26 


10° 52' 


46' OM" 


6' 6" 


3' 10^" 


8 


5.58 


10° 14' 


47' 5%" 


6' 6" 


3' 10^" 


9 


6.20 


9° 14' 


50' 1^" 


7' 0" 


4' 6 » 


10 


6.84 


8° 21' 


52' 8%" 


7' 0" 


4' ^M" 


11 


7.57 


7° 33' 


57' ^M" 


8' 0" 


5' 3 * 


12 


8.17 


7° 00' 


60' OM" 


9' 0" 


5' 11 • 



Split switches are made either stiff or with springs. 



TURNOUTS. SPLIT SWITCHES. 



1087 



The following is the practice of the Buda Manufacturing Co.: 
63. — Turnouts from Straight Track. Split Switches. 

, ^. Acfuar L ead (^Mchj^^^^l'-^'OO' 




(a).- 



Fig. 49. 
-Properties' of Turnouts. 



Frog 


Frog 


Radius of 


Degree of 


Lead. 


Length of 


Mid. Ord. 


Number. 


Angle. 


Curve. 


Curve. 




Curve. 


of Curve. 


4 


14° 15' 


121.841' 


48° 27' 


43' 93^" 


26' 91^" 


Sif" 


5 


11° 25' 


193.991' 


29° 52' 


50' 4^" 


33' 0%" 


8^" 


6 


9° 32' 


283.525' 


20° 19' 


56' 8 " 


38' 1 W 


8 " 


7 


8° 10' 


393.603' 


14° 36' 


62' 10^" 


44' 7if" 


1%" 


8 


7° 10' 


516.219' 


11° 8' 


68' 2}i" 


49' &%" 


73^' 


9 


6° 22' 


667.734' 


8° 36' 


73' 83^" 


54' lOH" 


%%" 


10 


5° 44' 


841.083' 


6° 49' 


78' lli^" 


59' 8^" 


6^" 



Frogs. 
rAcfualpoint 



--__ Poiri^ fJMnarwf. 




^.Spread 



Fig. 50. 

(b). — Properties of Frogs. 



Frog 
No. 


Angle. 


Length. 


Actual 


Theoretical 


Spread. 


Point to 
Heel. 


Point to 
Toe. 


Point to 
Heel. 


Point to 
Toe. 


Heel. 


Toe. 


4 
5 
6 
7 
8 
9 
10 


14° 15' 
11° 25' 
9° 32' 
8° 10' 
7° 10' 
6° 22' 
5° 44' 


6' 0" 

7' 0" 

8' 0" 

9'0" 

10' 0" 

ir 0" 

12' 0" 


3' 7 " 
4' 33^" 
5' " 
5' 8^" 
6' 2 " 
6' 103^" 
7' 7 " 


2' 5 " 
2' 81^" 
3' " 
3' 33^" 
3' 10 " 
4' ly^" 
4' 5 " 


3' 9" 
4' 6" 
5' 3" 
6' 0" 
6' 6" 
7' 3" 
8' 0" 


2' 3" 
2' 6" 
2' 9" 
3'0" 
3' 6" 
3' 9^^ 
4' 0" 


111^" 
lOH" 

103^" 

103^" 
9^" 


6^" 
6 " 

4H* 



Notation: 

g=gage; 

5 = length of switch rail; 
t = throw of switch ; 
a = switch angle; 
^ = frog angle; 
J^= radius of turnout curve; 

i?'=i^+i-; /=lead. 

All in feet. 



Formulas for Split Switches. 
(a) . — Assuming Frog to be Curved. 



T- 


'j-Uead. ^ 




~'s~-i'^^^^'k i " 




Fig. 61. 



10S8 



5^.— RAILROADS. 



Formulas (Approximate): 

t 



Sin a = 



l = s + 



g-t 



R' = 



tan ^ (<^+ a) 
g-t 



cos a — cos (f> 

(b) . — Assuming Frog to be Straight. 
Notation: 

/ = straight length of frog to P. F., in feet. 
Balance of notation as on preceding page. 

Formulas (Approximate): • " 

Sin a = — ; 



l=s + 



R' 



g- 



g — t — f cos 
tan i (^ + a) 
■t— f sin ^ 



+ / cos <^; 




tan i (0+ a) 

Note. — Switches can be planned graphically very easily by drawing 
them to scale and scaling the dimensions. After the switch length and 
switch angle are drawn for one rail, the frog, with angle <f> and length /, 
may be "slid" long the other "rail" until the semi-tangents, s-P, are equal. 

Wharton Switch. — ^The virtue of the Wharton switch lies in the fact that 
no frog is necessary for a turnout curve from the main line, for which it is 
especially designed, and that the main line rails are therefore unbroken. 
Fig. 53 is a general plan of the switch as patented April 2, 1901, showing it 
in position for main-line traffic. The switch rails are on a grade, rising from 
the points, and when thrown over and set for the siding the wheels of the 
train ride up on them, clearing the flanges from the main track. The trip 
rail is pivoted so that when the switch is set for the siding, the end G of 
guard, nearest the point of switch, swings outward and hugs the main rail. 
Hence, a train coming heel on, on the main line, crowds the trip rail inward 
and throws the switch automatically, for the main line. 



DJ_a_D?DJ 




V ■-■: -4'I0"- 

/•Clip „ ,„ 



.r< 47l >) "^ 



Fig. 53.— Wharton Switch. (See also Fig. 43.) 



WHARTON SWITCH. LADDER TRACKS. 



1089 



64. — Ladder Tracks. Spacing of Frogs. 
(Any Gage.) 
Part I. — ^Diagonal (Direct) Distances d between 
Points, [d in Feet.] 



Frog 




i-h^Fig. 54. 



Frog 
No 



Distances Between Track Centers, c. In Feet. 



General* 



10 



10.5 



11 



11.5 



12 



12.5 



13 



13.5 



14 



14.5 



15 



4 
5 

6 

6^ 
7 

7^ 
8 

SH 
9 

9^ 
10 

11 

11^ 

12 

12^ 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 



4.06250 C 

4.55555 

5.05000 C 

5.54546 C 

6.04167 

6.53846 

7.03571 

7.53333 

8.03125 

8.52941 

9.02778 

9.52632 

10.0250 

10.5238 

11.0227 

11.5217 

12.0208 

12.5200 

13.0192 

14.0179 

15.0167 

16.0156 

17.0147 

18.0139 

19.0132 

20.0125 C 

21.0119 C 

22.0114 c 

23.0109 c 

24.0104 c 



40.625 
45.556 
50.500 
55.455 
60.417 
65.385 
70.357 
75.333 
80.318 
85.294 
90.278 
95.263 
100.25 
105.24 
110.23 
115.22 
120.21 
125.20 
130.19 
140.18 
150.17 
160.16 
170.15 
180.14 
190.13 
200.13 
210.12 
220.11 
230.11 
240.10 



42.656 
47.833 
53.025 
58 227 
63.438 
68.654 
73.875 
79.100 
84.328 
89.559 
94.792 
100.03 
105.26 
110.50 
115.74 
120.98 
126.22 
131.46 
136.70 
147.19 
157.68 
168.16 
178.65 
189.15 
199.64 
210.13 
220.62 
231.12 
241.61 
252.11 



44.688 
50.111 
55.550 
61.000 
66.458 
71.923 
77.393 
82.867 
88.344 
93.824 
99.306 
104.79 
110.28 
115.76 
121.25 
126.74 
132.23 
137.72 
143.21 
154.20 
165.18 
176.17 
187.16 
198.15 
209.15 
220.14 
231.13 
242.13 
253.12 
264.11 



46.719 
52.389 
58.075 
63.773 
69.479 
75.192 
80.911 
86.633 
92.359 
98.088 
103.82 
109.55 
115.29 
121.02 
126.76 
132.50 
138.24 
143.98 
149.72 
161.21 
172.69 
184.18 
195.67 
207.16 
218.65 
230.14 
241.64 
253.13 
264.63 
276.12 



48.750 
54.667 
60.600 
66.546 
72.500 
78.462 
84.429 
90.400 
96.375 
102.35 
108.33 
114.32 
120.30 
126.29 
132.27 
138.26 
144.25 
150.24 
156.23 
168.21 
180.20 
192.19 
204.18 
216.17 
228.16 
240.15 
252.14 
264.14 
276.13 
288.12 



50.781 
56.944 
63.125 
69.318 
75.521 
81.731 
87.946 
94.167 
100.39 
106.62 
112.85 
119.08 
125.31 
131.55 
137.78 
144.02 
150.26 
156.50 
162.74 
175.22 
187.71 
200.20 
212.68 
225.17 
237.67 
250.16 
262.65 
275.14 
287.64 
300.13 



52.813 
59.222 
65.650 
72.091 
78.542 
85.000 
91.464 
97.933 
104.41 
110.88 
117.36 
123.84 
130.33 
136.81 
143.30 
149.78 
156.27 
162.76 
169.25 
182.23 
195.22 
208.20 
221.19 
234.18 
247.17 
260.16 
273.15 
286.15 
299.14 
312.14 



54.844 
61.500 
68.175 
74.864 
81.563 
88.269 
94.982 
101.70 
108.42 
115.15 
121.88 
128.61 
135.34 
142.07 
148.81 
155.54 
162.28 
169.02 
175.76 
189.24 
202.73 
216.21 
229.70 
243.19 
256.68 
270.17 
283.66 
297.15 
310.65 
324.14 



56.875 
63.778 
70.700 
77.636 
84.583 
91.538 
98.500 
105.47 
112.44 
119.41 
126.39 
133.37 
140.35 
147.33 
154.32 
161.30 
168.29 
175.28 
182.27 
196.25 
210.23 
224.22 
238.21 
252.19 
266.18 
280.18 
294.17 
308.16 
322.15 
336.15 



58.906 

66.056 

73.225 

80.409 

87.604 

94.808 

102.02 

109.23 

116.45 

123. 

130.90 

138.13 

145.36 

152.60 

159.83 

167.06 

174.30 

181.54 

188.78 

203.26 

217.74 

232.23 

246.71 

261.20 

275.69 

290.18 

304.67 

319.17 

333. 

348. 15 



60.938 
68.333 
75.750 
83. 182 
90.625 
98.077 
105.54 
113.00 
120.47 
127.94 
135.42 
142.89 
150.38 
157.86 
165.34 
172.83 
180.31 
187.80 
195.29 
210.27 
225.25 
240.23 
255.22 
270.21 
285.20 
300.19 
315.18 
330.17 
345.16 
360.16 



Part II. — Horizontal Distances h between Frog Points, [h in Feet.] 



Frog 
No. 



Distances Between Track Centers, c. In Feet. 



General t 



10 



10.5 



11 



11.5 



12 



12.5 



13 



13.5 



14 



14.5 



15 

59.063 
66.667 
74.250 
81.818 
89.375 
96.923 
104.46 
112.00 
119.53 
127.06 
134.58 
142.11 
149.63 
157.14 
164.66 
172.17 
179.69 
187.20 
194.71 
209.73 
224.75 
239.77 
254.78 
269.79 
284.80 
299.81 
314.82 
329.83 
344.84 
359.84 



4 

4^ 

5 

5^ 

6 

6^ 

7 

IH 

8 

SH 

9 

9^ 
10 
10^ 

11 

11^ 

12 

12J^ 
13 
14 
15 
16 
17 
18 
19 
20 
?.l 
22 
23 
24 
* 

t 



3.93750 c 
4.44444 c 
4.95000 c 
5.45455 c 
5.95833 C 
6.46154 C 
6.96428 C 
7.46667 C 
7.96875 C 
8.47059 c 
8.97222 c 
9.47368 
9.97500 c 
10.4762 c 
10.9773 
11.4783 c 
11.9792 
12.4800 
12.9808 
13.9821 
14.9833 
15.9844 
16.9853 
17.9861 
18 9868 c 
19.9875 c 
20.9881 c 
21.9886 c 
22.9891 C 
23.9896 c 



39.375 
44.444 
49.500 
54.545 
59.583 
64.615 
69.643 
74.667 
79.688 
84.706 
89.722 
94.737 
99.750 
104.76 
109.77 
114.78 
119.79 
124.80 
129.81 
139.82 
149.83 
159.84 
169.85 
179.86 
189.87 
199.88 
209.88 
219.89 
229.89 
239.90 



41.344 
46.667 
51.975 
57.273 
62.562 
67.846 
73.125 
78.400 
83.672 
88.941 
94.208 
99.474 
104.74 
110.00 
115.26 
120.52 
125.78 
131.04 
136.30 
146.81 
157.32 
167.84 
178.35 
188.85 
199.36 
209.87 
220.38 
230.88 
241.39 
251.89 



43.313 

48.889 
54.450 
60.000 
65.542 
71.077 
76.607 
82.133 
87.656 
93.177 
98.694 
104.21 
109.73 
115.24 
120.75 
126.26 
131.77 
137.28 
142.79 
153.80 
164.82 
175.83 
186.84 
197.85 
208.85 
219.86 
230.87 
241.87 
252.88 
263.89 



45.281 
51.111 
56.925 
62.727 
68.521 
74.308 
80.089 
85.867 
91.641 
97.412 
103.18 
108.95 
114.71 
120.48 
126.24 
132.00 
137.76 
143.52 
149.28 
160.79 
172.31 
183.82 
195.33 
206.84 
218.35 
229.86 
241.36 
252.87 
264.37 
275.88 



47.250 
53.333 
59.400 
65.455 
71.500 
77.539 
83.571 
89.600 
95.625 
101.65 
107.67 
113.68 
119.70 
125.71 
131.73 
137.74 
143.75 
149.76 
155.77 
167.79 
179.80 
191.81 
203.82 
215.83 
227.84 
239.85 
251.86 
263.86 
275.87 
287.88 



49.219 
55.556 
61.875 
68.182 
74.479 
80.769 
87.054 
93.333 
99.610 
105.88 
112.15 
118.42 
124.69 
130.95 
137.22 
143.48 
149.74 
156.00 
162.26 
174.78 
187.29 
199.81 
212.32 
224.83 
237.34 
249.84 
262.35 
274.86 
287.36 
299.87 



51.188 
57.778 
64.350 
70.909 
77.458 
84.000 
90.536 
97.067 
103.59 
110.12 
116.64 
123.16 
129.68 
136.19 
142.70 
149.22 
155.73 
162.24 
168.75 
181.77 
194.78 
207.80 
220.81 
233.82 
246.83 
259.84 
272.85 
285.85 
298.86 
311 



53.156 

60.000 

66.825 

73.637 

80.438 

87.231 

94.018 

100.80 

107.58 

114.35 

121.12 

127 

134.66 

141.43 

148.19 

154.96 

161.72 

168.48 

175.24 

188.76 

202.27 

215.79 

229.30 

242.81 

256.32 

269.83 

283.34 

296.85 

310.35 

323 



55.125 
62.222 
69.300 
76.364 
83.417 
90.462 
97.500 
104.53 
111.56 
118.59 
125.61 
132.63 
139.65 
146.67 
153.68 
160.70 
167.71 
174.72 
181.73 
195.75 
209.77 
223.78 
237.79 
251.81 
265.82 
279.83 
293.83 
307.84 
321.85 
335.85 



The number preceding c is the cosecant of the frog angle 0. 
The number preceding c is the cotangent of the frog angle 6 



57.094 
64.444 
71.775 
79.091 
86.396 
93.692 
100.98 
108.27 
115.55 
112.82 
130.10 
137.37 
144.64 
151.91 
159.17 
166.44 
173.70 
180.96 
188.22 
202.74 
217.26 
231.77 
246.29 
260.80 
275.31 
289.82 
304.33 
318.83 
333.34 
347.85 



1090 



^.—RAILROADS. 



65. — Crossovers. Spacing of Frogs. 
Gage = 4' SVz". 

Part I. — Lengths 5 (Feet) of Straight Track between 

Frog Points. 




Fig. 65. 



Frog 
No. 



Distances Between Track Centers, c. In Feet. 



General' 



10 10.5 11 



1.5 12 12.5 13 13.5 14 14.5 15 



4 

4^ 

5 

53^ 

6 

6^ 

7 

8 

SVz 

9 

9^ 
10 

10^ 
11 

11^ 
12 

12^ 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 



-37.667 
-42.375 
-47.083 
-51.792 

• 56.500 

■ 61.208 

• 65.917 

- 70.625 

• 75.333 

■ 80.042 

- 84.750 

■ 89.458 

- 94.167 
•98.875 
-103.583 
-108.292 
-113.000 
-117.708 
•122.417 
•131.833 
•141.250 
-150.667 
-160.083 
-169.500 
-178.917 
-188.333 
-197.750 
-207.167 
-216.583 
-226.000 



2.958 

3.181 

3.417 

3.663 

3.917 

4.176 

4.440 

4.708 

4.979 

5.252 

5.528 

5.805 

6.083 

6.363 

6.644 

6.925 

7.208 

7.492 

7.775 

8.346 

8.917 

9.490 

10.064 

10.639 

11.215 

11.792 

12.369 

12.947 

13.526 

14.104 



990 
458 
942 
435 
938 
446 
958 
475 
994 
517 
042 
568 
096 
625 
155 
686 
218 
752 
285 
355 
425 
497 
571 
646 
722 
798 
875 
953 
031 
109 



021 
736 
467 
208 
958 
715 
476 
242 
010 
782 
556 
332 
108 
887 
667 
447 
229 
012 
795 
364 
934 
505 
078 
653 
229 
804 
381 
959 
537 
114 



052 
014 
992 
981 
979 
984 
994 
008 
026 
046 
070 
095 
121 
149 
178 
208 
239 
272 
304 
373 
442 
513 
586 
660 
735 
810 
887 
964 
042 
120 



084 
292 
517 
754 
000 
253 
512 
775 
042 
311 
584 
858 
133 
411 
689 
969 
250 
532 
814 
382 
950 
521 
093 
667 
,242 
817 
393 
,970 
,548 



13.116 
14.569 
16.042 
17.526 
19.021 
20.522 
22.030 
23.542 
25.057 
26.576 
28.097 
29.621 
31.146 
32.673 
34.201 
35.729 
37.260 
38.792 
40.324 
43.391 
46.459 
49.528 
52.600 
55.674 
58.748 
61.823 
64.899 
67.976 
71.053 



I25I74! 130186 



147 

847 
567 
299 
042 
792 
548 
308 
073 
840 
611 
384 
158 
934 
,712 
490 
,270 
,052 
833 
400 
,967 
536 
108 
,681 
,255 
,829 
,405 
.982 
,558 
,135 



17.178 

19.125 

21.092 

23.072 

25.063 

27.061 

29.065 

31.075 

33.089 

35. 105 

37.125 

39.147 

41.171 

43.196 

45.223 

47.251 

49.281 

51.312 

53.343 

57.409 

61.475 

65.544 

69.615 

73 

77.762 

81.835 

85.911 

89.987 

94.064 

98.140 



19.209 
21.403 
23.617 
25.844 
28.083 
30.330 
32.583 
34.842 
37.104 
39.370 
41.639 
43.911 
46.183 
48.458 
50.735 
53.012 
55.291 
57.572 
59.852 
64.418 
68.984 
73.552 
78.123 
82.695 
87.268 
91.842 
96.417 
100.99 
105.57 
110.15 



21.241 
23.681 
26.142 
28.617 
31.104 
33.599 
36.101 
38.608 
41.120 
43.634 
46.153 
48.674 
51.196 
53.720 
56.246 
58.773 
61.302 
63.832 
66.362 
71.426 
76.492 
81.560 
86.630 
91.702 
96.775 
101.85 
106.92 
112.00 
117.08 
122.15 



23.272 
25.958 
28.667 
31.390 
34.125 
36.869 
39.619 
42.375 
45.136 
47.899 
50.667 
53.437 
56.208 
58.982 
61.757 
64.534 
67.312 
70.092 
72.871 
78.436 
84.000 
89.567 
95.137 
100.71 
106.28 
111.85 
117.43 
123.00 
128.58 
134.16 



Part II. — Horizontal Distances H (Feet) between Frog Points. 



Frog 
No. 



Distances Between Track Centers, c, in Feet. 



General t 



10 



10.5 



11 



11.5 



12 



12.5 



13 



13.5 



14 



14.5 



15 



4 

4^ 

5 

5^ 

6 

6^ 

7 

8 

9 

9^ 
10 
10^ 
11 

11^ 
12 

12J^ 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 



h- 37.667 
h- 42.375 
h- 47.083 
h- 51.792 
h- 56.500 
h- 61.208 
h- 65.917 
h- 70.625 
h- 75.333 
h- 80.042 
h- 84.750 
h- 89.458 
h- 94.167 
h- 98.875 
h-103.583 
h-108.292 
h-113.000 
h-117.708 
11-122.417 
h-131.833 
h-141.250 
h-150.667 
11-160.083 
h-169.500 
h-178.917 
11-188.333 
h-197.750 
h-207.167 
h-216.583 
h-226.000 



708 
069 
417 
754 
083 
407 
726 
042 
354 
664 
972 
279 
583 
887 
190 
491 
792 
092 
391 
988 
583 
177 
770 
361 
951 
542 
131 
719 
,308 
,896 



677 
292 
892 
481 
062 
638 
208 
775 
339 
899 
458 
016 
571 
125 
679 
230 
782 
332 
882 
979 
075 
170 
262 
354 
445 
535 
625 
714 
802 



646 
514 
367 
208 
042 
869 
690 
508 
323 
135 
944 
752 
558 
363 
167 
969 
771 
572 
372 
970 
566 
162 
755 
347 
938 
529 
119 
,708 
297 



9.583 
10.958 



9.84212.317 



935 
021 
100 
172 
242 
308 
370 
431 
489 
545 
601 
656 
708 
761 
812 
863 
961 
058 
154 
248 
340 
431 
523 
613 
702 
791 



891137.886 



13.663 
15.000 
16.331 
17.654 
18.975 
20.292 
21.605 
22.917 
24.226 
25.533 
26.839 
28.144 
29.447 
30.750 
32.052 
33.353 
35.952 
38.550 
41.146 
43.740 
46.333 
48.925 
51.517 
54.107 
56.696 
59.286 
61.875 



552 
181 
792 
390 
979 
561 
137 
708 
276 
840 
403 
964 
520 
077 
632 
186 
740 
292 
843 
943 
042 
138 
233 
326 
418 
510 
601 
691 
780 
,870 



521 
403 
267 
117 
958 
792 
619 
442 
261 
076 
889 
700 
508 
316 
121 
926 
730 
532 
334 
934 
533 
131 
726 
319 
912 
504 
095 
685 
275 
865 



490 
625 
742 
845 
938 
023 
101 
175 
245 
311 
375 
437 
495 
554 
610 
665 
719 
772 
824 
925 
025 
123 
219 
312 
405 
498 
589 
679 
,770 
,860 



17.458 
19.847 
22.217 
24.572 
26.917 
29.254 
31.583 
33.908 
36.230 
38.546 
40.861 
43.174 
45.483 
47.792 
50.098 
52.404 
54.709 
57.012 
59.315 
63.916 
68.517 
73.115 
77.711 
82.305 
86.899 
91.492 
96.083 
100.67 
105.26 
109.85 



19.427 
22.069 
24.692 
27.299 
29.896 
32.484 
35.065 
37.642 
40.214 
42.782 
45.347 
47.910 
50.470 
53.030 
55.587 
58.143 
60.698 
63.252 
65.805 
70.907 
76.008 
81.107 
86.204 
91.298 
96.392 
101.49 
106.58 
111.67 
116.76 
121.85 



21.396 
24.292 
27.167 
30.026 
32.875 
35.715 
38.547 
41.375 
44.199 
47.017 
49.833 
52.647 
55.458 
58.268 
61.076 
63.882 
66.688 
69.492 
72.295 
77.898 
83.500 
89.099 
94.697 
100.29 
105.89 
111.48 
117.07 
122.66 
128.25 
133.84 



*For values of d, see Table 64, Part I. fFor values of h, see Table 64, Part II. 



CROSSOVERS—FROG SPACING. 



1091 



65. — Crossovers. Spacing of Frogs. — Concluded. 
Gage = 4'' 8>^". 
Part III. — Direct Distances D (Feet) between Frog Points. 



Frog 






Distances Between Track Centers, c. 


In Feet. 






No. 


General. 


10 


10.5 


11 


11.5 


12 


12.5 13 


13.5 


14 


14.5 


15 


4 




5.56 


6.86 


8.45 


10.20 


12.04 


13.93 


15.86 


17.81 


19.78 


21.75 


23.74 


m 




5.68 


7.21 


9.06 


11.06 


13.16 


15.31 


17.49 


19.70 


21.91 


24.14 


26.38 


5 




5.82 


7.58 


9.69 


11.96 


14.31 


16.72 


19.15 


21.61 


24.08 


26.56 


29.05 


5% 




5 96 


7.97 


10.34 


12.87 


15.49 


18.15 


20.84 


23.55 


26.27 


29.00 


31.74 


6 




6.12 


8.38 


11.01 


13.81 


16.68 


19.59 


22.54 


25.50 


28.47 


31.46 


34.45 


6^ 




6.29 


8.81 


11.70 


14.75 


17.88 


21.05 


24.25 


27.47 


30.69 


33.93 


37.17 


• 7 


6.47 


9.25 


12.40 


15.72 


19.10 


22.53 


25.98 


29.44 


32.92 


36.41 


39.90 


7^ 


.^ 

b 


6.66 


9.69 


13.12 


16.69 


20.33 


24.01 


27.71 


31.43 


35.16 


38.89 


42.64 


8 


6.85 


10.15 


13.84 


17.66 


21.56 


25.50 


29.45 


33.42 


37.40 


41.39 


45.38 


SH 


7.05 


10.62 


14.56 


18.65 


22.80 


26.99 


31.20 


35.42 


39.65 


43.89 


48.13 


9 


ej^ 


7.26 


11.09 


15.30 


19.64 


24.05 


28.49 


32.95 


37.42 


41.90 


46.39 


50.88 


9H 




Cs> 


7.47 


11.57 


16.04 


20.64 


25.30 


29.99 


34.70 


39.43 


44.16 


48.90 


53.64 


10 




1 


7.69 


12.05 


16.78 


21.64 


26.55 


31.50 


36.46 


41.44 


46.42 


51.41 


56.41 


10^ 




^ 


7.92 


12.54 


17.53 


22.64 


27.81 


33.01 


38.23 


43.45 


48.69 


53.93 


59.17 


11 




+ 


8.14 


13.04 


18.28 


23.65 


29.07 


34.52 


39.99 


45.47 


50.95 


56.44 


61.94 


11^ 






8.37 


13.53 


19.04 


24.66 


30.34 


36.04 


41.76 


47.49 


53.22 


58.96 


64.71 


12 


> 


8.61 


14.03 


19.80 


25.67 


31.60 


37.56 


43.53 


49.51 


55.49 


61.48 


67.48 


12^ 


8.85 


14.54 


20.56 


26.69 


32.87 


39.08 


45.30 


51.53 


57.76 


64.01 


70.25 


13 


II 


9.09 


15.04 


21.32 


27.71 


34.14 


40.60 


47.07 


53.55 


60.04 


66.53 


73.02 


14 




^ 


9.58 


16.06 


22.85 


29.75 


36.69 


43.65 


50.62 


57.60 


64.59 


71.58 


78.58 


15 




4- 


10.08 


17.09 


24.39 


31.79 


39.23 


46.70 


54.17 


61.66 


69.15 


76.64 


84.13 


16 




^ 


10.59 


18.12 


25.94 


33.84 


41.79 


49.75 


57.75 


65.71 


73.70 


81.70 


89.69 


17 


-> 


11.11 


19.16 


27.48 


35.88 


44.34 


52.81 


61.29 


69.77 


78.26 


86.76 


95.25 


18 


11.64 


20.20 


29.04 


37.95 


46.90 


55.87 


64.85 


73.84 


82.83 


91.82 


100.82 


19 


11 


12.16 


21.25 


30.59 


40.01 


49.47 


58.94 


68.42 


77.90 


87.39 


96.89 


106.38 


20 


Q 


12.70 


22.30 


32.15 


42.07 


52.03 


62.00 


71.98 


81.97 


91.96 


101.96 


111.95 


21 




13.23 


23.35 


33.71 


44.14 


54.60 


65.07 


75.55 


86.04 


96.53 


107.02 


117.52 


22 




13.78 


24.41 


35.27 


46.20 


57.17 


68.14 


79.12 


90.11 


101.10 


112.10 


123.09 


23 




14.32 


25.47 


36.84 


48.271 59.73 


71.21 


82.70 


94.18 


105.67 


117.17 


128.67 


24 






14.87 


26.53 


38.40 


50.34 


1 62.30 


74.28 


86.26 


I 98.25 


110.25 


122.24 


134.24 



EXCERPTS AND REFERENCES. 

Train Resistance Formulas (By J. G. Crawford. Eng. News, Oct. 31, 
1901). — ^Also diagrams of train resistance curves representing various 
formulas. 

Holbrook's Spiral Curves (Eng. News, Jime 13 and Aug. 15, 1901).— 

Formulas and tables. 

Transition Curves (By W. B. Lee. Trans. A. S. C. E., Vol. XLVI). 

Gravity Yards, Switches and Frogs of the Chicago Transfer and 
Clearing Co. (Eng. News, Jan. 2, 1902). — Illustrated. 

The Weehawken Inclined Railway (By C. L. Duenkel. Eng. News, 
Oct. 16, 1902).— Illustrated. 

The Rutland=Canadian Railway and Its Structures (By J. W. Burke. 
Eng. News, Jan. 15, 1903). — Illustrations of tiuntable machinery for swing 
bridge, details of overhead steel highway bridge, masonry cattle pass, 
masonry box culvert, surface cattle-guard. 

Largest Capacity Gondola Cars; Chicago & Alton Ry. (Eng. News, 
Feb. 26, 1903).— Illustrated. Capacity, 80,000 lbs.; weight. empty, 31,000 
to 32,000 lbs.; length over end sills, 36 ft. 

An Automatic Mine Car Tipple (Eng. News, May 21, 1903). — Illus- 
trated. 

Largest Freight Car in the World (Eng. News, July 2, 1903).— Illus- 
trated. Capacity, 300,000 lbs.; weight, 196,420 lbs.; length of car over 
couplers, 103 ft. lOi ins. 

Steel Ties on Bessemer fi: Lake Erie R. R. (Eng. News, Nov. 5, 1903). 
— Illustrated, 



1092 SQ.^RAILROADS. 

Cost of Railway Ballast (W. C. Gushing. Eng. News, Mar. 31, 1904). 
— Record of experimental test of three different sizes of broken stone ballast 
on B. & O. Ry., with cost of each: Average costs per ft. of double track, 
$1. to $1.50. 

A Conduit Electric Railway in London (Eng. News^ April 21, 1904). 
— Illustrated. 

Standard Qirder=Rail Track Construction for City Streets, Penn. 
R. R. (Eng. News, Aug. 11, 1904). — Illustrated: Cross-section of rail, and 
details of track construction. 

Screw Spikes for Railway Track (Eng. News, Aug. 25, 1904).— 

Illustrated: 7 forms of spikes, and machine for driving. 

Reinforced=Concrete Ties, Ulster & Delaware Ry. (Eng. News, 
Oct. 6, 1904).— Illustrated. 

Transfer Tables Without Pits and Traveling on Curves (Eng. News, 
Oct. 6, 1904).— Illustrated. 

Cost of Electric Railway Power Production and Transmission in 
the State of Indiana (By A. S. Richey. Eng. News, Feb. 16, 1905).— 
Efficiencies: Step-up transformers, 94%; transmission lines, 97%; step- 
down transformers, 93%; rotary converters, 80%; direct-current distri- 
bution, 80%; combined efficiency, 54%. 

The Cost for Concrete Fence Posts (Eng. News, Mar. 9, 1905). 

The San Pedro, Los Angeles & Salt Lake Ry. (Eng. News, June 22, 
1905). — Illustrations of standard roadbed cross-sections. 

A Table of Turnout Curves and Crossings (By J. H. Milbum. Eng. 
News, July 13, 1905). — Including frog numbers 10, 12, 15, 16 and 20. 

Some Records of Maintenance=of=Way Costs on American Rail- 
ways (Eng. News, July 27, 1905).— Tables. 

Concrete Ties on the L. S. & M. S. Ry. (Eng. News, Aug. 17, 1905). 
— Illustrated. (See, also, Eng. News of Oct. 5, 1905, for descriptions and 
illustrations of the Campbell and Percival ties.) 

Electric Equipment and Reconstruction of the New York Terminal 
Lines and Grand Central Station, N. Y., C. & H. R. R. R. (Eng. News, Nov. 16. 
1905). — Numerous illustrations, with 2-page insert. 

Switch Leads for Narrow=Qage Track (Eng. News, Dec. 7, 1905). — ■ 

Tratman's formula. 

Reinforced=Concrete Fence Posts (By P. L. Wormeley, Jr. Eng. 
News, Jan. 18, 1906). — Illustrations of posts, molds for making, wire attach- 
ment, etc. 

Summit or Hump Yards for Gravity Switching (Eng. News, Mar. 22, 
1906). — Illustrated. 

Curving Rails by Power; Nashville, Chattanooga & St. Louis Ry. 

(By G. F. Blackie. Eng. News, May 31, 1906). — Illustration of mechanism. 

Track Construction of Underground Railways (Eng. News, Aug. 2, 
1906). — Illustrated. 

A New Snow Scraper for Use on Locomotives (Eng. News, Aug. 9, 
1906). — ^The Root scraper — illustrated. 

Some Tables and Other Data for Railway Locating Engineers (By 

C. P. Howard. Eng. News, Sept. 13, 1906). — Formulas: Weights of bridges; 
spiral curves. Tables: Spiral curves; excavation tables, _ embankment 
tables; box culverts; areas for waterways; contents of retaining walls and 
abutments; weights of trusses, plate girders and viaducts; etc. 

Track Construction for Railway Tunnels (Eng. News, Sept. 20, 
1906).— Illustrated. 

Devices to Keep Railroad Switches From Becoming Clogged With 
Snow and Ice (By F. G. Shaw. Eng. News, Oct. 18, 1906). — ^Two systems 
described: Gas heating, and oil heating. 

Bumping Posts are illustrated and described in Eng. News, Oct. 25, 
1906. 

A Simplified Method of Laying Out Transition Curves (By T. A. 

Ross. Eng. News, Nov. 15, 1906). — Transition curve table suitable for 
speeds of about 30 miles per hoiu:. 



MISCELLANEOUS DATA. 1093 

Wooden Ties Buried in Concrete, illustrated in Eng. News, Jan. 17, 
1907. See, also, discussions of Jan. 31 and Feb. 7, 1907. 

A Noise=Deadening Experiment on the Chicago Elevated Loop (Eng. 
News, Feb. 21, 1907).— Illustrated. 

Tests of Holding Power of Railway Spikes (Eng. News, Mar. 7, 1907). 
— ^Tests made by Mr. Roy I. Webber, Instr. of Civ. Eng., Univ. of 111., and 
the results are given in Bulletin No. 6, issued by the Experiment Station. 
Screw spikes and plain spikes; direct pull and lateral displacement. 

A New System of Block Signaling on the P. R. R. (Eng. News, 
May 9, 1907).— Illustrated. 

New Rails for the Chicago Street Railways (Eng. News, May 23, 
1907). — Specifications. Illustrated section of 129-lb. grooved girder rail. 

Experience With Wide=Base Rails on the A., T. & S. F. Ry. (Eng. 
News, June 13, 1907). — Illustrated section of 101-lb. rail, 5|" high and 6f' 
width of base. The rail was designed with the idea of giving a larger bear- 
ing upon the ties, and using it without tie-plates. The rail was not easy to 
roll, and the mills experienced great difficulty with it. The results are 
said to have been satisfactory as far as the effect upon the ties was con- 
cerned, but it has been found that the rails break very readily in the base. 
The rail joints have 26" splice bars, with four |" bolts spaced 5" c. to c. 
These bars weigh about 75 lbs. per pair. 

A New 100=lb. Rail Section (Eng. News, June 27, 1907).— Illus- 
trated (6'' high, and 5i'' width of base). Table of dimensions of heavy 
rails compares this rail with the A. S. C. E. (100-lb.), Dudley or N. Y. C. R. R. 
(100-lb.), and the A., T. & S. F (101-lb.— see above). 

Notes on Recent Rail Design (Eng. News, July 25, 1907). — Illus- 
trated section of 90-lb. rail; hi" high, 5f" width of base. 

Standard Turntable Pit; Seaboard Air Line Ry. (By Philip Aylett. 
Eng. News, Aug. 15, 1907). — For 70-ft. turntable. Illustrated details. 

Single=Phase Electric Traction on the Rochester Division of the 
Erie R. R. (By W. N. Smith. Eng. News, Oct. 17, 1907).— Illustrated. 

A Tracklaying Machine With Rail Carriers (Eng. News, Nov. 28, 
1907). — Illustrated. Performance: Rails 33 ft. long, 2 to 2^ miles of track 
laid per day with 1 foreman, 4 men to operate the^ machine and feed ties 
and rails, 6 men to distribute and space ties, 8 spikers, 4 nippers, and 1 
spike peddler. 

New Interlocking Plant, Hoboken Terminal Yard, D., L. & W. Ry. 

(Eng. News, Jan. 30, 1908). — Illustrated. 

General Formulas for Simple Curves (By J. G. Locke. Eng. News, 
Mar. 26, 1908). — Diagram and numerous formulas, prepared and used on 
subway track work. 

Train Resistance (Eng. News, Mar. 26, 1908). 

New Steel Rail Specifications of P. R. R. (Eng. News, April 16, 1908). 

New Rail Sections and Specifications of the Am. Ry. Assn. (Eng. 
News, May 14, 1908). — ^Two types illustrated: A and B, eack rolled at 60-, 
70-, 80-, 90-, and 100-lb. Tables of properties. 

Standards of Track Construction on American Railways (Eng. News, 
June 4, 1908). — Four large tables comprising tabulated data from 59 rail- 
roads: Table 1. — Standard Practice as Applied to Rails and Rail Joints 
(Rail; weight per yard; length; type of section; lightest in main track. 
Rail Joints; type; square or broken; number of bolts; size of bolts; spac- 
ing of bolts; nutlock; nuts. Splice bars; length; weight per pair. Special 
or patented joints used). Table 2. — Standard Practice as to Ties and Plates 
(Ties; wood, and where obtained; average life in main track; cost; re- 
newed for wear or decay mainly; size, length, thickness, width; number 
per 33-ft. rail; spacing at joints; preservative process; number of treated 
ties in use. Tie-=plates; make; size; weight; where used). Table 3. — 
Standard Practice as to Frogs and Switches (Frogs; standard pattern; 
spring or rigid; numbers, for main track; numbers, for yards; fiangeway 
for guard rail, ins. Switches; standard pattern; length of switch rail. 
Miscellaneous). Table 4. — Standard Practice as to Spikes, Ballast and 
Curve Protection (Spikes; style; size; screw spikes used. Ballast; kind 



1094 ^.—RAILROADS, 

used; size of stone; depth under tie. Guard rails for curves; on what 
curves; flangeway, ins.; are tie rods or bars used to hold gage on sharp 
ciurves ?) 

Readjustment of Curves and Tangents in Maintenance=of-Way 
Work (By W. H. Wilms. Eng. News, Sept. 17, 1908).— Illustrated. 

Some Special Designs of Rails and Tie=Plates (Eng. News, Oct. 15, 
1908). — Illustrated sections of 85-lb. rails for Western Pac. Ry. (5i" x 5i'0 
and Great Nor. Ry. (5'' x 5'0; also plans of four types of metal tie-plates 
in use on different railways. 

Cast Manganese=Steel Rails on Curves, Boston Elev. Ry. (By H. M. 

Steward, Eng. News, Oct. 22, 1908). — ^Table showing comparative life of 
rails of ordinary Bessemer, high carbon Bessemer, nickel-steel, manganese- 
steel and open-hearth. 

A Study of Rail Pressures and Stresses in Track Produced by Dif- 
ferent Types of Locomotives on Curves (By E. E. Stetson. Bulletin No. 104, 
Oct., 1908, Am. Ry. Assn.; Eng. News, Nov. 26. 1908). — Extensive table; 
and analytic discussion of pressiu-es and stresses. 

A Wedge Rail Fastening for Steel Ties (Eng. News, Dec. 24, 1908).— 
Illustrated. It requires no bolts or clamps; the width of gage can be 
varied as required by wear; neatness of gage can be obtained, even if the 
rail section is not exact; gives greater resistance t6 shearing, as compared 
with bolts; the fastening can be insulated; and it is impossible for derailed 
wheels to destroy the fastening. 

New Rail Specifications for the P. R. R. (Eng. News, Jan. 14, 1909), 

A 1200=Volt Direct=Current Electric Railway (Eng. News, Jan. 21, 
1909).— The Pittsburg, Harmony, Butler & New Castle Ry. Table of 
horse powers. 

The Disadvantages of Concrete Foundations for Railway Crossings 

(By R. P. Black. Eng. News, April 29, 1909). — (1) The concrete gives too 
rigid a bearing under frogs, causing an anvil blow at frogs which soon wears 
down the points, especially at fiat crossings, on account of the greater dis- 
tance between points. (2) This anvil or solid blow is hard on bolts, especially 
when high speed is maintained over frogs, (3) In a high speed track con- 
crete cracks away from timbers on account of the excessive jar. (4) It is 
hard to maintain a good surface on the track, as the concrete footing does 
not permit of raising, (5) The rigid bearing is hard on equipment. (6) The 
concrete foundation for crossing is a failure. The only case where it can be 
used to good advantage is where traffic is light, speed low and angle of 
crossing 90° more or less. 

Earthwork Calculations for Side Hill Work (By R. S. Beard. Eng. 
News, Jime 10, 1909. — Diagrams, formulas and tables. 

A Railway Transfer Table Without a Pit (By H. V. Miller. Eng. News, 
July 15, 1909).— Illustrated. 

Standard. Specifications for Structural Steel for Buildings (Proc. A. S. 
T. M., Vol. IX„ 1909).— Adopted Aug. 16. 1909. 

Standard Specifications for Bessemer and for Open=Hearth Steel Rails 

(Proc. A. S. T. M., Vol. IX„ 1909).— Adopted by Aug. 16. 1909. 

Taper Curves Used on Southern Pacific R. R. (Oct. 28, 1909).— -Tables, 

formulas and illustrations. 

Special Type of Track Construction for Tunnels and Subways (Eng. 
News, Aug. 19, 1909). — Illustrated, with tables of cost. 

Tie=Plates and Braces for Guard Rails on Sharp Curves ; Natal Gov't Rys. 

(Eng. News, Nov. 11, 1909). — Illustrated description. 

Street Railway Track Construction and Paving (Report of Comm. on 
Way, Am. St. and Interurban Ry. Eng'g Assn., Oct. 4 to 8, 1909; Eng. 
News, Nov. 11, 1909). — Use of T=Rails in Paved Streets: — Recommenda- 
tions: (1) For track construction where type of pavement will permit, as in 
macadam or other shallow pavement, the T-rail weighing not less than 80 



MISCELLANEOUS DATA. 1095 

lbs. per yd. adopted as "recommended practice" by the Am. Ry. Eng. and 
M. of W. Assn., 1908. (2) For heavy service in connection with deep block 
pavements, a T-rail 7 ins. high with a 6-in. base, 17/32-in. web and a head 
of 2f ins. wide and 1-11/16 ins. deep, weighing about 100 lbs. per yd., as 
illustrated (not reproduced here). (3) For light service, in connection with 
deep block pavement, a T-rail 7 ins. high, 6-in. base, 7/1 6-in web, 2 J by 
1-9/32 in. head, and weighing 80 lbs. per yd., as illustrated (not reproduced 
here). (4) For heavy service in connection with deep block pavement on 
streets where traffic is confined to the railway strip, etc. (cities of large 
class), the half-grooved (or "Trilby") section recommended in 1907. Other 
Subjects Discussed: — Track in paved streets; Opinions of City Engineers, 
etc., as to St. Ry, track construction; Widening gages on curves; Cost and 
life of steel ties; Wear of gage lines of rails on tangents; Creosoted wood- 
block pavement; Spacing of ties; Setting of concrete foundation; Life of 
rail joints on paved streets; Efficiency of electrically brazed and soldered 
rail bonds; Discussions on rails, ties, foundations, paving. 

Train Resistance by Various Formulas (Eng. Rec, June 4, 1910). — 
Table of train resistance in pounds per ton by various formulas: (1) Am. 
Ry. Eng. & M. W. Assn., Bulletin 114, p. 4; (2) Ditto, Bulletin 84, p. 100; 
(3) Am. Loco. Co., Bulletin 1001, p. 3; (4) Am. Ry. Eng. & M. W. Assn., 
Bulletin 120, p. 26. Experiments indicate that train resistance increases 
with the speed. 

Elementary Theory of the Gyroscope in the Brennan Monorail Car (By 

E. V. Huntington. Eng. News, July 21, 1910). — Discussion of: (a) Steady 
precession; (b) Accelerated precession; (c) Application to the monorail car; 
(d) Efficiency of the Brennan apparatus; (e) Proportions of the car; (f) 
Proofs of theorems. Illustrated. 

The Track and Line Construction of Electric Railways (Eng. News, Oct. 
27, 1910). — Tabular results of inquiries as to general lines of practice on 18 
interurban railways and 17 street railways in various parts of the country. 
The tables are: (I) Length, track arrangement and ballasting; (II) Grades 
and curves; (III) Rails and rail joints; (IV) Ties and tie-plates; (V) Frogs, 
switches and switch-stands; (VI) Pole equipment; (VII) Line and car equip- 
ment, signals, etc. These tables are accompanied by a general discussion 
of prevailing practices. 

Mountain Rack Railways, and the Jungfrau Ry. in Switzerland (By E. 

L. Corthell. Eng. News, Oct. 27, 1910j. — Includes a table of 57 railways 
in Europe, Asia and Australia, and North and South America, giving date 
of building, gage of track, length, grades of adhesion and rack, kind of 
traction, minimum curve on rack, number and weight of locomotives, and 
train weight. 

Track Construction on the Chicago Street Railways (Eng. News, Nov. 
3, 1910). — "The complete reconstruction of the street railway tracks in 
Chicago was one of the requirements of the new system of municipal regu- 
lation and the agreement between the city and the companies, which went 
into effect about three years ago, and by which the city exercises a strong 
control over the construction, equipment, operation and financial affairs 
of the street railways." The supervision of this work is in the hands of a 
Board of Supervising Engineers, composed of Bion J. Arnold, George Weston, 
Harvey B, Fleming, John Z. Murphy and A. L. Drum. This article gives 
illustrated descriptions of: — (1) Types of track construction (grooved 
girder rails on steel ties embedded in concrete; same with wooden ties; 
grooved girder rails on wooden ties on broken stone ballast; T-girder rails 
with brick paving; tram-head girder rails on wooden ties in macadam). 
(2) Track material (rails, with table; rail joints; ties; tie-plates; spikes and 
tie bars; switches and frogs). (3) Foundation and paving (broken stone; 
concrete; paving; track spacing; track grade and street grade; curves at 
street intersections). (4) Methods of construction. (5) Track deflection. 

The Electric Bolt Lock as Applied to Interlocking (Report of Committee 
on Power Interlocking, presented at annual meeting of Railway Signal 
Assn., Oct. 11-13, 1910; Eng. News, Nov. 3, 1910).— The principal merit is 
safety. 

Gravity Freight Classification Yard for the P. R. R. at Northumberland, 

Pa. (By W. A. MacCart. Eng. News, Nov. 17, 1910). — Illustrated des- 
cription. 



1096 



59.— RAILROADS, 



Important Illustrations for Reference. 

Description. 
Railway ditching machine for small cuts and fills 
Ties: rein. -cone, steel-and-wood, steel-and-concrete 
Track construction with 114-lb. rails, Belgium 
120-ft. turntable for Malet locomotives, A. T. & S. F. Ry. 
American electric locomotives (tables) 
The design of the electric locomotive 

How to run transition curves without tables (C. P. Howard) 
Concrete and timber snow sheds on Gt. Nor. Ry. 
A deflection recorder for track switches, No Y., O. & W. Ry. 

Street railway tract construction in Charlotte, N. C. 
Curves showing relation of train resistance to velocity 
Track details on German railroads; anti-creeper 
Cross-sections of single and double track roadbed 
Elevated car storage yard, Interborough Rap. Tr. Co., N. Y 



Eng. News. 

Jan. 20. '10. 
Feb. 17, '10 
Apr 14, '10. 
June 23, '10. 
Aug. 4, '10. 
Aug. 4, '10, 
Oct. 13, '10. 
Dec. 15, '10. 
Dec. 22. '10. 

Eng. Rec. 
Apr. 24, '09. 
31, '09. 
29, '10. 
19. '10. 
15. '10. 



[uly 
Jan. 
Feb. 
Oct. 



60.— HIGHWAYS. 



A.— TRACTION. 

Power of a Horse. — It is generally estimated that an average horse 
weighing 1200 lbs. can exert a force of 100 lbs. for a day of 10 hours at the 
rate of 2^ miles per hr., on a fairly level grade; and for a short haul he can 
exert a force of 2| times the above, or 250 lbs. A constant force of 100 lbs. 
at 2^ miles (13200 ft.) per hr. is equal to just two-thirds of a horsepower 
(H, P.), and for 10 hrs. it is equal to 6f horsepower-hours, or 13 200 000 
ft.-lbs. of work. 

Effect of Road Surfaces on Traction. — ^The tractive force required to 
move one ton of 2000 lbs. on various kinds of level roads is approximately 
as follows: 

F 

Earth road 100 lbs. per ton of 2000 lbs. = 1:20 

Macadam road 40 *' 

Granite blocks 30 " 

Brick pavement.. . . 25 " 

Asphalt pavement.. 20 ** 

Fromf 8 " 

Steel rails \ 7V' 

To...( 7 •• 

Experiments in Iowa showed the following tractive resistances: Brick 
pavement, 25.4 to 58 lbs. per ton; asphalt pavement, 23.3 to 67.8 lbs. per ton. 
(See page 1142.) 

Effect of Grades on Traction. — In the above case, if we let F=the 



W F:W = 


A 


)00 lbs. = 1:20 = 


.05 


" " =1:50 = 


.02 


•• " =1:661 = 


.015 


" •• =1:80 = 


.0125 


•• '• =1:100 = 


.01 


" " =1:250 = 


.004 


" " =l:266t = 


.00375 


" " =1:2851 = 


.0035 



tractive force, and W = th.e load, then willrrr 

W 



A, the tangent* of the angle 



of repose, or grade of the road at which the load would just begin to slide 
or descend of its own weight. Hence, if F lbs. are required to move a load 
W on a level, it is clearly evident that 2F lbs. would be required to haul it 
up a grade 6^ = A, 3F lbs. up a grade G = 2A, etc., approximately. This propo- 
sition is often erroneously neglected, the usual formula given being, F = WG, 
in which F== tractive force, W = load, and 6^ = grade. The correct formulas 



are; 



whence 



W (A + G) in ascending.. 
W iA — G) in descending. 

in ascending . , 



W 



(1) 

(2) 

(3) 



A + G 

Records of actual tests appear in various works stating that a horse 
which can pull 1000 lbs. on a level road, can pull only 900 lbs. up a 1% 
grade, 810 lbs. up a 2% grade, 750 lbs. up a 2i% grade, 720 lbs. up a 2^% 
grade, 640 lbs. up a 3i% grade, 540 lbs. up a 4% grade, 500 lbs. up a 4^% 
grade, 400 lbs. up a 5% grade, and 250 lbs. up a 10% grade. 

Problem. — On an extremely bad earth road it requires a constant force 
F=100 lbs. to pull 1000 lbs. on a level. ^ What would be the maximum allow- 
able grade, assuming that F could be increased to 250 lbs. while ascending 
the grade ? 

Solution. — ^Transposing equation (3), above, we have, since .4=100-5-1000, 

Maximum grade, G=F-^W—A 
_ 250 100 

1000 1000 
=0.15=15%. Ans. 



* Approximately; see Sec. 59, Railroads, page 992. 

1097 



1098 00.— HIGHWAYS. 

B.— ROADS AND STREETS. 

Definitions. — A Street is a public way in a city or town, and consists 
generally of a roadway and two sidewalks. A Road consists essentially of a 
roadway or public thoroughfare through a county district, and with or with- 
out sidewalks. City streets are usually paved, while roads, and roadways of 
streets in small towns, are merely surfaced. Roads and streets may be classed 
according to the kind of surface or pavement, the selection of which will 
depend upon the kind and amount of traffic, grades, cleanliness desired, 
material available, climate, allowable first cost, cost of maintenance, etc. 

Dirt Roads, sometimes dignified by the name Earth Roads, are the 
pioneers in any new country. Dirt from the sides is simply thrown up into 
the center, forming a sort of crown for lateral shedding of rain water. In 
the Middle West, dirt roads are constructed very rapidly and cheaply by 
plowing' one or two furrows on either side, and using scrapers in casting this 
material up for the crown of the road. Where extensive road-making is 
contemplated, it is well to figure on regular road-making machines. 

Corduroy Roads are probably among the first in any new country, and 
in thinly populated sections generally, to supplant the common dirt road in 
low, marshy, wet ground. The typical corduroy road consists of round 
sticks of wood a few inches in diameter, laid transversely across the road. 
These are sometimes supplemented with half-round sticks, or slabs from 
saw-mills. At best they are but makeshifts, and give way sooner or later to 
plank- or other construction of surface of a smoother character and calling 
for less tractive power in hauling. Corduroy roads are often improved by 
crowning them with gravel, using sticks or poles as a foundation. 

Plank Roads are usually the first form of improvement in timbered 
sections where there is much rainfall. On many of our old maps, in various 
sections of the country," we may see the "Old Plank Road" shown in dotted 
lines. The typical plank streets now being constructed in our small towns of 
the Pacific Northwest are composed of planking 3'' to V thick, laid trans- 
versely, and spiked (or not) to longitudinal wooden stringers of size say i" 
by l(y\ more or less, spaced about 4-ft. centers. 
The sidewalk planking is usually about 2" thick, sfdewa/k 
laid with a slope of about Y per ft. toward the ^^^^g^ 
curb. A simple gutter is shown in Fig. 1. For Hp ^ 

the roadway, the planking is sometimes laid ^j^ ^oaetwa^ 

level from gutter to gutter, and sometimes 
crowned in the form of a parabola, the quarter 
points being f the height of the middle. The ^. 

method sometimes adopted of laying the middle .rig- 1. 

third of the roadway level, and the sides sloping, is particularly objectionable 
because of the two continuous longitudinal joints formed along the edges 
of the level portion. If the street is on a steep grade, it is generally best to 
have the longitudinal stringers under the planking "broken" and not "con- 
tinuous," in order to prevent excessive wash of the soil. Two nails are used 
at each intersection of plank with stringer. Barbed wire nails are most 
commonly preferred. Sometimes the planks, if very heavy, are simply laid 
on the stringers without spiking, but this method is objectionable. The 
stringers should be imbedded firmly in the ground. 

Gravel Roads are excellent when properly made. Angular, pit gravel is 
the best; smooth, sea-polished gravel will never bind properly unless a 
binder is added, and that increases the expense. After the gravel has been 
screened, \" to \\" , it is spread on the ground in Z" or 4" layers and thor- 
oughly compacted, after sprinkling, with a steam road roller weighing from 
3 to 10 tons. A small quantity of clay added to the gravel acts as a binder 
and it is still further improved if mixed with crushed gravel or small broken 
limestone. Sand should not be used. 

Gravel Walks are constructed practically in the same manner. The 
main thing to look out for in gravel construction is good drainage, as the 
clay binder is easily softened when saturated. Tile or box under-drains 
should be laid to carry off the surplus water, and their proper use greatly 
decreases the cost of repairs. Coal tar is sometimes used with gravel in 
making foot-walks, but the result is usually unsatisfactory. 




I 5u^i>r 



ROADS AND STREETS. PAVEMENTS. 1099 

Broken«Stone Pavement has undergone rapid improvement since the 
advent of the rock crusher and the steam road-roller. The original Telford 
and Macadam methods, about 1825, have been supplanted by more 
modem methods of construction. In providing for a good pavement, under- 
drainage should be provided when required ; a sub-grade should be prepared 
by removing all perishable matter and the top soil; the under-soil should 
be compacted and a foundation of gravel prepared when necessary, if the 
best results are expected. The broken-stone should not be larger than 2" 
for the softer rock or lY' for the harder, and if clean it should not (generally) 
be screened, as the stone dust and chips make good binders. A soft stone 
foundation and a hard stone surface are the best. Sand and gravel may be 
employed to fill the voids, but very little if any clay or loam should be 
used. Crushed granite should never be used. Trap or basalt is best for the 
surface, and limestone for the bottom. The material is spread in layers of 
about V to 4 1", sprinkled sufficiently, and compacted with steam road 
rollers weighing 5 to 10 tons. It will roll to about to the spread thickness. 
The thickness of broken stone pavement is usually from &' to 10", although 
4'' is quite common. It is to be noted that a thin pavement laid on a good 
gravel foundation is often superior to a much thicker pavement laid directly 
on the soil. The rolling should begin at the sides and work toward the 
crown of the road. 

Hydraulic-Cement Pavement consists of a 3" to 6" concrete base under- 
lying a wearing surface composed of one part hydraulic cement to two parts 
finely crushed rock, 1^" or more in thickness. The concrete base may rest 
on a gravel or cinder bed. The finished surface may be flushed with a 1 to 1 
sand and cement mixture. 

Cement Sidewalks usually consist of a 1" to \Y wearing surface com- 
posed of a 1 to 1 sand and cement mixture, overlaying a 3" to 4" concrete 
base resting on cinders. The wearing surface is often flushed with a pure, 
or nearly pure, cement mortar. 

Wood-Block Pavements have given good results when properly con- 
structed, but the expense of preparing good foundations necessary to keep 
the blocks in even surface is considerable. The best foundation is a layer of 
concrete, say V or more in thickness, and this is the practice in Europe 
where this kind of pavement has reached its highest perfection. It is also 
becoming standard in many of the principal cities of the United States. 
Experiments have been made in New York City and elsewhere in this country 
with varying degrees of success, but it can be stated that, generally, the 
trend is toward some more permanent form of construction. This is the 
case also in many of our western cities, even those of the Northwest where 
timber is very plentiful but where the wood block is being supplanted by 
asphalt, brick and Belgian block pavements. In section, the blocks may be 
of various shapes, round, square, rectangular, etc., but the rectangular 
section is perhaps the most common. The blocks should be laid with the 
fibers or grain vertical and with close joints. In the cheaper construction, 
in some parts of the West, they are laid on a bed of sand resting on a gravel 
foundation, or on planking of single or double thickness, and with joints 
packed with sand. In the better construction, they are laid on a bed of 
mortar spread on a concrete foundation, with joints smeared with tar, 
asphalt or cement, and expansion joints provided at intervals and along 
the curb. Many engineers prefer to lay the blocks so the course of joints 
will be diagonal to direction of traffic. If laid square they are more easily 
loosened by the corks of the horses' shoes. A common objection to wooden 
block pavement is made on the ground of cleanliness or sanitation. The 
blocks should be treated with some preservative, as creosote, but this is not 
always done. Blocks so treated expand much less after being laid, and 
less provision need be made therefor; but if laid untreated they will swell 
with the absorption of moisture, and careful provision must be made for 
expansion, at frequent intervals. The writer has seen"blisters" raised on 
the surface because of lack of provision in this respect. The harder woods, 
as oak, are not generally used in this country, as they wear too slippery. 

Cobblestone Pavement consists of cobblestones from 4" to 6" in longest 

diameter, set vertically, with the fattest end up, in abed of gravel, thoroughly 
rammed, and with joints filled with gravel. This pavement is but a make- 
shift for roadways, and is being supplanted by Belgian blocks or other 
pavement of better quality. It is frequently used for gutters. 




1100 m.—HIGHWAYS, 

Belgian Block Pavement consists of liard trap or basalt blocks of stone 
laid in parallel courses and rammed into a bed of sand, with sand joints. 
The blocks are usually about 7 ins. high and 5 or 6 ins. square. 

Granite Block Pavement is supplanting the Belgian blocks as the latter 
did the cobblestones, and it is now considered the best pavement for heavy- 
traffic. The blocks may be, say 7" deep, 3'' wide and 10" long, laid with con- 
tinuous parallel joints or courses at right angle to the direction of traffic, 
on a concrete foundation not less than 4" thick; 6" to 8" is better for un- 
usually heavy traffic. They are laid directly on a thin layer of sand, well 
rammed to a firm bed, and the joints filled with bituminous cement. 

Brick Pavement has become quite popular in recent years in certain 
sections of the country where a moderately durable pavement is required 
for all-around traffic at not too great expense. The brick should be uni- 
formly hard-burned and tough, and subjected to rigid inspection before being 
laid. They should be laid on edge in parallel courses (with broken joints) 
at right angle with the direction of traffic, on Portland cement concrete i" 
or more in thickness, with a cushion layer of sand, and with small joints so 
the sand will work up into the joints during rolling. The bricks may be of 
the ordinary size, or vitrified blocks may be used. 
At street intersections, the courses may be diagonal 
with either street. There is no advantage in laying 
bricks herring-bone fashion as in Fig. 2, except 
perhaps to please the eye, as in sidewalk fancies. 
Straight, transverse courses give a better foothold Fig. 2. 

for the horses. Bituminous cement, or, better, Portland cement grout 
should be used in filling the joints after the bricks are laid. Sand is far 
inferior and should never be used. 

Asphalt Pavement takes first rank for combined general serviceability, 
low tractive resistance, cleanliness and hygienic properties. It consists, in 
general, of about a 4" base of hydraulic cement concrete or bituminous con- 
crete, for a foundation. On this base is laid a i'' (finished) cushion coat or 
binder which contains about 3% more asphalt cement than the surface 
coat. The surface or wearing coat is laid with a finished thickness of about 
2". It is composed of asphaltic cement (85 parts pure asphalt and 15 parts 
heavy petroleum oil) 15%, sand and stone dust 80%, and crushed carbonate 
of lime 5%, more or less; the proportions often being varied. A bituminous 
base is composed of broken stone coated or mixed with coal-tar cement. 
It usually calls for a slightly thicker cushion coat than the above, say 1" to 
iy\ and also a lY' wearing coat. 

Asphalt Paving Blocks are made in a variety of shapes, and laid on 
sand, gravel, or concrete foundation. 

Bituminous-Rock Pavement is made from bituminous sandstone or 
limestone, of which extensive quarries are worked in California, Kentucky, 
and elsewhere. The quarried rock is broken up, melted, and rolled while 
hot. A proper amount of asphalt is added if necessary to give the required 
proportions. The product is commonly called rock asphalt. This is heated 
to about 200° F. and spread in a finished layer, after rolling, of about 2i" 
in thickness. 



PAVEMENTS. ROAD SPECIFICATIONS. 1101 

C— PAVEMENT SPECIFICATIONS. 

ALLEGHENY COUNTY (PA.) ROAD SPECIFICATIONS. 

(Geo. T. Bamsley, Chf. Rd. Engr.) 

Work by Contractor. — Do clearing, grubbing, leveling, grading, surfac- 
ing; make excavations, embankments, ditches, drains, gutters; construct 
masonry, stonework; build fences and protection railings required. In fact, 
complete road and structures. Excavation. — Straight classification, and in- 
cludes trees and clearing. Embankments rolled in layers not over 12" thick. 
Earthwork measured and paid for by cu. yd. in excav. When required, top 
soil to be removed from road surface and deposited as directed. May 
require clay and spongy material to be removed to any depth and replaced 
with gravel or coarse stone. Where possible, embankment slopes to be 
covered with 3" of surface loam. No work on covered drains, paved gutters 
or foundations to be paid for in excavation. Ordinarily, no allowance for 
excavation beyond lines of cross-section. Clearing. — Trees, stumps, bushes, 
roots, etc., to be removed; and no perishable matter allowed under em- 
bankments. Drainage. — Where required, a trench 12" wide at bottom and 
15" wide at top to be excavated at least 30" below sub-grade; at least 3" of 
gravel to be placed in the bottom, and on this lay salt-glazed vitrified drain- 
pipe as directed, with bell and spigot joints, laid with open joints, ordi- 
narily; then fill to 6" above pipe with gravel between 1" and i" screening; 
then fill to top of trench with stone between _ 3" and 2" screening. Open 
ditches paid for as excavation; covered drains, by the lin. ft., price to 
include all expenses of trenching, furnishing and laying pipe, refilling, etc. 
Dry Rubble. — For small culverts, ordinarily, and for retaining walls. Spalls 
used only where needed for leveling or pinning; large stones for foundation 
courses, and for heads and faces of culverts. Covering-stones of culverts to 
be not less than 12" thick, laid close together, and cracks closed with pin- 
ners. All walls to be of coursed rubble, laid with best bed down; breaking 
joints at least 1 ft., and to have no pinners on the face; joints not over f". 
End walls of culverts, and all retaining walls, to be capped with roughly- 
scabbled coping stones 16" thick, at least 24" wide, and as long as possible. 
This masonry paid for by cu. yd., actual measurement. Fencing. — Posts to 
bo straight locust, not less than 6" dia., with knots hewn down to face; 
spaced 8 ft. apart and set 3 ft. in ground, with 3^ ft. above surface. Top 
rail 4" sq. and notched into top of post so all surfaces will incline 45°; in 
addition to spiking, it shall be held with a j^" by 11" iron strap 30" long, 
securely nailed. Side rail, 2" by 6", notched into inside of posts and spiked. 
• Paid for by lin. ft. in place. Shaping Sub=grades. — Before the foundation is 
laid, the roadbed shall be shaped and rolled with a steam roller of at least 
10 tons, all resulting depressions filled, and the surface again rolled. 
Foundations. — Upon the sub-grade so prepared, a foundation to be laid 
according to method a, b, or c: (a) For clay or wet soil, a Telford founda- 
tion, with stones 6" to 8" deep, not over 4" wide on top, and 6" to 16" long; 
laid by hand with the broadest edges down and longest side across the 
road, on the sub-grade. Not over 10% to be less than 7" deep. Stones to 
break joints; projecting points to be broken off with hammer; wedging 
stones to be driven until foundation is to grade and 8" thick. Foundation 
then to be rolled with not less than 10-ton roller, (b) For soil of medium 
resistance, spread evenly to finished depth of 8" broken stone between 3i" 
and 2Y' ring dia.; roll this with roller of not less than 10 tons until the 
whole mass is firmly imbedded into the earth sub-way and the top is 4" 
below finished grade, (c) Where nature of soil will permit, spread broken 
stone 6" finished depth, and roll as in b. Surfacing. — Upon the foundation 
prepared according to a, b or c, spread two layers of broken, close-grained, 
trap rock, granite, ligonier or limestone, free from dirt or dust, and broken 
in fairly uniform or regular cubes, and comparatively free from fiakes or 
splinters; crushing strength of not less than 20000 lbs. per sq. in. Con- 
tractor required to furnish certified copies of railroad weights for macadam 
material shipped and placed, or cu. yds. of same when furnished by local 
crusher. The first layer to consist of 2" to 2^" broken stone and to be 2" 
thick when consolidated; rolled with at least a 10-ton roller, all depressions 
filled with stone of same quality, and again rolled to finished surface 2" 
below finished grade. The second layer to consist of 2" in consolidated 
thickness of %" to IK" broken stone to which may be added a proportionate 
amount of J^" to ^" screenings, free from dust; the whole surface then to be 



1102 GO.--HIGHWA YS. 

rolled to finished grade. If ordered, dust from the crusher shall be laid on as 
a binding course, just sufficient to bond the top and make the surface smooth, 
and in no case thicker than I''; then sprinkled and the whole rolled until 
mud flushes to the surface, and until roller causes no wave in surface. 
Engineer may vary macadam surfacing from 6'' for heavy traffic, to 4" for 
light traffic; and increase width of macadam from 14 ft. to 22 ft., or decrease 
it to 10 ft. Mile Stones. — Cut granite, 10" sq. at top and 12" sq. at bottom; 
5i ft. long, 3 ft. above ground. 

BOSTON (MASS.) PAVEMENT SPECIFICATIONS. 

(William Jackson, City Engineer.) 

Granite Block Pavement — Brick Sidewalk. 
Covers, etc. — City to reset and repair catch-basin and manhole covers 
and other structures to be left in street. Preparing Site. — Ground to be 
brought to proper sub-grades; bottom of excavation for sidewalks and 
edgestones to be rammed and rolled, and bottom for roadway to be watered 
thoroughly, made solid and of even surface, with heavy steam road roller, 
places not accessible to be tamped with hand rammers; unsuitable bottoms 
to be excavated and refilled. Edgestones. — (a) New edgestones, including 
circulars and comers, to be of Quincy, Cape Ann, or other equally good 
granite, all of same color, cut in lengths not less than 6 ft., free from bunches 
and depressions, and have horizontal beds; ends to entire depth to be square 
with top, and set with mortar joints not over |"; to be out of wind; ham- 
mered surfaces to be full to line; to be 7" wide on top and 20" deep; to be 
hammered on top, fine pointed 3" down on the back, squared with the top, 
^nd fine pointed 10" down on the face; remainder of face to be straight split; 
face to be cut square with top. (b) Excavation to be 18" wide, and its bot- 
tom to be at the sub-grade of 24" below top of finished edgestone. (c) Upon 
this bottom, the foundation to consist of clean coarse gravel 4" thick when 
rammed; then more gravel to be spread, the edgestone laid thereon, with 
closed joints, spaces under stone thoroughly filled with gravel and tamped 
firmly to grade, (d) Excavation on each side of edgestone then filled to the 
sub-grade of roadway and sidewalk, respectively, with clean gravel, laid in 
4" layers, each rammed and tamped under and around the edgestone, and 
joints carefully pointed, top, front and back, with mortar, of equal parts 
Natural hydraulic cement and clean, sharp sand, (e) Good clean gravel then 
to be laid, without ramming, against and up to top of edgestone on side- 
walk side; after which the roadway paving is to be laid and rammed, care 
being taken not to disturb the grade of stone; and thereafter all gravel and 
other material on the sidewalk side to be excavated to depth of 12" below 
top of edgestone, and replaced by gravel in 4" layers, rammed , to sub-grade 
of sidewalk. Roadway. — (Either gravel base or concrete base.) Gravel Base. 
— Sub-grade to be 12" below finished surface of roadway; and on this lay 
gravel base, consisting of coarse-screened paving gravel, not larger than f", 
thoroughly rammed into a solid layer, 4" thick when completed. Concrete 
Base. — Sub-grade to be 16" below finished surface of roadway; and on this 
lay concrete base, consisting of 1 part Portland cement, 3 parts screened 
coarse sharp sand, and 7 parts broken stone, not larger than 2^" and very 
few smaller than i", and evenly graded between these sizes, (b) Templates 
and other forms used to hold concrete in place, to be set true to lines and 
grades, and secured firmly, (c) Cement and sand, just before concrete is to 
be used, to be mixed dry, then add only enough water to make a paste, 
thoroughly worked with hoes or other tools; broken stone is then wet, 
after which the materials are handled rapidly to the end; paste is spread 
evenly over pile of stones, on platform, and the whole turned over at least 
twice, thoroughly mixed, put in place at once, and thoroughly rammed so 
that interstices between stones are filled with mortar and water flushes to 
surface, made true and parallel with finished roadway, (d) After ramming, 
concrete allowed to set; defects remedied with good concrete, (e) When 
Portland cement is specified it is to be Dykerhoff, Star Stettin, Alsen, Alpha, 
Lehigh, Vulcanite or Atlas; when Natural cement is specified it is to be 
Natural hydraulic cement, equal to best Rosendale; other brands can only 
be substituted when approved. No cement will be tested in cars, or in 
course of transportation, or on the street. Granite Block Paving. — (a) 
Standard granite blocks, 3i" to 4" wide, 7i" to 8" deep, and 9" to 14" long 
(average not less than 11^"); edges to be sharp and straight, right angle 
both horizontally and vertically; faces to be straight-split, free from bimches 



PAVEMENT— GRANITE, WOOD. BRICK WALK. 1103 

and depressions exceeding Y, and carefully piled, (b) Upon the gravel or 
concrete base, spread a layer of clean, coarse-screened bedding sand, on 
which lay the blocks in courses of uniform depth, at right angle with street 
line (ordinarily), with close joints, the longitudinal joints broken by lap of 
at least 2", sufficient sand being used to bring blocks to grade, after thor- 
ough ramming; then covered, and covering raked and swept until joints are 
filled, blocks then thoroughly rammed to unyielding bed, with surface 
parallel with grade and crown required ; then again covered and raked and 
swept as before; blocks again rammed until solid and secure at grade and 
crown of finished roadway; no ramming done within 15 ft. of paving being 
laid ; one rammer to each paver, (c) If blocks are laid with Gravel Joints, 
cover blocks (after being rammed) with clean, coarse-screened sand, dried 
by artificial heat if necessary, and rake and sweep until joints are filled; 
entire area covered with 1" layer, (d) If blocks are laid with Pitch Joints, 
lay as above, but the covering is to be washed pebbles, equal to best Long 
Island white pebbles, ^" to f", thoroughly heated and raked or swept, 
filling joints to within \" of surface; joints then filled with paving cement of 
proper consistency, flush to grade and crown of finished roadway; the 
cement left upon top of blocks to be covered with dry sand, sufficient to 
absorb the cement if required. The paving cement to be obtained by the 
direct distillation of coal tar, and kept at a temperature of 300° F. while 
being used. Flagging Crosswalks. — (a) Granite flagging stone, each exactly 
2 ft. wide, not less than 4 ft. long, of same thickness as the others, not less 
than &' nor more than 7'',^ of best grade and quality, uniform color, top 
rough pointed, and ends jointed and square-cut to full depth of stone. 

(b) Upon the gravel or concrete base, spread a layer of clean, coarse-screened 
bedding sand, in which lay the flagging crosswalks; to be rammed and 
tamped to a solid and unyielding bed, sufficient sand being used to bring 
surface of flagging to grade and crown of finished roadway, after ramming. 

(c) If crosswalks are to be laid with Gravel Joints, fill joints with sand as for 
block paving with gravel joints, (d) If crosswalks are to be laid with Pitch 
Joints, fill joints with paving cement as for block paving with pitch joints. 
Brick Sidewalks. — (a) Bricks to be burned hard entirely through, straight- 
edged, of compact texture, regular and uniform in shape and size; bricks 
which after being thoroughly dried and then immersed in water for 24 hrs. 
absorb more than 16% of their volume, may be rejected; any edge of a 
brick sidewalk not against a curb or buildings is to be supported by a con- 
tinuous spruce plank, 2" by 8", held by 2" by 4'' spruce stakes driven in the 
ground, (b) Excavation for sidewalk is to have its bottom brought to sub- 
grade 8" below finished surface of walk, and on this lay foundation consisting 
of coarse-screened paving gravel, not larger than f, rolled and rammed so as 
to be 4" thick when completed, (c) On this foundation, spread a layer at 
least 2" thick of clean, sharp sand, parallel with finished grade of walk; on 
this, the bricks are to be laid on their widest side, in courses of uniform 
width and depth, at right angle with street, or in herring-bone fashion, with 
close joints, all longitudinal joints broken by at least 2"\ the bricks then to 
be covered with clean, fine, dry sand, using screen of 20 meshes to an inch; 
and upon the bricks, a plank, covering several courses, is to be placed and 
rammed carefully with a heavy hammer to a firm bed, with surface to proper 
grade; then spread fine sand over surface and sweep or rake so as to fill 
joints. 

Wood Block Pavement — Brick Sidewalk. 
Covers, etc. — (Same as for granite block pavement.) Preparing Site. — 
(Same as for granite block pavement.) Edgestones. — (Same as for granite 
block pavement.) Roadway. — (Either gravel base or concrete base.) Gravel 
Base. — (Same as for granite block pavement.) Concrete Base. — (Same as for 
granite block pavement, except that sub-grade is to be 10^'' below finished 
surface of roadway for wood block pavement.) Flagging Crosswalks. — (Same 
as for granite block pavement.) Wood Block Pavement. — (a) Southern long- 
leaf yellow pine, not less than 90% of heart, texture permitting satisfactory 
treatment; sticks inspected at works before being sawed into blocks, 
(b) Blocks to be of sound timber, free from bark, loose or rotten knots, or 
other defects which would be detrimental to life of block or interfere with 
laying; no second growth timber allowed, (c) Blocks to be well made, 
rectangular and of uniform size; depth (parallel to fiber) 4"; length not less 
than 8"; width not less than 4''; depth and width to be exact, (d) The 
Method of Treatment to conform to the best and most advanced knowledge 
of the art, the purpose being to allow contractors to manufacture block by 



1104 eo.— HIGHWAYS. 

following any preferred detail and by use of any process which may properly 
be adapted to secure the results demanded, namely, that all parts of each 
block shall be thoroughly impregnated with the preservative (an antiseptic 
and water-proofing oil), not less than 20 lbs. per cu. ft. of wood; the block 
not to split or warp, and to have a specific gravity greater than that of 
water, (e) The preservative to have a specific gravity not less than 1.12 at 
68° F. When distilled in a retort, with the thermometer suspended not less 
than V above the oil, it is to lose not more than 35% up to 315° C. and not 
more than 50% up to 370° C. Oil to be free from adulteration or foreign 
material, (f) After treatment the blocks are to show such waterproof 
qualities that, after being dried in an oven at a temperature of 100° for a 
period of 24 hours, weighed, and then imniersed in clear water for a period 
of 24 hours and weighed, the gain in weight is not to be greater than 3%. 
(g) Material and blocks may be rejected if not satisfactory, (h) Upon the 
surface of the concrete foundation is to be spread a bed of cement mortar 
Y' thick, the surface to be composed of slow-setting Portland cement and 
clean, sharp sand, free from pebbles over i'' diameter, 1 part cement to 4 
parts sand ; this mortar top to be thoroughly ramnied into place with con- 
crete rammers until all unevenmess in the concrete is taken up, and is then 
to be "struck" to a true surface parallel to top of finished pavement, (i) On 
this mortar surface, lay the blocks with the grain vertical and at such an 
angle with the curb as may be directed; to be laid in parallel courses with 
tight joints, firmly imbedded in the mortar bed so as to form a true and even 
surface, (j) Joints then to be filled with cement grout, 2 parts clean sand 
and 1 part Portland cement, mixed to a liquid form, and the surface of the 
block slushed with same and the joints swept until completely filled; ex- 
pansion joints, filled with a paving cement of proper consistency, to be 
»made next the edgestones. Surface then covered with ^^ of screened sand, 
(k) Where grade of street is more than 3%, the blocks are to be not less 
than 8'' nor more than 10''' long, and the upper edge of each block is to be cut 
away for a width of Y\ and a depth of 1", to provide a transverse groove 
between each course of blocks when laid in place ; or such other construction 
is to be used as will provide an equally good foothold for horses. Brick 
Sidewalks. — (Same as granite block pavement.) 

Asphalt Pavement. 

Bituminous Concrete Binder. — (a) On the concrete base, covered with 
Trinidad asphalt, lay the binder, (b) In making, use 15 gallons Trinidad 
asphaltic cement and 1 cu. yd. of crushed stone, not over i", heated and 
thoroughly mixed, (c) In using, this binder, while hot and plastic, is to be 
evenly spread and thoroughly rolled until the roller ceases to make any 
impression, the compressed binder to be at least 1^'' thick, on which an 
asphalt wearing surface is to be laid. Trinidad Wearing Surface. — For 
Trinidad asphalt pavement, the wearing surface is to be composed of (1) 
Trinidad Lake asphalt, specially refined and brought to a uniform standard 
of purity and gravity; (2) heavy petroleum oil, freed from all impurities 
and brought to a specific gravity of 18° to 22° Baum^, and a fire test of 
250° F.; (3) sand entirely free from clay or other objectionable material, of 
such size that none of it will pass through a No. 80 screen, and all through a 
No. 10 screen; (4) powdered carbonate of lime of such degree of fineness 
that 15% by weight will pass through a No. 100 screen, and all through a 
No. 26 screen, (b) In making, 100 parts of the asphalt and 15 to 20 parts 
of the petroleum oil are to be made into an asphaltic cernent, which is to 
have fire test of 250° F., and a temperature of 60° F. is to have a 
specific gravity of 1.19; this cement and the sand are to be kept heated 
separately to about 300° F., and the carbonate of lime is to be kept 
cold; 70 to 83% of the sand so heated and 5 to 15% of the lime while cold 
are to be thoroughly mixed together, but the lime may be reduced or 
omitted if the sand is satisfactory in quality and quantity; to this mixture, 
while so heated, is to be added 12 to 15% of asphaltic cement at a tempera- 
ture of about 300° F., kept at that temperature, and thoroughly mixed in 
a suitable apparatus, (c) In using, spread the above mixture, at about 
250° F., evenly on the concrete binder by hot iron rakes, to produce a uni- 
form surface; then compress with tamping irons and hand rollers and sweep 
a small amount of dry hydraulic cement over it; then thoroughly compact 
by steam roller, not less than 5 tons, until roller fails to make any impression 
on surface, the finished wearing surface not to be less than 1^'' thick. Sicilian 
Wearing Surface. — (a) For Sicilian asphalt pavement, the wearing surface is 



PAVEMENT—ASPHALT, BITUUTHIC, MACADAM. 1105 

to be composed of (1) natural bituminous limestone rock, mined by the 
"United Limmer & Verwohle Rock Asphalt Co., Limited," at Ragusa, 
Sicily; (2) limestone rock mixed by said company at Vorwohle, Germany, 
(b) In making, 3 or 4 parts of the Ragusa rock are to be thoroughly mixed 
with 1 part of the Verwohle rock, and this mixture to be crushed, pulver- 
ized to a powder and passed through a fine sieve, nothing being added or 
taken from the powder, (c) In using, spread the above mixture, at about 
160° F., evenly upon the concrete base, and compress evenly by heated hand 
rollers and rammers, smooth by heated smoothers and roll for 2 or 3 days 
with a heavy iron roller until it ceases to make any impression, the finished 
wearing surface not to be less than 2" thick. Coal Tar Painting. — The wearing 
surface, to a width of 24" from curb, is to be painted with coal tar distil- 
late and ironed with hot smoothing irons, to make a continuous layer without 
holes. 

BiTULiTHic Pavement. 

Sub^Qrade. — 6" below finished surface of roadway . Crushed Stone Founda- 
tion. — Upon the sub-grade, lay a foundation consisting of a layer of hard 
crushed stones to a depth of 6", and compress with heavy steam road-roller. 
Upon these stones, spread a thin layer of Warren's No. 1 Puritan brand 
bituminous semi-liquid cement, to be flexible and to unite freely with the 
cold stones. Upon this cement, spread a heavy coating consisting of 1 gallon 
of Warren's No. 24 Puritan brand hard bituminous cement to each sq. yd. 
of surface, the wearing surface immediately spread thereon, and the stones 
firmly bound together and with the wearing surface by this coating. Wearing 
Surface. — Hard crushed trap rock to be heated in a rotary mechanical dryer 
to a temperature of about 250° F. This material then to be elevated, passed 
through a rotary screen having 6 sections, each with a different sized opening, 
the largest 1 f " and the smallest xo'' diameter, the materials to be separated 
by these sections into 6 lots, each lot consisting of the materials passing 
through one of the sections of the screen into a separate compartment or 
bin. The materials in each lot are then weighed separately and mixed with 
the materials of each of the other lots into batches, in the proportions which 
shall have been determined by laboratory tests to give the best results, 
that is, the most dense mixture of mineral aggregate having inherent 
stability; and if the fine crushed rock does not provide the best proportion 
of fine-grained particles there must be supplied not more than 15% of hy- 
draulic cement, pulverized stone or very fine sand. Each batch is then 
passed into a "Twin Pug" or other approved form of mixer, and then mixed 
with a sufficient quantity of Warren's No. 21 Puritan brand bituminous 
waterproof cement to thoroughly coat all the materials and fill all voids; the 
cement when used is to be at a heat between 200° and 250° F., and the 
amount used with each batch is to be accurately weighed and used in such 
proportions as shall have been determined by laboratory tests to give the 
best results, the mixing to continue until mixture is a uniform bituminous 
concrete, that will when cold have as closely as practicable the solidity and 
density of solid stone. This concrete is immediately after mixing to be 
hauled to the street, spread on the No. 24 cement, and compressed with a 
steam road-roller to finished thickness of 2". Surface Finish. — On the wearing 
surface, a thin coating of Warren's quick-drying bituminous flush-coat com- 
position is to be so spread over the surface that any unevenness or honey- 
combing in the concrete is filled. A thin layer of stone chips is then to be 
rolled into the surface so it will be gritty and not slippery. In General. — 
Each layer of the work to be kept as free as possible from dirt so that the 
layers will unite. Bituminous cement used shall be free from water, petro- 
leum oil, water gas or process tars, and all light oil, naphthalin and other 
crystalline matter susceptible to atmospheric influences removed by refining. 

Macadam Roadway — Crushed-Stone Sidewalk. 

Granite Block Paving. — To be used for gutters and brows for crosswalks; 
blocks to be 3^ to W wide, 7" to 8'' deep, and 7" to 12'' long (average not less 
than lO'O- Sub-grade 12" below finished surface; on this, the gravel base 
4" thick when completed. On this foundation, spread a layer of bedding 
sand, and in this lay the blocks, in courses of uniform width and depth at 
right angle with street line (ordinarily), with close joints, longitudinal joints 
lapping at least 2", enough sand used to bring blocks to grade ; usual spread- 
ing of gravel on surface, sweeping and ramming. Brick Block Paving. — To be 
used for gutters and crosswalks; blocks to be re-pressed, hard, tough, com- 



1106 m.— HIGHWAYS, 

pletely vitrified, rectangular in shape, uniform in size and color, homogeneous 
in texture, and substantially 3" wide, V deep, 9" long. Rattler Test for 
abrasion to be as specified by Natl. Brick Manfrs'. Ass'n*; average loss in 
weight not to exceed 20%, and no one brick to lose more than 24%. Absorp- 
tion Test, using 5 bricks previously subjected to rattler test, or bricks broken 
in half; bricks to be dried 48 hrs. at 230° to 250° F., then weighed and im- 
mersed in water 48 hrs., wiped and weighed, increase in weight not to exceed 
4%. Sub-grade for brick blocks to be 10'' below finished surface; on this, 
lay V layer of paving gravel. On this foundation, spread a layer of bedding 
sand, and in this lay the bricks, in courses of uniform width and depth at 
right angle with street line (ordinarily), with close joints, longitudinal joints 
lapping at least 2", enough sand used to bring blocks to grade, usual spreading 
of gravel on surface, sweeping and ramming. Macadam Surface. — Sub-grade 
6" below finished surface; on this lay the macadam surface, made as follows: 
Hard, durable, broken stones, either of the best quality of broken trap or 
Roxbury conglomerate, or of acceptable field stone, 2|" to V diameter screen, 
free from round or other ill-shaped or improper stones, to be spread over whole 
surface of base, and thoroughly rolled and packed with 15-ton steam road- 
roller, until surface is i" below finished roadway; spaces between stones then 
to be filled with fine screenings or binding gravel applied in at least 3 layers, 
each layer thoroughly worked in by wetting and rolling aforesaid before the 
next layer is applied, and during the operation the surface to be brought, 
with the broken stone, to a finished grade. Crushed=Stone Sidewalk. — Sub- 
grade 5" below finished surface of walk; on this, spread 2^" to V broken 
stone, making a 3" layer after rolling and ramming. On this, a layer of No. 2 
crushed stone to be spread and thoroughly rolled to 2" thick. On this, spread 
* a layer of fine screenings, trimmed, watered and rolled with steam, horse or 
hand rollers so as to make a hard, compact sidewalk at required grade. Edges 
to be supported by spruce plank if required. 

THE PROPER CONSTRUCTION OF BRICK STREET PAVEMENTS. 

(Will P. Blairt.) 

Sub=Qrade. — Must be drained, graded, compacted and parallel with 
grade of finished street; not essentially different than required for other 
pavements. A depression here and there, a spot of loose earth, a lack of 
thorough compaction, or a wet condition due to improper drainage will be 
followed by disaster to the street as a whole. Foundation. — (a) Impossible 
to define the proportions of cement, sand, broken stone or gravel that shall 
compose the mix, because of the varied qualities of these materials; but in 
order to secure maximum strength they must be mixed dry in the first in- 
stance, and then thoroughly mixed after the water is applied, (b) Either 
in the machine or hand mixing, an intelligent supervision is worth while at 
all times. The value of the concrete is often reduced at least 50% by care- 
lessness, by ignorance or indifference, by application of too much or too 
little water, by lack of proper proportion of some one or another of the 
other ingredients composing the foundation, resulting in 1, 2, or 3 sq. yds. of 
the concrete foundation being of no more value than merely loose piles of 
stone or gravel, (c) The concrete surface as it is put in place must have a 
uniform surface with grade of finished street, and the surface must be smooth. 
This cannot be accomplished by the eye; the grade stakes should be set at 
no greater distance apart than 4 or 5 ft. If any stone used in the concrete 
exceeds 2" in largest diameter it will be next to impossible to accomplish the 
condition desired. Sufficient water should be used in the mixing so that one 
man can smooth the top with an ordinary dirt shovel — never should it be so 
stiff as to call into use a rammer. When we say "smooth surface" we mean 
that a greater variation than 1" shall not be allowed. Sand Cushion. — Must 
be 2" thick; if less, it will not afford a sufficient relief from the vibration 
created by the impact of travel; if more, it cannot be sufficiently compacted 
to afford a support to the load coming upon the brick street, and prevent 
cracking and crushing of the joints of the cement filler which is required in 
finishing the street. Thus, this cushion must be of such a thickness that 
will afford relief from the impact and weight, slight though it be, yet suffi- 
ciently unyielding to furnish the support for the load it must bear. Expan=» 
sion Cushion. — This must be provided, after the sand cushion is spread, by 

* See page 507. 

t Secretary National Paving Block Manufacturers' Association. 



PAVEMENT— BRICK (STREET), BOULDER, 1107 

placing next the curb a board of sufficient width to extend above the height 
of the brick; and in order that it may be drawn readily, it is advisable that 
a wedge be dropped at intervals of 3 or 4 ft. behind this board and extending 
above it from 3'' to 4''; the wedges to be Y' thick at top; the thickness of 
board varying with width of street, providing sufficient thickness ranging from 
V to IV*' Laying. — The brick should be placed in the street with the best 
edge up. This is a rule universally required of brick construction in masonry 
work. In order that this shall be done, the brick should be delivered to the 
person who drops them into the street with the face placed to suit the hand 
operation of such person, called the brick layer. The brick should not be 
laid in place in close contact with one another. Such practice will result in 
the brick being chipped, and it will be impossible to put in the cement filler 
properly in the interstices. Inspection. — After the bricks are placed, they " 
should be inspected before being rolled so that as few bricks as possible will 
be disturbed after the rolling. Rolling. — The roller should be a light one, 
from 4 to 5 tons; one that is easily handled, and can move rapidly upon the 
surface of the brick. The rolling should proceed from each side along the 
curb, working toward the center of the street; then cross-rolling at angles 
of 45°; again rolling longitudinally and cross-rolling as before, continuing 
this process until the bricks are thoroughly compacted into the sand, so that 
the grade of the pavement shall be as intended and the inequalities of the 
cushion ironed out by the sand being pushed up in the interstices of the 
brick, a condition always found in the case of properly rolled streets by an 
uneven amount pressed upwards in the interstices running from ^'' to f" and 
possibly V in some cases. (The use of the horse roller and the 8 to 10-ton 
steam roller should be prohibited). Wetting. — After rolling, the bad bricks 
should be replaced with good ones, the street swept clean, and then sprinkled, 
by the use of a nozzle either upon a sprinkling can or a hose which will 
permit but the finest spray of water to come upon the street. Cement Filler. — 
The sand to be clean, sharp and dry; the mixing, not over i bu. of sand and 
same amount of Portland cement, to be placed in box and mixed dry until 
mass is of even shade; water then added, forming mix. like thick cream. 
Must be kept in constant motion from time of mixing until floated into 
joints. Mix. to be removed from the box to the street surface with a scoop 
shovel; box to be 3^ to 4 ft. long, 27'' to 30" wide and H'^ deep, with one 
corner low, and 8" to 10'' above pavement. The mix., from the moment it 
touches the bricks, shall be thoroughly swept into the joints. Two boxes 
to be provided where street is under 20 ft. wide; over 20 ft., 3 boxes. This 
work of filling should be carried forward in line until an advance of 15 to 20 
yards has been made, when the same force shall be turned back and cover 
the same space in like manner, except that the proportions for the second- 
coat shall be § Portland cement and J sand. To avoid possibility of thickening 
at any point, there should be a man with a sprinkling can, the head per- 
forated with small holes, sprinkling gently the surface ahead of the sweepers. 
Within i to f hour after the second coat is applied, and grout between joints 
has fully subsided, and initial set is taking place, the whole surface is to be 
slightly sprinkled and all surplus mixture left on the tops of the bricks 
swept into the joints, bringing them up flush and full. Then, after sufficient 
time for evaporation has taken place, a Y layer of sand shall be spread over 
the^ whole surface, and if under a hot summer sun, the sand should be 
sprinkled occasionally for a few days. 

CINCINNATI (OHIO) PAVEMENT SPECIFICATIONS. 

Boulder Pavement. 
Sub-Grade. — Brought to even surface, parallel with grade proposed for 
pavement, making necessary excavation and embankment. Soft or spongy 
earth, etc., to be removed, and space filled with broken stone, rammed or 
rolled. Sub-grade surface to be compacted by rolling with steam roller 
pressing not less than 250 lbs. per lin. in. of roller; portions not accessible 
to be rammed. Finished sub-grade to be 14" below surface of pavement. 
Foundation. — Upon the sub-grade thus prepared the entire surface of the 
roadway between the gutters will be spread evenly with a layer of sound, 
hard hill limestone, broken into fragments as nearly regular as practicable, 
not over 2^" dia.; the layer to be of such thickness that when thoroughly 

* This cushion to be composed of pitch or asphaltum composition, 
filling f the allotted space, the remaining top third to be filled with sand. 



1108 m.— HIGHWAYS. 

compacted its surface shall be 6" above true surface of sub-grade, using roller 
above described. Where additional material is required, after rolling, to bring 
surface to proper grade, the rolled surface must be loosened to depth of 2" 
to receive the new material, and afterward rerolled. Gravel Layer. — On the 
broken-stone foundation, spread a layer of gravel, loose, and of sufficient 
depth in which to pave the boulders. The gravel must be clean and free 
from animal or vegetable matter or refuse; it must not contain more than 
15% of clay or loam, nor pebbles exceeding 1" longest diameter. Paving 
(General). — In paving, the foundation work shall be kept laid to proper grade, 
rolled or rammed into proper slope or shape at least 100 ft. ahead of paving; be 
laid in sections of not less than 100 ft. in length, entirely free from gravel, 
rubbish, etc., and thoroughly swept, ready for inspection. Boulders. — The 
boulders will be laid down between the gutter-flagging; to be of good shape 
free from flaws or breaks, of hard, imperishable substance, no sandstone or 
limestone boulders to be used, no stone to measure less than 4'' nor more 
than 6'' in longest diameter; the stones to be carefully assorted and so placed 
that the largest shall be next the gutter-flagging and gradually diminish in 
size to the center. Laying. — No stone when set in an upright position, to 
show a horizontal diameter of less than 3" or more than 6'' in any direction, 
and they must be set firmly on the foundation in a perfectly upright position, 
with small ends down, and as closely and compactly together as possible; 
none to be laid flat or on side edge. When boulders have been set for a 
distance of 60 ft., the first 50 ft. must be lightly rammed, after which a 
covering of gravel, sufficient only to fill the interstices, will be spread over 
the surface and thoroughly broomed in, when the whole will be thoroughly 
rammed with not less than 40-lb. rammers. When two sections (aggregat- 
ing 110 ft.) have been treated, the first 100 ft. will be again covered with 
gravel, broomed, rammed, and ready for inspection. As soon as each 
section of 100 ft. is accepted, a final covering of 2" of gravel will be spread 
over the entire surface. 

DETROIT (MICH.) PAVEMENT SPECIFICATIONS. 

General. 

Old Pavement and curbing to be measured to contractor as excavation; 
all old material and rubbish, including surface dirt, to be removed, city 
reserving curbing, etc. for new pavement, same to be delivered free by con- 
tractor to nearest city yard or for distance of 1 mile, if required. Contractor 
to use care in removal of old cusjiion sand to prevent mixture with other 
material. Grading. — ^After excavation to sub-grade, should there be places 
in street which are not firm, the earth must be taken out and the space re- 
filled with crushed stone and rolled. Before laying the concrete foundation 
and after the curbstone has been set, the sub-grade shall be rolled with 7-ton 
roller furnished and operated by City, at cost to contractor of ^c. per sq. yd. 
After this rolling, high places shall be brought to sub-grade and depressions 
filled with concrete at expense of contractor. Sub-grade to be properly 
planked before teaming is allowed. Curb Trench. — ^Trench to be excavated 
on each side of roadway to depth sufficient to set curb on concrete base 6" 
deep and of such width as to allow concrete backing to the curb of 4'' thick- 
ness; bottom to be smoothly trimmed parallel to curb grade. Stone Curbing. 
— Old curbing taken up shall be rejointed, edges rounded, retopped,_refaced 
and reset wherever directed, as per specifications for jointing, facing and 
setting new curb. New curb to be of best quality of granite, Medina, North 
River Blue, Elyria, Berea, or other curb as may be bid upon and ordered. 
The stone shall be 4'' thick (ordinarily), at least 3 ft. long, and 18'' deep; 
upper corner next to roadway to be rounded with radius of 1^". Top and 
face of above-named stone curb to be dressed to what is known as 4-cut 
work, true and even, and the softer curb to be crandall dressed, true and 
smooth, all with close joints at the ends of at least 7'' below top of c\irb, 
and a joint of not to exceed ^'^ for the remaining 18" depth of curb; stones 
to have straight and even face on gutter side to depth of T' below top. 
Back of curb to be dressed 3'' down from top. Top to be dressed to a straight 
line, and to i" bevel in 5", and to uniform thickness or 4'' (ordinarily). 
Stones to be taken out of wind, set with close joints to street line and grade, 
on concrete bed 6" deep, and to full width of trench, and backed up with 
concrete to within 4" of top of curb; the remaining 4'' behind curb to be 
filled with suitable earth well compacted. The crushed stone for concrete 
to be i'' to V. Concrete Curbing.-— Concrete for cement curb, plain or rein- 



CURBING. BRICK PAVEMENT ON CONCRETE. 1109 

forced with metal, shall consist of not more than 4 parts of V to Y broken 
stone or slag, 2 parts sharp sand, and 1 part Portland cement; curbing to 
be of approved construction and finish. Foundation. — When the roadbed 
has been prepared, it shall be covered with a layer of concrete not less than 
-&' thick, and rammed until the surplus cement mortar appears on the sur- 
face, which shall be smooth and parallel to roadbed. No teaming allowed 
until set and covered with plank; defects to be repaired before work pro- 
ceeds. Concrete. — Broken stone, 2'' to Y', may be from boulders, granite, 
syenite, slag, or hard limestone; it must be clean, screened if necessary to 
free it from dirt or stone refuse, and wetted before being placed on mixing 
boards. The concrete shall consist of 1 part natural^ cement, 2 parts sand, 
and 4 parts stone or slag; or 1 part Portland cement, 3 parts sand, and 6 
parts stone or slag; depending upon which cement is specified. 

Brick Pavement on Concrete Foundation. 

Cushion. — Coat of clean, sharp, bank, lake or river sand, well screened, 
to be spread over concrete foundation to depth of \Y' when compacted. 
Brick Paving. — Shall consist of best quality of sound, hard, burned paving 
brick, or cement brick, made especially for street paving purposes, and to 
stand all reasonable tests as to durability and fitness, to which paving 
material is usually, subjected. Bricks to be round or bevel-edged, straight, 
free from cracks and other defects, of uniform size, and of approved quality, 
equal to approved sample in office. Handling and Piling Brick. — Brick to be 
handled with brick tongs, carefully, to avoid breakage or chipping, and 
piled on street in rectangular piles, with uniform tiers or courses to aid 
counting. Manner of Laying. — Upon the cushion, the pavement to be laid 
with a single layer of brick on edge, end to end, in right angle or diagonal 
courses across the street, as may be directed, except at street intersections 
and along street railway tracks where the courses are to be placed at such 
angles as may be designated. Bricks to be set in straight courses, with body 
of bricks close together, sides and ends touching, and breaking joints at 
least 2" with the bricks in adjoining courses; to be set perpendicular to 
grade of street, and to height of from Y to V, or as may be directed, above 
the true grade and crown of street when finished, to allow for settlement 
in pounding and rolling. Whole bricks to be used, except in starting a 
course or in making a closure, when not less than half bricks may be used in 
breaking joints, tight and close at ends. Rolling and Tamping. — The paving 
when laid, and before filling of the joints and top dressing is put on, shall be 
rolled three or more times lengthwise of street, with not less than 7-ton 
roller, furnished and operated by City at |c. per sq. yd. Parts which cannot 
be rolled shall be rammed. Tar Filling. — ^Whenever tar filler is specified, the 
joints to be filled to the bottom with paving cement obtained from the 
direct distillation of coal tar, and shall be residuum thereof, such as is 
ordinarily numbered 6 and 6 at the manufactory, or any other approved 
composition; quality and temperature to be approved. Extra material 
and care to be used at gutters, catch-basins, etc., to prevent leakage of water 
to sub-roadway. Grout Filling. — A joint, Y' in width, next to and parallel 
to the curbstone, to be filled to top with a composition of coal tar cement, 
mixed with at least 10% of refined asphalt, and the whole mixed with 
sufficient still wax to prevent softening or brittleness in hot or cold weather. 
On streets with car tracks, three rows of brick to be laid along outside of 
rails in form of stretchers, with broken joints, and all joints filled with 
above composition. Balance of pavement to be filled with grout, composed 
of 1 part Portland cement and 1 part sand. The grout to be prepared in small 
quantities, stirred while being applied to pavement, and swept into joints 
with proper brooms; no settlings or residue to be used. Filling to be done 
by two or more applications of grout; the first I" in depth from the bottom 
to be filled with grout somewhat thinner than required for the remainder; 
the balance with a thicker grout, and if necessary refilled; brick to be pre- 
viously wet. Teaming and traffic prohibited for about one week. Top 
Dressing. — Surface of paving to be covered with Y top dressing of sand. 
Retaining Stone. — At intersections with paved streets and alleys having a 
different surface, the pavement shall be finished up to a Medina stone 
header i" thick, not less than 16'' deep, and not less than 30" long, to be set 
on a concrete bed 6'' deep, 8" wide and backed with 4" of concrete to within 
4'' of top. Stones to be dressed on top, pointed down on both sides to bot- 
tom of surfacing material, having good joint for depth of Z" from top, and 
set to grade. 



1110 m.— HIGHWAYS. 

Sheet Asphalt Pavement on Concrete Foundation. 

Binder. — Upon the concrete bed a binder course to be laid, composed of 
clean, broken stone, varying in size from fine to coarse, all to pass a 1^" ring 
in its larger dimensions. Stone after being heated shall not contain less 
than 5% nor more than 15% of material passing a No. 10 screen. Stone to 
be heated not higher than 350° F. in suitable appliances; then thoroughly- 
mixed by machinery with asphalt ic cement, such as is acceptable for surface 
cement; penetration, 60 to 90, at 77° F., City standard, at such tempera- 
tures and in such proportions that the resulting binder will have life and 
gloss without an excess of cement. Should it appear dull, from overheating 
or lack of cement, it will be rejected. While hot, it will be hauled upon the 
work and spread upon the base, so that when compacted it will be at least 
li'' thick, and immediately rammed and rolled until it is cold. Wearing Sur- 
face. — Upon the binder course will be laid the wearing surface, or pavement 
proper, the binding material of which must be a cement prepared from 
asphalt, refined until free from water and volatile oils. This surface to be 
composed of asphalt ic cement, clean, sharp-grained sand and fine absorbent 
mineral dust. The Asphaltic Cement must be prepared from refined asphalt 
of one of the following brands: Trinidad, Bermudez, Obispo, or any other 
equally as good. The refined asphalt shall be softened into a proper asphaltic 
cement by the addition of a suitable flux. The flux must be either a resi- 
duum from eastern petroleum oil, Texas petroleum oil, or a maltha from 
which the light oils and water have been removed by distillation. The 
asphaltic cement to be satisfactory, practically free from water, and within 
the range of 40 and 70 penetration (amount of penetration to be fixed by 
Department of Public Works), at 77° F., City standard. The Sand to be 
hard grained and moderately sharp. On sifting, it should have at least 15% 
caught on a 40 mesh to the inch screen; 25% pass an 80 mesh, 10% of 
which must pass a 100 mesh screen. If the sand used does not contain the 
desired fine material, mineral dust can be added to make up the deficiency, 
and in any case at least 5% of such mineral dust shall be used. The Mineral 
Dust shall be fine, absorbent inorganic dust, not acted upon by water, the 
whole of which shall pass a 30 mesh screen, and at least 75% pass a 100 
mesh screen. The Asphalt Paving Mixture to be composed of above ma- 
terials mixed in proportions by weight, depending upon their character and 
the street traffic and character of asphalt, and will be determined by the 
inspector; but the per cent bitumen in any mixture, soluble in carbon 
di-sulphide, shall not exceed the limits 9 to 13 per cent. Proportions of 
mixture must not be varied from those specified. The sand, or the mixture 
of sand and stone dust, and the asphaltic cement, shall be heated separately 
to about 300° F. The dust, if limestone, will be mixed while cold with hot 
sand, in the required proportions, and then mixed with the asphaltic cement 
at the required temperature and in the proper proportion, in a suitable ap- 
paratus, so as to eftect a thoroughly homogeneous mixture. The mixture 
thus prepared will be brought to the street in carts, at a temperature of not 
less than 230° nor more than 350° F., depending on the asphalt in use; 
canvas covers to be used if the temperature of the air is less than 60° F. 
It is then to be spread to a thickness of at least 3" by means of hot rakes, 
to a uniform grade, so that when compressed it will have a finished thick- 
ness of at least 2". The surface to be compressed by rolling, after which a 
small amount of hydraulic cement will be swept over it, and it will then be 
thoroughly compressed by a steam roller weighing not less than 175 lbs. 
to the inch run, the rolling being continued for not less than 5 hours for each 
1000 yds. of surface. Contractor to furnish a 10-year guarantee. Retaining 
Stone. — (Same as for brick pavement.) 

Cedar Block Pavement on Concrete Foundation. 

Cushion. — (Same as for brick pavement.) Cedar Blocks to be i" long, of 
best quality of sound, selected, live timber, stripped of all bark and free from 
traces of rot or indications of decay, and not less than 4^" nor more than 9' 
in diameter and so selected in size as to make a close-jointed pavement. 
Filling. — Spaces between blocks to be filled with screened gravel or crushed 
granite or boulders of size varying from Y' to 1" diameter and free from dust, 
sand, loam or thin stone, screened, when necessary, through a wire screen 
set at an angle of 60°, with meshes of not less than 8'' lengthwise by Y' in 
width; tamped with iron tamping bars as required, and then the surface 
well rolled by City roller at cost to contractor of ic. per sq. yd. After rolling, 



ASPHALT. WOOD. MACADAM. TELFORD. CURB, 1111 



spaces between gravel or stone filling of the blocks to be completely filled 
from bottom to top with paving cement obtained from the direct distillation 
of coal tar, and shall be the residuum thereof, such as is ordinarily numbered 
5 or 6. Quality and temperature to be approved. Extra care at gutters, 
catch-basins, etc. Top Dressing.— (Same as for brick pavement.) Retaining 
Plank. — Where cedar pavement is laid, the pavement at intersections of 
unpaved streets and alleys to be finished up to a piece of timber 3" thick 
and 12" deep, set on 6" of concrete, and extending across the proposed 
width of roadway of such street or alley. For all other kinds of pavement 
a stone header will be used, similar to, and set as stated for retaining stone. 
Retaining Stone. — (Same as for brick pavement.) 

EASTON (PA.) PAVEMENT SPECIFICATIONS. 

(John McNeal, City Engineer.) 
Macadam and Telford Roads. 

Work. — Done by City force; not by contract. Grading. — Completed 
grade to have slope of from i" to 1" per ft. from center to sides, according to 
percentage of grade of street. Roadbed. — Must be rolled firm with steam 
road roller; depressions formed by rolling to be filled and rolled again to 
finished sub-grade. Macadam Foundation. — On the sub-grade, place 3" to 
I" crushed stone, spread evenly, and roll with road roller until none of the 
stones move under the roller; all material to be added dry, but water added 
ahead of the roller. This course to be 5" thick. Telford Foundation. — On the 
sub-grade, place the bottom course composed of stones 8" to 12" long, 3" to 
4" wide, and 5" deep, vertically by hand on their broadest edges and pointed 
at the top; stones to be laid in lengthwise courses across the street, and all 
interstices filled with broken stone, wedged with a hammer; projecting 
points to be broken off to surface grade. This course to be thoroughly rolled 
until stones do not rock under the roller. Clay may be used as a binder on 
this course if directed. Second Course. — (Same for either macadam or tel- 
ford.) On the foundation, lay a 3" course of crushed stone, 1^' to f", and 
roll until firm and solid, water being applied ahead of the roller. Binder. — 
The binder for the bottom and second course shall be limestone screenings,' 
applying water ahead of the roller if necessary. Surface. — On the second 
course, a coat of 50% of f" stone and 50% of screenings, proper^'- mixed, 
and about 1" thickness, shall be applied dry and rolled once before wetting, 
then alternate watering and rolling until finally completed, when the sur- 
face must be uniform to shape and grade. (The several courses of material 
must be of required depth after rolling, allowance for compression being at 
least one-half.) Rolling. — Each course to be rolled with the utmost thor- 
oughness, the roller starting from the sides and working toward the center. 



Concrete Curbs, Gutters and Sidewalks. 



Vifrified Block 



xjt"' Concrefp t-,. i .Concrete / 

Cinders ^ 




Fig. 3. — Concrete Curb, Gutter and Sidewalk. 

After sidewalk is excavated and shaped to proper depth and grade, the 
cement curb, gutter and sidewalk shall be constructed in place, upon a bed 
of gravel or cinders 8" to 10" deep, well consolidated by ramming to an even 



1112 eo.— HIGHWAYS. 

surface, and moistened before the concrete is placed thereon. The curb, 
gutter and sidewalk to be composed of concrete formed by mixing dry, 
1 part Portland cement, 2 parts coarse, clean sand and 4 parts clean screened 
limestone or trap rock, crushed to pass through a 1^'' mesh screen, to which 
shall be added sufficient water to form a concrete that when placed in the 
templets and thoroughly rammed, free mortar will appear on the surface. 
The ramming of the concrete in the forms shall be done with the proper 
tamping bars and other tools to insure a compact mass with full square 
comers. All exposed surfaces to be covered with a finished coat V' thick, 
composed of 1 part cement, 1 part clean, fine hard stone screenings, and 1 
part clean, coarse sand, applied before concrete has hardened. Top facing of 
curb, gutter and sidewalk to be thoroughly troweled to insure perfect con- 
tact; when sufficiently hard it shall be troweled and floated to a smooth 
true surface. Concrete curb to be 6" thick at top, 8'' at base, and 2V deep, 
exclusive of foundation. Concrete gutter to be 6" deep, 3 ft. wide on all 
streets more than 20 ft. wide, and 2 ft. wide on narrower streets; surface of 
outside edge to be grooved with 4" squares for width of 2 ft. on the 3-ft. 
gutters, and 1 ft. on 2-ft. gutters, to prevent slipping of horses. Concrete 
sidewalk to be at least 5'' deep. For the entire depth of curb, gutter and 
sidewalk, joints to be made with tar paper or by means of removable plates 
to form expansion joints or planes of weakness; joints to be not more than 
10 ft. apart. 

REINFORCED CONCRETE FOUNDATIONS. 

See article entitled "Reinforced Concrete Foundations over Excavations 
on Paved Streets," in Trans. A. S. C. E., Vol. LX, p. 217 (1908). 

EL PASO (TEX.) PAVEMENT SPECIFICATIONS. 

(F. H. Todd, City Engineer.) 

Petrolithic Pavement. 

Preparing Roadway. — ^The roadway shall be so excavated or filled that 
after thorough rolling, or tamping with hand rammers at such points as can- 
not be well done with roller, its surface shall be 2'' below and approximately 
parallel with surface of finished street. Soft and boggy places not affording 
a firm foundation shall be dug out, refilled and thoroughly tamped with good 
sound earth, cinders, gravel, slag, stone or concrete, as may be directed; 
the contractor to be paid for this excavation the same price as other exca- 
vation, if the soft and boggy place was not caused by him, but if caused by 
him, then at his own proper cost and expense. The entire roadway shall 
then be ploughed to a depth of not less than 6" nor more than 9", and 
thoroughly pulverized by cultivating, harrowing, or such other method as 
will accomplish the result. Foundation. — On this thoroughly pulverized 
roadway, including the intersections of all streets and alleys up to the prop- 
erty lines of the street being improved, the roadway shall be evenly coated 
with liquid asphalt, ^ gal. per sq. yd. of surface; it shall then be thoroughly 
cultivated to a depth of 6" until the liquid asphalt which has been applied, 
is thoroughly mixed with the soil; then a second application of liquid 
asphalt, i gal. per sq. yd., shall be made and the area again well and thor- 
oughly cultivated to a depth of 6'^ until the liquid asphalt and the material 
comprising the surface of the street are well and thoroughly mixed; then 
the third application of liquid asphalt, ^ gal. per sq. yd., shall be evenly 
spread over the entire roadway and the area for a third time well and thor- 
oughly cultivated in such a manner that the liquid asphalt shall become 
thoroughly mixed with the street surface to a depth of 6''. The street shall 
be thoroughly watered after each application of the liquid asphalt. The 
surface of the street shall then be brought to a grade approximately parallel 
to grade of finished street. Tamping. — ^The street shall then be tamped 
with a petrolithic rolling tamper until it is solid to within 2'' of the surface. 
When the tamping of the road with the rolling tamper is begun, the rolling 
tamper shall be immediately followed by a cultivator set so as not to disturb 
the sub-base already tamped; said cultivator being reset as the tamping 
progresses, so as to cultivate to shallower depths. The cultivator shall be 
used continually during the tamping, the purpose being to prevent a too 
rapid solidification whereby the road way_ would be solidified without being 
compacted from the bottom up. Upon this base prepared as above specified, 
shall be applied liquid asphaltum, i gal. per sq. yd. of street surface. Wearing 



PETROLITHIC PA VEMENT. 1113 

Surface. — On the foundation, shall be placed a layer of hard, durable, 
crushed stone, 2" to V screen, spread evenly and to such thickness that 
after it has been thoroughly sprinkled, and rolled with a roller weighing not 
less than 10 tons, its surface shall be parallel to and \" below surface of 
finished street. Liquid asphaltum shall then be applied, I gal. per sq. yd. of 
surface; then a layer of crushed rock, \\" to \" screen, shall be spread evenly 
to depth of f'; then the surface shall be thoroughly watered and rolled, 
followed by a coating of liquid asphaltum applied at rate of \ gal. per sq. yd. 
then a light coating of rock screenings \" and under in size, and in sufficient 
quantity to absorb all the surface liquid asphaltum and produce a uniform 
surface. The pavement shall then be watered and rolled until it becomes 
hard, smooth, true to grade and cross-section, free from all hollows and 
other irregularities, until but slight movement takes place under the action 
of the roller. In streets or avenues having a street-railway track, the street 
shall be excavated to a point Q" below the ties and 6'' beyond the end of the 
ties, and this space filled with broken stone or slag to the upper surface of the 
ties; this filling to be so well tamped with stones of assorted sizes, from 3" in 
greatest- to h" in least dimension, that but slight movement will take place 
under the ties during the passage of the electric car The space between the 
rails shall then be filled in the same manner as is provided for wearing sur- 
face on other portions of the street. It will be necessary, however, to pro- 
vide a roller so formed as to make the flange-ways similar in form to that 
approved by the City Council, Jan. 9, 1908. With this roller, the surface of 
the street between the rails must be so compressed that but slight move- 
ment takes place under a 10-ton road roller. That portion of the roadway 
outside of the rails, shall be treated in a manner similar to the wearing surface 
for the remaining portion of the street. In case rock is encountered in exca- 
vating, it will be necessary to remove that to a depth of at least 3'' below 
finished surface of street. Liquid Asphaltum. — The liquid asphaltum used 
shall contain not less than 75% of asphaltum at 80° penetration when tested 
at a temperature of 77° F.^ The specific gravity shall not be lower than 10° 
nor higher than 11° Baume, at a temperature of 60° F., and shall not contain 
more than 2% of water and sediment. In all cases the liquid asphaltum shall 
be applied at a temperature between 200° and 250° F. It shall be applied to 
the roadway by an approved form of sprinkler, such as will give a uniform 
distribution over the entire surface of the roadway. Concrete. — The cement 
must be equal in quality to the best American Portland cement, and oppor- 
tunity shall be provided to test it for 30 days before it is used in the work, in 
order to prove its strength and soundness. The Sand, for mortar, must be 
clean, sharp, and have grains of different sizes so proportioned as to make a 
dense aggregate. The Concrete, for pavement foundation or street rail- 
way foundation, must be made from 1 part by measure of Portland cement, 
3 parts sand, and 7 parts clean, sound, hard, broken rock, or clean gravel, of 
various sizes from 2^" in greatest- to ^" in least dimension, and these sizes so 
proportioned as to give the greatest density. Measuring Boxes shall be pro- 
vided, if required. After the cement, sand and rock or gravel have been 
thoroughly mixed dry, either by hand or machinery, it shall be wet in such 
manner as not to wash away the cement, and at the same time the mixing 
shall be done so that the wet particles of cement and sand are thoroughly 
incorporated as mortar, and each stone covered with mortar, the whole mass 
having in it just enough water to make the concrete - when in place, slightly 
"sloppy." From the time water is first applied to the batch until the con- 
crete is thoroughly rammed in place, the work must proceed rapidly and the 
ramming must be so well done that no voids remain in the concrete, and the 
water flushes to the surface. Before the concrete sets, but after it is rammed 
in place, enough hard, clean, broken stone, 2^'^ to 1^", to about cover \ of 
the surface shall be evenly spread on it. These stones shall then be rammed 
far enough into the concrete to hold them firmly and yet leave it rough 
enough to securely hold the petrolithic wearing surface. The Concrete 
Gutter shall be composed of 1 part Portland cement, 2\ parts sand, and 
5 parts clean, hard, durable broken stone, \" to 2" in greatest dimension and 
in such proportions as to give the densest mixture, and placed in the same 
manner as specified for concrete foundation. After the concrete has been 
thoroughly rammed in the gutter, and while it is yet soft, \Y of mortar com- 
posed of 1 part Portland cement and 2 parts sand and as much fine crushed 
hard stone as will make the densest aggregate, shall be placed on top and well 
rammed, after which the surface shall be finished with the proper tools to 
make it accurately conform to the given lines and grades. Proper forms 
must be provided for all concrete work. Expansion Joints of roofing paoer 



1114 ^.—HIGHWAYS, 

and bitumen, or of bitumen alone, must be provided and placed at such points 
and in such manner in the concrete as may be required, but not nearer 
together than 20 ft. All concrete work shall be covered with earth and sand 
after it has set and shall be kept wet for at least 1 week and shall be protected 
from injury for at least 10 days after laying, to allow it to set properly. 
Marginal Curb. — Whenever a paved street is joined to an unpaved one, a 
marginal curb of hard, durable stone at least 15" deep, 5" thick, and 2 ft. 
long and with top surface broken to a straight line, must be provided and 
set, if required. This curb shall be set in 6'' of the same class of concrete as 
specified for foundation, and backed up with the same to within &' of the 
top. The upper surface must conform to the cross-section of the street. 
General. — ^The price bid per sq. yd. for complete petrolithic pavement must 
include the excavation or filling required to bring the street to its established 
grade and surface, and further, must include the furnishing, placing and 
manipulation of all material necessary for the construction of this pavement, 
including all labor and necessary implements. No wearing surface shall be 
laid when the temperature of the air is below 40° F., and preferably, the 
asphaltum for the foundation and surface shall be applied diiring warm, 
dry weather. 

LOS ANGELES (CAL.) PAVEMENT SPECIFICATIONS, 

(Homer Hamlin, City Engineer.) 

Graveled Streets. 
(Oiled.) 
Sub-Grade. — For the roadway, shall be V below surface of finished work, 
unless otherwise indicated or directed. Grading. — Shall include all filling, 
excavation, shaping and trimming required to bring surface of street to 
grade and cross-section. Mud and other soft material, to a depth of 2 ft,, 
shall be taken out and the space filled with good earth or gravel. All filling' 
to be with good sound earth ; the embankment to be carried up of full width 
in horizontal layers, not over 1 ft. thick, the teams to travel as evenly as 
possible over the whole surface of each layer, both going and coming. After 
the street has been brought to the required grade and cross-section, the surface 
shall be thoroughly moistened and rolled with a roller weighing not less than 
250 lbs. to the inch width of tire, until it is unyielding. Depressions made by 
the rolling shall be leveled up with good earth and again rolled. Such portions 
of the street as cannot be reached by the roller, and all places excavated below 
grade and refilled, and all pipe trenches and other places that cannot be 
properly compacted by the roller, shall be tamped solid, and in case of wet 
weather or soft or muddy ground, making the use of the roller unsafe or 
impracticable, the rolling shall not be undertaken until the ground has be- 
come sufficiently dry. The sub-grade shall then be tested for grade, cross- 
section and condition. Surfacing Roadway. — Upon the sub-grade, spread a 
layer of good gravel, to have a thickness of V (ordinarily) after rolling. 
The surface of this layer for a depth of 1" is to be raked free from all stones 
larger than V in greatest dimension. If no gutters are provided, these 
larger stones shall be raked to the curb and distributed over a strip 2 ft. in 
width next to the curb; if gutters are provided, the stones are to be dis- 
tributed on a strip 2 ft. wide next to the gutter. This layer of gravel is to be 
uniformly spread on the roadway, and well moistened; then well rammed 
for at least 1 ft. from the gutters, should these be paved; or if not paved, 
then 1 ft. from the curb. The remaining portion of the roadway shall then 
be rolled with a roller weighing not less than 250 lbs. to the inch width of 
tire. The rolling of roadway shall commence at the rammed portion. All 
depressions must be promptly filled, moistened, and again rolled. The 
sprinkling and rolling must continue until the surface is uniformly hard, 
compact, and in such condition that it will not yield or cut up under the 
wheels of a heavily loaded wagon. Oiling. — Oil shall then be distributed 
evenly over the entire surface of roadway, 1 gal. per sq. yd. Coarse, sharp 
sand shall then be sprinkled over the entire surface of roadway until no free 
oil can be seen. After a lapse of not less than 12 hrs., oil shall again be dis- 
tributed over entire surface, i gal. per sq. yd. Entire surface of roadway 
shall again be sprinkled with coarse, sharp sand until the oil is completely 
absorbed, and then rolled with a roller weighing not less than 250 lbs. to the 
inch width of tire until the surface is unyielding. In all cases, sufficient sand 
shall be used to prevent the oil material from picking up. Total amount of 
oil used shall not be less than 1^ gals, per sq. yd. ot street surface. In process 



OILED GRAVEL ST. BITUM.-BRICK GUTRS, 1115 

of rolling, care must be taken not to soil the curbs or walks. After the oiling 
has begun, it shall be carried on diligently and continually to its completion. 
Sand used in covering the oil must be distributed in piles along the sides of 
the street before the oil is applied, and must be spread quickly and in suffi- 
cient quantity to prevent the oiled surface from picking up. Oil shall not be 
applied to the surface of a street while in a wet condition. During and im- 
mediately after rolling, the surface of the street shall be gone over with 
brooms or rakes and all irregularities removed. 

Oil. — (a) The oil used shall be a natural oil treated to remove water or 
sediment, or one from which the volatile material has been removed by dis- 
tillation. It must not have been injured by over-heating, and it must not be 
obtained by adding solid asphalt to lighter oils, or by cutting asphalt with 
distillates, (b) Temperature. All oil must be delivered at the point required 
for sprinkling, at a temperature not less than 150° F. (c) Measurement. In 
determining the quantity of oil delivered, the correction for expansion by 
heat shall be as follows: 60° F. shall be considered normal temperature; 
subtract 0.0004 of measured volume for each °F. above 60° F., as a correc- 
tion for expansion by heat, (d) Volatility. The oil shall not contain more 
than 8% of matter volatile when said oil is heated slowly to 220° F. and 
maintained at that temperature for 15 minutes, (e) Asphalt. After being 
freed from water and sediment, the oil shall contain not less than 70% of 
asphalt, having a temperature of 77° F., a penetration of 80°, District of Col- 
umbia standard. The percentage of asphalt shall be determined by heating a 
weighed amount of said oil in an evaporating oven to a temperature of 400° F. 
until it has reached the proper consistency, when the weight of the residue 
shall be determined and the per cent calculated, (f) Water and Sediment. 
Deduction will be made for water and sediment in exact proportion to the 
percentage of water and sediment found therein, which must not exceed 2%. 
(g) Tank wagons. All tank wagons used for the delivery of this oil must first 
be submitted to the Department of Oil Inspection, which will gauge and 
stamp into the steel heads of said tanks the capacity in gallons, which shall 
be the official rating, (h) All oil used shall be tested by the Department of 
Oil Inspection. 

Surfacing Sidewalk Areas. — In cases where the plans provide for cement 
sidewalks on portions of the street to be improved, and do not provide for 
such walks over its entire length, then there shall be constructed at the 
excepted places gravel walks 2" deep and of a width and location correspond- 
ing to those for the cement walk provided for. In cases where the plans do 
not provide for cement walks on the street to be improved, then gravel side- 
walks 2" deep and 5 ft. wide shall be constructed, except, however, in cases 
where the total width of the sidewalk area is less than 5 ft., in which event 
the total area of the sidewalk is to be graveled. In the construction of the 
graveled walks, the same quality of material as that used in the roadbed 
may be employed. It shall be raked free from large stones, sprinkled and 
rolled until firm. 

BiTUMiNizED Brick Gutters. 
Sand Cushion. — Upon the concrete base (the surface of which is 6'' below 
finished grade, and thoroughly watered for at least 48 hours, before receiving 
cushion coat, and swept free from all dirt and rubbish) shall be spread a layer 
of sand 2" deep. The sand need not necessarily be sharp, but it must be 
screened, dry and free from more than 3% of loamy matter. It shall be 
spread by the aid of a templet and made to conform smoothly to the true 
slope of the gutter. There shall be no disturbance of the surface of the 
cushion coat previous to laying the bituminized brick thereon. Laying 
Bituminized Brick. — Upon the cushion coat shall be laid the bituminized 
brick, vertically on edge, and in close contact with each other. Brick in 
adjoining rows must be laid so as to break joints at least 2" . No bats or 
parts of bricks shall be used except for the purpose of closure or for breaking 
joints in starting courses. After the bricks are laid they shall be thoroughly 
inspected and all warped, spalled and chipped brick removed and replaced by 
more perfect ones. The edge of the gutter next the curb shall then be care- 
fully tamped by hand and the whole gutter shall then be rolled until all 
bricks are thoroughly bedded and the tops lie in a smooth surface conforming 
to grade and cross-section of gutter. Asphalt of a composition hereinafter 
described, and heated to a temperature of 300° F., shall then be poured into 
the joints until they are full and remain full to height of top of brick. Surplus 
asphalt shall then be removed, before it has become stiff, from the surface of 



1116 eO.— HIGHWAYS. 

• 

the gutter, and fine sand shall be swept over the top until all stickiness is 
removed. Bituminized Brick. — Shall be obtained by subjecting ordinary- 
brick of the quality specified hereinafter to a bath of asphalt, also of the 
quality hereinafter described, and heated to a temperature of from 300° to 
325° F., until at least 80% of the cross-section of the brick shall have become 
saturated with asphalt. They shall be free from warps, cracks, chips or 
other flaws, and the surfaces shall be free from superfluous asphalt and in 
condition to lay closely together. All bituminized brick will be subject to 
the following Abrasion Test. — This test shall be made in a foundry rattler 
whose inside diameter is 28'' and inside length is 20^'. Such a number of 
whole, dry brick that their total volume shall equal, as nearly as possible, 
8% of the cubic contents of the rattler, shall be placed therein. There shall 
then be added an abrasive charge of 300 lbs. of cast iron blocks as follows: 
10 blocks about 2^' square and 4^" long, with edges rounded to about Y' 
radius and weighing 7^ lbs. each, and 225 lbs. of cubical blocks about lY' on 
a side and with square comers and edges. The rattler shall be revolved 1800 
times at a speed of from 28 to 30 rev. per min. The loss by abrasion during 
such test shall not exceed 20% of the original weight of the brick. Ordinary 
Brick. — Shall be whole, sound brick with smooth, rectangular surfaces and 
straight edges and must give a clear ringing sound when struck together. 
They shall be uniform in quality, free from laminations, and shall run in size 
from 8'' to 8^' long, 4" wide and from 2" to 2Y' thick and burned to a medium 
degree of hardness. All brick shall be culled or sorted by the contractor 
before being treated to an asphalt bath, and will be subject to the following 
test: Three or more bricks shall be broken across, thoroughly dried, weighed, 
then immersed in water for 24 hours and weighed again. The absorption 
shall be determined by the difference between the two weights, and it shall 
not exceed 15% nor be less than 12% of the dry weight of the brick; other- 
wise the brick from which the tested samples were selected, shall be rejected. 
Asphalt. — ^This must be prepared from California products. It shall be a 
mixture of refined liquid asphalt with a refined solid asphalt or be an oil 
asphalt, and must be free from admixture with any residues obtained by the 
artificial distillation of coal, coal tar or paraffine oil. The asphalt must be 
homogeneous and its consistency at the time of its use in the bath must fall 
within the limits of 60° and 80° penetration by the District of Columbia 
standard. It must be adhesive and ductile and also slightly elastic at a tem- 
perature of 32° F. When 20 grams are heated to a temperature of 300° F. 
for 5 consecutive hours in an uncovered cylindrical dish 3^ cm. high by 5i cm. 
in diameter, it must not lose more than 1% in weight, and its penetration 
must not be reduced, as a result of such heating, more than 50%. It must, 
when ready for use, contain at least 90% of bitumen soluble in carbon di- 
sulphide. It shall be soluble in cold carbon tetrachloride to the extent of 
at least 97%. Not less than 70% shall be soluble in 86° naphtha. It shall not 
contain more than 15% of fixed carbon on ignition. When the asphalt is 
prepared by mixing a solid oil asphalt with a liquid asphalt, the solid oil 
asphalt shall be prepared by distilling the crude oil until the asphaltic 
residuum has a penetration not less than 50° by the District of Columbia 
standard, and shall not be prepared by mixing or fluxing a more solid asphalt 
with a liquid or softer asphalt. The refined liquid asphalt used in softening 
a solid asphalt must be a stiff residuum of petroleum oil with an asphalt base. 
It must be free from water and from light oils volatile at less than 250° F. 
When 20 grams are heated to a temperature of 300° F. for 5 consecutive 
hours in an uncovered cylindrical dish 3^ cm. high by 5i cm. in diameter, it 
must not lose more than 5% in weight. It must contain not less than 99% 
of bitumen soluble in carbon disulphide. 

MARYLAND STATE HIGHWAY SPECIFICATIONS. 

(Maryland Geological Survey.) 

Macadam Construction. 

Class A, B and C. — ^Thickness after rolling, to be as follows: 

Class A. 1st course, 3''; 2nd, 3"; 3rd, as described later. 

ClassB. " 5",in21ayers; " 3''; " 

Class C. " 5", gravel; " 3", stone; " 

Roadbed. — Natural earth bed prepared and rolled until firm and hard ; 
a small amount of clay to be added if sandy or other soil will not compact 
readily under roller. In Cuts and Fills, roadbed is to be graded 24 ft. wide. 



MACADAM CONSTRUCTION. CEMENT WALK. 1117 

Roadbed prepared for broken stone siirface to be 14 ft. wide* and rolled firm 
and hard; depressions filled with earth and rerolled. Old earth roadbed, 
where there is no change in grade, is to be shaped to proper cross-section, ele- 
vations and depressions removed, and surface rolled hard and smooth. The 
portion of the roadbed prepared for the broken stone, is to be below the 
sides by an amount equal to thickness of 1st course of stone, to prevent 
spreading at sides. Roadbed to have cross-slope of f' to 1 ft. First Course. — 
Sound broken stone, 3'' to 2", known as "No. 1" size. If approved it may be 
gravel, S" to 1", with not more than 25% less than V. No layer of crushed 
stone to be spread thicker than Q" before being thoroughly rolled. Broken 
stone or gravel for 1st course to be rolled with steam roller weighing not less 
than 10 tons, until compacted firm and smooth; sprinkling with water or 
lightly spreading with sand if needed; rolling to begin at sides and work 
toward center. Unevenness or depressions to be remedied. Shoulders. — 
After 1st course is made, construct shoulders along each side for width of 
at least 5 ft.; against these shoulders, spread broken stone for second course; 
the shoulders with the 14 ft. of broken stone will make a total width of 24 ft., 
to be cross-sloped |" to 1 ft. Second Course. — Same width as first course. 
Broken stone 1" to 2'\ known as No. "2" size. Unless otherwise specified the 
stone for this course shall be trap rock with a "coefficient of wear" as deter- 
mined by tests made at the laboratory of the Highway Division of the Mary- 
land Geological Survey, of not less^than 15, or limestone with a "coefficient 
of wear" of not less than 10. The broken stone to be spread upon the 1st 
course, to a uniform thickness, and rolled with not less than a 10-ton roller, 
sprinkling with water or lightly spreading with sand or other material if 
necessary, until surface is hard and smooth; cross-slope of surface, |'' to 1 ft. 
Unevenness and depressions to be remedied. Third Course. — ^Trap rock 
screenings, from V to dust; other material may be used if approved; lime- 
stone screenings to be used with a limestone 2nd course. Upon the 2nd 
course, and in quantity just enough to cover it, the screenings are to be 
spread dry, then sprinkled with a sprinkling cart, and rolled with not less 
than a 10-ton roller, beginning at the sides. If after rolling the screenings, 
the No. 2 stone appears at the surface, use additional screenings. Rolling 
and watering to continue until the water flushes to the surface; the rolling 
to extend over whole width or road and shoulders. Unevenness and depres- 
sions to be remedied. 

NATIONAL ASSOCIATION OF CEMENT USERS. 

(Philadelphia, Pa.) 

Portland Cement Sidewalk Specifications. 

(Adopted January, 1908.) 

Materials. — (1) Cement shall meet requirements of specification for Port- 
land cement of the A. S. T. M., and adopted by this association (Spec. No. 1), 
January, 1906. (2) Sand shall pass a No. 4 screen and be free from foreign 
matter, except loam and clay up to 5% when not occurring as a coating on 
the sand grains. Not more than 40% shall be retained on a No. 10 sieve; or 
35% pass a No. 10 and be retained on a No. 20; or 35% pass a No. 20 and be 
retained on a No. 30; or 35% pass a No. 30 and be retained on a No. 40; or 
35% pass a No. 40 and be retained on a No. 50. Not more than 20% shall 
pass a No. 50 sieve; or 70% pass a No. 10 and be retained on a No. 40; or 
70% pass a No. 20 and be retained on a No. 50 sieve. (3) Stone shall be 
crushed from clean, sound, hard, durable rock, be screened dry through a 
i" mesh, and be retained on a I" mesh. (4) Screenings from the crushed 
stone, if they meet the requirements for sand, may be used as sand if approved. 
(5) Gravel shall be clean, hard, and vary in size from i'' to f'' screening; un- 
screened gravel shall be clean, hard, and contain no particles larger than f', 
the proportions of fine and coarse to be determined and corrected to agree 
with requirements for concrete. (6) Water to be clean, free from oil, 
sulphuric acid and strong alkalies. Forms. — (7) Lumber, free from warp, 
and not less than If" thick; all mortar and dirt to be removed from forms 
previously used. (8) Setting. — ^The forms shall be well staked to the estab- 
lished lines and grades, and their upper edges shall conform with finished 



* Mr. E. F. Ruggles, First Assistant Engineer, writes the author as fol- 
lows: "As a rule, our roads are built with a width of 12 ft. for the macadam; 
although wo have built, where the travel requires it, a good deal of 14-ft. 
macadam." 



1118 &b.— HIGHWAYS. 

grade of sidewalk, which shall have sufficient rise from curb to provide prop- 
er drainage; this rise not to exceed ^" per ft., except where such rise shall 
parallel length to walk. (9) Cross Forms, at each block division, shall be 
put in the fuil width of walk and at right angle to side forms. (10) Expan- 
sion Joints. A metal parting strip y thick shall take the place of the cross- 
forms at least once in every 50 lin. ft. of sidewalk. When sidewalk has 
become hard, this parting strip shall be removed and joint filled with suit- 
able material prior to opening the walk to traffic. Similar joints shall be 
provided where new sidewalks abut curbing or other artificial stone sidewalk. 
(11) Wetting. All forms to be thoroughly wetted before any material is de- 
posited against them. Size and Thickness of Blocks. — (12) In Business 
Districts, blocks shall be so divided that no dimension shall be greater 
than 6 ft.; thickness of sidewalk shall correspond directly with the greatest 
dimension of the walks as follows: 6" thick for block 6 by 6 ft.; 5^' for 
block 5 by 5 ft.; 5'' for block 4i by 4^ ft.; V for block 4 by 4 ft. (13) In 
Residence Districts, thickness of sidewalk shall be as follows: 6'' thick for 
block 6 by 6 ft . ; 5" for block 5 by 5 ft. ; V for block 4 by 4 ft. ; r for block 
3 by 3ft.; it being permissible to lay sidewalks with a thickness at the 
edges 25% less than at center. (14) Minimum Thickness of walk to be 3'' 
in any case. Sub=Base. — (15) Preparation. Sub-base to be thoroughly 
rammed, and all soft spots removed and replaced by suitable hard material. 
(16) Fills. When a fill exceeding 1 ft. thick is required, it shall be thoroughly 
compacted by flooding and tamping in layers not over 6'' thick, and shall 
have a slope of not less than 1 to 1^. The top of all fills shall extend at least 
12" beyond the sidewalk. (17) Wetting. While compacting, the sub-base 
shall be thoroughly wetted and shall be maintained in that condition until 
the concrete is deposited. Base. — (18) Proportions. The concrete for the 
base shall be so proportioned that the cement shall overfill the (19) Voids* 
in the sand by at least 5%, and the mortar shall overfill the voids in the 
stone by at least 10%. The proportions shall not exceed 1 part cement to 
8 parts of the other materials. When the voids are not determined, the 
concrete shall be: 1 part cement, 3 parts sand or screenings, and 5 parts 
stone or gravel. A sack of cement (94 lbs.) shall be considered to have a 
volume of 1 cu. ft. (20a) Hand Mixing. Spread sand evenly on level water- 
tight platform; spread cement upon sand; mix thoroughly dry to uniform 
color; add water in a spray, and turn mass until homogeneous mortar of 
even consistency is obtained; to this mortar, add the required amount of 
stone or gravel previously drenched, and mix the whole until the aggregate 
is thoroughly coated with mortar. When unscreened gravel is used, the 
cement and gravel shall be thoroughly mixed dry until no streaks of cement 
are visible ; water shall be added with a spray in sufficient quantity to render 
when thoroughly mixed, a concrete equal to that specified above. Water 
may be added during the process of mixing, but the concrete shall be turned 
at least once immediately after its addition. (20b) Mechanical Mixing. Ma- 
chine mixing will be acceptable when a concrete equal in quality to that speci- 
fied above is obtained; mixing to be thorough. (21) Retempering will not be 
permitted. (22) Depositing. The concrete shall be deposited within 1 hour 
after being mixed, and shall be transferred to the forms in water-tight 
wheelbarrows ; the barrows not to be filled so full as to allow mortar to slop 
out, and shall not be run over freshly laid concrete. The concrete to be 
spread evenly and tamped until water flushes to the top. (23) Separation of 
Blocks shall be done with a tool not over 6" wide and i" thick, and to insure 

* To determine voids, fill a vessel with sand and let net weight of sand 
equal B. Fill same vessel with water and let net weight of water equal A. 

Per cent voids = ^^2.65 ^ ^^^' 

■ This formula may also be used in determining voids in crushed stone 
and screenings by substituting for 2.65 the specific gravity of the stone. 

The following is a more simple method of determining voids in coarse 
aggregate: Fill a vessel with the aggregate and let net weight equal B. 
Add water slowly until it just appears on the surface, and weigh. Let net 
weight equal A. Fill same vessel with water and let net weight equal C. 

A —B 
Per cent voids = ^ X 100. 

Use a vessel of not less than one-half (i) cubic foot capacity. The larger 
the vessel, the more accurate the results. 



CEMENT WALK. GRANITE BLOCK PAVEMENT. 1119 

complete separation the groove should be cut through into the sub-base. 
Fill the groove with dry sand before the top coat is spread, and the top coat 
should be cut through to the sand after flofiting and troweling and a jointer 
run in the groove; then again draw a trowel through the groove, so as to 
insure a complete separation of the block. (24) Protection. Workmen not 
permitted to walk on freshly laid concrete, and where sand or dust collects 
on the base it shall be carefully removed before the wearing surface is applied. 
Wearing Surface. — (25) Thickness. I". (26) Mixing. The mortar to be mixed 
in the same manner as the mortar for the base, but using 1 part cement to 

2 parts of sand or screenings, and it shall be of such consistency as will not 
require tamping, but will be readily floated with a straight-edge. (27) De- 
positing. Spread mortar on the base within 30 minutes after mixing, and in 
no case shall more than 50 minutes elapse between the time that the concrete 
for the base is mixed and the time that the wearing course is floated. Float 
a thin coat of mortar on the base before spreading the wearing surface. 
(28) Marking. After being worked to an approximately true surface, the block 
markings shall be made directly over the joints in the base with a tool which 
shall cut clear through to the base and completely separate the wearing 
courses of adjacent blocks. (29) Edges. All surface edges of blocks to be 
rounded to radius of not less than \". (30) Troweling. When partially set, 
the surface shall be troweled smooth. (31) Roughening wearing surface. On 
grades exceeding 5%, the surface shall be roughened, by using a grooving 
tool, toothed roller, brush, wooden float or other suitable tool, or by working 
coarse sand or screenings into the surface. (32) Color. If color is desired, 
only mineral colors shall be used, which shall be incorporated with the 
entire working surface. Single Coat Work. — (23) Proportions. Single coat 
work shall be composed of 1 part cement, 2 parts sand, 4 parts gravel or 
crushed stone, and the blocks separated as provided for in the specifications 
for two-coat work. (34) Finishing. The concrete shall be thoroughly com- 
pacted by tamping and evenly struck off and smoothed to the top of mold. 
Then, with a suitably grooved tool the coarser particles of the concrete 
tamped to the necessary depth so as to finish the same as two-coat work. 
Protection and Grading. — (35) Protection. When completed, the sidewalk 
shall be kept moist and protected from traffic and the elements for at least 

3 days; the forms to be removed with great care, and when removed, earth 
shall be banked against the edges of the walk. (36) Grading, after the 
walks are ready for use, should be on the curb side of the sidewalk, l^*' 
lower than the sidewalk, and not less than \" per ft. fall toward the curb or 
gutter. On the property side of the walk, the ground should be graded 
back at least 2 ft. and not lower than the walk; this will insure the frost 
throwing the walk alike on both sides. 

MANHATTAN (N. Y. CITY) BOROUGH PAVEMENT SPECIFICATIONS. 

Granite Block Pavement. 

Blocks. — Shall be of a durable, sound and uniform quality of granite; 
8" to 12'' long, 3F to 4^' wide, and T to 8" deep; same quality -as to hard- 
ness, color and grain. No outcrop, soft, brittle or laminated stone accepted. 
Blocks to be rectangular on top and sides, uniform in thickness, to lay closely, 
and with fair and free surfaces, free from bunches. Other dimensions of 
blocks may be used for special construction. Stone from each quarry shall 
be piled and laid separately in different sections of the work; no mixing of 
stones from different quarries. Sand Cushion. — On the concrete foundation 
(previously prepared ^^ thick) place a layer of clean, course, dry sand to 
such a depth (not less than \Y) as may be necessary to bring the surface 
of pavement when thoroughly rammed, to the proper grade. Laying. — On 
this sand bed, and to grade and crown specified, lay the blocks at right 
angle to line of street, or at such angle as may be directed; each course to 
be straight and regular, with the end joints by lap of at least 3''; stones of 
different width not to be laid in the same course, except on 'curves; joints to 
be close, except where gravel filling is used the joints between courses shall 
not exceed \". After the blocks are laid, they shall be covered with clean, 
hard and dry gravel (previously heated and dried), to be brushed in until 
all the joints are filled therewith to within 3" of the top ; the gravel to be 
washed white quartz, free from sand or dirt, and ^^ to \" mesh screenings. 
Ramming. — Blocks must then be rammed and ramming repeated until they 
are brought to an unyielding bearing with a uniform surface, true to even 
grade and crown; no ramming to be done within 20 ft. of face of work being 



1120 m.— HIGHWAYS. 

laid. Joints. — After ramming, the pavement cement heated to 300° F. shall 
then be poured into the joints until same are full and remain full to top of 
gravel. Hot gravel shall then be poured along the joints flush with top of 
blocks; and paving cement again poured in joints, filling all voids. Paving 
Cement. — Shall be composed of 20 parts of refined asphalt and 3 parts of 
residuum oil, mixed with 100 parts of coal-tar pitch such as is ordinarily 
numbered 4 at the manufactory, the proportions to be determined by 
weight. 

Wood Block Pavement. 

Foundation. — 6" thick, including 5^" of concrete proper and i'^ of mortar 
top surface, ordinarily. Blocks. — (a) Either of southern long-leaf yellow 
pine, southern black gum, Norway pine or tamarack, not less than 90% of 
heart; texture permitting satisfactory treatment; inspection at works, in 
the stick, before being sawed into blocks, (b) All blocks shall be of sound 
timber, free from bark, loose or rotten knots, or other defects detrimental 
to life of blocks or to laying; no second-growth timber allowed, (c) Blocks 
shall be well made, rectangular and of uniform dimensions: Depth (parallel 
to fiber) 3F, length 6'' to 10'', width 3" to V; in any one contract, blocks to 
be of same timber, and depth and width shall not vary more than i''. (d) 
Blocks to be treated with an antiseptic and waterproof mixture, not more 
than 75% per cent of which shall be creosote or heavy oil of coal tar, and at 
least 25% of which shall be resin; all parts of each block to be thoroughly 
treated, injecting not less than 20 lbs. per cu. ft. (e) Treated pine blocks 
shall weigh as much as water; treated gum blocks, at least 59 lbs. per cu. ft.; 
any other wood, at least 20 lbs. per cu. ft. more than its recognized weight 
untreated. Blocks cut from the several classes of timber will require different 
treatment, hence the exact methods of applying the mixture will not be 
specified, but must conform in every respect to the best and most advanced 
knowledge of the art. (f) The creosote oil at 68° F. shall have a specific grav. 
of not less than 1.12; when distilled in a retort with the thermometer sus- 
pended not less than 1'' above the oil, it shall lose not more than 35% up to 
315° C, and not more than 50% up to 370° C. Oil to be free from adultera- 
tion or foreign material, (g) The resin to be solid resin obtained from pine; 
and reduced to a fine dust by grinding and then incorporated with the hot 
creosote oil in a suitable mixing tank until the proper proportions are se- 
cured, (h) After treatment, blocks not to gain more than 3^% in weight 
after being oven-dried at 100° for 24 hours and then immersed in water 
24 hours. Analysis of Treated Block. — Fine turnings from the block shall be 
placed in an extraction apparatus and the oil completely extracted therefrom 
with ether or carbon bisulphide; the oil then placed in a still and distilled; 
the portion up to 120° C, consisting of the solvent, is to be collected apart; 
the oil then distilled up to 370° C. The oil thus obtained must conform in 
all respects to the requirements of (h), above. Mortar Bed. — On concrete 
foundation, spread i'' layer of mortar composed of 1 part Portland cement 
to 4 parts clean, sharp sand, free from pebbles over \" diameter; the mortar 
top to be "struck" 3^'' below and parallel to top of finished pavement. The 
mortar bed to be laid as follows: On surface of concrete foundation, before 
mortar bed is laid, set strips of wood 4" wide by Y' thick, or strips of steel 4^^ 
by i'', and of convenient length; these strips to be set parallel and about 8 
to 10 ft. apart, running from curb to curb, and imbedded in mortar so that 
top surface shall be 3^" below grade of finished pavement; the space between 
two strips having been filled with mortar, a true and even top surface shall be 
struck by using an iron-shod straight-edge on the strips as a guide, the strips 
to be removed and the places filled with mortar as the blocks are laid. 
Laying. — On this mortar surface the blocks are laid, with the grain vertical, 
in parallel courses, at angles as directed, tight joints as possible, each block 
being firmly imbedded in the mortar bed so as to form a true and even 
surface. Expansion joints, Y\ shall be used along each curb, and across the 
street every 100 ft. The joints shall then be filled with cement grout (2 parts 
sand and 1 part Portland cement, mixed to a perfectly liquid form) and the 
surface of the blocks shall be slushed with same and joints swept until com- 
pletely filled; surface then covered with ^ of screened sand. Grooved 
Blocks. — Where grade exceeds 3%, the blocks shall be between 6" and lO'' 
long, the upper edge of each block to be cut away for a width of Y' and depth 
of 1", so as to provide transverse grooves between each course (as a foothold 
for horses) ; or other equally good construction. Blocks to be laid (in usual 
manner), V lap, whole blocks used, and covered with sand when laid. 



PAVEMENT— WOOD, IRON-SLAG, BRICK. 1121 

RICHMOND (N. Y. CITY) BOROUGH PAVEMENT SPECIFICATIONS. 

(Louis L. Tribus, Commissioner.) 
Iron Slag Block Pavement. 

Blocks. — Iron slag blocks, 8" to 9'' long, 3F wide, 3" deep; shall be 
hard, durable and perfect; upper edges to be chamfered. On grades of 6% 
or over, the joints between blocks are to be left open to receive hot, clean 
gravel, with paving cement; for grades below 6%, the joints will be laid 
close without gravel but filled with paving cement. Sand Cushion. — On the 
foundation (concrete) place about a 2'' layer of clean, dry sand to bring 
surface of pavement, when rolled, to proper grade; sand to be screened and 
free from stones and rubbish; cushion to be brought to required form and 
crown by means of template resting on curbs and drawn forward a few feet, 
ahead of laying. Laying. — Blocks laid on edge at right angle to curb line, 
except at street intersections where they shall be laid as required. End 
joints to be broken by lap of half length of block. At every 4th course, or 
as often as directed, blocks shall be closed up by hammering and the course 
straightened. End joints to be closed by means of crowbar applied at ends 
next the curbs before the closures are made. Whole blocks used except in 
beginning or closing a course, or as directed. Rolling. — As soon as street 
block has been laid , the pavement shall be swept clean and rolled with 5-ton 
roller until all blocks are thoroughly imbedded in the sand cushion; all de- 
pressed surfaces to be relaid; pavement then rerolled until finished surface 
is smooth and even, to required grade and crown. Broken or chipped blocks 
are to be replaced. Expansion Joints. — Before laying the blocks, laths f" 
thick shall be placed next each curb. The spaces thus formed shall be filled 
with hot paving cement composed as follows: Paving Cement. — Shall be 
bituminous material either natural or artificial, free from coal tar or any 
products of coal tar distillation. It shall be waterproof, free from water or 
decomposition products, remain ductile and pliable at all climatic tempera- 
tures to which it may be subjected in actual use, and shall not run in the 
joints in the hottest temperature in summer, nor become hard and brittle 
through the action of frost. It shall conform to the following requirements: 
99% or over by weight shall be soluble in carbon bisulphide; specific gravity 
at 66° F. to be not less than 1; 100 grams of this cement not to lose more 
than 10% weight when maintained at a uniform temperature of 400° F. for 
7 hours in a cylindrical vessel S^" diameter and 1" high; amount of fixed 
carbon not to be more than 12% and it shall show a flashing point, open oil 
tester, of more than 510° F., and shall not contain more than 2J% of paraf- 
fine scale; if obtained by mix of bituminous materials, it shall be homo- 
geneous, free from water and light oils, obtained by agitation with hot air 
at a temperature of not more than 400° until all the mass is blended com- 
pletely, and shall be free from granular accumulations. Penetration test: 
At 32° F. with No. 2 needle and 100 grams weight for 5 seconds, shall not be 
less than 1 millimeter; at 115° F., No. 2 needle, 50 grams, not less than 8 
nor more than 15 mm.; | gram of the material when made into a ball shall 
not melt and drip through an aperture 1 mm. in diameter at less than 220° F. 
The paving cement shall be heated on the work to a temperature between 
400° and 450° F., in quantities to allow this temperature to be maintained in 
the kettles during progress of pouring; none with temperature below 400° to 
be used. It shall then be put in a conical can and poured in the interstices 
of the blocks till the filler is flush with top of blocks;^ repeating filling if 
necessary. All joints between blocks shall be filled with this hot paving 
cement, pouring from center to sides, but no flushing of the pavement will 
be permitted. Where girder rails are used, the space between the web of rail 
and adjoining blocks shall be filled with mortar, composed of Portland cement 
1, sand 4. Underdraining. — Pipes shall be vitrified, salt-glazed stoneware 
pipe, fitted with proper collars, pipes to be 4" inside diameter and 12'' to 24" 
long. 

Vitrified Brick Pavement. 

Blocks. — The carriageway to be paved with best quality of repressed 
vitrified paving blocks made of shale or clay, and with J" lugs on the sides; 
uniform in size, color and quality and of same make; blocks to be from 2^" 
to 3i" wide, 7" to 10" long, and 4" to 4^" deep, exclusive of projection. Selected 
samples (34 blocks of average quality from each 50000 or less) to be sub- 
jected to the following tests: Bricks to be free from lime and magnesia in 
the form of pebbles and shall show no signs of cracking or spawling on re- 
maining in water 96 hours; when subjected to tests for abrasion, loss in 



1 122 GO.—HIGHWA YS. 

weight not to be more than 20%; shall have a specific gravity of not less 
than 2.3; shall not absorb more than 3% of water when dried at 212° F. for 
48 hours and afterward imniersed for 48 hours in water (test to be made on 
blocks which have been subjected to abrasion test) ; for transverse test, they 
shall show a modulus of rupture of not less than 2000 lbs. per sq. in. when 
tested on edge as laid in the pavement, the modulus to be computed by the 
formula R= dlw-^2bd^, in which R is the modulus of rupture, / the length 
between supports ( = 6), b and d breadth and depth, all in inches, and w the 
load, in lbs., producing rupture. Sand Cushion, Laying, Rolling, Expansion 
Joints, Paving Cement, Underdraining. — (Same as for iron slag block pave- 
ment.) 

Asphalt Block Pavement. 

Blocks. — ^The asphalt blocks shall be 4^ to 5 i" wide, llf'to 12^'' long, 
and 2i" to 3i'' deep; and composed of 6 to 8 parts asphaltic cement, 86 to 
82 parts crushed trap rock, and 8 to 10 parts inorganic stone dust. The 
Asphaltic Cement shall be composed of refined and natural asphalt, or as- 
phaltum, fluxed with liquid petroleum residuum or refined maltha or liquid 
asphalt; no residuum of petroleum other than that contained in the flux, to 
be usedj the refined asphalt and flux shall be mixed in proper and approved 
proportions; the bituminous flux to be free from impurities and brought to 
a specific gravity of from 18° to 22° Beaume, and afirst test not less than 300° 
F., and shall contain no appreciable amount of light oils, or matter volatile 
under 250° F.; the distillate of the petroleum oil, if used at 400° F. for 30 
hours, shall not exceed 10%. The Crushed Trap Rock shall not exceed y'; 
the size to be nearly cubical as possible, and graded from maximum size to 
dust, so as to give the mineral aggregate with a minimum percentage of 
voids. The Inorganic Dust shall be pulverized stone, free from loam, clay 
or other earthy material; no weathered rock or dust from same to be used. 
Blocks when laid shall have a specific gravity of not less than 2.45, and when 
dried for 1 day at a temperature of 150° F., and then immersed in water 
7 days, they shall not absorb more than 1 % water. Laying. — On the concrete 
surface, after same has been swept and wetted, spread a layer of cement 
mortar (composed of 1 part slow-setting Portland cement and 4 parts clean 
sharp sand, free from gravel over i" diameter) to such thickness that when 
struck to a surface 3" below and parallel to the grade of completed pavement 
its depth shall be nowhere less than ^" ; the spreading and surfacing to be as 
follows: On the surface of the concrete, set strips of wood 4'' wide by i" 
thick, or strips of steel 4" by I", and of greatest convenient length; these 
strips to be set parallel and about 8 or 10 ft. apart, running from curb to 
curb, and imbedded in mortar throughout their length so top surface shall 
be 3" below and parallel to grade of finished pavement; the space between 
two strips having been filled with mortar, an even surface shall be struck by 
using an ironshod straight-edge on the strips as a guide, and as soon as the 
bed has been struck, the strip which would interfere with laying the block 
shall be removed and its place filled with mortar with a trowel. On this 
mortar surface, the blocks are to be laid, in courses at right angle to line of 
street (ordinarily), each course to be of uniform width and depth, laid to 
proper crown and grade, with close joints, and end joints broken by a lap 
of at least 4". The surface shall present no greater variation than i" between 
adjoining blocks. Blocks fractured or broken shall be replaced. Sand Joints 
and Covering. — When laid, the blocks shall be covered with clean, fine sand. 




Fig 4. — Curb on Concrete Foundation. 

entirely free from loam or earthy matter, perfectly dry and screened through 
a sieve having not less than 20 meshes per lin. in.; the sand to be swept and 
brushed into the joints and left on the surface until such time when the pave- 
ment shall be swept clean for final inspection, and defects remedied. 



ASPHALT BLOCK. STREET CROWNS. BRICK. 



1123 



RICHMOND (IND.) STREET CROWNING. 

(H. L. Weber, City Engineer.) 

H 4 

Crowning is parabolic: hi = — ; h2 = ihi = ~ H. 

y y 



Leyel ofCurb 
^euiter 



r 



f 



>♦< b — — >k c ->k d J"^ 

Half Widfhof Street - -X^^ 



Fig.^ 


1. — Diff. between g 


andi:f = 


diff. in 


elev. bet. < 


:iirb and crown levels. 


Width 




















of 


a 


6 


c 


d 


G 


g 


H 


h2 


hi 


Street. 
Ft. 




















Ft. Ins. 


Ft. Ins. 


Ft. Ins. 


Ft. Ins. 


Ins. 


Ins. 


Ins. 


Ft. 


Ins. 


Ft. 


Ins. 


20 


2 4 


2 6i 


2 6i 


2 7 


6 


4i 


2^ 


.083 


1 


.021 


^ 


24 


2 4 


3 2i 


3 2i 


3 3 


6 


4i 


3 


.112 


1t^ 


.026 


^ 


30 


2 4 


4 2i 


4 2^ 


4 3 


6 


4i 


4 


.146 


u 


.036 


i 


30 


2 4 


4 2h 


4 2i 


4 3 


4^ 


2i 


2i 


.083 


1 


.021 


i 


36 


2 4 


5 2i 


5 2i 


5 3 


6 


4i 


5 


.188 


2| 


.047 


40 


2 4 


5 10^ 


5 lOi 


5 11 


6 


41 


6 


.224 


\i 


.052 


1 


*45 


(2 4) 


(5 2) 


7 6 


7 6 


6 


(4i) 


(6) 


.288 


.073 


t45 


(2 4) 
(2 3) 


(5 2) 


7 6 


7 6 


6 


(41) 


(4i) 


.222 


n 


.056 


^f 


. #45 


(6 9) 


6 9 


6 9 


6 (4i) 


(4i) 


.157 


li 


.039 


h 


48 


2. 4 


7 2i 


7 2i 


7 3 


6 4i 


7 


.260 


H 


.062 


f 



* Crowning is calculated for H='ll" at curb line; and level of crown is 
If" above level of curb. 

t Half width of street to be used on side next to stone curb. Crowning 
is' calculated for H = &' at curb line; and crown is level with curb. 

# Half width of street to be used on side next to cement curb; and 
crown is level with curb. 



SYRACUSE (N. Y.) PAVEMENT SPECIFICATIONS. 

(H. C. Allen, City Engineer.) 

General. 

Excavation. — Includes earth, rock or other material necessary to be 
removed from work, to depth required to reach sub-grade, and work con- 
nected with adjusting street intersections and grading slopes back of curb 
line. Excavations below sub-grade shall be made up with cement concrete. 
Surplus material shall be deposited within an average distance of 2000 ft., or 
be disposed of by contractor. E mbankment. — Shall start from a well-prepared 
base, mellowed or stepped on sloping ground, and be carried up in horizontal 
layers not over 4'' thick, each layer carefully rammed or rolled, and well 
watered. Rolling and Ramming. — After the sub-grade has been brought to 
lines prescribed, it shall be rolled with steam roller of not less than 5 tons, 
until surface is firm and compact. Portions inaccessible to roller shall be 
rammed, and all depressions, defects, etc., made by the contractor, and 
again rolled and rammed. 

Vitrified Brick Pavement. 
Concrete Foundation.— Upon the sub-grade thus prepared, lay a bed of 
Portland cement concrete 6'' deep, of 1 part cement, 3 parts sand, 6 parts 
broken stone. Use best quality of Portland cement; sand to be clean, coarse 
and sharp, free from foreign matter; broken stone to consist of hard durable 
stone, varying in size from 2Y to \" in diameter, and free from dust or dirt. 
Cushion. — Upon the concrete foundation, thoroughly set and dry, carefully 
spread Y layer of clean, coarse, sharp sand, or fine screened gravel, free from 
loam, dirt or vegetable matter; leaving surface true and even and parallel 
to grade. Paving Bricks. — Shall be made and burned especially for street 



1124 60.— HIGHWAYS, 

paving purposes, and shall stand all reasonable tests as to durability and 
fitness, to which paving material is usually subjected; the material to be 
biimed in down-draught kilns or furnaces. Bricks to be square and strsCight, 
with sharp or slightly beveled edges, free from cracks or other defects, and 
of uniform size and pattern, and approved quality, equal to approved sam- 
ples; specific gravity not less than 2. Size may be: length 7^' to 8}", width 
21'^ to 2^", depth 3^^'' to 4". The absorption of water by any one brick to be 
not greater than 3% ; average absorption of all bricks tested not to exceed 
2% of their dry weight; tests to be made on either abraded or broken bricks 
by drying them for 12 hours in an oven and then soaking them 12 hours in 
water. At least three bricks shall be used for this test. The bricks shall also 
be tested in the standard rattler* and by the method of the National Brick 
Manf .Assn. and Am. Soc. of Municipal Improvements. Sample Bricks. — Three 
or more bricks of the kind or quality to be used on the paving shall be fur- 
nished with each proposal; the bricks to be labeled with bidder's and maker's 
names and addresses. Manner of Laying. — Upon the cushion, the pavement 
to be constructed with a single layer of bricks, laid on edge, end to end, in 
courses at right angles with the curb line, except at street intersections, 
where courses are to be placed at such angles as directed. Bricks to be set 
in courses across the street, which must be kept true and parallel, with the 
body of the bricks close together, sides and ends touching, and breaking 
joints at least 3'' with bricks in adjoining courses; they are to be set perpen- 
dicular to grade of street, and to a height of from I" to f" (ordinarily) above 
the true grade and crown of street when finished, to provide for settlement 
in pounding. Whole bricks to be used except in starting and closing courses 
at curbs, catch-basins and street structures, when not less than half bricks 
may be used in breaking joints, which shall be tight and close at ends. 
Ramming and Tamping. — ^The paving when laid, and either before or after 
the filling of the joints and top dressing is put on, as may be directed, shall 
be thoroughly rammed not less than three times with a paver's rammer of 
90-lb. weight; the blows of the rammer must not be made directly on the 
bricks, but upon a 2" plank not less than 10 ft. long and 12'' wide, which will 
be laid upon the surface of the pavement, which must conform to true grade 
and crown of street. Grout, composed of equal parts Portland cement and 
sand, with proper amount of water, to be poured upon the pavement and 
swept to and fro until every joint is filled flush with the surface of the pave- 
ment, and continued until joints are entirely filled. Wet sand then to be 
spread over the entire pavement i" thick, and kept wet until ordered re- 
moved. 

Sandstone Block Pavement. 
Concrete Foundation. — (Same as for vitrified brick pavement.) Cushion. 
— Upon the concrete foundation, thoroughly set hard, carefully spread 2" 
layer of clean, coarse sand, free from loam, dirt or vegetable matter; leaving 
surface true and even and parallel to grade. Sandstone Blocks. — Shall be of 
best quality of Medina or Potsdam sandstone, not less than 3'' nor more than 
6'' thick, not less than 6'' nor more than 7" deep, and from 7" to 12'' long; 
must be sufficiently dressed to present rectangular faces with straight edges 
on top, bottom and sides, and all blocks whose faces vary more than ^" from 
the rectangular shape will be rejected. ^ The sides and ends of the blocks 
must be so dressed that they will make joints not to exceed i" in width. If 
necessary to obtain a satisfactory surface, the top surface of the blocks to 
be cut or "axed" off smooth, sides and ends to receive similar treatment, 
when necessary to secure the i" joint. The stones to be set tight together, 
perpendicular to grade, so as to break joints at least 2", in uniform rows 
across the street at right angles to line of curb, except at street intersections 
and other places as may be directed. When laid, the blocks to be carefully 
rammed as may be directed, with a paver's rammer, no iron being allowed on 
its lower surface to come in contact with the pavement, which is to be sur- 
faced by using a long straight-edge; shall conform to established grade and 
crown. Grout Filling. — Pour into joints a Portland cement grout composed 
of 2 parts clean, sharp sand to 1 part Portland cement, of approved quality, 
together with enough water to make proper grout, which will iDe poured upon 
the pavement and swept to and fro until every joint is filled flush with sur- 
face of pavement, operation to be continued until joints are entirely filled. 
Wet sand then to be spread over entire pavement 1" thick, and kept wet until 
ordered removed. 



* See page 507. 



PAVEMENT— SANDSTONE BLOCK, ASPHALT, 1125 

Asphalt Sheet Pavement. 

Upon the concrete foundation, thoroughly set and hard, shall be laid 
the wearing surface, which is divided into two classes of standard grade 
sheet asphalt: Refined Asphalt and Rock Asphalt. It is intended to admit 
the use of any asphalt of reputation which can be made into a suitable paving 
mixture. 

Refined Asphalt Sheet Pavement will consist of (1) refined asphalt (an 
asphalt after it has been freed wholly or in part from water or organic matter 
by being heated) ; (2) a hydro-carbon flux or softening agent; (3) sand and 
fine powder of carbonate ot lime. Asphaltic Flux. — Properties required are: 
It shall contain no material volatile under 300° F. ; shall be chemically stable 
and not lose its fluidity by molecular change; shall dissolve the asphalt and 
not simply form a mechanical mixture with it; must not volatilize more 
than 5% when heated for 7 hours at 325° F. The flash points shall be taken 
in a N. Y. State closed oil tested. Asphaltic Cement. — ^The refined asphalt is 
to be combined with the flux to produce asphaltic c'ement, to be tested as 
follows: Must not flash below 350° F.; must contain no water; must not 
show more than 4% loss by weight on being heated at 325° F ; and must not 
show a loss of more than 8|% in weight on being heated 7 hours at 350° F. 
It must not contain more than 4|% of carbonaceous matter insoluble in 
carbon bi-sulphide, and must not show more than 15% of fixed carbon, and 
must not contain more than 3% of paraffine scale. The test for consistency 
of penetration of the asphaltic cement shall be the distance expressed in Vioo 
of a centimeter that a No. 2 needle will penetrate into it at 25° C. (77° F.) 
under a weight of 100 grams in 5 seconds of time, the needle to penetrate 
direct without friction. Sand. — Sand used for body of wearing surface shall 
be clean and sharp and composed of grains not easily crushed. Shall be 
graded in size of grains to reduce voids to a minimum, to secure which, a 
quantity of powdered carbonate of lime, from 5 to 15%, shall be added. The 
sand grains to be graded in size about as follows: Retained on 10 mesh per 
lineal inch, 3% ; on 20 mesh, 5% ; 40 mesh 25% ; 60 mesh 25% ; 80 mesh 12% ; 
100 mesh 18%; passed 100 mesh, 12%; total 100%. Mixing.— The wearing 
surface of graded aggregate and sufficient asphaltic cement to fill the voids 
when laid shall contain no trace of coal-tar, water, appreciable amount of 
light oils, no matter volatile at a temperature of 250° F. It shall yield, when 
extracted with bisulphide of carbon and after evaporation of the solvent, not 
less than 9^% nor more than 13% of bitumen, and at least 68% of the ex- 
tracted bitumen shall be soluble in petroleum naphtha. The sand and as- 
phaltic cement to be heated separately to about 300° F. The pulverized 
carbonate of lime, while cold, will be mixed with the hot sand in the required 
proportion and then mixed with the asphaltic cement at the required tem- 
perature, and in proper proportion, in a suitable apparatus, so as to effect a 
thoroughly homogeneous mixture. Sand boxes and tar and asphalt gauges 
will be weighed up daily. Laying. — Pavement mixture thus prepared will 
be laid on the foundation (same as for vitrified brick pavement) in two coats: 
The first, or cushion coat, will contain 2 to 4% more asphaltic cement than is 
required for the surface mixture, and laid so as to give a thickness of i'' after 
being consolidated by rollers. The second, or surface coat, prepared as 
required, to be laid on the cushion coat; it will be brought to the ground in 
carts at a temperature not less than 250° nor more than 300° F., and if the 
temperature of the air is less than 50° the contractor must provide canvas 
covers for use in transit; the mixture to be spread with hot iron rakes to 
uniform grade, and to have thickness of 2" after ultimate compression. The 
surface to be compressed by a hand roller, after which a small amount of 
hydraulic cement will_„be swept over it, then thoroughly compressed with a 
heavy steam roller, continued as long as it makes an impression on the sur- 
face, at least 5 hours for each 1000 sq. yds. of surface. Gutters. — Strip of 
pavement 12" wide next to curb, to be coated with hot pure asphalt, and 
smoothed with hot smoothing irons, price to be included in price for pave- 
ment. 

Rock Asphalt Pavement, — Shall consist of one or more natural bituminous 
limestones or bituminous sandstone rocks. If necessary, they are to be mixed 
together or a quantity of natural asphalt added to secure proper proportion 
of bitumen, between 9 and 10%. A bituminous limestone shall be coarse- 
grained, as nearly as possible a pure carbonate thoroughly and evenly im- 
pregnated with asphalt, with no more impurities than the standard German 
rock asphalt of Limmer or Vorwohle. Laying Pavement. — (1) The lumps of 
rock to be crushed and pulverized and the powder passed through a fine 



1126 GO.— HIGHWAYS. 

sieve. (2) This powder to be heated in a suitable apparatus to a temperature 
of about 200° F., and brought to the ground at such temperature, in carts, 
and spread on the concrete foundation (same as for vitrified brick pavement) 
previously prepared. (3) Then skillfully compressed by heated hand rollers 
and rammers until it shall have required thickness of 2^". (4) Surface then 
made even by heated smoothers. (5) Finally, after completion, to be rolled 
with a heavy roller at least 5 hours for each 1000 sq. yds., a small amount of 
hydraulic cement to be swept over the surface during rolling. 

Creosoted Wood Block Pavement. 
Concrete Foundation. — (Same as for vitrified brick pavement.) Mortar Bed. 
— On the foundation, spread a ^" bed of Portland cement mortar, 1 part 
cement and 2 parts sand, surfaced true and smooth and parallel to finished 
grade and cross-section. Blocks. — Long-leaf yellow pine, 50% heart, treated 
as described below; blocks to be of sound timber, free from bark, sap wood, 
loose or rotten knots, etc.; no second-growth timber to be allowed. Blocks 
not less than 3" wide, 5'' to 9'' long and 4'' deep, uniform in depth and thick- 
ness; to be treated throughout with an antiseptic and water-proofing mixture 
as follows. Treatment of Blocks. — Mixture shall contain 50% of dead oil of 
tar, known as creosote oil; its specific gravity shall not be less than 1.12 at 
68° F., it shall lose not more than 40% when distilled in a flask or retort for 
30 minutes up to 600° F. The distillate to contain 4% tar acids and at least 
12% naphthalin. Specific gravity of residue obtained by distilling the mix- 
ture up to 600° F. must be at least 1.15. After treatment, specific gravity of 
block shall be greater than that of water; and shall show such waterproof 
qualities that, after being dried in an oven at a temperature of 100° for 24 hrs. , 
weighed and then immersed in water for 24 hours, the gain in weight not to 
be greater than 3%. Blocks to be treated as follows: Blocks to be placed 
in an air-tight cylinder, and when doors are closed the dry heat is to be raised 
to 215° F. without pressure, for 1 hour, to get rid of moisture. Then heat 
to be increased, pressure to be applied and both are to be raised gradually to 
avoid injury to fiber, for 2 hours, until heat has reached about 285° and 
pressure about 90 lbs., and both are to be held there for 1 hour. The heat is 
then to be shut off and the tanks allowed to cool gradually for 1 hour; then 
heat reduced to 250° and pressure to about 40 lbs. Pressure is then blown 
off and heat still further reduced. Vacuum is then applied until about 26" is 
raised, and while under vacuum the creosote mixture (which shall contain 
no tar, petroleum, or petroleum residue) is to be run into the cylinders at a 
temperature of 175° to 200°, and hydraulic pressure is to be applied, reaching 
200 lbs. per sq. in., and kept at this point until 20 lbs. of the mixture per cu. 
ft. has been absorbed. The liquid is then run off, and the wood placed in 
another cylinder and milk of lime at a temperature of about 150° is run in and 
hydraulic pressure of about 200 lbs. applied for from ^ to 1 hour. The anti- 
septic and waterproof mixture shall not contain more than 2% water at any 
time. The quantity specified for each tar acid, naphthalin and residue is the 
minimimi for each. Laying Blocks. — Blocks to be set immediately upon the 
cement mortar bed, before it has set, and driven together as closely as pos- 
sible. Pavement to be constructed with a single layer of the blocks laid on 
edge, with grain vertical, end to end in courses at right angle with the curb 
line except at street intersections where the courses are to be placed as di- 
rected. Blocks to be set in courses across the street; must be kept true and 
parallel, sides and ends touching, and breaking joints at least 3'' with blocks 
in adjoining courses. Only whole blocks to be used except in starting and 
closing courses. Expansion Joints of bituminous cement to be placed at curb 
lines and across the street at intervals of 50 ft. Gutter joints to be V and 
cross joints i''. To make these, a plank shall be inserted and the blocks laid 
against it; the plank then removed, and the crack thus left filled with bitu- 
minous cement which shall have a temperature of at least 300° F. Pavement 
then to be rolled with a hand roller until tops of blocks are even. Bituminous 
Cement used shall not flow at 120° F., and shall not become brittle at 0° F.; 
shall be proof against street liquids, and pliable rather than rigid. Filling 
Joints. — After pavement is rolled, bituminous cement heated to at least 300° 
F. shall be poiired along, filling each crack, and only when blocks are dry. 

BiTULiTHic Pavement. 

Foundation. — Bituminous, or concrete. Bituminous Foundation. — On the 

sub-foundation, crushed hard limestone which will pass a 3^'' ring, to be 
spread to a depth of 6", and compressed with heavy steam road roller. On 



CREOSOTED WOOD BLOCK P, BITULITHIC P. 1127 

this, after rolling, spread a heavy coating of Puritan brand bituminous 
cement, to make foundation unite with bitulithic wearing surface : One gallon 
of the cement to each sq. yd. of foundation. Concrete Foundation. — (Same 
as for vitrified brick pavement.) Wearing Surface. — On the foundation, lay 
the wearing surface, composed of carefully selected, sound, hard, crushed trap 
or syenite rock, mixed with bitumen, as follows: After heating stone in rotary 
mechanical mixer to temperature of about 250° F., it is elevated and passed 
through a rotary screen, having sections with various size openings; differ- 
ence in width of openings in successive sections not to exceed i" in sections 
with openings less than Y\ and not to exceed i'' in sections with openings 
more than Y'. The several sizes of stone thus separated by the screen sections 
shall pass into a bin with sections corresponding to screen sections. From 
these, the stone is drawn into a weigh box, resting on a scale having seven 
beams; the stone to be weighed, using the proportions previously determined 
by laboratory tests to give the best results; that is, the most dense mixture 
of mineral aggregate, and one having inherent stability. If the crushed 
stone in the wearing surface does not provide the best proportions of fine- 
grained particles, such deficiency must be supplied by the use of not to ex- 
ceed 25% hydraulic cement, pulverized stone, or very fine sand. From the 
weigh -box each batch of mineral aggregate, of different sizes weighed as 
above, shall pass into a "twin pug," or other approved form of mixer. In 
this mixer shall be added a sufficient quantity of Puritan brand bituminous 
waterproof cement, to coat all the particles of stone and to fill all voids in 
the mixture. Before mixing, the bituminous cement shall be heated to 
between 200° and 250° F., and the amount in each batch shall be weighed 
and used in such proportions as have been determined. Mixing to continue 
until the combination is a uniform bituminous concrete. In this condition it 
is to be hauled to the street, and spread on the prepared foundation to such 
a depth that, after compression with a steam road roller, it shall have a thick- 
ness of 2''. The proportioning shall be such that the compressed mixture 
shall have the density of solid stone, as nearly as practicable. Surface Finish. — 
After rolling the wearing surface, spread over it, while it is still warm, a thin 
coating of quick-drying bituminous flush coat composition, by means of a 
suitable flush coat spreading machine provided with a flexible spreading 
band and adjustable device for regulating the quantity and uniformity of 
the composition. On grades of over 4% a mineral flush coat may be used in 
place of the liquid flush coat. While the flush coat is still warm, spread over 
it, in at least two coats, fine particles of hot crushed stone, in sufficient quan- 
tity to cover the surface of the pavement ; these stone chips to be spread by 
means of a suitable stone spreading machine, so designed as to provide a 
storage receptacle of at least 5 cu. ft. capacity and to rapidly and uniformly 
cover the surface of the pavement properly. The hot stone chips to imme- 
diately be rolled into the surface until it has become cool. Patents. — Agree- 
ment of Warren Bros. Co., on file with City Engineer, to license all contractors 
desiring to bid for the work to lay bitulithic pavement in accordance with 
its patents. 

TORONTO (ONT.) PAVEMENT SPECIFICATIONS. 

(C. H. Rust, City Engineer.) 

A. — Grading. 
Excavation. — Levels and cross-sections may be varied to conform to sills 
of buildings, grades of intersecting streets, lanes, carriage ways, etc. Trenches, 
Etc. — Trenches or excavations that have been made for or in connection 
with sewers, private drains, gas or water pipes, telephone or electric wires, 
pipes or conduits, street or other railway works, or any other lawful purpose, 
and which are not thoroughly settled, shall be opened out and re-filled with 
gravel, well pounded, in layers of not over V\ no extra allowance to be made 
to contractor. Defective Places. — Soft, boggy, wet, muddy or defective places 
must be wholly removed and filled with gravel, as in above clause, with no 
extra allowance. Boulders, Trees, Etc. — Boulders, stones, rocks, stumps, trees, 
roots, etc., to be removed when directed, without extra pay. Excavations 
Below Grade. — Where excavation is made below proper level of sub-grade, it 
shall be made up with concrete where foundation is concrete, or with gravel 
in all other cases, without extra pay. Rolling, Etc. — Sub-grade to be rolled 
with steam roller weighing not less than . . . tons ; to be omitted when engineer 
shall consent in writing. Portions inaccessible to roller shall be rammed. 
Settlements, etc., to be repaired, and again rolled or rammed. Engineer 



1128 GO.—HIGHWAYS. 

reserves privilege of testing sub-grade with City Roller. Slopes. — In cuttings, 
excavation to be made for a sufficient distance above and behind curbing, 
to form a slope of 2 horizontal to 1 vertical. Oil Macadam. — Where a ma- 
cadam or broken stone roadway has to be excavated, the old material must 
be picked out and screened separately, and the stone delivered by contractor 
where directed, within 1 mile, without pay; extra haul at ^c. per cu. yd. per 
100 ft. 

B. — Cedar Block Pavement. 
Character of Pavement. — When gravel, sand or broken stone is used for 
foundation, it must be watered, rolled, rammed and consolidated, luitil 
quite hard and compact, using a 12i^-ton roller, unless a lighter one is allowed. 
Blocks. — Shall be of first-growth, sound white cedar, stripped of bark. No 
pin hole more than V diameter, and not more than 3 pin holes allowed on 
wearing surface of block. Blocks from 5" to 11'' in diameter, and shall not 
show more than i" of sap wood at any part of outside edge. Laying. — After 
blocks are laid, they shall be rammed with a pounder of 80 lbs. or more, 12" 
diameter and fiat on bottom. The whole surface then rolled with a roller 
weighing at least 5 tons. When pavement has been brought to surface, it 
shall (unless other filling is called for) be covered with a sufficient quantity 
of gravel to fill all joints, being worked in with suitable brooms. Surplus 
material, if any,, then to be swept off, and pavement to be rammed as before. 
To be repeated if necessary. Finally, pavement to be covered with |" layer 
of good clean gravel. Board Bed. — Shall be laid upon the foundation bed if 
required. Shall be pine, 9" to 12" wide, perfectly sound, etc.; and thor- 
oughly swabbed on both sides and ends with approved composition, or 
dipped in same, or other preservative process. Where double boards are 
used, the lengthwise boards shall be 12 and 16 ft. in length, respectively, 
laid so as to break joint. Concrete Bed. — When a concrete bed is required it 
shall be laid as specified under "C. — Concrete," below. After the bed has 
set hard, spread a layer of clean, sharp sand so that it will be V thick after 
the blocks are laid upon it and rammed. Grouting. — When grout filling is 
required between the blocking it shall be composed of 1 part Portland cem., 
3 parts coarse, sharp, clean sand, thoroughly mixed and flooded, and swept 
into all joints; repeating same until joints are full and flush with surface. 
Surface then to be covered with at least f" layer of approved gravel. Tar 
Filling. — When tar composition is required for filling, it shall be composed of 
1 part coal-tar to 2 parts pitch, boiled and freed from moisture, and applied 
at a temperature of 250 to 275° F., filling all joints. A paving pitch, when 
approved, may be used instead, and applied similarly, or as directed. Clean 
gravel, i" to f", dried and heated, must be used with above composition; 
the gravel to be swept into the interstices hot, and the composition applied 
at once, completely filling joints and fiushing to surface. Composition and 
Gravel. — With board foundation, use not less than 2 gallons (imperial) of 
composition per sq. yd. of pavement; without board foundation, use not 
less than 3 gallons, and as much more as needed to fill flush to top of pave- 
ment. After ramming, pavement to be swept clean, and covered with hot 
composition, upon which shall immediately be spread heated gravel or 
stone chippings, i" to Y\ at least Y' deep. 

C. — Concrete. 

Proportions. — Upon the sub-grade, lay a bed of concrete, .... ins. thick, 
composed of 1 part best Portland cement, 3 parts clean, sharp sand, 7 parts 
broken stone or furnace slag; the proportions may be varied to 1 part 
cement and 10 parts of the sand, broken stone and slag. Gravel may be 
required instead of broken stone. 

D. — Asphalt Pavement. 

Kinds of Asphalt. — Use best quality of following asphalt: Trinidad, 
Bermudez, Venezuela, Natural Rock, California, or other equally as good, 
and approved. No coal-tar, or any other product thereof, or any other in- 
ferior products, will be accepted. Cushion Coat. — When light asphalt is 
specified no binder or cushion coat will be required. When heavy asphalt is 
specified a binder course 1" thick will be laid directly on the concrete founda- 
tion, but it will not be required where Natural Rock asphalt is used. This 
underlayer to be rolled and consolidated, while fresh and hot, with a steam 
roller weighing not less than 5 tons, to a finished thickness of V. No binder 
asphalt to be laid during wet weather. Should Rock asphalt be used it 
should be at least 2" thick. Surface Coat. — Shall be 2" thick after ultimate 
compression. 



CEDAR BLOCK, ASPHALT, MACADAM, ETC, 1129 

E. — Brick Pavement. 

Foundation. — A base of Portland cement concrete 4/' or 6" in depth. 
Cushion. — Where no thickness is shown, the depth of sand will be at least H". 
Paving Bricks. — Must not be less than 2Y' by 8^'' by 4". Paving Blocks. — 
Must not be less than BY' by 8^" by 4''. Tests shall be made for absorption 
and abrasion. Laying. — (Usual manner.) Ramming and Rolling. — First with 
80-lb. rammer, then with 5-ton roller. Grout Filling. — Composed of equal 
parts best Portland cement and sand, mixed; water added, stirred and im- 
mediately used, filling joints to no more than half their depth. Then thicker 
grout, 2 cement to 1 sand, for balance of joint. Sand Coating. — Not less 
than i*. 

F. — Macadam Roadway. 

Foundation Course. — Upon the sub-grade, lay a course of stone, 5" in 
thickness; stones to be laid by hand, largest sides down, and in line at right 
angle to the curb, and breaking joint as much as practicable. The upper 
surface of stones not to be less than 6" nor more than 9" in width, nor less 
than 12" nor more than 15'' in length. The stones to be set close together 
and bound by wedging in small stones and filling the interstices with stone 
chipping so as to form a compact bed. Stones projecting above the surface 
must be broken off, great care being used not to loosen the foundation. No 
wedging is to be done within 20 ft. of the work being laid. Next, spread 
evenly over this course, clean gravel to fill all interstices, and then roll and 
re-roll until thoroughly consolidated. Intermediate Course. — Upon the 
foundation, place a layer of broken stone, as granite, trap, or hard limestone, 
of approved quality. Not more than 5% of the stone shall be less than If" 
and no particle of stone shall be more than 3" in breadth. After this course 
has been evenly spread over the street (and raked if necessary) so as to 
present a uniform surface, a layer of coarse, clean sand is to be spread upon 
it, sufficient to fill voids, and the surface then watered (if necessary) and rolled, 
more sand and stone being applied where and as required, and the rolling 
and watering continued until an even, hard and uniform surface is obtained, 
after which the sand is swept up and removed from the surface. Top 
Course. — Consists of broken stone, .... ins. in depth, as uniform in size as possi- 
ble, no particle to be more than 2^', and not more than 5% to be less than 1^", 
in length or breadth. The surface shall be raked evenly, watered, rolled, 
repaired, brought fully up to grade, and then re-rolled until firm, compact 
and true. A sufficient layer of good, coarse, clean sand is then spread over 
the surface, rolled, and flooded with water, to carry the sand into all inter- 
stices; more sand to be added as rolling and watering progresses, so as to 
have finished thickness of i" to i". 

K. — Concrete Sidewalk. 

Foundation. — After the street has been graded, a foundation shall be 
laid, composed of coarse gravel or suitable soft coal cinders, to a depth of i", 
after being consolidated by pounding or rolling with a suitable and approved 
roller, weighing at least 1 ton, and the whole brought to an even surface. 
Whilst pounding, a small quantity of water may be used through a sprinkler, 
if directed. Templates. — When required, the contractor must furnish 
wooden templates, cut to exact form and slope of the walk, for use by the 

1 7 Concrete Surface 
"F T 







•4 Cinders or Gravel 



Z THe 
Drain- 

Fig. 6. — Cross-section. 

inspectors. Concrete Base. — Upon this foundation, a layer of concrete, 3i* 
thick, shall be laid, composed of 1 part Portland cement (of approved qual- 
ity), 2 parts of clean, sharp, coarse sand, and 5 parts of approved furnace 



1130 ^.—HIGHWAYS. 

slag, broken stone or screened gravel, thoroughly free from stone over 2* 
diameter, and free from clay, loam, dirt or other impurities. The concrete 
thus made shall be rammed with iron rammers into one solid mass, and until 
it has a straight and even surface. Divisions. — Every 6 ft. a cut shall be 
made completely through the concrete before it is set, with an iron for that 
purpose, not less than i" in -width. The opening shall then be filled in with 
clean, sharp sand. A clear space not less than i'' must be left between back 
of curbing and abutting ends of sidewalks to allow room for expansion, 
excepting where the walk and curb are combined. Heavy Surface. — On the 
concrete base, before it has had time to set, lay the wearing surface, H" 
thick, composed of 1 part Portland cement, 1 part clean, sharp, coarse sand, 
and 3 parts crushed granite or quartzite. Light Surface. — On 'the concrete 
base, before it has had time to set, lay the wearing surface 1'' thick, com- 
posed of 1 part Portland cement, 1 part clean, sharp, coarse sand, and 3 
parts of screened pea gravel, crushed granite, quartzite or suitable hard 
limestone. (See Fig. 6.) 



TARS FOR ROAD SURFACES. 1131 

D.— CARE OF ROAD SURFACES. 

DUST PREVENTIVES.* 

Classification. 
Two classes: 1st, water, salt solutions, certain light oils and tars, and 
oil and tar emulsions; 2nd, the heavier oils, tars, semi-solid and solid ma- 
terials. Salt solutions are valuable because the dissolved salt has a con- 
siderable affinity for water and keeps the road moist long after a surface 
treated with water alone would have become dry. The light oils and tars, 
and oil and tar emulsions, leave upon the road surface a comparatively 
small amount of true binding base after the volatile products have evapor- 
ated. The heavy oils and tars contain a greater amount of binding base, 
hence more lasting. The semi-solid and solid preparations usually contain a 
still greater amount of binder, and also other materials of a solid nature, 
such as rocks, sand or clay. With some few exceptions, all the true binders 
are bitumens, either natural or artificial. 

Tars, Their Manufacture and Properties. 

Coal Tars. — Coal is by far the most important source of tar. (a) Tar 
from Coke Ovens is made as follows: The coal is charged into long narrow 
chambers or retorts of about 4 or 5 tons capacity and heated by means of 
flues set in the retort walls; volatile matter held in the coal passes out 
through an opening in the top and is conducted through a series of washers 
and scrubbers, as in gas manufacture, to remove the tar and ammonia; 
the purified gas is then allowed to pass into a holder from which it is drawn 
as needed for burning under the retorts, (b) Tar from Gas Plants is un- 
avoidable in the manufacture of illuminating gas; the bituminous 'coal is 
placed in fire-clay retorts about 8 ft. long, 15" high and 18" wide; 6 or 8 
retorts set together in a furnace and forming a "bench;" a number of these 
benches built together is called a "stack;" the retorts are heated by means 
of a coke fire or by generator gas. The tar which collects in the hydraulic 
main, the condensers, and the tar towers, is run into large wells where it is 
allowed to settle; the accompanying ammoniacal liquor, being lighter than 
the tar, rises and is drawn off; the crude oil tar which remains is a black 
viscid fluid with peculiar odor, and with specific gravity from 1.1 to 1.2. It 
represents about 5% of weight of coal. The true tarry products are artifi- 
cial bitumens; the natural bitumens being found in various mineral oils and 
asphalts. The nature and value of tar vary with the coal used, and with the 
temperature and other conditions under which it is produced. Free carbon, 
having no binding value, will prove detrimental, (c) Refined Coal Tar is 
obtained by fractional distillation for the separation of certain constituents 
used in the arts; the residue left in the still is known as coal-tar pitch, and is a 
thick viscous material while hot. It represents the true binding base of the 
tar, and if the tar is produced at comparatively low temperature the residue 
is composed mainly of bitumens. After cooling for a few hours it is run out 
of the still and is graded as soft, medium or hard, according to its condi- 
tion when cold. The dead oils, or heavier distillation products, and of less 
value than the other volatile distillates, are often run back into -the still 
before, the pitch is drawn off, in which case the pitch is liquid when cold. 
In preparing a tar for dust prevention, most of the valuable products are 
removed by fractionation, and the least valuable — as some of the carbolic 
and all the dead oils — are run back into the pitch until it reaches about the 
consistency of a heavy crude tar, sometimes adding dead oils from previous 
distillation, if necessary. These oils give life to the tar, and if percentage of 
pitch is not reduced too low the mixture has certain advantages oyer the 
crude tar, and it is comparatively free from naphthalin and anthracine and 
contains none of the volatile oils and ammoniacal liquor found in the latter. 
(d) Dehydrated Tar (crude) is sometimes prepared for dust prevention, the 
idea being to remove all water, ammonium compounds, and some of the 
light oils. The absence of water makes it easier to handle when applied hot, 
and probably allows of a better absorption of the tar by the road surface. 
Water in tar hastens disintegration of the heavy binding materials; the 
ammoniacal liquor may saponify some of the oily products, mix with the 
water and wash out. Dehydrated tar may be prepared by boiling the crude 
material in open kettles until its boiling point lies between 105° and 110° C. 

* Digest of Bulletin No. 34, Office of Public Roads, U. S. Dept. of Agric. 
Prevost Hubbard, Assistant Chemist. 



1132 



GO.-^HIGHWAYS. 



Water-Gas Tar has been used to some extent as a dust layer and road 
preservative. It is first obtained by admitting steam into a chamber called 
a generator which contains coke heated to incandescence; the water vapor 
reacts with this coke to form certain products and the mixture of gases is 
led into another chamber called the carburetor, where it meets a spray of 
hot oil, which is thus volatilized and carried to another chamber known as 
the superheater, where most of the hydrocarbons combine to form a per- 
manent gas. The gas thus produced is washed with water and passed 
through extractors and scrubbers in much the same manner as ordinary coal 
gas, in order to remove the tarry products. The product is entirely different 
from ordinary coal-tar and contains a relatively small amount of heavy 
bitumens; the base is more or less thin, and of poorer binding quality than 
that of good coal tar; it may be used to advantage in certain instances, 
being cheap and easily handled. It compares quite favorably with the 
lighter oils, and oil and tar emulsions. 

Composition of Tars. — ^The following Table shows some of the properties 
of crude coal tar, refined coal tar and water-gas tar, previously described. 
The notes to the table refer to the condition of distillates and residues when 
cold: 

Specific Gravity and Composition of Tar Products. 



Kind of Tar. 


Specific 
Gravity. 


Ammo- 
niacal 
Water. 


Total 
Light Oils 
to 170°C. 


Total 
Dead Oils 
170°-270°C. 


Residue 
(by differ- 
ence) . 


Water-gas tar. . . . 
Crude coal tar. . . . 
Refined coal tar . . 


1.041 
1.210 
1.177 


Per cent. 
2.4 
2.0 
0.0 


Per cent. 
a21.6 
dl1.2 
M2.8 


Per cent. 
652.0 
^26.0 
g47.6 


Per cent. 
c24.0 
/54.8 
/39.6 


a Distillate 
b Distillate 
c Pitch ver 
d Distillate 


mostly li( 
all liquid 
y brittle, 
mostly so 


luid. 
lid. 


e Distillate one-half solid. 
/ Pitch hard and brittle. 
g Distillate one-third solid. 



The Application of Tars. 

Application to finished road surfaces. — The primitive method in this 
country is: Road surface first thoroughly swept to remove all dust; hot tar 
then spread on and thoroughly broomed in; road then, if possible, closed to 
traffic 12 hours to allow tar to soak in; at end of that time, or sooner, a coat 
of clean sand or stone chips applied to absorb excess of tar, and surface then 
rolled several times to bring it to proper condition quickly. The tar is 
heated in an open kettle preferably mounted on wheels and fitted with a 
portable fire-box. It is usually brought to its boiling point — about 190° F. 
(If temperature of crude tar is raised above 190° F. when heated it is very 
likely to foam up, boil over and catch fire.) — before being spread upon the road, 
although a lower temperature is sometimes sufficient; and if the ket1>le is of 
the above type the tar may be run out upon the road by means of a hose, 
the kettle being kept just in advance of the work; two kettles will allow 
continuous working, one being charged and heated while the other is in use; 
kettles to hold easily 9 barrels or about 450 gallons. Application of tar by 
mechanical means is also being used, notably in England. 

Use of tar in road construction. — When a New Road is under construc- 
tion, or an old road being resurfaced, the road should first be shaped and 
consolidated as well as possible without the use of water. The voids should 
be filled well with clean, fine stone chips free from dust, but an excessive 
amount of rolling should be avoided because if the roller is used too freely 
the larger stones will become rounded and covered with dust, thus preventing 
the tar from adhering properly. Hot tar may be applied to all the courses 
if desired, but sometimes only the upper course is so treated. After the tar 
has been applied, a dressing of fine material is spread on and the whole road 
well rolled. The tar may be spread by hand, but it is most economical to 
use a tar spreader: the spraying apparatus is mounted on wheels and is so 
arranged that the tar is forced from the tank in which it is heated, into an air 
receiver under a pressure of 150 to 350 lbs. per sq. in.; the necessary power 



TARS AND OILS FOR ROAD SURFACES. 1133 

for pumping the air and the liquid into the receiver being obtained by means 
of chain drive from the road wheel. From the receiver the tar is distributed 
upon the road by means of specially designed spraying nozzles. 

Amount and cost of materials. — According to conditions and methods 
of application, a surface-treated road will require from 0.35 to 0.70 gallons of 
tar per sq. yd. when applied by hand; and as small as 0.21 gallon has been 
used with good results when applied by machine, for first treatment. With 
either method the application of tar must be repeated from time to time, 
though less is required at each successive application. If tar is applied as 
road is built, as much as 1.5 gallons per sq. yd. are often consumed if spread 
by hand; but by means of devices like the pneumatic tar sprayers, it is 
claimed that the road stones to a depth of 3^" may be well covered with 
about 0. 6 gallon per sq. yd. Crude coal tar can ordinarily be purchased from 
gas or coke companies at from 3 to 5 cts. per gallon; renned tars run from 
6 to 12 cts. and even higher. The cost of treatment in France, by machine, 
will average about 3 cts., and by hand 5 cts.; in this country, where it is 
generally applied by hand, the cost ranges from about 6 to 12 cts. or more 
per sq. yd. 

Oils, Their Classification and Properties. 
Oil fields. — There are seven distinct oil fields in the United States: 1st 
the Appalachian (including New York, Penn., W. Va., southeastern Ohio, 
and parts of Ky. and Tenn.) produces oils known as eastern oils or paraffin 
petroleums, and which are therefore of use only as temporary binders in dust 
suppression; 2nd the Ohio-Indiana field produces oils much like those of the 
Appalachian and are also classed as paraffin oils; 3rd the Colorado field, 
similar to above; 4th the Wyoming field with oils varying from the lighter 
oils, to the heavy asphaltic oils which are found principally in California; 
5th the California field produces oils of the most varied character, consisting 
mainly of more or less dense asphaltic hydrocarbons, none of the components 
being of the paraffin series, the percentage of asphaltic residue usually high and 
of good binding character, the oils being considered the best for use as perma- 
nent binders; 6th the Texas field contains oils of a mixed character, with 
some paraffin as well as a greater or less amount of asphaltic residue, some 
having been used successfully as dust preventives, with others unfit for this 
purpose; 7th the Kansas field (including Oklahoma) produces oils quite 
similar to those from Texas. The same is true of Louisiana. In general, 
the eastern oils are of the paraffin type and useless as permanent binders; 
the western oils are of asphaltic character and of great value as permanent 
binders; while the southern oils are of a mixed character, their value as dust 
preventives lying in the relative amount of asphalt base contained. 

Refining. — Although crude oil is used to a great extent in the West as a 
dust preventive, it is often customary in the East to partially distill oils 
containing asphaltic residues before using them, thus recovering many of the 
more valuable constituents and producing residual oils having a much better 
binding quality because they contain a larger percentage of asphalt base. 
Crude petroleum is an oily liquid, of unpleasant odor, with specific gravity 
from 0.73 to 0.97, according to locality from which it is derived; color from 
greenish brown to nearly black, often reddish brown or orange when viewed by 
transmitted light; sometimes fluorescent. The crude petroleum is refined 
by means of fractional distillation, somewhat similar to that for crude coal 
tar. The most valuable products are the kerosene, or burning oils, and the 
method called "cracking" is employed to increase their yield; this consists 
in modifying the fire, during process of distillation, so that only the bottom 
of the still is intensely heated, while top and sides, being exposed to the air, 
become somewhat cooled; thus the heavy oil vapors are condensed within 
the still itself, and upon dropping back into the residuum, which is much hotter 
than their boiling point, break up into lighter oils with lower boiling points, 
with a separation at the same time of free carbon or coke, which is deposited 
in the residuum. The paraffin petroleum residuums contain a large amount 
of paraffin hydrocarbons and paraffin scale or crude paraffin, and are unsuit- 
able for road surface treatment. The base held by the California petroleums 
is composed of bitumens resembling asphalt; the residuum contains no par- 
affin and, if cracking has not been employed in its preparation, carries but 
little free carbon; both the crude oil and the residuums, if properly prepared, 
are excellent binders and give the best results of any oils which have been 
used as dust preventives. The semi-asphaltic oils, as from Texas, carry an 
asphaltic base, but also a considerable amount of paraffin hydrocarbons and 



1134 



-HIGHWAYS, 



1% or more of paraffin scale; are somewhat inferior to the Calif orina prod- 
ucts but often give good results. 

Comparisons of crude oils and residuums. — ^The two following Tables 
show some results obtained from an examination of various crude and 
refined petroleums in the New York Testing Laboratory. 

Results of Tests of Crude Petroleums. 



Kind of Oil. 


Spec. 
Grav. 


Flash- 
ing 
Point. 


Volatil'y 

at 110°C. 

7 hours. 


Volatil'y 
at 160°C. 
7 hours. 


Volatil'y 

at 205°C. 

7 hours. 


Residue. 


Pennsylvania, paraffin. 
Texas, semi-asphaltic. . 
California, asphalt ic. . . 


0.801 
.904 
.939 


26 


Per cent. 
47.3 
20.0 


Per cent. 
58.0 
27.0 


Per cent. 
68.0 
49.0 

(i42.7 


Per cent. 
632.0 
c51.0 
^57.3 










a Ordinary temperature. d Volatility at 200°, 7 hours. 
b Soft. e Soft maltha; sticky. 
c Quick flow. 

Results of Tests of Petroleum Residuums. 


Kind of Oil. 


Spec. 
Grav. 


Flash- 
ing 
Point. 


Volatil'y 
at 200°C. 

7 hours. 


Residue. 


Solid 
Paraffin. 


Fixed 
Carbon. 


Pennsylvania, paraffin. 
Texas, semi-asphaltic. 
California, asphaltic . . . 


0.920 

.974 

1.006 


°C. 
186 
214 
191 


Per cent. 

14.2 

6.2 

17.3 


Per cent. 
a85.8 
a93.8 
a82.7 


Per cent. 
11.0 
1.7 
0.0 


Per cent. 
3.0 
3.5 
6.0 



a Soft. 



The Application of the Heavier Oils. 



To macadam surfaces. — Holes and inequalities should be repaired; not 
necessary to remove all dust as in case of tar, but sticks, leaves, etc., should 
be removed; crude oil either hot or cold, according to its viscosity and 
ability to penetrate the road surface; much cheaper applied cold. A cover- 
ing of sharp sand or Y stone screenings should be applied after the oil has 
been allowed to penetrate as much as possible, in order to take up all excess, 
and the surface well compacted by rolling, additional sand or screenings 
being thrown on wherever the oil shows a tendency to force its way to the 
surface and produce a sticky condition. Sometimes 2 or 3 courses of oil and 
screenings are applied. 

During construction of macadam road. — The greatest success has been in 
California where the heaviest asphaltic oils are found; and the residuums 
obtained from the partial distillation of these oils have, so far, given the 
best results. The treatment is essentially the same as with tar, above de- 
scribed. The macadam is built in the usual manner and each course thor- 
oughly rolled until the whole road is consolidated. A road constructed in 
this manner will require from f to 1^ gallons of oil per sq. yd. 

To gravel roads. — A gravel road is oiled in much the same way whether 
it is an old road or one under construction, as only the upper course is treated 
in either case. Good drainage is very essential. The oil may be applied 
either hot or cold, according to its viscosity, by any method previously de- 
scribed. Where the treated surface is loose and contains a considerable 
amount of clay, the oil may be worked into the upper course by raking, in- 
suring an equal distribution. After application of oil, the road should be 
rolled until properly compacted, adding fresh material as the oil works to 
the surface. Condensed specifications from biennial report (1906) by Com- 
missioner of Department of Highways of California are: Sub-grade to be 
thoroughly rolled; then 2 layers of gravel, bottom layer h" and top layer 3" 
after being rolled; 1st layer containing not larger than 2Y stone; gravel to 



SPECIFICATIONS— FOR COAL TARS; OILS, 1135 

be evenly spread, well moistened, rammed 1 ft. from gutter or curb, and 
remaining portion rolled; depressions filled, moistened, and again rolled to 
unyielding surface; on this surface place the top layer of gravel, no stones 
over I" diameter, and compacted in same manner. Oil should then be evenly 
distributed over the entire surface, i gallon per sq. yd., and covered with 
clean, sharp sand until no oil can be seen; after 12 hours, another applica- 
tion of oil and sand in same manner, and rolled to unyielding siirface. Use 
crude oil applied at temperature between 150° and 190° F. 

To earth roads. — (a) The oil is simply sprinkled on the road, laying the 
dust and incidentally hardening the surface. Alkali soils disintegrate the 
oil and destroy its binding qualities. A sandy loam is the most suitable for 
treatment, usually giving good results when properly treated with an oil of 
good binding quality. Clay is probably the worst of all, as it does not 
absorb the oil well and exhibits a tendency to ball up and give trouble ; sand 
should therefore be added to the clayey surface. Special attention should 
be paid to drainage, the roadbed to be dry when the oil is applied, (b) An- 
other method: The road is first plowed to depth of 6'' and properly crowned; 
all clods and lumps broken up by means of a harrow, and roadway well 
sprinkled with water; a specially constructed rolling tamper is then used by 
which the lower portion of the loose earth is compacted to depth of about 2'\ 
except in cases where sub-grade is unusually firm. After the lower portion is 
made firm, a heavy asphaltic oil is applied, 1^ gallons per sq. yd., and a 
cultivator passed over the road until the oil and earth are thoroughly mixed. 
The tamper is then used again, and the road is further compacted until only 
ly of loose material remain on top. Oil is again applied, and surface rolled 
with the tamper until firm, and finally it is ironed down with an ordinary 
roller, additional applications of earth being made wherever necessary to 
take up any excess of oil. A road constructed in this manner will require 
from 2^ to 3 gallons of oil per sq. yd. It is hard and dustless and resembles 
asphalt. California oils are the best. Texas and Kentucky oils cost from 
4 to 8 cts. per gallon. The residuums and special preparations vary from 
2 to 12 cts. 

Specifications for Coal Tars. 

I. When used as temporary binder it is usually employed in form of an 
emulsion; specifications difficult. 

II. When used as a semi-permanent binder it is necessary that sufficient 
binding base be present to last through dusty season; and for economy, 
material to be sufficiently fluid to apply cold; hence — 

1. Coal tar to be formed at low temperature, such as produced from 
by-product coke ovens or by gas plants. 

2. A crude tar may be employed because of cheapness. 

3. If not sufficiently fluid to apply cold, enough water-gas tar may 
be added to bring it to proper consistency, but proportion of latter should 
not exceed 60% of mix. 

III. When used as a permanent binder for surface application, either a 
crude or a refined product may be employed, preferably the latter. 

1. If a crude tar, it should have the following properties: 

(a) Same as II, 1. 

(b) Its specific gravity to be between 1.15 and 1.19. 

(c) It should be free from water-gas tar and oil tar. 

2. If a refined tar, the following might be specified: 

(a) .Same as II, 1. 

(b) Its specific gravity to be between 1.17 and 1.20. 

(c) No water nor ammoniacal liquor to be present; boiling poirit 

above 110° C. (230° F.). 

(d) Upon distillation, at least 40% by volume of pitch should re- 

main after all oils have been driven off below 270° C. (518°F.). 

IV. When used as a permanent binder in road construction it is neces- 
sary that more binding base be present than is usually found in the crude 



1136 GO.— HIGHWAYS. 

product. A refined product should therefore be employed; either prepared 
from a crude tar by the contractor or a special preparation purchased. 

1. In the former case, a mixture may be prepared as follows from a 
crude tar, which meets the specifications set forth in III, 1: 

(a) The tar should be heated until boiling point is raised to at 

least 110° C. (230° F.). 

(b) One-tenth part or more of good soft pitch should then be dis- 

solved in the tar while hot; the quantity added should 
be sufficient to produce a pitch residue of at least 50% by 
volume after all oils have been driven off under 270° C. 
(518° F.). 

2. A refined tar for this purpose should meet the requirements as 
suggested for III, 2, except that the pitch remaining after volatile oils 
under 270° C. have been driven off should amoimt to at least 50% by 
volume. 

In some cases, especially where climate is warm throughout the year, a 
tar considerably thicker in pitch may be preferred. 

Specifications for Mineral Oils. 

(a) The oil shall have a specific gravity of not less than 0.95. (b) Its 
flash point shall not be lower than 300° F. (c) It shall be free from water 
as determined by the gasoline test, (d) When heated to 400° F. its loss in 
weight should not be over 35%. The character of the residue should be 
smooth and nearly solid when cold, but not so hard that it may not be 
easily dented with the finger, and when soft it should pull to a long, thin 
thread, (e) The oil shall be soluble in carbon disulphide to the extent of 
98%, and in 88° naphtha to at least 88%. 

EXPERIMENTS WITH DUST PREVENTIVES. 

Note. — Following is from circular No. 89, Office of Public Roads, U. S. 
Dept. of Agric, issued April 20, 1908. During the past few years a number 
of preparations for laying and preventing dust have appeared on the market 
in competition with crude materials, such as coal tar and petroleum, and it 
was therefore decided by the Office of Public Roads to carry on a series of 
experiments during the summer of 1907 with a view to determining, if pos- 
sible, the relative value of these preparations and crude products and their 
adaptability to different conditions. Experiments were conducted at Way- 
land, Mass., Washington, D. C, and Bowling Green, Ky. Following are 
results: 

Experiments at Wayland, Mass. — The water-gas tar was obtained from 
a local gas company at $1.50 per barrel of 50 gallons, delivered. The crude 
coal tar, in 50-gal. bbls. at $2 per bbl.,del. It had been produced at a low 
temperature and contained a good pitch base. The special coal-tar product 
was supplied free in 50-gal. bbls., the Office paying the freight from Boston 
to Wayland. It contained no water, was free from the extremely volatile 
oils present in the crude tar, and held a good pitch base. The other proper- 
ties are shown in comparison with the water-gas and coal tars in preced- 
ing Table. Labor cost per 8-hour day was as follows: Common labor, $1.50 to 
$1.85; single teams, $3; double teams, $5; foreman, $3; steam roller, $12. 
The cost of repairs per sq. yd. of road surface was from 2.6 to 3.3 cts. When 
gravel was used it was obtained from pits near the road, costing the commis- 
sion $1.08 percu. yd. Clean trap screenings, 1|'", or pea stone, however, was 
used whenever it could be obtained and was furnished at $1 .10 per ton by a rock- 
crushing plant located about 4 miles from nearest section of work. Thir- 
teen experiments were made as summarized in the two following Tables. 



DUST PREVENTIVES— TAR EXPERIMENTS. 



1137 



Cost Data of Tar Experiments. 



o' 


Material Supplied. 


Cost of 

repairs 

per square 

yard. 


Cost of 
application 

per 
square yard. 


Cost of removing waste 

material per square 

yard. 


0^ 

u 

8 


+3 

g 


P! 

1 


1 




1 


•a 


1 

CO 


o 

1 


1 


Water-cas tar 


$0,022 
.023 


$0,010 
.010 


$0,049 
.049 
.006 
.013 
.061 
.056 
.044 
.058 
.030 
.037 
.046 
.148 
.058 


$0,027 
.011 
.009 
.008 
.024 
.028 
.017 
.033 
.015 
.017 
.026 


$0,008 
.008 


$0,013 
.013 


$0,129 
.114 
.015 
.033 
.129 
.129 
.088 
.137 
.062 
.072 
.102 
.316 
.127 


$220 74 


2 




19 03 




II 


7 30 


/^ 


It 






"!668 
.009 
.014 
.010 
.007 
.008 
.017 
.033 
.019 


.012 
.010 
.010 
.013 
.010 
.010 
.010 
.013 
.012 
.010 


6 60 


5 


Coal tar 


.018 
.018 


.008 
.008 


32.46 


6 




129.14 


7 


•1 


52 82 


8 
g 


Water-gas and 


coal- 
. tars 


.018 


.008 


57.49 
58 26 


10 






119.89 


11 


Special tar mixti 
Special tar prepa 


j.re. . . 






27.27 


12 
13 


ration 


.023 


.010 


31.64 
103.45 




Total 










72.28 


36.00 


382.07 


203.38 


82.26 


90.10 




866.09 









Miscellaneous Data of Tar Experiments. 





. 




A a'd 


i w 


li.^r^ 




o 












to 


c 




Surface 


'^it 




•^ 0) S 


S^ 


I 


Material Applied. 


Applica- 
tion. 


m 




> (U 

8i 


X 






;i ft 








W 






^ 


& 


< 


1 


Water-gas tar 


Gravel 


0.90 


0.0079 


0.009 


1,700 


2 


11 


None 

Gravel*. \ \ '. 


.38 
.30 
.25 
.60 
.70 
.42 
.90 


.0080 


.009 


167 


3 


«« 


500 


4 


<( 


".ooso" 

.0080 
.0133 
.0096 


.009 
.009 
.009 
.009 
.009 


200 


5 


Coal tar 


250 


6 


11 


1,000 


7 


<( 


600 


8 


Water-gas an^ coal tars. . . . 


417 


9 




Pea stone. 
Gravel. . . . 
Pea stone . 


.43 
.48 
.42 


.0086 
.0072 
.0202 


.009 
.009 
.009 


933 


10 


<i 


1,667 


11 


Special tar mixture.. ..'.... 


267 


12 


Special tar preparation 


1' 


1.50 


.0400 


.009 


100 


13 


*« 


" 


.67 


.0226 


.009 


817 




Total . . 














11,118 

















1138 QO.—HIGHWA YS, 

Experiments at Washington, D. C. — ^With calcium chloride, as a dust 
preventive; tested on portion of macadam driveway in Agricultural Dept. 
grounds, Washington, D. C. The roadway is built of trap rock, held in posi- 
tion by a soft limestone binder; the screenings of the binder pulverized 
rapidly under traffic, forming a light dust continually raised by passing 
vehicles, and carried away by the wind ; the road was thus becoming stripped 
of its binding material. In preparing for treatment, all dust and dirt were 
scraped from surface of roadway ; a solution was prepared by mixing 300 lbs. 
of commercial calcium chloride (granular, containing 75% calcium chloride 
and 25% moisture) with 300 gallons of water in an ordinary street sprinkler, 
agitating the liquid thoroughly before applying. It was then applied from 
one sprinkling head, and the sprinkler passed slowly back and forth over the 
road to facilitate the complete absorption of the solution; each application 
consisting of 600 gallons over an area of 1582 sq. yds., or 0.38 gallon per 
sq. yd. The first application was made July 13, 1907, followed by a similar 
one July 15, to increase the efficacy of the treatment. The effect was marked. 
The texture of the road surface was completely changed: before treatment, 
raveling was excessive in spots and the whole surface seemed loosely knit 
together; after treatment of July 15, this condition changed and the road 
surface became smooth, compact and resilient. The third treatment, 
August 3rd, was given because certain points exposed to the most severe 
wear were showing signs of raveling; the results of this treatment, and of 
successive ones, were most satisfactory and not unlike those attending the 
first two treatments. The calcium chloride is charged at rate of $16 per ton, 
f. o. b. cars at Baltimore; freight 13 cts. per 100 lbs. Specific gravity of 
solutions, 1.053 to 1.060. Following is Table of cost of applying: 

Cost of Applying Calcium Chloride. 



Item. 



Cost. 



600 pounds calcium chloride, at $18.60 per ton. . 

3 men for 1^ hours, at 15 cents per hour 

1-horse sprinkling wagon for 1^ hours, at 35 cents per hour. 

Total cost of 1,582 square yards 

Cost per square yard at this rate 

Total cost of five applications 

Cost per square yard of five applications 



$5.58 
.675 
.525 
6.78 
.0043 

33.90 
.0235 



Experiments at Bowling Green, Ky. — Materials used were Kentucky 
rock asphalt tested for its fitness as a binder in macadam construction, crude 
Kentucky oil, and special preparation of residuum oils, the last two of which 
were used as dust preventives. 

(a) Rock Asphalt Experiment. — The rock asphalt used is a natural product 
formed in the Chester group of subcarboniferous rocks over a course extending 
through Breckenridge, Grayson, Edmonson, Logan, and Warren counties, 
marking the edge of the coal fields lying in the western part of Kentucky. 
It is a fine-grained sandstone impregnated with mineral pitch or bitumen, 
the latter averaging from 6 to 8%, with a maximum of 12%. After quarry- 
ing and crushing, 2" size, it is further passed between steel rollers, the finished 
product being a mass of individual grains of sand, each thoroughly coated 
with a film of mineral pitch, adhering to surrounding grains and packing 
very firmly if subjected to pressure; with a rich dark brown color with a 
slight luster which gradually disappears as the bitumen hardens and dries. 
If chilled when compacted, a lump becomes very hard and tough; if warmed 
in the hand, the bitumen becomes soft and semi-fluid and the individual 
grains of sand fall from the mass of their own weight. The test was made 
on Cemetery pike, a main thoroughfare. The form of construction originally 
adopted was a 20-ft, Telford road. When the surface had been worn away 
exposing the foundation, it was repaired and brought to grade with a sharp 
gravel containing about 20% sand and clay, the layer being about 8" thick 
when compacted. Previous to the experiment, it was loosened to a depth 
of 4" by means of a spiked roller and a heavy harrow, and shoveled out by 
hand. The sub-grade was then made to conform to crown of roadway, 
planned to be ^Y in 9 ft., or an average of Y per ft. After rolling the sub- 
grade the wearing course of stone was laid ; it consisted of crushed limestone. 



EXPER.— CALCIUM CHL., ROCK ASPH., OIL. 



1139 



1" toli", spread in a 4'' layer, rolled once to turn down the sharp edges of 
the stone and form a smooth, even surface. The rock asphalt was then 
thrown on with shovels and spread to a depth of 1^\ care being taken to 
break all lumps and to work all the asphalt rock possible into the interstices 
of the stone without disturbing the latter. As the work progressed, the 
roller was kept moving back and forth parallel to axis of roadway, working 
from outer edge to crown as in ordinary macadam construction. The in- 
advisability of working the material when chilled and damp was apparent, 
for the portion of the road laid at a temperature of 65° F. failed to become 
hard and firm for several hours after subsequent applications had compacted 
satisfactorily. The cost of the work is shown in the following Table. Labor 
ranged from $1.20 to $1.25, and teams cost $3 per day of 10 hours; roller 
was loaned, cost of operation being $2,50 per day for engineer plus cost of 
fuel; stone was delivered on roadway at $1.20 per cu. yd. and was spread 
4'' thick uncompacted, making cost per sq. yd. delivered 13 cts; cost of 
asphalt charged at market price of $5 per ton f. o.b. cars at Bowling Green; 
it was spread about 1^" thick, or at rate of 24.5 sq. yds. per ton. 

Cost Data of Rock Asphalt Experiment. 



Item. 


Cost 

per square 

yard. 


Total cost. 


Percentage 
of total. 


Shaping sub-grade 


Cents. 

5.66 

13.71 

.78 

.09 

23.77 

1.44 

2.18 


Dollars. 

43.60 

105.60 

5.97 

.67 

183.10 

11.061 

16.78i 


Per cent. 
11.8 


Stone on work 


28.8 


Spreading stone 


1.7 


Rolling stone 


.3 


Asphalt on work 


50.0 


Spreading asphalt 


3.1 


Rolling asphalt 


4.3 






Total 


47.63 


366.79 


100.0 







(b) Oils. — In connection with work on rock asphalt, experiments were 
made to determine the comparative value of a residuum oil preparation and 
crude oil, as dust preventives. The oil preparation is a patented mixture of 
residuum oils, combined with a view to obtaining such proportions of as- 
phaltic and lighter oils as shall be best fitted for immediate dust laying and 
permanent improvement of the roadway. The tests were made on Cemetary 
pike beyond the point where the rock asphalt work ended. The general 
condition of the gravel roadway was unsatisfactory for the purposes of such 
a test; the cross section of the roadway was quite flat, the crown having 
been completely worn down, so that lateral drainage was defective; also 
there were extensive pockets of loose material, characteristic of roads made 
of gravel containing a large percentage of sand and clay. Heavy rains had 
fallen for several days preceding the application of the oil, and although the 
road surface was quite dry there was a large amount of moisture in the road- 
bed, thus prevening the rapid absorption of the oil by the gravel before the 
evaporation of the lighter oils took place. The oils were applied over a 
width of 12 ft. of the roadway by means of an oil sprinkler adapted to the 
uniform spreading of heavy liquids. A tank load of the oil preparation was 
spread over an area of 841 sq. yds., or an average of 0.903 gallon per sq. yd.; 
the temperature of the oil 87° F. due to its exposure to the sun; it was heavy 
and was absorbed very slowly by the gravel, about 4 days being required to 
cause it to become thoroughly incorporated into the surface of the road; 
the gravel becoming very compact and showing few traces of wheel marks 
except at points where repairs had been made, in which cases the cementing 
process took place more slowly. In the case of the crude oil, 5 tank wagons, 
or a total of 3712 gallons, were applied to 4416 yards of surface, making an 
average of 0.84 gallon per sq. yd. An experiment was also made with crude 
oil on macadam road ; there was about Y' oi dust on roadway consisting 
largely of powdered limestone; the roadway was given an application of 
755 gallons over an area of 1452 sq. yds., or at rate of 0.52 gallon per sq. yd. 
The following Table gives a statement of the cost of repairs, materials, and 
application of the oils. 



1140 



GO.—HIGHWAYS. 



Cost Data of Oil Experiments. 



Experiment. 


Item. 


Cost per 
square 
yard. 


Total 
cost. 




Repairs and ditching. . . 
Oil 


Cents. 

0.57 

.13 

.47 


Dollars. 

4.79 

110.00 


Special oil preparation 


Application 


3.93 




Total cost 






1.17 


118.72 




Repairs and ditching. . . 
Oil 






.33 

4.46 

.16 


14.38 
196.94 


Crude oil on gravel road 


Application 


7.28 




Total cost 






4.95 


218.60 




roii 






2.75 
.11 


40.00 


Crude oil on macadam road . . . 


Application 


1.64 




Total cost 






2.86 


41.64 









ASPHALT AND BITUMINOUS ROCK DEPOSITS. 



1141 



EXCERPTS AND REFERENCES. 

The Asphalt and Bituminous Rock Deposits of the U. S. (By G. H, 

Eldridge; U. S. Geol. Surv.; Eng. News, June 5, 1902).— 

Table I. — Classification of Natural Hydrocarbons. 



Gaseous 



Bituminous. 



i 



Resinous 
Cereous . 



r Marsh gas. 

\ Natural gas. 

Fluid f Naphtha. 

\ Petroleum. 

( Maltha. 
Viscous (malthite) . f Mineral tar. 

I Brea. 

[Chapapote. 
T71 +• /Elaterite (mineral caoutchouc), 

^l^stic (Wurtzilite. 

^Albert it e. 

Impsonite. 

Grahamite. 

Nigrite. 
^Uintaite (gilsonite). 

Lignite. 

B it uminous coal . 

Semi-bituminous coal. 

Anthracite coal. 
* Succinite (amber). 

Copalite. 
^Amberite, etc. 
r Ozocerite. 



Solid 



'Asphaltic , 



Coal. 



\ Hatcjietite, etc. 

^ , ,,. jFichtelite. 

Crystallme iHartite. etc. 

Table II. — Grouping of Natural and Artificial Bituminous 

Compounds. 

Mixed with limestone fSeyssel, Val de Travers, Lobsan, Illinois, 
(asphaltic limestone). .\ Utah, etc. 

''iSdTisScr.'' . .jCalifomia. Kentucky. Utah. etc. 



Natural. 



''ief(alph\lt1c^^^ C-b-' California. Utah. , 

Bituminous schists {^^Jtc!^^' ^^^'^°'^^^' ^^^^^^ky. Virginia. 



Fluid. 



fThick oils from distilled petroleum, 
\ "residuum." 



Artificial. 



Viscous . 



Gas tar. 

Pitch. 

Refined Trinidad asphaltic earth, 
q 1.-I I Mastic of asphaltite. 

^°"^ ^ Gritted asphaltic mastic. 

Paving compounds. 



The Adjustment of Macadam Road Design to Various Subgrade Soils 

(Report of Mass. Highway Commission; Eng. News, Sept. 4, 1902). — Deals 
with Sand and Gravel, Clay, and Sandy Loam. 

Paving a Country Road With Brick (By Sam. Houston. Paper, 
Ohio Soc. C. E. & Surv., Jan. 20, 1902; Eng. News, Sept. 25, 1902).— Illus- 
trated details in cross-section. Specifications and discussion under the 
following headings: Underdrains; Foundation of Broken Vitrified Pipe; 
Vitrified Clay Curbing; Crown of Pavement and Curve of Summer Road; 
Brick Pavement. Cost data is given showing that 5479 ft. of road cost 
$8,367.72; the total width of road between ditches is 26 ft., with 10. ft. 
width of pavement proper; etc. 



1142 QO.— HIGHWAYS. 

An Experimental Steel Trackway in N. Y. City (Eng. News, Dec. 4, 
1902). — Illustrated cross-section of trackway, and details of rail section', 
used in Murray St. Rails are special channel section, 40 ft. long, and 
weigh 25 lbs. per lin. ft. Approximate cost of trackway, laid, complete. 
$7,500 per mile. 

The Design of a Bituminous Macadam Road for Salem Co., N. J. 

(Eng. News, Sept. 24, 1903). — Illustrated cross-section of 30-ft. road. 

Data on Roads and Pavements in Iowa (By A. Marston. Report, 
Iowa Eng. Soc; Eng. News, Feb. 9, 1905). — Table of traction tests on 
brick and asphalt pavements in various stages of condition. Traction 
resistance: Brick pavement, 25.4 to 58 lbs. per ton. Asphalt pavement, 
23.3 to 67.8 lbs. per ton. 

Experience With Various Pavements on Streets With Heavy Grades 
(By C. G. Anderson. Paper, 111. Soc. of Engrs. and Surv., Jan., 1907; 
Eng. News, Mar. 14, 1907). — Illustrated cross-sections showing different 
styles of paving employed. 

Bituminized Dirt Roads at Santa Monica, Cal. (Eng. News, May 16, 
1907). — Illustration of latest type of road-tamping roller. 

Street Railway Track and Paving at Fort Wayne, Ind. (Eng. News, 
June 20, 1907). — Illustrations of track and paved streets, nose block for 
street railway track, and reinforced concrete poles. 

Notes on Tar Macadam (By C. F. Wike, England; Eng. News, 
Aug. 8, 1907). — Initial cost of tar macadam roads in England is about 55 
to 60 cents per sq. yd., exclusive of foundation; and the annual charge 
(including initial cost) for a period of 14 years has averaged 8 cents per 
sq. yd. — down to 5 cents. 

The Use of T=Rails for Street Railway Tracks in Cities (By C. G. 
Reel, Paper, Am. St. & Int. Ry.f Assn., Oct., 1907; Eng. News, Oct. 31, 
1907). — Illustrations: Standard T-rail construction in Milwaukee; latest 
construction in Kingston, N. Y., showing special brick outside of rail. 

Cost of Brick Pavements and Cement Mortar Curbs at Centerville, 
Iowa (By M. G. Hall. Eng. News, April 2, 1908).— Tables of costs. 

Cost of Oiling Roads in N. Y. State (Eng. News, May 14, 1908).— 
Table showing costs of experimental road oiling, for macadam road, sand 
road, and gravel road. 

Concrete Paving for Streets (Eng. News, Aug. 20, 1908). — Costs, 
and illustrated sections of Streets. Specifications. 

Specifications and Notes on Macadam Road Construction (By A. N. 
Johnson. Paper, West. Soc. of Engrs", Oct., 1908; Eng. News, Nov. 5, 
1908). 

The Use of Asphaltic Flux for Coating Macadam Roads at Palo 
Alto, Cal. (By J. F. Byxbee. Eng. News, May. 13, 1909).— Specification 
and description of method. Total cost for material and labor, 5^ cents per 
sq. yd. 

An "Accelerated Test" of Road Wear by Automobile Traffic in Germany 
(Eng. News, July 8, 1909).— Illustrated. 

Method of Keeping Data Relating to Street Lines and Grades, in Brook- 
line, Mass. (By H. A. Varney. Eng. News, July 8. 1909).— Illustrated 
Sample pages of data sheets. 

Sampittic Surfacing (By W. W. Crosby. Trans. A. S. C. E., Vol. LXIV.. 
Sept. 1909). — Specifications. 

Inverted Macadam Road Construction (Eng. Rec, Jan. 8, 1910). — 
Inverted macadam road construction has been adopted in a number of 
cases by Mr. A. N. Johnson, highway engineer of Illinois, as a means of 
meeting conditions which obtain in many parts of the State. The specifi- 
cations proposed to cover this method of construction and the reasons for 
placing the fine material on the bottom and the large pieces of stone at the 
top were described in a paper Mr. Johnson presented before the Western 
Soc. of Engrs. This paper was printed in the Eng. Rec. of Nov. 7, 1908. 

Vitrified Clay Curbing for Streets and Roads (Eng. News, Jan. 13, 1910).— 
Illustrated. The blocks are hollow and form continuous drains, so that the 
usual line of broken stone for drainage of the roadbed is not required where 
this form of curb is used. Has been used for eight years on the country 



CHICAGO-CEMENT WALKS, STREET CROWNS. 1143 

road between Toledo and Calumet, O., and is said to be in excellent con- 
dition, showing practically no wear. 

Sidewalk Practice in Chicago (By N. E. Murray. Paper 111, Soc. Engrs. 
and Survrs., Jan. 26. 28, 1910; Eng. News, Feb. 17, 1910). — The average 
cost of cement walks laid in Chicago from 1901 to 1908, inclusive, based on 
the total cost ($8,918,278) divided by the total mileage (1862) was $4,787 per 
mile, or 15.11 cents per sq. ft. This price was for walks complete, including 
filling, which in many instances was from 2 ft. to 6 ft. in depth and is not a 
fair average cost for the ordinary cement walk. Average Chicago prices 
for labor and material, give following average cost of material delivered on 
the work: Cinders, 50 cts. per cu. yd.; cement, $1.20 per bbl, (3.8 cu. ft.); 
sand, $1.75 per cu. yd.; gravel, $1.50 per cu. yd. An ordinary concrete 
sidewalk gang in Chicago is usually composed of six men paid as follows (for 
8 hours): 1 finisher at 65 cts., $5.20; 1 helper at 47i cts., $3.80; 4 laborers at 
37^ cts., $12; total, $21. Assuming that this gang of six men can construct 
600 square feet of walk per day (a fair assumption, borne out by experience), 
we have, as total cost, 13.51 cents per sq. ft.; thus: — 

Cinders (20% shrinkage), 20.83 cu. yds. at 50 cts $10.42 

Base. 4iins. (1 : 2^ : 5): 

Cement, 9.77 bbls. at $1.20 $11.72 

Sand, 3.47 cu. yds. at $1.75 6.07 

Gravel, 6.85 cu. yds. at $1.50 10.28 28.07 

Wearing coat, f-in. (2:3): 

Cement, 5.56 bbls. at $1.20 6 . 67 

Sand, 1.17 cu. yds. at $1.75 2 .04 8.71 

Water, 1 mill per sq. f t .60 

Labor, 1 gang one day 21 . 00 

Use of tools, waste of material, etc., at 2% 1.37 

Supt. and office expenses, at 5% , 3.51 

Profit at 1 % 7.36 

Total 600 sq. ft. at 13.51 cts. per sq. ft $81 .04 

Paving Practice in Chicago (By P. E. Green. Trans. A. S. C. E., Vol. 
LXVI., Mar 1910). — Crown of roadway: Chicago ordinance calls for arc of 
circle, but the parabola is mostly used in construction; the formula is y — 
hx^-^a^, in which 6 = depth of gutter below grade of center or roadway, 
a = half roadway, :t: = hor. dist. from center of roadway, and 3; = vert. dist. 
below the grade. Above formula applies to roadways up to 100 ft. and to 
pavements having a rigid wearing surface; for wider roadways, and with 
macadam surfaces, the curve should approach a straight line from center of 
roadway to gutters. The Rosewater (Omaha) formula for height of crown 
is if = H^(100-4P)-^5000, in which ii' = height of crown, W = width of 
roadway, and P = percentage of grade ; this formula is best adapted to park 
roads or boulevards, or for streets having stiff grades where little crown is 
necessary. For residence streets a good formula is 7? = 0.021^. Illus- 
trations of girder rail and pavement. 

Two Years' Experience in Dust Suppression on New Jersey Roads (By 

James Owen. Paper before State Sanitary Assn., of N. J.; Eng. Rec, Dec. 
10, 1910). — Mr. Owen records some of his failures in road construction, and 
draws certain conclusions as the results of his experience: *Tn construction, 
use no penetration of heavy oil, but construct the road as of ordinary mac- 
adam and provide after complete consolidation of surface a coating of 45% 
oil, using about one-half gallon per sq. yd., and in the maintenance contract 
for the year provide for two applications in that period. With this practice 
entire satisfaction has been realized." It is apparent that an ordinary 
macadam surface, even when oiled, is not satisfactory, but satisfaction has 
been obtained with various patent pavements as Amiesite, Filbertine, 
Warrenite and what is known as road asphalt. These all use the broken 
stone with a plastic mixture injected, in most cases asphalt. Cost. — Before 
the automobile era, the cost of maintaining a good surface upon the Essex 
County roads was 3 cts. per sq. yd. per annum. This was increased in later 
years to about 5 cts., and including the oiling, amounts to 6 cts. The 
patent pavements alluded to cost from 80 cts. to $1.20 per sq. yd. It is 
obvious that those pavem.ents must last 15 to 20 years to be on the same 
monetary basis as ordinary repairs. 



1144 



00.— HIGHWAYS, 



Tests of Various Road Surfacing Materials by the Ohio State Highway 
Department (Bulletin No. 12, O. S. H. D.; Eng. News, Nov. 10, 1910). — 
The following table shows the wear on Nelson Avenue experimental road, 
one year after its construction: 



Section. 



1. Glutin 

2. Standard Asphalt . . 

3. Pioneer Asphalt 

4. Tarvia "X" 

5. Tarvia "B" 

6. Indian Asphalt 

7. Ugite 

8. Fairfield Asphalt . . 

9. Asphaltoilene 

10. Rock Asphalt 

11. Carbo-Via 

12. Concrete Macadam. 

13. Taroid 

14. Petrolithic 

15. Limestone Concrete 

16. Gravel Concrete 

17. Water-bound Mac- 

adam 



8 ft. 


4 ft. 


Center 


4 ft. 


East. 


East. 


line. 


West. 


.07 ft. 


.07 ft. 


.00 ft. 


.00 ft. 


.02 ft. 


.07 ft. 


.06 ft. 


.08 ft. 


.07 ft. 


.07 ft. 


.05 ft. 


.03 ft. 


.02 ft. 


.04 ft. 


.04 ft. 


.05 ft. 


.08 ft. 


.04 ft. 


.03 ft. 


.00 ft. 


.01 ft. 


.03 ft. 


.01 ft. 


.03 ft. 


.04 It. 


.05 ft. 


.05 ft. 


.03 ft. 


.04 ft. 


.04 ft. 


.04 ft. 


.05 ft. 


.05 ft. 


.04 ft. 


.04 ft. 


.05 ft. 


.00 ft. 


.00 ft. 


.02 ft. 


.03 ft. 


.03 ft. 


.09 ft. 


.07 ft. 


.08 ft. 


.04 ft. 


.05 ft. 


.05 ft. 


.01 ft. 


.02 ft. 


.01 ft. 


.03 ft. 


.00 ft. 


.03 ft. 


.01 ft. 


.06 ft. 


.00 ft. 


.04 ft. 


.06 ft. 


.02 ft. 


.00 ft. 


.00 ft. 


.00 ft. 


.00 ft. 


.00 ft. 


.01 ft. 


.03 ft. 


.03 ft. 


.01 ft. 



8 ft. 
West. 



00 ft. 

09 ft. 

03 ft. 

03 ft. 

00 ft. 

02 ft. 
05 ft. 
00 ft. 

03 ft. 

04 ft. 
,07 ft. 

00 ft. 

00 ft. 

00 ft. 

00 ft. 

01 ft. 

01 ft. 



Illustrations and Specifications. 

Description. 
Specifications for b^uminous concrete paving 
Specifications for sheet asphalt pavement 
Specifications for concrete sidewalk, curbs, street pavement 
Suggestions for street pavement crowns, with formulas 
General plans for location of street conduits and pipes, Seattle 
Formulas for street crowns — curbs at different elevations 

Combined concrete and gutter in Salt Lake City 

Typical sections of covered conduits (10' x 20') at Jones Falls 



Eng. News. 
Mar. 17, '10. 
Mar. 17, '10. 
Mar. 17, '10. 
May 5, '10. 
May 12, '10. 
June 30, '10. 
Eng. Rec. 
Oct. 15, '10. 
Dec. 3 '10. 



61.— HYDROSTATICS. 

Hydrostatics, in its broadest sense, treats of the conditions of equilibrium 
and pressure of -fluids* at rest. As the term "fluid" comprehends both 
liquids and gases our discussion will be confined to the former, and especially 
to water. To the engineer, Water may be considered as practically friction- 
less and incompressible . In other words, we assume it to be a perfect fluid 
possessing no statical friction; and not subject to increased density under 
pressure, to any perceptible degree. f From this it follows (1) that the 
intensity of pressure (in lbs. per sq. in., or lbs. per sq. ft.) at any given point 
in the liquid is equal in all directions; (2) that, neglecting the weight of the 
atmosphere above, the intensity of pressure is directly proportional to 
the depth below the surface; (3) that the intensity of pressure is directly 
proportional to the density ( = mass of a unit of volume) or to the weight, 
of a unit volume of the liquid; (4) that the pressure is always normal to 
any plane, pressed surface; (5) that the pressure on a curved surface may 
be resolved into one or more resultant pressures each acting normal to a 
tangent plane projected on the curved portion. 

The reader is referred to the subject of Dams, pages 845, etc., for many of 
the elementary principles of hydrostatics which bear particularly on that 
subject, and they will not be repeated here. 

Atmospheric Pressure may be neglected in most hydrostatic calculations 
because its effects are usually balanced. For instance, the effect of the atmos- 
pheric pressure on the up-stream water surface at a dam exerts so much 
additional force tending to overturn the structure, but it must be remem- 
bered that an equal and opposite pressure is exerted against the down- 
stream face of the dam, tending to preserve equilibrium, and hence the 
effect is neutralizedj. Thfe height of the atmosphere above sea level ia 
variously estimated at from 40 to 200 miles. Whatever the height may be 
it is certain that, beginning with the maximum density at sea level, it be- 
comes exceedingly rarefied above an elevation of 30 to 40 miles. A column 
of air at sea level will balance a column of water 34 ft. in height or a column 
of mercury 2^ ft. (30 ins.) in height, each column exerting a pressure of 
14.7 lbs. per sq. in. This is based on a cubic ft. of water weighing about 
62.5 lbs., and the specific gravity of mercury at 13.6. 

Air, dry, at atmospheric pressure, and at a temperature of 56° F., 
weighs exactly ^^ of a pound per cubic foot. (See Table of weight of air, 
page 463.) 

Water is 773 times as heavy as the denser air at 0° C. A column of water 
1 sq. in. in section and 1 ft. high weighs about 0.434 lb.; or, in other words, 
1-ft. "head" corresponds to a "pressure" of about 0.434 lb. per sq. in. 
Hence a pressure of 1 lb. per sq. in. corresponds to a head of about 2,304 ft. 
These values should be committed to memory.] | Table No, 1, following, 
when used decimally will give corresponding pressures in lbs. per sq. in. for 
any given heads in ft,, and vice versa, 

* ''Incompressible'* fluids, only, are included in the modem acceptation 
of the term hydrostatics. 

t Under one atmosphere (14.7 lbs. per sq, in,) fresh water is compressed 
to 0.99995 its original volume, amounting to an increased density of 0.0032 
lb. per cu. ft.; salt water to 0.999956 its original volume. Sea water one 
mile in depth below the surface, equal to 156 additional atmospheres or 
2292 lbs, per sq. in,, is compressed only to 0.99995566 its original volume; 
hence the additional 156 atmospheres increases its density only about f of 1% 
more than does one atmosphere. 

J This is only partly true, as a partial vacuum is frequently formed at the 
down-stream face when the water is flowing over the crest of the dam. 

II The refinement sometimes employed by using about 62.424 to 62.428 
lbs. as the weight of a cu. ft, of water at its maximum density, is unnecessary 
for ordinary engineering problems. The value 62.5, used above, involves 
an error of only 0.5 lb.* pressure per sq, in. for a 1000-ft, head, equivalent to 
about Vio of 1 per cent on the side of safety. But see Tables 3, 4 and 5, 
following. 

1145 



1146 



QU— HYDROSTATICS. 



Hydrostatic Pressure. — ^There are two Pressure Units in general use in 
the United States, as follows: 

(a) "Lbs. per sq. in." corresponding to head in ft., as given in Tables 1, 3 

and 4, is used in the design of water pipes, tanks, sewers, etc. 

(b) "Lbs. per sq. ft." corresponding to head in ft., is used in the design of 

dams. See Tables 2 and 5. 
Let P = total pressure in lbs. on any submerged plane surface; 
^' = pressure in lbs. per sq. ft.; 
J?* = pressure in lbs. per sq. in.; 
h = height of column of water or "head," in ft.; 
a' = area of submerged surface acted upon, in ft.; 
a" = area of submerged surface acted upon, in ins.; 

«/' = wt. of a column of water 1 ft high and 1 ft. sq , in lbs. = 62.5 lbs.; 
M;" = wt. of a column of water 1 ft. high and 1 in. sq., in lbs. = .434 lb. 
Then, neglecting atmospheric pressure (14.7 lbs. per sq in ), we have, 

P = w' a' /j= 62.5 a' h (1) 

=w'' a" /t = .434 oTh (2) 

p'^w' h=Q2.5h (3) 

p''=^<u;''h = A3^h (4) 

h^^^ mep' (5) 

= 1^^ = 2. 304 ^^ (6) 

1. — Head and Pressure Equivalents. — Lb. per Sq. In. 
(Water assumed at 62.5 lbs. per cu. ft.) 



Head. 


Pressure. 


Pressure. 


Head. 


Feet. 


Lbs. per Sq. In. 


Lbs. per Sq. In. 


Feet. 


1 


0.434 


1 


2.304 


2 


0.868 


2 


4.608 


3 


1.302 


3 . 


6 912 


4 


1.736 


4 


9.216 


5 


2.170 


5 


11.520 


6 


2.604 


6 


13.824 


7 


3.038 


7 


16.128 


8 


3.472 


8 


18.432 


9 


3.906 


9 


20.736 


10 


4.340 


10 


23.040 



Example. — What pressure in lbs. Solution - 

per sq. in. corresponds to a head of (From above 
104.8 ft.? table.) 



100=43.4 
4= 1.736 
.8= .347 



Ans. 45.483 lbs. 

2. — Head and Pressure Equivalents. — Lbs. per Sq. Ft. 
(Water assumed at 62.5 lbs. per cu. ft.) 





Head. 


Pressure. 


Pressure. 


Head. 




Feet. 


Lbs. per Sq. Ft. 


Lbs. per Sq. Ft. 


Feet. 




1 


62.5 


1 


.016 




2 


125. 


2 


.032 




3 


187.5 


3 


048 




4 


250. 


4 


.064 




5 


312.5 


5 


.080 




6 


375. 


6 


.096 




7 


437.5 


7 


.112 




8 


500. 


8 


.128 




9 


562.5 


9 


.144 




10 


625. 


10 


.160 



Example — Pressure correspond- 
ing to 82.4 ft. head = 5000+ 125+ 25 
= 5150 lbs. per sq. ft. Or, =62.5X 
82.4 ( = ego X 82.4), by formula. 



Example. — Head corresponding 
to pressure of 1250 lbs. per sq. ft.= 
16+3.2+.8 = 20ft. 

Or, = .016 X 1250, by formula. 



PRESSURES REDUCED TO HEADS. H, TO P. 



1147 



3. — Head op Water, in Feet 

Corresponding to 

Given Pressures in Lbs. Per Sq In. 

Note. — Weight of water assumed at 62.424 lbs. per cu. ft. 

Decimal point may be moved simultaneously to right or left for 

Pressure and Head. 

[Head of Water, in Feet.] 



P.P. 

2.3068 



.231 

.461 
.692 
.923 
1.153 
1.384 
1.615 
1.845 
2.076 



Pres- 
sure. 
Lbs.pr 
Sq.In. 



Units. 



2. 



4. 



7. 



8. 



1-. 

2-. 
3-. 

4-. 
5-. 
6-. 

7-. 
8-. 

§ 10-. 

1 11-. 

•5 12-. 
« 13-. 
•g 14-. 

S}^ 

o 16-. 

!t 17-. 

tl 18-. 

o 19-. 

5 20-. 

W) 21-. 

« 22- 

•■S 23-. 

2 24-. 
Z 25-. 

5 26-. 
te 27-. 
^ 28-. 
^ 29-. 
= 30-. 
g 31-. 

3 32-. 
" 33-. 
^ 34-. 
o 35-. 
"^ 36-. 
j£ 37-. 
w 38-. 
^ 39-. 

>'fr- 

.Q 41-. 
-d 42-. 
.2 43-. 

6 44-. 
g 45-. 
« 46-. 
S 47-. 
S 48-. 
a 49-. 
w 50-. 

51-. 
52-. 
53-. 
54-. 
55-. 
56-, 
57-. 
58-. 
59-. 
60- 



23.068 

46.136 

69.204 

92.272 

115.340 

138.408 

161.476 

184.544 

207.612 

230.680 

253.748 

276.816 

299.884 

322.952 

346.020 

369.088 

392.156 

415.224 

438.292 

461.360 

484.428 

507.496 

530.564 

553.632 

576.700 

599.768 

622.838 

645.904 

668.972 

692.040 

715.108 

738.176 

761.244 

784.312 

807.380 

830.448 

853.516 

876.584 

899.652 

922.720 

945.788 

968.856 

991.924 

1014.992 

1038.060 

1061.128 

1084.196 

1107.264 

1130.332 

1153.400 

1176.468 

1199.536 

1222.604 

1245.672 

1268.740 

1291.808 

1314.876 

1337.944 

1361.012 

1384.080 



. 2. 

25. 

48. 
71, 
94. 
117, 
140, 
163, 
186, 
209, 
232, 
256. 
279, 
302, 
325. 
348, 
371. 
394. 
417, 
440, 
463 
486, 
509, 
532 
555. 
579. 
602. 
625, 
648, 
671, 
694, 
717, 
740. 
763. 
786. 
809. 
832. 
855. 
878. 
901. 
925. 
948. 
971. 
994. 

1017. 

1040. 

1063. 

1086. 

1109. 

1132. 

1155. 

1178. 

1201. 

1224. 

1247. 

1271. 

1294. 

1317. 

1340. 

1363. 

1386 



307 

375 

443 

511 

579 

647 

715 

783 

851 

91 

987 

055 

123 

191 

259 

327 

395 

464 

531 

599 

667 

735 

803 

871 

939 

007 

075 

143 

211 

279 

347 

415 

483 

551 

619 

687 

755 

823 

891 

959 

027 

095 

163 

231 

299 

367 

435 

503 

571 

639 

707 

775 

843 

911 

979 

047 

115 

183 

251 

319 

387 



4. 
27. 
50. 
73. 
96. 

119. 

143. 

166. 

189. 

212. 

235. 

258. 

281. 

304. 

327. 

350. 

373. 

396. 

419, 

442. 

465. 

489. 

512. 

535. 

558. 

581. 

604. 

627. 

650. 

673. 

696. 

719. 

742. 

765. 

788. 

811. 

835. 

858. 

881. 

904. 

927. 

950. 

973. 

996. 
1019. 
1042. 
1065. 
1088. 
1111. 
1134. 
1158. 
1181. 
1204. 
1227. 
1250. 
1273. 
1296. 
1319. 
1342. 
1365. 
1388 



614 
682 
750 
818 
886 
954 
022 
090 
158 
226 
294 
362 
430 
498 
566 
634 
702 
770 
838 
906 
974 
042 
110 
.178 
246 
314 
382 
450 
518 
586 
654 
722 
790 
858 
926 
994 
062 
130 
198 
266 
334 
402 
470 
538 
606 
674 
742 
810 
878 
946 
014 
082 
150 
218 
286 
354 
422 
490 
558 
626 
694 



29. 
53. 
76. 
99 
122. 
145. 
168, 
191. 
214, 
237 
260 
283 
306 
329 
352 
376 
399 
422 
"445. 
468 
491. 
514. 
537 
560 
583 
606 
629. 
652. 
675 
698. 
722, 
745. 
768. 
791. 
814. 
837. 
860. 
883. 
906. 
929 
952 
975 
998 

1021 

1044. 

1068. 

1091. 

1114. 

1137. 

1160. 

1183. 

1206. 

1229. 

1252. 

1275, 

1298. 

1321. 

1344. 

1367. 

1391 



920 
.988 
,056 
,124 

192 
.260 
.328 
.396 
.464 
.532 
.600 
.668 
.736 
.804 
.872 
.940 
.008 
.076 
.144 
.212 
.280 
.348 
.416 
.484 
.552 
.620 
.688 
.756 
.824 

892 
.960 
,028 
.096 
.164 

232 



32 

55 
78 
101 
124 
147 
170 
193 
216 
239 
262 
286 
309 
332 
355 
378 
401 
424 
447 
470 
493 
516 
539 
562 
585 
608 
632 
655 
678 
701 
724 
747 
770 
793 
300 816 
368' 839 
436 862 
504 885 



572 
640 
708 
776 
844 
912 
980 
048 
116 
184 
252 
320 
388 
456 
524 
592 
660 
728 
796 
864 
932 
000 



908. 

931. 

955. 

978. 
1001. 
1024. 
1047. 
1070. 
1093. 
1116. 
1139. 
1162. 
1185. 
1208. 
1231. 
1254. 
1277. 
1301. 
1324. 
1347. 
1370 
1393 



.227 
.295 
.363 
.431 
.499 
.567 
.635 
.703 
.7W 
,839 
.907 
.975 
.043 
.111 
.179 
.247 
.315 
.383 
.451 
.519 
.587 
.655 
.723 
791 
.859 
.927 
.995 
.063 
.131 
.199 
.267 
.335 
403 
471 
.539 
.607 
.675 
.743 
.811 
.879 
.947 
.015 
.083 
.151 
.219 
287 
355 
423 
491 
559 
627 
695 
763 
831 
899 
967 
035 
103 
171 



11. 

34. 
57. 
80. 

103. 

126. 

149. 

173. 

196. 

219. 

242. 

265. 

288. 

311. 

334. 

357. 

380. 

403. 

426. 

449. 

472 

495 

519, 

542. 

565. 

588, 

611, 

634, 

657, 

680. 

703, 

726. 

749. 

772. 

795. 

818. 

841. 

865. 

888. 

911. 

934. 

957 

980. 
1003. 
1026. 
1049. 
1072. 
1095. 
1118. 
1141. 
1164. 
1188. 
1211. 
1234. 
1257. 
1280. 
1303. 
1326. 
1349. 
2391372. 
30711395. 



534 

602 

670 

738 

806 

874 

942 

010 

078 

146 

214 

282 

350 

418 

486 

554 

622 

690 

758 

826 

894 

962 

030 

(198 

166 

234 

302 

370 

438 

506 

574 

642 

710 

778 

846 

914 

982 

050 

118 

1 

254 



13. 
36. 
59. 
83. 
106. 
129. 
152. 
175. 
198. 
221. 
244. 
267, 
290, 
313, 
336, 
359, 
382, 
405. 
429, 
452, 
475, 
498, 
521, 
544 
567. 
590 
613 
636 
659 
682 
705 
728 
752 
775 
798. 
821. 
844. 
867. 
890. 
913. 
936. 
322 959. 
390 982 
4581005 
5261028. 



594 
662 
730 
798 
866 
934 
002 
070 
138 
206 
274 
342 
410 
478 
546 
614 



1051. 
1074. 
1098. 
1121. 
1144. 
1167. 
1190. 
1213. 
1236. 
1259. 
1282. 
1305. 
1328. 
1351. 
1374. 
1397 



841 
909 
977 
045 
113 
181 
249 
317 
385 
453 
521 
589 
657 
725 
793 
861 
929 
997 
065 
133 
201 
269 
337 
405 
473 
541 
609 
677 
745 
813 
881 
949 
017 
085 
153 
221 
289 
357 
425 
493 
561 
629 
697 
765 
833 
901 
969 
037 
105 
173 
241 
309 
377 
445 
513 
581 
649 
717 
785 
853 
921 



16. 
39. 
62. 

85. 

108. 

131. 

154. 

177. 

200. 

223. 

246. 

269. 

292. 

316. 

339. 

362. 

385. 

408. 

431. 

454. 

477. 

500. 

523. 

546. 

569. 

592. 

615. 

638. 

662. 

685. 

708. 

731. 

754, 

777 

800, 

823. 

846. 

869. 

892. 

915. 

938. 

961. 

985. 
1008. 
1031. 
1054. 
1077. 
1100. 
1123. 
1146. 
1169. 
1192. 
1215. 
1238. 
1261. 
1284. 
1307 
1331 
1354. 
1377. 
1400 



148 
216 
284 
352 
420 
488 
556 
624 
692 
760 
828 
896 
964 
032 
100 
168 
236 
304 
372 
440 
508 
576 
644 
712 
780 
848 
916 
984 
052 
120 



18. 

41. 

64. 

87. 
110. 
133. 
156. 
179. 
202. 
226. 
249. 
272. 
295. 
318. 
341. 
364. 
387. 
410. 
433. 
456. 
479. 
502. 
525. 
549. 
572. 
595. 
618. 
641 
664 
__-, 687. 
188 710. 



256 

324 

392 

4G0 

528 

596 

664 

732 

800 

868 

936 

004 

072 

140 

208 

276 

344 

412 

480 

548 

61 

684 

752 

820 

888 

956 

024 

092 

160 

228 



733. 

756. 

779. 

802. 

825. 

848. 

871. 

895. 

918. 

941. 

964. 

987. 
1010. 
1038. 
1056. 
1079. 
1102. 
1125. 
1148. 
1171 
1194. 
1217. 
1241. 
1264. 
1287 
1310. 
1333. 
1356. 
1379. 
1402 



454 

522 

590 

658 

726 

794 

862 

930 

998 

066 

134 

202 

270 

338 

406 

474 

542 

610 

678 

746 

814 

882 

950 

018 

086 

154 

222 

290 

358 

426 

494 

562 

630 

698 

766 

834 

902 

970 

038 

106 

174 

242 

310 

378 

446 

514 

582 

650 

71 

786 

854 

922 

990 

058 

126 

194 

26 

330 

398 

466 

534 



20.761 

43.829 

66.897 

89.965 

113.033 

136.101 

159.169 

182.237 

205.305 

228.373 

251.441 

274.509 

297.577 

320.645 

343.713 

366.781 

389.849 

412.917 

435.985 

459.053 

482.121 

505.189 

528.257 

551.325 

574.393 

597.461 

620.529 

643.597 

666.665 

689.733 

712.801 

735.869 

758.937 

782.005 

805.073 

828.141 

851.209 

874.277 

897.345 

920.413 

943.481 

966.549 

989.617 

1012.685 

1035.753 

1058.821 

1081.889 

1104.957 

1128.025 

1151.093 

1174.161 

1197.229 

1220.297 

1243.365 

1266.433 

1289.501 

1312.569 

1335.637 

1358.705 

1381.773 

1404.841 



Example. — A pressure of 45.3 lbs. per sq. in. is equivalent to (moving the 
decimal point one place to left in 453 and 1044.98) 104.5 ft. head. 



1148 



61.— H YDROSTA TICS, 



4. — Pressures in Lbs. per Sq. In. 

Corresponding to 

Given Heads of Water, in Feet. 

Note. — Weight of v/ater assumed at 62.424 lbs. per cu. ft. 

Decimal point may be moved simultaneously tc right or left for 

Head and Pressure. 

[Pressure in Lbs. per Sq. In.] 



Q> 


1 


h 


2 


•M 


3 


o 


4 


S 


5 


^ 


6 


a> 


7 


H 


8 
9 



P.P. 

.4335 
.0434 
,0867 
.1301 
.1734 
.2168 
.2601 
.3035 
.3468 
.3902 



Units. 



3. 



5. 



6. 



3350 
6700 
0050 
3400 
6750 
0100 
3450 
6800 
0150 
3500 
6850 
0200 
3550 
6900 
0250 
3600 
6950 
0300 
3650 
7000 
0350 
3700 
7050 
0400 
3750 
7100 
0450 
3800 
7150 
0500 
3850 
7200 
0550 
3900 
7250 
0600 
3950 
7300 
0650 
4000 
7350 
0700 
4050 
7400 
0750 
4100 
7450 
0800 
4150 
7500 
0850 
4200 
7550 
0900 
4250 
7600 
0950 
4300 
7650 
1000 



13. 

17. 

22. 

26. 

30. 

35. 

39. 

43. 

48. 

52. 

56. 

61. 

65. 

69. 

74. 

78. 

82. 

87. 

91. 

95. 
100. 
104. 
108. 
113. 
117. 
121. 
126. 
130. 
134. 
139. 
143. 
147. 
152. 
156. 
160. 
165. 
169. 
173. 
178. 
182. 
186. 
191. 
195. 
199. 
204. 
208. 
212. 
217. 
221. 
225. 
230. 
234. 
238. 
243. 
247. 
251. 
256. 
260. 



4335 

7685 
1035 
4385 
7735 
1085 
4435 
7785 
1135 
4485 
7835 
1185 
4535 
7885 
1235 
4585 
7935 
1285 
4635 
7985 
1335 
4685 
8035 
1385 
4735 
8085 
1435 
4785 
8135 
1485 
4835 
8185 
1535 
4885 
8235 
1585 
4935 
8285 
1635 
4985 
8335 
1685 
5035 
8385 
1735 
5085 
8435 
1785 
5135 
8485 
1835 
5185 
8535 
1885 
5235 
8585 
1935 
5285 
8635 
1985 
5335 



5. 
9. 

13. 

18. 

22. 

26. 

31. 

35. 

39. 

44. 

48. 

52. 

57. 

61. 

65. 

70. 

74. 

78. 

83. 

87. 

91. 

96. 
100 
104. 
109. 
113. 
117. 
122. 
126. 
130. 
135. 
139. 
143. 
148. 
152. 
156. 
161. 
165. 
169. 
174. 
178. 
182. 
187. 
191. 
195. 
200. 
204. 
208. 
213. 
217. 
221. 
226. 
230. 
234. 
239. 
243. 
247. 
252. 
256. 
260. 



8670 
2020 
5370 
8720 
2070 
5420 
8770 
2120 
5470 
8820 
2170 
5520 
8870 
2220 
5570 
8920 
2270 
5620 
8970 
2320 
5670 
9020 
2370 
5720 
9070 
2420 
5770 
9120 
2470 
5820 
9170 
2520 
5870 
9220 
2570 
5920 
9270 
2620 
5970 
9320 
2670 
6020 
9370 
2720 
6070 
9420 
2770 
6120 
9470 
2820 
6170 
9520 
2870 
6220 
9570 
2920 
6270 
9620 
2970 
6320 
9670 



3005 
6355 
9705 
3055 
6405 
9755 
3105 
6455 
9805 
3155 
6505 
9855 
3205 
6555 
9905 
3255 
6605 
9955 
3^05 
6655 
0005 
3355 
6705 
0055 
3405 
6755 
0105 
3455 
6805 
0155 
3505 
6855 
0205 
3555 
6905 
0255 
3605 
6955 
0305 
3655 
7005 
0355 
3705 
7055 
0405 
3755 
7105 
0455 
3805 
7155 
0505 
3855 
7205 
0555 
3905 
7255 
0605 
3955 
7305 
0655 
4005 



10 

14. 

19. 

23. 

27. 

32. 

38. 

40. 

45. 

49. 

53. 

58. 

62. 

66. 

71. 

75. 

79. 

84. 

88. 

92. 

97. 
101. 
105. 
110. 
114. 
118. 
123. 
127. 
131. 
136. 
140. 
144. 
149. 
153. 
157. 
162. 
166. 
170. 
175. 
179. 
183. 
188. 
192. 
196. 
201. 
205. 
209. 
214. 
218. 
222. 
227 
231 
235 
240 
244 
248 
253 
257 
261 



7340 
0690 
4040 
7390 
0740 
4090 
7440 
0790 
4140 
7490 
0840 
4190 
7540 
0890 
4240 
7590 
0940 
4290 
7640 
0990 
4340 
7690 
1040 
4390 
7740 
1090 
4440 



7790119 



1140 
4490 
7840 



1190136 



.4540 
.7890 
.1240 
.4590 
.7940 
.1290 
.4640 
.7990 
.1340 
.4690 
.8040 
.1390 
.4740 
.8090 
.1440 
.4790 
.8140 
.1490 
.4840 
.8190 
.1540 
.4890 
.8240 
.1590 
.4940 
.8290 
.1640 
.4990 
.8340 



40. 
145. 
149. 
153, 
158. 
162. 
166. 
171, 
175. 
179. 
184, 
188, 
192, 
197, 
201, 
205, 
210, 
214, 
218. 
223. 
227, 
231, 
236, 
240, 
244, 
249, 
253, 
257. 
262. 



1675 
5025 
8375 
1725 
5075 
8425 
1775 
5125 
8475 
1825 
5175 
8525 
1875 
5225 
8575 
1925 
5275 
8625 
1975 
5325 
8675 
2025 
5375 
8725 
2075 
5425 
8775 
2125 
5475 
8825 
2175 
5525 
8875 
2225 
5575 
8925 
2275 
5625 
8975 
2325 
5675 
9025 
2375 
5725 
9075 
2425 



11. 

15. 

19. 

24. 

28. 

32. 

37. 

41. 

45. 

50. 

54. 

58. 

63. 

67. 

71. 

76. 

80. 

84. 

89. 

93. 

97. 
102. 
106. 
110. 
115. 
119. 
123. 
128. 
132. 
136. 
141. 
145. 
149. 
154. 
158. 
162. 
167. 
171. 
176. 
180. 
184. 
189. 
193. 
197. 



5775 202 



9125 
2475 
5825 
9175 
2525 
5875 
9225 
2575 
5925 
9275 



206, 
210, 
215, 
219, 
223, 
228, 
232, 
236, 
241, 
245, 



6010 
9360 
2710 
6060 
9410 
2760 
6110 
9460 
2810 
6160 
9510 
2860 
6210 
9560 
2910 
6260 
9610 
2960 
6310 
9660 
3010 
6360 
9710 
3060 
6410 
9760 
3110 
6460 
9810 
3160 
6510 
9860 
3210 
6560 
9910 
3260 
6610 
996U 
3310 
6660 
0010 
3360 
6710 
0060 
3410 
6760 
0110 
3460 
6810 
0160 
3510 
6860 
0210 
3560 
6910 
0260 



2625249 
5975254 
93251258 
26751262 



3610245 
6960250 
0310254 
3660258 
70101263 



.0345 

.3695 

.7045 

.0395 

,3745 

,7095 

,0445 

.3795 

.7145 

.0495 

.3845 

.7195 

.0545 

.3895 

.7245 

.0595 

.3945 

.7295 

.0645 

.3995 

.7345 

.0695 

.4045 

.7395 

.0745 

.4095 

.7445 

.0795 

.4145 

.7495 

.084 

.4195 

.7545 

.0895 

.4245 

.7595 

.0945 

.4295 

.7645 

.0995 

. 4345 

.7695 

. 1045 

.4395 

.7745 

.1095 

.4445 

.7795 

.1145 

.4495 

.7845 

.1195 

.4545 

.7895 

.1245 

.4595 

.7945 

.1295 

.4645 

.7995 

.1345 



8030 
1380 
4730 
8080 
1430 
4780 
8130 
1480 
4830 
8180 
1530 
4880 
8230 
1580 
4930 
8280 
1630 
4980 
8330 
1680 
5030 
8380 
1730 
5080 
8430 
1780 
5130 
8480 
1830 
5180 
8530 
1880 
5230 
8580 
1930 
5280 
8630 
1980 
5330 
8680 
2030 
5380 
8730 
2080 
5430 
8780 
21.30 
5480 
8830 
2180 
5530 
8880 
2230 
5580 
8930 
2280 
5630 
8980 
2330 
5680 



8. 

12. 

16. 

21. 

25. 

29. 

34. 

38. 

42. 

47. 

51. 

55. 

60. 

64. 

68. 

73. 

77. 

81. 

86. 

90. 

94. 

99. 
103. 
107. 
112 
116 
120 
125 
129 
133 
138 
142 
146 
151 
155 
159 
164 
168 
172 
177 
181 
185 
190 
194 
198 
203 
207 
211 
216 
220 
224 
229 
233 
237 
242 
246 
250 
255 
259 
264 



9015 
2365 
5715 
9065 
2415 
5765 
,9115 
.2465 
.5815 
.9165 
.2515 
.5865 
.9215 
.2565 
.5915 
.9265 
.2615 
.5965 
.9315 
.2665 
.6015 
.9365 
.2715 
.6065 
.9415 
.2765 
.6115 
.9465 
.2815 
.6165 
.9515 
.2865 
.6215 
.9565 
.2915 
.6265 
.9615 
.2965 
.6315 
.9665 
.3015 
.6365 
.9715 
.3065 
.6415 
.9765 
.3115 
.6465 
.9815 
.3165 
.6515 
.9865 
.3215 
.6565 
.9915 
.3265 
.6615 
.9965 
.3315 
.6665 
.0015 



Example. — At a depth of 362.4 ft. below the surface, the water pressure 
is 156.9270+. 1734= 157.1004 lbs. per sq. in. 



HEADS REDUCED TO PRESSURES. 



1149 



5. — Pressures in Lbs. per Sq. Ft. 

Corresponding to 

Given Heads of Water, in Feet. 

Note. — Weight of water assumed at 62.424 lbs. per cu. ft. 

Decimal point may be moved simultaneously to right or left for 
Head and Pressure. 

[Pressure in Lbs. per Sq. Ft.] 



P.P. 

62.424 

1 
2 



6.24 
12.48 
18.73 
24.97 
31.21 
37.45 
43.70 
49.94 
56.18 



Head. 
Feet 



1-. 
2-. 
3-. 
4-. 
5-. 
6-. 
. 7-. 
8-. 
9-. 

d lo- 

a 11-- 

§ 12-. 

"3 13-. 

" 14-. 

'S 15-. 

o 17-. 

:: 18-. 

t3 19-. 

« 20-. 

S 21-. 

tJ3 22-. 

"« 23-. 

•3 24-. 

2 25-. 

S 26-. 

5 27-. 

i 28-. 

1 30-. 

'■a 31-. 

a 32-. 

a 34-. 

^ 35-. 

o 36-. 

^ 37-. 

A 38-. 

S§ 39-. 

■d 40-. 

>» 41-. 

-Q 42-. 

•2 44-. 
g 45-. 
§ 46-. 
o 47-. 
cu 48-. 

a 50-. 
«2 51-. 

52-. 

53-. 

54-. 

55-. 

56-. 

57-. 

58-. 

59-. 

60-. 



Units. 



624 

1248 

1872 

2496 

3121 

3745 

4369 

4993 

5618 

6242 

6866 

7490 

8115 

8739 

9363 

9987 

10612 

11236 

11860 

12484 

13109 

13733 

14357. 

14981 

15606 

16230 

16854 

17478. 

18102 

18727 

19351 

19975. 

20599. 

21224 

21848 

22472. 

23096. 

23721 

24345 

24969. 

25593 

26218. 

26842. 

27466. 

28090. 

28715. 

29339. 

29963. 

30587. 

31212. 

31836. 

32460. 

33084. 

33708. 

34333. 

34957. 

35581. 

36205. 

36830. 

37454. 



62 
686 
1310 
1935 
2559 
3183 
3807 
4432 
5056 
5680 
6304 
6929 
7553 
8177 
8801 
9426 
0050 

10674, 

11298, 

11922, 

12547. 

13171. 

13795. 

14419. 

15044. 

15668. 

16292. 

16916. 

17541. 

18165. 

18789. 

19413. 

20038. 

20662. 

21286. 

21910. 

22535. 

23159. 

23783. 

24407. 

25032. 

25656. 

26280. 

26904. 

27528. 

28153. 

28777. 

29401. 

30025. 

30650. 

31274. 

31898. 

32522. 

33147. 

33771. 

34395. 

35019. 

35644. 

i6268. 

36892. 

37516. 



124 
749 
1373 
1997 
2621 
3246 
3870 
4494 
5118 
5743 
6367 
6991 
7615 
8239 
8864 



3. 



10112 
10736 
11361 
11985 
12609 
13233 
13858. 
14482. 
15106. 
15730. 
16355. 
16979. 
17603. 
18227. 
18852. 
19476. 
20100. 
20724. 
21349. 
21973. 
22597. 
23221. 
23845. 
24470. 
?!5094. 
25718. 
26342. 
26967. 
27591. 
28215. 
28839. 
29464. 
30088. 
30712. 
31336. 
31961. 
32585. 
33209. 
33833. 
34458. 
35082. 
35706. 
36330. 
36955. 
37579. 



187 
811, 
1435, 
2059, 
2684, 
3308 
3932, 
4556. 
5181. 
5805. 
6429. 
7053. 
7678. 
8302. 
8926. 
9550. 

10175. 

10799. 

11423. 

12047. 

12672. 

13296. 

13920. 

14544. 

15169. 

15793. 

16417. 

17041. 

17665. 

18290. 

18914. 

19538. 

20162. 

20787. 

21411. 

22035. 

22659. 

23284. 

23908. 

24532. 

25156. 

25781. 

26405. 

27029. 

27653. 

28278. 

28902. 

29526. 

30150. 

30775. 

31399. 

32023. 

32647. 

33271. 

33896. 

34520. 

35144. 

35768. 

36393. 

37017. 

37641. 



249, 
873. 

1498, 

2122, 

2746, 

3370, 

3995. 

4619. 

5243. 

5867. 

6492. 

7116. 

7740. 

8364. 

8989. 

9613. 
10237. 
10861. 
11486. 
12110. 
12734. 
13358. 
13982. 
14607. 
15231. 
15855- 
16479 
17104. 
17728. 
18352. 
18976. 
19601. 
20225. 
20849- 
21473. 
22098. 
22722. 
23346. 
23970. 
24595. 
25219. 
25843. 
26467. 
27092. 
27716. 
28340. 
28964. 
29588. 
30213. 
30837. 
31461. 
32085. 
32710. 
33334. 
33958. 
34582. 
35207. 
35831. 
36455. 
37079. 
37704 



5. 



312 
936 
1560 
2184 
2809 
3433 
4057 
4681 
5306 
5930 
6554 
7178 
7803 
8427 
9051 
9675 

10299 

10924. 

11548. 

12172. 

12796. 

13421. 

14045. 

14669. 

15293. 

15918. 

16542. 

17166. 

17790. 

18415. 

19039. 

19663. 

20287. 

20912. 

21536. 

22160. 

22784. 

23409. 

24033. 

24657. 

25281. 

25905. 

26530. 

27154. 

27778. 

28402. 

29027. 

29651. 

30275. 

30899. 

31524. 

32148. 

32772. 

33396. 

34021. 

34645. 

35269. 

35893. 

36518. 

37142. 

37766. 



374 
998 
1623 
2247 
2871 
3495 
4119 
4744 
5368 
5992 
6616, 
7241, 
7865, 
8489, 
9113. 
9738. 

10362, 
2010986, 

11610. 

12235, 

12859. 

13483. 

14107. 

14732. 

15356. 

15980. 

16004. 

17229. 

17853. 

18477. 

19101. 

19725. 

20350, 

20974. 

21598. 

22222. 

22847. 

23471. 

24095. 

24719. 

25344. 

25968. 

26592. 

27216. 

27841. 

28465. 

29089. 

29713. 

30338. 

30962. 

31586. 

32210. 

32835. 

33459. 

34083. 

34707. 

35331. 

35956. 

36580. 

37204. 

37828. 



7. 



436 
1061 
1685 
2309 
2933 
3558 
4182, 
4806, 
5430, 
6055, 
6679. 
7303, 
7927, 
8552, 
9176. 



10424 

11049 

11673 

12297 

12921 

13546 

14170 

14794 

15418. 

16042 

16667 

17291 

17915 

18539 

19164. 

19788 

20412. 

21036. 

21661. 

22285. 

22909. 

23533. 

24158. 

24782. 

25406. 

26030. 

26655. 

27279. 

27903. 

28527. 

29152. 

29776. 

30400. 

31024. 

31648. 

32273. 

32897. 

33521. 

34145. 

34770. 

35394. 

36018. 

36642. 

37267. 

37891. 



499 

1123 

1747 

2372 

2996 

3620 

4244 

4869 

5493 

6117 

6741 

7366 

7990 

8614, 

9238, 

9862. 

10487, 

11111, 

11735, 

12359, 

12984. 

13608 

14232 

14856 

15481 

16105 

16729 

17353 

17978 

18602. 

19226. 

19850 

20475 

21099 

21723. 

22347. 

22972. 

23596. 

24220. 

24844. 

25468. 

26093. 

26717. 

27341. 

27965. 

28590. 

29214. 

29838. 

30462. 

31087 

31711 

32335. 

32959. 

33584 

34208 

34832. 

35456. 

36081 

36705 

37329 

37953 



561.82 

1186.06 

1810.30 

2434.54 

3058.78 

3683.02 

4307.26 

4931.50 

5555.74 

6179.98 

6804.22 

7428.46 

8052.70 

8676.94 

9301.18 

9925.42 

10549.66 

11173.90 

11798.14 

12422.38 

13046.62 

13670.86 

14295.10 

14919.34 

15543.58 

16167.82 

16792.06 

17416.30 

18040.54 

18664.78 

19289.02 

19913.26 

20537.50 

21161.74 

21785.98 

22410.22 

23034. 46 

23658.70 

24282.94 

24907.18 

25531.42 

26155.66 

26779.90 

27404.14 

28028.38 

28652.62 

29276.86 

29901.10 

30525.34 

31149.58- 

31773.82 

32398.06 

33022.30 

33646.54 

34270.78 

34895.02 

35519.26 

36143.50 

36767.74 

37391.98 

38016.22 



Ex.- 



-At a depth of 344 ft. below the surface, the pressure is 21473.80 lbs. per sq. ft. 
At a depth of 34.4 ft. below the surface, the pressure is 2147.386 lbs. per sq. ft. 



1 1 50 61.— H YDROSTA TICS. 

The Center of Pressure on any submerged plane surface is the point of 

resultant pressure on that surface. We have explained, under Dams, page 
846, the position of the center of pressure on rectangular surfaces. We will 
now present a general formula for finding the center of pressure on any plane 
surface, and whether vertical or inclined. 

Let t b, Fig. 1, represent the edge of a sub- 
merged plane figure of any shape (as . 
a rectangle, circle, triangle, etc.) ; A i,mTer Surface . 
a, the angle which this plane makes with -v^- V^ ''^ 
the water surface; \''"vs\ > h" « 
Dx, the inclined distance (ft.) from A to the \^ \ <^L 

center of gravity, or to the horizon- \k- \ I • ?. >. /, 

tal neutral axis, of the figure t h\ ^\\V f^"-^^ ^''f'^- 

Do. the inclined distance to the center of \.^\^Cen.ofPfes. 

pressure ; ^<^ 

hn, the head in feet on the center of gravity; Fig. 1. 

ho, the head in feet on the center of pressure; 
a', the area in sq. ft. of the surface t h. Also — 
Let 7x = the moment of inertia of the figure t b about a horizontal neutral 
axis passing through its center of gravity; 
/A=the moment of inertia of the figure tb about a horizontal axis 
passing through A, 

S = the statical moment about A, 
= a' Dx; 

^, n Ik I^-ha'D^ 

ThenDo = ^=-^^^-.... (7) 

A ^ 7 n • /x + a'Dx2 . 

And ho = Do sm a = sm a (8) 

CL Ux 

Having obtained ho we can easily find the total pressure P from formulas 
(1) and (8); thus, making h = ho, 

P= 62.5 a' h = 62.5 a' /zq^ 62.5 i^^^-^) sin a (9) 

If the submerged figure tb is rigid, the pressure P may be considered as a 
resultant pressure acting normal to the surface tb at the center of pressure 
distant ho ft. vertically (or Do ft. inclined) below the surface of the water. 
Moreover, if the figure tb is vertical, a = 90° and sin a= 1 in equations (8) 
and (9); hence, ho = Do. 

The practical application of equations (8) and (9) consists in substituting 
the proper values for Zx, ci' , Dx and sin a in the second members of the 
equations. The angle a, the distance A t, and the shape and dimensions of 
the figure t b will usually be given; then find Ix and Dx — At, from the tables 
in Section 29, page 524, etc. 

Problem. — Let t b represent the section of a triangular plane of altitude 
tb= 12 ft. and horizontal base at bottom 6 = 8 ft. Let it be submerged so 
that At=10 ft., and a= 60°. Find the head ho on the center of pressure? 
Find the total pressure P? 

(12)3 
Solution. — From Table 6, next page, we deduce /x= 8X „^ = 384, and 

3o 

D^= 10+1X12= 18. Area of triangle a' = 12X4= 48; sin a = 0.866. Then, 
(8). ;?o= To^Co X0.866= 181X0.866= 15.973 ft.; and P=62.5X48X 

4o X lo 

15.973=47,919 lbs. Ans. 

Any other shaped figure may be treated in the same manner. The 
center of pressure will always be found below the center of gravity of the 
figure. 

If the figure tb is a. rectangle, in a vertical plane, and with the top tatA, 
just touching the surface of the water, then Do = ^o=^ tb (Fig. 1). 

The position of the center of pressure is essential in designing large 
hydraulic gates, valves, etc. 



CENTER OF PRESSUkE. 



1151 



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o p 




►5i 


►-H 


►Si ?s 



1152 



61.—;? YDROSTA TICS, 



(A 




^ 


Wqrer_. 


XSurfac^ 
6 


8 




c i 



Pressure in Pipes, Tanks, etc. — In Fig. 2, let h be the total head in ft., 

at any level, on each diameter d of sections 
A, B, and C. .4 is a water pipe leading 
from the water tank B, which in turn is 
connected by the tube t with the tank C 
through the vertical tube T. Then it follows 
that the water surfaces in B and T must 
remain at the same level. From formulas 
(3) and (4) we have for either section .A, B 
or C, at a depth h below the surface: 

Pressure in lbs. per sq. ft. = 62.5 h'. 

Pressure in lbs. per sq. in. = .434 /t. F^S- 2. 

If (i'=the diameter in feet, and c?'' = the diameter in inches, then A, B 
and C will each have to resist a bursting pressure of ^l.hhd' lbs. per lin. ft., 
or .434 h d" lbs. per lin. in. of pipe or tank; and each side would have to 
resist one-half that pressure. 

The combination of the tube T inserted in the tank C (with i omitted) 
illustrates what is called the "hydrostatic paradox." The unit pressure at o 
due to the head h remains the same no matter what the diameter of T. 
Clearly, the diameter of T does not affect the unit pressure; the height h 
does. Thus, a heavy cask, as C, may be made to burst if even a small tube, 
as r, is filled with water so as to give the required bursting headi 

Flotation. — ^The weight of a substance is proportional to its density; 
and the relative density of a substance, referred to water, is called its specific 
gravity.* Hence, of two substances of equal weight, that one having the 
least volume has the greater specific gravity. As the volume of a body is 
affected by heat, it follows that the specific gravity of substances decreases 
with a rise in temperature, the formation of ice being a phenomenal 
exception to this law. Pure water at 4° C, its maximum density, is 
assumed to have a specific gravity of 1: and all other substances, at 0° C, 
are referred to that standard. A porous substance will increase in density 
by absorbing water; thus, water-soaked logs is an instance of this kind. 

Any solid with specific gravity greater than unity will sink; if less than 
unity it will float. Any floating body, whether solid or not, will displace a 
volume of water whose weight is equal to the weight of the body; if the 
body sinks, the weight of the volume of water displaced will be less than 
that of the body. 

The Depth of Flotation depends upon the specific gravity of the body, if 
solid; upon the average specific gravity of the volume, if hollow; and upon 



VJater 




Surface. 



Fig. 3. 

the surface form of the body. Assuming the average specific gravity of 
volume to be less than unity, let cf = depth of flotation, 

5 = average specific gravity of volume of 
body ; 
then for any rod, bar, tube, cylinder, etc., of uniform cross-section and of 
length /, floating vertically, 

d = sl (10) 

For the same, if lying horizontal, d can be obtained by fixing the water sur- 
face on the end section at such elevation (Fig. 3) that 

shaded area below water surface ,, -. 

5 = z z — -, : (11) 

total area of the section 

Buoyancy. — Let Figs. 4, 5 and 6 represent oval sections, and Fig. 7 a 
circular section, of any body floating in water, with the surface at S\ 
then — 

Stable equilibrium is represented by Fig. 4, 

Unstable equilibrium, by Fig. 6; 

Neutral equilibrium, by Fig. 7. 



* For discussion of Specific Gravity, see page 460, etc. 



PRESSURE. FLOTATION. BUOYANCY. 



1153 



The position in Fig. 5 can obtain only when some outside force is applied, 
as will be seen from the following discussion: The center of buoyancy B in 
each figure below is the center of gravity of the displaced water; and G is the 





Fig. 4. 



Fig. 5. 



Fig. 6. 



Fig. 7. 



center of gravity of the body. Then, as long as G and B are in the same 
vertical line there is some kind of equilibrium, as stable, unstable, or neutral, 
because the resultant downward weight of the body, acting in a vertical 
line passing through its center of gravity G, must be equal and opposite to 
the resultant upward pressure R of the water below the body. Furthermore, 
the upward resultant pressure R will not be changed in position, direction or 
amount if we imagine the body to be removed from the "depression" in the 
water and that depression refilled with the displaced water; for equilibrium 
would be maintained by the equal weight of the displaced water, its re- 
sultant passing vertically through its center of gravity B. Hence if R, 
acting vertically, is equal and opposite to the resultant weight of the body 
and also to the displaced water, acting through their respective centers of 
gravity G and B, then it follows that G and B are in the same vertical line 
when the body is in equilibrium. 

The equilibrium of a floating body may be tested by noting the position 
of the "metacenter" M when the body is slightly disturbed in any direction 
from a position of rest. (M is the intersection of the "equilibrium axis" 
a — a with a vertical through B.): 

(a) When M rises above G it indicates that the body was in stable equili- 

brium. 

(b) When M falls below G it indicates that the body was in unstable equili- 

brium. ... 

(c) When M coincides with G it indicates neutral equilibrium. 

In the above. Fig. 5 shows that Fig. 4 is in stable equilibrium. In the 
same way it can be proved that Fig. 6 is unstable, and Fig. 7 neutral. 

The following rules apply not only to floating bodies but to supported 
bodies in general. 

(1) A body is in stable equilibrium when a slight change tends to raise its 

center of gravity; 

(2) A body is in unstable equilibrium when a slight change tends to lower 

its center of gravity; 

(3) A body is in neutral equilibrium when a slight change neither raises nor 

lowers its center of gravity. 



62.— HYDRAULICS. 

Hydraulics embraces the application of the principles of both hydro- 
statics and hydrokinetics;* for a fluid at rest, as treated by hydrostatics, 
is but the lower limit of a condition of motion. It therefore treats of the 
laws governing the pressure, flow and energy of water (and other liquids) 
with the accompanying phenomena. These laws, however, are but imper- 
fectly understood and, like all other branches of engineering, hydraulics is 
not an exact science. Theoretical hydraulics assumes no loss of energy 
during the flow, a condition which can never obtain in practice; but its 
great value, in the investigation of any problem, lies in fixing the upper 
limit of efficiency which can ever be expected. Practical hydraulics is 
founded on theoretical hydraulics, but takes into consideration the losses of 
energy during the flow. These losses are deduced from experiments. 

Theory of Flow. — Under hydrostatics we have discussed the relation 
between the static head h and the pressure p of still water. In hydraulic 
computations it is convenient to reduce all pressures, velocities and losses of 
every description to equivalent heads and losses of head. Moreover, the 
unit of pressure used is, in English units, the "lb. per sq. in." Thus, referring 
particularly to a pipe line, we have: 

^ = pressure in lbs. per sq. in. = 0.434 h^; 
H = hydrostatic head, or simply static nead = 2.304 p\ 
^e = entry head, or loss of head at entry; 
/t = velocity head, or head due to velocity at given section; 
^a = "velocity of approach" head, or gain in head a given section due to 

velocity of approach above the given section; 
ht —iriction head, or loss of head due to friction; 
^a = "suction" head, or head due to suction, acting in the direction of the 

flow; 
ho = "curvature" head, or loss of head due to curves or bends; 
^p= pressure head, or head due to the resultant pressure at a given sec- 
tion (i. e., piezometer head); 
A, = "expansion head," or loss of head due to expansion of section; 
^k = "contraction head," or loss of head due to contraction of section; 
h\ = total loss of head above a given section. 

Then, using the same sub-notation for the velocity, we have, by theory: 
V = theoretic velocity due to H, or V^=2gH; v^=2g h^\ 
v^=2gh; v^^=2gK\ V{^=2ghr, Vs^=2ghs: 

Vc^==2ghc; Vy'^=2gh^\ v.^=2gh.', _V),^=^2gh^; 

v\^=2gh\\ in which the gravity acceleration g= 32.16, and V2g = say 8.02. 
Considering the gains and losses in the pipe line, the following relations 
exist: 

Gains Losses 



Pres. Stat. Appr. Suet. Veloc. Entry Fric. Curv. Expan. Contr. 
Ap = H + /J« + /t. - h - (/te + /tf + K + h. + hv) .. (1) 

2g~ 2g"^ 2g "^ 2g 2g V2g "^ 2g "^ 2g '^ 2g ^ 2g/ ' ' ^ ^ 
If we neglect the velocity of approach and the suction head, and let h 
represent the losses in head (mostly friction), equations (1) and (2) reduce to 

k^ =H -h -hi (3) 

vl Yl __t^ _vf^ 

2g^2g 2g 2g ^^^ 

Combining (3) and (4), we have the general equation (5). following. 

* The term hydrodynamics was formerly defined as the science which 
treats of the motion of liquids (now included under hydrokinetics) , but it now 
has a broader acceptation: The science which treats of the laws of jorce as 
applied to fluids. Hence it comprises hydrostatics and hydrokinetics. 

1154 



THEORETIC— FLOW', VELOCITIES FOR HEADS H. 1155 



Velocity and Discharge. — The velocity of discharge v at any section 

may be obtained from equation (1), by substituting jj^ for h, and trans- 
posing, as follows: 



2g 



v = \/2g (H-h,-hO (5) 

If the pressure head h^ equals zero for any given section, and we also assume 
no losses hu then 

v = V¥g H = 8.02VH (6) 

The following is a table of theoretic velocities for various heads, calcu- 
lated from equation (6). See, also, table on page 283. 

1. — ^Theoretic Velocities v for Various Heads h. (Equation 6.) 





, 


? ^ 


, 


• t-> 




,? ^ 




,.• (h 




. (4 




. f- 




s3 -M 


ISci 


"i+J 




a +^ 


2, ag 


i^ 


2. as 


cj >^ 




i^ 


g^d 




r 


>e- 


S^ 


«'- 


>£^ 


S^ 


>i^ 


s^ 








• 


.005 


.57 


.37 


4 88 


.93 


7.73 


3.1 


14 1 


18.5 


34.5 


68 


66.1 




.010 


.80 


.38 


4.94 


.94 


7.78 


3.2 


14.3 


19.0 


35.0 


69 


66.6 


.015 


.98 


.39 


5.01 


.95 


7.82 


3.3 


14.6 


.5 


35.4 


70 


67.1 


1 


.020 


1.13 


.40 


5.07 


.96 


7.86 


3.4 


14.8 


20.0 


35.9 


71 


67.6 


•53 


.025 


1.27 


.41 


5.14 


.97 


7.90 


3.5 


15.0 


.5 


36.3 


72 


68.0 


> 


.030 


1.39 


.42 


5.20 


.98 


7.94 


3.6 


15.2 


21.0 


36.8 


73 


68.5 


5 


.035 


1.50 


.43 


5.26 


.99 


7.98 


3.7 


15.4 


.5 


37.2 


74 


69.0 


1 


.040 


1.60 


.44 


5.32 


1.00 


8.02 


3.8 


15.6 


22.0 


37.6 


75 


69.5 


.045 


1.70 


.45 


5.38 


1.02 


8 10 


3.9 


15.8 


.5 


38.0 


76 


69.9 


B 


.050 


1.79 


.46 


5.44 


1.04 


8.18 


4.0 


16.0 


23.0 


38.5 


77 


70.4 


1 


.055 


1.88 


.47 


5.50 


1.06 


8.26 


.2 


16.4 


.5 


38.9 


78 


70.8 


.060 


1.96 


.48 


5 56 


1.08 


8.33 


.4 


16.8 


24.0 


39.3. 


79 


71.3 


■a 


.065 


2.04 


.49 


5.61 


1.10 


8.41 


.6 


17.2 


.5 


39.7 


80 


71.7 


"eS 


.070 


2.12 


.50 


5.67 


1.12 


8.49 


.8 


17.6 


25 


40.1 


81 


72.2 


d 


.075 


2.19 


.51 


5.73 


1.14 


8.56 


5.0 


17.9 


26 


40.9 


82 


72.6 


"S 


.080 


2.27 


.52 


5.78 


1.16 


8.64 


.2 


18.3 


27 


41.7 


83 


73.1 


'S 


.085 


2,34 


.53 


5.84 


1.18 


8.71 


.4 


18.6 


28 


42.4 


84 


73.5 


■>-* 


.090 


2.41 


.54 


5.89 


1.20 


8.79 


.6 


19.0 


29 


43.2 


85 


73.9 


•M 


.095 


2.47 


.55 


5.95 


1.22 


8.86 


.8 


19.3 


30 


43.9 


86 


74.4 


o 


.100 


2.54 


.56 


6.00 


1.24 


8.93 


6 


19.6 


31 


44.7 


87 


74.8 




.105 


2.60 


.57 


6.06 


1.26 


9.00 


.2 


20.0 


32 


45.4 


88 


75.2 


a 


.110 


2.66 


.58 


6.11 


1.28 


9.07 


.4 


20.3 


33 


46.1 


89 


75.7 


.a 


.115 


2.72 


.59 


6.16 


1.30 


9.14 


.6 


20.6 


34 


46.8 


90 


76.1. 


o 


.120 


2.78 


.60 


6.21 


1.32 


9.21 


.8 


20.9 


35 


47.4 


91 


76.5 


03 


.125 


2.84 


.61 


6.26 


1.34 


9.28 


7.0 


21.2 


36 


48.1 


92 


76.9 


a 


.130 


2.89 


.62 


6.31 


1.36 


9.35 


.2 


21.5 


37 


48.8 


93 


77.3 


QQ 


.135 


2.95 


.63 


6.37 


1.38 


9.42 


.4 


21.8 


38 


49.4 


94 


77.8 


1 


.140 


3.00 


.64 


6.42 


1 40 


9.49 


.6 


22.1 


39 


50.1 


95 


78.2 


.145 


3.05 


.65 


6.47 


1.42 


9.56 


.8 


22.4 


40 


50.7 


96 


78.6 


: 


.150 


3.11 


.66 


6.52 


1.44 


9.62 


8.0 


22.7 


41 


51.4 


97. 


79.0 


i 


.155 


3.16 


.67 


6.56 


1.46 


9.69 


.2 


23.0 


42 


52.0 


98 


79.4 


.160 


3.21 


.68 


6.61 


1.48 


9.76 


.4 


23.3 


43 


52.6 


99 


79.8 


.165 


3.26 


.69 


6.66 


1.50 


9.82 


.6 


23.5 


44 


53.2 


100 


80.2 


i 


.170 


3.31 


.70 


6.71 


1.52 


9.89 


.8 


23.8 


45 


53.8 


110 


84.1 


.175 


3.35 


.71 


6.76 


1.54 


9.95 


9.0 


24.1 


46 


54.4 


120 


87.9 


•+* 


.180 


3.40 


.72 


6.80 


1.56 


10.02 


.2 


24.3 


47 


55.0 


130 


91.4 


d 


.185 


3.45 


.73 


6.85 


1.58 


10.08 


.4 


24.6 


48 


55.6 


140 


94.9 


s 


.190 


3.50 


.74 


6.90 


1.60 


10.14 


.6 


24.8 


49 


56.1 


150 


98.2 


1 


.195 


3.54 


.75 


6.95 


1.65 


10.30 


.8 


25.1 


50 


56.7 


175 


106.1 


.200 


3.59 


.76 


6.99 


1.70 


10.46 


10.0 


25.4 


51 


57.3 


200 


113.4 


.21 


3.68 


.77 


7.04 


1.75 


10.61 


.5 


26.0 


52 


57.8 


225 


120.3 


1 


.22 


3.76 


.78 


7.08 


1.80 


10.76 


11.0 


26.6 


53 


58.4 


250 


126.8 


.23 


3.85 


.79 


7.13 


1.85 


10.91 


.5 


27.2 


54 


58.9 


275 


133.0 


s 


.24 


3.93 


.80 


7.17 


1.90 


11.05 


12.0 


27.8 


55 


59.5 


300 


138.9 


'0 


.25 


4.01 


.81 


7.22 


1.95 


11.20 


.5 


28.4 


56 


60.0 


325 


144.6 


cq 


.26 


4.09 


,82 


7.26 


2.00 


11.34 


13.0 


28.9 


57 


60.6 


350 


150.0 


O 


.27 


4.17 


.83 


7.31 


2.10 


11.62 


.5 


29.5 


58 


61.1 


375 


155.3 


S. 


.28 


4.24 


.84 


7.35 


2.20 


11.90 


14.0 


30.0 


59 


61.6 


400 


160.4 




.29 


4.32 


.85 


7.39 


2 30 


12.16 


.5 


30.5 


60 


62.1 


450 


170.1 




.30 


4.39 


.86 


7 44 


2.40 


12.43 


15.0 


31.1 


61 


62.6 


500 


179.3 


.31 


4.47 


.87 


7.48 


2 50 


12 68 


.5 


31.6 


62 


63.1 


550 


188.1 


<lj 


.32 


4.54 


.88 


7.52 


2.60 


12.93 


16.0 


32.1 


63 


63.7 


600 


196.4 


1 


.33 


4.61 


.89 


7.57 


2 70 


13.18 


.5 


32.6 


64 


64.2 


700 


212.2 


i 


.34 


4.68 


.90 


7.61 


2.80 


13.42 


17.0 


33.1 


65 


64.7 


800 


226.8 


t 


.35 


4.74 


.91 


7 65 


2.90 


13.66 


.5 


33.5 


66 


65.2 


900 


240.6 


z 


.36 


4.81 


.92 


7.69 


3.00 


13.89 


18.0 


34.0 


67 


65.6 


1000 


253.6 



1156 ^2.— HYDRAULICS. 

The theoretic velocity, therefore, is the same as that which would be 
acquired by the water (or any other body) falling freely in vacuo through 
the height H. Also, from equation (6), we have 

/f = . 01555 z;2 (7) 

The Discharge through a pipe, when the velocity is known, is obtained 
from the following simple formula: 

q = av (8) 

In which <7 = discharge in cu. ft, per *sec.; 

a = area of cross-section of flowing water, in sq. ft.; 
z; = mean velocity of flow, in ft, per sec. 
Then, from (8) and (5) we have, for any practical case, 



g= 8.02 a VH-h -hi (9) 

which takes into consideration all the losses of head. If there are no losses 
in head, and no pressure head at the section considered, then we have for 
the theoretical discharge, since, hp = 0, hi=0, 

q=8.02a VH (10) 

in which 8.02 \/H is the theoretic velocity (equation 6) whose values are 
given in Table 1, preceding. 

fTable 2, following, gives the areas of pipes in square feet for various 
diameters in feet and inches: 

Problem 1. — What is the least diameter of pipe that could possibly be 
used for discharging 300 cubic ft. per sec. under a 14-ft. head? 

Solution,— Neglecting friction and other losses we have from Table 1, 
page 1155, that the theoretic velocity is 30 ft. per second. Without the use 

of any formulaj we know that the area of pipe required = -ttt- =10 sq. ft.; 

and, from Table 2, page 1157, the corresponding diameter is 3 ft. 6| ins. 
Hence, we know that the diameter would have to be larger than 3 ft. 6j ins. 
to take care of the friction- and other losses. 

In practice, after a pipe has been properly designed to meet the con- 
ditions of the problem and take care of all losses of head, it is customary to 
increase the diameter of the pipe somewhat: (1) to provide for future in- 
creased demands on the supply, and (2) to anticipate the roughening of the 
inner surface of the pipes from rust or vegetable growth. Sewer pipes are 
increased usually about 2 ins. in dia,; water mains, about the same; and 
small pipes, 10 to 50% in area. 



* To find cubic feet per minute, multiply g by 60 

" hour, " " 3,600 

" 24 hours, " " 86,400 

" 30 days, " " 2,592,000 

" 365 days, " " 31,536,000 

To reduce cubic feet to gallons, multiply by - ""tvTT ='^•^^0^2. 
t See, also, tables of circles, pages 230-235, 
% Or, from equation (5) , g = a t;, we have a = — = -r^r- . 



PIPES— DISCHARGE; DIAMETERS TO AREAS. 



1157 



2, — Areas of Pipes in Sq. Ft. for Diameters in Feet and Inches. 



Diam- 
eter. 






Fraction of an 


Inch. 








Ft. Ins. 


g 


* 


i 


f 


h 


1 


f 


I 








.000085 


.00034 


.00077 


.00137 


.00213 


.00307 


-.00418 


1 


.00545 


.00690 


.00852 


.0103 


.0123 


.0144 


.0167 


.0192 


2 


.0218 


.0246 


.0276 


.0308 


.0341 


.0376 


.0412 


.0451 


3 


.0491 


.0533 


.0576 


.0621 


.0668 


.0717 


.0767 


.0819 


4 


.0873 


.0928 


.0985 


.1044 


.1104 


.1167 


.1231 


.1296 


5 


.1364 


.1433 


.1503 


.1575 


.1650 


.1726 


.1803 


.1883 


6 


.1964 


.2046 


.2131 


.2217 


.2304 


.2394 


.2485 


.2578 


7 


.2673 


.2769 


.2867 


.2967 


.3068 


.3171 


.3276 


.3382 


8 


.3491 


.3601 


.3712 


.3826 


.3941 


.4056 


.4176 


.4296 


9 


.4418 


.4541 


.4667 


.4794 


.4922 


.5053 


.5185 


.5319 


10 


.5454 


.5591 


.5730 


.5871 


.6013 


.6157 


.6303 


.6450 


11 


.6600 


.6750 


.6903 


.7057 


.7213 


.7371 


.7530 


.7691 


1 


.7854 


.8019 


.8185 


.8352 


.8522 


.8693 


.8866- 


.9041 


1 


.9218 


.9396 


.9575 


.9757 


.9940 


1.013 


1.031 


1.050 


2 


1.069 


1.088 


1.108 


1.1-27 


1.147 


1.167 


1.187 


1.207 


3 


1.227 


1.247 


1.268 


1.289 


1.310 


1.331 


1.353 


1.374 


4 


1.396 


1.418 


1.440 


1.462 


1.485 


1.507 


1.530 


1.553 


5 


1.576 


1.599 


1.623 


1.646 


1.670 


1.694 


1.718 


1.742 


6 


1.767 


1.792 


1.817 


1.842 


1.867 


1.892 


1.917 


1.943 


7 


1.969 


1.995 


2.021 


2.047 


2.074 


2.100 


2.127 


2.154 


8 


2.182 


2.209 


2.237 


2.264 


2.292 


2.320 


2.348 


2.376 


9 


2.405 


2.434 


2.463 


2.492 


2.521 


2.550 


2.580 


2.610 


10 


2.640 


2.670 


2.700 


2.730 


2.761 


2.792 


2.823 


2.854 


11 


2.885 


2.916 


2.948 


2.980 


3.012 


3.044 


3.076 


3.109 


2 


3.142 


3.174 


3.207 


3.240 


3.274 


3.307 


3.341 


3.375 


1 


3,409 


3.443 


3.477 


3.512 


3.547 


3.581 


3.616 


3.651 


2 


3.687 


3.722 


3.758 


3.794 


3.830 


3.866 


3.903 


3.939 


3 


3.976 


4.013 


4.050 


4.087 


4.125 


4.162 


4.200 


4.238 


4 


4.276 


4.314 


4.353 


4.391 


4.430 


•4.469 


4.508 


4.547 


5 


4.587 


4.626 


4.666 


4.706 


4.746 


4.786 


4.827 


4.868 


6 


4.909 


4.950 


4.991 


5.032 


5.074 


5.115 


5.157 


5.199 


7 


5.241 


5.283 


5.326 


5.369 


5.412 


5.455 


5.498 


5.541 


8 


5. 585 


5.629 


5.673 


5.717 


5.761 


5.805 


5.850 


5.895 


9 


5.940 


5.985 


6.030 


6.075 


6.121 


6.167 


6.213 


6.259 


10 


6.305 


6.351 


6.398 


6.445 


6.492 


6.539 


6.586 


6.633 


11 


6.681 


6.729 


6.777 


6.825 


6.874 


6.922 


6.971 


7.020 


3 


7.069 


7.118 


7.167 


7.216 


7.266 


7.316 


7.366 


7.416 


1 


7.467 


7.517 


7.568 


7.619 


7.670 


7.721 


7.773 


7.824 


2 


7.876 


7.928 


7.980 


8.032 


8.084 


8.137- 


8.190 


8.243 


3 


8.296 


8.349 


8.403 


8.456 


8.510 


8.564 


8.618 


8.672 


4 


8.727 


8.781 


8.836 


8.891 


8.946 


9.001 


9.057 


9.112 


5 


9.168 


9.224 


9.280 


9.336 


9.393 


9.450 


9.507 


9.564 


6 


9.621 


9.678 


9.736 


9.794 


9.852 


9.910 


9.968 


10.026 


7 


10 085 


10.144 


10.203 


10.262 


10.321 


10.380 


10.440 


10.499 


8 


10.559 


10.619 


10.680 


10.740 


10.801 


10.862 


10.923 


10. 984 


9 


11.045 


11.106 


11.168 


11.229 


11.291 


11.353 


11.416 


11.478 


10 


11.541 


11.604 


11.667 


11.730 


11.793 


11.856 


11.920 


11.984 


11 


12.048 


12.112 


12.177 


12.241 


12.306 


12.371 


12.436 


12.501 


4 


12.566 


12.632 


12.698 


12.764 


12.830 


12.896 


12.962 


13.028 


1 


13.095 


13.'162 


13.229 


13.296 


13.364 


13.431 


13.499 


13.567 


2 


13.635 


13.703 


13.772 


13.840 


13.909 


13.978 


14.047 


14.116 


3 


14.186 


14.256 


14.326 


14.396 


14.466 


14.536 


14.606 


14.677 


4 


14.748 


14.819 


14.890 


14.962 


15.033 


15.105 


15.176 


15.249 


5 


15.321 


15.393 


15.465 


15.538 


15.611 


15.684 


15.758 


15. 831 


6 


15.904 


15.978 


16.052 


16.126 


16.200 


16.274 


16.349 


16.424 


7 


16.499 


16.574 


16.649 


16.724 


16.800 


16.876 


16.952 


17.028 


8 


17.104 


17.181 


17.257 


17.334 


17.411 


1.7.488 


17.565 


17.643 


9 


17.721 


17.799 


17.876 


17.954 


18.033 


18.111 


18.190 


18.269 


10 


18.348 


18.427 


18.506 


18.586 


18 665 


18.745 


18.826 


18.906 


11 


18.986 


19.067 


19.147 


19.228 


19.309 


19.390 


19.472 


19.553 



Note. — Areas of pipes are proportional to the squares of their diameters. 

Example. — The area of pipe 2' 6^^ in dia. = 4.909 sq. ft. Then, for a pipe 
5' 0"dia., = 4.909X4- for T 6" dia., a = 4. 909X9; for 10' 0'' dia., a = 4.909 
X 16; etc. See, also, Tables 13-15, pages 230-235. 



1158 &i.^HYDRAULlCS, 

Velocity inversely proportional to a and to d^. — When a pipe of variable 
cross-section is discharging a constant, full volume of water, as per Fig. 1, 
we have, 

q = av = aiVi = a2V2 = a3 Vs, etc (11) 

But a =-r\ ai = —r I a2 = -j- ; etc. Then g = -j-?^ = -7-2^1= -7-1/2, etc. .(12) 

ctj CI2 Oi a 

V3 Vg V, V 

Fig. 1. 

Whence v= — ; Wi = — ; 2/2 = — ; etc (13) 

a ai a2 

Or. v=^2' ^1 = -^' z;2=-^2' etc (14) 

In which q = discharge in cubic feet per second ; 
d, di, d2, dz = diameter of pipe in feet at different sections; 
a, ax, 02, 03 = area of pipe in sq. ft. at sections d, dx, d2, d^', 
V, ^ii ^2. ^3 = velocity of flow in ft. per sec. at sections a, ai, 02, a^; 

7r= 3.1416; j= 0.7854. 

It is to be noted that the above formulas represent the practical rela- 
tions which may exist in any pipe line regardless of friction and loss of head. 
These formulas may be transposed in various ways. If it is desired to 
substitute the "head" of water in place of the velocity we must be careful 
to use the velocity head h and not the static head H. Thus, from (6), (13) 
and (14) we have 

q = ^X 8.02\/¥ = 6.298908 ^2 V~h (15) 

whence d^^J ^—^ = 0.3984--^ (16) 

\8.02;:VT \/h 

and if all friction and other losses are neglected, h = H, 

and d = 2 J ^—— = 0. 3984 -4: (17) 

\8.02;r^'^ "'" 



:VH ' Vh 



Hence it is seen that the diameter of the pipe is directly proportional 
to the square root of the discharge, and inversely proportional to the fourth 
root of the "head." Solving Problem 1 by equation (16), we obtain 

£^=0.3984^^7^ = 3.57 ft. = 3 ft. 6Hns. 
\/l4 
We will now take up the question of "losses" of energy, head and 
pressure, which occur during flow. 

Losses During Flow. — In the following discussion reference is made to 
Fig. 2, showing water discharging from the upper reservoir U. R., through 
the pipe line o d, into the lower reservoir L. R, 

= orifice, intake or inlet end of pipe, at which a gate is placed; 
ci = discharge end, or outlet of pipe, at which a gate is placed; 
t V r = Venturi water meter, inserted in the pipe line for measuring 
the discharge; 
1^ = Venturi itself, or the contracted section of the meter; 
t or c = contracted section of the pipe line; 
T or ^ = expanded section of the pipe line; 
6 = bend in the pipe line. 
During flow, loss of head will occur at all these points; and throughout 
the whole line, due (1) to the friction of the water along the sides of the 
pipe, and (2) to the lateral or radial forces* set up by the impingement of 

* Usually termed viscosity. 



FLOW IN PIPES— VELOCITIES AND LOSSES. 



1159 



the water particles against each other in oblique directions to the flow. 
But (1) and (2) are usually grouped together under the symbol ht= loss of 
head due to friction (see page 1160). 

There are two other losses of head which have to be considered, namely, 
(1) loss due to "drooping" head (decrease of H) in the upper reservoir, 
and (2) loss due to "rising" head, above d, in the lower reservoir (decrease 
of Hs) ; the static head H at d, is H2+hf when the water surface in L. R. is 
below d\ and Hz when above d. 



.^J^§9C^SE^3_li)^rau^^^ \_ 




Fig. 2. — Pipe Line Between Reservoirs. 

Explanation of Figure 2. — A hydraulic grade line is a line of "no pres- 
sure." That is, if a vertical tube or piezometer is inserted in the top of a 
pipe line at any point it will be found that the water will rise in the piezo- 
meter tube to the hydraulic grade line and thus indicate the pressure head h^ 
at that point of the line. Hence, a H. G. L. is very useful in the study of 
any pipe line as it gives directly the pressure head at any point. The above 
Figure shows five H.G.L.'s, (a), (6), (c), {d) and (^), which will be explained: 

(a). — ^This can obtain only when a gate is closed in the pipe line as at d, 
or some other point. As soon as the gate is opened, water will begin to dis- 
charge at d and the end of the H. G. L. at L will drop. 

(6). — ^This condition will obtain with the gate at d, partly open, when 
the H. G. L. would drop to M and the pressure head h^ at d would equal 
H2 + h' — Hx if the siurface of water in L. R. is below d\ or H3—H1 if surface 
of water is above d. The total loss of head is Hi — h, the latter being the 
velocity head. Note that the H. G. L. 5 M is straight only when the loss of 
head is uniform or proportional to the length of the line. The discharge at 

d will issue with a velocity *z;= 8.02 \/h, h forming a fractional part of Hi. 

(c). — ^This is the usual condition for a pipe line of uni- 
form cross-section discharging freely, that is, all gates open, 
and the water-level below d in L. R. Note that there is no 
pressure head above the top of the pipe at d because the efflux 
does not quite fill the discharge end; but the actual mean 
pressure head at d is shown in Fig. 3, by h\ the depth below 
the free surface, and the velocity of discharge at outlet is 
v= S.02^yh'-h velocity of approach. 

(d). — ^The hydraulic grade line is here shown as a very broken or ir- 
regular line, and marks the elevation at which the water would rise in the piez- 
ometer tube if inserted in the pipe line at any point. The pipe is as shown 
in Fig. 2, and not uniform in section as was assumed with (6) and {c). A 
sharp drop (to the right) in the H. G- L. indicates a greater rate of loss of 
head, per lin. ft. of pipe, than one of flatter slope. Thus, the grade drops 
sharply, by comparison, at c and e, points of abrupt change in diameter; at 
G, the converging tube leading to the venturi V; and at the bend b. The 
curved form of the H. G. L. at section A is due to a loss of head at the 
orifice o forming a downward (first) part of curve; while the upward (second) 
part of the curve is due to suction by vacuum just beyond the orifice. A 
similar suction occurs just beyond the venturi V causing an upward H. G. L. 
at section H. Note that the H. G. L. at section D is sharper than at sections 
B, F, I and K because the area of the pipe is less, therefore the velocity, 
friction and rate of loss of head are greater. The conditions at the outlet d 

* Treated as a weir without end contraction; see Weirs, page 1177, etc. 




Fig. 3. 



1160 62.— HYDRAULICS. 

are the same as described in (c) . This case (d) represents a usual occurrence 
in practice, and the losses are described in detail in the following pages. 
It may be remarked here, however, that the pressure head hp at any point 
in the pipe line is the vertical distance from the H. G. L. to the center of 
pressure of the water section in the pipe. The center of pressure is usually 
assumed at the center of a circular pipe, although in reality it is a little 
below the center. 

(e). — Here, it is assumed that the water in the lower reservoir has risen 
above the discharge end d of the pipe, to the elevation of s'. The theoretic 
head is therefore H^. Now we know that the total loss of head cannot ex- 
ceed the theoretic head H3, therefore the H« G. L. must slope to some point 
X, above or at s\ Moreover, we know that x must be above s^, otherwise 
there would be no pressure head (x s') to cause a flow into L. R. We can 
reasonably assume x 5' to be less than h' (say about half the diameter of the 
pipe) and to diminish as s^ rises in elevation, so that when Hs = o. xs^ = o 
also. 

Total Head is usually sub-divided into entry head, velocity head and 
friction head. That is, the force exerted by the pressure due to the total 
head, as in a reservoir, is expended partly (1) in overcoming the resistance 
at the entrance of the pipe; (2) in producing velocity of flow or discharge; 
(3) in producing friction (and viscosity). There is a fourth resistance which 
has to be overcome, namely, curvature head*, but this is usually included in 
an ordinary pipe line, under friction head. The combined entry and velocity 
heads seldom exceed one foot even when the entrance is sharp-edged, in 
which case they are about equal. The entry head reduces with increased 
length of pipe line, becoming inappreciable when the length exceeds about 
1000 diameters. Even for short pipe lines, less than 1000 diameters, the 
entry head almost entirely disappears with the use of the bell-shaped or 
flaring entrance. It is thus seen that the main considerations in any ordi- 
nary pipe line are velocity head and friction head. Moreover, one is a 
function of the other so that, for a given smoothness of wetted surface and 
a given diameter of pipe, they both increase with the hydraulic slope. The 
relations between velocity head (or rather velocity due to the velocity 
head), friction head, smoothness of wetted surface, and hydraulic slope, con- 
stitute the laws of flow and when thoroughly known, may be incorporated 
in a working formula. 

Loss of Head due to Friction. — In long pipe lines this is really the only 
loss that is practically considered, or at least other incidental losses are 
included in "friction" losses. The "dropping" head as in emptying a 
reservoir may be reduced to "mean average" head or, with more exactness 
to several mean average heads at successive elevations of water surface. 
The loss of head at entrance becomes almost a negligible quantity; more- 
over, the orifice is usually a converging tube, thereby reducing the loss to a 
minimum. Similarly, converging and diverging "reducers" are inserted in a 
line joining pipes of different diameters, thereby greatly reducing the loss of 
head which would obtain if the change were abrupt as at c and e. Fig.- 2. 
Where such changes of section occur fre- _ 

quently, as in lap-jointed riveted steel I t %" " qc/ % r" 7 

pipe (Fig. 4), it is customary to use the •? a . ''^y'^ * I 

smaller diameter d in computing the flow ' -^ S a t S 

The losses due to contraction of sections at 

c, and the expansion at e, together with Fig 4. — Riveted Pipe. 

the loss effect due to rivet heads are considerable, and it is a question 
whether a uniform pipe of diameter d would not show the greater capac- 
ity. The losses due to bends are seldom computed unless the curvature is 
considerable. Short curves, moderately sharp, are the best. 

Friction losses are deduced from experiments on pipe lines actually 
constructed, and from laboratory experiments. The former point to a safe 



* The loss of head due to curvature of pipe, as short curves, bends, etc., 
is but imperfectly understood. Theoretical formulas, as those of Weisbach, 
are not reliable. Roughly speaking, the loss of head in a short 90° bend 
may be assumed at 3 to 5 times what it would be in the same length of 
straight pipe. For a good discussion of this subject see Paper No 911 in 
Trans. Am. Soc. C. E., for April, 1902, by Gardner S. Williams, Clarence W. 
Hubbell and George H. Fenkell. 



FLOW IN PIPES— LOSSES OF HEAD— FRICTION, 1161 

precedent for similar designs, while the latter lead to the establishing of 
general laws of flow, in a relative sense. It would hardly be safe, however, 
to design large pipe lines based on formulas derived from laboratory experi- 
ments alone. Many physicists have attempted to formulate laws of flow 
and of friction based almost wholly on theory without considering the 
viscosity of the flowing liquid. It is perhaps needless to say that all such 
formulas, while interesting, are practically useless to the engineer, and they 
are positively dangerous to the young student. The class of formulas which 
the hydraulic engineer is using to-day, and will contiune to use, are founded 
on experimentation. Experiments rteveal to us that — 

(a) Friction F of water in pipes is proportional to some variable power of 
the velocity v, that is, F oc v^. In pipes of commercial size the values 
of X have been found to range from about x=1.7bto x=2, depending 
on the diameter of pipe and roughness of wetted surface, or perimeter. 

{h) Friction is proportional to the length of the pipe, so far as at present 
known. 

(c) Friction is less in curves of short radius, down to 2^ diameters, than in 
curves of long radius, for the same total angle of deflection. (See 
Trans. Am. Soc. C. E., Vol. XLVII, pages 183, 191.) 

{d) Friction is inversely proportional to some power of the diameter of the 
pipe or the hydraulic radius r of the conduit. The index of the 
power may be assumed as 1.2, within small limits of error. 

{e) Friction increases rapidly with the roughness of the wetted perimeter. 
A newly laid conduit with a smooth surface can often deteriorate so 
that the velocity of discharge will decrease from 2 to 5 per cent 
per annum for a number of years. 

Author's Hydraulic Formula. — ^The following general formula is based 
on the most reliable existing experimental data, and is applicable to the 
flow of water in pipes, sewers, flumes, etc. 

Notation: 

a = area of cross-section of water in pipe, flume, etc., in sq. ft.; 

?; = velocity of discharge in ft. per second {v = — 1 ; 

g = discharge in cubic ft. per second {q = a v); 
H = \oss of head in friction, etc., in ft. per 1000 ft. (i^= 1000 s); 
5== hydraulic slope or sine of angle of slope (5 =.001 H); 
^ = wetted perimeter of cross-section of pipe, etc., in ft.; 

•7-= hydraulic (mean) radius in ft. =— -. (For circular pipe ^==x) » 

ff = diameter of circular pipe in ft. (cf = 4r) ; 

c = coefficient of smoothness of wetted surface; also alinement; 

:*; = index of the power of the velocity, proportional to friction; 

y = index of the power of d or 4r, inversely proportional to friction. 

* As the velocity of discharge v increases with /^^^ t l^ 

the mean radius r, it is desirable to design the section ^ ' " ^^ 
of the conduit so that r will be a maximum for the 

area a, other things equal. The circular section is Fig. 5. 

ideal in this respect, and the value of r is equal to -r- 

whether the circular pipe is full, as in Fig. 5, or only 

half full, as in Fig. 6. When not full, but more than Fig. 7. 

half full ,r>-T-: r= 0.3043 (its maximum value) when _ 

the depth of water in the circular pipe is 0. 8 Id. When I ^^^ ^ ~ ""^z^Z ^^J 

the pipe is less than half full, ?"< -r- ■^^^- ^ shows the ^^ 

section of a flume, in which r has its maximum value •^^' °* 

when fei = 2^1. Practical considerations however usually call for a less pro- 
portionate depth. Fig. 8 is a section of a trapezoidal canal in which r is a 
maximum when 62= 2 (ic — 3;), or 62= 2cf2 (cosec — cot 6). In practice, canals 
and ditches are usually designed with such slopes as will probably be main- 
tained without wash during the flow of water. 




1162 Q2— HYDRAULICS, 

Formulas: 
Hyd. Radius, r. Diameter, d. 

General: ^ - h ' ^y "'h'W "> 

But X generally varies from 1 .75 to 2.00 — somewhat nearer the former value; 
and y varies usually from 1 . 15 to 1 .25. Assuming average values of x and y, we 

^"^^ « = 1.82; y = 1.2. 

1 t)l-82 1 1,1.82 

Hence, ■^= 7 " 3(4^ ^=7 ' FJi^ • ■<*> 

Equation (2) is the author's formula for loss of head in feet per 1000 feet. 
The various values of c, v, r, d, and H, raised to the proper powers, are given 
in the subjoined Tables 3, 4, 5, 6 and 7. Other forms of (2) which are useful 
in practice are the following: 

Hydraulic slope, .s= .001 H (3) 

Velocity. z;= (3c)0-« H^^^ (4r)o ee = (3c)0-55 h^.ss ^o.ee (4) 

Diameter. "^ ^ (3^)ol3 ' :^oli ' v'-^-^ (6) 

Hydraulic radius, *' "= T" • -- (6) 

^Coefficient. ^ = 3^(4^)1-2 ^ SHd^^ ^^ 

Note that 7/ = hydraulic slope mult, by 1000. 

Logarithmic Solution of Above Hydraulic Formulas: 

Equations (2) , (4) , (5) and (7) may also be solved by logarithmic formulas 
(2a), (4a), (5a) and (7a), following: 

Friction Head: Log /? = 1 .82 log z; - (1 .2 log 4r + log c + log 3) (2a) 

Log J? = 1.82 log V- (1.2 log (i + logc +log3) (2a) 

Velocity: Log z' =0.55 (log 3c +logH) +0.66 log 4r (4a) 

Log z; = 0.55 (log 3c + logH) +0.66 logi (4a) 

Diameter: Log J = 1.50 log i; -0.83 (log 3a + log i?) (5a) 

Log 4r = 1 .50 log z; - 0.83 (log 3c-\-\ogH) (5a) 

Coefficient: Log c = 1 . 82 log z; - (1 . 2 log 4^ + log 3 + log ff) . . . . (7a) 

Log c = 1.82 logzf- (1.2 log d + log3+ logH) (7a) 

* Formula (7) is used in determining the coefficient c of smoothness and 
alinement in pipe lines actually built and tested. The values given in Table 3 
are close averages from actual experiments. 



HYDRAULIC FORMULAS. 



1163 



c o 









2 io 
4) w 



e3 



CQ 



02 



^ " 






2 2^ 

In. w-> (^ 






222 



22 






as2 
a|a 



CI "^ 



8=? 



CO 6.*^ 53 



^6c 



O, 

Q, — 

o +3 o 






^1 



^ 



o-d S'g 



o a> 



CI 

o 

ft 

^ cQ H5 



Otfpq 



ftO 
ft Oj M 
ft^« 






Wh^ 






O 



1164 



62,^HYDRAULICS. 



-Values op v^-^ for Various Values op v 
In Author's Formula, preceding. 



Ve- 
loc- 
ity 

V 




Tenths. 


.0 


.1 


.2 


.3 


.4 


.5 


.6 


.7 


.8 


.9 



1 
2 
3 
4 


0.000 
1.00 
3.53 
7.39 
12.47 


0.015 
1.19 
3.86 
7.84 
13.04 


0.053 
1.39 
4.20 
8.31 
13.62 


0.112 
1.61 
4.55 
8.78 
14.22 


0.189 
1.84 
4.92 
9.27 
14.83 


0.283 
2.09 
5.30 
9.78 
15.45 


0.395 

2.35 

5.69 

10.29 

16.08 


0.523 

2.63 

6.10 

10.82 

16.72 


0.666 

2.91 

6.51 

11.36 

17.37 


0.826 

3.22 

6.94 

11.91 

18.04 


5 
6 
7 
8 
9 


18 71 
26.08 
34.52 
44.02 
54.54 


19.40 
26.87 
35.42 
45.02 
55.65 


20.10 
27.68 
36.34 
46.04 
56.77 


20.81 
28.50 
37.26 
47.07 
57.89 


21.53 
29.33 
38.19 
48.10 
59.03 


22.26 
30.16 
39.14 
49.15 
60.18 


23.00 
31.01 
40.09 
50.21 
61.34 


23.75 
31.87 
41.06 
51.28 
62.51 


24.52 
32.75 
42.03 
52.35 
63.69 


25.29 
33.63 
43.02 
53.44 
64.87 


10 


66.07 


67.28 


68.50 


69.72 


70.96 


72.21 


73.46 


74.73 


76.01 


77.29 




Ex.— 


3.1 1-82 = 


= 7.84; 


6.31.82 = 


= 28.50; 


91.82 = 54.54. 









5. — Values op *v^-^+ for Various Values of v 
In Author's Formula, preceding. 



Ve- 
loc- 
ity 






Tenths. 












.0 


.1 


.2 


.3 


.4 


.5 


.6 


.7 


.8 


.9 





0.000 


0.030 


0.087 


0.161 


0.249 


0.350 


0.461 


0.582 


0.713 


0.852 


1 


1.00 


1.16 


1.32 


1.49 


1.67 


1.85 


2.04 


2.24 


2.44 


2.65 


2 


2.86 


3.08 


3.31 


3.54 


3.77 


4.01 


4.26 


4.51 


4.77 


5.03 


3 


5.29 


5.56 


5.84 


6.12 


6.40 


6.69 


6.98 


7.27 


7 57 


7.88 


4 


8.19 


8.50 


8.82 


9.14 


9.46 


9.79 


10.12 


10.46 


10.80 


11.14 


5 


11.48 


11.83 


12.19 


12.55 


12.91 


13.27 


13.64 


14.01 


14.38 


14.76 


6 


15.14 


15.53 


15.92 


16.31 


16.70 


17.10 


17.50 


17.90 


18.31 


18.72 


7 


19.13 


19.55 


19.97 


20.39 


20.81 


21.24 


21.67 


22.11 


22.54 


22.98 


8 


23.43 


23.87 


24.32 


24.77 


25.22 


25.68 


26.14 


26.60 


27.07 


27.54 


9 


28.01 


28.48 


28.96 


29.43 


29.92 


30.40 


30.89 


31.38 


31.87 


32.36 


10 


32.86 


33.36 


33.87 


34.37 


34.87 


35.38 


35.90 


36.41 


36.93 


37.45 



*1.50+ = 1.82-J-1.2. 



HYDRAULIC FORMULAS, 



1165 



6. — ^Values of (4r)0-66, rfo.ee^ and -77-:, for Various Values op d 
In Author's Formula, preceding. 



Equiv. 


d 


^0.66 = 
(4/-)0-66 


1 


Equlv. 


d 


(4r)0.b6 


1 


d 


d0.66 = 

(4r)0.66 




Dia.in 
Ins. 


( = 4r) 
Ft. 


dl.2 


Dia. in 
Ins. 


( = 4r) 
Ft. 


d"r2 


(=4r) 

Ft. 


dl.2 


.25 


.0208 


.0777 


104.11 




1.45 


1.278 


.640 


10 


4.571 


.0631 


.375 


.0312 


.1015 


64.00 


18 


1.50 


1.307 


.615 


11 


4.868 


.0563 


.50 


.0417 


.1228 


45.32 




1.60 


1.364 


.569 


12 


5.155 


.0507 


.625 


.0521 


.1422 


34.67 


20 


1.667 


1.401 


.542 


13 


5.435 


.0461 


.75 


.0625 


.1604 


27.86 




1.70 


1.419 


.529 


14 


5.708 


.0421 


.875 


.0729 


.1776 


23.15 




1.80 


1.474 


.494 


15 


5.973 


.0388 


1.00 


.0833 


.1940 


19.73 




1.90 


1.528 


.463 


16 


6.233 


.0359 


1.25 


.1042 


.2248 


15.09 


24 


2.00 


1.580 


.435 


17 


6.488 


.0334 


1.375 


.1146 


.2393 


13.46 




2.10 


1.632 


.411 


18 


6.737 


.0312 


1.50 


.1250 


.2535 


12.13 




2.20 


1.683 


.388 


19 


6.982 


.0292 


1.75 


.1458 


.2806 


10.08 




2.30 


1.733 


.368 


20 


7.222 


.02746 


2.00 


.1667 


.3065 


8.586 




2.40 


1.782 


.350 


21 


7.459 


.02591 


2.25 


.1875 


.3305 


7.454 


30 


2.50 


1.831 


.333 


22 


7.691 


.02450 


2.50 


.2083 


.3551 


6.569 




2.60 


1.879 


.3177 


23 


7.920 


.02322 


2.75 


.2292 


.3782 


5.859 




2.70 


1.926 


.3036 


24 


8.146 


.02207 


3. 


.2500 


.4005 


5.278 




2.80 


1.973 


.2907 


25 


8.368 


.02101 


3.5 


.2917 


.4434 


4.387 




2.90 


2.019 


.2787 


26 


8.588 


.02007 


4. 


.3333 


.4843 


3.737 


36 


3.00 


2.065 


.2676 


27 


8.804 


.01916 


6. 


.4167 


.5611 


2.859 




3.20 


2.155 


.2476 


28 


9.018 


.01834 


6. 


.5000 


.6329 


2.297 




3.40 


2.243 


.2303 


29 


9.230 


.01758 


7. 


.5833 


.7007 


1.909 


42 


3.50 


2.286 


.2224 


30 


9.438 


.01688 


8. 


.6667 


.7652 


1.627 




3.75 


2.393 


.2047 


35 


10.45 


.01403 


9. 


.7500 


.8271 


1.412 


48 


4.0 


2.497 


.1895 


40 


11.41 


.01195 


10. 


.8333 


.8866 


1.245 


54 


4.5 


2.698 


.1645 


45 


12.33 


.01038 


11. 


.9167 


.9442 


1.110 


60 


5.0 


2.893 


.1450 


50 


13.22 


.00915 


12. 


1.0000 


1.000 


1.000 


66 


5.5 


3.081 


.1293 


55 


14.08 


.00816 


13. 


1.083 


1.054 


.908 


72 


6.0 


3.263 


.1165 


60 


14.91 


.00735 




1.150 


1.097 


.846 


78 


6.5 


3.440 


.1058 


65 


15.72 


.00668 


14- 


1.167 


1.107 


.831 


84 


7.0 


3.612 


.0968 


70 


16.51 


.00611 




1.200 


1.128 


.803 


90 


7.5 


3.780 


.0891 


75 


17.28 


.00562 


15. 


1.250 


1.159 


.765 


96 


8.0 


3.945 


.0825 


80 


18.03 


.00520 




1.300 


1.189 


.730 


102 


8.5 


4.106 


.0770 


85 


18.77 


.00484 


16. 


1.333 


1.209 


.708 


108 


9.0 


4.264 


.0716 


90 


19.49 


.00452 




1.350 


1.219 


.698 


114 


9.5 


4.419 


.0674 


95 


20.20 


.00423 




1.400 


1.249 


.668 


120 


10.0 


4.571 


.0631 


100 


20.89 


.00398 



Ex.— For a pipe 18'' dia., (i= 1.6, r= 1.5^4, rfo-66= 1.307, l-J-cfi-2=0.615. 



1166 



62.— HYDRAULICS. 



7. — Values op H^-^ and 



FOR Various Values op H 



In Author's Formula, preceding. 



"" 
























..S- 


HO.K 


1 


K- "=> •♦3 


HO.bB 


1 


I 

^ o • 


H0.55 


1 




^0.55 


1 


wsS 


H0.83 + 


^2S 


H0-83 + 




H0.83 + 


I 


H0.S3+ 


.000 


.000 


00 


.10 


.282 


6.81 


1.0 


1.000 


1.000 


10 


3.55 


.147 


.002 


.033 


177.5 


.11 


.297 


6.29 


1.1 


1.054 


0.924 


11 


3.74 


.136 


.004 


.048 


99.6 


.12 


.312 


5.88 


1.2 


1.105 


.859 


12 


3.92 


.126 


.006 


.060 


71.0 


.13 


.326 


5.48 


1.3 


1.155 


.804 


13 


4.10 


.118 


.008 


.070 


55.9 


.14 


.339 


5.15 


1.4 


1.203 


.755 


14 


4.27 


.111 


.010 


.079 


46.4 


.15 


.352 


4.86 


1.5 


1.250 


.713 


15 


4.44 


.105 


.012 


.088 


39.9 


.16 


.365 


4.60 


1.6 


1.295 


.676 


16 


4.59 


.0992 


.014 


.096 


35,1 


.17 


.377 


4.38 


1,7 


1.339 


.643 


17 


4.76 


.0943 


.016 


.103 


31.4 


.18 


.389 


4.17 


1.8 


1.382 


.613 


18 


4.90 


.0899 


.018 


.110 


28.4 


.19 


.401 


3.99 


1.9 


1.423 


.586 


19 


5.05 


.0860 


.020 


.116 


26.05 


.20 


.413 


3.82 


2.0 


1.464 


.561 


20 


5.19 


.0824 


.022 


.123 


24.06 


.22 


.435 


3.53 


2.2 


1.543 


.518 


22 


5.48 


.0760 


.024 


.129 


22.38 


.24 


.456 


3.28 


2.4 


1.619 


.482 


24 


5.74 


.0708 


.026 


.134 


20.93 


.26 


.477 


3.07 


2.6 


1.691 


.451 


26 


6.00 


.0662 


.028 


.140 


19.68 


.28 


.497 


2.89 


2.8 


1.762 


.424 


28 


6.25 


.0622 


.030 


.145 


18.58 


.30 


.516 


2.73 


3.0 


1.830 


.400 


30 


6.49 


.0588 


.032 


.151 


17.62 


.32 


.534 


2.58 


3.2 


1.896 


.379 


32 


6.73 


.0557 


.034 


.156 


16.74 


.34 


.552 


2.46 


3.4 


1.960 


.361 


34 


6.96 


.0529 


.036 


.161 


15.96 


.36 


.570 


2.34 


3.6 


2.023 


.344 


36 


7.18 


.0505 


.038 


.166 


15.26 


.38 


.587 


2.24 


3.8 


2.084 


.329 


38 


7.39 


.0483 


.040 


.170 


14.62 


.40 


.604 


2.15 


4.0 


2.144 


.315 


40 


7.61 


.0463 


.042 


.175 


14.04 


.42 


.621 


2.06 


4.2 


2.202 


.302 


42 


7.81 


.0444 


.044 


.179 


13.50 


.44 


.637 


1.98 


4.4 


2.259 


.291 


44 


8.02 


.0427 


.046 


.184 


13.01 


.46 


.652 


1.91 


4.6 


2.315 


.280 


46 


8.21 


.0412 


.048 


.188 


12.56 


.48 


.668 


1.84 


4.8 


2.370 


.271 


48 


8.41 


.0398 


.050 


.193 


12.14 


.50 


.688 


1.78 


5.0 


2.42 


.262 


50 


8.60 


.0385 


.055 


.203 


11.21 


.55 


.720 


1.65 


5.5 


2.55 


.242 


55 


9.06 


.0355 


.060 


.213 


10.43 


.60 


.755 


1.53 


6.0 


2.68 


.225 


60 


9.51 


.0330 


.065 


.222 


9.76 


.65 


.789 


1.43 


6.5 


2.80 


.210 


65 


9.93 


.0308 


.070 


.232 


9.17 


.70 


.822 


1.35 


7.0 


2.92 


.198 


70 


10.35 


.0290 


.075 


.241 


8.66 


.75 


.854 


1.27 


7.5 


3.03 


.187 


75 


10.75 


.0274 


.080 


.249 


8.20 


.80 


.885 


1.20 


8.0 


3.14 


.177 


80 


11.14 


.0259 


.085 


.258 


7.80 


.85 


.914 


1.15 


8.5 


3.25 


.168 


85 


11.51 


.0247 


.090 


.266 


7.44 


.90 


.944 


1.09 


9.0 


3.35 


.160 


90 


11.88 


.0235 


.095 


.274 


7.11 


.95 


.972 


1.04 


9.5 


3.45 


.153 


95 


12.24 


.0225 


.100 


.282 


6.81 


1.00 


1.000 


1.00 


10.0 


3.55 


.147 


100 


12.59 


.0215 



Ex.— For hydraulic slope 5 = .004, H = i.O, H0-«= 2.144, and 



HO.M 



CHEZY'S FORMULA, KUTTER'S FORMULA, 1167 

Problems in Use of Author's Hydraulic Formula, 
(See page 1161, etc.) 

Problem 1. — What loss of head per 1000 ft. would occur in an ordinary 
cast iron pipe line 20 ins. in diameter, if the velocity is 3 ft. per second? 

Solution. — Use formula (2); then — , from Table 3, =1.50; v^'^, from 

Table 4, =7.39; -~, from Table 6, =0.542; hence, i7= 1.50X7.39X iX 

0.542 = 2.00 ft. per 1000 ft. Ans. 

Problem 2. — "What velocity of discharge would be expected in an un- 
clean, egg-shaped, brick sewer, whose hydraulic radius is 0.75, and hydraulic 
slope 0.00125? 

Solution.— Use formula (4); then from the Tables, (3c)0-«= 1.29; 
H0'55= 1.250 55= 1.13: (4r)0-66 = (i0-66= 30-66= 2.065; hence, i;= 1.29X 1.13X 2.07 
= 3.02 ft. per sec. Ans. 

Problem 3. — What diameter of riveted-steel pipe would be required to 
discharge 2.4 cu. ft. per second, after it has been in use about 12 years, if 
the hydraulic slope is 0.01225? 

Solution. — Use formula (4) by trial method — assuming d, obtaining v, 
and then q ( = a z;= 2.4). First, assume pipe to be 12 ins. in diameter; then 
ti=l, ^=1. 29X3.97X1 = 5.12; <7 = a7; = 0.7854X5.04 = 4.02; hence, as 4.02 
> 2.4, a 12-in. pipe is too large. Second, assume pipe to be 10 ins. in diameter; 
then ci = 0.833; 7^=1.29X3.97X0.887 = 4.54; g = az; = 0. 5454 X 4.54 = 2.48, 
the required discharge being 2.4; hence a 10-in. pipe is required. Ans, 

Problem 4. — A conduit whose hydraulic radius is 0.875, and whose 
hydraulic slope is .001375 has a velocity of discharge of 4.5 ft. per second. 
What is the coefficient c of smoothness and alinement of the conduit? 

Solution. — Use formula (7), in which i; = 4.5, H= 1.375, and 4r=3.5; 

15 45X0 2224 
then by use of accompanying Tables, c= — ' ^ „-, — = 0.83, which places 

it under Class C in Table 3. 

Chezy's Hydraulic Formula assumes the velocity to be proportional to 
the square root of both the hydraulic radius and the hydraulic slope; or 

v = cV7s (1) 

in which c is a coefficient to be determined by experiment. (See Kutter's 
formula, following.) 

Kutter's Formula, so-called, is really the Chezy formula (preceding), but 

with values of the coefficient c supplied so as to give it a general application. 
These values of c were deduced from experiments made by Ganguillet and 
Kutter, and were primarily intended to apply to the flow of water in streams 
and canals. They have, however, received general application, to a greater 
or less extent, in the design of water mains, sewers, flumes, ditches, etc. 
The value of the coefficient in English* measure is as follows: 



Mil +41.65+^55281 



(2) 



l+_^(4i.65+:5??81\ 

in which r and s are the mean radius and the slope, respectively, as explained 



* In metric measure, 



(3) 



1 + 






1168 62.— HYDRAULICS, 

on page 1161; and « is a coefficient depending upon the rotighness of the 

Wetted surface of the conduit, sewer, canal, or stream. Thus, 
« = .009 for well-planned timber, perfectly aligned.* 
= .010 for neat (pure) cement; glazed or enameled surfaces generally; 

smooth cast iron or iron pipes; planed timber.* 
= .011 for cement with one-third sand, in good condition; well-jointed 

pipes of iront, cement, and terra cotta. 
= .012 for unplaned timber in good alinement, as flumes. 

■=.013 for ashlar and brickwork, well laid; ordinary metal; earthen-, 
cement-, stoneware- and terra cotta pipe not well pointed nor in 
first-class order; cement plaster and planed timber in second-class 
condition; generally, the materials for w = .010 when imperfect in 
quality or condition. 

= .015 for unclean surfaces in pipes and sewers; second-class or rough 
brickwork; stonework, well dressed; iron-, stoneware- and terra cotta 
pipes with imperfect joints and in bad condition. 

= .017 for rubble masonry in good order; brickwork, stoneware and 
ashlar in poor condition; generally, the materials for w = . 013 when 
imperfect in quality or condition; tuberculated iron pipes. 

= .020 for canals in very firm gravel, and carefully trimmed ; inferior 
rubble in cement; coarse, dry rubble. 

= .0225 for coarse, dry rubble in bad condition. 

= .025 for canals and rivers free from stones and weeds, and in good 
order. 

= .030 for canals and rivers having some stones and weeds. 
= .035 for canals and rivers in bad order, with great quantities of stones 
and weeds. 

= .040 for rivers in extremely bad condition. 

These values of n are to be inserted in equation (2) for finding the value 
of c used in equation (1). The correct use of Kutter's (or Chezy's) formula 
depends upon the proper selection of the values of n for the roughness of the 
wetted surface. The values n==.010 to n=.016 will cover all conditions for 
good water mains — the lower values for the smooth pipes, and the higher 
values for the pipes with rough surfaces. (See, also, page 1188.) 

In order to simplify the calculations of the coefficients c in Kutter's 
formula, equations (1) and (2), the following reductions of form of equation 
(2) are given for various values of roughness n. It remains only to substitute 
the proper values of — 

r, the hydraulic radius of the pipe, sewer or canal, in ft.; 

dt the diameter of the circular pipe, in ft.; 

St the average slope of the pipe, or rate of fall of the hydraulic grade line. 

Using Hyd. Rad r. Using Diameter d, 

u. AAo \^(242.872 5+. 00281) \/7(242.872 5-h. 00281) ,„ 

Forn = .009, c = ;=: = —i^ .(4) 

5 (Vr-h. 37485)4-. 0000253 5 (V7-h. 7497) H-. 0000506 

For n= 010 c = ^^(^^^-^g 5+ .00281) \/7(222.75 s+ .p0281) 

5(V7+. 4165) + .0000281 5(VJ+ .833) + .0000562 * 

17 niAK \/7(21 4.135+. 002 81) V^(214.135+. 00281) ,^, 

For « = .0105, c = -—" = -zr ^.(6) 

5 (V7 + . 437325) + .0000295 ^(Vci +.87465) + . 000059 

For«=011 v^(206.29 5+. 00281) \/7(206.29 5+. 00281) 

5(V7+. 45815)+. 0000309 5(V7+. 9163)+. 0000618 * 



* The values « = .010 and » = .0105 are often used for wood-stave pipe 
well laid, with dressed timber. 

f The values « = .014 and « = .016 are often used for lap- jointed steel- 
riveted pipe. For foul and tuberculated iron pipe, use w = .016+. 



KUTTER'S FORMULA— VALUES OF N. 1169 

Using Hyd. Rad r. Using Diameter d, 

\/7(192.57 5+. 00281) V^(192.57 5+ .00281) ,„ 

For» = .012, c =- 7=^ — = ;= ^.(8) 

s(Vr +.4998) + .0000337 5 (V^+. 9996) +.0000674 

_ ^^7(180.96 5+.00281) _ V7(180.96 5+. 00281) 
Forn-. 1 . ^-^(^7_j_ 54i45)^_ QQ00365 5 (V7+ 1.0829) + .0000 7 30* 

V7(171.0l5 +. 00281) V 7(171. 01 5+. 0281) ,,^, 

For n = . 014, c = 7= = ;= (10) 

siVr +.5831) + .0000393 5 (\/(i+ 1.1662) + .0000786 

\/7(162.385+. 00281) \/7(162.38 5+.00281) ,,,, 

For» = .015, c = ' 7ZZ ==- — 7= -(11) 

5(Vr+. 62475) + . 0000422 5(V7+1.2495)+. 0000844 

\/7(148.18 5+.00281) \/ 7(148.18 5+. 0281) ,,., 

Forn = .017, c = ^ — = 7=^ ^(12) 

s(Vr +.70805) + .0000478 ^(v^+1.4161) + .0000956 

\/7(132.205+. 00281) V7( 132,20s +.00281) ,,^, 

Forn = .020. c = ^ — = 7=:^ ^(13) 

siVr +.833) + .0000562 s(V^+l,666)+.0001124 

««or V7(122.14s+. 00281) V7(122.14 5+. 00281) ,,,, 

Forn = .0225,c = ^ —=- ^ ^(14) 

siVr +.93713)+. 0000632 sCVci +1.87426) +.0001264 

V7(114.09 5+. 00281) V7(114.09 5+. 00281) ,,,, 

For« = .025, c = 7== = 7= -(15) 

5(V7+1.04125)+. 0000703 s(V7+2.0825)+. 0001406 

^^(102.02 5+. 00281) \/7(102.02s+. 00281) ,,,, 

Forn = .030, c = -^zzz == -=z -(16) 

s(Vr +1.2495) + .0000843 s(V7+ 2. 499) +.0001686 

_ ^^^ VT(93.39 5+ . 00281) v^(93. 39 s+. 00281) ,,^ 

Forn = .035, c = 7= = 7= (17) 

s(Vr +1.45775)+. 0000984 sCVti +2.9155) +.0001968 

-- ^.^ \/ 7(86.925 5+.00281) V7(86.925s+. 00281) ,,^, 

For» = .040, c = — -;= = -z=^ ^(18) 

5(V7+ 1.666) + .0001124 s(Vd + 3. 332) +.0002248 



1170 



^2.— HYDRAULICS. 



Coefficients c in Kutter's formula, for various values of roughness «, 
mean radius r, and slope s, are given in Table 8, following. Intermediate 
values of c, for values of n, r and 5 not given in the Table, may be obtained 
by interpolation, by simple proportion. In cases where the slope 5 is greater 
than 5=.01, section 7, the values of c as deduced from section 7 may be 
considered sufficiently accurate — a little too large for values of r greater than 
3.28 (1 meter), and a little too small for values of r less than 3.28. Note 
that the value of c is independent of the slope 5 in all cases where the mean 
radius 7-= 3.28 (1 meter) ; and for this reason the "one meter" line is shown 
in Italics in each of the 7 sections. Above the one-meter line the values of c 
increase with the slope s\ below, they decrease. 

For explanation of the use of Kutter's formula and Table 8, see Practical 
Examples following the table. 



8. — Coefficients c in Kutter's Formula, v==cVrs, English Measure 
For various values of s, r and n. 



Hyd. 
Rad.r 


Coefficients n of Roughness. 


Ft. 


.0091.010 1.01051 .0111 .012 1 .013|.014| .015| .017 | .020|.0225| .025| .030|.035| .040 




(1) For Slope s = .000025 = 1 in 40,000 = .132 ft. per mUe. 


.1 


66.0 


57.1 


53.4 


50.1 


44.6 


40.0 


36.3 


33.1 


28.1 


?;2.8 


19.6 


17.2 


13.7 


11.4 


9.7 


.2 


86.6 


75.4 


70.7 


66.6 


59.4 


53.6 


48.7 


44.5 


38.0 


31.0 


26.8 


23.6 


18.9 


15.8 


13.5 


.3 


100.6 


87.9 


82.6 


77.9 


69.7 


63.0 


57.4 


52.6 


45.1 


36.9 


32.0 


28.2 


22.7 


19.0 


16.3 


.4 


111.3 


97.5 


91.8 


84.9 


76.2 


68.9 


62.9 


59.0 


50.7 


41.7 


36.2 


31.9 


25.8 


21.6 


18.6 


.6 


127.3 


112.1 


105.7 


100.0 


90.1 


81.9 


74.9 


68.9 


59.5 


49.1 


42.9 


37.9 


30.8 


25.8 


22.3 


.8 


139.3 


123.1 


116.3 


110.1 


99.5 


90.6 


83.1 


76.6 


66.3 


55.0 


48.1 


42.7 


34.8 


29.3 


25.3 


1 


148.8 


131.9 


124.8 


118.3 


107.1 


97.7 


89.8 


82.9 


72.0 


59.9 


52.5 


46.7 


38.1 


32.2 


27.8 


1.5 


166.6 


148.4 


140.7 


133.7 


121.6 


111.4 


102.7 


95.1 


83.0 


69.6 


61.2 


54.6 


44.2 


38.1 


33.0 


2 


179.4 


160.4 


152.3 


145.0 


132.2 


121.5 


112.3 


104.3 


91.3 


77.0 


68.0 


60.8 


52.0 


42.7 


37.2 


3 


196.0 


177.4 


168.9 


161.1 


147.6 


136.1 


126.3 


117.7 


103.7 


88.0 


78.2 


70.2 


58.5 


50.0 


43.7 


3.28 


201.2 


181.1 


112.5 


164.6 


151.0 


139.3 


129.4 


120.7 


106.5 


90.6 


80.5 


72.4 


60.4 


51.7 


45.3 


4 


209.8 


189.3 


180.5 


172.5 


158.5 


146.6 


136.4 


127.4 


112.8 


96.3 


85.8 


77.4 


64.8 


55.7 


48.8 


6 


226.8 


205.7 


196.7 


188.4 


173.8 


161.4 


150.8 


141.3 


125.9 


108.3 


97.1 


88.0 


74.3 


64.3 


56.7 


8 


238 4 


217.0 


207.7 


199.3 


184.5 


171.8 


160.9 


151.2 


135.3 


117.1 


105.4 


95.9 


81.4 


70.8 


62.7 


10 


247.0 


225.4 


216.0 


207.5 


192.5 


179.7 


168.5 


158.7 


142.5 


123.9 


111.9 


102.1 


87.1 


76.1 


67.6 


12 


253.7 


232.0 


222.6 


214.0 


198.9 


185.9 


174.5 


164.8 


148.4 


129.5 


117.3 


107.2 


91.9 


80.5 


71.7 


16 


262.3 


242.0 


232.5 


223.9 


208.6 


195.5 


184.2 


174.1 


157.4 


138.2 


125.7 


115.4 


99.5 


87.6 


78.5 


20 


271.2 


249.3 


239.8 


231.1 


215.8 


202.7 


191.2 


181.1 


164.3 


144.8 


132.1 


121.7 


105.4 


93.3 


83.8 


30 


283.5 


261.6 


252 1 


243.4 


228.0 


214.9 


203.4 


193.2 


176.3 


156.5 


143.7 


133.0 


116.3 


103.7 


93.8 


50 


297.0 


275.2 


265.7 


257.1 


241.8 


228.6 


217.2 


207.1 


190.1 


170.4 


157.4 


146.6 


129.7 


116.7 


106.5 


75 


30C.2 


284.5 


275.1 


266.5 


251.3 


238.3 


226.9 


216.9 


200.1 


180.4 


167.5 


156.7 


139.8 


126.8 


116.5 


100 


312.0 


290.4 


281.1 


272.5 


257.4 


244.4 


233.2 


223.2 


206.5 


187.0 


174.2 


163.5 


146.6 


133.7 


123.3 




<2) For Slope s = .00005 = 1 in 20,000 = .264 ft. per mUe. 


.1 


79.0 


68.1 


63.6 


59.6 


52.8 


47.2 


42.6 


38.7 


32.6 


26.2 


22.4 


19.5 


15.4 


12.6 


10.7 


.2 


100.7 


87.5 


82.0 


77.1 


68.6 


61.7 


55.9 


51.0 


43.3 


35.0 


30.1 


26.3 


20.9 


17.3 


14.7 


.3 


114.7 


100.1 


94.0 


88.5 


79.2 


71.4 


64.9 


59.4 


50.6 


41.2 


35.5 


30.5 


24.9 


20.6 


17.6 


.4 


125.0 


109.5 


103.0 


97.2 


87.1 


78.8 


71.8 


65.8 


56.3 


46.0 


39.8 


35.0 


27.5 


22.8 


19.5 


.6 


139.9 


123.3 


116.2 


109.9 


98.9 


89.8 


82.1 


75.5 


64.9 


53.4 


46.4 


40.9 


33.0 


27.6 


23.6 


.8 


150.7 


133.2 


125 8 


119.1 


107.6 


97.9 


89.8 


82.7 


71.5 


59.1 


51.5 


45.6 


37.0 


31.0 


26.6 


1 


159.0 


141.0 


133.3 


126.4 


114.4 


104.4 


95.9 


88.5 


76.7 


63.7 


55.7 


49.4 


40.2 


33.8 


29.1 


1.5 


174.0 


155.1 


147.0 


139.7 


127.0 


116.4 


107.3 


99.4 


86.6 


72.5 


63.7 


56.8 


46.6 


39.4 


34.1 


2 


184.3 


164 9 


156.6 


149.1 


135.9 


124.9 


115.5 


107.2 


93.9 


79.0 


69.8 


62.4 


51.4 


43.7 


38.0 


3 


198.3 


178.3 


169.7 


161.9 


148.3 


136.8 


126.9 


118.3 


104.2 


88.5 


78.5 


70.6 


58.7 


50.2 


43.9 


3.28 


201.2 


181.1 


112.5 


164.6 


151.0 


139.3 


129.4 


120.7 


106.5 


90.6 


80.5 


72.4 


60.4 


51.7 


45.3 


4 


207.6 


187.3 


178.6 


m.7 


156.8 


145.0 


134.9 


126.0 


111.6 


95.2 


84.9 


76.6 


64.1 


55.0 


48.4 


6 


220.0 


199.3 


190.5 


182.4 


168.2 


156.1 


145.8 


136.6 


121.7 


104.7 


93.9 


85.2 


72.0 


62.4 


55.1 


8 


228.1 


207.3 


198.3 


190.2 


175.8 


163.6 


153.1 


143.9 


128.7 


111.4 


100.3 


91.3 


77.6 


67.7 


60.0 


10 


233.9 


213.0 


204.1 


195.9 


181.4 


169.1 


158.6 


149.3 


133.9 


116.4 


105.1 


96.0 


82.1 


71.8 


64.0 


12 


238.4 


217.5 


208.5 


200.3 


185.8 


173.5 


162.9 


153.5 


138.1 


120.4 


109.0 


99.8 


85.6 


75.2 


67.2 


16 


245.1 


224.1 


215.1 


206.8 


192.3 


180.0 


169.3 


159.9 


144.3 


126.5 


115.0 


105.7 


91.3 


80.6 


72.3 


20 


249.9 


228.9 


219.8 


211.6 


197.1 


184.7 


174.0 


164.5 


149.0 


131.1 


119.5 


110.1 


95.5 


84.7 


76.3 


30 


257.6 


236.7 


227.6 


219.4 


204.9 


192.5 


181.8 


172.4 


156.8 


138.8 


127.2 


117.7 


103.0 


92.0 


83.5 


50 


266.0 


245.0 


236.0 


227.8 


213.4 


201.0 


190.4 


181.0 


165.4 


147.6 


136.0 


126.5 


111.8 


100.8 


92.1 


75 


271.5 


250.6 


241.7 


233.5 


219.1 


206.8 


196.2 


186.9 


171.4 


153.7 


142.2 


132.8 


118.1 


107.2 


98.6 


100 


274.9 


254.1 


245.1 

1 


237.0 


222.6 


210.4 


199.9 


190.6 


175.2 


157.6 


146.2 


136.8 


122.3 


111.4 


102.6 



KUTTERS FORMULA— VALUES OF Go 



1171 



8.— COEPPICIENTS 


C IN 


Kutter's Formula, English Measure — Cont'd. 


Hyd. 
Rad.r 


Coefficients n of Roughness. 


Ft. 


.009 ( .0101.01051 .011 1 .0121 .0131 .0141 . 015 | .017 | .020 |.0225| .025| .030| .035 1 .040 




(3) For Slope s = .0001 = 1 in 10,000 = .528 It. per mile. 


.1 


90.8 


78.2 


73.1 


68.4 


60.5 


54.1 


48.7 


44.2 


37.1 


29.6 


25.2 


21.8 


17.1 


13.9 


11.7 


.2 


112.7 


98.0 


91.8 


86.3 


76.9 


69.1 


62.6 


57. C 


48.3 


38.9 


33.3 


29.0 


22.9 


18.8 


15.9 


.3 


126.2 


110.3 


102.8 


97.6 


87.3 


78.7 


71.6 


65.4 


55.7 


45.2 


38.9 


34.0 


27.0 


22.3 


18.9 


.4 


135.2 


119.3 


112.3 


105.9 


95.0 


85.9 


78.3 


71.7 


60.8 


50.0 


43.2 


37.8 


30.2 


25.0 


21.3 


.6 


149.7 


132.0 


124.5 


117.8 


106.1 


96.3 


88.1 


81.0 


69.6 


57.2 


49.7 


43.8 


35.2 


29.3 


25.0 


.8 


159.2 


140.9 


133.2 


126.2 


114.0 


103.8 


95.2 


87.8 


75.8 


62.6 


54.5 


48.2 


39.0 


32.6 


27.9 


1 


166.5 


147.8 


139.8 


132.6 


120.1 


109.7 


100.8 


93.1 


80.6 


66.9 


58.5 


51.8 


42.1 


35.3 


30.4 


1.5 


179.1 


159.9 


151.6 


144.1 


131.1 


120.1 


110.8 


102.7 


89.6 


74.9 


65.9 


58.7 


48.0 


40.6 


35 1 


2 


187.7 


168.0 


159.6 


152.0 


138.6 


127.4 


117.8 


109.5 


95.9 


80.7 


71.2 


63.7 


52.5 


44.6 


38,7 


3 


198.9 


178.8 


170.2 


162.4 


148.8 


137.2 


127.3 


118.7 


104.6 


88.8 


78.8 


70.8 


58..9 


50.4 


44.1 


3.28 


201.2 


181.1 


172.5 


164.6 


151.0 


139.3129.4 


120.7 


106.5 


90.6 


80.5 


72.4 


60.4 


51.7 


45 3 


4 


206.2 


186.0 


lll.Z 


169.4 


155.6 


143.9 


133.8 


125.0 


110.7 


94.4 


84.2 


76.0 


63.6 


54.7 


48.0 


6 


215.7 


195.3 


186.5 


178.5 


164.5 


152.6 


142.4 


133.5 


118.8 


102.1 


91.6 


83.1 


70.2 


60 8 


53.8 


8 


221.7 


201.2 


192.4 


184.4 


170.3 


158.3 


148.0 


139.0 


124.2 


107.4 


96 6 


88.0 


74.8 


65.2 


57.9 


10 


226.1 


205.5 


196.7 


188.6 


174.5 


162.5 


152.1 


143.1 


128.2 


111.2 


100.4 


91.6 


78.3 


68.6 


61.1 


12 


229.4 


208.8 


200.0 


191.9 


177.7 


165.7 


155.3 


146.3 


131.3 


114.3 


103.4 


94.6 


81.1 


71.3 


63.7 


16 


234.2 


213.6 


204.7 


196.7 


182.5 


170.4 


160.1 


151.0 


136.0 


118.8 


107.9 


99.0 


85.4 


75.4 


67.8 


20 


237.6 


217.0 


208.1 


200.1 


185.9 


173.8 


163.4 


154.4 


138.4 


122.2 


111.2 


102.3 


88.6 


78.0 


70.8 


30 


243.1 


222.5 


213.7 


205.6 


191.4 


179.4 


169.0 


159.9 


144.9 


127.8 


116.8 


107.8 


94.2 


84.0 


76.2 


50 


248.9 


228.3 


219.5 


211.4 


197.3 


185.3 


175.0 


165.9 


151.0 


133.9 


123.0 


114.1 


100.4 


90.3 


82.5 


75 


252.7 


232.2 


223.3 


215.3 


201.2189.3 


178.9 
181.4 


169.9 


155.0 


138.1 


127.2 


118.4 


104.8 


94.8 


87.0 


100 


255.0 


234.5 


225.7 


217.7 


203.6191.7 


172.4 


157.6 


140.7 


129.9 


121.1 


107.6 


97.6 


89.9 




(4) For Slope s = .0002 = 1 in 5.000 = 1.056 ft. per mUe. 


.1 


99.4 


85.8 


80.1 


75.0 


66.4 


59.3 


53.4 


48.4 


40.6 


32.3 


27.4 


23.7 


18.5 


15.0 


12.6 


.2 


121.1 


105.5 


98.9 


93.0 


82.8 


74.5 


67.5 


61.5 


52.0 


41.9 


35.8 


31.1 


24.5 


20.0 


16.9 


.3 


134.1 


117.5 


110.4 


104.0 


93.1 


84.0 


76.4 


69.8 


59.4 


48.2 


41.4 


36.2 


28.7 


23.6 


19.9 


.4 


143.3 


125.5 


118.6 


111.9 


100.5 


90.9 


82.5 


76.0 


65.0 


53.0 


45.4 


40.0 


31.9 


26.3 


22.3 


.6 


156.0 


137.7 


130.0 


123.0 


110.9 


100.8 


92.2 


84.9 


73.0 


60.0 


52.0 


45.8 


36. rf 


30.5 


26.0 


.8 


164.6 


145.9 


138.0 


130.8 


118.3 


107.8 


98.9 


91.2 


78.8 


65.1 


56.7 


50.1 


40.5 


33.8 


28 9 


1 


171.1 


152.1 


144.0 


136.6 


123.9 


113.1 


104.0 


96.1 


83.3 


69.2 


60.4 


53.6 


43.5 


36 4 


31.3 


1.5 


182.3 


162.8 


154.4 


146.9 


133. 9 


122.6 


113.1 


104.9 


91.5 


76.6 


67.3 


60.0 


49.1 


41.5 


35.8 


2 


189.7 


169.9 


161.4 


153.7 


140.3 


129.0 


121.6 


110.9 


97.0 


81.8 


72.2 


64.6 


53.2 


45.2 


39.2 


3 


199.2 


179.2 


170.6 


162.8 


149.1 


137.5 


127.6 


119.0 


104.9 


89.0 


79.0 


71.0 


59.1 


50.5 


44.2 


3.28 


201.2 


181.1 


112.5 


164.6 


151.0 


139.3 


129.4 


120.7 


106.5 


90.6 


80.5 


72.4 


60.4 


51.7 


45.3 


4 


205.4 


185.2 


176.6 


168.7 


154.9 


143.2 


133.1 


124.4 


110.1 


93.9 


83.7 


75.5 


63.2 


54.4 


47.8 


6 


213.3 


192.9 


184.2 


176.3 


162.3 


150.5 


140.4 


131.5 


117.0 


100.5 


90.1 


81.7 


69.0 


59.7 


52.9 


8 


218.2 


197.8 


189.1 


181.1 


167.1 


155.2 


145.1 


136.2 


121.5 


104.9 


94.4 


85.9 


73.0 


63.6 


56.5 


10 


221.8 


201.3 


192.6 


184.6 


170.6 


158.7 


148.4 


139.5 


124.8 


108.1 


97.5 


89.0 


75.9 


66.5 


59.2 


12 


224.4 


204.0 


195.2 


187 2 


173.2 


161.3 


151.1 


142.1 


127.4 


110.7 


100.0 


91.4 


78.3 


68.7 


61 5 


16 


228.3 


207.9 


199.1 


191.1 


177.0 


165.1 


154.9 


145.9 


131.2 


114.4 


103.7 


95.0 


81.9 


72.2 


64.9 


20 


231.0 


210.6 


201.8 


193.8 


179.8 


167.8 


157.6 


148.7 


133.9 


117.2 


106.4 


97.7 


84.5 


74.8 


67.4 


30 


235.4 


214.9 


206.2 


198.2 


184.2 


172.2 


162.0 


153.1 


138.3 


121.5 


110.8 


101.7 


88.9 


79.2 


71.8 


50 


239.9 


219.5 


210.7 


202.8 


188.8 


176.9 


166.7 


157.8 


143.1 


126.3 


115.7 


107.1 


93.9 


84.2 


76.8 


75 


242.9 


222.5 


213.7 


205.8 


191.8 


180.0 


169.8 


160.9 


146.2 


129.6 


119.0 


110.4 


97.3 


87.7 


80.3 


100 


244.7 


224.3 


215.6 


207.6 


193.7 


181.8 


171.7 


162.8 


148.2 


131.1 


121.0 


112.5 


99.5 


89.9 


82.6 




(5) For Slope s = .0004 = 1 in 2,500 = 2.112 ft. per mUe, 


.1 


104.8 


90.5 


84.5 


79.2 


70.1 


62.6 


56.4 


51.2 


42.9 


34.1 


28.9 


25.0 


19.4 


15.7 


13.1 


.2 


126.2 


110.0 


103.2 


97.1 


86.6 


77.8 


70.5 


64.3 


54.4 


43.8 


37.6 


32.5 


25.6 


20.9 


17.5 


.3 


138.8 


121.7 


114.4 


107.9 


96.6 


87.2 


79.3 


72.6 


61.8 


50.1 


43.1 


37.6 


29.7 


24.4 


20.6 


.4 


147.6 


129.8 


122.3 


115.5 


101.7 


92.0 


85.6 


78.4 


67.2 


54.8 


47.3 


41.4 


33.0 


27.2 


22.9 


.6 


159.6 


141.1 


133.2 


126.1 


113.8 


103.5 


94.7 


87.2 


75.0 


61.7 


53.5 


47.1 


37,8 


31.4 


26.7 


.8 


167.7 


148.8 


140.7 


133.4 


120.7 


110.1 


101.1 


93.3 


80.6 


66.7 


58.1 


51.3 


41.4 


34.6 


29.6 


1 


173.8 


154.5 


146.4 


138.9 


126.0 


115.1 


105.9 


97.9 


84.9 


70.5 


61.7 


54.6 


44.3 


37.1 


31.9 


1.5 


184.1 


164.4 


156.0 


148.4 


135.1 


122.8 


114.4 


106.1 


92.6 


77.6 


68.2 


60.7 


49.7 


42.0 


36.3 


2 


190.8 


170.9 


162.4 


155.1 


141.3 


129.9 


120.1 


111.7 


97.9 


82.5 


72.8 


65.1 


53.6 


45.2 


39.5 


3 


199.5 


179.4 


170.8 


162.9 


149.3 


137.7 


127.8 


119.2 


105.0 


89.1 


79.1 


71.1 


59.2 


50.6 


44.2 


3.28 


201.2 


181.1 


172.5 


164.6 


151.0 


139.3 


129.4 


120.7 


106.5 


90.6 


80.5 


72.4 


60.4 


51.7 


45.3 


4 


205.0 


184.8 


176.1 


168.3 


154.5 


142.8 


132.8 


124.1 


109.8 


93.6 


83.5 


75,3 


63.0 


54.2 


47.6 


6 


212.0 


191.7 


183.0 


175.1 


161.2 


149.4 


139.3 


130.5 


116.0 


99.6 


89.3 


80.9 


68.3 


59.2 


52.3 


8 


216.4 


196.0 


187.3 


179.4 


165.4 


153.6 


143.5 


134.6 


120.1 


103.6 


93.1 


84.7 


71.9 


62.7 


55 6 


10 


219.5 


199.1 


190.4 


182.4 


168.5 


156.6 


146.5 


137.6 


123.0 


106.5 


95.9 


87.5 


74.6 


65.3 


58.1 


12 


221.8 


201.5 


192.7 


184.8 


170.8 


159.0 


148.8 


139.9 


125.3 


108.7 


98.1 


89.6 


76.7 


67.3 


60.1 


16 


225.2 


204.8 


196.1 


188.1 


174.2 


162.3 


152.1 


143.3 


128.6 


112.0 


101.4 


92.9 


79.9 


70.4 


63.2 


20 


227.6 


207.2 


198.5 


190.5 


176.5 


164.7 


154.5 


145.6 


131.0 


114.3 


103.8 


95.2 


82.2 


72.7 


65.5 


30 


231.4 


211.0 


202.3 


194.3 


180.4 


168.5 


158.3 


149.5 


134.8 


118.2 


107.6 


99.1 


86.1 


76.6 


69.3 


50 


235.3 


215.0 


206.2 


198.3 


184.4 


172.5 


162.4 


153.5 


138.9 


122.4 


111.8 


103.3 


90.4 


80.9 


73.7 


75 


237.9 


217.5 


208.8 


200.9 


187.0 


175.2 


165.0 


156.2 


141.7 


125.2 


114.7 


106.2 


93.3 


83.9 


76.7 


100 


239.4 


219.1 


210.4 


202.5 


188.6 


176.8 


166.7 


157.9 


143.3 


126.9 


116.4 


108.0 


95.2 


85.8 


78.6 



1172 



Q2,— HYDRAULICS. 



8. — Coefficients c in Kutter's Formula, English Measure. — Concrd. 



Hyd. 

Rad.r 

Ft. 



Coeflacients n of Roughness. 



0091 .0101.01051 .011 I .0121 .013| .014| .015| .017 | .020|.0225| .025] .030 









(6) For Slope i 


• = .001 = 


= 1 in l.OOC 


= 5.28 ft 


per 


mile. 






.1 


108.4 


93.8 


87.6 


82.1 


72.7 


65. C 


58.6 


53.1 


44.5 


35.4 


30.0 


25. S 


20.1 


16.2 


13. £ 


.2 


129.7 


113.1 


106.1 


99. £ 


89.1 


80.2 


72.7 


66.3 


56.1 


45.2 


38.6 


33.5 


26.3 


21.5 


18.C 


.3 


142.0 


124.5 


117.1 


110.5 


99. C 


89.4 


81.4 


74.5 


63.4 


51.5 


44.2 


38.6 


30.5 


25.0 


21.1 


A 


150.5 


132.5 


124.8 


117. S 


106. C 


96. C 


86.fi 


80.4 


68.8 


56.1 


48.4 


42.4 


33.7 


27.8 


23. J 


.6 


162.0 


143.3 


135.4 


128.2 


115.7 


105.2 


96.4 


88.fi 


76.4 


62.9 


54.5 


48.0 


38.5 


32.0 


27.2 


.8 


169.7 


150.7 


142.5 


135.2 


122.4 


111.6 


102.5 


94.6 


81.8 


67.7 


59.0 


52.2 


42.1 


35.1 


30. C 


1 


175.5 


156.] 


147.9 


140.4 


127.4 


116.5 


107.1 


99.1 


86.0 


71.5 


62.5 


55.4 


44.9 


37.6 


32.2 


1.5 


185.2 


165.5 


157.1 


149.4 


136.1 


124.9 


115.3 


106.9 


93.4 


78.2 


68.8 


6T.3 


50.2 


42.4 


36.6 


2 


191.5 


171.6 


163.1 


155.4 


141.9 


130.5 


120.7 


112.3 


98.4 


82.9 


73.2 


65.5 


53.9 


45.8 


39.7 


3 


199.6 


179.5 


170.9 


163.1 


149.4 


137.8 


127.9 


119.3 


105.1 


89.2 


79.2 


71.2 


59.2 


50.7 


44.3 


3.28 


201.2 


181.1 


172.5 


164.6 


151.0 


139.3 


129.4 


120.7 


106.5 


90.6 


80.5 


72.4 


60.4 


51.7 


45.3 


4 


204.7 


184.5 


175.9 


168.0 


154.2 


142.6 


132.6 


123.9 


109.6 


93.5 


83.3 


75.1 


62.9 


54.1 


47.5 


6 


211.2 


190.9 


182.2 


174.3 


160.4 


148.7 


138.6 


129.8 


115.4 


99.1 


88.7 


80.4 


67.9 


58.8 


52. C 


8 


215.2 


194.9 


186.2 


178.3 


164.4 


152.6 


142.5 


133.7 


119.1 


102.7 


92.3 


83.9 


71.2 


62.1 


55.1 


10 


218.1 


197 8 


189.0 


181.1 


167.2 


155.4 


145.2 


136.4 


121.9 


105.4 


94.9 


86.5 


73.7 


64.5 


57.4 


12 


220.2 


199.9 


191.2 


183.2 


169.3 


157.5 


147.3 


138.5 


123.9 


107.4 


97.0 


88.5 


75.7 


66.4 


59.3 


16 


223.3 


203.0 


194.3 


186.3 


172.4 


160.6 


150.4 


141.6 


127.0 


110.5 


100.0 


91.5 


78.6 


69.3 


62.1 


20 


225.5 


205.2 


196.4 


188.5 


174.6 


162.7 


152.6 


143.8 


129.2 


112.6 


102.1 


93.6 


80.7 


71.4 


64.2 


30 


229.0 


208.6 


199.9 


192.0 


178.0 


166.2 


156.1 


147.3 


132.7 


116.1 


105.7 


97.2 


84.3 


74.9 


67.7 


50 


232.5 


212.2 


203.5 


195.6 


181.7 


169.9 


159.8 


151.0 


136.4 


119.9 


109.5 


101.0 


88.2 


78.8 


71.7 


75 


234.8 


214.5 


205.8 


197.9 


184.0 


172.3 


162.2 


153.4 


138.9 


122.4 


112.0 


103.6 


90.8 


81.5 


74.4 


100 


236.2 


216.0 


207.3 


199.4 


185.5 


173.7 


163.6 


154.9140.4 


124.0 


113.6 


105.2 


92.5 


83.2 


76.2 




(7) For Slope s = .01 = 1 in 100 = 52.8 ft. per mUe. 


.1 


110.9 


95.1 


89.6 


84.0 


74.4 


66.5 


60.0 


54.4 


45.6 


36.3 


30.7 


26.5 


20.6 


16.6 


13.8 


.2 


131.9 


115.1 


108.0 


101.7 


90.7 


81.7 


74.1 


67.6 


57.2 


46.1 


39.4 


34 2 


26.8 


21.9 


18.4 


.3 


144.0 


126.3 


118.9 


112.1 


100.5 


90.1 


82.7 


75.7 


64.5 


52.3 


45.0 


39.3 


31.0 


25.5 


21.5 


.4 


152.3 


134.1 


126.4 


119.5 


107.4 


97.3 


88.0 


81.6 


69.8 


57.0 


49.1 


43.0 


34.2 


28.2 


23.8 


.6 


163.5 


144.7 


136.7 


129.5 


116.9 


106.4 


97.4 


89.8 


77.3 


63.6 


55.2 


48.6 


39.0 


32.4 


27.6 


.8 


171.0 


151.8 


143.7 


136.3 


123.4 


112.6 


103.4 


95.5 


82.6 


68.4 


59.6 


52.7 


42 5 


35.5 


30.3 


1 


176.5 


157.1 


148.9 


141.4 


128.3 


117.3 


107.9 


99.9 


86.7 


72.1 


63.0 


55.8 


45.3 


38.0 


32.6 


1.5 


185.9 


166.1 


157.7 


150.1 


136.7 


125.4 


115.8 


107.5 


93.8 


78.6 


69.2 


61.6 


50.5 


42.2 


36.8 


2 


191.9 


172.0 


163.5 


155.8 


142.2 


130.8 


121.0 


112.6 


98.7 


83.2 


73.4 


65.7 


54.1 


46.0 


39.9 


3 


199.7 


179.6 


171.0 


163.1 


149.4 


137.9 


127.9 


119.3 


105.2 


89.3 


79.3 


71.2 


59.3 


50.7 


44.3 


3.28 


201.2 


181.1 


172.5 


164.6 


151.0 


189.3 


129.4 


120.7 


106.5 


90.6 


80.5 


72.4 


60.4 


51.7 


45.3 


4 


204.6 


184.4 


175.7 


167.9 


154.1 


142.4 


132.4 


123.7 


109.5 


93.3 


83.2 


75.0 


62.8 


54.0 


47.4 


6 


210.7 


190.4 


181.7 


173.8 


160.0 


148.2 


138.2 


129.4 


115.0 


98.7 


88.4 


80.1 


67.6 


58.6 


51.8 


8 


214.5 


194.2 


185.5 


177.6 


163.7 


152.0 


141.8 


133.1 


118.6 


102.2 


91.8 


83.4 


70.8 


61.7 


54.7 


10 


217.2 


196.9 


188.2 


180.3 


166.4 


154.6 


144.5 


135.7 


121.2 


104.7 


94.3 


85.9 


73.2 


64.0 


57.0 


12 


219.3 


198.9 


190.2 


182.3 


168.4 


156.6 


146.5 


137.7 


123.1 


106.2 


96.2 


87.8 


75.0 


65.8 


58.8 


16 


222.2 


201.9 


193.2 


185.2 


171.3 


159.5 


149.4 


140.6 


126 


109.5 


99.1 


90.6 


77.8 


68.5 


61.5 


20 


224.2 


203.9 


195.2 


187.3 


173.3 


161.5 


151.4 


142.6 


128.1 


111.6 


101.1 


92.7 


79.8 


70.5 


63.4 


30 


227.5 


207.2 


198.5 


190 5 


176.6 


164.8 


154.7 


145.9 


131.4 


114.9 


104.4 


96.0 


83.2 


73.9 


66.8 


50 


230.8 


210.5 


201.8 


193.9 


180.0 


168.3 


158.2 


149.4 


134.9 


118.4 


108.0 


99.6 


86.4 


77.6 


70.5 


75 


233.0 


212.7 


204.0 


196,1 


182.3 


170.5 


160.4 


151.6 


137.2 


120.8 


110.4 


102.0 


89.3 


80.1 


73.1 


100 


234.3 


214.1 


205.4 


197.5 


183.6 


171.9 


161.8 


153.0 


138.6 


122.2 


111.9 


103.5 


90.9 


81.7 


74.7 



Practical Examples in Use op Kutter's Formula and Table 8. 

(1) Given, the average or mean cross-section of a stream; to find the 
mean radius r? 

Calculate or scale the area a in sq. ft. of the section of flowing water; 
also, the wetted perimeter ^ in ft., or transverse length of (sectional) contact 

of flowing water with the stream bed. Then, ?" = — . 

(2) Given, a stream, canal, or pipe line, etc., of average uniform section; 
to find the hydraulic slope s? 

5 = the average fall in ft. of the stream or canal, per foot of length; or 
the fall in ft. of any hydraulic grade line, per ft. of length. 

(3) Given, the character of a stream, canal, or pipe line; to find the 
roughness «? 

Consult the tabular values, page 1168. 

(4) Given, the co-efficient c, mean radius r, and slope s; to find the 
velocity v? 



KUTTERS FORMULA, THE VENTURI METER. 1173 

From Chezy's formula, v = cVrs. Note that the roughness n is not con- 
sidered directly because c and n are functions of each other; but as c is given, 
n can be obtained if desired. (The discharge in cu. ft. per sec.=-q = av, in 
which a= the area of cross-section of the discharging volume, in sq. ft.) 

(5) Given, roughness «==.014, mean radius r=18, slope 5=.0003; to 
find velocity vl 

First, find c from sections 4 and 5 of Table 8: c=i(154.9 4- 157.6+ 152.1 
+ 154.5) = 154.8. Then z; = c\/^= 154.8X4.243X. 01732= 11.38 ft. per sec 

(6) Given, velocity v = 1 . 2 ft. per sec, mean radius r= 1.5, slope 5= .0004; 
to find nl 

First find c, = — ■== = '■ = 49.0. Then from section 5 of pre- 

VVs 1.2247X.02 

ceding Table, on line with r= 1.5, we find that a coefficient c of 49.7 corres- 
ponds to n = .030; .'. use » = .030. 

(7) Given, roughness n==.0l7, slope j = .001, velocity z;=2.2 ft. per sec; 
to find the mean radius r? 

1 v^ 
Use "cut and try" method: assume r in the equation, t= — • — o = 

1000 — !• 1st, let r = 1, then from section 6 of Table 8, — • —^ = 1000 X ~~ 
c^ s c^ (86 )2 

= 0.65; 2nd, let r = 0.8, then— •—2 = 1000X7^^2 = 0.72; 3rd, let r= 0.75, 

then - . -0 = 1 000X7^^4^2 = 0-75. .'. r=0.75. Checking, v^cWs^^ 
s c^ (80.45)'* 

80.45X0.866X0.03162 = 2.2, the given velocity. 

(8) Given, mean radius r=6, roughness « = .020, and required velocity 

2;=3; to find the slope st 

1 v^ \ v"^ 
Use "cut and try" method: assume 5 in the equation 5= — * -i;=-2- • — i- 

r c^ c^ 

1 ?;2 1 9 

1st, let 5=.0001, then from section 6 of Table 8, — * — ^= -^ • TTTrTTTT^ =" 

r c^ (lUii.l}'' 

.000144; 2nd, let 5=. 00015, theni.—^=^--rx|-^r^ = . 00015. .-.5=. 00015. 



Checking, v = cVr5= 101. 3X 2. 45X. 01225= 3.04. If greater accuracy is 
required, assume a new slope slightly less than 5=. 00015 and try again. 

The Venturi Meter. — ^This is an apparatus for determining the rate of 
discharge of water through pipes, by inserting the meter in the pipe line 

Wafer- f/ff/ff Sree/ Charn^ 
Confy//)/m Me ter RecrJsfer JJ \nlnnRri nnHfir.<tirifiwfTHt 

^Fressure^ 
pip 




\/enfuri Meter Tude 
L^ < t i 
Scale of Feet 

Fig. 9. — ^Venturi Meter. 

and connecting it, by pressure pipes, to a register. The principle on which it 
is based was discovered by J. B. Venturi, and his experiments were followed 
by Bossut, Casteh Herschel and others. The apparatus is illustrated in 
Fig. 9, and consists of a short pipe V, of comparatively small diameter, called 
the venturi, joining a converging frustum of a cone or mouthpiece M , with a 
diverging conical frustum D\ these three pipes bein^ inserted in the pipe 
line PP, with the water flowing in the direction indicated by the arrows. 
Surrounding the venturi F, is an air chamber, the cast-iron shell separating 
the air chamber from the venturi being pierced with a few small holes, care- 
fully drilled, leaving sharp, square edges on the inside face of the venturi. 



1174 



Q2.— HYDRAULICS, 



Venturi discovered that when water flows through such a converging 
mouthpiece and a narrow throat, V, and then expands into a diverging tube 
or bell, D, there is a decided decrease of water pressure against the inside 
of the conduit from the beginning of the mouthpiece to the venturi, at which 
point it is a minimum, and then a decided increase in pressure from the 
venturi to the end of the diverging tube D. He correctly interpreted the 
cause of this reduction of pressure at V as due to a partial vacuum at o, 
just beyond the venturi, caused by the jet of water expanding from the 
throat along the tube D. In fact, not only is the pressure at the venturi 
greatly reduced, as in Fig. 11, but with high velocity of discharge the pres- 
sure changes, sometimes, from positive to negative, actually causing a 
suction in the air chamber at the venturi. 

Mr. Herschel has clearly demonstrated that the difference in pressure at 
the beginning of the mouthpiece and at the venturi increases with the velocity 
of flow, and has established, by numerous experiments, a relation or rela- 
tions between the two so that, knowing the difference in pres- 
sures at the two points, for a given meter, the discharge can be 
determined. 

The Meter Register, indicated in Fig. 9, automatically records 
the discharge directly, in gallons, cubic feet or pounds — per 
second, minute, hour or day. The register is controlled by the 
pressures, or difference of pressures, in the pressure pipes leading 
from the meter. 

The Manometer (Fig. 10) is a cheap device and not auto- 
matic as a constant recorder. It is a portable instrument that 
can be attached to any meter tubes on the pipe line and the rate 
of flow determined by reading the gauge, which has been gradu- 
ated to "cubic feet per second," or "gallons per day," etc., 
based on difference of pressure in the connecting pressure Fig. 10. 
tubes. Manometer. 

Piezometer tubes form a very simple device for measuring the pressure 
and flow in pipe lines and venturi meters. These are glass and iron tubes 
inserted at the required points along the line by "tapping" the pipe so that 
water will rise in the piezometers to the hydraulic gradient, or line of no 
pressure. Let PM, PV and PD be glass piezometers inserted at points M, 
F, and D, Figs. 11 and 12. Each figure shows the hydraulic grade line (1) 






-^-Z^Z'TZ^ 




Fig. 11. 
Pressure at the venturi. 



Fig. 12. 
Suction at the venturi. 



for a uniform section of pipe, and (2) with the venturi meter inserted in the 
pipe line. If there is no meter in the pipe line, that is, no contraction of area, 
the H. G. L. will be a straight line on a falling grade in the direction of the 
flow, but may be considered as "level" for such a small distance as the 
length of the meter would occupy, as far as the present discussion is con- 
cerned; in other words, the loss of head for this distance would be practically 
zero. If now the meter is inserted in the pipe line, the H. G. L. will drop 
with a descending grade from M to V, and with an ascending grade from V 
to D', but it will be noticed that it does not rise to its former level at D, 
there being a loss of head L in the flow through the meter from M to D. As 
the areas of the pipe at the piezometer PM and PD are equal, the velocities 
at these points are equal, hence the loss of pressure head L represents a total 
loss of head between those points, no part of it being compensated for by an 
increase in velocity. On the other hand, the loss of pressure head between 
M and V, represented by the difference in water levels in the piezometers 
PM and PV, is nearly all compensated for by the increased velocity of flow 



VENTURI METER, ORIFICES, JETS, ETC, 1175 

at y over that at M, or in other words by the increased velocity head h,, Fig. 
11. Experiments on standard meters show that this velocity head represents 
about 98 per cent of the lost pressure head shown by the piezometers, the 
other 2 per cent being a total loss. 

Where there is negative pressure or suction at the venturi (Fig. 12) the 
piezometer tube is bent in the form of a siphon and the amount of negative 
pressure head is measured by the height hs of the column of water in the 
tube. This distance is laid off below the center of the venturi, as fixing the 
lowest point of depression in the hydraulic grade line. The velocity head is 
determined by the total change of pressure head as in the preceding descrip- 
tion, taking into account of course the small percentage of actual loss. 

Fig. 13 shows the standard dimensions of the venturi meter in terms of 
the length of the venturi y, assumed as unity; and it is claimed that, based 



Fig. 13. — Standard Proportions of Venturi Meter. 

on these proportions, for such a meter, with the diameter of the pipe ranging 
from one to nine feet, and with a velocity through the pipe ranging from 
one-half to six feet per second, the total loss of head in the mouthpiece will 
be about 2 per cent of the difference in pressure at these points. That is to 
say, the coefficient of velocity at the venturi closely approxmates 98 per cent. 
The standard meter is proportioned with one main point in view, namely, to 
produce a marked increase in velocity at the venturi, with as little total loss 
as possible in the whole meter. In principle, it shows that if a pipe line is 
contracted at any point, the pressure head is reduced at that point, being 
partly converted into velocity head; and conversely, if it is expanded there 
is a tendency toward reconversion of velocity head into pressure head. 

Leaving out the velocity of approach, the makers use the following 
forrnula for discharge: 



q = ca\/{im±)h. (1) 

In which <j = discharge in cu. ft. per sec, or any other units; 

c = a constant or coefficient of velocity, determined by experiment; 
a = area of cross-section of venturi V, in sq. ft.; 
(1.02 ±)/jv = difference in height of piezometers PM and PF, in ft.; 
/jv = increase in velocity head at V over that at M. 
Note that (1.02±)/jv is used for simplicity of illustration in keeping 
clearly in mind the relative value of the difference of velocity heads h-,. It 
is t o be noted also that the actual velocity at the t hroa t or venturi^ Fi, 

«V2 g ^ + the velocity of approach, and not simply V2 g h. 

Orifices, Tubes, Nozzles and Jets. — The theoretic discharge from an 
orifice is obtained by integrating or taking the summation of the discharge 
of each infinitesimal layer of water of transverse length x, thickness dy, and 
head y between the proper limiting values of the head. Thus, in computing 
Table 9, following, we have — 



y = H 



For rectangular orifice, submerged, Q= \ x dy\/2 gy (2) 

in which x = h', hence, theoretically, 

q=l b ^-2j{m-hh ..(3) 

For rectangular weirs, the upper head h = 0, H = d, bd = a; .'. q=-^a \^2 g d. 
Other forms of orifices are calculated in the same manner. The acttmt 
discharge and velocity are closely approximated by using coefficients of area 
for contraction, velocity and discharge. Let — 

Ca = coefficient of contraction of jet; 

c, = coefficient of velocity of jet; 

Cq = coefficient of discharge. 



1176 



&!,— HYDRAULICS. 



Then, Actual area of issuing jet at its smallest section = Cao; (4) 

Actual veloc. of issuing jet at its smallest section = CvZ;= 8. 02 CyV^; (5) 

Actual discharge of issuing jet = Cq q= 8.02 Ca a Cv V^; (6) 

Coefficient of discharge = (7q = Ca Cv (7) 

The values of Ca, Cv and Cq for the various shapes and conditions of ori- 
fices, nozzles, jets, etc., are determined by experiment. The coefficient of 
contraction of area, c^, for small standard orifices may be assumed at about 
0.63; the coefficient of velocity, Cv, at about 0.98; and the coefficient of 
discharge Cq = Ca c, = 0.63X0.98 = 0.62. These values of the coefficients sub- 
stituted in equations (4), (5) and (6) will give, approximately, the results 
which may be expected in practice. 

Table 9, following, shows the discharge, g, from rectangular, triangular 
and circular orifices: 

9. — Table of Discharges Ffom Orifices. 
(a) Orifices Submerged. 



Function, 



(1) 



Waf erSur face 

T-T 



1^ 



y±JL 



J 



Fig. 14. 
(2) 



Wafer Surface 

I 



Fig. 15. 
(3) 



\tlafer Surface 
rr — T~ 

Fig. 16. 
(4) 



Area. 



Discharge. 



hd 



hd 
2 

Not important 




And when the tops of the orifices touch the surface of the water, h be- 
comes zero and we have, in each case, the following values: 

(b) Water Surface at Tops of Orifices. 



Discharge 



c^la V2g d 



^%bd\/2gd 



Cq;rr2V2gr(0.96) 



= .,f V2i ( 



Hi d^ H-i d^ H- 



H-l d^ 



2 6 32 80 / 

Comparison of Orifices and Tubes. 

The standard orifice (Fig. 17) has sharp, vertical edges. From data above, 
Ca = 0.625; Cv = 0.98; c^= 0.6125. The discharge can be increased by insert- 
ing the standard tube. 

The standard tube (Fig. 18) is made just long enough so the expanding 




Fig. 17. Fig. 18. Fig. 19. Fig. 20. Fig. 21. 

jet completely fills its extremity. Around the contracted portion of the jet 
there is a partial vacuum, which may be tested by tapping the tube and 
inserting a glass piezometer, P, with its lower end immersed in a basin of 
water. The water will rise in the piezometer, indicating suction. The 
vacuum is produced by the particles of water or spray of the jet forcing the 



DISCHARGE FROM ORIFICES, ETC. WEIRS. 



1177 



air out of the tube. The effect is to increase the velocity and discharge, 
over that of the standard orifice. The values of Cq and c^ are about 0.81 or 
0.82, average. 

The conical nozzle (Fig. 19) is designed to increase the energy of the jet. 
The angle of divergence of the sides of the cone is about 13° or 14° for maxi- 
mum discharge, when Cq=0.95, nearly. Sometimes the outer end of the 
cone is provided with a short straight tube. The coefficient of velocity 
increases with the angle of the cone. 

The Venturi meter is simply a compound tube (Fig. 20) and is explained 
on page 1173, etc. 

The flaring or bell-shaped mouthpiece (Fig. 21) decreases the loss of head 
at entrance, usually called Entry Head (see page 1160) and is recommended 
for use at the intake end of pipe lines. 

Weirs. — ^The weir is simply a water meter. Next to the Venturi meter 

' (see page 1173) the standard weir is probably the most accurate device for 

measuring the flow of water. It is practically adapted to gaging the flow of 

small streams, creeks, canals, ditches, and the discharge from pipe lines and 

sewers. 

The standard weir is a rectangular, vertical opening through which the 
surface water is allowed to discharge. The edges of the 
opening should be chisel-edged, the "crest" perfectly 
level, and the sides perfectly plumb. Fig. 22 represents a 
section of a standard weir 



Level 



D >ri 

Surface Levef • ^ 



-fi 





Fig. 23. 
eir, because 



with the water flowing over 
the crest. Note that the sur- 
face of the discharging vol- 
ume describes a curve from 
a point of tangency T , distant 
D from the crest of the dam, 
the drop at the crest being 5. 
Fig. 23 is a plan of the weir 
showing the length of crest L, 
and the side contractions e 
and e' of the flow, for the 
"end contractions" Sand E' . 

The standard weir is a weir ^. ^^ o ^- 1-1^1 
with end contractions; the ^^^g- 22.— Sectional Elevation, 
weir without end contractions is called the "suppressed" 
the end contractions are suppressed. 

The theoretic discharge through a rectangular weir, neglecting velocity 
of approach, is obtained from the equations on pages 1175 and 1176, reduc- 
ing to — 

q = ^L\/2g H^ (no velocity of approach) (8) 

in which • q = discharge in cu. ft. per second; 
L = length of weir, in feet (Fig. 23); 

ii/ = "surface level" head on crest, in feet (Fig. 22). 

If it is desired to include velocity of approach,* v^ ( = \^2g h»), this 
value must be inserted in equation (2), page 1175, before integrating; thus, 

J*y = H ^ 
X dy V2gyTvJ (9) 
y = h 

whence g=f L\^2g[(H-+-h»)^— (h+ha)"^ (including velocity of approach) .(10) 
in which /i„ = velocity head of approach; and h = 0. 

It is to be noted that these formulas, (8) and (10), are purely theoretical, 
and that little or no importance can be attached to them as they stand. 
But like all theoretic expressions, each forms a skeleton or groundwork 
upon which to build the more or less "emperical" formulas which are used 
in practice. The practical formulas are deduced from the above by supply- 
ing the proper coefficients, which are determined by experiments. 

*The velocity of approach should be measured at the point T, Fig. 22. 
It should comprise generally the average velocity for some depth below the 
surface and not the "surface" velocity, which is often used; the latter may 
be close enough, however, in many instances. The surface velocity may be 
obtained by the surface-float method (Fig. 29) ; the average velocity, by the 
integrating float rod (page 1183), or by the current meter (page 1185), or by 
the Pitot tube (page 1183). 



1178 Q2.— HYDRAULICS. 

(a) Francis' Weir Formulas. 
General formula applicable to any sharp-crested, surface weir: 

q= 3,33 (L- 0.1 n H) [{H^-h.y'-hh (11) 

in which q = discharge in cu. ft. per second; 
L = length of weir in feet ; 
« = number of end contractions, as 0, 1 or 2; 
// = "surface level" head on crest, in feet; 
ft.= velocity head in feet = velocity of approach v. 

2g 
Formula (11) may be simplified to meet the following special cases: 
With velocity of approach — 

Contracted |^°^^ ^^^^- «= ^'^^ (L-0.2H) [ (H-hh.)^^-0] (12) 

[One end: q=3.SS (L-O.IH) [(H + h.)^-hu^, (13) 

Suppressed: q= 3.33 L[(H+h.)^-h^^] (14) 

Without velocity of approach (i. e., ho = 0) — 

Contracted 1^°^^ ^^^^- «= 3.33 (L- 0.2 H) H^.. (15) 

iOne end: g= 3.33 (L- 0.1 H) H^ , . (16) 

Suppressed: q=S.33H^ (17) 

The above formulas are very extensively used in the United States. 
They were deduced by Mr. J. B. Francis, in 1854, from very elaborate experi- 
ments which he conducted at Lowell, Mass., on large weirs about 10 feet long. 
The heads ranged usually from 0.4 to 1.6 feet. The formulas are simple, 
logical and fairly reliable within the range of the experiments. The heads 
H were measured 6 ft back from the crest (Fig. 22). 

(b) Bazin's Weir Formula. 
General formula applicable to shaip-crested, suppressed, surface weirs is 

*,= (0.405 + M^) [1 + 0.55 (^)']lHv/2F« (18) 

in which :^ = height of crest above the bottom (Fig. 22) and the other 
notation as per Francis' formulas, preceding. This formula (18) takes into 
consideration the velocity of approach in the height of crest p, hence the 
former does not require any special treatment when p is known. It should 
be remarked that the formula was deduced from very careful experiments 
with standard weirs from 0.656 ft. (0.2 meter) to 6.56 ft. (2 meters) in length 
and with heads ranging from 2.50 to 1.00 ft., the lov/er heads for the longer 
weirs. Care should be used in extending the use of the formula beyond the 
above range for length L and head H. It should be noted that Bazin 
measured the head H at points distant Z)= 16.4 ft. back from the crest. (See 
Fig. 22.) 

The above general formula (18) can be simplified for practical use by 
letting 



m 



= (0.405.M-l)[,,0.55(^)^]. 



hence q = m L H V2gH = m VTg L H Vh (19) 

Either of these two forms (19) may be used in connection with the fol- 
lowing table which gives the values of m (and of wV2g) for given values of 
p and H. 

The second form, q = {m-y/Yg) L H y/H will probably be found more 
convenient as the table gives average values of {ni\/2g) for each five successive 
values of H, and intermediate values may be interpolated. If e xtrem e 

accuracy is required the first form is preferred, remembering that \/2gH= 
8.O2VH. Note that the last column in Table 10 gives the limiting value of 
m for p = oo, whence m = I 0.405 H — '—Jt — I. the first part of the coefficient, 

as the latter part reduces to unity. Hence this value of m may be used 
when there is no velocity of approach, that is, when p is very large. 



* The standard weir provides for free admission of air under the falling 
sheet of water. 



WEIR FORMULAS— FRANCIS, BAZIN. 



1179 



10. — Values op the Coefficient m in the Formula q = m L H \/2gH, 
FOR A Sharp-Crested Weir without Lateral Contraction, 
THE Air being Admitted Freely Beneath 
THE Overflowing Sheet of Nappe. 



•s^» 
















^^8 


lU 


Values of the coefficient m corresponding to the height v of the 


.Is" 


weir above the bottom of the channel. 




B^t 


ow-= 
















5gs 


(1) 


(2) 


(3) 


(4) 


(5) 


(6) 


(7) 


(8) 


(9) 


(10) 


(11) 


Value 


1 




















ofp, in 


0.65( 


5 0.984 


1.312 


1.640 


1.968 


2.624 


3.280 


4.920 


6.560 


CO 


Feet. 






















0.164 


0.458 


0.453 


0.451 


0.450 


0.449 


0.449 


0.449 


0.448 


0.448 


0.4481 


0.197 


0.456 


0.450 


0.447 


0.445 


0.445 


0.444 


0.443 


0.443 


443 


0.4427 


0.230 


0.455 


0.448 


0.445 


0.443 


0.442 


0.441 


0.440 


0.440 


0.439 


0.4391 


0.262 


0.456 


0.447 


0.443 


0.441 


0.440 


0.438 


0.438 


437 


0.437 


0.4363 


0.295 


0.457 


0.447 


0.442 


0.440 


0.438 


0.436 


0.436 


0.435 


0.434 


0.4340 


Means 


0.456 


0.449 


0.446 


0.444 


0.443 


0.442 


0.441 


0.441 


0.440 


0.4400 


my/2g 


3.66 


3.60 


3.58 


3.56 


3.55 


3.54 


3.54 


3.53 


3.53 


3.53 


0.328 


0.459 


0.447 


0.442 


0.439 


0.437 


0.435 


0.434 


0.433 


0.433 


0.4322 


0.394 


0.462 


0.448 


0.442 


0.438 


0.436 


0.433 


0.432 


0.430 


0.430 


0.4291 


0.459 


0.466 


0.450 


0.443 


0.438 


0.435 


0.432 


0.430 


0.428 


0.428 


0.4267 


0.525 


0.471 


0.453 


0.444 


0.438 


0.435 


0.431 


0.429 


0.427 


0.426 


0.4246 


0.591 


0.475 


0.456 


0.445 


0.439 


0.435 


0.431 


0.428 


0.426 


0.425 


0.4229 


Means 


0.467 


0.451 


0.443 


0.438 


0.436 


0.432 


0.431 


0.429 


0.428 


0.4271 


m^2g 


3.74 


3.62 


3.56 


3.52 


3.50 


3.47 


3.46 


3.44 


3.44 


3.43 


0.656 


0.480 


0.459 


0.447 


0.440 


0.436 


0.431 


0.428 


0.425 


0.423 


0.4215 


0.722 


0.484 


0.462 


0.449 


442 


0.437 


0.431 


428 


0.424 


0.423 


0.4203 


0.787 


0.488 


0.465 


0.452 


0.444 


0.438 


0.432 


0.428 


0.424 


0.422 


0.4194 


0.853 


0.492 


0.468 


0.455 


446 


0.440 


0.432 


0.429 


0.424 


0.422 


0.4187 


0.919 


0.496 


0.472 


0.457 


0.448 


0.441 


0.433 


0.429 


0.424 


0.422 


0.4181 


Means 


0.488 


0.465 


0.452 


0.444 


0.438 


0.432 


0.428 


0.424 


0.422 


0.4196 


ms/1g 


3.92 


3.73 


3.63 


3.56 


3.52 


3.46 


3.44 


3.40 


3.39 


3.37 


0.984 


0.500 


0.475 


0.460 


0.450 


0.443 


0.434 


0.430 


0.424 


0.421 


0.4147 


1.050 


0.500 


0.478 


0.462 


0.452 


0.444 


0.436 


0.430 


0.424 


0.421 


0.4168 


1.116 


0.500 


0.481 


0.464 


0.454 


0.446 


0.437 


0.431 


0.424 


0.421 


4162 


1.181 


0.500 


0.483 


0.467 


0.456 


0.448 


0.438 


0.432 


0.424 


0.421 


0.4156 


1.247 


0.500 


0.486 


0.469 


0.458 


0.449 


0.439 


0.432 


0.424 


0.421 


0.4150 


Means 


0.500 


0.481 


0.464 


0.454 


0.446 


0.437 


0.431 


0.424 


0.421 


0.4162 


m\/Zg 


4.01 


3.86 


3.73 


3.64 


3.58 


3.50 


3.46 


3.40 


3.38 


3.34 


1.312 


0.500 


0.489 


0.472 


0.459 


0.451 


0.440 


0.433 


0.424 


0.421 


0.4144 


1.378 


0.500 


0.491 


0.474 


0.461 


0.452 


0.441 


0.434 


0.425 


0.421 


0.4139 


1.444 


0.500 


0.494 


0.476 


0.463 


0.454 


0.442 


0.435 


0.425 


0.421 


0.4134 


1.509 


0.500 


0.496 


0.478 


0.465 


0.456 


0.443 


0.435 


0.425 


0.421 


0.4128 


1.575 


0.500 


0.496 


0.480 


0.467 


0.457 


0.444 


0.436 


0.425 


0.421 


0.4122 


Means 


0.500 


0.493 


0.476 


0.463 


0.454 


0.442 


0.435 


0.425 


0.421 


0.4133 


7n^2g 


4.01 


3.96 


3.82 


3.72 


3.64 


3.55 


3.49 


3.41 


3.38 


3.32 


1.640 


0.500 


0.496 


0.482 


0.468 


0.459 


0.445 


0.437 


0.426 


0.421 


0.4118 


1.706 


0.500 


0.496 


0.483 


0.470 


0.460 


0.446 


0.438 


0.426 


0.421 


0.4112 


1.772 


0.500 


0.496 


0.485 


0.472 


0.461 


0.447 


0.438 


0.426 


0.421 


0.4107 


1.837 


0.500 


0.496 


0.487 


0.473 


0.463 


0.448 


0.439 


0.427 


0.421 


0.4101 


1.S03 


0.500 


0.496 


0.489 


0.475 


0.464 


0.449 


0.440 


0.427 


0.421 


0.4096 


1.S39 


0.500 


0.496 


0.490 


0.476 


0.466 


0.451 


0.441 


0.427 


0.421 


0.4092 


Means 


0.500 


0.496 


0.486 


0.472 


0.462 


0.448 


0.436 


427 


0.421 


0.4104 


m^Zg 


4.01 


3.98 


3.90 


3.79 


3.71 


3.60 


3.50 


3.42 


3.38 


3.29 



1180 



—HYDRAULICS. 



Problem. — What is the discharge through a weir whose crest is 7 ft. long 
and 2.5 ft. above the bottom; the measured head, H, being 0.75 ft.? 

Solution. — The value of m in the preceding table for p=2.5 and H = 0. 75, 
is 0.433; then from first equation (19), q=0.433X 8.02X 7X0.75X0.866= 
15.8 cu. ft. per second. Ans. 

(c) Fteley and Stearns' Weir Formulas. 
General formula for sharp-crested, suppressed, surface weirs, is 



1 



No velocity of approach ... (20) 



7e-} 



(21) 



Q=: 0.4125 L // V2g /f +0.007 

= 3.31 LH Vh-\-0.007L 
and, if there is velocity of approach, 

^ ( Including ve- 

9=3.31 L (H+1.5h.) VH+1.5 ha+ 0.007 L] locity 

( approach. 

Use notation for Francis' formula, page 1 1 78. The above formulas were 
deduced from experiments which they made in Boston, Mass., 1877-9, with 
weirs 5 to 19 ft. long and under heads ranging from 0.065 up to 1.6 feet. 
The heads H were measured 6 ft. back from the crest. 

(d) *Parmley*s Weir Formula. 
General formula for sharp-crested, surface weirs, either contracted or 
suppressed, taking into account velocity of approach, if any: 

q = C\K[L- 0.1 nH]H^ (22) 

in which 9 = discharge in cu. ft. per second; 

L = length of weir, in ft.; 

n = number of end contractions, as 0, 1 or 2; 
H = "surface level" head on crest, in ft.; 
C' = constant for given value of H, from Table 11; 

K= constant for given value of -j-, from Table 12; 

a=(L— 0.1 n H) H = contracted area of discharge; 
A =area of water section in channel of approach. 

11. — Values of Coefficient C for Given Values of H. 
(Equation 22.) 



H 


C 


H 


C" 


H 


C 


H 


C 


H 


C 


Ft. 




Ft. 




Ft. 




Ft. 




Ft. 




0.10 


3.580 


0.40 


3.385 


0.70 


3.351 


1.00 


3.334 


1.60 


3.301 


0.15 


3.520 


0.45 


3.376 


0.75 


3.349 


1.10 


3.329 


1.70 


3.296 


0.20 


3.478 


0.50 


3.368 


0.80 


3.346 


1.20 


3.324 


1.80 


3.290 


0.25 


3.444 


0.55 


3.362 


0.85 


3.343 


1.30 


3.319 


1.90 


3.285 


0.30 


3.420 


0.60 


3.358 


0.90 


3.340 


1.40 


3.313 


2.00 


3.280 


0.35 


3.400 


0.65 


3.354 


0.95 


3.337 


1.50 


3.307 








12. 



-Values of Coefficient K for Given Values of H. 
(Equation 22.) 



a 


K 


a 


K 


a 


A 




A 




A 


0.01 


1.0001 


0.13 


1.0093 


0.25 


0.02 


1.0002 


0.14 


1.0108 


0.26 


0.03 


1.0005 


0.15 


1.0124 


0.27 


0.04 


1.0009 


0.16 


1.0141 


0.28 


0.05 


1.0014 


0.17 


1.0159 


0.29 


0.06 


1.0020 


0.18 


1.0178 


0.30 


0.07 


1.0027 


0.19 


1.0198 


0.31 


0.08 


1.0035 


0.20 


1.0220 


0.32 


0.09 


1.0044 


0.21 


1.0243 


0.33 


0.10 


1.0055 


0.22 


1.0266 


0.34 


0.11 


1.0066 


0.23 


1.0291 


0.35 


0.12 


1.0079 


0.24 


1.0317 


0.36 



K 


1.0344 


1.0372 


1.0401 


1.1431 


1.0463 


1.0495 


1.0529 


1.0563 


1.0599 


1.0636 


1.0674 


1.0713 



a 


K 


a 


K 


A 




A 




0.37 


1.0753 


0.49 


1.1321 


0.38 


1.0794 


0.50 


1.1375 


0.39 


1.0837 


0.51 


1.1431 


0.40 


1.0880 


0.52 


1.1487 


0.41 


1.0925 


0.53 


1.1545 


0.42 


1.0970 


0.54 


1.1604 


0.43 


1.1017 


0.55 


1.1664 


0.44 


1.1065 


0.56 


1.1725 


0.45 


1.1114 


0.57 


1.1787 


0.46 


1.1164 


0.58 


1.1850 


0.47 


1.1215 


0.59 


1.1915 


0.48 


1.1267 


0.60 


1.1980 



* Discussion by W. C. Parmley, Trans. Am. Soc. C. E., Vol. XLIV, p. 351. 



WEIRS— SURFACE AND SUBMERGED— FORMULAS. llSl 

Parmley's formula (22) was not deduced from any particular set of 
experiments, but is based on the experiments and formulas of Bazin, 
Francis, and Fteley and Steams. The result obtained is a formula com- 
prehensive in character and giving average values. Bazin's experiments 
were made very carefully under ideal conditions in a long, smooth canal 
lined with cement, while those of Francis and Fteley and Steams were con- 
ducted under conditions more nearly approaching those to be found in 
practice. The roughness of the stream or canal must necessarily produce 
an appreciable effect on the flow. 

Triangular and trapezoidal weirs have been proposed in place of the 
standard, rectangular section, on account of the very slight variation of the 
coefficient of discharge for different heads H. They have come into practical 
use in Utah, Nevada and other States. 

The Submerged Weir, so-called because the water level below the weir 
rises higher than the crest, is shown in Fig. 24. 



-Si^Ct 




Fig. 24. — Submerged Weir. 

(a) Fteley and Steams' formula* for submerged weirs, without end con- 
tractions is. 



q==mL(H+jj (H-h)^. 



(23) 



in which <? = discharge in cu. ft. per second; 

// = up-stream head on crest, in feet; 
^ = down stream head on crest, in feet; 
L = length of weir, in feet ; 

w = coefficient for given values of jj, as in the following Table. 



13. — Values of Coefficient m for Given Values of 



H' 











(Equat 


ion 23.; 










h 
H 


m 


n 

IT 


m 


h 


m 


n 

H 


m 


n 

H 


w. 


0.00 
0.04 
0.08 


3.33 
3.35 
3.37 


0.12 
0.16 
0.20 


3.35 
3.32 
3.28 


0.30 
0.40 
0.50 


3.21 
3.15 
3.11 


0.60 
0.70 
0.80 


3.09 
3.09 
3.12 


0.90 
1.00 


3.19 
3.33 









(6) Herschel's formula t for submerged weirs, based on experiments 
made by Francis and by Fteley and Steams, is as follows: 



g=3.33L(c//)2 



(24) 



in which c = coefficient for given values of 77 in the following Table, the 

ti 

balance of the notation same as for the Fteley and Steams' formula, preced- 
ing. 



* Trans. Am. Soc. C. E., Vol. XII, page 103. 

t Discussion by Clemens Herschel, Trans. Am. Soc. C. E., Vol. XIV. 
page 194, 



1182 



^2.— HYDRAULICS. 



14. — Coefficients c for Given Values of 77. 
(Equation 24.) 



(Fig. 24.) 



h 




h 




h 












H 




H 




H 


0.00 


1.000 


0.18 


0.989 


0.38 


.01 


1.004 


.20 


0.985 


.40 


.02 


1.006 


.22 


0.980 


.42 


.04 


1.007 


.24 


0.975 


.44 


.06 


1.007 


.26 


0.970 


.46 


.08 


1.006 


.28 


0.964 


.48 


.10 


1.005 


.30 


0.959 


.50 


.12 


1.002 


.32 


0.953 


.52 


.14 


0.998 


.34 


0.947 


.54 


.16 


0.994 


.36 


0.941 


.56 



0.935 
0.929 
0.922 
0.915 
0.908 
0.900 
892 
0.884 
0.875 
0.866 



0.58 
.60 
.62 
.64 
.66 
.68 
.70 
.72 
.74 
.76 



0.856 
0.846 
0.836 
0.824 
0.813 
0.799 
0.787 
0.771 
0.755 
0.738 



0.78 
.80 
.82 
.84 



.90 

.92 

.96 

1.00 



0.722 

0.703 

0.681 

0.659 

0.634 

0.606 

0.574 

0.54 

0.45 

^.000 



Hydraulic Measurements. — Measurements of flowing 
water may be made by several methods, depending upon the 
circumstances and requirements of the particular case: 

(1) By tank measurement; 

(2) By venturi meter; 

(3) By weir measurement; 

(4) By pi tot-tube meter; 

(5) By floats; 

(6) By current meters. 

These are discussed in the order named. 

Tank measurement. — This method is the most exact and 
can be used where the discharge is small. The discharge 
from a pressure-pipe line leading to a high service reser- 
voir may be measured by using the latter as a tank and 
shutting off the outlet while it is being filled. The capacity 
of the reservoir must be determined accurately, or it may 
be calibrated to gauge readings above the bottom. 

Venturi meter.— This is described fully on page 1173, etc. 
This meter is particularly adapted to measuring the dis- 
charge in pipes, and can be had in sizes up to 6 ft. or more 
in diameter. Results can be obtained usually within 2 or 
3 per cent of the actual discharge. In ordering, it is neces- 
sary to state the diameter of the pipe and the average 
velocity of discharge so that the cones and venturi may be 
proportioned correctly, for accurate results. The velocity 
through the venturi must be accelerated sufficiently to cause 
a marked decrease in pressure, and this is done by giving the 
diameter of the venturi the proper ratio to that of the main 
pipe. It should be large for high velocities and small for 
low velocities. A register should accompany each meter. 

Weir measurement. — Weir formulas are discussed on page 
1177, etc. In constructing a standard weir it is necessary to 
have the crest perfectly level and the sides vertical. The 
inner edges of the opening should be sharp and chisel-edged 
or square cornered. If square cornered the boards or parti- 
tion should be thin or they may be beveled on the down- 
stream side, thus, ft. The crest is often formed of a thin 

sheet of metal fastened on the inside of the wooden parti- 
tion. Special care must be taken to insure free access of 
air under the falling sheet of water below (down stream 
from) the crest, otherwise a partial vacuum will form there, 
draw the sheet inward toward the weir, and affect the dis- 
charge. In measuring the head H on the crest, it is neces- 
sary to take the elevation of the water-surface at a suffi- 
cient distance D (Fig. 22) above the weir to reach the still- 
water level, or practically so. Francis used Z) = 6 ft., while 




Fig. 25. 






zr— 



HOOK GAGE, PITOT TUBE METER. FLOATS. 1183 

Bazin used jD=16.4 ft. Hence the distance D may depend somewhat on 
the formula to be used (see pages 1177 and 1178). There ^e three methods 
in use for determining the head H. One is by setting a reading gage in the 
stream, using the elevation of crest as the datum plane. The second method 
consists in suspending a plumbbob on the end of a steel tape supported 
from some point E, Fig. 22, at a known elevation h above the crest, and 
measuring the vertical distance d to the water surface. Then the required 
head H = h — d. In order to get the measurement d accurately, the plumb- 
bob may be allowed to swing gently, and raised until it just ceases to cause 
a ripple on the water surface. The third method is by the Hook Gage, 
shown in Fig. 25. The point of the hook at the bottom of the rod (the hook 
may be attached to a leveling rod, reading to thousandths of a ft.) is raised 
until it pierces the "skin" of the surface, raising a slight pimple. ^ The hook 
is then lowered until the pimple "just" disappears. The elevation is read 
by the vernier, which should be set at zero when the end of the hook is 
at the elevation of the crest of the weir. 

Pilot tube meter. — The Pitot tube, in its primitive conception, consists 
essentially of a bent tube (of glass) inserted in a cur- 
rent of water. Fig. 26, with the open end of the lower 
arm squarely facing the current. Then, theoretically, 

will the water rise in the tube to a height h^=-^, in 

which t; = velocity at that particular point in ft. per 

second ; "^ 

and /?. = velocity head in ft. due to v. Fig. 26. 

Hence, by mea suring the height hs of the column of water in the tube, the 
velocity v{==V2gh.) is obtained. 

For measuring the velocity of flow under pressure, as in pipes, the Pitot 
tube, complete, comprises two pipes, crudely 
shown in Fig. 27. One of these pipes termi- 
nates as in Fig. 26, and records the velocity ff 
head /tv+ the pressure head /^p. The other ter- ""*— 
minates as a piezometer tube in an orifice at 
right angle to the direction of flow, and 
records the pressure head h^ alone. It is 
clearly evident, then, that the difference in 
elevation of the water levels in the tubes is 
the velocity head h . and that, theoretically, 
the velocity z; = V2g /^v as above. ^. <,_ 

Fig. 28 shows one of the forms of Pitot ^^S- ^'• 

tubes used in the Detroit experiments,* to receive the velocity and pressure 
heads. For accurate reading, the tops of both of the tubes (Fig. 27) are 
connected by pipes to a differential gauge consisting of two parallel glass 
tubes with a sliding scale between. The dift'erence in elevation is reduced 
to hv as above, or to any other equivalent expression. Mercurial or oil gauges 
may be used. (See Trans. Am. Soc. C. E., Vol. XLVII, pp. 72-3). 

Floats. — ^These are classed as surface -floats, sub-surface floats and rod 
floats. 

Surface floats (Fig. 29) are the least accurate. Prof. Dwight Porter 
of the Mass. Inst. Tech. says: "These are of little value when run alone, 
since they are easily affected by wind, and the relation between surface and 
mean velocity is uncertain. As a rough approximation, the mean velocity 
in a vertical may be assumed as from 0.9 to 1.0 times the surface velocity 
in a vertical, and the mean velocity for the entire cross-section of the 
stream as about 0.8 times the maximum surface velocity." 

Sub-surface floats consist of small cylindrical boxes, say 8 or 9 ins. in 
diameter, and weighted at the bottom section with a small cube of lead. 
These are suspended from small surface floats to any depth at which it is 
required to measure the velocity. 

Rod floats are the most exact. They are particularly adapted to measur- 
ing the flow in canals or in streams where the depth of water is fairly con- 
stant. The length of rod should be equal to nearly the depth of the water, 
should project slightly above the surface and reach nearly to the bottom. 
They are used frequently in depths exceeding even 30 feet. The advantage 
of the road float is that the mean velocity in the vertical is obtained directly 

* See Trans. Am. Soc. C. E., Vol. XLVII, p. 12. 



1184 



,—HYDkAUUCS. 





20erman Silver sz 
Fig. 28. — Pitot Tube Meter. (See page 1183.) 



PITOT TUBE, FLOATS, CURRENT METERS. 



1185 



I 



k 6 — ^ 

«W^ V/ew 




Top View 

Fig. 29. 



for any vertical section of the stream or canal in which it is allowed to float 
at a "timed" rate between two cross-sections of the stream at a given dis- 
tance apart. The floats may be made of long, tin 
cylinders about 2 to 2^ ins. in diameter, loaded at the 
bottom with lead, accurately weighted or adjusted 
with pebbles, and closed at the top with cork. In 
measuring the discharge in a canal, the cross-section 
of the latter is divided into vertical strips of given 
area, the mean velocity for each of these strips is 
determined by the rod float, which velocity multiplied 
by its area gives the discharge for that strip. The 
total discharge divided by the total area of cross- 
section gives the mean velocity. The rods are floated 
through the middle of the strips at the dotted lines 
shown in Fig. 30, which represents a section of the 
canal. 

Current meters. — Current meters may be expected 
to give results with error ranging from 3 to 10 per 
cent. The greatest accuracy will be obtained in 
canals with a moderately high velocity of flow; the 
least accuracy in turbulent streams with cross cur- 
rents, also when the velocity of the water is very 
slight, or when it contains suspended matter, as in 
sewers. 

There are two principal types of current 
meters, namely, the cup-wheel meter and 
the propeller-wheel meter. Fig. 31 illus- 
trates the former type and Fig. 32 the lat- 
ter. The turns of the wheel which indicate 
the velocity of the water are registered by 
an electric meter similar to that shown in 
Fig. 33. All meters should be tested or 
rated before being 
used, as they are 
sometimes subject to 
fluctuations which 
may seriously affect 
the result. The me- 
ter is rated by moving 
it through still water 
at given speeds, plot- 
ting the results on 
cross - section paper, 
with "revolutions per 
minute" as abscissa, 
and "velocity in feet 
per second" as ordi- 
nates. The curve 
drawn through the 
plotted points is the 
"rating line" for the 
meter for immediate 
use. 

Fig. 31. — Price Meter. 



Fig. 30. 





Fig. 32.— Haskell Meter. 



1186 



62.— HYDRAULICS. 



The current meter may be used in two ways, namely, by the "spot" 
method and by the vertical "integration" method. The former is preferable 
because more accurate, and consists in holding the meter in a given position 
or point ot the cross-section of the stream, that is, the center of a given area, 
for a definite length of time and noting the register of the meter for that 
interval. The velocity per second and the discharge for that area can be 
obtained, and likewise for the other areas. The total discharge of the stream 
divided by the total area of cross-section gives the mean velocity, Vm. The 
integration method is quicker but somewhat less accurate unless very care- 




Fig. 33. — Meter Register. 

fully done. It consists in lowering the meter from the surface of the water 
to the bottom and then back again to the surface through the center of each 
given vertical strip as shown by the dotted lines. Fig. 30. The movement 
must be slow and constant and the reading of the Register must be for the 
actual time consumed, beginning and ending with the caps or vanes of the 
meter well under the surface. The result deduced is the velocity and dis- 
charge for that particular vertical strip. The other strips are treated in the 
same manner, and the resulting mean velocity obtained. 



CURRENT METER REGISTER. MISCELLANY, 1187 

EXCERPTS AND REFERENCES. 

Instructions for Installing Weirs, Measuriiig=Flunies and Water 
Registers (By C. T. Johnston, and Elwood Mead, Paper, U. S. Dept. of 
Agnculttire, and Bulletin 86, Irrigation Investigations; Eng. News, Aug. 29, 
1901). — Illustrations: Fig. 1, arrangement of Cippoletti weir; Fig. 2, meas- 
uring flume and register; Figs. 3 and 4, arrangement of pulleys to magnify 
the record of water registers; Figs. 3a and 4a, arrangement of pulleys to 
reduce record of water registers; Fig. 5, the Mead stream heights or water 
register; Fig. 6, the Friez stream heights or water register; Fig. 7, the 
Leitz stream heights or water register; Fig. 8, the standard stream heights 
or water register. Descriptions and discussions. 

Measurement of the Flow of Water in the Sudbury and Cochituate 
Aqueducts (By W. W. Patch. Eng. News, June 12, 1902).— Illustration of 
current meter apparatus; also diagrams. 

Current Meter and Weir Discharge Comparisons (By E. C. Murphy. 
Trans. A. S. C. E., Vol. XL VII). 

The Effect of Long Lengths of Hose on Fire Streams (By S. A. Charles. 
Paper, Am. W, W. Assn., June 12, 1902; Eng. News, July 3, 1902).— Dia- 
gram showing heights, volumes and velocities of fire streams for various 
lengths of hose, size of nozzle and pounds of hydrant or steamer and nozzle 

Pressure — from experiments by S. A. Charles, combined with those af 
. R. Freeman. See discussions in Eng. News of July 17 and 31, and Dec. 3, 
1902. 

The Flow of Water in Wood Pipes (By Theron A. Noble. Trans. 
A. S. C. E., Vol. XLIX). 

Methods of Measuring the Flow of Streams (By J. C. Hoyt. Eng. 
News, Jan. 14, 1904). — Interesting diagram showing vertical velocity curves 
for Susquehanna River; also table of velocity determinations. See, also, 
Eng. News, of Aug. 4, 1904. 

Notes on the Computation of Stream Qagings (By O. V. P. Stout. 
Paper, Soc. of Eng. and Mech., 12th Natl. Irrig. Cong., Nov. 15 to 18, 1904; 
Eng. News, Dec. 8, 1904). — Formulas and diagrams. 

The Hydraulic Plant of the Puget Sound Power Company (By E. H. 
Warner. Trans. A. S. C. E., Vol. LV).— Table 1, Make-up of steel pipe; 
Table 3, Experiments on the hydraulics of the timljer flume. 

Depth of Thread of Mean Velocity in Rivers (By F. W. Hanna. Eng. 
News, Jan. 11, 1906). — In the measurement of the flow of rivers it is often 
assumed that the thread of mean velocity lies at approximately 0.6 of the 
total depth from the surface, varying for extreme cases between the limits 
of 0.5 and 0.7. There seems to be a general impression that this assumption 
is not capable of theoretical demonstration. Mr. Hanna proceeds to demon- 
strate "rationally" that these values are correct, from the assumption that 
the vertical velocity curve is sensibly a parabola. 

An Experiment to Determine *'N'* in Kutter*s Formula (By C. W. 

Babb. Eng. News, Feb. 1, 1906). — In connection with the work of the 
Reclamation Service of the St. Mary project in Montana, the values of "n" 
in earth canal were found to be: .027, .024, .023, .023, .021; average .0236, 
or practically .025. 

Some Experiments on the Frictionless Orifice (By Horace Judd and 
R. S. King. Paper, Am. Assn. for the Advan. of Science, July, 1906; Eng. 
News, Sept. 27, 1906). — Diagrams and tables. 

Additional Information on the Durability of Wooden Stave Pipe 

(By A. L. Adams. Trans. A. S. C. E., Vol. LVIII).— Refers to the 1\ miles 
of wood-stave pipe laid for the Astoria city water works ten years previously. 
As a result of examination of the pipe line, the following conclusions are 
drawn: (1) Staves which are constantly subject to water pressure from 
within and are buried in the ground, may be very short-lived. (2) The 
magnitude of the water pressure, beyond a moderate head, has but little 
or no influence in preserving the timber. (3) The pipe laid above ground 
has not deteriorated to any considerable extent, nor has the pipe laid in 
the trenches leading from the distributing reservoir. (4) Where buried, its 
durability has depended upon the soil conditions and the depth of back- 
fill. (5) When the depth of backfill has exceeded 2 ft. above the pipe, and 
the material has been free from vegetable matter, and has been of a fine 



1188 



Q2.— HYDRAULICS, 



and impervious character, much less deterioration has taken place. (6) 
Whenever the staves have been in contact with loamy earth or earth con- 
taining vegetable matter, or wherever they have been covered with porous 
material, or to a depth less than 2 ft., rapid decay has resulted. (7) De- 
cayed staves have been found all around the pipe. (8) Sound staves have 
been frequently found contiguous to badly decayed staves. (9) The char- 
acter of the grain, whether slash or grain edge, has not influenced the 
diirability. (10) The bruising of the staves during the process of erecting 
seems to have been one of the chief agencies in hastening decay. (11) De- 
cay has not been confined to the outside of the pipe. (12) The pipe has not 
usually shown leakage as long as sound wood has remained in excess of i" 
in thickness. (13) The malleable cast band fastenings have been found to 
be in good condition. (14) The bands, i^'' in size, have been considerably 
corroded save where secured by the nut, but all have been used again by 
placing the nut in its original position. (15) "The pipe in the 2nd and 3rd 
Sections, 2^ miles, is nearly all buried in fine-grained sand, and will last 
perhaps 10 years more by giving it a general repairing, say 5 years hence, 
but the greater part of the 1st and 4th Sections will have to be replaced 
in about 4 years. — Superintendent." For Mr. Adam's original paper, see 
Trans. A. S. C. E., Vol. XXXVI; see also Trans., Vol. XLI. 

The Kinetic Energy of Flowing Water (By L. F. Harza. Eng. News, 
Mar. 7, 1907). — ^Formulas for the energy in rivers, streams, pipes, etc. 

Observations to Determine Values of "C" and "N" as Used in 
Kutter's Formula (By J. B. Lippincott). — Eight experiments described: 
Exper. No. 1. — ^Tunnel 446 ft. long, floor grade 0.00096, rectangular section 
4^' wide X 4' deep with semi-circular arch and finished with a 1:3 cement 
mortar plaster. Exper. No. 2. — Tunnel 318 ft. long, floor grade 0.00095, 
section and finish same as No. 1. Exper. No. 5.— -Canal, 800-ft. length, 
trapezoidal section with slopes 1:1 and bottom width 11.5 ft., concrete 
lined with 1:3 mortar plaster, bottom filled up with 1.5 to 2.5 ft. of fine 
sand. Exper. No. 4-— Canal, 600-ft. length, trapezoidal section with side 
slopes 2 on 1 and bottom width 8 ft., concrete lined with 1:3 mortar spread 
roughly, partially cleaned before measurements were made. Exper. Nos. 6 
and ^.—Conduit, 700-ft. length, concrete lined with 1:3 cement mortar in 
smooth condition, grade of floor same as water surface. Exper. No. 7.— - 
Canal, 1000-ft. tangent, lining of concrete without plaster, bottom free from 
sand and gravel, sides and bottom covered with thin coat of moss. Exper. 
No. 8. — Conduit, 1000-ft. length, lining of concrete tamped in behind boards 
and not plastered, several inches of sand and gravel on bottom, some moss 
and grass on sides. 

Tabular Results of Experiments: 



Ex 


Dis- 


Area 


Mean 


Hyd. 


Wetted 


Hydraul. 


Coeffi- 


Coef . of 


pen- 


charge. 


Water 


Veloc- 


Mean 


Perim. 


Grade. 


cient. 


Rough- 


ment. 




Section. 


ity. 


Radius. 








ness. 


No. 


Q- 


a. 


V. 


r. 


P- 


s. 


c. 


n. 


1 


70.60 


14.08 


5.01 


1.31 


10.71 


0.00126 


123.3 


0.0128 


2 


74.64 


15.73 


4.74 


1.37 


11.50 


0.00082 


141.6 


0.0113 


3 


60.52 


30.92 


1.96 


1.49 


20.74 


0.00092 


52.9 


0.0284 


4 


8.50 


6.96 


1.22 


0.703 


9.89 


0.00063 


58.0 


0.0218 


5 


13.55 


5.00 


2.71 


0.817 


6.12 


0.00051 


132.6 


0.0111 


6 


14.51 


5.16 


2.81 


0.830 


6.22 


0.00051 


136.7 


0.0108 


7 


15.34 


6.75 


2.27 


0.982 


6.875 


0.0007 


86.7 


0.0166 


8 


26.79 


10.21 


2.62 


0.817 


12.50 


0.00106 


89.2 


0.0157 



The use of the following coefficients in Kutter's formula was suggested 
by the board of consulting engineers, consisting of John R. Freeman, 
Frederic P. Stearns and Jas. D. Schuyler: (1) For open masonry conduits 
of cement or smoothly plastered masonry, « = 0.018; (2) for concrete-lined 
tunnels, or covered masonry conduits, « = 0.014; (3) for steel pipe with 
rivet heads and seams projecting on the interior, « = 0.016; (4) for earth 
canals with bottom left by dredging, n = 0.0275. — {Eng. News, June 6, 1907.) 

The Effect of Changes in Canal Cross-Sections Upon the Rate of 
Flow (By F. W. Hanna. Eng. News, June 6, 1907). — In the construction 



MISCELLANEOUS DATA. 1189 

of a canal it is usually necessary to provide for carrying its waters through 
culverts, flumes, siphons and other important changes of cross-section of 
waterway; and in order to compute properly its capacity, some estimate 
of the effects of such changes on the flow must be made. The mathematical 
discussion in this article deals with this problem. 

A Solution of the Problem of Determining the Economic Size of 
Pipe for High=Pressure Water=Power Installation (By A. L. Adams. Trans. 
A. S. C. E., Vol. LIX). — Rule: "That pipe fulfills the requirements of 
greatest economy wherein the value of the energy annually lost in frictional 
resistance equals four-tenths (0.4) of the annual cost of the pipe line." 
Discussed by formulas. 

The Flow of Water Through Submerged Tubes; Results of Experi- 
ments at the University of Wisconsin (By C. B. Stewart. Bulletin, Univ. of 
Wisconsin. Eng. News, Jan. 9, 1908). — Very extensive article with illus- 
trations and diagrams. 

A Logarithmic Diagram for the Flow of Water in Open Channels 

(By G. T. Prince. Eng. News. Feb. 6, 1908). 

Coefficient of Discharge through Circular Orifices (Eng. News, 
July 9, 1908). — ^Tables of coefficients; also seven conclusions stated by the 
writer, Mr. H. J. Bilton. 

Bazin's Hydraulic Formula ("Annales des Fonts et Chaussees" for 
the fourth quarter of 1897; Eng. News, Au^. 13, 1908). — Given in metric 
measure, as follows: — z; = 87^/r5-^(l+m-^^/ 5), the value of m varying with 
the surface of the channel as follows: (1) Very smooth (cement, planed 
wood, etc.), m = 0.06; (2) smooth (plank, brick, cut stone, etc.), m = 0.16; 
(3) rough masonry, w = 0.46; (3a) mixed, or intermediate, very regular 
earth excavation, paved slopes, etc., w = 0.85; (4) ordinary earth channels, 
m= 1.30; (5) earth channels in bad condition, w= 1.75. 

A diagram for Bazin's formula for flow in open channels, prepared by 
O. von Voigtlander, is published in Eng. News of April 15, 1909. 

A Collection of Formulas for Water=Pressure and Moments in Sub- 
merged Beams (By D. N. Showalter. Eng. News. May 27, 1909). — Formu- 
las, illustrations and tables. 

Ratings of a Pitot Tube (By E. C. Murphy. Eng. News, Aug. 12, 1909.)— 
Illustrated. Rating curve diagrams. 

Huge (625 Cu. Ft. per Sec.) Venturi Meters, India (Eng. News, Nov. 25, 
1909).— Illustrated. 

The Use and Care of the Current Meter, as Practised by the U. S. Qeol. 
Survey (By J. C. Hoyt. Trans. A. S. C. E., Vol. LXVL, Mar., 1910).— 
Illustrations and descriptions of various types of meters and recording 
devices, rating of meters, and methods of making hydraulic measurements. 

Friction of Air in Small Pipes (By E. G. Harris. Univ. of Missouri 
Bulletin; Eng. Rec, Dec. 3 1910). — Experiments to determine the value 
of the coefficient c in the formula: 

in which / = loss of pressure in lbs. per sq. in.; Z=length of pii)e in ft.; y=» 
the cu. ft. of free air passing per second; <i = diam. of pipe in ins.; r= ratio 
of compression to atmospheres; c = the experimental coefficient. The 
result of the experiments indicated that for pipes of i" to 12" in diameter 
the value of c= 0.076— 0.00188ci, so that the above formula would reduce to: 

/=(0.076~0.00188d)^. 



63.— WATER SUPPLY. 

Source and Distribution. — Water supply is derived primarily from rain, 
snow, hail, sleet and dew, generally termed rainfall or precipitation. The 
amount of rainfall in any locality is the depth of the precipitation in inches, 
when melted. In some localities dew forms no inconsiderable proportion of 
the total. The "dew-point" is that temperature at which the air begins to 
deposit (more) moisture (than it takes up). It is not a fixed temperature. 

The Distribution of Rainfall and Source of Water Supply may be grouped 
as follows: Rainfall — 

if Streams. 
Surface Water | Reservoirs { Art^fidL?^^^^'^ * 
Evaporation. 



Seepage 



GroundWater{Ve«^t,-«-,. 



Underground 
Water 



' Galleries. 

Springs. 

f Shallow. 

Wells { -n^^^ A^-.r^^ / Artesian.* 

I Deep driven | p^^pi^g. 



Use in Various Branches of Engineering. — The consideration of water 

supply is pertinent to the sections which follow, namely, 

64. Water Works page 1202. 

65. Sanitation page 1295. 

66. Irrigation page 1313. 

67. Waterways page 1320. 

68. Water Power page 1332. 

70. Electric Power and Lighting page 1379. 

Rainfall. — Engineers are concerned mainly with ( 1) the average monthly 
precipitation, (2) the monthly rainfall for the driest years, and (3) the maxi- 
mum rates of rainfall which may be expected, in any given locality. The 
first two are for the consideration of Supply; the last, for Discharge* 

*Mr. Myron L. Fuller, Geologist in Charge of Underground Water 
SuppHes, U. S. Geol. Surv. (Water Supply & Irrigation Paper No. 160), 
states that there is much looseness in the use of the word "artesian," and 
great variation of its use in different parts of the country. After a careful 
canvass of the leading geologists of the country engaged in hydraulic 
studies, he has proposed the following definitions of terms; 

Artesian Principle. — The hydrostatic principle in virtue of which water 
confined in the materials of the earth's crust tends to rise to the level of the 
water surface at the highest point from which water is transmitted. Gas 
as an agent in causing the water to rise is expressly excluded from this 
definition. 

Artesian Pressure. — The pressure exhibited by water confined in the 
earth's crust at a level lower than its static head. 

Artesian Water. — That portion of the undergroiind water which is under 
artesian pressure and will rise if encountered by a well or other passage 
affording an outlet. 

Artesian System. — Any combination of geologic structures, such as 
basins, planes, joints, faults, etc., in which waters are confined under 
artesian pressure. 

Artesian Basin. — A basin of porous bedded rock in which, as a result of 
the synclinal structure, the water is confined under artesian pressure. 

Artesian Slope. — A monoclinal slope of bedded rocks in which water is 
confined beneath relatively impervious covers owing to the obstruction to 
its downward passage by the pinching out of the porous beds, by^ their 
change from a pervious to an impervious character, by internal friction, or 
by dykes or other obstructions. 

Artesian Area. — An area underlain by water under artesian pressure. 

Artesian Well. — Any well in which the water rises under artesian pressure 
when encountered. 

1190 



RAINFALL. ARTESIAN NOMENCLATURE. 



1191 



1, — ^Average Monthly Precipitation in the U. S., in Inches. 
From Time of Establishment of Station to End of 1904. 



Locality. 



Alabama. 

Birmingham (50.94) 

Mobile (61.37) 

Montgomery (50.85) 

Arizona. 

Flagstaff (21.32) 

Phoenix ( 4.47) 

Yuma ( 2.84) 

Arkansas. 

Fort Smith (41.27) 

Little Rock (49 68) 

California. 

Eureka ..(47.67) 

Fresno ( 8.78) 

Independence ( 3.66) 

Los Angeles (15.36) 

Mount Tamalpais (30.25) 

Point Reyes Light (25.32) 

Red Bluff (25.80) 

Sacramento (20.20) 

San Diego ( 9.72) 

San Francisco (22.39) 

San Luis Obispo (19.66) 

Colorado. 

Denver (13.88) 

Grand Junction . . . ( 7.61) 

Peuble (11.84) 

Connecticut. 

New Haven (47 06) 

District of Columbia 
Washington (42.89) 

Florida. 

Jacksonville (53.21) 

Jupiter (59.19) 

Key West (37.57) 

Pensacola (56.33) 

Tampa (53.99) 

Georgia. 

Atlanta (49.12) 

Augusta (48.10) 

Macon (46.28) 

Savannah (50.62) 

Idafio. 

Boise (12.11) 

Lewlston (13.46) 

Pocatello (11.82) 

Illinois. 

Cairo (41.52) 

Chicago (33.37) 

Springfield (38.67) 

Indiana. 

Evansville (40.77) 

Indianapolis (42.00) 

Iowa. 

Charles City (30.73) 

Davenport (32.82) 

Des Moines (32.41) 

Dubuque (34.62) 

Keokuk (35.57) 

Sioux City (25.41) 



2.42 
4.93 

7.62 

1.57 

0.92 

2.64 

4.75 

4.90 

4.72 

3.94 

1. 

4.75 

4 

0.48 



0.35 

3.96 

3.38 

3.12 
3.81 
2.05 
3.97 
3.06 

5.20 
4.12 
3.61 
3.12 

1.50 
0.86 
0.89 

3.69 
2.05 
2.45 

3.13 
2 

0.96 
1.65 
1.19 
1.50 
1.80 
0.51 



3.57 



5.95 
7.21 
6.26 

1.81 
0.28 
0.27 

3.68 

4. 

7.06 
1.56 
0.42 
2.87 
5.93 
3.69 
3.15 
2.86 
1.52 
3.23 
3.48 

0.91 
0.70 
0.69 

4.48 

4.04 

3.41 
3.20 
1.27 
5.53 
2.76 

5.94 
4. 

5.69 
3.66 

1.70 
1.70 
2.20 

3.95 
2.59 
3.09 

5.32 
3.97 



5.13 
4.36 
4.53 

1.63 
0.20 
0.07 

3.97 
4.29 

4.17 

0.70 

14 

15 

00 



0. 

1. 

2. 

1.68 

2.08 

2.13 

0.67 

1.80 

1.56 

1.98 
0.72 
1.22 

3.56 

3.21 

2.82 
2.59 
1.21 
3.07 
1.91 



3.69 
3.48 
3.70 
3.15 



1.23 
0.82 
1.51 

3.63 
2.74 
3.63 

2.95 
3.46 

2.51 
2.71 
3.00 
2.92 
3.29 
2.78 



2.93 
4.10 
3.84 

1.21 
0.03 
0.03 

4.94 
5.11 



2.58 
0.72 
1.92 

3- 
3. 

3.94 
4.94 
2.77 
2.84 
2.62 

3.26 
3.22 
3.09 
2.81 

0.99 
1.47 
1.15 

3. 

3.45 

4.71 

3.39 
4.03 

4.02 
4.34 
4.78 
4.38 
4.32 
4.04 



4.17 
5.98 
4.34 

0.55 

0.13 

T. 

4.08 
3.97 

1.06 
0.12 
0.06 
0.09 
0.11 
0.20 
0.51 
0.14 
0.05 
0.19 
0.10 

1.49 
0.37 
1.40 

2. 

4.04 

5.51 
6.60 
4.14 
5.10 
9.23 

4.03 
4.82 
4.13 
6.04 

0.47 
1.18 
0.42 

4.40 
3.63 
4.55 

4.59 
4.35 

5.16 
4.00 
4.88 
4.56 
4.37 
3.92 



5.74 
6.70 
4.34 

1.79 
1.21 
0.13 

3.65 
4.02 

0.13 

T. 

0.04 

0.01 

0.02 

0.10 

0.02 

T. 

0.04 

0.02 

T. 

1.65 
0.51 
2.07 

4.92 

4.52 

6.19 
5.02 
3.64 
6.85 
8.00 

4.86 
5.05 
4.69 
6.06 

0.22 
1.01 
0.23 

3.47 
3.62 
3.10 

4.03 
4.17 

3.96 
3.65 
3.83 
4.58 
4.26 
3.67 



4.81 
6.99 
4.46 

3.30 
0.99 
0.32 

3.49 
3.57 



2.10 
4.85 

2:70 



09 

13 

0.17 



1.36 
1.05 
1.61 

4.92 

3.93 

6.06 
5.17 
4.72 
7.99 
8.51 

4.52 
5. 

4.73 

7.77 

0.20 
0.57 
0.37 

2.62 
2.88 
2.46 

2.31 
3.18 

2.83 
3.64 
3.57 
2.98 
3.15 
3.05 



3.13 
3.34 

1.26 
0.28 
0.14 
0.08 
0.78 
1.06 
0.77 
0.33 
0.05 
0.44 
0.39 

0.86 
0.84 
0.48 

3.67 

3.59 

8.16 
9.66 
6.91 
4.61 

7.84 

3.55 
3.68 
2.87 
5.67 

0.44 
0.89 
0.32 

2.52 
2.94 
3.22 

2.67 
3.35 

3.21 
3.21 
2.99 
3.91 
4.07 
2.42 



2.42 
3.16 
2.27 

1.39 
0.32 
0.23 

2.87 
2.40 

2.83 

0.53 

0.33 

0.81 

2.58 

2.19 

1.48 

1.08 

0, 

1.32 

1.63 

0.90 
0.83 
0.73 

3.84 

3.04 

5.21 
10.08 
5.31 
3.65 
2.98 

2.26 
2.40 
2.00 
3.57 

1.70 
1.05 
0.92 

2.60 
2.55 
2.71 

2.77 
2.74 

2. 

2.38 

2.75 

2.59 

2.62 

1.77 



3.04 
3.51 
3.27 

1.52 
0.68 
0.28 

3.26 
4.54 



,79 

.50 

4.53 



0.53 
0.49 
0.31 

3.64 

2.71 

2.45 
2.85 
2.25 
4.08 
1.87 

3.44 
2.96 
3.02 
2.44 

1.02 
1.28 
0.75 



2.53 
3.01 

3.81 
3.62 

1.39 

1. 

1.45 

1.78 

1.91 

0.74 



§.X 



0.69 
0.43 
0.49 

2.84 
4.09 

7.21 
1.45 
0.33 
2.99 
2.91 
3.34 
5.26 
3.79 
1.87 
4.21 
2.04 

0.64 
0.34 
0.52 

3.53 

3.06 

2.98 
2.41 
1.66 
4.11 
1.75 

4.35 
3.35 
4.18 
3.03 

1.03 
1.35 
1.14 

3.34 
2.09 
2.60 

3.33 
3.03 

1.39 
1.62 
1.36 
1.74 
1.78 
0.80 



* Figures enclosed in parentheses are totals in inches for the 12 months, 
and indicate average yearly precipitation for the period of years stated in the 
last column. 

Note. — "T" means trace — too small to measure. 



1192 



63.— WATER SUPPLY. 



1. — Average Monthly Precipitation in the U. S., in Inches. — 

Continued. 



Locality. 



Kansas. 

Concordia (27.35) 

Dodge (18.78) 

Topeka (34.04) 

Wichita (30.31) 

Kentucky. 

Lexington (41.64) 

Louisville (43.95) 

Louisiana. 

New Orleans (57.09) 

Shreveport (45.72) 

Maine. 

Eastport (42.98) 

Portland (42.89) 

Maryland. 
Baltimore (43,13) 

Massachusetts. 

Boston (43.64) 

Nantucket (37.33) 

Michigan. 

Alpena (33.38) 

Detroit (32.13) 

Escanaba (31.61) 

Grand Haven (33.70) 

Houghton (34.24) 

Marquette (32.43) 

Port Huron (30.88) 

Sault Ste. Marie (32.33( 

Minnesota. 

Duluth (29.77) 

Minneapolis (28.69) 

Moorhead (24.68) 

St. Paul (28.36) 

Mississippi. 

Meridian (51.84) 

Vicksburg (53.35) 

Missouri. 

Columbia (38.42) 

Hannibal (34.49) 

Kansas City (37.29) 

St. Louis (37.04) 

Springfield (43.00) 

Montana. 

Havre (14.18) 

Helena (13.06) 

Kalispell (15.50) 

Miles City (12.28) 

Nebraska. 

Lincoln (27.63) 

North Platte (18.07) 

Omaha (30.80) 

Valentine (19.42) 

Nevada. 

Carson City (10.75) 

"VS^innemucca ( 8.36) 

New Hampshire. 
Concord (40.30) 

New Jersey. 
Atlantic City (41.11) 

New Mexico. 
Santa Fe (14.19) 



2.23 
91 
49 
64 
82 

.01 
92 

,37 



0.68 
0.46 
0. 
0.78 

3.87 
3.86 

4.53 
4.44 

3.20 
3.71 

3.19 

3.84 
3.25 



0.98 
0.65 
0.70 
0.90 

4.89 
5.49 

2.14 
2.13 
1.25 
2.22 
2.45 

0.76 
1.06 
0.98 
0.41 

0.67 
0.42 
0.62 
0.57 

1.97 
1.05 

3.33 

3.44 

0.58 



1.44 
0.85 
2.09 
2.06 

4.77 
4.35 

5.14 
4.48 

4.54 
3.97 



4.27 
3.95 

1.95 
2.46 
1.88 
2.39 
2.94 
1.97 
2.54 
1.79 



1.56 
1.77 
1.12 
1.61 



5.36 
6.22 

3.12 
2.79 
2.70 
3.46 
3.93 

0.55 
0.84 
0.89 
1.34 

1.21 
0.76 
1.42 
1.37 

1.43 
0.93 

3.52 

3.64 

0.71 



2.40 
1.84 
2.74 
2.82 

3.23 
3.97 

4.98 
4.60 

2.99 
3.14 

3.24 

3.46 
2.88 

2.19 
2.22 
2.03 
2.47 
2.18 
2.11 
2.03 
2.05 

2.13 
2.46 
2.49 
2.40 

4.02 
5.13 

4.08 
3.18 
28 
3.42 
3.82 

0.97 
1.12 
0.80 
1.09 

2.67 

2.10 

03 

2.46 

0.57 
0.85 

2.91 

3.12 

0.75 



4.78 
3.19 

5.28 
4 

3.54 
3.60 

4.01 
3.87 

3.65 
3.66 

3.58 

3.45 
2.67 

3.28 
3.30 
3.48 
3.27 
2.93 
3.21 
3.25 
3.08 

3.45 
3.34 
2.60 
3.28 

3.94 
4.43 

4.96 
5.35 
5.47 
4.30 
5.73 



2.06 
1.98 
2.39 
2.06 

4.59 
2.80 
4.44 
2.24 

0.75 
1.03 

3.14 

2.93 

1.15 



4.77 
2.72 
4.76 
4.91 

4.11 
4.24 

6.19 
3.83 

3.37 
3.23 

3.77 

02 
2.17 

61 
83 
3.64 
3.38 
2.38 
3.43 
3.27 
2.91 

4.39 

3.75 
4.08 
4.35 

4.87 
4.43 

4.85 
3.67 
4.45 
4.62 
5.09 



65 
2.20 

4.36 
3.26 
5.18 
3.38 

0.25 
0.60 

3.25 

2.95 

1.04 



3.68 
2.47 
4.90 
3.41 

3.92 
3.73 

6.36 



3.48 
3.34 

4.66 

3.47 
3.42 

3.12 
3.49 
3.36 
2.83 
3.48 
3.16 
2.70 
.00 

3.75 
4.22 
3.89 
3.51 

4.92 
4.46 

4.24 
4.17 
5.01 
3.57 
18 

2.08 
1.16 
1.24 
1.29 

4.13 
2.69 
4.53 
3.15 

0.11 
0.18 

3.95 

3.74 

2.70 



2.90 
2.52 
4.46 
3.09 

3.71 
3.55 

5.68 
2.14 

3.19 
3.54 

4.14 

4.06 
2.86 



2.41 
1.48 
3.37 

2.87 

2.44 
2.66 

4.63 
3.42 

3.05 
3.22 

3.91 

3.19 
2.61 



39 3.54 



3.43 

3.80 
2.93 
3.44 

5.12 
3.33 

3.34 
3.07 
4.52 
2.44 
3.91 

1.32 
0.64 
1.10 
0.93 

39 
2.37 
3.56 

53 

0.22 
0.17 

3.68 

4.40 

2.43 



2.63 
3.62 
3.61 
4.20 
3.63 
2.73 
3 

3.55 
3.17 
2.30 
3.26 

3.01 
3.33 

3.53 
3.36 
3.59 
2.85 
3.37 



1.11 
1.14 
1.70 
0.: 



2.14 
1.37 
2.90 
1.10 

0.27 
0.33 

3.21 

3.08 

1.64 



2.19 

1.46 
1.99 
2.40 

2.03 
2.49 

2.96 
3.10 

4.00 
3.79 

3.01 

3.96 
3.66 

3.51 
2.38 
3.24 
3.02 
3.16 
3.16 
2.65 
3.27 

2.73 

2.75 
2.07 
2.40 

1.90 
2.44 

2.09 

1.48 
2.37 
2.19 

2.77 

0.64 
0.77 
0.67 
0.76 

2.07 
1.00 
2.47 
0.87 

0.46 
0.56 

3.46 

3.52 

1.05 



0.82 
0.50 
1.08 
0.90 

3.42 



3.74 
4.12 

4.05 
3.76 

2.96 

4.16 
3.45 



2. 
2. 
2. 
2. 
3. 
2. 
2. 
3. 

1. 
0. 

0.90 
1.21 

3.04 
4.22 

2.10 
1.91 
1.72 
2.84 
2.68 

0.74 
0.78 
1.65 
0.57 

0.77 
0.39 
0.99 
0.59 

21 
0.67 



3.20 
0.68 



0.50 
0.59 
0.93 
0.89 

3.23 
3.70 

4.25 
4.46 

3 
3.62 

3.08 

3.26 
3.42 

2.20 
2.35 
1.68 
2.50 
3.18 
2.44 
2.20 
2.46 



1.00 

0.79 

12 

4.61 
5.11 

1.81 
1.55 
1.33 
2.26 
2.60 

0.54 
0.80 
1.27 
0.32 

0.76 
0.47 
0.94 
0.49 

1.79 
1.07 

3.34 

3.76 

0.72 



* Figures enclosed in parentheses are totals in inches for the 12 months, 
and indicate average yearly precipitation for the period of years stated in the 
last column. 



AVERAGE MONTHLY PRECIPITATION. 



1193 



1. — Average Monthly Precipitation in the U. S.. in Inches. — 

Continued. 



Locality. 



New York. 

Albany (36.84) 

Binghamton (31.26) 

Buffalo (32.64) 

New York (44.65) 

Oswego (36.06) 

Rochester (34.47) 

North Carolina. 

AsheviUe (42.45) 

Charlotte (49.24) 

Hatteras (61.17) 

Raleigh (49.74) 

Wilmington (50.97) 

North Dakota. 

Bismarck (17.67) 

Williston (14.96) 

Ohio. 

Cincinnati (38.13) 

Cleveland (34.90) 

Columbus (36.98) 

Sandusky (34.14) 

Toledo (30.81) 

Oklahoma. 

Oklahoma (31.44) 

Oregon. 

Baker City.. (13.27) 

Portland (45.48) 

Roseburg (35.65) 

Pennsylvania. 

Erie (39.45) 

Harrlsburg (37.56) 

Philadelphia (40.54) 

Pittsburg (36.59) 

Scranton (41.62) 

Rhode Island. 

Block Island (44.81) 

Narragansett (47.98) 

South Carolina. 

Charleston (52.57) 

Columbia (47.16) 

South Dakota. 

Huron.... (20.44) 

Pierre (16.24) 

Rapid City (16.49) 

Yankton (25.85) 

Tennessee. 

Chattanooga (50.89) 

KnoxviUe (49.68) 

Memphis (50.81) 

Nashville • (48.30) 

Texas 

Abilene.......* (24.19) 

AmariUo (21.03) 

Corpus Christl (27.03) 

El Paso (9.48) 

Fort Worth (28.79) 

Galveston (47.51) 

Palestine (43.96) 

San Antonio (27.01) 

Taylor (34.53) 



2.60 
72 
21 



14 
3.24 

3.04 

4.23 

5.11 

.56 

.62 

0.53 
0.59 

28 

48 

03 

2.14 

1.95 

1.36 

1.31 
6.70 
5.86 

05 
2.78 
3.28 
2 
2.32 

4.02 
4.62 

3.32 
3.41 



57 
2.00 
2.99 
3.84 
2.64 
2.85 

3.57 
4.54 
4.50 
4.41 
3.35 

0.64 
0.50 

3.42 
2.73 
3.25 
2.58 
2.08 

0.95 

1.49 
6.00 
4. 



10 



3.39 
2.79 
2.63 



49 

48 
0.36 
0.500.73 



4.32 
4.47 

3.37 

4.75 

0.48 
0.49 
0.55 



83 
2. 

70 
4.05 
2.96 
3.07 

4.14 
4.71 
5.35 
4.23 
3.59 

1.00 
0.70 

3.72 

2.69 

3.28 

2 

2.21 

2.19 

1.34 
5.19 
3.90 

2.87 
3.18 
3.40 
3.07 
3.22 

4.43 
4.84 

3.68 
4.13 

0.98 



5.63 
5.05 
5.40 
4.80 

0.89 
0.56 
2.22 
0.50 
1.87 
3.62 
4.15 
1.87 
0.79 



5.17 
5.15 
4.98 
4.68 



1.32 
1.24 
1.24 



6.31 
5.61 
5.67 
5.29 

1.34 

0.39 

1.80 

0.33 

2. 

3.09 

3.81 

1.60 

1.47 



3.08 
3.45 
4.25 
3.42 
2.72 

2.22 
1.26 

2 

2.27 

2.62 

2.44 

22 

2.48 

0.97 
3.16 
2.61 

2.71 
2.44 
2.89 
2.91 
2.64 

3.76 
3.97 

3.10 
2.90 

2.73 
2.11 
2.20 
3.08 

4.48 
4 

4.95 
4.51 

2.24 

1.58 
1.72 
0.21 



832 
3 
4 
2 
2 



00 

69 

14 

3.16 

77 
02 

3.62 
84 
4.16 
5.01 
3.93 

2.58 
2.13 

3.41 
3.24 
3.83 
3.30 
3.37 

1.24 

1.73 

35 

2.04 



3.77 
3.24 
3.52 
3.27 
3.31 
.05 

4.08 
4.55 
4.09 
4.73 
5.55 

2.58 
3.50 

3 94 
3.64 
3.55 
3.90 
3.42 

3.19 



62 
10 
3.16 
35 
58 



3.71 
4.48 

3.52 
3. 

2.66 
1 

2.76 
3.84 

3.76 
3.77 
4.29 
3.50 



1.16 
1.70 
1.15 



3.75 
3.45 
3.18 
3.73 
4.92 

2.75 
2.58 

5.38 
4.36 

3.67 
3.18 
3.59 
3.98 



4.11 
4.15 
4.47 
4.25 

3.07 
3.14 
2.61 
0.55 
3.54 
4.57 
4.07 
2.80 
2.65 



89 
11 
42 
4.46 
3.33 
16 

4.84 
5.31 
.28 
6.12 
6.71 

2.57 
2.08 

38 
3.63 
3.59 
3.67 
3.25 

3.30 

0.44 
0.58 
0.37 

13 
3.91 
4.19 
4.59 
4. 

3.21 
3.19 

39 
5.74 

2 

2.39 
1.91 
3.76 

3.65 
4.09 
3.32 
4.40 



3.98 
3.35 
3.07 
4.59 
2.65 
2.90 



63 

D.25 
6.10 
5. 
6. 

2.12 
1.28 

3.34 
3.01 
3.06 
3.27 
2.73 

2.75 

0.44 
0.64 
0.35 

3.17 
4.11 
4.50 
3.12 

5.22 

3.50 
3.99 

7.31 
6.64 

2.60 
1.87 
1.41 
3.16 

3.84 
4.08 
3.39 
3.39 



2.012.00 
2 



27 
82 
3.15 
3.50 
2.84 
2.37 

2.95 
3.27 
5.56 
40 
5.32 

1.04 
0.85 

2.35 
3.34 

2.54 

2 

2.44 

2.69 

0.77 

76 

1.12 

3.57 
2.77 
3.41 
2.46 
3.30 

3.08 
66 

5.48 
3.74 

1.60 
1.02 
0.91 
2 

3.31 
2.92 
2.99 
3.63 



2.72 
2.33 
1.65 
1.80 
5.08 
2.36 
2.86 
1.29 



4^ 


> 


6 


O 


o 


o 


O 


"^ 


Q 


3.06 


2.78 


2.69 


2.98 


2.04 


2.27 


3.33 


3.34 


3.37 


3.71 


3.57 


3.39 


3.23 


3.39 


3.62 


2.77 


2.72 


2.89 


2.61 


2.76 


3.13 


3.37 


2.95 


3.77 


6.03 


4.62 


5.12 


3.72 


2.42 


2.92 


3.78 


2.39 


3.05 


1.10 


0.63 


0.66 


0.86 


0.59 


0.62 


2.21 


3.18 


3.02 


2.60 


2.75 


2.52 


2.33 


3.20 


2.70 


2.37 


2.78 


2.32 


2.24 


2.63 


2.27 


1.94 


2.52 


1.83 


0.99 


1.12 


1.51 


3.56 


6.48 


7.36 


2.70 


4.43 


6.14 


3.70 


3.71 


3.07 


2.87 


2.46 


2.61 


3.01 


3.18 


2.95 


2.32 


2.59 


2.80 


4.07 


1.75 


4.08 


4.14 


4.06 


3.83 


4.23 


4.04 


3.91 


3.99 


2.92 


3.11 


2.55 


2.44 


3.02 


1.29 


0.55 


0.59 


0.67 


0.48 


0.55 


0.66 


0.41 


0.49 


1.38 


0.66 


0.88 


2.68 


3.62 


4.33 


2.52 


3.61 


4.07 


2.63 


4.37 


4.35 


2.28 


3.77 


3.80 


2 13 


1.18 


1.23 


1.64 


0.75 


0.86 


1.99 


2.23 


1.40 


1.05 


0.42 


0.49 


2.72 


1.65 


1.20 


4.19 


4.01 


3.76 


3.48 


3.73 


3.62 


1.52 


1.70 


1.62 


2.88 


2.36 


1.17 



l^' 



* Figures enclosed in parentheses are totals in inches for the 1 2 months, 
and indicate average yearly precipitation for the period of years stated in the 
last column. 



1194 



Q3— WATER SUPPLY. 



1. — Average Monthly Precipitation in the U. S., in Inches. — 

Concluded. 



Locality. 



Utah. 

Modena (7.92) 

Salt Lake City (15.94) 

VejTnont. 
Northfield (33.03) 

Virginia. 

Lynchburg (43.59) 

Norfolk (50.25) 

Richmond (43.28) 

Wytheville (39.72) 

Washington. 



Port Crescent . . 

Seattle 

Spokane 

Tacoma 

Tatoosh Island. . 
Walla Walla .... 

West Virginia. 

Elklns (42.21) 

Parkersburg (39.32) 

Wisconsin 

Green Bay (30.97) 

La Crosse (30.82) 

Milwaukee (30.88) 

Madison (31.82) 

Wyoming. 

Cheyenne (12.31) 

Lander (13.34) 

Yellowstone Park . . (19.27) 



(44.97) 
..(37.15) 
..(18.13) 
. . (47.34) 
. . (92.79) 
..(17.61) 



2.65 



1.45 
1.40 

2.31 

3.71 
3.80 
3.79 
3.54 

5.64 
4.03 
1.99 
5.87 
9.37 
1.62 

3.48 
3.60 

1.70 
1.05 
1.86 
1.56 

0.40 
0.68 
1.92 



0.58 
1 

2.99 

3.90 
4.62 
4.22 
3.78 

4.61 
3.31 
1.44 
4.31 
8.43 
1.86 

4.23 
3.78 

2.32 

1.58 
2.67 
2.18 

0.70 
1.59 
2 



0.35 
2.13 

2.16 

3.18 

3.90 

3.56 

92 



53 
97 

1.29 
48 
93 

1.78 



3.20 
2.99 

2.49 
2 

2.63 
2. 

1.39 

2.45 
1.23 



0.74 
1.97 

2.67 

3.97 
4.26 
3.09 
3.83 

1.95 
2.26 
1.40 
2.10 
4.31 
1.66 

4.20 
3.34 

47 

49 

3.32 

3.40 

2.25 

2 

1.94 



0.16 
0.73 

3.12 

3.81 
4.21 
3.63 
4.64 

1.45 
1.60 
1.48 
1.96 
4.04 
1.12 

5.72 
4.74 

3.46 
4.41 
3.44 
4.19 

1.53 
1.30 
1.65 



4.07 
5.81 
4.87 
3.78 



0.54 
0.52 

2.58 



0.57 
0.80 
0.69 
0.73 
2.08 
0.43 

4.72 
4.65 

3.46 
4.14 
3.11 
4.13 

1.79 

0. 

1.23 



1.55 
0.74 

3.91 

4.14 
5.87 
4.67 
4.41 

0.83 
0.50 
0.50 
0.68 
2.41 
0.41 

3.16 
3.38 

2.63 
3.30 
2.83 
3.10 

1.47 
0.53 
1.07 



1.08 
0.80 

2.83 

3.75 
4.19 
3.61 
3.00 

1.50 
2.12 
0.99 
2.29 
6.56 
0.94 

2.57 
2.74 

37 
4.05 
2.97 
3.24 

0.99 
0.94 
0.99 



0.74 
1.50 

2.34 

3.33 
3 

3.19 
2.60 

3.29 
2.96 
1.34 
2.76 
8.55 
1.49 

1.80 
2.23 

2.63 
2.52 
2.21 
2.48 

0.67 
1.03 
1.09 



0.30 
1.40 

2.75 

2.82 
2.87 
2.52 
1.99 

8.91 
6.31 
2.27 
9.49 
12.17 
2.07 

2.26 
2.86 

1.90 
1.48 
1.94 
1.76 

0.37 
0.52 
1.59 



0.16 
1.43 

2.72 

3.12 
3.45 
3.46 
2.79 

7.30 
5.96 
2.40 
7. 
15.02 
2.06 

3.34 

2.74 

1.86 
1.31 
1.88 
1.74 

0.38 
0.77 
1.86 



1^ 



* Figures enclosed in parentheses are totals in inches for the 12 months, 
and indicate average yearly precipitation for the period of years stated in the 
last column. 

In connection with the preceding table, the following Table, No. 2, shows 
in respective order the percentages of rainfall to average rainfall: First, for 
the driest year; Second, for the two driest years; Third, for the three driest 
years; closing with 1896. 

Ex. — For Pittsbiirg, the average rainfall, from Table 1, is 36.59 ins. 
Then, using the percentages in Table 2, following, we have: First, for the 
driest year, 36.59X0.70=25.61 ins.; Second, for the two driest years, 36,59 
X 0.78 = 28.54 ins.; Third, for the three driest years, 36.59X 0.85= 31.10 ins. 



RAINFALL— MONTHLY: HIGH INTENSITIES, 1195 

2. — *Percentage op Rainfall to Average Rainfall. 

(To be used in connection with preceding table. See Example, p. 1194.) 

North Atlantic: Boston, 60, 70, 80; New Haven, 74, 78, 82; New York, 62, 

77, 80; Philadelphia, 70, 75, 80; Washington, 69, 71, 74. 
South Atlantic: Wilmington, 75, 80, 81; Charleston, 48, 55, 62; Augusta, 

81, 88, 87; Jacksonville, 74, 77, 83; Key West, 54, 61, 73. 
Gulf and Lower Mississippi: Shreveport, 67, 75, 75; Montgomery, 76, 80, 83; 

Mobile, 68, 75, 78; New Orleans, 64, 75, 77; Galveston, 50, 65, 72; 

Nashville, 67, 73, 83; Vicksburg, 70, 74, 83. 
Ohio Valley: Pittsburg, 70, 78, 85; Cincinnati, 60, 72, 71; Indianapolis, 59, 

76, 82; Louisville, 74, 81, 85; Cairo, 62, 75, 81. 
Lake Region: Marquette, 69, 75, 81; Detroit, 65, 72, 79; Cleveland, 71, 74, 

81; Duluth, 65, 81, 88. 
Upper Mississippi Valley: St. Louis, 55, 65, 75; Davenport, 56, 68, 73; 

Chicago, 66, 80, 86; Milwaukee, 66, 74, 73; Madison, 39, 58, 68; 

LaCrosse, 57, 78, 79; St. Paul, 53, 54, 75. 
The Plains: Omaha, 57, 63, 70; Dodge City, 51, 58, 73; North Platte, 56, 

67, 72; Denver, 59, 71, 77; Cheyenne, 39, 62, 75. 
The Plateau: Yuma. 25, 50, 46; Phoenix, 52, 88, 90; Tucson, 44, 79, 80; 

Santa Fe, 53, 63, 66; Carson City, 57, 63, 72; Salt Lake City, 55, 64, 

74; Spokane, 73, 84, 84; Walla Walla, 46. 81, 86. 
Pacific Coast: Astoria (average rainfall 77 ins.), 64, 68, 77; Portland, 67, 76, 

79; Red Bluff, 52, 64, 58; Sacramento, 42, 67, 84; San Francisco, 51, 

73, 78; Los Angeles, 33, 48, 59; Fresno, 61, 65, 74; San Diego, 30, 

54. 61. 

High Intensities of Rainfall. — Flood discharges in rivers, creeks, sewers, 
etc., naturally occur at times of high intensities of rainfall, and hence it is 
important to know, for particular localities, the maximum rates of down- 
pour which have been recorded, and which are liable to take place in the 
future. Considerable confusion has arisen as to the meaning of the various 
terms "intensity of rainfall," "maximum rate of rainfall," etc.: for even a 
single shower has various intensities of downpour, while the recorded 
"intensity" often comprises the ratio of the total downpour to the total 
time, omitting, perhaps, the critical data most desired for the design of the 
sewer, namely, a higher intensity of downpour for a shorter period of time, 
during the shower. In order to clarify this subject, the following notation is 
used: 

Let (i = the depth of rainfall in ins., for any given time T in hrs.; 

7" = time in hours corresponding to depth of precipitation d in ins.; 
r = average rate of downpour in inches per hour during the whole 

shower = -=, designating the length of time by a sub-numeral 

in the denomination of hours, thus — 
d 4 

r* = -:=:= say -J- =^ average rkte of downpour during the whole 

4 hours of 'the shower; 
i= maximum average rate of downpour in inches per hour 

during a protracted period of highest intensity = -= , 

designating the length of time by a sub-numeral in 
the denomination of hours, thus — 
d Z 

*i-5= yr = say :j-r = maximum average rate for 1.5 hours; 

I = maximum, maximorum rate of downpour in inches per 

hour during the short time or greatest intensity = -^^ 

designating the length of time by a sub-numeral in 
the denomination of hours, thus — 
d 75 

/o-26 =-^ = say ^== highest intensity, for i hour. 
1 .lo 



* Abstract from Table No. 6, "Public Water Supplies" by Tumeaureand 
Russell: John Wiley & Sons, 1901. Although not strictly applicable to pre- 
ceding table closing with 1904, these percentages may be used with fairly 
good results. Published by permission. 



1196 



-WATER SUPPLY, 



Thus, from the above, say for any given shower, we have the complete 
data: r4= 1, ii.5= 2, Io-25— 3. This means that it rained 4 ins. in 4 hours., or 
at the average rate of 1 in. per hour; 2 ins. per hour for 1.5 hours of greatest 
downpour; and 0.75 in. in 15 minutes, or at the rate of 3 ins. per hour for 
the short period of greatest intensity. In current Hterature it is often proble- 
matical whether r, i or I is intended as the rate of downpour. 

Formulas for Maximum Intensity of Downpour, i. 

Notation. 

t = average rate of downpour in ins. per hour for the time t in mins.; 
t = time or duration of downpour in minutes. , 

Formulas. 

A. N. Talbot's "Maximum formula for Eastern U. S*. 

360 .. 

'^JTTo •••••^^> 

A. N. Talbot's "Ordinary" formula for Eastern U. S*. 

-<-^ ^^> 

E. Kuichling's formula for New York City and vicinity: 

-F^ ^3) 

tC. W. Sherman's formula for Eastern U. S., when t< 180: 

• J20 ,.. 

* = /T30 ^*> 

But when O180 the value of i is too small. 
fC. W. Shermans "Maximum" formula for Boston: 

fC. W. Sherman's "Ordinary" formula for Boston: 

• 2512 ... 

* = ToT^ <**> 

E. S. Dorr's "Ordinary" formula for Boston: 

150 ,„ 

'-TTsb ^^> 



Standard Rain Gage. — For a complete description of the measurement 
of precipitation, see Weather Bureau Paper "W. B., No. 285" U. S. Dept. 
of Agriculture, by C. F. Marvin, 1903, 
from which the following cut (standard 
8-in. gage) is reproduced. A is the 
receiver; B the overflow attachment; 
C the measuring tube; D the meas- 
uring stick. The diameter of the re- 
ceiving tube a is just 8 ins. (82=64) 
and of the measuring tube C is 2.53 
ins. (2.532=6.4); hence the rainfall 
is magnified in the latter 10 times, 
which facilitates accurate measure- 
ment. The measuring stick D is a 
strip of straight-grained cedar 0.08 in. 
thick, 5 in. wide, and 24 ins. long. 
When the tube C is full it overflows 
at d into B, and is later poured into C 
and measured. If the precipitation is 
snow or hail it is melted before meas- 
uring. Snow will occupy from 7 to 
34 times its depth melted. 




Horizontal ^^ 
Section CF. 



OI2JiS6_789IO 12 



Inchex 



Fig. 1. 



Scate 
Rain Gage. 



* Territory east of the Rocky Mountains. 

t See Trans A. S. C. E., Vol. LIV. pp. 178 and 212. 



RAINFALL FORMULAS. RAIN GAGE. RUNOFF, 



1197 



3. — Maximum Rates of Rainfall by Preceding Formulas. 

For periods of time ranging from 5 minutes to 3 hours. 

[Rates of downpour in inches per hour for time t.] 





Formula. 


Duration 


or time of Downpour. 




t=5m. 


t=10m. 


r=30m. 


t=QOm. 


r=180m. 


<» -4^0- 


Talbot's maximum; 
for Eastern U. S. 


8.57 


7.50 


5.00 


3.33 


1.43 


<^> -i^. 


Talbot's ordinary; 
tor Eastern U. S. 


5.25 


4.20 


2.33 


1.40 


0.54 


<3> -a- 


Kulchllng's: lorN. Y. 
City and vicinity. 


4.80 


4.00 


2.40 


1.50 


0.60 


<« -^• 


Shermans; for Eastern 
U. S.. when K 180. 


12.00 


10.50 


7.00 


4.67 


2.00 


(« i=f#. 


Sherman's maximum; 
for Boston. 


12.79 


7.94 


3.73 


2.32 


1.09 


m i 25.12 


Sherman's ordinary; 
for Boston. 


8.31 


5.16 


2.43 


1.51 


0.71 


(7) i- ^^\ 
^'^ * r+30' 


Dorr's ordinary; 
for Boston. 


4.29 


3.75 


2.50 


1.67 


0.71 



Note. — ^To obtain the total depth of rainfall for the duration of time t 

given: Multiply the rates or intensities * given in the table, by rr ( = 7', the 

time in hours) ; thus, from formula (3) the total rainfalls which it is possible 
to expect in 5-, 10-, 30-, 60- and 180 minutes of downpour are, respectively, 

4.80X ^ = 0.4 in., 4.00X ^^ = 0.67 in., 2.40X |5 = 1.20 in., 1.50X ^ = 1.50 in., 

180 
and 0.60 X-^ = 1.80 in. In calculating the maximum rate of discharge 

from a catchment area into a stream or sewer, it is necessary to estimate the 
time required for the rain to reach such outlet, say 30 minutes, more or less, 
for sewers, and a longer time for streams, generally. See Run-off, below. 

Run-off. — The term "run-off" is rather indefinite, but in general it may 
be defined as the surface discharge from a catchment area or basin. Each 
stream has its catchment area. The discharge from a stream may be 
estimated in three ways: 

1st. By Kutter's formula (p. 1167), with sectional area, mean radius 

and hydraulic slope known; 
2nd. By actually gaging the stream (see page 1182, etc.); 
3rd. By estimating the percentage of run-off to total rainfall. 

All three methods should be used if possible, for checking. 

Formulas for run-off are founded on actual stream gagings, but must be 
used with care for the following reasons: 

(a). Each stream, strictly spreaking, should have its own run-off formula 
(or formulas — see 6, c and d). 

(b) . The run-off increases with the amount and intensity of rainfall. 

(c) . It is greater in the spring, when the ground is frozen and the snows 
are melting, than during the summer. 

(d). It may vary considerably at different points in the same stream, the 
variation being due almost wholly to seepage discharge or influx from springs. 

(e). The run -off varies generally with the compactness of the soil and 
sub-soil, being very slight in sand. 

(/). It is influenced more or less by the slope of the catchment basin 
(contrary to some authorities). 

In addition to the above, it is to be noted that the time of run -off affects 
largely any water proposition. If the run-off is immediate the waters usually 



1198 63.— WATER SUPPLY. 

have to be stored unless the supply is very much in excess of the demand. 
Storage may obtain in several ways, namely, (1) by high altitude of the 
catchment basin, delaying the melting of the snow; (2) by forest growth in 
the catchm.ent basin, decreasing evaporation: (3) by natural lakes in the 
catchment area; (4) by artificial storage reservoirs: (5) by sub-soil storage. 
Sub-surface flow accompanies every stream and may be considered part 
of the run-off, under certain circumstances, if readily available. It is not 
included in ordinary stream measurements, but may be detected in "dry" 
streams by digging test pits. It may be brought to the surface and become 
available by the construction of tight coffer-dams extending to bed rock, or 
by the construction of ordinary dams with tight foundation. Many creeks 
apparently "dry" in summer may thus be made to yield a considerable dis- 
charge. 

Formulas for Run-off. — ^The following notation is for the subjoined 
formulas: 

a = area of catchment basin, in acres; 
A = area of catchment basin, in square miles; 
<i = precipitation on catchment per month, in inches; 
I? = total precipitation for any period, in inches; 
c = coefficient of discharge: 
Q = discharge in cu. ft. per second per square mile of catchment for 

precipitation d. 
Q = total discharge in cu. ft. per acre of catchment for precipitation D. 
Then for total run-off per acre of catchment, 

= c • j2 • 43560 a = 3630 cDa (1) 

in which c may vary from to 100; almost always between 0.05 and 0.75; 
generally between 0.20 and 0.60; rough average 0.40. (But see page 1197.) 
The value of c in formula (1) can usually be derived from the coefficient for 
some other catchment, near the locality, for which the run-off has been 

gauged. Then, by transposition, c= Qgon r) " " The accuracy of this method 

is enhanced more or less if the two catchments are very dissimilar in character 
of soil, geology, topography, area, etc. It is far superior however to any 
blind assumption of percentage based on averages in distant localities. 

The monthly discharge may be obtained by substituting d for D in 
formula (1). 

Dicken's formula assumes the maximum discharge ^max in cubic feet per 
second to be proportional to \^A ; thus, 

q^.. = c' VA (2) 

in which c' is a coefficient depending upon rainfall, character of soil, topo- 
graphy, geology, etc. The value of c' in formula (2) may often be obtained 
from similar catchments in the same locality, in about the same manner as 
described for c in formula (1). When c' is known this formula is useful in 
approximating the flood discharge from a catchment for the design of 
wasteways for dams. 

The writer has seen it stated repeatedly that run-off is independent of 
the degree of slope of the catchment area. But his experience and investi- 
gations are to the contrary. Both the declivity of the ground surface and 
its geological formation have considerable effect on the time and amount of 
run-off, especially if measured near the headwaters of the effluent stream. 

Run-off formulas must be used with great caution and only in connection 
with gagings of streams, unless the merest approximation is desired. Even 
then the gagings should cover a long period, by months, for a number of 
years, as the precipitation varies from year to year in both amount and 
intensity, greatly affecting the run-off. In general, the records of 20 years 
are desirable if the demands approach closely the minimum annual supply. 
The minimum as well as the average discharge should be sought. Much 
valuable data relating to discharge of streams may be obtained from the 
Water Supply Papers issued by the U. S. Geological Survey, Dept. of the 
Interior. 

For a good discussion of an additive method of computing run-off to 
sewers, see article in Eng. News of March 11, 1909, from Paper by Carl H. 
Nordell. 



RUNOFF FORMULAS. EVAPORATION. 



1199 



Evaporation. — Evaporation is a physical change from a solid or liquid 
to a vaporous or gaseous state. It is primarily due to heat. All substances 
emit vapors to a greater or less extent, depending largely upon the tempera- 
ture and also upon many other conditions, including humidity. 

Evaporation from Ice and Snow takes place at all temperatures. The 
mean evaporation from the surface of ice may be assumed at about 0.04 in., 
and from snow at about 0.02 in., per day of 24 hours. 

Evaporation from Water surface goes on continually day and night, at 
all temperatures. But when the temperature is below the dew-point the ad- 
ditional moistiire from condensation may more than offset the loss due to 
evaporation. The rate of evaporation increases with — 

(1) Temperature of water surface; 

(2) Dryness of air at water surface; 

(3) Velocity of the wind at water surface. 

The maximum evaporation from surface of running water exposed to 
hot sun and dry, hot wind in the tropics may reach 0.50 inch per day of 
24 hours. But in the southern part of the United States it will rarely if 
ever exceed 0.40 inch per day. The average rate of evaporation from water 
surface of reservoirs during the summer months may be assumed safely not 
to exceed 0.35 inch per day in the south and southwest, and 0.18 inch in the 
northern states, unless the reservoirs are unusually exposed or shallow. The 
average rate per day for the year may be assumed at about one-half these 
figures, that is, 0.18 and 0.09 inches respectively. Running water evapor- 
ates more rapidly than still water; and water in a shallow pan, exposed to 
the sun, will evaporate more rapidly than in a reservoir. 

In a catchment basin the evaporation from the land surface continues 
only for a definite period after each shower until the land dries, unless there 
is vegetation in which case it continues at a greater or less rate. Evapora- 
tion from a meadow of luxuriant grass has been found to be two or three 
times as rapid as from a still water surface. 



-Monthly Evaporation in Inches in the United States 
From Water Surfaces (Approximate). 



Locality. 


i 




I 


< 






>> 
3 


1 


4i 
ft 


o 
O 


> 
o 


S 


An- 
nuaL 


North Atlantic Coast 


1.0 


1.3 


1.7 


2.5 


2.5 


3.4 


3.4 


3.4 


2.9 


2.5 


2.0 


1.4 


28 


Middle Atlantic Coast 


1.7 


1.8 


2.3 


3.8 


3.8 


5.0 


4.8 


4.5 


3.7 


3.5 


3.0 


2.1 


40 


South Atlantic Coast 


2.5 


2.6 


3.3 


4.3 


4.1 


4.8 


4.4 


4.3 


4.1 


3.7 


3.3 


2.6 


44 


Gulf Coast 


2.3 


2.7 


3.7 


4.5 


4.5 


4.5 


4.9 


5.0 


5.0 


4.6 


3.9 


2.4 


48 


Ohio Valley..-: 


1.8 


2.2 


2.8 


5.1 


4 7 


5.3 


6.0 


6.0 


5.5 


4.2 


3.4 


2.0 


49 


Great Lakes Region 


0.7 


1.0 


1.2 


2.3 


3.1 


4.1 


4.8 


4.7 


3.3 


2.7 


1.9 


1.2 


31 


Upper Mississippi Valley. . . 


0.6 


1.0 


1.7 


3.2 


3.5 


4.5 


6.2 


5.4 


3.8 


3.1 


2.1 


0.9 


36 


M ddle Mississippi Valley. . 


1.2 


1.6 


2.5 


5.3 


4.6 


4.7 


6.4 


7.0 


5.2 


4.3 


3.5 


1.7 


48 


Missouri Valley 


0.8 


1.3 


1.5 


4.0 


3.8 


4.6 


6.2 


4.4 


4.0 


3.5 


2.7 


1.2 


38 


Northern Rocky Mt. Slope. 


1.1 


2.3 


1.8 


5.0 


5.0 


6.0 


7.2 


6.1 


5.4 


3.6 


2.9 


1.6 


48 


Middle Rocky Mt. Slope. . . 


2.0 


2.6 


2.8 


5.1 


4.6 


6.3 


6.7 


6.3 


5.2 


4.4 


3.8 


2.2 


52 


Southern Rocky Mt. Slope . 


3.2 


3.6 


4.4 


6.0 


7.5 


8.3 


8.8 


8.9 


5.3 


4.7 


4.2 


3.1 


68 


Northern Plateau 


1.1 
1.4 


2.4 

2.5 


3.8 
3.6 


5.6 
6.3 


6.5 
6.5 


6.0 
8.1 


9.2 
9.2 


8.0 
9.2 


5.6 

7.4 


3.7 

5.5 


2.2 
3.9 


1.9 
2.4 


56 


Middle Plateau 


66 


Southern Plateau 


3.4 


4.0 


5.2 


7.8 


9.2 


11.7 


9.8 


9.4 


7.1 


6.7 


5.2 


3.5 


83 


North Pacific Coast 


1.1 


1.2 


2.2 


2.5 


3.4 


3.0 


3.2 


3.0 


2.6 


2.0 


1.6 


1.2 


27 


Middle Pacific Coast 


2.5 


3.5 


4.3 


4.5 


4.7 


5.4 


6.4 


6.3 


6.6 


7.6 


4.2 


3.0 


59 


South Pacific Coast 


3.3 


2.5 


2.8 


3.9 


4.1 


4.5 


5.4 


5.7 


4.5 


5.0 


3.3 


3.0 


48 



Note that the above table shows increasing evaporation toward the 
South for any section where conditions other than temperature are about 
equal. Evaporation is less in regions adjacent to large bodies of water (as 
the Atlantic, Pacific and Great Lakes regions) for the same latitude, because 
there is greater humidity in the atmosphere. Conversely, it is to be noted 
that the great dry plateau region in the West furnishes the greatest evapora- 
tion. Evaporation on the North Pacific Coast is low because of the almost 
incessant rains for six months of the year. But south of Oregon the Pacific 
Coast shows greater evaporation than the Atlantic Coast of the same lati- 
tude, because of dryness of the atmosphere and higher mean temperature. 



1200 63.— WATER SUPPLY, 

Seepage. — ^The seepage in a catchment area is equal to the total rain- 
fall minus the run -off and evaporation. When the ground is frozen the 
seepage is practically naught, while in deep, sandy soil it may equal nearly 
the total precipitation. It is usual to estimate the seepage and evaporation 
together as the total loss in a catchment, reservoir, stream or canal. If the 
latter has locks or gates, leakage is also included. The loss due to seepage 
and evaporation combined, for a canal in an earthen bed, will vary generally 
from li to 2^ inches per day, depending upon size of canal, character of 
soil, etc. The rate of seepage often decreases with the age of the canal, as 
the fine particles pack into the soil bed and decrease the voids. Experiments 
with models have shown that the ratio of annual seepage to annual evapora- 
tion for total rainfall (with no run-off), in temperate climates is, for earth 
bed, about 1 : 2.4; and for sandy bed, about 5 : 1. These figures, however, 
cannot apply to reservoirs and canals where the process of evaporation is 
continuous, which was not the case with the experiments. Roughly speak- 
ing, the comparative rates of seepage in various soils is about as follows, 
assuming that for sandy soil to be unity: Minute gravel, 100; coarse sand, 
10; fine sand, 2; sandy soil, 1; sandy clay, 0.5; clay, 0.25 to 0.00. 

Mr. Elwood Mead states* that the loss from seepage and evaporation as 
determined from a large number of measurements made by the U. S. Dept. 
of Agriculture, was 2.47 per cent per mile in 1900, and 1.45 per cent per 
mile in 1901 — grouped as follow: 

Loss per mile. 
Capacity of Canals. Per cent. 

(a). Canals carrying 100 cubic ft. per second or more . 98 

(b). Canals carrying between 50 and 100 cu. ft. per second 2 .67 

(c). Canals carrying between 25 and 50 cu. ft. per second 5 . 22 

(d). Canals carrying less than 25 cubic feet per second 7.48 

It is interesting to note here that the above table may be generalized by 
the following law: The total loss per mile for canals carrying from 25 to 200 
cubic feet per second was, in 1900 and 1901, from 1.9 to 2.0 cubic feet per 
second. 

From evaporation alone it is probable that the loss rarely exceeds 1 per 
cent per mile in irrigation canals, when the velocity is 2 i to 3 feet per second. 
Reservoir losses from the same cause may vary from 2 to 7i feet in depth, 
annually; generally from 3 to ^ft.; rough average, 4 ft. (see Table 4, pre- 
ceding page). 

* "Irrigation Institutions," pp. 123-4. 



SEEPAGE AND EVAPORATION, MISCELLANY, 1201 



EXCERPTS AND REFERENCES. 

Relation of Rainfall to Runoff in California (By J. B. Lippincott and 
S. G. Bennett. Eng. News, June 5, 1902). — ^The lollowing basins are dis- 
cussed: Sacramento River Basin, San Matea Creek, Salt Springs Valley 
Watershed, Stanislaus River Basin, Tuolumne River Basin, San Joaquin 
and Kings River Basin, Mojave River, Cuyamaca Reservoir Watershed, 
Sweetwater Reservoir Basins. A diagram shows the annual and mean run- 
off from the above watersheds. A table gives the number of "acre-ft. per 
sq. mile" for "depth of runoff in ins." for depths advancing by hundredths 
of an inch up to one inch. The table is based on V depth of runoff = 5 3i 
acre-ft. per square mile=640^12. 

Formulas for Computing the Cost of Impure Water Supplies (Eng. 
News, Nov. 15. 1906). 

Works for the Purification of the Water Supply of Washington, D. C. 

(By Allen Hazen and E. D. Hardy. Trans. A. S. C. E., Vol. LVII). 

Railways and Water Pollution, With Special Reference to the Water 
Supply of Seattle (Eng. News, Dec. 27, 1906). 

Report of the Proposed 226=Mile Aqueduct for the Water Supply of 
Los Angeles, Cal. (Eng. News, Jan. 24, 1907). — Board composed of J. R. 
Freeman, F. P. Steams, J. D. Schuyler. Total estimated cost, $24,485,000. 

Length of Time Required to Determine Rainfall or Stream Flow Within 
a Given Percentage of Error (By J. C. Stevens. Eng. News, Sept. 17, 1908). 
— Diagram. 

A New Water Supply for the City of Vancouver, B. C. (By H. M. 
Burwell. Eng. News, April 1, 1909). — Cost data on wood-stave pipe and 
steel pipe. 

The Purification of Ground Waters Containing Iron and Manganese 
(By R. S. Weston. Trans. A. S. C. E., Vol. LXIV). 

Illustrations of Water Supply Works: — 

Description. Eng. News. 

Cross-section of concrete water supply conduit, Los Angeles Aug. 27, 1903. 
The California or "stovepipe" method of well construction Nov. 12, '03. 

Brick and cast-iron infiltration conduits, Columbus, O. Feb. 11, '04. 

Water-filter plant at Danville, 111.; 11 illustrations Aug. 28, '04. 

Settling tank, and sinking shoe for pump well, Ithaca April 20, '05. 

Plans and details of sedimentation basin, Charlestown, W. Va. June 7, '05. 
Cross-sections of new conduit for Vienna water supply Oct. 11, '06. 

Cross-section of reinforced-concrete pressure conduit that fail'd Oct. 29, '08. 
Aqueduct, 7 x 40 ft.; steel frame, rein.-conc. lining Feb. 24, '10. 

Eng. Rec. 
Proposed inclined wells at shore of Lake Superior June 19, '09. 

Catskill Aqueduct, cut-and-cover construction Jan. 8, '10. 

Cross-section of 6i-ft. rein.-conc. pressure pipe line Feb. 19, '10. 

Details of anchorage for 42-in. pipe line Feb. 19, '10. 

Section (14' 2" dia.) Moodna cone, pressure tunnel June 4, '10. 

Rein.-conc. siphon for 65-ft. head, Los Angeles Aqueduct July 9, '10. 

7-ft. dia. rein.-conc. coaduit under 12 to 16-ft. head July 23, '10, 



64.— WATER WORKS. 

This subject is treated under five main headings, as follows: 

A. — Consumption of Water Page 1202. 

B. — Purification of Water Page 1204. 

C. — Reservoirs Page 1205. 

D. — Conduits Page 1207. 

E. — Distributing System Page 1280. 

A.— CONSUMPTION OF WATER. 

The "consumption" of water in cities and towns is measured in gallons 
per capita per day, of population supplied,* and includes the total amount 
of water "used" and wasted. The amount of water actually needed for 
domestic supply could undoubtedly be restricted to 20 gallons per capita 
per day, but there is a great deal of careless and wanton waste, as well 
as use for fire protection, manufacturing purposes, etc. Roughly spreaking, 
it is customary to estimate the probable consumption at 50, 60 or 80 gallons 
for small cities and towns about to be supplied, and based on a population 
20 years in the future. For large cities, see Tables 1 and 2 following. 

Water meters are most frequently used in connection with manufacturing 
plants; and where water is not plentiful they are useful in restricting undue 
waste in domestic supply, otherwise they should be omitted, as a free use 
of water promotes cleanliness. 



1. — Population in Thousands op Persons in Various Cities 

OF THE U. S. 
(To accompany Table 2, following.) 





Boston. 


1 


1 


i 
1 




03 

1 

a 


1 
1 


+2 

1 

(0 




f 

.•a 



1 


> 

2 





i 


a 

c3 


13 

a 
>> 







i 


2 . 

IS 


at 


i 


>^ 


5 


S 


3 


P4 


w 


01 





d 


P 


^ 


>A 


Ph 


H^ 


f^ 





^ 


!< 


M 




(1) 


(2) 


(3) 


(4) 


(5) 


(6) 


(7) 


(8) 


(9) 


(10) 


(11) 


(12) 


(13) 


(14) 


(15) 


(16) 


(17) 


(18) 


i8P>n 


140 

178 




30 


121 
566 


97 
267 


78 
161 


115 
161 


17 
43 


21 
46 


20 
45 


43 

68 


42 
51 


33 
37 


12 
14 


15' 
26 


14 
19 


16 
22 


20 


i86n 




112 


22 


1870 


225 


87 


299 


674 


396 


311 


216 


93 


80 


71 


101 


69 


41 


27 


40 


28 


21 


24 


1880 


306 


108 


503 


847 


567 


351 


255 


160 


116 


116 


124 


105 


59 


49 


53 


38 


27 


28 


1890 


410 


118 


1209 


1047 


839 


496 


305 


262 


206 


204 


112 


141 


78 


75 


70 


56 


42 


31 


1900 






2008 
2060 


1294 
1438 


1065 
1313 


595 
714 


332 
358 


397 
462 


289 
359 


285 
343 


163 
178 


187 
214 


95 
95 


108 
107 


92 
97 


74 
83 


63 
75 


36 










1905 


t732 


t596 


38 


1906 












750 






384 














85 







































* A source of error in gathering statistics on consumption of water is to 
include the total population of a town as being served from a certain supply, 
whereas a portion of the population, sometimes large, may be served from 
some other supply, often from wells. 

t Boston, Somerville, Chelsea and Everett. 

t Metropolitan Water Works (Boston only). 

1202 



CONSUMPTION OF WATER, 



120^ 



•*DAiLy Average Consumption op Water in Gallons per 
Capita per Day in Various Cities op the U. S. 
(See Population Statics, Table 1, preceding.) 





Boston. 


1 

id 


ft 


i 
2 


I 

4J 


«e 
d 
d 

•s 

d 


i 

f 


a 


i 

d 

1 


6 
1 


J 

1 


1 




i 
> 


05 

be 

1 


d 
d 


t 




^ear 


II 
P 


at 
II 


a 

0) 




o 


;^ 





P4 


PQ 


m 








Q 


§ 


H^ 


Ph 


A 


p^ 





>A 


"A 


III 




(1) 


(2) 


(3) 


(4) 


(5) 


(6) 


(7) 


(8) 


(9) 


(10) 


(H) 


(12) 


(13) 


(14) 


(15) 


(16) 


(17) 


(18) 


1850 


42 
64 
97 
66 
66 
60 
63 
72 
72 
69 
71 












20 
25 
30 
29 
48 
53 
54 
50 
55 
60 
68 
























1855 












"u 

22 
31 
36 
40 
43 
45 
44 
49 


■44' 

52 
55 
64 
76 
88 
98 
97 
120 
HI 




















1860 


"27"* 

44 
56 
70 
77 
73 
86 
80 


'43" 

42 
73 
72 
74 
88 
96 
100 
103 


"36* 
50 
55 
55 
54 
56 
58 
69 
























1865 


"29' 
47 
47 
53 
59 
54 
59 
57 


'45" 
51 
55 
61 
62 


"29' 
45 


'is' 

23 

21 
22 
22 
24 
24 
27 
















1870 








'44' 
43 
38 
48 
50 
57 
51 






'31* 


1 












38 


2 












45 


3 








42 
40 
32 




55 


4 






"12 
18 
25 


51 


1875 


.... 


'24' 
29 




6 




■59* 


7 


72 


75 




■58' 


59 


66 


64 


56 


111 


69 


29 


24' 


30 


26 


53 


32 




58 


8 


80 


76 


i23' 


64 


58 


67 


66 


51 


110 


85 


30 


26 


32 


25 


45 


32 




53 


9 


87 


88 




66 


60 


72 


68 


63 


125 


96 


33 


30 


35 


27 


47 


34 




55 


1880 


87 


87 


112" 


68 


54 


72 


76 


65 


130 


106 


42 


34 


38 


28 


46 


32 




55 


1 


94 


80 




71 


56 


76 


87 


77 


145 


109 


56 


35 


40 


30 


46 


32 


"86' 


59 


2 


95 


73 


iio* 


76 


58 


v6 


69 


68 


132 114 


47 


33 


43 


36 


45 


37 


82 


63 


3 


97 


74 




76 


58 


75 


66 


76 


146 108 


52 


37 


46 


31 


47 


36 


78 


61 


4 


73 


65 


114* 


74 


61 


63 


74 


83 


159 


101 


56 


34 


48 


26 


46 


39 


72 


60 


1885 


73 


68 


116 


72 


64 


67 


64 


93 


176 


105 


62 


37 


56 


26 


51 


42 


85 


62 


6 


74 


72 


118 


80 


65 


73 


74 


91 


176 


110 


65 


37 


59 


27 


56 


44 


86 


64 


7 


80 


72 


120 


89 


65 


73 


88 


96 


197 


117 


64 


37 


62 


25 


60 


48 


85 


77 


8 


87 


75 


119 


100 


67 


74 


99 


95 


204 


105 


62 


40 


69 


28 


63 


45 


88 


71 


9 


81 


69 


123 


110 


67 


73 


99 


99 


172 


116 


67 


41 


62 


27 


62 


43 


89 


68 


1890 


83 


71 


126 


132 


68 


78 


HI 


106 


161 


109 


100 


48 


69 


29 


62 


45 


98 


69 


1 


90 


75 


132 


140 


70 


77 


138 


HI 


155 


112 


98 


50 


74 


31 


60 


49 


92 


87 


2 


96 


79 


135 


143 


79 


83 


123 


118 


154 


101 


103 


55 


73 


28 


66 


53 


88 


81 


3 


107 


86 


157 


150 


86 


83 


124 


130 


165 


108 


104 


60 


79 


27 


75 


54 


99 


66 


4 


101 


89 


152 


159 


85 




129 


113 


156 


108 


97 


64 


75 


27 


69 


61 


85 


67 


1895 


104 


83 


158 


162 


84 


'84' 


135 


137 


163 


101 


115 


57 


81 


36 


72 


68 


84 


62 


6 


117 


88 


157 


168 


86 


86 


110 


129 


141 


96 


101 


56 


82 


36 


76 


68 


89 


66 


7 


118 


88 


153 


185 


90 


85 


108 


136 


134 


89 


102 


52 


78 


36 


76 


70 


95 


64 


8 






147 


196 


84 


85 


95 


138 


143 


85 


100 


54 


78 


32 


86 


70 


102 


65 


9 




156 


199 


85 


.... 


98 


153 


159 


85 


100 


53 


81 


35 


87 


79 


107 


70 


1900 


K 


161 


221 


86 


106 


115 


169 


162 


83 


92 


54 


83 


35 


79 


65 


101 


75 


1 


|l 


192 


211 


83 


. . . . 


121 


169 


163 


80 


102 


55 


74 


34 


81 


62 


91 


78 




©"S^ 


%o 


































9. 






194 


232 


83 


HI 


125 


168 


162 


81 


98 


58 


57 


40 


85 


62 


91 


83 


3 




196 


237 


84 
88 


123 
131 


130 

137 


142 
139 


157 
179 


80 
84 


103 
101 


65 
67 


55 
53 


38 
36 


90 
90 


63 
64 


96 
96 


86 
85 


4 


140 


149 


203 


234 


1905 


140 


151 


200 


227 


91 


101 


124 


131 


168 


86 


103 


68 


58 


41 


93 


59 


95 


88 


6 


" 
















160 

























































* Data in Tables 1 and 2 up to the year 1890 are from Paper No. 758 of 
Trans. A. S. C. E., Vol. XXXIV, entitled "Consumption and Waste of 
Water" by Dexter Brackett. From 1890, the data have been gathered by 
the writer, from official sources. 



1204 Qi.— WATER WORKS. 

B.— PURIFICATION OF WATER. 

The purification of water for domestic use has received considerable 
attention in the United States during the past 15 or 20 years, and in Europe 
for a much longer period. Speaking generally, there are three main steps 
in the purification of surface water, from streams or reservoirs, namely, (1) 
screening; (2) sedimentation; (3) filtration. The last named is often 
omitted.* 

Screening. — At the approach to the intake end of the pipe there should 
be placed a grating composed of vertical bars of wood, iron or steel to 
screen out the leaves and prevent the drift generally from entering the pipe. 
These bars are placed edgewise to the direction of flow and are stayed 
laterally by small rods and fillers. 

Sedimentation. — ^The screened water is next allowed to pass into a settling 
basin of greater or less capacity to deposit, while at rest, the sedimentary 
matter which it held in suspension while in an agitated state. Turbid waters, 
those suspending sand, silt and clay, should necessarily thus be treated. If 
the particles of clay are minute, that is, as fine or finer than bacteria, it 
becomes necessary to add a coagulant to the water before it enters the 
basin. Various compounds of alum, lime and iron are used as coagulants. 
Perhaps sulphate of alumina is the most common. If a fair amount of 
coagulant is used, the time required for sedimentation will be about 24 
hours, depending upon various conditions. It is to be noted that sedimenta- 
tion with coagulation may be efficient in removing not only the fine particles 
of suspended matter above referred to, but a large percentage of bacteria 
as well. 

Settling basins may be single or multiple and provided for continuous 
or intermittent flow. 

"Slow" Sand Filtration. — ^This is essentially the so-called English system, 
having been employed almost exclusively in England since the early part 
of the Nineteenth Century. The filter beds consist of one or more water- 
tight reservoirs each with an area of 10,000 to 80,000 square feet. On the 
bottom of each reservoir is laid a system of drains for carrying off the 
filtered water. The materials composing the beds proper are broken stone, 
gravel and sand, placed in successive layers and graduated to size, with the 
coarser material at the bottom. The top bed of sand usually varies from 
24 to 60 inches in thickness and should be composed of fine grains of uniform 
size. 

The rate of filtration usually averages from 6 to 9 feet of water column 
per day of 24 hours with good results. Care must be used to see that the 
drains carry off the filtered water freely. The water may be delivered to 
the filter beds by gravity or by pumping. If it contains much sediment it 
should first be passed through the settling basin as previously described 
under Sedimentation. Two systems are employed — the continuous and the 
intermittent. 

The result of sand filtration when highly efficient is to remove from the 
water its impurities and render it potable. These impurities are injurious 
bacteria, vegetable organic compounds (sometimes coloring the water 
highly) and animal pollution, as sewage. The last named is the most easily 
removed. A good sand filter usually removes from 98.5 to 99.5 per cent of 
bacteria. To eliminate vegetable organic coloring effectively, coagulants 
are employed — compounds of lime, alum, iron, etc. 

During filtration a thin film forms on the surface of the sand and acts 
as a fine strainer, collecting most of the bacteria and other impurities. 
Although the process is (purely?) mechanical in theory yet it is a fact that 
water is often rendered purer chemically by having passed through the filter. 
The oxygen of the air (and of the water) is the great purifier. 

"Rapid" Sand Filtration. — This system is distinctly American and is 
generally termed"mechanical" filtration. Mechanical filters are patented 

^Another process, the Copper Sulphate treatment, is being used con- 
siderably where algae is present in the water. Many articles descriptive of 
copper sulphate as an algaecide, for reservoir treatment, etc., have appeared 
in current technical literature during the past ten years ; also, of sulphate 
of alumina, hypochlorite of lime, etc. 



PURIFICATION OF WATER. RESERVOIRS. 1205 

devices for increasing the rate of filtration, by agitating the sand mechanic- 
ally or by compressed air, and by the use of coagulants. They have generally 
given satisfaction and bid fair under many conditions to compete with 
ordinary sand filters in beds. The first cost of mechanical filters is less than 
for sand filters, but the cost of operation is greater. 

C— RESERVOIRS. 

Reservoirs may be classed as follows: 
(a) Storage Reservoirs, comprising large natural basins with dam (see 

Dams, page 844), and wasteway constructed at outlet. 
(6) Distributing Reservoirs (artificial) constructed in earth by excavation 
and embankment, with paved inner slopes: 
High -service reservoirs; 
Low-service reservoirs. 
(cr) Stand-Pipes or water towers of steel, reinforced-concrete or wood. 

(a) Storage Reservoirs* of greater or less extent are commonly to be 
had on most running streams, generally near the source. At the lower end 
of the natural basin and at a contracted point of the out-flowing stream, as 
in a canon, the dam is constructed of sufficient height to guarantee a storage 
supply sufficient for the dry or summer months. If the supply is for domestic 
use the storage will be estimated in millions or gallons; if for irrigation it 
will be estimated in acre ft. (one acre-foot = 43,560 cubic feet); if for 
water power it will be estimated in cubic feet or millions of cubic feet. 

(6) Distributing Reservoir sites should be selected with great care. A 
side-hill location should be examined very carefully for any indications of 
sliding land, a serious condition for reservoir construction. A slide usually 
reveals a wet or marshy spot (sump) at its upper end. The water collects there, 
filters to an inclined sub-stratum (generally clay) and by moistening or 
lubricating it the coefficient of friction is reduced to such an extent that 
sliding takes place. If, unfortunately, it is discovered that a reservoir has 
been constructed on such a slide, the hill-side above and around^ the reser- 
voir should be sub-drained thoroughly at the sliding plane to keep it as dry as 
possible. t This sub-drainage may be accomplished by driving small tun- 
nels, with branches, and laying drain tile to carry away the water. 

Distributing Reservoirs situated immediately at the townsite are con- 
venient and economical. Incidentally they serve as secondary settling 
basins, and hence provision should be made for "blowing" them out when- 
ever the bottoms accumulate much sediment. The waste pipe leads from a 
depression in the bottom of the reservoir, and small water pipes are laid for 
occasional washing and cleansing. Economically, distributing reservoirs 
serve to lower the static head on the distributing system of pipes through 
the town, by receiving the supply direct, through the pressure-pipe line, 
from the storage reservoir or headworks. By this means the pressure on the 
system may often be reduced several hundred feet. If, however, the storage 
reservoir is not at too high an altitude the pressure-pipe line may be con- 
nected directly with the mains and allowed to discharge into them, but 
usually only in case of fire when high pressure is needed. The high-service 
and low service reservoirs are primarily designed to supply the high and the 
low sections of the town, respectively, each connected with its own distribut- 
ing system. 

It is to be noted that considerable power for lighting and pumping can 
often be developed at the distributing reservoirs by utilizing the available 
head from the source above. By this means water may be pumped to a 
stand-pipe for supplying an isloated section of the town at considerable 
elevation and distance. 

The outlet pipe from a reservoir should be so arranged that water may 
be taken at different elevations above the bottom of the reservoir, in order 
to avoid suspended .matter and sediment. 

* See Paper entitled "Lake Cheesman Dam and Reservoir" by Messrs. 
Harrison and Woodard, in Trans. Am. Soc. C. E., Vol. LIII (Dec, 1904) 
page 89, for a very complete description of a typical storage reservoir, with 
construction of dam, wasteway, outlet, etc. 

t See Paper entitled "A Phenomenal Land Slide" by D. D. Clarke, in 
Trans. Am. Soc. C. E., Vol. LIII (Dec, 1904), page 322. 



1206 Gi.— WATER WORKS, 

Reservoir Linings. — ^The bottom may be lined with 6 inches or more of 
concrete laid with expansion joints; on this, from | to ^ inch of cement 
mortar; next, a coat of liquid asphalt; and finally, a harder coat of asphalt. 
The side slopes may be lined with 6 inches or more of concrete laid with 
expansion joints; a coat of asphalt; a layer of brick dipped in hot asphalt 
and laid flat; and thena finishing coat of asphalt, filling up all the joints. 
Instead of the brick lining, a coating of asphalt or asphalt cement is some- 
times employed. Expansion joints are spaced from 10 to 20 feet, depending 
upon the climate. 

It is generally advisable to discharge the water into the reservoir through 
an aerating fountain. 

(c) Stand-Pipes are simply upright riveted steel pipes with the lower 
end capped, resting on a foundation usually of concrete, and thoroughly 
anchored. The plates, joints and rivets will have to resist various com- 
plicated stresses if the stand-pipe is in a freezing climate: 

434 h r 

(a) For static pressure, t=— 7 — [See notation below.] (1) 

• ^ t 

(6) If the water at surface of tank is frozen and pumps are acting to 
force water into the tank there is an added pressure which would 
have to be reduced to equivalent head. 

(c) Or, if the water subsides from the ice there is a minus pressure pro- 

duced by the resifiting vacuum, which may tend to collapse the 
tank. 

(d) The wind pressure is a very considerable item which has to be 

reckoned within the design of large stand-pipes. The resisting 
moment Mk" in inch-lbs., for a cylindrical section of stand-pipe, 
due to bending, is 

M^" =^-=fnr^ te (2) 

where / = allowable fiber stress in lbs. per sq. in.; 

/ = moment of inertia of cross-section, in ins.; 
y = r = radius of pipe or tank, in ins.; 
;r=3.1416; 

^ = thickness of metal shell, in ins.; 
^ = efficiency of riveted joint, say 0.50 to 0.80. 
But the bending moment Mb' in ft. -lbs., is 
^^ , 6P dh^ 

^-=10 --2- (2> 

where P = wind pressure in lbs. per sq. ft for flat surfaces; 

6P 

—r- =wind pressure in lbs. per sq. ft for cylindrical diame- 

1" ters; 

c^ = diameter of tank, in feet; 

/t = depth of section considered, in feet, below top of tank. 
Equating the resisting moment (2) with the bending moment (3), 
multiplying the latter by 12 to reduce to inch-lbs., we have 

fvcr^ te^ Z.QPdh^ (4) 

^ S.6Pdh'i 0.03183 P/i2 

^^^^^^ ^=-;^WT7 = — itj— <5> 

and if P is taken at 50 lbs. per sq. ft., we have 

--dTT ^^> 

. , 1.5915/t2 
^^^ ^= dfe ^^ 

(e) In order to provide for possible corrosion, the calculated thickness 

of shell may be increased by say 1^ in., as with riveted steel 

water pipe. It should be noted that the thickness of shell at top 

of tank should not be less than j inch. 

A common method employed in the design of stand-pipes is to calculate 

for static pressure (a) , wind pressure (d) , and allow for corrosion (e) . If soft 

steel is used, with an ultimate strength of 60,000 lbs., / may be taken at 



RESERVOIRS AND STANDPIPES. CONDUITS. 1207 

15,000 lbs. where no freezing is expected. But otherwise this should be 
reduced to 12,000 or 10,000 lbs., depending upon the climate. The tank 
should be well stiffened against collapse. 

An Elevated Tank is a tank supported at an elevation, usually on tower 
posts, hence the name water-tower. The whole structure is sometimes 
enclosed in a circular masonry shell. 

Stand-pipes and elevated tanks are more economical in certain localities 
than are ordinary reservoirs. They are useful in furnishing a supply to 
districts where the population is small or isolated; or to sections above the 
altitude of the main reservoirs. They are often used in towns equipped 
with the direct pumping systern, to provide safe pressure in case of fire; 
and are connected with the main line, as close to the pumping station as 
possible. 

D.— CONDUITS. 

After the water has been purified to a greater . or less degree at the 
Head works by screening, sedimentation, fiUff^Hon, etc., it is delivered by the con- 
duit to the high- or low-service reservoirs or to stand-pipes, situated as near the 
town as proper altitude and selection of site will permit. 

Conduits (or aqueducts) may be classified as follows: 
/. Open conduits, or those which follow the hydraulic grade line. 

(a) Canals. 

(b) Flumes. 

//. Closed conduits which follow relatively the hydraulic grade line. 

(a) Bored wooden pipe, not banded. 

(b) Salt glazed vitrified pipe. 

(c) Masonry aqueducts, including cement pipe. 
///. Pressure pipe lines. 

(a) Bored wooden pipe, banded. 

(b) Wood stave pipe (banded). 

(c) Cast iron pipe. 

(d) Wrought iron pipe. 

(e) Steel pipe. 

(/) Attachments. 
(g) Specials. 

la.— CANALS. 
Where the topography will admit, considerable saving of cost may often 
be made by employing open channel construction from the headgate to 
the settling basin instead of the more expensive pipe-line construction 
usually adopted. But the sanitation must also be considered, as well as 
the possibility of freezing. 

lb.— FLUMES. 

Wooden flumes may be substituted for canals (above) for small towns 
where a cheap construction is desired, as in the West. Flumes may be 
V-shaped, rectangular, or semi-circular (see Irrigation, page 1317). It is to 
be noted that no open conduit construction is to be allowed below the 
point where the water is purified. Steel flumes are usually semicircular. 

Ila— BORED WOODEN PIPE. 

Bored Wooden Pipe has been used to a small extent in the West in 
certain localities where timber is cheap. Its use, however, has been con- 
fined mainly to the smaller sizes, say 6 inches or less in internal diameter. 
Joints are made by using wooden or cast iron hubs to couple the connecting 
ends. When not banded they are seldom subjected to very much 
static pressure. The writer has seen specimens of bored pipe 5 inches in 
diameter which had been removed after being in service about 40 years. 

lib— SALT GLAZED PIPE. 

Salt Glazed Vitrified Pipe is specially adapted to economic con- 
struction on small work where the topography of the ground will admit of 
the pipe line being laid fairly close to the hydraulic grade line, and where 
the static pressure is inconsiderable. It has been used successfully up to 
20 inches in diameter.* 



* See paper read before Am, W. W. Ass'n at Cleveland, O., April 7, 1888, 
by Mr. Stephen E. Babcock. 



1208 



Qi.— WATER WORKS. 



lie— MASONRY AQUEDUCTS. 

Masonry Aqueducts are particularly economical in conveying large 
supplies of water under little or no pressure. Various shapes of section are 
used, as circular, egg shape, rectangular with arch, and tunnel shape. The 
first and last named are typical of the New Croton aqueduct, the circular 
section being 14 feet in diameter. The ordinary section is shaped like a 
horse-shoe resting on an inverted arch. Rubble masonry, concrete and 
brick are variously employed, the first named being confined to the side 
walls, and the last named to the lining. The brick lining is laid generally in 
one, two or three layers. Concrete may replace the rubble and brick to a 
greater or less extent. Reinforced concrete is used for high pressures. 




Fig. 1. — Reinforced Concrete Aqueduct. 

Fig. 1 is a section of reinforced concrete aqueduct for the City of Mexico, 
J. D. Schuyler, consulting engineer (see Eng. News, April 19, 1906). 

Ilia.— BORED WOODEN PIPE, BANDED. 

Bored Wooden Pipe (Ila) if Banded may be used under pressure, but 
it should be avoided generally for better construction. The cheaper form 
of banding is by spiral wound wiring. 

Illb.— WOOD STAVE PIPE. 

Wood Stave Pipe has had a variable reputation and has been the 
subject of much discussion among engineers. The writer has laid a great 



'5/?oe 




Fig. 2. Fig. 2a. 

many miles of this pipe and can recommend it for cheapness of first cost 
and for carrying capacity. But it yet remains to be demonstrated to what 



MASONRY AQUEDUCTS. WOOD STAVE PIPE. 



1209 



extent it will compare in economy with other kinds of pipe when its lasting 
qualities are better known. 

Fig. 2 shows a half -section of stave pipe with staves, tongues, band 
(with head, nut and washer) and shoe. Fig. 2a is a view of an ordinary- 
stave showing the saw-kerf at one end, and the metallic tongue 
inserted at the other. On the nearest edge is shown a beaded projection 
left by the planer on the finished stave to insure water-tightness, but as this 
bead is subject to injury it is often omitted. Note that the tongue projects 
somewhat beyond either edge of the stave and is embedded in adjoining 
staves when the pipe is cinched tightly. The metallic tongues are usually, 
but not always, galvanized before being placed in the pipe. Fig. 3 is a plan 




5": >s 

Length u.h. 

Fig. 3. 

of i-inch band (upset) with head, nut and washer. Note that the upset is 
very long to allow for proper cinching, and that the length is "under head." 
Bands should be protected from rust before the pipe is back-filled. A 
common method in the West is to dip them in hot asphalt after bending. 
The bending and dipping plants may be erected on the ground where the 
pipe is laid if the work is of considerable magnitude; otherwise they should 
leave the mill ready for laying. Fig. 4 illustrates the ordinary malleable 
iron shoe or coupling for the bands, and explains itself. • 




Table 3, following, was prepared by the writer some years ago, and 
will be found useful to those who desire to make estimates on wood stave 
pipe construction. 



1210 



H,— WATER WORKS, 



3. — Wood Stave Pipe and Details — 



Internal diameter of pipe, ins 

*Thlckness t of staves . . |< W ->[ . . . 

Finished width W, ins. .•^H--^^'*'^??>^l . . . 
Partial depth TV, ins. . . %^*^^^^l^^^^ ■ • • 
Partial depth O, ins. . . ^s^-y^'^'^ ^^^ 
Finished width w, ins.. . K"- - ^ - — ->i . . . 

Size of rough stave, ins 

No. of staves to the circle 

Ft. B. M. of rough lumber per 100' of pipe . . . . 

Size of tongue — width and B. W. G 

No. of tongues per 100' of pipe (average) 

Wt. of tongues per 100' of pipe, lbs 

Size of band, ins , . . . 

Length of band (under head), ft. and ins . 

Wt. of band and upset, nut and head, lbs 

Value of s in lbs. for band section, factor of 4, 

Allowable sin lbs. to avoid crushing staves 

Closest band spacing practicable, ins 

Band spacing for 100' head, ins 

No. of bands per 100' of pipe for 100' head . . . 
Wt. of bands per 100' of pipe for 100' hd., lbs. 

Wt. of one shoe, lbs 

Wt. of shoes per 100' of pipe for lOO'hd., lbs 

Wt. of washers per 100' of pipe for 100' hd., 
lbs 

Size of band, Ins 

r- 

Length of band (under head), ft. and ins. ... . 
Wt. of band and upset, nut and head, lbs 

Value of s in lbs. for band section, factor of 4. 

Allowables in lbs. to avoid crushing staves 

Closest band spacing practicable, ins 

Band spacing for 100' head, ins 

No. of bands per 100' of pipe for 100' head , . . . 

Wt.ofbandsperlOO'ofpipeforlOO'hd.,lbs 

Wt. of one shoe, lbs 

Wt. of shoes per 100' of pipe for 100' hd., lbs. . . . 

Wt. of washers per 100' of pipe for 100' hd., 

lbs 

Areaofpipe,sq. ft.=cu. ft.perft.of length 

Gallons of water per lin. ft. of pipe 

Velocity V in ft. per sec. forn=. 0105 ;s=. 0001 

" " " " s=.001 

" " " " s = .01 

Dischargeinmilliongals.per24hrs.;s=.0001 

s=.001 

s=.01 



3^ 

A 
if 

2f 

1^x4 
9 
450 

r.#i6 

45 
3 



10' 



3^ 
\ 
1 

913 

lix4 

11 

550 

rx^ie 

55 
4 



12" 



li 

3ii 

\ 

\h 

3^ 

lix4 

12 

600 

I'xlie 

60 
4 



14' 



lire 

3f 

t\ 

1 5 

3i 

lix4 
14 
700 

70 
5 



16* 



1 7 

4i 

2x6 

11 

1100 

rx#i( 

55 
6 



18" 



If 
51 

f^ 

4f 

2x6 

12 

1200 

l''x#16 

60 

6 



2-111 
1.19 



3-6^ 
1.40 



4-li 
1.60 



4-7f 
1.80 



5-2i 



5-9i 
2.22 



1611 



1035 

li 
4A+ 
277 
330 

h 
139 

21 



1255 


1475 


1611 


1611 


li 


li 


li 


H 


41+ 


4tV- 


4i- 


3f+ 


274 


271 


284 


318 


384 


434 


511 


633 


i 


i 


i 


f 


137 


136 


142 


239 


21 


21 


22 


24 



1611 

31- 
358 
795 

f 
269 

27 



f " round. 



2-111 
1.23 



3-61 
1.44 



4-li 
1.65 



4-71 
1.86 



5-2| 
2.06 



5-9f 
2.28 



1656 



700 

2t6^ 

409 

503 

i 

205 

31 



34907 

2.611 

.355 

1.30 

4.20 

.08 

.29 

.95 



849 
li 
3 

405 
583 

203 

31 



54542 

4.080 

.420 

1.54 

4.97 

.15 

.54 

1.76 



998 
li 
3 

401 
662 

200 

31 



78540 

5.875 

.490 

1.77 

5.70 

.25 

.90 

2.88 



1146 
li 
3 

399 

742 
\ 

200 

31 



1.0690 
7.997 

.550 
1.98 
6.37 

.38 
1.37 
4.40 



1291 
li 
3 

396 

816 
I 

297 

30 



1.3963 

10.44 

.612 

2.19 

6.99 

.55 

1.98 

6.30 



1452 

li 
3 

397 
905 

f 

298 



1.7672 
13.22 

.673 
2.38 
7.59 
.77 
2.72 
8.67 



♦Finished thickness t of staves, in inches. 



WOOD STAVE PIPE AND DETAILS, 



1211 



— For Pipe Diameters 8 to 38 Inches.* 



20" 


22* 


.24" 


26" 


28" 


30" 


32" 


34" 


36" 


38" 




If 


If 


If 


If 


li^ 


H 


H 


1^ 


1t% 


1t^ 


1 


5^ 


5^ 




5f 


5f 


5f 


5i 


5i 


5^ 


5f 


2 


S 


■h 


iV 


i 


i 


i 


^ 


-h 


A 




3 


lA 


Hi 


Hi 


m. 


1^1 


n- 


U- 


n- 


li^ 


1^— 


4 


m 


^ 


5 


H 


H 


5i 


5 


0^ 


5^+ 


5^ 


5 


2x6 


2x6 


2x6 


2x6 


2x6 


2x6 


2x6 


2x6 


2x6 


2x6 


6 


13 


14 


15 


16 


17 


18 


20 


21 


22 


23 


7 


1300 


1400 


1500 


1600 


1700 


1800 


2000 


210G 


2200 


2300 


8 


li'x^H 


li''x#i4 


li^x^U 


irx#14 


ir-#i4 


ir^lfu 


li"x#14 


l^x^H 


li"-#14 


lj"x#14 


9 


65 


70 


75 


80 


85 


90 


100 


105 


110 


115 


10 


11 


12 


13 


14 


15 


19 


20 


21 


22 


23 


11 




t 


" round 








^" round. 




6-4 


6-lOi 


7-4i 


7-lOf 


8-5i 


9-li 


9-7f 


lO-lf 


10-8i 


ll-2f 


12 


2.48 


2.68 


2.88 


3.07 


3.28 


4.84 


6.11 


5.38 


5.64 


5.93 


13 




1656 




2254 


14 


1592 


1656 


1656 


1656 


1656 


2254 


2254 


2254 


2254 


2254 


15 


H 


li 


li 


li 


li 


iVk 


1^ 


lA 


1t^ 


1t^ 


16 


h\-^ 


2H+ 


2|i 


m 


2f 


3 


ni 


2H 


2-A 


2^ 


17 


395 


411 


442 


473 


508 


399 


423 


445 


470 


493 


18 


980 


1101 


1273 


1452 


1666 


1931 


2162 


2394 


2651 


2923 


19 


!• 


f 


1 


i 


1 


1 


1 


1 


1 


1 


20 


296 


308 


332 


355 


381 


399 


423 


445 


470 


493 


21 


30 


31 


34 


36 


39 


31 


32 


34 


36 


38 


22 




1 


Y round. 






i" round. 




6-5i 


6-1 If 


7-5f 


8-0 


8-6f 


9-li 


9-7f 


10-2 


10-8f 


11-3 


23 


3.47 


3.74 


4.01 


4.28 


4.56 


6.39 


6.74 


7.09 


7.46 


7.81 


24 






2254 






2944 




25 


1740 


1893 


2049 


2199 


2254 


2722 


2888 


2944 


2944 


2944 


26 


lA 


1^ 


l-l^ 


iVk 


1^ 


If 


If 


If 


If 


If 


27 


3i^ 


3ik 


3f 


3f 


3.V. 


3f 


3f 


HI 


3^ 


3t^ 


28 


361 


359 


358 


357 


373 


330 


329 


341 


359 


381 


29 


1253 


1343 


1436 


1528 


1701 


2109 


2217 


2418 


2678 


2976 


30 


1 


1 


1 


1 


1 


li 


li 


li 


li 


li 


31 


361 


359 


358 


357 


373 


413 


411 


427 


449 


476 


32 


28 


28 


27 


27 


29 


33 


33 


34 


36 


38 


33 


2.1817 


2.6398 


3.1416 


3.6870 


4.2761 


4.9087 


5.5851 


6.3050 


7.0686 


7.8758 


34 


16.32 


19.75 


23.50 


27.58 


31.99 


36.72 


41.78 


47.16 


52.88 


58.92 


35 


.729 


.785 


.850 


.913 


.947 


.996 


1.05 


1.09 


1.13 


1.18 


36 


2.55 


2.74 


2.93 


3.07 


3.26 


3.40 


3.56 


3.73 


3.86 


4.00 


37 


8.20 


8.80 


9.34 


9.96 


10.39 


10.83 


11.35 * 


11.87 


12.30 


12. 73 


38 


1.03 


1 34 


1.72 


2.18 


2.62 


3.16 


3.79 


4.43 


5.16 


6.02 


39 


3.60 


4.67 


5.95 


7.32 


9.00 


10.8 


12.8 


15.2 


17.6 


20.4 


40 


11.5 


15.0 


19.0 


23.8 


28.7 


34.3 


40.9 


48.3 


56.5 


64.8 


41 



*The first five items are dimensions of finished staves (see Fig.). 



1212 



6i.— WATER WORKS. 





3. — Wood Stave Pipe and Details 


(Concluded)- 


— 




i 


Internal diameter of pipe, ins . 


40" 


42" 


44" 


46" 


48" 


50" 


M 






1 


♦Thickness t of staves. |^. y\/ 1 

Finished width W, ms.-7^-f--^-^^-__ -^ 


li^ 


If 


If 


If 


m 


IH 


2 


5f 


51^ 


^^ 


5H 


5f 




3 


Partial depth N, ins. .^-'^^^yf^^ 




■h 


-h 


1^- 


1^- 


^— 


4 


Partial depth O, Ins.. . ^^J^p^^ '^^^^<<^ 


1^ — 


1*- 


If- 


If- 


1+*- 


l^ — 


5 


Finished width w, ins. k— • *v ^' 


5i- 


5i 


5t^- 


^16 '■ 


5f 


514- 


6 


Size of rough stave, ins. 


2x6 

24 

2400 


2x6 

25 

2500 


2x6 

26 

2600 


2x6 

27 

2700 


2x6 

28 

2800 


2x6 


7 


No. of staves to the circle 


29 


8 


Ft. B. M. of rough lumber per 100' of pipe 


2900 


9 


Size of tongue — width and B. W. G 


lY^m 


W^fn 


li"x#l2 


li"x#12 


H"x#l2 


ir^fiz 


10 


No. of tongues per 100' of pipe (average) . . 


120 


125 


130 


135 


140 


145 


11 


Wt. of tongues per 100' of pipe, lbs 


32 


33 


35 


36 


38 


46 




Size of band, Ins 


Y round. 












12 


Length of band (under head), ft. and ins. 


ll-9i 


12-4 


12-lOi 


13-4^ 


13-lli 


14-5 J 


13 


Wt. of band and upset, nut and head, lbs. 
Value of s ini bs. for band section , factor of 4 
Allowable 5 In lbs. to avoid crushing staves. 


8.16 


8.53 


8.88 


9.23 


9.59 


9.95 


14 


2944 


15 


2944 


2944 


2944 


2944 


2944 


2944 


16 


Closest band spacing practicable. Ins 


1^ 


l\ 


1* 


\h 


1* 


li 


17 


Band spacing for 100' head, ins 


35^2 


m 


2|i 


nh 


2t^ 


Hi 


18 


No. of bands per 100' of pipe for 100' head 


395 


413 


432 


450 


470 


487 


19 


Wt. of bands per 100' of pipe for 100' hd.,lbs. 


3223 


3523 


3836 


4154 


4507 


4846 


20 


Wt. of one shoe, lbs. (x2=2 shoes per band) 


li 


li 


li 


li 


li 


li 


21 


Wt. of shoes per 100' of pipe for 100' hd.. lbs. 


494 


516 


540 


562 


588 


609 


22 


Wt. of washers per 100' of pipe for 100' hd.. 
















lbs 


39 


41 


43 


45 


47 


48 










Size of band, ins 






f " round. 












23 


Length of band (under head), ft. and Ins. . 


ll-9f 


12-4i 


12-lOi 


13-4^ 


13-1 1^ 


14-5f 


24 


Wt. of band and upset, nut and head, lbs. 
Value of 5 in lbs. for band section, factor of 4 
Allowable s In lbs. to avoid crushing staves. 


12.96 


13.54 


14.08 


14.63 


15.21 


15.75 


25 


4602 


26 


4312 


4525 


4602 


4602 


4602 


4602 


27 


Closest band spacing practicable. Ins 


li 


1^ 


1^ 


If 


If 


If 


?8 


Band spacing for 100' head. Ins 


4,^ 
270 


^■h 


4U 


4A 

288 


4 


312 


29 


No. of bands per 100' of pipe for 100' head 


270 


277 


300 


30 


Wt. of bands per 100' of pipe for 100' hd., lbs. 


3499 


3656 


3900 


4213 


4563 


4914 


31 


Wt. of one shoe, lbs. (x2= 2 shoes per band) 


\h 


li 


H 


H 


H 


li 


32 


Wt. of shoes per 100' of pipe for 1 00' hd. , lbs. 


405 


405 


416 


432 


450 


468 


33 


Wt. of washers per 100' of pipe for 100' hd., 
















lbs.. . . 


32 


32 


32 


34 


35 


36 








34 


Area of pipe, sq. ft.=cu. ft. per ft. of length 


8.7267 


9.6211 


10.559 


11.541 


12.566 


13.635 


35 


Gallons of water per lin. ft. of pipe 


65.28 


71.97 


78.99 


86.33 


94.00 


102.0 


36 


Velocltyt;lnft.persec.forn=.0l05;s=.0001 


1.23 


1.27 


1.31 


1.36 


1.40 


1.44 


37 


" " " " s=.0101 


4.16 


4.29 


4.42 


4.55 


4.68 


4.81 


38 


s=.01.. 


13.24 


13.66 


14.08 


14.49 


14.90 


15.31 


39 


Discharge In million gals. per24hrs.:5=.0001 


6.95 


7.88 


8.93 


10.1 


11.4 


12.7 


40 


' 5 = .001. 


23.5 


26.6 


30.1 


33.9 


38.0 


42.3 


41 


" " " 5 = .0l.. 


74.7 


84.2 


96.0 


108.0 


121.0 


134.8 



* Finished thickness t of staves, in inches. 



WOOD STAVE PIPE AND DETAILS, 



1213 



— For Pipe Diameters 40 to 72 Inches.* 



52" 


54' 


56" 


58" 


60" 


62" 


64" 


66" 


68" 


70" 


72' 


M 


IH 


H 


2i 


2i 


2i 


2i 


2i 


2A 


2 A 


2-h 


2f 


1 


51+ 


7f- 


7^ 


7f 


7f- 


7f+ 


7f- 


m 


7f 


7f- 


7A+ 


2 


i+ 


i 


i- 


i- 


i - 


i- 


1^+ 


1^+ 


A+ 


1^+ 


A- 


3 


m- 


2i- 


2^- 


2^- 


2i- 


2h- 


2i- 


2i^- 


2i\- 


2i^- 


2f- 


4 


5t^ 


7+ 


7+ 


6^+ 


7f 


l-h 


7i^ 


7i 


7i- 


7iV+ 


7tV- 


5 


2x6 


2^x8 


2^x8 


2ix8 


3x8 


3x8 


3x8 


3x8 


3x8 


3x8 


3x8 


6 


30 


24 


25 


26 


26 


27 


28 


29 


30 


31 


32 


7 


3000 


4000 


4167 


4333 


5200 


5400 


5600 


5800 


6000 


6200 


6400 


8 


li''x/12 


If's^lO 


ir^fflO 


IfXiflO 


ir^#io 


ir'^#io 


\r-m 


ir^#io 


If" ^#10 


2"x#10 


2"xj^l0 


9 


150 


120 


125 


130 


130 


135 


140 


145 


150 


155 


160 


10 


47 


61 


63 


64 


67 


70 


73 


75 


76 


95 


97 


11 


y round. 






i" rount 


i. 




f " round. 




U-llf 


15-8i 


16-3i 


16-9f 


17-1 If 


18-5f 


19-0 


19-7i 


20-lf 


20-7f 


21-2f 


12 


10.29 


10.79 


17.63 


18.17 


20.02 


20.57 


21.12 


31.31 


32.09 


32.89 


33.72 


13 


2944 


4602 


6627 


14 


2944 


2944 


4602 


4602 


4602 


4602 


4602 


6627 


6627 


6627 


6627 


15 


li 


li 


If 


If 


If 


If 


If 


2 


2 


2 


2 


16 


2f 


2i 


3^^ 


3.«, 


3f 


3^ 


2M 


4f 


4iV 


^% 


m 


17 


505 


534 


353 


364 


382 


393 


405 


290 


297 


305 


314 


18 


5196 


5762 


6223 


6614 


7648 


8084 


8554 


9080 


9531 


10031 


10588 


19 


li 


li 


li 


n 


l|x2 


If x2 


lfx2 


2x2 


2x2 


2x2 


2x2 


20 


631 


668 


530 


546 


1337 


1375 


1417 


1160 


1188 


1220 


1413 


21 


50 


53 


41 


42 


45 


46 


47 


47 


48 


49 


51 


22 


f * round. 






1" roun< 


a. 




I'rc 


und. 




15-Oi 


15-9i 


16-3f 


16-10 


17-1 If 


18-6i 


19-Oi 


19-7f 


20-lf 


20-8f 


21-2f 


23 


16.31 


17.09 


25.44 


26.22 


28.90 


29.68 


30.61 


42.69 


43.76 


44.82 


45.95 


24 


4602 


6627 






9( 


)20 


25 


4602 


4602 


6627 


6627 


6627 


6627 


6627 


9020 


9020 


9020 


9020 


26 


If 


If 


H 


If 


If 


If 


If 


2f 


2f 


2i 


2i 


27 


^^. 


H 


4^.1 


41 


ihl 


4f 


4.^^ 


5f 


5i 


5f 


5^ 


28 


323 


342 


245 


253 


265 


273 


281 


213 


217 


224 


231 


29 


5268 


5845 


6233 


6634 


7659 


8103 


8601 ■ 


9093 


9496 


10040 


10614 


30 


n 


n 


H 


If 


2x2 


2x2 


2x2 


2fx2 


2ix2 


2fx2 


2ix2 


31 


485 


513 


429 


443 


1060 


1092 


1124 


958 


976 


1008 


1040 


32 


38 


40 


40 


41 


43 


44 


45 


53 


54 


56 


58 


33 


14.74? 


15.904 


17.104 


18.34J 


19.635 


20.966 


22.340 


23.758 


25.220 


26.725 


28.274 


34 


110.3 


119.0 


127.9 


137.3 


146.9 


156.8 


167.1 


177.7 


188.7 


199.9 


211.5 


35 


1.48 


1.52 


1.56 


1.59 


1.64 


1.67 


1.71 


1.75 


1.79 


1.82 


1.85 


36 


4.94 


5.03 


5.16 


5.28 


5.41 


5.50 


5.62 


5.75 


5.87 


5.96 


6.08 


37 


15.72 


16.02 


16.42 


16.82 


17.22 


17.50 


17.90 


18.29 


18.57 


18.96 


19.35 


38 


14.1 


15.6 


17.2 


18.9 


20.8 


22.5 


24.7 


26.8 


29.2 


31.2 


33.8 


39 


47.0 


51.7 


57.0 


62.6 


68.5 


74.0 


81.2 


88.0 


95.5 


103.0 


111.0 


40 


150.0 


165.0 


181.0 


199.0 


218.0 


235.0 


259.0 


280.0 


300.0 


325.0 


350.0 


41 



* The first five items are dimensions of finished staves (see Fig.). 



1^14 



6i.— WATER WORKS. 



NoTETs ON Table 3, preceding. 

Thickness of staves given is the maximum. Some engineers limit the thick- 
ness to 2 ins. even for the large sizes of pipe. 

Staves can be planed readily by grinding the blades of the sticker to the 
proper shapes. 

Sizes of rough staves are merchantable. The lumber must be clear and 
seasoned without checking, and before planing, otherwise the subsequent 
shrinkage will greatly affect the finished diameter of pipe. 

The metal tongues should be galvanized smoothly and evenly to prevent 
leaky pipes; but are sometimes left plain. 

Length of band may be obtained from following formulas: 

For single shoes, Li =;r (D + 2i + —) -f- ci + upset length — i. 



For double shoes, Lj 



7t {D+2t+-^)-\-2{d + upset length - \) . 



in which Li = length of single band under head, in ins. 

L2 = total length of double band under head, in ins., 

n =3.1416, 
D = internal diameter of pipe in ins., 

t = thickness of stave in ins., 

d = diameter of band in ins., 
Care must be used in ordering bands. It is better to have them a little 
long than short. Wood stave pipe is apt to form up with a diameter 
from i" to Y' too large, which must be taken into consideration when 
ordering hands. 

Bands are proportioned to not exceed a hydrostatic-pressure stress of 15,000 
lbs. per sq. in. Small bands, especially on pipes of small diameter, 
are apt to crush the wood, hence a lower stress than 15,000 is assumed 
to meet this contingency, in necessary cases, as shown by the preceding 
table. 



4. — Discharge in Million Gallons per 24 Hours through Wood 

Stave Pipe. New, Clean and in First-Class Condition. 

(Value of roughness w = .010) 

[Million Gallons per 24 Hours.] 



Grade 
or Slope 






Diameter of Pipe, 


in Inches. 






s. 


12 


16 


20 


24 


30 


36 


48 


60 


72 


.0001 


0.26 


0.60 


1.09 


1.78 


3.33 


5.48 


11.9 


21.8 


35.6 


.00015 


0.30 


0.74 


1.38 


2.27 


4.16 


6.85 


14.9 


27.0 


43.8 


.0003 


0.50 


1.11 


2.05 


3.35 


6.12 


9.96 


21.6 


38.9 


63.0 


.0005 


0.67 


1.47 


2.69 


4.40 


8.03 


13.10 


28.2 


50.9 


82.6 


.0008 


0.85 


1.86 


3.40 


5.58 


10.20 


16.50 


35.6 


64.6 


104.0 


.001 


0.95 


2.07 


3.81 


6.25 


11.40 


18.50 


39.8 


72.2 


117.0 


.003 


1.68 


3.65 


6.68 


10.90 


19.90 


33.40 


69.5 


126.0 


203.0 


.005 


2.17 


4.74 


8.70 


14.20 


25.70 


42.00 


90.2 


162.0 


263.0 


.008 


2.75 


6.00 


11.00 


18.00 


32.50 


53.10 


114.0 


206.0 


332.0 


.010 


3.07 


6.71 


12.30 


20.10 


36.30 


59.30 


127.0 


230.0 


371.0 


.016 


3.88 


8.50 


15.50 


25.40 


46.00 


75.10 


161.0 


291.0 


469.0 


.020 


4.34 


9.49 


17.40 


28.40 


51.40 


83.90 


180.0 


325.0 


524.0 



Note. — For designing ordinary wood stave pipe lines it is safer to use a 
coefficient of roughness n = .0105 (Table 3) rather than .010 as above. 
Compare discharges in Table 4 with those in Table 3. 

lllc— CAST IRON PIPE. 

Cast Iron Pipe is more durable than wood-stave-, wrought-iron-, 
or steel pipe, but its first cost is greater for the ordinary sizes required in a 
pressure-pipe line or in a distributing system. It especially commends 
itself for use in large cities where the increased first cost can be borne 
easily; also for certain portions of any pipe line demanding fairly permanent 
construction, that is, where cost of renewal would be excessive; and gener- 
ally where "specials" are required. Before deciding on the kind of pipe to 



WOOD STAVE PIPE. CAST IRON PIPE. i2l5 

use, chemical analyses should be made of the soil and the water, and the 
question of electrolysis also should be considered. Cast iron pipe should be 
dipped in hot coal tar or asphalt or a mixture of the two before being laid 
in the ground. 

Formulas for Designing Cast Iron Pipe are numerous, but they all a^ree 
in taking into consideration (1) the static pressure, (2) water ram, and (3) 
liability to breakage from rough handling before and during the laying. 

Notation. 

i== thickness of pipe shell, in inches; 

c? = inside diameter of pipe, in inches; 

^ = pressure head in feet ( = 2.304 p) ; 

^ = pressure of water, in lbs. per square inch (=0.434^); 

j^"" = allowance for water rani, in lbs. per square inch; 

r = internal radius of pipe, in ins.; 

5 = allowable tensile stress, in lbs. per square inch; 
Based on a factor of safety of 5, s is assumed at 3,200 to 3,600 lbs. 
For static pressure alone, we have, 

.^Pd _ 0.217 hd 

Practical working formulas are as follows: 

Formula used by Metropolitan Water-works, of Boston: — 

,^(^0<i^0.25 = (M34_M:£Oi^0.25 (2) 

25 6600 

assuming 5= 3300. Allowances for water ram are — 

^'=120 lbs. (277 ft. hd. ) ford= 3 to 10 ins.; 

= 110 " (254 " " ) " c/=12or 14ins.; 

= 100 " (231 " " ) " c/= 16 or 18 ins.; 

= 90 " (208 " " ) " (i=20ins.; 

= 85 " (196 " " ) " cf = 24ins.; 

= 80 " (185 " " ) " d=SOms.; 

t= 75 " (173 " " ) " cf=36ins.; 

= 70." (162 " " ) " i = 42to60ins. 
Formula recommended by R. D. Wood & Co., Philadelphia: — 
,_(P±]md / d\ 

Here, s= 3600 (or 2^= 7200); ^'is assumed at the constant value 100; and 
the added thickness to allow for rough handling is represented by the last 
term, a variable dependent on the diameter. But formula (3) easily reduces 

'-^^m^-o.^^^ (4) 

which is readily comparable with (2). 

Kinds of Pipe Joints. — Cast iron pipe 3 ins. or more in diameter comes 
in a standard laying length of 12 feet. There are four principal types of 
joints, namely, (1) bell and spigot; (2) turned, page 1235; (3) flanged, page 
1235; and (4) flexible joint, page 1238. These are discussed as follows. 

(1) Bell and Spigot Joint Pipe (see p. 1220) is the most common, being 
almost universally employed when cast iron pipe is to be laid in trenches 
and back-filled. In laying the pipe it is customary to begin with some 
"special" as a gate or tee, etc., thrusting the spigot end of each pipe into the 
bell of the piece previously placed, "bell holes" having been dug in the 
trench where each joint is to come. When blocked or tamped to proper line 
and grade the joints are caulked at the inner or spigot end with oakum, jute 
or hemp, and then the balance of the joint is run with hot lead, and caulked 
tightly. In order to confine the molten lead so it will fill the joint flush 
with the end of the bell, a gasket is clamped around the entering pipe 
tightly against the bell of the other, leaving an opening at the top for pour- 
ing. Improvised gaskets may be made wholly out of clay, but manufac- 
tured gaskets composed of layers of rubber and hemp cloth, and backed by 
steel springs are universally employed. The ends are clamped nearly 
together, a pouring hole is made with clay, and the metal poured. (See 
page 1219 for notes on pipe laying; page 1280 for lead-melting furnace.) 



1216 



6i.— WATER WORKS. 






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C. I. PIPE— WEIGHT, DISCHARGE, LEAD, HEMP. 



1217 



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1218 



Qi.— WATER WORKS, 



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C. I. PIPE— DISCHARGE, LA YING, WEIGHTS. 1219 

Examples in use of Table 6, preceding. 

(1) The maximum discharging capacity, for instance, of 8500 ft. of 6* 
straight cast iron pipe, under a head of 195 ft., may be found as follows: 
First ascertain the frictional head for 1000 ft., thus: 195 ft. -^ 8.5 = 22.94 as 
the fric. head in ft. per 1000 ft. By referring to the table, 1000 ft. of 6" 
pipe under 23.01 ft. head will discharge 500 galls, per min. 

(2) To find the diam. of pipe for a given discharge. — To deliver, say, 
4,250,000 galls, per 24 hrs., the dist. being 20,000 ft. and the head 130 ft. 
The fric. head per 1000 ft. is 1304-20=6.5 ft., and the table shows that 
under 6.19 ft. head per 1000 ft. a 16'' pipe will discharge 4,320,000 galls. 

(3) To ascertain the pressure at any point in a line of water main, given 
its diam., rate of disch. and static head against which the water is forced. 
Assume that 500 galls, of water are to be forced per minute, through 5000 ft. 
of 8" cast iron pipe, laid on an incline to a height of 75 ft.; what would be 
the varying pressure at each 1000 ft. from the pumps? The table shows a 
fric. head of 5.64 ft. per 1000 ft. of 8" pipe, discharging 500 galls, per min. 
If the dist. is 5000 ft., the pressure required to overcome fric. is 5X5.64 = 
28.20 ft. head, and the total resistance head at pumps is 28.2+75 ft. = 103.2 
ft., and for every 1000 ft. from the pumps, this head is diminished 5.64 ft.4- 
the vert, ascent; that is, at 3000 ft. from the pumps the resistance head is 
61.92 ft. Resistance head in ft. -^ 2. 3 = pres. in lbs. per sq. in. 

r4) The volume of water -flowing in a pipe may be found by ascertaining 
the loss of pressure or fric. resistance per 1000 ft. of pipe. To this end, if 
two accurate gages, entirely similar in all respects, and placed 1000 ft. apart 
on a line of level 12" pipe, show a loss of 0.5 lb., the nearest figures indicate 
a flow of 600 galls, per min.; if the loss is 1\ lbs., a flow of 1400 galls, per 
min., etc. Due allowance should be made for diff. in elev. of points of 
observation. 

Pipe-Laying Notes. 

The following Notes on Pipe-Laying relate to the re-construction of the 
distributing system at The Dalles, Oregon, in the winter of 1898-9, and are 
taken from the writer's note book: 

1 yamer can yam 1000 1. f. of 6" to 10" C. I. pipe per day. 

2 caulkers can caulk 1000 1. f. of 6" to 10" C. I. pipe per day. 
Gang laid 700 1. f. 8" pipe on 4th Street one day = l|c. per 1. f. 
Gang took up 2400 1. f. 8" pipe for $60.00, at rate of 2|c. per 1. f. 
Cost of cutting 4", 6", 8", 10", 12" pipe = 6c, 9c, 12c, 15c, 18c. 
Cheaper to place all pipe over 10" with derrick unless ground is sandy 

and soft; handle 8" and 10" pipe with rope at each end. 
4" and 6" pipe can be rolled in trench if soft; otherwise by hand. 
4" pipe @ 20#= 240 lbs.; two men can carry on shoulders. 
6" pipe @ 32# = 384 lbs. ; three men can carry with two bars ( 1 behind.) 
8" pipe @ 45#= 540 lbs.; four men can carry with two bars. 
10" pipe @ 65#= 780 lbs.; six men can carry with three bars. 
12" pipe @ 75#= 900 lbs.; eight men can carry with four bars. 
12" pipe @ 85# = 1020 lbs. ; nine men can carry with five bars (1 behind) . 
Trucked 15-4"(3600#), 9-6" (3456#), 7-8" (3780#), 4-10" (3120#), 
3-12" (2700 to 3060#) pipe per load, two horses, two men to unload. 
Required \ cord wood for melting 1000# lead (forlOOO ft. 8" pipe). 
Required \ cord wood for melting 250 ft. of laid, 8" pipe for taking up. 
Bell-holes were dug around the joints and fires kindled. 

Weights and Dimensions of Cast Iron Pipe and Specials. — ^The following 
tables are mainly from a pamphlet issued by the Engineering Department of 
the Metropolitan Water Board of Boston. The weight of castings is based 
on one cubic inch of cast iron weighing 0.2604 lb. These tables are selected 
because they are based on methods and details quite generally approved 
not only in New England but elsewhere. Various foundries furnish details 
differing more or less from these and from each other, and hence the sub- 
joined data of weights, etc., must necessarily be approximate. Five classes 
of pipe are recognized as standards: Class A for heads up to 115ft. (50 lbs.); 
class B for heads up to 150 ft. (65 lbs.); class C for heads up to 200 ft. (87 lbs); 
class D for heads up to 250 ft. (109 lbs.) ; and class E for heads up to 300 ft. 
(130 lbs. per sq. inch). (See formula (2), page 1215.) 



1220 



6i.— WATER WORKS. 



7. — Straight Cast-Iron Pipes with Bell and Spigot.* 
(Met. Water Works.) 



"K 


^T" "1 


<> 


lf-i:v.< - 


'1 '°-^' 11 \ firf 




k d ->K IZ'O- >i 

Fig. 7. 



i. 




Dimensions in Inches. 


Weight in Pounds. 












1 














Per 

Length. 


Per Ft. of 


a 


b 


c 


d 


t 


3 


Straight 
Pipe. 


4 


D 


1.50 


1.30 


0.65 


3.00 


0.40 


0.40 


230 


17.3 


4 


E 










0.45 




255 


19.7 


6 


D 




1.40 


0.^70 




0.46 




380 


29.2 


6 


E 










0.50 




415 


31.9 


8 


D 




1.50 


0.^75 


3.50 


0.52 




565 


43.5 


8 


E 










0.55 




600 


46.2 


10 


D 










0.60 




800 


62.4 


10 


E 










0.63 




840 


65.7 


12 


B 




1.60 


0.80 




0.57 




910 


70.3 


12 


C 










0.61 




970 


75.5 


12 


D 










0.65 




1030 


80.7 


12 


E 










0.69 




1095 


86.0 


14 


B 




1.^70 


0.^85 




0.61 




1130 


87.5 


14 


C 










0.65 




1200 


93.5 


14 


D 










0.70 




1290 


101.0 


14 


E 










0.75 




1380 


108.6 


16 


B 


1.75 


1.80 


0.90 


4.00 


0.65 


0.50 


1380 


106.2 


16 


C 










0.70 




1485 


114.8 


16 


D 










0.75 




1590 


123.3 


16 


E 










0.81 




1715 


133.7 


20 


B 




2.^00 


1.00 




0.73 




1930 


148.6 


20 


C 










0.79 




2080 


161.2 


20 


D 










0.85 




2235 


174.0 


20 


E 










0.92 




2415 


188.9 


24 


B 


2.00 


2.10 


1.05 




0.80 




2525 


194.8 


24 


C 










0.87 




2740 


212.4 


24 


D 










0.95 




2985 


232.7 


24 


E 










1.03 




3230 


253.1 


30 


B 




2.^30 


1.15 


4.^50 


0.92 




3625 


279.3 


30 


C 










1.00 




3930 


304.3 


30 


D 




2.50 


1.25 




1.10 




4335 


335.8 


30 


E 








«« 


1.20 




4720 


367.5 


36 


A 










0.93 




4400 


337.1 


36 


B 










1.03 




4850 


374.4 


36 


C 










1.13 




5300 


411.9 


36 


D 




2.^80 


1.^40 




1.25 




5900 


457.1 


36 


E 










1.36 




6400 


498.8 


42 


A 








5.^00 


1.01 




5610 


426.4 


42 


B 










1.14 




6290 


482.8 


42 


C 










1.27 




6975 


539.4 


42 


D 




3.20 


1.60 




1.40 




7750 


596.5 


48 


A 




3.00 


1.50 




1.15 




7270 


554.9 


48 


B 










1.25 




7870 


604.3 


48 


C 










1.40 




8760 


678.9 


48 


D 


2.25 


3.50 


1.75 




1.55 




9820 


753.9 


54 


A 


2.^25 


3.^10 


1.^55 


5.50 


1.23 




8770 


666.9 


54 


B 










1.35 




9570 


733.5 


54 


C 




3.90 


1.95 




1.53 




11030 


834.0 


60 


A 




3.20 


1.60 




1.35 




10630 


813.0 


60 


B 










1.50 




11750 


905.6 


60 


C 




4.20 


2.10 




1.70 




13590 


1029.7 



See also Tables 26, 27, 28 and 29, following. 



CAST IRON PIPE WITH BELL AND SPIGOT. 



1221 



Special Castings are Grouped as follows: 
I. — 16 ins. and smaller, only Class E. f Thickness of metal (except 

II.-20 to 24 ins., inclusive. Classes C and E.^^- tmTclafse?^^' strT&S 
III; — 48, 54 and 60 ins.. Classes B, C and E. Ipipe, Table 7. 



8. — Bells of Special Castings. 
(Met. W. W.) 



azf-r 




Fig. 8. 



NoTTiinal 
Diameter- 
Inches. 


Class. 




Dimensions in Inches. 




a 


b 


c 


d 


J 


3 


E 


1.50 


1.20 


0.60 


4.00 


0.40 


4 






1.30 


0.65 






6 






1.40 


0.70 






8 






1.50 


0.75 






10 








«« 


4.50 




12 






1.60 


0.80 






14 






1.70 


0.85 






16 




1.75 


1.80 


0.90 


5.00 


0.50 


20 


C 




2.00 


1.00 






20 


E 




•• 


• • 






24 


C 


2.00 


2.10 


1.05 






24 


E 




•* 


•• 






30 


C 




2.30 


1.15 






30 


E • 




2.50 


1.25 






36 


C 




" 


•« 






36 


E 




2.80 


1.40 






42 


C 












42 


E 




3.20 


1.60 






48 


B 




3.00 


1.50 






48 


C 


" 










48 


E 


2.25 


3.50 


1.75 






54 


B 




3.10 


1.55 






60 


B 




3.20 


1.60 







1222 



M.— WATER WORKS. 



9. — Straight Cast-Iron Pipes, Bell and Spigot. 

(Met. W. W.) 

Standard, Maximum and Minimum Weights per Length and per Inch; 

Weight of Lead Joints and Gaskets. All Weights in Lbs. 







St'd Wt. 1 




Max. 


Wt. 


Min. 


Wt. 


Wt. of Lead 


^^ 


CO 

c 








d . 
.3^ 










per Joint. 


3 ft 

►-5 , 


M 


Per 


Per 


Per 


Per 


Per 


Per 




c 






25 . 


P 




length 


Inch 


C o 


length 


Inch 


length 


Inch 


With 
Gasket 


Solid 
Lead 


ll« 


D 


of 
12 Feet. 


of Str. 
Pipe. 


>< 
% 


of 
12 Feet. 


of Str. 
Pipe. 


of 
12 Feet. 


of Str. 
Pipe. 


^0^ 


4 


230 


1.4 


4 


239 


1.5 


221 


1.4 


7 


9.25 


0.10 


4 


E 


255 


1.6 




265 


1.7 


245 


1.6 


( < 






6 


D 


380 


2.4 




395 


2.5 


365 


2.3 


9.75 


12.75 


0.15 


6 


E 


415 


2.7 




432 


2.8 


398 


2.6 


* 1 


< < 




8 


D 


565 


3.6 




588 


3.8 


542 


3.5 


12.5 


18.75 


0.^25 


8 


E 


600 


3.9 




624 


4.0 


576 


3.7 


< ( 






10 


D 


800 


5.2 




832 


5.4 


768 


5.0 


15.25 


23.25 


0.30 


10 


E 


840 


5.5 




874 


5.7 


806 


5.3 


* 


' • 




12 


B 


910 


5.9 




946 


6.1 


874 


5.6 


17.75 


26.75 


0.35 


12 


C 


970 


6.3 




1009 


6.5 


931 


6.0 


< < 


' • 




12 


D 


1030 


6.7 




1071 


7.0 


989 


6.5 


18 


27.25 




12 


E 


1095 


7.2 




1139 


7.5 


1051 


6.9 


( < 


*• 




14 


B 


1130 


7.3 




1175 


7.6 


1085 


7.0 


20.5 


31 


0.40 


14 


C 


1200 


7.8 




1248 


8.1 


1152 


7.5 


*♦ 


« « 




14 


D 


1290 


8.4 




1342 


8.8 


1238 


8.1 


20.75 


31.5 




14 


E 


1380 


9.1 




1435 


9.4 


1325 


8.7 


< 


« < 




16 


B 


1380 


8.9 




1435 


9.2 


1325 


8.5 


31 


50 


0.65 


16 


G 


1485 


9.6 




1544 


10.0 


1426 


9.2 


31 


50 25 




16 


D 


1590 


10.3 




1654 


10.7 


1526 


9.9 


31.25 


50.5 




16 


E 


1715 


11.1 




1784 


11.6 


1646 


10.7 


31.5 


51 




20 


B 


1930 


12.4 




2007 


12.9 


1853 


11.9 


38 


61 


0.80 


20 


C 


2080 


13.4 




2163 


14.0 


1997 


12.9 


38 25 


61.5 




20 


D 


2235 


14.5 




2324 


15.1 


2146 


13.9 


38.5 


62 




20 


E 


2415 


15.7 




25i2 


16.4 


2318 


15.1 


39 


62.5 




24 


B 


2525 


16.2 


3^ 


26 l3 


16.8 


2437 


15.7 


45 


73 


0.^95 


24 


C 


2740 


17.7 




2836 


18.3 


2644 


17.1 


45.25 


73.5 




24 


D 


2985 


19.4 




3089 


20.1 


2881 


18.7 


45.5 


74 




24 


E 


3230 


21.1 




33-13 


21.8 


3117 


20.4 


46 


74.5 




30 


B 


3625 


23.3 


3^ 


3734 


24.0 


3516 


22.6 


56 


100.5 


1.^55 


30 


C 


3930 


25.4 




4048 


26.1 


3812 


24.6 


56.25 


101 




30 


D 


4335 


28.0 




4465 


28.8 


4205 


27.1 


56.5 


101.5 




30 


E 


4720 


30.6 




4862 


31.5 


4578 


29.7 


57 


102 




36 


A 


4400 


28.1 




4532 


28.9 


4268 


27.2 


66.5 


120 


1.85 


36 


B 


4850 


31.2 




4995 


32.1 


4705 


30.2 


, 67 


120.5 




86 


C 


5300 


34.3 




5459 


35.4 


5141 


33 3 


67.5 


121 




36 


D 


5900 


38.1 




6077 


39.2 


5723 


36.9 


68 


122 




36 


E 


6400 


41.6 




6592 


42.8 


6208 


40.3 


68.5 


122 5 




42 


A 


5610 


35.5 




5778 


36.6 


5442 


34.5 


77.5 


153 


2.^60 


42 


B 


6290 


40.2 




6479 


41.4 


6101 


39.0 


77.5 


154 




42 


C 


6975 


45.0 




7184 


46.3 


6766 


43.6 


78 


155 




42 


D 


7750 


49.7 




7983 


51.2 


7518 


48.2 


78.5 


156 




48 


A 


7270 


46.2 




7488 


47.6 


7052 


44.9 


88 


175 


3.00 


48 


B 


7870 


50.4 




8106 


51.8 


7634 


48.8 


88.5 


176 




48 


C 


8760 


56.6 




9023 


58.3 


8497 


54.9 


89 


177 




48 


D 


9820 


62.8 




10115 


64.7 


9525 


61.0 


89.5 


178 




54 


A 


8770 


55.6 




9033 


57.2 


8507 


53.9 


99 


214.5 


3.95 


54 


B 


9570 


61.1 




9857 


63.0 


9283 


59.3 


99.5 


215 




54 


C 


11030 


69.5 




11361 


71.6 


10699 


67.4 


100 


215.5 




60 


A 


10630 


67.8 




10949 


69.8 


10311 


65.7 


110 


238 


4.40 


60 


B 


11750 


75.5 




12103 


77.7 


11398 


73.2 


110.5 


239 




60 


C 


13590 


85.8 




13998 


88.4 


13182 


83.2 


111 


241 





C. /. PIPE WITH BELL AND SPIGOT. BELL. 



1223 



10. — Properties op Bells op 

Straight Pipes and op 

Special Castings. 

(Met. W. W.) 



Center ofOivivrty 

-1 


















^i 




Fig. 10. 








Bells of Straight Pipes. 


Bells of Special Castings. 


i 




Area 






Diam. 


Rad. 

of 
Bell. 
Feet. 




Area 






Diam. 






« 




(Shad- 






of Bell 


Wt. 
Lbs. 


(Shad- 






of Bell 


Wt. 
Lbs. 


5 


1 




ed). 
Sq. 
Ins. 


g 


q 


Open- 
ings. 
Ins. 


ed). 
Sq. 
Ins. 


g 


q 


Open- 
ings. 
Ins. 


3 


E 














3.90 


0.40 


1.50 


4.7 


15 


E 


4 


D 


"siei" 


'6!46* 


'i!55' 


'**5.'66* 




"'19' 














4 


E 




• • 


" 


5.70 


0.35 


20 


4.26 


0.44 


1.55 


5.7 


23 


E 


6 


D 


3.92 


0.49 


1.65 


7.72 




31 














6 


E 




< < 


' • 


7.80 


0.44 


31 


4.62 


0.47 


1.65 


7.8 


36 


E 


8 


D 


4.65 


0.51 


1.75 


9.84 


0.54 


41 














8 


E 




• * 


« « 


9.90 


< c 


42 


5.03 


0.50 


1.75 


9.9 


45 


E 


10 


D 




c < 


' • 


12.00 


0.63 


50 














10 


E 


" 


• ' 


*' 


12.06 


« • 


50 


5.40 


0.49 


1.75 


12.1 


58 


E 


12 


B 


5.02 


0.55 


1.85 


13.94 


0.71 


62 














12 


C 




' ' 


♦ • 


14.02 


0.72 


62 














12 


D 




' ' 


' ' 


14.10 


' ' 


62 














12 


E 




• ' 


' « 


14.18 


'♦ 


63 


5.82 


0.53 


1.85 


14.2 


73 


E 


14 


B 


5.40 


0.59 


1.90 


16.02 


0.80 


76 














14 


C 




' • 


' ' 


16.10 


0.81 


76 














14 


D 




• * 


' ' 


16.20 


0.82 


77 














14 


E 




• • 


" 


16.30 


*• 


77 


6.25 


0.57 


1.90 


16.3 


89 


E 


16 


B 


6.59 


0.61 


2.10 


18.30 


0.91 


105 














16 


C 




' • 


' ' 


18.40 


0.92 


106 












■ 


16 


D 


< < 


' ' 


' ' 


18.50 


( ( 


106 














16 


E 




• ' 


' ' 


18.62 


0.93 


107 


7.49 


0.59 


2.10 


18.6 


121 


E 


20 


B 


7.43 


0.67 


2.30 


22.46 


1.10 


145 














20 


C 




' ' 


' * 


22.58 


1.11 


145 


8.43 


0.65 


2.30 


22.6 


165 


C 


20 


D 




' ' 


' • 


22.70 


• ' 


146 














20 


E 




• ' 


' ' 


22.84 


1.12 


147 


8.43 


0.65 


2.30 


22.8 


167 


E 


24 


B 


8.^15 


0.74 


2.40 


26.60 


1.28 


187 














24 


C 




« « 


' ' 


26.74 


1.29 


188 


9.20 


0.72 


2.40 


26.6 


212 


C 


24 


D 




' ' 


' ' 


26.90 


1.30 


189 














24 


E 




' ' 


1 « 


27.06 


( ( 


190 


9.20 


0.72 


2.40 


26.9 


214 


E 


30 


B 


9.65 


0.79 


2.55 


32.84 


1.56 


272 














30 


C 




• ' 


• ' 


33.00 


1.57 


273 


10.23 


0.78 


2.55 


32.9 


289 





30 


D 


10.^63 


0.85 


2.75 


33.20 


1.59 


304 














30 


E 




' ' 


' ' 


33.40 


1.60 


305 


11.26 


0.84 


2.75 


33.2 


321 


E 


36 


A 


10.60 


0.87 


' • 


38.86 


1.83 


352 














36 


B 




• * 


♦ ' 


39.03 


1.84 


354 














36 


C 




' ' 


' ' 


39.26 


' • 


356 


11.23 


0.86 


2.75 


39.1 


376 


C 


36 


D 


12.09 


0.95 


3.00 


39.50 


1.88 


410 














36 


E 




• ' 


' ' 


39.72 


1.89 


412 


12.79 


0.94 


3.00 


39.5 


433 


E 


42 


A 


12.^80 


' ' 


' ' 


45.02 


2.11 


491 














42 


B 




• ' 


' ' 


45.28 


2.12 


494 














42 


C 




' ' 


•* 


45.54 


2.13 


497 


12.80 


0.95 


3.00 


45.3 


494 


C 


42 


D 


15.01 


1.07 


3.35 


45. 80 


2.18 


589 














42 


E 




' ' 


' ♦ 


46.06 




592 


15.01 


1.07 


3.35 


45.8 


589 


E 


48 


A 


13.93 


1.02 


3.15 


51.30 


2.39 


608 














48 


B 




• ' 


' • 


51.50 


2.40 


610 


13.93 


1.02 


3.15 


51.5 


610 


B 


48 


C 








51.80 


2.41 


614 


' ' 


' • 


' • 


51.8 


614 


C 


48 


D 


17.13 


1.23 


3.60 


52.10 


2.46 


765 














48 


E 




' • 




52.40 




769 


17.13 


1.23 


3.60 


52.4 


769 


E 


54 


A 


15.62 


1.07 


3.25 


57.46 


2.65 


762 














54 


B 




' ' 


• • 


57.70 


2.66 


765 


15.62 


1.07 


3.25 


57.7 


765 


B 


54 


C 


20.45 


1.34 


3.95 


58.06 


2.74 


1016 














60 


A 


16.20 


1.13 


3.35 


63.70 


2.92 


874 














60 


B 


' ' 


" 


< ( 


64.00 


2.93 


878 


16.20 


1.13 


3.35 


64.0 


878 


B 


60 


C 


22.39 


1.43 


4.20 


64.40 


3.03 


1232 















1224 



Qi.— WATER WORKS. 



11. — Properties op ^ Curves.* 
(Met. W. W.) 




Diam- 
eter. 


Dimensions in Inches. 


Class. 


Weight. 
Lbs. 












Ins. 





t 


r 


■ k 


S 






4 


5.7 


0.45 


16 


22.6 


8 


E 


80 


6 


7.8 


0.50 


♦ ■ 








125 


8 


9.9 


0.55 


< « 


• < 


] 







185 


10 


12.1 


0.63 


< ( 




] 


2 




265 


12 


14.2 


0.69 


•• 










340 


14 


16.3 


0.75 


18 


25.5 








455 


16 


18.6 


0.81 


24 


34.0 








680 


20 


22.6 


0.79 


•• 


« ' 






C 


835 


20 


22.8 


0.92 


( • 


< < 






E 


950 


24 


26.6 


0.87 


30 


42.4 






C 


1260 


24 


26.9 


1 03 


' 


" 






E 


1465 



* See also Table 31, following. 
11a. — Properties op 3^ Curves. f Hb. — Properties op i^ Curves. f 



(Met. W. W ) 




Fig. 11a: 








Note. — Dimensions are in 


inches 


; weights, in lbs. 




1 










H Curves. 


^ Curves. 


^f^ 


Class. 





t 












po 








r 


k 


s 


Weight 


r 


k 


Weight 


4 


E 


5.7 


0.45 


24 


18 4 


4 


60 


48 


18.7 


55 


6 




7.8 


0.50 


' ' 




•• 


100 


Ik 


• ' 


85 


8 




9.9 


0.55 


•• 




' • 


135 


' • 


• t 


120 


10 




12.1 


0.63 


' ' 




( < 


185 


" 


• • 


165 


12 




14.2 


0.69 


' ' 




•• 


240 


' • 


•* 


210 


14 




16.3 


0.75 


36 


27.6 




345 


72 


28 1 


345 


16 




18.6 


0.81 


' ' 






440 


' • 




440 


20 


C 


22.6 


0.79 


48 


36.7 




675 


96 


37.5 


675 


20 


E 


22.8 


0.92 


' ' 






760 




' 


760 


24 


C 


26.6 


0.87 


60 


45.9 




1050 


120 


46 8 


1050 


24 


E 


26.9 


1,03 


•* 






1215 


«< 


' 


1215 


30 


•C 


32.9 


1.00 








1490 


< 1 


• 


1490 


30 


E 


33.2 


1.20 


' ' 






1780 


" 


' ' 


1780 


36 


C 


39.1 


1.13 


90 


68.9 




2810 


180 


70.2 


2810 


36 


E 


39.5 


1.36 


* ' 






3400 


' • 


' ' 


3400 


42 


c 


45.3 


1.27 


• ' 






3700 


' ' 


«# 


3700 


42 


E 


45.8 


1.53 


' • 






4470 


' 


'• 


4470 


48 


B 


51.5 


1.25 


' ' 






4200 


' ' 


** 


4200 


48 


c 


51.8 


1.40 


" 






4650 


*' 




4650 


48 


E 


52.4 


1.70 


' ' 






5700 


" 




5700 


54 


R 


57.7 


1.35 


« « 






5100 


• 




5100 


60 


B 


64.0 


1.50 


" 






6250 


' ' 


f • 


6250 



t See also Table 32, following. 



CAST IRON PIPE— CURVES AND BRANCHES, 



1225 



12. — Properties of Branches — L's, T's and Crosses.* 
(Met. W. W.) 



'^. 



x-./.x 



-^S^. .^^ 




Fig. 12. 



Dimensions in Inches. 


i 


Weights, in Lbs. 






















Three Way 


Four Way 


e 


f 


1 


P 


S 


Oe 


Of 


X 


y 


g 





Branch. 


Branch. 




2BeIls 


3BeIIs 


3BeIls 


4Bells 


4 


4 


11 


11 


23 


5.70 


5.70 








E 


120 


125 


155 


160 


6 


4 


11 


12 


' ' 


7.80 


* * 










160 


165 


200 


205 




6 


12 


( < 


24 




7.80 










190 


195 


245 


250 




4 


11 


13 


23 


9.90 


'5.70 








E 


215 


210 


250 


245 


" 


6 


12 




24 




7.80 


• 








245 


240 


300 


295 


" 


8 


13 




25 




9.90 










270 


265 


345 


340 


10 


4 


11 




23 


12.1 


5.70 








E 


285 


275 


320 


310 




6 


12 




24 




7.80 










315 


305 


370 


360 


• ' 


8 


13 




25 




9.90 










345 


335 


415 


405 


< < 


10 


14 




26 




12.1 










380 


370 


480 


470 


12 


4 


11 




23 


14.2 


5.7 








E 


335 


340 


395 


380 




6 


12 




24 




7.8 










390 


375 


440 


425 


«« 


8 


13 




25 




9.9 










420 


405 


490 


475 


•' 


10 


14 




26 




12.1 










460 


445 


550 


535 


< < 


12 


15 




27 




14.2 


1.25 


1.62 


2.50 




525 


510 


665 


650 


14 


4 


11 




23 


16.^3 


5.7 








E 


435 


415 


475 


455 


" 


6 


12 




24 




7.8 










475 


455 


525 


505 


> ( 


8 


13 




25 




9.9 










510 


490 


575 


555 


•« 


10 


14 




26 




12.1 










550 


530 


640 


620 


" 


12 


15 




27 




14.2 


1.25 


1.62 


2.50 




615 


595 


755 


735 


" 


14 


16 




28 




16.3 


' * 


♦ • 


• • 




665 


645 


835 


815 


16 


4 


11 




23 


18.6 


5.7 








E 


545 


530 


580 


565 


*' 


6 


12 




24 




7.8 










580 


565 


630 


615 


•• 


8 


13 




25 




9.9 










620 


605 


685 


670 


a < 


10 


14 




26 




12.1 










665 


650 


750 


735 


«• 


12 


15 




27 




14.2 


1.25 


1.62 


2.50 




730 


715 


865 


850 


•• 


14 


16 




28 




16.3 


' ' 


• ' 


• ' 




780 


765 


945 


930 


•• 


16 


17 




29 


•• 


18.6 


•• 


'• 


' ' 




870 


855 


1100 


1085 


20 


6 


12 




24 


22.6 


7.8 








C 


710 


710 


760 


760 


" 


t c 


•' 




< < 


22.8 


( f 








E 


795 


770 


845 


820 


" 


8 


13 




25 


22.6 


9.9 








C 


750 


750 


820 


820 


«• 




* « 




' ' 


22.8 


' ' 








E 


840 


815 


905 


880 


• < 


10 


14 




26 


22.6 
22.8 


12.^1 








C 
E 


800 
890 


800 

865 


890 
975 


890 
950 


;; 


12 


15 




27 


22.6 
22.8 


14.2 


1.25 


1.^62 


2.50 


c 

E 


870 
965 


870 
940 


1005 
1095 


1005 
1070 


< < 


14 


16 




28 


22.6 


16.3 








C 


925 


925 


1090 


1090 


" 


' ' 


' ' 




• ' 


22.8 


• • 








E 


1025 


1000 


1180 


1155 


< < 


16 


17 




29 


22.6 
22.8 


18.^6 










E 


1000 
1120 


1000 
1095 


1205 
1335 


1?05 
1310 


< < 


20 


19 




31 


22.6 


22.6 




" 




C 


1125 


1125 


1400 


1400 


: « 


' ' 


' ' 


" " ' 22.8 


22.8 


' ' 




E 


1260 


1235 


1555 


1530 



* See also Table 35, following. 



1226 



Qi.— WATER WORKS. 



12.- 


-Properties op 


Branches — L 


s. T 


's AND Crosses. — Continued. 


Dimensions in Inches. 


B 


Weights, m Lbs 


• 


e 


1 


1 


P 


s 


Oe 


0" 


X 


y 


s 


3 Way Branch 


4 Way Branch 


2 Bells 


3 Bells 


3 Bells 


4 Bells 


24 


6 


12 


21 


24 


26 6 


7.8 








c 


915 


910 


960 


955 


• < 


■ « 


♦ • 






26.9 


< 








E 


1035 


995 


1080 


1040 


« t 


8 
« > 


13 




25 

4 4 


26.6 
26.9 


9 9 








C 
E 


960 
1090 


955 
1050 


1025 
1150 


1020 
1110 




10 


14 




26 


26.6 
26.9 


12 1 








C 

E 


1020 
1150 


1015 
1110 


1105 
1230 


1100 
1190 


• • 


12 


15 




27 


26.6 


14.2 


1.25 


1.62 


2^50 


C 


1090 


1090 


1230 


1225 


• • 


< 1 


* ■ 




• 


26.9 


•' 


'* 


" 




E 


1240 


1200 


1360 


1320 




14 


16 
• < 




28 


26.6 
26.9 


16.3 








C 

E 


1160 
1300 


1155 
1260 


1315 
1450 


1310 
1410 


«♦ 


16 


17 




29 


26.6 


18.6 








C 


1255 


1250 


1475 


1470 


• • 


•• 


4 4 




'• 


26.9 


4 ( 








E 


1400 


1360 


1610 


1570 


24 


20 


19 


21 


31 


26.6 


22.6 


1.25 


1.62 


2.50 


C 


1375 


1370 


1640 


1635 




• t 


< < 




*■ 


26.9 


22.8 




|] 




E 


1560 


1520 


1830 


1790 


«• 


24 


21 




33 


26.6 
26.9 


26.6 
26.9 








C 
E 


1520 
1730 


1515 
1690 


1860 
2080 


1855 
2040 


30 


12 
• • 


15 
« i 


24 


27 

»4 


32.9 
33.2 


14.2 


1.25 


1.62 
• • 


2^50 


C 
E 


1490 
1740 


1470 
1690 


1620 
1860 


1600 
1810 


< * 


14 


16 




28 


32.9 
33.2 


16.3 
( < 




',. 




C 

E 


1570 
1820 


1550 
1770 


1720 
1960 


1700 
1910 


«« 


16 


17 




29 


32.9 
33.2 


18 6 








C 

E 


1680 
1940 


1660 
1890 


1880 
2130 


1860 
2080 




20 
* < 


19 
< « 




34 


32.9 
33.2 


22.6 
22.8 








C 

E 


1900 
2210 


1810 
2070 


2140 
2470 


2050 
2330 


** 


24 


21 




36 


32.9 


26.6 








C 


2060 


1970 


2370 


2280 


• < 


'• 


• " 




" 


33.2 


26.9 








E 


2410 


2270 


2730 


2590 


• « 


30 


24 




41 


32.9 


32 9 


1 50 


2.00 


3.00 


C 


2410 


2270 


2870 


2730 


• • 


< 1 






< « 


33.2 


33 2 








E 


2840 


2640 


3320 


3120 


36 


12 


15 


27 


27 


39.1 


14.2 


1.25 


1 62 


2^50 


C 


1960 


1920 


2070 


2030 




'• 


4 4 






39 5 


( < 




;; 




E 


2310 


2250 


'2410 


2350 


• • 

• • 


14 


16 




?? 


39.1 
39.5 


16.3 








C 

E 


2050 
2410 


2010 
2350 


2190 
2540 


2150 
2480 


•• 


16 


17 




29 


39.1 


18.6 








C 


2170 


2130 


2360 


2320 


< • 


i < 


4 < 






39.5 


< ( 








E 


2550 


2490 


2720 


2660 


• i 


20 


19 




34 


39.1 
39.5 


22.6 
22.8 








C 

E 


2450 
2890 


2310 
2710 


2670 
3120 


2530 
2940 


«f 


24 
1 1 


21 
< < 




36 

< 4 


39.1 
39.5 


26.6 
26.9 








C 

E 


2640 
3110 


2500 
2930 


2920 
3*00 


2780 
3220 


• « 


30 


24 

4 4 




41 


39.1 
39.5 


32.9 
33.2 


1 50 


2 00 


3.^00 


C 

E 


3040 
3610 


2830 
3340 


3450 
4030 


3240 
3860 


• • 


36 


27 




44 


39.1 


39.1 








C 


3390 


3180 


3940 


3730 


• • 


" 


'• 




<< 


39.5 


39.5 








E 


4050 


3770 


4680 


4400 


42 


12 


15 


30 


27 


45.3 


14.2 


1.^25 


1.62 


2.^50 


C 


2520 


2470 


2630 


2580 




4 4 


4 4 






45.8 


• 4 








E 


30i0 


2950 


3110 


3050 


• « 


14 


16 




28 


45.3 


16.3 








C 


2630 


2580 


2760 


2710 


f 


• ' 


•• 






45.8 


' * 








E 


3140 


3080 


3250 


3190 


< • 


16 


17 




29 


45.3 
45.8 


18.6 

4 4 








C 

E 


2780 
3290 


2730 
3230 


2960 
3460 


2910 
3400 


*i 


20 


19 




34 


45.3 


22.6 








C 


3120 


2940 


3330 


3150 


«« 


• « 


• 4 




• • 


45.8 


22.8 








E 


3720 


3490 


3930 


3700 




24 


21 




?5 


45.3 
45.8 


26.6 
26.9 








G 

E 


3350 
3990 


3170 
3760 


3600 
4240 


3420 
4010 


« • 
f < 


30 


24 




41 
t < 


45.3 
45.8 


32 9 
33.2 


1.^50 


2.^00 


3.^00 


C 
E 


3820 
4570 


3550 
4230 


4180 
4980 


3910 
4640 




36 


27 




44 


45.3 
45.8 


39.1 
39.5 








C 
E 


4210 
5040 


3940 
4700 


4690 
5590 


4420 
5250 


• « 


42 


30 




47 


45.3 


45.3 








C 


4700 


4420 


5410 


5130 


ft 


* 4 


4 4 




• 4 


45.8 


45.8 








E 


5650 


5310 


6460 


6120 


48 


16 


17 


33 


29 


51.5 
51.8 


18.^6 


1.^25 


1.^62 


2.^50 


B 
C 


3140 
3420 


3140 
3350 


3320 
3600 


3320 
3530 


• 


• « 


< < 




<( 


52.4 


«• 








E 


4150 


4090 


4300 


4240 




20 


19 




34 


51.5 


22.6 








B 


3510 


3360 


3710 


3560 


• * 


< 


• < ' '« 1 « « 


51.3 


■ • 








C 


3840 


3600 


4040 


3800 


, • 


« t 


•'1 •• 1 " 


52.4 


22.8 








E 


4670 


4400 


4860 


4590 



CAST IRON PIPE—BRANCHES, 



1227 



12. — ^Properties op Branches — L's, T's and Crosses.— Concluded. 



Dimensions in Incbes. 





Weights 


. m Lbs. 




f 


1 










X 






3 Way Branch 


4 Way Branch 


e 


p 


s 


o» 


Of 


y 


S 
































2 Bells 


3 Bells 


3 Bells 


4 Bells 


48 


24 


21 


33 


36 
• « 

<• 


51.5 
51.8 
52.4 


26.6 
< • 

26.9 


1.25 


1.62 


2,50 


B 
C 

E 


3750 
4110 
4980 


3600 
3870 
4710 


4000 
4350 
5210 


3850 
4110 
4940 




30 


t( 




41 


51.5 
51.8 
52 4 


32.9 
< < 

33.2 


1.50 


2.00 


3.00 


B 
C 
E 


4250 
4650 
5660 


4000 
4300 
5250 


4600 
5000 
6010 


4350 
4650 
5600 




36 






44 


51 5 
51 8 


39.1 
• < 








B 
C 


4650 
5100 


4400 
4750 


5080 
5510 


4830 
5160 










*• 


52.4 


39.5 








E 


6200 


5790 


6660 


6250 




42 






48 


51.5 


45.3 








B 


5200 


4900 


5800 


5500 










< < 


51.8 


< < 








C 


5680 


5280 


6300 


5900 










< ( 


52.4 


45.8 








E 


6900 


6420 


7600 


7120 




48 






50 
i * 


51.5 
51.8 
52.4 


51.5 
51.8 
52.4 








B 
C 
E 


5620 
6150 
7520 


5370 
5800 
7110 


6400 
6950 
8490 


6150 
6600 
8080 



13. — Properties of Y Branches.* 
(R. D, Wood & Co.) 




Fig. 13. 







Dimensions in Inches. 




Approximate 
Weight 




A 


B 


C 


D 


Pounds. 




3 


3 


9 


12 


80 




4 


3 


10 


13 


105 




5 


3 


12 


15 


140 




6 


3.5 


13.5 


17 


180 




8 


4 


16 


20 


250 




10 


4.5 


18.5 


23 


360 




12 


5 


21 


26 


495 




14 


5.5 


24.5 


30 


700 




16 


6.5 


27.5 


34 


905 




18 


7 


30 


37 


1090 




20 


7.5 


32.5 


40 


1310 




24 


8.5 


37.5 


46 


1920 



* See also Table 37, following. 



1228 



H.— WATER WORKS. 



14. — ^Properties op Y Branches.* 
(Met. W. W.) 




Fig. 14. 











Dimensions in Inches. 






1^ 






















o§ 


^.^ 


e 


t 


s 


P 


V 


w 


n 


r 





ti 


t2 


t3 


Z 


X 


y 


g 







20 


20 


18 


34 


13.5 


16.4 


1.25 


30 


22.6 


0.79 


1.15 


0.79 


1.^25 


1.25 


1.^62 


2.^50 


C 


1740 


•* 


" 


• * 






" 


1.45 




22.8 


0.92 


1.35 


0.92 










E 


1940 


24 


*• 


12 






< < 


1.25 




22.6 


0.87 


1.15 


0.79 










c 


1690 




< i 








•« 


1.45 




22.8 


1.03 


1.35 


0.92 










E 


1910 




24 


18 




15.^25 


19.3 


1.35 




26.6 


0.87 


1.25 


0.87 










C 


2320 




< • 








< < 


1.60 




26.9 


1.03 


1.50 


1.03 










E 


2660 


30 


• ♦ 


12 






•• 


1.35 




26.6 


1.00 


1.25 


0.87 










C 


2300 


< t 


" 


' ' 






•« 


1.60 




26.9 


1.20 


1.50 


1.03 










E 


2650 




30 


18 




18.0 


23.7 


1.55 




32.9 


1.00 


1.45 


1.00 




1.^50 


2.^00 


3.00 


C 


3780 




• • 


< < 






♦* 


1.90 




33.2 


1.20 


1.75 


1.20 










E 


4490 


36 


•• 


10 






•♦ 


1.55 




32.9 


1.13 


1.45 


1.00 










C 


3630 




" 


« < 






•* 


1.90 




33.2 


1.36 


1.75 


1.20 










E 


4310 




36 


18 




21.0 


28.2 


1.80 


24 


39.1 


1.13 


1.65 


1.13 


1.50 








C 


5540 




• ' 


• ' 






•• 


2.15 




39.5 


1.36 


2.00 


1.36 










E 


6480 




30 


6 




18.0 


23.7 


1.55 


30 


32.9 


1.27 


1.45 


1.00 


1.25 








C 


3630 












« « 


1.90 




33.2 


1.53 


1.75 


1.20 










E 


4290 




36 


10 




21.0 


28.2 


1.80 


24 


39.1 


1.27 


1.65 


1.13 


1.50 








C 


5240 




• • 


• ' 






'* 


2.15 




39.5 


1.53 


2.00 


1.36 










E 


6240 




42 


18 




25.0 


33.1 


1.95 




43.5 


1.27 


1.80 


1.27 










C 


7500 




• • 


•' 






•• 


2.45 




45.8 


1.53 


2.25 


1.53 










E 


9130 


48 


36 


2 




21.0 


^28.2 


1.55 




39.1 


1.25 


1.45 


1.03 










B 


4540 






;; 








1.80 




< • 


1.40 


1.65 


1.13 










C 


5040 




< < 








(« 


2.15 




39.5 


1.70 


2.00 


1.36 










E 


5880 




42 


10 




25.0 


33.1 


1.75 




45.3 


1.25 


1.60 


1.14 










B 


6590 




" 


• ' 






•• 


1.95 




• < 


1.40 


1.80 


1.27 






<( 




C 


7230 




«• 


• • 






«< 


2.45 




45.8 


1.70 


2.25 


1.53 










E 


8780 




48 


18 


68.5 


28.0 


37.6 


1.95 




51.5 


1.25 


1.80 


1.25 










B 


9370 




•• 








•• 


2.20 




51.8 


1.40 


2.05 


1.40 




" 






C 


10400 


•• 


•• 


* • 






•• 


2.75 




52.4 


1.70 


2.55 


1.70 


•• 


•• 


, , 




E 


12750 


60 


• < 


6 






• < 


1.95 




51.5 


1.50 


1.80 


1.25 


• • 


• < 






B 


8900 



* See also Table 36, following. 



C. /. PIPE—Y BRANCHES; HYDRANT BRANCHES. 1229 



15. — Properties op Hydrant Branches. 
(R. D. Wood & Co.) 




* 


1 


■- 


M 


^ 




Fig. 15. 







Dimensions in Inches. 






Approximate 
Weight. 


A 


B 


C 


D 


E 


L 


R 


Lbs. 


8 


6 


12 


24 


8 


36 


5 


240 


10 


6 


12 


24 


11 


36 


5 


315 


12 


6 


12 


24 


12 


36 


5 


385 


14 


6 


12 


24 


14 


36 


5 


490 


16 


6 


12 


24 


15 


36 


5 


580 


18 


6 


12 


24 


16 


36 


5 


670 • 


20 


6 


12 


24 


17 


36 


5 


770 


24 


6 


12 


24 


19 


36 


5 


1000 



1230 



6i.— WATER WORKS. 



16. — Properties of Blow-ofp Branches.* 
(Met. W. W.) 






^. 


W' 


1 


r^ 




m 


1> 



HJ* 



Fig. 16. 









Dimensions in Inches. 


Class. 


Weight 
m Lbs. 
























e 


f 


1 


P 


Oe . 


Of 


ti 


t2 


X 


y 


g 






8 


4 ] 





7 
8 


9.9 
12.1 


5.7 


0.55 
0.63 


0.^45 








E 


195 
255 


10 








12 


• < 




10 


14.2 




0.69 










• * 


320 


U 


•• 




11 


16.3 




0.75 










• • 


390 


16 


c • 




12 


18.6 




0.81 










« t 


500 


7.0 


6 1 




U 


22.6 
22.8 


7.8 


0.79 
0.92 


0.50 








C 
E 


700 
765 


• f 








24 


" 




16 


26.6 




0.87 










C 


910 


i« 


• « 






26.9 




1.03 










E 


1000 


30 
«< 


• 12 ] 




20.5 


32.9 
33.2 


14.2 


1.00 
1.20 


0.69 


1.25 


1.62 


2.50 


C 

E 


1380 
1580 


36 


< < 




23.5 


39.1 




1.13 










C 


1800 


• « 


• ' 






39.5 




1.36 










E 


2100 


42 






26.5 


45.3 
45.8 




1.27 
1.53 










C 
E 


2510 
2990 


48 


12 




29.5 


51.5 




1.25 


0.69 


1.25 


1.62 


2.50 


B 


3120 


r< 


• • 






51.8 
52.4 




1.40 
1.70 




" 






C 

E 


3340 
4090 


• • 

• • 
«( 


16 






51.5 
51.8 
52.4 


18.6 


1.25 
1.40 
1.70 


0.^81 








B 
C 
E 


3180 
3400 
4150 



* See also Table 38, following. 



C. I, PIPE— BLOW-OFF BRANCHES, WITH M. H. 1231 



17.- 



-Properties op Blow-opp Branches* 
WITH Manholes. 
(Met. W. W.) 



'* 



!^Dfam. 







Fig. 17. 



Dimensions In Inches. 


. 


h 


































Bolts. 


e 


t 1 


P 


o„ 


Of 


ti 


t? 


T 


V 


g 


V n 


ta 


m 


u 


k 


z 








































No. 


Dia 


o 


30 


121 


7 20.5 


32.9 


14.2 


1.00 


0.69 


1.2 


51.6 


2 2.50 2 


21 


1.2 


51.7 


5 1.7 


1. 


25. 


5 20 


Wa 


c 


?030 


« < 


1 1 ( 


< . < 


33.2 


«. 


1.20 










. (< 
















E 


2250 


36 


• 1 « 


•23.5 


39.1 




1.13 










• 24 
















C 


?500 


« < 


• « . 


. . « 


39.5 




1.36 










. . . 
















E 


2840 


42 


•• • 


'26.5 


45.3 




1.27 










' 27 



















3110 


1 1 


I . . 




45.8 




1.53 










( < ( 
















E 


3590 


48 


" • 


•29.5 


51.5 




1.25 










• 30 
















B 


3520 


* • 


t < • 


. .. 


51.8 




1.40 










. . • 



















3740 


a t 


(< . 


. ( . 


52.4 




1.70 










. « . 
















E 


4460 


t • 


16 • 


. •< 


51.5 


18.6 


1.25 


0.81 








' ' ' 
















B 


3600 


• • 


<< « 


. <i 


51.8 




1.40 










" " 
















c 


3800 


« < 


... 




52.4 




1.70 


























E 


4510 



See also Table 39, following. 



1232 



Qi,— WATER WORKS. 



18. — Properties op Manhole Pipes.* 
(Met. W. W.) 




Fig. 18. 



See also Table 40, following. 



19. — Properties of Sleeves. f 
(R. D. Wood & Co.) 




Fig. 19. 
Note. — Dimensions are in inches; weigh tsln lbs. 



1 















Dimensions in Inches. 


























Is 


























Bolts. 


m 


e 





V 


1 


s 


n 


ti 


ts 


m 


k 


u 


z 




O 


^h3 


No. 


Dia 


^ 


30 


32.9 


20 


17 


29 


21 


1.00 


1.25 


1.75 


1.0 


1.7 


25.5 


20 


IM 


c 


1900 


< < 


33.2 






< c 




• • 


1.20 












" 










E 


2150 


36 


39.1 






• « 




24 


1.13 












< < 








" 


C 


2380. 


« • 


39.5 






« 5 




• ' 


1.36 












" 










E 


2740 


42 


45.3 










27 


1.27 












« ' 










C 


2980 


( < 


45.8 






' ' 




' * 


1.53 












< ( 










E 


3490 


48 


51 . 5 






' ' 




30 


1.25 












< * 










B 


3330 




51.8 






' ' 




" 


1.40 






















C 


3610 


• • 


52.4 






' ' 




< • 


1.70 












' « 










E 


4320 


60 


64.0 






18 




37 


1.50 






















B 


4820 





D 


Dimensions ii 


1 Inches. 


4^ 

ft 
1 




i 




Dimensions in Inches. 




A 


a 


b 


1 


0' 


t 


a 


b 


1 


0' 


t 


$ 


3 


E 


1.50 


1.20 


10 


4.8 


0.60 


35 


36 





2.00 


2.50 


15 


39.7 


1.25 


820 


4 






1.30 


< < 


5.8 


0.65 


45 


' ' 


E 






" 


( < 




1.50 


910 


6 






1.40 


• ' 


8.0 


0.70 


67 


42 


C 






2.80 


" 


45.5 


1.40 


1050 


8 






1.50 


12 


10.1 


0.75 


100 










*♦ 


20 


< ( 


( < 


1320 


10 








« « 


12.2 


•♦ 


120 




E 






( ( 


15 


46.0 


1.65 


1170 


12 






1.60 


14 


14.3 


0.80 


170 




• ' 






** 


20 


" 


" 


1490 


14 






1.70 


15 


16.4 


0.85 


220 


48 


B 






3.00 


15 


51.7 


1.50 


1280 


16 




1.75 


1.80 


' • 


18.7 


0.90 


275 










<( 


20 




( < 


1600 


20 


C 




2.00 


" 


23.0 


1.00 


370 











' * 


15 


52.0 


1.50 


1300 


" 


E 




' ' 


' ' 


( < 


1.05 


385 










• ' 


20 


« « 


'* 


1625 


24 


C 
E 


2.^00 


2.10 


;; 


27.0 


1.15 


470 
495 




E 






*' 


15 
20 


52.6 
< < 


1.85 
< < 


1460 
1870 


30 


C 




2.30 


'• 


33.4 


< ( 


630 


60 


B 


2. 


25 


3.20 


15 


64.2 


1.60 


1740 


< < 


E 










1.30 


680 


< ( 








20 






2170 



t See also Table 43, following. 



C. I. PIPE— MANHOLE: SLEEVES, REDUCERS. 



1233 



20. — Properties op Reducers — Type 1.* 
(R. D. Wood & Co.) 



ri:: 




Fig. 20. 



Dimensions in Inches 




Approximate Weight 
in Pounds. 


A 


B 


L 


No. 1. 


4 


3 


20 


55 


5 


4 


21 


65 


6 • 


4 


23 


75 


8 


6 


23 


110 


10 


6 


• 30 


155 


10 


8 


28.6 


180 


12 


6 


32 


185 


12 


8 


30.6 


210 


12 


10 


28.8 


240 



* See also Table 41, following. 



21. — Properties of Reducers — ^Type 2.t 
(Met. W. W.) 




Fig. 21. 



Dimensions in Inches. 


rn 


S ,. 


Dimensions in Inches. 


zn 


S .5 






^ 


•E-a-^ 










^ 


.Sf.fl^ 


e 


f 
10 


V 


s 


ti 


t2 


O 


^ ^ 


e 


f 


V 


s 


ti 


t2 


O 


^ ^ 


14 


20 


e 


0.75 


0.63 


E 


260 


36 


30 


32 


8 


1.13 


1.00 


c 


1440 


16 


' • 






0.81 


* ' 


' « 


300 




' ' 


' ' 




1.36 


1.20 


E 


1740 


" 


12 






' • 


0.69 


• ' 


330 


42 


' ' 


' ' 




1.27 


1.00 





1690 


20 


' ♦ 






0.79 


' • 


C 


435 




' ' 


' ' 




1.53 


1.20 


E 


2040 




• ♦ 






0.92 


' ' 


E 


480 




36 


" 




1.27 


1.13 


C 


1920 


- 4 


16 






0.79 


0.70 





485 




' ' 


' ' 




1.53 


1.36 


E 


2320 


" 


' ' 






0.92 


0.81 


E 


565 




' ' 


66 




1.27 


1.13 


C 


3280 


24 


" 






0.87 


• ' 


C 


610 




' • 


' ' 




1.53 


1.36 


E 


3970 


< 


" 






1.03 


< ( 


E 


675 


48 


** 


32 




1.25 


1.03 


B 


1970 


«« 


20 






0.87 


0.79 





655 




' ' 


' ♦ 




1.40 


1.13 


C 


2200 


(< 








1.03 


0.92 


E 


775 




' ' 


' ' 




1.70 


1.36 


E 


2660 


30 








1.00 


0.79 





810 




42 


' ' 




1.25 


1.14 


B 


2200 


• < 


' ' 






1.20 


0.92 


E 


970 




' ' 






1.40 


1.27 





2460 


•• 


24 






1.00 


0.87 


C 


905 




' ' 


• « 




1.70 


1.53 


E 


3GC0 


f 


*• 






1.20 


1.03 


E 


1090 




36 


132 




1.25 


1.03 


B 


60C0 


36 


•• 


32 




1.13 


0.87 





1240 










1.40 


1.13 


C 


67C0 


«< 


• < 






1.36 


1.03 


E 


1490 










1.70 


1.36 


E 


8130 



t See also Table 42, following. 



1234 



Qi.— WATER WORKS, 



22. — ^Properties op Caps.* 
(Met.W. W.) 




pL, ^^''"°^^'^' Plan. ■ SectidnonAB. 

I4inch and Smafiec eoinch and Larger, 

Fig. 22. 
Note. — Dimensions are in inches; weights in lbs. 



s 


01 












Dimensions In Inches. 










g 


Ph 


e 
















1 


5 


a 


b 


c 


d 


0' 


h 


t 


m 


k 


z 


r 


g 


X 


y 


4 


F 


1.50 


1.3 


0.65 


3.0 


5.8 


3.60 


0.60 
















25 


6 




* • 


1.4 


0.70 


•' 


8.0 


3.65 


0.65 
















40 


8 




«« 


1.5 


0.75 


3.5 


10.1 


4.25 


0.75 
















60 


10 




« « 




( 1 




12.2 


( < 


' ' 


1.50 


0.75 












80 


12 




'• 


1.6 


0.80 


•• 


14.3 


< ( 


( ( 


1.75 


• ' 












105 


14 




•« 


1.7 


0.85 


" 


16.4 


4.40 


0.90 


1.90 


' ' 












140 


16 




1.75 


1.8 


0.90 


4.0 


18.7 


5.00 


1.00 


2.00 


* • 






2.50 


1.25 


1.62 


240 


20 




♦ • 


2.0 


1.00 


♦ ' 


23.0 


5.25 


«( 


2.50 


1.00 


1.75 


22.35 


" 


" 


' ♦ 


350 


24 




2.00 


2.1 


1.05 


4.5 


27.0 


6.00 


1.05 


3.00 


' • 


1.80 


28.65 


• ' 


•* 


•• 


490 


30 




< < 


2.3 


1.15 


« • 


33.4 


•• 


1.15 


2.85 


'• 


2.00 


44.60 


3.00 


1.50 


2.00 


690 


36 




•• 


2.5 


1.25 


< ( 


39.7 


c ( 


1.25 


4.00 


1.25 


' ' 


44.55 


' • 


* • 


• ' 


970 


42 





• ' 


2.8 


1.40 


5.0 


45.5 


7.00 


1.50 


*• 


1.50 


•• 


63.25 


« • 


• ' 


• • 


1510 


< < 


E 


«' 




" 


' ' 


46.0 


•• 


1.60 


3.90 




2.25 




* ' 


« • 




1570 


48 


B 


" 


3.0 


1.50 


5.25 


51.7 


8.00 


1.75 


4.00 


*' 


3.00 


90.0 


♦ • 


• ' 




2120 


•' 


C 


' ' 


• • 


•• 


" 


52.0 


' ' 


1.90 


3.85 


f t 


3.15 


" 


• • 


• ' 


" 


2200 


f • 


E 








< < 


52.6 




2.00 


3.75 


• 


3.45 










2280 



* See also Table 44, following. 



23. — Properties op Plugs. f 
(R. D. Wood & Go.) 

I 1? 



i_J; 



Fig. 23. 
Note. — Dimensions are in inches. 



Diameter of Pipe. 


A 


D 


Approximate Weight 
In Pounds. 


3" 


3.8" 


5.5" 


4 


4 


4.9 


5.5 


6 


5 


6.0 


5.5 


8 


6 


7.0 


5.5 


10 


8 


8.1 


5.5 


13 


10 


11.2 


6.0 


28 


12 


13.2 


6.0 


38 



t See also Table 45, following. 



C. /. PIPE'-CAPS, PLUGS, OFFSETS, 



1235 



24. — ^Properties op Offsets.* 
(R. D. Wood & Co.) 

C H 




Fig. 24. 



Pipe 






Dimensions In Inches 






Approx. 


Dlam. 












Weight m 


Ins. 


R 


A 


B 


C 


L 


T 


Pounds. 


3 


8 


2 


4.6 


13.86 


10 


0.40 


65 


3 


14 


2 


4.6 


24.25 


10 


0.40 


75 


4 


8 


2 


5.8 


13.86 


10 


.0.45 


90 


4 


14 


2 


5.8 


24.25 


10 


0.45 


110 


5 


8 


2 


6.8 


13.86 


10 


0.48 


115 


5 


14 


2 


6.8 


24.25 


10 


0.48 


140 


6 


8 


2 


7.8 


13.86 


10 


0.50 


140 


6 


14 


2 


7.8 


24.25 


10 


0.50 


175 


8 


10 


2 


9.9 


17.32 


10 


0.55 


215 


8 


15 


2 


9.9 


25.98 


10 


0.55 


255 


10 


12 


2 


12.0 


20.78 


10 


0.60 


310 


10 


18 


2 


12.0 


• 31.18 


10 


0.60 


375 


12 


14 


2 


14.1 


24.25 


10 


0.65 


430 


12 


20 


2 


14.1 


34.61 


10 


0.65 


510 



* See also Table 34, following. 



(2) Turned and Bored Joint Pipe (see page 1215) is rarely used in the 
United States. During extremely cold weather the joints are liable to pull 
apart. If there is no danger from this cause, this kind of pipe will perhaps 
be economical if the cost of boring the bell end and turning the spigot end 
does not exceed the cost of lead joint for the ordinary bell and spigot pipe. 

(3) Flanged Joint Pipe (see page 682) or simply flange pipe is used for 
special connections and open work, generally at the ends of pipe lines, as in 
pump-houses, gate chambers, etc. The pipes come in 12-ft. lengths, and the 
flanges are bolted together, with rubber or other packing between. The 
following represents the practice of R. D. Wood & Co., of Philadelphia, 
adopting the standard of National Ass'n of Master Steam and Hot Water 
Fitters and the Am. Soc. of Mechanical Engineers, in flange diameters and 
drilling: 



Note. — ^Flexible Joint Pipe is described on page 1238, 



1236 



Qi.— WATER WORKS, 



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Weight 

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i-i«oc<ioo»0'<»«csi«oeoo>ooesjot^cooo-HCO<»'*eo 
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ooi«oot^»ncoesio>o>'^o>cvioo»eo«o<y>e<iccj-Hrq 

«oe>qoomcooo»t-?ooo050'!j'«30ot^oo>e<i-<o 

'^c<ie<aco'<*<ir5ir3?oooo^eoint-ooo'*is)»«o«o 

^^,,H^^c>ac«acO'<j«»r5«5c^ 


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ococooob-otoio-^osooeooooo-^ocMOOc^at^oo 


_,^^H.^C^JeO■«4*'9»»0 


Thick- 
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oscDecl^^*«ot^o>-H»^J<ooo-Hlorr-cc^^n«D«oco^~oo 
co-^-^j<'!}<-^-<jt-Ti<mioin)<»«oto«oi>.i>.c»osoo^ 


dici<=ici<z><:ici<sooociS<:i<:><^<zi<6r-t'^'^ 


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STANDARD CAST FLANGE PIPE. 



1237 



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Weight 

of Bolts 

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per 

Joint. 


BS. 

of that 


Diam. 

of 
Bolts. 


^^ 


Weight 

of 
Single 
Flange 


I. — Pressures 100 an 

ge.) 

ressure should not ex( 

its are approximate on 


1:4 

1 


1 
1 


Thick- 
ness 
of 
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rE Pipe. Part I 

3 on preceding pa 

tor steam the p 

tits in lbs. Weig 


1 

I 

1 

•c' 

1 


Weight 

of Bolts 

and Nuts 

per 

Joint. 


Diam. 

of 
Bolts. 


OF Standard Flang 

(See Notei 

en pipes are required 

ns are in inches ; weig] 


Weight 

of 
Single 
Flange 




t 

1 


perties 

ter; wh 
imensio 


Thick- 
ness 
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5.— Pro 

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ro^>o«ot-ooo>oe^^»««ocooeaj!;ooo£200 



1238 



6i.— WATER WORKS. 



(4) Flexible Joint Pipe (page 1215) is particularly advantageous and eco- 
nomical when it is necessary to carry the pipe line across a stream of any 
considerable size. If the stream is navigable and vessels are liable to 
anchor, the bed of the stream must be dredged on the line of the pipe, and 
a line of guide piles is often driven for this purpose. These piles also serve 
later in laying the pipe in the dredged trench. The process of laying the 
pipe is comparatively simple. Beginning at one shore, connection is made 
with the line laid up to that point (in a gradually desending deep trench 
toward the stream). An improvised scow with a run-way or incline is well 
suited to the purpose. The pipes are laid successively on the run-way, the 
joints are leaded, and the scow pulled ahead, allowing the pipe to gradually 
settle to the bottom of the trench without straining the joints excessively. 




Fig. 25. 

In pouring the lead for any joints the two adjacent pipes must be in a 
straight line. Sometimes trestle work is employed, in shallow water, and 
the pipes are lowered into place by block and tackle. The method of assem- 
bling the pipe on the bank, and then dragging the whole connected line into 
the stream is very liable to strain the joints and even pull them apart, as 
sometimes happens. 

• Flexible joint pipe is manufactured in pipe length, including pipe and 
joint, with bell-end machined inside; or the cast knuckle joints with flange 
ends or bell and spigot ends may be ordered separately for connecting with 
ordinary pipe. 

Flexible joints are used not only with cast iron pipe but with wrought 
iron and steel pipe as well.* 



* For design of flexible joint in connection with 52-in. riveted steel pipe, 
see Trans. Am. Soc. C. E., Vol. XXXIV. p. 23. 



FLEX. JOINT PIPE. C. I. PIPE SPECIFICATIONS. 1239 

Amertcan Water Works Association 

* STANDARD SPECIFICATIONS 

FOR 

CAST IRON PIPE AND SPECIAL CASTINGS 

With Tables of Dimensions, 

Thicknesses and Weights. 

Adopted May 12, 1908. 

STANDARD SPECIFICATIONS. 
Description of Pipes. 

Section \^ The pipes shall be made with hub and spigot joints, and shall 
conform accurately to the dimensions given in Tables 26 and 27. They shall 
be straight and shall be true circles in section, with their inner and outer 
surfaces concentric, and shall be of the specified dimensions in outside 
diameter* They shall be at least 12 feet in length, exclusive of socket. 

Pipes with thickness and weight intermediate between the classes in 
Table 27 shall be made of the same outside diameter as the next heavier 
class Pipes with thickness and weight less than shown by Table 27 shall be 
made of the same outside diameter as the Class A pipe; and pipes with 
thickness and weight more than shown by Table 27 shall be made of the 
same outside diameter as the Class D pipe. 

All pipes having the same outside diameter shall have the same inside 
diameter at both ends. The inner diameter of the lighter pipes of each 
standard outside diameter shall be gradually increased for a distance of 
about 6 inches from each end of the pipe so as to obtain the required standard 
thickness and weight for each size and class of pipe. 

For pipes of each size from 4-inch to 24-inch inclusive, there shall be 
two standards of outside diameter, and for pipes from 30-inch to 60-inch 
inclusive, there shall be four standards of outside diameter, as shown by 
Table 26. 

For pipes 4-mch to 12-inch inclusive, one class -of special castings shall 
be furnished, made from Class D pattern* Those having spigot ends shall 
have outside diameters of spigot ends midway between the two standards of 
outside diameter as shown by Table 26, and shall be tapered back for a 
distance of 6 inches. 

For pipes from 14-inch to 24-inch inclusive, two classes of special castings 
shall be furnished; Class B special castings with Classes A and B pipes, 
and Class D special castings with Classes C and D pipes; the former shall 
have cast on them the letters '*AB* and the latter "CD." For pipes 30-inch 
to 60-inch inclusive, four classes of special castings shall be furnished, one 
for each class of pipe, and shall have cast on them the letter of the class to 
which they belong. 

Allowable Variation in Diameter op Pipes and Sockets. 

Section 2. Special care shall be taken to have the sockets of the re- 
quired size. Tha sockets and spigots will be tested by circular gages, and 
no pipe will be received which is defective in joint room from any cause. 
The diameters of the sockets and the outside diameters of the spigot ends 
of the pipes shall not vary from the standard dimensions by more than 0.06 
of an inch foi pipes 16 inches oi less in diameter; 0.08 of an inch for 18-inch, 
20-inch and 24-inch pipes, 0.10 of an inch for 30-inch, 36-inch and 42-inch 
pipes; 0.12 of an inch for 48-inch, and 0.15 of an inch for 64-inch and 60-inch 
pipes. 

Allowable Variation in Thickness. 

Section 3. For pipes whose standard thickness is less than 1 inch, the 
thickness of metal in the body of the pipe shall not be more than 0.08 of an 
inch less than the standard thickness, and for pipes whose standard thick- 
ness is 1 inch or more, the variation shall not exceed 0.10 of an inch, except 



' Mr. J. M. Diven, Sec'y Am. W. W. Ass'n, writes the author as follows, 
under date of Sept. 3, 1908: "The specifications are not absolutely perfect 
yet, nor has there been any agreement between the various associations as 
to a universal standard. This is something greatly to be desired, and which 
I trust will be brought about in the next two or three years." 



1240 M.— WATER WORKS. 

that for spaceo not exceeding 8 inches in length in any direction, variations 
from the standard thickness of 0.02 of an inch in excess of the allowance 
above given shall be permitted. 

For special castings of standard patterns a variation of 50 per cent 
greater than allowed for straight pipes shall be permitted. 

Defective Spigots May be Cut. 

Section 4. Defective spigot ends on pipes 12 inches or more in diameter 
may be cut off in a lathe and a half-round wrought-iron band shrunk into a 
groove cut in the end of the pipe. Not more than 12 per cent of the total 
number of accepted pipes of each size shall be cut and banded, and no pipe 
shall be banded which is less than 11 feet in length, exclusive of the socket. 

In case the length of pipe differs from 12 feet, the standard weight of 
the pipe given in Table 27 shall be modified in accordance therewith. 

Special Castings. 

Section 5. All special castings shall be made in accordance with the 
cuts and dimensions given in the tables forming a part of these specifications. 

The diameters of the sockets and the external diameters of the spigot 
ends of the special castings shall not vary from the standard dimensions by 
more than 0.12 of an inch for castings 16 inches or less in diameter; 0.15 of 
an inch for 18-inch, 20-inch and 24-inch; 0.20 of an inch for 30-inch, 36-inch 
and 42-inch, and 0.24 of an inch for 48-inch, 54-inch and 60-inch. These 
variations apply only to special castings made from standard patterns. 

The flanges on all manhole castings and manhole covers shall be faced 
true and smooth, and drilled to receive bolts of the sizes given in the tables. 
The manufacturer shall furnish and deliver all bolts for bolting-on the man- 
hole covers, the bolts to be of the sizes shown on plans and made of the best 
quality of mild steel, with hexagonal heads and nuts and sound well-fitting 
threads. 

Marking. 

Section 6. Every pipe and special casting shall have distinctly cast upon 
it the initials of the maker's name. When cast especially to order, each 
pipe larger than 4-inch may also have cast upon it figures showing the year 
in which it was cast and a number signifying the order in point of time in 
which it was cast, the figures denoting the year being above and the number 
below, thus: 

1908 1908 1908 

12 3 

etc., also any initials, not exceeding four, which may be required by the 
purchaser. The letters and figures shall be cast on the outside and shall not 
be less than 2 inches in length and 3^ of an inch in relief for pipes 8 inches 
in diameter and larger. For smaller sizes of pipes the letters may be 1 inch 
in length. The weight and the class letter shall be painted conspicuously in 
white on the inside of each pipe and special casting after the coating has 
become hard. 

Allowable Percentage of Variation in Weight. 

Section 7. No pipes shall be accepted the weight of which shall be less 
than the standard weight by more than 5 per cent for pipes 16 inches or 
less in diameter, and 4 per cent for pipes more than 16 inches in diameter, 
and no excess above the standard weight of more than the given percent- 
age for the several sizes shall be paid for. The total weight to be paid for 
shall not exceed for each size and class of pipe received the sum of the 
standard weights of the same number of pieces of the given size and class 
by more than 2 per cent. 

No special casting shall be accepted the weight of which shall be less 
than the standard weight by more than 10 per cent for pipes 12 inches or 
less in diameter, and 8 per cent for larger sizes except that curves, Y-pieces 
and breeches pipe may be 12 per cent below the standard weight, and no 
excess above the standard weight of more than the above percentages for 
the several sizes will be paid for. These variations apply only to castings 
made from the standard patterns. 

Quality of Iron. 

Section 8. All pipes and special castings shall be made of cast iron of 
good quality, and of such character as shall make the metal of the castings 



CAST IRON PIPE SPECIFICATIONS, 1241 

strong, tough and of even grain, and soft enough to satisfactorily admit of 
drilling and cutting The metal shall be made without any admixture of 
cinder iron or other inferior metal, and shall be remelted in a cupola or air 
furnace. 

The contractor shall have the right to make and break three bars from 

each heat or run of metal, and the test shall be based upon the average 

results of the three bars. Should the dimensions of the three bars differ 

' from those given below, a proper allowance therefor shall be made in the 

results of the tests. 

Tests of Material. 

^Section 9. Specimen bars of the metal used, each being twenty-six inches 
long by two inches wide and one inch thick, shall be made without charge 
as often as the engineer may direct, and in default of definite instructions 
the contractor shall make and test at least one bar from each heat or run 
of metal. The bars when placed flatwise upon supports twenty-four inches 
apart, and loaded in the center, shall support a load of 2,000 pounds, and 
show a deflection of not less than 0.30 of an inch before breaking; or if 
preferred, tensile bars shall be made which will show a breaking point of 
not less than 20,000 pounds per square inch. 

Casting of Pipe. 

Section 10. The straight pipes shall be cast in dry sand molds in a vertical 
position. Pipes 16 inches or less in diameter shall be cast with the hub end 
up or down, as specified in the proposals. Pipes 18 inches or more in dia- 
meter shall be cast with the hub end down. 

The pipes shall not be stripped or taken from the pit while showing color 
of heat, but shall be left in the flasks for a sufficient length of time to prevent 
unequal contraction by subsequent exposure. 

Quality of Castings. 

Section 11. The pipes and special castings shall be smooth, free from 
scales, lumps, blisters, sand holes and defects of every nature which unfit 
them foi the use for which they are intended. No plugging or filling will be 
allowed. 

Cleaning and Inspection. 

Section 12. All pipes and special castings shall be thoroughly cleaned 
and subjected to a careful hammer inspection. No casting shall be coated 
unless entirely clean and free from rust, and approved in these respects by 
the engineer immediately before being dipped. 

Coating. 

Section 13 Every pipe and special casting shall be coated inside and 
out with coal-tar pitch varnish. The varnish shall be made from coal tar. 
To this material sufficient oil shall be added to make a smooth coating, 
tough and tenacious when cold, and not brittle nor with any tendency to 
scale off. 

Each casting shall be heated to a temperature of 300 degrees Fahrenheit 
immediately before it is dipped, and shall not possess less than this tempera- 
ture at the time it is put in the vat. The ovens in which the pipes are 
heated shall be so arranged that all portions of the pipe shall be heated to 
an even temperature. Each casting shall remain in the bath at least five 
minutes. 

The varnish shall be heated to a temperature of 300 degrees Fahrenheit 
(or less if the engineer shall so order), and shall be maintained at this tem- 
perature during the time the casting is immersed. 

Fresh pitch and oil shall be added when necessary to keep the mixture 
at the proper consistency and the vat shall be emptied of its contents and 
refilled with fresh pitch when deemed necessary by the engineer. After 
being coated the pipe shall be carefully drained of the surplus varnish. Any 
pipe oi special casting that is to be recoated shall first be thoroughly 
scraped and cleaned. 

Hydrostatic Test. 

Section 14 When the coating has become hard, the straight pipes shall 
be subjected to a proof by hydrostatic pressure, and, if required by the 
engineer, they shall also be subjected to a hammer test under this pressure. 

* Pipe may be made under higher metal tests when desired. Stock 
pipe may be made under metal tests as low as 1,800 pounds. 



1242 



^.— WATER WORKS, 



The pressures to which the different sizes and classes of pipes shall be 
subjected are as follows: 



20-Inch Diameter 

and Larger. 

Pounds per 

Square Inch. 



Less than 20- Inch 

Diameter. 

Pounds per 

Square inch. 



Class A Pipe 
Class B Pipe 
Class C Pipe 
Class D Pipe 



150 
200 
250 
300 



300 
300 
300 
300 



Weighing. 
Section 15. The pipes and special castings shall be weighed for payment 
under the supervision of the engineer after the application of the coal-tar 
pitch varnish. If desired by the engineer, the pipes and special castings 
shall be weighed after the delivery, and the weights so ascertained shall be 
used in the final settlement, provided such weighing is done by a legalized 
weighmaster. Bids shall be submitted and a final settlement made upon 
the basis of a ton of 2,000 pounds. 

Contractor to Furnish Men and Materials. 
Section 16. The contractor shall provide all tools, testing machines, 
materials and men necessary for the required testing, inspection and weigh- 
ing at the foundry, of the pipe and special castings; and should the purchaser 
have no inspector at the works, the contractor shall, if required by the engi- 
neer, furnish a sworn statement that all of the tests have been made as 
specified, this statement to contain the results of the tests upon the test 
bars. 

Power op Engineer to Inspect. 

Section 1 7. The engineer shall be at liberty at all times to inspect the 
material at the foundry, and the molding, casting and coating of the pipes 
and special castings. The forms, sizes, uniformity and condition of all 
pipes and other castings herein refered to shall be subject to his inspection 
and approval, and he may reject, without proving, any pipe or other casting 
which is not in conformity with the specifications or drawings . 

Inspector to Report. 

Section 18. The inspector at the foundry shall report daily to the 
foundry office all pipes and special castings rejected, with the causes for 
rejection. 

Castings to be Delivered Sound and Perfect. 

Section 19. All the pipes and other castings must be delivered in all 
respects sound and conformable to these specifications. The inspection 
shall not relieve the contractor of any of his obligations in this respect, and 
any defective pipes or other castings which may have passed the engineer 
at the works or elsewhere shall be at all times liable to rejection when dis- 
covered, until the final completion and adjustment of the contract; pro- 
vided, however, that the contractor shall not be held liable for pipes or 
special castings found to be cracked after they have been accepted at the 
agreed point of delivery. Care shall be taken in handling the pipes not to 
injure the coating, and no pipes or other material of any kind shall be 
placed in the pipes during transportation or at any time after they have 
received the coating. 

Definition of the Word "Engineer." 
Section 20. Wherever the word "engineer" is used herein it shall be 
understood to refer to the engineer or inspector acting for the purchaser 
and to his properly authorized agents, limited by the particular duties 
intrusted to them. 



CAST IRON PIPE WITH BELL AND SPIGOT. 



1243 



26. — Dimensions of Cast Iron Pipe. 
(A. W. W. A.) 

Classes A, B, C, D. 



Ur-a-H 




itzf] 






/S'O'-' 



X= ^^ on 3" to 6" inclusive. 
V= A" on 3" to 6" 
X= l"on 8" to 84" 
¥«= M"on 8''to 84" 







Fig. 


26. 














Nom- 




Actual 


DIam. of Sockets. 


Depth of S'kets. 








inal 
DIam 


Classes. 


Outside 
Diam. 










a 


b 




Pipe. 
Inches. 


Special 


Pipe. 
Inches. 


Special 


c 


Inches. 




Inches. 


Cast'gs. 
Inches. 


Cast'gs. 
Inches 








4 


A 


4.80 


5.60 


5.70 


3.50 


4.00 


1.5 


1.30 


.65 


4 


B-O-D 


5.00 


5.80 


5.70 


3.50 


4 00 


1.5 


1.30 


.65 


6 


A 


6.90 


7.70 


7.80 


3.50 


4.00 


1.5 


1 40 


.70 


6 


B-C-D 


7.10 


7.90 


7.80 


3.50 


4.00 


1.5 


1.40 


.70 


8 


A-B 


9.05 


9.85 


10.00 


4.00 


4.00 


1.5 


1.50 


.75 


8 


C-D 


9.30 


10.10 


10.00 


4 00 


4.00 


1.5 


1.50 


.75 


10 


A-B 


11.10 


11.90 


12.10 


4 00 


4.00 


1.5 


1.50 


.75 


10 


C-D 


11.40 


12.20 


12.10 


4.00 


4.00 


1.5 


1.60 


.80 


12 


A-B 


13.20 


14.00 


14.20 


4.00 


4.00 


1 5 


1.60 


.80 


12 


C-D 


13.50 


14.30 


14.20 


4 00 


4 00 


1.5 


1.70 


.85 


14 


A-B 


15.30 


16.10 


16.10 


4 00 


4.00 


1.5 


1.70 


.85 


14 


C-D 


15.65 


16.45 


16.45 


4.00 


4 00 


1 5 


1-80 


.90 


16 


A-B 


17.40 


18.40 


18-40 


4 00 


4.00 


1.75 


1.80 


.90 


16 


C-D 


17.80 


18.80 


18.80 


4 00 


4.00 


1-75 


1,90 


1.00 


18 


A-B 


19.50 


20.50 


20.50 


4 00 


4.00 


1.75 


1.90 


.95 


18 


O-D 


19.92 


20,92 


20.92 


4.00 


4.00 


1.75 


2.10 


1.05 


20 


A-B 


21.60 


22.60 


22.60 


4.00 


4.00 


1.75 


2.00 


1.00 


20 


C-D 


22.06 


23.06 


23.06 


4.00 


4.00 


1.75 


2 30 


1.15 


24 


A-B 


25.80 


26.80 


26.80 


4.00 


4.00 


2.00 


2.10 


1.05 


24 


C-D 


26.32 


27 32 


27.32 


4.00 


4.00 


2.00 


2.50 


1.25 


30 


A 


31.74 


32 74 


32 74 


4.50 


4.50 


2.00 


2.30 


1.15 


30 


B 


32.00 


33.00 


33.00 


4.50 


4.50 


2. 00 


2.30 


1.15 


30 


C 


32.40 


33.40 


33.40 


4.50 


4.50 


2.00 


2.60 


1.32 


30 


D 


32.74 


33.74 


33.74 


4.50 


4.50 


2.00 


3.00 


1.50 


36 


A 


37.96 


38 96 


38.96 


4.50 


4.50 


2.00 


2.50 


1.25 


36 


B 


38.30 


39.30 


39.30 


4 50 


4.50 


2.00 


2.80 


1.40 


36 


C 


38.70 


39.70 


39.70 


4.50 


4.50 


2.00 


3.10 


1 60 


36 


D 


39.16 


40.16 


40.16 


4.50 


4.50 


2.00 


3 40 


1.80 


42 


A 


44.20 


45.20 


45.20 


5.00 


5.00 


2 00 


2 80 


1.40 


42 


B 


44.50 


45.50 


45.50 


5.00 


5.00 


2.00 


3.00 


1.50 


42 


C 


45.10 


46.10 


46.10 


5.00 


5.00 


2.00 


3.40 


1.75 


42 


D 


45.58 


46.58 


46.58 


5.00 


6.00 


2.00 


3.80 


1.95 


48 


A 


50.50 


51.50 


51.50 


5.00 


5 00 


2.00 


3.00 


1.50 


48 


B 


50.80 


51.80 


51.80 


5 00 


5.00 


2.00 


3.30 


1.65 


48 


C 


51.40 


52.40 


52.40 


5.00 


5.00 


2.00 


3.80 


1.95 


48 


D 


51.98 


52.98 


52.98 


5.00 


5.00 


2.00 


4.20 


2.20 


54 


A 


56.66 


57.66 


57 66 


5.50 


5 50 


2.25 


3.20 


1.60 


54 


B 


57.10 


58. 10 


58.10 


5.50 


5 50 


2.25 


3.60 


1.80 


54 


C 


57.80 


58.80 


58.80 


5.50 


5.50 


2.25 


4.00 


2.15 


54 


D 


58.40 


59.40 


59.40 


5.50 


5.50 


2.25 


4.40 


2.45 


60 


A 


62 80 


63.80 


63.80 


5.50 


5.50 


2.25 


3.40 


1.70 


60 


B 


63 40 


64.40 


64.40 


5.50 - 


5.50 


2.25 


3.70 


1.90 


60 


C 


64.20 


. 65.20 


65.20 


5.50 


5.50 


2.25 


4.20 


2.25 


60 


D 


64.82 


65.82 


65.82 


.5.50 


5.50 


2.25 


4.70 


2.60 


72 


A 


75.34 


76.34 


76.34 


5.50 


5.50 


2.25 


3.80 


1.87 


72 


B 


76.00 


77.00 


77.00 


5 50 


5 50 


2.25 


4.20 


2.20 


72 


C 


76.88 


77.88 


77.88 


5.50 


5.50 


2.25 


4.60 


2.64 


84 


A 


87-54 


88-54 


88.54 


5.50 


5.50 


2.50 


4.10 


2.10 


84 


B 


88-54 


89-54 


89.54 


5.50 


5.50 


2.50 


4.50 


2.60 



1244 



6i.— WATER WORKS. 



N 






P. 












Oi 




J 


Z 






O 




C 


(4 




•»i 


H4 




0} 


Yn 




,c! 


< 




.^ 


O 




<u 


Pd 




Q '^ 


o 


,^-^ 




OT 


< 


<J I 


H 




- t£3 


1 


^ 
^ 


pq - 


Q 


< 


M 0} 


S: 




r5 W 


•< 




" s 








CO 




to 


H 




C 


^ 
Ul 




4) 

a 


2 

» 




Q 

1 


^ 




1 

w 

H 


r^ 




O 


« 




^ 



Nominal 

Inside 

Diameter. 

Inches. 




^^ooo C2;*020 S^^^S SSKS 




CLASS D 

400-Feet Head. 

173 Pounds Pressure. 




tt) 


o CO t- cq oioooiri oooooo oca • . 

co-^<oo> c>a lo oj CO i>. «o 'O' in os co —i ^ • • 

,-iT_i_ic^CM coiat«o>cv4 coo . . 


*s 


oooooc^ ocacot-cq t-oooo t^eo . . 

irtooirjco ooJod'-^'oi cdd»o»no ^*co I ' 
,-1 _i _i ,-( ca co-^cooo^ CO lo . . 


Thick- 
ness, 
Inches. 


dddd dddd^* ^^^^'^' esicsi '- '. 


CLASS C 

300-Feet Head. 

130 Pounds Pressure. 


bc 

1 


be 
►-1 


CM-^eooo ^^t>.T^»« coooiocoo* t- -^ oo . 
__^(Mcsa co-^coooo cocoevi . 


■*3 

§ 
Pm 


cooo^oo c^t^oooco cqooot^co t«t«c<i . 

coificdd »-<<oco»r>od ddincood ^-^-^ '. 

,.H T-l ,-1 . 


Thick- 
ness, 
Inches. 


TttiaiOfo cot-QOQoos oeMcoioi>. ooco . 




CLASS B 

200-Feet Head. 

86 Pounds Pressure. 


1 


5 

be 
a 


ooo»a iftoooo ooooo oooo 

coot^co oocoooo ooinoo o »o « in 

CM "* »o c^ osc^air500-H oo o -^ —i e^ c<i c<i in c<i 

,-i-H^c<i c^a -«*< ITS t- o> -Hcoooio 


o 


t^comoo »-<ioooo cococ^it^o cocqoocg 

'-i CO i>I CO eo* cq kfi d iri co co •^" -h d co -^ »n -^ 

cq CO ■^i< «o ooocqint^ co co in os in co o -^ ^ 

■^^^^ cQCO-^mc* Oi^m-^ 


Thick- 
ness, 
Inches. 


inoo.^t^ cqcooino o»coinooe^ int^mcq 
■^i< ■>* in in CO CO t^ c^ oo oo o »-i ca •»»» in co o> ca 

ddoo ci<s<::^<^ci d ^' ^ -^ r-^ ^ ,>; ^' est 




CLASS A 

100-Feet Head. 

43 Pounds Pressure. 


P. 


s 

be 

c 

3 


ooinin oinooo ooooo oooo 

•«i<t--Hoo t-t-oino inoomo oooo 

cvjcomco GO o CO in oo ti< in c^ -h o co o -r*" co 

^^^.M caco-*cooo O4^ino> 


■1^ 


oooo>»H m to CO e^a o cq t^ ^^ in t* ob--^-"** 

d d cvi b-I e>ci OS 00 oi d ■^" ^' -h cq d d d eo co 

e>q CO -^ in t» oo o evi in o oi oi ^ co o-hooco 

i-<^^ csje^acomco oo o» cm co 


Thick- 
ness, 
Inches. 


(M^too ■^c^O'<i<t^. cooocnoco m o> cq cm 
T»< ■rj« ■»»« in inmcococo t~ooos.-ie»q cococot.- 

o d d d c><6<6<6d odd'^'^ ^^^^ 




1«l 






•^cooo^ c<i-*cooo^ -^^coevaoo •>* o c<« ■*«« 
_i .-1 ,^ ,.^ ,>H c^ c4CQco'^-<4* incot^-oo 



CAST IRON PIPE TABLE. BELL AND SPIGOT. 



1245 



28. — Dimensions of Pipe for High Pressure Service. 

(A. W. W. A.) 
Classes E, F, G, H. 




S^'C-tL 



Fig. 28. 

Note.— Dimensions are in inches. 



Nomi- 
nal 
Dlam. 


Classes. 


Actual 

Outside 

Diam- 


Dlam. of 
Sockets. 


Depth of 
Sockets. 


a 


b 


C 


R 


Nomi- 
nal 
Dlam. 






Inches 




Inches 


Pipe and 
Specials. 


Pipe and 
Specials. 










Inches 


6 


E-F 


7.22 


8.02 


4.00 


1.50 


1.75 


0.75 


l.iO 


6 


6 


G-H 


7.38 


8.18 


4.00 


1.50 


1.85 


0.85 


1.10 


6 


8 


E-F 


9.42 


10.22 


4.00 


1.50 


1.85 


0.85 


1.10 


8 


8 


G-H 


9.60 


10.40 


4.00 


1.50 


1.95 


0.95 


1.10 


8 


10 


E-F 


11.60 


12.40 


4.50 


1.75 


1.95 


0.95 


1.10 


10 


10 


G-H 


11 84 


12.64 


4 50 


1 75 


2.05 


1.05 


1.10 


10 


12 


E-F 


13.78 


14 58 


4.50 


1.75 


2.05 


1.05 


1.10 


12 


12 


G-H 


14.08 


14 88 


4.50 


1.75 


2.20 


1.20 


1.10 


12 


14 


E-F 


15.98 


16 78 


4 50 


2.00 . 


2.15 


1.15 


1 10 


14 


14 


G-H 


16.32 


17.12 


4.50 


2.00 


2.35 


1 35 


1.10 


14 


16 


E-F 


18.16 


18.96 


4.50 


2.00 


2 30 


1.25 


1 15 


16 


16 


G-H 


18 54 


19 34 


4.50 


2.00 


2.55 


1 45 


1.15 


16 


18 


E-F 


20 34 


21.14 


4 50 


2.25 


2.45 


1.40 


1.15 


18 


18 


G-H 


20 78 


21 58 


4.50 


2.25 


2.75 


1.65 


1.15 


18 


20 


E-F 


22 54 


23.34 


4.50 


2.25 


2.55 


1.50 


1.15 


20 


20 


G-H 


23.02 


23.82 


4.50 


2.25 


2.85 


1.75 


1.20 


20 


24 


E-F 


26.90 


27.90 


5.00 


2 25 


2.85 


1.70 


1.20 


24 


30 


E 


33.10 


34.10 


5.00 


2.25 


3.25 


1 80 


1.50 


30 


30 


F 


33.46 


34 46 


5 00 


2.25 


3.50 


2.00 


1.55 


30 


36 


E 


39.60 


40 60 


5.00 


2.25 


3 70 


2 05 


1.70 


36 


36 


F 


40.04 


41.04 


5 00 


2.25 


4.00 


2.30 


1.80 


36 



1246 



H.—WATER WORKS. 



n 

H ^ 

& > 

H CO 

t/i CO 

^ W 

o « 

o w ^ 

^ £^ 

I. fe 



o 



> d 



W 



O 



^5" 



WO) (D 
SP4 









^ 



Sec 



osoooeq -^Oi-^eo 



«oot-oo t>.>At><<0 



1-1 CM C<I CO 



^ 






»r5 00 evj CO c> CO c<i 00 
1-1 »H e<i c<i CO CO 



Y-iooo-^ NpqT-HOO 



■^c^oco t- 1-1 CO CM 
1-1 1-1 »-i cq eqco 



OOOO ,-1 1-1 1-1 1-1 



^ 






e^iosoco 05-^coco 1-1 c<i '«« 

»fl t>. 1-1 "<t« OOCOOOi* i>.ooo 

»-i i-< i-ie^e^ico •^c~o> 



eo ko e»a «M t- lo oo co «m lo o 
■^ CO o> eg looeooo o> oo e<i 

1-1 1-1 1-1 pq C<I CO lO 00 



CO t^ 00 oo 
o o o o 



•d* ^ 


rd 


5 

W) 

c 

3 


O"<fcoco ij«in-i.t<io ^coo 

^^^22 ^^S;;3 5SSSo 


CLASS 
iOO-Feet H 
Pounds Pr 


1 




SI 


Thick- 
ness, 
I nches. 


o o d o d d -H* rH ^' ^* »H 



©5 oj a 



COOOOCM «4< (O 00 



OCM «4< (OOOO 



00 ;3 



fr. ft) 

r/i CO 

gM 
w C 

'd 0) 
aS 

CO 0) 

w o 

^^ 

as 



o 
o 

(0 W HI 

u O o 

j5 cfl C 

c 4So 

Cj o CO 

g la 

- ^§ 



CO 7 

o 






6 S • 
s CO u 

■ or. 



.« ctf 



Ph y 



-SS 






CAST IRON PIPE, HIGH PRESSURE. LUGS. 



1247 



30. — Lugs. 
Number and Weights of Lugs on Outlets of Different Sizes. 
(A.W. W. A). 



4 Lugs, 12-14 inches 
8 I^ugs, 42-60 inches 




Fig. 30a. 




-6 Lug?, 16-86 inches 



Nominal 

Diameter 

Outlet, 

Inches. 


Number 

of Pairs of 

Lugs. 


Approximate 

Weight Lugs 

on One Bell, 

Pounds. 


Nominal 
Diameter 

Outlet. 

Inches. 


Number 

of Pairs of 

Lugs. 


Approximate 

Weight Lugs 

on One Bell, 

Pounds. 


12 
14 
16 
18 
20 
24 


4 
4 
6 
6 
6 
6 


32 
32 
56 
56 
56 
56 


30 
36 
42 
48 
54 
60 


6 
6 
8 
8 
8 
8 


80 
80 
111 
114 
134 
137 



Two pairs of lugs are placed on the vertical axis of each bell, the others 
at equal distances around circumference. ^ = depth of bell on all sizes. 

(7=2.50 inches, X=1.25 inches, F=1.63 inches, for 12 to 24 inches 
inclusive. 

6^=3.00 inches, AT =1.50 inches, y=2.00 inches, for 30 to 60 inches 
inclusive. 



1248 



H.— WATER WORKS. 



31, 32. — Properties of Curves, Bell and Spigot, }i, }4, ^. 










Fig. 32a. Fig. 32b. 

Table 32. 



U Curves. 


H Curves. 


^ Curves. 


1' 


5 


Dimensions, 
Inches. 


Approx 
Weight 
Pounds 


■4 




Dimensions, 
Inches. 


Approx 
Weight 
Pounds 


Dimensions, 
Inches. 






t 


r 


k 


t 


r 


k 


r 


k 


§^£ 


4 


D 


0.52 


16 


22.60 


82 


4 


D 


0.52 


24 


18.40 


66 


48 


18.70 


66 














6 


D 


0.55 


24 


18.40 


105 


48 


18.70 


105 


6 


D 


0.55 


16 


22.60 


130 


8 


D 


0.60 


24 


18.40 


150 


48 


18.70 


150 














10 


D 


0.68 


24 


18.40 


202 


48 


18.70 


202 


8 


D 


0.60 


16 


22.60 


200 


12 


D 


0.75. 


24 


18.40 


265 


48 


18.70 


265 


10 


D 


0.68 


16 


22.60 


278 


14 


B 


0.66 


3.6 


27.60 


359 


72 


28.10 


312 














14 


D 


0.82 


36 


27.60 


442 


72 


28.10 


382 


12 


D 


0.75 


16 


22.60 


366 


16 


B 


0.70 


36 


27.60 


445 


72 


28.10 


388 














16 


D 


0.89 


36 


27.60 


558 


72 


28.10 


484 


14 


B 


0.66 


18 


25.50 


406 


18 


B 


0.75 


36 


27.60 


533 


72 


28.10 


464 


14 


D 


0.82 


18 


25.50 


504 


18 


D 


^.96 


36 


27.60 


663 


72 


28.10 


574 














20 


B 


0.80 


48 


36.70 


758 


96 


37.50 


676 


16 


B 


0.70 


24 


34.00 


594 


20 


D 


1.03 


48 


36.70 


964 


96 


37.50 


858 


16 


D 


0.89 


24 


34.00 


750 


24 


B 


0.89 


60 


45.90 


1181 


120 


46.80 


1072 














24 


D 


1 16 


60 


45.90 


1515 


120 


46.80 


1372 


18 


B 


0.75 


24 


34.00 


710 


30 


A 


0.88 


60 


45.90 


1475 


120 


46.80 


1342 














30 


B 


1.03 


60 


45 90 


1684 


120 


46.80 


1528 


18 


D 


0.96 


24 


34.00 


888 


30 


C 


1.20 


60 


45.90 


1983 


120 


46.80 


1800 














30 


D 


1.37 


60 


45.90 


2291 


120 


46.80 


2080 


20 


B 


0.80 


24 


34.00 


840 
































36 


A 


99 


90 


68.90 


2472 


180 


70.20 


2472 


20 


D 


1.03 


24 


34.00 


1070 


36 


B 


1.15 


90 


68.90 


2916 


180 


70.20 


2916 














36 


C 


1.36 


90 


68.90 


3430 


180 


70.20 


3430 


24 


B 


0.89 


30 


42.40 


1290 


36 


D 


1.58 


90 


68.90 


4012 


180 


70.20 


4012 














42 


A 


1.10 


90 


68.90 


3286 


180 


70.20 


3286 


24 


D 


1.16 


30 


42.40 


1656 


42 


B 


1.28 


90 


68.90 


3778 


180 


70.20 


3778 














42 


c 


1.54 


90 


68.90 


4600 


180 


70.20 


4600 


30 


A 


0.88 


36 


50.90 


1814 


42 


D 


1.78 


90 


68.90 


5360 


180 


70.20 


5360 


30 


B 


1,03 


36 


50.90 


2082 


48 


A 


1.26 


90 


68.90 


4230 


180 


70.20 


4230 














48 


B 


1.42 


90 


68.90 


4820 


180 


70.20 


4820 


30 


C 


1.20 


36 


50.90 


2454 


48 


C 


1.71 


90 


68.90 


5796 


180 


70.20 


5796 














48 


D 


1.96 


90 


68.90 


6750 


180 


70.20 


6750 


30 


D 


1.37 


36 


50.90 


2836 


54 


A 


1.35 


90 


68.90 


5180 


180 


70.20 


5180 














54 


B 


1.55 


90 


68.90 


5990 


180 


70.20 


5990 


36 


A 


0.99 


48 


67.90 


2964 


54 


C 


1.90 


90 


68.90 


7330 


180 


70.20 


7330 


36 


B 


1.15 


48 


67.90 


3500 


54 


D 


2.23 


90 


68.90 


8620 


180 


70.20 


8620 














60 


A 


1.39 


90 


68.90 


5990 


180 


70.20 


5990 


36 


C 


1.36 


48 


67.90 


4120 


60 


B 


1.67 


90 


68.90 


7130 


180 


70.20 


7130 














60 


C 


2.00 


90 


68.90 


8590 


180 


70.20 


8590 


36 


D 


1.58 


48 


67.90 


4820 


60 


D 


2.38 


90 


68.90 


10240 


180 


70.20 


10240 



5= 10 inches on sizes 8 inches. 



5= 8 inches on sizes 4 and 6 inches. 

5=12 inches on sizes 10 to 36 inches. 

5=6 inches on }-i Curves on sizes 4 to 30 inches inclusive. 

5=6 inches on ^ Curves on sizes 4 to 12 inches inclusive. 

All weights are approximate. 



C. /. PIPE— CURVES— BELL AND SPIGOT. 



1249 



33, 34. — ^Properties of Curves, Bell and Spigot. — Offsets. 
(A. W. W. A.) 





1< ^'-H 



Fig. 33b. 




Fig. 34. 



All dimensions are in inches. 
Table 33. Table 34. 





^V 


Curves. 




♦ 


gij Curves.* 


Offsets. 


o3h-i 










« 4i OV 






w^im 










a 




6 


t 


r 


k 


Appro: 
Weigh 
Pound 


r 


k 


Appro: 
Weigh 
Pound 


1 


r 


I 




4 


D 


0.52 


120 


23.52 


66 








4 


D 


8 


35.85 


91 


6 


D 


0.55 


120 


23.52 


104 


















8 


D 


0.60 


120 


23.52 


150 








6 


D 


14 


46.25 


183 


10 


D 


0.68 


120 


23.52 


192 


















12 


D 


0.75 


120 


23.52 


250 








8 


D 


15 


48.00 


280 


14 


B 


0.66 


180 


35.28 


364 


















14 


D 


0.82 


180 


35.28 


450 








10 


D 


16 


49.70 


390 


16 


B 


0.70 


180 


35.28 


453 


















16 


D 


0.89 


180 


35.28 


570 








12 


D 


17 


51.45 


530 


18 


B 


0.75 


180 


35.28 


542 


















18 


D 


0.96 


180 


35.28 


674 








14 


B 


18 


53.70 


555 


20 


B 


0.80 


240 


47.05 


808 


480 


47.10 


808 












20 


D 


1.03 


240 


47.05 


1028 


480 


47.10 


1028 


14 


D 


18 


53.70 


695 


24 


B 


0.89 


240 


47.05 


1080 


480 


47.10 


1080 












24 


D 


1.16 


240 


47.05 


1380 


480 


47.10 


1380 


16 


B 


19 


55.40 


708 


30 


A 


0.88 


240 


47.05 


1350 


480 


47.10 


1350 










• 


30 


B 


1.03 


240 


47.05 


1540 


480 


47.10 


1540 


16 


D 


19 


55.40 


900 


30 
30 


C 
D 


1.20 
1.37 


240 
240 


47.05 
47.05 


1810 
2090 


480 
480 


47.10 
47.10 


1810 
2090 
























36 


A 


0.99 


240 


47.05 


1790 


480 


47.10 


1790 












36 


B 


1.15 


240 


47.05 


2100 


480 


47.10 


2100 


II 


S 


t 


k 


S 


n 


36 


C 


1.36 


240 


47.05 


2470 


480 


47.10 


2470 


c3 










36 


D 
A 


1.58 
1.10 


240 
240 


47.05 
47.05 


2880 
2380 


480 
480 


47.10 
47.10 


2880 
2380 


;z;Q 


o 










42 


4 


D 


0.52 


13.85 


10.00 


2.00 


42 


B 


1.28 


240 


47.05 


2720 


480 


47.10 


2720 














42 


C 


1.54 


240 


47.05 


3310 


480 


47.10 


3310 


6 


D 


0.55 


24.25 


10.00 


2.00 


42 


D 


1.78 


240 


47.05 


3850 


480 


47.10 


3850 














48 


A 


1.26 


240 


47.05 


3150 


480 


47.10 


3150 


8 


D 


0.60 


26.00 


10.00 


2.00 


48 


B 


1.42 


240 


47.05 


3480 


480 


47.10 


3480 














48 


C 


1.71 


240 


47.05 


4170 


480 


47.10 


4170 


10 


D 


0.68 


27.70 


10.00 


2.00 


48 


D 


1.96 


240 


47.05 


4860 


480 


47.10 


4860 














54 


A 


1.35 


240 


47.05 


3750 


480 


47.10 


3750 


12 


D 


0.75 


29.45 


10.00 


2.00 


54 


B 


1.55 


240 


47.05 


4330 


480 


47.10 


4330 














54 


C 


1.90 


240 


47.05 


5290 


480 


47.10 


5290 


14 


B 


0.66 


31.20 


10.00 


2.50 


54 


D 


2.23 


240 


47.05 


6220 


480 


47.10 


6220 














60 


A 


1.39 


240 


47.05 


4340 


480 


47.10 


4340 


14 


D 


0.82 


31.20 


10.00 


2.50 


60 


B 


1.67 


240 


47.05 


5140 


480 


47.10 


5140 














60 


C 


2.00 


240 


47.05 


62U0 


480 


47.10 


6200 


16 


B 


0.70 


32.90 


10.00 


2.50 


60 


D 


2.38 


240 


47.05 


7400 


480 


47.10 


7400 


16 


D 


0.89 


32.90 


10 00 


2 50 



*First three columns, for 20'' to 60'' diam., apply also to ^^ curves. 



1250 



Qi.— WATER WORKS. 



35. — ^Properties op Branches. — 3- Way and 4-Way. 
(A.W.W. A.) 









l=!=f 






m 






ftiiJ/'IftLf 


M 


m 






CUr*:^. 




J 










4-.L 


,._ 


'••7*:-""-4- 












!5^ 


'""•' 


..J 


.„..._^^ 


' 1 


41 








ii^^^ 


^■i" 


I- -jT^ 










■^pj^t 


0-^D 










Fi 


J j 1 








» 1 * 
?. 35. 




Nominal DIam. 
Inches. 




Dimensions, Inches. 


Approximate Weights. Pounds. 




Class. 










A 


B 


H 


J 


I 


3-Way Branches 


4-way Branches 




2 Bells. 


3 Bells. 


3 Bells. 


4 Bells. 


4 


3 


D 


11 


23 


11 


121 


120 


153 


153 


4 


4 


D 


11 


23 


11 


125 


128 


164 


166 


6 


3 


D 


12 


24 


12 


173 


170 


207 


204 


6 


4 


D 


12 


24 


12 


185 


183 


223 


221 


6 


6 


D 


12 


24 


12 


203 


200 


259 


257 


8 


4 


D 


13 


25 


13 


262 


255 


301 


294 


8 


6 


D 


13 


25 


13 


278 


270 


333 


325 


8 


8 


D 


13 


25 


13 


301 


294 


378 


372 


10 


4 


D 


14 


26 


14 


356 


338 


395 


377 


10 


6 


D 


14 


26 


14 


371 


352 


424 


406 


10 


8 


D 


14 


26 


14 


389 


371 


461 


443 


10 


10 


D 


14 


26 


14 


414 


395 


511 


493 


12 


4 


D 


15 


27 


15 


473 


445 


514 


486 


12 


6 


D 


15 


27 


15 


486 


458 


540 


512 


12 


8 


D 


15 


27 


15 


502 


474 


573 


545 


12 


10 


D 


15 


27 


15 


519 


491 


605 


577 


12 


12 


D 


15 


27 


15 


540 


512 


651 


623 


14 


4 


B 


16 


28 


16 


485 


480 


535 


530 


14 


4 


D 


16 


28 


16 


614 


588 


666 


641 


14 


6 


B 


16 


28 


16 


500 


495 


560 


555 


14 


6 


D 


16 


28 


16 


634 


608 


730 


700 


14 


8 


B 


16 


28 


16 


515 


510 


600 


595 


14 


8 


D 


16 


28 


16 


662 


636 


787 


761 


14 


10 


B 


16 


. 28 


16 


535 


525 


635 


625 


14 


10 


D 


16 


28 


16 


679 


653 


822 


796 


14 


12 


B 


16 


28 


16 


560 


550 


680 


670 


14 


12 


D 


16 


28 


16 


698 


672 


860 


834 


14 


14 


B 


16 


28 


16 


575 


569 


723 


715 


14 


14 


D 


16 


28 


16 


750 


724 


938 


963 


16 


4 


B 


17 


29 


17 


615 


610 


675 


670 


16 


4 


D 


17 


29 


17 


783 


760 


864 


841 


16 


6 


B 


17 


29 


17 


630 


625 


695 


690 


16 


6 


D 


17 


29 


17 


802 


779 


902 


879 


16 


8 


B 


17 


29 


17 


645 


640 


730 


725 


16 


8 


D 


17 


29 


17 


831 


808 


961 


938 


16 


10 


B 


17 


29 


17 


660 


655 


760 


755 


16 


10 


D 


17 


29 


17 


872 


849 


1042 


1019 


16 


12 


B 


17 


29 


IT 


685 


680 


805 


800 


16 


12 


D 


17 


29 


17 


884 


861 


1066 


1043 


16 


14 


B 


17 


29 


17 


695 


690 


825 


820 



CAST IRON PIPE— BRANCHES 



1251 



36. — Branches. — 3- Way and 4- Way. — Continued. 



Nomtnal Diam. 
Inches. 




Dimensions, Inches 


Approximate Weights, 


Pounds. 




Class. 




































3-Way Branches 


4-way Branches 


A 


B 




H 


J 


I 


























2 Bells 


3 Bells. 


3 Bells. 


4 Bells. 


16 


14. 


D 


17 


29 


17 


903 


880 


1104 


1082 


16 


16 


B 


17 


29 


17 


729 


727 


904 


901 


16 


16 


D 


17 


29 


17 


991 


969 


1282 


1259 


18 


4 


B 


18 


30 


18 


755 


750 


820 


815 


18 


4 


D 


18 


30 


18 


953 


927 


1046 


1020 


18 


6 


B 


18 


30 


18 


765 


760 


840 


835 


18 


6 


D 


18 


30 


18 


968 


942 


1075 


1049 


18 


8 


B 


18 


30 


18 


780 


775 


870 


865 


18 


8 


D 


18 


30 


18 


1000 


974 


1140 


1114 


18 


10 


B 


18 


30 


18 


795 


790 


900 


895 


18 


10 


D 


18 


30 


18 


1038 


1012 


1216 


1190 


18 


12 


B 


18 


30 


18 


815 


810 


940 


935 


18 


12 


D 


18 


30 


18 


1075 


1049 


1290 


1264 


18 


14 


B 


18 


30 


18 


825 


820 


955 


950 


18 


14 


D 


18 


30 


18 


1083 


1057 


1306 


1280 


18 


16 


B 


18 


30 


18 


855 


850 


1020 


1015 


18 


16 


D 


18 


30 


18 


1108 


1082 


1356 


1330 


18 


18 


B 


18 


30 


18 


895 


889 


1101 


1096 


18 


18 


D 


18 


30 


18 


1170 


1144 


1480 


' 1454 


20 


4 


B 


19 


31 


19 


923 


916 


1006 


999 


20 


4 


D 


19 


31 


19 


1172 


1148 


1273 


1248 


20 


6 


B 


19 


31 


19 


930 


920 


1010 


1000 


20 


6 


D 


19 


31 


19 


1188 


1164 


1304 


1280 


20 


8 


B 


19 


31 


19 


945 


935 


1035 


1025 


20 


8 


D 


19 


31 


19 


1212 


1188 


1352 


1328 


20 


10 


B 


19 


31 


19 


955 


945 


1060 


1050 


20 


10 


D 


19 


31 


19 


1252 


1227 


1431 


1407 


20 


12 


B 


19 


31 


19 


975 


965 


1100 


1090 


20 


12 


D 


19 


31 


19 


1288 


1263 


1502 


1479 


20 


14 


B 


19 


31 


19 


980 


970 


1110 


1100 


20 


14 


D 


19 


31 


19 


1342 


1318 


1613 


1588 


20 


16 


B 


19 


31 


19 


1010 


1000 


1170 


1160 


20 


16 


D 


19 


31 


19 


1347 


1323 


1622 


1597 


20 


18 


B 


19 


31 


19 


1035 


1025 


1225 


1215 


20 


18 


D 


19 


31 


19 


1365 


1341 


1658 


1634 


20 


20 


B 


19 


31 


19 


1077 


1070 


1314 


1307 


20 


20 


D 


19 


31 


19 


1462 


1438 


1852 


1828 


24 


6 


B 


21 


33 


21 


1309 


1289 


1425 


. 1405 


24 


6 


D 


21 


33 


21 


1670 


1637 


1809 


1775 


24 


8 


B 


21 


33 


21 


1323 


1303 


1453 


1433 


24 


8 


D 


21 


33 


21 


1697 


1664 


1863 


1830 


24 


10 


B 


21 


33 


21 


1341 


1321 


1489 


1469 


24 


10 


D 


21 


33 


21 


1732 


1699 


1933 


1900 


24 


12 


B 


21 


33 


21 


1362 


1342 


1532 


1511 


24 


12 


D 


21 


33 


21 


1768 


1735 


2005 


1972 


24 


14 


B 


21 


33 


21 


1402 


1381 


1609 


1589 


24 


14 


D 


21 


33 


21 


1810 


1777- 


2088 


2055 


24 


16 


B 


21 


33 


21 


1443 


1423 


1694 


1673 


24 


16 


D 


21 


33 


21 


1858 


1825 


2185 


2151 


24 


18 


B 


21 


33 


21 


1460 


1440 


1727 


1706 


24 


18 


D 


21 


33 


21 


1885 


1852 


2238 


2205 


24 


20 


B 


21 


33 


21 


1474 


1454 


1756 


1736 


24 


20 


D 


21 


33 


21 


2025 


1991 


2518 


2484 



1252 



Qi.— WATER WORKS. 



35. — Branches. — 3- Way and 4-Way. — Continued. 



Nominal DIam. 
Inches. 




Dimensions, Inches 


Approximate Weights. 


Pounds. 




aass. 








































3-Way Branches 


4-way Branches 


A 


B 




H 


J 


I 




























2 Bells. 


3 Bells. 


3 Bells. 


4 Bells. 


24 


24 


B 


21 


33 


21 


1523 


1503 


1854 


1834 


24 


24 


D 


21 


33 


21 


2146 


2113 


2727 


2694 


30 


6 


A 


13 


25 


24 


1272 


1300 


1407 


1434 


30 


6 


B 


13 


25 


24 


1433 


1417 


1580 


1563 


30 


6 


C 


13 


25 


24 


1693 


1673 


1870 


1850 


30 


6 


D 


13 


25 


24 


1934 


1920 


2113 


2099 


30 


8 


A 


14 


26 


24 


1318 


1346 


1453 


1481 


30 


8 


B 


14 


26 


24 


1482 


1466 


1624 


1609 


30 


8 


C 


14 


26 


24 


1765 


1745 


1953 


1934 


30 


8 


D 


14 


26 


24 


2004 


1990 


2182 


2168 


30 


10 


A 


15 


27 


24 


1369 


1396 


1512 


1540 


30 


10 


B 


15 


27 


24 


1538 


1521 


1685 


1668 


30 


10 


C 


15 


27 


24 


1857 


1837 


2075 


2056 


30 


10 


D 


15 


27 


24 


2108 


2094 


2319 


2306 


30 


12 


A 


15 


27 


24 


1395 


1420 


1555 


1580 


30 


12 


B 


15 


27 


24 


1555 


1540 


1715 


1700 


30 


12 


C 


15 


27 


24 


1911 


1891 


2184 


2164 


30 


12 


D 


15 


27 


24 


2154 


2140 


2411 


2398 


30 


14 


A 


18 


30 


26 


1547 


1575 


1737 


1764 


30 


14 


B 


18 


30 


26 


1805 


1789 


2085 


2069 


30 


14 


C 


18 


30 


26 


2159 


2140 


2497 


2477 


30 


14 


D 


18 


30 


26 


2567 


2553 


3026 


3013 


30 


16 


A 


19 


31 


26 


1648 


1675 


1805 


1832 


30 


16 


B 


19 


31 


26 


1899 


1883 


2200 


2184 


30 


16 


C 


19 


31 


26 


2272 


2253 


2662 


2642 


30 


16 


D 


19 


31 


26 


2692 


2678 


3206 


3192 


30 


18 


A 


20 


34 


26 


1757 


1741 


2024 


2007 


30 


18 


B 


20 


34 


26 


2044 


1976 


2387 


2318 


30 


18 


C 


20 


34 


26 


2434 


2353 


2862 


2781 


30 


18 


D 


20 


34 


26 


2805 


2791 


3361 


3348 


30 


26 


A 


21 


36 


26 


1857 


1818 


2157 


2118 


30 


20 


B 


21 


36 


26 


2182 


2088 


2584 


2490 


30 


20 


C 


21 


36 


26 


2667 


2555 


3237 


3126 


30 


20 


D 


21 


36 


26 


3041 


2921 


3657 


3538 


30 


24 


A 


23 


38 


26 


1979 


1940 


2312 


2274 


30 


24 


B 


23 


38 


26 


2313 


2219 


2742 


2648 


30 


24 


C 


23 


38 


26 


2847 


2736 


3474 


3362 


30 


24 


D 


23 


38 


26 


3290 


3170 


4014 


3895 


30 


30 


A 


26 


43 


26 


2212 


2129 


2602 


2520 


30 


30 


B 


26 


43 


26 


2599 


2453 


3106 


2960 


30 


30 


c 


26 


43 


26 


3310 


3137 


4110 


3937 


30 


30 


D 


26 


43 


26 


3850 


3660 


4799 


4609 


36 


8 


A 


14 


26 


27 


1751 


1777 


1938 


1963 ' 


36 


8 


B 


14 


26 


27 


2055 


2073 


2268 


2287 


36 


8 


c 


14 


26 


27 


2421 


2433 


2679 


2691 


36 


8 


D 


14 


26 


27 


2780 


2780 


3038 


3039 


36 


10 


A 


15 


27 


27 


1810 


1835 


1996 


2021 


36 


10 


B 


15 


27 


27 


2128 


2147 


•2345 


2364 


36 


10 


C 


15 


27 


27 


2534 


2546 


2822 


2834 


36 


10 


D 


15 


27 


27 


2903 


2902 


3188 


3188 


36 


12 


A 


16 


28 


27 


1884 


1909 


2084 


2109 


36 


12 


B 


16 


28 


27 


2219 


2238 


2458 


2477 


36 


12 


C 


16 


28 


27 


2644 


2656 


2962 


2973 



CAST IRON PIPE— BRANCHES. 



1253 



36. — Branches. — 3-Way and 4-WAy. — Continued. 



Nominal DIam. 
Inches 


Class 


Dimensions, Inches 


Approximate Weights, Pounds. 












3-Way Branches 


4-way Branches 


A 


B 




'H 


J 


I 




























2 Bells. 


3 Bells. 


3 Bells. 


4 Bells. 


36 


12 


D 


16 


28 


27 


3032 


3033 


3349 


3350 


36 


14 


A 


18 


30 


29 


2039 


2065 


2279 


2304 


36 


14 


B 


18 


30 


29 


2415 


2433 


2709 


2728 


36 


14 


C 


18 


30 


29 


2872 


2883 


3251 


3263 


36 


14 


D 


18 


30 


29 


3470 


3470 


4033 


4033 


36 


16 


A 


19 


31 


29 


2135 


2160 


2410 


2436 


36 


16 


B 


19 


31 


29 


2521 


2540 


2853 


2872 


36 


16 


C 


19 


31 


29 


3003 


3014 


3431 


3442 


36 


16 


D 


19 


31 


29 


3618 


3617 


4231 


4230 


36 


18 


A 


20 


34 


29 


2279 


2246 


2581 


2548 


36 


18 


B 


20 


34 


29 


2701 


2650 


3073 


3022 


36 


18 


C 


20 


34 


29 


3206 


3136 


3673 


3604 


36 


18 


D 


20 


34 


29 


3852 


3755 


4506 


4409 


36 


20 


A 


21 


36 


29 


2409 


2346 


2752 


2689 


36 


20 


B 


21 


36 


29 


2885 


2800 


3336 


3251 


36 


20 


C 


21 


36 


29 


3537 


3426 


4212 


4101 


36 


20 


D 


21 


36 


29 


4050 


3905 


4757 


4612 


36 


24 


A 


23 


38 


29 


2451 


2513 


2844 


2907 


36 


24 


B 


23 


38 


29 


3099 


3014 


2624 


3539 


36 


24 


C 


23 


38 


29 


3806 


3695 


4585 


4474 


36 


24 


D 


23 


38 


29 


4511 


4366 


5307 


5161 


36 


30 


A 


26 


43 


29 


2830 


2708 


3242 


3120 


36 


30 


B 


26 


43 


29 


3594 


3438 


4335 


4179 


36 


30 


C 


26 


43 


29 


4248 


4055 


5140 


4947 


36 


30 


D 


26 


43 


29 


5160 


4918 


6192 


5950 


36 


36 


A 


29 


46 


29 


3067 


2946 


3539 


3418 


36 


36 


B 


29 


46 


29 


4046 


3891 


4956 


4800 


36 


36 


C 


29 


46 


29 


4788 


4595 


5867 


5673 


36 


36 


D 


29 


46 


29 


5810 


5567 


7099 


6857 


42 


12 


A 


16 


28 


30 


2507 


2577 


3467 


3537 


42 


12 


B 


16 


28 


30 


2670 


2889 


3131 


3170 


42 


12 


C 


16 


28 


30 


3478 


3507 


3830 


3860 


42 


12 


D 


16 


28 


30 


3971 


3989 


4307 


4325 


42 


14 


A 


18 


30 


32 


2671 


2739 


2942 


3010 


42 


14 


B 


18 


30 


32 


3075 


3114 


3400 


3440 


42 


14 


C 


18 


30 


32 


3747 


3776 


4147 


4177 


42 


14 


D 


18 


30 


32 


4590 


4609 


5288 


5306 


42 


16 


A 


19 


31 


32 


2778 


2846 


3080 


3148 


42 


16 


B 


19 


31 


32 


3196 


3235 


3552 


3592 


42 


16 


C 


19 


31 


32 


3891 


3920 


4325 


4354 


42 


16 


D 


19 


31 


32 


4754 


4772 


5487 


5506 


42 


18 


A 


20 


34 


32 


2950 


2941 


3268 


3258 


42 


18 


B 


20 


34 


32 


3407 


3357 


3794 


3744 


42 


18 


C 


20 


34 


32 


4393 


4312 


5108 


5028 


42 


18 


D 


20 


34 


32 


5049 


4939 


5819 


5709 


42 


20 


A 


21 


36 


32 


3104 


3056 


3459 


3411 


42 


20 


B 


21 


36 


32 


3582 


3486 


4009 


3913 


42 


20 


C 


21 


36 


32 


4615 


4479 


5387 


6251 


42 


20 


D 


21 


36 


32 


5297 


5123 


6122 


5948 


42 


24 


A 


23 


38 


32 


3314 


3266 


3724 


3676 


42 


24 


B 


23 


38 


32 


3852 


3756 


4370 


4274 


42 


24 


C 


23 


38 


32 


4965 


4829 


5866 


5730 


42 


24 


D 


23 


38 


32 


5709 


5535 


6579 


6405 



1254 



M.— WATER WORKS. 



35. — Branches. — 3-Way and 4-Way. — Concluded. 



Nominal Dlam. 
Inches. 




Dimensions. Inches. 


Approximate Weights, 


Pounds. 




Class. 
























3-Way Branches 


4-way Branches 


A 


B 




H 


J 


I 










2 Bells. 


3 Bells. 


3 Bells. 


4 Bells. 


42 


30 


A 


26 


43 


32 


3679 


3553 


4144 


4018 


42 


30 


B 


26 


43 


32 


4554 


4370 


5416 


5230 


42 


30 


C 


26 


43 


32 


5649 


5402 


6675 


6428 


42 


30 


D 


26 


43 


32 


6561 


6258 


7729 


7426 


42 


36 


A 


29 


46 


32 


4076 


3950 


4705 


4579 


42 


36 


B 


29 


46 


32 


4903 


4718 


5845 


5659 


42 


36 


C 


29 


46 


32 


6150 


5904 


7261 


7015 


42 


36 


D 


29 


46 


32 


7187 


6884 


8512 


8209 


42 


42 


A 


32 


49 


32 


4393 


4267 


5109 


4983 


42 


42 


B 


32 


49 


32. 


5533 


5348 


6641 


6455 


42 


42 


C 


32 


49 


32 


7001 


6755 


8392 


8146 


42 


42 


D 


32 


49 


32 


8158 


7855 


9803 


9500 


48 


12 


A 


17 


29 


33 


3266 


3319 


3853 


3707 


48 


12 


B 


17 


29 


33 


3752 


3804 


4107 


4160 


48 


12 


C 


17 


29 


33 


4510 


4576 


4940 


5007 


48 


12 


D 


17 


29 


33 


5564 


5624 


6376 


6436 


48 


14 


A 


18 


30 


35 


3422 


3476 


3762 


3815 


48 


14 


B 


18 


30 


35 


4173 


4226 


4836 


4889 


48 


14 


C 


18 


30 


35 


4965 


5030 


5712 


5778 


48 


14 


D 


18 


30 


35 


5754 


5815 


6596 


6656 


48 


16 


A 


19 


31 


35 


3565 


3619 


3947 


4001 


48 


16 


B 


19 


31 


35 


4046 


4098 


4466 


4519 


48 


16 


C 


19 


31 


35 


5055 


5121 


5755 


5821 


48 


16 


D 


19 


31 


35 


5967 


6028 


6860 


6921 


48 


18 


A 


20 


34 


35 


3775 


3729 


4166 


4120 


48 


18 


B 


20 


34 


35 


4287 


4225 


4718 


4655 


48 


18 


C 


20 


34 


35 


5479 


5407 


6328 


6256 


48 


18 


D 


20 


34 


35 


6328 


6227 


7259 


7158 


48 


20 


A 


21 


36 


35 


3956 


3860 


4378 


4282 


48 


20 


B 


21 


36 


35 


4500 


4380 


4973 


4853 


48 


20 


C 


21 


36 


35 


5745 


5604 


6652 


6511 


48 


20 


D 


21 


.36 


35 


6607 


6425 


7574 


7392 


48 


24 


A 


23 


38 


35 


4221 


4125 


4706 


4609 


48 


24 


B 


23 


38 


35 


5028 


4908 


5798 


5678 


48 


24 


C 


23 


38 


35 


6193 


6052 


7272 


7131 


48 


24 


D 


23 


38 


35 


7064 


6882 


7994 


7812 


48 


30 


A 


26 


43 


35 


4748 


4553 


5361 


5166 


48 


30 


B 


26 


43 


35 


5685 


5451 


6653 


6418 


48 


30 


C 


26 


43 


35 


7042 


6762 


8265 


7985 


48 


30 


D 


26 


43 


35 


8051 


7708 


9303 


8960 


48 


36 


A 


29 


46 


35 


5150 


4953 


5859 


5662 


48 


36 


B 


29 


46 


35 


6322 


6088 


7382 


7148 


48 


36 


C 


29 


46 


35 


7603 


7323 


8915 


8635 


48 


36 


D 


29 


46 


35 


8830 


8487 


10336 


9993 


48 


42 


A 


32 


49 


35 


5503 


5307 


6266 


6069 


48 


42 


B 


32 


49 


35 


6821 


6587 


7973 


7739 


48 


42 


C 


32 


49 


35 


8278 


7999 


9750 


9470 


48 


42 


D 


32 


49 


35 


9644 


9301 


11367 


11024 


48 


48 


A 


35 


52 


35 


6043 


5846 


7U43 


6846 


48 


48 


B 


35 


52 


35 


7659 


7424 


9076 


8841 


48 


48 


C 


35 


52 


35 


9229 


8950 


11006 


10726 


48 


48 


D 


35 


52 


35 











CAST IRON PIPE— BRANCHES. 



1255 



5. — Properties op Y 
Branches, Type 1. 

(A. W. W. A.) 





J'sd, 













m^ 




* Figs. S 


16. * 




k 




Nominal 
Diam. Ins. 
















Thickness, 


Ins. 


ii4 






1 


s 


P 


V 


w 


n 

■ 


r 








^•g§ 


e 


t 


ti 


t2 


t3 


§^S 


12 


12 


D 


16 00 


21.50 


8.00 


9.79 


1.17 


30 


0.75 


1.08 


0.75 


687 


14 


14 


B 


16.00 


24.00 


9.00 


11.30 


1.08 


30 


0.66 


0.99 


0.66 


738 


U 


14 


D 


16.00 


24.00 


9.00 


11.30 


1.32 


30 


0.82 


1.22 


0.82 


894 


16 


16 


B 


17.00 


27.50 


10.50 


13.00 


1.12 


30 


0.70 


1.03 


0.70 


942 


16 


16 


D 


17 00 


27.50 


10.50 


13.00 


1.39 


30 


0.89 


1.29 


0.89 


1275 


18 


18 


B 


18.00 


30.00 


12.00 


14.70 


1.17 


30 


0.75 


1.08 


0.75 


1266 


18 


18 


D 


18.00 


30.00 


12.00 


14.70 


1.46 


30 


0.96 


1.36 


0.96 


1607 


20 


20 


B 


18.00 


34.00 


13.50 


16.40 


1.26 


30 


0.80 


1.16 


0.80 


1635 


20 


20 


D 


18.00 


34.00 


13.50 


16.40 


1.57 


30 


1.03 


1.46 


1.03 


2296 


24 


20 


B 


12.00 


34.00 


13.50 


16.40 


1.26 


30 


0.89 


1.16 


0.80 


1663 


24 


20 


D 


12.00 


34.00 


13.50 


16.40 


1.57 


30 


1.16 


1.46 


1.03 


2393 


24 


24 


B 


18.00 


38.00 


15.25 


19.30 


1.36 


30 


0.89 


1.26 


0.89 


2300 


24 


24 


D 


18.00 


38.00 


15.25 


19.30 


1.75 


30 


1.16 


1.63 


1.16 


2957 


30 


24 


A 


12 00 


38.00 


15.25 


19.30 


1.36 


30 


0.88 


1.26 


0.89 


2171 


30 


24 


B 


12.00 


38.00 


15.25 


19.30 


1.36 


30 


1.03 


1.26 


0.89 


2217 


30 


24 


C 


12.00 


38.00 


15.25 


19.30 


1.75 


30 


1.20 


1.63 


1.16 


2717 


30 


24 


D 


12.00 


38.00 


15.25 


19.30 


1.75 


30 


1.37 


1.63 


1.16 


2811 


30 


30 


A 


18.00 


48.00 


18.00 


23.70 


1.32 


30 


0.88 


1.22 


0.88 


3153 


30 


30 


B 


18.00 


48.00 


18.00 


23.70 


1.59 


30 


1.03 


1.47 


1.03 


3687 


30 


30 





18.00 


48.00 


18.00 


23.70 


1.88 


30 


1.20 


1.74 


1.20 


4285 


30 


30 


D 


18.00 


48.00 


18.00 


23.70 


2.17 


30 


1.37 


2.01 


1.37 


4941 


36 


30 


A 


10.00 


48 00 


18.00 


23.70 


1.32 


30 


0.99 


1.22 


0.88 


3343 


36 


30 


B 


10.00 


48.00 


18.00 


23.70 


1.59 


30 


1.15 


1.47 


1.03 


3874 


36 


30 


C 


10.00 


48.00 


18.00 


23.70 


1.88 


30 


1.36 


1.74 


1.20 


4486 


36 


30 


D 


10.00 


48.00 


18.00 


23.70 


2.17 


30 


1.58 


2.01 


1.37 


5189 


36 


36 


A 


18.00 


56.00 


21.00 


28.20 


1.50 


24 


0.99 


1.39 


0.99 


4949 


36 


36 


B 


18.00 


56.00 


21.00 


28.20 


1.79 


24 


1.15 


1.66 


1.15 


5858 


36 


36 


C 


18.00 


56.00 


21.00 


28.20 


2.13 


24 


1.36 


1.98 


1.36 


6804 


36 


36 


D 


18.00 


56.00 


21.00 


28.20 


2 48 


24 


1.58 


2.31 


1.58 


8082 


42 


30 


A 


6.00 


48.00 


18.00 


23.70 


1.32 


30 


1.10 


1.22 


0.88 


3368 


42 


30 


B 


6.00 


48.00 


18.00 


23.70 


1.59 


30 


1.28 


1.47 


1.03 


3890 


42 


30 


C 


6.00 


48.00 


18.00 


23.70 


1.88 


30 


1.54 


1.74 


1.20 


4543 


42 


30 


D 


6.00 


48.00 


18.00 


23.70 


2.17 


30 


1.78 


2.01 


1.37 


5241 


42 


36 


A 


10.00 


56.00 


21 00 


28 20 


1.50 


24 


1.10 


1.39 


0.99 


4904 


42 


36 


B 


10 00 


56.00 


21.00 


28.20 


1.79 


24 


1.28 


1.66 


1.15 


5789 


42 


36 


C 


10.00 


56.00 


21.00 


28.20 


2.13 


24 


1.54 


1.98 


1.36 


6761 


42 


36 


D 


10.00 


56.00 


21 00 


28.20 


2.48 


24 


1.78 


2.31 


1.58 


8025 


42 


42 


A 


18.00 


66.00 


25.00 


33.10 


1.72 


24 


1.10 


1.60 


1.10 


7394 


42 


42 


B 


18.00 


66 00 


25.00 


33.10 


2.05 


24 


1.28 


1.90 


1.28 


8417 


42 


42 


C 


18.00 


66.00 


25.00 


33.10 


2.46 


24 


1.54 


2.28 


1.54 


10377 


42 


42 


D 


18 00 


66.00 


25.00 


33.10 


2.85 


24 


1.78 


2.64 


1.78 


12072 


48 


36 


A 


2.00 


56.00 


21.00 


28.20 


1.50 


24 


1.26 


1.39 


0.99 


4727 


48 


36 


B 


2.00 


56.00 


21 00 


28.20 


1.79 


24 


1.42 


1.66 


1.15 


5584 


48 


36 


C 


2.00 


56.00 


21.00 


28.20 


2.13 


24 


1.71 


1.98 


1.36 


6494 


48 


36 


D 


2.00 


56.00 


21.00 


28.20 


2.48 


24 


1.96 


2.31 


1.58 


7731 


48 


42 


A 


10.00 


66.00 


25.00 


33.10 


1.72 


24 


1.26 


1.60 


1.10 


7345 


48 


42 


B 


10.00 


66.00 


25.00 


33.10 


2.05 


24 


1.42 


1.90 


1.28 


8338 


48 


42 


C 


10.00 


66.00 


25.00 


33.10 


2.46 


24 


1.71 


2.28 


1.54 


10249 


48 


42 


D 


10.00 


66.00 


25.00 


33.10 


2.85 


24 


1.96 


2.64 


1 78 


11924 


48 


48 


A 


18.00 


76.00 


28.00 


37.60 


1.99 


24 


1.26 


1.86 


1.26 


10200 


48 


48 


B 


18.00 


76.00 


28.00 


37.60 


2.32 


24 


1.42 


2.15 


1.42 


12132 


48 


48 


C 


18.00 


76.00 


28.00 


37.60 


2.78 


24 


1.71 


2.57 


1.71 


14716 


48 


48 


D 


18.00 


76.00 


28.00 


37.60 


3.20 


24 


1.96 


2.95 


1.96 


16965 



Note. — All dimensions are in inches. 



1256 



U.^WATER WORKS. 



37. — Properties of Y Branches 
Type 2. 

(A. W. W. A.) 





1;/ t2 

^fo 20 Inches. 



W% \ 



%. 






Figs. 37. 



i \ 



24to4Slnches. 



Nominal 
Diam. Ins. 



w 



Thickness. 
Inches. 



t2 



4 

6 
8 

10 
12 

14 
14 

16 
16 
18 

18 
20 
20 
24 
24 

24 
24 
80 
30 
30 

30 
36 
36 
36 
36 

42 
42 
42 
42 

42 
42 

48 
48 

48 
48 
48 
48 



30 
30 
36 
36 

30 
30 
36 
36 

42 
42 
36 



D 
D 
D 
D 
D 

B 
D 
B 
D 
B 

D 
B 
D 
B 
D 

B 
D 
A 
B 
A 

B 
A 
B 

A 
B 

A 
B 
A 
B 

A 
B 

A 
B 

A 
B 
A 
B 



11.50 
13.00 
14.00 
15.50 
15.50 

16.00 
16.00 
17.50 
17.50 
18.00 

18.00 
18.75 
18.75 
18.75 
18.75 

19.75 
19.75 
17.00 
17.00 
22.75 

22.75 
19.75 
19.75 
24.00 
24.00 

16.75 
16.75 
21.00 
21.00 

25.25 
25.25 
18.00 
18.00 

22.25 
22.25 
26.50 
26.50 



10.50 
13.00 
16.00 
18.50 
21.50 

24.00 
24.00 
31.00 
31.00 
34.00 

34.00 
37.00 
37.00 
40.00 
40.00 

42.00 
42.00 
49.50 
49.50 
52.50 

52.50 
56.00 
56.00 
60.00 
60.00 

63.00 
63 00 
66.00 
66.00 

69.00 
69.00 
71.00 
71.00 

74.00 
74.00 
77.00 
77.00 



7.18 

9.27 

11.85 

13.94 

16.54 

18.62 
18.62 
25.20 
25.20 
28.00 

28.00 
30.75 
30.75 



6.64 
7.46 
8.30 
9.12 
9.92 

ia.76 

10.76 
11.60 
11.60 
12.00 

12.00 
12.50 
12.50 



2.18 
3.27 
3.85 
4.94 
4.54 



0.52 
0.55 
0.60 
0.68 
0.75 

0.66 
0.82 
0.70 
0.89 
0.75 

0.96 
0.80 
1.03 
0.89 
1.16 

0.89 
1.16 
0.88 
1.03 
0.88 

1.03 
0.99 
1.15 
99 
1.15 

1.10 
1.28 
1.10 
1.28 

1.10 
1.28 
1.26 
1.42 

1.26 
1.42 
1.26 
1.42 



0.64 
0.67 
0.72 
0.83 
0.93 

0.84 
1.00 
1.03 
1.29 
1.12 

1.44 
1.20 
1.50 
0.80 
1.03 

0.89 
1.16 
0.89 
0.89 
0.88 

1.03 
0.88 
1.03 
0.99 
1.15 

0.88 
1.03 
0.99 
1.15 

1. 10 
1.28 
0.99 
1.15 

1.10 
1.28 
1.26 
1.42 



Note. — All dimensions are in inches. 



CAST IRON PIPE— BRANCHES. 



1267 



. — Properties op Blow-off Branches, 
(A. W. W. A.) 





-bigs. 



Nominal 
Diam. 
Incbes. 




■ 




Thickness. 
Inches. 


m 


Nominal 
Diam. 
Inches. 








Thickness. 
Inches. 








1 


P 












O 




P 






a§§ 


e 


f 


t. 


t2 


e 


f 


ti 


t2 




8 


4 


D 


12 


7 


0.60 


0.52 


111 


36 


12 


A 


13 


23 


0.99 


0.75 


1702 


10 


4 


D 


12 


8 


0.68 


0.52 


286 


36 


12 


B 


13 


23 


1.15 


0.75 


1972 


10 


6 


D 


12 


8 


0.68 


0.55 


300 


36 


12 


C 


13 


23 


1.36 


0.75 


2285 


12 


4 


D 


12 


10 


0.75 


0.52 


365 


36 


12 


D 


13 


23 


1.58 


0.75 


2627 


12 


6 


D 


12 


10 


0.75 


0.55 


379 


42 


12 


A 


15 


26. 


1.10 


0.75 


2432 


14 


4 


B 


12 


11 


0.66 


0.52 


400 


42 


12 


B 


15 


26 


1.28 


0.75 


2728 


14 


4 


D 


12 


11 


0.82 


0.52 


471 


42 


12 


C 


15 


26 


1.54 


0.75 


3271 


14 


6 


B 


12 


11 


0.66 


0.55 


415 


42 


12 


D 


15 


26 


1.78 


0.75 


3768 


14 


6 


D 


12 


11 


0.82 


0.55 


486 


42 


16 


A 


15 


26 


1.10 


0.70 


2489 


16 


4 


B 


12 


12 


0.70 


0.52 


497 


42 


16 


B 


15 


26 


1.28 


0.70 


2786 


16 


4 


D 


12 


12 


0.89 


0.52 


597 


42 


16 


C 


15 


26 


1.54 


0.89 


3365 


16 


6 


B 


12 


12 


0.70 


0.55 


513 


42 


16 


D 


15 


26 


1.78 


0.89 


3862 


16 


6 


D 


12 


12 


0.89 


0.55 


613 


48 


12 


A 


17 


30 


1.26 


0.75 


3274 


18 


4 


B 


12 


13 


0.75 


0.52 


586 


48 


12 


B 


17 


30 


1.42 


0.75 


3699 


18 


4 


D 


12 


13 


0.96 


0.52 


704 


48 


12 


C 


17 


30 


1.71 


0.75 


4417 


18 


6 


B 


12 


13 


0.75 


0.55 


603 


48 


12 


D 


17 


30 


1.96 


0.75 


5107 


18 


6 


D 


12 


13 


0.96 


0.55 


720 


48 


16 


A 


17 


30 


1.26 


0.70 


3337 


20 


4 


B 


12 


14 


0.80 


0.52 


687 


48 


16 


B 


17 


30 


1.42 


0.70 


3762 


20 


4 


D 


12 


14 


1.03 


0.52 


850 


48 


16 


C 


17 


30 


1.71 


0.89 


4523 


20 


6 


B 


12 


14 


0.80 


0.55 


705 


48 


16 


V 


17 


30 


1.96 


0.89 


5214 


20 


6 


D 


12 


14 


1.03 


0.55 


867 


54 


12 


A 


19 


33 


1.35 


0.75 


4287 


24 


6 


B 


12 


16 


0.89 


0.55 


916 


54 


12 


B 


19 


33 


1.55 


0.75 


4945 


24 


6 


D 


12 


16 


1.16 


0.55 


1149 


54 


12 


C 


19 


33 


1.90 


0.75 


5981 


24 


8 


B 


12 


16 


0.89 


0.60 


935 


54 


12 


D 


19 


33 


2.23 


0.75 


7002 


24 


8 


D 


12 


16 


1.16 


0.60 


1170 


54 


16 


A 


19 


33 


1.35 


0.70 


4355 


30 


8 


A 


13 


20 


0.88 


0.60 


1269 


54 


16 


B 


19 


33 


1.55 


0.70 


5013 


30 


8 


B 


13 


20 


1.03 


0.60 


1382 


54 


16 


C 


19 


33 


1.90 


0.89 


6096 


30 


8 


C 


13 


20 


1.20 


0.60 


1616 


54 


16 


D 


19 


33 


2.23 


0.89 


7126 


30 


8 


D 


13 


20 


1.37 


0.60 


1867 


60 


12 


A 


21 


36 


1.39 


0.75 


5263 


30 


12 


A 


13 


20 


0.88 


0.75 


1315 


60 


12 


B 


21 


36 


1.67 


0.75 


6159 


30 


12 


B 


13 


20 


1.03 


0.75 


1426 


60 


12 


C 


21 


36 


2.00 


0.75 


7418 


30 


12 


C 


13 


20 


1.20 


0.75 


1658 


60 


12 


D 


21 


36 


2.38 


0.75 


8798 


30 


12 


D 


13 


20 


1.37 


0.75 


1913 


60 


16 


A 


21 


36 


1.39 


0.70 


5336 


36 


8 


A 


13 


23 


0.99 


0.60 


1653 


60 


16 


B 


21 


36 


1.67 


0.70 


6233 


36 


8 


B 


13 


23 


1.15 


60 


1922 


60 


16 


C 


21 


36 


2.00 


0.89 


7542 


36 


8 


C 


13 


23 


1.36 


0.60 


2234 


60 


16 


D 


21 


36 


2.38 


0.89 


8927 


36 


8 


D 


13 


23 


1.58 


0.60 


2576 



















Note. — All dimensions are in inches. 



1258 



Qi.— WATER WORKS. 



39.— Properties of Blow-off Branches with Manhole. 
(A. W. W. A.) 

Approximate Weight of Cap, 290 Pounds. 





Figs. 39. 



Nomln'l 
Diam. 
Inches. 


i 


I 


P 


n 


Thickness, 
Inches. 


73 

1 


Non 
Dia 
Incl 

e 


lin'I 

m. 

les. 

f 


5 


1 


P 


n 


Thickness, 
Inches. 




e 


f 


ti 


t2 


ti 


t2 


30 


8 


A 


17 


20 


21 


0.88 


0.60 


1628 


48 


12 


A 


17 


30 


30 


1.26 


0.75 


3391 


30 


8 


B 


17 


20 


21 


1.03 


0.60 


1758 


48 


12 


B 


17 


30 


30 


1.42 


0.75 


3803 


30 


8 


c 


17 


20 


21 


1.20 


0.60 


2015 


48 


12 


C 


17 


30 


30 


1.71 


0.75 


4497 


30 


8 


D 


17 


20 


21 


1.37 


0.60 


2290 


48 


12 


D 


17 


30 


30 


1.96 


0.75 


5167 


30 


12 


A 


17 


20 


21 


0.88 


0.75 


1672 


48 


16 


A 


17 


30 


30 


1.26 


0.70 


3454 


30 


12 


B 


17 


20 


21 


1.03 


0.75 


1803 


48 


16 


B 


17 


30 


30 


1.42 


0.70 


3866 


30 


12 


C 


17 


20 


21 


1.20 


0.75 


2057 


48 


16 


C 


17 


30 


30 


1.71 


0.89 


4604 


30 


12 


D 


17 


20 


21 


1.37 


0.75 


2335 


48 


16 


D 


17 


30 


30 


1.96 


0.89 


5274 


36 


8 


A 


17 


23 


24 


0.99 


0.60 


2045 


54 


12 


A 


19 


33 


33 


1.35 


0.75 


4390 


36 


8 


B 


17 


23 


24 


1.15 


0.60 


2351 


54 


12 


B 


19 


33 


33 


1.55 


0.75 


5032 


36 


8 


c 


17 


23 


24 


1.36 


0.60 


2690 


54 


12 


C 


19 


33 


33 


1.90 


0.75 


6039 


36 


8 


D 


17 


23 


24 


1.58 


0.60 


3071 


54 


12 


D 


19 


33 


33 


2.23 


0.75 


7033 


36 


12 


A 


17 


23 


24 


0.99 


0.75 


2094 


54 


16 


A 


19 


33 


33 


1.35 


0.70 


4458 


36 


12 


B 


17 


23 


24 


1.15 


0.75 


2395 


54 


16 


B 


19 


33 


33 


1.55 


0.70 


5100 


36 


12 


C 


17 


23 


24 


1.36 


0.75 


2741 


54 


16 


C 


19 


33 


33 


1.90 


0.89 


6154 


36 


12 


D 


17 


23 


24 


1.58 


0.75 


3122 


54 


16 


D 


19 


33 


33 


2.23 


0.89 


7157 


42 


12 


A 


17 


26 


27 


1.10 


0.75 


2726 


60 


12 


A 


21 


36 


36 


1.39 


0.75 


5357 


42 


12 


B 


17 


26 


27 


1.28 


0.75 


3033 


60 


12 


B 


21 


36 


36 


1.67 


0.75 


6230 


42 


12 


C 


17 


26 


27 


1.54 


0.75 


3595 


60 


12 


C 


21 


36 


36 


2.00 


0.75 


7462 


42 


12 


D 


17 


26 


27 


1.78 


0.75 


4109 


60 


12 


D 


21 


36 


36 


2.38 


0.75 


8810 


42 


16 


A 


17 


26 


27 


1.10 


0.70 


2783 


60 


16 


A 


21 


36 


36 


1.39 


0.70 


5429 


42 


16 


B 


17 


26 


27 


1.28 


0.70 


3090 


60 


16 


B 


21 


36 


36 


1.67 


0.70 


6304 


42 


16 


C 


17 


26 


27 


1.54 


0.891 3689 


60 


16 1 C 


21 


36 


36 


2.00 


0.89 


7587 


42 


16 


D 


17 


26 


27 


1.78 


89| 4203 


60 1 16 1 D 


21 


36 


36 


2.38 


0.89 


8939 



Note. — ^AU dimensions are in inches. 



C.I. BLOW-OFFS, MANHOLE P., REDUCERS. 



1259 



40. — Properties op Manhole Pipe. 

(A. W. W. A.) 
Note. — ^All dimensions are in inches. 




Fig.. 40 



Approximate weight of cap, 
290 Pounds. 



minal 

iam. 

ches. 


05 






4^ M 


minal 

iam. 

ches. 


w 






Sfl 


% 


n 


t 


t§ 


c3 


n 


t 






u 






&£ 


|W^ 


u 






^S 


30 


A 


21 


0.88 


1536 


48 


A 


30 


1.26 


3194 


30 


B 


21 


1.03 


1711 


48 


B 


30 


1.42 


3610 


30 


c 


21 


1.20 


1973 


48 


C 


30 


1.71 


4292 


30 


D 


21 


1.37 


2245 


48 


D 


30 


1.96 


4968 


36 


A 


24 


0.99 


1953 


54 


A 


33 


1.35 


4006 


36 


B 


24 


1.15 


2260 


54 


B 


33 


1.55 


4598 


36 


C 


24 


1.36 


2614 


54 


c 


33 


1.90 


5578 


36 


D 


24 


1.58 


3012 


54 


D 


33 


2.23 


6522 


42 


A 


27 


1.10 


2535 


60 


A 


36 


1.39 


4750 


42 


B 


27 


1.28 


2869 


60 


B 


36 


1.67 


5606 


42 


C 


27 


1.5^ 


3445 


60 


C 


36 


2.00 


6720 


42 


D 


27 


1.78 


3971 


60 


D 


36 


2.38 


7959 



/ = 17 inches on 30 inches to 48 inches; 19 inches on 54 inches; 21 inches 
on 60 inches diameter. 



41. — Properties op Reducers and Increasers, Type No. 1. 
(A. W. W. A.) 




Figs. 41. 
Note. — All dimensions are in inches. 



Diam.. 


[nches. 


k 


m 




Thickness, 
Inches. 


Weights, Pounds. 




f 




ti 


t2 


Large 


Small 










End Bell. 


End Bell. 


6 


4 


3.30 


14 70 


3 


0.55 


0.52 


99 


88 


8 


4 


5.30 


12.70 


4 


0.60 


0.52 


131 


108 


8 


6 


3.90 


14.10 


4 


0.60 


0.55 


149 


138 


10 


4 


7.10 


10.90 


5 


0.68 


0.52 


164 


132 


10 


6 


6.00 


12.00 


5 


0.68 


0.55 


181 


160 


10 


8 


4.40 


13.60 


5 


0.68 


0.60 


205 


195 


12 


6 


7.90 


10.10 


6 


0.75 


0.55 


225 


191 


12 


8 


6.60 


11.40 


6 


75 


0.60 


246 


224 


12 


10 


4.80 


13.20 


6 


0.75 


0.68 


271 


260 



Class D. 6x4 inches to 12 x 10 inches. On all sizes « = 2 inches. 
On all sizes / = 30 inches and s= 10 inches. 



1260 



Qi.— WATER WORKS. 



42. — Properties op Reducers and Increasers, Type No. 2. 
(A. W. W. A.) 




Fig. 42a. — 6 x 4 inches to 60 x 54 inches. 



Nominal Dlam 
Inches. 




Thickness, Inches. 




Weights, Poonds. 




V 

Ins. 






Class. 




e 


f 


ti 


t2 


Spigot 
Ends. 


Large 
End Bell. 


Small 
End Bell. 


6 


4 


18 


0.55 


0.52 


D 


82 


104 


97 


8 
8 


4 
6 


18 
18 


0.60 
0.60 


0.52 
0.55 


D 


104 
121 


132 
150 


119 
143 


10 
10 
10 


4 
6 
8 


18 
18 
18 


0.68 
0.68 
0.68 


0.52 
0.55 
0.60 


D 
D 
D 


131 
150 
170 


162 
180 
201 


146 
169 
198 


12 
12 

12 
12 


4 

6 

8 

10 


18 
18 
18 
18 


0.75 
0.75 
0.75 
0.75 


0.52 
0.55 
0.60 
0.68 


D 
D 
D 
D 


163 
181 
202 
229 


201 
218 
240 
267 


179 
202 
231 
261 


14 
14 
14 
14 


6 
6 
8 

8 


20 
20 
20 
20 


0.66 
0.82 
0.66 
0.82 


0.55 
0.55 
0.60 
0.60 


B 
D 
B 
D 


194 
234 
220 
260 


249 
288 
275 
314 


216 
256 

. 248 
288 


14 
14 
14 
14 


10 
10 
12 
12 


20 
20 
20 
20 


0.66 
0.82 
0.66 
0.82 


0.68 
0.68 
0.75 
0.75 


B 
D 
B 
D 


250 
290 
284 
324 


305 
344 
339 
378 


279 
320 
321 
360 


16 
16 
16 
16 


6 
6 

8 
8 


20 
20 
20 
20 


0.70 
0.89 
0.70 
0.89 


0.55 
0.55 
0.60 
0.60 


B 
D 
B 
D 


226 
278 
252 
304 


300 
355 
326 
381 


248 
300 
280 
332 


16 
16 
16 
16 


10 
10 
12 
12 


20 
20 
20 
20 


0.70 
0.89 
0.70 
0.89 


0.68 
0.68 
0.75 
0.75 


B 
D 
B 
D 


282 
334 
317 
368 


356 
410 
391 
445 


312 
364 
353 
405 


16 
16 


14 
14 


20 
20 


0.70 
0.89 


0.66 
0.82 


B 
D 


315 
407 


389 
484 


370 
461 


18 
18 
18 
18 


8 
8 

10 
10 


20 
20 
20 
20 


0.75 
0.96 
0.75 
0.96 


0.60 
0.60 
0.68 
0.68 


B 
D 
B 
D 


287 
345 
317 
375 


374 
438 
404 
468 


315 
373 
347 
405 


18 
18 
18 
18 


12 
12 
14 
14 


20 
20 
20 
20 


0.75 
0.96 
0.75 
0.96 


0.75 
0.75 
0.66 
0.82 


B 
D 
B 
D 


352 
410 
350 
448 


438 
602 
437 
541 


388 
446 
406 
502 


18 
18 


16 
16 


20 
20 


0.75 
0.96 


0.70 
0.89 


B 
D 


383 
492 


469 

585 


457 
569 


20 
20 


10 
10 


26 
26 


0.80 
1.03 


0.68 
0.68 


B 
D 


414 
499 


516 
615 


445 
529 



On all sizes 5 = 8 inches. 



CAST IRON PIPE— REDUCERS, INCREASERS. 



1261 



42. — Reducers and Increasers, Type No. 2. — Continued. 
(See Fig. 42a, preceding page.) 



Nominal Diam. 
Indies 




Thickness, Inches. 




Weights, Pounds. 










Class. 








f 


V 


ti 


t2 


Spigot 


Large 


Small 


e 


Ins. 




Ends. 


End Bell. 


End Bell. 


20 


12 


26 


80 


0.75 


B 


455 


556 


491 


20 


12 


26 


1 03 


0.75 


D 


539 


656 


576 


20 


14 


26 


0.80 


0.66 


B 


453 


554 


508 


20 


14 


26 


1.03 


0.82 


D 


583 


700 


638 


20 


16 


26 


0.80 


0.70 


B 


490 


592 


564 


20 


16 


26 


1.03 


0.89 


D 


635 


751 


711 


20 


18 


26 


80 


0.75 


B 


531 


633 


617 


20 


18 


26 


1.03 


0.96 


D 


683 


800 


776 


24 


14 


26 


89 


0.66 


B 


552 


680 


607 


24 


14 


26 


1.16 


0.82 


D 


710 


866 


764 


24 


16 


26 


0.89 


0.70 


B 


589 


717 


663 


24 


16 


26 


1.16 


0.89 


D 


762 


917 


838 


24 


18 


26 


0.89 


0.75 


B 


630 


758 


717 


24 


18 


26 


1.16 


0.96 


D 


810 


965 


901 


24 


20 


26 


0.89 


0.80 


B 


675 


803 


776 


24 


20 


26 


1.16 


1.03 


D 


871 


1027 


987 


30 


18 


26 


0.88 


0.75 


A 


710 


903 


796 


30 


18 


26 


1.03 


0.75 


B 


791 


969 


878 


30 


18 


26 


1.20 


0.96 


C 


956 


1166 


1048 


30 


18 


26 


1.37 


0.96 


D 


1054 


1305 


1146 


30 


20 


26 


0.88 


0.80 


A 


754 


947 


856 


30 


20 


26 


1.03 


0.80 


B 


836 


1014 


937 


30 


20 


26 


1.20 


1.03 


C 


1018 


1227 


1134 


30 


20 


26 


1.37 


1.03 


D 


1115 


1366 


1232 


30 


20 


66 


0.88 


9.80 


A 


1468 


1661 


1569 


30 


20 


66 


1.03 


0.80 


B 


1626 


1804 


1728 


30 


20 


66 


1.20 


1.03 


C 


1981 


2190 


2098 


30 


20 


66 


1.37 


1.03 


D 


2172 


2423 


2289 


30 


24 


26 


0.88 


0.89 


A 


854 


1047 


981 


30 


24 


26 


1.03 


0.89 


B 


935 


1113 


1063 


30 


24 


26 


1.20 


1.16 


C 


1144 


1354 


1300 


30 


24 


26 


1.37 


1.16 


D 


1242 


1493 


1398 


30 


24 


66 


0.88 


0.89 


A 


1661 


1921 


1869 


30 


24 


66 


1,03 


0.89 


B 


1820 


1998 


1946 


30 


24 


66 


1.20 


1.16 


C 


2228 


2438 


2384 


30 


24 


66 


1.37 


1.16 


D 


2419 


2670 


2575 


36 


20 


32 


0.99 


0.80 


A 


1039 


1286 


1141 


36 


20 


32 


1.15 


0.80 


B 


1170 


1450 


1272 


36 


20 


32 


1.36 


1.03 


C 


1417 


1739 


1534 


36 


20 


32 


1.58 


1.03 


D 


1589 


1951 


1705 


36 


20 


66 


0.99 


0.80 


A 


1771 


2018 


1872 


36 


20 


66 


1.15 


0.80 


B 


1994 


2274 


2095 


36 


20 


66 . 


1.36 


1.03 


C 


2416 


2738 


2533 


36 


20 


66 


1.58 


1.03 


D 


2710 


3072 


2827 


36 


24 


32 


0.99 


0.89 


A 


1153 


1339 


1280 


36 


24 


32 


1.15 


0.89 


B 


1283 


1564 


1411 


36 


24 


32 


1.36 


1.16 


C 


1562 


1884 


1718 


36 


24 


32 


1.58 


1.16 


D 


1734 


2096 


1890 



On all sizes 5 = 8 inc hes. 



1262 



QL— WATER WORKS. 



42. — Reducers and Increasers, Type No. 2. — Continued. 
(See Fig. 42a, page 1260.) 



Nominal Diarn. 
Inches. 




Thickness, Inches. 




Weights. Pounds. 




V 

Ins. 






Class. 






e 


f 


ti 


t2 


Spigot 
Ends. 


Large 
End Bell. 


Small 
End Bell. 


36 
36 
36 
36 


24 
24 

24 
24 


66 
66 
66 
66 


0.99 
1.15 
1.36 
1.58 


0.89 
0.89 
1.16 
1.16 


A 
B 
C 
D 


1964 
2188 
2664 
2957 


2211 
2468 
2985 
3319 


2091 
2314 
2820 
3113 


36 
36 
36 
36 


30 
30 
30 
30 


32 
32 
32 
32 


0.99 
1.15 
1.-36 
1.58 


0.88 
1.03 
1.20 
1 37 


A 
B 
C 
D 


1243 
1467 
1730 
2013 


1490 
1747 
2051 
2375 


1436 
1645 
1939 
2264 


36 
36 
36 
36 


30 
30 
30 
30 


66 
66 
66 
66 


0.99 
1.15 
1.36 
1.58 


0.89 
1.03 
1.20 
1.37 


A 
B 
C 
D 


2119 
2502 
2950 
3434 


2366 
2783 
3271 
3796 


2312 
2680 
3159 
3684 


42 
42 
42 
42 


20 
20 
20 
20 


32 
32 
32 
32 


1.10 
1.28 
1.54 
1.78 


0.80 
0.80 
1.03 
1.03 


A 
B 
C 

D 


1262 
1413 
1753 
1975 


1602 
1768 
2168 
2445 


1364 
1515 
1869 
2092 


42 
42 
42 
42 


20 
20 
20 
20 


66 
66 
66 
66 


1.10 
1.28 
1.54 
1.78 


0.80 
0.80 
1.03 
1.03 


A 
B 
C 
D 


2152 
2410 
2989 
3369 


2491 
2764 
3405 
3839 


2254 
2511 
3106 
3486 


42 
42 
42 
42 


24 
24 
24 
24 


32 
32 
32 
32 


1.10 
1.28 
1.54 
1.78 


0.89 
0.89 
1,16 
1.16 


A 
B 
C 
D 


1576 
1527 
1898 
2120 


1715 
1881 
2313 
2590 


1504 
1654 
2053 
2276 


42 
42 
42 
42 


24 
24 
24 
24 


66 
66 
66 
66 


1.10 
1.28 
1.54 
1.78 


0.89 
0.89 
1.16 
1.16 


A 
B 
C 
D 


2346 
2603 
3237 
3616 


2685 
2958 
3652 
4086 


2472 
2730 
3392 
3772 


42 
42 
42 
42 


30 
30 
30 
30 


32 
32 
32 
32 


1.10 
1.28 
1.54 
1.78 


0.88 
1.03 
1.20 
1.37 


A 
B 
C 
D 


1467 
1711 
2065 
2399 


1806 
2065 
2480 
2869 


1660 
1889 
2275 
2605 


42 
42 
42 
42 


30 
30 
30 
30 


66 
66 
66 
66 


1.10 
1.28 
1.54 
1.78 


0.88 
1.03 
1.20 
1.37 


A 
B 
C 
D 


2500 
2917 
3523 
4093 


2839 
3271 
3938 
4563 


2693 
3095 
3732 
4344 


42 
42 


36 
36 


32 
32 


1.10 
1.28 


0.99 
1.15 


A 
B 


1645 
1926 


1984 
2281 


1891 
2207 



On all sizes 5=8 inches. 




Fig. 42b. — Long Increaser. 48 to 30 inches x 132 inches V. 



CAST IRON PIPE— REDUCERS, INCREASERS. 



1263 



42. — Reducers and Increasers, Type No. 2. — Continued. 
(See Fig. 42a, page 1260.) 



Nominal Diam. 
Inches. 




Thickness, Inches 




Weights, Pounds. 




V 






Class. 






















e 


f 




ti 


t2 




Spigot 


Large 


Small 


Ins. 




Ends. 


End Bell. 


End Bell 


42 


36 


32 


1.54 


1.36 


C 


2320 


2735 


2642 


42 


36 


32 


1.78 


1.58 


D 


2714 


3184 


3076 


42 


36 


66 


1.10 


0.99 


A 


2803 


3143 


3050 


42 


36 


66 


1.28 


1.15 


B 


3285 


3639 


3565 


42 


36 


66 


1.54 


• 1.36 


C 


3958 


4373 


4279 


42 


36 


66 


1.78 


i.58 


D 


4631 


5101 


4993 


48 


30 


66 


1.26 


0.88 


A 


2975 


3381 


3168 


48 


30 


66 


1.42 


1.03 


B 


3428 


3883 


3606 


48 


30 


66 


1.71 


1.20 


C 


4092 


4641 


4801 


48 


30 


66 


1.96 


1.37 


D 


4762 


5388 


5013 


48 


30 


132 


1.26 


0.88 


A 


5363 


5769 


5556 


48 


30 


132 


1.42 


1.03 


B 


6180 


6635 


6359 


48 


30 


132 


1.71 


1.20 


C 


7379 


7928 


7588 


48 


30 


132 


1.96 


1.37 


D 


8588 


9214 ■ 


8839 


48 


36 


66 


1.26 


0.99 


A 


3278 


3684 


3525 


48 


36 


66 


1.42 


1.15 


B 


3796 


4252 


4077 


48 


36 


66 


1.71 


1.36 


C 


4527 


5076 


4849 


48 


36 


66 


1.96 


1.58 


D 


5300 


5925 


5662 


48 


36 


132 


1.26 


0.99 


A 


5909 


6316 


6156 


48 


36 


132 


1.42 


1.15 


B 


6844 


7299 


7125 


48 


36 


132 


1.71 


1.36 


C 


8164 


8713 


8485 


48 


36 


132 


1.96 


1.58 


D 


9558 


10184 


9920 


48 


42 


66 


1.26 


1.10 


A 


3659 


4066 


3998 


48 


42 


66 


1.42 


1.28 


B 


4212 


4667 


4564 


48 


42 


66 


1.71 


1.54 


C 


5100 


5649 


5516 


48 


42 


66 


1.96 


1.78 


D 


5959 


6585 


6429 


48 


42 


132 


1.26 


1.10 


A 


6597 


7003 


6936 


48 


42 


132 


1.42 


1.28 


B 


7594 


8049 


7948 


48 


42 


132 


1.71 


1.54 


C 


9197 


9746 


9612 


48 


42 


132 


1.96 


1.78 


D 


10747 


11373 


11217 


54 


36 


66 


1.35 


0.99 


A 


3722 


4228 


3969 


54 


36 


66 


1.55 


1.15 


B 


4330 


4925 


4610 


54 


36 


66 


1.90 


1.36 


C 


5259 


5953 


5580 


54 


36 


66 


2.23 


1.58 


D 


6181 


6995 


6543 


54 


36 


132 


1.35 


0.99 


A 


6710 


7216 


6957 


54 


36 


132 


1.55 


1.15 


B 


7806 


8401 


8087 


54 


36 


132 


1.90 


1.36 


C 


9484 


10178 


9805 


54 


36 


132 


2.23 


1.58 


D 


11148 


11962 


11510 



On all sizes 5 = 8 inches. 




Fig. 42c. — Short Increaser. 48 to 30 x 66 inches v. 



1264 



6i.— WATER WORKS. 



42. — Reducers and Increasers, Type No. 2. — Concluded. 
(See Fig. 42a, page 1260.) 



Nominal Diam. 
Inches. 




Thickness, Inches. 




Weights, Pounds. 






V 






Class. 




















e 


f 




ti 


t2 




Spigot 


Large 


Small 


Ins. 




Ends. 


End Bell. 


End Bell 


54 


42 


66 


1.35 


1.10 


A 


4103 


4609 


4442 


54 


42 


66 


1.55 


1.28 


B 


4745 


5340 


5100 


54 


42 


66 


1.90 


1.54 


c 


5832 


6526 


6247 


54 


42 


66 


2.23 


1.78 


D 


6841 


7655 


7310 


54 


42 


132 


1.35 


1.10 


A 


7398 


7903 


7737 


54 


42 


132 


1.55 


1.28 


B 


8556 


9151 


8910 


54 


42 


132 


1.90 


1.54 


C 


10517 


11211 


10932 


54 


42 


132 


2.23 


1.78 


D 


12338 


13152 


12807 


54 


48 


66 


1.35 


1.26 


A 


4578 


5083 


4984 


54 


48 


66 


1.55 


1.42 


B 


5256 


5851 


5711 


54 


48 


66 


1.90 


1.71 


C 


6401 


7095 


6950 


54 


48 


66 


2.23 


1.96 


D 


7512 


8326 


8137 


54 


48 


132 


1.35 


1.26 


A 


8253 


8759 


8660 


54 


48 


132 


1.55 


1.42 


B 


9478 


10073 


9933 


54 


48 


132 


1.90 


1.71 


C 


11544 


12239 


12093 


54 


48 


132 


2.23 


1.96 


D 


13550 


14364 


14175 


60 


36 


66 


1.39 


0.99 


A 


4096 


4711 


4342 


60 


36 


66 


1.67 


1.15 


B 


4906 


5576 


5186 


60 


36 


66 


2.00 


1.36 


C 


5867 


6692 


6189 


60 


36 


66 


2.38 


1.58 


D 


6960 


7934 


7322 


60 


36 


132 


1.39 


0.99 


A 


7384 


7999 


7631 


60 


36 


132 


1.67 


1.15 


B 


8846 


9516 


9126 


60 


36 


132 


2.00 


1.36 


C 


10581 


11405 


10902 


60 


36 


132 


2.38 


1.58 


D 


12554 


13527 


12916 


60 


42 


66 


1.39 


1.10 


A 


4477 


5092 


4816 


60 


42 


66 


1.67 


1.28 


B 


5321 


5991 


5676 


60 


42 


66 


2.00 


1.54 


C 


6440 


7264 


6855 


60 


42 


66 


2.38 


1.78 


D 


7619 


8593 


8089 


60 


42 


132 


1.39 


1.10 


A 


8072 


8687 


8411 


• 60 


42 


132 


1.67 


1.28 


B 


9595 


10265 


9950 


60 


42 


132 


2.00 


1.54 


C 


11614 


12439 


12030 


60 


42 


132 


2.38 


1.78 


D 


13743 


14716 


14213 


60 


48 


66 


1.39 


1.26 


A 


4957 


5572 


5363 


60 


48 


66 


1.67 


1.42 


B 


5832 


6502 


6287 


60 


48 


66 


2.00 


1.71 


C 


7006 


7830 


7555 


60 


48 


66 


2.38 


1.96 


D 


8285 


9259 


8910 


60 


48 


132 


1.39 


1.26 


A 


8938 


9552 


9344 


60 


48 


132 


1.67 


1.42 


B 


10517 


11187 


10972 


60 


48 


132 


2.00 


1.71 


C 


12634 


13458 


13183 


60 


48 


132 


2.38 


1.96 


D 


14943 


15917 


15568 


60 


54 


66 


1.39 


1.35 


A 


5404 


6019 


5910 


60 


54 


66 


1.67 


1.55 


B 


6348 


7018 


6961 


60 


54 


66 


2.00 


1.90 


C 


7750 


8574 


8444 


60 


54 


66 


2.38 


2.23 


D 


9178 


10152 


9992 


60 


54 


132 


1.39 


1.35 


A 


9745 


10360 


10251 


60 


54 


132 


1.67 


1.55 


B 


11462 


12132 


12075 


60 


54 


132 


2.00 


1.90 


C 


13979 


14803 


14673 


60 


54 


132 


2.38 


2.23 


D 


16557 


17530 


17371 



On all sizes s=8 inches. 



CAST IRON PIPE— REDUCERS AND SLEEVES, 



1265 



43. — Properties of Sleeves. 
(A. W. W. A.) 




>»^K 



Fig. 43. 
For dimensions a and b see Table No. 26. 



3m1nal 

iameter. 

ches. 


i 


D 

Ins. 


L 
Ins. 


T 
Ins. 


Approx. 
Weight. 
Pounds. 


ominal 

iameter. 

iches. 


i 


D 

Ins. 


L 

Ins. 


T 
Ins. 


Approx. 
Weight. 
Pounds. 


ZQa 


o 








';^Q^ 


B 








4 


D 


5.80 


10 


0.65 


47 


36 


B 


39.40 


15 


1.40 


943 


4 


D 


5.80 


15 


0.65 


61 


36 


C 


39.80 


15 


1.60 


1077 


6 


D 


7.90 


10 


0.70 


68 


36 


D 


40.20 


15 


1.80 


1217 


6 


D 


7.90 


15 


0.70 


87 


36 


A 


39.00 


24 


1.25 


1202 


8 


D 


10.10 


12 


0.75 


104 


36 


B 


39.40 


24 


1.40 


1362 


8 


D 


10.10 


15 


0.75 


119 


36 


C 


39.80 


24 


1.60 


1563 


10 


D 


12.20 


12 


0.80 


123 


36 


D 


40.20 


24 


1.80 


1772 


10 


D 


12.20 


18 


0.80 


176 


42 


A 


45.30 


15 


1.40 


1097 


12 


D 


14.30 


14 


0.85 


174 


42 


B 


45.60 


15 


1.50 


1184 


12 


D 


14.30 


18 


0.85 


223 


42 


c 


46.20 


15 


1.75 


1381 


14 


B 


16.20 


15 


0.85 


220 


42 


D 


46.70 


15 


1.95 


1561 


14 


B 


16.20 


18 


0.85 


249 


42 


A 


45.30 


24 


1.40 


1577 


14 


D 


16.50 


15 


0.90 


240 


42 


B 


45.60 


24 


1.50 


1702 


14 


D 


16.50 


18 


0.90 


280 


42 


C 


46.20 


24 


1.75 


1997 


16 


B 


18.50 


15 


0.90 


274 


42 


D 


46.70 


24 


1.95 


2262 


16 


B 


18.50 


24 


0.90 


391 


48 


A 


51.60 


15 


1.50 


1337 


16 


D 


18.90 


15 


1.00 


305 


48 


B 


51.90 


15 


1.65 


1481 


16 


D 


18.90 


24 


1.00 


443 


48 


C 


52.50 


15 


1.95 


1752 


18 


B 


20.60 


15 


0.95 


321 


48 


D 


53.10 


15 


2.20 


1986 


18 


B 


20.60 


24 


0.95 


462 


48 


A 


51.60 


24 


1.50 


1922 


18 


D 


21.00 


15 


1.05 


360 


48 


B 


51.90 


24 


1.65 


2129 


18 


D 


21.00 


24 


1.05 


518 


48 


C 


52.50 


24 


1.95 


2532 


20 


B 


22.70 


15 


1.00 


374 


48 


D 


53.10 


24 


2.20 


2879 


20 


B 


22.70 


24 


1.00 


532 


54 


A 


57.70 


15 


1.60 


1612 


20 


D 


23.10 


15 


1.15 


440 


54 


B 


58.20 


15 


1.80 


1835 


20 


D 


23.10 


24 


1.15 


625 


54 


C 


58.90 


15 


2.15 


2156 


24 


B 


26.90 


15 


1.05 


477 


54 


D 


59.50 


15 


2.45 


5450 


24 


B 


26.90 


24 


1.05 


680 


54 


A 


57.70 


24 


1.60 


2316 


24 


D 


27.40 


15 


1.25 


583 


54 


B 


58.20 


24 


1.80 


2634 


24 


D 


27.40 


24 


1.25 


821 


54 


C 


58.90 


24 


2.15 


3126 


30 


A 


32.80 


15 


1.15 


648 


54 


D 


59.50 


24 


2.45 


3571 


30 


B 


33.10 


15 


1.15 


652 


60 


A 


63.90 


15 


1.70 


1906 


30 


C 


33.50 


15 


1.32 


760 


60 


B 


64.50 


15 


1.90 


2127 


30 


D 


33.80 


15 


1.50 


876 


60 


C 


65.30 


15 


2.25 


2491 


30 


A 


32.80 


24 


1.15 


943 


60 


D 


65.90 


15 


2.60 


2895 


30 


B 


33.10 


24 


1.15 


949 


60 


A 


63.90 


24 


1.70 


2731 


30 


C 


33.50 


24 


1.32 


1088 


60 


B 


64.50 


24 


1.90 


3058 


30 


D 


33.80 


24 


1.50 


1262 


60 


C 


65.30 


24 


2.25 


3601 


36 


A 


39.00 


15 


1.25 


833 


60 


D 


65.90 


24 


2.60 


4231 



1266 



6i.— WATER WORKS. 



44. — Properties of Caps. 
(A. W. W. A.) 




r.t:^~1 




-Tap FOR 2 WiPiPB 
Figs. 44. 
Bosses A and B cast on only when so ordered. 



Nom'l 


















Approx. 


Diam. 


Class. 


d 





1 


t 


m 


k 


r 


Weight. 


Inches. 


















Pounds. 


4 


D 
D 
D 
D 


4.00 
4.00 
4.00 
4.00 


5.70 

7.80 

10.00 

12.10 




0.60 
0.65 
0.75 
0.75 








26 


6 








40 


8 








S9 


10 


1.50 


0.75 


16.20 


81 


12 


D 


4.00 


14.20 




0.75 


1.75 


0.75 


18.70 


104 


U 


B 


4.00 


16.10 




0.90 


1.90 


0.75 


22.40 


140 


14 


D 


4.00 


16.45 




0.90 


1.90 


0.75 


22.40 


149 


16 


B 


4.00 


18.40 




1.00 


2.00 


0.75 


27.00 


183 


16 


D 


4.00 


18.80 




1.00 


2.00 


0.75 


27.00 


198 


18 


B 


4.00 


20.50 




1.00 


2.00 


1. 00 


32.00 


226 


18 


D 


4.00 


20.92 




1.00 


2.00 


1.00 


32.90 


242 


20 


B 


4.00 


22.60 




1.00 


3.00 


1.00 


18.20 


278 


20 


D 


4.00 


23.06 




1.00 


3.00 


1.00 


18.20 


308 


24 


B 


4.00 


26.80 


"2!56'" 


1.05 


3.50 


1.00 


23.50 


392 


24 


D 


4.00 


27.32 


2.50 


1.05 


3.50 


1.00 


23.50 


442 


30 


A 


4.50 


32.74 


2.62 


1.15 


3.50 


1.15 


34.80 


589 


30 


B 


4.50 


33.00 


2.62 


1.15 


3.50 


1.15 


34.80 


596 


30 


C 


4.50 


33.40 


2.62 


1.15 


3.50 


1.15 


34.80 


647 


30 


D 


4.50 


33.74 


2.62 


1.15 


3.50 


1.15 


34.80 


704 


36 


A 


4.50 


38.96 


3.12 


1.25 


4.00 


1.25 


44.00 


849 


36 


B 


4.50 


39.30 


3.12 


1.30 


3.95 


1.25 


44.00 


918 


36 


C 


4.50 


39.70 


3.12 


1.35 


3.90 


1.25 


44.00 


998 


36 


D 


4.50 


40.16 


3.12 


1.40 


3.85 


1.25 


44.00 


1084 


42 


A 


5.00 


45.20 


3.37 


1.40 


4.00 


1.40 


63.50 


1300 


42 


B 


5.00 


45.50 


3.37 


1.50 


3.90 


1.40 


63.50 


1388 


42 


C 


5.00 


46.10 


3.37 


1.60 


3.80 


1.40 


63.50 


1539 


42 


D 


5.00 


46.58 


3.37 


1.70 


3.70 


1.40 


63.50 


1679 


48 


A 


5.00 


51.50 


3.62 


1.70 


4.00 


1.50 


76.50 


1772 


48 


B 


5.00 


51.80 


3.62 


1.90 


3.80 


1.50 


76.50 


1943 


48 


C 


5.00 


52.40 


3.62 


2.00 


3.70 


1.50 


76.50 


2144 


48 


D 


5.00 


52.98 


3.62 


2.10 


3.60 


1.50 


76.50 


2341 


54 


A 


5.50 


57.66 


3.87 


1.90 


4.50 


1.50 


82.00 


2329 


54 


B 


5.50 


58.10 


3.87 


2.00 


4.40 


1.50 


82.00 


2519 


54 


C 


5.50 


58.80 


3.87 


2.10 


4.30 


1.50 


82.00 


2770 


54 


D 


5.50 


59 40 


3.87 


2.20 


4.20 


1.50 


82.00 


3009 


60 


A 


5.50 


63.80 


4.12 


2.00 


4.50 


1.50 


99.00 


2868 


60 


B 


5.50 


64.40 


4.12 


2.10 


4.40 


1.50 


99.00 


3082 


60 


C 


5.50 


65.20 


4.12 


2.20 


4.30 


1.50 


99.00 


3388 


60 


D 


5.50 


65.82 


4.12 


2.30 


4.20 


1.50 


99.00 


3687 



Note. — ^All dimensions are in inches. 



CAST IRON PIPE— CAPS AND PLUGS. 



45. — Properties of Plugs. 
(A. W. W. A.) 



1267 




Figs. 45. 
4 to 20 ins. Bosses a and b cast on only when so ordered. 




42 to 60 ins* 



Nominal 

Diameter. 

Inches. 


i 


e 


f 


a 


■ 


m 


Thickness. 
Inches. 




o.d'O 
(H bed 


t 


t2 


t3 


^^S 


4 
6 


D 
D 
D 
D 
D 

E 

B 
D 
B 
D 
B 
D 
B 

I> 
A 
B 
C 
D 
A 
B 
C 
D 
A 
B 
C 
D 
A 
B 
C 
D 
A 
B 
C 
D 
A 
B 
C 
D 


4.90 
7.00 
9.15 
11.20 
13.30 
15.30 
15.65 
17.40 
17.80 
19.50 
19.92 
21.60 
22.06 
25.92 
26.44 
31.86 
32.12 
32.52 
32.86 
38.08 
38.42 
38.82 
39.28 
44.32 
44.62 
45.22 
45.70 
50.62 
50.92 
51.52 
52.10 
56.78 
57.22 
57.92 
58.52 
62.92 
63.52 
64.32 
64.94 


5.28 
7.38 
9.65 
11.70 
13.80 
15.80 
16.15 
17.90 
18.30 
20.00 
20.42 
22.10 
22.56 
26.30 
26.82 
32.24 
32.50 
32.90 
33.24 
38.46 
38.80 
39.20 
39.66 
44.70 
45.00 
45.60 
46.08 
51.00 
51.30 
51.90 
52.48 
57.16 
57.60 
58.30 
58.90 
63.30 
63.90 
64.70 
65.32 




5.50 
5.50 
5.50 
6.00 
6.00 
6.00 
6.00 
6.50 
6.50 
6.50 
6.50 
6.50 
6.50 
8.00 
8.00 
8.00 
8.00 
8.00 
8.00 
8.00 
8.00 
8.00 
8.00 
9.00 
9.00 
9.00 
9.00 
9.00 
9.00 
9.00 
9.00 
9.00 
9.00 
9.00 
9.00 
9.00 
9.00 
9.00 
9.00 


"2!66' 

2.00 
2.00 
2.00 
2.00 
2.00 
2.00 
2.50 
2.50 
2.75 
2.75 


0.50 
0.60 
0.60 
0.70 
0.75 
0.70 
0.75 
0.70 
0.80 
0.75 
0.85 
0.85 
1.00 
0.89 
1.16 
0.88 
1.03 
1.20 
1.37 
0.99 
1.15 
1.36 
1.58 
1.10 
1.28 
1.54 
1.78 
1.26 
1.42 
1.71 
1.96 
1.35 
1.55 
1.90 
2.23 
1.39 
1.67 
2.00 
2.38 


0.40 
0.40 
0.40 
0.50 
0.50 
0.50 
0.50 
0.50 
0.60 
0.60 
0.60 
0.60 
0.60 


0.20 
0.20 
0.20 
0.20 
0.20 
0.20 
0.20 
0.30 
0.30 
0.30 
0.30 
0.30 
0.30 


"2" 
2 
2 
2 
2 
3 
3 
3 
3 
3 
3 

4 
4 


8 
14 


8 




24 


10 




38 


12 




50 


14 




63 


14 




65 


16 




90 


16 




96 


18 




111 


18 




121 


20 




151 


20 




156 


24 


25.68 
26.20 
31.62 
31.88 
32.28 
32.62 
37.84 
38.18 
38.58 
39.04 
44.08 
44.38 
44.98 
45.46 
50.38 
50.68 
51.28 
51.86 
56.54 
56.98 
57.68 
58.28 
62.68 
63.28 
64.08 
64.70 


375 


24 






472 


30 






481 


30 






556 


30 






641 


30 






723 


36 






682 


36 






786 


36 






914 


36 






1050 


42 






991 


42 






1138 


'4?, 






1353 


42 






1551 


48 






1349 


48 






1506 


48 






1800 


48 






2047 


54 






1697 


54 
54 
54 






1945 
2356 
2733 


60 






2045 


60 






2434 


60 






2904 


60 






3397 



Note. — All dimensions are in inches. 



1268 ^.— WATER WORKS, 

Hid.— WROUGHT IRON PIPE. 

Wrought Iron Pipe corrodes more rapidly than cast iron, and less 
rapidly than steel. Its first cost, in the smaller sizes, is less than the 
former; and in all sizes it is greater than the latter. Between cast iron on 
the one hand and the steel on the other the use of wrought iron in pressure 
pipe lines has become limited, but it yet finds a demand in the smaller sizes 
of pipe for distributing systems. It remains to be seen whether the few 
remaining maniifacturers of wrought iron pipe will be able to offer convinc- 
ing proofs of the superior merit and ultimate economy of their article, as 
claimed. More attention is paid now to suitable coating for pipes than 
formerly, and a steel pipe well coated when laid should last from 25 to 40 
years in ordinary soil, and if occasionally cleaned and painted it will last 
much longer. Wrought iron pipe may be manufactured in the same forms 
as steel pipe (Ille), but it is usually lap-welded, seldom riveted. 

IIIe.—STEEL PIPE. 

Steel Pipe is usually designated by the kind of "seam" or "joint." 
The latter term is generally used with reference to the (end) connection 
between two pipes, and frequently also to the longitudinal seam at the 
joining of plates in the pipe itself. Seams or joints in a pipe may be lap- 
welded, spiral riveted, longitudinal riveted, or locking-bar. Riveted (longi- 
tudinal) joints may be lap- or butt-, and be (single-), double- or triple- 
riveted, with (single or) double straps. Single riveting or single straps are 
seldom used for longitudinal joints. The end joints for pipes may be screw-, 
bell-and-spigot- (rarely used), flange- (welded, riveted, or screw), single- 
riveted-, patent-locking-, etc. A few of these will be described as follows: 

Riveted Steel Pipe is especially economical for large diameters, sub- 
jected to pressure head of 300 ft. or more. For heads between 200 and 300 
ft. riveted steel has to compete with cast iron pipe; and for heads under 
200 ft. both riveted steel and cast iron pipe have to compete with wood 
stave pipe, especially in the West where lumber is plentiful. 

The thickness of steel plates is proportioned by the formula — 

-^1^- a) 

where / = thickness of steel shell, in inches; 
(i = inside diameter of pipe, in inches; 
/t = pressure head in ft. ( = 2 . 304 p) ; 

^ = pressure of water in lbs. per square inch ( = 0.434 ^); 
/>' = allowance for water ram, in lbs. per square inch; 
s = allowable tensile stress of steel, in lbs. per square inch; 
^ = efficiency of riveted joint, say 0.50 to 0.80; 
c = thickness to be added to plate for corrosion, etc. 
Based on a safety factor of 4, 5 is assumed at 15,000 (to 16,000) lbs; 
and c may be assumed at say rs in. 

The practical working formula will result as follows, assuming p' at 
80 lbs. <see page 1215) : 

(p + p^)d _ (0.434 ;z+ 80)^ 

^ = -300007"^^- 30000^ + ^-^^2^ (2) 

If h = the static head in feet, we have, from (2), 

h = ^~-^ (t-^)-18i.S2 ; (3) 

In designing the riveting of longitudinal seams, we have, if F = total 
tension per lin. inch of seam, due to p and p', 

F=^(^p+p') = ff(0.217 /j + 40) = 1500 {t-c)e (4) 

in which F should not exceed the bearing- nor the shearing resistance of the 
rivets. The bearing value may be assumed at 16,000 lbs. per square inch 
on the thickness of plate t — c, as it is assumed that c is the allowance for 
corrosion; while the shearing value may be assumed at say 10,000 lbs. per 
square inch of rivet section. The diameter of rivet is approximately twice 
the thickness of the plate. 



WROUGHT IRON PIPE, STEEL PIPE. 



1269 



The following is a table of pitches and dimensions of rivets adopted in 
the manufacture of the 42-in. pipe for the Seattle (Wash.) Water-works, 
constructed in 1900: 



Thickness of Steel, in inches 


-h 


H 


A 


H 


A 


Diameter of rivets, in inches 


V2 


^ 


% 


M 


if 


Diameter of rivet holes, in inches 


^ 


% 


H 


H 


% 


Pitch in double riveted seams, in ins . . 


1.75 


2.00 


2.25 


2.50 


2.50 


Pitch in single riveted seams, in ins . . . 


1.40 


1.60 


2.00 


2.25 


2.25 


Distance between rows on double 
riveted seams, in inches 


1.50 


1.75 


2.00 


2.00 


2 00 






Lap from center of rivet to edge of 
plate, in inches 


.750 


.875 


1.00 


1.00 


1.125 







The edges of all plates are bevel sheared for caulking. Shop- riveting 
and caulking are invariably done by machine. For thick plates the rivets 
may be countersunk on inside of pipe. The field- riveting and caulking are 
usually done by hand, but machine riveters and caulkers are sometimes 
employed, not always to advantage. The field work consists in joining the 
pipes together in the trench and making them water-tight under pressure. 
If the end joints are lap-riveted there must necessarily be alternate courses 
with two separate diameters, but the internal diameters of the smaller 
course shall be "the diameter" of the pipe. The internal diameter of the 
larger course must equal exactly the external diameter of the smaller course. 
Pipes of ordinarily large diameter usually leave the shop made up in four 
courses, and in length from 20 to 30 ft., after being tested to a sufficient 
hydrostatic pressure, and suitably coated (see page 358, etc.). 

In shipment it is well to bear in mind that a "full" carload cannot 
always be made up with pipes of the same diameter if large. 

When laid in the trench the longitudinal seams should be at the top of 
the pipe, staggered not less than 6 inches. 

* Locking«Bar Joint Pipe is designed to supplant the riveted joint. 
Fig. 46 shows a transverse section of the (longi- 
tudinal) bar and joint and also the (end) joint 
ring. In manufacture, the edges of the plate 
are upset and inserted in the grooves on either 
side of the locking bar. The bar is then sub- 
jected to a great hydraulic pressure, tightly lock- 
ing the plates in a highly efficient manner. 
No riveted longitudinal joints are used. Sleeves 
or joint rings are used for end joint connections 
of pipes and are run with lead; or the end joints 
may be riveted. 

Lap-Welded Pipe is seldom used in the pressure pipe line: it is better 
adapted to use in the distributing system (see page 1280) . The lour principal 
types used are those with screw ends or flange ends (Section 36), Converse 
Joint (p. 1282), and Matheson Joint (p. 1281). 

Spiral Riveted Pipe (p. 680) is used largely in hydraulic mining. 




Bar 



Fig. 46. 



Illf.— PRESSURE PIPE ATTACHMENTS. 

Brief mention will be made of some of the more important items in 
connection with an ordinary pressure- pipe line. The varying conditions 
are so great in different lines that the merest hints only will be given. 

* This form of joint was brought prominently to the attention of Ameri- 
can engineers in 1898 by an article published in Eng. News, Vol. XXXIX, 
g. 373, describing its use in connection with the construction of a large pipe 
ne in Australia. 



1270 



Qi.— WATER WORKS. 



Fig. 47 shows a Sluice=Qate for the inlet end of the pipe at the head- 
works, and Fig. 48 shows Stand and Wheel for operating same. Each may 
be fastened to either masonry (by stone-bolts) or to timber (by screw-bolts) . 
This pattern is suitable for large inlets, 30 ins. or more in diameter. The 
gear-wheel is often horizontal instead of vertical. 





Fig. 47. — Sluice-Gate for Headworks. 



Fig. 48.— Stand and Wheel for 
Operating Sluice-Gate. 



Air Relief Valves, or "Air Valves," are placed at summits of pipe lines 
to afford relief from air presstire or from vacuum. As the pipe fills with 
water the valve operates so as to allow the air to escape, but closes against 
the outlet of the water. As the water in the pipe subsides, the valve allows 
the air to enter the pipe and prevent collapse from vacuum. If air is allowed 




Fig. 49. 



Fig. 50. 



to collect at summits without air valves it is clearly seen that the water 
area of the pipe at those points, and hence the flow, will be decreased ac- 
cordingly; and also that the pipe is more liable to leak and burst (from air- 



PRESSURE PIPE ATTACHMENTS. VALVES, 



1271 



than from water pressure). Air valves may be single, or they may be 
arranged in clusters to operate consecutively. There is an advantage in 
having a stop-valve between the air valve and the top of the main pipe, so 
the water can be shut off during repairs. Valves should be protected by 
wood- or metal casings and be out of reach of the frost. 

Fig. 49 illustrates the Ludlow automatic lever and float air valve, and 
Fig. 50 the globe air valve. 

Fig. 51 shows section and plan of Metropolitan Water Works air valve, 
with table of dimensions. 



ffir/r foifanf righf 
to open , 




tL.Li...U-— ^ 



Sectional Elevation. 

Fig. 51. 



Half PIgn of Cap C 



46.— M. W. W. Air Valves. (Fig. 51.) 
Dimensions are in inches; weights, in pounds. 





h 


m 


d 


r 


/ 


f 


g 


e 


h 


t 


n 


n' 


Stem s 


Diam. 


Wt. 
Lbs. 




Diam. 


Thds. 


1 

1^ 


1* 


1 


15 


l3\ 

1^ 


tl 


¥: 




IH 
13^ 


2% 


3^ 


1^ 
IM 


IH 


H 


12 
12 


Vs 
% 


i 



Standpipes (small) are sometimes erected at the summits of pipe lines 
to serve both as air valves, and also as piezometers for determining the 
hydraulic grade line (see p. 1159). They are often preferred to air valves 
where the static head is not great. 

Gate Valves, stop valves, gates, or valves, as they are variously termed 
are of numerous designs, suited to special purposes. The best manufac- 
turers are beginning to standardize their product so that there is very little 
difference in quality of material and service. But there is a vast difference 
in length of service between these standard makes and most of the cheaper 
valves thrown upon the market. Waterworks engineers, generally, fully 
realize the importance of securing at all times the best article even at the 
greater first cost. It is economy in the end. The best gates are bronze- 
mounted. 



1272 



6i,— WATER WORKS. 



Fig. 65, page 1285, shows a section of the Chapman valve with wedge- 
shaped gate. i^ig. 52, below, shows a section of a double gate valve of the 
Ludlow type; and Fig. 53, views of the gates and wedges. The pipe con- 

A— Case. 



B — Cover or Bonnet. 

C — Stem or Spindle. 

D— Packing Plate or 
StuflSngBox. 

B— Stuffing Box Gland 
or FoOower. 

F-StemNut 




Fig. 52.— Ludlow Bronze Mounted Double Gate Valve 
With Bolted Stuffing Box. 




Fig. 53. — Style of Gates and Wedges for Double Gate Valves, 



GATE VALVES— GEARED AND UNGEARED. 



1273 



nections may be bell- (as shown in the illustrations), flange- or screw-, as 
required. Gates are designed for either vertical or horizontal (the larger 
sizes) position, and for operating by hand or "power," with screw, piston, 
lever, or gearing (gates under 16 ins. in size are seldom geared). They are 
also manufactured with or without by-pass relief. The by-pass is useful in 
filling long pipe lines without impact, and in equalizing the pressure on both 
sides of a gate when being opened or closed. 

47. — M. W. W. Gate Valves. 
Principal Dimensions, in Inches; Weights, in Pounds. 

























-r, 


03 ^ 


+3 


^BP 


a 


b 


d 


1 


g 


c 


h 


e 





t 


s 




& o 




Weig 
of Co 
posit 


12 


12 


4 


46tV 


35H 


26 




103^ 


UH 


% 


IH 


2 


58 


1175 


76 


16 


13 


4^ 


54-^ 


421^ 


30^ 




12t^ 


18% 


H 


2 


2 


74 


1625 


100 


20 


16 


4^. 


52A 

57^, 


35 


8 


14^^ 


23 


1 


2H 


6 


*203 


2675 


180 


24 


21 


4^ 


73^ 


38 


8 


15% 


27 


m 


2H 


6 


*230 


3325 


215 


30 


36 


5 


110^4 


75M 


44f^, 


545^ 


28M 


33 


1^ 


3H 


5 


254 


7000 


600 


36 


iO% 


5^ 


125 


87 


53^ 


66% 


32 


39M 


1% 


3^ 





306 


10250 


710 



* Using gears. With wrench on main stem, 
20-inch, and 76^ turns the 24-inch valve. 



67^ turns will open the 




Secf/ona/ Elevation 
Gears for iO''& 24" Valves. 

Pics. 54 



March. 1837. 



1274 



Qi.— WATER WORKS. 




:^^^ ^ yt 



^:^ 









CO 



CO "^ 



e^a 00 -^ -^ 00 c^a e^q io o 



CM c<a CO CO -* -H e^ -^ e>a 



i;^:^ 



;:^' ^^-h;^=;§^:^:^^:^ 



<:=> c<i CO a> 1-, t£> ] CO Oi-^ y-< I CM 
cq e^a cq egor-iT-ioo m 



^ ^':^ ^^;^^x^ 



:^;^^^':^^Hs;^^ 



|<M C<1 M < 



toooNco^i-H i oscom-H I, 
-< ,-H c^ c>a«5 i-HCNj 00 



xoooc^a<Nii-io i05eo-«*<— i-h 
1-1 th cva cvi to 1-1 c<i to 



lOOOCCli-irtOO I OSCOCO.-I I ^ 

M i-ieq 00 



1-1 i-< CM 



<< ar^m WoQW fe 



e3 
Ph <» 
I M 

>>^ 
«^ 
^ >. 

g« 
^^ 

bJC bC 

c a 

-d-d 
WW 
o o 
o o 

03 o3 

WW 

o o 
03 c3 
WW 



W^OO 
Tj+j a> 



§1^ 



w^5 



S 



^ P c> pa 






DOUBLE GATE LUDLOW VALVES. 



1276 



0> 0> CO «o »o »-< 
coco lO «o»o 



:^:^ :^::^^ ^i 



t^ t^ '«1< Tj< T-t 



^«^fil CON 

}4<o«000»>. 



^ 



CO fO«0 lOtO ■* 



;^ ^:^ 



\a c» 



lOicn co»oo>oo^ 

CO CO CO CO 



CO CO ■'H »0 «0 (Nl 
COCOtJI in-* 



;:r' ^:^ 



«£><=> I I Q0<0 

^-4 CO cq C<I •^ CO 

cot- CO CO 



:f! 



to CO OJ XOO c» 
MCOCO oco 



o th e^q 00 CO to o I I »-i to 

cocqos^Ht>.i— icocacvjt-io 

CO t~rt*-«^ 



;^^. 



^cq cq 

CO CO OS ""a* K-Tf" 'tIi 

ca 1-1 1- ^-H CO 



:^s! 



00 

e^aco CO coco 

^ "^ CO CO 



0000 

I cq 0000 
,CO ^ T-H 



•»J* cq 1-4 CO 00 O 
cqcofo -^co 



iri CO CO -^ CO (rq ^ I |mi 

cai— it-1— icoi-tcqcococoi 



i-c*kOC3C5cqcocococ^io 
cq t- CO CO cq cq CQ CO cq 



•<*< e>q CO 00 o> o t- t- I I t^ ( 
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T-t-^OSt-COOJCO-^lOCqcO 

cq CO "^ »o 1-1 cq cq CO cq 



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osPQ 



c3(^id'd'33— -03 

•S'S.S.S'dd^o 

>^^4Jddg>. 

0303C^C3ddcL03 



«M iM 'r fH <S c3 

g§sl§§ 

^- - d d <» cQ 

> > .2 5 g a 

03 03 C3 o3 :3 p 



m 



1276 



Qi.— WATER WORKS. 



49. — Weights of Ludlow Gates and Valves, in Lbs. 
(^Abbreviations.) 



List No. l.{«:^- 


and under 
and above: 


T. 
T. 


P.W.= 

P. w.= 


150 lbs. 
100 lbs. 


;R. 
R. 


W. 
W. 


=75 lbs 
= 50 lbs 


;R.S.= 
;R. S.= 


=50 lbs. 
= 35 lbs. 




sizes 


1" 


IM" 


13^" 


IW 


2" 


2%" 


3' 


3^" 


4' 


4^" 


5' 


6' 


7" 


8' 


9' 


10" 


12' 


Sc. Soc 






8 




13^ 
223^ 
23 


22^ 
31^ 


29^ 

42 

65 


54 
68 


59 
73 

78 


. . . . 


124 
129 
141 


151 
158 
173 


204 
206 
208 


231 
234 
234 




365 
358 
361 




Fig 






452 


Hub 










458 


Spig 












O.S.& Y.,ex. . 


































88 


L. F., ex.. . 














25 




28 




35 


40 


50 


65 


85 


100 


130 




















T ;c^- M^ 9 r20-in. and under: T. P.W. = 100 lbs.; R. W.=50 lbs.; R. S. = 351bs. 
l^ist INO. ^•\24-in.and above: T. P.W.= 75 lbs.; R. W.=401bs.; R. S. = 251bs. 


Sizes 


14" 


15' 


16' 


17' 


18' 


20' 


22" 


24" 


27' 


30' 


36' 


40' 


42" 


48" 


50" 


60" 


72' 


Fig 


558 
542 




680 
692 




8301.905 




1400 
1305 


.... 


2950 
2783 
















Hub 


833 


956 


4100 














Spig 














Sp. Gr., ex . . 


105 
114 





105 
144 




105 
114 


105 
114 




67 
68 





160 
146 


160 














Bv. Gr., ex. . . 














By-Ps., ex. . . 





















































ListNo.3.{|«t 


. and under 
. and above 


: T. 
: T 


P.W.= 
P.W.= 


100 lbs. 
75 lbs. 


;R. 
;R. 


W. 
W. 


= 60 lbs 
= 50 lbs 


;R. 
.;R. 


s.= 


45 lbs. 
= 30 lbs. 


Sizes 


14' 


15" 


16" 


17" 


18' 


20' 


22' 


24' 


27' 


30' 


36' 


With Gearing. 


40' 


42' 


48' 


50" 


60' 


72' 


Fig 


605 
578 


640 
675 


770 
740 


.... 


875 
875 


1057 
1057 


1590 
1600 


1712 
1580 


.... 


3130 
2900 


4530 
4145 


6615 
6850 












Hub 




11850 








Spig 








Sp. Gr., ex . 


105 
114 


105 
114 


105 
114 




105 
114 


105 
114 


.... 


67 
68 





160 
146 
















Bv. Gr., ex. . . 
















By-Ps., ex. . . 





















































T ;cf M^ ± /6-in. and under: T. P. W. = 300 lbs.; R.W. = 1501bs.; R. S. =85 lbs. 
1.1st iNo. 4-|7.in. and above: T. P.W. = 300 lbs.; R.W. = 150 lbs.; R. S. =70 lbs. 



Sizes 


^4" 


"% 


I" 


IM" 


1^" 


2' 


2^" 


3' 


m" 


4' 


i'A" 


5' 


6' 


7' 


8' 


9' 


10' 


12' 


Sc. Soc 










8 
14 


14 
23 
25 


32 
28 


If 

67 


55 
693^ 


62 
76 
80 
80 
26 
28 


103 
111 


130 
131 
142 


160 
166 
180 
180 
45 

.0 


210 
220 
208 

■■56 


239.... 
236 ... . 
244 - 


391 
406 
384 
420 
90 
100 


530 


Fig 










513 


Hub 










490 


Spig. .... 












251 
60 
65 


■*85 


5;^o 


0. S. & Y., ex 




















.... 


43 
35 


102 


L. F., ex 
















25 




130 





















*The following abbreviations are used in Table 49: 

For connections, etc.: Sc. Soc. = Screw Socket; F/g. = Flanged; Spig.= 
Spigot; O. S. and Y., ^^. =Outside Screw and Yoke, extra; L. F., ex.= 
Loose Flanges, extra; Sp. Gr., ex. = Spnr Gear, extra; Bv. Gr., ex.= 
Bevel Gear, extra; By-Ps., ^a:. = By-Pass, extra; P. F., R. 0. = Plain 
Frame, round opening; P. F., S. 0. = Plain Frame, square opening; 
R. O. with Spig. = 110x1116. Opening with Spigot; R. O. with F. & N.= 
Round Opening with Flange and Neck Piece; E. F., R. 0. = Extension 
Frame, round opening; E. F., S. O. = Extension Frame, square opening; 
5. O. with F. & A/'. = Square Opening with Flange and Neck Piece. 

For Test- and Working Pressures: T. P. W. =Test Pressure, Water; R. W. = 
Recommended for Water. — Working pressures not to exceed the number 
of pounds given in Table headings; R. 5. = Recommended for Steam — 
Working Pressures not to exceed number of lbs. given in Table headings. 



WEIGHTS OF LUDLOW GATES AND VALVES, 



1277 



49. — ^Weights of Ludlow Gates and Valves, in Lbs. — Continued. 

T-.^M^ K /6-in. and under: r.P.T^. = 3501bs.;i?.l^. = 2001bs.;i?.5. = 1001bs. 
^^^^^°-^-l7-in.andabove:r.P.iy. = 3501bs.;i?.l^.=2001bs.;i?.5.= 85 lbs. 



Sizes .... 


m" 


2" 


2y2" 


3" 


31/^" 


4" 


iV^" 


5" 


6" 


7" 


8" 


9" 


10" 


12" 






Sc. Soc 








55 
64 
67 




75 
87 
93 


102 
111 


130 
131 
140 


191 
202 
204 


212 
220 
221 


270 
282 
280 


350 
374 


437 
463 
409 


520 


Fig 








576 


Hub 








547 


Spig 










O. S, & Y., ex 
















43 
35 


45 
45 


'50 


60 
65 


■*85 


90 
100 


102 


L. F.. ex 
















130 





















List No. 53^. r. P. T/F. = 400 1bs.; i?. 5. = 150 lbs. 



Sizes 


iVz" 


2" 


2W 


3" 


m" 


4" 


m" 


5" 


6" 


7" 


8" 


9" 


10" 


12" 


Flansced 






38 


70 
64 


75 


123 
107 




187 
185 


257 
237 


300 


379 
370 





548 


725 


Screwed . .... 























List No. 5K. T. P. W 


. = 600 lbs.; 


R. 


W. 


= 400 lbs.; 


P. 5.= 


250 lbs 






Sizes. 


m" 


2" 


2y2" 


3" 


3ya" 


4" 


m" 


5" 


6" 


7" 


8" 


9" 


10" 


12" 


Sc. Soc 






37 
46 


77 
80 


80 
110 


104 

120 

105 

30 

40 


"m 


200 
230 


268 

285 

275 

59 

75 


280 
354 

"96 


390 

410 

410 

90 

94 


'525 


632 
690 
650 
115 
150 


800 


Fis: . 






930 


Hub 






870 


O. S. & Y.. ex. 

L. F., ex 






12 


17 




— 


50 
60 


130 
170 


By-Ps., ex 










































* * 


Sizes 


14" 


15" 


16" 


18" 


20" 


24" 


30" 


36" 


Remarks. 


Fig 






2050 




3000 










Hub. c. . . , 






5180 






The 24" is bevel gear. 


O S & Y., ex 
















L. F., ex 










320 










By-Ps., ex 
























, 

























f20-inandunder:T. P. l¥. = 3001bs.;i?.TF. = 2001bs.;i?.5. = 851bs. 

List No. 6.^24-36-ins.: T. P. IF. = 300 lbs.; i?. P1^. = IGOlbs.; i?. 5. = 75Jbs. 

[40-in.andabove:T.P.VF. = 3001bs.;i?.Vr. = 1401bs.;i?.5. = 651bs 



Sizes 


14" 


15" 


16" 


17" 


18" 


20" 


22" 


24" 


30" 


With Gearing. 




36" 


40" 


42- 


48" 


50" 


54" 


60" 


72" 


Fig 


770 
705 


890 
820 


963 

875 




1260 
1203 


1650 
1585 




2580 
2461 


4260 
4425 


8600 
8500 


11500 


11650 
12160 


18970 
18550 










Hub 










Splg 










Sp. Gr., ex. . . 


105 
114 


105 
114 


105 
114 




67 
68 


67 
68 




103 
104 


200 
188 


















Bv. Gr., ex. . . 


















0. S. & Y.,ex. 


















L. F.. ex 






























■ * 






By-Ps., ex. . . 









































































Notes.— 16", Hub, #6, Bevel Gear, and 2' By-Pass, 1230 lbs.; 30'', Hub, 
#6, Bevel Gear, and 6'' By-Pass, DD Stem, 5650 lbs.; 36", Hub, #6, Bevel 
Gear, and By-Pass, 9500 lbs. 



1278 



M.— WATER WORKS. 



49. — Weights of Ludlow Gates and Valves, in Lbs. — Continued. 

re-in. and under: T. P. T^. = 3001bs.; i?. 1^. = 200 lbs.; R. S. 
(3-in.) = 150 1bs. 
List No. 7. \ 7-in. and above: T. P. W. = 250 lbs.; R. W. = 125 lbs.; R. S. 
(3i-6-in., inc.) = 125 1bs. 
7-12-in., inc.: R. 5. (7-12-in., inc.) = 85 lbs. 



Sizes 


M" 


%" 


Vz" 


M" 


V 


IH" 


1^" 


m" 


2" 


2^" 


3" 


3f^" 


4" 


43^" 


5" 


fi" 






Sc. Soc 








4J& 


tk 


m 
1% 


loft 


.... 


153^ 


15^ 
223^ 


18% 
27 


30 
39^ 


443^ 
583^ 


77^4 


67 
88 


95 


Fig 






118 










Sizes 


7" 


8" 


9" 


10" 


11" 


12" 


14" 


16" 


18" 


20" 


22" 


24" 


27" 


30" 










Sc. Soc 


































Flff 






290 




351 


583 
















































List No. 8. 


r. P. 1^. = 1500 lbs.; i?. T1^. = 750 1bs. 


Sizes 


1" 


\W 


2" 


2^" 


3" 


3^" 


4" 


4^" 


5" 


6" 


7" 


8" 


9" 


10" 


1?" 






Sc. Soc 






51 


"124 
14 


101 

139 

30 




160 

198 

61 






405 

435 

95 












Fig 










.... 


815 
160 








L. F., ex 














. . . . 


















List No. 9. 


r. P. T^. = 2000 1bs.; i?. 1^. = 2000 lbs.; i?. 5. = 1200 lbs. 


Sizes 


1" 


\Mt" 


2" 


2^" 


3" 


33^^" 


4" 


4^" 


5" 


6" 


7" 


8" 


9" 


10" 


12" 


Sc. Soc 








123 


158 
205 


.... 


420 


















Fig 












1062 
173 












L. F., ex 


























































List No. 10. Brass Hor. C'ks: T. P. T^. = 300 lbs. ; R. T^. =150 lbs. ; R. S. =1001bs. 


Sizes 


H" 


^^ 


H" 


1" 


IM" 


1^" 


2" 


2W 


3" 


3^" 


4" 


5" 


6" 


7" 


8" 


9" 


10" 


Sc. Soc 




1 


VA 


li^ 


m 


3i^ 


5^ 


83^ 




















Fig 








































' 



















Horizontal 
Check 
Valves. 



List No. 11. 



List No. 12. 



6-in. and under: T. P. "M;^. = 300 lbs.; R. W.= 

150 lbs.; i?. 5. = 100 lbs. 
7-in. and above: T. P. W. = SOO lbs.; R. W.=- 

= 150 lbs.; i?. 5. = 85 lbs. 
20-in. and under: T. P. W.^SOO lbs.; R. W.= 

200 lbs. i?. 5. = 85 lbs. 
24-36-in..inc.: T. P. ■py. = 300 lbs.; i^. 1^. = 150 

lbs.; P. 5. = 75 lbs. 
40-in. and above: T. P. 1^. = 300 lbs;. R. W.= 

140 lbs.; P. 5. = 65 lbs. 



Sizes 








2" 


2^" 


3" 


33^" 


4" 


4^" 


5" 


6" 


7" 


8" 


9" 


10" 


1?" 






Sc. Soc 












45 

47 
55 


.... 


80 
92 
95 





120 
133 
150 


133 

150 
160 


*236 


225 
242 
245 






650 


Fig 












.... 


490 
483 

50" 


6^0 


Hub 












fiSO 
















Sizes 


14" 


15" 


16" 


18" 


20" 


22" 


24" 


28" 


30" 


36" 


40" 


42" 


44" 


48" 


60" 






Fig 


880 
885 


.... 


960 
990 


1200 


1800 
1880 


.... 


2700 
2710 




7250 
7370 
















Hub.. 


11360 





























WEIGHTS OF LUDLOW GATES AND VALVES. 



1279 



49. — ^Weights op Ludlow Gates and Valves, in Lbs. — Concluded. 
Vertical Check Valves. List No. 13. T. P. W= 300 lbs.; R. W. = 150 lbs. 



Sizes 


2" 


2^" 


3" 


V^A" 


4" 


4^" 


5" 


6" 


7" 


8" 


9" 


10" 


12" 


Sc. Soc 










58 






113 
152 




155 




280 

287 




Fie: . . . 




36 










415 


Hub 


















































Sizes 


14" 


15" 


16" 


18" 


20" 


24" 


28" 


30" 


36" 


40" 


42" 


44" 


48" 


Hg 


685 














7080 












Hub 




















































Vertical Foot ^ 


halves. List No. 14. T. P. W .-■ 


= 200 lbs.; JR. T^. = 100 1bs. 


Sizes 


2" 


2^" 


3" 


3^" 


4" 


41/^" 


5" 


6" 


7" 


8" 


9" 


10" 


12" 


Sc. Soc 










60 
58 










153 
160 
132 








Flff. . . 














105 




.... 


240 
275 


340 


Hub. 














410 
























Sizes 


14" 


15" 


16" 


18" 


20" 


24" 


28" 


30" 


36" 


40" 


42" 


44" 


48" 


Fig 


519 
562 




660 





875 






3350 












Hub 











































Flume Valves. / ^^'^^^ ^^^ ^^^^^- ^- ^' ^' = ^^ ^^^^ ^' ^' = ^^ '''^^' 
List No. 15. \ 3Q_.^^ ^^^ above: T. P. IF. = 30 lbs.; R. W. = 25 lbs. 



Sizes 


14" 


15" 


16" 


18" 


20" 


22" 


24" 


28" 


30" 


36" 


40" 


With Gearing. 


42" 


44" 


48" 


50" 


54" 


60" 


72" 


Fig 


435 




518 
492 


700 


735 
730 


.... 


1100 
1070 


.... 


1837 
1790 


2700 
2750 












8890 






Hub 




5900 


.... 


6255 








Sp. Gr., ex. . . 














Bv. Gr., ex. . 













































































Sluice Gates. List No. 21. 



Sizes 


10" 


12" 


14" 


15" 


16" 


18" 


20" 


24" 


28" 


30" 


36" 


40" 


42" 


44" 


48" 


Rem'ks 


P. F.. R. O. . 




215 


320 


340 


435 


630 




750 


1080 


1150 


1535 


2350 






2764 


. . 


P. F., S. O. . 






^o 


R.O. with 
Spig 
































h^ 


Pv. O. with 
F. & N 
































M°* 

-s 


E. F.. R. O. . . 




.... 




























s* 


E. F., S. O . 
































fl». 


S. O. with 
F. & N 
































c- 




1 




























<s 



1280 



■WATER WORKS. 



Blow-Offs are placed on the bottom of pipes at depressions in the pipe 
line for cleaning out or emptying the conduit. They consist essentially of 
a small pipe, say from 4 to 16 inches in diameter, leading to some suitable 
point where the waste can be discharged. A gate is inserted near the main 
pipe line. See Tables 16, 17, 38 and 39, preceding. 

Illg.— "SPECIALS." 

"Specials" are usually of cast iron and include bends, reducers, tees, 
wyes, and in fact many other shapes that enter also into the distributing 
system (see preceding and following tables). Fig. 56 shows the method of 
connecting wood-stave pipe with a cast special. The socket or bell of the 
casting is usually about 6 inches deep, with offset equal to thickness of 
staves. Fig. 57 is a front half view of special casting for connecting a small 






Fig. 56. Fig. 57, Fig. 58. Fig' 59. 

branch or a blow-off with a wood-stave pipe . Note the attached lugs (bosses) 
or shoes s for holding the ends of the steel bands which cinch around the 
opposite side of the pipe. These shoes are shown in side view by Fig. 58, 
and in the end view by Fig. 59, the band passing between the prongs. 



E.— DISTRIBUTING SYSTEM 



Cast Iron Pipe is by far the best that 
can be used, although the first cost is 
greater than that of lap-welded pipe. 
For sizes, details and weights of cast iron 
pipe, see tables under IIIc, page 1214 and 
following. 

It is to be noted especially that there 
should be no "dead ends" in a distributing 
system, i. e., cross connection should be 
made at terminal points, as at ends of 
streets. 

Fig. 60 is a Portable Lead=MeIting 

Furnace and ladle for pouring the joints. 
(See page 1215 for description and use of 
gasket.) 

Matheson Patent Lock Joint Pipe (steel 
lap-welded) is manufactured in lengths 
up to about 20 feet (average about 17 to 
18 feet over all, according to size) and is 
tested to 500 lbs. hydraulic pressure per 
square inch. 

Tables 50-54, on following page, give 
sizes and weights of pipe and specials; 
also, lead required per joint. 




Fig. 60. 



BLOW-OFFS. SPECIALS. MATHESON PIPE. 



1281 



50. — Matheson Pipe. 



=1 




o 


Approximate 


B6 


JS 


Weights. 


a 




m 






inal 
Dia 
pe. 


gS) 


Lead 


Com- 


B (upu 


^§2 


ap^ 


per 


plete 




gfflg 




Joint. 


per Foot 


Ins. 




Ins 


Lbs. 


Lbs. 


2 


13 


Ax ^ 


1.07 


1.93 


3 


12 


^^H 


1.77 


3.36 


4 


11 


lixH 


2.67 


4.94 


5 


10 


Hx'H 


3.50 


6.56 


6 


10 


Axl 


4.67 


8.38 


7 


9 


Axl 


5.62 


10.32 


8 


9 


%Xl 


6.90 


12.42 


9 


m 


%xl 


8.38 


14.74 


10 


8 


1^X1 


9.82 


17.26 


12 


7 


>^xiM 


13.20 


23.26 


13 


63^ 


^xlM 


15.30 


26.44 


14 


6 


%xlM 


17.20 


30.07 


15 


5H 


%xlM 


19.20 


33.81 


16 


5 


%xlM 


21.80 


37.92 


17 


4^ 


5^X1^ 


23.70 


42.45 


18 


3/^ 


%xiM 


25.60 


47.23 


19 


3 


%xlM 


28.80 


52.61 


20 


2^ 


MxiM 


31.10 


58.34 


22 


1 


^xl3^ 


40.20 


70.86 


24 


.330" 


Mxl^ 


48.10 


84.88 


26 


.362" 


Mxl3^ 


55.30 


100.69 


28 


.396" 


ixl^ 


64.70 


119.02 


30 


.432" 


ixl^ 


74.60 


138.85 



Furnished Asphalted only, and Kala- 
mein and Asphalted. 



51. — Tees. 



Size. 


Wt. 


Size. 




Wt. 


Ins. 


Lbs. 


Ins. 




Lbs. 


2x2x2 


11 


6x 6x 


4 


96 


3x3x3 


19 


6x 6x 


3 


93 


3x3x4 


35 


6x 4x 


4 


100 


4x4x4 


35 


6x 3x 


6 


90 


4x4x4 


39 


7x 7x 


7 


.... 


4x4x3 


35 


7x 7x 


6 


115 


4x4x3 


35 


8x 8x 


8 


159 


4x4x2 


37 


8x 8x 


6 


173 


4x4x2 


36 


8x 8x 


4 


172 


4x4x1 


34 


8x 6x 


8 


176 


4x4x6 


98 


9x 9x 


9 


.... 


4x3x4 


35 


lOx lOx 


10 


256 


5x5x5 


41 


lOx lOx 


8 


270 


5x5x4 


58 


lOx lOx 


6 


268 


5x5x4 


58 


lOx lOx 


4 


285 


5x3x5 


56 


llx llx 11 


353 


6x6x6 


91 


12x 12x 


12 


. . . . 



Heavy- faced figures indicate openings 
tapped for Standard Pipe. 




Fig. 61. — Matheson Joint. 
52. — Plugs. 



Size. 


Wt., 


Size. 


Wt., 


Size. 


Wt., 


Ins. 


Lbs. 


Ins. 


Lbs. 


Ins. 


Lbs. 


2 


1 


6 


7 


10 


23 


3 


2 


7 


13 


12 




4 


3 


8 


15 


14 


53 


5 


5 


9 




16 


88 



53. — Crosses. 



Size. 
Ins. 



2x 2x 
3x3x 
4x 4x 
4x 4x 
4x 4x 
4x 4x 
4x 4x 
4x3x 
5x 5x 
5x 5x 
5x 5x 
5x 4x 
6x6x 
6x6x 
6x 6x 
6x 4x 



2x2 
3x3 
4x4 
4x0 
3x3 
2x2 
2x2 
3x3 
5x5 
5x4 
4x4 
5x5 
6x6 
4x4 
4x3 
4x4 



Wt. 
Lbs. 



13 

28 

42 

43 

46 

45 

43 

45 

66 

69 

74 

72 

108 

117 

120 

127 



Size. 
Ins. 



6x4 

7x7 

7x7 

8x8 

8x8 

8x8 

8x8 

8x8 

8x6 

8x6 X 

8x6 X 

8x4 X 

9x 9 X 
lOx lOx 
lOx lUx 
]2x 12x 



3x 

7x 
6x 
8x 
8x 
8x 
4x 



X 

X 
X 
X 
X 

X 14x 16 

X 



«x b 

8x 4 

3x 3 

4x 4 

9x 9 

lOx 10 

lOx 8 

12x12 



Wt., 
Lbs. 



125 
135 
153 
200 
229 
230 
209 
1190 
220 
235 
238 
218 

*337 
339 



Heavy-faced figures indicate openings 
tapped for Standard Pipe. 



54. — Reducers. 



Size. 
Ins. 


Wt., 
Lbs. 


Size. 
Ins. 


Wt., 
Lbs. 


Size. 
Ins. 


Wt., 
Lbs 


3x2 
4x3 
4x3 

4x2 
5x5 
5x4 
5x3 
6x5 
6x4 


...... 

14 
12 
19 
17 

"22" 


6x4- 

6x3 

6x3 

7x6 

7x5 

8x7 

8x6 

8x4 


21 
"25** 

"39** 
43 


9x8 
9x7 
9x 6 
10x9 
10x8 
10x6 
10x4 
12x10 


"56 
46 
52 

75 



Heavy-faced figures indicate openings 
tapped for Standard Pipe. 



1282 



6i,— WATER WORKS. 



Converse Patent Lock Joint Pipe (steel lap-welded) is manufactured in 
average lengths of about 18 ft. and tested to 500 lbs. per square inch. A 
hub is leaded to each length of pipe at mill to receive the spigot end of the 
adjoining pipe when laid. 



55. — Converse Pipe, 



Size. 


Approxima 


te Weig 


ht. 


^1 








1 




Nominal O 
side Diame 
of Pipe. 


-J 




Hub. 


p 

SI 


P 

6S 


Ins. 


In. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


2 


.094 


1.91 


5. 


1.00 


2.24 


3 


.108 


3.33 


8.50 


2.15 


3.92 


4 


.118 


4.89 


12.50 


2.75 


5 73 


5 


.125 


6.51 


19. 


3.00 


7.72 


6 


.132 


8.27 


21. 


3.75 


9.64 


7 


.139 


10.20 


32. 


5.50 


12.26 


8 


.146 


12.25 


37. 


6.50 


14.66 


9 


.154 


14.55 


37.50 


7.50 


17.03 


10 


.162 


17.02 


41. 


7.75 


19.72 


11 


.171 


19.78 


50. 


8.60 


23.00 


12 


.181 


22.85 


58. 


9.50 


26.59 


13 


.190 


26.00 


65. 


10.75 


30.21 


14 


.200 


29.48 


73. 


12. 


34.16 


15 


.210 


33.18 


85. 


15. 


38.68 


16 


.221 


37.85 


132. 


17.5 


45.4? 


17 


.233 


41.73 


140. 


23.75 


51.38 


18 


.245 


46.46 


149. 


30. 


56.31 


19 


.258 


51.65 


183. 


34. 


63.65 


20 


.272 


57.32 


217. 


38. 


71.35 


22 


.300 


69.53 


280. 


50. 


87.68 


24 


.330 


83.43 


342. 


58.5 


105.46 


26 


.362 


99.13 


380. 


70. 


123.88 


28 


.396 


116.76 


430. 


85. 


145.09 


30 


.432 


136.44 


475. 


100. 


168.07 




SB^H ^^^^^i^^" 



Fig. 62. — Converse Joint. 

(Cast Hub.) 

For details of specials see National 
Tube Works catalog. - 



Furnished Asphalted Only, and Kalameln 
and Asphalted. 



Pipe-Dipping Tank. — ^The writer has found it convenient in some cases 
to order the lap-welded pipe from the manufacturer uncoated, and to coat 
it in the field before laying. For this purpose a dipping tank is used as shown 
in Fig. 63. The tank should be about 20 ft. long, 2 ft. wide and 2 feet deep. 
Such a tank constructed of No. 12 gauge steel will weight about 700 lbs. 
It is set over an improvised brick furnace, cheaply constructed and pro- 
vided with a smoke stack. Hard and liquid asphaltum are mixed in it in 




Fig. 63. — Pipe-Dipping Tank. 

the proper proportion and the pipe is dipped when it has reached the proper 
temperature. Approximately, the number of pounds of asphalt required 
per hundred feet of pipe is equal to 5.5 X diameter of pipe in inches. 



CONVERSE PIPE. TANK. TAPPING MACHINE 



1283 



Tapping Machines are employed for tapping mains for service connec- 
iions. There are many styles, more or less expensive. The Mueller tap- 




Fig. 64. — Mueller Tapping Machine. 

ping machine is illustrated in Fig. 64. For water mains, the machine com- 
plete includes: 

1 each, Combined Drill and Tap — }4, %, % and 1 inch. 

1 each, Screw or Hexagon, Plug — 3^, ^, 5^ and 1 inch. 

4 Malleable Iron Saddles; any size. 

1 Chain for any size of Pipe. 



1284 



Qi.—WATER WORKS. 



Black or Galvanized Pipe of "standard'* weight as manufactured by the 
National Tube Co. is shown in the following Table. For the "extra strong" 
pipe the 1-in. size is 0.182 in. thick, and 12-in. size 0.500 in.; while the "double 
extra" is 0.364 in. and 0.875 in. thick, respectively. 

56. — Dimensions, IN Inches, and Weights of Black Pipe (Standard) 



Diameter. 


1 


Transverse Areas. 


Nom. 
Wt. per 

Foot. 
Pounds 


11 


Wgt. of 
Coup- 
ling, 

Pounds 


Nom. 


Exter'l. 


Inter'l 


Extern '1 


Internal. 


Metal. 


% 


0.405 


0.269 


0.068 


0.1288 


0.0568 


0.0720 


0.241 


27 


031 


H 


0.540 


0.364 


0.088 


0.2290 


0.1041 


0.1249 


0.42 


18 


0.046 




0.675 


0.493 


0.091 


0.3578 


0.1909 


0.1669 


0.559 


18 


0.078 


^ 


0.840 


0.622 


0.109 


0.5542 


0.3039 


0.2503 


0.837 


14 


0.124 


H 


1.050 


0.824 


0.113 


0.8659 


0.5333 


0.3326 


1.115 


14 


0.250 


1 


1.315 


1.047 


0.134 


1.3581 


0.8609 


0.4972 


1.668 


11^ 


0.456 


m 


1.660 


1.380 


0.140 


2.1642 


1.4957 


0.6685 


2.244 


11^ 


0.562 


m 


1.900 


1.610 


0.145 


2.8353 


2.0358 


0.7995 


2.678 


11^ 


0.800 


2 


2.375 


2.067 


0.154 


4.4301 


3.3556 


1.074 


3.609 


11^ 


1.250 


2^, 


2.875 


2.467 


0.204 


6.4918 


4.7800 


1.712 


5.739 


8 


1.757 


3 


3.500 


3.066 


0.217 


9.6211 


7.3827 


2.238 


7.536 


8 


2.625 


3^, 


4.000 


3.548 


0.226 


12.566 


9.886 


2.680 


9.001 


8 


4.000 


4 


4.500 


4.026 


0.237 


15.904 


12.730 


3.174 


10.665 


8 


4.125 


4^. 


5.000 


4.508 


0.246 


19.635 


15.960 


3.675 


12.34 


8 


4.875 


5 


5.563 


5.045 


0.259 


24.306 


19.985 


4.321 


14.502 


8 


8.437 


6 


6.625 


6.065 


0.280 


34.472 


28.886 


5.586 


18.762 


8 


10.625 


7 


7.625 


7.023 


0.301 


45.664 


38.743 


6.921 


23.271 


8 


11.270 


8 


8.625 


7.981 


0.322 


58.426 


50.021 


8.405 


28.177 


8 


15.150 


9 


9.625 


8.937 


0.344 


72.760 , 


62.722 


10.04 


33.701 


8 


17.820 


10 


10.750 


10.018 


0.366 


90.763 


78.822 


11.94 


40.065 


8 


27.700 


11 


11.750 


11.000 


0.375 


108.43 


95.034 


13.40 


45.95 


8 


33.250 


12 


12.750 


12.000 


0.375 


127.68 


113.09 


14.59 


48.985 


8 


43.187 



57. — Discharge in Gallons per Minute Through Small Pipes. 

(Values are approximate.) 
H =head in ft. ; L = length of pipe in ft. ; numbers in first column are 

values of H-^L. 
[Gallons per Minute.] 



Head 










Diameter of Pipe, 


In Inches. 








H 


% 


u 


1 


IH 


IH 


2 


2^ 


3 


4 


5 


6 


0. lOOZ, 


2.0 


3.5 


5.4 


11.2 


19.5 


30.8 


63.2 


110.4 


174.5 


358.1 


624.7 


985 5 


O.IUI; 


2.1 


3.6 


5.7 


11.8 


20.6 


32.5 


66.6 


116.4 


183.9 


377.5 


658.5 


1039. 


0.1251. 


2.2 


3.9 


6.1 


12.5 


21.8 


34.4 


70.7 


123.5 


195.1 


399.0 


698.5 


1102. 


0.1431/ 


2.4 


4.1 


6.5 


13.4 


23.3 


36.8 


75.6 


132.0 


208.5 


428.0 


746.7 


1178. 


0.1671/ 


2.6 


4.4 


7.0 


14.4 


25.2 


39.8 


81.6 


142.6 


225.2 


463.2 


806.5 


1272. 


0.200L 


2.8 


4.8 


7.7 


15.8 


27.6 


43.9 


89.4 


156.2 


246.7 


506.5 


883.5 


1384. 


0.2501/ 


3.1 


5.5 


8.6 


17.7 


30.9 


48.7 


100.0 


174.6 


275.8 


566.2 


987.7 


1558. 


0.3331. 


3.6 


6.8 


9.9 


20.4 


35.6 


56.2 


115.4 


201.6 


317.8 


653.8 


1141, 


1799. 


0.5001/ 


4.4 


7.7 


12.2 


25.0 


43.7 


68.7 


141.4 


246.9 


390.1 


800.8 


1394. 


2204. 


0.7501/ 


5.4 


9.5 


14.9 


80.6 


53.5 


84.3 


173.1 


302.4 


477.1 


979.3 


1711. 


2693. 


L 


6.3 


10.9 


17.2 


35.3 


61.7 


97.4 


199.9 


349.2 


555.5 


1133. 


1976. 


3116. 


1.251/ 


7.0 


12.2 


19.3 


39.5 


69.0 


108.9 


223.5 


390.4 


615.9 


1264. 


2209. 


3484. 


1.501/ 


7.7 


13.4 


21.1 


43.2 


75.6 


119.3 


248.8 


427.7 


674.8 


1385. 


2420. 


3817. 


1.751/ 


8.3 


14.4 


22.8 


46.8 


81.6 


128.8 


264.4 


462.0 


728.8 


1496. 


2613. 


4122. 


21/ 


8.8 


15.4 


24.3 


50.0 


87.3 


137.7 


282.7 


493.0 


780.2 


1602. 


2791. 


4407. 


81/ 


10.8 


18.9 


29.8 


61.2 


106.9 


168.7 


346.3 


604.9 


955.5 


1962. 


3406. 


5391. 


iL 


12.5 


21.8 


34.4 


70.7 


123.4 


194.8 


399.8 


698.5 


1103. 


2265. 


3951. 


6233. 


bL 


14.0 


24.4 


38.5 


79.0 


138.0 


217.7 


447.0 


780.9 


1234. 


2532. 


4417. 


6968. 


6L 


15.3 


26.5 


42.2 


86.6 


151.2 


238.5 


488.1 


855.4 


1351. 


2774. 


4839. 


7633. 


71/ 


16.5 


28.9 


45.6 


93.5 


163.3 


257 6 


528.9 


924.0 


1460. 


2996. 


5227. 


8245. 


81/ 


17.7 


30.1 


48.7 


100.0 


174.6 


275.4 


566.4 


987.8 


1560. 


3203. 


5588. 


8814. 


91/ 


18.7 


32.7 


51.7 


106.0 


185.2 


292.1 


599.7 


1048. 


1651. 


3397 


5928. 


9349. 


101/ 


19.8 


34.5 


54.4 


1 111.8 


195.2 


308.0 


632.2 


1104. 


1745. 


3581. 


6247. 


9855. 



Ex. — What is the capacity of a pipe 134 in. dia., and 150 ft. long, the head 

50 
being 50 ft.? Solution. — //= rT^L = .333L; hence, for a IMin. pipe the 

discharge is 35.6 gallons per min. 



BLACK OR GALVANIZED PIPE, GATE VALVES. 



1286 



Gate Valves should be of the best quality. 

Fig. 65 shows a section of standard bronze mounted babbitt seat valve, 
up to 15 ins., as manufactured by the Chapman Valve Co. of Indian 
Orchard, Mass. 

Fig. 66 shows a section of the Eddy gate valve. 

Notation of Parts: — 
A— Stem Nut. G— Body. 

B — Stem . H — Body and Coyer Bolts, 

C — Follower. I — Ball and Carrier. 

D— Follower Bolts. J— Gate. 
E— Stuffing Box. K— Gate Ring. 
F — Cover. L — Case Ring. 





Fig. 65. — Chapman. 



Fig. 66.— Eddy. 



1286 



Qi.— WATER WORKS. 







LUDLOW GATE VALVES, SMALLER SIZES, 



1287 



h 



^:^:t^ ^ ^ 



l-l\riJSi4\r4\?-(S 

-* lo T-i 00 lo T-H eq 



•73 8: 



I 1-1 lO OO CO -^ CO lO -xj* -^ CO <0 CO 
■^ ^ CO CO CO »-< 1-H ,-(CO»-4 



_ __ i-i\i-Ni-Ki-Nr<\ 

Ti< »o i-H CO lo T-H ca 






^^;^is ;:^ ;:2eH:s ;^;:^:^ 



I CO coco ^ 1-1 i-<e^ 



ocq-^iotoost^evj 
■^io i-ico •* oa 






\ T-i e<i -^ <o 1-M o> -^ e^a •«*< CO ST) ^ 



i-H CO CO CO r-l ■ 



OS t^ cq 



T i-H e^q o «o r:< OS km T-H -rj< CO 00 o 



ico CO eg i-< -M i-( e^a 1-1 



00 00-^ 
CO -^ 1-1 



00'<»<eQ 



m:^::1s :^:^ :^ ;;§^ ;^ 



ICQ eviC>3 1-1 11 



CO "^ T-H CO CO CQ 



1-1 OCOlOlO-'i'COCO'^IMt-OO 

o i-ioa cacQi-i r-i eq 



;:^::^ ::^::^;^;^^ 



^Hs:^;^ :^ ::^;;i^^^:^ 



OiCO«MCOOJ«OOi-( 

ca CO ^ c^a e^a e^q 






»0005Cqci)CO-Hi-(«, 
eva ^ ^ n oa 



OO-^O • -lO OOi-i 

CNJCO >H . . 1-1 



;:^ 



1-1 OS lO »« OS oa 1-1 «3 00 OS cq lO 
ob ^ ^ ^ ^ oq 



I ^H t>. -^ Ti< CO CO OS A eg t^ — < Tj* 
<o ^ ^ ^ c>q 



i:^^:^;:^^ 



::^ 



I r-i t>.«>qi!t*iOiOOS100qt-CD'«** 



OS-«4< C 



I «0 CM -^ -^ T-< 00 "*< eg XO <0 CO 

CO ^ ^ ^ OCI 



C0C^«010CT)00Oi-l 



;^ 



I »o e<i . ^H i~t t^ oo c^ lo in CO 



i:^ 



CO l>- CO -<J< t^ CO OS 



*f, 



HUSsoSSi-Ni-iS 



ooost~cooq-^oooq 



e^ci lo CO ^ CO CO CO 



^ 



00 00 QO e<i e»a -^ 



Od CO 



:^ 



coo 



;;i^:^;^^ 



oo CO • c- 



W fe H OOZ^'Obi, 



Ph h3 M l-^ M ^ ^ .M 



2-S 






O O fl i^'O 









o O p o P 

rt rt i? "3 .«? 






O bfi O O ^ 

^ tH Ul O 

gOPHfl.'g 

lira's 



p 



a 



O M >^ t> 02 c3 i-:i > 



• P • 4) P 

• ft : o a 

:o^-oo 

I P (U p p 
O) jp O) O) 



III 



ft 
O 

OPl( 



p 



alaai:;^ 

(D o3 <D o ^3 7; 
+j hH +3 +3 ro o 
M W 02 02 ^-^ -O Sis 
<^^ kJ «M .w 2 P ^ 

opoo5iBf=< 



o o o o 



O Q; (H O 

^ c3 P 

p ■— QJ 

woo 



S 



CU -^ <U OJ <U (U j_ 



1288 



M.— WATER WORKS. 



59. — Weights of Ludlow Gate- or Valve Boxes. 
[Weight in Lbs.] 



Size of Valve 


2" 


Z" 


4" 


5" 


6" 


8" 


10" 


12" 


14" 


16" 


18" 


20" 


24" 


30' 


36" 


2% ft. Trench 






















. . . , 










3 " " 




80 
85 
105 
110 
112 
121 
126 


80 
85 
105 
110 
112 
121 
126 


.... 


71 
80 
100 
105 
110 
119 
121 


65 

80 

80 

95 

105 

113 

119 


56 
71 
80 
90 
95 
103 
113 


















ZU " 




56 
65 
80 
80 
95 
103 




' 












4 " " 


















iU. " 




80 
80 
85 
95 


60 
80 
80 
85 


.... 


56 
56 
74 
80 








5 " •• 










514" " 










6 " " : 










6^ •• " 










n ** «< 




. . . . 





























































Gate Boxes, Fig. 68, are telescopic casings of cast iron to place over th^ 
gate valves and protect them from surrounding earth or 
back-fill; and render them accessible to operate from the 
street in opening and closing. The cover is a circular cast 
plug which fits in the bell of the top casting or telescope 
portion of the box. The telescope is provided with a flange, 
placed at the middle or at the lower end to hold it at the 
proper elevation when placed. The main box is enlarged 
at the bottom^ to fit over the body of the gate. Caution 
rnust be used in designing gate boxes as the outside dimen- 
sions of gates by different makers vary considerably. 
Thus, a gate box which would fit a 12-in. gate manufac- 
tured by A, would perhaps be too small for the "same size" 
gate manufactured by B. See Table 59 for weights. 

Check Valves, Fig. 69, are placed in pipe lines or mains 
(either in a vertical, inclined or horizontal position) to 
prevent back flow or excessive back pressure. They are 
usually located at or near the pumping station. 

Pressure Relief Valves are specially designed to relieve 
excessive pressure within a main due to water hammer. 
They allow some of the water to escape, and operate auto- 
matically. 

Hydrants or "fire plugs" are designated by the size 
of valve opening; by the number of nozzles, as single-, 
double,- etc.; and by the kind of discharge nozzles, as 
steamer-, or hose-. The 4, 4^, 5 and 6-inch (valve opening) Fig. 68. 
fire hydrants are the most common. The essential principal of a hydrant is 
that the valve be placed below the reach of frost, and that when it is closed 





Fig. 69.— Check Valve, 
the water remaining above it shall be allowed to "drip" out to prevent its 
freezing. For this reason the hydrants should be set on a bed of firm, loose 
rock to provide for the drip. Figs. 70 and 71 show sections of the Chapman 
and Ludlow types, respectively. Figs. 72 and 73 show sections of valves 
and seats for the Mathews' patent hydrant, the former being single-acting 
and the latter double-acting. 



GATE BOXES, VALVES, HYDRANTS. 



1289 



Hydrants are connected with transverse pipes leading from the street 
mams, and hence it will be seen that there is considerable hydrostatic pres- 
sure actmg horizontally against them at the bottom. This should be 
resisted by a strong stone backing when the hydrant is set. 





Fig. 70. 



A— Hydrant Barrel or 
stand Pipe. 

B— Bottom. 

C— Stem, 

D— Top Nut. 

E — Bronze Sleeve. 

F— Stuffirig Box Nut. 

G— Stuffing Box. 

H— Follower. 

I — Dome. 

J— Dome Bolts. 

K — Bronze Nozzle. 

L — Nozzle Cap. 

M— Upper Valve i 

Plate. ' 

N— Valve. ; 

— Lower Valve 
Plate. 

P — Valve Rubbers. 

Q — Bronze Seats, 

R — Bronze Nut. 

S — Bronze Lock Nut. 

T — Bronze Corrugated 
Drip Piece. j 

V— Drip Rubber. ' 

X— Bronze Drip Nut, 

Y — Bronze Drip Cup- 

Z— Flange Bolts. 



IQ 



Fig. 71. 




A— BrQti'zia Rod Ntit 

B— Brouze Tap Bplts 

C— Bronze PacHng: Nut 

D— Bronze Stuffing^-bo:^ 

'E— Iron Post Cap ' 

H -Bronze Hose Nozzle 

I —Bronze Steamer Nozzle 

J— Hydrant Post 

K— Standpipe or Barrel 

L— Hydrant Body 

M -Iron Extension Rod 

N —Bronze Spindle 

0— Gate or Plug 

P — Apron 

0— Frost Case 

R— Bronze Face on Plug 

S —Babbitt or Brbnze Seat 

T —Bronze Bushing in Plug 

U— Babbitted WedgingSurf ace 

V— Rib or Spline 

W— Drip Hole 



Pig. 72. 




Fig. 73 



1290 Qi.— WATER WORKS. 

60. — Weight of Ludlow Hydrants, in Lbs. 



Yard Hydrants. "1 Test Pressure 
List No. 25. J Water = 100 lbs. 




Wash Hydrants. \ Test Pressure 
List No. 25^. } Water = 100 lbs. 




Sizes 


H" 


H" 


1* 


Sizes 


H" 


%" 


1* 








5-f t. Length 


45 
3 


55 
5 


75 
6 


6-ft. Length 




40 
5 


50 


6-in. Length, extra 


6-in. Length, extra 




6 











Fire Hydrants. List No. 75. 


Test Pressure Water = 300 lbs. 














Frost 




Stand 


Seat 


Pipe (Con- 


Number and 


Wt. Standard 


Weight 


Case for 


Weight 


Pipe. 


Ring. 


nection. 


Size of 
Nozzles. 


Length 
5 Feet. 


per Ft. 


5-Ft. 
Length. 


per Ft. 


Ins. 


Ins. 


Ins. 


Ins. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


3 


2 


2 or 3 


one 2 


130 


12 






4^ 


3 


3 or 4 


one 2^ 


310 to 315 


26 


84 


21 


5M 


4 


3 or 4 


one 2^ 


365 to 370 


30 


96 


24 


5M 


4 


4 or 6 


two 2^ 


37 5 to 380 


30 


96 


24 


6M 


4^ 


5 or 6 


two 23^ 


437 to 442 


34 


106 


26 


Q% 


4^ 


5 or 6 


three 2^ 


448 to 453 


34 


106 


26 


7 


5 


5 or 6 


/one steamer "1 
land two 29^ J 


475 to 486 


36 


116 


29 


8 


6 


6 or 8 


Jone steamer 1 
land two 23^ / 


590 to 600 


40 


144 


36 


10 


8 


8 or 10 


six 2^ 


1200 to 1220 


73 


219 






Notes on List No. 75: Add for a 6-in. Hub or Bell over Hydrant with 
4-in. Hub or Bell, 16 lbs.; add for each additional 2^-in. Nozzle, including 
Cap and Chain, 11 lbs.; add for each additional 4-in. Steamer Nozzle, includ- 
ing Cap and Chain, 25 lbs. The above weights are based on a length of 5 feet 
measuring from the surface of ground to bottom of connecting pipe. 

Hydrants with Crane Attachment. List No. 76. Test Pressure Water= 300 lbs. 
Extra to add to weight of Hydrant: 300 lbs. for 3 In.; 310 lbs. for 4-In. 



Water Cranes. List No. 77. Test Pressure Water= 300 lbs. 




3-in. Length, 5 feet, 587 lbs. Each 6-ins. Stand Pipe and Box, 


13 lbs. 


4-in. Length, 5 feet, 600 lbs. Each 6-ins. Stand Pipe and Box. 


13 lbs. 


Flush Hydrants. List No. 78. Test Pressure Water= 300 lbs. 




Seat 


Stand 


Pipe 


Nozzles. 


Weight 5-ft. 


Weight 


Ring. 


Pipe. 


Connection. 




Length. 


per Ft. 


Ins. 


, Ins. 


Ins. 


Ins. 


Lbs. 


Lbs. 


2 


3 


2 


one 2 


90 


12 


3 




3 or 4 


one 23^ 


290 


26 


4 


m 


4 or 6 


one 2J^ 


360 


30 












1 



HYDRANTS. MISCELLANEOUS DATA, 1291 

EXCERPTS AND REFERENCES. 

The New Water-Works Reservoir at Trenton, N. J. (By C. A. Hague. 
Eng. News, June 13, 1901). — Six illustrations: Section of reservoir, cross- 
section of wall and embankment; view of gate house; plans and sections of 
outlet pipe. 

Iron or Wooden Lock Gates ; Cost of Construction and Maintenance 
(Eng. News, Oct. 9, 1902). 

A Stress Diagram for Water-Tank Hoops (By Ballinger & Perrot, 
Eng. News, Mar. 5, 1903). 

Four Systems of Softening Water for Industrial Purposes (Eng. News, 
July 2, 1903). — Described and illustrated. 

A Proposed HigIi»Pressure Water=Supply System for Fire Protection 
in Chicago (Eng. News, Mar. 3, 1904). — Illustrations: Main conduit; details 
of steel and cast-iron pipe ; details of fire-boat connection ; section of main 
conduit lateral; maps of high-pressure water supply systems for fire pro- 
tection in different cities. 

Experience in Thawing Water Pipes By Electricity (Eng. News, 
Mar. 17, 1904). 

The Five Dams and Wood=Stave Conduit of the Southern California 
Mountain Water Co. (Eng. News, April 7, 1904). — Fifteen illustrations; and 
13 tables, including cost data. 

The Philadelphia Filtration System (Eng. News, Dec. 8. 1904).— 

Numerous illustrations. 

Cost of Laying a 12-in. Water Pipe Across a River (Eng. News, 
Mar. 2, 1905). — The labor cost of building. 44 A-frames, placing and caulking 
the 516-ft. pipe line, was $122. 

A Diagram for Estimating the Yardage of a Trench (Eng. News, 
July 6, 1905). — Quantities in cu. yds. for various widths in ft., up to 20 ft., 
and various depths in ft., up to 30 ft. 

Purification of Water by Copper Sulphate (By D. D. Jackson. . Eng. 
News, Sept: 21, 1905). — For other papers and discussions, see Eng. News 
of Nov. 30, 1905. 

The Baiseleys, Springfield, Forest Stream, and Hempstead, Filter 
Plants, Borough of Brooklyn, New York (Eng. News, Aug. 23, 1906).— Table 

showing the net amount of water filtered at two mechanical and at two 
slow sand filter plants, during the year 1905; following totals are repro- 
duced: 

Baiseleys (mechanical) 1,435.5 million gals. @ $6.53 p. m. g. 

Springfield (mechanical) 694.6 ' 9.58 

Hempstead (slow sand) 416.8 " " " 2.89 

Forest Stream (slow sand) 1.075.3 " " *' 2.28 

The Cost of Clearing and Grubbing a Reservoir Site (By J. Griggs. 
Paper, A. S of Mun. Imp., Oct., 1906; Eng. News, Nov. 29, 1906). 

Disjointing Cast-iron Water Mains (Eng. News, Oct. 10, 1907).— 

Methods used. 

Repairing a Remarkable Leak in a Reservoir Embankment at Provi- 
dence, R. I. (Eng. News, Oct. 17, 1907.)— Described and illustrated. 

Diagrams for Computing Thickness of Steel Pipe Shells for Different 
Joint Efficiencies (By R. Muller. Eng. News, April 2, 1908).— The dia- 
grams are based on the following assumptions: Tensile strength of plate, 
50 000 lbs per sq. in of section; factor of safety, 4. Percentage of effi- 
ciency of riveted joints, assumed at: single riveted joint, 56%; double- 
riveted joint, 69%; triple riveted-joint, 75%; double-weld butt joint, 87%; 
quadruple-riveted joint, 95%. 

Method and Cost of Hauling a Water Main Across Channel at Van- 
couver, B. C. (Eng. News, May 14, 1909).— Illustrated: method, and section 
through flexible-joint. 

Data on Silting Up of Reservoirs (By R. H. Bolster. Eng. News, 

Repairs to a 72-in. Reinforced-Concrete Jacketed Steel Conduit 
Under 30-Ft. of Water (By A. W. Cuddeback. Paper, A. W. W. Assn., 
May, 1908; Eng. News, Aug. 6, 1908).— Illustrated. 



1292 ^.^WATER WORKS, 

The Design of Elevated Tanks and Stand-Pipes (By C. W. Birch-Nord. 
Trans. A. S. C. E., Vol. LXIV., Sept., 1909).— Specifications. 

Watsr-Works Valuation (By Leonard Metcalf. Trans, A. S. C. E., Vol. 
LXIV., Sept., 1909). — Numerous bond-, compound-interest-, and sinking- 
fund formulas; a bibliography of water-works valuation; various kinds of 
"values" discussed. 

Cost of Clearing Water in Settling Basins (By S. Bent Russell. Paper, 
Central States W. W. Assn., Columbus, O., Sept. 28, 1909. Eng. News. 
Oct. 14, 1909). — Tables of cost of reservoirs and settling water. 

The Purification of the Water Supply of Steelton, Pa. (By J. H. Fuertes. 
Trans. A. S. C. E., Vol. LXVI., Mar., 1910).— Cost data; illustrations; 
discussions. 

The Use of Sulphate of Alumina and Hypochlorite of Lime in the Storage 
and Distributing Reservoir of the Nashville Water- Works (By George Reyer. 
Eng. News, Apr. 7, 1910). — ^The cost of sulphate of alumina (the ordinary 
is used) is $1.07^ per 100 lbs. and of the hypochlorite of lime (it contains 
about 36% of chlorine) is $1.50 per 100 lbs. The cost per 1,000,000 gals, of 
water treated is about $1.75 for sulphate of alumina and $1.05 for cost of 
hypochlorite of lime, making the combined cost of chemicals $2.80. The 
water comsumption for the year 1909 averaged about 14,000,000 gals., or 
some 107 gals, per capita. 

This article is followed by five others, namely: A 20,000,000-Gal. Hypo- 
chlorite Water-Disinfecting Plant at Minneapolis, Minn. (By J. A. Jensen). 
The Use of Hypochlorite of Lime to Disinfect the Water-Supply of Montreal, 
P. Q. (Geo. Janin, C. E.). The Use of Hypochlorite of Lime in Connection 
with the Mechanical-Filtration Plant of Harrisburg, Pa. (G. C. Kennedy, 
Supt.). Hypochlorite of Lime as an Adjunct to Mechanical Water Filtra- 
tion at Quincy, 111. (By W. R. Gelston. Paper, Bl. Water Supply Assn., 
Mar. 8, 9, 1910). — Experiments with Hypochlorite of Lime as a Water 
Disinfectant at Hartford, Conn. (By Ermon M. Peck, Engr.). 

The Groined Arch in Filter and Covered Reservoir Construction (By 
Thomas H. Wiggin. Paper,' Nat'l Assn. Cement Users, Feb. 21-26, 1910; 
Eng. News, April 7, 1910). — Discussions of: Volumes of elliptical groined 
arch units; Methods of design; Computation of groined arch as cantilever; 
Effect of steel in groin arch; Shrinkage and temperature changes; Aid given 
by earth covering; Construction stresses; Floors of filters and reservoirs; 
Side walls and division walls; Comparison of groined arch roof with rein- 
forced -concrete slab and beam construction. Illustrations. Table con- 
taining data on groined arch roofs for filters and reservoirs. 

The Improved Water and Sewerage Works of Columbus, O. (By J. H. 
Gregory. Trans. A. S. C. E., Vol. LXVIL, June, 1910).— Illustrations; 
Scioto river storage dam; pumping station; offices and laboratories; settling 
basins; details of filter gallery; details of filters; details of filtered -water 
reservoirs, etc. 

Determination of the Resultant Angle in Laying Out Combined Bends 
for Pipe Lines (By C. A. Jackson. Eng. News, Aug. 11, 1910). — Graphical 
and analytical methods. Example given. 

Concrete Tower Enclosing a Water-Works Tank at Gary, Ind. (Eng. 
News, Oct. 20, 1910). — ^Tower is octagonal in plan, 34 ft. diam., inside the 
faces, and 133 ft. high from the grade line. Fully described and illustrated. 

Steel Pipes for Water-Works (By Emil Kuichling. Paper, Jour. Am. 
W. W. Assn., presented Sept. 21-23, 1910; Eng. News, Oct. 27, 1910).— 
Discussion of wrought-iron and of steel water mains and protective coatings, 
strength of pipes, vacuum relief valves, capacity of pipe. Cost of steel and 
cast-'iron pipe. — "The fact that lock -bar and welded pipe can develop the 
full strength of the plate, while a riveted seam has only about 70% efficiency, 
influences the comparative costs. This difference is likely to be made up by 
a countervailing difference in unit-prices, however. Empirical formulas 
for cost of steel and cast-iron pipe have been devised by the author (Mr. 
Kuichling) as below. They assume (1) a water-pressure of 100 lbs., (2) 
an equivalent pressure, used in design, 50% greater than the static pressure 
to allow for water hammer and (3) an addendum of i" thickness for cast- 
iron pipe and 1-16" for steel pipe. Taking d as the nominal dia. of pipe, in 
ins., the cost in cents per lin. ft. is: Cost of steel pipe, 0Al25d'^; cost of cast- 
iron pipe, 19-l-d (0.4845£i— 0.851). Subtracting the latter from the former 
^ives the saving due to using steel: Steel pipe cheaper by 19+(i (0.072d— 



MISCELLANEOUS DATA. 1293 

0.851), which in case of a 36-in. pipe, for example, amounts to 81.7 cts. per 
ft. For a complete comparison, however, the shorter life of the steel pipe 
must be taken into account." 

Depreciation in Water-Works Operation and Accounting (By Leonard 
Metcalf. Paper read before the N. E. W. W. Assn., Sept. 22, 1910; Eng. 
News, Nov. 3, 1910). — Contains sinking-fund and depreciation diagrams 
and tables. . 

Pneumatic Caulking of Mains with Lead Wool (By C. C. Simpson, Jr. 
Paper read before the Am. Gas Institute; Eng. Rec, Nov. 12, 1910). — ^The 
Cons. Gas Co. of New York, in an effort to reduce the cost of caulking mains 
by hand, carried on extensive tests with pneumatic tools and lead wool, 
and the results were so satisfactory that the pneumatic method was finally 
adopted on a considerable portion of its work, which includes lines of 48, 36 
and 30-in. mains. The article gives detailed costs. 

Forms for Concrete (By J. D. Stevenson. Paper before Eng'rs Soc. of 
Western Penn., Sept. 20, 1910; Eng. Rec, Dec. 10, 1910). — Discussion of 
experience of three years in construction of the Pittsburg filtration works 
(placing about 334,000 cu. yds. of concrete) under the following heads:— 
Material, design of forms, support of forms, buoyancy of forms, joints, care 
while filling, and removing forms. For labor costs on forms, see Eng. Rec. 
of Dec. 17. 1910. 

Waterproofing the New Ulm (Minn.) Reservoir (By H. F. Blomguist. 
Eng. Rec, Dec 17, 1910). — Reservoir is 75 ft. in diam., 30 ft. deep, capacity 
1,000,000 gals., built of reinforced concrete, with a conical reinforced 
concrete roof. Hard clay, well drained naturally, upon which a 12-in. 
layer of stone having the voids filled with a wet fine-grained concrete formed 
the, foundation for the 10-in. floor slab proper, which was reinforced near 
the surface with 16-gauge, 3-in. mesh expanded metal. To instire water 
tightness, special care was taken during construction to grade the concrete 
aggregate. Pebbles varying in size from \ to 2^ in., screened from a gravel 
bank, were used in the upper floor and walls, as experiments had shown 
that these pebbles made a denser concrete than broken stone. To reduce 
the permeability of the concrete to a minimum 20 lbs. of hydrated lime was 
used to every barrel of cement, and after the forms were removed the walls 
were brushed and cleaned with steel brushes, and two coats of 1 : 2 cement 
plastering about i in. thick were applied. The mortar for the plastering 
contained the waterproofing compound to the extent of 10% of the cement 
used. A slush coat of 1 : 2 cement mortar was floated over the surface of 
the concrete floor at the time it was laid and a brush coat of cement grout 
applied to the slushed surface after it had set. After water was let in, some 
leakage took place and cracks developed. The reservoir then received 
further treatment, and was rendered watertight. 

Illustrations of Important Works: — 

Description. Eng. News. 

Intake caisson for Cincinnati Water works, showing break May 30, 1901. 
Intake tunnel of Cleveland water works, showing accident Aug. 29, '01. 

Water tank at Fairhaven, Mass., showing failure Nov. 21, '01. 

An 8" gate valve for River Rogue, Michigan April 17, '02. 

Concrete-lined reservoir with concave slopes, Aurora, 111. May 22, '02. 

Slow sand filters of Hudson, N. Y., water works Aug. 14, '02. 

Cross-section of a fire hydrant with a balanced valve Nov. 20,* 02. 

Water tank with hemispherical bottom, Chicago Dec. 25, '02. 

Plan of Klein's classifier for fine material Jan. 8, '03. 

Intake tunnel for the Champion Mill on Lake Superior Oct. 1. '03. 

Plan of scow for submarine pipe-laying Dec. 24, '03. 

New type of water tower, Victoria railways, Australia Dec. 31, *03. 

A tool for removing broken taps in water pipes, etc. Jan. 7, '04. 

Cast-iron pipe, 48" dia., flattened under back-fill Dec. 15, '04. 

Reinforced-concrete waterworks standpipe 81' high Feb. 25, '04. 

Sectional plan of 30" "Premier" water meter May 19, '04. 

Failure of a water tank designed by an architect May 19, '04. 

Sections of core walls for reservoir embankments June 30, '04. 

Laying submerged pipe at N. Tonawanda, N. Y. Aug. 4, '04. 

Details of coagulating plant, St. Louis settling basins ' Oct. 27, '04. 

New water tank and tower, E. Providence, R. I. Nov. 10, '04. 

Small brick reservoir, showing failure Nov, 17, *04, 



1294 



M.— WATER WORKS, 



Automatic regulating valve for reservoirs and standpipes Mar. 9, *06. 

The water softening plant at Oberlin, O. Sept. 21, '05. 

A 75 000-gal. rein.-conc. cistern for fire protection Sept. 28, '05. 

Details of elevated water tank and supporting tower Oct. 26, *05. 

Plans of rein.-conc. reservoir at Ft, Meade, So. Dak. Dec. 28, '05. 

Steel pipe line supported across gulch by arched girder Feb. 8, '06. 

Reinforced-concrete filter-bed walls and roofs April 26, '06. 

Plans of water purification plant, Paris, Ky. May 3, '06. 

Plans of a slow sand water filter for the home Aug. 9, '06. 

Reinforced-concrete mechanical filter plant, Germany Sept. 6, '06. 

Details of novel reinforced-concrete conduit, 42" dia. Oct. 4, '06. 

Apparatus for regulating discharge from reservoir Oct. 25, '06. 

Plans of slow sand filtration plant, Wash. D. C. Nov. 8, '06. 

Steel water tank (150' dia. x 20' high), vertical bracing Nov. 15, '06. 

Sand chutes for canal of Bijou Irrigation District Jan. 3, '07. 

Reinforced-concrete standpipe, Attleboro, Mass. Feb. 21, '07. 

A new design for balanced gate valve Oct. 17. '07. 

Forbes water-sterilizing apparatus Oct. 31, '07. 

Plan and section of gate house, reservoir, Portland, Ore. Feb. 6, '07. 

Automatic controlling valve for reservoirs, tanks, etc. Mar. 12, '08. 

Experimental water-filter plant, Oakland, Cal. May 21, '08. 

Reinforced-concrete reservoir, Indianapolis, Ind. Oct. 15, '08. 

Reinforced-concrete tank supported by hollow shaft Jan. 7, '09. 
Alternate designs of hand- and mach. washed slow sand filters May 20, '09. 

Fire-proof temporary crib, water tunnel, Chicago June 17, '09. 

Design of an 8-mile head hydraulic conduit Aug. 5, '09. 

Steel penstocks — expansion joints, elbow, saddle Feb. 10, '10. 

Brick water tower, 133 ft. high, 28 ft. diam. Mar. 24, '10. 

Forms for concrete vaulted roofs, water filters Apr. 14, '10. 

10, 000, 000-gal. pressure mechanical filter plant Apr. 14, '10. 

Sliding sluice-gates, Charles River Basin, Boston May 26, '10. 
Rein.-conc. pumping cistern, 42' high x 20' dia., sunk in 

ground Aug. 25, '10. 

A new filter-sand washing machine, in France Oct. 13, '10. 

Rein.-conc. water tank with dome-shaped bottom Dec. 16, '10. 

Large (102-in.) venturi meter for Montreal W. W. Dec. 15, '10. 

New filtration plant at Portsmouth, Eng. Dec. 15, '10. 

Eng. Rec 

An endless screen for a water-works intake May 29, '09. 

Rate controller and valve for Cincinnati filters June 19, '09. 

Plan of aerator, settling basin, filter, inlet, Dover, N. H. I^^® 1^» '^^* 

Details pipe line construction. Canyon City water supply Dec. 4, '09. 
Collapsible form for concrete conduit, Los Angeles aqueduct Jan. 1, '10. 

10, 500, 000-gal. covered, rein.-conc. reservoir, Mexico Aug. 6, '10. 

Reinforced concrete water tower. Westerly, R. I. Sept. 24, '10. 

Details of concrete lining, Seattle reservoirs Sept. 24, '10. 

Ultra- violet ray sterilizer for sterilizing water; cost Dec. 10, '10. 

Illustrated details of the Toledo (O.) filtration plant Nov. 26, '10. 
Tall rein.-conc. block water tower (28,250 cu. ft. capac.) near 

Brussels Dec. 10. '10- 



65.— SANITATION, 



The Disposal of Refuse from buildings in cities and towns may be 
classified as follows: 



Dry Refuse 



Paoer / ^^ large cities, waste paper is collected and sold 



Ashes. 



Garbage 



Sewage 
(House-drainage) 



Sewers 



to paper mills. 

r Ashes are collected and used in extensive filling 
\ operations. 

Dumned I ^^ ^^^' ^^^^ objectionable — ^not sanitary. 
pe I ^^ ggg^ Expensive; not always sanitary. 

Crematory- or reduction plants (incinerators). Often self-sup- 
porting (from fuel properties of oil and grease), in fur- 
nishing light and power. 

Cesspools. Economical for isolated buildings, as factories, 
and for suburban districts. 

Farm irrigation; largely employed in France. 

Septic tanks ; for inland towns. 

Chemical process — sludge. 

Rivers. Raw sewage emptying into streams 

is becoming prohibitive. 
Ocean. Boston sewage reaches Moon Island 

by submarine tunnel where it is pumped 

into the Bay. 

Storm water also becomes sewage when mixed with the latter. 

House Drainage is effected by — 

Waste Pioes I Which receive and convey the waste water from baths, 
waste iripes ^ basins, sinks, wash tubs, etc., but no human excreta. 

! Which receive and convey the human excreta, faeces (solid 
matter) and urine, from closets and urinals; also, general 
waste. 
Drain pipes connect with the waste- and soil pipes at the building line, and 
convey the waste and soil to the cesspool or sewer. 

Generally speaking, the waste- a,nd soil pipes are, in the main, vertical^ 
extending from top to bottom inside of the building, and connected by 
branch pipes with the various discharge fixtures; while the drain pipes arc 




^ ^enf 





Fig. 2. 



Fig. 3. 



practically horizontal and outside of the building. Soil- and waste pipes 
should be extended vertically up through the roof of the building as shown 
in Fig. 1, to provide for the escape of sewer gas. In order to prevent the 
latter from finding its way into the rooms, a trap is placed on each 

1295 



1296 Q5.—S ANITA TION. 

pipe leading from the fixture to the soil- or waste pipe. These traps are 
designated by various letters of the alphabet, but the S-trap (Fig. 2) is 
perhaps the most common. Their efficiency is reduced when the water 
evaporates or becomes siphoned out of the trap. Fig. 3 shows a section 
of Waring's check valve, which remains efficient unless foreign matter 
lodges between the valve and its seat, or the bearings become too much 
worn. 

Cesspools are excusable in sandy or gravelly soil in new additions to 
townsites where the water supply is not derived from wells in the immediate 
locality; or for isolated buildings where cost of sewer would be excessive. 
A notable example of the latter is the cesspool recently constructed for the 
discharge of soil and waste from the new boathouse pavilion in Prospect 
Park, Brooklyn. It is a large cylindrical well with a sand bottom and with 
sides lined with concrete. 

Sewers. — Figs. 4 to 8, in Tables 4 to 8, illustrate five standard sections 
of sewers, namely. Circular, Catenary, Basket-Handle, Gothic, and Egg or 
Egg-shape. The proportional dimensions are based on the diametrical 
height of unity, in each case. 

For the Circular section. Fig. 4, there are annexed diagrams of relative 
area a, velocity v,^ and discharge q, for relative depths of_ flow. Thus, if the 
conduit or sewer is full, a, v and q are assumed to be unity; if the depth is 
0.8, a=0.86, i;=1.14, g=0.98; if half full, a = 0.5, z;=1.0, <?=0.5, etc. Note 
that V and q are based on the velocity v being proportional to VrTin which 
r = hydraulic radius in feet (see Hydraulics, page 1161). Similar diagrams 
of area, velocity and discharge may be drawn for the other sections — 
Catenary (Fig. 5), Basket-Handle (Fig. 6), Gothic (Fig. 7), and Egg (Fig. 8). 
The Circle, and the Horseshoe with invert, are the most common forms for 
tunnel construction. 

The Egg-shape (Fig. 8) possesses the merit of maintaining a compara- 
tively high value of r (and hence v) for small depths of flow; and it is to 
be noted that when flowing f full depth, r (and v) are greater than when 
flowing full. (See Table 8.) 

Table 1, following, gives the value of r, \/r"and aVr"for Circular sections, 
advancing by inches up to 20 ft. 11 ins. dia. These properties are useful in 
designing, i. e., in connection with the use of Kutter's formula, being inde- 
pendent of the grade or slope s. 

Table 2, used in connection with Table 1, enables us to find the values 
of r, Vr and a's/rior the Catenary, Basket-Handle, Gothic and Egg sections. 

Table 3 gives the velocities in Circular brick sewers, unclean, using 
value of roughness n = 0.015. 

Tables 4, 5, 6, 7 and 8 show properties of Circular, Catenary, Basket- 
Handle, Gothic, and Egg sections respectively, with relative diameters, 
both vert, and hor., for equivalent areas. These tables will be found useful 
for comparison of the difi'erent sections in designing. 

Table 9 gives the velocities in Egg-shaped sewers which have become 
somewhat fouled, using value of roughness w = 0.015. 

Size and Grade of Sewers. — ^The design of sewers may be by either of 
three methods, namely, (1) by formulas; (2) by tables; (3) by diagrams. 
For discussion of formulas, see Hydraulics, page 1167. In using Kutter's 
formula the value of c may be determined from n = 0.013* for ordinary brick 
sewers, and w = 0.015 for large sewers with unclean surfaces. As there is 
more or less sand or scouring matter, the velocity of flow should generally 
be not greater than 5 or 6 ft. per sec. 

*For ordinary brick sewers, new and well laid; but it is safer to use 
»=0.014. See, also, pages 1168 and 1188. 



CIRCULAR SEWERS— VELOCITY, DISCHARGE. 



1297 



1. — ^Values of r, ^r and aVr for ^Circular- Conduits or Sewers. 

For Formulas: 

(1) Hydraulic radius r=diam. in ft. -5-4. 

(2) Velocity v (in ft. per sec.) = 

(3) Discharge q (in cu. ft. per sec.) 

av = c a\/r V^5 



4. 



Values of 



(1) r_ 

(2)\/r 

(3)av7 



in Feet. 
in Feet. 

in Feet. 





Decimals of a Foot. 


DIa. 


.000 1.083 1.167 1.250 1.333 1.417 1.500 1.583 1.667 1.750 | . 833 |.917 


in 
Ft. 


Inches. 


0" 


1" 


2" 


3" 


4" 


5" 


6" 


7" 


8" 


9" 


10" 


11" 




0.000 


0.021 


042 


0.063 


0.083 


0.104 


0.125 


0.146 


0.167 


0.188 


0.208 


0.229 


o] 


0.000 


0.144 


0.204 


0.250 


0.289 


323 


0.354 


0.382 


0.408 


0.433 


0.456 


0.479 




.0000 


.0008 


.0045 


.0123 


.0252 


.0440 


.0694 


.1021 


.1425 


.1913 


.2489 


.3159 




0.250 


0.271 


0.292 


0.313 


0.333 


0.354 


0.375 


0.396 


0.417 


0.438 


0.458 


0.479 


1 \ 


0.500 


0.520 


0.540 


0.559 


0.577 


0.595 


0.612 


0.629 


0.645 


0.661 


677 


0.692 




.3927 


.4797 


.5773 


.6860 


.8061 


.9381 


1.082 


1.239 


1.408 


1.591 


1.787 


1.997 




0.500 


0.521 


0.542 


0.563 


0.583 


0.604 


0.625 


0.646 


0.667 


0.688 


0.708 


0.729 


2i 


0.707 


0.722 


0.736 


0.750 


0.764 


0.777 


0.791 


0.804 


0.816 


0.829 


0.842 


0.854 




2.221 


2.460 


2.714 


2.982 


3.266 


3.565 


3.881 


4.212 


4.560 


4.925 


5.306 


5.705 




0.750 


0.771 


0.792 


0.813 


833 


0.854 


0.875 


0.896 


0.917 


0.938 


0.958 


0.979 


si 


0.866 


0.878 


0.890 


0.901 


0.913 


0.924 


0.935 


0.946 


0.957 


0.968 


0.979 


0.990 




6.122 


6.556 


7.008 


7.478 


7.966 


8.474 


9.000 


9.545 


10.11 


10.69 


11.30 


11.92 




1.000 


1.021 


1.042 


1.063 


1.083 


1. 104 


1.125 


1.146 


1.167 


1.188 


1.208 


1.229 


4i 


1.000 


1.010 


1.021 


1.031 


1.041 


1.051 


1.061 


1.070 


1.080 


1.090 


1.099 


1.109 




12.57 


13.23 


13.92 


14.62 


15.35 


16.10 


16.87 


17.66 


18.47 


19.31 


20.17 


21.05 




1.250 


1.271 


1.292 


1.313 


1.333 


1.354 


1.375 


1.396 


1.417 


1.438 


1.458 


1.479 


si 


1.118 


1.127 


1.137 


1.146 


1.155 


1.164 


1.173 


1.181 


1.190 


1.199 


1.208 


1.216 




21.95 


22.88 


23.83 


24.80 


25.80 


26.82 


27.86 


28.93 


30.02 


31.13 


32.27 


33.44 




1.500 


1.521 


1.542 


1.563 


1.583 


1.604 


1.625 


1.646 


1.667 


1.688 


1.708 


1.729 


ei 


1.225 


1.233 


1.242 


1.250 


1.258 


1.267 


1.275 


1.283 


1.291 


1.299 


1.307 


1.315 




34.63 


35.84 


37.08 


38.35 


39.64 


40.96 


42.30 


43.67 


45.06 


46.49 


47.93 


49.41 




1.750 


1.771 


1.792 


1.813 


1.833 


1.854 


1.875 


1.896 


1.917 


1.938 


1.958 


1.979 


ri 


1.323 


1.331 


1.339 


1.346 


1.354 


1.362 


1.369 


1.377 


1.384 


1.392 


1.399 


1.407 




50.91 


52.44 


53.99 


55.58 


57.19 


58.83 


60.49 


62.19 


63.91 


65.66 


67.44 


69.25 




2.000 


2.021 


2.042 


2.063 


2.083 


2.104 


2.125 


2.146 


2.167 


2.188 


2.208 


2.229 


si 


1.414 


1.422 


1.429 


1.436 


1.443 


1.451 


1.458 


1.465 


1.472 


1.479 


1.486 


1.493 




71.09 


72.95 


74.85 


76.77 


78.72 


80.71 


82.72 


84.76 


86.83 


88.94 


91.07 


93.23 




2.250 


2.271 


2.292 


2.313 


2.333 


2.354 


2.375 


2.396 


2.417 


2.438 


2.458 


2.479 


9- 


1.500 


1.507 


1.514 


1.521 


1.528 


1.534 


1.541 


1.548 


1.555 


1.561 


1.568 


1.575 




95.43 


97.65 


99.91 


102.2 


104.5 


106.9 


109.2 


111.6 


114.1 


116.6 


119.1 


121.6 




2.500 


2.521 


2.542 


2.563 


2.583 


2.604 


2.625 


2.646 


2 667 


2.688 


2.708 


2.729 


10 i 


1.581 


1.588 


1.594 


1.601 


1.607 


1.614 


1.620 


1.627 


1.633 


1.639 


1.646 


1.652 




124.2 


126.8 


129.4 


132.1 


134.8 


137.5 


140.3 


143.1 


145.9 


148.8 


151.7 


154.6 




2.750 


2.771 


2.792 


2.813 


2.833 


2.854 


2.875 


2.896 


2.917 


2.938 


2.958 


2.979 


11. 


1.658 


1.665 


1.671 


1.677 


1.683 


1.689 


1.696 


1.702 


1.708 


1.714 


1.720 


1.726 




157.6 


160.6 


163.6 


166.7 


169.8 


172.9 


176.1 


179.3 


182.6 


185.8 


189.2 


192.5 




3.000 


3.021 


3.042 


3.063 


3.083 


3.104 


3.125 


3.146 


3.167 


3.188 


3.208 


3.229 


12 i 


1.732 


1.738 


1.744 


1.750 


1.756 


1.762 


1.768 


1.774 


1.780 


1.785 


1.791 


1.797 




195.9 


199.3 


202.8 


206.3 


209.8 


213.3 


216.9 


220.6 


224.2 


227.9 


231.7 


235.5 




3.250 


3.271 


3.292 


3.313 


3.333 


3.354 


3.375 


3.396 


3.417 


3.438 


3.458 


3.479 


13 i 


1.803 


1.809 


1.814 


1.820 


1.826 


1.831 


1.837 


1.843 


1.848 


1.854 


1.860 


1.865 




239.3 


243.1 


247.0 


251.0 


254.9 


258.9 


263.0 


267.0 


271.2 


275.3 


279.5 


283.7 


( 


3.500 


3.521 


3.542 


3.563 


3.583 


3.604 


3.625 


3.646 


3.667 


3.688 


3.708 


3.729 


14] 


1.871 


1.876 


1.882 


1.887 


1.893 


1.899 


1.904 


1.909 


1.915 


1.920 


1.926 


1 931 




288.0 


292.3 


296.6 


301.0 


305.4 


309.9 


314.4 


318.9 


323.5 


328.1 


332.8 


337.5 




3.750 


3.771 


3.792 


3.813 


3.833 


3.854 


3.875 


3.896 


3.917 


3.938 


3.958 


3.979 


15 i 


1.936 


1.942 


1.947 


1.953 


1.958 


1.963 


1.969 


1.974 


1.979 


1.984 


1.990 


1.995 




342.2 


347.0 


351.8 


356.6 


361.5 


366.5 


371.4 


376.5 


381.5 


386.6 


391.7 


396.9 




4.000 


4.021 


4.042 


4.063 


4.083 


4.104 


4.125 


4.146 


4.167 


4.188 


4.208 


4.229 


16 i 


2.000 


2.005 


2.010 


2.016 


2.021 


2.026 


2.031 


2.036 


2.041 


2.046 


2.051 


2.056 




402.1 


407.4 


412.7 


418.0 


423.4 


428.8 


434.3 


439.8 


445.3 


450.9 


456.5 


462.2 




4.250 


4.271 


4.292 


4.313 


4.333 


4.354 


4.375 


4.396 


4.417 


4.438 


4.458 


4.479 


17 i 


2.062 


2.067 


2.072 


2.077 


2.082 


2.087 


2.092 


2.097 


2.102 


2.107 


2.111 


2 116 




467.9 


473.7 


479.5 


485.3 


491.2 


497.1 


503.1 


509.1 


515.2 


521.3 


527.4 


533.6 




4.500 


4.521 


4.542 


4.563 


4.583 


4.604 


4.625 


4.646 


4.667 


4.688 


4.708 


4.729 


isi 


2.121 


2.126 


2.131 


2.136 


2.141 


2.146 


2 151 


2.155 


2.160 


2.165 


2.170 


2.175 




539.8 


546.1 


552.4 


558.7 


565.1 


571.6 


578.1 

4.875 


584.6 


591.2 


597.8 


604.5 


611.2 




4.750 


4.771 


4.792 


4.813 


4.833 


4.854 


4.896 


4.917 


4.938 


4.958 


4.979 


19- 


2.179 


2.184 


2.189 


2.194 


2.198 


2.203 


2.208 


2 213 


2.217 


2.222 


2.227 


2.231 




617.9 


624.7 


631.6 


638.5 


645.4 


652.4 


659.4 


666.5 


673.6 


680.7 


687.9 


695.2 




5.000 


5.021 


5.042 


5.063 


5.083 


5.104 


5.125 


5.146 


5.167 


5.188 


5.208 


5.229 


20 j 


2.236 


2.241 


2.245 


2.250 


2.255 


2.259 


2.264 


2.268 


2.273 


2.278 


2.282 


2.287 




702.5 


709.8 


717.2 


724.6 


732.1 


739.6 


747.2 


754.8 


762.5 


770.2 


778.0 


785.8 



Ex. — For_a circular pipe 3' 2" (=3.167 ft.) dia., the hydraulic radius 
^ = 0.792, \/r = 0.890, a\/r= 7.008. (a = area in sq. ft.; see table, p. 1157.) 
* See also Table 2 for other sections than Circular. 



1298 



Q5.— SANITATION. 



-Use op Table 1, preceding, for Other Sections than Circular. 
(See also Tables 4, 5, 6, 7 and 8.) 







To 








Section. 


Relation of 
Diameters. 


Find 

Value 

of— 


Mult. Value of— 


Multi- 
plier. 


Logar- 
ithm. 




Vert. dia.=dia. circle. 


r 


r In Table 1 by 


0.92688 


9.967 0235 


Catenary, 

Flowing < 
Full Depth. 


" = * 


VT 


VT " 


0.96275 


9.983 5117 


" = " 


a\/r 


a^/r 


0.86146 


9.935 2350 


Hor. dia. = " 


r 


T " " 


1.0427 


0.018 1760 


" = " 


vr 


vr " 


1.0211 


0.009 0880 




= 


a\/r 


a\/r 


1.1564 


0.063 1163 




Vert. dla.= 


r 


r " " 


0.98572 


9.993 7536 


Basket-handle, 
Flowing < 
Full Depth. 


= " 


VE 




0.99283 
0.99386 


9.996 8768 

9.997 3255 


Hor. dia. = " 


r 


r " " 


1.0442 


0.018 7816 


" __ «» 


\/T 


VT " 


1.0219 


0.009 3908 




= 


a\/r 


as/r 


1.1479 


0.059 8955 




Vert. dla.= " 


T 


T " " 


0.90760 


9.957 8945 


Gothic. 
Flowing 
Full Depth. 


« __ i< 


VT 


s/T " 


0.95268 


9.978 9472 


" = " 


a\/T 


a\/r 


0.79487 


9.900 2975 


Hor. dia. = 


r 


r " *' 


1.0948 


0.039 3400 


" = *• 


vr 


x/F •' 


1.0463 


0.019 6700 




= " 


o\/r 


a\/T " 


1.2703 


0.103 9113 




Vert. dia. = ''^ 


r 


r " " 


0.77253 


9.887 9172 


Egg-shape. 
Flowing 
FuU Depth. 




^/T 


x/r •• 


0.87894 


9.943 9586 


" = " 


a\/r 


a'^T " 


0.57125 


9.756 8265 


Hor. dia. = 


T 


r 


1.1588 


0.064 0085 




vr 


VT " 


1.0765 


0.032 0042 




• _. <» 


a\/^ 


a\/T 


1.5742 


0.197 0547 




'Vert. dia.= " 


T 


J. .1 .1 


0.84187 


9.925 2433 


Egg-shape. 
Flowing 


" =r " 


\/T 


x/r " 


0.91753 


9.962 6216 


" =r= " 


a\/T 


ay/r 


0.39244 


9.593 7703 


Hor. dia. = 


r 


r " " 


1.2628 


0.101 3346 


f Full Depth. 




vr 


VT " 


1.1237 


050 6673 




" = " 


a-s/r 


a\/T 


1.0814 


0.033 9986 




'Vert. dia.= 


r 


r " " 


0,55093 


9.741 0990 






vr 


vr " 


0.74225 


9.8J0 5495 


Egg-shape, 


•• = " 


a\/T 


ay/r 


0.11929 


9.076 5953 


Flowing 


Hor. dia. = 


r 


r " " 


0.82640 


9.917 1903 


i Full Depth. 


" = " 


vr 


VT " 


0.90907 


9.958 5951 




" = " 


a\/T 


a\/r 


0.32872 


9.516 8235 



Examples in Use of Table 2, above. 

Ex. 1. — From Tables 2 and 1, find the hydraulic radius r, \/r, and a\/r 
(a = area of section considered) for the Catenary, (1) whose vertical diameter 
is 8 ft. 3 ins., and (2) whose horizontal diameter is 7 ft. 4 ins.? 

Solution.— For (1), vert. dia. of 8 ft. 3 ins., we have, r=2.063X.9269 = 
1.912. Vr=1.436X. 9628= 1.383, aV7=76.77X.8615=66J4; and for (2), 
hor. dia. of 7 ft. 4 ins., we have, r= 1.833X 1.043= 1.912, \/r= 1.354X 1.0211 
-1.383, a\/7=57.19X 1.1564 = 66.13 + . 

Comparing the above results it is found that they are equal, simply 
because 7 ft. 4 ins. ( = 88 ins.) is the hor. dia. of a catenary whose vert. dia. 
is 8 ft. 3 ins. ( = 99 ins.); i.e., the hor. dia. of catenary = f vert. dia. See 
Table 5 following. 

Ex. 2. — In Kutter's formula^ 7; = <:Vr5 = c\/r Vs, it is required to find 
the velocity v of flow in an Egg-shaped sewer whose hor. dia, is 4 ft. 6 ins., 
flowing f full depth, and on a grade of 5 = .0004, assuming the value of 
coefficient c = 1 00 ? 

Solution. — From Tables 1 and 2, preceding, the value of Vr= 1.061 X 
1.1237 = 1.192; hence the velocity of flow is ?;= lOOX 1.192X .02 = 2.384 ft. 
per sec. 



VELOCITIES IN CIRCULAR BRICK SEWERS, 



1299 



Ex. 3. — In Kutter's formula, q = av = c . as/r . \/s, it is required to find 
the discharge from a Basket-Handle conduit whose vert. dia. is 8 ft., flowing 
full, and on a grade of .000225, assuming the value of roughness n = .015. 

Solution.— s = . 000225, and \/J=.015. From Tables 1 and 2, r=2X 
•9857= 1.9714, and aVr= 71 09 X .9939= 70.66. The value of c is obtained 
from Table 8, page 1171, and =111. Hence, the discharge $=111X70.66X 
.015= 117.6 cu. ft. per sec. 

3. — Velocities in Feet per Second in Circular Brick Sewers.* 
By Kutter's Formula, using w = 0.015. 

(Slope s = value in first column-^ 100.) 



Fall in 
Ft. per 




Diameter of Pipe or Sewer, in Feet. 


vr 


100 Ft, 






(100 s) 


0.5 


1 


2 


3 


4 


5 


6 


7 


8 


10 


12 




.002 


0.09 


0.16 


0.27 


0.36 


0.44 


0.52 


0.59 


0.65 


0.71 


0.82 


0.92 


.00447 


.004 


0.13 
0.16 


0.22 


0.38 
0.47 


0.51 
0.63 


0.63 
0.77 


0.73 
0.90 


0.83 
1.02 


0.92 
1.13 


1.00 
1.23 


1.16 
1.42 


1.31 
1.60 


.00632 


.006 


0.27 


.00775 


.008 


0.18 


0.32 


0.54 


0.72 


0.89 


1.03 


1.17 


1.30 


1.42 


1.64 


1.85 


.00894 


.01 


0.20 


0.35 
0.43 


0.60 
0.74 


0.81 
0.99 


0.99 
1.21 


1.16 
1.42 


1.31 
1.60 


1.45 
1.78 


1.59 
1.94 


1.84 
2.25 


2.06 
2.53 


.01 


.015 


0.25 


.01225 


.02 


0.29 


0.50 


0.85 


1.14 


1.40 


1.64 


1.85 


2.05 


2.24 


2.60 


2.92 


.01414 


.025 


0.32 


0.56 


0.95 


1.28 


1.57 


1.83 


2.07 


2.30 


2.51 


2.90 


3.26 


.01581 


.03 


0.35 


0.61 


1.04 


1.40 


1.72 


2.00 


2.27 


2.52 


2.75 


3.18 


3.58 


.01732 


.035 


0.38 


0.66 


1.12 


1.51 


1.85 


2.16 


2.45 


2.72 


2.97 


3.44 


3.86 


.01871 


.04 


0.40 


0.71 


1.20 


1.62 


1.98 


2.31 


2.62 


2.91 


3.17 


3.67 


4.13 


.02 


.045 


0.43 


0.75 


1.27 


1.71 


2.10 


2.45 


2.78 


3.08 


3.37 


3.90 


4.38 


.02121 


.05 


0.45 


0.79 


1.34 


1.81 


2.22 


2.59 


2.93 


3.25 


3.54 


4.11 


4.62 


.02236 


.06 


0.50 


0.87 


1.47 


1.98 


2.43 


2.83 


3.21 


3.56 


3.89 


4.50 


5.06 


.02449 


.07 


0.53 


0.94 


1.59 


2.14 


2.62 


3.06 


3.47 


3.84 


4.20 


4.86 


5.46 


.02646 


.08 


0.57 


1.00 


1.70 


2.28 


2.80 


3.27 


3.70 


4.11 


4.49 


5.20 


5.84 


.02828 


.09 


0.61 


1.06 


1.80 


2.42 


2.97 


3.47 


3.93 


4.36 


4.76 


5.51 


6.19 


.03 


.10 


0.64 


1.12 


1.90 


2.55 


3.13 


3.66 


4.14 


4.59 


5.02 


5.81 


6.53 


.03162 


.12 


0.70 


1.23 


2.08 


2.80 


3.43 


4.01 


4.54 


5.03 


5.50 


6.36 


7.15 


.03464 


.14 


0.76 


1.32 


2.25 


3.02 


3.71 


4.33 


4.90 


5.43 


5.94 


6.87 


7.73 


.03742 


.16 


0.81 


1.42 


2.40 


3.23 


3.96 


4.63 


5.24 


5.81 


6.35 


7.35 


8.26 


.04 


.18 


0.86 


1.50 


2.55 


3.43 


4.20 


4 91 


5.56 


6.17 


6.73 


7.79 


8.76 


.04243 


.20 


0.90 


1.58 


2.69 


3.61 


4.43 


5.17 


5.86 


6.50 


7.10 


8.22 


9.23 


.04472 


.25 


1.01 


1.77 


3.00 


4.04 


4.96 


5.79 


6.55 


7.27 


7.94 


9.19 


10.3 


.05 


.30 


1.11 


1.94 


3.29 


4.42 


5.43 


6.34 


7.17 


7.96 


8.69 


10.1 


11.3 


.05477 


.35 


1.20 


2.09 


3.55 


4.78 


5.86 


6.84 


7.75 


8.60 


9.39 


10.9 


12.2 


.05916 


.40 


1.28 


2.24 


3.80 


5.11 


6.27 


7.32 


8.29 


9.19 


10.0 


11.6 


13.1 


.06325 


.50 


1.43 


2.50 


4.25 


5.71 


7.01 


8.18 


9.26 


10.3 


11.2 


13.0 


14.6 


.07071 


.60 


1.57 


2.74 


4.65 


6.26 


7.68 


8.96 


10.1 


11.3 


12.3 


14.2 


16.0 


.07746 


70 


1.69 


2.96 


5.03 


6.76 


8.29 


9.68 


11.0 


12.2 


13.3 


15.4 


17.3 


.08367 


.80 


1.81 


3.17 


5.37 


7.22 


8.86 


10.3 


11.7 


13.0 


14.2 


16.4 


18.5 


.08944 


.90 


1.92 


3.36 


5.70 


7.66 


9.40 


11.0 


12.4 


13.8 


15.1 


17.4 


19.6 


.09487 


1.00 


2.02 


3.54 


6.00 


8.08 


9.91 


11.6 


13.1 


14.5 


15.9 


18.4 


20.6 


.1 


1.20 


2.21 


3 88 


6.58 


8.85 


10.9 


12.7 


14.4 


15.9 


17.4 


20.1 


22.6 


.10955 


1.40 


2.39 


4.19 


7.11 


9.56 


11.7 


13.7 


15.5 


17.2 


18.8 


21.7 


24.4 


.11832 


1.60 


2.56 


4.48 


7.60 


10.2 


12.5 


14.6 


p16.6 


18.4 


20.1 


23.2 


26.1 


.12649 


1.80 


2.71 


4.75 


8.06 


10.9 


13.3 


15.5 


17.6 


19.5 


21.3 


24.6 


27.7 


.13416 


2.00 


2.86 


5.01 


8.50 


11.4 


14.0 


16.4 


18.5 


20.5 


22.4 


26.0 


29.2 


.14142 


to 


0.196 


0.785 


3.142 


7.068 


12.566 


19.635 


28.274 


38.485 


50.266 


78.540 


113.10 




r 


0.125 


0.250 


0.500 


0.750 


1.000 


1.250 


1.500 


1.750 


2.000 


2.500 


3.000 




cvr. 


20.21 


35.40 


60.08 


80.77 


99.10 


115.7 


131.0 


145.3 


158.7 


183.7 


206.5 




ac\/T 


3.9604 


27.803 


188.77 


570.90 


1245.3 


2272.7 


3702.3 5591.6 


7978.3 


14426 


23352 





*For second-class brick, dressed stone, or tuberculated iron. (See Table 9.) 
Note. — Velocities in table are equal to cVr X VF; cVr is in the next 

line to bottom for the particular diameter of pipe, and VJ is foimd in the 
last column. 

fa = area of section in sq. ft.; r = hydraulic radius in ft.; c = coefficient in 
Kutter's formula, using mean value of 5 = .001. Velocity in ft. per sec. =«;=» 
c\/rs = c\/r \/s. Discharge in cu. ft. per sec. = g = oz; «= ac\/Ts = ac\/T Vj. 



1800 



Q5.-^SANITATI0N. 



4. — Properties op Circular- Conduits or Sewers. 

(a) General. 



When flowing full: 

= 0.7854^2. 

p = Tzd, 
= 3.1416d. 
d 

When flowing | full: 

nd^ 

^ = T' 

= 0.3927(i2. 

7:d 

= 1.5708(J. 
d 

In which 

a = area in sq. ft. 
p = wetted perimeter in ft. 
r = hydraulic radius in ft. 
Log 7z = 0.4971499. 



^ J.U--^d 1 • i^vl 


?AQ "=:^ h:^^^ -,^ A 


Eo.y s, H-^-fi^- 7 / 


;^ nft ^ -,X^^ ^^)^ 


^ ^-^ \ ^2%^^^ 


•^A7 >,3^I^S3 i 


^ ^--^ I ^'^^df^ L 


**nc. ^ y^^S^ 2 


li- 0.6 >^'^ ^ "^ 


"feoq .^ z: 


"^0-5 ^^-^ t:7 


:£«>, ^2 A^y 


t.0.4 ,,o'^ / , rf / 


^n7 ^c^<2i^ / .iMi^ 


pO.3 cj>7^^,'- J^^ 


*feoo J^^# t ^^"^ 


2 0.2 ^7;^=^ z ^^"^ 


~A, 7Z ^^'^ 


•^01 01 ^^^ 


iS Cti^'^ 



Ql 02 0.3 0.4 05 0.6 0.7 Q8 09 1.0 I;) 



Fig. 4. — Ratios of a, v, and g for Filled 
Segments. 
(a, t; or g = unity when section is full.) 



Log J 



9.8950899. 



(6) Comparison of Circle with Other Sections of Equal Area, 
(See also Tables 5, 6, 7 and 8.) 
d = dia. of Circle in ft. Logarithm. 

= . 9407 X vert. dia. of Catenary of equiv. area 9 . 973 4667 

^ 0.024 6192 

0.000 4345 

0.025 4625 

9.956 4020 

0.037 8475 

9.906 4340 

0.082 6263 



= 1.0583Xhor. 


" 


" 


= 1.001 Xvert. 


<t 


BasketHandle 


= 1.060 Xhor. 


•• 




= 0.9045 Xvert. 


«( 


Gothic 


= 1.091 Xhor. 


<( 


" 


= 0.8062 Xvert. 


<« 


Egg-shape 


= 1.2093 Xhor. 


4« 





SEWERS— CIRCULAR AND CATENAR Y SECTIONS. 1 301 



5. — Properties of Catenary- Conduits or Sewers. 



(a) General. 




a = area in sq. ft. 


Logarithm. 


= 0.70277 (iv2 
= 0.88944 cih2 


9.846 8132 
9.949 1182 


^= wetted perimeter in ft 


. 


= 3.03284 c^v 
= 3.41195 c^h 


0.481 8497 
0.533 0022 


r=hydraiilic radius in ft. 




= 0.23172 ^v 
= 0.260685 £/h 


9.364 9635 
9.416 1160 


Where c^.=vert. dia. in ft. 
dh=lior. dia. in ft. 





••V ^"|'""*V 


AQ £, 1 X- ^<r- 


0-9^ I .'^^^ 


no ^^^ S 


^•^ ^# \ 


n-7 dS-v 3 


0.7 X-^r; r 


nc -^^ "^^ 4 


O-o^ _ \l. ^L. 




0-5 X"^!. T 


C\A ^Tv 


0.4 ^ ^ ^ 


A"7 "'^Y "^ 


0.3 •--■ 7i=tog'^^j.^^ ----'"■' 


n? ^^ S"^^ 


U.L ^;j \ [y 


n 1 v^ \ J 


"•1 ^ ;, ^-^ 


A-,:!: .-=--^ 



0.1 0.2 0.3 0.4 0.5 
Fig. 5. — Catenary. 



(6) Comparison of Catenary with Other Sections of Equal A 

(See also Tables 4, 6, 7 and 8.) 
cfv = vert. dia. of Catenary in ft. 

= 1 . 063 X dia. of Circle of equivalent area 

= 1 . 125 Xhor. dia. of Catenary of equiv. area 

= 1.064 IX vert. " Basket-Handle 

= 1.1272 Xhor. '* 

= 0. 9615 X vert. " Gothic 

= 1.1598Xhor. " 

= 0. 8570 X vert. " Egg-shape 

= 1.2855 Xhor. " 

£iu=hor. dia. of Catenary in ft. 

= . 9449 X dia. of Circle of equivalent area. ...... 

. 8889 X vert. dia. of Catenary of equiv. area 



9458 X vert. 
1.0019 Xhor. 
0. 8546 X vert. 
1.0309 Xhor. 
0. 7618 X vert. 
1.1426Xhor. 



Basket-Handle 

Gothic 

Egg-shape 



rea. 

Logarithm. 
0.026 5333 
0.051 1525 
0.026 9678 
0.051 9958 
9.982 9353 
0.064 3808 
9.932 9673 
0.109 0586 



9.975 3808 
9.948 8475 
9.975 8153 
0.000 8433 
9.931 7828 
0.013 2283 
9.881 8148 
0.057 9061 



1302 



Q5.'^SANITATI0N. 



6. — Properties op Basket-Handle- Conduits or Sewers. 



(a) General, 
a = area in sq. ft. 

= 0.78621 £fv2 

= 0.88226 fih2 
/> = wetted perimeter in ft. 

= 3.19040 d. 

= 3.37966 cih 
r = hydraulic radius in ft. 

= 0.24643 dv 
= 0.26105 dh 
Where dv = vert. dia. in ft. 
(ih = hor. dia. in ft. 



Logarithm. 

9.895 5386 
9.945 5946 



0.503 8450 
0.528 8730 



9.391 6936 
9.416 7216 



1.0 
0.9 
0.8 
0.7 
0.6 
O.944-,05- 
0.4 
0.3 
0.2 
0.1 




""--^ 




i 


^. 


\ _, 


T^^ 


^1 l\ 1 1 1 


i:^ 


;5 


^^ 


t 




A 


-^^i 


\ 






.^^^ i 




_^Hj_ 




=> 




s 








"'-^^ 






•'''Ms> 




^7 




rzg'^ 




I^^ 


^ 


-=^ 



0.1 0.2 0.3 0.4 05 



Fig. 6. — Basket-Handle. 

(6) Comparison of Basket-Handle with Other Sections of Equal Area, 
(See also Tables 4, 5, 7 and 8.) 
dv = vert. dia. of Basket-Handle in ft. Logarithm. 

= . 999 X dia. of Circle of equivalent area 9 . 999 5655 

= . 9398 X vert. dia. of Catenary of equiv. area 9.973 0322 



= 1.0573Xhor. ' 


• a 




<t 


...0.024 1847 


= 1.059 X " 


Basket-Handle 




4« 


...0.025 0280 


= 0. 9036 X vert. ' 


Gothic 




<( 


...9.955 9675 


= 1.0900Xhor. ' 


t << 




<t 


...0.037 4130 


= . 8054 X vert. ' 


' Egg-shape 




«« 


...9.905 9995 


= 1.208lXhor. ' 








...0.082 0908 


= hor. dia. of Bask 


et-Handle in ft. 








= 0,943 Xdia. of( 


;!^ircle of equivalent area. . 





...9.974 5375 


= 0. 8872 X vert, di 


a. of Catenary of 


squiv. 


area . . 


...9.948 0042 


= 0.998lXhor. ' 


i ti 


" 


" 


...9.999 1567 


= 0.944 Xvert. ' 


Basket-Handle 


<t 


it 


...9.974 9720 


= 0.8530X " 


Gothic 


«• 




...9.930 9375 


= 1.0289Xhor. * 


t <t 


<« 


«« 


...0.012 3850 


= 0.7603 Xvert. ' 


• Egg-shape 


«« 




...9.880 9715 


= 1.1404Xhor. ' 








....0.067 0628 



SEWERS— BASKET-HANDLE AND GOTHIC SECTIONS. 1303 



7. — Properties op Gothic- Conduits or Sewers. 



(a) General. 
a = area in sq. ft. 
= 0.6553 £f 2 
= 0.9535 (ia2 

/>= wetted perimeter in ft. 
= 2.8881 ci. 
= 3.4838 cih 
f= hydraulic radius in ft. 
= 0.2269 (i, 
= 0.2737 dh 
Where (iv = vert. dia. in ft. 
c?h=hor. dia. in ft. 



Logarithm. 
9.816 4402 
9.979 3312 



0.460 6057 
0.542 0512 

9.355 8345 
9.437 2800 



1.0 
0.9 
0.8 
0.7 
0.6 
0.5 
0.4 
0.3 
0.2 
0.1 



^'-: T 


M ^v 


IK. ^^ 


^ V 


S ^ 


-5 ^ 


^ A 


^ ± 




r Tiri'y r^ 




L- - -P: 0AI4 "-^ - 




;; r 


*■' I . 


t i 


'$. /- 


L_ 4 


! ^ 


«iL__-==:_ 



O.l 0.e 0.3 0.4 



Fig. 7.— Gothic. 



(6) Comparison of Gothic with Other Sections of Equal Area, 
(See also Tables 4, 5, 6 and 8.) 
vert. dia. of Gothic in ft . Logarithm. 

1 . 1056 X dia. of Circle of equivalent area 0.043 5980 

1.040 IX vert. dia. of Catenary of equiv. area 0,017 0647 

" 0.0682172 

Basket-Handle 



= 1.170lXhor 

= 1.1 067 X vert. 

= 1.1724Xhor. 

= 1.2063X " " Gothic 

= 0.8913Xvert. '* Egg-shape 

= 1.3370Xhor. " 

dh=hor. dia. of Gothic in ft. 

= .9165 X dia. of Circle of equivalent area 9 .962 1525 

= . 8622 X vert. dia. of Catenary of equiv. area 9 . 936 6192 



0.044 0325 
0.069 0605 
0.081 4455 
9.950 0320 
0.126 1233 



0.9700Xhor. 
0. 91 75 X vert. 
0.9719Xhor. 
0. 8290 X vert. 
0.7389X " 
1.1084Xhor. 



Basket-Handle 

Gothic 
Egg-shape 



9.986 7717 
9.962 5870 

9.987 6150 
9.918 5545 
9.868 5865 
0.044 6778 



1304 



Q5.^SANITATI0N, 



8. — ^Properties of Egg (-Shaped)- Conduits or Sewers. 
(a) General. 



Wetted 
Section. 



Area a. 



Log = 



Area a. 



Log = 

Perimeter p 
Log = 

Perimeter p 
Log = 

Hyd. rad. r 
Log = 

Hyd. rad. r 
Log = 



Flowing 
Full Depth, 



0.510455^.2 
9.707 9578 

1.148525ci,2 
0.060 1404 

2.6433(iv 
0.-422 1409 

3.9649£fh 
0.598 2322 

0.19313(fv 
9.285 8572 

0.2897Jh 
9.461 9485 



Flowing f 
Full Depth. 



0.335922(f.2 
9.526 2386 

0.755825dh2 
9.878 4212 

1.59607tf. 
0.203 0510 

2.3941Jh 
0.379 1423 

0.21047£/v 
9.323 1833 

0.3157Jh 
9.499 2746 



Flowing \ 
Full Depth 



0.126222tiv2 
9.101 1357 

0.284(ih2 
9.453 3183 

0.91647d. 
9.962 1166 

1.3747c?h 
0.138 2079 

0.13773dv 
9.139 0390 

0.2066(ih 
9.315 1303 



1.0 
0.9 
0.8 



R'l 



0:1 
0.6 
0.5 
0.4 



'a/O.S 



6.Z 
0.1 



Wffl 


kzzzzy. 


-Si- 






t 






:::::/:: 


2 


f 


ym 



0.1 0.2 0.3 0.4 
Fig. S.—Egg. 



Where d,= vert. dia. in ft. 
dh= hor. dia. in ft. 

(6) Comparison of Egg-Shape with Other Sections of Equal Area, 
(See also Tables 4, 5, 6 and 7.) 
c?7 = vert. dia. of Egg-shape in ft. Logarithm. 

= 1 . 2404 X dia. of Circle of equivalent area . 093 5660 

= 1.1 669 X vert. dia. of Catenary of equiv. area 0.067 0327 



= 1.3128Xhor. 
= 1. 2417 X vert. 
= 1.3153Xhor. 
= 1.121 9 X vert. 
= 1. 3534 X hor. 
= 1. 5000 X hor. 



Basket-Handle 
Gothic 



0.118 1852 
0.094 0005 
0.119 0285 
0.049 9680 
0.131 4135 
0.176 0913 



Egg-shape 
iih=hor. dia. of Egg-shape in ft. 

= 0. 8269 X dia. of Circle of equivalent area 9.917 4747 

= . 7779 X vert. dia. of Catenary of equiv. area 9 . 890 9414 



= 0. 8752 X hor. 
= 0. 8278 X vert. 
= 0. 8769 X hor. 
= 0. 7480 X vert. 
= 0. 9022 X hor. 
= 0. 6667 X vert. 



Basket-Handle 

Gothic 

Egg-shape 



9.942 0939 
9.917 9092 
9.942 9372 
9.873 8767 
9.955 3222 
9.823 9087 



EGG-SHAPED SEWERS. 



1306 



9. — Velocities in Feet per Second in Egg-shaped Sewers. 

By Kutter's Formula, using w = 0.015. 

(Slope 5 = value in first column-^ 100.) 

[Velocities, v, in Feet per Second.] 



Fall in Ft. per 


Greatest Transverse or Horizontal Diameter tfh In Feet. 


1 


00 Ft. 
100 s) 






{ 


1 


2 


3 


4 


5 


6 


7 


8 


10 


12 


V« 




f .01 


0.40 


0.67 


0.90 


1.10 


1.28 


1.45 


1.60 


1.75 


2.02 


2.27 


.01 




.02 


0.56 


0.95 


1.27 


1.55 


1.81 


2.04 


2.27 


2.47 


2.85 


3.21 


.01414 




.04 


0.79 


1.34 


1.79 


2.20 


2.56 


2.89 


3.20 


3.50 


4.04 


4.54 


.02 


i 


.06 


0.97 


1.64 


2.20 


2.69 


3.13 


3.54 


3.92 


4.28 


4.95 


5.56 


.02449 


.10 


1.25 


2.12 


2.84 


3.48 


4.05 


4.57 


5.07 


5.53 


6.38 


7.17 


.03162 


§• 


.14 


1.48 


2.50 


3.36 


4.11 


4.79 


5.41 


5.99 


6.54 


7.56 


8.49 


.03742 


•S 


.20 


1.77 


2.99 


4.01 


4.91 


5.72 


6.47 


7.16 


7.82 


9.03 


10.1 


.04472 


53 


.40 


2.51 


4.23 


5.67 


6.95 


8.10 


9.15 


10.1 


11.1 


12.8 


14.3 


.06325 


^\ 


.60 


3.07 


5.18 


6.95 


8.51 


9.91 


11.2 


12.4 


13.5 


15.6 


17.6 


.07746 


tfi 


.80 


3.54 


5.99 


8.02 


9.83 


11.4 


12.9 


14.3 


15.6 


18.1 


20.3 


.08944 


B 


1.00 


3.96 


6.69 


8.97 


11.0 


12.8 


14.5 


16.0 


17.5 


20.2 


22.7 


.1 


^ 


1.40 


4.68 


7.92 


10.6 


13.0 


15.1 


17.1 


19.0 


20.7 


23.9 


26.8 


.11832 


o 


2.00 


5.60 


9.46 


12.7 


15.5 


18.1 


20.4 


22.7 


24.7 


28.5 


32.1 


.14142 




a 


1.148 


4.594 


10.337 


18.376 


28.713 


41.347 


56.278 


73.506 


114.85 


165.39 






r 


.2897 


.5794 


.8691 


1.159 


1.449 


1.738 


2.028 


2.318 


2.897 


3.476 






I Cy/T 


39.62 


66.93 


89.70 


109.9 


128.0 


144.6 


160.2 


174.8 


201.9 


226.8 




• 


.01 


0.42 


0.71 


0.95 


1.16 


1.35 


1.53 


1.70 


1.85 


2.13 


2.39 


.01 




.02 


0.60 


1.01 


1.35 


1.65 


1.92 


2.16 


2.40 


2.61 


3.02 


3.39 


.01414 




.04 


0.85 


1.43 


1.91 


2.33 


2.71 


3.06 


3.39 


3.70 


4.27 


4.79 


.02 


g 


.06 


1.04 


1.75 


2.34 


2.85 


3.32 


3.75 


4.16 


4.53 


5.23 


5.86 


.02449 


% 


.10 


1.34 


2.26 


3.01 


3.68 


4.28 


4.84 


5.36 


5.85 


6.74 


7.57 


.03162 


^ 


.14 


1.59 


2.67 


3.57 


4.36 


5.07 


5.73 


6.35 


6.92 


7.98 


8.96 


.03742 


.20 


1.90 


3.19 


4.26 


5.21 


6.06 


6.85 


7.58 


8.27 


9.54 


10.7 


.04472 


5 


.40 


2.68 


4.52 


6.03 


7.37 


8.57 


9.68 


10.7 


11.7 


13.5 


15.1 


.06325 


•-' . 


.60 


3.28 


5.53 


7.38 


9.02 


10.5 


11.9 


13.1 


14.3 


16.5 


18.5 


.07746 


•*< 


.80 


3.79 


6.39 


8.53 


10.4 


12.1 


13.7 


15.2 


16.5 


19.1 


21.4 


.08944 




1.00 


4.24 


7.14 


9.53 


11.6 


13.5 


15.3 


17.0 


18.5 


21.3 


23.9 


.1 


1.40 


5.02 


8.45 


11.3 


13.8 


16.0 


18.1 


20.1 


21.9 


25.2 


28.3 


.11832 


2.00 


6.00 


10.1 


13.5 


16.5 


19.2 


21.6 


24.0 


26.1 


30.2 


33.9 


.14142 


^ 


a 


0.756 


3.023 


6.802 


12.093 


18.895 


27.210 


37.035 


48.373 


75.583 


108.84 






r 


0.316 


0.631 


0.947 


1.263 


1.579 


1.894 


2.210 


2.526 


3.157 


3.788 






cs/r 


42.40 


71.42 


95.33 


116.5 


135.5 


153.1 


169.6 


184.9 


213.3 


239.4 






\ .01 


0.30 


0.52 


0.70 


0.87 


1.01 


1.15 


1.28 


1.40 


1.62 


1.83 


.01 




.02 


0.43 


0.74 


1.00 


1.22 


1.43 


1.63 


1.81 


1.98 


2.29 


2.59 


.01414 




.04 


0.61 


1.04 


1.41 


1.73 


2.03 


2.30 


2.56 


2.80 


3.24 


3.66 


.02 


4 


.06 


0.74 


1.28 


1.73 


2 12 


2.48 


2.82 


3.13 


3.43 


3.97 


4.48 


.02449 




.10 


0.96 


1.65 


2.23 


2.74 


3.20 


3.64 


4.04 


4.42 


5.13 


5.79 


.03162 


0) 


.14 


1.14 


1.95 


2.64 


3.24 


3.79 


4.30 


4.79 


5.24 


6.07 


6.85 


.03742 




.20 


1.36 


2.33 


3.15 


3.87 


4.53 


5.14 


5.72 


6.26 


7.25 


8.19 


.04472 


3 


.40 


1.92 


3.29 


4.46 


5.48 


6.41 


7.27 


8.09 


8.85 


10.3 


11.6 


.06325 


«-i 


.60 


2.36 


4.03 


5.46 


6.71 


7.85 


8.91 


9.91 


10.8 


12.6 


14.2 


.07746 


Hn 


.80 


2.72 


4.66 


6.30 


7.75 


9.06 


10.3 


11.4 


12.5 


14.5 


16.4 


.08944 




1.00 


3.04 


5.21 


7.05 


8.66 


10.1 


11.5 


12.8 


14.0 


16.2 


18.3 


.1 


1 


1.40 


3.60 


6.16 


8.34 


10.2 


12.0 


13.6 


15.1 


16.6 


19.2 


21.7 


.11832 


2.00 


4.30 


7.37 


9.97 


12.2 


14.3 


16.3 


18.1 


19.8 


22.9 


25.9 


.14142 


pc< 


a 


0.284 


1.136 


2.556 


4.544 


7.100 


10.224 


13.916 


18.176 


28.400 


40.892 






r_ 


0.207 


0.413 


0.620 


0.826 


1.033 


1.240 


1.446 


1.653 


2.066 


2.479 






> c\/r 


30.41 


52.09 


70.48 


86.61 


101.3 


115.0 


127.9 


139.9 


162.1 


183.1 





Notation. — a = sectional area of wetted section of sewer, in square feet; 
f = hydraulic mean radius; c = coefficient in Kutter's formula, using mean 

value of 5 = .001. Velocity u = c'\/rX V^ for the particular diameter dh and 
the particular grade or slope s. 



1306 65.— SANITATION. 

Example in Use of Table 9, preceding. 

Example. — ^What size of egg-shaped sewer, running two-thirds full depth, 
on grade of one-tenth of one per cent., will carry 16 cu. ft. per second, 
assuming w = 0.015? 

Solution. — From the preceding table for "Flowing two-thirds full depth" 
we select, if possible, the width d\, whose velocity multiplied by the corres- 
ponding area a will give the discharge q ( = av) = 16, for a grade of 0.1 ft. per 
100 ft. Now for cih = 2 ft., g= 3.023X2.26=6.83 cu. ft. per sec; and for 
dh = 3 ft., g = 6.802 X 3.01 = 20.47 cu. ft. per sec. Hence, by proportion, for 
g= 16, <ih = 2.67 ft. or, say, 2 ft. 8 ins.s and the height d» = UJh = 4 ft. (See 
Table 8.) In practice it is customary to increase the calculated diameter 
by about 2 ins., more or less. 

Thickness of Brick Sewer Walls. — No fixed rule can be used, but for brick 
sewers under 30 ins. in diameter a single 4-in. ring will generally suffice. 
Two rings (2X4 ins.) of brick are required for diameters from 2^ to 6 ft.; 
and three rings from about 6 to 10 ft. Above 10 ft. diameter treat sewer 
above spring line as a masonry arch, and keep the line of resultant pressure 
well within the middle third. Baldwin Latham gives the following formula 
for thickness: 

'=^ (1) 

where <= thickness of brick wall, in feet; 
(i = depth of excavation, in feet; 
r = radius of sewer, in feet. 

It is essential that sewer linings be water tight. 

Sewer Foundations. — Many sewers are rendered leaky and unsanitary by 
reason of their having been constructed on improper foundations. An 
economical foundation for sewers projected throught soft, marshy ground 
consists of two-pile bents driven say 4 or 5 feet apart, capped with suitable 
cross timbers, on top of which are laid the thick longitudinal floor planking, 
forming a platform for the sewer proper. Sometimes the piling supporting 
the platform is omitted. Timber foundations should always be below the 
ground-water level to prevent decay. 

10. — Some References to Illustrated Sewer Construction 

IN "Engineering News." 

June 6, 1901. Cast iron pipe outfall sewer, with frost casing, pile foundation, 

etc. 
Oct. 10, 1901. Sewers in Brooklyn; circular, 13' 6" dia. with 16'' brick arch; 

deep manhole construction. 
Jan. 1, 1903. Sewer construction in Flatbush, Brooklyn, showing various 
types, with foundations. 

Nov. 19, 1903. Method and cost of constructing a 30^ concrete sewer with 
brick arch, at Medford, Mass.; illustrated, with forms. 

Dec. 17, 1903. Pear-shaped concrete sewer with brick arch 12* thick; vert, 
dia. = 72'', hor. dia. = 72". 

Feb. 4, 1904. Wear and repairs of inverts in St. Louis sewers; with diagram 
of sewer; vert. dia= 15', hor. dia. = 20'; arch ring of cut stone. 

Feb. 18, 1904. A novel form of centering for 5-ft. egg-shaped sewer at 
Washington, D. C. 

Oct. 20, 1904. A steel center or form for constructing concrete sewers. 

Jan. 5, 1905. A new form of reinforced concrete-block sewer construction; 
illustration of sewer 42" dia. with 4" thickness of shell. 

Sept. 26, 1907. A 36" steel lap-welded pipe, f" metal, 18' lengths, used for 
extending sewer outfalls at Blackpool, England. 

Nov. 11, 1907. A new concrete-pipe joint; hot asphalt poured into grooves 
at ends of abutting cast concrete pipes. 

Mar. 26, 1908. Intercepting and outfall sewer at Waterbury; vert. dia. 4' 5*, 
hor. dia. 4' 6", crown thickness 6"; reinforced concrete. 

July 30, 1908. Sewers at St. Louis, constructed with collapsible steel center- 
ing: (1) vert. dia. 18' 6", hor. dia. 29' 0", crown thickness about 13"; 
(2) vert. dia. about 17' 0", hor. dia. 25' 0", crown thickness about 
13"; reinforced concrete. 



SEWER-WALLS, -FOUNDATIONS, -PIPE. 



1307 



Sewer Pipe. — Vitrified clay pipe is admirably adapted for sewers. 
It comes in lengths of 2 ft. 6 ins. for the smaller sizes (8 to 18 ins.). 




Fig. 9. 

while the large diameters, are made in 3-ft. lengths. The joints are 
cemented in place. Fig. 9 shows a section of three lengths. 

11. — Standard Salt-Glazed Vitrified Pipe. 
(Manufactured by Blackmer & Post Pipe Co., St. Louis.) 



Branches 

Curves, 

etc. 

Each. 



$16.25 
17.50 
20.00 
25.00 
30.00 



Inside 
Dia- 
meter. 


Thick- 
ness of 
Shell. 


Depth of 
Socket. 


Annular 
Space. 


Length 

of 
Sections . 


Weight 

per 

Foot. 


Car Load 
14 Tons. 


Price 

per 

Foot. 


Ins. 


Ins 


Ins. 


In. 


Ft. 


Lbs. 


Ft. 




27 


2H 




M 


3 


215 


129 


$3.25 


28 


2 3-32 




H: 


3 


230 


120 


3.50 


30 


234 




Vs 


3 


270 


108 


4.00 


33 


2% 


43^ 


1 


3 


320 


90 


5.00 


36 


23^ 




^ 


3 


365 


81 


6.00 



The above price list is subject to large discounts. 

12. — Extra Heavy Sewer and Culvert Pipe. 
(Manufactured by Evans & Howard, St. Louis.) 



Weight, per foot 

Thickness in inches 

Depth of Socket in inches 

No. of feet in carload of 1 5 tons. 
Price, per foot 





Diameter of Pipe in Inches 






12 


15 


18 


21 


24 


27 


30 


36 


52 


76 


103 


136 


170 


240 


305 


390 


m 


IH 


m 


m 


2 


2H 


2H 


2M 


3 


m 


3M 


4 


4 


43^ 


4^. 


534 


577 


400 


291 


220 


181 


125 


98 


77 


$0.75 


$1.00 


$1.50 


$2.00 


$2.50 


$3.25 


$4.00 


$6.00 



Sizes 18-in. and under in 2^-foot lengths, sizes 21-in. and over in 3-foot 
lengths, improved corrugated deep sockets and corrugated ends. 
Discount on application. 

Kinds of Sewers. — Sewers may be classified according to the special 
purposes for which they are designed or according to the material entering 
into their construction. Thus we have branch sewers, main-line sewers, 
intercepting sewers, trunk -line sewers, and outfall sewers. Trunk-line 
sewers are often projected through several towns, the expense being borne 
proportionately. An outfall sewer is the main stem of discharge to the 
outlet. Small sewers are built of terra cotta pipe, cast iron pipe, concrete 
pipe, etc.; the larger sewers are constructed or brick, concrete, reinforced 
concrete, etc. Wood-stave pipe enters into the design of the outfall sewer 
of the City of Los Angeles, projected as follows: 2400 ft. of 52-in , circular 
brick conduit; 4,400 ft. of 40-in. circular brick conduit; 16,900 ft. of in- 
verted syphon of 38-in. wood-stave pipe; 1,900 ft. of 40-in. circular brick 



1308 



05.— SANITATION. 



conduit; 5,800 ft. of 6-ft. oval brick and concrete tunnel ; 800 ft. of 40-in. 
circular brick conduit; 19,100 ft. of inverted syphon of 36-in. wood-stave 
pipe; 12,100 ft. of 40-in. circular brick conduit; 900 ft. of 6-ft. oval brick 
and concrete tunnel; 600 ft. of 40-in. circular brick conduit; 1,300 ft. of 6-ft. 
oval brick and concrete tunnel; 1,100 ft. of 24-in. cast iron pipe, to outlet 
in Pacific Ocean. 

Location of Sewers. — Sewers are most commonly located under the 
centers of streets, or on either side of broad avenues. Some cities however 
prefer to build only one sewer near one curb, in streets of ordinary width. 
In cities where alleys exist, the latter are often selected for the sewer lines. 

Manholes are street openings to sewers; they are built for inspection 
and cleaning of sewers, and for ventilation. Those systems of sewers which 
have been built without manholes are certainly at a disadvantage. For 
large sewers the manholes are built tangent to one side of sewer, in the form 
of a conic frustum; or, an entrance to side of sewer may be made by a 
slanting shaft with steps for descent, with a manhole entrance at top of 
steps. Manholes should be situated at bends of both alinement and grade 
of the sewer, so that a clean sight may be had through the sewer from one 
manhole to another. If this is impracticable by reason of the expense, 
lamp-holes may be substituted for them 
occasionally. These latter consist of 
a small circular shaft from the street 
to the top of sewer through which a lamp 
may be suspended to be sighted at from 
the nearest manhole in either direction. 
Fig. 10 shows a section of a circular man- 
hole frame and cover, to be set flush with t^. ^ ^ , , . , 
the street surface. ^ig- 10— Manhole. 

Catch Basins are placed along the gutters at the sides of streets, and 
especially at street corners, to act as settling basins for surface waters con- 
taining sand and dirt. The outlet from the catch basin to the sewer is near 
the top of the basin. They should be cleaned out before the deposit becomes 
too great. The styles used are innumerable. That shown in Fig. 11 is 





•'^^ 






Fig. 11. — Catch Basin. 

•nanufactured by James B. Clow & Sons, of Chicago, and is designed for 
street corners. 



MANHOLES, CATCH BASINS, MISCELLANY. 



1309 



EXCERPTS AND REFERENCES. 

The Sanitary Protection of the Water Supply of Baltimore, Md. 

(Eng. News, Dec. 5, 1901). — Illustrations: Standard designs for cesspools, 
privies and catchbasins in the drainage area of the Baltimore water supply. 

Electric Sewage Pumps, Septic Tanks and Contact Beds at Fond du Lac, 

Wis. (By G. S. Pierson._ Eng. News, May 22, 1902). — Illustrations: General 
view of works; nearer view of pump house and main carrier; general plan of 
sewage disposal works and sections of contact filter beds; storm water 
overflows in intercepting sewers; grit and screen chamber near sewage 
pumping station; plan and section of pumping station; sewage distribut- 
ing and effluent receiving chamber for contact beds. 

Treatment of Sewage in a Large Open Septic Tank at Worcester, 
Mass. (Eng. News, May 29, 1902).— Table. 

The Chicago Intercepting Sewer System (Eng. News, May 28, 1903). 

— Illustrations: Map of the system; driving sheet piling for trench of 16-ft. 
sewer; swinging-derrick with orange-peel bucket ; bricklaying in 1 6-ft. sewer; 
trenching machine on intercepting sewer; train of dump cars on sewer 
work; shield^ and details of shield for 20-ft. sewer; plan of intake and 
pumping station; sections of channels at pumping station; details of screen 
chamber or sand catcher at intake. 

The Sanitary Disposal of Municipal Refuse (Trans. A. S. C. E., 
Vol. L). 

Sewage Disposal for Country Residences (Eng. News, July 2, 1903). 
— Illustration of septic tank, the estimated cost of which is: 3 casks at 
75 cents each = $2.25; sewage siphon, $14.00; 100 ft. of farm tile, $1.00; 
pipe and plank, $2.50; labor and cement, $6.00; total, $25.75. 

The Northwestern Ave. Sewer at Indianapolis, Ind. (Bv W. Buehler. 
Paper, Ind. Eng. Soc.; Eng. News, July 16, 1903). — Illustrations: Map; 
overflow section and junction with main sewer; dam in combined sewer for 
diverting sewage flow into separate sewer; construction of concrete floor 
and sides of brick and concrete overflow sewer; forms and I-beams in place 
for roof of concrete and steel rectangular sewer. 

A Table Giving Quantities of Cement and Sand and of Cement Mortar 
for Sewer Pipe Joints (By J. N. Hazlehurst. Eng. News, Feb. 25. 1904). — 



Quantities op Cement, Sand and op Cement Mortar for Sewer 

Pipe Joints. 

For Each 100 ft. of Sewer (With Portland Cement 375 lbs. 
net per bbl.) 





L'gth, 


Mortar 




Proportions 


: 1 Cement to 




Size 

of 

Pipe. 


1 Sand. 


2 Sand. 




ft. 


yds. 






No. 






No. 








Cement, 


Sand, 


ft. to 


Cement, 


Sand, 


ft. to 








bbls. 


cu. yd. 


bbl. 
Cemt 


bbls. 


cu. yd. 


bbl. 
Cemt 


6-in 


2i 


0.003 


0.01248 


0.00201 


803 


0.00855 


0.00252 


1,168 


8-in 


2- 


0.038 


0.15808 


0.02546 


633 


0.10830 


0.03192 


923 


10-in 


2h 


0.058 


0.24128 


0.03886 


410 


0.16530 


0.04872 


605 


12-in 


2 


0.089 


0.37024 


0.05963 


270 


0.25365 


0.07476 


394 


15-in 


2 


0.123 


0.51268 


0.08241 


195 


0.35055 


0.10332 


285 


18-in 


2 


0.167 


0.69472 


0.11189 


144 


0.47595 


0.14018 


210 


20-in 


2i 


0.237 


0.98592 


0.15879 


101 


0.67545 


0.19908 


148 


24-in 


n 


0.299 


1.24384 


0.20033 


80 


0.85215 


0.25116 


117 


27-in 


3 


0.492 


2.04672 


0.32964 


49 


1.40220 


0.41328 


71 


30-in 


3 


0.548 


2.27968 


0.36716 


44 


1.56180 


0.46032 


64 


36-in 


3 


0.849 


3.53184 


0.56883 


29 


2.41965 


0.71316 


41 



1310 



^.--SANITATION, 



The Wear of Sewer Inverts (By E. A. Hermann. Eng. News, Feb. 4, 
1904). — ^The materials most commonly used for sewer construction are 
vitrified clay pipe, vitrified brick, common building brick (more or less 
unbumed), and concrete. In the sewers of St. Louis, which are on the 
combined system, the grades range from 0.2% to 2%, average about 0.5% 
for sewers more than 5 ft. in diameter and about 1% for the smaller sewers. 
The vitrified clay pipes show no appreciable wear after about 35 years' use ; 
these sewers are mostly laterals, and have, of course, the smallest discharge, 
though some of those in the business and manufacturing _ sections of the 
city carry a constant stream from 1 to 3 ins. deep, containing more or less 
acids, scalding hot water and steam. The pipes vary from 12 to 24 ins. in 
dia., except a few lines, which are 30 to 36 ins. in dia. The vitrified brick 
(in use about 12 years) also shows no appreciable wear. The inverts of 
sewers built of common brick begin to show some wear after about 3 years 
of service, and after about 30 years' use the first ring of brick is worn away 
from 2 ins. to nearly its whole depth of 4 ins. This wear varies greatly in 
sewers of different size, grade, quantity and quality of sewage, and hard- 
ness of brick; the average life of such brick appears to be about 40 years. 
Illustrations: Method employed in repairing badly worn sewer inverts; 
section of large sewer at St. Louis, Mo. 

A New Jointing Material — Sulphur and Sand — for Sewer Pipes (By 

Alex. Potter. Eng. News, Mar. 10, 1904). — The following table gives infor- 
mation concerning the cost and amount of material in making sulphur-sand 
joints: 

Approximate Costs. 





Amt. of 












Per foot 


Size of 


Mixture 


Mixture 


Gasket. 


Fuel. 


Labor. 


Total. 


Lengths. 


Pipe. 


Lbs. per 




















Joint. 












3-Ft. 


2-Ft. 


24-in. 


10.0 


$0,125 


$0.02 


$0.02 


$0.13 


$0,295 


$0.10 


$0.15 


22-in. 


9.0 


.1125 


.02 


.02 


.13 


.282 


.095 


.14 


20-in. 


8.0 


.10 


.02 


.02 


.12 


.260 


.09 


.13 


18-in. 


7.0 


.087 


.02 


.02 


.11 


.247 


.08 


.12 


15-in. 


5.5 


.069 


.01 


.01 


.10 


-.187 


.065 


.095 


12-in. 


4.2 


.052 


.01 


.01 


.09 


.162 


.055 


.08 


10-in. 


3.3 


.041 


.01 


.01 


.08 


.141 


.045 


.07 


8-in. 


2.5 


.031 


.01 


.01 


.07 


.121 


.04 


.06 



The exclusion of ground water will increase the capacity of the sewer 
from 10% to 100% through territory subject to the admission of ground 
water, so that the increase in cost becomes a trifling matter, especially 
when it is considered that tree roots are kept out of the sewer. 

Refuse Destructor Combined With Electric Light Plant at West- 
mount, P. Q. (Eng. News, May 24, 1906). — Table of estimated costs. 

Specifications for Refuse Destructor, Borough of Richmond, New 
York City (Eng. News, Dec. 6, 1906).— The first thorough-going, if not 
absolutely the first, specifications and call for bids for a refuse destructor 
designed to produce heat for lighting or power purposes in the U. S. 

Cost of Shallow and Deep Sewer Trenches (By J. G. Palmer. Eng. 
News, June 25, 1908). — Cost -data tables. 

Principles of Sewage Purification on Land (By Rudolph Hering. 
Eng. News, May 27 and June 3. 1909). 

Rainfall, and Run=Off in Storm^Water Sewers (By C. C. Gregory. 
Trans. A. S. C. E., Vol. LVIII).— Formulas, diagrams and tables. Table 1, 
(Comparison of rainfall and discharge from the 6th Avenue sewer, N. Y. City, 
the drainage area of which is 221 acres; Table 2, Short -time storms; 
Table 3, Long-time storms. 

Cost of a 66=Inch Brick Sewer at Gary, Ind. (By E. M. Scheflow. Eng. 
Rec,. Jan. 2, 1909). — ^Total cost of sewer per lin. ft.: Excavation by machine; 
to. 50 3; excavation by hand, $1.25; pumping, $1.61; sheeting, $0,368, 



MISCELLANEOUS DATA. 1311 

laying, $1.32; backfilling by machine, $0,406; backfilling by hand, $0,136; 
hauling material, $0,607; superintendence and general, $0.30; materials, 
$1.81; depreciation of machinery, repairs and the like (estimated), $1.50; 
total. $10,122. 

Louisville Sewerage Improvements (By R. DeL. French. Eng. News, 
Oct. 14, 1909). — Illustrated: — Standard sections of circular concrete 
sewers, with table of dimensions and quantities, for hard and soft ground; 
Typical horseshoe section of reinforced-concrete sewer, 15 ft. high; Typical 
semi-elliptical section of reinforced concrete sewer, 10 ft. high; Typical 
junction chamber, reinforced-concrete sewers: Combined sewer and drain. 

Aerial Distribution of Sewage over Percolating Filters (By Wm. Gavin 
Taylor. Eng. News, Nov. 11, 1909). — Illustrations of nozzle; diagrams of 
distribution effected by same; pressure undulating valves, described and 
illustrated. 

The Improved Water and Sewerage Works of Columbus, O. (By J. H. 

Gregory. Trans. A. S. C. E., Vol. LXVIL, June, 1910).— Illustrations: 
Lime saturators; mixing tanks; sewage pumping station; purification works; 
septic tanks; gate house; etc. 

Modern Procedure in District Sewer Design (By W. W. Homer. Eng. 
News, Sept. 29, 1910). — Diagrams: (a) Rainfall curves for St. Louis; (b) 
Approximate curves for designing vitrified pipe and circular egg-shaped 
sewers under 60 ins. mean diameter; (c) Approximate curves for designing 
circular sewers 4 to 24 ft. in diameter; (d) Capacities of standard horse-shoe 
sewer. Illustrations : Details of sewer inlets. Table : Data for branch sewer. 

The Design of Storm=Water Drains (Eng. Rec, Oct. 29, 1910).— Tables 

of heavy precipitations, and discussion of typical precipitation curves. 

Illustrations of Interesting Designs: — 

Description. Eng. News. 

Adjustable basin and manhole covers July 11, 1901. 

A centrifugal separator for shavings and dust Sept. 26, '01. 

Plan of small sewer system, at Lake Bluff, 111. June 6, '01. 

Deep manhole construction on 60th St. sewer tunnel, B'klyn Oct. 10, '01. 

Septic tank and double contact filter beds, Glencoe, 111. Oct. 24, '01. 
Proposed light refuse crematory on dumping pier, N. Y. City April 17, '02. 

Steel plate mine ventilating fan, Modoc C. & M. Co., Ohio June 19, '02. 

Plan and details sewage purification works, Depew, N. Y. June 26, '02. 
Sewage disposal wks., filter beds and septic tk., Id. Pk. Resort July 24, '02. 

Method of anchoring a rein.-coiic. lining for septic tank Aug. 21, '02. 

Couplings for rods used in cleaning sewers, conduits, etc. Sept. 4, '02. 

A new German automatic flush tank Nov. 27, '02. 

The 64th St. sewer tunnel and outlet tunnel, Brooklyn Jan. 1, '03. 
General arrangement of sewage sludge treatment apparatus, 

Cassel, Germany Jan. 15, '03. 

Cross-section of 72" concrete and brick sewer. Corning, N. Y. Dec. 17, '03. 

A novel center and form for concrete sewer work Feb. 18, '04. 

Catch pit, septic tank and automatic flush tank for a jail Mar. 3, '04. 

A steel form for concrete sewers Oct. 20, '04. 

A sewer pipe centering device Feb. 8 , '06. 

Sprinkler motor for percolating sewage filter, England May 2, '07. 

Rein. -cone, intercepting and outfall sewer, Waterbury, Conn. Mar. 26, '08. 

Septic tank and percolating filters, Univ. of Minn. June 25, '08. 

Plans of Winston-Salem intercepting sewer, and details June 25, '08. 

Septic tanks and percolating filters, Washington, Pa. July 16, '08. 

Steel centering for Harlem Creek sewer July 30, '08. 

Adjustable metal forms for construction of large sewer Oct. 8, '08. 

Septic tank and filter bed for residence, Philippines Oct. 8, '08. 

A household cesspool and overflow, with details Dec. 17, '08. 

Reinforced-concrete outlet sewer, 10^ ft. dia. Jan. 28, '09. 

Details of 5 J and 6-ft. concrete-block sewers, Toledo, O. Feb. 4, '09. 

10-ft. sewers, rein. -cone, pile found., Kansas City Jan. 27, '10. 

Plans and cross-section of Milwaukee refuse incinerator July 21, '10. 



1312 ^,— SANITATION, 

Description. Eng. Rec. 

Section of Stony Brook conduit, and cableway carriage Feb. 6, *09. 

Design of a 15-ft. drop in a large sewer Feb. 6, '09. 

Sections of rein. -cone, sewers, Louisville, Ky. May 1, '09. 

Typical manholes, U. S. Military Academy May 8, '09. 

13.12-ft. dia., circular rein. -cone, conduit drain, Mexico June 5, '09. 

Ventilating manhole cover with dust-catching box June 26, '09. 

Molded concrete pipe and storm drains, Newark, N. J. Nov. 6, '09. 

Storm water drain, 3 ft. diam., with cost Dec. 18, '09. 

Details, septic tanks and contact beds. Grand Canyon Jan. 29, '10. 

Cost of catch basins in Boston: illustrated Mar. 12, '10. 

Cross-section rein. -cone, storm sewer (13 x 11 ft.), on piers Mar. 19, '10. 

Rein. -cone, sewers in soft ground, Seattle, Wash. Oct. 22, '10. 
Sewer sections, canal crossing and siphon shaft, N. trunk 

sewer, Seattle, Wash. Dec. 10, '10. 

New method of handling sewage sludge, in Germany Dec. 10, *10. 

Intercepting sewer and outfall at New Bedford Dec. 17, '10. 

Heavy plain concrete sewers (2H ft. rad., 8'' ring) at Albany Dec. 17, '10. 



66.— IRRIGATION. 

General Discussion. — Those who are specially interested in irrigation 

matters will find much valuable information in the Bulletins issued by the 
U. S. Department of Agriculture, and in the Water-Supply and Irrigation 
Papers, Annual Reports, etc., of the U. S. Geological Survey. Lists of the 
various Documents may be had on application, and the papers are gen- 
erally for free distribution. 

The Problems of Irrigation are broad and intricate, the local conditions 
of soil, water, climate, crops, etc., requiring much study, often extending 
over considerable periods of time. The amount of water required will 
depend upon the kind of crops, as well as upon the soil, and to a certain 
extent upon the climate; also upon the amount of seepage and evaporation 
before it reaches the land to be irrigated ; and last of all upon the method of 
irrigation, which may be more or less wasteful. If there is any rainfall, the 
records of monthly precipitation, extending over say 15 to 20 years, should 
be examined for one-, two- and three- year periods of greatest drought, for 
a possible reduction in the artificial supply. (See Water Supply, page 1194.) 

The Unit of Land Area in irrigation is the acre, equal to 43560 sq. ft. 
A square 1-acre tract is 208.71 ft. square; a square 2-acre tract is 295.16 
ft. square; a square 3-acre tract is 361.50 ft. square; a square 4-acre tract 
is 417.42 ft. square; a square 5-acre tract is 466.69 ft. square; a square 
10-acre tract is 660 ft. square = i mile square; a square 20-acre tract is 
933.38 ft. square; a square 40-acre tract is 1320 ft. square = i mile square; 
a square 80-acre tract is 1866.76 ft. square; a square 160-acre tract is 2640 
ft. square = ^ mile square. 

The Units of Flow of water are the "inch" (miner's inch) and the cubic 
foot per second. 

The "Inch" is the volume of water (say in cubic feet) which will flow 
through a vertical standard orifice one-inch square (say in one minute) 
under a given pressure head (say 5 to 6^ inches, and fixed by State law). 
This amounts to about 1.5 cubic feet per minute (see Table 1). The "inch" 
is convenient in delivering water to small users because the quantity being 
delivered is apparent at a glance. Where the delivery calls for more than 
one "inch" a box is constructed with a long horizontal slot one inch or 
more in height and provided with a slide and scale.* By manipulating the 
slide, the slot may be elongated so as to deliver the number of inches re- 
quired; but as the end contractions remain constant it is apparent that 
the rate of discharge increases faster than the scale readings indicate. 

Table 1 is based on one "inch" discharging 1.5 cu. ft. per min. 

1. — Equivalents of Discharge op 1 "Inch," at 1.5 Cu. Ft. per Min. 



Duration of Flow 


U. S. 
Gallons. 


Cubic 
Feet. 


Tons of 
2000 Lbs. 


Acre-Ins. 


Acre-Ft. 


1 Second. 


.187 
11.221 
673.25 
16157.9 
484738. 


0.025 
1.5 
90. 
2160. 
64800. 


.00078 
.0469 
2.8125 
67.5 
2025. 


.0000069 
.0004132 
.024793 
.59504 
17.851 


00000057 


1 Minute 


.00003444 


1 Hour 


.0020661 


1 24-Hour Day 


.049587 


1 30-Day Month 


1.4876 



Note. — 1 cu. ft. = 7.48052 U. S. liquid gallons = .03125 tons (at 62.5 lbs. 
per cu. ft) = .000 275 4821 acre-in. = .000 022 956 84 acre-ft. 

The Cubic Foot per Second is a unit rate of flow, used mainly in gaging 
streams and canals. It is definite, whereas the "Inch" is more or less 
indefinite. Note that both the "Inch" and the Cu. Ft. per Sec. are rates of 

* See Foot-note on following page. 

1813 



1314 



m.—IRRIGATIOl^ , 



*flow, and that where definite quantities are demanded the element of time 
should be included, as 10 "inches" for one month, or 0.25 cu. ft. per sec. for 
one month, etc. (See Tables 1 and 2.) Contracts for water based on any 
fixed number of cu. ft. per sec. throughout the season are unfair to the 
user, or wasteful, or both. There are many and various recording instru- 
ments on the market for determining the rates of discharge. They are self- 
registering and consist usually of a fluctuating line drawn on profile paper 
wound around a cylinder, by a pen- or pencil point traveling longitudinally 
to represent the time, and by the cylinder oscillating as the water rises or 
falls in the measuring chamber, in which a float is connected with the 
instrument. 

2. — Equivalents of Discharge of 1 Cubic Foot per Second. 

(1 cu. ft. per sec.=about 40 "inches.") 



Duration of Flow. 


U. S. 
Gallons. 


Cubic 
Feet. 


Tons of 
2000 Lbs. 


Acre-Ins. 


Acre-Ft. 


1 Second 


7.48 
448.83 
26 929.9 
646 317. 
19 389 508. 


1 

60 

3 600 

86 400 

2 592 000 


.031 

1.875 
112.5 
2 700. 
81 000. 


.00028 
.01653 
.99173 
23.801 
714.05 


.000023 


1 Minute 


.001377 


1 Hour 


08264 


1 24-Hour Day 

1 30-Day Month 


1.98347 
59.504 



Note.— 1. cu. ft. = 7.48052 U. S. liquid gallons = .03125 tons (at 62.5 lbs. 
per cu. ft.) = .000 275 4821 acre-in. = .0000 229 5684 acre-ft. 

The Units of Volume of water are the Acre-Foot and the Acre-Inch, 
the latter being only occasionasly used, and ts of the former. 

The Acre-Foot is the volume of water which will cover an acre 1 ft. 
deep = 43560 cu. ft. Mr. Elwood Mead says of this unit: 

It is a convenient unit for selling stored water, since the capacity of reservoirs 
can be measured by the same unit. Contracts in which the acre-foot is used provide 
for the delivery of water on the demand of the irrigator, or at intervals rather than In 
continuous flow, and canal companies have hesitated about adopting this unit 
because of a fear that satisfactory arrangements for delivery could not be made, 
but that more water would be called for at some time than the canal could supply, 
while at other times the entire volume would run to waste. Wherever the acre-foot 
has been adopted it has proved acceptable to irrigators, because they share in the 
benefit resulting from care and skill in distribution. 

Table 3 gives the rates of flow equivalent to an acre-ft. in a 24-hour 
day, and Table 4 gives the rates of flow equivalent to an acre-ft. in a 30-day 
month. 

3. — URATES OF Flow for Discharge op 1 Acre-Foot in 1 Day 

(24 Hours). 
(For 1 acre-inch, divide by 12.) 



Flow— 


U.S. 
Gallons. 


Cubic 

Feet. 


Tons of 
2000 Lbs. 


Acre-Ins. 


Acre-Ft. 


Per Second 


3.77 
226.29 
13577.14 
325851.5 
9775544. 


.504 
30.25 
1815. 
43560. 
1306800. 


.0158 
.9453 
56.7188 
1361.25 
40837.5 


.000139 
.008333 
.5 
12. 
360. 


.0000116 


" Minute 


.0006944 


' ' Hour 


.04167 


" 24-Hour Day 

" 30-Day Month 


1. 
30. 



* Water sold by the inch by any individual or corporation shall be 
measured as follows, to wit: Every inch shall be considered equal to an 
inch-square orifice under a five-inch pressure, and a five-inch pressure 
shall be from the top of the orifice of the box put into the banks of the ditch 
to the surface of the water. Said boxes or any slot or aperture through 
which such water shall be measured shall in all cases be six inches perpen- 
dicular, inside measurement, except boxes delivering less than twelve inches, 
which may be square, with or without slides. All slides for the same shall 
move horizontally, and not otherwise, and said box put into the banks of 
ditch shall have a descending grade from the water in ditch of not less than 
one-eight of an inch to the foot. (General Statutes of Colorado, sec. 3472.) 



UNITS AND DUTY OF WATER. 



1316 



4. — Rates op Flow for Discharge of 1 Acre-Foot in 1 Month 

(30 Days). 
(For 1 acre-inch, divide by 12.) 



Flow— 


U. S. 
Gallons. 


Cubic 
Feet. 


Tons of 
2000 Lbs. 


Acre-Ins, 


Acre-Ft. 


Per Second 


.1257 
7.543 
452.57 
10861.7 
325851.5 


.0168 
1.0083 
60.5 • 
1452. 
43560. 


.000525 
.03151 
1.8906 
45.375 
1361.25 


.00000463 
.0002778 
.016667 
.4 
12. 


000000386 


" Minute 


.00002315 


" Hour 


.0013889 


•• 24-Hour Day... 
•• 30-Day Month. . 


.033333 



The Duty of Water is the amount of land which a unit volume of water, 
properly applied, can irrigate; or, conversely, it is measttred by the amount 
of water required to irrigate one acre of land. This amount of water may 
be measured by volume, as in storage, or by (average) flow, as in distribution. 

The (miner's) "Inch" is often cited by many irrigators as the amount 
required for one acre, and by others as the amount required for two, or 
even three, acres of land. By referring to Table 1, perceding, it can readily 
be seen that a flow of 1 "inch" is equal to about 1.5 acre-ft. per month. 
If flowing constantly this would amount to about 6 acre-ft. for an irrigation 
period of 4 months (say from May 15 to September 15), and about 7.5 
acre-ft. for an irrigation period of 5 months (say from May 1 to October 1). 
For shorter periods it would be proportionately less. In the Rocky Mountain 
regions the irrigation season ranges from 100 to 150 days; in parts of 
Southern California it is used throughout the year; while in the colder 
climates, as in some parts of Montana, the period is often shortened to two 
months. 

Assuming that a continuous (average) flow of one "inch," throughout 
the irrigation season, will irrigate If acres, then from Table 1 this is equiva- 



lent to 1 cu. ft. per sec. for ( If X 'no^^) 70 acres of land. If the length 

(59 504 X 4 



70 



') 



3.4 



acre-ft. per acre. Similarly, the equivalents in cu. ft. per sec. and in acre-ft, 
can be found for any other assumed duty in acreage per "inch." 

The Cu. Ft. per Sec. is another unit of flow in measuring the duty of 
water. Assuming 1 cu. ft. per sec. will irrigate 60 acres, we see by Table 2 
that this is equivalent to 59.5 acre-ft. per month, or about 1 acre-ft. per 
acre per month, or 4 acre-ft. per acre for 4-months period of irrigation. 



Again, assuming 1 cu. ft. per sec. for 80 acres, we would require 



59.504 
80 



or 



about M acre-ft. per acre per month, or 3 acre-ft. per acre for 4-months 
period of irrigation. 

The A.cre-Foot is now generally conceded to be the best duty unit by 
most experts in irrigation, but it must be admitted that the "Inch" and the 
Cu. Ft. per Sec. have some merits as supplementary factors. The acre-ft. is 
a definite quantity both in volume (43560 cu. ft.) and in depth of equivalent 
precipitation (12 ins.); and by the use of the preceding tables, it is easily 
converted into other units. It is well to remember that 1 cu. ft. per sec. == 
ahout 40 "inches" = about 2 acre-ft. per 24-hour day = about 60 acre-ft. per 
month. One great advantage of the acre-ft. is that it is convenient to apply 
in measuring the storage or source of water supply, as well as the direct 
quantity required in irrigation, the contents of reservoirs being stated usually 
in acre-ft. = total storage capcaity in cu. ft. divided by 43560. 

Tables 5, 6, 7 and 8, following, are from Experiment Stations Bulletin 86, 
U. S. Dept. of Agriculture. The first three tables show the duty of water 
under varying conditions, while Table 8 shows the length of the irrigation 
seasons. 



1316 



-IRRIGATION. 



5. — Duty of Water where Measurements Were Made on Small 
Canals or Laterals. 



Location. 



Acre-Ft. 



Location. 



Acre-Ft. 



Cronqulst farm, Utah 2.60 

Long farm, Idaho 2.40 

Gage Canal, Cal 2.24 

Canal No. 2, Wyo 2.53 

Vance farm, Ariz 2.82 



Biles Lateral, Colo i . 82* 

Middle Cr. Ditch, Mont 2.10 

Daggett farm, Nebr 2 47 

Mean of all above 2.37 



* Low duty due in part to scanty supply of water. 



6. — Duty of Water where Measurements Were Made at Margin 

OF Fields. 



Location. Acre-Ft. 

J lateral, Wyoming, oats 1.55 

J lateral, Wyoming, corn 70 

Farm, Edgar Wilson, Idaho 1.48 

Lowest division. Gage Canal 1,78 

Mean of measurements at Bozeman, Mont., Exp. Sta 1 . 20 



7. — Duty op Water where Losses in Main Canal Are Included. 



Name of Canal. 



Acre-Ft. 



Name of Canal. 



Acre-Ft. 



Pecos Canal, New Mexico 6.61 

Mesa Canal, Ariz 3.81 

Butler Ditch, Utah 6.24 

Brown and Sandford Ditch, 

Utah 5.32 



Upper Canal, Utah 6.30 

Amity Canal, Colo 4.92 

Rust Lateral. Idaho 5.06 

Average 5.47 



8. — ^Duration of Irrigation Period on Some Main Canals. 



State. 


Canal. 


Days. 


State. 


Canal. Days, 


Cal 


. . Gage Canal 

. . Mesa Canal 

Pecos Canal. 


365 

365 

. 175 


Utah. . . 
Idaho . . 
Colo. . . . 
Utah. . . 
Nebr.... 
Mont. . . 
Wyo... 


. . Lower Canal 160 


Ariz 

N. Mex 


. . Boise & Nampa Canal . . 155 
. . Amity Canal 140 


Utah. . . . 
Utah 


. . Brown & Sanford Ditch 175 
. . Butler Ditch 175 


. . Logan & Richmond Canal 125 
. . Gothenburg Canal 120 


Utah. . . 


. . Upper Canal 


165 


. . Middle Creek Ditch 95 


Utah 


. . Big Ditch 


165 


. . Canal No. 2, Wheatland. . 60 


Utah 


. . Green Ditch 


165 













DUTY OF WATER. CANALS, CONDUITS. 



1317 



Table 9, following, is from King's "Irrigation and Drainage," the 
second column having been adopted by King from Wilson's "Manual of 
Irrigation Engineering," and includes all losses from the head of the canals: 

9. — Duty of 1 Cu. Ft. per Sec, and No. of "Inches" in 10 Days, 
In Various Countries. 



Name of Country. 



No. of Acres 
per Second-Foot, 



No. of Inches 
per 10 Days. 



Northern India 

Italy 

Colorado 

Utah 

Montana 

Wyoming 

Idaho 

New Mexico 

Southern Arizona . . . 
San Joaquin Valley. . 
♦Southern California 



60 to 150 

65 to 70 

80 to 120 

60 to 120 

80 to 100 

70 to 90 

60 to 80 

60 to 80 

100 to 150 

100 to 150 

150 to 300 



3.967 

3.661 

2.975 

3.967 

2.975 

3.4 

3.967 

3.967 

2.38 

2.38 

1.587 



to 1.587 
to 3.4 
to 1 . 983 
to 1.983 
to 2.38 
to 2.644 
to 2.975 
to 2.975 
to 1.587 
to 1.587 
to 0.793 



* The comparitively high duty of water in California is due to the extra 
care taken in construction of conduits to prevent losses, and to economy of 
distribution. 

Canals. — Canals in earth are usually made wide and shallow with side 
slopes varying from ^ to 2 horizontal on 1 vertical. It is best to select such 
side slopes, depending on the earthy material, as will be maintained during 
flow. The cross-section of a canal is determined often by the allowable 
velocity which may not exceed, ordinarily, about 2| feet per second in ordi- 
nary soils. If the soil is light 2 ft. is about the maximum, while if hard and 
gravelly 3 or 4 feet may be exceeded. The velocity of flow in a canal is 
dependent on the hydraulic radius, the grade or vslope, and the roughness n 
of wetted surfaces (see Kutter's formula, page 1167). Canals with rubble-, 
concrete- or cement lining are often used where water is scarce and valuable. 

In estimating the flow by Kutter's formula, use w = .0225 for ordinary 
earthen canal with clean bed, and n==.OZb for bed in bad order having stones 
and weeds in great quantities. (See page 1168.) 

Wilson, in his Manual of Irrigation Engineering, gives the following data 
relative to some great perennial canals: 

10. — Data on Some Great Perennial Canals. 
(See Wilson's Manual of Irrigation for full and extended data.) 



Name of Canal. 



Locality 


L'ngth 
Miles. 


Capacity 
Sec.-Ft. 


Grade. 


Bed- 
Width; 
Feet. 


Utah 


150 


1000 


1 In 5280 


50 


Idaho 


70 


2585 


1 " 2640 


40 


N. Mex. 


75 


1100 


1 " 6707 


45 


Cal. 


93 


1500 


\ " 5280 


70 




67 


600 


1 •• 5280 


32 


•' 


32 


700 


1 " 6600 


80 


Ariz. 


41 


1000 


1 " 2640 


36 


Colo. 


70 


1184 


1 •• 3000 


40 




50 


2400 


1 " 2112 


65 



Depth, 
Feet. 



Bear River Canal 

Idaho Min. & Irrig. Co. Canal. . 

Pecos Canal 

Turlock Canal , 

King's R. & San Joaquin Canal 

Calloway Canal 

Arizona Canal 

Highllne Canal 

Del Monte Canal 



7 

10 

6 

7.5 
4.5 
3.5 
7.52 
7 
5.5 



Conduits and Flumes frequently take the place of canals even in locali- 
ties other than at crossing or streams. The flumes are usually constructed 
of wood (sometimes of steel), generally rectangular in cross-section, and 
supported on trestle work of the same material as the flume. The V-shaped 
flume is seldom used in connection with large irrigation projects, unless 
combined with logging. The semi-circular flume, of wood-stave pipe con- 
struction, or of steel, has the merit of giving the maximum velocity of 



1318 m.— IRRIGATION, 

flow for a given area of wetted cross-section. Pipe lines of wooden staves 
or of steel are frequently used as inverted syphons for conveying water 
across valleys where cost of flumes on high trestles would be very expensive. 
Wooden flumes may be made tight (1) by tangential compressive forces as 
v/ith adjustable rods or wedges;^ (2) by caulking and tarring the joints; 
(3) by using two thicknesses of siding and bottom plank with tarred paper 
between. Steel flumes are caulked similarly to riveted steel pipe. 

EXCERPTS AND REFERENCES. 

Noteworthy Water Storage and Irrigation Works of Southern Cali- 
fornia (By R- Fletcher. Eng. News, Aug. 22, 1901). — Descriptions of the 
following: System of the San Diego Land and Town Co.; Southern Cali- 
fornia Mountain Water Co.; The Morena Dam and Reservoir; The Barrett 
Dam and Reservoir; The Lower Otay Dam and Reservoir; The Upper 
Otay Dam and Reservoir: 

Irrigation System of the Arkansas Valley Sugar Beet & Irrigated 
Land Co., Colo. (By W. P. Hardesty. Eng. News, Nov. 13, 1902).— De- 
scriptions of Fort Lyon Canal System (canal, dam, head-gates, regulating 
gates, stream crossings, division gates); Kicking Bird Canal; Santava 
Canal; Lone Wolf Canal (reservoir system); Comanche Canal; Pawnee 
Canal; Amity Canal System (canal, dam, head-gate, secondary head-gate); 
Buffalo Canal. Ten illustrations of structures. 

Establishing Irrigation Canal Tangents So Cut and Fill Will Balance 

(Eng. News, Aug. 10, Sept. 21, 1896).— Illustrated. 

A Diagram to Aid the Location of Small Irrigation Canals (By P. Mc- 

Geehan. Eng. News, Feb. 1, 1906). — Also table for laying grade line on 
small irrigation canals. 

An Underflow Canal Used for Irrigation at Ogalalla, Nebr. (By S. C. 

Slichter- Eng. News, July 5, 1906). — Map. Table of flow. 

The Truckee=Carson Project of the U. S. Reclamation Service (By 

W. P. Hardesty. Eng. News, Oct. 18, 1906). — Illustrations: Details of 
diversion dam and head-gates; lining in rock cut; junction of earth and 
concrete-lined tunnel; method of lining tunnel; details of waterway, with 
Tain tor gates and mechanism for operation; a simple form of fall or drop 
in canal; details of headworks of distributing system; details of combined 
fall and waterway. Tables of cost data. 

Effect of Changes in Canal Grades on Rate of Flow (By F. W. Hanna. 

Eng. News, Nov. 21, 1907). 

Lining Ditches and Reservoirs to Prevent Seepage Losses (Eng. News, 
Dec. 5, 1907). — Illustrated methods of lining. 

Cost of Canal Work on the Lower Yellowstone Project (Eng. News, 
July 16, 1908). 

Depth of Water in Irrigation Canals (By C. E. Grunsky. Eng. 
News, Sept« 10, 1908). — ^Table 1: Effect of water depth upon required 
amount of excavation for irrigation canals. Table 2: Required fall for 
canals of varying capacity. 

Cost Data of the Cold Springs Reservoir Construction, Oregon, U. S. 
Reel. Serv. (Eng. News, Nov. 12, 1908). 

Earthwork Diagrams for Estimating Quantities in Small Irrigation 
Canals in Level Sections (Eng. News, June 10, 1909). 

Cost of a Large Irrigation Canal (Eng. Rec. Jan. 2. 1909).— Table of 

unit costs on about 4,400 ft. of the south canal of the Uncompahgre Proiect 
of the U. S. Reel. Serv. The canal has a bottom width of 40 ft., side slopes 
of 2 to 1, and a depth of water of 8.3 ft. 

Field Work in Locating Irrigation Ditch and Canal Lines (By A. B. 

Bartlett. Eng. News, May 26 1910). — Small Ditches: Ditches designed to 
irrigate 100 to 200 acres may be laid out by setting stakes on the lower side 
of the ditch, determined by the level only. Ditches 5 to 10 ft. on the bottom 



MISCELLANEOUS DATA. 1319 

and 1 to 3 ft. deep require more care: the slope stakes on the lower side of 
the ditch are set first for about 1,500 ft., after which they are lined up by eye 
to tangents and curves nearly conforming to the contour which the slope 
stake on the lower side of the ditch would naturally follow, but improving 
on this alinement by increasing or decreasing the cut in order to have good 
curves and tangents, etc. Ditches over 10 to 12 ft. on the bottom generally 
require the methods of railway location and cross-sectioning. Side Slopes. 
Lower side of ditch, from 1 on 1^ to 1 on 3 in excavation, and from 1 on 2 to 
1 on 4 for fill on lower bank. On side-hill work, cut on upper side of ditch 
is made from 1 on 1 to 1 on 2, for earth. Greatest velocities, in feet per 
second, permissible in different classes of material are as follows: Sandy soil, 
1.3; loose gravelly soil, 2.5; firm soil and firm sandy loam, 3.0; gravel, 3.6; 
firm gravelly soil, 5.3; loose rock and shale, 6.0 to 7.0; solid rock, 7.0 to 15.0. 
Kutter's formula: Use n==0.03 or 0.025 for clean ditches in ordinary earth. 

Drainage of Irrigated Lands with Special Reference to Experiments in 
Utah (By C. F. Brown. Farmers' Bulletin No. 371, U. S. Dept. or Agric, 
Oct. 2, 1909; Eng. News, Oct. 13, 1910).— Illustrations: (1) Method of 
grading drainage trench; (2) Plan of drainage; (3) Box drains with and 
without bottoms; (4) Relief well and method of laying box drain through 
soft ground. The tract drained consisted of 31.5 acres. Cost. — ^Tile (1000' 
of 10" @ $0.18i= $182.50, 1012' of 8" @ $0.12= $121.14, 330' of 6" @ $0.06 = 
$19.80, 650' of 5" @ $0,058 = $37.70, 2 6" on 8" wyes @ $0.72= $1.44) 
$362.88; Hauling (30 tons U miles @ $0.30 per ton-mile) $13.50; Digging 
(134 rods @ $0.52 = $69.68, and 42 rods @ $0.40 = $16.80) $86.48; Laying 
tile, $18.75; Filling trench, $10.00. Total cost, $491.61 = $15.60 per acre. 
Labor, $2 per 10 hours; tile layers, $2.50; man with team, $3 per day. 

Illustrations of Irrigation Structures: — 

Description. Eng. News. 

Automatic drop shutters for irrigation dam, India June 4, 1903. 

Details of masonry and steel head-gate for irrigation canal Aug. 13, '03. 

Reinforced-concrete siphon on an irrigation canal, Spain Aug. 1, '07. 

Pumping plant of the Hvmtley irrigation project Sept. 3, '08. 

Eng. Rec. 
Construction of "cut-and-cover" canal, Mojave desert Apr. 3, *09. 

Economic sections for earth canals July 31, '09. 



67.— WATERWAYS.* 

The Suez Canal, connecting the Mediterranean and Red Seas, was begun 
in 1859 and completed in 1869. Its total length is 90 miles, of which about 
two- thirds is through shallow lakes. The material excavated was usually 
sand, though in some cases strata of solid rock from 2 to 3 feet in thickness 
were encountered. The total excavation was about 80,000,000 cubic yards 
under the original plan, which gave a depth of 25 feet. In 1895 the canal 
was so enlarged as to give a depth of 31 feet, a width at bottom of 108 feet 
and at the surface of 420 feet. The original cost was $95,000,000, and for 
the canal in its present form slightly in excess of $100,000,000. The canal 
is without locks, being at the sea level the entire distance. The length of 
time occupied in passing through the canal averages about 18 hours. By 
the use of electric lights throughout the entire length of the canal, passages 
are made at night with nearly equal facility to that of the day. The tolls 
charged are 8.50 francs per ton net register, "Danube measurement," which 
amounts to about $2 per ton United States measurement. Steam vessels 
passing through the canal are propelled by their own power. Since Jan. 1, 
1902, the minimum draft of water has been raised to 26' Z" (8 meters). 

Table 1, next page, shows the number and tonnage of vessels which 
passed through the Suez Canal, together with the transit receipts for same, 
and also the number of passengers carried, from its opening until 1903. 

The Cronstadt and St. Petersburg Canal, connecting the bay of Cronstadt 
with St. Petersburg, is a work of great strategic and commercial importance 
to Russia. The canal and sailing course in the bay of Cronstadt are about 
16 miles long, the canal proper being about 6 miles and the bay channel 
about 10 miles, and they together extend from Cronstadt, on the Gulf of 
Finland; to St. Pertersburg. The canal was opened in 1890 with a navigable 
depth of 20i feet, the original depth having been about 9 feet. The width 
ranges from 220 to 350 feet. The total cost is estimated at about $10,000,000. 
There are no locks. 

The Corinth Canal, connecting the Gulf of Corinth with the Gulf of 
^gina, reduces the distance from Adriatic ports about 175 miles and from 
Mediterranean ports about 100 miles. Its length is about 4 miles, a part of 
which was cut through granitic soft rock and the remainder through soil. 
There are no locks, as is also the case in both the Suez and Cronstadt canals, 
described above. The width of the canal is 72 feet at bottom and the depth 
26i feet. The work was begun in 1884 and completed in 1893, at a cost 
of about $5,000,000. The average tolls are 18 cents per ton and 20 cents 
per passenger. 

The Manchester Ship Canal, connecting Manchester, England, with the 
Mersey River, Liverpool, and the Atlantic Ocean, was opened for traffic 
January 1, 1894. The length of the canal is 35 J miles, the total rise from 
the water level to Manchester being 60 feet, which is divided between four 
sets of locks, giving an average to each of 15 feet. The minimum width is 
120 feet at the bottom and averages 175 feet at the water level, though in 
places the width is extended to 230 feet. The minimum depth is 26 feet, 
and the time required for navigating the canal from 5 to 8 hours. The total 
amount of excavation in the canal and docks was about 45,000,000 cubic 
yards, of which about one-fourth was sandstone rock. Th6 lock gates are 
operated by hydraulic power. Railways and bridges crossing the route of 
the canal have been raised to give a height of 75 feet to vessels traversing 
the canal, and an ordinary canal whose route it crosses is carried across by 
a springing aqueduct composed of an iron caisson resting upon a pivot 
pier. The total cost of the canal is given at $75,000,000. 



* Much of the information relating to large ship canals is from Great 
Canals of the World, 1905, Department of Commerce and Labor, Washington, 
D. C. Those who are particularly interested in this subject should consult 
the "List of Works Relating to Deep Waterways from the Great Lakes to 
the Atlantic Ocean with some other related works," published by Supt. of 
Documents, Washington. D. C. 

1320 



IMPORTANT CANALS, SUEZ CANAL DATA. 



1321 



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1322 67,— WATERWAYS, 

The Kaiser Wilhelm Canal. Two canals connect the Baltic and North 
Seas through Germany; the first, known as the Kaiser Wilhelm Canal, 
having been begun in 1887 and completed in 1895 and constructed largely 
for military and naval purposes, but proving also of great value to general 
mercantile traffic. The length of the canal is 61 miles, the terminus in the 
Baltic Sea being at the harbor of Kiel. The depth is 29^- feet, the width at 
the bottom 72 feet, and the minimum width at the surface 190 feet. The 
route lies chiefly through marshes and shallow lakes and among river val- 
leys. The total excavation amounted to about 100,000,000 cubic yards, 
and the cost to about $40,000,000. 

The Elbe and Trave Canal, the second canal connecting the Baltic and 
North Seas through Germany, is smaller than the Kaiser Wilhelm Canal. 
It was opened in 1900, with a length of about 41 miles and a depth of about 
10 feet. It is described in the International Yearbook, 1900, as follows: 

"The Elbe and Trave Canal, in Germany, was opened by the Emperor of Germany 
on Jime 16, 1900. It has been under construction for five years, and has cost about 
$5,831,000. of which Prussia contributed $1,785,000 and the old Hanse town of 
Lubek $4,046,000. The length of the new canal is about 41 miles, and is the second 
to join the North Sea and the Baltic, following the Kaiser Wilhelm Canal (or Kiel 
Canal), built about five years ago at a cost of $37,128,000. The breadth of the new 
canal is 72 feet; breadth of the locks, 46 feet; length of locks, 261 feet, depth of 
locks, 8 feet 2 inches. It is crossed by 29 bridges, erected at a cost of $1,000,000. 
There are seven locks, five being between Lubek and the MoUner See (the summit 
I)olnt of the canal) and two between Mollner See and Fauenberg-on-the-Elbe. At 
this point it may be noted that the Germans began experiments during 1 900 with 
electric towing on the Finow Canal between Berlin and Stettin. A track of 1-meter 
gauge was laid along the bank of the canal, having one 9-pound and one 1 8-pound 
rail laid partly on cross ties and partly on concrete blocks. The larger rail serves 
for the return current, and has bolted to It a rack which gears with a spur wheel on 
the locomotive. The locomotive is 6 feet 10 inches by 4 feet 10 Inches, mounted on 
four wheels, with a wheel base of 3 feet 6 inches, and weighing 2 tons. It is fitted 
with a 1 2-horsepower motor, current for which is furnished by a 9-kilowatt dynamo, 
driven by a 1 5-horsepower engine. The current Is 500 volts, and Is transmitted by 
a wire carried on wooden poles 23 feet high and about 120 feet apart. The boats are 
about 132 feet long and 15 feet 6 Inches beam, and carry from 150 to 175 tons on a 
draft of 4 feet 9 inches. During 1900 the Stettin-Swlnemund Canal, with a length of 
35 miles, has been dredged throughout, and Is now open to steamers drawing 22 feet 
of water. Swlnemund Is on the Baltic Sea. Among the various projects for European 
canals may be mentioned one connecting the Danube a little below Vienna, Austria, 
with the Adriatic Sea at Trieste, a distance of about 3 1 9 miles. Herr Wagenfahrer, 
of Vienna, Is said to have the concession for this canal, the construction of which 
will cost some $120,000,000. Late in 1 900 a canal from Liege to Antwerp, In Belgium, 
was being seriously discussed, in order to connect the prosperous city of Liege with 
the sea, and make it, like the city of Manchesffer, England, a seaport. The original 
promoter of the scheme was Mr. Joseph Redonti, who is now dead. Mr. Redonti's 
plans have recently been put In practical shape by Louis Hubln and Gaston Delvllle, 
who propose a canal 84 miles long, 200 feet wide, and 23 feet deep from Antwerp to 
Liege, with locks at Liege, Hasselt, Herenthals, and Antwerp. The difference In 
level to be overcome by locks would be 175 feet, and It Is thought that thirteen 
single locks and one double lock would be sufficient. The total estimated cost of 
the work Is $25,200,000.** 

The Welland Canal connects Lake Ontario and Lake Erie on the Cana- 
dian side of the river. It was constructed in 1833 and enlarged in 1871 and 
again in 1900. The length of the canal is 26| miles, the number of locks 25, 
the size of locks 270 by 45 feet, the total rise of lockage 326f feet, and the 
total cost about $25,000,000. The annual collection of tolls on freight, 
passengers, and vessels averaged about $225,000 and the canal is open on 
an average about 240 days in a year. By order in council dated April 27, 
1903, the levy of tolls for passage through Dominion canals has been 
abolished for a period of two seasons of navigation. 

The Sault Ste. Marie Canals, at Sault Ste. Marie, Mich., and Ontario 
are located adjacent to the falls of the St. Marys River, which connects 
Lake Superior with Lake Huron, and lower and raise vessels from one level 
to the other, a height of 17 to 20 feet. 

The canal belonging to the United States was begun in 1853 by the 
State of Michigan and opened in 1855, the length of the canal being 5,674 
feet, and provided with two tandem locks, each being 350 feet in length 
and 70 feet wide, and allowing passage of vessels drawing 12 feet, the 
original cost being $1,000,000. The United States Government, by consent 
of the State, began in 1870 to enlarge the canal, and by 1881 had increased 



IMPORTANT CANALS. CANADIAN SYSTEMS. 



1323 



its length to 1.6 miles, its width to an average of 160 feet, and its depth to 
16 feet; also had built a single lock 515 feet long and 80 feet wide, with a 
depth of 17 feet on the sills, which was located 100 feet south of the State 
locks. The State relinquished all control of the canal in March, 1882. 
In 1887 the State locks were torn down and replaced by a single lock 800 feet 
long, 100 feet wide, and a depth of 22 feet of water on the sills. This lock 
was put in commission in 1896. The canal was also deepened to 25 feet. 

The canal on the Canadian side, on the north side of the river, is 1 
miles long, 150 feet wide, and 22 feet deep, with lock 900 feet long, 60 feet 
wide, with 22 feet on the miter sills, and was built during the years 1888 to 
1895. 

Canadian Canal Systems. — ^The canal systems of the Dominion, under 
Government control, in connection with lakes and navigable rivers, are as 
follows: 

First. The through route between Montreal and the head of Lake Superior, 
14 feet navigation: 



Name of Waterway. 


Canal. 


River. 


Remarks. 


1. Lachlne Canal 


Miles. 


Miles. 

"'zm' 
""5*" 
"'ibii' 

""'m' 

"236^" 

594 
"266"* 




River St. Lawrence 




2. Soulanges Canal 


14 




River St. Lawrence 




3. Cornwall Canal 


11 




River St. Lawrence , . . 


The Farrans 


4. Farrans Point Canal 


1 


Point, Rapide 


River St. Lawrence 


Plat, and Ga- 


5. Rapide Piat Canal 


z% 


lops canals are 


River St. Lawrence 


collectively 


6. Galops Canal 


iVs 


known as the 


River St. Lawrence and Lake Ontario. 


Williamsburg 


7. Welland Canal 


2m 


canals. 


Lake Erie, Detroit River, Lake St. Clair, 
and St. Marys River 




8. Sault Ste. Marie Canal 


""iH 




Lake Superior to Port Arthur 




Lake Superior to Duluth, 390. 






Total 


73% 


1.165J^ 









Second. Ottawa to Lake Champlain:^ Greenville, Carillon, St. Annes, 
Chambly, St. Ours Canals. 

Third. Ottawa to Kingston (and Perth) : Rideau River and Canal. (Con- 
nects with Perth by Tay Canal.) 

Fourth. Lake Ontario at Trenton to Lake Huron at mouth of river Severn: 

Trent Canal (not completed). 

Fifth. Ocean to the Bras d'Or Lakes: St. Peters Canal. 

The Lake Borgue, Louisiana, Canal was formerly opened in August, 
1901. It opens continuous water communication with lakes Maurepas, 
Ponchartrain, and Borgue, the Mississippi Sound, Mobile, and the Alabama 
and Warrior Rivers, and the entire Mississippi River system, and has an 
important bearing as a regulator of freight rates between these sections. 
The effects of the canals may be briefly summed up as: Shortening the dis- 
tance between New Orleans and the Gulf points east of the Mississippi; 
bringing shipments from the Gulf coast direct to the levees at New Orleans; 
saving the transshipment of through freights, with a consequent reduction 
in freight rates; enabling sea-going vessels, drawing 10 to 12 feet of water, 
to come within 20 miles of New Orleans, saving all such craft the cost of 
towage and shortening, by 60 miles, direct water communication between 
New Orleans and the deep water of the Gulf. In addition to these effects 
may be enumerated the cheapening of coal for consumption at New Orleans. 
Coal has hitherto been floated down the rivers from Pittsburg, a distance of 
2100 miles. The canal opens up the coal fields in the interior of Alabama 
for New Orleans consumption and reduces coal prices considerably, giving 



1324 m^—WATERWAYS. 

an additional advantage to domestic industries and to steamers purchasing 
bunker coal. The canal is 7 miles long and from 150 to 200 feet wide. 
Bayou Dupree forms a portion of the canal. The lock chamber is 200 feet 
long, 50 feet wide and 25 feet deep, and connects the canal with the Missis- 
sippi River. 

The Chicago Sanitary and Ship Canal connects Lake Michigan at 
Chicago with the Illinois River at Lockport, a distance of 34 miles. The 
canal was cut for the purpose of giving to the city of Chicago proper drainage 
facilities by reversing the movement of water, which formerly flowed into 
Lake Michigan through the Chicago River, and turning a current from Lake 
Michigan through the Chicago River to the Illinois River at Lockport, and 
thence down the Illinois River to the Mississippi. The minimum depth of 
the canal is 22 feet, its width at bottom 160 feet, and width at the top 
from 162 to 290 feet, according to the class of material through which it is 
cut. The work was begun September 3, 1892, and completed and the water 
turned into the channel January 2, 1900. The flow of water from Lake 
Michigan toward the gulf is now at the rate of 360,000 cubic feet per minute, 
and the channel is estimated to be capable of carrying nearly twice that 
amount. The total excavation in its construction included 28,500,000 
cubic yards of glacial drift and 12,910,000 cubic yards of solid rock, an 
aggregate of 41,410,000 cubic yards. In addition to this the construction of 
a new channel for the Desplaines River became necessary in order to permit 
the canal to follow the bed of that river, and the material excavated in 
that work amounted to 2,068,659 cubic yards, making a grand total dis- 
placement in the work of 43,478,659 cubic yards of material which, according 
to a statement issued by the trustees of the sanitary district of Chicago, 
would, if deposited in Lake Michigan in 40 feet of water, form an island 
1 mile square with its surface 12 feet above the water line. 

All bridges along the canal are movable structures. The total cost of 
construction, including interest account, aggregated $34,000,000, of which 
$21,379,675 was for excavation and about $3,000,000 for rights of way 
and $4,000,000 for building railroad and highway bridges over the canal. 
The city and State authorities by whom the canal was constructed are now 
proposing to Congress to make this canal a commercial highway in case 
Congress will increase the depth of the Illinois and Mississippi Rivers to a 
depth of 14 feet, with locks for fleets of barges from Lockport, the terminus 
of the drainage canal, to St. Louis. This, it is argued, would give through 
water transportation from Lake Michigan to the Gulf by way of the drainage 
canal, the Illinois River, and the Mississippi River, and would enable the 
United States in case of war to quickly transport light-draft war vessels 
from the Gulf to the lakes. This work of deepening the Illinois River would 
also give through water connection from Rock Island, on the Upper Mis- 
sissippi River, to Lake Michigan via the Illinois and Mississippi Canal, 
which extends from Rock Island, on the Mississippi River, to Hennepin, 
on the Illinois River. The estimate of the Chicago sanitary district trustees 
of the cost of deepening the Illinois and Mississippi Rivers from the terminus 
of the ship canal to St. Louis to a depth of 14 feet is $25,000,000, including 
.five locks and dams. 

Proposed American Isthmian Canal. — The construction of a waterway to 
connect the Atlantic and Pacific across the American isthmus has been a 
subject of consideration for nearly 400 years. Vasco Nunez de Balboa, 
governor of a province in Darien, in 1513 crossed the isthmus and discovered 
the Pacific, and from that time forward efforts were almost constantly 
made to discover a water connection between the oceans at this point, 
which it was hoped nature had supplied. When this hope was abandoned 
the construction of an artificial water route was immediately proposed, 
and Charles V. is said to have directed, as early as 1520, that the Isthmus 
of Panama be surveyed with the purpose of selecting a route for the con- 
struction of an artificial waterway to connect the two oceans. Another 
decree was issued in 1534, authorizing a careful examination by experienced 
men for this purpose ; but the governor having reported that such work was 
impracticable, and that no king, however powerful, was capable of forming 
a junction of the two seas, and the suggestion having been made that the 
opening of a canal through the isthmus would be "in opposition to the will 
of the Almighty, who had placed this barrier in the way of navigation 
between the two oceans," the project was temporarily abandoned. Further 
examinations were made, in 1771 in the hope of finding a continuous water- 



CHICAGO CANAL. ISTHMIAN CANAL. 1325 

way, as statements had been made that a river had been found flowing 
from ocean to ocean, but an examination proved the inaccuracy of this 
statement. In 177& a survey was made to determine the practicability of 
connecting the oceans by way of Lake Nicaragua, but the report was not 
encouraging. From that date forward, however, numerous examinations 
and surveys of the isthmus were made at various points, the latest being 
that of the American Commission appointed in 1899, and which has recently 
presented its full report to Congress. That report concludes as follows: 

"The investigations of this Commission have shown that the selection 
of 'the most feasible and practicable route' for an isthmian canal must be 
made between the Nicaragua and Panama locations. Furthermore, the 
complete problem involves both the sea-level plan of canal and that with 
locks. The Panama route alone is feasible for a sea-level canal, although 
both are entirely practicable and feasible for a canal with locks. The time 
required to complete a sea-level canal on the Panama route, probably more 
than twice that needed to build a canal with locks, excludes it from favor- 
able consideration, aside from other serious features of its construction. 
It is the conclusion of this Commission, therefore, that a plan of canal with 
locks should be adopted. 

"A comparison of the principal physical features, both natural and 
artificial, of the two routes reveals some points of similarity. Both routes 
cross the continental divide less than ten miles from the Pacific Ocean, 
the Panama summit being about double the height of that in Nicaragua. 
For more than half its length the location of each route on the Atlantic 
side is governed by the course of a river, the flow from whose drainage basin 
is the only source of water supply for the proposed canal; and the summit 
levels, differing but about 20 feet in elevation, Panama being the lower, 
are formed by lakes, natural in the one case and artificial in the other, 
requiring costly dams and wasteways for their regulation and for the im- 
pounding of surplus waters to reduce the effect of floods and to meet 
operating demands during low-water seasons. 

"The investigations made in connection with the regulation of Lake 
Nicaragua have demonstrated that that lake affords an inexhaustible water 
supply for the canal by that route. The initial proposition, on the other 
hand, for the Panama route is to form Lake Bohio so as to yield a water 
supply for a traffic of 10,000,000 tons, which can be supplemented when 
needed by an amount sufficient for more than four times that traffic, by 
means of the Alhajuela reservoir. For all practical purposes this may be 
considered an unlimited supply for the Panama route. So far as the practical 
operation of a ship canal is concrened, therefore, the water supply features 
on both lines are satisfactory. 

"The difficulties disclosed and likely to be encountered in the construc- 
tion of the dams are less at Conchuda on the Nicaragua line than at Bohio 
on the Panama route. Both dams, however, are practicable, but the cost 
of that at Bohio is one-half more than at Conchuda. A less expensive dam 
at Bohio has been proposed, but through a portion of its length it would be 
underlaid by a deposit of sand and gravel pervious to water. The seepage 
might not prove dangerous, but the security of the canal is directly dependent 
upon this dam, and the policy of this Commission has been to select the 
more perfect structure even at a somewhat greater cost. The wasteways at 
both locations present no serious difficulties. The advantages in the design 
and construction of the dams are in favor of the Nicaragua route. 

"The system of regulation at Lake Bohio consists only of the discharge 
of water over the crest of a weir, as the lake level rises under the influence 
of floods in the Chagres River. The plan of regulating the level of Lake 
Nicaragua is less simple, though perfectly practicable. It involves the 
operation of movable gates at such times and to such extent as the rainfall 
on the lake basin may require. The experience and judgment of the operator 
are essential elements in the effective regulation of this lake. The regula- 
tion of Lake Bohio is automatic. 

"The only means of transportation now found on the Nicaragua route 
are the narrow-gauge Silico Lake Railroad, about 6 miles in length, and the 
limited navigation of the San Juan River and the lake, but the Nicaraguan 
Government is now building a railroad along the beach from Greytown to 
Monkey Point, about 45 miles to the northward, where it proposes to estab- 
lish a commercial port. By means of a pier, in the area protected by the 
point, goods and material for canal purposes can readily be landed and 
transported by rail to Greytown. Such piers are in constant use on our 
Pacific coast. This railroad and port would be of great value during the 



1326 Q7.— WATERWAYS. 

period of preparation and harbor construction, and should materially 
shorten that period. A well-eqiupped railroad is in operation along the 
entire length of the Panama route, and existing conditions there afford 
immediate accommodation for a large force of laborers. 

"The Nicaragua route has no natural harbor at either end. At both the 
Atlantic and Pacific termini, however, satisfactory harbors may be created 
by the removal of material at low-unit prices, and by the construction of 
protective works of well-established design. An excellent roadstead, 
protected by islands, already exist at Panama, and no work need be done 
there for either harbor construction or maintenance. At Colon, the Atlantic 
terminus of the Panama route, a serviceable harbor already exists. It has 
afforded harbor accommodations for many years, but it is open to northers, 
which a few times in each year are liable to damage ships or force them to 
put to sea. Considerable work must be done there to create a suitable 
harbor at the entrace of the canal, which can easily be entered, and will 
give complete protection to shipping lying within it. The completion of 
the harbors as planned for both routes would yield but little advantage to 
either, but the balance of advantages, including those of maintenance and 
operation, is probably in favor of the Panama route. 

"The existence of a harbor at each terminus of the Panama route, and 
a line of railroad across the Isthmus, will make it practicable to commence 
work there, after the concessions are acquired, as soon as the necessary 
plant can be collected and put in place, and the working force organized. 
This period of preparation is estimated at one year. In Nicaragua this 
period is estimated at two years, so as to include also the construction of 
working harbors and terminal and railroad facilities. 

"The work of excavation on the Nicaragua route is distributed; it is 
heaviest near Conchuda, at Tamborcito, and in the divide west of the lake. 
On the Panama route it is largely concentrated in the Culebra and Emperador 
cuts, which are practically one. As a rule distributed work affords a greater 
number of available points of attack, contributing to a quicker completion; 
but in either of these cases such difficulties as may exist can be successfully 
met with suitable organization and efficient appliances. 

"The time required for constructing the Nicaragua Canal will depend 
largely on the promptness with which the requisite force of laborers can be 
brought to Nicaragua, housed and organized at the locations of heaviest 
work along the route. The cut through the divide west of the lake will 
probably require the longest time of any single feature of construction. 
It contains about 18,000,000 cubic yards of earth and rock excavation, or 
a little less than 10 per cent of the total material of all classes to be removed. 
With adequate force and plant this Commission estimates that it can be 
completed in four years. This indicates, under reasonable allowance for 
ordinary delays, that if force and plant enough were available to secure a 
practically concurrent execution of all portions of work on the route, the 
completion of the entire work might be expected within six years after its 
beginning, exclusive of the two years estimated for the period of preparation. 

"The securing and organizing of the great force of laborers needed, 
largely foreigners, so as to adjust the execution of the various portions of 
the work to such a definite programme of close-fitting parts in a practically 
unpopulated tropical country, involves unusual difficulties and would pro- 
long the time required for completion. 

"The greatest single feature of work on the Panama route is the excava- 
tion in the Culebra section, amounting to about 43,000,000 cubic yards of 
hard clay, much of which is classed as soft rock, or nearly 45 per cent of all 
classes of material to be removed. It is estimated that this cut can be 
completed in eight years, with allowance for ordinary delays, but exclusive 
of a two-year period for preparation and for unforseen delays, and that the 
remainder of the work can be finished within the same period. The great 
concentration of work on this route and its less amount will not require so 
great a force of laborers as on the Nicaragua route; hence the difficulties 
and delays involved in securing them will be correspondingly diminished. 

"The total length of the Nicaragua route from sea to sea is 183.66 miles, 
while the total length of the Panama route is 49.09 miles. The length of 
standard canal section and in harbors and entrances is 73.78 miles for the 
Nicaragua route and 36.41 miles for the Panama route. The length of sailing 
line in Lake Nicaragua is 70.51 miles, while that in Lake Bohio is 12.68 
miles. That portion of the Nicaragua route in the canalized San Juan is 
39.37 miles. 

"The preceding physical features of the two lines measure the magnitude 



PROPOSED AMERICAN ISTHMIAN CANAL. 1327 

of the work to be done in the construction of waterways along the two 
routes. The estimated cost of constructing the canal on the Nicaragua 
route is $45,630,704 more than that of completing the Panama Canal, 
omitting the cost of acquiring the latter property. This sum measures the 
difference in the magnitude of the obstacles to be overcome in the actual 
construction of the two canals and covers all physical considerations, such 
as the greater or less height of dams, the greater or less depth of cuts, the 
presence of absence of natural harbors, the presence or absence of a railroad, 
and the amount of work remaining to be done. 

"The estimated annual cost of maintaining and operating the Nicaragua 
Canal is $1,300,000 greater than the corresponding charges for the Panama 
Canal. 

"The Panama route would be 134.57 miles shorter from sea to sea than 
the Nicaragua route. It would have less summit elevation, fewer locks, 
1,568 degrees and 26.44 miles less curvature. The estimated time for a 
deep-draft vessel to pass through is about 12 hours for Panama and thirty- 
three hours for Nicaragua. These periods are practically the measure of 
the relative advantages of the two canals as waterways connecting the two 
oceans, but not entirely, because the risks to vessels and the dangers of de- 
lay are greater in a canal than in the open sea. 

"Except for the items of risks and delays, the time required to pass 
through the canals need be taken into account only as an element in the 
time required by vessels to make their voyages between terminal ports. 
Compared on this basis, the Nicaragua route is the more advantageous for 
all transisthmian commerce except that originating or ending on the west 
coast of South America. For the commerce in which the United States is 
most interested, that between our Pacific ports and Atlantic ports, European 
and American, the Nicaragua route is shorter by about 1 day. The same 
advantage exists between our Atlantic ports and the Orient. For our Gulf 
ports the advantage of the Nicaragua route is nearly two days. For com- 
merce between North Atlantic ports and the west coast of South America 
the Panama route is shorter by about two days. Between Gulf ports and 
the west coast of South America the saving is about one day. 

"The Nicaragua route would be the more favorable one for sailing ves- 
sels because of the uncertain winds in the Bay of Panama. This is not, 
however, a material matter, as sailing ships are being rapidly displaced by 
steamships. 

"A canal by the Panama route will be simply a means of communication 
between the two oceans. That route has been a highway of commerce for 
more than three hundred years, and a railroad has been in operation there 
for nearly fifty years, but this has effected industrial changes of but little 
consequence, and the natural features of the country through which the 
route passes are such that no considerable development is likely to occur as 
a result of the construction and operation of a canal. 

"In addition to its use as a means of communication between the two 
oceans, a canal by the Nicaragua route would bring Nicaragua and a large 
portion of Costa Rica and other Central American States into close and 
easy communication with the United States and with Europe. The intimate 
business relations that would be established with the people of the United 
States during the period of construction by the expenditure of vast sums 
of money in these States and the use of American products and manufac- 
tures would be likely to continue after the completion of the work, to the 
benefit of our manufacturing, agricultural, and other interests. 

"The Nicaragua route lies in a region of sparse population and not in 
a pathway of much trade or movement of people: conditions productive of 
much sickness do not exist. On the other hand, a considerable population 
has long existed on the Panama route, and it lies on a pathway of compara- 
tively large trade, along which currents of moving people from infected 
places sometimes converge, thus creating conditions favorable to epidemics. 
Existing conditions indicate hygienic advantages for the Nicaragua route, 
although it is probable that no less effective sanitary measures must be taken 
during construction in the one case than in the other" 

The relative Estimated cost of the Nicaragua and Panama canals is as 
follows: Nicaragua $189,864,962; completing the Panama canal $144,233,- 
358+ $40,000,000 for acquiring the rights and property of the New Panama 
Canal Co., making a total of $184,233,358. [Actual cost will be double.] 

Estimates of annual cost of maintenance are; Nicaragua canal, $3,300,- 
000; Panama canal $2,000,000. 



1328 



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ISTHMIAN CANAL DISTANCES. COST DATA. 1329 

The Harlem River Ship Canal, connecting the Hudson River and Long 
Island Sound, by way of Spuyten Duyvil Creek and Harlem River, was 
opened for traffic on June 17, 1895, and cost about 12,700,000. 

Cost of Maintenance and Operation of Canals. — In order to form an 
estimate of the cost of maintaining and operating the Isthmian Canal, the 
Isthmian Canal Commission obtained data bearing on this point from the 
Suez, Manchester, Kiel, and St. Marys Falls canals, as follows: 

There are no locks on the Suez Canal, but the channel is through drifting 
sand for a great part of its length. The entrance to the harbor of Port Said 
on the Mediterranean intercepts the drift of sand discharged from the Nile 
and carried along the coast by the easterly current. The maintenance of 
the Suez Canal therefore requires a large amount of dredging and consists 
mainly of this class of work. The operating expenses are also large, the 
great traffic involving heavy costs for pilotage. The general expenses for 
administration have necessarily been greater for the Suez Canal than for 
the Kiel or Manchester Canals, on account of the distance of the work from 
the point of central control, a disadvantage which would also attend the 
operation of the Isthmian Canal. The annual cost of maintenance and 
operation of the Suez Canal is about $1,300,000, or about $13,000 per mile. 

The annual cost of maintenance and operation of the Kiel Canal is $8,600 
per mile. The cost of maintenance only of the Manchester Canal is $9,500 
per mile. These canals have locks and other mechanical structures, and 
therefore might be expected to have a higher cost of maintenance than the 
Suez Canal, which has none, but this appears to be more than offset by 
reduced- cost of maintaining the prism and more economical central control. 
The traffic being light on these canals, the cost of pilotage and port service 
is small. The mechanical structures are now nearly new, and will soon 
require larger annual outlays for maintenance, while, with the increase of 
traffic, operating expenses will become larger. 

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is remarkable by reason of its short length, large proportion of mechanical 
structures, and immense traffic. Its length is about li miles. Its annual 
traffic, limited by the severity of the winter to a period of about eight 
months, is nearly three times that of the Suez Canal, eight times that of 
the Kiel Canal, and ten times that of the Manchester Canal. Both mainte- 
nance and operating expenses are therefore very large, amounting to from 
$70,000 to $90,000 per year, or $46,000 to $60,000 per mile. The annual 
cost per mile of maintenance and operation, however, for comparison with 
other canals, should be determined by considering the 18^ miles of dredged 
channel ways in St. Marys River as part of the canal. Then for the 20 miles 
of canal and canalized river the expenses per mile would be from $3,000 to 
$5,000 annually. Tolls were collected by the State from 1855 to 1881. 
Since its ownership by the government no tolls have been charged. 

The Principal Commercial Canals of the U. S., showing cost of construc- 
tion (which includes cost of improvements), date of completion, length, 
number of locks, and navigable depth, are given in Table 3, on following 
page, which is reproduced, by permission, from the New York World 
Almanac. 



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Q7,— WATERWAYS. 



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COMMERCIAL CANALS OF THE U. S. 1331 



EXCERPTS AND REFERENCES. 

The Relation of Depth of Water to Speed and Power of Ships (Eng. 
News, Mar. 16, 1905). — Diagrams. Contains other references. 

The Relation of the Depth of Harbor Channels to Modern Shipping 

(Eng. News, Dec. 27, 1906).— Table. 

Construction and Unit Costs of Concrete Lock, Rough River, Ky. 

(Eng. News, Jan. 9, 1908). — Illustrated, with tables of costs. 

The Design of Emergency Movable Dams for Canal Locks (Eng. 

News, June 24, 1909).— Illustrated. 

Water Supply for the Lock Canal at Panama (By Julio F. Sorgano, 
Trans. A. S. C. E., Vol. LXVIL, June, 1910). 

The Siphon Lock on the New York State Barge Canal (By D. A. Watt, 
Eng. News, Nov. 17, 1910). Illustrated description of the general design 
and operation of the siphon lock at Oswego, N. Y. 

Cost Data on the Panama Canal ("Canal Record," Nov. 9, 1910; Eng. 
News, Nov. 24, 1910). — General statement of construction expenditures, to 
Sept. 30, 1910; dam construction; lock and spillway construction. 

Illustrations of Some Important Works: — 

Description. Eng. News. 

Plan and section of concrete lock for small craft, Madison, Wis. Dec. 10, 1903. 

Suggested new type of lock, Sault Ste. Marie Canal June 24, '09. 

The lock gates of the Panama canal Sept. 16, '09. 

Typical sections and lock on N. Y. State barge canal June 9, '10. 

Design and cons, of movable dam and lock, Lockport Oct. 6, '10. 

Emergency gates on the Illinois and Mississippi Canal Dec. 15, '10. 

Eng. Rec. 

Plan, profile, section, Cape Cod ship canal, also breakwater July 24, '09. 
Siphon lock on barge canal at Oswego July 30, '10. 

Plans of new canal gates at Sault Ste. Marie Dec. 10, '10. 



68.— WATER POWER. 

Definitions and Formulas. — Power is the rate of work (see page 291); 
and Water Power is the rate of work available from stored or flowing water, 
through the agency of gravity. The unit of mechanical power is the Horse- 
Power {H. P.), equivalent to the energy expended in raising 550 lbs. one ft. 
high in one second, or to the energy of 550 lbs. falling one ft. in one second. 
Hence, the theoretic horse-power of a stream is — 

H. p. = ^^fj'' = 0.113i Qh (1) 

650 

where 62.37=weight of a cu. ft. of water at 60°F., in lbs.; 
Q = discharge of stream, in cu. ft. per sec; 
h = ia,\\ of water, or available head, in ft. 

TTTt. ^ S.82XH.P. 

Whence Q = -, (2) 

n 

. , 8.82XH.P. 
and h== ^ (3) 

* 'Horse-Power Hours" is a term used to denote a certain amount of 
work or energy, and not the rate of work. It is the maintenance of one horse- 
power for one hour, or of ten horse-power for six minutes, etc.; and may 
be applied to definite quantities or volumes of stored or running water. 
Thus, 

No. of H.-P. hours- ^^ = 0.0000315 FA (4) 

where y = volume of stored water, in cu. ft.; 
3600 = No. of seconds in one hour. 

_,„ -. 31746 X No. of H.-P. hours 

Whence V= r (5) 

• 31746XNO. of iJ.-P. hours 
and /j= ■■ y. (6) 

At the Prime Motor the available horse-power is always less than the 
theoretic horse-power, the decrease being due to loss of head, leakage or 
waste, evaporation, etc., depending upon the character of conduit or canal. 
Equation (1) will give the available horse-power at the prime motor if — 
G = actual discharge, in cu. ft. per sec, at prime motor; 
/j = effective head, in ft., at prime motor. 

Economic Design of Penstock. — For a high-pressure water-power pipe, 
the volume of discharge being assumed fixed or constant, the following eco- 
nomic relation holds true: 

"That pipe fulfills the requirements of greatest economy wherein the 
value of the energy annually lost in f rictional resistance equals four-tenths 
(0.4) of the annual cost of the pipe line."* 

Thus, if L=value of energy annually lost in frictional resistance, 
and C=annual cost of pipe line; 
thenL=0.4C, for economic design of pipe. 



*Mr. A. L. Adams, in Trans. Am. Soc C. E., Vol. LIX, page 177. 

1332 



HORSE-POWER PER CUBIC FOOT PER SECOND, 



1333 



1. — ^Number of Horse-Power, Equivalent to Flow of 1 Cu. Ft. 

PER Sec. Under Various Effective Heads. (Equa. 1.) 

(To find the total H. P., mult, values in Table by No. of cu. ft. per sec.) 

[Horse-Power per Cu. Ft. per Sec. of Flow.] 



Co 03 

* 



Units. 



0. 



2. 



8. 



0-. 

2~; 

3-. 

4-. 

5-. 

6-. 

7-. 

8-. 

9-. 

10-. 

11-. 

.12- 

014-. 
5l5-. 
816- . 
^17-. 
018-. 
ftl9-. 
220-. 

=322-. 
£23-. 
Ei24-. 
a25-. 
♦a26-. 
a 27-. 
P28-. 
XJ29-. 
£30-. 
^31-. 
^•32-. 
l33-. 
^34-. 
0.35- . 
^36-. 
T337-. 
«i38-. 
&39-. 
§40-. 
§41-. 
i42-. 
243-. 
D.44-. 
«45-. 
M6-. 
'ft47-. 
§•48-. 
§49-. 

50-. 

51-. 

52-. 

53-. 

54-. 

55-. 

56-. 

57-. 

58-. 

59-. 

60-. 



.0000 
1.1340 
2.2680 
3.4020 
4.5360 
6.6700 
6.8040 
7.9380 
9.0720 
10.2060 
11.3400 
12.4740 
13.6080 
14.7420 
15.8760 
17.0100 
18. 1440 
19.2780 
20.4120 
21.5460 
22.6800 
23.8140 
24.9480 
26.0820 
27.2160 
28. 3500 
29. 4840 
30 6180 
31.7520 
32.8860 
34.0200 
35. 1540 
36.2880 
37.4220 
38.5560 
39.6900 
40.8240 
41.9580 
43.0920 
44.2260 
45.3600 
46.4940 
47.6280 
48.7620 
49.8960 
51.0300 
52.1640 
53.2980 
54.4320 
55.5660 
56.7000 
57.8340 
58.9680 
60.1020 
61.2360 
62.3700 
63.5040 
64.6380 
65.7720 
66.90C0 
68.0400 



.1134 

1.2474 

•2.3814 

3.5154 
4.6494 
5.7834 
6.9174 
8.0514 
9. 1854 
10.3194 
11.4534 
12.5874 
13.7214 
14.8554 
15.9894 
17.1234 
18.2574 
19.3914 
20.5254 
21.6594 
22.7934 
23.9274 
25.0614 
26.1954 
27.3294 
28.4634 
29.5974 
30.7314 
31.8654 
32.9994 
34.1334 
35.2674 
36.4014 
37.5354 
38.6694 
39.8034 
40.9374 
42.0714 
43.2054 
44.3394 
45.4734 
46.6074 
47.7414 
48.8754 
50.0094 
51.1434 
52.2774 
53.4114 
54.5454 
55.6794 
56.8134 
57.9474 
59.0814 
60.2154 
61.3494 
62.4834 
63.6174 
64.7514 
65.8854 
67.0194 
68.1534 



.2268 
1.3608 
2.4948 
3.6288 
4.7628 
5.8968 
7.0308 
8.1648 
9.2988 
10.4328 
11.5668 
12.7008 
13.8348 
14.9688 
16.1028 
17.2368 
18.3708 
19.5048 
20.6388 
21.7728 
22.9068 
24.0408 
25.1748 
26.3088 
27.4428 
28.5768 
29.7108 
30.8448 
31.9788 
33.1128 
34.2468 
35.3808 
36.5148 
37.6488 
38.7828 
39.9168 
41.0508 
42.1848 
43.3188 
44.4528 
45.5868 
46.7208 
47.8548 
48. 9888 
50.1228 
51.2568 
52.3908 
53.5248 
54.6588 
55.7928 
56.9268 
58.0608 
59.1948 
60.3288 
61.4628 
62.5968 
63.7308 
64.8648 
65.9988 
67.13 
68.2668 



.3402 
1.4742 
2.6082 
3.7422 
4.8762 
6.0102 
7.1442 
8.2782 
9.4122 
10.5462 
11.6802 
12.8142 
13.9482 
15.0822 
16.2162 
17.3502 
18.4842 
19.6182 
20.7522 
21 8862 
23.0202 
24.1542 
25.2882 
26.4222 
27.5562 
28.6902 
29.8242 
30.9582 
32.0922 
33.2262 
34.3602 
35.4942 
36.6282 
37.7622 
38.8962 
40.0302 
41.1642 
42.2982 
43.4322 
44.5662 
45.7002 
46.8342 
47.9682 
49. 1022 
50.2362 
51.3702 
52.5042 
53.6382 
54.7722 
55 9062 
57.0402 
58.1742 
59.3082 
60.4422 
61.5762 
62.7102 
63.8442 
64.9782 
66.1122 
67.2462 
68.3802 



.4536 
1.5876 
2.7216 
3.8556 
4. 9896 
6.1236 
7.2576 
8.3916 
9.5256 
10.6596 
11.7936 
12.9276 
14.0616 
15.1956 
16.3296 
17.4636 
18.5976 
19.7316 
20.8656 
21.9996 
23.1336 
24.2676 
25.4016 
26.5356 
27.6696 
28 8036 
29.9376 
31.0716 
32.2056 
33.3396 
34.4736 
35.6076 
36.7416 
37.8756 
39.0096 
40. 1436 
41.2776 
42.4116 
43. 5456 
44.6796 
45.8136 
46. 9476 
48.0816 
49.2156 
50.3496 
51.4836 
52.6176 
53.7516 
54.8856 
56.0196 
57.1536 
58.2876 
59.4216 
60.5556 
61.6896 
62.8236 
63.9576 
65.0916 
66.2256 
67.3596 
68.4936 



.5670 
1.7010 
2.8350 
3.9690 
5.1030 
6.2370 
7 3710 
8.5050 
9.6390 
10.7730 
11.9070 
13.0410 
14. 1750 
15.3090 
16.4430 
17.5770 
18.7110 
19.8450 
20.9790 
22.1130 
23.2470 
24.3810 
25.5150 
26.6490 
27.7830 
28.9170 
30.0510 
31.1850 
32.3190 
3o. 4530 
34. 5870 
35.7210 
36.8550 
37.9890 
39.1230 
40.2570 
41.3910 
42.5250 
43.6590 
44.7930 
45.9270 
47.0610 
48.1950 
49.3290 
50.4630 
51.5970 
52.7310 
53.8650 
54.9990 
56.1330 
57.2670 
58.4010 
59 5350 
60.6690 
61.8030 
62.9370 
64.0710 
65.2050 
66.3390 
67.4730 
68.6070 



6804 
1.8144 
2.9484 
4.0824 
5.2164 
6.3504 
7.4844 
8.6184 
9.7524 
10.8864 
12.0204 
13.1544 
14.2884 
15.4224 
16.5564 
17.6904 
18.8244 
19.9584 
21.0924 
22.2264 
23.3604 
24.4944 
25.6284 
26.7624 
27.8964 
29.0304 
30.1644 
31.2984 
32.4324 
33.5664 
34.7004 
35.8344 
36.9684 
38.1024 
39.2364 
40. 3704 
41.5044 
42.6384 
43.7724 
44. 9064 
46.0404 
47.1744 
48.3084 
49. 4424 
50.5764 
51.7104 
52.8444 
53.9784 
55.1124 
56.2464 
57.3804 
58.5144 
59.6484 
60.7824 
61.9164 
63.0504 
64.1844 
65.3184 
66.4524 
67.5864 
68.7204 



.7938 
1.9278 
3.0618 
4.1958 
5.3298 
6.4638 
7.5978 
8.7318 
9.8658 
10.9998 
12.1338 
13.2678 
14.4018 
15.5358 
16.6698 
17.8038 
18.9378 
20.0718 
21.2058 
22.3398 
23.4738 
24.6078 
25.7418 
26.8758 
28.0098 
29.1438 
30.2778 
31.4118 
32.5458 
33.6798 
34.8138 
35.9478 
37.0818 
38.2158 
39.3498 
40.4838 
41.6178 
42.7518 
43.8858 
45.01 
46. 1538 
47.2878 
48.4218 
49.5558 
50.6898 
51.8238 
52.9578 
54.0918 
55.2258 
56.3598 
57,4938 
58.6278 
59 7618 
60.8958 
62.0298 
63.1638 
64.2978 
65.4318 
66.5658 
67.6998 
68.8338 



.9072 
2.0412 
3.1752 
4.3092 
5.4432 
6.5772 
7.71121 
8.8452 
9.9792 
11.1132 
12.2472 
13.3812 
14.5152 
15.6492 
16.7832 
17.91/2 
19.0512 
20.1852 
21.3192 
22.4532 
23. 5872 
24.7212 
25.8552 
26.9892 
28.1232 
29.2572 
30.3912 
31.5252 
32.6592 
33.7932 
34.9272 
36.0612 
37.1952 
38.3292 
39.4632 
40.5972 
41.7312 
42.8652 
43.9992 
45.1332 
46.2672 
47.4012 
48.5352 
49.6692 
50.8032 
51.9372 
53.0712 
54.2052 
55.3392 
56.4732 
57.6072 
58.7412 
59.8752 
61.0092 
62 1432 
63.2772 
64.4112 
65. 5452 
66 6792 
67.8132 
68.9472 



1.0206 
2.1546 
3.2886 
4.4226 
5.5566 
6 6906 
7.8246 
8. 9586 
10.0926 
11.2266 
12.3606 
13.4946 
14.6286 
15.7626 
16.8966 
18.0306 
19. 1646 
20.2986 
21.4326 
22.5666 
23.7006 
24.8346 
25.9686 
27.1026 
28.2366 
29.3706 
30.5046 
31.6386 
32.7726 
33.9066 
35.0406 
36.1746 
37,3086 
38.4426 
39.5766 
40.7106 
41.8446 
42.9786 
44.1126 
45..2466 
46 3806 
47.5146 
48.6486 
49.7826 
50.9166 
52.0506 
53.1846 
54.3186 
55.4526 
56.5866 
57.7206 
58.8546 
59. 9886 
61.1226 
62.2566 
63.3906 
64.5246 
65.6586 
66.7926 
67.9266 
69.0606 



* Effective head and actual horse power are proportional; hence the 
decimal point may be moved in each, a corresponding number of places to 
right or left. 

Ex. — ^The equivalent to a flow of 1 cu. ft. per sec. under a head of 14 ft. 
is 1.5876 H. P.; under a 140-ft. head, 16.876 H. P,\ under a 1400-ft. head, 
168.76 H. P. 



1334 



68.— WATER POWER, 



2. — ^NuMBER OP Horse-Power Hours, Equivalent to Storage op 
1 000 000 Cu. Ft. Under Various Effective Heads. (Equa. 4.) 
(To find total H.-P. H., mult, values in Table by No. of millions of 
cu. ft. stored.) 

[Horse-Power Hours per each million cu. ft.] 



it 




Units. 










Kg 


0. 


1. 


2. 


3. 


4. 


5 


6. 


7. 


8. 


9. 


0-. 





31 5 


63 


94.5 


126 


157.5 


189 


220.5 


252 


283.5 


1-. 


315 


346.5 


378 


409.5 


441 


472.5 


504 


535.5 


567 


598.5 


2-. 


630 


661 5 


693 


724.5 


756 


787.5 


819 


850.5 


882 


913.5 


3-. 


945 


976.5 


1008 


1039.5 


1071 


1102.5 


1134 


1165.5 


1197 


1228.5 


4-. 


1260 


1291.5 


1323 


1354.5 


1386 


1417.5 


1449 


1480.5 


1512 


1543.5 


5-. 


1575 


1606 5 


1638 


1669.5 


1701 


1732.5 


1764 


1795.5 


1827 


1858.5 


6-. 


1890 


1921.5 


1953 


1984.5 


2016 


2047.5 


2079 


2110.5 


2142 


2173.5 


7-. 


2205 


2236.5 


2268 


2299.5 


2331 


2362.5 


2394 


2425.5 


2457 


2488.5 


8-. 


2520 


2551.5 


2583 


2614.5 


2646 


2677.5 


2709 


2740.5 


2772 


2803.5 


9-. 


2835 


2866.5 


2898 


2929.5 


2961 


2992 5 


3024 


3055.5 


3087 


3118.5 


10-. 


3150 


3181.5 


3213 


3244. 5 


3276 


3307.5 


3339 


3370.5 


3402 


3433.5 


gll-. 


3465 


3496 5 


3528 


3559.5 


3591 


3622.5 


3654 


3685.5 


3717 


3748.5 


gl2-. 


3780 


3811.5 


3843 


3874,5 


3906 


3937.5 


3969 


4000.5 


4032 


4063.5 


5 13-. 


4095 


4126.5 


4158 


4189.5 


4221 


4252 5 


4284 


4315.5 


4347 


4378.5 


Oil- 


4410 


4441.5 


4473 


4504.5 


4536 


4567.5 


4599 


4630.5 


4662 


4693.5 


4725 


4756.5 


4788 


4819.5 


4851 


4882.5 


4914 


4945.5 


4977 


5008.5 


016-. 


5040 


5071.5 


5103 


5134.5 


5166 


5197.5 


5229 


5260.5 


5292 


5323.5 


0.17-. 


5355 


5386.5 


5418 


5449.5 


5481 


5512.5 


5544 


5575.5 


5607 


5638.5 


018-. 
**19-. 


5670 


5701.5 


5733 


5764.5 


5796 


5827 5 


5859 


5890.5 


5922 


5953.5 


5985 


6016.5 


6048 


6079.5 


6111 


6142.5 


6174 


6205.5 


6237 


6268.5 


^20-. 
«21-. 


6300 


6331.5 


6363 


6394.5 


6426 


6457.5 


6489 


6520.5 


6552 


6583.5 


6615 


6646.5 


6678 


6709.5 


6741 


6772.5 


6804 


6835.5 


6867 


6898.5 


§22-. 

lit: 


6930 


6961.5 


6993 


7024.5 


7056 


7087.5 


7119 


7150.5 


7182 


7213.5 


7245 


7276.5 


7308 


7339.5 


7371 


7402.5 


7434 


7465.5 


7497 


7528.5 


7560 


7591.5 


7623 


7654.5 


7686 


7717.5 


7749 


7780.5 


7812 


7843.5 


4^25-. 


7875 


7906. 5 


7938 


7969.5 


8001 


8032.5 


8064 


8095.5 


8127 


8158.5 


"226-. 


8190 


8221.5 


8253 


8284.5 


8316 


8347.5 


8379 


8410.5 


8442 


8473.5 


3 27-. 


8505 


8536.5 


8568 


8599.5 


8631 


8662.5 


8694 


8725.5 


8757 


8788.5 


£,28-. 


8820 


8851 5 


8883 


8914.5 


8946 


8977.5 


9009 


9040.5 


9072 


9103.5 


§29-. 


9135 


9166.5 


9198 


9229.5 


9261 


9292.5 


9324 


9355.5 


9387 


9418.5 


^30-. 


9450 


9481. 5 


9513 


9544.5 


9576 


9607.5 


9639 


9670 5 


9702 


9733.5 


.SI- 


9765 


9796.5 


9828 


9859.5 


9891. 


9922.5 


9954 


9985.5 


10017 


10048.5 


'S 32-. 


10080 


10111.5 


10143 


10174.5 


10206 


10237 5 


10269 


10300.5 


10332 


10363.5 


§§33-. 


10395 


10426 5 


10458 


10489.5 


10521 


10552.5 


10584 


10615.5 


10647 


10678.5 


•034-. 


10710 


10741.5 


10773 


10804.5 


10836 


10867.5 


10899 


10930.5 


10962 


10993.5 


>.35-. 


11025 


11056.5 


11088 


11119.5 


11151 


11182.5 


11214 


11245.5 


11277 


11308.5 


^36-. 


11340 


11371.5 


11403 


11434 5 


11466 


11497.5 


11529 


11560.5 


11592 


11623.5 


^37-. 


11655 


11686.5 


11718 


11749.5 


11781 


11812.5 


11844 


11875.5 


11907 


11938.5 


S38-. 


11970 


12001.5 


12033 


12064 5 


12096 


12127.5 


12159 


12190.5 


12222 


12253.5 


&39-. 


12285 


12316.5 


12348 


12379.5 


12411 


12442.5 


12474 


12505.5 


12537 


12568.5 


§40-. 


12600 


12631.5 


12663 


12694 5 


12726 


12757.5 


12789 


12820.5 


12852 


12883.5 


|41-. 


12915 


12946.5 


12978 


13009.5 


13041 


13072.5 


13104 


13135.5 


13167 


13198.5 


°42-. 


13230 


13261.5 


13293 


13324.5 


13356 


13387.5 


13419 


13450.5 


13482 


13513.5 


g43-. 


13545 


13576.5 


13608 


13«39.5 


13671 


13702.5 


13734 


13765.5 


13797 


13828.5 


g44-. 


13860 


13891.5 


13923 


13954.5 


13986 


14017.5 


14049 


140«0.5 


14112 


14143.5 


5*45- 


14175 


14206.5 


14238 


14269.5 


14301 


14332.5 


14364 


14395.5 


14427 


14458.5 


§47-. 


14490 


14521.5 


14553 


14684.5 


14616 


14647.5 


14679 


14710.5 


14742 


14773.5 


14805 


14836.5 


14868 


14899 5 


14931 


14962.5 


149f>4 


15025.5 


15057 


15088.5 


§48-. 


15120 


15151.5 


15183 


15214.5 


15246 


15277.5 


15309 


15.^40.5 


15372 


15403.5 


§'49-. 


15435 


15466.5 


15498 


15529.5 


15561 


15592.5 


15624 


156.^5.5 


15687 


15718.5 


0550-. 


15750 


15781.5 


15813 


15844.5 


15876 


15907.5 


15939 


15970.5 


16002 


16033.5 


51-. 


16065 


16096.5 


16128 


16159.5 


16191 


16222.5 


16254 


16^85.5 


16317 


16348.5 


52-. 


16380 


16411.5 


16443 


16474.5 


16506 


16537.5 


16569 


16600.5 


16632 


16663.5 


53-. 


16695 


16726.5 


16758 


16789.5 


16821 


16852.5 


16884 


16915.5 


16947 


16978.5 


54-. 


17010 


17041.5 


17073 


17104.5 


17136 


17167.5 


17199 


17230.5 


17262 


17293.5 


55-. 


17325 


17356.5 


17388 


17419.5 


17451 


17482.5 


17514 


17545.5 


17577 


17608.5 


56-. 


17640 


17671.5 


17703 


17734.5 


17766 


17797.5 


17829 


17860.5 


17892 


17923.5 


57-. 


17955 


17986.5 


18018 


18049.5 


18081 


18112.5 


18144 


18175.5 


18207 


18238.5 


58- 


18270 


18301.5 


18333 


18364.5 


18396 


18427.5 


18459 


18490.5 


18522 


18553.5 


59-. 


18585 


18616.5 


18648 


18679.5 


18711 


18742.5 


18774 


18805.5 


18837 


18868.5 


60-. 


18900 


18931.5 


18963 


18994.5 


19026 


19057.5 


19089 


19120,5 


19152 


19183.5 



* Effective head and actual horse-power hours are proportional; hence 
the decimal point may be moved in each, a corresponding number of places 
to right or left. 

Ex. — ^The equivalent to a storage of 1 000 000 cu. ft. with a head of 
14 ft. is 441 H.-P. H\ with a 140-ft. head, 4410 H.-P. H.\ with a 1400-ft. 
head, 44100 H.-P. H. 



CU. FT, AND ACRE-FT. TO H.-P. HOURS. 



1335 



3. — Number op Horse-Power Hours, Equivalent to Storage op 

1 Acre-Foot. Under Various Effective Heads. (Equa. 4.) 

(To find the total H.-P. H., mult, values in Table by No. of acre-feet stored.) 

Note. — 1 acre-foot = 43560 cubic feet. 

[Horse-Power Hours per each acre-foot.] 



Units. 



0. 



3. 



7. 



0.0000 
13.721 
27.443 
41 164 
54.886 
68.607 
82.328 
96.050 
109.77 
123.49 
137.21 
150.94 
164.66 
178.38 
192.10 
205.82 
219.54 
233.26 
246.99 
260.71 
274.43 
288.15 
301.87 
315.59 
329.31 
343.04 
356,76 
370.48 
384.20 
397.92 
411.64 
425.36 
439.08 
452.81 
466.53 
480.25 
493.97 
507.69 
521.41 
535.13 
548.86 
562.58 
576.30 
590.02 
603.74 
617.46 
631 18 
644.91 
658.63 
672.35 
686.07 
699.79 
713.51 
727.23 
740.96 
754.68 
768.40 
782.12 
795.84 
809.56 
823.28 



1.3721 
15.094 
28.815 
42.536 
56.258 
69.979 
83.701 
97.422 
111.14 
124.86 
138.59 
152.31 
166.03 
179.75 
193.47 
207.19 
220.91 
234.64 
248.36 
262.08 
275.80 
289.52 
303.24 
316.96 
330.69 
344.41 
358.13 
371.85 
385.57 
399.29 
413.01 
426.74 
440.46 
454.18 
467.90 
481.62 
495.34 
509.06 
522.79 
536.51 
550 23 
563.95 
577.67 
591.39 
605.11 
618.84 
632.56 
646.28 
660.00 
673.72 
687.44 
701.16 
714.88 
728.61 
742.33 
756.05 
769.77 
783.49 
797.21 
810.93 
824.66 



2.7443 
16.466 
30.187 
43.908 
57.630 
71.351 
85.073 
98.794 
112.52 
126.24 
139.96 
153.68 
167.40 
181.12 
194.85 
208.57 
222.29 
236.01 
249.73 
263.45 
277.17 
290.89 
304.62 
318.34 
332.06 
345.78 
359.50 
373.22 
386.94 
400.66 
414.39 
428.11 
441.83 
455.55 
469.27 
482.99 
496.71 
510.44 
524.16 
537.88 
551.60 
565.32 
579.04 
592.76 
606.49 
620.21 
633.93 
647.65 
661.37 
675.09 
688.81 
702.54 
716.26 
729.98 
743.70 
757.42 
771.14 
784.86 
798.59 
812.31 
826.03 



4.1164 
17.838 
31.559 
45.281 
59.002 
72.723 
86.445 
100.17 
113.89 
127.61 
141.33 
155.05 
168.77 
182.49 
196.22 
209.94 
223.66 
237.38 
251.10 
264.82 
278.54 
292.27 
305.99 
319.71 
333.43 
347.15 
360.87 
374.59 
388.32 
402.04 
415.76 
429.48 
443.20 
456. 92 
470. 64 
484.37 
498.09 
511.81 
525. 53 
539.25 
552.97 
566.69 
580.42 
594.14 
607.86 
621.58 
635.30 
649.02 
662.74 
676.47 
690.19 
703.91 
717.63 
731.35 
745.07 
758.79 
772.51 
786.24 
799.96 
813.68 
827.40 



5.4886 
19.210 
32.931 
46.653 
60.374 
74.096 
87.817 
101.54 
115.26 
128.98 
142.70 
156.42 
170.15 
183.87 
197.59 
211.31 
225.03 
238.75 
252.47 
266.20 
279.92 
293.64 
307.36 
321.08 
334. 80 
348.52 
362.24 
375.97 
339. 69 
403.41 
417.13 
430.85 
444. 57 
458.29 
472.02 
485.74 
499.46 
513.18 
526.90 
540.62 
554.34 
568.07 
581.79 
595.51 
609.23 
622.95 
636.67 
650.39 
664.12 
677.84 
691.56 
705.28 
719.00 
732.72 
746.44 
760.17 
773.89 
787.61 
801.33 
815.05 
828.77 



6.8607 
20.582 
34.304 
48.025 
61.746 
75.468 
89.189 
102.91 
116.63 
130.35 
144.07 
157.80 
171.52 
185.24 
198.96 
212.68 
226.40 
240.12 
253.85 
267.57 
281.29 
295,01 
308.73 
322.45 
336.17 
349.90 
363.62 
377.34 
391.06 
404.78 
418.50 
432.22 
445. 95 
459.67 
473.39 
487.11 
500.83 
514.55 
528.27 
542.00 
555.72 
569.44 
583.16 
596.88 
610.60 
624.32 
638.05 
651.77 
665.49 
679.21 
692.93 
706.65 
720.37 
734.09 
747.82 
761.54 
775.26 
788.98 
802.70 
816.42 
830.14 



8.2328 
21.954 
35.676 
49.397 
63.118 
76.840 
90.561 
104.28 
118.00 
131.73 
145.45 
159.17 
172.89 
186.61 
200.33 
214.05 
227.78 
241.50 
255.22 
268.94 
282.66 
296.38 
310.10 
323.83 
337.55 
351.27 
364.99 
378.71 
392.43 
406.15 
419.87 
433.60 
447.32 
461.04 
474.76 
488.48 
502.20 
515.92 
529.65 
543.37 
557.09 
570.81 
584.53 
598.25 
611.97 
625.70 
639.42 
653.14 
666.86 
680.58 
694.30 
708.02 
721.75 
735.47 
749.19 
762.91 
776.63 
790.35 
804.07 
817.80 
831.52 



9. 6050 
23.326 
37.048 
50.769 
64.491 
78.212 
91.933 
105.65 
119.38 
133.10 
146.82 
160.54 
174.26 
187.98 
201.70 
215.43 
229.15 
242.87 
256.59 
270.31 
284.03 
297.75 
311.48 
325.20 
338. 92 
352.64 
366.36 
380.08 
393.80 
407.53 
421.25 
434.97 
448.69 
462.41 
476.13 
489.85 
503.58 
517.30 
531.02 
544.74 
558.46 
572.18 
585.90 
599.63 
613.35 
627.07 
640.79 
654.51 
668.23 
681.95 
695.67 
709.40 
723.12 
736.84 
750.56 
764.28 
778.00 
791.72 
805.45 
819.17 
832.89 



10.977 
24.699 
38.420 
52.141 
65.863 
79.584 
93. 306 
107.03 
120.75 
134. 47 
148.19 
161.91 
175.63 
189.36 
203.08 
216.80 
230.52 
244.24 
257.96 
271.68 
285.41 
299.13 
312.85 
326.57 
340.29 
354.01 
367.73 
381.45 
395.18 
408.90 
422.62 
436.34 
450.06 
463.78 
477.50 
491.23 
504.95 
518.67 
532.39 
546.11 
559.83 
573.55 
587.28 
601.00 
614.72 
628.44 
642.16 
655.88 
669.60 
683.33 
697.05 
710.77 
724.49 
738.21 
751.93 
765.65 
779.38 
793.10 
806.82 
820.54 
834.26 



12.349 
26.071 
39. 792 
53.513 
67.235 
80.956 
94. 678 
108.40 
122.12 
135.84 
149.56 
163.28 
177.01 
190.73 
204.45 
218.17 
231.89 
245.61 
259.33 
273.06 
286.78 
300.50 
314.22 
327.94 
341.66 
355.38 
369.11 
382.83 
396.55 
410.27 
423.99 
437.71 
451.43 
465.16 
478.88 
492.60 
506.32 
520.04 
533.76 
547.48 
561.21 
574.93 
588.65 
602.37 
616.09 
629.81 
643.53 
657.26 
670.98 
684.70 
698.72 
712.14 
725.86 
739.58 
753.30 
767.03 
780.75 
794.47 
808.19 
821.91 
835.63 



* Effective head and actual horse-power hours are proportional; hence 
the decimal point may be moved in each, a corresponding number of places 
to right or left. 

Ex. — ^The equivalent to a storage of 1 acre-foot with a head of 14 ft. is 
19.210 H.-P. if.; with a 140-ft.head, 192.10 H. -P. H.; with a 1400-ft. head, 
1921.0 H.-P. H, 



1336 • Q8.— WATER POWER. 

Water Motors. — ^The energy of flowing water, by virtue of its velocity, 
weight and pressure, all acting together to a greater or less extent, may be 
transmitted to mechanical motors and transformed into various kinds or 
forms of energy. The principal types of water motors comprise many and 
various forms of wheels, among which are the following:* 

The Current Wheel is the simplest form. It consists essentially of a 
paddle wheel with blades, somewhat similar to the side wheel of a steamer, 
and is mounted on a horizontal shaft, supported over the current of water 
at the bank of the stream, and adjustable to height. The principal use of the 
current wheel in the West is for raising water for domestic supply and irri- 
gation purposes. Attached to, or near, the periphery of the wheel are open 
metal buckets which scoop up the water from the stream and then deliver 
it into a flume whence it is conveyed into reservoirs, or directly on the land. 
A small wheel will irrigate several acres. Prof F. H. King, in describing the 
wheels used in Bavaria on the River Regnitz, a branch of the Main, where 
he counted no less than 20 in a distance of li to IJ miles, says:t 

These wheels have a diameter of 1 6 ft. and carry upon one or both sides a row of 
24 churn-lilve buckets each lifting out of the stream, and to a height of 12 ft., not 
less than 3 galls, of water, from which it is conveyed to the bank through a conduit 
hewn from a log. The wheel under consideration was making at the time of the 
writer's visit four revolutions each minute, so that the water lifted was not less than 
288 gallons per minute, and probably exceeded 300 gallons. Another wheel with a 
row of buckets on each side was making three revolutions and discharging not less 
than 450 gallons per minute. The first of these wheels was pumping water at a rate 
sufficient to irrigate, to a depth of 4 inches every 10 days, 38 acres, and the second 
60 acres. 

The ordinary current wheel Is sometimes called an "undershot wheel," although 
this Is a misnomer. 

The Undershot Wheel of approved design has curved blades or buckets. 
The current is accelerated by the construction of a flum.e with a decided 
grade and perhaps also by inserting a headgate giving increased head. 
The principle is somewhat similar to that of the current wheel, the water 
flowing beneath it. The efficiency is very low, rarely exceeding 40 per cent. 

The Breast Wheel has pronounced buckets and acts as a dam, backing 
the water up nearly to its top so that all the buckets on the up-stream side 
are filled with water. Being thus continually unbalanced it is revolved. 
The efficiency ranges at about 60 to 65 per cent. 

The Overshot Wheel has a greater efficiency than either of those previously 
described, reaching in some cases 75 per cent. The water is conveyed by 
flume over the top of the wheel, filling the buckets on the down-stream side, 
producing an unbalanced force on the horizontal shaft, which is connected 
up with the working machinery. 

Impulse Wheels and Turbine Wheels are the most common forms of water 
motors, and are discussed below. 

Impulse Water Wheels. — This type is the most efficient and economical 

of any of the pure types of water wheel on the market. It also compares 
favorably with turbines for moderate heads of water, and surpasses them 
for high heads. In principle, the impulse water wheel consists essentially 
of a wheel mounted on a horizontal shaft. Attached to the circumference 
of the wheel are numerous double-cup-shaped buckets into which one or 
more tangential jets of water are played from nozzles. The greatest amount 
of energy of the jet has been imparted to the wheel when the velocity of the 
jet has been totally destroyed, and this is accomplished by the peculiar shape 
of the buckets which split the jet in the middle and reverse either half in 
direction by 180°. When the water simply falls from the buckets, by the 
action of gravity, the wheel is working with the greatest jet efficiency. 

Fig. 1 illustrates the Pelton water wheel with needle and single deflect- 
ing nozzle, for economic regulation. The needle nozzle consists of a tapered 
needle which may be operated so as change the discharge area of the nozzle. 



* The hydraulic ram and the hydraulic pressure engine are not included 
in this discussion. 

t See Farmer's Bulletin No. 46, U. S. Dept. of Agriculture. 



WATER MOTORS, IMPULSE WATER WHEELS. 



1337 



The deflecting nozzle is simply a cast-iron nozzle provided with a ball and 
socket joint. It can be arranged automatically to raise and throw the jet 
on the buckets or to lower and throw it off. Intermediate positions are 
regulated by change of "load." 




Fig. 1. 

The number of nozzles (one, two, or more) which it is advisable to use 
per wheel depends upon the head of water and also upon the amount of 
power required. As a general rule it may be stated that the power devel- 
oped is proportional to the amount of water or number of nozzles used, 
the head of water and size of nozzles remaining constant. Double nozzles 
are often used on small wheels which are called for by certain speed require- 
ments and where the single nozzle would not give the required power. 
The quintex nozzle wheel is specially designed for very low head. (See 
Table 5, page 1342.) 



1338 



I— WATER POWER. 



4. — Single Nozzle Pelton Water Wheel Data. 
(Suitable for high heads.) 

Note. — Compare with Table 5 for low heads, making due allowance for 
5 nozzles in that table. 

The Calculations for Power in these Tables are based upon the application 
of one stream to the wheel and on effective heads. In using these tables 
liberal allowance should be made to cover the friction loss in pipe, elbows, 
gates, etc. The smaller figures under those denoting the various heads 
give the equivalent pressure in pounds per square inch, and spouting velocity 
of water in feet per minute. The water measurement is also based on the 
flow per minute. 



*Head 

in 
Feet. 


Size of Wheels. 


6 

Inch 


12 

Inch. 


15 

Inch. 


18 
Inch. 


24 

Inch. 


3 

Foot. 


4 

Foot. 


5 

Foot. 


6 

Foot. 


20 

8 Lbs. 

2151.97 


Horse Power . . . 

Cubic Feet 

Miner's Inches. . 
Revolutions 


.05 
1.67 
1.04 

684 


.12 
3.91 
2.44 

342 


.20 
6.62 
4.00 

274 


.37 
11.72 

7.82 
228 


.66 
20.83 
'18.88 

171 


1.50 

46.93 

31.28 

114 


2.64 

83.32 

55.52 

85 


4.18 

130.60 

86.90 

, 70 


6.00 

187.72 

125.12 

57 


30 

13 Lbs. 

2635.62 


Horse Power. . . . 

Cubic Feet 

Miner's Inches. . 
Revolutions 


.10 
2.05 
1.28 

837 


.23 
4.79 
2.9$ 

418 


.38 
8.11 
5.06 

335 


.69 

14.36 

9.57 

279 


1.22 

25.51 

17.00 

209 


2.76 

57.44 

38.28 

139 


4.88 

102.04 

68.00 

104 


7.69 

159.66 

106.44 

83 


11.04 

229.76 

153.12 

69 


40 

17 Lbs. 
3043.39 


Horse Power.... 

Cubic Feet 

Miner's Inches.. 
Revolutions 


.15 
2.37 
1.48 

969 


.35 
5.53 
3.45 

484 


.59 
9.37 

5.85 
387 


1.06 

16.59 

11.06 

323 


1.89 

29.46 

19.64 

242 


4.24 

66.36 

44.24 

161 


7.58 

107.84 

78.56 

121 


11.85 

184.36 

122.91 

96 


16.96 

265.44 

176.96 

80 


50 

21 Lbs. 
3402.61 


Horse Power.... 

Cubic Feet 

Miner's Inches.. 
Revolutions 


.21 
2.64 
1.65 
1083 


.49 
6.18 
3.86 

541 


.84 

10.47 

6.54 

433 


1.49 

18.54 

12.36 

361 


2.65 

32.93 

21.95 

270 


5.98 

74. 17 

49.45 

180 


10.60 

131.72 

87.80 

135 


16.63 

206.13 

137.42 

108 


23.93 

296.70 

197.80 

90 


60 

26 Lbs. 
3727.37 


Horse Power. . . . 

Cubic Feet 

Miner's Inches. . 
Revolutions 


.28 
2.90 
1.81 
1185 


.65 
6.77 
4.23 

592 


1.10 

11.47 

7.16 

473 


1.96 

20.31 

13.54 

395 


3.48 

36.08 

24.05 

296 


7.84 

81.25 

54.16 

197 


13.94 

144.32 

96.20 

148 


21.77 

225.80 

150.52 

118 


31.36 

325.00 

216.64 

98 


70 

30 Lbs. 

4026.00 


Horse Power 

Cubic Feet 

Miner's Inches. . 
Revolutions 


.35 
3.13 
1.95 
1281 


.82 
7.31 
4.56 

640 


1.39 
12.39 

7.74 
512 


2.47 

21.94 

14.63 

427 


4.39 

38.97 

25.98 

320 


9.88 

87.76 

58.52 

213 


17.58 

155.88 

103.92 

160 


27.51 

243.89 

162.60 

130 


39.52 

351 04 

234 08 

106 


80 
34 Lbs. 
4303.99 


Horse Power. . . . 

Cubic Feet 

Miner's Inches.. 
Revolutions 


.43 
3.35 
2.09 
1368 


1.00 

7.82 

4.88 

684 


1.70 

13.25 

8.28 

546 


3.01 

23.46 

15.64 

456 


5.36 

41.66 

27.77 

342 


12.04 

93.84 

62.56 

228 


21.44 

166.64 

111.08 

171 


33.54 

260.73 

173.82 

137 


48.16 

375.36 

250.24 

114 


90 

39 Lbs. 
4565.04 


Horse Power. . . . 

Cubic Feet 

Miner's Inches.. 
Revolutions 


.51 
3.55 
2.22 
1452 


1.20 

8.29 

5.18 

726 


2.03 

14.05 

8.78 

581 


3.60 

24.88 

16.58 

484 


6.39 

44.19 

29.46 

363 


14.40 

99.52 

66.32 

242 


25.59 

176.75 

117.83 

181 


40.04 

276.55 

184.36 

145 


57.60 

398.08 

265.28 

121 


100 

43 Lbs. 

4812.00 


Horse Power. . . . 

Cubic Feet 

Miner's Inches.. 
Revolutions 


.60 
3.74 
2.33 
1530 


1.40 

8.74 

5 46 

765 


2.32 

14.81 

9.25 

612 


4.21 

26.22 

17.48 

510 


7.49 

46.58 

31.05 

382 


16.84 

104.88 

69.93 

255 


29.93 

186.32 

124.21 

191 


46.85 

291.51 

194.34 

152 


67.36 

419.52 

279.72 

127 


110 

48 Lbs. 
5046.87 


Horse Power. . . . 

Cubic Feet 

Miner's Inches.. 
Revolutions 


.69 
3.92 
2.45 
1605 


1.62 

9.16 

5.72 

802 


2.74 

15.53 

9.70 

642 


4.86 

27.50 

18.33 

535 


8.64 

48.85 

32.56 

401 


19.44 

110.00 

73.33 

267 


34.58 

195.41 

130.27 

200 


54.11 

305.73 

203.82 

160 


77.76 

440.00 

293.32 

133 



* Theoretic head. The data in table, except in first column, are based 
on effective heads at 85 per cent of the theoretic heads; or in other words, 
the efficiency of wheel is assiimed at 85 per cent. 



SINGLE NOZZLE PELTON WHEELS. 



1339 



4. — Single Nozzle Pelton Water Wheel Data. — Continued. 



♦ Head 

m 

Feet. 


Size of Wheels. 


6 

Inch 


12 

Inch. 


15 

Inch. 


18 

Inch. 


24 

Inch. 


3 

Foot. 


4 

Foot. 


5 

Foot. 


6 

Foot. 


120 

52 Lbs. 

5271.30 


Horse Power — 

Cubic Feet 

Miner's Inches.. 
Revolutions 


.79 
4.10 
2.56 
1677 


1.84 

9.57 

5.98 

838 


3.12 

16.21 

10.13 

671 


5.54 

28.72 

19.15 

559 


9.85 

51.02 

34.01 

419 


22.18 

114.91 

76.60 

279 


39.41 

204.10 

136.06 

209 


61.66 

319.33 

212.89 

167 


88.75 

459.64 

306.43 

139 


130 

56 Lbs. 

5486.54 


Horse Power. . . . 

Cubic Feet 

Miner's Inches.. 
Revolutions 


.89 
4.27 
2.66 
1746 


2.08 

9.96 

6.22 

873 


3.53 

16.89 

10.55 

698 


6.25 
29.90 
19.93 

582 


11.11 

53.10 

35.40 

436 


25.02 

119.60 

79.73 

291 


44.46 

212.43 

141.62 

218 


69.53 

332.37 

221.58 

174 


100.08 

478.41 

318.94 

145 


140 

60 Lbs. 

5693.65 


Horse Power. . . . 

Cubic Feet 

Miner's Inches.. 
Revolutions 


.99 
4.43 

2.76 
1812 


2.33 

10.34 

6.46 

906 


3.94 

17.53 
10.95 

725 


6.99 

31.03 

20.68 

604 


12.41 

55.11 

36.74 

453 


27.96 

124.12 

82.72 

302 


49.64 

220.44 

146.96 

226 


77.71 

344.92 

229.94 

181 


111.85 

496.48 

330.88 

151 


150 

65 Lbs. 

5893.44 


Horse Power. . . . 

Cubic Feet 

Miner's Inches. . 
Revolutions 


1.10 

4.55 
2.84 
1875 


2.58 

10.70 

6.68 

937 


4.37 
18.14 
11.33 

750 


7.75 

32.11 

21.41 

625 


13.77 

57.04 
38.03 

468 


31.-01 

128.47 

85.64 

312 


55.08 

228.19 

152.12 

234 


86.22 

357.02 

238.05 

187 


124.04 

513.90 

342.59 

156 


160 

69 Lbs. 

6086.74 


Horse Power — 

Cubic Feet 

Miner's Inches. . 
Revolutions 


1.22 
4.73 
2.95 
1938 


2.84 

11.05 

6.90 

969 


4.82 
18.74 
11.71 

775 


8.54 

33.17 

22.11 

646 


15.17 
58.92 
39.28 

484 


34.16 

132.68 

88.46 

323 


60.68 

235.68 

157.12 

242 


94.94 

.368.73 

245.82 

193 


136.65 

530.75 

353. 84 

161 


180 

78 Lbs. 
6455.97 


Horse Power. . . . 

Cubic Feet 

Miner's Inches.. 
Revolutions 


1.45 
5.02 
3 13 
2049 


3.39 

11.72 

7.32 

1024 


5.75 

19.87 

12.41 

820 


10.19 

35.18 

23.45 

683 


18.10 

62.49 

41.66 

513 


40.77 

140.74 

93.82 

342 


72.41 

249.97 

166.64 

256 


113.30 

391.10 

260.73 

206 


163.08 

562.96 

375.29 

171 


200 

87 Lbs. 
6805.17 


Horse Power. . . . 

Cubic Feet 

Miner's Inches.. 
Revolutions 


1.70 
5.29 
3.30 
2160 


3.97 
12.36 

7.72 
1080 


6.74 

20.94 

13.08 

864 


11.93 

37.08 

24.72 

720 


21.20 

65.87 

43.91 

540 


47.75 

148.35 

98.90 

360 


84.81 

263.49 

175.66 

270 


132.70 

412.25 

274.83 

216 


191.00 

593.40 

395.60 

180 


220 

95 Lbs. 
7137.35 


Horse Power. . . . 

Cubic Feet 

Miner's Inches . . 
Revolutions 


1.96 
5.55 
3.46 
2268 


4.59 

12.96 

8.10 

1134 


7.77 

21.96 

13.72 

906 


13.77 

38.89 

25.93 

756 


24.46 

69.08 

46.05 

567 


55.09 

155.59 

103.72 

378 


97.85 

276.35 

184.23 

283 


153.10 

432.38 

288.25 

226 


220.36 

622.36 

414.91 

189 


240 

105 
Lbs. 

7454.70 


Horse Power — 

Cubic Feet 

Miner's Inches.. 
Revolutions .... 


2.24 
5.80 
3.62 
2370 


5.23 

13.54 

8.46 

1185 


8.86 

22.93 

14.33 

948 


15.69 

40.62 

27.08 

790 


27.87 

72.16 

48.10 

592 


62.77 

162.50 

108.34 

395 


111.50 

288.64 

192.42 

296 


174.45 

451.60 

301.07 

237 


251.10 

650.03 

433.36 

197 


260 

113 
Lbs. 

7759. 10 


Horse Power. . . . 

Cubic Feet 

Miner's Inches. . 
Revolutions 


2.52 
6.04 
3.77 
2466 


5.89 

14.09 

8.80 

1233 


10.05 

23.88 

14.92 

986 


17.69 

42.28 

28.19 

822 


31.43 

75.10 

50.07 

617 


70.78 

169.14 

112.76 

411 


125.72 

300.43 

200.28 

308 


196.71 

470.04 

313.36 

247 


283.15 

676.59 

451.05 

206 


280 
121 
Lbs. 

8052.01 


Horse Power. . . . 

Cubic Feet 

Miner's Inches. 
Revolutions .... 


2.82 
6.26 
3.91 
2562 


6.59 

14.62 

9.13 

1281 


11.16 

24.79 

15.49 

1025 


19.77 
43.88 
29.25 

854 


35.12 

77.94 

51.29 

639 


79.11 
175.53 
117.02 

427 


140.51 

311.77 

205.18 

319 


219.84 

487.79 

325.19 

255 


316.44 

702.12 

468.06 

213 


300 

130 

Lbs. 

8334.62 


Horse Power. . . . 

Cubic Feet 

Miner's Inches.. 
Revolutions 


3.13 

6.48 
4.05 
2652 


7.31 

15.13 

9.45 

1326 


12.38 

25.66 

16.03 

1060 


21.93 
45.42 
30.28 

884 


38.95 

80.67 

53.78 

663 


87.73 

181.69 

121.12 

442 


155.83 

322.71 

215.14 

331 


243.82 

504.91 

336.60 

265 


350.94 

726.76 

484.51 

221 


320 

139 
Lbs. 

8607.94 


Horse Power. . . . 

Cubic Feet 

Miner's Inches, . 
Revolutions 


3.45 
6.70 
4.18 
2739 


8.05 

15.63 

9.76 

1369 


13.64 

26.50 

16.56 

1095 


24.16 

46.91 

31.27 

913 


42.91 

83.32 

55.55 

685 


96.65 

187.65 

125.10 

456 


171.68 

333.29 

222.19 

342 


268.60 

521.46 

347.64 

274 


386.62 

750.60 

500.40 

228 



*See Note and Foot-note on preceding page. 



1340 



,— WATER POWER, 



4. — Single Nozzle Pelton Water Wheel Data. — Continued. 



♦Head 

In 
Feet. 


Size of Wheels. 


6 

Inch 


12 

Inch. 


15 

Inch. 


18 
Inch. 


24 

Inch. 


3 

Foot. 


1 ' 

I Foot. 


5 

Foot 


6 

Foot. 


340 

147 
Lbs. 

8872.89 


Horse Power. . . . 

Cubic Feet 

Miner's Inches. . 
Revolutions 


3.78 
6.90 
4.31 
2823 


8.82 
16.12 
10.07 

1411 


14.94 

27.31 

17.06 

1130 


26.46 

48.35 

32.24 

941 


47.00 

85.88 

57.26 

706 


105.86 

193.42 

128.98 

470 


188.02 

343.55 

229.04 

353 


294.18 

537.51 

358.34 

282 


423.44 

773.71 

515.93 

235 


360 

156 

Lbs. 

9130.14 


Horse Power. . . . 

Cubic Feet 

Miner's Inches. . 
Revolutions .... 


4.10 
7.10 
4.43 
2907 


9.61 
16.58 
10.36 

1453 


16.28 

28.10 

17.56 

1161 


28.83 

49.75 

33.17 

969 


51.21 

88.37 

58.91 

726 


115.34 

199.03 

132.68 

484 


204.86 

353.51 

235.64 

363 


320.52 

553.10 

368.73 

290 


461.36 

796.14 

530.75 

242 


380 

165 

Lbs. 

9380.32 


Horse Power. . . . 

Cubic Feet 

Miner's Inches. . 
Revolutions 


7^30 
4.56 
2985 


10.42 

17.04 

10.65 

1492 


17.66 

28.88 

18.03 

1194 


31.27 

51.12 

34.08 

995 


55.54 

90.80 

60.53 

746 


125.08 

204.48 

136.32 

497 


222.16 

363.20 

242.13 

373 


347.60 

568.25 

378.83 

298 


500.33 

817.95 

545.29 

248 


400 

173 
Lbs. 

9624.00 


Horse Power. . . '. 

Cubic Feet 

Miner's Inches. . 
Revolutions 


4.82 
7.49 
4.68 
3063 


11.25 

17.48 

10.92 

1531 


19.07 

29.63 

18.51 

1225 


33.77 

52.45 

34.96 

1021 


59.98 

93.16 

62.10 

765 


135.08 

209.80 

139.84 

510 


239.94 

372.64 

248.40 

382 


375.40 

583.02 

388.68 

306 


540.35 

839.20 

559.35 

255 


420 

182 

Lbs. 

9861.66 


Horse Power 

Cubic Feet 

Miner's Inches.. 
Revolutions. . . . 


5.19 
7.67 
4.79 
3141 


12.11 

17.91 

11.19 

1570 


20.52 

30.36 

18.93 

1255 


36 33 

53.74 

35.83 

1047 


64.54 
95.46 
63.64 

785 


145.34 

214.98 

143.32 

523 


258.16 

381.84 

254.56 

392 


403.91 

597.41 

398.28 

313 


581.39 

859.93 

573.28 

261 


440 

191 
Lbs. 

10093.74 


Horse Power. . . . 

Cubic Feet 

Miner's Inches. . 
Revolutions 


5.56 
7.85 
4.90 
3213 


12.98 

18.33 

11.45 

1606 


22.01 

31.07 

19.41 

1285 


38.96 

55.01 

36,66 

1071 


69.20 

97.70 

65.13 

803 


155.85 

220.04 

146.64 

535 


276.82 

390.82 

260.53 

401 


433.11 

611.47 

407.65 

320 


623.40 

880.16 

586.56 

267 


460 

200 
Lbs. 

10320.58 


Horse Power 

Cubic Feet 

Miner's Inches. . 
Revolutions. . . . 


5.95 
8.03 
5.01 
3285 


13.88 

18.74 

11.71 

1642 


23.53 

31.77 

19.79 

1313 


41.65 

56.24 

37.50 

1095 


73.97 

99.90 

66.60 

821 


166.60 
224.98 
150.00 

547 


295.91 

399.61 

266.40 

410 


462.97 

625.22 

416.80 

327 


666.40 

899.95 

600.00 

273 


480 

208 Lbs. 
10542.56 


Horse Power. . . 
Cubic Feet . ; . . . 
Miner's Inches. . 
Revolutions 


6.34 
8.20 
5.12 
3357 


14.79 
19.15 
11.96 

1678 


25.07 

32.45 

20.28 

1343 


44.39 

57.45 

38.30 

1119 


78.85 

102.05 

68.00 

839 


177.58 

229.82 

153.20 

559 


315.42 

408.20 

272.12 

419 


493.49 

638.66 

425.78 

335 


710.33 

919.29 

612.80 

279 


500 

217 Lbs. 
10759.96 


Horse Power. . . . 

Cubic Feet 

Miner's Inches. . 
Revolutions 


6.74 
8.37 
5.23 
3426 


15.73 

19.54 

12.21 

1713 


26.66 

33.12 

20.72 

1370 


47.20 

58.64 

39.09 

1142 


83.83 

104.15 

69-41 

856 


188.80 

234.56 

156.36 

571 


335.34 

416.62 

277.64 

428 


524.66 

651.83 

434.56 

342 


755.20 

938.25 

625.44 

285 




Horse Power. . . , 












200.22 

239.21 

159.47 

582 


355.62 

424.87 

283.24 

436 


556.39 

664.74 

443.16 

349 


800.88 


520 


Cubic Feet 












956 84 


226 Lbs. 

10973.04 


Miner's Inches. . 












637.88 


Revolutions. . . . 












291 


















540 

234 Lbs. 

11182.07 


Horse Power 












211.88 

243.76 

162.51 

593 


376.33 

432.96 

288.64 

445 


588.80 

677.41 

451.61 

356 


847.52 


Cubic Feet 












975.07 


Miner's Inches. . 












650 04 


Revolutions 












296 




















Horse Power. . . . 












223.76 

248.24 

165.49 

604 


397.43 

440.91 

293.94 

453 


621.82 

689.84 

459.89 

362 


895 04 


560 

243 Lbs. 
11387.26 


Cubic Feet 












992 96 


Miner's Inches 












661 96 


Revolutions. . . . 












302 




















Horse Power. . 












235.86 

252.63 

168.42 

615 


418.92 

448.71 

299.14 

461 


6o5.43 

702.04 

468.03 

369 


943 44 


580 

252 Lbs. 

11588.83 


Cubic Feet 












1010 54 


Miner's Inches. . 












673 69 


Revolutions .... 












307 



















*See Note and Foot-note on second page preceding. 



StNCLE NOZZLE PELTON WHEELS. 



1341 



4. — Single Nozzle Pelton Water Wheel Data. — Concluded. 



♦Head 

In 
Feet. 


Size of Wheels. 


6 

Inch 


12 

Inch. 


15 

Inch. 


18 

Inch. 


24 

Inch. 


3 

Foot. 


4 

Foot. 


5 
Foot. 


6 

Foot. 




Horse Power. . . 












248.16 

256.95 

171.30 

625 


440.77 

456.38 

304.24 

469 


689.63 

714.05 

476.03 

375 


992.65 


600 


Cubic Feet 












1027.80 


260 Lbs. 


Miner's Inches 












685 20 


11786.94 


Revolutions . . . 












312 




















Horse Power. . 












279.82 

267.44 

178.29 

651 


497.01 

475.02 

316.68 

488 


777.62 

743.21 

495.47 

390 


1119 29 


650 


Cubic Feet 












1069.77 


282 Lbs. 


Miner's Inches . 












713.18 


12268.24 


Revolutions 












325 




















Horse Power. . . 












312.73 

277.54 

185.02 

675 


555.46 

492.95 

328.63 

506 


869.06 

771.26 

514.18 

405 


1250.92 


700 


Cubic Feet 












1110.16 


304 Lbs. 


Miner's Inches. . 












740.09 


12731.34 


Revolutions. . . . 












337 




















Horse Power. . . . 












346.83 

287.28 

191.52 

699 


616.03 

510.25 

340.16 

524 


963.82 

798.33 

532.22 

419 


1387.34 


750 


Cubic Feet 












1149.13 


326 Lbs. 


Miner's Inches. . 












766 09 


13178.19 


Revolutions 












349 




















Horse Power. . 












382.09 

296.70 

197.80 

722 


678.66 

526.99 

351.32 

542 


1061.81 

824.51 

549.68 

433 


1528 36 


800 


Cubic Feet 












1186.81 


348 Lbs. 


Miner's Inches . 
Revolutions 












791.21 


13610.40 












361 


















Horse Power. . . . 












455.94 

314.70 

209.80 

766 


809.82 

558.96 

372.64 

574 


1267.02 

874.53 

583.02 

459 


1823 76 


900 


Cubic Feet 












1258 81 


391 Lbs. 


Miner's Inches. . 












839.20 


14436.00 


Revolutions . . . . 












383 




















Horse Power. . . . 












534.01 

331.72 

221.15 

807 


948.48 

589.19 

392.79 

605 


1483.97 
921.83 
614.56 

484 


2136.04 


1000 


Cubic Feet . . 












1326.91 


434 Lbs. 


Miner's Inches . . 












884.61 


15216.89 


Revolutions. . . . 












403 



















*See Note and foot-note on third page preceding. 



1342 



QS.— 'WATER POWER, 



5. — QuiNTEX Nozzle Pelton Water Wheel Data. 
(Five nozzles, suitable for low heads.) 
Note. — Compare with Table 4, preceding, making due allowance for 6 
nozzles in this table. 

Revolutions are number per minute. 



II 


Size of 
Wheels. 


18 

In. 


24 

In. 


36 

In. 


42 

In. 


48 

In. 




Size of 
Wheels. 


18 

In. 


24 

In. 


36 

In. 


42 

In. 


48 
In. 


10 


Horse Power 
Revolutions. 


.91 
141 


1.33 
10& 


2.95 
71 


5.17 
60 


8.17 
53 


60 


Horse Power 
Revolutions. 


13.3 
345 


19.3 
259 


43.5 
173 


77.4 
148 


120. 
129 


15 


Horse Power 
Revolutions. 


1.68 
172 


2.42 
129 


5.43 
86 


9.68 
74 


15.0 
65 


65 


Horse Power 
Revolutions. 


15.1 
359 


21.9 
269 


49.1 
180 


86.3 
154 


136. 
135 


20 


Horse Power 
Revolutions. 


2.58 
199 


3.72 
149 


8.35 
100 


14.9 
85 


23.1 

75 


70 


Horse Power 
Revolutions. 


16.9 
373 


24.4 
279 


54.8 
186 


97.4 
160 


152. 
140 


25 


Horse Power 
Revolutions. 


3.60 
223 


5.20 
167 


11.7 
111 


20.8 
95 


32.3 
84 


75 


Horse Power 
Revolutions. 


18.7 
386 


26.3 
289 


60.7 
193 


108. 
165 


168. 
145 


30 


Horse Power 
Revolutions. 


4.72 
244 


6.83 
183 


15.3 
122 


27.3 
103 


42.5 
92 


80 


Horse Power 
Revolutions. 


20.5 
399 


29.7 
299 


66.7 
199 


119. 
171 


184. 
149 


35 


Horse Power 
Revolutions. 


5.90 
263 


8.62 
197 


19.3 
132 


34.4 
113 


53.5 
99 


85 


Horse Power 
Revolutions. 


22.6 
410 


32.5 
308 


73.3 
205 


130. 
176 


202. 
154 


40 


Horse Power 
Revolutions. 


7.30 
282 


10.5 
212 


23.6 
141 


42.2 
121 


65.4 
105 


90 


Horse Power 
Revolutions. 


24.6 
423 


35.4 
317 


80.0 
211 


142. 
181 


221. 

158 


45 


Horse Power 
Revolutions. 


8.68 
299 


12.6 
224 


28.2 
179 


50.2 
128 


78.0 
112 


95 


Horse Power 
Revolutions. 


26.8 
434 


38.5 
325 


86.7 
218 


154. 
186 


239. 
163 


50 


Horse Power 
Revolutions. 


10.2 
315 


14.7 
236 


32.9 
158 


58.7 
135 


91.3 
118 


100 


Horse Power 
Revolutions. 


28.8 
446 


41.6 
334 


93.5 
223 


166. 
19^1 


258. 
167 


55 


Horse Power 
Revolutions. 


11.7 
331 


16.9 
248 


38.2 
165 


67.7 
142 


105. 
124 




Weight, lbs.. 


750- 
1000 


1000- 
1500 


2400- 
3400 


3500- 
4300 


5000- 
6000 



Turbine Water Wheels. — Turbines differ from the wheels, described 
above, in this respect: That in the case of the wheels the water acts only 
upon a portion of their circumference at any one instant, while with the 
turbines the water acts symmetrically and uniform.ly upon the moving parts. 
Turbines consist essentially of two main parts, namely, (1) the wheel 
(runner) with vanes arranged around the circumference, and revolving on 
a vertical or horizontal shaft; and (2) the casing, provided v/ith fixed 
guide vanes to give direction to the flow of water before it reaches the 
vanes of the wheel. Turbines are sometimes classified as "radial," "axial" 
and "combined or mixed;" and again, as reaction- and impulse turbines. 
In a radial turbine the water may flow from the circumference inward to 
the center, or the reverse; in an axial turbine, from the top downward 
or from the bottom upward; and in a combined turbine, inward and down 
or up, or outward and down or up. Turbines are most commonly mounted 
on vertical shafts with water flowing simply inward, outward or downward. 
The difference in principle between a reaction turbine and an impulse 
turbine is this: A reaction turbine is driven by the dynamic force of the 
flowing water augmented by static pressure to a greater or less extent; 
when there is no static pressure it becomes an impulse turbine. Standard 
makes of turbines will give an efficiency of from 65 to 85 per cent. An 
efficiency of 80 per cent can be counted upon for the best makes. 

Nomenclature of Terms. — Mr. John W. Thurso suggests the following, 
for uniformity. 

For "water wheel" say "turbine" whenever a turbine is meant. 

For "Jonval" turbine say "reaction" turbine. For Girard, Pelton, 
impulse or free deviation turbine say "action" turbine. For Fourneyron 
or Boyden turbine say "radial outward flow reaction" turbine, or else 
simply "outflow reaction" turbine. 



QUINTEX NOZZLE WHEELS. TURBINES. 1343 

For Francis turbine say "radial inward flow reaction" turbine, or sim- 
ply "inflow reaction" turbine, as this type is usually understood by the 
name Francis turbine. While it is not likely that the term Francis turbine 
will go out of use, the term "inflow reaction" turbine should always be used 
where an exact expression is of importance; as in contracts, for the reason 
that Mr. Francis also designed outflow reaction turbines. 

For "parallel flow" turbine say "axial" flow turbine, or axial turbine. 
For "segmental feed" turbine, say "partial feed" turbine, or "partial" 
turbine. 

For one, two, three, four; five or six turbines on one shaft say single, 
double, triple, quadruple, quintuple or sextuple turbine. 

For "draft chest" or "camelback" say "draft-tee." For "butterfly 
gate" say "wing-gate." For "feeder-pipe" or "water feeder" say "pen- 
stock." 

For open flume say "turbine -chamber" whenever a turbine chamber is 
meant, leaving the term "flume" to mean a water conductor only, built of 
wood, steel or masonry, and carrying water not under pressure. 

For the distance to which a vertical draft tube reaches below the surface 
of the tailwater, say dip of draft tube. 

For speed of water in the cylindrical part of the penstock say "penstock 
speed." For speed of water, while leaving or quitting lower end of draft 
tube, say "draft tube speed." 

For the part of the head that is above the turbine say "pressure head." 
F5r the part of the head that is utilized by means of a draft tube, say 
"draft head." 

The water available under the head utilized should be called "power 
water," corresponding in meaning to the term live steam. The water 
having descended either through the turbines or over the falls, should be 
called "tail-water," corresponding in meaning to the term exhaust steam. 

The terms horizontal or vertical turbine should always mean a turbine 
on a horizontal or vertical shaft, and never one revolving in a horizontal or 
vertical plane, as it is now sometimes understood. 

The area left open or clear by the regulating gate or gates for the passage 
of the water should be called "gate opening." At present the term "gate 
opening" or "gate" is nearly always used to mean the amount of water 
flowing through the gate opening, but this amount should be designated by 
discharge; for example, instead of saying: "This turbine, with five-eighths 
gate opening, gave an efficiency," etc., should be said; "This turbine, with 
five-eighths discharge, gave an efficiency," etc., whenever the discharge is 
meant. 

The motor furnishing the power for actuating the regulating gates of a 
turbine, and usually controlled by a speed governor, should be called a 
relay. In Europe the term servomotor is used for such auxiliary machines. 

Losses of Energy in Turbines. — ^These losses may be segregated as 
follows: 

(1) The hydraulic loss in the casing, from the penstock flange to the 

entrance to the guides (guide vanes) ; this is dependent to a large 
extent on the velocity of flow in the casing, it being remembered 
that for a "given velocity" the percentage of loss decreases with 
the increase of head. 

(2) The hydraulic loss in the guides and runners; this is affected by the 

type of runner (wheel) and the design of the guides, but the 
percentage of hydraulic loss remains practically_constant for any 
one type if the speed is allowed to vary as V/j, in which h — the 
head, in feet. 

(3) The hydraulic loss due to the leakage around the runner; this, as 

well as the discharge, varies as the vX hence the ratio of leakage 
to discharge remains constant. 

(4) The hydraulic loss in the draft-tube. 

(5) The mechanical loss due to the friction of the revolving parts; this 

increases with the head, h, but the percentage of mechanical loss 
decreases with the head. 

Efficiencies of Turbines. — The term "efficiency" is often used loosely, 
and it is very important that in turbine tests, and in specifications and 



1344 Q^.— WATER POWER. 

contracts, the specific kind of efficiency which is meant should be stated. 
There are four kinds of efficiencies which may be employed, namely: 

(a) The hydraulic efficiency, embracing losses (1), (2) and (3), above. 

(b) The mechanical effxciency, embracing loss (5), above 

(c) The efficiency of the turbine as a whole, being the product of the 

hydraulic and mechanical efficiencies, (a) and (b). 

(d) The efficiency of the plant as a whole, comprising the turbine 

efficiency (c) and embracing the draft-tube loss (4) and the pen- 
stock loss. 

Theoretic Horse-Power of Turbines. — ^The theoretic horse-power of a 
turbine varies as h^-^, in which h^the head of water in feet. This is true 
since the horse-power is proportional to the pressure of the water (which in 
turn varies with h, being equal to about 62.37a/z, see Table 1, preceding), 

and to the velocity_of flow per second (which in turn varies with \/'h, being 

equal to v=8.02\/h). Thus, 

Theoretic H. P. ^^lllSlL^^^l^ ^ 0.909468 ah^s (7) 

In which a = area of discharge in square feet; 
h = head in feet. 

Formula (1), preceding, is the same as formula (7), but expressed 'in 
terms of discharge Q instead of area a. 

When the efficiency of the turbine (or plant) has been determined, the 
actual H. P. 'is obtained by multiplying the values of equations (1) or (7) 
by said efficiency, expressed in per cent. 

The Transmission of Power from water motors to machinery is through 
the main shaft, and may be by belt, by gearing or by coupling. If the power 
is for electric transmission it is customary to connect the main shaft of the 
wheel or turbine directly with the electric dynamo or generator. Hence the 
term "direct-connected generator." 



TURBINES. MISCELLANEOUS DATA. 1345 



EXCERPTS AND REFERENCES. 

Modern Turbine Practice and the Development of Water Powers (By 

J. W. Thurso. Eng. News, Dec. 4, 1902 and Jan. 8, 1903). — Illustrations 
of turbines, hydraulic governors, and power-house installation. 

An Analysis of the "Commerciar* Value of Water Power per Horse- 
Power per Annum (By A. F. Nagle. Paper, Am. Soc. Mech. EngrS.; Eng. 
News, Jan. 22, 1903). — Discussion and tables of cost. 

Water Power Development at Chaudiere Falls, P. Q. (Eng. News, 
May 7, 1903). — Illustrations: Sections of main and wing dams, bulkhead 
wall, steel framing for gates and screens, operating mechanism for gates 
and screens, supports and anchorages for penstocks on side hill, plan and 
section of power house, traveling platform for building main dam, etc. 

The Niagara Power Plant of the Electrical Development Co., of Ont. 
(Eng. News, Nov. 9 and 30, 1905). — Fully illustrated with details. 

Power Plant of the Chicago Drainage Canal (Eng. News, Jan. 18, 
906). — Illustrated. 

Theory of Determining the Principal Dimensions of Water=Turbine 
Runners (By S. J. Zowski. Eng. News, Jan. 6, 1910). — ^Formulas, illus- 
trations and diagrams. 

Characteristics of the Modern Hydraulic Turbines (By C. W. Lamer. 
Trans. A. S. C. E., Vol. LXVI., Mar., 1910).— Tables of tests, of 28-in., 30-in., 
31-in. and 32-in. turbines. Formulas for power, speed, etc. 

Some Points in the Design of Impulse=Wheei Buckets (By Geo. M. 
Peck. Eng. News, May 5, 1910). — Illustrated. 

Hydro=Electric Development of the Michoacan Power Co., Mexico (By 
I. C. McBride. Eng. Rec, Aug. 27, 1910). — Illustrations: Details of rating 
flume; penstock intake; plan and details of sand trap; plan and details of 
headgate structure, plan and section of power house substructure. 

Important Designs for Reference: — 

Description. Eng. News. 

3,000-H. P. turbine for a Niagara water-power plant Nov. 14, 1901. 

Concrete dam with automatic flashboards June 12, '02. 

Section through power house and wheel pit, Niagara Plant July 3, '02. 

A water-wheel governor of novel construction Nov. 13, '02. 

Designs of buckets for impulse water wheels Oct. 8, '03. 

Cross-section of power house, Puyallup power development Sept. 29, '04. 

Section of flume and power station, De Sabla, Gal. Aug. 10, '05. 

Turbines at Sewalls Falls, under low and variable heads Jan. 18, '06. 

10,000-H. P. single-wheel turbine. Snoqualmie Falls Mar. 29, '06. 

Section of flume and power house, Cazadero, Ore. June 27, '07. 

Plan and section of power house. High Falls, Ont. July 18, '07. 

Sections of canal and flume, Centerville, Gal. Mar. 19, '08. 

Herschel's "Fall-increaser" for utilizing waste water July 11, '08. 

Flumes, gates, power station, etc., Kern River, Gal. Dec. 24, '08. 

Comparison of Am. high-speed runners for turbines Jan. 28, '09. 

Designs of intakes for hydro-electric plants April 8, '09. 

Eng. Rec. 
Sections of dam, sluice gate, powerhouse, hydro-elec. plant Mar. 27, '09. 
Expansion -joint details and reinforcement of concrete conduit May 1, '09. 
A low, 20-ft. head hydro-elec. development May 15, '09. 

Plans and sections of Hennepin power plant May 29, '09. 

Details turbine flumes, and their reinforcement, Schnec. Power 

Go. July 24, '09. 

Sections showing turbine settings, Gent. Ga. Power Co. May 14, '10. 

Details header pipe and upper end of penstock, Gt. W. Power 

Co, June 18, '10. 

Plan and section power house, Boulder hydro-elec. devel. July 30, '10. 



69.— STEAM AND GAS POWER. 

A.— HEAT. 

Matter and Energy. — In the light of modem science, all natural phe- 
nomena are due to matter and energy. 

Matter may be defined as anything capable of entering into chemical 
combination, and hence is made up of the chemical elements, of which the 
total known number, now about 90, may vary with future discoveries. By 
the Law of the Conservation of Matter, as deduced from chemistry, we learn 
that matter is indestructible: it may appear in various forms, yet the con- 
constituent elements are never actually lost or destroyed. Matter may 
exist in a solid, liquid, gaseous, or ethereal state. 

Energy is matter in motion, and is measured by the formula, 

Energy = i mass X (velocity)2; orE=hMV'^ (1) 

in which mass M = force F X acceleration a, 

= weight W-h gravity acceleration g. 

The Law of the Conservation of Energy, proven by various experiments, 
teaches us that energy is indestructible: it can appear and disappear in 
various forms, yet none of it is actually lost or destroyed. 

From the foregoing laws of the conservation of matter and energy, and 
from equation (1), we deduce the following: That the summation of each 
individual, infinitesimal sub-atom of matter in the universe, multiplied by 
the square of its velocity, at any single instant, is equal to twice the total 
energy in the universe and is a constant. Or, applied to any independent 
system protected from outside influences, the same statement holds true; 
hence in any such system, we have, 

i" M F2 = a constant (2) 

without regard to any change in form of the matter or its motion. 

Kinds of Energy. — ^There are two kinds of energy generally referred to, 
namely, 'Kinetic or actual ("moving") energy, and Potential or stored (static) 
energy. Strictly speaking, potential energy is simply a term used to express 
the amonut of kinetic energy which would be expended by a mass in chang- 
ing itself from one state, condition or position which it has assumed, to 
another state, condition or position which is predetermined. The energy 
expended may be in one or more forms. 

Forms of Energy. — ^There are four principal forms or outward mani- 
festations of energy: 

Examples. 
Mass (?) or "I Potential: A raised weight. 
Mechanical. [Kinetic: the falling of a raised weight. 
Molecular (?) j Potential: the latent heat of liquefaction. 
or Thermal. /Kinetic: the release of latent heat. 
Atomic (?) or 1 Potential: the energy stored in dynamite. 
Chemical. (Kinetic: the discharge of dynamite. 

Ethereal (?) or 1 Potential: the electric charge in a leyden jar. 
Electrical. J Kinetic: the discharge of a leyden jar. 

Other manifestations of energy are light, sound, magnetism, etc. 

Transformation of Energy. — Energy which is manifest in one form may 
appear in another forms: 

Examples. 
Mechanical to Thermal: (1) Heat resulting from impact of a projectile. 

*• Chemical: (2) [No direct transformation wholly kinetic?] 

" Electrical: (3) Electric current generated from a dynamo. 

1346 



THERMODYNAMICS. 



1347 



Thermal to Chemical: (4) Chemical building up of plant life by heat. 
Electrical: (5) Current generated by heating a thermo- 
electric pile. 
" Mechanical: (6) The mechanical work of a steam engine. 

Chemical to Electrical: (7) Current generated from a voltaic battery. 
Mechanical: (8) Energy of all animal and vegetable life. 
'_* Thermal: (9) Heat from oxidation, as combustion of fuel. 

Electrical to Mechanical: (10) Energy of an electric motor or fan. 

Thermal: (11) Heat evolved by current through non-con- 
ductor. 
** Chemical: (12) Electrolytic action, as electroplating, etc. 

Thermal Energy. — Heat and its effects may be considered as related to 
molecular action; and the transference of heat, simply as the transference 
of molecular action from one body to another, or from one medium to 
another. 

First Law of Thermodynamics : Heat and mechanical energy are mutu- 
ally convertible; and heat requires for its production, and produces by its 
disappearance, a certain definite number of units of work for each thermal 
unit. 

Thermal Units. — There are three units as follows: 

The British unit of heat or British thermal unit (B. T. U.) is equivalent 
to 778 foot-pounds of work, and may be defined as the quantity of heat 
required to raise the temperature of one pound of pure water 1° Fahrenheit, 
at or near its maximum density at 39.1° F. (most authors); or at the more 
prevailing temperature of 62° F. (Professor Peabody). 

The French thermal unit or calorie (Cal.) is equivalent to 426.84 meter- 
kilograms of work, and may be defined as the quantity of heat required to 
raise the temperature of one kilogram of water 1° centigrade, at about 4° C. 
(39.1° F.). 

The British-French unit of heat or "pound-calorie" (Lb.-Cal.) is equiva- 
lent to 1400.4 foot-pounds of work, and may be defined as the quantity of 
heat required to raise the temperature of one pound of water 1° C, at about 
4°C. 

Mechanical Equivalent of Heat, J. — One Joule* = 7=778 ft.-lbs.= 
1 British thermal unit {B. T. U.), is generally used in the United States as 
the mechanical equivalent of heat (see above). Hence, /= 1 B.T. U.= 
778 ft.-lbs.= l. 41^^45 horse-power seconds = 0.023^^57 h.-p. minute = 

0.00039^29 h.-p. hour; and y-=l ft.-lb. = 0.001285 B. T. U. One horse- 
power hour = 1,980,000 ft. -lbs. = 2545 heat units. One horse-power =2545 
heat units per hour = 25,450 heat units per day of ten hours. 

Tables 1 and 2, following, will be found useful in the conversion of 
mechanical and thermal power and work. 



" Joule's original experiments gave 7= 772 ft.-lbs. 



1348 



l—STEAM AND GAS POWER, 



—Thermal-Unit Equivalents. 
(See also Table 2, following.) 





Heat Units. 




Equivalent Mechanical Work. 








British Therm. 


French Therm. 


Br.-Fr. Therm . 




Units. 
(1 Lb. 1° F.) 


Units. 
(1 Kg. 1° C.) 


Units. 
(1 Lb. 1° C.) 




Foot- 


Meter- 


B. T. U. 


Cal. 


Lb.-Cal. 


pounds. 


Kilograms. 


1 


.252 


.5556 


778. 


107.57 


lis 


.4536 


I. 


1400.4 


193.63 


2. 


.504 


1.1111 


1556. 


215.14 


3. 


.756 


1.6667 


2334. 


322.71 


3.6 


.9072 


2. 


2800.8 


387.26 


3.968 


I. 


2.2046 


3087. 


426.84 


4. 


1.008 


2.2222 


3112. 


430.28 


5. 


1.260 


2.7778 


3890. 


537.85 


5.4 


1.3608 


3. 


4201.2 


580.89 


6. 


1.512 


3.3333 


4668. 


645.42 


7. 


1.764 


3.8889 


5446. 


752.99 


7.2 


1.8144 


4. 


5601.6 


774.52 


7.936 


2. 


4.4092 


6174. 


853.68 


8. 


2.016 . 


4.4444 


6224. 


860.56 


9. 


2.268 


5. 


7002. 


968.13 


10. 


2.520 


5.5556 


7780. 


1075.70 


10.8 


2.7216 


6. 


8402.4 


1161.78 


11.904 


3. 


6.6138 


9261. 


1280.52 


12.6 


3.1752 


7. 


9802.8 


1355.41 


14.4 


3.6288 


8. 


11203.2 


1549.04 


15.872 


4. 


8.8184 


12348. 


1707.36 


16.2 


4.0824 


9. 


12603.6 


1742.67 


18.0 


4.5360 


10. 


14004. 


1936.30 


19.840 


5. 


11.0230 


15435. 


2134.20 


23.808 


6 


13.2276 


18522. 


2561.04 


27.776 


7. 


15.4322 


21609. 


2987.88 


31.744 


8. 


17.6368 


24696. 


3414.72 


35.712 


9. 


19.8414 


27783. 


3841.56 


39.68 


10. 


22.046 


30870. 


4268.40 



Ex.— 9 B. T. U = 2.268 Cal. = 5 Lb.-Cal. 
kilograms. 



7002 ft.-lbs. = 968.13 meter- 



Caution. — In the above Table it is to be noted that the equivalents are 
for the same amount of work in each case; thus, 1 B. T. U. and 0.252 Calorie 
and 0.5556 Lb.-Cal.' are each equal to 778 ft. -lbs. of work. But using com- 
pound units we have: 

1 B. T. U. per pound = f Cal. per kilogram = | Lb.-Cal. per pound. 
1 .8 B. T. U. per pound = 1 Cal. per kilogram = 1 Lb.-Cal. per pound. 



THERMAL AND MECHANICAL EQUIVALENTS. 



1349 



2. — Mechanical Equivalents of Heat (B. T. U.). 



J^ 


t 


Thermal. 


Mechanical. 


o 


[eat Units 


Rate of Work in Foot-Pounds. 


Amt. of Work in Horse-Power- 


^ 




(B. T. U.) 
Per Hour 
)r 1 Hour. 
















Jc 


Per Min. 


Per Hour. 


Per Day. 
(24 Hours.) 


Minutes. 


Hours. 


Days. 
(24 Hrs.) 


rH 




.O453556 


.00069^4 


.041^6 


I. 


.Or. 1^2 6 


.O721044 


.0987682 






.O3IO711 


.00138^8 


.083^3 


2. 


.0,2^52 


.O742O88 


.0817536 


o 




.O3I6O67 


.00208^3 


.125''0 


3. 


.0,3^78 


.O763132 


.0826305 


«+-! 


-^ 


.O321422 


.00277^7 


.166^6 


4. 


.055^05 


.0784176 


.0835073 




4^ 
(.4 


.0326778 


.00347^2 


.208''3 


5. 


.056^31 


.0610522 


.0843841 


o 


g 


.O332134 


.00416^6 


.250''0 


6. 


.057^'57 


.0612626 


.0852609 


^ 


.0337489 


.00486^1 


.291^6 


7. 


. 058^83 


.0614731 


.0^61378 


u 

CD 




.0342845 


.00555^5 


.333''3 


8. 


.0410^10 


.0616835 


.0870146 


a 




.0348200 


.00625'"0 


. 375^0 


9. 


.0411^^36 


.0618940 


.0878914 






.001285 


.016''6 


1. 


24. 


.00003^03 


.065^05 


.O72IO44 


'S 




.002571 


. 033^^3 


2. 


48 


00006^06 


.Osl^'Ol 


.O742O88 


Ji 




.003856 


.050^0 


3. 


72. 


00009^09 


.051^51 


.O763132 


% 


e<I 


.005141 


.066^6 


4. 


96. 


.00012''12 


.052^02 


.0784176 


<u 


s 


.006427 


.083''3 


5. 


120. 


.00015^15 


052^52 


.0610522 


ffi 


fi 


.007712 


.100^0 


6. 


144. 


OOOIS'^IS 


.053^03 


.0612626 


- 


.008997 


.116^6 


7. 


168. 


.00021^21 


.053''53 


.0614731 


: o 




,010283 


.133''3 


8. 


192. 


.00024^^24 


.054^04 


.0616835 




.011568 


.ISO'^O 


9 


216. 


.00027''27 


.054^54 


.0618940 




.07712 


1. 


60. 


1440 


.001^^81 


.00003^03 


-05^26 




.15424 


2. 


120. 


2880. 


.003^^63 


.00006^06 


052^52 




.23136 


3. 


180. 


4320. 


.005^45 


.00009''09 


.053^78 


CO 


.30848 


4 


240. 


5760. 


.007^^27 


00012^12 


.055^05 


^'^ 


4^ 
(-1 


.38560 


5. 


300 


7200 


.oog'^oo 


,00015''15 


.056^31 


g^ 


£ 


.46273 


6. 


360. 


8640. 


.010^90 


.00018^^18 


.057^57 


^S 


.53985 


7. 


420. 


10080. 


.012^72 


.00021''21 


.058^83 


^"^ 




.61697 


8. 


480. 


11520. 


.014''54 


.00024^24 


.04lO''10 


SJ 




.69409 


9. 


540. 


12960. 


.016^36 


.00027^27 


.04ll''36 






1. 


12.96^^6 


778. 


18672. 


.023^57 


.00039^29 


.O4I637 




2. 


25.93^3 


1556. 


37344. 


.047''15 


.00078^58 


.043274 




3. 


38.90^0 


2334. 


56016. 


.070^72 


.001^78 


.044912 


2£ 


•^ 


4. 


51.86''6 


3112. 


74688. 


.094^^30 


.0015^71 


.046549 


"Sj 


12 


5. 


64.83^3 


3890. 


93360. 


.117^^87 


.0019^64 


.048186 


;Scg 


s 


6. 


77.80^0 


4668. 


112032. 


.141^45 


.0023^57 


.049823 




7. 


90.76^6 


5446. 


130704. 


.165''03 


.0027^50 


.031146 




8. 


103.73^3 


6224. 


149376. 


.188^60 


.0031^43 


.O3I31O 


^.-^ 




9. 


116.70''0 


7002. 


168048. 


.212''18 


.0035^36 


.O3I473 


gi 




42.41''6 


550. 


33000. 


792000. 


I. 


.016^6 


.00069^4 


o ^ 




84.83^3 


1100. 


66000. 


1584000. 


2. 


.033^3 


.00138''8 


•ro 




127.25^0 


1650. 


99000. 


2376000. 


3. 


.050^0 


.00208^3 


^■s 


lO 


169.66'^6 


2200. 


132000. 


3168000. 


4. 


.066^6 


.00277^7 


a§ 


(-1 


212.08^3 


2750. 


165000. 


3960000. 


5. 


.083^^3 


.00347^2 


i2 2 


s 


254.50^0 


3300. 


198000. 


4752000. 


6. 


.100^0 


.00416^6 


•^6 
§5^ 


296.9^6 


3850. 


231000. 


5544000. 


7. 


.116^6 


.00486^^1 




339.33^3 


4400. 


264000. 


6336000. 


8. 


.133^^3 


.00555^5 






381.75^0 


4950. 


297000. 


7128000. 


9. 


.150^0 


.00625^0 




2545. 


33000. 


1980000. 


47520000. 


60. 


1 


.041^6 




5090. 


66000. 


3960000. 


95040000. 


120. 


2. 


.083^3 






7635. 


99000. 


5940000. 


142560000. 


180. 


3. 


.125^0 


1:5 'S. 


«d 


10180. 


132000. 


7920000. 


190080000. 


240. 


4. 


.166''6 


4^ 


12725. 


165000. 


9900000. 


237600000. 


300. 


5. 


.208^3 




S 


15270. 


198000. 


11880000. 


285120000. 


360. 


6. 


.250^0 


II 


17815. 


231000. 


13860000. 


332640C00. 


420. 


7. 


.29^6 




20360. 


264000. 


15840000. 


380160000. 


480. 


8. 


.333^3 




22905. 


297000. 


17820000. 


427680000. 


540. 


9. 


.375^0 






61080. 


792000. 


4752x104 


114048x104 


1440. 


24. 


1 




122160. 


1584000. 


9504x104 


228096x104 


2880. 


48. 


2! 


.Sffi 




183240. 


2376000. 


14256x104 


342144x104 


4320. 


72. 


3. 


■^' 


^ 


244320. 


3168000. 


19008x104 


456192x104 


5760. 


96. 


4. 


< a 


4i 
t-t 


305400. 


3960000. 


23760x104 


570240x104 


7200. 


120. 


S. 


& 


366480. 


4752000. 


28512x104 


684288x104 


8640, 


144. 


6. 


6 


427560. 


5544000. 


33264x104 


798336x104 


10080. 


168. 


7. 


•{fl 




488640. 


6336000. 


38016x104 


912384x104 


11520. 


192. 


8. 






549720. 


7128000. 


42768x104 


1026432x104 


12960. 


216. 


9 



1350 



I— STEAM AND GAS POWER. 



Examples in the Use of Table 2, preceding. 
Part 1. — 1 000 000 ft. -lbs. per day is at the rate of 694.44 ft.-lbs. per min., 
and equivalent to 53.556 heat units per hour. 53.556 heat units represent 
an amount of work equal to 1.2626 horse-power min., or .021 h.-p hour, or 
.000877 h.-p. day. 

Part 2.-5 000 000 tt.-lbs. per hour = 6427 heat units per hour. 5 000 000 
ft.-lbs., or 6427 heat units, are equal to 2.5252 horse-power hours, or 1 horse- 
power for 2.5252 hours, or 2.5252 horse-power for 1 hour. 

Part 3. — 33 000 ft.-lbs. per min. = 2313.6+ 231.36=2545 heat units per 
hour. 

Part 4' — One heat unit for any time T is equal to 778 ft.-lbs. in time T; 
hence Columns 1 and 3 may be used independently of time, throughout the 
Table, to give Equivalent Work in Ft. -Lbs. and Heat Units. 

Part 6. — 60 horse-power min. = 2545 heat units. 

Part e.—Zbl horse-power hours= 763500+ 127250+ 17815 = 908565 heat 
units, and equivalent to 706 860 000 (= 70686 XlO^) ft.-lbs. (per hour for 
1 hour.) 

Part 7. — Six horse-power days =366480 heat units; and 366480 heat 
units per hour represent work at the rate of 6 842 880 000 ft.-lbs. per day. 

B.— FUEL. 

Heating Power of Fuels. — The amount of heat energy, iriB. T. U., con- 
tained in coal, wood and other fuels may be determined by three methods: 
(a) by chemical analysis; (b) by burning the fuel in a calorimeter; (c) by 
practical test in connection with a steam boiler. 

(a) Chemical Analysis. — There are two kinds of analysis of coal (or fuel), 
namely, "proximate analysis" and "ultimate analysis." 

Proximate Analysis separates the coal into moisture, volatile matter, 
fixed carbon, and ash. In making this analysis a pulverized sample is 
weighed carefully and then heated to a temperature not to exceed 280° F. 
When, after repeated weighings, it ceases to lose more weight, the loss in 
weight by this heating is recorded as "moisture." The heating is then con- 
tinued in a crucible with a lid cover, and after the temperature is raised 
gradually to a red heat and continued for a few minutes until gas ceases to 
be driven off, the crucible is cooled in a dessicator, to prevent absorption of 
moisture from the air, and weighed, the loss in weight by this heating being 
recorded as "volatile matter." i?he heating is then continued and the 
crucible raised to a white heat, the lid being partly open to admit air to 
bum the coal, and when the carbon is burned away, leaving nothing but the 
ash, the latter is weighed when cooled and the loss in weight by this heating 
is recorded as "fixed carbon." The weight of the ash is of course recorded 
as "ash". If the weighings are recorded in percentages the sum of the four 
constituent parts will add up to 100 per cent. Sometimes "sulphur" is 
included in the proximate analysis, either as part of the 100 per cent above 
mentioned or separately, the percentage of sulphur being determined by 
separate analysis, and 40 to 50% assumed to have escaped with the volatile 
matter and 60 to 50% with the fixed carbon. 

The proximate analysis, above, indicates the character of the coal. 
Omitting the moisture and the ash, and letting the sum of the "volatile 
matter" and "fixed carbon" equal 100 per cent of the "combustible," we 
have the following classification: 

3. — Coals Classified by Relative Percentages of Carbon and 

volatiles. 



Kind of Coal. 


Per cent 

Fixed 

Carbon. 


Per cent. 

Volatile 

Matter. 


Heating Value. 

B. T. U. 

per Lb. Coal. 


Relative 

Combustible 

Value. 


Anthracite 


100 to 92 
92 to 87 
87 to 75 
75 to 60 
65 to 50 

under 50 


to 8 
8 to 13 
13 to 25 
25 to 40 
35 to 50 
over 50 


14600 to 14800 
14700 to 15000 
15500 to 16000 
14800 to 15200 
13500 to 14800 
11000 to 13500 


93 


Semi-Anthracite 

Semi-Bituminous 

Bituminous, Eastern 

Bituminous, Western 

Lignite 


94 

100 

95 

90 

77 



COAL AS FUEL--CHEMICAL ANALYSIS. 



1351 



4. — Proximate Analysis and Heating Values op U. S. Coals. 
Note. — See Table 5 for Ultimate Anaylses of Fuels. 



Kind of Coal. 



Anthracite. 

Northern coal field 

East Middle coal field 

West Middle coal field 

Southern coal field 

Semi-Anthracite. 

Loyalsock field 

Bemice basin 

Semi-Bituminous. 

Broad Top, Pa 

Clearfield County, Pa 

Cambria County, Pa 

Somerset County, Pa 

Cumberland, Md 

Pocahontas, Va, 

New River, W. Va. . 

Bituminous. 

Connellsville, Pa 

Youghiogheny, Pa 

Pittsburg, Pa 

Jefferson County, Pa 

MiddleKittanning seam. Pa 

Up.Freeport seam,Pa.and O 

Thacker, W. Va 

Jackson County, O 

Brier Hill, O 

Hocking Valley, O 

Vanderpool, Ky 

Muhlenberg County, Ky 

Scott County, Tenn 

Jefferson County. Ala 

Big Muddy, 111 

Mt. Olive, 111 

Streator, 111 

Missouri 

Lignites and Lignitic Coals. 

Iowa 

Wyoming 

Utah 

Oregon Lignite 



15 



4. 

3.08 
3.72 
4.28 

8.10 
9.40 

15.61 
22.52 
19.20 
16.42 
17.30 
21.00 
17.88 



83.27 

86.40 

81.59 

.81 

83.34 
83.69 

77.30 
71.82 
71 12 
71.51 
73.12 
74.39 
77.64 



37.09 
38.72 
41.97 
42.98 



35.60 
41.83 
44.37 
33.32 



8 

6.22 

10.65 

8.18 

6.23 
5.34 

5.40 

3 

7.04 

8.62 

7.75 

3.03 

3.36 

8.23 
2.61 
8.02 
4.27 
7.18 
9.10 
6.27 
6.50 
4.30 
8.00 
7.30 
4.95 
8.02 
2.62 
8.00 
13.00 
14.00 
05 

18.86 

11.26 

3.20 

7.11 



.73 

.58 
.50 
.64 

1.63 
.91 

.90 
.91 
1.70 
1.87 
.74 
.58 
.27 

.78 
.81 
1.80 
1.00 
1. 
2.89 
1.28 



1.59 



1.57 
1.80 
1.42 



a'3 

■B-i 
16 



13,160 
13,420 
12,840 
13,220 

13,920 
13,700 

14,i 

14,950 

14.450 

14.200 

14.400 

15.070 

15,220 

14,050 
14,450 
13.410 
14,370 
13,200 
13,170 
14,040 
13,090 
13,010 
12.130 
12.770 
13,060 
13,700 
13,770 
12,420 
10,490 
10.580 
12,230 

8.720 
10.390 
11.030 

8,540 



1 

o « 

> 



5.00 
3.44 
4.36 
4.85 

8.86 
10.98 

17.60 
24.60 
22.71 
20.37 
19.79 
22.50 
18.95 

34.03 
38.73 
41.61 
35.47 
40.27 
43.59 
39.33 
35.76 
38.20 
42.81 
38.50 
38.86 
34.17 
37.63 
36.30 
47.00 
45.00 
43.94 

51.03 
48.07 
48.60 
54.95 






n 

t> BO 

lis 



14.900 
14,900 
14,900 
14.900 

15,500 
15,500 

15.800 
15.700 
15.700 
15.800 
15.800 
15.700 
15,800 

15.300 

15,000 

14.800 

15.200 

14.500 

14.800 

15,200 

14,600 

14,300 

14,200 

14.400 

14,400? 

15.100? 

14.400? 

14,700 

13,800 

14.300 

14,300? 

12,000? 
12.900? 
12.600? 
11,000? 



O TO 

8.1 



^a 

ap 

03O 



15.42 
15.42 
15-42 
15.42 

16 05 
16.05 

16.36 
16 25 
16.25 
16.36 
16.36 
16.25 
16.36 

15.84 
15.53 
15.32 
15.74 
15.01 
15.32 
15.74 
15.11 
14.80 
14.70 
14.91 
14.91 
15.63 
14.91 
15.22 
14.29 
14.80 
14.80 

12.42 
13.35 
13.04 
11.39 



Note. — ^The following values are given for Anthracite coal from one mine: 
Egg coal (screen 2^"-lf'0 88.49% carbon, 5.66% ash; stove coal (screen 
ir-li'O 83.67% carbon, 10.17%, ash; chestnut coal (screen lV'-¥') 80.72% 
carbon, 12.67% ash; pea coal (screen f'-D 79.05% carbon, 14.66% ash; 
buckwheat coal (screen Y'-i") 76.92% carbon, 16.62% ash. 



Ultimate Analysis reduces the "combustible" constitutents of the fuel, 
^. ^., the "volatile matter" and "fixed carbon" (but not the moisture and 
ash) to the ultimate chemical elements. 



1352 



h— STEAM AND GAS POWER 



5. — Chemical Composition op Several Kinds op Solid Fuels. 

(Ultimate Analyses.) 
Note. — See Table 4 for Proximate Analyses of Coals. 



Kind of Fuel. 


Moist- 
ure. 


Car- 
bon. 


Hydro- 
gen 


Oxy- 
gen. 


Nitro- 
gen. 


Sul- 
phur. 


Ash. 


B.T.U. 
per Lb. 

of 
Fuel. 


W 


ood, dry, average 

" Ash 

















10.0 
20.0 
40.0 
30.0 
12.0 
16.0 

1.0 

10 

1.4 

7.5 
11.0 
17.4 
15, 8 


49.5 

49.18 

49.06 

48.88 

48.99 

50.36 

50.16 

50.31 

44.5 

39.6 

29.7 

40.6 

84, 

36. 

86. 

84. 

75. 

67. 

56. 

50. 

55. 


6.1 

6.27 

6.11 

6.06 

6.20 

5.92 

6.02 

6.20 

5.5 

4.9 

3.7 

4.2 

1.0 

5.0 

1.0 

4.2 

5.0 

4.8 

5.0 

4.0 

4.5 


43.8 
43.91 
44.17 
44.67 
44.25 
43.39 
43.36 
43.08 
39.4 
35.0 
26.2 
21.7 
0. 
38.0 
1.0 
3.4 
8.0 
10.0 
11.0 
14.0 
16.9 


0.1 

07 

0.09 

0.10 

0.06 

0.05 

0.09 

0.04 

0.1 

0.1 

0.1 




0.5 

0.57 

0.57 

0.29 

0.50 

0.28 

0.37 

0.37 

0.5 

0.4 

0.3 

3.5 

3. 

5. 

'?: 

8. 

8. 
13. 
13.6 

5.6 


8800 
8480 




" " Beech 

" Birch.., 

" Elm 




8591 






8586 






8510 




" Fir 




9063 




•• Oak 




9316 




" Pine 




9153 




10% moisture.av 
20% 

40% " 
at 












Pe 

Ch 
Sti 
Co 

Br 

Lig 






arcoal 

'aw 














al. Anthracite 

Semi-Bituminous. . . 

• Bituminous Pitts'g.. 
" Hocking Val., O 
•• Illinois 

own Coal, Pac. Coast. . . 

fnite, Pacific Coast 


0.5 
0.8 
1.0 
1.2 
1.0 
1.0 
1.1 




1 

1 
3 


5 
6 
6 
5 






1 


1 






Green wood contains from 25 to 50 per cent moisture. Air-dried wood 
contains from 10 to 20 per cent moisture — usually 12 to 15 per cent. 

Calculations of the Heat of Combustion of a fuel, based on the ultimate 
analysis, are usually performed by means of Dulong's formula, or by means 
of other formulas which resemble, or are more or less modifications of, 
Dulong's formula. Thus, 

The total heat units ^B. T. U.) per lb. of coal — 

= 14650 C+ 62100 {H-\0) (Dulong) (1) 

= 14650 C+ 62100 H - 5400 {.0 + N) (Mahler) (2) 

In which C, H, O and N represent the relative parts of carbon, hydrogen, 
oxygen and nitrogen in the coal. 

Example. — ^The ultimate analysis of a certain coal gave, in parts (per- 
centages expressed in parts of a unit), carbon (C) = 0.8566, hydrogen {H) = 
0.0278, nitrogen (A^) = 0.0077, oxygen (O) = 0.0287, ash = 0.0733, volatile 
sulphur =0.0059. How many heat units would one pound of this coal be 
expected to supply, or in other words, what is the total heat of combustion ? 

Solution.— ?Tom Dulong's formula, total heat = 14650X .8566+ 62100 
(. 0278- iX. 0287) = 14052 B. T. U.\ and from Mahler's formula, total heat 
= 14079 5. T. U. 

Mr. Henry J. Williams* gives the following values for heat of combus- 
tion, as calculated by Dulong's formula: — Anthracites: Lehigh, 13963 B. T. 
U., Lykens Valley, 13954; Drifton, Pa,, 14171. Semi-bituminous: Poca- 
hontas, 14805; Georges Creek, 14484; Clearfield, Pa., 14448; New River, 
W. Va., 14607. Bituminous: Connellsville (coking), 14043; Big Muddy, 
Carterville. 111., 12561; Dominion, Cape Breton, 13755; Pittsburg (steam- 
ing), 13719. But compare these heating values with those in Table 4, pre- 
ceding. 

(b) Calorimeter. — ^The "bomb calorimeter" is an instrument used to 
determine the actual heating value of coal. Mahler's instrument consists of 
a strong steel vessel immersed in water. One gram of the coal is placed in 
a platinum vessel within the bomb, oxygen gas is introduced, and the coal 



* See page 41, Steam Boilers, by Peabody and Miller; John Wiley & Sons, 
New York. 



FUELS. TESTS—CALORIMETER; BOILER. 



lUz 



ignited by an electric spark. The resultant heat of combustion is radiated 
into the surrounding water and is measured by the rise in temperature of 
the water, making the necessary corrections for absorption of heat by the 
instrument itself. 

This method agrees usually within 2 per cent of the calculations from 
chemical analysis, when the tests are carefully made. 

(c) Practical (Boiler) Test. — ^The following Table gives the summary of 
nine tests made on one 250 H. P. Cahall boiler at factory of the Armstrong 
Cork Co., Pittsburg, Pa., 1896; 



6. — Average op Nine Boiler Tests — Coal as Fuel. 



Duration of test , hours. . 

No. of boilers 

Average Pressure of Steam in Boiler J)y Gage 

Average Temperatures. 

Of feed-water entering boiler deg. F . . 

Of steam in boiler deg. F. . 

Fuel (Nut, Nut and Slacls, Run of mine, etc.) Coal. 

Cost pel ton of 2,000 pounds, delivered 

Calorific power by Calorimeter B. T. U . . 

Total quantity consumed lbs . . 

Total asli, clinkers and unburned coal lbs . . 

Proportion of ash, etc., to coal per cent . . 

Total combustible burned lbs. . 

Combustion per Hour. 

Coal actually consumed lbs . . 

Combustible actually consumed lbs . . 

Per sq. ft. grate surface — coal lbs . . 

Per sq. ft. grate surface — combustible lbs. . 

Per sq ft. tieating surface — coal lbs . . 

Per sq. ft. heating surface— combustible lbs . . 

Water. 

Amount apparently evaporated lbs . . 

Factor of evaporation . . 

Equivalent evaporation into dry steam from and at 212°F. . .lbs. . 
Economic Evaporation — per pound of coal. 

Water actually evaporated bis . . 

Equivalent from and at 2 1 2° F lbs . . 

Per pound of combustible — water actually evaporated lbs. . 

Equivalent from and at 212° F lbs. . 

Evaporation per Hour. 

Water actually evaporated lbs . . 

Equivalent from and at 2 1 2° F lbs . . 

Per sq. ft. heating surface — water actually evaporated. ... lbs . . 

Equivalent from and at 212° F lbs. . 

Per sq. ft. grate surface — water actually evaporated lbs . . 

Equivalent from and at 2 1 2° F lbs . . 

Efficiency. 

Percentage of total calorific power utilized, or efficiency. ...%.. 

Water evaporated per $1.00 worth of fuel lbs. . 

Cost of evaporating 1,000 lbs. or water cents. . 

Coal consumed per horse power per hour . lbs . . 

Cost of same cents . . 

Horse Power. 

Actually developed on basis of 34^ lbs. of water evaporated per 
hour from and at 212° F 

Commercial rating 

Proportion capacity developed is of commercial rating . . . % . 

Heating surface required to develop one horse power. . . .sq.,ft. . 



1.8 



62.6 
334.2 

$1.01 

12,963.5 

10.132.8 

975.06 

9.69 

8,961.94 

1,190.29 

1,082.79 

31.03 

32.54 

.47 

.43 

86,199.08 

1.20 

103,007.73 

8.735 
10.43 

9.1106 
11.5869 

10,448.57 

12,582.82 

4.03 

4.9571 

298.53 

320.68 

77.867 

19,858.47 
4.98 
3.35 
0.1724 



364.708 

250.0 

145.859 

7.3602 



1354 m.STEAM AND GAS POWER. 

C— STEAM. 

General Discussion. — If one pound of ice at absolute zero (= — 273.7° C. 
«=» — 460.66° F.) is gradually heated, its temperature will rise about directly 
in proportion to the amount of heat it receives (say approx. 1* C. per each 
Calorie, or 1° F. for each B. T. U.) until it reaches the melting point or 
fusion point ( = 0° C. = + 32° F.) ; then its temperature wiH remain constant 
during melting or until it has received 143 additional E. T. U., making a 
total of about 460.66+ 32+ 143= 635.66 B. T..U. to change one pound of ice 
at absolute zero into one pound of water at the temperature of freezing.* 

The 143 B. T. U. required to change one pound of ice at + 32° F. into 
one pound of water at 32° F. is called the latent heat of fusion of ice, and 
represents the work required to change the molecular condition of ice to the 
molecular condition of water, since there is no rise in temperatiire. Con- 
versely, one pound of water at 32° F. will give out 143 B. T. U. in changing 
to ice at 32° F. 

If one pound of water is gradually heated in an open vessel, under an 
atmospheric pressure of 14.7 pounds per square inch, from freezing point at 
32° F. up to the boiling point at 212° F., its temperature will rise almost 
directly in proportion to the amount of heat received, i. e., 1° F, for each 
B. T. U., or a total of 180 B. T. U.; then its temperature will remain con- 
stant during the process of boiling away, called vaporization. But if the at- 
mospheric pressure on the surface of the water had been increased, its 
boiling point would have been raised. This is shown clearly in the first two 
columns of Tables 7 and 8, following. Thus, from Table 8, an absolute 
pressure of 90 pounds per square inch would raise the boiling point tempera- 
ture to 320° F. and it would remain at that temperature until completely 
vaporized. In this case, the pound of water would become the receptacle of 
about 320—212=108 B. T U. more than it could receive under ordinary- 
atmospheric pressure. It is thus seen that the temperature under which 
steam is generated, depends upon the pressure of the liquid. 

Starting with one pound of water at 32° F., and heating it, we find from 
Column 3, Tables 7 and 8, the number of heat units received at the various 
boiling-point temperatures, under corresponding pressures; and if the heat- 
ing is continued at the boiling point. Column 5 shows the number of heat 
units required to convert the water into steam at the same temperature, 
called the heat of vaporization; and Column 4 shows the total heat in B. T. 
U., of the steam above that which the water contained at 32° ¥. Thus, for 
a temperature of 320° F., the total heat = 1179.5 = 290.0+ 889.5 = heat of the 
liquid + heat of vaporization. 

It is to be remembered that the heat of vaporization, or latent heat of 
vaporization, represents the number of heat units consumed in changing 
one pound of water into steam at the same temperature. Now as the 
temperature is not raised, it is evident that some other work is being per- 
formed. In fact, the latent heat is converted into the work of separating 
the molecules of the water: (1) against molecular attraction, and (2) against 
external resistance or pressure. The 1st is called the heat equivalent of 
internal work (Table 7, Column 6) , and the 2nd is called the heat equivalent 
of external work (Column 7). It is evident, then, that the value in Column 5 
of Table 7, for any definite temperature or pressure, is equal to the sum of 
the corresponding values in Columns 6 and 7. 

Steam generated in a closed vessel is necessarily in contact with the 
water and must have the same temperature and pressure as the water. In 
this condition it is called wet saturated steam or "wet steam." Its specific 
volume, or volume in cubic feet of one pound, and its density, or weight in 
pounds per cubic foot, are each constant for any definite temperature and 
pressure, as will be seen from the last two columns of Table 7. Just at the 
point when the water is completely evaporated the steam is known as dry 
saturated steam or "saturated steam." If further heated, it becomes super- 
heated steam. Summarizing, we have wet steam from the time water 
begins to evaporate up to the time of complete evaporation ; saturated steam 
at the moment of complete evaporation; and superheated steam when 
further heated. 

Furthermore, saturated steam is dry steam of a temperature due to its 
pressure; that is, it contains no moisture, nor is it superheated. When it 
is heated above a temperature due to its pressure it becomes superheated 

* This illustration is given simply to convey graphically to the mind the 
general effect of heat; the exact values are not important. 



STEAM— FORMULAS. 1355 

steam. Dry steam may be either saturated or superheated. If water is 
injected into superheated steam it will immediately vaporize, reducing the 
temperature of the latter; and if sufficient water is injected the steam will 
be reduced to saturated steam. If cold water is injected into saturated 
steam some of the steam will be condensed, and the temperature and pres- 
sure of the remainder will be lowered. Wet steam contains mist or globules 
of moisture, but its temperature and pressure bear the same relation as 
that for saturated steam. Saturated steam, then, can have but one tempera- 
ture for any given pressure, Mobile superheated steam can have any higher 
temperature. 

Superheated Steam tends to follow the laws of perfect gases, the more 
nearly so the farther it is removed from the point of saturation. From the 
laws of gases, we have, 

pv = RT (1) 

In which :^ = pressure in lbs. per sq. in.; 
I' = volume in cu. ft.; 

T = temp. in degrees F. above absolute zero; 
R = a, constant, depending upon the gas; / 

= foot-pounds of external work done in altering the temp. 1° 

under constant pressure; 
= 53.2 for air; 

= for any gas, 5 3. 2-;- the specific gravity of the gas, referred to 
air. 

Saturated Steam. — The laws of saturated steam do not follow closely 
the laws of perfect gases, yet have a close analogy to them. The following 
notation and formulas are mainly those used in Thermodynamics of the 
Steam Engine * by Prof C. H. Peabody (under whom the writer studied), 
and are used here with his permission. See also Tables 7 to 10. 

Notation and Formulas, 

F = fahrenheit (absolute zero= — 460.66 F.); 1 Observe the 
C = centigrade (absolute zero = — 273. 7 C.) ; algebraic 

it = temperature in degrees F. = §]fo+ 32; f signs: 

/o = temperature in degrees C. = | (i— 32) ; | +for above 0°; 
r = absolute temperature in degrees F. = 460.66 + ^; —for below 0°. 

To = absolute temperature in degrees C. = 273.74-^; J 
^ = pressure (above vacuum) of saturated steam in lbs. per square inch. 



Log 



D n ( In which A = 6.1007; 
^ = A-^--^ ] log 5 = 3.43642; log L> = 5.59873; 
^ ^ / and using r = ^-|- 461.2° F. 



'heat of the liquid; that is, the quantity of heat, in French units (num- 
ber of heat units of 426.84 meter-kgs.), required to raise the tem- 
perature of one kilogram of water from freezing (0° C.) to a given 
temperature to', 

'^0+ 0.000002 ^o2+ 0.0000003^3. 



■/' 



c dt, in English units. 



A = total heat; that is, the quantity of heat (number of heat units of 
778 ft. -lbs.) required to raise one pound of water from freezing 
(32° F.) to a given temperature t, and to entirely vaporize it under 
the pressure due to that temperature; 
= 1091.7 + 0.305 (t- 32). But see Marks and Davis formula, page 1378. 
r = heat of vaporization, which may be defined* as the number of heat 
units, of 778 ft. -lbs., required to vaporize one pound of water at a 
given temperature under the corresponding pressure; 
ip total heat — heat of liquid = X—q. 
/o = heat equivalent of internal work; that is, the internal latent heat or 
the heat units, of 778 ft. -lbs., required to do the disintegration 
work during the vaporization of one pound of water; 
= r—A p u = r — A p {s— d)\ 

where A = reciprocal of mechanical equivalent of heat, 

= j= ^ = 0.00128535. and 



* Published by John Wiley & Sons, New York. 



1356 69.— STEAM AND GAS POWER, 

s= specific volums of. saturated steam; that is, the volume 
in cubic feet of one pound of satiirated steam. 
5 = specific volume of the water. 

pu = p (s— d) = external work. 
Apu = hea,t equivalent of external work; that is, the external latent heat or 
the heat units, of 778 ft. -lbs., required to overcome the external 
pressure, and do the work of increasing the volume from d to 5. 

<: = specific heat of the liquid == -rr. 

at 

I -:=• = entropy of the liquid ; the term entropy is used to denote a quality 
J ^ or condition of the liquid, increasing when heat is added, and de- 

creasing when heat is subtracted; the entropy remains constant 
when no heat is added or subtracted; it is a very useful property in all 
thermodynamic calculations * 
5 = specific volume, or volume in cubic feet of one pound of steam. 
r = density, or weight in pounds of one cubic foot of steam; 
1 



Saturated-Steam Tables. — Tables 7 and 8, following, are the old 
tables which have been used by engineers for the past ten years. These 
are followed by Tables 9 and 10, which are calculated from more recent ex- 
perimental data. The older tables are reproduced here simply for methods 
of comparison, and Tables 9 and 10 should be used because the values are 
more correct. 

The formulas given above are the old formulas. The new formula for 
heat of vaporization, in English units, is 

r = 141.124 (689 — °'^^*9. 



* Entropy Diagrams are very useful for this purpose. They represent, 
graphically, the successive thermal changes in a body due to the simultane- 
ous variations of Temperature and Entropy — two of the coordinates 
characterizing the conditions of the body. In this connection it may be 
stated that Total Heat represents energy or Work. Now, Work is made up 
of two factors, force and distance (W=fd). Similarly, the Toial Heat may be 
considered as made up of two factors, temperature a.nd entropy (H =Te or A= 
TO); or, graphically, the total heat is the diagram area whose ordinate at 
the extreme right of the area considered is the temperature T, laid off from 
the abscissa whose value is the entropy 6; the total heat being the sum- 
mation of the vertical strips. 



SATURATED STEAM— FORMULAS; TABLES. 



1357 



7. — Saturated Steam — English Units.* 
(Condensed from Prof. Peabody's Tables.) 
Note. — See preceding notation; also Table 8, following. 







i 






4 


IS 




« 


Density. 


A 


1 


i 

1 


w 




li 

■PHH 




1 




=1 




fi 


w 


H 


» 


w 


M 


GQ 


^ 


t 


P 


Q. 


A 


r 


p 


Apu 


ff 


s 


r 


32 


0.0890 


0. 


1091.7 


1091.7 


1035.9 


55.8 


0.0000 


3387 


0.0002952 


34 


0.0963 


2.01 


1092.3 


1090.3 


1034.3 


56.0 


0.0041 


3138 


0.0003187 


36 


0.1042 


4.03 


1092.9 


1088.9 


1032.8 


56.1 


0.0081 


2910 


0.0003436 


38 


0.1126 


6.04 


1093.5 


1087.5 


1031.3 


56.2 


0.0122 


2700 


0.0003704 


40 


0.1216 


8.06 


1094.1 


1086.0 


1029.6 


56.4 


0.0162 


2506 


0.0003990 


45 


0.1471 


13.08 


1095.7 


1082.6 


1025.8 


56.8 


0.0262 


2087 


0.0004792 


50 


0.1773 


18.10 


1097.2 


1079.1 


1021.8 


57.3 


0.0361 


1745 


0.0005731 


55 


0.2128 


23.11 


1098 7 


1075.6 


1017.9 


57.7 


0.0459 


1465 


0.0006829 


60 


0.2545 


28.12 


1100.2 


1072.1 


1014.0 


58.1 


0.0555 


1234 


0.0008104 


70 


0.3602 


38.11 


1103.3 


1065.2 


1006.2 


59.0 


0.0745 


885.0 


0.001130 


80 


0.5027 


48.09 


1106.3 


1058.2 


998.3 


59.9 


0.0932 


643.8 


0.001553 


90 


0.6925 


58.04 


1109.4 


1051.4 


990.6 


60.8 


1114 


474.6 


0.002107 


100 


0.9421 


68.01 


1112 4 


1044.4 


982.7 


61.7 


0.1293 


354.0 


0.002824 


110 


1.2663 


78.0 


1115.5 


1037.5 


974.8 


62.7 


0.1470 


267.5 


0.003738 


120 


1.6828 


88.1 


1118.5 


1030.4 


966.7 


63.7 


0.1645 


204 4 


0.004892 


130 


2.2119 


98.1 


1121.6 


1023.5 


958.9 


64.6 


0.1817 


157.8 


0.006336 


140 


2.8774 


108.2 


1124.6 


1016.4 


950.8 


65.6 


0.1986 


123. 2 


0.008120 


150 


3.7063 


118.3 


1127.7 


1009.4 


942.8 


66.6 


0.2152 


97.03 


01031 


160 


4.7292 


128.4 


1130.7 


1002.3 


934.8 


67.5 


0.2316 


77.14 


0.01296 


170 


5.981 


138 5 


1133.8 


995.3 


926.8 


68.5 


0.2477 


61.85 


0.01617 


180 


7.500 


148.5 


1136.8 


988.3 


918.9 


69.4 


0.2636 


50.01 


0.02000 


190 


9.330 


158.6 


1139.9 


981.3 


911.0 


70.3 


0.2792 


40 73 


0.02455 


200 


11.520 


168.7 


1142.9 


974.2 


903.0 


71.2 


0.2946 


33.40 


0.02994 


210 


14.122 


178.8 


1146.0 


967.2 


895.2 


72.0 


0.3097 


27.57 


0.03628 


220 


17.186 


188.9 


1149.0 


960.1 


887.1 


73.0 


0.3246 


22.98 


0.04352 


230 


20.783 


198.9 


1152.1 


953.2 


879.4 


73.8 


0.3393 


19.20 


0.05208 


240 


24.982 


209.0 


1155.1 


946.1 


871.6 


74.5 


0.3538 


16.14 


0.06195 


250 


29.856 


219 1 


1158.2 


939.1 


863.8 


75.3 


3681 


13.65 


0.07327 


260 


35.483 


229.2 


1161.2 


932.0 


855.9 


76.1 


0.3822 


11.60 


0.08619 


270 


41.945 


239.3 


1164.3 


925.0 


848.1 


76.9 


0.3961 


9.918 


0.1008 


280 


49.328 


249.3 


1167.3 


918.0 


C40.4 


77.6 


0.4098 


8.521 


0.1173 


290 


57.72 


259.4 


1170.4 


911.0 


832.6 


78.4 


0.4233 


7.356 


0.1359 


300 


67.22 


269.5 


1173.4 


903.9 


824.7 


79.2 


0.4366 


6.380 


0.1567 


310 


77.93 


279.6 


1176.5 


896.9 


817,0 


79.9 


0.4498 


5.558 


0.1799 


320 


89.95 


290.0* 


1179.5 


889.5 


808.8 


80.7 


0.4633 


4.861 


0.2058 


330 


103.38 


300.5 


1182.6 


•882.1 


800.8 


81.3 


0.4766 


4.267 


0.2343 


340 


118.34 


310.9 


1185.6 


874.7 


792.7 


82.0 


0. 4897 


3.760 


0.2660 


350 


134.95 


321.4 


1188.7 


867.3 


784.7 


82.fi 


0.5027 


3.324 


0.3008 


360 


153.33 


331.8 


1191.7 


859.9 


776.7 


83.2 


0.5155 


2.949 


0.3391 


370 


173.60 


342.3 


1194.8 


852.5 


768.7 


83.8 


0.5282 


2.623 


0.3812 


380 


195.91 


352.8 


1197.8 


845.0 


760.8 


84.2 


0.5407 


2.338 


0.4276 


390 


220.39 


363.2 


1200.9 


837.7 


753.0 


84.7 


0.5531 


2.092 


0.4780 


400 


247.21 


373.7 


1203.9 


830.2 


745.2 


85.0 


«,5653 


1.874 


0.5336 


410 


276.54 


384.1 


1207.0 


822.9 


737.6 


85.3 


0.5774 


1.682 


0.5945 


420 


308. 57 


394.6 


1210.0 


815.4 


730.0 


85.4 


0.5893 


1.512 


0.661 


428 


336 26 


403.0 


1212.5 


809.5 


724.0 


85.5 


0.5988 


1.390 


0.719 



*See Revised Table, second page following. 



1358 



).— STEAM AND GAS POWER, 



8. — Saturated Steam — English Units.* 

(From Peabody, and Babock and Wilcox.) 

Note. — See Notation and Table 7, preceding. 



Pressure in Lbs. 
per Sqr. Inch 
above vacuum 


Temperature 
in Degrees, 
Fahrenheit. 


Heat in Liquid 
from 32° in 
Units. 


Total Heat in 
Heat Units 
From Water 
at 32°. 


Heat of Vapor- 
ization, or Lat- 
ent Heat in 
Heat UnitSo 


Factor of 
Equivalent 
Evaporation 
at 212°. 


Volume of One 
Pound in 
Cubic Feet. 


Density, or Wt. 
of Cubic Foot 
in Pounds. 


Total Pressure 
Above Vacu- 
um. 


P 


t 


Q 


A 


r 




s 


r 


P 


1 


101.99 


70.0 


1113.1 


1043.0 


.9661 


334.5 


0.00299 


1 


2 


126.27 


94.4 


1120-.5 


1026.1 


.9738 


173.6 


0.00576 


2 


3 


141.62 


109.8 


1125.1 


1015.3 


.9786 


118.5 


0.00844 


3 


4 


153.09 


121.4 


1128.6 


1007.2 


.9822 


90.33 


0.01107 


4 


5 


162.34 


130.7 


1131.5 


1000.8 


.9852 


73.21 


0.01366 


5 


6 


170.14 


138.6 


1133.8 


995.2 


.9876 


61.65 


0.01622 


6 


7 


176.90 


145.4 


1135.9 


990.5 


.9897 


53.39 


0.01874 


7 


8 


182.92 


151.5 


1137.7 


986.2 


.9916 


47.06 


0.02125 


8 


9 


188.33 


156.9 


1139.4 


982.5 


.9934 


42.12 


0.02374 


9 


10 


193.25 


161.9 


1140.9 


979.0 


.9949 


38.15 


0.02621 


10 


15 


213.03 


181.8 


1146.9 


965.1 


1.0003 


26.14 


0.03826 


15 


20 


227.95 


196.9 


1151.5 


954.6 


1.0051 


19.91 


0.05023 


20 


25 


240.04 


209.1 


1155.1 


946.0 


1.0099 


16.13 


0.06199 


25 


30 


250.27 


219.4 


1158.3 


938.9 


1.0129 


13.59 


0.07360 


30 


35 


259.19 


228.4 


1161.0 


932.6 


1.0157 


11.75 


5.08508 


35 


40 


267.13 


236.4 


1163.4 


927.0 


1.0182 


10.37 


0.09644 


40 


45 


274.29 


243.6 


1165.6 


922.0 


1.0205 


9.285 


0.1077 


45 


50 


280.85 


250.2 


1167.6 


917.4 


1.0225 


8.418 


0.1188 


50 


55 


286.89 


256.3 


1169.4 


913.1 


1.0245 


7.698 


0.1299 


55 


60 


292.51 


261.9 


1171.2 


909.3 


1.0263 


7.097 


0.1409 


60 


65 


297.77 


267.2 


1172.7 


905. 5 


1.0280 


6.583 


0.1519 


65 


70 


302.71 


272.2 


1174.3 


902.1 


, 1.0295 


6.143 


0.1628 


70 


75 


307.38 


276.9 


1175 7 


898.8 


1.0309 


5.760 


0.1736 


75 


80 


311.80 


281.4 


1177.0 


895.6 


1.0323 


5.426 


0.1843 


80 


85 


316.02 


285.8 


1178.3 


892.5 


1.0337 


5.126 


0.1951 


85 


90 


320.04 


290.0 


1179.6 


889.6 


1.0350 


4.859 


0.2058 


90 


95 


323.89 


294.0 


1180.7 


886.7 


1.0362 


4.619 


0.2165 


95 


100 


327.58 


297 9 


1181.9 


884.0 


1.0374 


4.403 


0.2271 


100 


105 


331.13 


301.6 


1182.9 


881.3 


1.0385 


4.205 


0.2378 


105 


110 


334.56 


305.2 


1184.0 


878.8 


1,0396 


4.026 


0.2484 


110 


115 


337.86 


308.7 


1185.0 


876.3 


1.0406 


3.862 


0.2589 


115 


120 


341.05 


312.0 


1186.0 


874.0 


1.0416 


3.711 


0.2695 


120 


125 


344.13 


315.2 


1186.9 


871.7 


1.0426 


3.571 


0.2800 


125 


130 


347.12 


318.4 


1187.8 


869.4 


1.0435 


3.444 


0.2904 


130 


140 


352.85 


324.4 


1189.5 


865.1 


1.0453 


3.212 


0.3113 


140 


150 


358.26 


330.0 


1191.2 


861.2 


1.0470 


3.011 


0.3321 


150 


160 


363.40 


335.4 


1192.8 


857.4 


1.0486 


2.833 


0.3530 


160 


170 


368.29 


340.5 


1194.3 


853.8 


1.0502 


2.^76 


0.3737 


170 


180 


372.97 


345.4 


1195.7 


850.3 


1.0517 


2.535 


0.3945 


180 


190 


377.44 


350.1 


1197.1 


847.0 


1.0531 


2.408 


0.4153 


190 


200 


381.73 


354.6 


1198.4 


843.8 


1.0545 


2.294 


0.4359 


200 


225 


391.79 


365.1 


1201.4 


836.3 


1.0576 


2.051 


0.4876 


225 


250 


400.99 


374.7 


1204.2 


829.5 


1.0605 


1.854 


0.5393 


250 


275 


409.50 


383.6 


1206.8 


823.2 


1.0632 


1.691 


0.5913 


275 


300 


417.42 


391.9 


1209.3 


817.4 


1.0657 


1.553 


0.644 


300 


325 


424.82 


399.6 


1211.5 


811.9 


i.0680 


1.437 


0.696 


325 


350 


431.90 


406.9 


1213.7 


806.8 


1.0703 


1.337 


0.748 


350 


375 


438.40 


414.2 


1215.7 


801.5 


1.0724 


1.250 


0.800 


375 


400 


445.15 


421.4 


1217.7 


796.3 


1.0745 


1.172 


0.853 


400 


500 


466.57 


444.3 


1224.2 


779.9 


1.0812 


.939 


1.065 


500 



*See Revised Table, second page follov/ing. 



SATURATED STEAM— TABLES. 



1359 



9. — Revised Saturated Steam Table — English Units. 

(Condensed from Prof. Peabody's Tables.) 

Note. — See also Table 10, following. 



2 . 

1 


Pressure, 
Pounds per 
Square Inch. 


1 
It 




Heat Equiva- 
lent of Inter- 
nal Work. 


Heat Equiva- 
lent of Exter- 
nal Work. 


•-I'd 


Entropy of 
Vaporiza- 
tion. 


Specific Vol- 
ume, Cubic 
Feet per 
Pound. 


Density, 
' Pounds per 
Cubic Foot. 


t 


P 


Q 


r 


P 


Apu 


e 


r 
T 


s • 


1 

s 


32 


0.0886 


0.0 


1071.7 


1017.5 


54.2 


0.0000 


2.1804 


3308. 


0.000302 


34 


0.0960 


2.0 


1070.7 


1016.3 


54.4 


0.0041 


2.1695 


3062. 


0.000327 


35 


0.0999 


3.0 


1070.2 


1015.6 


54.6 


0.0061 


2.1642 


2950. 


0.000339 


36 


0.1040 


4.0 


1069.7 


1015.0 


54.7 


0.0082 


2.1588 


2842. 


0.000352 


38 


0.1126 


6.1 


1068.7 


1013.8 


54.9 


0.0122 


2.1481 


2634. 


0.000379 


40 


0.1217 


8.1 


1067.6 


1012.5 


55.1 


0.0163 


2.1373 


2446. 


0.000409 


45 


0.1476 


13.1 


1065.0 


1009.4 


55.6 


0.0262 


2.1109 


2035. 


0.000491 


50 


0.1780 


18.1 


1062.3 


1006.2 


56.1 


0.0361 


2.0850 


1703. 


0.000587 


55 


0.2140 


23.1 


1059.7 


1003.1 


56.6 


0.0459 


2.0596 


1429. 


0.000700 


60 


0.2561 


28.1 


1057.0 


998.8 


57.2 


0.0556 


2.0347 


1207. 


0.000828 


65 


0.3054 


33.1 


1054.4 


996.7 


57.7 


0.0652 


2.0103 


1021. 


0.000979 


70 


0.3627 


38 1 


1051.8 


993.6 


58.2 


0.0747 


1.9863 


868. 


0.001152 


75 


0.4289 


43.1 


1049.2 


990.5 


58.7 


0.0841 


1.9629 


739. 


0.001353 


80 


0.5056 


48.1 


1046.5 


987.2 


59.3 


0.0934 


1.9398 


634. 


0.001577 


90 


0.6960 


58.1 


1041.2 


980.9 


60.3 


0.1117 


1.8948 


469.2 


0.002131 


100 


0.9461 


68.0 


1035.7 


974.4 


61.3 


0.1297 


1.8511 


350.8 


0.002851 


110 


1.271 


78 


1030.1 


967.7 


62.4 


0.1473 


1.8088 


265.2 


0.003771 


120 


1.689 


88 


1024.4 


961.0 


63.4 


0.1647 


1.7677 


203.0 


0.004926 


130 


2.220 


98.0 


1018.7 


954.2 


64.5 


0.1818 


1.7281 


157.1 


0.00637 


140 


2.885 


108.0 


1013.1 


947.5 


66.6 


0.1986 


1.6899 


122.8 


0.00814 


150 


3.715 


118.0 


1007.2 


940.6 


66.6 


0.2151 


1.6526 


96.9 


0.01032 


160 


4.738 


128.0 


1001.4 


933.7 


67.7 


0.2314 


1.6164 


77.2 


0.01296 


170 


5 990 


138.0 


995.5 


926.8 


68.7 


0.2475 


1.5814 


62.0 


0.01613 


180 


7.510 


148.0 


989.5 


919.8 


69.7 


0.2633 


1.5474 


50.2 


0.01993 


190 


9.339 


158.1 


983.4 


912.7 


70.7 


0.2789 


1.5141 


40.92 


0.02444 


200 


11.528 


168.2 


977.2 


905.5 


71.7 


0.2943 


1.4817 


33.62 


0.02974 


210 


14.125 


178.3 


970.9 


898.3 


72.6 


0.3095 


1.4502 


27.80 


0.03597 


212 


14.698 


180.3 


969.7 


896.9 


72.8 


0.3125 


1.4441 


26.78 


0.03734 


220 


17.188 


188.4 


964.6 


891.0 


73.6 


0.3244 


1.4196 


23.14 


0.04321 


230 


20.78 


198.5 


958.1 


883.6 


74.5 


0.3392 


1.3895 


19.37 


0.0516 


240 


24.97 


208 6 


951.4 


876.0 


75.4 


0.3538 


1.3602 


16.31 


0.0613 


250 


29.82 


218.8 


944 7 


868 5 


76.2 


0.3683 


1.3315 


13.82 


0.0724 


260 


35 42 


229.0 


937.8 


860.7 


77.1 


0.3825 


1.3034 


11.75 


0.0851 


270 


41.84 


239.1 


930.7 


852.8 


77.9 


0.3965 


1.2758 


10.05 


0.0995 


280 


49.19 


249.4 


923.6 


844.9 


78.7 


0.4104 


1.2489 


8.639 


0.1158 


290 


57.53 


259.6 


916.3 


836.9 


79.4 


0.4242 


1.2225 


7.454 


0.1349 


300 


66.98 


269.8 


908.9 


828.8 


80.1 


0.4378 


1.1967 


6.462 


0.1547 


310 


77.63 


280.1 


901.3 


820.5 


80.8 


0.4512 


1.1713 


5.622 


0.1779 


320 


89.59 


290.4 


893.7 


812.3 


81.4 


0.4644 


1.1465 


4.907 


0.2038 


330 


102.98 


300.6 


885.9 


803.8 


82.1 


0.4775 


1.1221 


4.312 


0.2319 


340 


117.91 


310.9 


878.0 


795.3 


82.7 


0.4905 


1.0982 


3.784 


0.2642 


350 


134.52 


321.3 


870 


786.8 


83.2 


0.5034 


1.0747 


3.342 


0.2992 


360 


152.89 


331.6 


861.8 


778.1 


83 7 


0.5161 


1.0516 


2.960 


0.3378 


370 


173.17 


341.9 


853.5 


769.3 


84.2 


0.5286 


1.C289 


2.626 


0.3808 


380 


195.52 


352.3 


845.1 


760.5 


84.6 


0.5411 


1.0066 


2.339 


0.4275 


390 


220.05 


362.7 


836.6 


751.6 


85.0 


0.5534 


0.9848 


2.088 


0.4789 


400 


246.9 


373.1 


827.9 


742.6 


85.3 


0.5656 


0.9633 


1.868 


0.535 


410 


276.3 


383.5 


819.1 


733.6 


85.5 


0.5777 


0.9420 


1.673 


0.598 


420 


308.5 


394.0 


810.1 


724.5 


85.6 


0.5896 


0.9211 


1.499 


0.667 


428 


336.2 


402.3 


802.9 


717.3 


85.6 


0.5991 


0.9047 


1.376 


0.727 



1360 



).— STEAM AND GAS POWER. 



10. — Revised Saturated Steam Table — English Units. 

(Condensed from Prof. Peabody's Tables.) 

(Note. — See also Table 9, preceding. 



Pressure, 
Pounds per 
Square Inch. 


Temperature, 
Degree 
Fahrenheit. 




Is 


Heat Equiva- 
lent of Inter- 
nal Work. 


Heat Equiva- 
lent of Exter- 
nal Work. 


^1 

n 


Entropy of 
Vaporiza- 
tion. 


Specific Vol- 
ume, Cubic 
Feet per 
Pound. 


Density, 
' Pounds per 
Cubic Foot. 


P 


t 


Q 


'• 


P 


Apu 





r 
T 


s 


1 


1 


101.84 


69.8 


1034.7 


973.1 


61.6 


0.1329 


1.8433 


333.1 


0.00300 


2 


126.15 


94.2 


1021.9 


957.8 


64.1 


0.1753 


1.7432 


173.1 


0.00578 


4 


153.00 


121.0 


1005.5 


938.6 


66.9 


0.2200 


1.6416 


90.4 


0.01106 


6 


170.07 


138.1 


995.5 


926.8 


68.7 


0.2476 


1.5812 


61.9 


0.01616 


8 


182.86 


150.9 


987.8 


917.8 


70.0 


0.2678 


1.5378 


47.26 


0.02116 


10 


193.21 


161.3 


981.4 


910.4 


71.0 


0.2838 


1.5036 


38.37 


0.02606 


12 


201.95 


170.1 


976.0 


904.1 


71.9 


0.2972 


1.4756 


32.40 


0.03088 


14 


209.55 


177.8 


971.2 


898.6 


72.6 


0.3088 


1.4516 


28.03 


0.03567 


14.7 


212.00 


180.3 


969.7 


896.9 


72.8 


0.3125 


1.4441 


26.78 


0.03734 


15 


213.03 


181.3 


969.1 


896.2 


72.9 


0.3140 


1.4409 


26.28 


0.03805 


20 


227.95 


196.4 


959.4 


885.1 


74.3 


0.3362 


1.3957 


20.09 


0.04978 


25 


240.07 


208.7 


951.4 


876.0 


75.4 


0.3593 


1.3600 


16.29 


0.0614 


30 


250.34 


219.1 


944.4 


868.2 


76.2 


0.3687 


1.3305 


13.74 


0.0728 


35 


259.29 


228.2 


938.2 


861.2 


77.0 


0.3815 


1.3054 


11.88 


0.0842 


40 


267.26 


236.4 


932.6 


855.0 


77.6 


0.3927 


1.2833 


10.49 


0.0953 


45 


274.46 


243.7 


927.5 


849.3 


78.2 


0.4027 


1.2638 


9.387 


0.1065 


50 


281.03 


250.4 


922.8 


844.1 


78.7 


0.4119 


1.2462 


8.507 


0.1176 


55 


287.09 


256.6 


918.4 


839.2 


79.2 


0.4202 


1.2302 


7.778 


0.1286 


60 


292.74 


262.4 


914.3 


834.7 


79.6 


0.4279 


1.2154 


7.166 


0.1395 


65 


298.00 


267.8 


910.4 


830.4 


80.0 


0.4351 


1.2018 


6.647 


0.1504 


70 


302.96 


272.9 


906.6 


826.3 


80.3 


0.4418 


1.1892 


6.199 


0.1613 


75 


307.64 


277.7 


903.1 


822.4 


80.7 


0.4480 


1.1772 


5 . 807 


0.1772 


80 


312.08 


282.2 


899.8 


818.9 


80.9 


0.4540 


1.1661 


5.466 


0.1829 


85 


316.30 


286.5 


896.6 


815.4 


81.2 


0.4595 


1.1557 


5.161 


0.1938 


90 


320.32 


290.7 


893.5 


812.1 


81.4 


0.4649 


1.1457 


4.886 


0.2047 


95 


324.16 


294.6 


890.5 


808.8 


81.7 


0.4699 


1.1363 


4.644 


0.2153 


100 


327.86 


298.5 


887.6 


805.7 


81.9 


0.4748 


1.1273 


4.432 


0.2256 


105 


331.42 


302.1 


884.8 


802.7 


82.1 


0.4794 


1.1187 


4.233 


0.2362 


110 


334.83 


305.6 


882.1 


799.7 


82.4 


0.4838 


1.1105 


4.047 


0.2471 


115 


338.14 


309.0 


879.5 


797.0 


82.5 


0.4881 


1.1026 


3 876 


0.2580 


120 


341.31 


312.3 


876.9 


794.2 


82.7 


0.4922 


1.0951 


3.723 


0.2686 


125 


344.39 


315.5 


874.5 


791.6 


82.9 


0.4962 


1.0878 


3.581 


0.2793 


130 


347.38 


318.6 


872.1 


789.0 


83.1 


0.5000 


1.0808 


3.451 


0.2898 


135 


350.27 


321.5 


869.8 


786.5 


83.3 


0.5037 


1.0741 


3.331 


0.3002 


140 


353.09 


324.4 


867.4 


784.0 


83.4 


0.5073 


1.0675 


3.220 


0.3106 


145 


355.83 


327.3 


865.2 


781.6 


83.6 


0.5108 


1.0612 


3.115 


0.3210 


150 


358.50 


330.0 


863.0 


779.3 


83.7 


0.5142 


1.0551 


3.014 


0.3318 


155 


361.09 


332.7 


860.9 


771.1 


83.8 


0.5175 


1.0491 


2.922 


0.3422 


160 


363.62 


335.3 


858.8 


774.9 


83.9 


0.5206 


1.0434 


2.834 


0.3528 


165 


366.09 


337.9 


856.8 


772.8 


84.0 


0.5237 


1.0378 


2.751 


0.3635 


170 


368.50 


340.4 


854.8 


770.6 


84.2 


0.5268 


1.0324 


2.673 


0.3741 


175 


370.86 


342.8 


852.8 


768.5 


84.3 


0.5297 


1.0270 


2.600 


0.3846 


180 


373.16 


345.2 


850.9 


766.5 


84.4 


0.5326 


1.0219 


2.531 


0.3951 


185 


375.41 


347.5 


849.0 


764.5 


84.5 


0.5354 


1.0169 


2.467 


0.4054 


190 


377.61 


349.8 


847.1 


762.6 


84.5 


0.5381 


1.0121 


2.405 


0.4158 


195 


379.78 


352.1 


845.3 


760.7 


84.6 


0.5408 


1.0071 


2.345 


0.4264 


200 


381.89 


354.3 


843.5 


758.8 


84.7 


0.5434 


1.0025 


2.288 


0.4371 


210 


386.02 


358.6 


840.0 


755.1 


84.9 


0.5485 


0.9935 


2.184 


0.4579 


220 


389.98 


362.7 


836.6 


751.6 


85.0 


0.5534 


0.9848 


2.088 


0.4789 


230 


393.80 


366.6 


833.3 


748.1 


85.2 


0.5580 


0.9755 


2.001 


0.4997 


240 


397.50 


370.5 


830.1 


744.8 


85.3 


0.5625 


0.9686 


1.921 


0.521 


250 


401.10 


374.2 


826.9 


741.5 


85.4 


0.5669 


0.9609 


1.845 


0.542 


260 


404.55 


377.8 


823.9 


738.5 


85.4 


0.5711 


0.9535 


1.775 


0.563 


270 


407.90 


381.3 


820.9 


735.4 


85.5 


0.5751 


0.9464 


1.711 


0.584 


280 


411.19 


384.8 


818.0 


732.5 


85.5 


0.5791 


0.9395 


1.652 


0.605 


290 


414.35 


388.1 


815.2 


729.6 


85.6 


0.5829 


0.9329 


1.595 


0.627 


300 


417.45 


391.3 


812.4 


726.8 


85.6 


0.5866 


0.9264 


1.542 


0.649 


310 


420.45 


394.4 


809.7 


724.1 


85.6 


0.5902 


0.9202 


1.492 


0.670 


320 


423.40 


397.5 


807.1 


721.5 


85.6 


0.5937 


0.9141 


1.446 


0.692 


330 


426.26 


400.5 


804.5 


718.9 


85.6 


0.5970 


0.9083 


1.402 


0.713 


338 


427.94 


402.2 


803.0 


717.4 


85.6 


0.5990 


0.9048 


1.377 


0.726 



STEAM— TABLE; FLOW; BOILERS. 



1361 



Flow of 

formula: 



Steam. — G. H. Babcock, in "Steam," gives the following 



Flow through pipes: 



W 



. \ riPx-p2)d^ 



(1) 



Where W = weight of steam which will flow per minute, in lbs.; 
J = diameter of pipe, in ins.; 

7- = density or weight per cubic foot for ^1, in lbs.; 
^1 = initial pressure, at entrance, in lbs. per sq. in.; 
^2= pressure at end or exit of pipe, in lbs. per sq. in.; 
L = length of pipe, in feet. 

11. — Flow of Steam through Pipes. 



Initial Pres- 


Diameter of Pipe, in Indies. Length of Each =240 Diameters. 


sure by- 
Gage, 


H 


1 


1^ 


2 


2^ 


3 


4 5 


6 


8 


10 


12 


15 


18 


Pounds per 
Sq. In. 


Weight of Steam per Minute, in Pounds, with One Pound Loss 


of Pressure. 


1 


1.12 


2 05 


5.65 


10 15 


15.26 


25 12 


46.27 


76.1 


111.6 


209.1 


336.3 


495.3 


792 


1160 


10 


1.38 


2.54 


6 98 


12.54 


18.85 


31 03 


57.15 


93.9 


137.9 


258.2 


415.3 


611.8 


979 


1433 


20 


1.62 


2.97 


8 18 


14.70 


22,09 


36 36 


66.97 


110.1 


161.6 


302.6 


486.7 


716.9 


1147 


1679 


30 


1.82 


2 34 


9.21 


16 54 


24.86 


40 92 


75.37 


123.9 


181.8 


340.6 


547.8 


806.9 


1291 


1889 


40 


2.01 


3.68 


10.12 


18.18 


27.34 


44.99 


82.87 


136.3 


199.9 


374.5 


602.3 


887.2 


1419 


2078 


50 


2.17 


3.98 


10.95 


19.67 


29.57 


48.67 


89.64 


147.4 


206.3 


404.9 


651.5 


959.7 


1535 


2248 


60 


2.32 


4.25 


11.71 


21.04 


31.63 


52.06 


95.89 


157.7 


231.3 


433.3 


696.9 


1026.5 


1642 


2404 


70 


2.46 


4.51 


12.42 


22.32 


33.55 


55.22 


101.71 


167.3 


245.4 


459.6 


739.3 


1088.9 


1742 


2550 


80 


2.59 


4.75 


13,09 


23. 52 


35.36 


58.19 


107.18 


176.3 


258.6 


484.3 


778.9 


1147.4 


1836 


2687 


90 


2.71 


4.96 


13.66 


24.55 


36.91 


60.74 


111.88 


183.9 


269.9 


505.5 


811.5 


1197.8 


1916 


2805 


100 


2.84 


5.20 


14.32 


25.73 


38 71 


63.66 


117.25 


192 8 


282.9 


529.8 


852.2 


1255.2 


2008 


2940 


120 


3.06 


5.61 


15.44 


27.75 


41.71 


68.64 


126.43 


207.9 


305.1 


571.3 


918.9 


1353.6 


2166 


3170 


150 


3.37 


6.16 


16.97 


30.49 


45.83 


75.42 


138.91 


228.4 


335.2 


627.7 


1009.6 


1487.2 


2379 


3483 



Note. — "For sizes of pipe below 6-inch, the flow is calculated from the 
actual areas of 'standard' pipe of such nominal diameters. For horse-power, 
multiply the figures in the table by 2. For any other loss of pressure, 
multiply by the square root of the given loss. For any other length of pipe, 
divide 240 by the givqn length expressed in diameters, and multiply the figures 
in the table by the square root of this quotient, which will give the flow for 1 lb. 
loss of pressure. Conversely, dividing the given length by 240 will give the 
loss of pressure for the flow given in the table." 

Steam Boilers. — The Efficiency of a steam boiler is the ratio of the heat 
utilized in heating the water and raising steam, to the total heat generated 
by the combustion of the fuel; and ranges from 50 to 75 per cent., the 
latter being rarely exceeded. Thus, if a pound of coal has a heating value 
of 14,000 B. T. U. it is theoretically capable of evaporating, "from and at"* 
212°F (see Table 7, preceding), 14000-^965.8=14.5 lbs. of water; while if 
the efficiency of the boiler is 75 per cent., the amount actually evaporated 
by the pound of coal will be 14.5 X .75= 10.87 lbs. of water. 

One commercial horse power of a boiler is variously defined as follows: 

(1) The evaporation of 30 lbs. of water per hour from a feed-water tempera- 

ture of 100 F into steam at 70 lbs. gage pressure (above atmos- 
■ phere); considered equivalent to 

(2) 34.5 (34.488 exact) units of vaporization at 212° F; considered equal 

to 

(3) 34.5 pounds of water evaporated from a feed-water temperature of 

212° F into steam at the same temperature. 

(5) From our steam table, 34.5 r= 34.5 X 965.8= 33,320 B. T. U. per hour; 

but 

(6) 33,305 British thermal units per hour is sometimes taken as a standard 

(=34.488 r, using r at 965.7). 
Conclusion: (1) is used in making actual tests of boilers; the others for 
purposes of calculation. 



* "From and at" 212° F, means that the vaporization takes place at 
212° F from feed water at the same temperature. 



1S62 69.— STEAM AND GAS POWER. 

Consumption of Coal per boiler horse-power hour may be determined from 
the above data. Assuming the boiler at 75 per cent efficiency, one pound 
of coal will generate say 1 4000 X .75= 10500 British heat units (if feed water 
is at 212° F). Then (1) the amount of coal per boiler horse-power per 

33 320 
hour= ..^ g/xr> = 3.17 lbs. This result is also obtained (2) by dividing 34.5 
10 oOO 

by 10.87; that is, the number of pounds of water evaporated per horse- 
power hour, divided by the number of pounds evaporated per pound of 
coal, as calculated above. 

In practice, we generally assume that one pound of coal will generate 
about 10 000 heat units, in which case it would require 3.332 pounds of 
coal to develop one boiler horse-power per hour, which is very close to the 
average performance of the Cahall boiler, Table 6, preceding. Now a boiler 
horse-power per hour (=33 320 heat units or 33 320X778=25,922,960 ft.- 
Ibs.) is very different from an engine horse-power hour (see page 1363) 
which is equal to 1,980,000 ft. -lbs. 

Thus, 1 boiler H.-P. hour= 13.0924 engine H.-P. hours, 
or 1 engine H.-P. hour= 0.07638 boiler H.-P. hours. 

Hence the number of pounds of coal consumption per engine horse- 
power, is approximately, 

3.332X0.07638 = 0.2545 ,^ . ,__, 

Efficiency of the Engine ^^^^ ^^^° ^^^^ ^^^^^ ^^^ 

provided there is no loss between the boiler and the engine. 

Kinds of Steam Boilers. — In the discussion of impulse water wheels, 
page 1336, it is stated that the greatest amount of the energy of the jet has 
been imparted to the wheel when the water loses its velocity on striking the 
buckets. The same principle holds good in the construction of boilers. 
They should be so designed that the feed water shall absorb the greatest 
amount practicable of the heat of combustion of the fuel, with the mini- 
mum waste of gases. 

Ordinary steam boilers may be classified as water-tube boilers, fire-tube 
boilers and flue boilers: 

(1) Water-tube boilers have a large number of tubes of moderate size 
connected together at their ends and also with a reservoir of water above; 
and are placed directly over the grate or in the path of the flames. The 
tubes may be horizontal, vertical, inclined, straight, curved, etc. There 
should be no "dead" ends. They are especially adapted to high steam- 
pressure. Notable examples: Babcock and Wilcox, Cahall, Thornycroft. 

(2) Fire-tube boilers consist of a large number of small tubes acting as 
flues and surrounded by the water contained in an outer shell, which requires 
special design. The circular tubular boiler is of this type, including loco- 
motive boilers. 

(3) Flue-boilers differ from the fire-tube boilers in that the tubes are 
reduced to one or more in number and greatly increased in size. If the 
furnace is outside the flues it is "externally fired;" if inside the flues it is 
"internally fired." Of the latter class the Cornish boiler has one flue, while 
the Lancaster has two. The Scotch marine boiler is a flue- and fire-tube 
boiler combined. 

Boiler Settings. — ^The following is from the notes of J. B. Stanwood: 

The brick-work about a boiler should be thick to prevent loss by radia- 
tion — a 2V' wall should be used if possible. All flues and surfaces exposed 
to action of heat should be lined with best fire-brick. It is not a good plan 
to convey gases back over top of boiler, unless there is space enough for ft 
man to enter and clean oft" soot. The distance from grate bars to lower 
portion of boiler shell should noc be less than 24''; 26" and 28" are not too 
great, and in large shells 30" can be employed. 

The bridge-wall should curve to conform to shape of boiler shell. Ten 
inches makes a good space between wall and shell. Back of bridge-wall the 
surface should be paved with hard brick, the surface dipping down to a 
depth at rear end of boiler of about 18" to 24" according to size of shell. 
The distance between back tube sheet and back wall should be 18" for a 48" 
shell; 24" for a 72" shell. 

Boiler walls will crack, and no form of construction seems to entirely 
prevent this. Walls with air spaces are as liable as those without, with the 
danger of leaking more air when they do crack. The best method to hold 



STEAM BOILERS. STEAM ENGINES. 1363 

boiler walls together is with "buck-staves" or "buck-bars." The best form 
is railway iron with ends mashed down under a hammer, to allow for drilling 
holes for tie-rod. Most builders do not supply "buck-staves" unless specially 
ordered. The cheapest form of fire-front is the so-called "half -arch," which 
does not cover any more of the front of the furnace than is absolutely decent. 
On small boilers it is employed as a support. For a good job a "flull flush 
front" should be used with damper plate and damper. 

Boilers, now-a-days, are not set in batteries, all to work together as a unit. . 
They are and should be set, so that each boiler is independent of the other 
in the battery. In this way any one can be shut down for cleaning or for 
repairs. This arrangement does away with the old-fashioned steam and 
mud-drums, which connected the boilers of a battery together. Do not 
buy either a mud-drum or steam drum; they are a source of trouble, danger 
and expense. The trade usually includes with the boiler front, the grate 
bars, a bearing -bar to support same, a soot or ash-door with frame, a 
brick arch plate or supporting bars, and a boiler stand for small boilers. 

Steam Engines are steam motors, or mechanisms for converting the 
thermal energy of steam into mechanical work. The efficiency of an engine 
is the ratio of the mechanical work done, to the energy of the steam con- 
sumed in doing it; or, the ratio of the heat changed into work, to the heat 
applied. A perfect heat engine,* working to absolute zero temperature, 
would of course have an efficiency of 100 per cent; that is, it would be able 
to convert all the applied heat into work. A ''perfect" steam engine, from a 
mechanical standpoint, can never be expected to exceed an efficiency of 
about 25 per cent. The best types of steam engines have an efficiency in 
actual practice of only about 18 per cent, because they cannot utilize a 
greater ratio of the energy of the steam delivered to them. Such engines 
are of the condensing- and multiple-expansion type. Non-condensing 
engines have efficiencies ranging from 5 to 10 per cent. Recent engines of 
moderate power and high superheat have attained an efficiency of 22%. 

Engine Horse-Power. — Steam engines are rated by the horse-power, the 
unit of work being the foot-pound. 1 H.-P. second = 550 ft. -lbs.; 1 H.-P. 
minute = 33,000 ft.-lbs.; 1 H.-P. hour= 1,980,000 ft.-lbs. 

Problem. — A steam engine working at an efficiency of 8% uses 40 lbs. of 
coal per hour. Assuming that each pound of coal generates 10 000 British 
thermal units in the steam, what horse-power is realized ? 

_ . . 40X10 000X778X.08 .^ ^„. „ r, ,, 

^^^^'*^^- 1980 000 = ^2.574 H.-P. Ans. 

In the above problem the coal consumed per horse-power per hour 
=a — ^jr— = 3.18 lbs. This result can also be obtained from equation (1); 

thus. -5^ = 3.18 lbs. 

Coal Consumption per horse-power per hour may vary from 2 lbs., for 
large efficient steam plants, to 4 lbs. for smaller plants, less efficient. It is 
well to estimate 6 or 7 lbs. for hoisting engines ordinarily used by contractors. 
Of course much depends upon the quality of the coal. If of poor quality 
the consumption will be greater. 

Value of Wood as fuel. — Assuming the wood to be thoroughly air-dried, 
one ton of coal is equivalent to 1 cord of Douglas (Oregon or Washington) 
fir; 1.05 cords of hickory; 1.1 cord of maple; 1.2 cord of white oak; 1.5 
cord of beech; 1.6 cord of black oak; 2.0 cords of elm; 2. 1 cords of chestnut; 
2.4 cords of pine. These values are necessarily approximate only. (See also 
page 1352.) 

Principle of the Steam Engine . — An engine has one or more "cylinders." 
A cylinder is simply a cylindrical barrel, with closed ends, in which is fitted 
a piston P (Fig. 1) made to move back and forth by the pressure of steam 
admitted alternately on either side of it. The piston rod is connected, 
perhaps indirectly, with a crank or wheel, giving the latter a circular 
motion, thus producing continuous power in one direction. 

* This is used in a distinct sense from the so-called "heat engines" which 
include gas- and oil engines. Such engines may have an efficiency of 20 per 
cent or more. 



1364 



l^STEAM AND GAS POWER, 




Fig. 1 shows a longitudinal section of the cylinder of a Corliss engine. 
The steam enters at the top of the cylinder through a steam pipe leading 
directly from the boiler, and under a pressure of say 100 lbs. (per sq. in.) 
more or less. (The exact pressure is registered by a pressure gage or Indi- 
cator.) It enters the cylinder proper through one of the steam valves S or s. 



Steam 




— Piston Stroke 

Bo/'/er Pressureiine-'^ 




Zero Line of Pressure-^ 
FIrt r> A= Admission 

•^ 'y* ^* A- B = Admission Line 

^-Compression 

and after it has performed its work of pressure and expansion it escapes 
through the parts E and e, at the bottom of the cylinder, into the exhaust. 
It may either be wasted, or condensed into hot feed-water for the boiler. 

Fig. 2 shows a typical double indicator diagram for a non-condensing 
engine, and illustrates graphically the action of the cylinder and the working 
pressure on the piston at every position. The full diagram is for the pres- 
sure on the right of the piston, and the dotted diagram for the pressure on 
the left; the two together illustrating a complete cycle, from the starting 
point at the right of the cylinder to its return to that position. 

Starting with the piston at the right. Figs. 1 and 2, and considering the 
steam pressure at the right of the piston: A is the point of admission of 
steam; A-B the admission line; B the point of initial pressure (the distance 
between B and the boiler pressure line shows the loss in pressure from the 
boiler to the engine); S-C the steam line (the valve S being open from 
A to C)\C the cut-off, at which point the valve 5 is closed and beyond which 
the steam works by expansion; C-R the expansion line, the pressure de- 
creasing as the volume of steam increases (this curve is very nearly a hyper- 
bola) ; R the point of release produced by the opening of the exhaust valve 
at e\ R-X the exhaust line; X-K the back-pressure line; K the point of 
cornpression; K-A the compression line. The area of this curve, A B C R 
X K A, divided by the horizontal length of figure, practically the piston 
stroke, gives the mean effective pressure (M. E. P.) per sq. in. on the right- 
hand side of the piston during its cycle. Similarly, the M. E. P. on the 
left-hand side of piston is obtained from the dotted diagram. Engine 
diagrams should be started at admission or during exhaust, otherwise they 
may not close. 



STEAM ENGINE— MEAN EFFECTIVE PRESSURE. 1365 



Mean Effective Pressure (M. E. P.)— The M. E. P. for any particular 
engine should be obtained by indicator diagrams as explained above. The 
following table gives average M. E. P's. for non-condensing engines: 

12. — Mean Effective Pressures for Varying Cut-offs and 
Initial Steam Pressures; Non-Condensing Engines. 



Initial 


i^« 


T^fu 


Vs 


H 


t"u 


T%^a 


1% 


^ 


Pressure: 


Cut-off. 


Cut-off. 


Cut-off. 


Cut-off. 


Cut-off. 


Cut-off. 


Cut-off. 


Cut-off. 


40 


3.65 


9.05 


13.46 


17.34 


20.75 


23.70 


26.22 


30.50 


45 


6.42 


11.32 


16.15 


20.39 


24.13 


27.32 


30.08 


34.75 


50 


7.19 


13.59 


18.85 


23.45 


27.50 


30.94 


33.95 


39.00 


55 


8.96 


15.86 


21.54 


26.50 


30.87 


34.56 


37.81 


43.25 


60 


10.73 


18.12 


24.24 


29.56 


34.24 


38.18 


41.68 


47.50 


65 


12.50 


20.39 


26.93 


32.61 


37.61 


41.80 


45.54 


51.75 


70 


14.27 


22.66 


.29.63 


35.67 


40.98 


45.42 


49.41 


56.00 


75 


16.04 


24.92 


32.32 


38.72 


44.35 


49.05 


53.27 


60.25 


80 


17.81 


27.19 


35.02 


41.78 


47.72 


52.68 


57.14 


64.50 


85 


19.58 


29.46 


37.71 


44.83 


51.09 


56.31 


61.00 


68.75 


90 


21.36 


31.72 


40.41 


47.89 


54.46 


59.94 


64.87 


73.00 


95 


23.13 


33.93 


43.10 


50.94 


57.83 


63.57 


68.73 


77.25 


. 100 


24.90 


36.26 


45.80 


54.00 


61.20 


67.20 


72.60 


81.50 



With throttling engines results are obtained dependent upon proportion 
of steam-ports, travel of valve, piston speed, size of governor, and design of 
engine. 

The theoretic mean effective pressure may be calculated by means of a 
formula involving the use of a table of hyperbolic logarithms. This formula 
assumes that the "expansion line" (see Fig. 2) is a hyperbola. It takes 
into consideration the point of cut-off. It is used principally in the design 
of engines, and only about 70 per cent of the theoretic M. E. P. is attained 
in practice. 

Horse-power from mean effective pressure (M. E. P,) — 

Notation. 
D = diameter of cylinder; 

A = area of cylinder = -j- = 0. 7864 D^; 

d = diameter of piston rod; 

o=area of piston rod=-j- 

5 = length of piston stroke; 



0.7854^2; 



)i pi 
of ] 



n = number of revolutions per minute. 
All dimensions in inches and square inches. 
Then the horse-power is — 
[(M. E. P. for head end) XA + (M. E. P. for crank end) X {A-a)]sn 



(2) 



33 000X12 

Problem. — A non-condensing engine is working at 3/10 cut-off under an 
initial pressure (at B — not boiler pressure) of 90 lbs., and at a speed of 130 
revolutions per minute, with a stroke of 24 inches. The diameter of cylinder 
is 16 ins., and of piston rod 2.5 ins. What horse-power is realized? 

Solution. — From Table 12, preceding, the M. E. P. (for both head- and 
crank end) is 64.46; hence, from equation (2), the horse power = 
64.46X(2yl-a) sn 



33 000X12 



= 170H. P. Ans. 



Economic Performance of Steam Engines. — ^The following is from the 
notes of J. B. Stanwood: 

Non-Condensing Engines. 

Slide-Valve Engine. — 75 to 80 lbs. of boiler pressure; stroke, long; mean 
effective pressure, 33 to 38 lbs. per sq. in; 25 to 100 H. P.; cut-off, ^stroke; 
about 40 lbs. steam per indicated H. P. per hour. When valves and piston 
are tight this has been reduced to 33 lbs. of dry steam per indicated horse- 
power (I. H. P.) per hour by careful test. 



1366 69.— STEAM AND GAS POWER. 

Automatic High-Speed Engines with single valves. — 75 to 80 lbs. of 
boiler pressure; stroke, about equal to piston dia.; M. E. P., 40 lbs. per sq. 
in.; 50 to 150 H. P.; cut-off, J^ stroke; about 40 lbs. steam per I. H. P. per 
hour. When valves and pistons are tight this has been reduced to 32 lbs. of 
dry steam per I. H. P. per hour. Valves difficult to keep tight. 

Automatic High Speed Engines with double valves. — 75 to 80 lbs. of 
boiler pressure; stroke 1^ to 2 times piston dia.; M. E. P., 40 lbs. per sq. in.; 
50 to 150 H. P.; cut-off, 34 stroke; about 35 lbs. steam per I. H. P. per hour. 
When valves and pistons are tight this has been reduced to 30 lbs. of dry- 
steam per I. H. P. per hour by careful test. 

Automatic Cut-off Engines, of the Corliss type. — Stroke, 2 to 3 times dia. 
of piston; 75 to 90 lbs. boilei pressure; M. E. P., 40 lbs. per sq. in.; under 
200 H. P.; cut-off Vs to M stroke, 29-30 lbs. steam per I. H. P. per hour; 
over 200 H. P., 27 lbs. steam per I. H. P per hour; when valves and piston 
are tight, this has been reduced to 233^ lbs. of dry steam per I. H. P. per 
hour by careful test. 

Compound Engines. — High speed, automatic cut-off, short stroke; 110 to 
120 lbs. boiler pressure; M. E. P., 25 to 27 lbs. per sq. in.; 6 expansions, 
100 to 250 H. P.; 27 lbs. steam per I. H. P. per hour. 

Condensing Engines. 

Automatic Cut-Off Engines, of the Corliss type. — Stroke, 2 to 3 times 
piston dia.; 70 to 80 lbs. boiler pressure; M. E. P., 40 lbs. per sq. in.; over 
200 H. P.; cut-off, Vs stroke, about 19-20 lbs. steam per I. H. P. per hour. 

Compound Engines', high speed, automatic cut-off; short stroke; 110 
to 120 lbs. boiler pressure; M. E. P., 27 to 30 lbs. per sq. in.; 9 expansions; 
200 to 500 H. P; 17 to 19 lbs. steam per I. H. P. per hour. 

Compound Automatic Cut-off Engines, of the Corliss type. — Stroke on 
high pressure cylinder, 2 to 3 times piston dia.; 135 to 110 lbs. boiler pres- 
sure; M. E. P., 24 to 14 lbs. persq. in.; over 400 H. P.; 16 to 20 expansions; 
14 to 17 lbs. of steam per I. H. P. per hour. One or two special cases, 13^ 
lbs. of steam per I. H. P. per hour has been obtained. 

Compound Engines. 

Compound Engines are devices by which high grades of expansion, and 
consequently high pressure of steam, can be successfully used; and the 
evils of leakage also can be reduced. 

By expanding steam partially in one, then in a .second, and perhaps a 
third cylinder, the internal condensation is kept small; for usually late cut- 
offs are employed, the advantage of which is that the surfaces of the cylin- 
ders presented for re-heating to the fresh charges of steam are small, com- 
pared with the volume of steam used ; and the difference in temperature 
between fresh and exhausted steam is slight. 

Leakage of steam with compound engines is not so serious a matter; the 
steam that leaks through one cylinder is caught by the second instead of 
being thrown away. As the cut-offs are frequently late, the slide-valve can 
be employed, which is the tightest of all valves. 

The Effect of Load Upon Economy of Steam Engines^ 
A large engine working with an extremely light load is wasteful of fuel 
and steam; it is worse than a small business with a large staff of expensive 
officers, for it has more than the fixed charges always with it. A small 
engine overloaded may be an annoyance and care, but if in good condition 
it may be economical in fuel. 

Steam Pumps. — Fig. 3 shows a longitudinal section of a Worthington 
direct-acting duplex steam pump. It is "direct-acting" because the plunger 
of the pump (on the right) is directly connected to the piston rod of the 
engine (on the left). It is "duplex" because two of the pumps are placed 
side by side, each one of which is connected by links and a rocker to the 
valve of the other. As the plunger works back and forth the water is 
sucked through the valves at the bottom, at either end, while the water 
previously sucked at the opposite end is at the same time forced through 
the upper valves into the discharge D. 

Duplex Pumps are durable and easy acting, and are used for all classes 
of pumping. 



STEAM ENGINES. STEAM PUMPS. 



1367 



Centrifugal Pumps consist essentially of a pump wheel or impeller with 
curved vanes which move tightly in an outer casing. The wheel is fixed 




Fig. 3. 



^ /y^vyy^/^/i^//^//////^^^^ 



-Worthington Steam Pump (p. 1366). 



on a shaft rotated by a belt running over a pulley. In a side-suction 
pump the water enters on either side at the center of the wheel and 
is forced out at the periphery through a tangential outlet from the casing. 
Centrifugal pumps are economical in raising water against low heads. 




Fig. 4. — Centrifugal Pump. 

say up to 30 ft., more of less; reciprocating pumps are more efficient for 
high heads. 

Rotary Pumps are operated by two rotating valves in "gear," each 
moving tightly in an outer casing. 

Duty of Pumps. — ^The duty of a pump was formerly tested and measured 
by the number of foot-pounds of work which it was capable of performing 
from the combustion of 100 pounds of coal used. As the quality of coal 
varies a new standard was proposed, in 1891, by a Committee of the A. S. 
C. E.:* 

j^ _ No. of foot-pounds of work done X 1 OOP 000 
Number of heat units consumed 

This would be equivalent to the old rule, provided 100 lbs. of coal will 
generate 1 000 000 (100X10 000) heat units (see page 1362) to the water 
in the boiler, a result which can generally be obtained. 



* See Transactions A.S.C.E., Vol. XII., page 530. 



1368 6^.— STEAM AND GAS POWER. 

D.— HEAT (INTERNAL-COMBUSTION) ENGINES. 

TESTS OF INTERNAL-COMBUSTION ENGINES ON ALCOHOL FUEL. 

(Digest of Bulletin 191, U. S. Dept. of Agric.) 

Introduction. — Recently in this country great interest has developed in 
the possibilities of alcohol as fuel, and the question of its being used as a 
substitute for the pertoleum fuels will become of increasing importance as 
time goes on. The supply of crude oil to be obtained in the United States 
must ultimately diminish, and the history of the past indicates that a con- 
stant increase in price of kerosene and gasoline may ultimately be expected. 
On the other hand, it is not improbable that the price of alcohol may fall, 
so that as regards cost alcohol may be used advantageously in comparison 
with the petroleum oils. 

Specific objects of the investigation. — First, to determine whether the 
gasoline and kerosene engines at present on the American market can run 
on alcohol as fuel; the manipulation to be followed in making the engines 
run on alcohol; the measurement of the relative maximum powers of the 
engines when using alcohol and the fuels for which they were originally 
made; and the relative consumptions of the different fuels. Second, to 
determine as far as possible the improvements which might be desirable in 
the design of engines manufactured especially for alcohol. 

Outline of the ground covered by the tests. — The engines used, and 
range of tests, were: 

No. 1. Gasoline engine, 15 h. p. at 280 rev. per min.; 2 cyl., single-acting, 
vertical, 4 cycle, 6^'' bore, 10'' stroke. 15 tests reported, giving 
consumption of alcohol and gasoline under different brake loads 
and with different initial compressions. Lowest consumptions ob- 
tained were 0.71 lb. (0.12 gal.) of gasoline and 1.12 lb. (0.16 gal.) 
of alcohol per brake h. p. hour. The highest working m. e. p. ob- 
tained was about 90 lbs. with both gasoline and alcohol, but at best 
consumption the m. e. p. were considerably lower 

No. 2. Gasoline engine, 6 h. p. at 300 rev. per min.; 1 cyl., water-cooled, 
horizontal, 4 cycle, 51'' bore, 9" stroke. 24 tests with gasoline and 
30 with alcohol. Highest mechanical efficiencies were 86% for 
gasoline and 90% for alcohol. 

No. 3. Gasoline engine, 6 h. p. at 340 rev. per min.; 1 cyl., horizontal, 4 cycle, 
5^" bore, 10" stroke. 18 tests with gasoline and 19 with alcohol. 
Best consumption with gaosline was 0.85 lb. (0.14 gal.) per brake 
h. p. hoiur; and with alcohol, at 320 r. p. m., 1.25 lb. (0.18 gal.). 

No. 4. Gasoline engine, 6 h. p. at 350 r. p. m., 1 cyl., vert., 4 cycle, 6* bore, 
8" stroke. 1 1 tests on gasoline and 26 on alcohol ; and also the effect 
of heating the air in advance of its entrance to the carburetter. 
Best alcohol consumption was 1.13 lb. (0.17 gal.) per brake h. p. 
hour; and with air entering the carburetter heated to 125° F., the 
engine would self -ignite, the m. e. p. at best consumption being 
93 lbs. 

No. 5. Kerosene engine, 6 h. p. at 360 r. p. m., 1 cyl., horizontal, 2 cycle with 
crank case compression, cyl. dia. 7" with 8" stroke. The engine has 
no carburetter, but is fitted with a separate vaporizing chamber. 
Oil is supplied to a pump on top of the engine, which delivers it 
directly through a pipe to the vaporizer lip, and has a hand- 
operated handle to deliver oil in starting. Four tests were made 
with kerosene and 5 with alcohol. The best consumption with kero- 
sene was 0.98 lb. (0.15 gal.), and with alcohol 1.60 lb. (0.23 gal.). 

No. 6. Automobile gasoline engine, 40 h. p. at 900 r. p. m., 4 cycle, 4 cylin- 
der, single acting, vertical, 4|" bore, 5V stroke. All valves are 
cam operated and the carburetter is of the constant level type. 
Two tests each were made with gasoline and alcohol. The result of 
these tests is as follows: 



TESTS OF HEAT ENGINES— ALCOHOL FUEL. 



1369 



No. 


Kind of 
Fuel. 


Brake Load. 


Revo- 
lutions 
per 
min- 
ute. 


Brake 
horse- 
Power. 


Dura- 
tion of 
Test. 


Fuel consumption 


Vacuum 


of 
Test 


W 


W. 


W-w 


Per 
Hour 


Per Horse- 
Power Hour 


in car- 
buretter. 


153 
154 
155 
156 


Gasoline 
...do... 
Alcohol 
...do... 


Lbs. 
215 
200 
218 
220 


Lbs. 
25 
22 
24 
26 


Lbs. 
190 
178 
194 
194 


660 
780 
680 
670 


26.3 
29.2 

27.7 
27.3 


MinSec 

2 19 

8 

10 

19 


Lbs. 
24.2 
29.9 
37.9 
39.2 


Lbs. 
0.92 
1.02 
1.37 
1.44 


Gallon. 

0.16 

.17 

.20 

.21 


Inch, of 

Mercury, 

ito i 

ito^ 

ftoli 



in which W- 
power. 



■w is the force used in computing the brake horse- 



No. 7. Automobile gasoline engine, 40 h. p. at 900 r. p. m., 4 cycle, 4 cylin- 
der, 4 1" bore, 5]^" stroke. Twelve tests with gasoline and 7 with 
alcohol. The best consumption with gasoline was 0.69 lb. (0.12 gal.) 
per brake h. p. hour; with alcohol 1.30 lb. (0.19 gal.). 

No. 8. Boat gasoline engine, 2 h. p. at 700 r. p. m., 1 cyl., vertical, 2 cycle, 
4" bore, 4" stroke. Ten tests with gasoline and 7 with alcohol. 
Best consumption on gasoline was 1.36 lb. (0.23 gal.) per brake h. p. 
hour: and with alcohol 2.52 lb. (0.37 gal.). 

Conclusions. — ^The following conclusions are drawn as a result of the 
investigations: 

(1) Any gasoline engine of the ordinary types can be run on alcohol fuel 
without any material change in the construction of the engine. The only 
difficulties likely to be encountered are in starting and in supplying a suffi- 
cient quantity of fuel, a quantity which, must be considerably greater than 
the quantity of gasoline required. 

(2) When an engine is run on alcohol its operation is more noiseless than 
when run on gasoline, its maximum power is usually materially higher than 
it is on gasoline and there is no danger of any injurious hapimering with 
alcohol such as may occur with gasoline. 

(3) For automobile air-cooled engines alcohol seems to be especially 
adapted as a fuel, since the temperature of the engine cylinder may rise 
much higher before auto-ignition takes place than is possible with gasoline 
fuel; and if auto-ignition of the alcohol fuel does occur no injurious hammer- 
ing can result. 

(4) The consumption of fuel in lbs. per brake H. P., whether the fuel is 
gasoline or alcohol, depends chiefly upon the H, P. at which the engine is 
being run and upon the setting of the fuel supply valve. It is easily possible 
for the fuel consumption per H. P. hour to be increased to double the best 
value, either by running the engine on a load below its full power or by a 
poor setting of the fuel supply valve. 

(5) These investigations also showed that the fuel consumption was 
affected by the time of ignition, by the speed, and by the initial compres- 
sion of the fuel charge. No tests were made to determine the maximum 
possible change in fuel consumption that could be produced by changing 
the time of ignition, but when near the best fuel consumption it was shown 
to be important to have an early ignition. So far as tested the alcohol fuel 
consumption was better at low than at high speeds. So far as investigated, 
increasing the initial compression from 70 to 125 lbs. produced only a very 
slight improvement in the consumption of alcohol. 

(6) It is probable that for any given engine the fuel consumption is 
also affected by the quantity and temp, of cooling water used and the nature 
of the cooling system, by the type of ignition apparatus, by the quantity 
and quality of lubricating oil, by the temp, and humidity of the atmosphere, 
and by the initial temp, of the fuel. 

(7) It seems probable that all well-constructed engines of the same size 
will have approximately the same fuel consumption when working under 
the most advantageous conditions. 



1370 



.—STEAM AND GAS POWER, 



(8) With any good small stationary engine as small a fuel consumption 
as 0.70 lb. of gasoline, or 1.16 lbs. of alcohol per brake H. P. hour may 
reasonably be expected under favorable conditions. These values corres- 
pond to 0.118 and 0.170 gallon respectively, or 0.95 pint of gasoline and 1.36 
pints of alcohol. Based on the high calorific values of 21,120 British ther- 
mal units per pound of gasoline and 11, 880 per pound of alcohol, these con- 
sumptions represent thermal efl&ciencies of 17.2% for gasoline and 18.5% 
for alcohol. 

But calculated on the basis of the low calorific values of 19,660 B. T. U. 
per pound of gasoline and 10,620 for alcohol, the thermal efficiencies 
become 18.5 for the former fuel and 20.7 for alcohol. The ratio of the high 
calorific values used above is, gasoline to alcohol, 1.78. The corresponding 
ratio of the low calorific values is 1.85. The ratio of the consumptions 
mentioned above is, alcohol to gasoline, 1.66 by weight, or 1.44 by volume. 

Properties of Liquid Fuels. — All liquid fuels available for commercial 
use are complicated mixtures of many different chemical substances, and 
hence are always liable to more or less change in chemical composition. 
Gasoline and kerosene are most easily examined by their specific gravities, 
but since each is a mixture of numerous lighter and heavier oils, a definite 
constant density is not a guarantee that the composition may not change 
sufficiently to affect the action of the fuel in an engine. 

Commercially pure grain or ethyl alcohol is sensibly pure except for the 
water which may be mixed with it. In this country alcohol is described 
according to its strength by stating the percentage of absolutely pure alcohol, 
by volume, which exists in the mixture of alcohol and water. Thus, 90% 
alcohol contains 90% alcohol and 10% water by separate volume. Since 
alcohol is lighter than water, the stronger the alcohol the lighter the specific 
gravity, and 90% alcohol contains less than 90% of alcohol by weight. 
Moreover, since when pure alcohol and water are mixed the volume of the 
mixture is less than the sum of the volumes of the water and alcohol before 
mixing, 90% alcohol contains more than 10% of water by volume. A 
U. S. "proof" gallon contains 50% alcohol by volume, the remainder of the 
mixture being water; hence a quantity of alcohol when stated in proof 
gallons is expressed by a number just twice as large as it would be if stated 
in gallons of 100% alcohol. 

The denatured alcohol which may be used in engines in the U. S. must 
be prepared as follows, according to the regulations of the Commissioner of 
Internal Revenue: To 100 volumes of ethyl or grain alcohol of a strength 
not less than 90% there must be added either 10 volumes of methyl or wood 
alcohol and i oi 1 volume of benzine or 2 volumes of methyl alcohol and 
^ of 1 volume of pyridin bases. The substances added to the grain alcohol 
will probably not be of uniform quality, and hence there will be some vari- 
ability in the properties of the denatured alcohol which will affect its use as 
a fuel. The following figures are fair average values of the different "fuels: 



Substance. 



Gasoline . 
Kerosene 



Spec. 
Grav. 



0.71 
0.80 



Lbs. 

per 

Gallon 



5.9 
6.7 



Substance. 



95% ethyl alcohol. 
90% ethyl alcohol. 



Spec. 
Grav. 



82 
0.83 



Lbs. 

per 

Gallon 



6.8 
6.9 



The two most important properties of a liquid fuel, which determine its 
availability or adaptability for use in an engine, are its heat of combustion 
and its volatility. 

Heat of Combustion. — For the various petroleum oils the heat of com- 
bustion varies between 19 000 and 21 000 B. T. U. per lb. of oil, and 20 000 
is an average value; for pure alcohol, about 12 700 B. T. U. per lb. 

All liquid fuels contain a considerable proportion of hydrogen, which 
when burned forms water in the condition of steam. When the fuel is 
burned in a calorimeter this steam is condensed by the cold water surround- 
ing the calorimeter and in this condensation contributes a considerable amount 
of heat to the total amount absorbed by the cold water. When a fuel is 
burned in an internal-combustion engine the products of combustion 



PROPERTIES OF LIQUID FUELS. 1371 

always leave the engine cylinder at a temperature much above the boiling 
point of water; hence the engine is unable to make use of the latent heat of 
condensation of the steam formed in combustion, although this latent heat 
is included in the heat measured by the calorimeter. On this account it is 
customary, in comparing fuels used in explosion engines, to calculate the 
heat of condensation of the steam in the products of combustion and to 
deduct this amount from the heat of combustion as measured in the calorim- 
eter, giving what is called the low value of the heat of combustion. Cor- 
respondingly, the heat as measured by the calorim'eter is called the high 
value of the heat of combustion. 

Air Necessary for Combustion. — When a fuel has a definite chemical 
composition, the air necessary for its combustion can be exactly calculated. 
Hence, this calculation can be made for pure methyl or ethyl alcohol, but 
can be made only approximately for a fuel like gasoline, which is a mixture 
in variable proportions of a large number of different chemical substances. 
By the formula for ethyl alcohol, C2 H5 OH, its molecular weight is 46: 
carbon 24 (= 1 2 X 2) + hydrogen 6 (= 1 X 6) + oxygen 16 ( = 16X1) = 46. For 
the complete combustion of 1 molecule of alcohol the two atoms of carbon 
require 4 atoms of oxygen to form carbon dioxide, and the 6 atoms of hy- 
drogen require 2 atoms of oxygen, in addition to the 1 atom present, to 
form steam, thus making 6 atoms of oxygen in all to be supplied. The 
weight of the 6 atoms is 6X16 = 96. Hence complete combustion of 1 lb. of 
C2 H5 OH requires 96 -f- 46= 2.086 lbs. of oxygen. In one pound of pure dry 
*air there is 0.230 lb. of oxygen, so that the combustion of 1 lb. of C2 Hry OH 
requires 2.086-^0.230 = 9.06 lbs. of air, or about 119 cu. ft. of pure air at a 
temp, of 60° and at sea level. If the alcohol contains water, 1 lb. of the 
alcohol-water mixture requires less air than that stated. If the air is moist, 
1 lb. of it contains slightly less than 0.230 lb. of oxygen and hence more air 
would be required.* In an actual engine the amount of air is proportioned to 
the amount of vapor, not by any exact measurement of either, but by trial 
to secure either the best results in maximum power or in minimum fuel 
consumption. 

Vaporization of Fuel. — Before any liquid fuel can be used in the usual 
form of explosion engine, it must be vaporized, and this vapor must be 
mixed with air in proper proportions. Thus the preparation of the com- 
bustible mixture involves three steps: First, vaporization of the fuel; 
second, mixture of the fuel vapor and air; and, third, the proper adjust- 
ment of the proportions of fuel and air. 

The differences in the devices used in engines to accomplish these objects 
constitute the widest variations in the detailed design of existing engines. 
In some of these devices the fuel is boiled in a separate chamber, called a 
vaporizer, from which the vapor flows into a stream of air entering the en- 
gine, the amount of vapor being regulated by a valve just as in the case of 
an engine using illuminating or producer gas. 

In another type of vaporizer the fuel is dropped on a hot plate over 
which the air flows, the proportion of fuel being regulated by the amount 
of liquid fuel forced against the plate. Kerosene requires a high heat to 
vaporize it completely, since its boiling point is high, and hence it is much 
used with vaporizers of the hot-plate type. Alcohol will work satisfactorily 
with a vaporizer of this type if the temperature of the hot plate is properly 
regulated. 

Gasoline is easily vaporized at ordinary atmospheric temperatures and 
hence requires no hot plate or heated vaporizing chamber. Usually the 
liquid gasoline is admitted directly into the air entering the engine through 
a device known as the carbureter, which is intended to regulate the propor- 
tion of fuel and to spray it uniformly through the mass of air so that as the 
liquid spray turns into vapor it will produce a homogeneous mixture of air 
and vapor. Alcohol can also be used in a gasoline carbureter. 

As with all substances which liquefy at ordinary temperatures, there is 
a definite limit to the amount of alcohol vapor which can exist in a cubic 
foot of space at any given temperature. Assuming the laws for perfect gases 
to hold, at any given constant temperature the weight of alcohol vapor 
present in a cubic foot of space is proportional to its vapor pressure and is 
usually measured or represented by this vapor pressure. This may be 

* Approx. results calculated in a similar manner for various petroleum 
fuels may be found in Sorel's Alcohol Engines. 



1372 m,— STEAM AND GAS POWER. 

illustrated by imagining a cylinder provided with a tight piston and contain- 
ing alcohol vapor at a pressure corresponding to 10 millimeters of mercury, 
and kept constantly throughout the experiment at a temperature of 70° F. 
If now the vapor is compressed by the piston until its volume is reduced 
one-half, the vapor pressure will rise to 20, there being of course twice as 
much vapor per cubic foot of space occupied as there was originally. If the 
volume is again halved the pressure will rise to 40. But if the compression 
is continued until the vapor pressure rises to 4 7 millimeters of mercury a change 
takes place in the action. The pressure will not rise above 47 if the temp, 
is kept at 70°. If the piston is moved a further amount, so as to reduce the 
volume still more, part of the alcohol vapor will be condensed into liquid, 
but the vapor pressure will remain the same and the amount of vapor per 
cu. ft. of space will remain constant. Hence there is a definite maximum 
amount of alcohol vapor which can exist in a cu. ft, of space at any given 
temp. It is important to remember that the space may contain any smaller 
amount with a correspondingly lower vapor pressure, but can not contain 
a greater amount than the quantity corresponding to the saturated state. 

The vapor pressure of saturation increases rapidly with the temp., and 
the values as determined by experiment for alcohol and some other sub- 
stances at various temperatures are given in Table 13, next page. 

When different gases or vapors exist simultaneously in the same space, 
if they have no chemical action on each other, each one acts by itself just 
as though no other gas were present. Thus if air were also present in the 
cylinder used in the illustration above, the oxygen and nitrogen would not 
interefere at all with the action of the alcohol vapor. But it is to be noted* 
that in such a case the pressure as measured by the barometer column or 
pressure gauge would be the sum of the separate pressures due to the air 
and due to the alcohol vapor. If moisture were presnet it would probably 
have some effect on the alcohol- vapor pressure, because water and alcohol 
have certain affinity for each other. On page 1371 it was shown that in a 
mixture of alcohol vapor and air the proportion for complete combustion ot 
the alcohol should be 9.06 lbs. of air to 1 lb. of alcohol. If less air is present 
the alcohol cannot be completely consumed and the excess passes off as 
alcohol or some substance formed by the partial decomposition of the 
alcohol molecule. If more air is present than the required amount no harm 
is done provided the excess is not too great, and it is in fact better so far 
as economical consumption of fuel is concerned to have some excess of air 
present. 

The vapor pressure of alcohol vapor in the theoretically best mixture of 
it with air may be calculated as follows: By Avagadro's law, for the same 
pressure and temp., the densities of gases are proportional to their molecular 
weights. Since the molecular weight of hydrogen is 2, the density of ethyl 
alcohol vapor is 46^2, or 23, compared with hydrogen. Hence 1 lb. of 
alcohol vapor occupying any stated volume has a vapor pressure equal to 

— of the vapor pressure of 1 lb. of hydrogen occupying the same volume. 

Likewise, since the density of air is 14.44 compared with hydrogen, 9.06 lbs. 
of air occupying the same stated volume has a vapor pressure equal to 

ttTa of the vapor pressure of 1 lb. of hydrogen occupying the same volume. 

Therefore the relative vapor pressures of the alcohol vapor and air are as 

— and Yf^,oras 0.0435 and 0.627, respectively. But 0.0435-1-0.627 = 

«o X4.44 

435 
0.670. Hence, of the total vapor pressure produced by the mixture, ^jwi. 

fi97 

or 6.5%, is due to alcohol vapor, and ^r=^, or 93.5%, is due to the air. 

If the mixture is under the ordinary atmospheric pressure of 14.7 lbs. 
per sq. in., or under 760 mm. of mercury, the vapor pressure of the alcohol 
in the combustible mixture is 0.065 times 14.7 or 0.955 lb. per sq. in., or 
0.065 times 760, or 49.4 mm. of mercury. Similarly, the pressure of the 
air is 0.935 times 14.7, or 13.74 lbs. persq. in., or 0.935 times 760, or 711 mm. 
of mercury. 

Table 13, following, contains the vapor pressure of saturation, in mm. 
or mercury, for pure ethyl or grain alcohol, pure methyl or wood alcohol, a 
sample of gasoline, and water, at various temperatures: 



PROPERTIES OF LIQUID FUELS. 



1373 



13. — Vapor Pressure of Saturation for 


Various Liquids.* 




erature. 


Vapor Pressure of Saturation In Millimeters of Mercury- 


Temp 














Pure Ethyl 


Pure Methyl 


Water. 


Gasoline. 






Alcohol. 


Alcohol. 






°C. 


op^ 










U 


32 


12 


30 


5 


99 


5 


41 


17 


40 


7 


115 


10 


50 


24 


54 


9 


133 


15 


59 


32 


71 


13 


154 


20 


68 


44 


94 


17 


179 


25 


77 


59 


123 


24 


210 


30 


86 


78 


159 


32 


251 


35 


95 


103 


204 


42 


301 


40 


104 


134 


259 


55 


360 


45 


113 


172 


327 


71 


422 


50 


122 


220 


409 


92 


493 


55 


131 


279 


508 


117 


561 


60 


140 


350 


624 


149 


648 


65 


149 


437 


761 


187 


739 



* The values for ethyl and methyl alcohol are taken from the Smith- 
sonian physical tables, the values for water from the steam tables in general 
use, and the values for gasoline, which were based upon tests of a sample of 
French commercial gasoline, are taken from Sorel's book on alcohol engines. 
Since gasoline is a variable substance, the values given for it are to be con- 
sidered as only generally representative. 

From the above table it is evident that ethyl alcohol can have a vapor 
pressure of 49 mm. only if its temp, is 72° F. or higher. Hence, a mixture 
of air and the alcohol vapor in the theoretical proportions for perfect com- 
bustion can not exist at a temp, below 72° F. A combustible mixture with 
some excess of air can exist at lower temperatures, as also a mixture in which 
a part of the alcohol is not in the form of vapor, but is carried with the air 
mechanically in the liquid form as a spray or fog. 

The table shows that methyl alcohol vaporizes much more readily than 
ethyl alcohol, and gasoline much more readily than either. A mixture of 
different fuels will usually have a higher vapor pressure than either of the 
separate ingredients unless one is present in too small a quantity to produce 
saturation. 

Since alcohol, as used commercially, is always mixed with some propor- 
tion of water, a combustible mixture formed by the vaporization of such 
alcohol may become saturated with the water vapor before it is saturated 
with alcohol, and this may retard the complete vaporization of the alcohol. 
Such a state is more likely to occur if the air originally contains a consider- 
able amount of water vapor — that is, if the relative humidity is high. In 
such a case a temp, higher than 72° would be necessary to maintain the 
required amount of alcohol vapor in the mixture. 

In order that alcohol may change from a liquid to a vapor it must 
receive a large amount of heat either from the air with which it mixes or 
from the metal parts of the carburetter with which it comes in contact. 
The hotter these parts the more quickly the alcohol can absorb the requisite 
amount of heat. But if the air is too hot there is danger that the mixture 
of air and alcohol vapor produced may be too rich in alcohol and some of 
the vapor must remain unburned. Still, if the air be moist, or the alcohol 
contain water, or the time allowed for vaporization be too short, the temp, 
of the air must be higher than 72° to form a proper explosive mixture. 

Air at any temp, will take up some alcohol vapor, and the higher the 
temp, the quicker it will take up the amount necessary for the best explo- 
sive mixture. In the case of incomplete vaporization, some of the fuel may 
be carried along as spray, which may or may not be vaporized in the cylinder 
on the compression stroke. If not, it certainly will be vaporized after the 
explosion of the rest. It would seem desirable, therefore, to heat consider- 
ably the air supplied to an alcohol carburetter. But too much heating of 
the air will bring about a bad effect on the engine, because it will make the 
charge hotter at the end of compression, and thus decrease the weight of 



1374 QQ.-^STEAM AND GAS POWER. 

the charge in the cylinder. The h. p. of the engine, other things being 
equal, will be decreased in direct proportion as the density of the charge is 
lowered by this heating, so that heating of the air before carburetting is 
good for complete vaporization, but bad if carried too far in its effects on 
power reduction. 

Methods of Testing. — Each engine tested was fitted with a suitable 
brake for absorbing the power developed. No. 1 was provided with a 
special water-cooled pulley, attached to the fly wheel. The pulley had an 
internal rim for retaining the cooling water and an external rim for retaining 
the brake in place. The latter was formed of wooden blocks, attached to a 
belt whose length could be adjusted by a screw. A wooden arm, connected 
to the belt, rested upon platform scales. On all the slow-speed engines 
similar wooden block band brakes, bearing upon platform scales, were used. 

Each automobile engine was fitted with a special water-cooled pulley, 
attached to its fly-wheel. The pulley was made by cutting out a disk of f" 
boiler plate, which was bolted to the fly wheel. The outside face of the disk 
was grooved to receive the rim of a standard cast-iron belt pulley 5" wide. 
The other edge of the pulley rim was fitted to another ring of boiler plate 
in a similar manner. Bolts passed from plate to plate. A rope break was 
used on this pulley, and each end of the rope was fastened to a spring scale, 
which in turn was suspended from the arm of a beam pivoted to a standard. 
The other end of the beam was held by a chain block. A movement of the 
chain block increased or decreased the tension on the rope. The rotation of 
the engine tended to pull one side tighter than the other, just as in the case 
with a windlass. A heavy scale, capable of recording 400 lbs., was attached 
to the beam near its center and carried the tight side of the rope. The other 
end of the rope was attached to a lighter scale, capable of recording a 
maximum of 24 lbs. With this brake a 200-lb. pull was ■ registered on the 
heavy scale, with only 8 lbs. on the smaller scale, and the tension could be 
quickly and rapidly varied by the chain block. 

The speed of the engines was obtained by actual counting, using a stop 
watch, by a hand speed counter and by a tachometer. Usually the speed 
was determined by more than one observer, so as to reduce the chances of 
error. 

The ordinary formula was used for computing the brake horse-power, 

2nRWN 
^^^- 33.000 ' 

in which i? = the brake arm in feet, W = the brake load in pounds, and 

iV = the number of revolutions per minute. 

On all the slow-speed engines, indicator cards were taken not only for 
the purpose of determining indicated horse-power but also for studying the 
characteristics of the combustion, compression, and other conditions in the 
cylinder. For this use a new outside spring indicator was loaned for these 
tests by the manufacturers of the indicator. 

It was difficult to determine the proper mean effective pressure to use 
in computing indicated horse-power, because successive strokes often gave 
indicator diagrams of very different size and shape. This was shown on some 
of the cards reproduced later. Hence it was impossible to determine 
exactly the average mean effective pressure for a stated period of time. 
This difficulty was avoided so far as possible by taking a large number of 
cards. When there were numerous differing cycles drawn on one sheet, 
usually the planimeter point was moved around on a line as near the average 
of the different cycles as could be determined by eye. This gave a sort 
of graphical average which seemed the best that could be done under the 
circumstances. In certain cases, explained in detail as they arise, cycles of 
different size were separately measured and the different areas were used in 
computing the indicated horse power. No attempt was made to measure 
the area of the lower or suction loop on the indicator cards with the plani- 
meter; the mean effective pressure is based upon the measured area of the 
upper loop only. 

The number of explosions per minute was determined in hit-and-miss 
governed engines both by counting the number of fuel admissions and also 
by listening to the exhausts. This dual method is necessary because, under 
certain circumstances, a charge may miss fire and an explosion may be 
recorded from observations of fuel admissions where one did not really 
occur. 



METHODS OF TESTING HEAT ENGINES. 1375 

The indicated horse-power was computed by the usual formula, 
jr.p PLAN 
''^ 33.000 • 

in which P = mean effective pressure in lbs. per sq. in., L = stroke of piston 
in feet, A = area of piston in sq. ins. and A/^ = number of explosions per 
minute. 

The engines were in all cases piped for alcohol fuel and also for either 
gasoline or kerosene, so that one could be switched on and the other off at 
any time. In addition a third connection was provided for the measuring 
tanks. Any variation in the adjustment of the engine was secured when 
using the permanent supply tanks before switching on the measuring tanks, 
and care was exercised to insure that there was no residue between the valve 
and the engine itself before measurements were taken. 

Two methods of measuring fuel were used: (1) by noting the drop in 
level in a glass gage attached to a tank; and (2) by the use of a large glass 
beaker on a small spring platform scale, which could be read to tenths of an 
ounce. 

Fuels used. — The alcohol used in tests was 'bought on competitive bids 
as 95% commerical alcohol and cost in barrels 33^ cents per wine gallon 
without the tax. 

An ultimate analysis of the alcohol gives the following composition: 

Per cent. 

Carbon 47.6 

Hydrogen 12.7 

Oxygen (by difference) 39.7 

As received the alcohol had a spec. grav. of 0.82 at 60° F., corresponding 
to about 91.1% by weight or 94% by volume, according to the Smithsonian 
physical tables. It was tested in the chemical laboratory in the usual way 
to determine the percentage of alcohol present after treatment to eliminate 
impurities, as follows: 

Twenty-six grams were diluted with water, redistilled, and the amount 
of alcohol in the distillate determined. From this the percentage by weight 
in the original sample was calculated. Using Richard's tables the result ob- 
tained was 93.1%. Using Morley's table, published in the journal of the 
American Chemical Society, October, 1904, the result obtained was 91.4%. 
The ultimate analysis indicates a slightly smaller proportion of alcohol, 
since a strength of 91.3% would have the following composition: 

Per cent. 

Carbon 47.6 

Hydrogen 12.9 

Oxygen 39 . 5 

These discrepancies, illustrating the difficulties of accurate determina- 
tions, even with suitable facilities and skillful observers, show the impossi- 
bility of obtaining more than approximate results except under exceptionally 
favorably circumstances. 

The percentage of alcohol found in a sample is always likely to be 
greater when determined chemically than when determined by the hydrom- 
eter, because the presence of impurities in the way of solids dissolved in 
the alcohol or as any of the series of higher alcohols tends to make the 
spec. grav. of the sample greater and hence indicate too low a percentage 
of alcohol. 

The calorific power or heat of combustion of the alcohol used in these 
tests was found by calorimeter determinations to be 11,880 British thermal 
units per pound, high value. The corresponding low value is 10,620. 

Using ihe customary values of calorific power of 14,500 for carbon and 62,000 
for hydrogen, the high heat value of combustion for pure alcohol would be 
by calculation 12,950. The Smithsonian physical tables give 12,930 as the 
value for pure alcohol on the authority of Favre and Silbermann, which 
would correspond to 11,800 for 91.3% alcohol. 

The gasoline used in the experiments, known as "motor gasoline," was 
bought in New York City at 15 cents a gallon by the barrel and had a spec. 
grav. of about 0.71 at 60° F. Its ultimate composition was as follows: 

Per cent. 

Carbon 85.0 

Hydrogen 14.8 

Total 99 . 8 



1376 



).— STEAM AND GAS POWER, 



Its heat o£ combustion, high value, as determined by the calorimeter, 
was 21,120 B. T. U. per pound. The corresponding low value is 19,660. 

Using, as before, 14,500 for carbon and 62,000 for hydrogen, the calcu- 
lated high heat value would be 21,500. 

To show the nature of the complex mixture forming the gasoline, 150 
cubic centimeters was distilled and collected in fractions of 10 cubic centi- 
meters each. The temperatures required for the distillation of the suc- 
cessive fractions are shown in the following table: 

14. — Fractional Distillation op Gasoline. 



Number 






Number 




of 


Temperatures. 


of 


Temperatures. 


Fraction. 






Fraction. 






°C. 


°F, 




">€. 


°F. 


1 


46 to 60 


114. 8to 140.0 


8 


97 to 100 


206.6 to 212.0 


2 


64 to 75 


147. 2to 167.0 


9 


100 to 104 


212.0 to 219.2 


3 


75 to 80 


167.0 to 176.0 


10 


104 to 108 


219.2 to 226.4 


4 


80 con- 


176 constant 


11 


108 to 112 


226. 4 to 233.6 




stant 




12 


112 to 120 


233.6 to 248.0 


5 


80 to 86 


176.0 to 186.8 


13 


120 to 126 


248. Oto 258.8 


6 


86 to 92 


186. 8to 197.6 


14 


126 to 140 


258. 8to 284.0 


7 


92 to 97 


197. 6to 206.6 


15 


140 to 155 


284.0 to 311.0 



From this table it appears that when distillation began the vapor showed 
a temp, of 46° C. or 114.8° F. and that the distillation was stopped at a 
temp, of 155° C. or 311° F., and that at this temp, there was left 5 cu. cm. 
that had not yet been vaporized. • * 

The kerosene used for testing had a spec. grav. of about 0.800 at 60° F. 
It was assumed to have the same heat of combustion as the gasoline used. 

In making computations on the results of the tests, the following 
weights were used. 



Substance. 


Spec. 
Grav. 


Lbs. 

per 

Gallon 


Lbs. 

per 

Pint. 


Pints 
per 
Lb. 


Substance. 


Spec. 
Grav. 


Lbs. 

per 

Gallon 


Lbs. 
Pint. 


Pints 


Water 

Alcohol.... 


1.000 
0.820 


8.33 
6.83 


1.04 
0.85 


0.96 
1.17 


Gasoline. . 
Kerosene.. 


0.710 
0.800 


5.91 
6.66 


0.74 
0.83 


1.35 
1.20 



HEAT-ENGINE FUELS. HEAT RESISTANCE, 



1377 



EXCERPTS AND REFERENCES. 

The Loomis Water-Qas and Producer-Qas Process (Eng. News, Sept. 
12, 1901). — Described and illustrated. 

Experiments on the Escape of Steam Through Orifices (By M. Rateau. 
Paper, Inter. Eng. Cong, at Glascow; Eng. News, Sept. 19, 1901). — Sketch 
of apparatus used in the experiments; and diagram illustrating the experi- 
ments. 

Heat Resistance, the Reciprocal of Heat Conductivity (Eng. News, 
Dec. 4, 1902). — Includes the following table: 

Heat Conducting and Resisting Values op Different 
Insulating Materials. 







Conductance 








B.T.U. per 


Coeffi- 






sq. ft. per day 


cient of 


No. 


Insulated Material. 


per degree 


heat. 






of difference 


resist- 






of 


ance. 






temperature. 








2iK* 


C* 


1 


f-in. oak board, 1-in. lampblack, Hn. pine b'd 








(ordinary family refrigerator) 


5.7 


4 21 


2 


f-in board, 1-in. pitch, -J-in. board 


4.89 


4 91 


8 


l-in. board, 2-in. pitch, |-in. board 


4.25 


5.65 


4 


|-in. board, paper, 1-in. hiineral wool, paper, 








|-in. board 


4.6 


5.22 


5 


l-in. board, paper, 2i-in. mineral wool, paper, 








i-in. board 


3.62 


6.63 


6 


|-in. board, paper, 2-in. calcined pumice, |-in. 








board 


3.38 
3.90 


7.10 


7 


Same as above, when wet 


6.15 


8 


1-in. board, paper, 3-in. sheet cork, |-in. board. 


2.10 


11.43 


9 


Two |-in. boards, paper, solid, no air space, 








paper, two |-in. boards 


4.28 


5.61 


10 


Two 1-in. boards, paper, 1-in. air space, paper. 








two i-in. boards 


3.71 


6.47 


11 


Two i-in. boards, paper, 1-in. hair felt, paper. 








two i-in. boards 


3.32 


7.23 


12 


Two |-in. boards, paper, 8-in. mill shavings. 








oaoer two ■S^-in boards 


1.35 


17 78 


13 


The same, slightly moist 


1.80 


13.33 


14 


The same, damp 


2.10 


11.43 


15 


Two 1-in. boards, paper, 3-in. air space, 4-in. 








sheet cork, paper, two |-in. boards 


1.20 


20.00 


16 


Same, with 5-in. sheet cork 


0.90 
1.70 


26.67 


17 


Same, with 4-in. granulated cork 


14.12 


18 


Same, with 1-in. sheet cork 


3.30 


7.27 


19 


Four double |-in. boards (8 boards), with paper 






bet. three 8-in. air spaces 


2.70 


8.89 


20 


Four 1-in. boards, with 3 quilts of i-in. hair 








bet. papers separating boards 


2.52 


9.52 


21 


f-in. board, 6-in. patented silicated strawboard 








finished inside with thin cement 


2.48 


9.68 









*iiL= conductance per hour; C=-^' 

Qas Engine Principles and Management (By E. W. Roberts. Eng. 
News, Sept. 17, 1903).— Illustrated. 

Notes on the Arrangement and Construction of Steam Pipes and 
Their Connections (By R. C. Monteagle. Paper, Soc. of Nav. Arch, and 
Ittar. Engr.; Eng. News, Nov. 26. 1903).— Illustrated. 



1378 m,— STEAM AND GAS POWER, 

Staybolts, Braces and Flat Surfaces in Boilers (By R. S. Hale. Paper, 
Am. Soc. M. E., Dec, 1904; Eng. News, Dec. 15, 1904).— ^Discussion and 
tables. 

The Use of Superheated Steam in Locomotive Boilers (By H. H. 

Vaughan. Paper, Am. Ry, M. M. Assn., June, 1905; Eng. News, Jtine 22, 
1905). 

Modern Problems in Gas Engineering (By Fred B. Wheeler. "Wis- 
consin Engineer," Dec, 1905; Eng. News, Jan. 11, 1906). — Includes table 
of standard gases. 

The Storage of Coal by Submergence in Salt Water (Eng. News, 
Aug. 23, 1906). 

The Use of Oil Fuels for Locomotives (Com. Rept. Trav. Engrs. 
Assn., Aug., 1906; Eng. News, Jan. 10, 1907). — Illustrations of the Booth, 
Sheedy and Lassoe-Lovekin oil burners. 

Comparative Cost of Gasoline, Gas, Steam and Electricity for Small 
Powers (By W. O. Webber. Eng. News, Aug. 15, 1907). — Diagram. See, 
also, in same issue, article by F. W. Ballard, entitled "Relative Economy 
of Steam and Gas Power Where Exhaust Steam is Used for Heating." 

The Solution of Steam Problems by the Use of a Diagram (By Lionel S. 
Marks, "Steam Tables and Diagrams." Eng. News, Aug. 26, 1909). — 
The diagram is 12 x 15 ins. and may be used in solving steam problems, of 
which numerous examples are given. The old equation (Regnault's) for 
the total heat of dry and saturated steam is, 

H= 1091.7+0.305 (i-32); 
The Marks and Davis equation for the range of temperature from 212° to 
400° F. is, 

H= 1150.3+0.3745 (i-212)-0.000550 (^-212)2. 
Below 212° the new values are well fixed by a number of individual deter- 
minations, but have not been represented by any simple equation. 

Unique Direct= Acting Explosion Pump (Eng. News, Dec 2, 1909). — 

Developed by H. A. Humphrey, an English engineer. This pump has 
neither piston nor cylinder, strictly speaking: as the water serves as the one, 
and the waterA^ay the other. The pumping is done with approx. 1 lb. of 
coal per water horse power in cornparison with 2 lbs. for compound high- 
duty steam engines, 1.7 lbs. for triple-expansion engines, and 1.6 lbs. for a 
gas-engine driving a centrifugal pump. Described and illustrated, with 
tables of duty. It is possible that this type of pump might prove highly 
efficient in connection with hydraulic dredging. 

Illustrations of Various Kinds: — 

Description. Eng. News. 

Front and side views of Emerson steam vacuum pump Oct. 31, 1901. 

Plans of the Northern Power Sta. of St. Louis Transit Co. April 10, '02. 

Raw Peat briquetting press, and coking retort June 12, '02. 

The Francke single-rotary-valve engine Feb. 5, '03. 

An adjustable staybolt for locomotive boilers July 23, '03. 

Malleable packings of various designs Aug. 6, '03. 

Steam turbines (4 articles), well illustrated June 6, '04. 
Long. vert, section of Mietz & Weiss oil eng. with evap. jacket Sept. 15, '04. 

Section through lower stories of Phipps power building, Pitts. Sept. 15, '04. 

300-H. P. 2-stage centrifugal pump, turbine driven Oct. 6, '04. 

Street railway power house in Kansas City Oct. 19, '05. 

Construction details of a modem gas-holder Oct. 26, '05. 

Centrifugal cinder separator for power plant smoke April 12, '06. 

Structural details of Long Island Power Station May 31, '06. 

New designs in flexible stay-bolts Jan. 21, '09. 

General arrangement of suction ash-conveying plant Aug. 8, '09. 

Tarless oil-gas producers, with test tables Dec. 9, '09. 

A commercial fuel-briquette plant Apr. 14, '10. 

Smyth, and Humphrey, direct-acting explosion pumps May 19, '10. 

Test of locottiotive using superheated steam, A. T. & S. F. Ry June 2, '10. 

The physical meaning of Entropy; illus. Sept. 1, '10. 



70.— ELECTRIC POWER AND LIGHTING. 

Electricity as a Form of Energy.— Under Steam and Gas Power (page 1347) 
we have seen that heat and mechanical work are mutually convertible; that 
is, 1 heat unit (B. T. U.) is equivalent to 778 ft.-lbs. of work, or 1 ft.-lb. of work 
= 0.001285 heat unit. We will now show that mechanical-, thermal-, and elec- 
trical work- and power units are mutually convertible, or have definite relations 
to each other. 

' The Electric Power Unit is the watt, equivalent to a f^ow or " current " of 
1 ampere under a " pressure " of 1 volt. Thus, 60 watts means 60 volt-amperes, 
or a current of 10 amperes at a pressure of 6 volts, or 15 amperes at 4 volts, etc. 

Watts (P) = volts (£) X amperes (C). Moreover, 1 watt = ;r— horse-power = 

74o 
0.7373 ft.-lb. per second = 44.238 ft.-lbs. per minute = 2654.28 ft.-lbs. per hour. 
Note that the watt does not represent work but the rate of work, or power. The 

watt- second, minute, or hour is equal to ■=rT^ horse-power second, minute or hour, 

74dj 

when applied to work. The watt unit, being small, is generally supplanted by 
the kilowatt in the design and rating of electric power plants and machines. (See 
Tables 34 and 35, Sec. 4, page 88, etc.; also Tables 1 and 2, Sec. 69, page 1348, etc.) 
The Kilowatt, K.-W. (= 1000 watts) = 1000 X 7373 = 737.3 ft.-lbs. per second 
= 44,238 ft.-lbs. per minute = 2,654,280 ft.-lbs. per hour; which is equivalent 
to 1.3405 horse-power. Conversely, 1 horse-power = 0.746 kilowatt. Compar- 
ing with units of mechanical work, it is evident from the above that 1 kilowatt 
hour = 1.3405 horse-power hour; and that 1 horse-power hour = 0.746 kilowatt 

hour. Comparing with units of thermal work, 1 kilowatt hour = — -— — = 

778 
3411.7 thermal units; and conversely, 1 thermal unit (B. T. U.) = 0.0002931 
kilowatt hour (= 0.2931 watt hour), or 10,000 B. T. U. (= amount of heat that 
may be assumed to be generated in a boiler from the combustion of 1 lb. of coal) 
= 2.9311 kilowatt hours. 

A Kilowatt is an " electric horse-power." 

Problem 1 (Steam- Electric) . — What power can be expected from a dynamo 
(generator) operated by a steam engine, with a consumption of 500 lbs. of coal 
per hour (assuming that each 100 lbs. generates 1,000,000 B. T. U. in the steam) ; 
the engine having an efficiency of 10 per cent, and the dynamo 93 per cent? 

Solution.— From the above we have 1,000,000 X 5 X 0.0002931 X 0.10 X 
0.93 = 136.292 kilowatts (or 182.7 horse-power). Ans. 

Problem 2 (Hydro-Electric). — A direct-connected dynamo is run by an 
impulse water-wheel under astatic head of 200 feet, and using water at the 
rate of 48 cubic feet per minute. Assuming the loss of head in the pipe 
line (from the storage to the wheel) to be 20 ft., the efficiency of the wheel 
85 per cent., and the generator 93 per cent*; what power will be furnished 
to the line? 

Solution. — ^Theoretically, the water power = 200 X 48 X 62.5= 600 000 ft.- 
lbs. per minute, or 18.182 horse-power. The efficiency of the pipe line is 
200—20 
— 27w) — or 90 per cent. Hence, power delivered to line is 18.182X0.90X 

0.85X0.93X0.746=9.65 kilowatts (or 18.182X0.90X0.85X0.93=12.94 
horse-power). 

What are Electrical Machines? — Electrical machines are machines for 
generating, controlling, transforming or converting electrical energy. 

Dynamos are machines for converting mechanical- into electrical power, 
that is, they generate electricity and hence the name "generator." An 

• Generators are constructed with efficiencies as high as 96 per cent. 

1379 



1380 70.— ELECTRIC POWER AND LIGHTING. 

electric "motor" differs from a generator in that it converts electrical- intcj 
mechanical power — a reverse operation. The essential principles of the 
motor are those of the dynamo. Dynamos and motors may be either 
alternate-current or continuous-ciirrent machines. 

Transformers are machines placed at either end (or at any point) of a 
transmission line to change the potential of the current. For instance, if it 
is desired to carry the current over the line at a higher potential than it is 
practicable for the dynamos to generate, the step-up transformer is used at 
the power-house end of the line; while at the delivery end the step-down 
transformer is installed. Transformers are either air-cooled, water-cooled 
or oil -cooled. In the latter case the whole transformer is placed in oil which 
penetrates the pores and insulates the wires. 

Converters, called rotary converters, are machines for converting alter- 
nate currents to direct currents, or vice versa; or they may be used for 
changing the voltage of continuous currents, or changing the voltage, phase 
or frequency of alternating currents. 

Boosters are machines inserted at distant points in the line, as at the 
outer ends of street-railway circuits, to compensate the drop in voltage in 
direct-current mains. A booster is a combination motor-generator or 
motor-dynamo or series motor driving an armature placed as a shunt across 
the mains. It can be arranged to either raise or lower the voltage, being 
generally used for the former purpose. 

Principles of Electricity and Magnetism. — Before proceeding immedi- 
ately with the discussion of electrical machines let us first consider the 
question — 

What is Electricity? — Scientists are pretty well agreed at the present 
day that electricity and magnetism, closely associated, are due to certain 
states of disturbance in the universal substance ( ?) called ether, which per- 
vades all space and gross matter. Other states of disturbance are called 
light and heat. Indeed, by some it is considered not improbable that gross 
matter itself is ethereal and electrical by nature, thus accoimting for the 
universal law of gravitation. It is not unlikely that the determination of the 
exact natiire of either of the above phenomena will reveal the nature of all 
the others. 

Ether, as a substance, necessarily has mass, and it has been estimated, 
from experiments made on the energy of waves of light, that a volume pf 
ether the size of the earth will weigh as much as 4 cubic feet of water. It 
is considered to be perfectly elastic and in a continual state of unrest. It 
is not difficult to imagine that it may be a fourth state of "matter" in the 
rising graduation from solid, liquid and gaseous to the ethereal; the first 
three constituting gross matter, with definite chemical composition, and the 
last, radiant or transcendental matter, whose composition is as yet unknown. 

Ether Waves vary greatly in length according to the phenomena pro- 
duced. The shortest wave lengths of which we have any knowledge (by the 
action of the photographic plate) number about 300,000,000,000,000,000 
vibrations per second (in a length of 186,500 miles or 11,816,640,000 inches) 
equivalent to less than one 25-millionth of an inch in length. These waves 
are able to penertate paper, wood and sheets of metal as ordinary light 
waves penetrate glass. They are far beyond the range of the eye as are 
also, but to a much less degree, the ultra-violet rays, estimated to be about 
one 70-thousandth of an inch in length. As the waves become longer they 
fall within the visible spectrum, the violet rays being about one 60- thous- 
andth of an inch, and the red rays (the longest rays visible), about one 
37-thousandth of an inch in length. Those rays from the violet to the red 
(including all the known colors) have the peculiar property, by virtue of the 
transverse vibrations, of acting on the retina of the eye (consisting of 
minute protuberances which are set in vibration) and producing the sensa- 
tion of light. As the waves increase in length we get first the heat wave 
and finally the electric wave, the latter being the longest. 

Electricity and Magnetism are mutually related, and it is owing to these 
peculiar properties or phenomena that we are enabled, to generate electricity 
and use it, commercially. 

Magnetic Field.— When a current of electricity flows "through" a wire 
in a closed circuit it induces around the wire a magnetic "field." Let 
Fig. 1 represent a section of such a circuit with the copper wire piercing a 



ELECTRICITY AND MAGNETISM, 



1381 



small sheet of paper, and the current flowing downward as shown by the 
vertical arrows. If, now, a few iron filings are sprinkled on the paper and 
the paper tapped gently, it will be found that the filings 
will arrange themselves in concentric rings as shown. If, 
further, a compass is brought close to the wire, the needle 
will be found to swing in a position tangent to the filing 
rings, with the N pole pointing in the direction of the 
arrows shown on the paper. Upon reversal of the current 
through the wire the magnetic needle will also reverse in 
direction by 180°. We may say, then, that the "lines of 
force'* in the induced magnetic field are right-handed or 
clockwise when looking in the direction of the circuit. 
The rapid reversal of the electric current in the circuit, 
producing what is called the alternating current (as against 
the continuous or direct current) induces, therefore, an alternating mag- 
netic field. The "strength' of the field is directly proportional to the cur- 
rent strength, and decreases with the distance from the wire. 

The Electro-Magnet. — ^The principle of the electro-magnet is based on 
the magnetic field, explained above. A By in each of the following Figs., is 
the primary circuit with the current flowing in the direction A jS, as shown 



I 

Fig. 1. 






B 



Fig. 2. Fig. 3. 



+ 
Fig. 4. 



Altract . ,_ Repulse ^ 
Fig. 5, Fig. 6. 



by the arrows. Now we have seen. Fig. 1, that when a current of electricity 
flows through the primary circuit, lines of force are set up, around the 
conductor, in a clock-wise or cock-screw direction. If the iron filings are 
replaced by a circular ring of metal, as in Fig. 2, lines of electro-magnetic 
force will be induced in the ring in the direction shown by the arrows. 
Fig. 3 shows the same phenomenon, but with the magnetic circuit forming 
merely a loop around the primary. Fig. 4 shows the circuit of the loop 
"broken" thus forming a primitive horse-shoe magnet, the 4- and — ends of 
which attract each other, tending to draw together and "close" the circuit. 
Figs. 5 and 6 show that two parallel conductors attract each other when the 
currents flow in the same direction, and r^^^/ each other when the currents 
flow in opposite directions. By reversing the direction of the current in 
A B, the direction of the magnetic circuit in Figs. 2 and 3 will also be re- 
versed, and in Fig. 4 the poles of the magnet will change from 4- to — , 
and from — to -f- . Repulsion will now take place between the parallel con- 
ductors in Fig. 5, and attraction in Fig. 6, assuming that the currents a h 
remain unchanged. Now assume a b, Figs. 5 and 6, to be uncharged from 
any secondary batteries, and assume the current in the primary conductor 
to flow in the direction A B; then, if the current is continuous there will be 
no induced current in a h, but if it is alternating there will be, during the 
time the current is increasing in strength or "setting up of the field, "^ a 
tendency toward generating currents in a 6. Such currents are called in- 
duced currents, and the general phenomenon is termed Induction. 




Core 




Fig. 7.* 



Fig. 8. 



Induction can, perhaps, best be explained by the theory that magnetism 
is merely electricity in a whirl, rotating about a conductor in the surround- 



1382 



10.— ELECTRIC POWER AND LIGHTING, 




ing ether. It is greatly assisted and intensified by the use of a soft-iron 
core around which a certain number of turns of each circuit is wound. 
Fig. 7 illustrates an induction coil or Transformer consisting essen- 
tially of a core around which is wound x turns of the primary circuit, 
P-Pt and then y turns of the secondary circuit, 5-5. From the number of 

turns of wire, we have, electro-motive force in S-S = -^X (electro-motive 

force in P-P), therefore, by increasing the ratio of turns in secondary coil to 
those in primary coil the E. M. F.* is increased. 

Faraday's ring, Fig, 8, consisted of a circular core with the primary 
and secondary coils on either side. The arrows on the ring indicate the 
direction of the electro-magnetic stresses. An alternating current in the 
primary coil P (Figs. 7 and 8) generates an alternating ctirrent in the 
secondary coil 5; and again, 5 may be used as the primary coil and P as 
the secondary coil. 

The Horse-Shoe Magnet is commonly made by bending a piece of steel 
in the form of a horse-shoe as shown in Fig. 9, winding it with fine copper 
wire (either partially as shown by the double lines 
or completely around the loop as shown by the single 
lines) and passing a current of electricity through the 
wire as shown by the arrows, "breaking" (shutting 
off) the current occasionally. The wire is now un- 
wound and it is found that the bar of steel has be- 
come a permanent magnet, capable of attracting 
magnetic substances as iron or steel. If soft iron is 
used for the "field core" instead of steel it will retain 
the properties of a magnet only when the current 
of electricity is "exciting" the field. Such a mag- 
net is called an electro-magnet. Note that the arma- 
ture A when placed within the "influence" of the 
magnetic poles of the field core is attracted by the 
latter, that is, the armature tends to "close" the -^iS- 9* 

magnetic circuit. The telegraph "sounder" is founded on this principle. 

Electro-Magnets are made in various forms, depending on the kind of 
work they are intended to do. We stated in the last paragraph that the 
armature A (Fig. 9) is attracted by the poles of the field core and tends to 
"clctse" the magnetic circuit. This is only partly true. As a matter of fact 
the magnetic circuit in Fig. 9 is already complete without the presence of 
the armature, but it is comparatively feeble and grasps the armature to 
form a more "permeable" path for its circuit than the air offers, much the 
same as a man uses a plank to cross a small stream instead of wading. The 
"Theory of least work" applies to all electric and magnetic phenomena as 
it does to mechanics. It is simply another name for the "Conservation of 
Energy." 

Principle of the Alternate=Current Dynamo. — A simple form of generator 
is illustrated in Fig. 10. The magnetic 
"field" between N and 5 is induced by 
either a permanent magnet or an electro- 
magnet (see Fig. 9). Between the poles , , , (~=^ — 1. '/CK,\ 
of the magnet is shown a loop of copper \\\ Q- |---4:^----) (b--/--^ 
wire which may be made to revolve on an 
axis ab\ we have also to imagine that 
there are lines^ tubes, or cylinders, of 
force stretching between the N and 5 Fig. 10. 
poles of the magnet. Now when the copper loop is revolved, say right- 
handed, it cuts these lines of force and alternate currents of electricity are 
generated in the copper wire circuit, the N pole inducing a current in the 
nearest parallel portion of the loop, toward a, and the 5 pole toward b. 
Both currents therefore flow around the coil in the same direction for each 
half turn, and then reverse. It is to be noted, however, that the compara- 
tive strength of the current depends on the number of lines of force cut per 
second, that is, the rate of transverse cutting of the lines of force. By 
referring to the end view in Fig. 10 it will be seen that this rate is propor- 
tional to sin a; when a= 90° the copper loop is cutting the lines of force at 
a maximum rate, unity; when a= 45° the rate is 0.707; when a = 0° no lines 




* Electro-motive force is voltage or difference of potential. E. M. F. is 
increased at the expense of current, where the power remains the same. 



ALTERNATE-CURRENT DYNAMOS, 



1383 




of force are being cut and hence no current is flowing in the circuit. This 
last position is also the point of alternation of the current, which continues 
in one direction for every half revolution of the coil and then reverses. 

The electricity generated in the 
revolving armature, for one-half 
turn, is conducted to the revolving 
"collecting"* ring, R (Fig. 11), 
thence by the fixed "brush" B to 
the line L, where it is made to per- 
form useful work such as running 
a motor, M. Passing through M it 
returns to the armature by way of 

L', B' and R', thus completing the . 

circuit. For the next half turn it •t'lg. 11. 

reverses in direction, thus alternating. If the circuit is disconnected or 
broken at any point no power can be furnished. The earth is some- 
times used for the return current as in telegraph circuits. In the case 
of electric railways the rails are used. If they are "bonded" the "resistance" 
is greatly lessened. 

Now instead of having a "revolving armature" composed of only one 
copper wire loop as shown in Fig. 10, we may increase the current output 
of the machine by using a cylindrical armature composed of a large number 
of loops; also, the field-magnet with only two poles may be increased so as 
to contain a large number of pairs of poles. 

The magnetic field of large commercial alternate-current dynamos is 
generally maintained by small constant-current generators called "exciters." 
One exciter will answer for one or more of the large alternators, and being 
continuous-current machines they are self -exciting (see page 1384). 



Classification op Alternate-Current Dynamos. 
(See also page 1384.) 

A. — With respect to stationary and moving parts: 

(a) Stationary field magnets and rotating armatures; 

(b) Stationary armatures and rotating field-magnets; 

(c) Stationary field-magnets and armatures, and revolving induc- 

tors. (Not common. Ex.: Stanley- Kelly inductor alter- 
nator.) 
B. — ^With respect to number of field-magnet poles: * 

(a) 2-pole (bipolar), 4-pole, 6-pole, 8-pole machines; 

(b) Multipolar machines include all above the bipolar. (Multipolar 

machines, with many poles, used for high power.) 

C. — ^With respect to form of armature: 

(a) Ring armatiire; 

(b) Drum armature; 

(c) Disk armature (core disks, sunk wound, are preferred to thin 

disks for high power machines, because of strength) ; 

(d) Pole armature. 

D. — ^With respect to armature coils: 

(a) Open coil; ) ( (A) In parallel 

>• Also < 

(b) Closed coil; ) ( (B) In series 



( For increase of 
( current. 
I For increase of 
( voltage. 



E. — With respect to armature windings: 

(a) Spiral or ring winding (ring armatiires) ; 

S^ ^^vrlSgH^-^^d disk armatures: 

(d) Sunk winding. 

F. — ^With respect to field-magnet windings: 

(a) Spiral winding; 

(b) Lap winding; Aicr.!^-^) In parallel, 
fc) Wave winding; ^^^^^ ( (B) In series, 
(d) Sunk winding. 

* Collecting rings or simply collectors are used on alternate-current 
machines, while "commutators" (see page 1384) are used on continuous- 
current dynamos. 



1384 



70.-^ELECTRIC POWER AND LIGHTING. 



G. — With respect to phase of machine: 

(a) Single-phase, 2-phase, 3-phase machines; 

(b) Polyphase machines include all above 2-phase. 

A single-phase alternator is one in which the currents act in 
imison, rising and falling together; while in a polyphase 
alternator the coils are arranged so that the impulses are 
produced closer together, thereby maintaining a more uni- 
form electro-motive force. 2-phase and 3-phase alternators 
are the most common. 

Principle of the ContinuouS"Current Dynamo. — Many of the principles 

of the alternate-current dynamos, just described, apply equally as well to 
the continuous-current machines. We have the field-magnet and also the 
armature; but instead of using collector rings or collectors as for the alter- 
nators, we use commutators, which deliver to the external circuit a continuous 
current, that is, a current in one direction through the external circuit. 

Fig. 12 shows a simple, two-part com- 
mutator, K and K\ joining the ends of a 
single-loop armature revolving between 
the poles of a magnet on the axis a b, and 
delivering current to the brush B. Note 
that the brushes B and B^ are fixed, and 
as the commutator revolves (with the 
armature) the K-part and the K'-part 
alternate in delivering current to B, and 
hence it always flows through the external Fig. 12. 

circuit, from B to B' , in one direction. 

Continuous-current dynamos are made with two, four, six or eight poles, 
the two-pole machines being the most common. The armature winding, 
however, is very complex. Instead of one loop, as in Fig., 12 there are 
many loops; and the commutator, instead of being composed of two shells 
or bars, as above, may be composed of many bars (see Fig. 13). Hard- 
drawn copper is best for the commutator bars, and also for the brushes.* 
All field and armature windings, as well as commutator bars, should be in- 
sulated. The latter are insulated with mica. 

Classification of Continuous-Current Dynamos. 
(See also page 1383.) 

I. — With respe(?t to excitation of field -magnets: 

(a) Separately-excited dynamo (Fig. 13), using a separate (small) 

dynamo as an exciter; 

(b) Series dynamo (Fig. 14), using the full current of the external 

circuit as an exciter; 

(c) Shunt dynamo (Fig. 15), using a small fraction of the external 

circuit (by a shimt circuit of thin wire) to excite the field; 

(d) Compound dynamo ( Fig. 16), using both shunt coils and series 

coils to excite the field. 





t MA»H 

CiRcyrj 
Fig. 13.* Fig. 14. Fig. 15. Fig. 16. 

Note. — In addition to the above we have also the Magneto dynamo, 
in which the field-magnets are permanent steel magnets; and 
the Separate-coil dynamo, in which the armature is wound 
with a separate coil to act as an exciter. 



* Carbon brushes are used for motors. 



CONTINUOUS-CURRENT DYNAMOS. TRANSMISSION. 1385 

II. — With respect to other characteristics see classification of alternate- 
current dynamos, page 1383, with the following comments 
for direct-current machines: 

A. — (a) Stationary field-magnets and rotating armatures, is the 
prevailing type. ' 

B. — Bipolar machines are generally used ; although 4-, 6-, and 8-pole 
machines are not uncommon. There is another type, namely, 
the homopolar (unipolar or 1-pole) machine which is being 
developed with more or less success. 

C. — '(b) Drum armatures are the most efficient and are generally used 
where high potential is required. Ring armatures give 
lower E. M. F. and less ctirrent for the same number of revo- 
lutions. 

D. — (b) Closed-coil armatiires, generally speaking, give greater 
commercial efficiency than open-coil machines; but the latter 
(whether of the ring or drum type) are better adapted to 
nigh electro-motive forces because of the methods of insulation 
of commutator bars. The Brush and the Thomson-Houston 
open-coil dynamos are especially adapted to electric lighting, 
and it may be said that they practically monopolize the field 
for this purpose. 

E. — (b) Lap winding is the most frequently used. 

F. — Wave winding has certain disadvantages and has not come into 
general use. 

G. — Phase applies to alternate-current machines. 

Electric Transmission of Power. — Important considerations come up in 
connection with the projection of any power plant, especially for long- 
distance transmission; and usually the combined efforts of the civil-, elec- 
trical-, mechanical- and hydraulic engineer are required for its successful 
installation. 

Uses of Electric Power. — Electric power can be used for every purpose 
for which power is required. Its practical utility as compared with the 
direct use of steam and water power, from which it is derived mainly, lies in 
the economic, convenient and sanitary distribution from a central station, 
where it can be generated economically. 

Sources of Electric Power. — Entropy, in thermodynamics, has been 
defined by some scientists as the "available" energy in any system, while 
others define it as the "non-available" energy in any system. Electricity 
may similarly be defined. It is only a question of constructing our machines 
so that the kind of energy which is usually lost may be increased, captured 
and turned into useful work at some distant point. The dynamo, the trans- 
mission line and the motor are specially designed to accomplish this result 
economically. 

The primary source of all electric- or other power is probably electricity 
itself. The immediate source appears to iis in the forms of chemical and 
physical actions, the latter through the agencies of heat and gravitation, 
acting on gross matter, either solid, liquid or gaseous. Thus, we get the 
battery, thermopile, steam-engine, so-called heat engine (including the gas 
engine), and the water wheel. No method has yet been discovered by 
which electricity can be generated directly from heat on a large commercial 
scale. The present indirect use of heat of fuel means a loss of 90 per cent or 
more of heat energy. 

For large electric plants the steam engine and the water wheel only can 
be considered as prime sources of power. The turbine wheel is best adapted 
to low heads, and the impulse (bucket) wheel for high heads. The exact 
dividing line between "low" and "high" heads is not well marked. They 
both overlap the 100-ft. head, sometimes to a considerable extent. 

Steam and Water-Power Compared. — ^The cost of fuel on the one hand, 
and the length of transmission, on the other, are prime factors in the com- 
parison. For example, the author has in mind a proposed change from 
steam- to water power m the generation of electric power for a street rail- 
way, demanding about 3000 horse-power. The fuel for the steam plant was 
very cheap, consisting of sawdust and slabs automatically fed from a 
neighboring saw-mill. This could have been supplanted by a transmission 



1386 70.— ELECTRIC POWER AND LIGHTING. 

30 miles long, and the installation of a power house with impulse wheels 
operating under a 600-ft. head. The low cost of fuel, however, favored the 
steam plant. With coal as a fuel the water-power proposition would have 
been cheaper. 

Alternating- vs. Continuous Current. — Although alternate-current motors 
may be used to convert altemate-ciirrent directly into mechanical work, 
yet, owing to certain disadvantages they have not come into general use. 
Even in lighting, the arc lamp on a continuous current circuit gives a bet- 
ter illumination, and it is also free from the humming sound produced 
when the lamps are on alternate-ciirrent circuits. With the incandescent 
lamp, however, there is less contrast in candle power. But there are other 
considerations which sometimes outweigh the above, and we find both the 
alternating- and continuous current in distributing systems for motive power 
and lighting. 

The electric transmission of power for moderate distances is almost always 
by direct current. This system involves line construction of the most simple 
character, namely, two lines of wire. The current is generated at the power 
station by continuous-current dynamos. Each machine may be designed 
to yield a voltage up to 2500 or even 4000 volts, although the latter is 
uncommon. In order to secure a high voltage in transmission (which 
means a saving in copper or aluminum wire) the dynamos are placed in 
"series." Thus, ten dynamos in series, each generating current at say 2000 
volts, will deliver about 10X2000 = 20,000 volts to the line. At the distri- 
buting end of the line the total voltage of the motors receiving the energy 
at anyone time must be equal to that of the generators, minus the "drop" 
in volts on the line. Continuous-current transmission may operate in three 
ways; 

(1) Constant voltage and variable ctirrent; 

(2) Constant current and variable voltage; 
• (3) Variable voltage and variable current. 

Long-Distance Transmission may be accomplished by either alternating 
or continuous current. In Europe the latter has many adherents, while in 
America "long-distance" is almost synonymous with "alternate-current" 
transmission. The "first cost" of installation, however, is usually greater 
for alternate-current transmission. 

Alternate-current transmission is carried out with three-phase three- 
wire, two-phase four-wire or single-phase two-wire circuits, the first named 
greatly predominating, and the last being seldom used. The number of 
wires varies from three, the lowest, to say six, a maximum per pole in good 
practice. High voltage for alternate-current transmission can be obtained 
only by step-up transformers installed at the power station because alter- 
nators cannot be connected up in series for higher voltage as can continuous- 
current dynamos. They may be designed, however, to deliver, individually, 
current at a much higher voltage, namely, up to 15,000 and possibly 20,000 
volts. The step-down transformers at the sub-stations need not be the same 
in number as the step-up transformers at the generating stations. Trans- 
formers can be wound for any rating of voltage of primary to secondary 
coils, but there are certain economic ratios related more or less to generator 
capacity. Static transformers are often used to change alternating current 
from 3-phase to 2-phase, and vice versa. 

The Transmission Line. — ^The material for line conductors has settled 
down to a choice mainly between aluminum and copper. In fact, the former 
metal has recently been used in many of the largest installations.* From 
an economic standpoint, considering the electrical and mechanical properties 
of the two metals, they possess equal merit when the price of aluminum 
wire, per pound, is about 2 to 2.1 times the price of copper; or when the price 
of copper wire is about 48 to 50 per cent of the price of aliuninum. Hence, 
with copper at 22 cents per pound, aluminum would be selected if imder 
46 cents per poundf. Aluminum cable is substituted ior wire ii the size called 
for is large, say larger than No. 1 B. & S. gage, and sometimes even for 
smaller sizes. The cables are made up of wires ranging in size usually from 
6 to 9 B. & S. gage, sometimes larger. Seven strands per cable are common, 
and as high as 37 have been used. 

* The use of aluminum has not been confined to main transmission lines. 
It is being used also in distributing power to sub-stations of electric rail- 
ways, and for city distribution of both light and power. tSee page 496. 



ELECTRIC TRANSMISSION OF POWER. 1387 

The Size of Conductors which it is advisable to use in a transmission 
line cannot be determined by any simple rule. As a broad proposition it 
may be stated that the whole power plant, from reservoir (or other source 
of power) to distribution, should be so designed that, when furnishing the 
required power, it is found that no slight increase or decrease of power could 
have been effected more economically at one part of the plant than at another. 
But there are many practical considerations. The "required" power varies 
from hour to hour of the day, from season to season of the year, and also 
from year to year. We may perhaps assume the "required" power to be 
the maximum or "peak" load, either present or prospective; or we may 
assume it to range somewhere between the peak load and the average load, 
generally near the former and a little below it. If storage batteries are 
installed to take care of the peak load the "required" power for the line 
may be lowered accordingly, but if this is not done the peak loads must of 
course be carried by bringing into action additional (reserved) units at the 
power house. Such units consist each of say a water wheel (or an engine) 
with the connected generator or generators. 

Let us assume that the "required" power, for which the transmission 

line is to be designed, has been determined as x watts ( ^ttt/^t^ kilowatts ) 

= C (amperes) XE (volts). Now we have as limiting guides, in determining 
the size of the wire, the following, namely, (1) that it is inadvisable to use a 
single copper wire of a size less than No. 4 B. & S. gage, for reasons of 
strength and stiffness; (2) a tentative calculation of the line wire should 
show a loss of say from 5 to 10 per cent of the "required" power delivered 
to the line, less for short- and perhaps greater for long transmission. Then 
by applying the above rule (in italics) see whether or not an incremental 
increase or decrease of power can be effected in the line (by changing the 
size of wire) at less cost than could be effected at the generator station 
and at other points. 

Transmission Line Problems. 

Problem 1. — What size of copper wire is required in a 25-mile continu- 
ous-current transmission receiving 3, 600 kilowatts at 30,000 volts pressure, 
allowing a "drop" in volts of 10 per cent on the line? 

Solution. — 3, 600 kilowatts =3,600,000 watts, hence the current in am- 
peres =120 amperes. With a 10 per cent drop in voltage the line loss = 

30,000X0.10=3,000 volts. Now applying Ohm's law ( C=^ ) we have, 

E SOOO 
resistance in ohms =^^=-^2^ = 25 ohms. As this resistance is distributed 

25 
over 25X2 (2-wire circuit) = 50 miles, the resistance per mile = -^7; = 0.50 

60 

ohm, which, from table 1, page 1389 corresponds to a No. 0, B. & S. gage, 
copper wire. If aluminum is used it would require a No. 000. In general, 
there is a difference of two numbers between copper and aluminum wire for 
the same resistance. Note in the above that E = the difference of potential 
between the two ends of the line. Also that the resistance is taken at 75° F., 
and that the resistance increases with increase of temperature (see Table 1, 
page 1389). 

The above method of solution applies to all continuous-current two-wire 
circuits, and can be used as well to find the weight of wire per unit of length, 
or the area in circular mils. It is a question simply of using different 
tables after the resistance has been obtained. Thus, a resistance of 0.5 ohm 
per mile is equal to 0.0947 ohm per 1,000 feet, or 10,560 feet per ohm; equiva- 
lent to 110,000 circular mils, or 3 feet per pound of wire, at 75° F. 

For alternating-current circuits the above method of solution may be 
used by noting the following: (1) The "virtual volts" and "virtual amperes" 
of the alternating current* (and these are what are commonly meant), 
such as are recorded by volt- and ampere meters, must be used in the calcu- 
lations. (2) There are two additional sources of loss in alternate-current 
lines that do not appear with the continuous current, namely, inductance 
(Continued on page 1392.) 



* In alternating currents the maximum volts and amperes rise to about 
1.414 times the virtual, alternating between that and zero, the virtual being 
about 0.707 times the maximum. 



1388 



10.— ELECTRIC POWER AND LIGHTING, 



1. — Copper Wire Table op the — 

(Supplement to Trans, of A. I. E. E., Oct., 1893.) 

The data from which this table has been computed are as follows: Mat- 
thiessen's standard resistivity, Matthiessen's temperature coefficient, specific 
gravity of copper = 8.89. Resistance in terms of the international ohm. 

Matthiessen's standard 1 meter-gram of hard drawn copper = 0.1469 
B. A. U. at 0° C. Ratio of resistivity, hard or soft copper, 1.0226. 

Matthiessen's standard 1 meter-gram of soft drawn copper = 0.14365 
B. A. U. at 0° C. 1 B. A. U. = 0.9866 international ohm. 

Matthiessen's standard 1 meter-gram of soft drawn .copper= 0J141729 
international ohms at 0° C. 

Temperature coefficients of resistance* for 20° C, 50° C, and 80° C. are 

(See Opposite Page for Areas in Square Mils.) 



Gages. 


Weight. 


Length. 


dec 


d«3 

pqcc 


^ a 


Area 
Circ. 
Mils. 


Lb.per 
Foot. 


Pounds per Ohm. 


Ft. per 
Lb. 


Feet per Ohm. 




@ 20°C. 
= ,68°F. 


@ 50°C. 
=122°F. 


@ 80°C. 
= 176°F. 


@ 20°C. 
= 68°F. 


@ 50°C. 
= 122°F. 


@ 80°C. 
= 176°F. 


0000 


0000 
000 


.460 
.454 
.425 


211600 
206100 
180600 


.6405 
.6239 
.5468 


13090 

12420 

9538 


11720 

11120 

8537 


10570 

10030 

7704 


1.561 
1.603 
1.829 


20440 
19910 
17450 


18290 
17820 
15620 


16510 
16080 
14090 


000 
00 


00 


.4096 

.380 

.3648 


167800 
144400 
133100 


.5080 
.4371 
.4028 


8232 
6096 
5177 


7369 
5456 
4634 


6647 
4924 
4182 


1.969 

2.288 
2.482 


16210 
13950 
12850 


14510 
12480 
11500 


13090 
11260 
10380 







1 


.340 

.3249 

.3000 


115600 

105500 

90000 


.3499 
.3195 
2724 


3907 
3256 
2368 


3497 
2914 
2120 


3155 
2630 
1913 


2.858 
3.130 
3.671 


11160 

10190 

8692 


9993 
9123 
7780 


9017 
8232 
7020 


1 


2 
3 


.2893 
.2840 
.2590 


83690 
80660 
67080 


.2533 
.2441 
.2031 


2048 
1902 
1316 


1833 
1702 
1178 


1654 
1536 
1063 


3.947 
4.096 
4.925 


8083 
7790 
6479 


7235 
6973 
5799 


6528 
6292 
5233 


2 
3 


4 


.2576 
.2380 
.2294 


66370 
56640 
52630 


.2009 
.1715 
.1593 


1288 
938.0 
810.0 


1153 
839.6 
725.0 


1040 

757.6 
654.2 


4.977 
5.832 
6.276 


6410 
5471 
5084 


5738 
4897 
4550 


5177 
4419 
4106 


4 


5 
6 


.2200 
.2043 
.2030 


48400 
41740 
41210 


.1465 
.1264 
.1247 


684.9 
509.4 
496.5 


613.0 
455.9 
444.4 


553.1 
411.4 
401.0 


6.826 
7.914 
8.017 


4675 
4031 
3980 


4184 
3608 
3562 


3775 
3256 
3215 


5 


7 
8 


.1819 
.1800 
.1650 


33100 
32400 
27230 


.1002 

.09808 

.08241 


320.4 
306.9 
216.7 


286.7 
274.7 
194.0 


258.7 
247.9 
175.0 


9. 980 
10.20 
12.13 


3197 
3129 
2629 


2862 
2801 
2354 


2582 
2527 
2124 


6 
7 


9 


.1620 
.1480 
.1443 


26250 
21900 
20820 


.07946 
.06630 
.06302 


201.5 
140.3 
126.7 


180.3 
125.6 
113.4 


162.7 
113.3 
102.3 


12.58 
15.08 
15.87 


2535 
2116 
2011 


2269 
1894 
1800 


2048 
1709 
1624 


8 


10 
11 


.1340 
.1285 
.1200 


17960 
16510 
14400 


.05435 
.04998 
.04359 


94.26 

79.69 
60.62 


84.37 
71.33 
54.26 


76.13 
64.36 
48.96 


18.40 
20.01 
23.94 


1734 
1595 
1391 


1552 
1427 
1245 


1401 
1288 
1123 


9 

10 


12 


.1144 
.1090 
.1019 


13000 
11880 
10380 


.03963 
.03596 
.03143 


50.12 
41.27 
31.52 


44.86 
36.94 
28.21 


40.48 
33.33 
25.46 


25.23 
27.81 
31.82 


1265 
1147 
1003 


1132 
1027 
897.6 


1021 
926.9 
809.9 


U 


13 
14 


.0950 

.09074 

.08300 


9025 
8234 
6889 


.02732 
.02493 
.02085 


23.81 
19.82 
13.87 


21.31 

17.74 
12.42 


19.23 
16.01 
11.21 


36.60 
40.12 
47.95 


871.7 
795.3 
665.4 


780.2 
711.8 
595.5 


704.0 
642.3 
637.4 


12 
13 


15 


.08081 
.07200 
.07196 


6530 
5184 
5178 


.01977 
.01569 
.01568 


12.47 
7.857 
7.840 


11.16 
7.032 
7.017 


10.07 
6.346 
6.332 


50.59 
63.73 
63.79 


630.7 
500.7 
500.1 


564.5 
448.1 
447.7 


509.4 
404.4 
404.0 


14 


16 

17 


.06500 
.06408 
.0580 


4225 
4107 
3364 


.01279 
.01243 
.01018 


5.219 
4.931 
3.308 


4.671 
4.413 
2.961 


4.215 
3.982 
2.672 


78.19 
80.44 
98.23 


408.1 
396.6 
324.9 


365.2 
355.0 
290.8 


329.6 
320.3 
262.4 



*Based on a resistance of unity at 0° C 



COPPER WIRE TABLE, 



1389 



—Am. Inst, op Electrical Engineers. 

1.07968. 1.20625, and 1.33681, respectively. 1 foot = 0.3048028 meter, 1 
pound = 453.59256 grams. 

Although the entries in the table are carried to the fourth significant 
digit, the computations have been carried to at least five figures. The last 
digit is therefore correct to within half a unit, representing an arithmetical 
degree of accuracy of at least one part in two thousand. The diameters of 
the B. & S. or A. W. G. wires are obtained from the geometrical series, in 
which No. 0000= 0.46 inch and No. 36 = 0.005 inch, the nearest fourth 
significant digit being retained in the areas and diameters so deduced. 

It is to be observed that while Matthiessen's standard of resistivity may 
be permanently recognized, the temperature coefficient of its variation which 
he introduced, and which is here used, may in future undergo slight revision. 
(See Opposite Page for Areas in Circular Mils.) 



Gages. 


Resistance. 


6m 


00 cc 


Diam. 
Ins. 


Area 
inSq. 
Mils.* 


Ohms per Lb. 


Ohms per Foot. 




@ 20° C. 

= 68° F. 


@ 50° C. 

= 122° F. 


@ 80° C. 

= 176° F. 


@ 20° C. 

= 68° F. 


@ 50° C. 

= 122° F. 


@ 80° C. 

= 176° F. 


0000 


0000 
000 


.460 
.454 
.425 


166190 
161883 
141863 


.00007639 
.00008051 
.0001048 


.00008535 
.00008996 
.0001171 


.00009459 
.00009969 
.0001298 


.00004893 
.00005023 
.00005732 


.00005467 
.00005612 
.00006404 


.00006058 
.00006220 
.00007097 


000 
00 


00 


.4096 

.380 

.3648 


131790 
113411 
104518 


.0001215 
.0001640 
.0001931 


.0001357 
.0001833 
.0002158 


.0001504 
.0002031 
.0002391 


.00006170 
.00007170 
.00007780 


.00006893 
.00008011 
.00008692 


.00007640 
.00008878 
.00009633 







1 


.340 

.3249 

.3000 


90792 
82887 
70686 


.0002560 
.0003071 
.0004223 


.0002860 
.0003431 
.0004718 


.0003169 
.0003803 
.0005228 


.00008957 
.00009811 
.0001150 


.0001001 
.0001096 
.0001285 


.0001109 
.0001215 
.0001424 


1 


2 
3 


.2893 
.2840 
.2590 


65732 
63347 
52685 


.0004883 
.0005258 
0007601 


.0005456 
.0005874 
.0008492 


.0006046 
.0006510 
.0009412 


.0001237 
.0001284 
.0001543 


.0001382 
.0001434 
.0001724 


.0001532 
.0001589 
.0001911 


2 
3 


4 


.2576 
.2380 
.2294 


52128 
44488 
41339 


.0007765 

.001066 

.001235 


.0008675 

.001191 

.001379 


.0009614 

.001320 

.001529 


.0001560 
.0001828 
.0001967 


.0001743 
.0002042 
.0002198 


.0001932 
.0002263 
.0002435 


4 


5 
6 


.2200 
.2043 
.2030 


38013 

32784 
32365 


.001460 
.001963 
.002014 


.001631 
.002193 
.002250 


.001808 
.002431 
.002494 


.0002139 
.0002480 
.0002513 


.0002390 
.0002771 
.0002807 


.0002649 
.0003071 
.0003111 


6 


7 
8 


.1819 
.1800 
.1650 


25999 
25447 
21382 


.003122 
.003258 
.004615 


.003487 
.003640 
.005156 


.003865 
.004034 
.005714 


.0003128 
.0003196 
.0003803 


.0003495 
.0003570 
.0004249 


.0003873 
. 0003957 
.0004709 


6 
7 


9 


.1620 
.1480 
.1443 


20618 
17203 
16351 


.004963 
.007129 
.007892 


.005545 
.007965 
.008817 


.006145 
.008827 
.009772 


.0003944 
.0004727 
.0004973 


.0004406 
.0005281 
.0005556 


.0004883 
.0005853 
.0006158 


8 


10 
11 


.1340 
.1285 
.1200 


14103 
12967 
11310 


.01061 
.01255 
.01650 


.01185 
.01402 
.01843 


.01314 
.01554 
.02042 


.0005766 
.0006271 
.0007190 


.0006442 
.0007007 
.0008033 


.0007140 
.0007765 
.0008903 


9 
10 


12 


.1144 
.1090 
.1019 


10283 
9331 
8155 


.01995 
.02423 
.03173 


.02229 
.02707 
.03545 


.02471 
.03000 
.03928 


.0007908 
.0008715 
.0009972 


.0008835 
.0009736 
.001114 


.0009791 

.001079 

.001235 


11 


13 
14 


.0950 

.09074 

.08300 


7088 
6467 
5411 


.04199 
.05045 
.07207 


.04692 
.05636 
.08052 


.05200 
.06246 
.08924 


.001147 
.001257 
.001503 


.001282 
.001405 
.001679 


.001420 
.001557 
.001861 


12 
13 


15 


.08081 
.07200 
.07196 


5129 
4072 
4067 


.08022 

.1273 

.1276 


.08962 

.1422 

.1425 


.09932 

1576 

.1579 


.001586 
.001997 
.001999 


.001771 
.002231 
.002234 


.001963 
.002473 
.002476 


14 


16 
17 


.06500 
.06408 
.0580 


3318 
3225 
2642 


.1916 
.2028 
.3023 


.2141 
.2266 
.3377 


.2373 
.2511 
.3742 


.002451 
.002521 
.003078 


.002738 
.002817 
.003439 


.003034 
.003122 
.003811 



*To find area in square inches, divide by 1,000,000. 



1390 



lO.^ELECTRIC POWER AND LIGHTING. 



1. — Copper Wire Table op the — 
(See Opposite Page for Areas in Square Mils.) 



Gages. 


Weight. 




Length. 




6m 


Op 

fqoQ 


5^ 


Area 
Circ. 
MUs. 


Lb. per 
Foot. 


Pounds per 


Ohm. 


Ft. 


Feet per Ohm. 




@ 20° C. 
= 68° F. 


@ 50° C. 
= 122°F. 


@ 80° C. 
= 176°F. 


@20° C. 
= 68°F. 


@ 50°C. 
=122°F 


@ 80°C. 
= 176°F. 


15 
16 


18 


.05707 
.05082 
.049 


3257 
2583 
2401 


.009858 
.007818 
.007268 


3.101 
1.950 
1.685 


2.776 
1.746 
1.509 


2.504 
1.575 
1.361 


101.4 
127.9 
137.6 


314.5 
249.4 
231.9 


281.5 
223.3 
207.6 


254.0 
201.5 
187.3 


17 

18 


19 


.04526 

.042 

.0403 


2048 
1764 
1624 


.006200 
.005340 
.004917 


1.226 
.9097 
.7713 


1.098 
.8143 
.6904 


.9906 
.7347 
.6230 


161.3 
187.3 
203.4 


197.8 
170.4 
156.9 


177.1 
152.5 
140.4 


159.8 
137.6 
126.7 


19 


20 
21 


.03589 

.035 

.032 


1288 
1225 
1024 


.003899 
.003708 
.003100 


.4851 
.4387 
.3066 


.4342 
.3927 
.2744 


.3918 
.3543 
.2476 


256.5 
269.7 
322.6 


124.4 
118.3 
98.90 


111.4 
105.9 
88.52 


100.5 
95.56 
79.88 


20 
21 


22 


.03196 
.02846 
.028 


1022 

810.1 

784.0 


.003092 
.002452 
.002373 


.3051 
.1919 
.1797 


.2731 
.1717 
.1608 


.2464 
.1550 
.1451 


323.4 
407.8 
421.4 


98.66 
78.24 
75.72 


88.31 
70.03 
67.78 


79.68 
63.19 
61.16 


22 
23 


23 


.02535 

.025 

.02257 


642.4 
625.0 
509.5 


.001945 
.001892 
.001542 


.1207 
.1142 
.07589 


.1080 
.1022 
.06793 


.09746 
.09224 
.06129 


514.2 
528.6 
648.4 


62.05 
60.36 
49.21 


55.54 
54.03 
44.04 


50.11 
48.75 
39.74 


24 


24 
25 


.022 

.0201 

.020 


484.0 
404.0 
400.0 


.001465 
.001223 
.001211 


.06849 
.04773 
.04678 


.06130 
.04272 
.04187 


.05531 
.03855 
.03778 


682.6 
817.6 
825.9 


46.75 
39.02 
38.63 


41.84 
34.93 
34.58 


37.75 
31.52 
31.20 


25 


26 

27 


.018 

.0179 

.016 


324.0 
320.4 
256.0 


.0009808 
.0009699 
.0007749 


.03069 
.03002 
.01916 


.02747 
.02687 
.01715 


.02479 
.02424 
.01548 


1020 
1031 
1290 


31.29 
30.95 
24.73 


28.01 
27.70 
22.13 


25.27 
24.99 
19.97 


26 
27 


28 


.01594 

.0142 

.014 


254.1 
201.5 
196.0 


.0007692 
.0006100 
.0005933 


.01888 
.01187 
.01123 


.01690 
.01063 
.01005 


.01525 

.009588 

.009071 


1300 
1639 
1685 


24.54 
19.46 
18.93 


21.97 
17.42 
16.94 


19.82 
15.72 
15.29 


28 


29 
30 


.013 

.01264 
.012 


169.0 
159.8 
144.0 


.0005116 
.0004837 
.0004359 


.008350 
.007466 
.006062 


.007474 
.006683 
.005426 


.006744 
.006030 
.004896 


1955 
2067 
2294 


16.32 
15.43 
13.91 


14.61 
13.82 
12.45 


13.18 
12.47 
11.23 


29 
30 


31 


.01126 
.01003 
.010 


126.7 
100.5 
100.0 


.0003836 
.0003042 
.0003027 


.004696 
.002953 
.002924 


.004203 
.002643 
.002617 


.003792 
.002385 
.002361 


2607 
3287 
3304 


12.24 
9.707 
9.658 


10.96 
8.688 
8.645 


9.886 
7.840 
7.800 


31 


32 
33 


.009 

.008928 
.008 


81.00 
79.70 
64.00 


.0002452 
.0002413 
.0001937 


.001918 
.001857 
.001197 


.001717 
.001662 
.001072 


.001549 
.001500 
.0009672 


4078 
4145 
5162 


7.823 
7.698 
6.181 


7.002 
6.890 
5.533 


6.318 
6.217 
4.992 


32 
33 


34 


.00795 
.00708 
.007 


63.21 
50.13 
49.00 


.0001913 
.0001517 
.0001483 


.001168 

.0007346 

.0007019 


.001045 

.0006575 

.0006283 


.0009436 
.0005933 
.0005669 


5227 
6591 
6742 


6.105 
4.841 
4.733 


5.464 
4.333 
4.236 


4.930 
3.910 
3.822 


34 
35 
36 


35 


.006305 
.005615 
.005 


39.75 
31.52 
25.00 


.0001203 

.00009543 

.00007568 


.0004620 
.0002905 
.0001827 


.0004135 
.0002601 
.0001636 


.0003731 
.0002347 
.0001476 


8311 
10480 
13210 


3.839 
3.045 
2.414 


3.436 
2.725 
2.161 


3.101 
2.459 
1.950 


37 
38 


36 


.004453 

.004 

.003965 


19.83 
16.00 
15.72 


.00006001 
.00004843 
.00004759 


.0001149 

.00007484 

.00007210 


.0001029 

.00006699 

.00006454 


.00009281 
. 00006045 
. 00005824 


16660 
20650 
21010 


1.915 
1.545 
1.519 


1.714 
1.383 
1.359 


1.547 
1.248 
1.226 


39 
40 




.003531 
.003145 


12.47 
9.888 


.00003774 
.00002993 


.00004545 
00002858 


. 00004068 
.00002559 


.00003671 
.00002309 


26500 
33410 


1.204 
.9550 


1.078 
.8548 


.9726 
.7713 



Note.— Sq. mils=circ. mils X 0.785398; circ. mils = sq. mils X 1.273240. 



COPPER WIRE TABLE, 



1391 



— Am. Inst, op Electrical Engineers.— Concluded. 
(See Opposite Page for Areas in Circular Mils.) 





Gages 




Resistance. 


doQ 


Oa3 
fQCQ 


Dlam. 
Tns. 


Area 
inSq. 
Mils.* 


Ohms per Lb. 


Ohms per Ft. 


^1 


@ 20°C. 
= 68°F. 


@ 50°C. 
= 1220F. 


@ 80°C. 
= 176°F. 


@ 20°C. 
= 68°F. 


@ 5U°C. 
= 122°F. 


@ 80°C. 
= 176''F. 


15 
16 


18 


.05707 
.05082 
.04900 


2558 
2029 
1886 


.3225 
.5128 
.5933 


.3603 
.5729 
.6629 


.3993 
.6349 
.7346 


.003179 
.004009 
.004312 


.003552 
.004479 
.004818 


.003936 
.004964 
.005339 


17 
18 


19 


.04526 
.04200 
.04030 


1609 
1385 
1276 


.8153 
1.099 
1.296 


.9109 
1.228 
1.448 


1.010 
1.361 
1.605 


.005055 
.005870 
.006374 


.005648 
.006558 
.007122 


.006259 
.007267 
.007892 


19 


20 
21 


.03589 
.03500 
.03200 


1012 
962.0 
804 2 


2.061 
2.279 
3.262 


2.303 
2.547 
3.644 


2.552 
2.822 
4.039 


.008038 
.008452 
.01011 


.008980 
.009443 
.01130 


.009952 

.01047 

.01252 


20 
21 


22 


.03196 
.02846 
.02.800 


802.0 
636.3 
615.8 


3.278 
5.212 
5.565 


3.662 
5.823 
6.217 


4.058 
6.453 
6.890 


.01014 
.01278 
.01321 


.01132 
.01428 
.01475 


.01255 
.01583 
.01635 


22 
23 


23 


.02535 

.0250 

.02257 


504.6 
490.9 
400.2 


8.287 
8.756 
13,18 


9.259 
9.783 
14.72 


10.26 
10.84 
16.32 


.01612 
.01657 
.02032 


.01801 
.01851 
.02271 


.01996 
.02051 
.02516 


24 


24 
25 


.0220 

.02010 

.0200 


380.1 
317.3 
314.2 


14.60 
20 95 
21.38 


16.31 
23.41 
23.88 


18.08 
25.94 
26.47 


.02139 
.02563 
.02588 


.02390 
.02863 
.02892 


.02649 
.03173 
.03205 


25 


26 
27 


.018 

.0179 

.016 


254.5 
251.7 
201.1 


32.58 
33.32 
52.19 


36.40 
37.22 
58.31 


40.34 
41.25 
64.62 


.03196 
.03231 
.04045 


.03570 
.03610 
.04519 


.03957 
.04001 
.05008 


26 

27 


28 


.01594 

.0142 

.014 


199.6 
158.3 
153.9 


52.97 
84.23 
89.04 


59.18 
94.11 
99.48 


65.59 
104.3 
110.2 


.04075 
.05138 
.05283 


.04552 
.05740 
.05902 


.05045 
.06362 
.06541 


28 


29 
30 


.013 

.01264 

.012 


132.7 
125.5 
113.1 


119.8 
133.9 
165.0 


133.8 
149.6 
184.3 


148.3 
165.8 
204.2 


.06127 
.06479 
.0719 


.06845 
.07239 
.08033 


.07586 
.08022 
.08903 


29 
30 


31 


.01126 
.01003 
.010 


99.53 
78.94 
78.54 


213.0 
338.6 
342.0 


237.9 
378.3 
382,1 


263.7 
419.3 
423.5 


.0817 
.1030 
.1035 


.09128 

.1151 

.1157 


.1012 
.1276 
.1282 


31 


32 
33 


.009 

.008928 

.008 


63.62 
62.60 
50.27 


521.3 
538.4 
835.1 


582.5 
601.6 
933.0 


645.5 
666 7 
1034. 


.1278 
.1299 
.1618 


.1428 
.1451 
.1807 


.1583 
.1608 
.2003 


32 
33 


34 


.00795 
.00708 
.007 


49.64 
39.37 
38.48 


856.2 
1361. 
1425. 


956.5 
1521. 
1592. 


1060. 
1685. 
1764. 


.1638 
.2066 
.2113 


.1830 
.2308 
.2361 


.2028 
.2558 
.2616 


34 

35 
36 


35 


.006305 
.005615 
.005 


31.22 
24.76 
19.64 


2165. 
3441. 
5473. 


2418. 
3845. 
6114. 


2680. 
4262. 
6776. 


.2605 
.3284 
.4142 


.2910 
.3669 
.4627 


.3225 
.4067 
.5129 


37 
38 


36 


.004453 

.004 

.003965 


15.57 
12.57 
12.35 


8702. 
13360. 
13870. 


9722. 
14930. 
15490. 


10770. 
16540. 
17170. 


.5222 
.6471 
.6585 


.5835 
.7230 
.7357 


.6466 
.8011 
.8154 


39 
40 




.003531 
.003145 


9.79 

7.77 


22000. 
34980. 


24580. 
39080. 


27240. 
43320. 


.8304 
1.047 


.9277 
1.170 


1.028 
1.296 



*To find area in square inches, divide by 1,000,000. 



1392 7Q,— ELECTRIC POWER AND LIGHTING, 

and impedance. The former may be assumed to be no greater than 
the "drop" in volts on the line (and from that down to about one-third of the 
amount), while the latter is usually so small as to be practically negligible, 
say 5 per cent of the inductance. Hence from 5 to 10 per cent may be 
added for losses in alternate-current circuits over those for continuous; 
in other words the areas of the wires may be increased by this amount. 
(3) In a single-phase two-wire circuit the size of wires is the same as for 
continuous current plus 5 to 10 per cent for inductance. (4) In a two-phase 
four-wire circuit the area of each wire is one-half that for the continuous 
current circuit plus 5 to 10 per cent for inductance; that is, the same weight 
of wire is required as for the single-phase two-wire alternating circuit. (5) In 
the three-phase three-wire circuit the area of each wire is one-half that 
for the continuous current circuit plus 5 to 10 per cent for inductance; that 
is, only three-fourths the weight of wire is required as for the 1-phase 2-wire, 
and 2-phase 4-wire, circuits. 

Problem 2. — In solving Problem 1 we find that the size of copper wire 
required is about 110,000 circular mills in area, corresponding to about 
No. B, & S. gage. Now, from the foregoing discussion, what sizes of 
wire would probably be installed for the three types of alternating circuits ? 
And what would be the theoretic relative weights of total amount of copper 
in the four systems? 

Continuous current 2-wire; each 110,000 circ. mills; total weight, 1.00 
AU..^o-h4^rr { 1-phase 2-wire; " 120,000 " " " 1.09 

Al4.^f^^ 2-phase 4-wire; " 60,000 " " " 1.09 

Current / s.phase 3-wire; " 60,000 " " " 0.82 

The 3-phase 3-wire system is the most commonly, used. 



1393 



'NATIONAL ELECTRIC CODE." 



Rules and Requirements of the National Board of Fire Underwriters 

for the Installation of Wiring and Apparatus (for Light, Heat 

and Power) as Recommended by the Underwriters' 

National Electric Association. 

Edition of 1907. 

The National Electric Code (originally drawn in 1897) is the result of 

the united efforts of the various electrical, architectural, insurance and 
other allied interests which, through the National Conference on Standard 
Electrical Rules, composed of delegates from various national associations, 
unanimously voted to recommend it to their respective associations for 
approval or adoption. 

The following is a list of the associations composing the National Con- 
ference on Standard Electrical Rules: — American Institute of Architects, 
American Institute of Electrical Engineers, American Society of Mechanical 
Engineers, American Institute of Mining Engineers, American Street and 
Interurban Railway Association, Associated Factory Mutual Fire Ins. Co's., 
Association of Edison Illuminating Companies, International Association of 
Municipal Electricians, National Board of Fire Underwriters, National 
Electric Light Association, National Electric Contractors' Association, 
National Electric Inspectors' Association, Underwriters' National Electric 
Association. 

GENERAL PLAN GOVERNING THE ARRANGEMENT OF RULES.* 

Class A. — Stations and Dynamo Rooms. Includes Central Stations; Dy- 
namo, Motor, and Storage-Battery Rooms; Transformer Sub- 
stations, etc. Rules 1 to 11. 

Class B. — Outside Work, all systems and voltages. Rules 12 to 13A, 
Class C. — Inside Work. Rules 14 to 39. Subdivided as follows: 

General Rules, all systems and voltages. Rules 14 to 17. 
Constant-Current Systems. Rules 18 to 20. 
Constant-Potential Systems: — 
General Rules, all voltages. Rules 21 to 23. 
Low-Potential Systems, 550 volts or less. Rules 24 to 34. 
High-Potential Systems, 550 to 3500 volts. Rules 35 to 37. 
Extra-High-Potential Systems, over 3500 volts. Rules 38 to 39. 
Class D. — Fittings, Materials, and Details of Construction, all systems and 

voltages. Rules 40 to 63. 
Class E. — Miscellaneous. Rules 64 to 67. 
Class F.— Marine Work. Rules 68 to 83. 



' General Suggestions. — In all electric work, conductors, however well 
insulated, should always be treated as bare to the end that under no condi- 
tions, existing or likely to exist, can a ground or short circuit occur, and so 
that all leakage from conductor to conductor, or between conductor and 
ground, may be reduced to a minimum. 

In all wiring, special attention must be paid to the mechanical execution 
of the work. Careful and neat running, connecting, soldering, taping of 
conductors, and securing and attaching of fittings, are specially conducive 
to security and efficiency, and will be strongly insisted on. 

In laying out an installation, except for constant current systems, every 
reasonable effort should be made to secure distribution centers located in 
easily accessible places, at which points the cut-outs and switches controlling 
the several branch circuits can be grouped for convenience and safety of 
operation. The load should be divided as evenly as possible among the 
branches, and all complicated and unnecessary wiring avoided. 

The use of wire-ways for rendering concealed wiring permanently acces- 
sible is most heartily endorsed and recommended; and this method of 
accessible concealed construction is advised for general use. 

Architects are urged, when drawing plans and specifications, to make 
provision for the channeling and pocketing of buildings for electric light or 
power wires, and also for telephone, district messenger and other signaling 
system wiring. 



1394 70.— ELECTRIC POWER AND LIGHTING, 

Class A.— STATIONS AND DYNAMO ROOMS. 

Includes Central Stations, Dynamo, Motor and Storage-Battery 
Rooms, Transformer, Sub-stations, Etc. 

1. Generators. — a. Must be located in a dry place. 

It Is recommended that water-proof covers be provided, which may be used In 
case of emergency. 

b. Must never be placed in a room where any hazardous process is 
carried on, nor in places where they would be exposed to inflammable gases 
or flyings of combustible materials. 

c. Must, when operating at a potential in excess of 550 volts, have their 
base frames permanently and effectively grounded. 

Must, when operating at a potential of 550 volts or less, be thoroughly 
insulated from the ground wherever feasible. Wooden base frames used for 
this purpose, and wooden floors which are depended upon for insulation 
where, for any reason, it is necessary to omit the base frames, n;ust be kept 
filled to prevent absorption" of moisture, and must be kept clean and dry. 

Where frame insulation is impracticable, the Inspection Department 
having jurisdiction may, in writing, permit its omission, in which case the 
frame must be permanently and effectively grounded. 

A high potential machine should be surrounded by an insulated platform. This 
may be made of wood, mounted on insulating supports, and so arranged that a man 
must always stand upon It in order to touch any part of the machine. 

In case of a machine having an insulated frame, if there is trouble from static 
electricity due to belt friction, it should be overcome by placing near the belt a 
metallic comb connected with the earth, or by grounding the frame through a re- 
sistance of not less than 300,000 ohms. 

d. Constant potential generators, except alternating current machines 
and their exciters, must be protected from excessive current by safety fuses 
or equivalent devices of approved design. 

For two-wire, direct-current generators, single pole protection will be considered 
as satisfying the above rule, provided the safety device is located in the lead not 
connected to the series winding. When supplying three-wire systems, the genera- 
tors should be so arranged that these protective devices will come In the outside 
leads. 

For three-wire, direct-current generators, a safety device must be placed In each 
armature, direct-current lead, or a double pole, double trip circuit breaker In each 
outside generator lead and corresponding equalizer connection. 

In general, generators should preferably have no exposed live parts and the 
leads should be well insulated and thoroughly protected against mechanical Injury. 
This protection of the bare live parts against accidental contact would apply also to 
any exposed, uninsulated conductors outside of the generator and not on the switch- 
board unless their potential is practically that of the ground. 

Where the needs of the service make the above requirements Impracticable, the 
Inspection Department having jurisdiction may. In writing, modify them. 

e. Must each be provided with a name-plate, giving the maker's name, 
the capacity in volts and amperes, and the normal speed in revolutions per 
minute. 

f. Terminal blocks when used on generators must be made of approved 
non-combustible, non-absorptive, insulating material, such as slate, marble 
or porcelain. 

2. Conductors. — From generators to switchboards, rheostats or other 
instruments, arid thence to oustide lines: — 

a. Must be in plain sight or readily accessible. 

Wires from generator to switchboard may, however, be placed in a conduit In 
the brick or cement pier on which the generator stands, provided that proper pre- 
cautions are taken to protect them against moisture and to thoroughly insulate them 
from the pier. If lead-covered cable is used, no further protection will be required, 
but it should not be allowed to rest upon sharp edges which in time might cut into 
the lead sheath, especially if the cables were liable to vibration. A smooth runway 
Is desired. If iron conduit is provided, double braided rubber-covered wire (see 
No. 47) will be satisfactory. 

b. Must have an approved insulating covering as called for by rules in 
Class "C" for similar work, except that in central stations, on exposed circuits, 
the wire which is used must have a heavy braided, non-combustible outer 
covering. 

Bus bars may be made of bare metal. 

Rubber Insulations Ignite easily and burn freely. Where a number of wires are 
brought close together, as Is generally the case in dynamo rooms, especially about the 



STATIONS AND DYNAMO ROOMS, 1395 

switchboard- tt Is therefore necessary to surround this Inflammable material with a 
tight, non-combustible outer cover. If this is not done, a fire once started at this 
point would spread rapidly a'ong the wires, producing intense heat and a dense 
smoke Where the wires have such a covering and are well insulated and supported, 
using only non-combustible materials, it is believed that no appreciable fire hazard 
exists, even with a large group of wires. 

Flame proofing should be stripped back on all cables a sufficient amount to give 
the necessary insulation distances for the voltage of the circuit on which the cable 
is used. The stripping back oi the flame proofing is necessary on account of the poor 
Insulating qualities of the flame proofing material now available. Flame proofing 
may be omitted where satisfactory fire-proofing is accomplished by other means, 
such as compartments, etc. 

c. Must be kept so rigidly in place that they cannot come in contact. 

d. Must in all other respects be installed with the same precautions as 
required by rules in Class "C" for wires carrying a current of the same 
volume and potential. 

e. In wiring switchboards, the ground detector, voltmeter, pilot lights 
and potential transformers must be connected to a circuit of not less than 
No. 14 B. & S. gage wire that is protected by an approved fuse, this circuit 
is not to carry over 660 watts. 

For the protection of Instruments and pilot lights on switchboards, approved 
N. E. Code Standard Enclosed Fuses are preferred, but approved enclosed fuses of 
other designs of not over two (2) amperes capacity, may be used. 

Voltmeter switches having concealed connections must be plainly marked, 
showing connections made. 

3 Switchboards. — a. Must be so placed as to reduce to a minimum 
the danger of communicating fire to adjacent combustible material. 

Special attention is called to the fact that switchboards should not be built down 
to the floor, nor up to the ceiling. A space of at least ten or twelve inches should be 
left between the floor and the board, except when the floor about the switchboard Is 
of concrete or other fifeproof construction, and a space of three feet, if possible, 
between the ceiling and the board, in order to prevent fire from communicating from 
the switchboard to the floor or ceiling, and also to prevent the forming of a partially 
concealed space very liable to be used for storage of rubbish and oily waste. 

b. Must be made of non-combustible material or of hardwood in skele- 
ton form, filled to prevent absorption of moisture. 

If wood is used all wires and all current carrying parts of the apparatus on the 
switchboard must be separated therefrom by non-combustible, non-absorptive In- 
sulating material. 

c. Must be accessible from all sides when the connections are on the back, 
but may be placed against a brick or stone wall when the wiring is entirely 
on the face. 

If the wiring Is on the back, there should be a clear space of at least eighteen 
inches between the wall and the apparatus on the board, and even If the wiring is 
entirely on the face, it is much better to have the board set out from the wall. The 
space back of the board should not be closed in, except by grating or netting either 
at the sides, top or bottom, as such an enclosure is almost sure to be used as a closet 
for clothing or for the storage of oil cans, rubbish, etc. An open space is much more 
likely to be kept clean, and is more convenient for making repairs, examinations, etc. 

d. Must be kept free from moisture. 

e. On switchboards the distances between bare live parts of opposite 
polarity must be made as great as practicable, and must not be less than those 
given for tablet-boards (see No. 53 A). 

4. Resistance Boxes and Equalizers. — (For construction rules, see No. 
60.) a. Must be placed on a switchboard, or if not thereon, at a distance of 
at least one foot from combustible material, or separated therefrom by a 
non-combustible, non-absortive insulating material such as slate or marble. 

This will require the use of a slab or panel of non-combustible, non-absorptive 
Insulating material such as slate or marble, somewhat larger than the rheostat, which 
shall be secured in position independently of the rheostat supports. Bolts for sup- 

Eorting the rheostat shall be countersunk at least i Inch below the surface at the 
ack of the slab and filled. For proper mechanical strength, slab should be of a 
thickness consistent with the size and weight of the rheostat, and In no case to be 
less than i inch 

If resistance devices are Installed In rooms where dust or combustible flyings 
woula be liable to accumulate on them, they should be equipped w th a dust-proof 
face plate. 

b. Where protective resistances are necessary in connection with auto- 
matic rheostats, incandescent lamps may be used, provided that they do not 



1396 IQ.-'ELECTRIC POWER AND LIGHTING, 

carry or control the main current nor constitute the regulating resistance of 
the device. 

When so used, lamps must be mounted in porcelain receptables upon non- 
combustible supports, and must be so arranged that they cannot have im- 
pressed upon them a voltage greater than that for which they are rated. 
They must in all cases be provided with a name-plate, which shall be perma- 
nently attached beside the porcelain receptacle or receptacles and stamped 
with the candle-power and voltage of the lamp or lamps to be used in each 
receptacle. 

c. Wherever insulated wire is used for connection between resistances 
and the contact plate of a rheostat, the insulation must be slow burning (see 
No. 43). For large field rheostats and similar resistances, where thecontact 
plates are not mounted upon them, the connecting wires may be run together 
in groups so arranged that the maximum difference of potential between any 
two wires in a group shall not exceed 75 volts. Each group of wires must 
either be mounted on non-combustible, non-absorptive insulators giving at 
least i inch separation from surface wired over, or, where it is necessary to 
protect the wires from mechanical injury or moisture, be run in approved 
lined conduit or equivalent. 

5. Lightning Arresters. — (For construction rules, see No. 63.) 

a. Must be attached to each wire of every overhead circuit connected 
with the station. 

It is recommended to all electric light and power companies that arresters be 
connected at intervals over systems in such numbers and so located as to prevent 
ordinary discharges entering (over the wires) buildings connected to the lines. 

b. ^ Must be located in readily accessible places away from combustible 
materials, and as near as practicable to the point where the wires enter the 
building. 

In all cases, kinks, coils and sharp bends in the wires between the arresters 
and the outdoor lines must be avoided as far as possible. 

The switchboard does not necessarily afford the only location meeting these 
requirements. In fact, if the arresters can be located in a safe and accessible place 
away from the board, this should be done, for, in case the arrester should fail or be 
seriously damaged there would then be less chance of starting arcs on the board. 

c. Must be connected with a thoroughly good and permanent ground 
connection by metallic strips or wires having a conductivity not less than 
that of a No. 6 B. & S. gage copper wire, which must be run as nearly in a 
straight line as possible from the arresters to the ground connection. 

(jround wires for lightning arresters must not be attached to gas pipes 
within the buildings. 

It Is often desirable to introduce a choke coil in circuit between the arresters and 
the dynamo. In no case should the ground wires from lightning arresters be put 
into iron pipes, as these would tend to impede the discharge. 

d. All choke coils or other attachments, inherent to the lightning pro- 
tection equipment, shall have an insulation from the ground or other con- 
ductors equal at least to the insulation demanded at other points of the cir- 
cuit in the station. 

6. Care and Attendance. — a. A competent man must be kept on duty 
where generators are operating, b. Oily waste must be kept in approved 
metal cans and removed daily. 

Approved waste cans shall be made of metal, with legs raising can three Inches 
from the floor, and with self-closing covers. 

7. Testing and Insulation Resistance. — a. All circuits except such as 
are permanently grounded in accordance with No. 13 A must be provided 
with reliable ground detectors. Detectors which indicate continuously and 
give an instant and permanent indication of a ground are preferable. Ground 
wires from detectors must not be attached to gas pipes within the building. 

b. Where continuously indicating detectors are not feasible, the circuits 
should be tested at least once a day, and preferably oftener. 

c. Data obtained from all tests must be preserved for examination by 
the Inspection Department having jurisdiction. 

These rules on testing to be applied at such places as may be designated by the 
Inspection Department having jurisdiction. 



STATIONS AND DYNAMO ROOMS, 1397 

8. Motors. — The use of motors operating at a potential In excess of 550 volts 
will only be approved when every practicable safeguard has been provided. Plans 
for such installations should be submitted to the Inspection Department having 
jurisdiction before any work is begun 

a. Must, when operating at a potential in excess of 550 volts, have no 
exposed live metal parts, and have their base frames permanently and effect- 
ively grounded. 

Meters operating at a potential of 550 volts or less must be thoroughly 
insulated from the ground wherever feasible Wooden base frames used for 
this purpose, and wooden floors, which are depended upon for insulation 
where., for any reason, it is necessary to omit the base frames, must be kept 
filled to prevent absorption of moisture, and must be kept clean and dry. 
Where frame insulation is impracticable, the Inspection Department having 
jurisdiction may, in writing, permit its omission, in which case the frame 
must be permanently and effectively grounded. 

A high-potential machine should be surrounded with an insulated platform. 
This may be made of wood, mounted on insulatine: supports, and so arranged that a 
man must stand upon it in order to touch any part of the machine. 

In case of a machine having an insulated frame, if there is trouble from static 
electricity due to belt friction, it should be overcome by placing near the belt a 
metallic comb connected to the earth, or by grounding the frame through a resistance 
of not less than 300.000 ohms. 

b. Motors operating at a potential of 550 volts or less must be wired 
with the same precautions as required by rules in Class "C" for wires carry- 
ing a current of the same volume. 

Motors operating at a potential between 550 and 3, 500 volts must be 
wired with appro ved_ multiple conductor, metal sheathed cable in approved 
unlined metal conduit firmly secured in place. The metal sheath must be 
permanently and effectively grounded, and the construction and installation 
of the conduit must conform to rules for interior conduits (see No. 25 
and No. 49 a, j, and k), except that at outlets approved outlet bushings 
shall be used. 

The motor leads or branch circuits must be designed to carry a current at least 
25 per cent greater than that for which the motor is rated, in order to provide for 
the inevitable occasional overloading of the motor and the increased current required 
In starting, without overfusing the wires; but where the wires under this rule would 
be overfused, in order to provide for the starting current, as in the case of many of the 
alternating current motors, the wires must be of such size as to be properly protected 
by these larger fuses. 

The insulation of the several conductors for high potential motors, where leaving 
the metal sheath at outlets, must be thoroughly protected from moisture and me- 
chanical injury. This may beaccomplished by means of a pot head or some equivalent 
method. The conduit must be substantially bonded to the metal casings of all 
fittings and apparatus connected to the inside high tension circuit. It would be 
much preferable to make the conduit system continuous throughout by connecting 
the conduit to fittings and motors by means of screw joints, and this construction is 
strongly recommended wherever practicable. 

High potential motors should preferably be so located that the amount of Inside 
wiring will be reduced to a minimum. Inspection Department having jurisdiction 
may permit the wire for high potential motors to be Installed according to the general 
rules for high potential systems when the outside wires directly enter a motor room 
(see Section /). Under these conditions there would generally be but a few feet of 
wire inside the building and none outside the motor room. 

c. Each motor and resistance box must be protected by a cut-out and 
controlled by a switch (see No. 17a), said switch plainly indicating whether 
*'on'' or "off." With motors of one-fourth horse power or less, on circuits 
where the voltage does not exceed 300, No. 21 d must be complied with, and 
single pole switches may be used as allowed in No. 22 c. The switch and 
rheostat must be located within sight of the motor, except in cases where 
special permission to locate them elsewhere is given, in writing, by the In- 
spection Department having jurisdiction. 

The use of circuit-breakers with motors Is recommended, and may be required 
by the Inspection Department having jurisdiction. 

Where the circuit-breaking device on the motor-starting rheostat disconnects 
all wires of the circuit, the switch called for in this section may be omitted. 

Overload-release devices on motor-starting rheostats will not be considered to 
take the place of the cut-out required by this section if they are Inoperative during 
the starting of the motor 

The switch is necessary for entirely disconnecting the motor when not In use, and 
the cut-out to protect the motor from excessive currents due to accidents or careless 
handling when starting. An automatic circuit-breaker disconnecting all wires of 
he circuit may. however, serve as both switch and cut-out. 

Id general, motors should preferably have no exposed live parts. 



1308 70.— ELECTRIC POWER AND LIGHTING. 

d. Rheostats must be so installed as to comply with all the require- 
ments of No. 4. Auto starters must comply with requirements of No. 4 c. 

Starting rheostats and auto starters, unless equipped with tight casings enclosing 
all current-carrying parts, should be treated about tlie same as knite switches, and 
In all wet, dusty or linty places, should be enclosed in dust-tight, fireproof cabinets. 
If a special motor room is provided, the starting apparatus and safety devices should 
be included within it. Where there is any liability of short circuits across their ex- 
posed live parts being caused by accidental contacts, they should either be enclosed 
In cabinets, or else a railing should be erected around them to keep unauthorized 
persons away from their Immediate vicinity. 

e. Must not be run in series-multiple or multiple-series, except on con- 
stant- potential systems, and then only by special permission of the Inspec- 
tion Department having jurisdiction. 

f. Must be covered with a waterproof cover when not in use, and, if 
deemed necessary by the Inspection Department having jurisdiction, must 
be enclosed in an approved case. 

When it is necessary to locate a motor in the vicinity of combustibles or in wet 
or very dusty or dirty places, it is generally advisable to enclose it as above. Such 
enclosures should be readily accessible, dust proof and sufficiently ventilated to 
prevent an excessive rise of temperature. The sides should preferably be made 
largely of glass, so that the motor may be always plainly visible This lessens the 
chance of Its being neglected, and allows any derangement to be at once noticed. 

The use of enclosed type motor is recommended in dusty places, being preferable 
to wooden boxing. From the nature of the question the decision as to details of 
construction must be left to the Inspection Department having jurisdiction to deter- 
mine In each instance. 

g. Must, when combined with ceiling fans, be hung from insulated hooks, 
or else there must be an insulator interposed between the motor and its 
support. 

h. Must each be provided with a name-plate, giving the makers* name, 
the capacity in volts and amperes, and the normal speed in revolutions per 
minute. 

i. Terminal blocks when used on motors must be made of approved non- 
combustible, non-absorptive, insulating material such as slate, marble or 
porcelain. 

j. Variable speed motors, unless of special and appropriate design, if 
controlled by means of field regulation, must be so arranged and connected 
that they cannot be started under weakened field. 

9. Railway Power Plants. — a. Each feed wire before it leaves the 
station must be equipped with an approved automatic circuit -breaker (see 
No. 62) or other device, which will immediately cut off the current in case 
of an accidental ground. This device must be mounted on a fireproof 
base, and in full view and reach of the attendant. 

10. Storage or Primary Batteries. — a. When current for light and power 
is taken from primary or secondary batteries, the same general regulations 
must be observed as apply to similar apparatus fed from dynamo generators 
delevoping the same difference of potential. 

b. Storage battery rooms must be thoroughly ventilated. 

c. Special attention is directed to the rules for wiring in rooms where 
acid fumes exist (see No. 24, i and j). 

d. All secondary batteries must be mounted on non-absorptive, non- 
combustible insulators, such as glass or thoroughly vitrified and glazed porce- 
lain. 

e. The use of any metal liable to corrosion must be avoided in cell con- 
nections of secondary batteries. 

11. Transformers. — (For construction rules, see No. 62.) (See also 
Nos. 13, 13A, 36.) a. In central or sub-stations the transformers must be 
so placed that smoke from the burning out of the coils or the boiling over 
of the oil (where oil filled cases are used) could do no harm. 

If the Insulation in a transformer breaks down, considerable heat Is likely to be 
developed. This would cause a dense smoke, which might be mistaken for a fire and 
result in water being thrown into the building, and a heavy loss thereby entailed. 
Moreover, with oil-cooled transformers, especially if the cases are filled too full, the 
oil may become Ignited and boil over, producing a very stubborn Are. 



OUTSIDE WORK— ALL SYSTEMS AND VOLTAGES, 1399 

b. In central or sub-stations, casings of all transformers must be per- 
manently and effectively grounded. 

Transformers used exclusively to supply current to switchboard Instruments 
need not be grounded, provided they are thoroughly insulated. 

Class B.— OUTSIDE WORK. 

(Light, Power and Heat. For Signaling Systems, See Class R.) 
ALL SYSTEMS AND VOLTAGES. 

12. Wires. — a. Line wires must have an approved weatherproof or 
rubber insulating covering (see No. 44 and No. 41). That portion of the 
service wires between the main cut-out and switch and the first support from 
the cut-out or switch on outside of the building must have an approved 
rubber insulating covering (see No. 41), but from the above-mentioned sup- 
port to the line, may have an approved weatherproof insulating covering 
(see No. 44), if kept free from awnings, swinging signs, shutters, etc. 

b. Must be so placed that moisture cannot form a cross connection be- 
tween them, not less than a foot apart, and not in contact with any substance 
other than their insulating supports. Wooden blocks to which insulators are 
attached must be covered over their entire surface with at least two coats of 
waterproof paint. 

c. Must be at least seven feet above the highest point of flat roofs, and 
at least one foot above the ridge of pitched roofs, over which they pass or to 
which they are attached. 

Roof structures are frequently found which are too low or much too light for the 
work, or which have been carelessly put up. A structure which is to hold the wires 
a proper distance above the roof in all kinds of weather must not only be of sufficient 
height, but must be substantially constructed of strong material. 

d. Must be protected by dead insulated guard irons or wires from pos- 
sibility of contact with other conducting wires or substances to which cur- 
rent may leak. Special precautions of this kind must be taken where sharp 
angles occur, or where any wires might possibly come in contact with electric 
light or power wires. 

Crosses, when ima voidable, should be made as nearly at right angles as possible. 

e. Must be provided with petticoat insulators of glass or porcelain. 
Porcelain knobs or cleats and rubber hooks will not be approved. 

f. Must be so spliced or joined as to be both mechanically and elec- 
trically secure without solder. The joints must then be soldered, to insure 
preservation, and covered with an insulation equal to that on the conductors. 

All joints must be soldered, unless made with some form of approved splicing 
device. This ruling applies to joints and splices In all classes of wiring covered by 
these rules. 

g. Must, where they enter buildings, have drip loops outside, and the 
holes through which the conductors pass must be bushed with non-com- 
bustible, non-absorptive, insulating tubes slanting upward toward the 
inside. 

For low potential systems the service wires may be brought into buildings 
through a single iron conduit. The conduit to be curved downward at its outer end 
and carefully sealed or equipped with an approved service-head to prevent the 
entrance of moisture. The outer end must be at least one foot from any wood-work 
and the inner end must extend to the service cut-out, and if a cabinet is required by 
the Code must enter the cabinet in a manner similar to that described in fine print 
note under No. 25&. 

h. Electric light and power wires must not be placed on the same cross- 
arm with telegraph, telephone or similar wires, and when placed on the 
same pole with such wires the distance between the two inside pins of each 
cross-arm must not be less than twenty-six inches. 

i. The metallic sheaths to cables must be permanently and effectively 
connected to "earth." 

Trolley Wires. — j. Must not be smaller than No.O B. & S. gage copper 
or No. 4 B. & S. gage silicon bronze, and must readily stand the strain upon 
them when in use. 

k. Must have a double insulation from the ground. In wooden pole 
construction the pole will be considered as one insulation. 



1400 10.— ELECTRIC POWER AND LIGHTING. 

I. Must be capable of being disconnected at the power plant, or of being 
divided into sections, so that, in case of fire on the railway route, the ciirrent 
may be shut off from that particular section and not interfere with the work 
of the firemen. This rule also applies to feeders. 

m. Must be safely protected against accidental contact where crossed 
by other conductors. 

Guard wires should be insulated from the ground and should be electrically 
disconnected in sections of not more than 300 feet in length. 

Ground Return Wires. — n. For the diminution of electrolytic corrosion of 
underground metal work, ground return wires must be so arranged that 
the difference of potential between the grounded dynamo terminal and any 
point on the return circuit will not exceed twenty-five volts. 

It is suggested that the positive pole of the dynamo be connected to the trolley 
line, and that whenever pipes or other underground metal work are found to be 
electrically positive to the rails or surrounding earth, that they be connected by 
conductors arranged so as to prevent as far as possible current flow from the pipes 
into the ground. 

12 A. Constant -Potential Pole Lines, Over 5,000 Volts. — (Over- 
head lines of this class unless properly arranged may increase the fire 
loss from the following causes: — Accidental crosses between such lines and 
low-potential lines may allow the high- voltage current to enter buildings over 
a large section of adjoining country. Moreover, such high-voltage lines, if 
carried close to buildings, hamper the work of firemen in case of fire in the 
building. The object of these rules is so to direct this class of construction 
that no increase in fire hazard will result, while at the same time care has been 
taken to avoid restrictions which would unreasonably impede progress in 
electrical development. 

It is fully understood that it is impossible to frame rules which will 
cover all conceivable cases that may arise in construction work of such an 
extended and varied nature, and it is advised that the Inspection Department 
having jurisdiction be freely consulted as to any modification of the rules in 
particular cases.) 

a. Every reasonable precaution must be taken in arranging routes so as 
to avoid exposure to contacts with other electric circuits. On existing lines, 
where there is a liability to contact, the route should be changed by mutual 
agreement between the parties in interest wherever possible. 

b. Such lines should not approach other pole lines.nearer than a distance 
equal to the height of the taller pole line, and such lines should not be on the 
same poles with other wires, except that signaling wires used by the Company 
operating the high-pressure system, and which do not enter property other 
than that owned or occupied by such Company, may be carried over the 
same poles. 

c. Where such lines must necessarily be carried nearer to other pole lines 
than is specified in Section h above, or where they must necessarily be carried 
on the same poles with other wires, extra precautions to reduce the liability 
of a breakdown to a minimum must be taken, such as the use of wires of ample 
mechanical strength, widely spaced cross-arms, short spans, double or extra 
heavy cross-arms, extra heavy pins, insulators, and poles thoroughly sup- 
ported. If carried on the same poles with other wires, the high-pressure 
wires must be carried at least three feet above the other wires. 

d. Where such lines cross other lines, the poles of both lines must be of 
heavy and substantial construction. 

Whenever it is feasible, end-insulator guards should be placed on the 
cross-arms of the upper line. If the high-pressure wires cross below the other 
lines, the wires of the upper line should be dead-ended at each end of the span 
to double-grooved, or to standard transposition insulators, and the line com- 
pleted by loops. 

One of the following forms of construction must then be adopted: — 
1. The height and length of the cross-over span may be made such that 
the shortest distance between the lower cross-arms of the upper line 
and any wire of the lower line will be greater than the length of 
the cross-over span, so that a wire breaking near one of the upper 
pins would not be long enough to reach any wire of the lower 
line. The high-pressure wires should preferably be above the 
other wires. 



OUTSIDE WORK— ALL SYSTEMS AND VOLTAGES. 1401 

2. A joint pole may be erected at the crossing point, the high-pressure 
wires being supported on this pole at least three feet above the other 
wires. Mechanical guards or supports must then be provided, so that 
in case of the breaking of any upper wire, it will be impossible for it 
to come into contact with any of the lower wires. 

Such liability of contact may be prevented by the use of suspension wires, 
similar to those employed for suspending aerial telephone cables, which will 
prevent the high-pressure wires from falling, In case they break. The sus- 
pension wires should be supported on high-potential Insulators, should have 
ample mechanical strength, and should be carried over the high-pressure 
wires for one span on each side of the joint pole, or where suspension wires 
are not desired guard wires may be carried above and below the lower wires 
for one span on each side of the joint pole, and so spread that a falling high- 
pressure wire would be held out of contact with the lower wires. 

Such guard wires should be supported on high-potential Insulators or should be 
grounded. When grounded, they must be of such size, and so connected 
and earthed, that they can surely carry to ground any current which may 
be delivered by any of the high-pressure wires. Further, the construction 
must be such that the guard wires will not be destroyed by any arcing at the 
point of contact likely to occur under the conditions existing. 

3. Whenever neither of the above methods is feasible, a screen of wire 
should be interposed between the lines at the cross-over. This 
screen should be supported on high tension insulators or grounded, 
and should be of such construction and strength as to prevent the 
upper wires from coming into contact with the lower ones. 

If the screen is grounded, each wire of the screen must be of such size and so 
connected and earthed that it can surely carry to ground any current which 
may be delivered by any of the high-pressure wires. Further, the construc- 
tion must be such that the wires of screen will not be destroyed by any arcing 
at the point of contact likely to occur under the conditions existing. 

e. When it is necessary to carry such lines near buildings, they must be 
at such height and distance from the building as not to interfere with firemen 
in event of fire; therefore, if within 25 feet of a building, they must be carried 
at a height not less than that of the front cornice, and the height must be 
greater than that of the cornice, as the wires come nearer to the building in 
accordance with the following table: — 

Distance of wire ?te.^l*^°^ °/ ^^S^ 



T^' 4.^ ^r. ^f -rrri^^ Elcvatiott of wirc 

^f^^^uJ^i^^ above cornice of 

from buildmg. building. 

Feet. Feet. 

25 

20 2 

15 4 



from building. ^^^^f^S^^^- °^ 

Feet. • Feet. 

10 6 

5 8 

2i 9 



It Is evident that where the roof of the building continues nearly in line with the 
walls, as In Mansard roofs, the height and distance of the line must be reckoned from 
some part of the roof Instead of from the cornice. 

13. Transformers. — (For construction rules, see No. 62.) (See also 
Nos. 11, 13 A and 36.) [Where transformers are to be connected to high- 
voltage circuits, it is necessary in many cases, for best protection to life and 
property, that the secondary system be permanently grounded, and pro- 
vision should be made for it when the transformers are built.] 

a. Must not be placed inside of any building, excepting central stations 
and sub-stations, unless by special permission of the Inspection Department 
having jurisdiction. 

An outsidelocatlon Is always preferable; first, because It keeps the high-voltage 
primary wires entirely out of the building, and second, for the reasons given in the 
note to No. 1 la. 

b. Must not be attached to the outside walls of buildings, unless separated 
therefrom by substantial supports. 

It Is recommended that transformers be not attached to frame buildings when 
any other location Is practicable. 

13 A. Grounding Low=Potential Circuits.— The grounding of low- 
potential circuits under the following regulations is only allowed when such 
circuits are so arranged that under normal conditions of service there will be 
no passage of current over the ground wire. 

Direct-Current S-Wire Systems. — a. Neutral wire may be grounded and 
when grounded the following rules must be complied with: — 



1402 70,-^ELECTRIC POWER AND LIGHTING. 

1. Must be grounded at the Central Station on a metal plate buried in 

coke beneath permanent moisture level, and also through all avail- 
able underground water and gas pipe systems. 

2. In underground systems the neutral wire must also be grounded at 

each distributing box through the box. 

3. In overhead systems the neutral wire must be grounded every 500 

feet, as provided in Sections c to g. 

Inspection Departments having jurisdiction may require grounding If they deem 
It necessary. 

Two-wire direct-current systems having no accessible neutral point are not to 
be grounded. 

Alternating-Current Secondary Systems. — b. Transformer secondaries of 
distributing systems should preferably be grounded, and when grounded, 
the following rules must be complied with: — 

1. The grounding must be made at the neutral point or wire, whenever 

a neutral point or wire is accessible. 

2. When no neutral point or wire is accessible one side of the secondary 

circuit may be grounded, provided the maximum difference of 
potential between the grounded point and any other point in the 
circuit does not exceed 250 volts. 

3. The ground connections must be at the transformer or on the indi- 

vidual service as provided in sections c to g, and when transformers 
feed systems with a neutral wire, the neutral wire must also be 
grounded at least every 250 feet lor overhead systems, and every 
500 feet for underground systems. 
Inspection Departments having jurisdiction may require grounding If they deem 
It necessary. 

Ground Connections. — c. When the ground connection is inside of any 
building, or the ground wire is inside of, or attached to any building (except 
Central or Sub-stations) the ground wire must be of copper and have an 
approved rubber insulating covering National Electric Code Standard, for 
from to 600 volts. (See No. 41.) 

d. The ground wire in direct-current 3-wire systems must not at Central 
Stations be smaller than the neutral wire and not smaller than No. 4 B. & S, 
gage elsewhere. The ground wire in alternating-current systems must never 
be less than No. 4 B. & S. gage. 

On three-phase system, the ground wire must have a carrying capacity equal to 
that of any one of the ttiree mains. 

e. The ground wire should, except for Central Stations and transformer 
sub-stations, be kept outside of buildings as far as practicable, but may be 
directly attached to the building or pole by cleats or straps or on porcelain 
knobs. Staples must never be used. The wire must be carried in as nearly 
a straight line as practicable, avoiding kinks, coils and sharp bends, and 
must be protected when exposed to mechanical injury. 

This protection can be secured by use of an approved moulding, and as a rule the 
ground wire on the outside of a building should be in moulding at all places where it 
is in within seven feet from the ground. 

f. The ground connection for Central Stations, transformer sub-stations, 
and banks of transformers must be made through metal plates buried in 
coke below permanent moisture level, and connection should also be made 
to all available underground piping systems including the lead sheath of 
underground cables. 

g. For individual transformers and building services, the ground con- 
nection may be made as in Section f, or may be made to water piping 
systems running into buildings. This connection may be made by carrying 
the ground wire into the cellar and connecting on the street side of meters, 
main cocks, etc. 

Where it is necessary to run the ground wire through any part of a build- 
ing it shall be protected by approved porcelain bushings through walls or 
partitions and shall be run in approved moulding, except that in basements 
it may be supported on porcelain. 

In connecting a ground wire to a piping system, the wire should be sweat into a 
lug attached to an approved clamp, and the clamp firmly bolted to the water pipe 
after all rust and scale have been removed; or be soldered Into a brass plug and the 
plug forcibly screwed into a pipe-fltting, or, where the pipes are cast iron, into a hole 
tapped into the pipe itself. For large stations, where connecting to underground 
pipes with bell and spigot joints, it is well to connect to several lengths, as the pipe 
joints may be of rather high resistance. 



INSIDE WORK— ALL SYSTEMS AND VOLTAGES. 1403 

Where groimd plates are used, a No. 1 6 Stubbs' gage copper plate, about 3x6 
feet in size, with about 2 feet of crushed coke or charcoal, about pea size, both under 
and over it, would make a ground of suflQcient capacity for a moderate-sized sta- 
tion, and would probably answer for the ordinary sub-station or bank of transformers. 
For a large central station, a plate with considerable more area might be necessary, 
depending upon the other underground connections available. The ground wire 
should be riveted to the plate in a number of places, and soldered for its whole length. 
Perhaps even better than a copper plate is a cast-iron plate with projecting forks, 
the idea of the fork being to distribute the connection to the ground over a fairly broad 
area, and to give a large surface contact. The ground wire can probably best be 
connected to such a cast-iron plate by soldering it into brass plugs screwed into holes 
tapped in the plate. In all cases, the joint between the plate and the ground wire 
should be thoroughly protected against corrosion by painting it with waterproof 
paint or some equivalent. 

CLASS C— INSIDE WORK. 

(Light Power and Heat. For Signaling Systems, see 

Class E.) 

ALL SYSTEMS AND VOLTAGES 

General Rules. 

14. Wires. — (For special rules, see Nos. 16, 18, 24, 35, 38 and 39.) 

a. Must not be of smaller size than No. 14 B. & S. gage, except as al- 
lowed under Nos. 24v and 45b. 

b. Tie wires must have an insulation equal to that of the conductors 
they confine. 

The use of some form of confining knob or insulator which will dispense with tie 
wires is recommended. 

c. Must be so spliced or joined as to be both mechanically and electrically 
secure without solder. The joints must then be soldered to insure preserva- 
tion, and covered with an insulation equal to that on the conductors. 

Stranded wires must be soldered before being fastened under clamps or 
binding screws, and whether stranded or solid, when they have a conduc- 
tivity greater than that of No. 8 B. & S. gage they must be soldered into 
lugs for all terminal connections. 

All jo'nts must be soldered unless made with some form of approved splicing 
device. This ruling applies to joints and splices in all classes of wiring covered by 
these rules. 

d. Must be separated from contact with walls, floors, timbers or parti- 
tions through which they may pass by non-combustible, non-absorptive 
insulating tubes, such as glass or porcelain, except as provided in No. 24u. 

Bushings must be long enough to bush the entire length of the hole in one con- 
tinuous piece, or else the hole must first be bushed by a continuous waterproof tube 
This tube may be a conductor, such as iron pipe, but in that case an insulating 
bushing must be pushed into each end of it, extending far enough to keep the wire 
absolutely out of contact with the pipe. 

e. Must be kept free from contact with gas, water or other metallic 
piping, or any other conductors or conducting material which they may 
cross, by some continuous and firmly fixed non-conductor, creating a per- 
manent separation. Deviations from this rule may sometimes be allowed 
by special permission. 

Where one wire crosses another wire the best and usual means of separating 
them is by a porcelain tube on one of the wires. The tubing must be prevented from 
moving out of place either by a cleat or knob on each end, or by taping it securely In 
place. 

The same method may be adopted where wires pass close to iron pipes, beams, 
etc., or, where the wires are above the pipes, as Is generally the case, ample protection 
can frequently be secured by supporting the wires with a porcelain cleat placed as 
nearly above the pipe as possible. 

This rule must not be construed as in any way modifying No. 24, Sections h and j. 

f. Must be so placed in wet places that an air space will be left between 
conductors and pipes in crossing, and the former must be run in such a way 
that they cannot come in contact with the pipe accidentally. Wires should 
be run over, rather than under, pipes upon which moisture is likely to 
gather or which, by leaking, might cause trouble on a circuit. 

g. The installation of electrical conductors in wooden moulding, or on 
insulators, in elevator shafts will not be approved, but conductors may be 
installed in such shafts if encased in approved metal conduits. 



1404 



10.— ELECTRIC POWER AND LIGHTING. 



15. Underground Conductors. — a. Must be protected against moisture 
and mechanical injury where brought into a building, and all combustible 
material must be kept from the immediate vicinity. 

b. Must not be so arranged as to shunt the current throiigh a building 
around any catch-box. 

c. Where underground service enters building through tubes, the tubes 
shall be tightly closed at outlets with asphaltum or other non-conductor, 
to prevent gases from entering the building through such channels. 

d. No underground service from a subway to a building shall supply 
more than one building except by written permission from the Inspection 
Department having jurisdiction. 

16. Table of Carrying Capacity of Wires. — a. The following table, 
showing the allowable carrying capacity ot copper wires and cables of 
ninety-eight per cent conductivity, according to the standard adopted by 
the American Institute of Electrical Engineers, must be followed in placing 
interior conductors , (See page 1388.) 

For insulated aluminum wire the safe carrying capacity is eighty-four per cent 
of that given in the following tables for copper wire with the same kind of insulation. 





Table A. 


Table B. 




Table A. 


Table B 




Rubber 


Other 






Concl'd. 


Concl'd. 




Insula- 


Insula- 




Circular 








tion. 


tions. 




Mils. 


Amperes. 


Amperes 




See 


See No. 42 










No. 41. 


to 44. 




200,000 


200 


300 


B. & S 






Circular 


300,000 


270 


400 


Gage. 


Amperes 


.Amperes. 


Mils. 


400,000 


330 


600 


18 


3 


5 


1,624 


500,000 


390 


590 


16 


6 


8 


2,583 


600,000 


450 


680 


14 


12 


16 


4,107 


700,000 


500 


760 


12 


17 


23 


6,530 


800,000 


550 


840 


10 


24 


32 


10.380 


900,000 


600 


920 


8 


33. 


46 


16,510 


1,000,000 


650 


1,000 


6 


46 


65 


26,250 


1,100,000 


690 


1,080 


6 


54 


77 


33,100 


1,200,000 


730 


1,150 


4 


65 


92 


41.740 


1,300,000 


770 


1,220 


3 


76 


110 


52,630 


1,400.000 


810 


1,290 


2 


90 


131 


66,370 


1,500,000 


850 


1,360 


1 


107 


156 


83,690 


1,600,000 


890 


1.430 





127 


185 


105,500 


1,700.000 


930 


1,490 


00 


150 


220 


133,100 


1,800,000 


970 


1,550 


000 


177 


262 


167,800 


1.900,000 


1.010 


1,610 


0000 


210 


312 


211,600 


2,000,000 


1,050 


1.670 



The lower limit is specified for rubber-covered wires to prevent gradual deterio- 
iiitlon of the high insulations by the heat of the wires, but not from fear of igniting 
the insulation. The question of drop is not taken into consideration in the above 

The carrying capacity of Nos. 1 6 and 18 B. & S. gage wire is given, but no smaller 
than No. 14 is to be used, except as allowed under Nos. 24v and 45b. 

17. Switches, Cut=Outs, Circuit=Breakers, Etc. — (For construction 
rules, see Nos. 51, 52 and 53.) a. On constant potential circuits, all service 
switches and all switches controlling circuits supplying current to motors or 
heating devices, and all cut-outs, unless otherwise provided (for exceptions 
as to switches see Nos. 8 c and 21a; forexceptionsas tocut-outssee No. 21 a 
and b) must be so arranged that the cut-outs will protect and the opening of 
the switch or circuit-breaker will disconnect all of the wires; that is, in the 
two-wire system the two wires, and the three-wire system the three wires, 
must be protected by the cut-out and disconnected by the operation of the 
switch or circuit-breaker. 

This, of course, does not apply to the grounded circuit of street railway S3rstems. 

b. Must not be placed in the immediate vicinity of easily ignitable stuff 
or where exposed to inflammable gases or dust or to flyings of combustible 
material. 



INSIDE WORK— CONSTANT-CURRENT SYSTEMS. 1405 

When the occupancy of a building is such that switches, cut-outs, etc., cannot be 
located so as not to be exposed to dust or flyings of combustible material they must be 
enclosed in approved dust-proof cabinets with self-closing doors, except oil switches 
and circuit-breakers which have dust-tight casings. 

c. Must, vv^hen exposed to dampness, either be enclosed in a moisture- 
proof box or mounted on porcelain knobs. 

The cover of the box should be so made, that no moisture which may 
collect on the top or sides of the box can enter it. 

d. Time switches, sign flashers and similar appliances must be of ap- 
proved design and enclosed in a steel box or cabinet lined with fire-resisting 
material. 

The cover of the box should be so made that no moisture which may collect on 
the top or sides of the box can enter it. 

If a steel box is used, the minimum thickness of the steel must be 0. 1 2 8 of an inch 
(No. 8 B. & S. gage). 

If a cabinet is used, it must be lined with marble or slate at least three-eighths 
of an inch thick, or with steel not less than 0.128 of an inch thick. Box or cabinet 
must be so constructed that when switch operates blade shall clear the door by at 
least one Inch. 

CONSTANT^-CURRENT SYSTEMS. 

Principally Series Arc Lighting. 

18. Wires. — (See also Nos. 14, 15 and 16.) a. Must has an ap- 
proved rubber insulating covering (see No. 41). 

b. Must be arranged to enter and leave the building through an approved 
double-contact service switch (see No. 51b), mounted in a non-combustible 
case, kept free from moisture, and easy of access to police or firemen. 

c. Must always be in plain sight, and never encased, except when re- 
quired by the Inspection Department having jurisdiction. 

d. Must be supported on glass or porcelain insulators, which separate 
the wire at least one inch from the surface wired over and must be kept 
rigidly at least 8 inches from each other, except within the structure of lamps, 
on hanger-boards or in cut-out boxes, or like places, where a less distance 
is necessary. 

e. Must, on side walls be protected from mechanical injury by a sub- 
stantial boxing, retaining an air space of one inch around the conductors, 
closed at the top (the wires passing through bushed holes), and extending 
not less than 7 feet from the floor. When crossing floor timbers in cellars, 
or in rooms where they might be exposed to injury, wires must be attached 
by their insulating supports to the under side of a wooden strip not less than 
one-half an inch in thickness. Instead of the running-boards, guard strips 
on each side of and close to the wires will be accepted. These strips to be 
not less than seven-eighths of an inch in thickness and at least as high as the 
insulators. 

Except on jolsted ceilings, a strip one-half of an inch thick Is not considered 
suflflciently stiff and strong. For spans of say eight or ten feet, where there is but 
little vibration, one-inch stock is generally sufficiently stiff; but where the span is 
longer than this or there is considerable vibration, still heavier stock should be used. 

19. Series Arc Lamps. — (For construction rules, see No. 57.) 

a. Must be carefully isolated from inflammable material. 

b. Must be provided at all times with a glass globe surrounding the arc, 
and securely fastened upon a closed base. Broken or cracked globes must 
not be used. 

c. Must be provided with a wire netting (having a mesh not exceeding 
one and one-fourth inches) around the globe, and an approved spark arrester 
(see No. 58), when readily inflammable material is in the vicinity of the 
lamps, to prevent escape of sparks of carbon or melted copper. It is recom- 
mended that plain carbons, not copper-plated, be used for lamps in such 
places. 

Outside arc lamps must be suspended at least eight feet above sidewalks. Inside 
arc lamps must be placed out of reach or suitably protected. 

Arc lamps, when used in places where they are exposed to flyings of easily 
inflammable material, should have the carbons enclosed completely in a tight globe 
in such manner as to avoid the necessity for spark arresters. 

"Enclosed arc" lamps, having tight inner globes, may be used, and the require- 
ments of Sections b and c above would, of course, not apply to them, except that a 



1406 Id.— ELECTRIC POWER AND LIGHTING. 

wire netting around the Inner globe may In some cases be required If the outer globe 
Is omitted. 

d. Where hanger-boards (see No. 56) are not used, lamps must be hung 
from insulating supports other than their conductors. 

e. Lamps when arranged to be raised and lowered, either for carboning 
or other purposes, shall be connected up with stranded conductors from the 
last point of support to the lamp, when such conductor is larger than 
No. 14 B. & S. gage. 

20. Incandescent Lamps in Series Circuits. — a. Must have the con- 
ductors installed as required in No. 18, and each lamp must be provided with 
an automatic cut-out. 

b. Must have each lamp suspended from a hanger-board by means of 
rigid tube. 

c. No electro-magnetic device for switches and no multiple-series or 
series-multiple system of lighting will be approved. 

d. Must not under any circumstances be attached to gas fixtures. 

CONSTANT-POTENTIAL SYSTEMS. 

General Rules — All Voltages. 

21. Automatic Cut=Outs (Fuses and Circuit=Breakers). — (See No. 17, 
and for construction, Nos, 52 and 53.) [Excepting on main switchboards, 
or where otherwise subject to expert supervision, circuit-breakers will not 
be accepted unless fuses are also provided.] 

a. Must be placed on all service wires, either overhead or underground, 
as near as possible to the point where they enter the building and inside the 
walls, and arranged to cut off the entire current of the building. 

Where the switch required by No. 22a is inside the building, the cut-out required 
by this section must be placed so as to protect it. 

For three-wire (not three-phase) systems the fuse in the neutral wire may be 
omitted, provided the neutral wire is of equal carrying capacity to the larger of the outside 
wires, and is grounded as provided for in No. 13 A. 

In risks having private plants, the yard wires running from building to building 
are not generally considered as service wires, so that cut-outs would not be required 
where the wires enter buildings, provided that the next fuse back is small enough to 
properly protect the wires inside the building in question. 

b. Must be placed at every point where a change is made in the size of 
wire [unless the cut-out in the larger wire will protect the smaller (see No. 16) J. 

For three-wire (not three-phase) systems the fuse in the neutral wire, except that 
called for under No. 2 Id, may be omitted, provided the neutral wire is of equal carrying 
capacity to the larger of the outside wires, and is grounded gw provided for in No. 13A. 

c. Must be in plain sight, or enclosed in an approved cabinet (see No. 54), 
and readily accessible. They must not be placed in the canopies or shells 
of fixtures. 

The ordinary porcelain link fuse cut-out will not be approved. Link fuses may 
be used only when mounted on slate of marble bases conforming to No. 52 and must 
be enclosed in dust-tight, flreproofed cabinets, except on switchboards located well 
away from any combustible material, as in the ordinary engine and dynamo room 
and where these conditions will be maintained. 

d. Must be so placed that no set of incandescent lamps requiring more 
than 660 watts, whether grouped on one fixture or on several fixtures or 
pendants, will be dependent upon one cut-out. 

Special permission may be given in writing by the Inspection Depart- 
ment having jurisdiction, for departure from this rule, in the case of large 
chandeliers. (For exceptions, see No. 31 A, b, 3[6] and 4 [6] for border lights, 
see List of Fittings for rules for electric signs ) All branches or taps from any 
three-wire system which are directly connected to lamp sockets or other 
translating devices, must be run as two-wire circuits if the fuses are omitted 
in the neutral, or if the difference of potential between the two outside 
wires is over 250 volts, and both wires of such branch or tap circuits must 
be protected by proper fuses. 

The above rule shall also apply to motors when more than one Is dependent on a 
single cut-out. 

The fuses in the branch cut-outs should not have a rated capacity greater than 
6 amperes on 110 volt systems, and 3 amperes on 220 volt systems. 

The idea is to have a small fuse to protect the lamp socket and the small wire 
used for fixtures, pendants, etc. It also lessens the chances of extinguishing a large 
number of lights If a short circuit occurs. 



INSIDE WORK— CONSTANT POTENTIAL SYSTEMS, 1407 

On open work in large mills approved link fused rosettes may be used at a voltage 
of not over 125 and approved enclosed fused rosettes at a voltage of not over 250, the 
fuse In the rosettes not to exceed 3 amperes, and a fuse of over 25 amperes must not 
be used in the branch circuit. 

e. The rated capacity of fuses must not exceed the allowable carrying 
capacity of the wire as given in No. 16. Circuit-breakers must not be set 
more than 30 per cent above the allowable carrying capacity of the wire, 
unless a fusible cut-out is also installed in the circuit. 

In the arms of fixtures carrying a single socket a No. 18 B. & S. gage wire sup- 
pl3ring only one socket will be considered as properly protected by a 6 ampere fuse. 

22. Switches. — (See No. 17, and for construction, No. 51.) 

a. Must be placed on all service wires, either overhead or underground, 
in a readily accessible place, as near as possible to the point where the wires 
enter the building, and arranged to cut off the entire current. 

Service cut-out and switch must be arranged to cut off current from all devices 
Including meters. 

In risks having private plants the yard wires running from building to building 
are not generally considered as service wires, so that switches would not be required 
In each building if there are other switches conveniently located on the mains or if 
the generators are near at hand. 

b. Must always be placed in dry, accessible places, and be grouped as 
far as possible. (See No. 17c.) Single-throw knife switches must be so 
placed that gravity will tend to open rather than close them. Double-throw 
knife switches may be mounted so that the throw will be either vertical or 
horizontal as preferred. 

When possible, switches should be so wired that blades will be "dead" when 
switch is open. 

If switches are used in rooms where combustible flyings would be likely to accu- 
mulate around them, they should be enclosed in dust-tight cabinets. (See note 
under No. 17 b.) Even in rooms where there are no combustible materials it is 
better to put all knife switches in cabinets, in order to lessen the danger of accidental 
short circuits being made across their exposed metal parts by careless workmen. 

Up to 250 volts and 30 amperes, approved indicating snap switches are advised 
In preference to knife switches on lighting circuits about the workrooms. 

c. Single pole switches must never be used as service switches nor placed 
in the neutral wire of a three-wire system, except in the two-wire branch or 
tap circuit described in 21 d. 

This, of course, does not apply to the grounded circuits of street railway systems. 
Three-way switches are considered as single-pole switches and must be wired so 
that only one pole of the circuit is carried to either switch. 

d. Where flush switches or receptacles are used, whether with conduit 
systems or not, they must be enclosed in boxes constructed of iron or steel. 
No push button for bells, gas-lighting circuits, or the like shall be placed in 
the same wall plate with switches controlling electric light or power wiring . • 

This requires an approved box in addition to the porcelain enclosure of the 
switch or receptacle. 

e. Where possible, at all switch or fixture outlets, a |-inch block must 
be fastened between studs or floor timbers flush with the back of lathing to* 
hold tubes, and to support switches or fixtures. When this cannot be done, 
wooden base blocks not less than f-inch in thickness, securely screwed to 
lathing, must be provided for switches, and also for fixtiires which are not 
attached to gas pipes or conduit. 

The above will not be necessary where outlet boxes are used which will give 
proper support for fixtures, etc. 

f. Sub-bases of non-combustible, non-absorptive insulating material, 
which will separate the wires at least i^-inch from the surface wired over, 
must be installed under all snap switches used in exposed knob and cleat 
work. Sub-bases must also be used in moulding work, but they may be 
made of hardwood. 

23. Electric Heaters. — It Is often desirable to connect In multiple with the 
heaters and between the heater and the switch controlling same, an incandescent 
lamp of low candle power, as it shows at a glance whether or not the switch is open, 
and tends to prevent its being left closed through oversight. Inspection Depart- 
ments having jurisdiction may require this provision to be carried out if they deem 
It necessary. 

a. Must be protected by a cut-out and controlled by indicating switches. 
Switches must be double pole except when the device controlled does not 
require more than 660 watts of energy. 



1408 10.— ELECTRIC POWER AND LIGHTING. 

b. Must never be concealed, but must at all times be in plain sight. 
Special permission may be given in writing by the Inspection Department having 

jurisdiction for departure from this rule in certain cases. 

c. Flexible conductors for smoothing irons and sad irons, and for all 
devices requiring over 250 watts must comply vi^ith No. 45 g. 

d. For portable heating devices the flexible conductors must be con- 
nected to an approved plug device, so arranged that the plug will pull out 
and open the circuit in case any abnormal strain is put on the flexible con- 
ductor. This device may be stationary, or it may be placed in the cord itself. 
The cable or cord must be attached to the heating apparatus in such manner 
that it will be protected from kinking, chafing or like injury at or near the 
point of connection. 

e. Smoothing irons, sad irons, and other heating appliances that are 
intended to be applied to inflammable articles, such as clothing, must con- 
form to the above rules so far as they apply. They must also be provided 
with an approved stand, on which they should be placed when not in use. 

An approved automatic attachment which will cut off the current when the iron 
Is not on the stand or in actual use is desirable. Inspection Departments having 
jurisdiction may require this provision to be carried out if they deem it advisable. 

f. Stationary electric heating apparatus, such as radiators, ranges, plate 
warmers, etc., must be placed in a safe location, isolated from inflammable 
materials, and be treated as sources of heat. 

Devices of this description will often require a suitable heat-resisting material 
placed between the device and Its surroundings. Such protection may best be 
secured by installine two or more plates of tin or sheet steel with a one-inch air space 
between, or by alternate layers of sheet steel and asbestos with a similar air space. 

g. Must each be provided with name-plate, giving the maker's name 
and the normal capacity in volts and amperes. 

CONSTANT=LOW=POTENTIAL SYSTEMS. 

550 Volts or Less. 

Any circuit attached to any machine, or combination of machines, which 
develops a difference of potential between any two wires, of over 
ten volts and less than 550 volts, shall be considered as a low- 
potential circuit, and as coming under this class, unless an approved 
transforming device is used, which cuts the difference of potential 
down to ten volts or less. The primary circuit not to exceed a 
potential of 3,500 volts unless the primary wires are installed in 
accordance with the requirements as given in No. 12 A, or are under- 
ground. 

For 550 volt motor equipments a margin of ten per cent above the 550 volt limit will 
be allowed at the generator or transformer. 

Before pressure is raised above 300 volts on any previously existing system 
of wiring, the whole must he strictly brought up to all of the requirements of the 
rules at date. 

24. Wires. — General Rules. (See also Nos. 14, 15 and 16.) 

a. Must he so arranged that under no circumstances will there he a, differ- 
ence of potential of over 300 volts between any bare m.etal parts in any distrtbut' 
ing switch or cut-out cabinet, or equivalent center of distribution. 

This rule Is not intended to prohibit the placing of switches or single pole cut- 
outs for motor systems of voltages above 300 in cabinets, but would require that the 
cabinets be divided by approved barriers so arranged that no one section shall con- 
tain more than one switch nor more than one single pole cut-out. 

b. Must not be laid in plaster, cement or similar finish, and must never 
be fastened with staples. 

c. Must not be fished for any great distance, and only in places where 
the inspector can satisfy himself that the rules have been complied with. 

d. Twin wires must never be used, except in conduits, or where flexible 
conductors are necessary. 

e. Must be protected on side walls from mechanical injury. When 
crossing floor timbers in cellars, or in rooms where they might be exposed to 
injury, wires must be attached by their insulating supports to the under side 
of a wooden strip, not less than one-half inch in thickness and not less than 
three inches in width. Instead of the running-boards, guard strips on each 



INSIDE WORK— CONSTANT-LOW-POTENTIAL. 1409 

side of and close to the wires will be accepted. These strips to be not less 
than seven-eighths of an inch in thickness, and at least as high as the insula- 
tors. 

Suitable protection on side walls should extend not less than five feet from the 
floor. This may be secured by substantial boxing, retaining an air space of one inch 
around the conductors, closed at the top (the wires passing through bushed holes) or 
by approved metal conduit, or pipe of equivalent strength. 

When metal conduit or pipe is used, the insulation of each wire must be rein- 
forced by approved flexible tubing extending from the insulator next below the pipe 
to the one next above it, unless the conduit is installed according to No. 25 (sections 
c and f excepted), and the wire used complies with No. 47. The two or more wires 
of a circuit each with its flexible tubing (when required), if carrying alternating current 
must, or if direct current, may be placed within the same pipe. 

In damp places the wooden boxing may be preferable because of the precautions 
which would be necessary to secure proper insulation if the pipe were used. With 
this exception, however, iron piping is considered preferable to the wooden boxing, 
and its use is strongly urged. It is especially suitable for the protection of wires 
near belts, pulleys, etc. 

f. When run in unfinished attics, will be considered as concealed, and 
when run in close proximity to water tanks or pipes, will be considered as 
exposed to moisture. 

In unfinished attics wires are considered as exposed to mechanical Injury, and 
must not be run on knobs on upper edge of joists. 

Special Rules. 
For Open Work — In dry places. 

g. Must have an approved rubber, slow-burning weatherproof, or slow- 
burning insulation (see Nos. 41, 42 and 43). 

A slow-burning covering, that is, one that will not carry fire, is considered good 
enough where the wires are entirely on insulating supports. Its main object is to 
prevent the copper conductors from coming accidentally into contact with each 
other or anything else 

h. Must be rigidly supported on non-combustible, non -absorptive in- 
sulators, which will separate the wires from each other and from the sur- 
face wired over in accordance with the following table: — 

Distance from Distance between 

Voltage. Surface. Wires. 

to 300 i inch 2^ inch 

301 to 550 1 " 4 •' 

Rigid supporting requires under ordinary conditions, where wiring along flat 
surfaces, supports at least every four and one-half feet. If the wires are liable to be 
disturbed, the distance between supports should be shortened, in buildings of mill 
construction, mains of not less than No. 8 B. & S. gage, where not liable to be dis- 
turbed, may be separated about six inches, and run from timber to timber, not break- 
ing around, and may be supported at each timber only. 

This rule not to be interpreted to forbid the placing of the neutral of an Edison 
three-wire system in the center of a three-wire cleat where the difference of potential 
between the outside wires is not over 300 volts, provided the outside wires are 
separated two and one-half inches. 

For Open Work — In damp places, or buildings specially subject to moisture 
or to acid or other fumes liable to injure the wires or their insulation. 
i. Must have an approved insulating covering. 

For protection against water, rubber insulation must be used. For protection 
against corrosive vapors, either weatherproof or rubber insulation must be used. 
(See Nos. 41 and 44.) 

j. Must be rigidly supported on non-combustible, non-absorptive in- 
sulators, which separate the wire at least one inch from the surface wired 
over, and must be kept apart at least two and one-half inches for voltages 
up to 300, and four inches for higher voltages. 

Rigid supporting requires under ordinary conditions, where wiring over flat 
surfaces, supports at least every four and one-half feet. If the wires are liable to be 
disturbed, the distance between supports should be shortened. In buildings of mill 
construction, mains of not less than No. 8 B. & S. gage, where not liable to be dis- 
turbed, may be separated about six inches, and run from timber to timber, not break- 
ing around, and may be supported at each timber only. 

For Moulding Work {Wooden and Metal) . (For construction rules see 
No. 50. See also No. 25 A.) 

k. Must have an approved rubber insulating covering. (For wooden 
moulding see No. 41, for metal moulding see No. 47.) 




1410 70.— ELECTRIC POWER AND LIGHTING. 

!. Must never be placed in either metal or wooden moulding in concealed 
or damp places, or where the difference of potential between any two wires 
in the same moulding is over 300 volts. Metal mouldings must not be used 
for circuits requiring more than 660 watts of energy. 

As a rule, wooden moulding should not be placed directly against a brick wall, 
as the wall is likely to "sweat" and thus introduce moisture back of the moulding. 

m. Must, for alternating current systems if in metal moulding, have the 
two or more wires of a circuit installed in the same moulding. 

It Is advised that this be done for direct current systems also, so that they may 
be changed to alternating systems at any time, induction troubles preventing such a 
change if the wires are in separate mouldings. 

For Conduit Work. 

n. Must have an approved rubber insulating covering (see No. 47). 

0. Must not be drawn in until all mechanical work on the building has 
been, as far as possible, completed. 

Conductors in vertical conduit risers must be supported within the con- 
duit system in accordance with the following table: — 

No. 14 to every 100 feet. 

No. 00 to 0000 every 80 feet. 

0000 to 350,000 C. M. every 60 feet. 

350,000 C. M. to 500,000 C. M. every 50 feet. 

500,000 C. M. to 750,000 C. M. every 40 feet. 

750,000 C. M. every 35 feet. 

A turn of 90 degrees in the conduit system will constitute a satisfactory 
support, as per above table. 

The following methods of supporting cables are recommended: — 

1. Jtinction boxes may be inserted in the conduit system at the re- 

quired intervals, in which insulating supports of approved type 
must be installed and secured in a satisfactory manner so as to 
withstand the weight of the conductors attached thereto, the boxes 
to be provided with proper covers. 

2. Cables may be supported in approved junction boxes on two or more 

insulating supports so placed that the conductors will be deflected 
at an angle of not less than 90 degrees, and carried a distance of 
not less than twice the diameter of the cable from its vertical 
position. Cables so suspended may be additionally sectired to 
these insulators by tie wires. 
Other methods, if used, must be approved by the Inspection Depart- 
ments having jurisdiction. 

p. Must, for alternating systems, have the two or more wires of a cir- 
cuit drawn in the same conduit. 

It Is advised that this be done for direct current S3'^stems also, so that they may 
be changed to alternating systems at any time, induction troubles preventing such 
a change if the wires are in separate conduits. 

The same conduit must never contain circuits of different systems, but may con- 
tain two or more circuits of the same system. 

For Concealed ''Knob and Tube'' Work. 

q. Must have an approved rubber insulating covering (see No. 41). 

r. Must be rigidly supported on non-combustible, non-absorptive in- 
sulators which separate the wire at least one inch from the surface wired 
over. Should preferably be run singly on separate timbers, or studdings, 
and must be kept at least five inches apart. 

Must be separated from contact with the walls, floor timbers and parti- 
tions through which they may pass by non-combustible, non-absorptive 
insulating tubes, such as glass or porcelain. 

Rigid supporting requires under ordinary conditions, where wiring along flat 
surface, supports at least every four and one-half feet. If the wires are liable to be 
disturbed the distance between supports should be shortened. 

At distributing centers, outlets or switches where space is limited and the five- 
Inch separation cannot be maintained, each wire must be separately encased in a 
continuous length of approved flexible tubing. 

Wires passing through timbers at the bottom of plastered partitions must be 
protected by an additional tube extending at least four inches above the timber. 

s. When in a concealed knob and tube system, it is impracticable to 
place the whole of a circuit on non-combustible supports of glass or porcelain, 
that portion of the circuit which cannot be so supported must be installed 
with approved metal conduit, or approved armored cable (see No. 24 t), ex- 



INSIDE WORK—CONSTANT-LOW-POTENTIAL, 1411 

cept that if the difference of potential between the wires is not over 300 volts, 
and if the wires are not exposed to moisture, they may be fished if separately 
encased in approved flexible tubing, extending in continuous lengths from 
porcelain support to porcelain support, from porcelain support to outlet, 
or from outlet to outlet. 

t. Mixed concealed knob and tube work as provided for in No. 24 s, 
must comply with requirements of No. 24 n to p, and No. 25, when conduit 
is used, and with requirements of No. 24 A, when armored cable is used. 

u. Must at all outlets, except where conduit is used, be protected by 
approved flexible insulating tubing, extending in continuous lengths from the 
last porcelain support to at least one inch beyond the outlet. In the case 
of combination fixtures the tubes must extend at least flush with outer end 
of gas cap. 

It is recommended but not required that approved outlet boxes or plates be 
Installed at all outlets in concealed "knob and tube" work, the wires to be protected 
by approved flexible insulating tubing, extending in continuous lengths from the 
last porcelain support into the box. 

For Fixture Work. v. Must have an approved rubber insulating cover- 
ing (see No. 46), and be not less in size than No. 18 B. &. S. gage. 

See No. 46 e, fine print note, for exceptions to the use of rubber-covered wire. 

W. Supply conductors, and especially the splices to fixture wires, must 
be kept clear of the grounded part of gas pipes, and, where shells or outlets 
boxes are used, they must be made sufficiently large to allow the fulfillment 
of this requirement. 

X, Must, when fixtures are wired outside, be so secured as not to be cut 
or abraded by the pressure of the fastenings or motion of the fixture. 

y. Under no circumstances must there be a difference of potential of more 
than 300 volts between wires contained in or attached to the same fixture. 

24 A. Armored Cables. — (For construction rules, see No. 48.) 
a- Must be continuous from outlet to outlet or to junction boxes, and 
the armor of the cable must properly enter and be secured to all fittings, and 
the entire system must be mechanically secured in position. 

In case of underground service connections and main runs, this involves running 
such armored cable continuously into a main cut-out cabinet or gutter surrounding 
the panel board, as the case may be. (See No. 54.) 

b. Must be equipped at every outlet with an approved outlet box or 
plate, as required in conduit work. (See No. 49 A.) 

Outlet plates must not be used where it is practicable to install outlet boxes. 

The outlet box or plate shall be so installed that it will be flush with the finished 
surface, and if this surface is broken it shall be repaired so that it will not show any 
gaps or open spaces around the edge of the outlet box or plate. 

In buildings already constructed where the conditions are such that neither 
outlet box nor plate ran be installed, these appliances may be omitted by special 
permission of the Inspection Department having jurisdiction, provided the armored 
cable Is firmly and rigidly secured in place. 

c. Must have the metal armor of the cable permanently and effectively 
grounded. 

It is essential that the metal armor of such systems be joined so as to afford 
electrical conductivity sufficient to allow the largest fuse of circuit-breaker in the 
circuit to operate before a dangerous rise in temperature in the system can occur. 
Armor of cables and gas pipes must be securely fastened in metal outlet boxes so as 
to secure good electrical connections. Where boxes used for centers of distribution 
do not afford good electrical connection the armor of the cables must be joined 
around them by suitable bond wires. Where sections of armored cable are installed 
without being fastened to the metal structure of buildings or grounded metal piping, 
they must be bonded together and joined to a permanent and efficient ground con- 
nection. 

d. When installed in so-called fireproof buildings in course of construc- 
tion or afterwards if concealed, or where it is exposed to the weather, or in 
damp places such as breweries, stables, etc., the cable must have a lead cover- 
ing at least one thirty-second inch in thickness placed between the outer 
braid of the conductors and the steel armor. 

e. Where entering junction boxes, and at all other outlets, etc., must be 
provided with approved terminal fittings which will protect the insulation 
of the conductors from abrasion, unless such junction or outlet boxes are 
specially designed and approved for use with the cable. 



1412 lO.—ELECTRIC POWER AND LIGHTING. 

f. Junction boxes must always be installed in such manner as to be 
accessible. 

g. For alternating current systems must have the two or more conductors 
of the cable enclosed in one metal armor. 

25. Interior Conduits. — (See also Nos. 24 n to p, and 49.) 
The object of a tube or conduit is to facilitate the insertion or extraction of the 
conductors and to protect them from mechanical injury. Tubes or conduits are to 
be considered merely as raceways, and are not to be relied upon for insulation between 
wire and wire, or between the wire and the ground. 

a. No conduit tube having an internal diameter of less than five-eighths 
of an inch shall be used. Measurements to be taken inside of metal con- 
duits. 

b. Must be continuous from outlet to outlet or to junction boxes, and the 
conduit must properly enter, and be secured to all fittings and the entire 
system must be mechanically secured in position. 

In case of service connections and main runs, this involves running each conduit 
continuously int© a main cut-out cabinet or gutter surrounding the panel board, as 
the case may be (see No. 54). 

c. Must be first installed as a complete conduit system, without the 
conductors. 

d. Must be equipped at every outlet with an approved outlet box or 
plate (see No. 49 A). 

Outlet plates must not be used where it is practicable to install outlet boxes. 

The outlet box or plate shall be so installed that it will be flush with the finished 
surface, and if this surface is brolten it shall be repaired so that it will not show any 
gaps or open spaces around the edge of the outlet box or plate. 

In buildings already constructed where the conditions are such that neither 
outlet box nor plate can be installed, these appliances may be omitted by special 
permission of the Inspection Department having jurisdiction, providing the conduit 
ends are bushed and secured. 

e. Metal conduits where they enter junction boxes, and at all other out- 
lets, etc., must be provided with approved bushings fitted so as to protect 
wire from abrasion, except when such protection is obtained by the use of 
approved nipples, properly fitted in boxes or devices. 

f. Must have the metal of the conduits permanently and effectually 
grounded. 

It is essential that the metal of conduit systems be joined so as to afford elec- 
trical conductivity sufficient to allow the largest fuse or circuit breaker in the circuit 
to operate before a dangerous rise in temperature in the conduit system can occur. 
Conduits and gas pipes must be securely fastened in metal outlet boxes so as to secure 
good electrical connection. Where boxes used for centers of distribution do not 
afford good electrical connection, the conduits must be joined around them by suit- 
able bond wires. Where sections of metal conduit are installed without being fas- 
tened to the metal structure of buildings or grounded metal piping, they must be 
bonded together and joined to a permanent and efflcieno ground connection. 

g. Junction boxes must always be installed in such manner as to be ac- 
cessible. 

h. All elbows or bends must be so made that the conduit or lining of 
same will not be injured. The radius of the curve of the inner edge of any 
elbow not to be less than three and one-half inches. Must have not more 
than the equivalent of four quarter bends from outlet to outlet, the bends at 
the outlets not being counted. 

25 A. Metal Mouldings. (See also Nos. 24 k to m, and 50.) 

a. Must be continuous from outlet to outlet, to junction boxes, or ap- 
proved fittings designed especially for use with metal mouldings, and must 
at all outlets be provided with approved terminal fittings which will protect 
the insulation of conductors from abrasion, unless such protection is afforded 
by the construction of the boxes or fittings. 

b. Such moulding where passing through a floor must be carried through 
an iron pipe extending from the ceiling below to a point five feet above the 
floor, which will serve as an additional mechanical protection and exclude the 
presence of moisture often prevalent in such locations. 

In residences, office buildings and similar locations where appearance Is an 
essential feature, and where the mechanical strength of the moulding Itself Is ade- 
quate, this ruling may be modified to require the protecting piping from the ceiling 
below to a point at least three inches above the flooring. 



INSIDE WORK— CONSTANT-LOW-POTENTIAL. 1413 

c. Backing must be secured in position by screws or bolts, the heads of 
which must be flush with the metal. 

d. The metal of the moulding must be permanently and effectively 
grounded, and must be so installed that adjacent lengths of moulding will 
be mechanically and electrically secured at all points. 

It is essential that the metal of such systems be joined so as to afford electric 
conductivity sufficient to allow the largest fuse in the circuit to operate before a 
dangerous rise of temperature in the system can occur. Mouldings and gas pipes 
must be securely fastened in metal outlet boxes, so as to secure good electrical con- 
nection. Where boxes used for center of distribution do not afford good electrical 
connection the metal moulding must be joined around them by suitable bond wires. 
Where sections are installed without being fastened to the metal structure of the 
building or grounded metal piping, they must be bonded together or joined to a 
permanent and effective ground connection. 

e. Must be installed so that for alternating systems the two or more 
wires of a circuit will be in the same metal moulding. 

It is advised that this be done for direct systems also, so that they may be changed 
to the alternating system at any time, induction troubles preventing such change 
If the wires must be in separate mouldings. 

26. Fixtures. — (See also Nos. 22 e, 24 v to y.) 

a. Must when supported from the gas piping or any grounded metal 
work of a building be insulated from such piping or metal work by means of 
approved insulating joints (see No. 59) placed as close as possible to the ceil- 
ing or. walls. 

Gas outlet pipes must be protected above the insulating joint by approved In- 
sulating tubing, and where outlet tubes are used they must be of sufficient length to 
extend below the insulating joint, and must be so secured that they will not be pushed 
back when the canopy is put in place. 

Where canopies are placed against plaster walls or ceilings in fireproof buildings, 
or against metal walls or ceilings, or plaster walls or ceilings on metallic lathing in 
any class of buildings, they must be thoroughly and permanently insulated from such 
walls or ceilings. 

b. Must have all burs or fins removed before the conductors are drawn 
into the fixture. 

c. Must be tested for "contacts" between conductors and fixture, for 
"short circuits" and for ground connections before it is connected to its 
supply conductors. 

d. All fixture arms made of tubing smaller than |-inch outside diameter, 
also the arms of all one-light brackets, must be secured after they are screwed 
into position by the use of a set-screw properly placed, or by soldering or 
cementing or some equally good method to prevent the arms from becoming 
unscrewed. Arms must not be made of tubing lighter than No. 18 B. & 
S. gage, and must have at screw joints not less than five threads all engaging. 
This rule does not apply to fixtures or brackets with cast or heavy arms. 

27. Sockets. — (For construction rules, see No. 55.) 

a. In rooms where inflammable gases may exist the incandescent lamp 
and socket must be enclosed in a vapor-tight globe, and supported on a 
pipe-hanger, wired with approved rubber-covered wire (see No. 41) soldered 
directly to the circuit. 

Key sockets contain a switch (see No. 17b). 

b. In damp or wet places "waterproof" sockets must be used. Unless 
made up on fixtures they must be hung by separate stranded rubber-covered 
wires not smaller than No. 14 B. & S. gage, which should preferably be twisted 
together when the pendant is over three feet long. 

These wires must be soldered direct to the circuit wires but supported in- 
dependently of them. 

c. Key sockets will not be approved if installed over specially inflam- 
mable stuff, or where exposed to flyings of combustible material. 

28. Flexible Cord.— 

a. Must have an approved insulation and covering (see No. 45). 

b. Must not be used where the difference of potential between the two 
wires is over 300 volts. 

The above rule does not apply to the grounded circuits in street railway property 

c. Must not be used as a support for clusters. 



1414 10.— ELECTRIC POWER AND LIGHTING. 

d. Must not be used except for pendants, wiring of fixtures, portable 
lamps or motors, and portable heating apparatus. 

The practice of making the pendants unnecessarily long and then looping them 
up with cord adjusters is strongly advised against. It offers a temptation to carry 
about lamps which are intended to hang freely in the air, and the cord adjusters 
wear off the insulation very rapidly. 

For all portable work, including those pendants which are liable to be moved 
about sufficiently to come in contact with surrounding objects, flexible wires and 
cables especially designed to withstand this severe service are on the market, and 
should be used. (See No. 45 f.) 

The standard socket is threaded for one-eighth inch pipe, and if it is properly 
bushed, the reinforced flexible cord will not go into it, but this style of cord may be 
used with sockets threaded for three-eighths inch pipe, and provided with substantial 
insulating bushings. The cable to be supported independently of the overhead 
circuit by a single cleat, and the two conductors then separated and soldered to the 
overhead wires. 

The bulb of an incandescent lamp frequently becomes hot enough to ignite paper, 
cotton and similar readily ignitible materials, and in order to prevent it from coming 
in contact with such materials, as well as to protect it from breakage, every portable 
lamp should be surrounded with a substantial wire guard. 

e. Must not be used in show windows except when provided with an 
approved metal armor. 

f. Must be protected by insulating bushings where the cord enters the 
socket. 

g. Must be so suspended that the entire weight of the socket and lamp 
will be borne by some approved device under the bushing in the socket, and 
above the point where the cord comes through the ceiling block or rosette, 
in order that the strain may be taken from the joints and binding screws. 

This is usually accomplished by knots in the cord inside the socket and rosette 

29. Arc- Lamps on Constant-Potential Circuits. — ^a. Must have a 
cut-out (see No. 17 a) for each lamp or each series of lamps. 

The branch conductors should have a carrying capacity about fifty per cent In 
excess of the normal current required by the lamp, to provide for heavy current re- 
quired when lamp is started, or when carbons become stuck without overfusing the 
wires. 

b. Must only be furnished with such resistances or regulators as are 
enclosed in non-combustible material, such resistances being treated as 
sources of heat. Incandescent lamps must not be used for this purpose. 

c. Must be supplied with globes and protected by spark arresters and 
wire netting around the globe, as in case of series arc lamps (see Nos. 19 
and 58). 

Outside arc lamps must be suspended at least eight feet above sidewalks. Inside 
arc lamps must be placed out of reach or suitably protected. 

d. Lamps when arranged to be raised and lowered, either for carbon- 
ing or other purposes, shall be connected up with stranded conductors from 
the last point of support to the lamp, when such conductor is larger than 
No. 14 B. 8c S. gage. 

30. Economy Coils. — a. Economy and compensator coils for arc 
lamps must be mounted on non-combustible, non -absorptive, insulating 
supports, such as glass or porcelain, allowing an air space of at least one inch 
between frame and support, and must in general be treated as sources 
of heat. ^ 

31. Decorative Lighting Systems. — a. Special permission may be 
given in writing by the Inspection Department having jurisdiction for the 
temporary installation of approved Systems of Decorative Lighting, pro- 
vided the difference of potential between the wires of any circuit shall not 
be over 150 volts and also provided that no group of lamps requiring more 
than 1,320 watts shall be dependent on one cut-out. 

No "System of Decorative Lighting" to be allowed under this rule which Is not 
listed in the Supplement to the National Electrical Code containing list of approved 
fittings. 

31 A. Theater Wiring. — (For rules governing Moving Picture Machine, 
see No. 65 A.) 

All wiring, apparatus, etc., not specifically covered by special rules herein 



INSIDE WORK—CONSTANT-LOW-POTENTtAU 1415 

given must conform to the Standard Rules and Requirements of the National 
Electrical Code. 

In so far as these Rules and Requirements are concerned, the term 
"theater" shall mean a building or part ol a building in which it is 
designed to make a presentation of dramatic, operatic or other per- 
formances or shows for the entertainment of spectators which is 
capable of seating at least four hundred persons, and which has a 
stage for such performances that can be used for scenery and other 
stage appliances. 

a. Services. — 1. Where source of supply is outside of building, there 
must be at least two separate and distinct services where practicable, fed 
from separate street mains, one service to be of sufficient capacity to supply 
current for the entire equipment of theater, while the other service must be 
at least of sufficient capacity to supply current for all emergency lights. 

By "emergency lights" are meant exit lights and all lights in lobbies, stairways, 
corridors and other portions of theater to which the public have excess which are 
normally kept lighted during the performance. 

2. Where source of supply is an isolated plant within same building, an 
auxiliary service of at least sufficient capacity to supply all emergency lights 
must be installed from some outside source, or a suitable storage battery 
within the premises may be considered the equivalent of such service. 

b. Stage. — 1. All permanent construction on stage side of proscenium 
wall must be approved conduit, with the exception of border and switchboard 
wiring. 

2. Switchboards. — Must be made of non-combustible, non-absorptive 
material, and where accessible from stage level must be protected by an 
approved guard rail to prevent accidental contact with live parts on the board. 

3. Footlights. — a. Must be wired in approved conduit, each lamp re- 
ceptacle being enclosed within an approved outlet box, the whole to be 
enclosed in a steel trough, metal to be of a thickness not less than No. 20 
gage, or each lamp receptacle may be mounted on or in an iron or steel box 
so constructed as to enclose all the wires and live parts of receptacles. 

b. Must be so wired that no set of lamps requiring more than 1,320 watts 
will be dependent on one cut-out. 

4. Borders. — a. Must be constructed of steel of a« thickness not less 
than No. 20 gage, treated to prevent oxidization, be suitably stayed and sup- 
ported by a metal framework, and so designed that flanges of reflectors will 
protect lamps. 

b. Must be so wired that no set of lamps requiring more than 1,320 watts 
will be dependent upon one cut-out. 

c» Must be wired in approved conduit, each lamp receptable to be en- 
closed within an approved outlet box, the whole to be enclosed in 
a steel trough, or each lamp receptacle may be mounted on or in 
the cover of a steel box so constructed as to enclose all the wires and 
the live parts of receptacles, metal to be of a thickness not less 
than No. 20 gage. 

d. Must be provided with suitable guards to prevent scenery or other 

combustible material coming in contact with lamps. 

e. Cables must be continuous from stage switchboard to border; conduit 

construction must be used from switchboard to point where cables 
must be flexible to permit of the raising and lowering of border, 
and flexible portion must be enclosed in an approved fireproof hose 
or braid and be suitably supported. 

Jimctlon boxes will be allowed on fly floor and rigging loft In existing 
theaters where the wiring has been completed and approved by Inspec- 
tion Department having jurisdiction, 

/. For the wiring of the border proper, wire with slow burning insulation 

should be used. 
g. Must be suspended with wire rope, same to be insulated from border 

by at least two approved strain insulators properly inserted. 
6. Stage Pockets. — Must be of approved type controlled from switch- 
board, each receptacle to be of not less than fifty amperes rating, and each 
receptacle to be wired with a separate circuit to its full capacity. 

6. Proscenium Side Lights. — Must be so installed that they cannot in 
terfere with the operation of or come in contact with curtain. 



1416 70.— ELECTRIC POWER AND LIGHTING. 

7. Scene Docks. — Where lamps are installed in Scene Docks, they must 
be so located and installed that they will not be liable to mechanical injury. 

8. Curtain Motors, — Must be of ironclad type and installed so as to 
conform to the requiremeijts of the National Electrical Code. (See No. 8.) 

9. Control for Stage Flues:- — 

a. In cases where dampers are released by an electric device, the electric 
circuit operating same must be normally closed. 

b. Magnet operating damper must be wound to take full voltage of 

circuit by which it is supplied, using no resistance device, and 
must not heat more than normal for apparatus of similar construc- 
tion. It must be located in loft above scenery, and be installed in 
a suitable iron box with a tight self-closing door. 

c. Such dampers must be controlled by at least two standard single pole 

switches mounted within approved iron boxes provided with self- 
closing doors without lock or latch, and located, one at the Elec- 
trician's station, and others. as designated by the Inspection De- 
partment having jurisdiction. 

c. Dressing Rooms. 

1. Must be wired in approved conduit, except that in existing buildings 
where it is impracticable to install approved conduit, approved SLTmoredcahle 
may be used, provided it is installed in accordance with No. 24 A. 

2. All pendant lights must be equipped with approved reinforced cord 
or cable. 

3. All lamps must be provided with approved guards. 

d. Portable Equipments. 

1. Arc lamps used for stage effects must conform to the following 
requirements: — 

o. Must be constructed entirely of metal except where the use of approved 
insulating material is necessary. 

h. Must be substantially constructed, and so designed as to provide for 
proper ventilation, and to prevent sparks being emitted from lamps 
when same is in operation, and mica must be used for frame in- 
sulation. 

c. Front opening must be provided with a self-closing hinged door frame 

in which wire gauze or glass must be inserted, excepting lens 
lamps, where the front may be stationary, and solid door be pro- 
vided on back or side. 

d. Must be provided with a one-sixteenth-inch iron or steel guard having 

a mesh not larger than one inch, and be substantially placed over 
top and upper half of sides and back of lamp frame; this guard to be 
substantially riveted to frame of lamp, and to be placed at a dis- 
tance of at least two inches from the lamp frame. 

e. Switch on standard must be so constructed that accidental contact 

with any live portion of same will be impossible. 

/. All stranded connections in lamp and at switch and rheostat must be 
provided with approved lugs. 

g. Rheostat, if mounted on standard, must be raised to a height of at 
least three inches above floor line, and in addition to being properly 
enclosed must be surrounded with a substantially attached metal 
guard having a mesh not larger than one square inch, which guard 
is to be kept at least one inch from outside frame of rheostat. 

h» A competent operator must be in charge of each arc lamp, except 
that one operator may have charge of two lamps when they are 
not more than ten feet apart, and are so located that he can prop- 
erly watch and care for both lamps. 

2. Bunches: — a. Must be substantially constructed of metal, and must 
not contain any exposed wiring. 

b. The cable feeding same must be bushed in an approved manner 
where passing through the metal, and must be properly secured to 
prevent any mechanical strain from coming on the connection. 

3. Strips. — a. Must be constructed of steel of a thickness not less than 
No. 20 gage, treated to prevent oxidization, and suitably stayed and sup- 
ported by metal framework. 

b. Cable feeding must be bushed in an approved manner where passing 
through the metal, and must be properly secured to prevent any 
mechanical strain from coming on the connections. 



INSIDE WORK— CONSTANT-LOW-POTENTIAL. 1417 

4. Portable Plugging Boxes. — Must be constructed so that no current 
carrying part will be exposed, and each receptacle must be protected by 
approved fuses mounted on slate or marble bases and enclosed in a fireproof 
cabinet equipped with self-closing doors. Each receptacle must be con- 
structed to carry thirty amperes without undue heating, and the bus-bars 
must have a carrying capacity equivalent to the current required for the 
total number of receptacles, allowing thirty amperes to each receptacle, 
and approved lugs must be provided for the connection of the master cable. 

5. Pin Plug Conductors— a. When of approved type may be used to 
connect approved portable lights and appliances. 

b. Must be so installed that the "female" part of plug will be on the live 
end of cable, and must be so constructed that tension on the cable 
will not cause any serious mechanical strain on the connections. 

6. Lights on Scenery. — Where brackets are used they must be wired 
entirely on the inside, fixture stem must come through to the back of the 
scenery and end of stem be properly bushed. 

7. String or Festoon Lights. — Wiring for same should be approved 
cable, joints where taps are taken from same for lights to be properly made, 
soldered and taped, and where lamps are used in lanterns or similar devices 
lamps must be provided with approved guards. Where taps are taken from 
cable, they should be so staggered that joints of different polarity will not 
come immediately opposite each other and must be properly protected 
from strain. 

' 8. Special Electrical Effects. — Where devices are used for producing 
special effects such as lightning, waterfalls, etc., the apparatus must be so con- 
structed and located that flames, sparks, etc., resulting from the operation 
cannot come in contact with combustible material. 

e. Auditorium. — 1. All wiring must be installed in approved conduit, 
except that in existing buildings where it is impracticable to install approved 
conduit, approved armored cable may be used, provided it is installed in 
accordance with No. 24 A. 

2. All fuses used in connection with lights illuminating all parts of the 
house used by the audience must be installed in fireproof enclosures so con- 
structed that there will be a space of at least six inches between the fuses 
and the sides and face of enclosure. 

3. Exit lights must not have more than one set of fuses between same 
and service fuses. 

4. Exit lights and all lights in halls, corridors or any other part of the 
building used by the audience, except the general auditorium lighting, 
must be fed independently of the stage lighting, and must be controlled 
only from the lobby or other convenient place in front of the house. 

5. Every portion of the theater devoted to the use or accommodation 
of the public, also all outlets leading to the streets and including all open 
courts, corridors, stairways, exits and emergency exit stairways, should be 
well and properly lighted during every performance, and the same should 
remain lighted until the entire audience has left the premises. 

32. Car Wiring and Equipment of Cars. — a. Protection of Car 
Body, etc. — 1. Under side of car bodies to be protected by approved fire- 
resisting, insulating material, not less than i-inch in thickness, or by sheet 
iron or steel, not less than .04-inch in thickness, as specified in Section a, 2, 
3 and 4. This protection to be provided over all electrical apparatus, such 
as motors with a capacity of over 75 H. P. each, resistances, contactors, 
lightning arresters, air-brake motors, etc., and also where wires are run, 
except that protection may be omitted over wires designed to carry 25 
amperes or less if they are encased in metal conduit. 

2. At motors of over 75 H. P. each, fire-resisting material or sheet iron or 
steel to extend not less than 8 inches beyond all edges of openings in motors, 
and not less than 6 inches beyond motor leads on all sides. 

3. Over resistances, contactors, and lightning arresters, and other 
electrical apparatus, excepting when amply protected by their casing, fire- 
resisting material or sheet iron or steel to extend not less than 8 inches 
beyond all edges of the devices. 

4. Over conductors, not encased in conduit, and conductors in conduit 



1418 70.— ELECTRIC POWER AND LIGHTING, 

when designed to carry over 25 amperes, unless the conduit is so supported 
as to give not less than -l-inch clear air space between the conduit and the 
car, fire-resisting material or sheet iron or steel to extend at least 6 inches 
beyond conductors on either side. 

The flre-reslsting insulating material or sheet iron or steel may be omitted over 
cables made up of flameproof braided outer covering when surrounded by i-inch 
flameproof covering, as called for by Section i, 4. 

5. In all cases fireproof material or sheet iron or steel to have joints 
well fitted, to be securely fastened to the sills, floor timbers and cross braces, 
and to have the whole surface treated with a waterproof paint. 

6. Cut-out and switch cabinets to be substantially made of hard wood. 
The entire inside of cabinet to be lined with not less than |-inch fire-resisting 
insulating material which shall be securely fastened to the woodwork, and 
after the fire-resisting material is in place the inside of the cabinet shall be 
treated with a waterproof paint. 

b. Wires, Cables, etc. — 1. All conductors to be stranded, the allowable 
carrying capacity being determined by Table "A" of No. 16, except that 
motor, trolley and resistance leads shall not be less than No. 7 B. & S. gage, 
heater circuits not less than No. 12 B. & S. gage, and lighting and other 
auxiliary circuits not less than No. 14 B & S. gage. 

The current used in determining the size of motor, trolley and resist- 
ance leads shall be the per cent of the full load current, based on one hour's 
run of motor, as given by the following table: — 

Size of each Motor Trolley Resistance 

Motor. Leads. Leads. Leads. 

75 H. P. or less 50% 40% 15% 

Over75H. P 45% 35% 15% 

Fixture wire complying with No. 46 will be permitted for wiring approved cluster. 

2. To have an insulation and braid as called for by No. 41 for wires 
carrying currents of the same potential. 

3. When run in metal conduit, to be protected by an additional braid 
as called for by No. 47. 

Where conductors are laid In conduit, not being drawn through, the additional 
braid will not be required 

4. When not in conduit, in approved moulding, or in cables surrounded 
by i-inch flameproof covering, must comply with the requirements of 
No. 41 (except that tape may be substituted for braid) and be protected 
by an additional flameproof braid, at least 1-32 of an inch in thickness, 
the outside being saturated with a preservative flameproof compound. 

This rule will be interpreted to include the leads from the motors. 

5. Must be so spliced or joined as to be both mechanically and electric- 
ally secure without solder. The joints must then be soldered and covered 
with an insulation equal to that on the conductors. 

Joints made with approved splicing devices and those connecting the leads at 
motors, plows or third-rail shoes need not be soldered. 

6. All connections of cables to cut-outs, switches and fittings, except 
those to controller connection boards, when designed to carry over 25 
amperes, must be provided with lugs or terminals soldered to the cable, and 
securely fastened to the device, by bolts, screws or by clamping; or, the 
end of the cable, after the insulation is removed, shall be dipped in solder 
and be fastened into the device by at least two set screws having check nuts. 

All connections for conductors to fittings, etc., designed to carry less 
than 25 amperes, must be provided with up-turned lugs that will grip the 
conductor between the screw and the lug, the screws being provided with 
flat washers; or by block terminals having two set screws, and the end of 
the conductors must be dipped in solder. Soldering, in addition to the 
connection of the binding screws, is strongly recommended, and will be 
insisted on when above requirements are not complied with. 

This rule will not be construed to apply to circuits where the maximum potential 
Is not over 25 volts, and current does not exceed 5 amperes. 

C. Cut-outs, Circuit-Breakers and Switches. — 1. All cut-outs and switches 
having exposed live metal parts to_ be located in cabinets. Cut-outs and 
switches, not in iron boxes or in cabinets, shall be mounted on not less than 
i-inch fire-resisting insulating material, which shall project at least i-inch 
beyond all sides of the cut-out or switch. 



INSIDE WORK'-CONSTANT'LOW-POTENTIAL. 1419 

2. Cut-outs to be of the approved cartridge or approved blow-out type. 

3. All switches controlling circuits of over 5 ampere capacity shall be 
of approved single-pole, quick-break or approved magnetic blow-out type. 

Switches controlling circuits of 5 ampere or less capacity may be of the 
approved single-pole, double-break, snap type. 

4. Circuit breakers to be of approved type. 

6. Circuits must not be fused above their safe carrying capacity. 

6. A cut-out must be placed as near as possible to the current collector, 
so that the opening of the fuse in this cut-out will cut off all current from 
the car. 

When cars are operated by metallic return circuits, the circuit breakers connected 
to both sides of the circuit, no fuses in addition to the circuit breakers will be required. 

d. Conduit. — [When from the nature of the case, or on account of the 
size of the conductors, the ordinary pipe and junction box construction is 
not permissible, a special form of conduit system may be used, provided 
the general requirements as given below are complied with.] 

1. Metal conduits, outlet and junction boxes to be constructed in accord- 
ance with Nos. 49 and 49 A, except that conduit for lighting circuits 
need not be over 5/16 inch internal diameter and ^-inch external diameter, 
and for heating and air motor circuits need not be over |-inch internal 
diameter and 9/16-inch external diameter, and all conduits where exposed 
to dampness must be water tight. 

2. Must be continuous between and be firmly secured into all outlet or 
junction boxes and fittings, making a thorough mechanical and electrical 
connection between same. 

3. Metal conduits, where they enter all outlet or junction boxes and 
fittings, must be provided with approved bushings fitted so as to protect 
cables from abrasion. 

4. Except as noted in Section i, 2, must have the metal of the conduit 
permanently and effectively grounded. 

5. Junction and outlet boxes must be installed in such a manner as to 
be accessible. 

6. All conduits, outlets or junction boxes and fittings to be firmly and 
substantially fastened to the framework of the car. 

e. Moulding.- — 1. To consist of a backing and a capping and to be 
constructed of fire-resisting insulating material, except that it may be made 
of hard wood where the circuits which it is designed to support are normally 
not exposed to moisture. 

2. When constructed of fire-resisting insulating material, the backing 
shall not be less than ^-inch in thickness and be of a width sufficient to 
extend not less than 1 inch beyond conductors at sides. 

The capping, to be not less than i-inch in thickness shall cover and 
extend at least |-inch beyond conductors on either side. 

The joints in the moulding shall be mitered to fit close, the whole 
material being firmly secured in place by screws or nails, and treated on 
the inside and outside with a waterproof paint. 

When fire-resisting moulding is used over surfaces already protected by i-inch 
fire-resisting insulating material no backing will be required. 

3. Wooden mouldings must be so constructed as to thoroughly encase 
the wire and provide a thickness of not less than |-inch at the sides and 
back of the conductors, the capping being not less than 3/16-inch in thick- 
ness. Must have both outside and inside two coats of waterproof paint. 

The backing and the capping shall be secured in place by screws. 

f. Lighting and Lighting Circuits. — 1. Each outlet to be provided with 
an approved porcelain receptacle, or an approved cluster. No lamp of over 
32 candle power to be used. 

2. Circuits to be run in approved metal conduit, or approved moulding. 

3. When metal conduit is used, except for sign lights, all outlets to be 
provided with approved outlet boxes. 

4. At outlet boxes, except where approved clusters are used, porcelain 
receptacles to be fastened to the inside of the box, and the metal cover to 
have an insulating bushing around opening for the lamp. 



1420 10.— ELECTRIC POWER AND LIGHTING. 

When approved clusters are used, the cluster shall be thoroughly insu- 
lated from the metal conduit, being mounted on a block of hard wood or 
fire-resisting insulating material. 

5. Where conductors are run in moulding the porcelain receptacles or 
cluster to be mounted on blocks of hard wood or of fireproof insulating 
material. 

g. Heaters and Heating Circuits. — 1. Heaters to be of approved type. 

2. Panel heaters to be so constructed and located that when heaters 
are in place all current-carrying parts will be at least 4 inches from all 
woodwork. 

Heaters for cross seats to be so located that current-carrying parts will 
be at least 6 inches below under side of seat, unless under side of seat is 
protected by not less than i-inch fire-resisting insulating material, of .04 
inch sheet metal with 1 inch air space over same, when the distance may be 
reduced to 3 inches. 

2. Circuits to be run in approved metal conduit, or in approved moulding, 
or if the location of conductors is such as will permit an air space of not 
less than 2 inches on all sides except from the surface wired over, they may 
be supported on porcelain knobs or cleats, provided the knobs or cleats 
are mounted on not less than ^-inch fire-resisting insulating material extend- 
ing at least 3 inches beyond conductors at either side, the supports raising 
the conductors not less than i-inch from the surface wired over, and being 
not over 12 inches apart. 

h. Air Pump Motor and Circuits. — 1. Circuits to be run in approved 
metal conduit or in approved moulding, except that when run below the 
floor of the car they may be supported on porcelain knobs or cleats, pro- 
vided the supports raise the conductor at least i-inch from the surface 
wired over and are not over 12 inches apart. 

2. Automatic control to be enclosed in approved metal box. Air pump 
and motor, when enclosed, to be in approved metal box or a wooden box 
lined with metal of not less than 1/32 inch in thickness. 

When conductors are run in metal conduit the boxes surrounding auto- 
matic control and air pump and motor may serve as outlet boxes. 

i. Main Motor Circuits and Devices. — 1. Conductors connecting between 
trolley stand and main cut-out or circuit breakers in hood to be protected 
where wires enter car to prevent ingress of moisture. 

2. Conductors connecting between third rail shoes on same truck, to 
be supported in an approved fire-resisting insulating moulding, or in approved 
iron conduit supported by soft rubber or other approved insulating cleats. 

3. Conductors on the under side of the car, except as noted in Section i, 4, 
to be supported in accordance with one of the following methods: — 

a. To be run in approved metal conduit, junction boxes being provided 

where branches in conduit are made, and outlet boxes where 
conductors leave conduit. 

b. To be run in approved fire-resisting insulating moulding. 

c. To be supported by insulating cleats, the supports being not over 

12 inches apart. 

4. Conductors with flameproof braided outer covering, connecting 
between controllers at either end of car, or controllers and contactors, may 
be run as a cable, provided the cable where exposed to the weather is en- 
cased in a canvas hose or canvas tape, thoroughly taped or sewed at ends 
and where taps from the cable are made, and the hose or tape enters the 
controllers. 

Conductors with or without flameproof braided outer covering connect- 
ing between controllers at either end of the car, or controllers and contactors, 
may be run as a cable, provided the cable throughout its entire length is 
surrounded by i-inch flameproof covering, thoroughly taped or sewed at 
ends, or where taps from cable are made, and the flameproof covering 
enters the controllers. 

Cables, where run below floor of car, may be supported by approved 
insulating straps or cleats. Where run above floor of car, to be in a metal 
conduit or wooden box painted on the inside with not less than two coats 
of flameproof paint, and where this box is so placed that it is exposed to 
water, as by washing of the car floor, attention should be given to making 
the box reasonably waterproof. 



INSIDE WORK— CONSTANT-LOW-POTENTIAL, 1421 

Canvas hose or tape, or flameproof material surrounding cables after 
conductors are in same, to have not less than two coats of waterproof 
insulating material. 

5. Motors to be so drilled that, on double truck cars, connecting 
cables can leave motor on side nearest to kingbolt. 

6. Resistances to be so located that there will be at least 6-inch air 
space between resistances proper and fire-resisting material of the car. 
To be mounted on iron supports, being insulated by non-combustible 
bushings or washers, or the iron supports shall have at least 2 inches of 
insulating surface between them and metal work of car, or the resistances 
may be mounted on hard wood bars, supported by iron stirrups, which 
shall have not less than 2 inches of insulating surface between foot of 
resistance and metal stirrup, the entire surface of the bar being covered 
with at least i-inch fire-resisting insulating material. 

The insulation of the conductor, for about 6 inches from terminal of 
the resistance, should be replaced, if any insulation is necessary, by a 
porcelain bushing or asbsetos sleeve. 

7. Controllers to be raised above platform of car by not less than 
1-inch hard wood block, the block being fitted and painted to prevent 
moisture working in between it and the platform. 

j. Lightning Arresters. — 1. To be preferably located to protect all 
auxiliary circuits in addition to main motor circuits. 

2. The ground conductor shall be not less than No. 6 B. & S. gage, 
run with as few kinks and bends as possible, and be securely grounded. 

k. General Rules. — 1. When passing through floors, conductors or 
cables must be protected by approved insulating bushings, which shall fit 
the conductor or cable as closely as possible. 

2. Moulding should never be concealed except where readily acces- 
sible. Conductors should never be tacked into moulding. 

3. Short bends in conductors should be avoided where possible. 

4. Sharp edges in conduit or in moulding must be smoothed to pre- 
vent injury to conductors. 

Z3, Car Houses. — ^a. The trolley wires must be securely supported on 
insulating hangers. 

b. The trolley hangers must be placed at such a distance apart that, 
in case of a break in the trolley wire, contact with the floor cannot be 
made. 

c. Must have an emergency cut-out switch located at a proper place 
outside of the building, so that all the trolley wires in the building may be 
cut out at one point, and line insulators must be installed, so that when 
this emergency switch is open, the trolley wire will be dead at all points 
within 100 feet of the building. The current must be cut out of the build- 
ing when not needed for use in the building. 

This may be done by the emergency switch, or If preferred, a second switch may 
be used that will cut out all current from the building, but which need not cut out 
the trolley wire outside as would be the case with the emergency switch. 

d. All lamps and stationary motors must be installed in such a way 
that one main switch may control the whole of each installation, lighting 
and power, independently of the main cut-out switch called for in 
Section c. 

e. Where current for lighting and stationary motors is from a grounded 
trolley circuit, the following special rules to apply: — 

1. Gut-outs must be placed between the non-grounded side and 

lights or motors they are to protect. No set or group of incan- 
descent lamps requiring over 2,000 watts must be dependent 
upon one cut-out. 

2. Switches must be placed between non-grounded side and lights 

and motors they are to protect. 

8. Must have all rails bonded at each joint with a conductor having 

a carrying capacity at least equivalent to No. 00 B. & S. gage 
annealed copper wire, and all rails must be connected to the out- 
side ground return circuit by not less than No. 00 B. & S. gage 
copper wire or by equivalent bonding through the track. All 



1422 10.— ELECTRIC POWER AND LIGHTING. 

lighting and stationary motor circuits must be thoroughly and 
permanently connected to the rails or to the wire leading to 
the outside ground return circuit. 

• f. All pendant cords and portable conductors will be considered as 
subject to hard usage (see 45, f). 

g. Must, except as provided in Section e, have all wiring and apparatus 
installed in accordance with the rules for constant potential systems. 

h. Must not have any system of feeder distribution centering in the 
building. 

i. Cars must not be left with the trolley in electrical connection with 
the trolley wire. 

34. Lighting and Power from Railway Wires. — a. Must not he per- 
mitted, under any pretense, in the same circuit 'with trolley wires with a 
ground return, except in electric railway cars, electric car houses and their 
power stations', nor shall the same dynamo he used for both purposes, 

CONSTANT-HIQH=POTENTIAL SYSTEMS. 

550 TO 3,500 Volts. 

Any circuit attached to any machine or combination of machines which 
develops a difference of potential between any two wires, of 
over 550 volts and less than 3,500 volts, shall be considered as a 
high-potential circuit, and as coming under that class, unless an 
approved transforming device is used, which cuts the difference 
of potential down to 550 volts or less. 
(See note following first paragraph under Low-Potential S3rstems, page 1408.) 

35. Wires. — (See also Nos. 14, 15 and 16.) a. Must have an a^^yoz;^^ 
rubber-insulating covering (see No. 41). 

b. Must be always in plain sight and never encased, except as provided 
for in No. 8 b, or where required by the Inspection Department having 
jurisdiction. 

c. Must (except as provided for in No. 8 b), be rigidly supported on 
glass or porcelain insulators, which raise the wire at least 1 inch from the 
surface wired over, and must be kept about 8 inches apart. 

Rigid supporting requires under ordinary conditions, where wiring along flat 
surfaces, supports at least about every 4^^ feet. If tlie wires are unusually liable to 
be disturbed, the distance between supports should be siiortened. 

In buildings of mill construction, mains of not less than No. 8 B. & S. gage, 
where not liable to be disturbed, may be separated about ten inches and run from 
timber to timber, not breaking around, and may be supported at each timber only. 

d. Must be protected on side walls from mechanical injury by a sub- 
stantial boxing, retaining an air space of 1 inch around the conductors, 
closed at the top (the wires passing through bushed holes) and extending 
not less than 7 feet from the floor. When crossing floor timbers, in cellars, 
or in rooms where they might be exposed to injury, wires must be at- 
tached by their insulating supports to the under side of a wooden strip 
not less than ^-inch in thickness. 

For general suggestions on protection, see note under No. 24 e. See also note 
under No. 1 8 e. 

36. Transformers. — (When permitted inside buildings under No. 13.) 
(For construction rules see No. 62.) (See also Nos. 13 and 18 A.) 

Transformers must not be placed Inside of buildings without special permission 
from the Inspection Department having jurisdiction. 

a. Must be located as near as possible to the point at which the 
primary wires enter the building. 

bo Must be placed in an enclosure constructed of fire-resisting material; 
the enclosure to be used only for this purpose, and to be kept securely 
locked, and access to the same allowed only to responsible parties. 

c. Must be thoroughly insulated from the ground, or permanently 
and effectually grounded, and the enclosure in which they are placed 
must be practically air-tight, except that it must be thoroughly ventilated 
to the outdoor air, if possilDle through a chimney or flue. There should 
be at least 6 inches air space on all sides of the transformer. 



INSIDE WORK—CONST.-POTEN, FITTINGS. 1423 

37 Series Lamps. — a. No multiple-series or series-multiple system 

of lighting will be approved. 

be Must not, under any circumstances, be attached to gas fixtures. 

CONSTANT EXTRA-HIQH-POTENTIAL SYSTEMS. 

Over 3.500 Volts. 

Any circuit attached to any machine or combination of machines which 
develops a difference of potential, between any two wires, of over 
3,500 volts, shall be considered as an extra-high-potential circuit, 
and as coming under that class, unless an approved transforming 
device is used, which cuts the difference of potential down to 
3,500 volts Of- less. 

38. Primary Wires. — a. Must not be brought into or over buildings, ex- 
cept power stations and sub-stations. 

39. Secondary Wires. — a. Must be installed under rules for high- 
potential systems when their immediate primary wires carry a current at a 
potential of over 3,500 volts, unless the primary wires are installed in 
accordance with the requirements as given in No. 12 A or are entirely under- 
ground, within city, town and village limits. 

Class D.— FITTINGS, MATERIALS AND DETAILS OF 
CONSTRUCTION. 

(Light, Power and Heat. For Signaling Systems, see Class E.) 

ALL SYSTEMS AND VOLTAGES. 

The following rules are but a partial outline of requirements. Devices 
or materials which fulfill the conditions of these requirements 
and no more, will not necessarily be acceptable. All fittings and 
materials should be submitted for examination and test before 
being introduced for use. 

Insulated Wires — Rules 40 to 43. 

40. General Rules- — a. Copper for insulated solid conductors of No. 4 
B. & S. gage and smaller must not vary in diameter more than .002 of an 
inch from the standard. On solid sizes larger than No. 4 B. & S. gage 
the diameter shall not vary more than one per cent from the specified 
standard. The conductivity of solid conductors shall not be less than 97% 
of that of pure copper of the specified size. 

In all stranded conductors the sum of the circular mils of the individual 
wires, shall not be less than the normal circular mils of the strand by more 
than one and one-half per cent The conductivity of the individual wires 
in a strand shall not be less than is given in the following table: — 

Number. 

23 
24 
25 
26 
27 
28 
29 
30 

The Standard for diameters and mileages shall be that adopted by the American 
Institute of Electrical Engineers. 

b. Wires and cables of all kinds designed to meet the following specifi- 
cations must have a distinctive marking the entire length of the coil so that 
they may be readily identified in the field. They must also be plainly 
tagged or marked as follows: — 

1, The maximum voltage at which the wire is designed to be used. 

2. The words "National Electrical Code Standard." 



Number. 


Per cent. 


14 and larger, 


. 97, 


.0 


15 


96 


.8 


16 


96 


.6 


17 


96 


.4 


18 


96 


.2 


19 


96 


.0 


20 


95 


.8 


21 


95 


.6 


22 


96 


.4 



Per cent. 


95 


.2 


95 


.0 


94 


.8 


94 


.6 


94 


.4 


94 


.2 • 


94 


.0 


93 


.8 



1424 IQ.-^ELECTRIC POWER AND LIGHTING, 

3. Name of the manufacturing company and, if desired, trade name of 

the wire. 

4. Month and year when manufactured. 

Wires described under Nos. 42, 43 and 44 need not have the distinctive marking, 
but are to be tagged. 

41. Rubber-Covered Wire. — a. Copper for conductors must be thor- 
oughly tinned. 

Insulation for Voltages, to 600 inclusive. — b. Must be of rubber or 
other approved substances, homogeneous in character, adhering to the con- 
ductor and of a thickness not less than that given in the following table: — 
B. & S. Gage. Thickness. 

18 to 16 1-32 inch. 

15 to 8 3-64 " 

7 to 2 1-16 •• 

1 to 0000 5-64 •• 

Circular Mils. 

250,000 to 500,000 3-32 " 

500,000 to 1,000,000 7-64 " 

Over 1,000,000 1-8 " 

Measurements of Insulating wall are to be made at the thinnest portion of the 
dielectric. 

c. The completed coverings must show an insulation resistance of at 
least 100 megohms per mile during thirty days' immersion in water at 
70° Fahrenheit (21° Centigrade). 

d. Each foot of the completed covering must show a dielectric strength 
sufficient to resist throughout five minutes the application of an electro- 
motive force proportionate to the thickness of insulation in accordance 
with the following table: — 

Thickness Breakdown Test Thickness Breakdown Test 

in 64ths inches. on 1 foot. in 64ths inches. on 1 foot. 



1 


3,000 Volts A. C. 


7 


16,500 Volts A. C 


2 


6,000 " 


8 


18,000 " 


3 


9.000 •• 


10 


21,000 " 


4 


11,000 " 


12 


23.500 •• 


5 


13,000 " 


14 


26,000 '• 


6 


15,000 " 


16 


28,000 •• 



The source of alternating electro-motive force shall be a transformer of 
at least one kilowatt capacity. The application of the electro-motive force 
shall first be made at 4,000 volts for five minutes, and then the voltage 
increased by steps of not over 3.000 volts, each held for five minutes, until 
the rupture of the insulation occurs. The tests for dielectric strength shall 
be made on a sample of wire which has been immersed in water for seventy- 
two hours. One foot of the wire under test is to be submerged in a conduct- 
ing liquid held in a metal trough, one of the transformer terminals being 
connected to the copper of the wire and the other to the metal of the trough. 

Insulations for Voltages, 601 to 3,500 inclusive. — e. The thickness of the 
insulating wall must not be less than that given in the following table: — 

B. & S. Gage. Thickness. 

14 to 1 3-32 inch. 

to 0000 3-32 " covered by tape or braid. 

Circular Mils. 

250.000 to 500,000 3-32" 

Over 500.000 1-8 " 

f. The requirements as to insulation and breakdown resistance for wiies 
for low-potential systems shall apply, with the exception that an insulation 
resistance of not less than 300 megohms per mile shall be required. 

Insulations for Voltages Over 3.500. — g. Wire for arc light circuits exceed- 
ing 3,500 volts potential must have an insulating wall not less than -rs of 
an inch in thickness, and shall withstand a breakdown test of at least 
23,500 volts, and have an insulation of at least 500 megohms per mile. 

The tests on this wire to be made under the same conditions as for low- 
potential wires. 



CONSTRUCTION— FITTINGS, MATERIALS, ETC, 1425 

Specifications for Insulations for alternating currents exceeding 3,500 volts have 
been considered, but on account of ttie somewliat complex conditions of such work, 
it has so far been deemed inexpedient to specify general insulations for this use* 

General. — h. The rubber compound or other approved substance used 
as insulation must be sufficiently elastic to permit all wires smaller than No. 7 
B. & S. gage and larger than No. 11 B. & S gage to be bent without injury 
to the insulation around a cylinder twice the diameter of the insulated 
wire measured over the outer covering. All wires No. 11 B. & S. gage and 
smaller to be bent without injury to the insulation around a cylinder equal 
to the diameter of the insulated wire measured over the outer covering. 

i. All the above insulations must be protected by a substantial braided 
covering, properly saturated with a preservative compound. This covering 
must be sufficiently strong to withstand all the abrasions likely to be met 
with in practice, and must substantially conform to approved samples 
submitted by the manufacturer. 

42. SIow=burning Weatherproof Wire. — [This wire is not as burnable 
as "weatherproof" nor as subject to softening under heat. It is not suitable 
for outside work.] 

a. The insulation must consist of two coatings, one to be fireproof in 
character and the other to be weatherproof. The fireproof coating must be 
on the outside and must comprise about 6/10 of the total thickness of the 
wall. The completed covering must be of a thickness not less than that 
given in the following table: — 

B. & S. Gage. Thickness. 

14 to 8 3-64 inch. 

7to 2 1-16 " 

1 to 0000 5-64 " 

Circular Mils. 

250,000 to 500,000 3-32 " 

500,000 to 1,000,000 7-64 " 

Over 1,000.000 1-8 " 

Measurements of Insulating wall are to be made at the thinnest portion. 

b. The fireproof coating shall be of the same kind as that required for 
"slow-burning wire," and must be finished with a hard, smooth surface. 

c. The weatherproof coating shall consist of a stout braid, applied and 
treated as required for "weatherproof wire." 

43. Slow-burning Wire. — a. The insulation must consist of three 
braids of cotton or other thread, all the interstices of which must be filled 
with the fireproofing compound or with material having equivalent resisting 
and insulating properties. The outer braid must be specially designed to 
withstand abrasion, and its surface must be finished smooth and hard. 
The completed covering must be of a thickness not less than that given in 
the table under No. 42 a. 

The solid constituent of the fireproofing compound must not be susceptible to 
moisture, and must not burn even when ground in an oxidizable oil, making a com- 
pound, which, while proof against fire and moisture, at the same time has considerable 
elasticity, and which when dry will suffer no change at a temperature of 2 50° Fahren- 
heit (121^ Centigrade), and which will not burn at even a higher temperature. 

This Is practically the old so-called "underwriters" insulation. It is especially 
useful in hot, dry places where ordinary insulations would perish, and where wires 
are bunched, as on the back of a large switchboard or in a wire tower, so that the 
accumulation of rubber Insulation would result in an objectionably large mass of 
highly inflammable material. 

44 Weatherproof Wire. — a. The insulating covering shall consist of 
at least three braids, all of which must be thoroughly saturated with a 
dense moisture-proof compound, applied in such a manner as to drive any 
atmospheric moisture from the cotton braiding, thereby securing a covering 
to a great degree waterproof and of high insulating power. This compound 
must retain its elasticity at 0° Fahrenheit (minus 18° Centigrade), and 
must not drip at 160° Fahrenheit (71° Centigrade). The thickness of insu- 
lation must not be less than that given in the table under No. 42 a, and the 
outer surface must be thoroughly slicked down. 

This wire is for use outdoors, where moisture is certain and where fire- 
proof qualities are not necessary. 



1426 n.— ELECTRIC POWER AND LIGHTING. 

45. Flexible Cord. — (For installation rules, see No. 28). a. Must, 
except as required for portable heating apparatus (see Section g), be made 
of stranded copper conductors, each strand to be not larger than No. 26 
or smaller than No. 30 B. & S. gage, and each stranded conductor must be 
covered by an approved insulation and protected from mechanical injury 
by a tough, braided outer covering. 

For Pendant Lamps. — [In this class is to be included all flexible cord 
which, under usual conditions, hangs freely in air, and which is not likely 
to be moved sufficiently to come in contract with surrounding objects. 

It should be noted that pendant lamps provided with long cords, so 
that they can be carried about or hung over nails, or on machinery, etc., 
are not included in this class, even though they are usually allowed to hang 
freely in air.] 

b. Each stranded conductor must have a carrying capacity equivalent 
to not less than a No. 18 B. & S. gage wire. 

c. The covering of each stranded conductor must be made up as fol- 
lows: — 

1. A tight, close wind of fine cotton. 

2. The insulation proper, which shall be waterproof. 

3. An outer cover of silk or cotton. 

The wind of cotton tends to prevent a broken strand puncturing the Insulation 
and causing a short circuit. It also keeps the rubber from corroding the copper. 

d. The insulation must be solid, at least 1/32 of an inch thick, and must 
show an insulation resistance of 50 megohms per mile throughout two 
weeks immersion in water at 70° Fahrenheit (21° Centigrade), and stand the 
tests prescribed for low-tension wires as far as they apply. 

e. The outer protecting braiding should be so put on and sealed in place 
that when cut it will not fray out, and where cotton is used it should be 
impregnated with a flameproof paint which will not have an injurious effect 
on the insulation. 

For Portables. — [In this class is included all cord used on portable lamps, 
small portable motors, or any device which is liable to be carried about.] 

f. Flexible cord for portable use, except in offices, dwellings or similar 
places, where cord is not liable to rough usage and where appearance is an 
essential feature, must meet all the requirements for flexible cord for "pend- 
ant lamps," both as to construction and thickness of insulation, and in 
addition must have a tough, braided cover over the whole. There must 
also be an extra layer of rubber between the outer cover and flexible cord, 
and in moist places the outer cover must be saturated with a moisture-proof 
compound, thoroughly slicked down, as required for "weatherproof wire." 
(See No. 44.) In offices, dwellings, or in similar places where cord is not 
liable to rough usage and where appearance is an essential feature, flexible 
cord for portable use must meet all of the requirements for flexible cord for 
"pendant lamps," both as to construction and thickness of insulation, and 
in addition must have a tough, braided cover over the whole, or, providing 
there is an extra layer of rubber between the flexible cord and the outer 
cover, the insulation proper on each stranded conductor of cord may be 
1/64 of an inch in thickness instead of 1/32 of an inch as requiried for 
pendant cords 

Flexible cord for portable use may, Instead of the outer coverings described 
above, have an approved metal flexible armor. 

For Portable Heating Apparatus. — [Applies to all smoothing and sad 
irons, and to any other device requiring over 250 watts.] 

g. Must be made up as follows: — 

1. Conductors must be of braided copper, each strand not to be larger 

than No. 30 or smaller than No. 36 B. «& S. gage. 
When conductors have a greater carrying capacity than No. 12 B. & S. gage they 
may be braided or stranded with each strand as large as No. 2 8 B. & S. 
gage. If stranded, there must be a tight, close wind of cotton between 
the conductor and the insulation. 

2. An insulating covering of rubber or other approved material not less 

than 1/64 inch in thickness . 

3. A braided covering of not less than 1/32 inch thick, composed of best 

quality long fiber asbestos, containing not over five per cent 
vegetable fiber. 



CONSTRUCTION— FITTINGS, MATERIALS, ETC. 1427 

4. The several conductors comprising the cord to be enclosed by an outer 
reinforcing covering not less than 1/64 inch thick, especially 
designed to resist abrasion, and so treated as to prevent the 
cover from fraying. 

46. Fixture Wire. — (For installation rules, see No. 24, v to y.) a. May 

be made of solid or stranded conductors, with no strands smaller than 
No. 30 B. & S. gage, and must have a carrying capacity not less than that 
of a No 18 B. & S. gage wire. 

b. Solid conductors must be thoroughly tinned. If a stranded con- 
ductor is used^ it must be covered by a tight, close wind of fine cotton. 

c. Must have a solid rubber insulation of a thickness not less than 
1/32 of an inch for Nos. 18 to 16 B. & S. gage, and 3/64 of an inch for Nos. 14 
to 8 B. & S. gage, except that in arms of fixtures not exceeding 24 inches 
in length and used to supply not more than one 1 6-candle-power lamp or 
its equivalent, which are so constructed as to render impracticable the use 
of a wire with 1/32 of an inch in thickness of rubber insulation, a thickness 
of 1/64 of an inch will be permitted. 

d. Must be protected with a covering at least 1/64 of an inch in thick- 
ness, sufficiently tenacious to withstand the abrasion of being pulled into 
the fixture, and sufficiently elastic to permit the wire to be bent around a 
cylinder with twice the diameter of the wire without injury to the braid. 

e. Must successfully withstand the tests specified in Nos. 41c and 41 d. 
In wiring certain designs of show-case fixtures, ceiling bulls-eyes and similar 

appliances in which the wiring is exposed to temperatures in excess of 120° Fahrenheit 
(49° Centigrade), from the heat of lamps, slow-burning wire may be used (see No 43). 
All such forms of fixtures must be submitted for examination, test and approval 
before being introduced for use. 

47.' Conduit Wire. — (For installation rules, see No. 24 n to p.) 

a. Single wire for lined conduits must comply with the requirements 
of No. 41. For unlined conduits it must comply with the same require- 
ments (except that tape may be substituted for braid) and in addition there 
must be a second outer fibrous covering, at least 1/32 of an inch in thick- 
ness and sufficiently tenacious to withstand the abrasion of being hauled 
through the metal conduit. 

b. For twin or duplex wires in lined conduit, each conductor must com- 
ply with the requirements of No. 41 (except that tape may be substituted 
for braid on the separate conductors), and must have a substantial braid 
covering the whole. For unlined conduit each conductor must comply 
with requirements of No. 41 (except that tape maybe substituted for braid), 
and in addition must have a braid covering the whole, at least 1/32 of an 
inch in thickness and sufficiently tenacious to withstand the abrasion of 
being hauled through the metal conduit. 

c. For concentric wire, the inner conductor must comply with the 
requirements of No. 41 (except that tape may be substituted for braid), 
and there must be outside of the outer conductor the same insulation as 
on the inner, the whole to be covered with a substantial braid, which, for 
unlined conduits, must be at least 1/32 of an inch in thickness, and suffi- 
ciently tenacious to withstand the abrasion of being hauled through the 
metal conduit. 

The braid or tape required around each conductor in duplex, twin and concentric 
cables is to hold the rubber insulation in place and prevent jamming and flattening. 
All the braids specified in this rule must be properly saturated with a preservative 
compound. 

48. Armored Cable. — rFor installation rules, see No. 24 A), a. The 

armor of such cables must nave at least as great strength to resist penetra- 
tion of nails, etc., as is required for metal conduits (see No. 49 b), and its 
thickness must not be less than that specified in the following table: — 



Nominal 


Actual 


Actual 




Internal 


Internal 


External 


Thickness 


[Diameter. 


Diameter. 


Diameter. 


of Wall. 


Inches. 


Inches. 


Inches. 


Inches. 


Vs 


.27 


.40 


.06 




.36 


.54 


.08 


^ 


.49 


.67 


.09 


H. 


.62 


.84 


.10 



1428 



70.— ELECTRIC POWER AND LIGHTING. 



Nominal 


Actual 


Actual 




Internal 


Internal 


External 


Thickness 


Diameter. 


Diameter. 


Diameter. 


of Wall. 


Inches. 


Inches. 


Inches. 


Inches. 


H 


.82 


1.05 


.11 


1 


1.04 


1.31 


.13 


ji^ 


1.38 


1.66 


.14 


1/^ 


1.61 


1.90 


.14 


2 


2.06 


2.37 


.15 


2H 


2.46 


2.87 


.20 


3 


3.06 


3.50 


.21 


3H 


3.54 


4.00 


.22 


4 


4.02 


4.50 


.23 


4J^ 


4.50 


5.00 


.24 


5 


5.04 


5.56 


.25 


6 


6.06 


6.62 


.28 



An allowance of .02 of an Inch for variation In manufacturing and loss of thick- 
ness by cleaning will be permitted. 

b. The conductors in same, single wire or twin conductors, must have 
an insulating covering as required by No. 41; if any filler is used to secure 
a round exterior, it must be impregnated with a moisture repellent, and the 
whole bunch of conductors and fillers must have a separate exterior covering. 

49. Interior Conduits. — (For installation rules, see Nos. 24 n to p and 
25.) a. Each length of conduit, whether lined or unlined, must have the 
maker's name or initials stamped in the metal or attached thereto in a 
satisfactory manner, so that inspectors can readily see the same. 

The use of paper stickers or tags cannot be considered satisfactory methods of 
marking, as they are readily loosened and lost off In tne ordinary handling of the 
conduit. 

Metal Conduits with Lining of Insulating Material — b. The metal cover- 
ing or pipe must be at least as strong as the ordinary commercial forms of 
gas pipe of the same size, and its thickness must be not lesss than that of 
standard gas pipe as specified in the table given in No. 48. 

c. Must not be seriously affected externally by burning out a wire 
inside the tube when the iron pipe is connected to one side of the circuit. 

d. Must have the insulating lining firmly secured to the pipe. 

e. The insulating lining must not crack or break when a length of the 
conduit is uniformly bent at temperature of 212° Fahrenheit (100° Centi- 
grade), to an angle of ninety degrees, with a curve having a radius of 15 ins., 
for pipes of one inch and less, and fifteen times the diameter of pipe for 
larger sizes. 

f. The insulating lining must not soften injiiriously at any temperature 
below 212° Fahrenhiet (100° Centigrade), and must leave water in which it 
is boiled practically neutral. 

g. The insulating lining must be at least 1/32 of an inch in thickness. 
The materials of which it is composed must be of such a nature as will not 
have a deteriorating effect on the insulation of the conductor, and be 
sufficiently tough and tenacious to withstand the abrasion test of drawing 
long lengths of conductors in and out of same. 

h. ^ The insulating lining must not be mechanically weak after three 
days' immersion in water, and when removed from the pipe entire, must 
not absorb more than ten per cent of its weight of water during 100 hours 
of submersion. 

i. All elbows or bends must be so made that the conduit or lining of 
same will not be injured. The radius of the curve of the inner edge of any 
elbow must not be less than 33^ inches. 



Unlined Metal Conduits. 
Trade Size. 
Inches. 



y2 



j. Pipe sizes to run as follows: — 

Approximate Internal Minimum Thickness 

Diameter. of Wall. 

Inches. Inches, 

.62 .100 

.82 .105 

1.04 .125 

1.38 .135 



CONSTRUCTION— FITTINGS, MATERIALS, ETC, 142§ 

Trade Size. Approximate Internal Minimum Thickness 

Inches. Diameter. of Wall. 

Inches. Inches. 

IH 1.61 .140 

2 2.06 .150 
2H 2.46 .200 

3 3.06 .210 
ZH 3.54 .220 

At no point (except at screw thread) shall the thickness of wall of finished 
conduit he less than the minimum specified In last column of above table. 

k. Pipe to be thoroughly cleaned to remove all scale. Pipe should be 
of siifficiently true circular section to admit of cutting true, clean threads, 
and should be very closely the same in wall thickness at all points with 
clean square weld. 

1. Cleaned pipe to be protected against effects of oxidation, by baked 
enamel, zinc or other approved coating which will not soften at ordinary 
temperatures, and of sufficient weight and toughness to successfully with- 
stand rough usuage likely to be received during shipment and installation; 
and of sufficient elasticity to prevent flaking when ^-inch conduit is bent in 
a curve the inner edge of which has a radius of 3i inches. 

m. All elbows or bends must be so made that the conduit will not be 
injured. The radius of the curve of the inner edge of any elbow not to be 
less than 3i inches. 

49 A. Switch and Outlet Boxes. — a. Must be of pressed steel having a 
wall thickness not less than .081 inch (No. 12 B. & S. gage), or of cast metal 
having a wall thickness not less than .128 inch (No. 8 B. & S. gage.) 

b. Must be well galvanized, enameled or otherwise properly coated, 
inside and out, to prevent oxidation. 

c. Must be so made that all openings not in use will be effectively closed 
by metal which will afford protection substantially equivalent to the walls 
of the box. 

d. Must be plainly marked, where it may readily be seen when installed, 
with the name or trade-mark of the manufacturer. 

e. Must be arranged to secvire in position the conduit or flexible tubing 
protecting the wire. 

This rule will be complied with If the conduit or tubing Is firmly secured In 
position by means of some approved device which may or may not be a part of the 
box. 

f. Boxes used with lined conduit must comply with the foregoing 
requirements, and in addition must have a tough and tenacious insulating 
lining at least 1/32 inch thick, firmly secured in position. 

g. Switch and outlet boxes must be so arranged that they can be securely 
fastened in place independently of the support afforded by the conduit 
piping, except that when entirely exposed, approved boxes, which are 
threaded so as to be firmly supported by screwing on to the conduit pipe, 
may be used. 

h. Switch boxes must completely enclose the switch on sides and 
back, and must provide a thoroughly substantial support for it. The re- 
taining screws for the box must not be used to secure the switch in position. 

i. Covers for outlet boxes must be of metal equal in thickness to that 
specified for the walls of the box, or must be of metal lined with an insu- 
lating material not less than 1/32 inch in thickness, firmly and permanently 
secured to the metal. 

50. Mouldings. — (For wiring rules, see No. 24 k to m.) 

Wooden Mouldings. — a. Must have, both outside and inside, at least 
two coats of waterproof material, or be impregnated with a moisture re- 
pellent. 

b. Must be made in two pieces, a backing and a capping, and must afford 
suitable protection from abrasion. Must be so constructed as to thoroughly 
encase the wire, be provided with a tongue not less than i inch in thickness 
between the conductors, and have exterior walls which, under grooves. 



1430 IQ.— ELECTRIC POWER AND LIGHTING. 

shall not be less than ^ inch in thickness, and on the sides not less than 

]4: inch in thickness. 

It is recommended that only hard-wood moulding be used. • 

Metal Mouldings. — (For wiring rules, see Nos. 24, k to m, and 25 A.) 

c. Each length of such moulding must have maker's name or trade- 
mark stamped in the metal, or in some manner permanently attached 
thereto, in order that it may he readily identified in the field. 

The use of paper stickers or tags cannot be considered satisfactory methods of 
marking, as they are readily loosened and lost off in ordinary handling of the mould- 
ing. 

d. Must be constructed of iron or steel with backing at least ,050 inch 
in thickness, and with capping not less than .040 inch in thickness, and 
so constructed that when in place the raceway will be entirely closed ; must 
be thoroughly galvanized or coated with an approved rust preventative 
both inside and out to prevent oxidation. 

e. Elbows, couplings and all other similar fittings must be constructed 
of at least the same thickness and quality of metal as the moulding itself, 
and so designed that they will both electrically and mechanically secure 
the different sections together and maintain the continuity of the raceway. 
The interior surfaces must be free from burrs or sharp comers which might 
cause abrasion of the wire coverings. 

f. Must at all outlets be so arranged that the conductors cannot come 
in contact with the edges of the metal, either of capping or backing. Specially 
designed fittings which will interpose substantial barriers between conductors 
and the edges of metal are recommended. 

g. When backing is secured in position by screws or bolts from the inside 
of the raceway, depressions must be provided to render the heads of the 
fastenings flush with the moulding. 

h. Metal mouldings must be used for exposed work only and must be 
so constructed as to form an open raceway to be closed by the capping or 
cover after the wires are laid in. 

50 A. Tubes and Bushings. — a. Construction. — Must be made straight 
and free from checks or rough projections, with ends smooth and rounded 
to facilitate the drawing in of the wire and prevent abrasion of its covering. 

b. Material and Test. — Must be made of non -combustible insulating 
material, which, when broken and submerged for 100 hours in pure water 
at 70° Fahrenheit (21° Centigrade), will not absorb over one-half of one per 
cent of its weight. 

c. Marking. — Must have the name, initials or trade-mark of the manu- 
facturer stamped in the ware. 

d. Sizes. — Dimensions of walls and heads must be at least as great as 
those given in the following table: — 



Diameter 


External 


Thickness 


External 


Length 


of 


Diameter. 


of 


Diameter 


of 


Hole. 




Wall. 


of Head. 


Head. 


Inches. 


Inches. 


Inches. 


Inches. 


Inches. 


T¥ 


■h 


H 


|| 


^ 


^ 


'H' 


# 


■ff 


V2 


% 


H 




\t 


¥2 


H 


lluT 


% 


lii 


/^ 


1 


lA 




lit 


% 


m 


llT 


3*2 


2ft 


6^ 


IV2 


2j^ 


ft 


H 


r 


2II 




3^ 


M 
M 


2M 


3j^ 


II 


3i4 


1 


2H 


3H 


11 


4A 


1 



An allowance of 1-64 of an inch for variation in manufacture will be permitted, 
except in the thickness of the wall. 

50 B. Cleats. — a. Construction. — Must hold the wire firmly in place 
without injury to its covering. 

Sharp edges which may cut the wire should be avoided. 



CONSTRUCTION— FITTINGS, MATERIALS, ETC, 1431 

b. Supports. — -Bearing points on the surface must be made by ridges 
or rings about the holes for supporting screws, in order to avoid cracking 
and breaking when screwed tight. 

c. Material and Test. — Must be made of non -combustible insulating 
material, which, when broken and submerged for 100 hours in pure water 
at 70° Fahrenheit (21° Centigrade), will not absorb over one-half of one 
per cent of its weight. 

d. Marking. — Must have the name, initials or trade-mark of the manu- 
facturer stamped in the ware. 

e. Sizes. — Must conform to the spacings given in the following table: — 

Distance from Wire Distance Between' 
Voltage. to Surface. Wires. 

0-300 I inch. 2i inches. 

This rule will not be interpreted to forbid the placing of the neutral of an Edison 
three-wire system in the center of a tliree-wire cleat where the difference of potential 
between the outside wires is not over 300 volts, provided the outside wires are sepa- 
rated 2^ inches. 

50 C. Flexible Tubing. — a. Must have a sufficiently smooth interior 
siirface to allow the ready introduction of the wire. 

b. Must be constructed of or treated with materials which will serve as 
moisture repellents. 

c. The tube must be so designed that it will withstand all the abrasion 
likely to be met with in practice. 

d. The linings, if any, must not be removable in lengths of over 3 feet. 

e. The i-inch tube must be so flexible that it will not crack or break 
when bent in a circle with 6-inch radius at 50° Fahrenheit (10° Centigrade), 
and the covering must be thoroughly saturated with a dense moisture- 
proof compound which will not slide at 150° Fahrenheit (65° Centigrade). 
Other sizes must be as well made. 

f . Must not convey fire on the application of a flame from Bunsen burner 
to the exterior of the tube when held in a vertical position. 

g. Must be sufficiently tough and tenacious to withstand severe tension 
without injury; the interior diameter must not be diminished or the tube 
opened up at any point by the application of a reasonable stretching force. 

h. Must not close to prevent the insertion of the wire after the tube has 
been kinked or flattened and straightened out. 

51. Switches. — (For installation rules, see Nos. 17 and 22.) 

General Rules. 

a. Must, when used for service switches, indicate, on inspection, whether 
the current be "on" or "oif." 

b. Must, for constant -current systems, close the main circuit and dis- 
connect the branch wires when turned "off;" must be so constructed that 
they shall be automatic in action, not stopping between points when started, 
and must prevent an arc between the points under all circumstances. They 
must indicate whether the current be "on" or "off." 

Knife Switches. 
Knife switches must be made to comply with the following speciflcations, except 
in those few cases wtiere peculiar design allows the switch to fulfil the general re- 

guirements In some other way. and where it can successfully withstand the test of 
ectlon i. In such cases the switch should be submitted for special examination 
before being used. 

c. Base. — Must be mounted on non-combustible, non-absorptive insulat- 
ing bases, such as slate or porcelain. Bases with an area of over 25 square 
inches must have at least four supporting screws. Holes for the supporting 
screws must be so located or countersunk that there will be at least \ an 
inch space, measured over the surface, between the head of the screw or 
washer and the nearest live metal part, and in all cases when between 
parts of opposite polarity must be countersunk. 

d. Mounting. — Pieces carrying the contact jaws and hinge clips must 
be secured to the base by at least two screws, or else made with a square 
shoulder, or provided with dowel pins, to prevent possible turnings, and 



1432 10.— ELECTRIC POWER AND LIGHTING, 

the nuts or screw-heads on the under side of the base must be countersunk 
not less than i inch and covered with a waterproof compound which will 
not melt below 150° Fahrenheit (65° Centigrade). 

e. Hinges. — Hinges of knife switches must not be used to carry current 
unless they are equipped with spring washers, held by lock-nuts or pins, or 
their equivalent, so arranged that a firm and seciure connection will be 
maintained at all positions of the switch blades. 

Spring washers must be of sufficient strength to take up any wear in the hinge 
and maintain a good contact at all times. 

f. Metal. — All switches must have ample metal for stiffness and to pre- 
vent rise in temperature of any part of over 50° Fahrenheit (28° Centigrade), 
at full load, the contacts being arranged so that a thoroughly good bearing 
at every point is obtained with contact surfaces advised for pure copper 
blades of about one square inch for each 75 amperes; the whole device 
must be mechanically well made throughout. 

g. Cross-Bars. — All cross-bars less than 3 inches in length must be 
made of insulating material. Bars of 3 inches or over, which are made of 
metal to insure greater mechanical strength, must be sufficiently separated 
from the jaws of the switch to prevent arcs following from the contacts to 
the bar on the opening of the switch under any circumstances. Metal bars 
should preferably be covered with insulating material. 

To prevent possible turning or twisting the cross-bar must be secured 
to each blade by two screws, or the joints made with square shoulders or 
provided with dowel-pins. 

h. Connections. — Switches for currents of over 30 amperes must be 
equipped with lugs, firmly screwed or bolted to the switch, and into which 
the conducting wires shall be soldered. For the smaller sized switches 
simple clamps can be employed, provided they are heavy enough to stand 
considerable hard usage. 

Where lugs are not provided, a rugged double-V groove clamp Is advised. A 
set-screw gives a contact at only one point, is more likely to become loosened, and 
is almost sure to cut into the wire. For the smaller sizes, a screw and washer con- 
nection with up-turned lugs on the switch terminal gives a satisfactory contact. 

i. Test. — Must operate successfully at 50 per cent overload in amperes 
and 25 per cent excess voltage, under the most severe conditions with which 
they are liable to meet in practice. 

This test is designed to give a reasonable margin between the ordinary ratlDg of 
the switch and the brealsing-down point, thus securing a switch which can always 
safely handle its normal load. Moreover, there is enough leeway so that a moderate 
amount of overloading would not injure the switch. 

j. Marking. — Must be plainly marked where it will be visible, when the 
switch is installed, with the name of the maker and the ctirrent and the 
voltage for which the switch is designed. 

Switches designed for use on Edison three-wire systems must be marked with 
both voltages, that is, the voltage between the outside wires and the neutral, and 
alsc that between the outside wires, followed by the ampere rating and the words 
"three-wire." (For example, "125-250 v. 30 a., three-wire.") 

k. Spacings. — Spacings must be at least as great as those given in the 
following table. The spacings specified are correct for switches to be used 
on direct-current systems, and can therefore be safely followed in devices 
designed for alternating currents. 

Minimum Separation of Minimum 

Nearest Metal Parts of Break- 

125 Volts or Less: Opposite Polarity. Distance. 

For Switchboards and Panel Boards: 

10 amperes or less ^ inch - - - H inch. 

11-30 amperes 1 " - - - M " 

31-50 " IM " - - - 1 " 

For Individual Switches: 

10 amperes or less 1 inch - - - M inch. 

11-30 amperes 1}4 " - - - 1 

31-100 " IM " - - - IH 

101-300 " 2M " - - - 2 

301-600 " 23^ " - - - 2K 

601-1000 " 3 •• - - - 2M 



CONSTRUCTION— FITTINGS, MATERIALS, ETC, 1433 

126 TO 250 Volts: Min. Min. 

For all Switches: Sep. B.-D. 

10 amperes or less 13^ inch - - - Ij/ inch. 

11-30 amperes IM " - - - IM " 

31-100 " 2M " - - - 2 

101-300 " 21^ " - - ' 2H '* 

301-600 " 2H " ' - ' 2}4 " 

601-1000 " 3 '• - - - 2M " 

For 100 ampere switches and larger, the above spaclnga for 250 volts direct 
current are also approved for 500 volts alternating current. Switches with these 
spacings intended for use on alternating-current systems with voltage above 250 
volts must be stamped "250-volt D. C," followed by the alternating-current voltage 
for which they are designed, and the letters "A. C." 

251 TO 600 Volts: 

For all Switches: 

10 amperes or less 33^ inch - - - 3 inch. 

11-35 amperes 4 " - - - 33^ " 

36-100 " 41^ " - . - 4 " 

Auxiliary breaks or the equivalent are recommended for switches designed for 
over 300 volts and less than 100 amperes, and will be required on switches designed 
/or use in breaking currents greater tban 100 amperes at a pressure of more than 300 
volts. 

For three-wire Edison systems the separations and break distances for plain 
three-pole knife switches must not be less than those required in the above table for 
switches designed for the voltage between the neutral and outside wires. 

Snap Switches. 

Flush, push-button, door, fixture and other snap switches used on constant- 
potential systems, must be constructed in accordance with the following specifica- 
tions. 

1. Base. — Current-carrying parts must be mounted on non-combustible, 
non -absorptive, insulating bases, such as slate or porcelain, and the holes 
for supporting screws should be countersunk not less than i of an inch. 
There must in no case be less than 3/64 of an inch space between supporting 
screws and current-carrying parts. 

Sub-bases of non-combustible, non-absorptive insulating material, which 
will separate the wires at least ^ inch from the surface wired over, must be 
furnished with all snap switches used in exposed or moulding work. 

m. Mounting. — Pieces carrying contact jaws must be secured to the 
base by at least two screws, or else made with a square shoulder, or provided 
with dowel-pins or otherwise arranged to prevent possible turnings; and 
the nuts or screw-heads on the under side of the base must be countersunk 
not less than i inch, and covered with a waterproof compound which will 
not melt below 150° Fahrenheit (65° Centigrade). 

n. Metal. — All switches must have ample metal for stiffness and to 
prevent rise in temperature of any part of over 50° Fahrenheit (28° Centi- 
grade) at full load, the contacts being arranged so that a thoroughly good 
bearing at every point is obtained. The whole device must be mechanically 
•well made throughout. 

In order to meet the above requirements on temperature rise without causing 
excessive friction and wear on current-carrying parts, contact surfaces of from 0.1 
to 0.15 square inch for each 10 amperes will be required, depending upon the metal 
used and the form of construction adopted. 

o. Insulating Material. — Any material used for insulating current- 
carrying parts must retain its insulating and mechanical strength when 
subject to continued use, and must not soften at a temperature of 212° 
Fahrenheit (100° Centigrade). It must also be non-absorptive. 

p. Binding Posts. — Binding posts must be substantially made, and the 
screws must be of such size that the threads will not strip when set up tight. 

A set-screw is likely to become loosened and Is almost sure to cut Into the wire. 
A binding screw under the head of which the wire may be clamped and a terminal 
plate provided with upturned lugs or some other equivalent arrangement, afford 
reliable contact. After July 1, 1908, switches with the set-screw form of contact 
win not be approved. 

q. Covers. — Covers made of conducting material, except face plates for 
flush switches, must be lined on sides and top with insulating, tough and 
tenacious material at least 1/32 inch in thickness, firmly secured so that it 



1434 1^.— ELECTRIC POWER AND LIGHTING, 

will not fall out with ordinary handling. The side lining must extend 
slightly beyond the lower edge of the cover. 

r. Handle or Button. — The handle or button or any exposed parts must 
not be in electrical connection with the circuit. 

s. Test. — Must "make" and "break" with a quick snap, and must not 
stop when motion has once been imparted by the button or handle. 

Must operate successfully at 50 per cent overload in amperes and at 
125 volt direct current, for all 125 volt or less switches, and at 250 volt 
direct cup-ent, for all 126 to 250 volt switches under the most severe condi- 
tions which they are liable to meet in practice. 

When slowly turned "on" and "off" at the rate of about two or three 
times per minute, while carrying the rated current at rated voltage, must 
"make" and "break" the circuit 6,000 times before failing. 

t. Marking. — Must be plainly marked, where it may be readily seen 
after the device is installed, with the name or trade-mark of the maker 
and the current and voltage for which the switch is designed. 

On flush switches these markings may be placed on the back of the 
face plate or on the sub-plate. On other types they must be placed on the 
front of the cap, cover or plate. 

Switches which indicate whether the current is "on" or "off" are recom- 
mended. 

52. Cut=Outs and Circuit Breakers. — (For installation rules, see Nos. 17 
and 21.) These requirements do not apply to rosettes, attachment plugs, car 
lighting cut-outs and protective devices for signaling systems. 

General Rules. 

a. Must be supported on bases of non-combustible, non-absorptive 
insulating material. 

b. Cut-outs must be of plug or cartridge type, when not arranged in 
approved cabinets, so as to obviate any danger of the melted fuse metal 
coming in contact with any substance which might be ignited thereby. 

c. Cut-outs must operate successfully on short-circuits, under the most 
severe conditions with which they are liable to meet in practice, at 25 per 
cent above their rated voltage, and for link fuse cut-outs with fuses rated 
at 50 per cent above the current for which the cut-out is designed, and for 
enclosed fuse cut-outs with the largest fuses for which the cut-out is de- 
signed. 

With link fuse cut-outs there Is always the possibility of a larger fuse being put 
into the cut-out than it was designed for, which is not true of enclosed fuse cut-outs 
classified as required under No. 52 q. Again, the voltage in most plants can, under 
some conditions, rise considerably above the normal. The need of some margin as a 
factor of safety to prevent the cut-outs from being ruined In ordinary service. Is 
therefore evident. 

The most severe service which can be required of a cut-out in practice is to open 
a "dead short-circuit" with only one fuse blowing, and It Is with these conditions that 
all tests should be made. (See Section j.) 

d. Circuit -breakers must operate successfully on short-circuits under 
the most severe conditions with which they are liable to meet in practice, 
at 25 per cent above their rated voltage and with the circuit -breaker set at 
the highest possible opening point. 

For the same reason as In Section c. 

e. Must be plainly marked where it will always be visible, with the name 
of the maker, and current and voltage for which the device is designed. 

Link-Fuse Cut-Outs. 
{Cut-outs of porcelain are not approved for link fuses.) 
The following rules are intended to cover open link fuses mounted on slate or 
marble bases, including switchboards, tablet-boards and single fuse-blocks. They 
do not apply to fuses mounted on porcelain bases, to the ordinary porcelain cut-out 
blocks, enclosed fuses, or any special or covered type of fuse. When tablet-boards 
or single fuse-blocks with such open link fuses on them are used in general wiring, 
they must be enclosed in cabinet boxes made to meet the requirements of No. 54. 
This is necessary, because a severe flash may occur when such fuses melt, so that they 
would be dangerous if exposed in the neighborhood of any combustible material. 

f. Base. Must be mounted on slate or marble bases. Bases with an 
area of over 25 square inches must have at least four supporting screws. 
Holes for supporting screws must be kept outside of the area included by 



CONSTRUCTION— FITTINGS, MATERIALS, ETC. 1435 

the outside edges of the fuse-block terminals, and must be so located or 
countersunk that there will be at least J^ inch space, measured over the 
surface, between the head of the screw or washer and the nearest live part. 

g. Mounting.—-'^nts or screw heads on the under side of the base must 
be countersunk not less than Ys inch, and covered with a waterproof com- 
pound Which will not melt below 150° Fahrenheit (65° Centigrade). 

h. Metal. — All fuse-block terminals must have ample metal for stiffness 
and to prevent rise in temperature of any part over 50° Fahrenheit (28° 
Centigrade) at full load. Terminals, as far as practicable, should be made 
of compact form instead of being rolled out in thin strips; and sharp edges 
or thin projecting pieces, as on wing thumb nuts and the like, should be 
avoided. Thin metal, sharp edges and projecting pieces are much more 
likely to cause an arc to start than a more solid mass of metal. It is a good 
plan to round all corners of the terminals and to chamfer the edges. 

i. Connections. — Clamps for connecting the wires to the fuse-block ter- 
minals must be of solid, rugged construction, so as to insure a thoroughly 
good connection and to withstand considerable hard usage. For fuses 
rated at over 30 amperes, lugs firmly screwed or bolted to the terminals 
and into which the conducting wires are soldered must be used. 

See note under No. 51 h. 

j. Test. — Must operate successfully when blowing only one fuse at a 
time on short circuits with fuses rated at 50 per cent above and with a 
voltage 25 per cent above the current and voltage for which the cut-out is 
designed. 

k. Marking. — Must be plainly marked, where it will be visible when the 
cut-out block is installed, with the name of the maker and the current 
and the voltage for which the block is designed. 

I. S pacings. — Spacings must be at least as great as those given in the 
following table, which applies only to plain, open link-fuses mounted on 
slate or marble bases. The spaces given are correct for fuse-blocks to be 
used on direct-current systems, and can therefore be safely followed in 
devices designed for alternating currents. If the copper fuse-tips overhang 
the edges of the fuse-block terminals, the spacings should be measured 
between the nearest edges of the tips. 

Minimum Separation of Minimum 

Nearest Metal Parts of Break 

125 Volts or Less: Opposite Polarity. Distance. 

10 amperes or less % inch- - - % inch. 

11-100 amperes 1 " - - - M " 

101-300 " 1 " - - - 1 

301-1000 •• IM " - - - IM " 

126 TO 250 Volts: 

10 amperes or less 13^ inch - - \yi inch. 

11-100 amperes IM " - - - IM " 

101-300 " „ 2 " - - - 13^ '• 

301-1000 " 21^ " - - - 2 

A space must be maintained between fuse terminals of the same polarity of at 
least h Inch for voltages up to 12 5, and of at least f Inch for voltages from 126 to 250. 
This is the minimum distance allowable, and greater separation should be provided 
when practicable 

For 250 volt boards or blocks with the ordinary front-connected terminals, 
except where these have a mass of compact form, equivalent to the back-connected 
terminals usually found in switchboard work, a substantial barrier of Insulating 
material, not less than i of an inch in thickness, must be placed in the "break" gap — 
this barrier to extend out from the base at least i of an inch farther than any bare 
live part of the fuse-block terminal. Including binding screws, nuts and the like. 

For three-wire systems cut-outs must have the break-distance required for 
circuits of the potential of the outside wires. 

Enclosed-Fuse Cut-Outs — Plug and Cartridge Type. 

nic Base. — Must be made of non-combustible, non-absorptive, insulating 
material. Blocks with an area of over 25 square inches must have at least 
four supporting screws. Holes for supporting screws must be so located or 
countersunk that there will be at least \ inch space, measured over the 
surface, between the screw-head or washer and the nearest live metal part, 
and in all cases when between parts of opposite polarity must be counter- 
sunk. 



1436 70.^ELECTRIC POWER AND LIGHTING. 

n. Mounting. — Nuts or screw-heads on the under side of the base must 
be countersunk at least i of an inch and covered with a waterproof compound 
which will not melt below 150° Fahrenheit (65° Centigrade). 

o. Terminals. — ^Terminals must be of either the Edison plug, spring 
clip or knife-blade type, of approved design, to take the corresponding 
standard enclosed fuses. They must be secured to the base by two screws or 
the equivalent, so as to prevent them from turning, and must be so made 
as to secure a thoroughly good contact with the fuse. End stops must be 
provided to insure the proper location of the cartridge fuse in the cut-out. 

p. Connections. — Clamps for connecting wires to the terminals must 
be of a design which will insure a thoroughly good connection, and must 
be sufficiently strong and heavy to withstand considerable hard usage. 
For fuses rated to carry over thirty amperes, lugs firmly screwed or bolted 
to the terminals and into which the connecting wires shall be soldered must 
be used. 

q. Classification. — Must be classified as regards both current and voltage 
as given in the following table, and must be so designed that the bases of 
one class cannot be used with fuses of another class rated for a higher 
current or voltage. 

0-250 Volts: 251-600 Volts: 

0- 30 amperes. 0- 30 amperes. 

31- 60 " 31- 60 

61-100 " 61-100 

101-200 •* 101-200 

201-400 *• 201-400 

401-600 
r. Design. — Must be of such a design that it will not be easy to form 
accidental short circuit across live metal parts of opposite polarity on the 
block or on the fuses in the block. 

s. Marking. — Must be marked, where it will be plainly visible when the 
block is installed, with the name of the maker and the voltage and range 
of current for which it is designed. 

53. Fuses. — (For installation rules, see Nos. 17 and 21.) 

Link Fuses. 

a. Terminals. — Must have contact surfaces or tips of harder nietal, 
having perfect electrical connections with the fusible part of the strip. 

The use of the hard metal tip is to afford a strong mechanical bearing for the 
screws, clamps or other devices provided for holding the fuse. 

b. Rating. — Must be stamped with about 80 per cent of the maximum 
current which they can carry indefinitely, thus allowing about 25 per cent 
overload before the fuse melts. 

"With naked open fuses, or ordinary shapes and with not over 500 amperes 
capacity, the minimum current which will melt them In about five minutes may be 
safely taken as the melting point, as the fuse practically reaches its maximum tem- 
perature in this time. With larger fuses a longer time Is necessary. This data Is 
given to facilitate testing. 

c. Marking. — Fuse terminals must be stamped with the maker's name 
or initials, or with some known trade-mark. 

Enclosed Fuses — Plug and Cartridge Type. 

These requirements do not apply to fuses for rosettes, attachment plugs, car 
lighting cut-outs and protective devices for signaling systems. 

d. Construction. — The fuse plug or cartridge must be sufficiently dust- 
tight so that lint and dust cannot collect around the fusible wire and be- 
come ignited when the fuse is blown. 

The fusible wire must be attached to the plug or cartridge terminals 
in such a way as to secure a thoroughly good connection and to make it 
difficult for it to be replaced when melted. 

e. Classification. — Must be classified to correspond with the different 
classes of cut-out blocks, and must be so designed that it will be impossible 
to put any fuse of a given class into a cut-out block which is designed for a 
current or voltage lower than that of the class to which the fuse belongs. 



CONSTRUCTION— FITTINGS, MATERIALS, ETC, 1437 

f, Terminal:. — The fuse terminals must be sufficiently heavy to insure 
mechanical strength and rigidity. The styles of terminals must be as follows: 

0-250 Volts: 

iA ( Cartridge fuse ) to (a, spring clip terminals. 
( (ferrule contact) j fit \b, Edison plug casings. 
B Approved plugs for Edison cut-outs. 
q-i flA «« 1 Cartridge fuse ) to (a, spring clip terminals. 

^ I (ferrule contact) j fit \b, Edison plug casings. 

61-100 ;; ] 

201-400 •• [Cartridge fuse (knife-blade contact). 

401-600 " J 

251-600 Volts: 

31-60 ^^^^^' } Cartridge fuse (ferrule contact). 

61-100 •• ) 

101-200 " [-Cartridge fuse (knife-blade contact). 

201-400 " ) 

g. Dimensions. — Cartridge enclosed fuses and corresponding cut-out 
blocks must conform to the dimensions given in the Table, on-next page. 

h. Rating. — Fuses must be so constructed that with the surrounding 
atmosphere at a temperature of 75° Fahrenheit (24® Centigrade) they will 
carry indefinitely a current 10 per cent greater than that at which they are 
rated, and at a current 25 per cent greater than the rating, they will open 
the circuit without reaching a temperature which will injure the fuse tube 
or terminals of the fuse block. With a current 50 per cent greater than 
the rating and at room temperature of 75° Fahrenheit (24° Centigrade), the 
fuses starting cold, must blow within the time specified below: — 

0- 30 amperes, 30 seconds. 

31- 60 " 1 minute. 

61-100 '• 2 minutes. 

101-200 *• 4 

201-400 " 8 

401-600 '* 10 

i. Marking. — Must be marked, where it will be plainly visible, with 
the name or trade-mark of the maker, the voltage and current for which 
the fuse is designed, and the words "National Electrical Code Standard." 
Each fuse must have a label, the color of which must be green for 250- 
volt fuses and red for 600-volt fuses. 

It will be satisfactory to abbreviate the above designation to "N. E. Code St'd'* 
where space is necessarily limited. 

j. Temperature Rise. — ^The temperature of the exterior of the fuse en- 
closure must not rise more than 125° Fahrenheit (70° Centigrade) above that 
of the surrounding air when the fuse is carrying the current for which it is rated. 

k. Test. — Must not hold an arc or throw out melted metal or sufficient 
flame to ignite easily inflammable material on or near the cut-out when 
only one fuse is blown at a time on a short circuit on a system of the voltage 
for which the fuse is rated. 

The normal capacity of the system must be in excess of the load on it 
just previous to the test by at least five times the rated capacity or the 
fuse under test. 

The resistance of the circuit up to the cut-out terminals must be such 
that the impressed voltage at the terminals will be decreased not more 
than one per cent, when a current of 100 amperes is passed between them. 

For convenience a current of different value may be used. In which case the per 
cent drop in voltage allowable would vary in direct proportion to the difference in 
current used. 

The above requirement regarding the capacity of the testing circuit is to guard 
against making the test on a system of so small a capacity that the conditions would 
be sufRciently favorable to allow really poor fuses to stand the test acceptably. On 
the other hand, it must be remembered that if the test is made on a system of very 
large capacity, and especially if there is but little resistance between the generators 
and fuse, the conditions may be more severe than are liable to be met with in practice 
outside of the large power stations, the result being that fuses entirely safe for general 
use may be rejected it such test is insisted upon. 



i4as 



10.— ELECTRIC POWER AND LIGHTING. 








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CONSTRUCTION— FITTINGS, MATERIALS, ETC, 1439 

53 A. Tablet and Panel Boards. — The following minimum distance 
between bare live metal parts (bus-bars, etc.) must be maintained: — 

Between parts of opposite polarity, except at Between parts of same 
switches and link fuses, polarity, 

When mounted on the When held free in At link 

same surface. the air. fuses. 

0-125 volts % inch ^ inch }/2 inch 

126-250 volts IM " M " % " 

At switches or enclosed fuses, parts of the same polarity may be placed as close 
together as convenience In handling will allow. 

It should be noted that the above distances are the minimum allowable, and It Is 
urged that greater distances be adopted wherever the conditions will permit. 

The spacings given In the first column apply to the branch conductors where 
inclosed fuses are used. Where link fuses or knife switches are used, the spacings 
must be at least as great as those required by Nos. 51 and 52. 

The spacings given In the second column apply to the distance between the 
raised main bars, and between these bars and the branch bars over which they pass. 
The spacings given In the third column are intended to prevent the melting of 
a link fuse by the blowing of an adjacent fuse of the same polarity. 

54. Cut-Out Cabinets. — a. Material. — Cabinets must be substantially 
constructed of non-combustible, non-absorptive material, or of wood. 
When wood is used the inside of the cabinet must be completely lined with 
a non -combustible insulating material. Slate or marble at least \ inch in 
thickness is strongly recommended for such lining, but, except with metal 
conduit systems, asbestos board at least \ inch in thickness may be used 
in dry places if firmly secured by shellac and tacks. 

With metal conduit systems the lining of either the box or the gutter 
must be of 1/16 inch galvanized, painted or enameled steel, or prefer- 
ably M inch slate or marble. 

The object of the lining of such cut-out cabinets or gutters is to render the same 
approximately fireproof in case of short circuit after the wires leave the protecting 
metal conduits. 

Two thicknesses of 1-32 inch steel may be used Instead of one 1-16 Inch. 

With wood cabinets the Wood should be thoroughly filled and painted before the 
lining is put into place. 

b. Door. — The door must close against a rabbet, so as to be perfectly 
dust-tight. Strong hinges and a strong hook or catch are required. Glass 
doors must be glazed with heavy glass, not less than \ inch in thickness, 
and panes should not exceed 300 square inches in area. A space of at least 
two inches must be allowed between the fuses and the door. This is necessary 
to prevent cracking or breaking by the severe blow and intense heat which 
may be produced under some conditions. 

A cabinet is of little use unless the door is kept tightly closed, and especial at- 
tention is therefore called to the Importance of having a strong and reliable catch or 
other fastening. A spring catch Is advised if a good one can be obtained, but most 
of those sold for use on cupboards, etc., are so small that they fail to catch when the 
door shrinks a little, or are so weak that they soon give out. 

It is advised that the bottoms of cabinets be given a decided slant to prevent 
their use as a shelf, as well as the accumulation of dust, etc. 

c. Bushings. — Bushings through which wires enter must fit tightly 
the holes in the box, and must be of approved construction. The wires 
should completely fill the holes in the bushings, using tape to build up the 
wire, if necessary, so as to keep out the dust. 

54 A. Rosettes. — Ceiling rosettes, both fused and fuseless, must be 
constructed in accordance with the following specifications: — 

a. Base. — Current-carrying parts must be mounted on non-combustible, 
non-absorptive, insulating bases. There should be no openings through 
the rosette base except those for the supporting screws and in the concealed 
type for the conductors also, and these openings should not be made any 
larger than necessary. 

There must be at least! inch space, measured over the surface, between 
supporting screws and current-carrying parts. The supporting screws 
must be so located or countersunk that the flexible cord cannot come in 
contact with them. 

Bases for the knob and cleat type must have at least two holes for 
supporting screws; must be high enough to keep the wires and terminals at 
least i inch from the surface to which the rosette is attached, and must 
have a porcelain lug under each terminal to prevent the rosette from being 



1440 n.—EL^CTRtC POWER AND LIGHTING, 

placed over projections which would reduce the separation to less than 
\ inch. 

Bases for the moulding and conduit box types must be high enough to 
keep tbe wires and terminals at least f inch from the surface wired over. 

b. Mounting. — Contact pieces and terminals must be secured in position 
by at least two screws, or made with a square shoulder, or otherwise arranged 
to prevent turning. 

The nuts or screw head on the under side of the base must be counter- 
sunk not less than \ inch and covered with a waterproof compound which 
will not melt below 150° Fahrenheit (65° Centigrade). 

c. Terminals. — Line terminal plates must be at least .07 inch in thickness, 
and terminal screws must not be smaller than No. 6 standard screw with 
about 32 threads per inch. 

Terminal plates for the flexible cord and for fuses must be at least 
.06 inch in thickness. The connection to these plates shall be by binding 
screws not smaller than No. 5 standard screw with about 40 threads per 
inch. At all binding screws for line wires and for flexible cord, up- turned 
lugs, or some equivalent arrangement, must be provided which will secure 
the wires being held under the screw heads. 

d. Cord Inlet. — The diameter of the cord inlet hole should measure 
13/32 inch in order that standard portable cord m.ay be used. 

e. Knot Space. — Ample space must be provided for a substantial knot 
tied in the cord as a whole. 

All parts of the rosette upon which the knot is likely to bear must be 
smooth and well rounded. 

f. Cover. — When the rosette is made in two parts, the cover must be 
secured to the base so that it will not work loose. 

In fused rosettes, the cover must fit closely over the base so as to prevent 
the accumulation of dust or dirt on the inside, and also to prevent any flash 
or melted metal from being thrown out when the fuses melt. 

g. Markings. — Must be plainly marked where it m.ay readily be seen 
after the rosette has been installed, with the name or trade-mark of the 
manufacturer, and the rating in amperes and volts. Fuseless rosettes may 
be rated 3 amperes, 250 volts; fused rosettes, with link fuses, not over 
2 amperes, 125 volts. 

h. Test. — Fused rosettes must have a fuse in each pole and must operate 
successfully when short-circuited on the voltage for which they are designed , 
the test being made with the two fuses in circuit. 

When link fuses are used the test shall be made with fuse wire which melts at 
about 7 amperes in one-inch lengths The larger fuse is specified for the test in order 
to more nearly approximate the severe conditions obtained when only one 2-ampere 
fuse (the rating of the rosette) is blown at a time. 

Fused rosettes equipped with enclosed fuses are much preferable to the link fuse 
rosettes. 

55. Sockets. — (For installation rules, see No. 27.) Sockets of all kinds, 
including wall receptacles, must be constructed in accordance with the follow- 
ing specifications: — 

a. Standard Sizes. — ^The standard lamp socket must be suitable for use 
on any voltage not exceeding 250 and with any size lamp up to fifty candle- 
power. For lamps larger than fifty candle-power a standard keyless socket 
may be used, or if a key is required, a special socket designed for the current 
to be used must be made. Any special sockets must follow the general 
spirit of these specifications. 

b. Marking. — All sockets must be marked with the manufacturer's 
name or trade-mark. The standard key socket m.ust also be plainly marked 
250 V. 50 c. p. Receptacles, keyless sockets and special sockets must be 
marked with the current and voltage for which they are designed. 

c. Shell. — Metal used for shells must be moderately hard, but not hard 
enough to be brittle or so soft as to be easily dented or knocked out of shape. 
Brass shells must be at least thirteen one-thousandths of an inch in thickness, 
and shells of any other material must be thick enough to give the same 
stiffness and strength as the required thickness of brass. 

d. Lining. — The inside of the shells must be lined with insulating 
material, which must absolutely prevent the shell from becoming a part of 



CONSTRUCTION-— FITTINGS, MATERIALS, ETC. 1441 

the circuit, even though the wires inside the sockets should become loosened 
or detached from their position under the binding screws. 

The material used for lining must be at least 1/32 of an inch in thick- 
ness, and must be tough and tenacious. It must not be injuriously affected 
by the heat from the largest lamp permitted in the socket, and must leave 
water in which it is boiled practically neutral. Is must be so firmly se- 
cured to the shell that it will not fall out with ordinary handling of the 
socket. It is preferable to have the lining in one piece. 

The cap must also be lined, and this lining must comply with the requirements 
for shell linings. 

The shell lining should extend beyond the shell far enough so that no part of the 
lamp base is exposed when a lamp Is In the socket. The standard Edison lamp base 
measures 15-16 inches in a vertical plane from the bottom of the center contact to 
the upper edge of the screw-shell. 

In sockets and receptacles of standard forms a ring of any material inserted 
between an outer metal shell of the device and the Inner screw shell for insulating 
purposes and separable from the device as a whole. Is considered an undesirable 
form of construction. This does not apply to the use of rings In lamp clusters or In 
devices where the outer shell is of porcelain, where such rings serve to hold the several 

Sorcelaln parts together, and are thus a necessary part of the whole structure of the 
evice. 

e. Cap. — Caps, when of sheet brass, must be at least thirteen one- 
thousandths of an inch in thickness, and when cast or made of other metals 
must be of equivalent strength. The inlet piece, except for special sockets, 
must be tapped with a standard \ inch pipe thread. It must contain suffi- 
cient metal for a full, strong thread, and when not in one piece with the cap, 
must be joined to it in such a way as to give the strength of a single piece. 

There must be sufficient room in the cap to enable the ordinary wire- 
man to easily and quickly make a knot in the cord and to push it into place 
in the cap without crowding. All parts of the cap upon which the i^knot is 
likely to bear must be smooth and well insulated. 

The cap lining called for In the note to Section d will provide a sufficiently 
smooth and well insulated surface for the knot to bear upon. 

Sockets with an outlet threaded for ^-inch pipe will, of course, be approved 
where circumstances demand their use. The size outlet is necessary with most stiff 
pendants and for the proper use of reinforced flexible cord, as explained In the note 
to No. 28 d 

f. Frame and Screws. — ^The frame which holds the moving parts must 
be sufficiently heavy to give ample strength and stiffness. 

Brass pieces containing screw threads must be at least six one-hundredths 
of an inch in thickness. 

Binding post screws must not be smaller than No. 5 standard screw 
with about 40 threads per inch. 

g. Spacing. — Points of opposite polarity must everywhere be kept not 
less than 3/64 of an inch apart, unless separated by a reliable insulation. 

h. Connections. — ^The connecting points for the flexible cord must be 
made to very securely grip a No. 16 or 18 B. & S. gage conductor. An up- 
turned lug, arranged so that the cord may be gripped between the screw 
and the lug in such a way that it cannot possibly come out, is strongly 
advised. 

i. Lamp Holder. — ^The socket must firmly hold the lamp in place so 
that it cannot be easily jarred out, and must provide a contact good enough 
to prevent undue heating with the maximum current allowed. The holding 
pieces, springs and the like, if a part of the circuit, must not be sufficiently 
exposed to allow them to be brought in contact with anything outside of 
the lamp and socket. 

j. Base. — With the exception of the lining, all parts of insulating 
material inside the shell must be made of porcelain. 

k. Key. — ^The socket key-handle must be of such a material that it 
will not soften from the heat of a fifty candle-power lamp hanging down- 
wards from the socket in the air at 70° Fahrenheit (21° Centigrade), and 
must be securely, but not necessarily rigidly, attached to the metal spin- 
dle which it is designed to turn. 

1. Sealing. — All screws in porcelain pieces, which can be firmly sealed 
in place, must be so sealed by a waterproof compound which will not melt 
below 200° Fahrenheit (93° Centigrade.) 

m. Putting Together. — ^The socket as a whole must be so put together 
that it will not rattle to pieces. Bayonet joints or an equivalent are recom- 
mended. 



1442 70.— ELECTRIC POWER AND LIGHTING. 

n. Test. — ^The socket, when slowly ttimed "on and off" at the rate ot 
about two or three times per minute, while carrying a load of one ampere at 
250 volts, must "make" and "break" the circuit 6,000 times before failing. 

o. Keyless Sockets. — Keyless sockets of all kinds must comply with the 
requirements for key sockets as far as they apply. 

p. Sockets of Insulating Material. — Sockets made of porcelain or other 
insulating material must conform to the above requirements as far as they 
apply, and all parts must be strong enough to withstand a moderate amount 
of hard usage without breaking. 

Porcelain shell sockets being subject to breakage, and constituting a hazard when 
broken, will not be accepted for use in places where they would be exposed to hard 



q. Inlet Bushing. — When_ the socket is not attached to a fixture, the 
threaded inlet must be provided with a strong insulating bushing having 
a smooth hole at least 9/32 of an inch in diameter. The edges of the bushing 
must be rounded and all inside fins removed, so that in no place will the 
cord be subjected to the cutting or wearing action of a sharp edge. 

Bushings for sockets having an outlet threaded for f-inch pipe should have a 
hole 1 3-32 of an inch in diameter, so that they will accommodate approved reinforced 
flexible cord. 

56. Hanger»boards for Series Arc Lamps.— a. Hanger-boards must be 
so constructed that all wires and current-carrying devices thereon will be 
exposed to view and thoroughly insulated by being mounted on a non- 
combustible, non-absorptive, insulating substance. All switches attached 
to the same must be so constructed that they shall be automatic in their 
action, cutting off both poles to the lamp, not stopping between points 
when started and preventing an arc between points under all circumstances. 

57. Arc Lamps. — (For installation rules, see Nos. 19 and 29.) a. Must 
be provided with reliable stops to prevent carbons from falling out in case 
the clamps become loose. 

b. All exposed parts must be carefully insulated from the circuit. 

c. Must, for constant-current systems, be provided with an approved 
hand switch, and an automatic switch that will shunt the current around 
the carbons, should they fail to feed properly. 

The hand switch to be approved, if placed anywhere except on the 
lamp itself, must comply with requirements for switches on hanger-boards 
as laid down in No. 56. 

58. Spark Arresters. — (For installation rules, see Nos. 19 c and 29 c.) 

a. Spark arresters must so close the upper orifice of the globe that it 
will be impossible for any sparks, thrown off by the carbons, to escape. 

59. Insulating Joints. — (See No. 26 a.) — a. Must be entirely made of 
material that will resist the action of illuminating gases, and will not give 
way or soften under the heat of an ordinary gas name or leak under a moderate 
pressure. Must be so arranged that a deposit of moisture will not destroy 
the insulating effect; must show a dielectric strength between gas-pipe 
attachments sufficient to resist throughout five minutes the application of 
an electro-motive force of 4,000 volts; and must be sufficiently strong to 
resist the strain to which they are liable to be subjected during installation. 

Insulating joints having soft rubber in their construction will not be approved. 

60. Rheostats. — (For installation rules, see Nos. 4 a and 8 c.) 

a. Materials. — Must be made entirely of non-combustible materials, 
except such minor parts as handles, magnet insulation, etc. All segments, 
lever arms, etc., must be mounted on non-combustible, non-absorptive, 
insulating material. 

Rheostats used in dusty or llnty places or where exposed to flyings of combus- 
tible material, must be so constructed that even if the resistive conductor be fused 
by excessive current, the arc or any attendant flame will be qulcltiy and safely 
extinguished. Rheostats used in places where the above conditions do not exist may 
be of any approved type. 

b. Construction. — Must be so constructed that when mounted on a 
plane surface the casing will make contact with such svirface only at the 



CONSTRUCTION— FITTINGS, MATERIALS, ETC. 1443 

points of support. An air space of at least J4 inch between the rheostat 
casing and the supporting surface will be required. 

The construction throughout must be heavy, rugged and thoroughly 
workmanlike. 

c. Connections. — Clamps for connecting wires to the terminals must 
be of a design that will insure a thoroughly good connection, and must be 
sufficiently strong and heavy to withstand considerable hard usage. For 
currents above fifty amperes, lugs firmly screwed or bolted to the terminals, 
and into which the connecting wires shall be soldered, must be used. 

Clamps or lugs will not be required when leads designed for soldered connections 
are provided. 

d. Marking. — Must be plainly marked, where it may be readily seen 
after the device is installed, with the rating and the name of the maker; 
and the terminals of motor-starting rheostats must be marked to indicate 
to what part of the circuit each is to be connected, as "line," "armature" 
and "field." 

e. Contacts. — ^The design of the fixed and movable contacts and the 
resistance in each section must be such as to secure the least tendency toward 
arcing and roughening of the contacts, even with careless handling or the 
presence of dirt. 

In motor-starting rheostats, the contact at which the circuit is broken 
by the lever arm when moving from the running to the starting position, 
must be so designed that there will be no detrimental arcing. The final 
contact, if any, on which the arm is brought to rest in the starting position 
must have no electrical connection. 

Experience has shown that sharp edges and segments of thin material help to 
maintain an arc, and it is recommended that these be avoided. Segments of heavy 
construction have a considerabie cooling effect on the air, and rounded corners tend 
to spread it out and thus dissipate it. 

It is recommended that the circuit-breaking contacts be so constructed as to 
"break" with a quick snap, independently of the slowness of movement of the oper- 
ator's hand, or that a magnetic blowout or equivalent device be used. For dial type 
rheostats the movable contact should be flexible in a plane at right angle to the 
plane of its movement, and for medium and larger sizes the stationary contacts should 
be readily renewable. 

f. No-Voltage Release. — Motor-starting rheostats must be so designed 
that the contact arm cannot be left on intermediate segments, and must be 
provided with an automatic device which will interrupt the supply circuit 
before the speed of the motor falls to less than one third of its normal value. 

g. Over load- Re lease. — Overload-release devices which are inoperative 
during -the process of starting a motor will not be approved unless other 
circuit-breakers or fuses are installed in connection with them. 

If, for Instance, the over-release device simply releases the starting arm and allowa 
It to fly back and break the circuit, it is inoperative while the arm is being moved 
from the starting to the running position. 

h. Test. — Must, after 100 operations under the most severe normal 
conditions for which the device is designed, show no serious burning of the 
contacts or other faults, and the release mechanism of motor-starting 
rheostats must not be impaired by such a test. 

Field rheostats, or main-line regulators intended for continuous use, 
must not be burned out or depreciated by carrying the full normal current on 
any step for an indefinite period. Regulators intended for intermittent use 
(such as on electric cranes, elevators, etc.) must be able to carry their 
rated current on any step for as long a time as the character of the apparatus 
which they control will permit them to be used continuously. 

61. Reactive Coils and Condensers. — a. Reactive coils must be made of 
non-combustible material, mounted on non-combustible bases and treated, 
in general, as sources of heat. 

b. Condensers must be treated like other apparatus operating with 
equivalent voltage and currents. They must have non-combustible cases 
and supports, and must be isolated from all combustible material and, 
in general, treated as sources of heat. 

62. Transformers. — (For installation rules, see Nos. 11, 13, 13 A and 36.) 

a. Must not be placed in any but metallic or other non-combustible 
cases. 



1444 70,— ELECTRIC POWER AND LIGHTING, 

On account of the possible dangers from biim-outs in the coils. (See 
,note under No. 11 a.) 

It is advised that every transformer be so designed and connected that 
the middle point of the secondary coil can be reached if, at any future time, 
it should be desired to ground it. 

b. Must be constructed to comply with the following tests: — 

1. Shall be run for eight consecutive hours at full load in watts under 

conditions of service, and at the end of that time the rise in tem- 
perature, as measured by the increase of resistance of the 
primary and secondary coils, shall not exceed 175** Fahrenheit 
(9 7<» Centigrade). 

2. The insulation of transformers when heated shall withstand con- 

tinuously for five minutes a difference of potential of 10,000 volts 
(alternating) between primary and secondary coils and between 
the primary coils and core, and a no-load "run" at double voltage 
for thirty minutes. 

63, Lightning Arresters. — (For installation rules, see No. 6.) a. Light- 
ning arresters must be of approved construction. (See list of Electrical 
Fittings.) 

Class E.~MISCELLANEOUS. 

64. Signaling Systems. — Governing wiring for telephone, telegraph, dis- 
trict messenger and call-hell circuits, fire and burglar alarms, and all similar 
systems which are hazardous only because of their liability to become crossed 
with electric light, heat or power ctrcuits. 

a. Outside wires should be run in underground ducts or strung on poles, 
and, kept off the roofs of buildings, except by special permission of the 
Inspection Department having jurisdiction, and must not be placed on the 
same cross-arm with electric light or power wires. They should not occupy 
the same duct, manhole or handhole of conduit systems with electric light or 
power wires. 

Single manholes, or handholes, may be separated into sections by means of par- 
titions of brick or tile so as to be considered as conforming wltli tlie above rule. 

The liability of accidental crossing of overhead signaling circuits with electric 
light and power circuits may be guarded against to a considerable extent by endeav- 
oring to keep the two classes of circuits on different sides of the same street. 

When the Entire Circuit prom Central Station to Building is 

Run in Underground Conduits, Sections b to m 

Inclusive do Not Apply. 

b. When outside wires are run on same pole with electric light or power 
wires, the distance between the two inside pins of each cross-arm must 
not be less than twenty-six inches. 

Signaling wires being smaller and more liable to break and fall, should generally 
be placed on the lower cross-arms. 

c. Where wires are attached to the outside walls of buildings they must 
have an approved rubber insulating covering (see No. 41), and on frame 
buildings or frame portions of other buildings shall be supported on glass 
or porcelain insulators, or knobs. 

d. The wires from last outside support to the cut-outs or protectors 
must be of copper, and must have an approved rubber insulation (see No. 41) ; 
must be provided with drip loops immediately outside the building and at 
entrance; must be kept not less than 2i inches apart, except when brought 
in through approved metal cables. 

e. Wires must enter building through approved non-combustible» non- 
absorptive, insulating bushings sloping upward from the outside. 

Installations where the Current-Carrying Parts of the Apparatus 
Installed are Capable of Carrying Indefinitely 
A Current of Ten Amperes. 

f. An all-metallic circuit shall be provided, except in telegraph systems. 

g. At the entrance of wires to buildings, approved single-pole cut-outs, 
designed for 251-600 volts potential and containing fuses rated at not over 
10 amperes capacity, shall be provided for each wire. These cut-outs must 
not be placed in the immediate vicinity of easily ignitible stuff, or where 



MISCELLANEOUS— SIGNALING SYSTEMS. 1446 

€ Kposed to inflammable gases or dust or to flyings of combustible material. 

h. The wires inside buildings shall be of copper not less than No. 16 
B. & S. gage, and must have insulation and be supported, the same as would 
be required for an installation of electric light or power wiring, 0-600 volts 
potential. 

i. The instruments shall be mounted on bases constructed of non- 
combustible, non-absorptive insulating material. Holes for the supporting 
screws must be so located, or countersunk, that there will be at least i inch 
space, measured over the surface, between the head of the screw and the 
nearest live metal part. 

Installations where the Current -Carrying Parts of the Apparatus 
Installed are Not Capable of Carrying Indefinitely 
a Current of Ten Amperes. 

j. Must be provided with an approved protective device located as 
near as possible to the entrance of wires to building. The protector must 
not be placed in the immediate vicinity of easily ignitible stuff, or where 
exposed to inflammable gases or dust or flyings of combustible materials. 

k. Wires from entrance to building to protector must be supported on 
porcelain insulators, so that they will come in contact with nothing except 
their designed supports. 

1. The ground wire of the protective device shall be run in accordance 
with the following requirements: — 

1. Shall be of copper and not smaller than No. 18 B. & S. gage. 

2. Must have an approved insulating covering as described in No. 41, 

for voltages from to 600, except that the preservative compound 
specified in No. 41, Section h may be omitted. 

3. Must run in as straight a line as possible to a good permanent ground. 

This may be obtained by connecting to a water or gas pipe 
connected to the street mains or to a ground rod or pipe driven in 
permanently damp earth. When connections are made to pipes, 
preference shall be given to water pipes. If attachment is made 
to gas pipe, the connection in all cases must be made between the 
meter and the street mains. In every case the connection shall 
be made as near as possible to the earth. 

When the ground wire is attached to water or gas pipes, these 
pipes shall be thoroughly cleaned and tinned with rosin flux 
solder, if such a method is practicable; the ground wire shall then 
be wrapped tightly around the pipe and thoroughly soldered to it. 

When the above method is impracticable, then, if there are 
fittings where a brass plug can be inserted, the ground wire shall 
be thoroughly soldered to it; if there are no such fittings, then the 
pipe shall be thoroughly cleaned and an approved ground clamp 
lastened to an exposed portion of the pipe and the ground wire 
well soldered to the ground clamp. 

When the ground wire is attached to a ground rod driven into 
the earth, the ground wire shall be soldered to the rod in a similar 
manner. 

Steam or hot water pipes must not be used for a protector 
ground. 

m. The protector to be approved must comply with the following 
requirements: — 

For Instrument Circuits of Telegraph Systems. — 1. An approved single- 
pole cut-out, in each wire, designed for 2,000 volt potential, and 
containing fuses rated at not over one ampere capacity. When 
main line cut-outs are installed as called for in Section g, the instru- 
ment cut-outs may be placed between the switchboard and the 
instrument as near the switchboard as possible. 

For All Other Systems. — 1. Must be mounted on non -combustible, non- 
absorptive, insulating bases, so designed that when the protector 
is in place, all parts which may be alive will be thoroughly insu- 
lated from the wall to which the protector is attached. 

2. Must have the following parts: — 

A lightning arrester which will operate with a difference of 
potential between wires of not over 500 volts, and so arranged that 
the chance of accidental grounding is reduced to a minimum. 




1446 70.— 'ELECTRIC POWER AND LIGHTING. 

A fuse designed to open the circuit in case the wires become 
crossed with light or power circuits. The fuse must be able to 
open the circuit without arcing or serious flashing when crossed 
with any ordinary commercial light or power circuit. 

A heat coil, if the sensitiveness of the instrument demands it, 
which will operate before a sneak current can damage the instru- 
ment the protector is guarding. 

Heat coils are necessary in all circuits normally closed through magnet 
windings, which cannot indefinitely carry a current of at least 5 amperes. 

The heat coil is designed to warm up and melt out with a current large 
enough to endanger the instruments if continued for a long time, but so 
small that it would not blow the fuses ordinarily found necessary for such 
Instruments. The smaller currents are often called "sneak" currents. 

3. The fuses must be so placed as to protect the arrester and heat coils, 
and the protector terminals must be plainly marked "line,'* 
"instrument," "ground." 

An easily read abbreviation of the above words will be allowed. 

The Following Rules Apply to All Systems whether the Wires 

FROM the Central Office to the Building are 

Overhead or Underground. 

n. Wires beyond the protector, or wires inside buildings where no pro- 
tector is used, must be neatly arranged and securely fastened in place in some 
convenient, workmanlike manner. They must not come nearer than 
6 inches to any electric light or power wire in the building imless encased 
in approved tubing so secured as to prevent its slipping out of place. 

The wires would ordinarily be insulated, i)ut the kind of Insulation is not speci- 
fied, as the protector is relied upon to stop all dangerous currents. Porcelain tubing 
or approved flexible tubing may be used for encasing wires where required as above. 

o. Wires where bunched together in a vertical run within any building 
must have a fire-resisting covering sufficient to prevent the wires from 
carrying fire from floor to floor unless they are run either in non-combustible 
tubing or in a fireproof shaft, which shaft shall be provided with fire stops 
at each floor. 

Signaling wires and electric light or power wires may be run in the same 
shaft, provided that one of these classes of wires is run in non-combustible 
tubing, or provided that when run otherwise these two classes of wires 
shall be separated from each other by at least 2 inches. 

In no case shall signaling wires be run in the same tube with electric 
light or power wires. 

Ordinary rubber Insulation Is Inflammable, and when a number of wires are 
contained in a shaft extending through a building they afford a ready means of 
carrying fire from floor to floor, unless they are covered with a flre-resistlng material, 
or unless the shaft Is provided with fire stops at each floor. 

65. Electric Gas Lighting. — a. Electric gas lighting must not be used 
on the same fixture with the electric light. 

The above rule does not apply to frictional systems of gas lighting. 

65 A. Moving Picture Machines. — a. Arc lamp used as a part of moving 
picture machines must be constructed similar to arc lamps of theaters, and 
wiring of same must not be of less capacity than No. 6 B. & S. gage. (See 
No. 31 A d. [1].) 

b. Rheostats must conform to rehostat requirements for theater arcs. 
(See No. 31 A d. [1].) 

c. Top reel must be encased in a steel box with hole at the bottom only 
large enough for film to pass through, and cover so arranged that this hole 
can be instantly closed. No solder to be used in the construction of this 
box. 

d. A steel box must be used, for receiving the film after being shown, 
with a hole in the top only large enough for the film to pass through freely, 
with a cover so arranged that this hole can be instantly closed. An opening 
may be placed at the side of the box to take the film out, with a door hung 
at the top, so arranged that it cannot be entirely opened, and provided 
with a spring-catch to lock it closed. No solder to be used in the construc- 
tion of this box. 



MISCELLANEOUS. MARINE WORK. 1447 

e. The handle or crank used in operating the machine must be secured 
to the spindle or shaft, so that there will be no liability of its coming off 
and allowing the film to stop in front of lamp. 

f. .A shutter must be placed in front of the condenser, arranged so as 
to be readily closed. 

g. Extra films must be kept in metal box with tight-fitting cover. 

h. Machines must be operated by hand (motor-driven will not be per- 
mitted.) 

i. Picture machine must be placed in an enclosure or house made of 
suitable fireproof material, be thoroughly ventilated and large enough for 
operator to walk freely on either side of or back of machine. All openings 
into this booth must be arranged so as to be entirely closed by doors or. 
shutters constructed of the same or equally good fire-resisting material 
as. the booth itself. Doors or covers must be arranged so as to be held 
normally closed by spring hinges or equivalent devices. 

66. Insulation Resistance. — ^The wiring in any building must test free 
from grounds; i. e., the complete installation must have an insulation 
between conductors and between all conductors and the ground (not including 
attachments, sockets, receptacles, etc.) not less than that given in the fol- 
lowing table: — 

Up to 5 amperes 4,000,000 ohms. 

10 " 2,000,000 " 

25 •• 800,000 •• 

50 " 400,000 •• 

100 " 200,000 •• 

200 •' • 100,000 •* 

400 " 50,000 " 

800 " 25,000 " 

" 1,600 " 12,500 " 

The test must be made with all cut-outs and safety devices in place. If 
the lamp sockets, receptacles, electroliers, etc., are also connected, only one- 
half of the resistances specified in the table will be required. 

67. Soldering Fluid. — a. The following formula for soldering fluid is 
suggested : — 

Saturated solution of zinc chloride 5 parts. 

Alcohol 4 parts. 

Glycerine 1 part. 

Class F.— MARINE WORK. 

68. Generators. — a. Must be located in a dry place. 

b. Must have their frames insulated from their bed -plates. 

c. Must each be proivded with a waterproof cover. 

d. Must each be provided with a name-plate, giving the maker's name, 
the capacity in volts and amperes, and the normal speed in revolutions per 
minute. 

69. Wires. — a. Must be supported in approved moulding or conduit, 
except at switchboards and for portables. 

Special permission may be given for deviation from this rule In dynamo rooms. 

b. Must have no single wire larger than No. 12 B. & S. gage. Wires to 
be stranded when greater carrying capacity is required. No single solid 
wire smaller than No. 14 B. & S. gage, except in fixture wiring, to be used. 

Stranded wires must be soldered before being fastened under clamps or binding 
screws, and when they have a conductivity greater than that of No. 8 B. & S. gage 
copper wire they must be soldered into lugs. 

c. Splices or taps in conductors must be avoided as far as possible. 
Where it is necessary to make them they must be so spliced or joined as to 
be both mechanically and electrically secure without solder. They must then 
be soldered, to insure preservation, covered with an insulating compound 
equal to the insulation of the wire, and further protected by a waterproof 
tape. The joint must then be coated or painted with a waterproof com- 
pound. 



1448 70.'-ELECTRlC POWER AND LIGHTING. 

All joints must be soldered unless made with some form of approved splicing 
device. 

For Moulding Work. — d. Must have an approved insulating covering. 

The insulation for conductors, to be approved, must be at least 3-32 of an inch Id 
thickness and be covered with a substantial waterproof braid. 

The physical characteristics shall not be affected by any change in temperature 
up to 200° Fahrenheit (93° Centigrade). After two weeks' submersion in salt water 
at 70° Fahrenheit (21° Centigrade), It must show an insulation resistance of 100 
megohms per mile after three minutes' electrification with 550 volts. 

e. Must have, when passing through water-tight bulkheads and through 
all decks, a metallic stuffing tube lined with hard rubber. In case of deck 
tubes, they must be boxed near deck to prevent mechanical injury. 

f. Must be bushed with hard rubber tubing, 3^ of an inch in thickness, 
when passing through beams and non-water-tight bulkheads. 

For Conduit Work. — g. Must have an approved insulating covering. • 
The Insulation for conductors, for use in lined conduits, to be approved, must be 
at least 3-32 of an inch in thickness and be covered with a substantial waterproof 
braid. The physical characteristics shall not be affected by any change In tem- 
perature up to 200° Fahrenheit (93° Centigrade). 

After two weeks' submersion In salt water at 70° Fahrenheit (21° Centigrade), 
It must show an Insulation resistance of 100 megohms per mile after three minutes' 
electrification with 550 volts. 

For unlined metal conduits, conductors must conform to the specifica- 
tions given for lined conduits, and in addition have a second outer fibrous 
covering at least 1/32 of an inch in thickness and sufficiently tenacious to 
withstand the abrasion of being hauled through the metal conduit. 

h. Must not be drawn in until the mechanical work on the conduit is 
completed and same is in place. 

i. Where run through coal bunkers, boiler rooms, and where they are 
exposed to severe mechanical injury, must be encased in approved conduit. 

70. Portable Conductors. — a. Must be made of two stranded conductors 
each having a carrying capacity equivalent to not less than No. 14 B. & S. 
gage, and each covered with an approved insulation and covering. 

Where not exposed to moisture or severe mechanical injury, each stranded 
conductor must have a solid insulation at least 1-32 of an Inch in thickness, and 
must show an Insulation resistance between conductors, and between either con- 
ductor and the ground, of at least 50 megohms per mile after two weeks' submersion 
In water at 70° Fahrenheit (21° Centigrade), and be protected by a slow-burning, 
tough-braided outer covering. 

Where exposed to moisture and mechanical Injury (as for use on decks, holds 
and fire-rooms), each stranded conductor shall have a solid Insulation, to be approved, 
of at least 1-3 2 of an inch in thickness and protected by a tough braid. The two 
conductors shall then be stranded together, using a jute filling. The whole shall then 
be covered with a layer of flax, either woven or braided, at least 1-32 of an Inch In 
thickness and treated with a non-inflammable, waterproof compound. After one 
week's submersion in water at 70° Fahrenheit (21° Centigrade), it must show an 
Insulation between the two conductors, or between- either conductor and the ground, 
of 50 megohms per mile. 

71. Bell or Other Wires. — a. Must never be run in same duct with 
lighting or power wires. 

72. Table of Capacity of Wires.— 





Area 




Size of 






Actual 


No. of 


Strands. 




B. & S. G. 


CM. 


Strands. 


B.«&S.G. I 


Vmpei 


19 


1,288 




,, 


,, 


18 


1,624 








3 


17 


2,048 








, , 


16 


2,583 








6 


15 


3,257 








, , 


14 


4,107 








12 


12 


6,530 








17 


• • 


9,016 


*7 


19 


21 


• • 


11,368 


7 


18 


25 


• • 


14,336 


7 


17 


30 


• • 


18,081 


7 


16 


35 


• • 


22,799 


7 


15 


40 


• • 


30,856 


19 


] 


L8 


50 



MARINE WORK. 1449 





Area 




Size of 






Actual 


No. of 


Strands 




B. & S. G. 


CM. 


Strands 


B. 8c S. G. 


Amperes, 




38,912 


19 


17 


60 




49,077 


19 


16 


70 




60,088 


37 


18 


85 




75,776 


37 


17 


100 




99,064 


61 


18 


12t) 




124,928 


61 


17 


145 




157,563 


61 


16 


170 




198,677 


61 


16 


200 




250,527 


61 


14 


235 




296,387 


91 


15 


270 




373.737 


91 


14 


320 




413,639 


127 


15 


340 



When greater conducting area than that of No. 12 B. & S. gage Is required, the 
conductor shall be stranded In a series of 7, 19. 37. 61, 91. or 127 wires, as may be 
required: the strand consisting of one central wire, the remainder laid around It 
concentrically, each layer to be twisted in the opposite direction from the preceding. 

73. Switchboards. — a. Must be made of non-combustible, non-absorp- 
tive insulating material, such as marble or slate. 

b. Must be kept free from moisture, and must be located so as to be 
accessible from all sides. 

c. Must have a main switch, main cut-out and ammeter for each gene- 
rator. 

Must also have a voltmeter and ground detector. 

d. Must have a cut-out and switch for each side of each current leading 

from board. 

e. Must be wired with conductors having an insulation as required for 
moulding or conduit work and covered with a substantial flame-proof 
braid. 

74. Resistance Boxes.-^(For construction rules, see No. 60.) a. Must 
be located on switchboard or away from combustible material. When not 
placed on switchboard they must be mounted on non-inflammable, non- 
absorptive insulating material. 

75. Switches. — (For construction rules, see No. 51 ) a. When exposed 
to dampness, they must be enclosed in a water-tight case. 

b. Must be of the knife pattern when located on switchboard. 

c. Must be provided so that each freight compartment may be separately 
controlled. 

76. Cut-Outs. — (For construction rules, see No. 52.) a. Must be 
placed at every point where a change is made in the size of the wire (unless 
the cut-out in the larger wire will protect the smaller). 

b. In such places as upper decks, holds, cargo spaces and fire-rooms, a 
water-tight and fire-proof cut-out may be used, connecting directly to mains 
when such cut-out supplies circuits requiring not more than 660 watts 
energy. 

c. When placed anywhere, except on switchboards and certain places, 
as cargo spaces, holds, fire-rooms, etc., where it is impossible to run from 
center of distribution, they must be in a cabinet lined with fire-resisting 
material. 

d. Except for motors, searchlights and diving lamps must be so placed 
that no group of lamps, requiring a current of more than 660 watts, shall 
ultimately be dependent upon one cut-out. 

77. Fixtures. — a. Must be mounted on blocks made from well-seasoned 
umber treated with two coats of white lead or shellac. 

b. Where exposed to dampness, the lamp must be surrounded by a 
vapor-proof globe. 

c. Where exposed to mechanical injury, the lamp must be surrounded 
by a globe protected by a stout wire guard. 



1450 10.— ELECTRIC POWER AND LIGHTING. 

d. Must be wired with same grade of insulation as portable conductors 
which are not exposed to moisture or mechanical injury. 

e. Ceiling fixtures over two feet in length must be provided with stay 
chains. 

78. Sockets. — (For construction rules, see No. 55.) 

79. Wooden Mouldings. — (For construction rules, see No. 50.) 

a. Where moulding is run over rivets, beams, etc., a backing strip 
must first be put up and the moulding secured to this. 

b. Capping must be secured by brass screws. 

80. Interior Conduits. — (For installation rules, see No. 25.) (For con- 
struction rules, see No. 49.) 

81. Signal Lights. — a. Must be provided with approved telltale board, 
located preferably in pilot house, which will immediately indicate a burned- 
out lamp. 

82. Motors. — a. Must be wired under the same precautions as with a 
current of same volume and potential for lighting. The motor and resist- 
ance box must be protected by a double-pole cut-out and controlled by a 
double-pole switch, except in cases where one-quarter horse power or less 
is used. 

The motor leads or branch circuits must be designed to carry a current at least 
25 per cent greater than that for which the motor is rated, in order to provide for 
the inevitable occasional overloading of the motor, and the increased current re- 
quired in starting, without overfusing the wires, but where the wires under this rule 
would be overfused, in order to provide for the starting current, as in the case of many 
of the alternating current motors, the wires must be of such size as to be properly 
protected by these larger fuses. 

In general, motors should preferably have no exposed live parts. 

b. Must be thoroughly insulated. Where possible, should be set on base 
frames made from filled, hard, dry wood and raised above surrounding 
deck. On hoists and winches they must be insulated from bed-plates by 
hard rubber, fiber or similar insulating material. 

c. Must be covered with a waterproof cover when not in use. 

d. Must each be provided with a name-plate, giving maker's name, the 
capacity in volts and amperes, and the normal speed in revolutions per 
minute. 

83. Insulation Resistance. — ^The wiring in any vessel must test free 
from grounds; i. e., the complete installation must have an insulation 
between conductors and between all conductors and the ground (not includ- 
ing attachments, sockets, receptacles, etc.), of not less than the following: — 

Up to 25 amperes 800,000 ohms. 

50 " 400,000 •• 

" 100 " 200,000 " 

•• 200 ** 100,000 " 

•• 400 *• 50,000 " 

•• 800 •* 25,000 " 

" 1,600 •• 12,500 " 

• All cut-outs and safety devices in place in the above. 

Where lamp sockets, receptacles and electroliers, etc., are connected, 
one-half of the above will be required. 



DEFINITIONS. 1461 

ELECTRICAL STANDARDIZATION. 

Standardization Rules op the Am. Inst, op Elec. Engrs. 

Approved by the Board of Directors, June 21, 1907. 

Accepted by the 24th Annual Convention, June 27, 1907. 

I.— DEFINITIONS AND TECHNICAL DATA. 

1 Note. — ^The following definitions and classifications are intended to be 

practically descriptive and not scientifically rigid. 

A.— DEFINITIONS. CURRENTS. 

2 A Direct Current is a unidirectional current. 

3 A Continuous Current is a steady, or non-pulsating, direct current. 

4 A Pulsating Current is a current equivalent to the superposition of 
an alternating current upon a continuous current. 

5 An Alternating Current is a current which, when plotted, consists bf 
half-waves of equal area in successively opposite directions from the 
zero line. 

6 An Oscillating Current is a current alternating in direction, and of 
decreasing amplitude. 

B.— DEFINITIONS. ROTATING MACHINES. 

7 A Generator transforms mechanical power into electrical power. 

8 A Direct-Current Generator produces a direct current that may or 
may not be continuous. 

9 An Alternator or Alternating-Current Generator produces alternating 
current, either single-phase or polyphase. 

10 A Polyphase Generator produces currents differing symmetrically in 
phase; such as two-phase currents, in which the terminal voltages on 
the two circuits differ in phase by 90°; or three-phase currents, in which 
the terminal voltages on the three circuits differ in phase by 120°. 

11 A Double-Current Generator produces both direct and alternating 
currents. 

12 A Motor transforms electrical into mechanical power. 

13 A Booster is a machine inserted in series in a circuit to change its 
voltage. It may be driven by an electric motor (in which case it is termed 
a motor-booster) or otherwise. 

14 A Motor -Generator is a transforming device consisting of a motor 
mechanically connected to one or more generators, 

15 A Dynamotor is a transforming device combining both motor and 
generator action in one magnetic field, with two armatures; or with an 
armature having two separate windings and independent commutators. 

16 A Converter is a machine employing mechanical rotation in changing 
electrical energy from one form to another. A converter may belong to 
either of several types, as follows: 

17 a. A Direct-Current Converter converts from a direct current to a 
direct current. 

18 6. A Synchronous Converter (commonly called a rotary converter) 
converts from an alternating to a direct current, or vice versa. 

19 c. K Motor-Converter is a combination of an induction motor with 
a synchronous converter, the secondary of the former feeding the arma- 
ture of the latter with current at some frequency other than the impressed 
frequency; i. e., it is a synchronous converter concatenated with an 
induction motor. 

20 d. A Frequency-Converter converts from an alternating-current sys- 
tem of one frequency to an alternating-current system of another 
frequency, with or without a change in the number of phases or in 
voltages. 

21 e. K Rotary Phase Converter converts from an alternating-current 
system of one or more phases to an alternating-current system of a dif- 
ferent number of phases, but of the same frequency. 

C.—DEFINITIONS. STATIONARY INDUCTION APPARATUS. 

22 Stationary Induction Apparatus change electric energy to electric 
energy through the medium of magnetic energy. They comprise 
several forms, distinguished as follows: 



1452 70.— ELECTRIC POWER AND LIGHTING. 

23 a. In Transformers the primary and secondary windings are in- 
sulated from one another. 

24 b. In Auto-Transformers, also called compensators, a part of the 
primary winding is used as a secondary winding, or conversely. 

25 c. In Potential Regulators a coil is in shunt and a coil is in series 
with the circuit, so arranged that the ratio of transformation between 
them is variable at will. They are of the following three classes: 

26 (a) Compensator Potential Regulators in which a number of turns 
of one of the coils are adjustable. 

27 (b) Induction Potential Regulators in which the relative positions 
of the primary and secondary coils are adjustable. 

28 {c) Magneto Potential Regulators in which the direction of the 
magnetic flux with respect to the coils is adjustable, 

29 d. Reactors, or Reactance Coils, formerly called choking coils, are 
a form of stationary induction apparatus used to produce reactance or 
phase displacement. 

D.— GENERAL CLASSIFICATION OF APPARATUS. 

30 Commutating Machines. Under this head may be classed the follow- 
ing: Direct current generators; direct-current motors; direct-current 
boosters; motor-generators; dynamotors; converters, compensators or 
balancers; closed-coil arc machines, and alternating-current commu- 
tating motors. 

31 Commutating machines may be further classified as follows: 

32 a. Direct-Current Commutating Machines, which comprise a magnetic 
field of constant polarity, a closed-coil armature, and a multisegmental 
commutator connected therewith. 

ZZ b. Alternating-Current Commutating Machines, which comprise a 
magnetic field of alternating polarity, a closed-coil armature, and a 
multisegmental commutator connected therewith. 

34 c. Synchronous Commutating Machines, which comprise synchronous 
converters, motor converters and double-current generators. 

35 Synchronous Machines, which comprise a constant magnetic field, 
and an armature receiving or delivering alternating-currents in synchron- 
ism with the motion of the machine; i.e., having a frequency equal to the 
product of the number of pairs of poles and the speed of the machine in 
revolutions per second. 

36. Stationary Induction Apparatus, which includes transformers, auto- 

transformers, potential regulators, and reactors or reactance coils. 

37 Rotary Induction Apparatus, or Induction Machines, which include 
apparatus wherein the primary and secondary windings rotate with re- 
spect to each other; i.e., induction motors, induction generators, fre- 
quency converters, and rotary phase converters. 

38 Unipolar or Acyclic Machines, in which the voltage generated in the 
active conductors maintains the same direction with respect to those 
conductors. 

39 Rectifying Apparatus, Pulsating-Current Generators. 

40 Electrostatic Apparatus, such as condensers, etc. 

41 Electrochemical Apparatus, such as batteries, etc. 

42 Electrothermal Apparatus, such as rheostats, heaters, etc. 

43 Protective Apparatus, such as fuses, lightning arresters, etc. 

44 Luminous Sources. 

E.— MOTORS. SPEED CLASSIFICATION. 

45 Motors may, for convenience, be classified with reference to their 
speed characteristics as follows: 

46 a. Constant-Speed Motors, in which the speed is either constant or 
does not materially vary; such as synchronous motors, induction 
motors with small slip, and ordinary direct-current shunt motors. 

47 b. Multispeed Motors (two-speed, three-speed, etc.), which can be 
operated at any one of several distinct speeds, these speeds being prac- 
tically independent of the load, such as motors with two armature wind- 
ings. 

48 c. Adjustable-Speed Motors, in which the speed can be varied grad- 
ually over a considerable range; but when once adjusted remains prac- 
tically unaffected by the load, such as shunt motors designed for a con- 
siderable range of field variation. 

49 d. Varying-Speed Motors, or motors in which the speed varies with 
the load, decreasing when the load increases; such as series motors. 



DEFINITIONS AND TECHNICAL DATA. 1453 

F.— DEFINITION AND EXPLANATION OF TERMS. 

(I) Load Factor. 

The Load Factor of a machine, plant or system is the ratio of the 
average power to the maximum power during a certain period of time. 
The average power is taken over a certain interval of time, such as a day 
or a year, and the maximum is taken over a short interval of the maxi- 
mum load within that interval. 

In each case the interval of maximum load should be definitely speci- 
fied. The proper interval is usually dependent upon local conditions and 
upon the purpose for which load factor is to be determined. 

(II) Non-Inductive Load and Inductive load. 

A non-inductive load is a load in which the current is in phase with 
the voltage across the load. 

An inductive load is a load in which the current lags behind the 
voltage across the load. A load in which the current leads the voltage 
across the load is sometimes called in anti-inductive load. 

(III) Power-Factor and Reactive-Factor. 

The Power-Factor in alternating-current circuits or apparatus is the 
ratio of the electric power in watts to the apparent power in volt-amperes. 
It may be expressed as follows; 

true power watts energy current _ energy voltage 

apparent power volt-amperes total current total voltage 
The Reactive-Factor is the ratio of the wattless volt-amperes (i.e.t 
the product of the wattless component of current by voltage, or wattless 
component of voltage by current) to the total amperes. It may be ex- 
pressed as follows: 

wattless volt-amperes _ wattless current _ wattless voltage 
total volt-amperes total current total voltage 

Power-Factor and Reactive-Factor are related as follows: 
lip = power-factor, q = reactive-factor, then with sine waves of voltage 
and current, 

^2 + ^2 == 1. 

With distorted waves of voltage and current, 
p2 + q2 = or < 1. 

(IV) Saturation-Factor. 

The Saturation-Factor of a machine is the ratio of a small percentage 
increase in field excitation to the corresponding percentage increase in 
voltage thereby produced. The saturation-factor is, therefore, a criterion 
of the degree of saturation attained in the magnetic circuits at any ex- 
citation selected. Unless otherwise specified, however, the saturation- 
factor of a machine refers to the excitation existing at normal rated 
speed and voltage. It is determined from measurements of saturation 
made on open circuit at rated speed. 

The Percentage of Saturation of a machine at any excitation may 
be found from its saturation curve of generated voltage as ordinates, 
against excitation as abscissas, by drawing a tangent to the curve at the 
ordinate corresponding to the assigned excitation, and extending the 
tangent to intercept the axis of ordinates drawn through the origin. The 
ratio of the intercept on this axis to the ordinate at the assigned excita- 
tion, when expressed in percentage, is the percentage of saturation and is 
independent of the scale selected for excitation and voltage. This ratio 
is equal to the reciprocal of the saturation-factor at the same excitation, 
deducted from unity. Thus, if / be the saturation-factor and p the per- 
centage of saturation ratio, 

(V) Variation and Pulsation. 

The Variation in Prime Movers which do not give an absolutely 
uniform rate of rotation or speed, as in reciprocating steam engines, is 
the maximum angular displacement in position of the revolving member 



1454 10.— ELECTRIC POWER AND LIGHTING. 

expressed in degrees, from the position it would occupy with uniform 
rotation, and with one revolution taken as 360°. 

60 The Pulsation in Prime Movers is the ratio of the difference between 
the maximum and minimum velocities in an engine-cycle to the average 
velocity. 

61 ^ The Variation in Alternators or alternating-current circuits in general 
is the maximum difference in phase of the generated voltage wave from a 
wave of absolutely constant frequency, expressed in electrical degrees 
(one cycle equals 360°) and may be due to the variation of the prime 
mover. 

62 The Pulsation in Alternators or alternating-current circuits, in gen- 
eral, is the ratio of the difference between maximum and minimum fre- 
quency during an engine cycle to the average frequency. 

63 Relation of Variation in prime m^over and alternator: 

64 If « = number of pairs of poles, the variation of an alternator is n 
times the variation of its prime mover, if direct-connected, and n/p times 
the variation of the prime mover if rigidly 'connected thereto in the ve- 
locity ratio p. 

II.— PERFORMANCE SPECIFICATIONS AND TESTS. 

A.— RATING. 

65 Rating by Output. All electrical apparatus should be rated by output 
and not by input. Generators, transformers, etc., should be rated by 
electrical output: motors by mechanical output. 

66 Rating in Kilowatts. Electrical power should be expressed in kilo- 
watts, except when otherwise specified. 

67 Apparent Power, Kilovolt-Amperes. Apparent power in alternating- 
current circuits should be expressed in kilovolt-amperes as distinguished 
from real power in kilowatts. When the power factor is 100 per cent,, 
the apparent power in kilovolt-amperes is equal to the kilowatts. 

68 The Rated {Full Load) Current is that current which, with the rated 
terminal voltage, gives the rated kilowatts, or therated kilovolt-amperes. 
In machines in which the rated voltage differs from the no-load voltage, 
the rated current should refer to the former. 

69 Determination of Rated Current. The rated current may be de- 
termined as follows: If P = rating in watts, or apparent watts if the 
power factor be other than 100 per cent., and E — full-load terminal 
voltage, the rated current per terminal is: 

P 

70 / =-^ in a direct-current machine or single-phase alternator. 

1 P 

71 /= — zr=, -E^ in a three-phase alternator, 

V 3 ^ 

1 P 

72 I =-^ -^ in 2k two-phase alternator. 

73 Normal Conditions. The rating of machines or apparatus should be 
based upon certain normal conditions to be assumed as standard, or to be 
specified. These conditions include voltage, current, power-factor, fre- 
quency, wave shape and speed; or such of them as may apply in each par- 
ticular case. Performance tests should be made under these standard 
conditions unless otherwise specified. 

74 a. Power Factor. Alternating-current apparatus should be rated in 
kilowatts, at 100 per cent power factor; i.e., with current in phase with 
terminal voltage, unless a phase displacement is inherent in the apparatus 
or is specified. If a power factor other than 100 per cent is specified, 
the rating should be expressed in kilovolt-amperes and power factor, at 
rated load. 

75 h. Wave Shape. In determining the rating of alternating-current ma- 
chines or apparatus, a sine wave shape of alternating current and 
voltage is assumed, except where a distorted wave shape is inherent to 
the apparatus. See Sees. 79-83. 

76 Fuses. The rating of a fuse should be tne maximum current which it 
will continuously carry. 

77 Circuit-Breakers. The rating of a circuit-breaker should be the max- 
imum current which it is designed to carry continuously. 



PERFORMANCE SPECIFICATIONS AND TESTS, 1465 

78 a. Note, In addition thereto, the maximum current and voltage at 
which a fuse or a circuit-breaker will open the circuit should be specified. 
It is to be noted that the behavior of fuses and of circuit-breakers is 
much influenced by the amount of electric power available on the circuit. 

B.— WAVE SHAPE. 

79 The Sine Wave should be considered as standard, except where a dif- 
ference in the wave form from the sinusoidal is inherent in the operation 
of the apparatus. 

80 A Maximum Deviation of the wave from sinusoidal shape not exceed- 
ing 10 per cent is permissible, except when otherwise specified. 

81 The Deviation of wave form from the sinusoidal is measured by de- 
termining the form by oscillograph or wave meter, computing therefro.m 
the equivalent sine wave of equal length, superposing the latter upon the 
observed wave in such a manner as to give least difference, and then 
dividing the maximum difference at any ordinate by the maximum value 
of the equivalent sine wave. 

82 The Equivalent Sine Wave is a sine wave having the same frequency 
and the same effective or r.m.s. (root of mean square) value as the actual 
wave. 

83 Non-Sine Waves. The phase displacement between two waves which 
are not sine waves, is that phase displacement between their equivalent 
sine waves which would give the same average product of instantaneous 
values as the actual waves; i.e.y the same electro-dynamometer reading. 

C— -EFFICIENCY. 
(I) — Definitions. 

84 The Efficiency of an apparatus is the ratio of its net power output 
to its gross power input. 

85 a. Note. An exception should be noted in the case of storage batteries 
or apparatus for storing energy in which the efficiency, unless otherwise 
qualified, should be understood as the ratio of the energy output to the 
energy intake in a normal cycle. An exception should also be noted in 
the case of luminous sources. 

86 ^ Apparent Efficiency. ^ In apparatus in which a phase displacement is 
inherent to their operation, apparant efficiency should be understood as 
the ratio of the net power output to volt-ampere input. 

87 a. Note. Such apparatus comprise induction motors, reactive syn- 
chronous converters, synchronous converters controlling the voltage of 
an alternating-current system, self -exciting synchronous motors, poten- 
tial regulators and open magnetic circuit transformers, etc. 

88 b. Note. Since the apparent efficiency of apparatus delivering electric 
power depends upon the power-factor of the load, the apparent efficiency, 
unless otherwise specified, should be referred to a load power-factor of 
unity. 

(II) — Determination of Efficiency. 

89 Methods. Efficiency may be determined by either of two methods, 
viz.: by measurement of input and output; or, by measurement of losses, 

90 a. Method of Input and Output. The input and output may both 
be measured directly. The ratio of the latter to the former is the 
efficiency. 

91 b. Method by Losses. The losses may be measured either collectively 
or individually. The total losses may be added to the output to derive 
the input, or subtracted from the input to derive the output. 

92 Comparison of Methods. The output and input method is preferable 
with small machines. When, however, as in the case of large machines, 
it is impracticable to measure the output and input; or when the per- 
centage of power loss is small and the efficiency is nearly unity, the 
method of determining efficiency by measuring the losses should be 
followed. 

93 Electric Power should be measured at the terminals of the appa- 
ratus. In tests of polyphase machines, the measurement of power should 
not be confined to a single circuit but should be extended to all the cir- 
cuits in order to avoid errors of unbalanced loading. 

94 Mechanical Power in machines should be measured at the pulley, 
gearing, coupling, etc., thus excluding the loss of power in said pulley 
gearing or coupling, but including the bearing friction and windage. The 



1456 IQ.— ELECTRIC POWER AND LIGHTING. 

magnitude of bearing friction and windage may be considered, with con- 
stant speed, as independent of the load. The loss of power in the belt and 
the increase of bearing friction due to belt tension should be excluded. 
Where, however, a machine is mounted upon the shaft of a prime mover, 
in such a manner that it cannot be separated therefrom, the frictional 
losses in bearings and in windage, which ought, by definition, to be included 
in determining the efficiency, should be excluded, owing to the practical 
impossibility of determining them satisfactorily. 

95 In Auxiliary Apparatus, such as an exciter, the power lost in the 

auxiliary apparatus should not be charged to the principal machine, but 
to the plant consisting of principal machine and auxiliary apparatus 
taken together. The plant efficiency in such cases should be distinguished 
from the machine efficiency. 

96- Normal Conditions. Efficiency tests should be made under normal 
conditions herein set forth and which are to be assumed as standard. 
These conditions include voltage, current, power-factor, frequency, wave 
shape, speed and barometric pressure, temperature, or such of them as 
may apply in each particular case. Performance tests should be made 
under these standard conditions imless otherwire specified. See Sees. 
73-75. 

97 a. Temperature. The efficiency of all apparatus, except such as may 
be intended for intermittent service, should be either measured at, or re- 
duced to, the temperature which the apparatus assumes under continuous 
operation at rated load, referred to a room temperature of 25° C. See 
Sees. 267-292. 

98 With apparatus intended for intermittent service, the efficiency 
should be determined at the temperature assumed under specified con- 
ditions. 

99 h. Power Factor. ^ In determining the efficiency of alternating-current 
apparatus, the electric power should be measured when the current is in 
phase with the voltage, unless otherwise specified, except when a 
definite phase difference is inherent in the apparatus, as in induction 
motors, induction generators, frequency converters, etc. 

100 c. Wave Shape. In electrical apparatus, the sine wave should be 
considered as standard, except where a difference in the wave form from 
the sinusoidal is inherent in the operation of the apparatus. See Sees. 
79-83. 

(Ill) — Measurement op Losses. 

101 Losses. The usual sources of losses in electrical apparatus and the 
methods of determining these losses are as follows: 

{A) — Bearing Friction and Windage. 

i02 The magnitude of bearing friction and windage (which may be con- 

sidered as independent of the load) is conveniently measured by driving 
the machine from an independent motor, the output of which may be 
suitably determined. See Sec. 94. 

{B) — Commutator Brush Friction. 

103 The magnitude of the commutator brush friction (which may be 
considered as indepednent of the load) is determined by measuring the 
difference in power required for driving the machine with brushes on 
and with brushes off (the field being unexcited) . 

(C) — Collector-Ring Brush Friction. 

104 Collector-ring brush friction may be determined in the same manner 
as commutator brush friction. It is usually negligible. 

{D) — Molecular Magnetic Friction and Eddy Currents. 

105 These losses include those due to molecular magnetic friction and 
eddy currents in iron and copper and other metallic parts, also the 
losses due to currents in the cross-connections of cross-connected 
armatures. 

106 In Machines these losses should be determined on open circuit and 
at a voltage equal to the rated voltage -f- / r in a generator, and — 
/ r in a motor, where / denotes the current strength and r denotes the 
internal resistance of the machine. They should be measured at the 
correct speed and voltage, since they do not usually vary in any definite 
proportion to the speed or to the voltage. 



PERFORMANCE SPECIFICATIONS AND TESTS. 1467 

107 Note. The Total Losses in bearing friction and windage, brush fric- 
tion, magnetic friction and eddy currents can, in general, be deter- 
mined by a single measurement by driving the machine with the field 
excited, either as a motor, of by means of an independent motor. 

108 Retardation Method. The no-load iron, friction, and windage losses 
may be segregated by the Retardation Method, in which the generator 
should be brought up to full speed (or, if possible, to about 10 per 
cent, above full speed) as a motor, and, after cutting off the driving 
power and excitation, frequent readings should be taken of speed and 
time, as the machine slows down, from which a speed-time curve can 
be plotted. A second curve should be taken in the same manner, but 
with full field excitation; from the second curve the iron losses may be 
found by subtracting the losses found in the first curve. 

109 The speed-time curves can be plotted automatically by belting a 
small separately excited generator (say 1/10 kw.) to the generator 
shaft and connecting it to a recording voltmeter. When the retardation 
method is not feasible, the frictional losses in bearings and in windage, 
which ought, by definition, to be included in determining the efficiency, 
may be excluded; but this should be expressly stated. 

(E) — Armature-Resistance Loss. 

110 This loss may be expressed by p P r; where r = resistance of one 
armature circuit or branch, / = the current in such armature circuit or 
branch, and p = the number of armature circuits or branches. 

(F) — Commutator Brush and Brush-Contact Resistance Loss. 

111 It is desirable to point out that with carbon brushes these losses may 
be considerable in low-voltage machines. 

(G) — Collector-Ring and Brush-Contact Resistance Loss. 

112 This loss is usually negligible, except in machines of extremely low 
voltage or in unipolar machines. 

(H) — Field Excitation Loss. 

113 With separately excited fields, the loss of power in the resistance of 
the field coils alone should be considered. With either shunt- or series- 
field windings, however, the loss of power in the accompanying rheostat 
should also be included, the said rheostat being considered as an 
essential part of the machine, and not as separate auxiliary apparatus. 

(/) — Load Losses. 

114 The load losses may be considered as the difference between the 
total losses under load and the sum of the losses above specified. 

115 a. In Commutating Machines of small field distortion, the load 
•losses are usually trivial and may, therefore, be neglected. When, 
however, the field distortion is large, as is shown, for instance, by the 
necessity for shifting the brushes between no load and full load, or with 
variations of load, these load losses may be considerable, and should be 
taken into account. In this case the efficiency may be determined 
either by input and output measurements, or the load losses may be 
estimated by the method of Sec. 116. 

116 b. Estimation of Load Losses. While the load losses cannot well 
be determined individually, they may be considerable and, therefore, 
their joint influence should be determined by observation. This can be 
done by operating the machine on short-circuit and at full-load current, 
that is, by determining what may be called the "short-circuit core loss." 
With the low field intensity and great lag of current existing in this case, 
the load losses are usually greatly exaggerated. 

117 One-third of the short-circuit core loss may, as an approximation, 
and in the absence of more accurate information, be assumed as the 
load loss. 

(IV) — Eppiciency op Different Types op Apparatus. 

(A) — Direct -Current Commutating Machines. 

118 In Direct-Current Commutating Machines the losses are: 

119 a. Bearing Friction and Windage. See measurement of Losses (i4). 
Sec. 102. 



1468 70.— ELECTRIC POWER AND LIGHTING. 

120 b. Molecular Magnetic Friction and Eddy Currents. See measure- 
ment of Losses {D) Sec. 105. 

121 c. Armature Resistance Losses. See Measuremeat of Losses {E), 
Sec. 110. 

122 d. Commutator Brush Friction. See Measurement of Losses (j5). 
Sec. 103. 

123 e. Commutator Brush and Brush Contact Resistance. See Meas- 
tirement of Losses {F), Sec. 111. 

124 /. Field Excitation Loss. See Measurement of Losses (/f), Sec. 113. 

125 g. Load Losses. See Measurement of Losses (/), Sec, 114. 

126 Note, b and c are losses in the armature or "armature losses"; 
d and e "commutator losses"; / "field losses." 

(B) — Alternating-Current Commutating Machines. 

127 In Alternating-Current Commutating Machines, the losses are: 

128 a. Bearing Friction and Windage. See Measurement of Losses (A), 
Sec. 102. 

129 b. Rotation Loss, measured with the machine at open circuit, the 
brushes on the commutator, and the field excited by alternating cur- 
rent when driving the machine by a motor. 

130 This loss includes molecular magnetic friction, and eddy currents, 
caused by rotation through the magnetic field, Pr losses in cross-con- 
nections of cross-connected armatures, Pr and other losses in armature- 
coils and arm.ature-leads which are short-circuited by the brushes as far 
as these losses are due to rotation. 

131 c. Alternating or Transformer Loss. These losses are measured by 
wattmeter in the field circuit, under the conditions of test b. They 
include molecular magnetic friction and eddy-currents due to the alter- 
nation of the magnetic field, Pr losses in cross-connections of cross-con- 
nected armatures, Pr and other losses in armature coil and commutator 
leads which are short-circuited by the brushes, as far as these losses are 
due to the alteration of the magnetic flux. 

132 The losses in armature coils and commutator leads short-circuited 
by the brushes, can be separated in b, and c, from the other losses, by 
running the machine with and without brushes on the commutator. 

133 d. Pr Loss, Other Load Losses in armature and compensating wind- 
ing and Pr loss of brushes, measured by wattmeter connected across the 
armature and compensating winding. 

134 e. Field Excitation Loss. SeeMeasurement of Losses (H), Sec. 113. 

135 /. Commutator Brush-Friction. See measurement of Losses (5), 
Sec. 103. 

(C) — Synchronous Commutating Machines. 

136 1. In Double-Current Generators, the efficiency of the machine 
should be determined as a direct -current generator, and also as an alter- 
nating-current generator. The two values of efficiency may be different, 
and should be clearly distinguished. 

137 2. In Converters the losses should be determined when driving the 
machine by a motor. These losses are: 

138 a. Bearing Friction and Windage. See Measurement of Losses (A), 
Sec. 102. 

139 b. Molecular Magnetic Friction and Eddy Currents. See Measure- 
ment of Losses (D), Sec. 105. 

140 c. Armature Resistance Loss. This loss in the armature is q Pr, 
where / = direct current in armature, r = armature resistance and q, a 
factor which is equal to 1.47 in single-circuit single-phase, 1.15 in double- 
circuit single-phase, 0.59 in three-phase, 0.39 in two-phase, and 0.27 in 
six-phase converters. 

141 d. Commutator-Brush Friction. See Measurement of Losses (JB). 
Sec. 103. 

142 e. Collector-Ring Brush Friction. See Measurement of Losses (C) 
Sec. 104. 

143 /. Commutator-Brush and Brush-Contact Resistance Loss. See 
Measurement of Losses (F), Sec. 111. 

144 g. Collector-Ring Brush-Contact Resistance Loss. See Measurement 
of Losses (G),Sec. 112. 

145 h. Field JExcitation Loss. See Measurement of Losses (H), Sec. 109 

146 i. Load Losses. These can generally be neglected, owing to the 
absence of field distortion. 



PERFORMANCE SPECIFICATIONS AND TESTS. 1459 

147 3. The Efficiency of Two Similar Converters may be determined by- 
operating one machine as a converter from direct to alternating, and 
the other as a converter from alternating to direct, connecting the alter- 
nating sides together, and measuring the difference between the direct- 
current input, and the direct-current output. This process may be 
modified by returning the output of the second machine through two 
boosters into the first miachine and measuring the losses. Another modi- 
fication is to supply the losses by an alternator between the two ma- 
chines, using potential regulators. 

(D) — Synchronous Machines. 

148 In Synchronous Machines the losses are: 

149 a. Bearing Friction and Windage. See Measurement of Losses (A), 
Sec. 102. 

150 b. Molecular Magnetic Friction and Eddy Currents. See Measure- 
ment of Losses (D), Sec. 105. 

151 c. Armature Resistance Loss. See Measurement of Losses (E)t 
Sec. 110. 

152 d. Collector-Ring Brush Friction. See Measurement of Losses (Q, 
Sec. 104. 

153 e. Collector-Ring Brush Contact Resistance Loss. See Measurement 
of Losses (6^), Sec„ 112. 

154 /. Field Excitation Loss. See Measurement of Losses (H), Sec. 113, 

155 g. Load Losses. See Measurement of Losses (/), Sec. 114. 

(E) — Stationary Induction Apparatus. 

156 In Stationary Induction Apparatus, the losses are: 

157 a. Molecular Magnetic Friction and Eddy Currents measured at 
open secondary circuit, rated frequency, and at rated voltage — 7 r, 
where / = rated current, r = resistance of primary circuit. 

158 b. Resistance Losses, the sum of the Pr losses in the primary and 
in the secondary windings of a transformer, or in the two sections of the 
coil in a compensator or auto-transformer, where 7 = rated current in 
the coil or section of coil, and r = resistance. 

159 c. Load Losses, i. e , eddy currents in the iron and especially in the 
copper conductors, caused by the current at rated load. For practical 
purposes they may be determined by short-circuiting the secondary of 
the transformer and impressing upon the primary a voltage sufficient 
to send rated load current through the transformer. The loss in the 
transformer u^der these conditions measured by wattmeter gives the 
load losses -f-7V losses in both primary and secondary coils. 

160 In Closed Magnetic Circuit Transformers, either of the two circuits 
may be used as primary when determining the efficiency. 

161 In Potential Regulators, the efficiency should be taken at the maxi- 
mum voltage for which the apparatus is designed, and with non-induct- 
ive load, unless otherwise specified. 

(F) — Rotary Induction Apparatus, or Induction Machines. 

162 In Rotary Induction Apparatus, the losses are: 

163 a. Bearing Friction and Windage. See Measurement of Losses (A), 
Sec. 102. 

164 b. Molecular Magnetic Friction and Eddy Currents in iron, copper 
and other metallic parts; also Pr losses which may exist in multiple- 
circuit windings, a and b together are determined by running the 
motor without load at rated voltage, and measuring the power input. 

165 c. Primary PR Loss, which may be determined by measurement of 
the current and the resistance. 

166 d. Secondary PR Loss, which may be determined as in the primary, 
when feasible; otherwise, as in squirrel-cage secondaries, this loss is 
measured as part of e. 

167 e. Load Losses', *". ^., molecular magnetic friction, and eddy currents 
in iron, copper, etc., caused by the stray field of primary and secondary 
currents, and secondary PR loss when undeterminable under {d) . These 
losses may for practical purposes be determined by measuring the total 
power, with the rotor short-circuited at standstill and a current in the 

• primary circuit equal to the primary energy current at full load. The 
loss in the motor under these conditions may be assumed to be equal to 
the load losses 4- 72r losses in both primary and secondary coils. 



1460 70.— ELECTRIC POWER AND LIGHTING, 

(G) — Unipolar or Acyclic Machines. 

168 In Unipolar Machines, the losses are: 

169 (a) Bearing Friction and Windage. See Measurement of Losses (A), 
Sec. 102. 

170 {h) Molecular Magnetic Friction and Eddy Currents. See Measure- 
ment of Losses {E), Sec. 106. 

171 {c) Armature Resistance Losses. See Measurement of Losses (£), 
Sec. 110. 

172 (d) Collector Brush Friction. See Measurement of Losses (Q, 
Sec. 104. 

173 (e) Collector Brush Contact Resistance. See Measurement of Losses 
(6^), Sec. 112. 

174 (/) Field-Excitation as in Sec. 113. See Measurement of Losses (H), 
Sec. 113. 

175 (g) Load Losses. See Measurement of Losses (/), Sec. 114. 

(H) — Rectifying Apparatus, Pulsating-Current Generators. 

176 This Division Includes: open-coil arc machines and mechanical and 
other rectifiers. 

177 ^ In Rectifiers the most satisfactory method of determiining the effi- 
ciency is to measure both electric input and electric output by watt- 
meter. The input is usually inductive, owing to phase displacement 
and to wave distortion. For this reason the power factor and the 
apparent efficiency should also be considered, since the latter may be 
much lower than the true efficiency. The power consumed by auxiliary 
devices, such as the synchronous motor or cooling devices, should be 
included in the electric input. 

178 In Constant-Current Rectifiers, transforming from constant potential 
alternating to constant direct current, by means of constant-current 
transforming devices and rectifying devices, the losses in the transform- 
ing devices are to be included in determining the efficiency and have to 
be measured when operating the rectifier, since in this case the losses 
may be greater than when feeding an alternating secondary circuit. 
In constant-current transforming devices, the load losses may be con- 
siderable, and, therefore, should not be neglected. 

179 .In Open Coil Arc Machines, the losses are essentially the same as in 
direct-current (closed coil) commutating machines. In this case, how- 
ever, the load losses are usually greater, and the efficiency should prefer- 
ably be measured by input- and output-test, using wattmeters for 
measuring the output. In alternating-current rectifiers, the output 
should, in general, be measured by wattmeter and not by voltmeter and 
ammeter, since owing to pulsation of current and voltage, a considerable 
discrepancy may exist between the watts and volt-amperes. If, how- 
ever, a direct-current and an alternating-current meter in the rectified 
circuit (either a voltmeter or an ammeter) give the same reading, the 
output may be measured by direct-current voltmeter and ammeter. 
The type of alternating-current instrument here referred to should 
indicate the effective or root-of -mean -square value and the type of 
direct-current instrument the arithmetical mean value, which would 
be zero on an alternating-current circuit. 

(/) — ^Transmission Lines. 

180 The Efficiency of transmission lines should be measured with non- 
inductive load at the receiving end, with the rated receiving voltage and 
frequency, also with sinusoidal impressed wave form, except where 
expressly specified otherwise, and with the exclusion of transformers or 
other apparatus at the ends of the line. 

(J) — Phase-Displacing Apparatus. 

181 In Apparatus Producing Phase Displacement as, for example, 
synchronous compensators, exciters of induction generators, reactors, 
condensers, polarization cells, etc., the efficiency should be understood 
to be the ratio of the volt-amperes minus power loss to the volt-amperes. 

182 The Efficiency may be calculated by determining the losses, sub- 
tracting them from the volt-amperes, and then dividing the difference 
by the volt-amperes. 

183 In Synchronous Compensators and exciters of induction generators, 
the determination of losses is the same as in other synchronous ma- 
chines. 



PERFORMANCE SPECIFICATIONS AND TESTS. 1461 

184 In Reactors the losses are molecular magnetic friction, eddy, losses 
and Pr loss. They should be measured by wattmeter. The efficiency of 
reactors should be determined with a sine wave of impressed voltage 
except where expressly specified otherwise. 

185 In Condensers, the losses are due to dielectic hysteresis and leakage, 
and should be determined by wattmeter with a sine wave of voltage. 

186 In Polarization Cells, the losses are those due to electric resistivity 
and a loss in the electrolyte of the nature of chemical hysteresis. These 
losses may be considerable. They depend upon the frequency, voltage 
and temperature, and should be determined with a sine wave of im- 
pressed voltage, except where expressly specified otherwise. 

D.— REGULATION. 
(I) — Definitions. 

187 Definition. The regulation of a machine or apparatus in regard to 
some characteristic quantity (such as terminal voltage, current or 
speed) is the ratio of the deviation of that quantity from its normal 
value at rated load to the normal rated load value. The term "regula- 
tion," therefore, has the same meaning as the term "inherent regula- 
tion," occasionally used. 

188 Constant Standard. If the characteristic quantity is intended to re- 
main constant (e. g., constant voltage, constant speed, etc.) between 
rated load and no load, the regulation is the ratio of the maximum varia- 
tion from the rated-load value to the no-load value. 

189 Varying Standard. If the characteristic quantity is intended to 
vary in a definite manner between rated load and no load, the 
regulation is the ratio of the maximum variation from the specified 
condition to the normal rated-load value. 

190 (a) Note. — If the law of the variation (in voltage, current, speed, 
etc.) between rated-load and no-load is not specified, it should be 
assumed to be a simple linear relation; i. e., one undergoing uniform 
variation between rated-load and no-load. 

191 (b) Note. — The regulation of an apparatus may, therefore, differ 
according to its qualification for use. Thus, the regulation of a com- 
pound-wound generator specified as a constant-potential generator, 
will be different from that which it possesses when specified as an over- 
compounded generator. 

192 In Constant-Potential Machines, the regulation is the ratio of the 
maximum difference of terminal voltage from the rated-load value 
(occurring within the range of rated load to open circuit) to the rated- 
load terminal voltage. 

193 In Constant-Current Machines, the regulation is the ratio of the 
maximum difference of current from the rated-load value (occurring 
within the range from rated-load to short-circuit, or minimum limit of 
operation), to the rated-load current. 

194 In Constant- Power Apparatus, the regulation is the ratio of maxi- 
mum difference of power from the rated-load value (occurring within 
the range of operation specified) to the rated power. 

195 In Constant-Speed Direct-Current M^ors and Induction Motors the 
regulation is the ratio of the maximum variation of speed from its 
rated-load value (occurring within the range from rated-load to no-load) 
to the rated-load speed. 

196 The regulation of an induction motor is, therefore, not identical with 
the slip of the motor, which is the ratio of the drop in speed from 
synchronism, to the synchronous speed. 

197 In Constant-Potential Transformers, the regulation is the ratio of 
the rise of secondary terminal voltage from rated non-inductive load to 
no-load (at constant primary impressed terminal voltage) to the 
secondary terminal voltage at rated load. 

198 In Over-Compounded Machines, the regulation is the ratio of the 
maximum difference in voltage from a straight line connecting the no- 
load and rated-load values of terminal voltage as function of the load 
current, to the rated-load terminal voltage. . 

199 In Converters, Dynamotors, Motor-Generators and Frequency Con- 
verters, the regulation is the ratio of the maximum difference of terminal 
voltage at the output side from the rated-load voltage, to the rated- 
load voltage on the output side. 

200 In Transmission Lines, Feeders, etc., the regulation is the ratio of 



14'62 70.— ELECTRIC POWER AND LIGHTING, 

the maximum voltage difference at the receiving end, between rated 
non-inductive load and no-load to the rated-load voltage at the re- 
ceiving end (with constant voltage impressed upon the sending end). 

201 In Steam Engines, the regulation is the ratio of the maximum 
variation of speed in passing slowly from rated-load to no-load (with 
constant steam pressure at the throttle) to the rated-load speed. For 
variation and pulsation, see Sees. 59-64. 

202 In a Hydraulic Turbine or Other Water-Motor, the regulation is the 
ratio of the maximum variation of speed in passing slowly from rated- 
load to no-load (at constant head of water; i. e., at constant difference 
of level between tail race and head race), to the rated-load speed. For 
variation and pulsation, see Sees. 59-64. 

203 In a Generator -Unit, consisting of a generator united with a prime- 
mover, the regulation should be determined at constant conditions of 
the prime-mover; i. e., constant steam pressure, head, etc. It includes 
the inherent speed variations of the prime-mover. For this reason the 
regulation of a generator-unit is to be distinguished from the regulation 
of either the prime-mover, or of the generator contained in it, when 
taken separately. 

(II) — Conditions for and Tests of Regulation. 

204 Speed. The Regulation of Generators is to be determined at con- 
stant speed, and of alternating apparatus at constant impressed fre- 
quency. 

205 Non-inductive Load. In apparatus generating, transforming or 
transmitting alternating currents, regulation should be understood to 
refer to non-inductive load, that is, to a load in which the current is in 
phase with the e.m.f. at the output side of the apparatus, except where 
expressly specified otherwise. 

206 Wave Form. In alternating apparatus receiving electric power, 
regulation should refer to a sine wave of e.m.f., except where expressly 
specified otherwise. 

207 Excitation. In commutating machines, rectifying machines, and 
synchronous machines, such as direct-current generators and motors, 
alternating-current and polyphase generators, the regulation is to be 
determined under the following conditions: 

(1) At constant excitation in separately excited fields. 

(2) With constant resistance in shunt-field circuits, and 

(3) With constant resistance shunting series-field circuits; i. e., the 
field adjustment should remain constant, and should be so chosen as to 
give the required full-load voltage at full-load current. 

208 Impedance Ratio. In alternating-current apparatus, in addition to 
the non-inductive regulation, the impedance ratio of the apparatus should 
be specified; i. e., the ratio of the voltage consumed by the total 
internal impedance of the apparatus at full-load current, to its rated 
full-load voltage. As far as possible, a sinusoidal current should be 
used. 

209 Computation of Regulation. When in synchronous machines the 
regulation is computed from the terminal voltage and impedance 
voltage, the exciting ampere-turns corresponding to terminal voltage 
plus armature-resistance-drop, and the ampere-turns at short-circuit 
corresponding to the armature-impedance-drop, should be combined 
vectorially to obtain the resultant ampere-turns, and the corresponding 
internal e. m. f. should be taken from the saturation curve. 

E.— INSULATION. 
(I) — Insulati6n Resistance. 

2 1 Insulation Resistance is the ohmic resistance offered by an insulating 
coating, cover, material or support to an impressed voltage, tending to 
produce a leakage of current through the same. 

211 Ohmic Resistance and Dielectric Strength. The ohmic resistance ot 
the insulation is of secondary importance only, as compared with the 
dielectric strength, or resistance to rupture by high voltage. Since the 
ohmic resistance of the insulation can be very greatly increased by 
baking, but the dielectric strength is liable to be weakened thereby, it 
is preferable to specify a high dielectric strength rather than a high 
insulation resistance. The high- voltage test for dielectric strength 
should always be applied. 



PERFORMANCE SPECIFICATIONS AND TESTS. 1463 

212 Recommended Value of Resistance. The insulation resistance of 
complete apparatus should be such that 'the rated voltage of the 

apparatus which will not send more than i nnn nnn of the rated-load 

current, at the rated terminal voltage, through the insulation. Where 
the value found in this way exceeds 1 megohm, it is usually sufficient. 

213 Insulation Resistance Tests should, if possible, be made at the pres- 
sure for which the apparatus is designed. 

(II) — Dielectric Strength. 

{A) — Test Voltages. 

214 Definition. The dielectric strength of an insulating wall, coating 
cover or path is measured by the voltage which must be applied to it 
in order to effect a disruptive discharge through the same. 

215 Basis for Determining Test Voltages. The test voltage which should 
be applied to determine the suitability of insulation for commercial 
operation is dependent upon the kind and size of the apparatus and 
its normal operating voltage, upon the nature of the service in which it 
is to be used, and the severity of the mechanical and electrical stresses 
to which it may be subjected. The voltages and other conditions of test 
which are recommended have been determined as reasonable and 
proper for the great maajority of cases and are proposed for general 
adoption, except when specific reasons make a modification desirable. 

216 Condition of Apparatus to he Tested. Commercial tests should, in 
general, be made with the completely assembled apparatus and not 
with individual parts. The apparatus should be in good condition and 
high-voltage tests, unless otherwise specified, should be applied before 
tljie machine is put into commercial service, and should not be applied 
when the insulation resistance is low owing to dirt or moisture. High- 
voltage tests should, in general, be made at the temperature assumed 
under normal operation. High- voltage tests considerably in excess of 
the normal voltages to determine whether specifications are fulfilled 
are admissible on new machines only. 

217 Points of Application of Voltage. The test voltage should be suc- 
cessively applied between each electric circuit and all other electric 
circuits including conducting material in the apparatus. 

218 The Frequency of the alternating-current test voltage is, in general, 
immaterial within commercial ranges. When, however, the frequency 
has an appreciable effect, as in alternating-current apparatus of high 
voltage and considerable capacity, the rated frequency of the apparatus 
should be used. 

219 Table of Testing Voltages. The following voltages are recomrnended 
for testing all apparatus, lines and cables, by a continued application 
for one minute. The test should be with alternating voltage having an 
effective value (or root mean square referred to a sine wave of voltage) 
given in the table and preferably for tests of alternating apparatus at 
the normal frequency of the apparatus. 

Rated Terminal Voltage of Rated Testing 

Circuit. Output. Voltage. 

220 Not exceeding 400 volts Under 10 kw 1,000 volts. 

400 " 10 kw. and over 1,500 " 

400 and over, but less than 800 volts. Under 10 kw 1,500 " 

400 '• " 800 " 10 kw. and over 2,000 " 

800 " •• 1,200 " Any 3,500 " 

1,200 " '• 2,500 " Any 5,000 " 

2,500 ** Any . . Double the normal rated 

Voltages. 

221 Exception. — Transformers. Transformers having primary pressures 
of from 550 to 5,000 volts, the secondaries of which are directly con- 
nected to consumption circuits, should have a testing voltage of 10,000 
volts, to be applied between the primary and secondary windings, and 
also between the primary winding and the core. 

222 Exception. — Field Windings. The tests for field windings should be 
based on the rated voltage of the exciter and the rated output of the 
machine of which the coils are a part. Field windings of synchronous 
motors and converters, which are to be started by applying alternating 



1464 70— ELECTRIC POWER AND LIGHTING. 

ciirrent to the armature when the field is not excited and a high voltage 
is induced in the field windings, should be tested at 5,000 volts. 

223 Rated Terminal Voltage. — Definition. The rated terminal voltage 
of circuit in the above table, means the voltage between the conductors 
of the circuit to which the apparatus to be tested is to be connected, — 
tor instance, in three-phase circuits the delta voltage should be taken. 
In the following specific cases, the rated terminal voltage of the circuit 
is to be determined as specified in ascertaining the testing voltage: 

224 (a) Transformers. The test of the insulation between the primary 
and secondary windings ot transformers, is to be the same as that 
between the high- voltage windings and core, and both tests should be 
made simultaneously by connecting the low-tension winding and core 
together during the test. If a voltage equal to the specified testing 
voltage be induced in the high-tension winding of a transformer it may 
be used for insulation tests instead of an independently induced voltage. 
These tests should be made first with one end and then with the other 
end of the high-tension winding connected to the low-tension winding 
and to the core. 

225 (6) Constant-Current Apparatus. The testing voltage is to be based 
upon a rated terminal voltage equal to the maximum voltage which 
may exist at open or closed circuit. 

226 (c) Apparatus in Series. For tests of machines or apparatus to be 
operated in series, so as to employ the sum of their separate voltages 
the testing voltage is to be based upon a rated terminal voltage equal 
to the sum of the separate voltages except where the frames of the 
machines are separately insulated, both from the ground and from 
each other, in which case the test for insulation between machines 
should be based upon the voltage of one machine, and the test between 
each machine and ground to be based upon the total voltage of the series. 

{B) — Methods of Testing. 

227 Classes of Tests. Tests for dielectric strength cover such a wide 
range in voltage that the apparatus, methods and precautions which 
are essential in certain cases do not apply to others. For convenience, 
the tests will be separated into two classes: 

228 Class 1. This class includes all apparatus for which the test voltage 
does not exceed 10 kilovolts, unless the apparatus is of very large static 
capacity, e. g.,a, large cable system. This rlass also includes all apparatus 
of small static capacity, such as line insuiators, switches and the like, 
for all test voltages. 

229 Method of Test for Class 1. The test voltage is to be continuously 
applied for the prescribed interval, — (one minute, unless otherwise 
specified). The test voltage may be taken from a constant-potential 
source and applied directly to the apparatus to be tested, or it may be 
raised gradually as specified for tests under Class 2. 

230 Class 2. This class includes all apparatus not included in Class 1. 

231 Method of Test for Class 2. The test voltage is to be raised to the 
required value smoothly and without sudden large increments and is 
then to be continuously applied for the prescribed interval, — (one 
minute, unless otherwise specified), and then gradually decreased. 

232 Conditions and Precautions for Class 1 and Class 2. The following 
apply to all tests: 

233 The Wave Shape should be approximately sinusoidal and the 
apparatus in the testing circuits should not materially distort this wave. 

234 The Supply Circuit should have ample current-supply capacity so 
that the charging current which may be taken by the apparatus under 
test will not materially alter the wave form nor materially affect the 
test voltage. The circuit should be free from accidental interruptions. 

235 Resistance or Inductance in series with the primary of a raising 
transformer for the purpose of controlling its voltage is liable seriously 
to affect the wave form, thereby causing the maximum value of the 
voltage to bear a different and unknown ratio to the root mean square 
value. This method of voltage adjustment is, therefore, in general, 
undesirable. It may be noted that if a resistance or inductance is 
employed to limit the current when burning out a fault, such resistance 
or inductance should be short-circuited during the regular voltage test. 

236 The Insulation under test should be in normal condition as to dry- 
ness and the temperature should, when possible, be that reached in 
normal service. 



PERFORMANCE SPECIFICATIONS AND TESTS. 1465 

237 Additional Conditions and Precautions for Class 2. The following 
conditions and precautions, in addition to the foregoing, apply to tests 
of apparatus included in Class 2. 

238 Sudden Increment of Testing Voltage on the apparatus under test 
should be avoided, particularly at high voltages and with apparatus 
having considerable capacity, as a momentarily excessive rise in testing 
voltage will result. 

239 Sudden Variations in Testing Voltage of the circuit supplying the 
voltage during the test should be avoided as they are likely to set up 
injurious oscillation. 

240 Good Connections in the circuits supplying the test voltage are 
essential in order to prevent injurious high frequency disturbances from 
being set up. When a heavy current is carried by a small water rheo- 
stat, arcing may occur, causing high-frequency disturbances which 
should be carefully avoided. 

241 Transformer Coils. In high-tension transformers, the low-tension 
coil should preferably be connected to the core and the ground when 
the high-tension test is being made, in order to avoid the stress from 
low-tension to core, which would otherwise result through condenser 
action. The various terminals of each winding of the high-tension 
transformer under test should be connected together during the test in 
order to prevent undue stress on the insulation between turns or 
sections of the winding in case the high-voltage test causes a break- 
down. 

(C) — Methods for Measuring the Test Voltage. 

242 For Measuring the Test Voltage, two instruments are in common 
use, (1) the spark gap, and (2) the voltmeter. 

243 1. The Spark Gap is ordinarily adjusted so that it will break down 
with a certain predetermined voltage, and is connected in parallel with 
the insulation under test. It ensures that the voltage applied to the 
insulation is not greater than the break-down voltage of the spark gap. 
A given setting of the spark gap is a measure of one definite voltage, 
and, as its operation depends upon the maximum value of the voltage 
wave, it is independent of wave form and is a limit on the maximum 
stress to which the insulation is subjected. The spark gap is not con- 
veniently adapted for comparatively low voltages. 

244 In Spark-Gap Measurements, the spark gap may be set for the 
required voltage and the auxiliary apparatus adjusted to give a voltage 
at which this spark gap just breaks down. This spark gap should then 
be adjusted for, say, 10 per cent higher voltage, and the auxiliary 
apparatus again adjusted to give the voltage of the former breakdown, 
'which is to be the assumed voltage for the test. This voltage is to be 
maintained for the required interval. 

245 The Spark Points should consist of new sewing needles, supported 
axially at the ends of linear conductors which are each at least twice the 
length of the gap. There should be no extraneous body near the gap 
within a radius of twice its length. A table of approximate striking 
distances is given in Appendix D. This table should be used in con- 
nection with tests made by the spark-gap methods. 

246 A Non-inductive Resistance of about one-half ohm per volt should be 
inserted in series with each terminal of the gap so as to keep the discharge 
current between the limits of one-quarter ampere and 2 amperes. The 
purpose of the resistance is to limit the current in order to prevent the 
surges which might otherwise occur at the time of break-down. 

247 2. The Voltmeter gives a direct reading, and the different values of 
the voltage can be read during the application and duration of the test. 
It is suitable for all voltages, and does not introduce disturbances into 
the test circuit. 

248 In Voltmeter Measurements, the voltmeter should, in general, derive 
its voltage from the high-tension testing circuit either directly or through 
an auxiliary ratio transformer. It is permissible, however, to measure 
the voltage at other places, — for example, on the primary of the trans- 
former, provided the ratio of transformation does not materially vary 
during the test; or that proper account is taken thereof. 

249 Spark Gap and Voltmeter. The spark gap may be employed as a 
check upon the voltmeter used in high-tension tests in order to de- 
termine the transformation ratio of the transformer, the variation from 
the sine wave form and the like. It is also useful in conjunction with 



1466 70.— ELECTRIC POWER AND LIGHTING, 

voltmeter measurements to limit the stress applied to the insulating 

material. 

(D) — Apparatus for Supplying Test Voltage. 

250 The Generator or Circuit supplying voltage for the test should have 
ample current-carrying capacity, so that the current which may be 
taken for charging the apparatus to be tested will not materially alter 
the wave form nor otherwise materially change the voltage. 

The Testing Transformer should be such that its ratio of trans- 
formation does not vary more than 10 per cent when delivering the 
charging current required by the apparatus under test. (This may be 
determined by short-circuiting the secondary or high voltage winding 
testing transformer and supplying 1/10 of the primary voltage to the 
primary under this condition. The primary current that flows under 
this condition is the maximum which should be permitted in regular 
dielectric tests.) 

251 The Voltage Control may be secured in either of several ways, 
which, in order of preference, are as follows: 

252 1. By generator field circuit. 

253 2. By magnetic commutation. 

254 3. By change in transformer ratio. 

255 4. By resistance or choke coils. 

256 In Generator Voltage Control, the voltage of the generator should 
preferably be about its approximate normal rated-load value when the 
lull testing voltage is attained, which requires that the ratio of the 
raising transformer be such that the full testing voltage is reached when 
the generator voltage is normal. This avoids the instability in the gen- 
erator which may occur if a considerable leading current is taken 
from it when it has low voltage and low field current. 

257 In Magnetic Commutation, the control is effected by shunting the 
magnetic flux through a secondary coil so as to vary the induction 
through the coil and the voltage induced in it. The shunting should be 
effected smoothly, thus avoiding sudden changes in the induced voltage. 

258 In Transformer Voltage Control, by change of ratio, it is neces- 
sary that the transition from one step to another be made without inter- 
ruption of the test voltage, and by steps sufficiently small to prevent 
surges in the testing circuit. The necessity of this precaution is greater 
as the inductance or the static capacity of the apparatus in the testing 
circuit under test is greater. 

259 When Resistance Coils or Reactors are used for voltage control, it 
is desirable that the testing voltage should be secured when the con- 
trolling resistance or reactance is very nearly or entirely out of circuit in 
order that the disturbing effect upon the wave form which results may 
be negligible at the highest volatge. 

F.— CONDUCTIVITY. 

260 Copper. The conductivity of copper in electric wires and cables 
should not be less than 98% of Matthiessen's standard of conductivity, 
as defined in the Copper Wire Table of the American Institute of Electri- 
cal Engineers. [See Table 1, page 1388.] 

G.— RISE OF TEMPERATURE. 
(I) — Measurement of Temperature. 
{A) — Methods. 

261 There are two methods in common use for determining the rise in 
temperature, viz. : (1) by thermometer, and (2) by increase in resistance 
of an electric circuit. 

262 ^ 1. By Thermometer. The following precautions should be observed 
in the use of thermometers: 

263 a. Protection. The thermometers indicating the room temperature 
should be protected from thermal radiation emitted by heated bodies, 
or from draughts of air or from temporary fluctuations of tempera- 
ture. Several room thermometers should be used. In using the thermo- 
meter by applying it to a heated part, care should be taken so to pro- 
tect its bulb as to prevent radiation from it, and, at the same time, not 
to interfere seriously with the normal radiation from the part to which 
it is applied. 



PERFORMANCE SPECIFICATIONS AND TESTS, 1467 

264 b. Bulb. When a thermometer is applied to the free surface of a 
machine, it is desirable that the bulb of the thermometer should be 
covered by a pad of definite area. A convenient pad may be formed of 
cotton waste in a shallow circular box about one and a half inches in 
diameter, through a slot in the side in which the thermometer bulb is 
inserted. An unduly large pad over the thermometer tends to interfere 
with the natural liberation of heat from the surface to which the ther- 
mometer is applied. 

265 2. By Increase in Resistance. The resistance may be measured 
either by Wheatstone bridge, or by drop-of potential method. A tem- 
perature coefficient of 0.42 per cent per degree C, from and at 0°C., 
may be assumed for copper. 

The temperature-coefficients from and at each degree cent, between 
0°C. and 50°C. are given in Appendix E. The temperature rise may be 
determined either (1) by dividing the percentage increase of initial re- 
sistance by the temperature-coefficient for. the initial temperature ex- 
pressed in per cent; or (2) by multiplying the increase in per cent of the 
initial resistance by 238.1 plus the initial temperature in degrees C, and 
then dividing the product by 100. 

266 3. Comparison of Methods. In electrical conductors, the rise of tem- 
perature should be determined by their increase of resistance where prac- 
ticable. Temperature elevations measured in this way are usually in 
excess of temperature elevations measured by thermometers. In very 
low resistance circuits, thermometer measurements are frequently more 
reliable than measurements by the resistance method. Where a ther- 
mometer applied to a coil or winding, indicates a higher temperattire ele- 
vation than that shown by resistance measurement, the thermometer 
indication should be accepted. 

{B) — Normal Conditions for Tests. 

267 1. Duration of Tests. The temperature should be measured after a 
run of sufficient duration for the apparatus to reach a practically con- 
stant temperature .^ This is usually from 6 to 18 hours, according to the 
size and construction of the apparatus. It is permissible, however, to 
shorten the time of the test by running a lesser time on an overload in 
current and voltage, then reducing the load to normal, and maintaining 
it thus until the temperature has become constant. 

268 2. Room Temperature. The rise of temperature thould be referred 
to the standard condition of a room temperature of 25°C. 

269 Temperature Correction. If the room temperature during the test 
differs from 25°C., correction on account of difference in resistance 
should be made by changing the observed rise of temperature by one- 
half per cent for each degree C. Thus with a room temperature of 35°C., 
the observed rise of temperature has to be decreased by 5 per cent, and 
with a room temperature of 15°C., the observed rise of temperature has 
to be increased by 5 per cent. In certain cases, such as shunt-field 
circuit without rheostat, the current strength will be changed by a 
change of room temperature. The heat-production and dissipation may 
be thereby affected. Correction for this should be made by changing the 
observed rise in temperature in proportion as the PR loss in the re- 
sistance of the apparatus is altered owing to the difference in room 
temperature. 

270 3. Barometric Pressure. Ventilation. A barometric pressure of 760 
mm. and normal conditions of ventilation should be considered as 
standard, and the apparatus under test should neither be exposed to 
draught nor enclosed, except where expressly specified. The baro- 
metric pressure needs to be considered only when differing greatly from 
760 mm. 

271 Barometric Pressure Correction. When the barometric pressure 

differs greatly from the standard pressure of 760 mm. of mercury, as at 

high altitudes, a correction should be applied. In the absence of more 

accurate data, a correction of 1% of the observed rise in temperature for 

each 10 mm. deviation from the 760 mm. standard is recommended. 

For example, at a barometric pressure of 680 mm. the observed rise of 

• * t, J J 1, 760-680 _^ 
temperattire is to be reduced by r^r ==8%. 



1468 70.— ELECTRIC POWER AND LIGHTING. 

(II) — Limiting Temperature Rise. 

272 General. The temperature of electrical machinery under regular 
service conditions, should never be allowed to remain at a point at 
which permanent deterioration of its insulating material takes place. 

273 ^ Limits Recommended. It is recommended that the following max- 
imum values of temperature elevation, referred to a standard room tem- 
perature of 25° centigrade, at rated load under normal conditions of 
ventilation or cooling, should not be exceeded. 

{A) — Machines in General. 

274 In commutating machines, rectifying machines, pulsating-current 
generators, synchronous machines, synchronous commutating machines 
and unipolar machines, the temperature rise in the parts specified 
should not exceed the following: 

275 Field and armature, 50°C. 

276 Commutator and brushes, by thermometer, 55°C. 

277 Collector rings, 65°C. 

278 Bearings and other parts of machine, by thermometer, 40X. 

(J5) — Rotary Induction Apparatus. 

279 The temperature rise should not exceed the following: 

280 Electric circuit, 50°C., by resistance. 

281 Bearings and other parts of the machine 40°C., by thermometer. 

282 In squirrel-cage or short-circuited armatures, 55°C., by thermo- 
meter, may be allowed. 

(Q — Stationary Induction Apparatus. 

283 a. Transformers for Continuous Service. The temperature rise 
should not exceed 50° centigrade in electric circuits, by resistance; 
and in other parts, by thermometer. 

284 h. Transformers for Intermittent Service. In the case of trans- 
formers intended for intermittent service, or not operationg continu- 
ously at rated load, but continuously in circuit, as in the ordinary case of 
lighting transformers, the temperature elevation above the surrounding 
air-temperature should not exceed 50°C., by resistance in electric cir- 
cuits and by thermometer in other parts, after the period corresponding 
to the term of rated load. In this instance, the test load should not be 
applied until the transformer has been in circuit for a sufficient time to 
attain the temperature elevation due to core loss. With transformers 
for commercial lighting, the duration of the rated-load test may be 
taken as three hours, unless otherwise specified. 

285 c. Reactors, induction- and magneto-regulators — electric circuts by 
resistance and other parts by thermometer, 50°C. 

286 a. Large Apparatus. Large generators, motors, transformers, or 
other apparatus in which reliability and reserve overload capacity are 
important, are frequently specified not to rise in temperature more than 
40° centigrade under rated load and 55° centigrade at rated overload. 
It is, however, ordinarily undersirable to specify lower temperature 
elevations than 40° centigrade at rated load, measured as above. 

{D) — Rheostats. 

287 In Rheostats, Heaters and other electrothermal apparatus, no com- 
bustible or inflammable part or material, or portion liable to come in 
contact with such material, should'rise more than 50°C. above the sur- 
rounding air under the service conditions for which it is designed. 

288 ^ a. Parts of Rheostats. Parts of rheostats and similar apparatus rising 
in temperature, under the specified service conditions, more than 50°C., 
should not contain any combustible material, and should be arranged or 
installed in such a manner that neither they, nor the hot air issuing from 
them, can come in contact with combustible material. 

{E) — Limits Recommended in special Cases. 

289 a. Heat Resisting Insulation. With apparatus in which the insu- 
lating materials have special heat-resisting qualities, a higher tempera- 
ture elevation is permissible. 

290 b. High Air Temperature. In apparatus intended for service in 
places of abnormally high temperature, a lower temperature elevation 
should be specified. 



PERFORMANCE SPECIFICATIONS AND TESTS. 1469 

291 c. Apparatus Subject to Overload. In apparatus which by the nature 
of its service may be exposed to overload, or is to be used in very 
high voltage circuits, a smaller rise of temperature is desirable than 
in apparatus not liable to overloads or in low-voltage apparatus. In 
apparatus built for conditions of limited space, as railway motors, a 
higher rise of temperature must be allowed. 

292 d. Apparatus for Intermittent Service. In the case of apparatus 
intended for intermittent service, except railway motors, the tempera- 
ture elevation which is attained at the end of the period corresponding 
to the term of rated load, should not exceed the values specified for 
machines in general. In such apparatus the temperature elevation, in- 
cluding railway motors, should be measured after operation, under as 
nearly as possible the conditions of service for which the apparatus is 
intended, and the conditions of the test should be specified. 

H.— OVERLOAD CAPACITIES. 

293 Performance with Overload. All apparatus should be able to carry 
the overload hereinafter specified without serious injury by heating, 
sparking, mechanical weakness, etc., and with an additional tempera- 
ture rise not exceeding 15°C., above those specified for rated loads, the 
overload being applied after the apparatus has acquired the tempera- 
ture corresponding to rated load continuous operation. Rheostats to 
which no temperature rise limits are attached are naturally exempt from 
this additional temperature rise of 15°C. under overload specified in 
these rules. 

294 Normal Conditions. Overload guarantees should refer to normal con- 
ditions of operation reguarding speed, frequency, voltage, etc., and to 
non-inductive conditions in alternating apparatus, except where a phase 
displacement is inherent in the apparatus. 

295 Overload Capacities Recommended . The following overload capaci- 
ties are recommended: 

296 a. Generators. Direct-current generators and alternating-current 
generators, 25 per cent for two hours. 

297 b. Motors. Direct-current motors, induction motors and synchronous 
motors, not including railway and other motors intended for intermittent 
service, 25 per cent for two hours, and 50 per cent for one minute. 

298 c. Converters. Synchronous converters, 25 per cent for two hours, 
50 per cent for one-half hour. 

299 d. Transformers and Rectifiers. Constant-potential transformers and 
rectifiers, 25 per cent for two hours; except in transformers connected 
to apparatus for which a different overload in guaranteed, in which case 
the same guarantees shall apply for the transformers as for the apparatus 
connected thereto. 

300 e. Exciters. Exciters of alternators and other synchronous machines, 
10 per cent more overload than is required for the excitation of the syn- 
chronous machine at its guaranteed overload, and for the same period 
of time. All exciters of alternating-current, single-phase or polyphase 
generators should be able to give at its rated speed, sufficient voltage 
and current to excite the alternator, at the rated speed, to the full-load 
terminal voltage, at the rated output in kilo volt-amperes and with 
60 per cent power factor. 

301 /.A Continuous-Service Rheostat, such as an armature- or field- 
regulating rheostat, should be capable of carrying without injury for 
two hours, a current 25 per cent greater than that at which it is rated. 
'It should also be capable of carrying for one minute a current 50 per 
cent greater than its rated load current, without injury. This excess of 
capacity is intended for testing purposes only, and this margin of capa- 
city should not be relied upon in the selection of the rheostat. 

302 g. An Intermittent Service or Motor -Starting Rheostat is used for 
starting a motor from rest and accelerating it to rated speed. Under 
ordinary conditions of service, and unless expressly stated otherwise, a 
motor is assumed to start in fifteen seconds and with 150% of rated 
current strength. A motor-starter should be capable of starting the 
motor under these conditions once every four minutes for one hour. 

303 (a) This Test may be carried out either by starting the motor at 
four-minute intervals, or by placing the starter at normal temperature 
across the maximum voltage for which it is marked, and moving the 
lever uniformly and gradually from the first to the last position dxiring a 



1470 n.— ELECTRIC POWER AND LIGHTING, 

period of fifteen seconds, the current being maintained substantially 
constant at said 50% excess by introducing resistance in series or by 
other suitable means. 

304 (b) Other Rheostats for Intermittent-Service are employed under 
such special and varied conditions, that no general rules are applicable 
to them. 

III.— VOLTAGES AND FREQUENCIES. 

A.— VOLTAGES. 

305 Direct-Current Generators. In direct-current, low-voltage gener- 
ators, the following average terminal voltages are in general use and 
are recommended : 

125 volts. 250 volts. 650 to 600 volts. 

306 Low-Voltage Circuits. In direct-current and alternating-current low- 
voltage circuits, the following average terminal voltages are in general 
use and are recommended: 

110 volts. ' 220 volts. 

307 Primary Distribution Circuits. In alternating-current, constant- 
potential, primary-distribution circuits, an average voltage of 2,200 
volts, with step-down transformer ratios 1/10 and 1/20, is in general use, 
and is recommended. 

308 Transmission Circuits. In alternating-current constant-potential 
transmission circuits, the following average voltages are recommended. 

6,600 11,000 22,000 33.000 44,000 66,000 88,000 

309 Transformer Ratio. It is recomm.ended that the standard transfor- 
mer ratios should be such as to transform between the standard voltages 
above named. The ratio will, therefore, usually be an exact multiple 
of 5 or 10, e. g., 2,200 to 11,000; 2,200 to 44,000. 

310 Range in Voltage. In alternating-current generators, or generating 
systems, a range of terminal voltage should be provided from rated 
voltage at no load to 10 per cent in excess thereof, to cover drop in trans- 
mission. If a greater range than ten per cent is specified, the generator 
should be considered as special. 

B.— FREQUENCIES. 

311 In Alternating-Current Circuits, the following frequencies ;ire 
standard: 

25^ 60^ 

312 These frequencies are already in extensive use and it is deemed ad- 
visable to adhere to them as closely as possible. 

IV.— GENERAL RECOMMENDATIONS. 

313 Name Plates. All electrical apparatus should be provided with a 
name plate giving the manufacturer's name, the voltage and the current 
in amperes for which it is designed. Where practicable, the kilowatt 
capacity, character of current, speed, irequency, type, designation and 
serial number should be added. 

314 Diagrams of Connections. All electrical apparatus when leaving the 
factory should be accompanied by a diagram showing the electrical 
connections and the relation of the different parts in sufficient detail 
to give the necessary information for proper installation. 

315 Rheostat Data. Every rheostat should be clearly and permanently 
marked with the voltage and amperes, or range of amperes, for which 
it is designed. 

316 Colored Indicating Lights. When using colored indicating lights 
on switch-boards, red should denote danger such as "switch closed,'' 
or "circuit alive;" green should denote safety, such as "switch open," 
or "circuit dead." 

317 When white lights are used a light turned on should denote danger, 
such as "switch closed" or "circuit alive;" while the light out should 
denote safety, such as "switch open," or "circuit dead." Low-effi- 
ciency lamps should be used. 

318 The use of colored lights is recommended, as safer than white lights. 



VOLTAGE, FkEQUENCY. GENERAL. APPENDIX. 1471 

319 Grounding Metal Work. It is desirable that all metal work near 
high potential circuits be grounded. 

320 Circuit Opening Devices. The following definitions are recommended. 

321 a. A Circuit-Breaker is an apparatus for breaking a circuit at the 
highest current which it rnay be called upon to carry. 

322 b. A Disconnecting Switch is an apparatus designed to open a circuit 
only when carrying little or no current. 

323 c. An Automatic Circuit-Breaker is an apparatus for breaking a 
circuit automatically under an excessive strength of current. It should 
be capable of breaking the circuit repeatedily at rated voltage and at 
the maximum current which it may be called upon to carry. 



v.— APPENDICES AND TABULAR DATA. 

APPENDIX A.— NOTATIONo 

The following notation is recommended: 

324 E, e, voltage, e.m.f., potential difference 
/, «', current 

P, power 

<P, magnetic flux 

B, By magnetic density 

R, r, resistance 

Xy reactance 

Zt z, impedance 

L, /, inductance 

C c, capacity 

y, y, admittance 

bt susceptance 

G, g, conductance 

Vector quantities when used should be denoted by capital italics. 

APPENDIX B.— RAILWAY MOTORS. 

(I) — Rating. 

325 Introductory Note on Rating. Railway motors usually operate in 
a service in which both the speed and the torque developed by the motor 
are varying almost continually. The average requirements, however, 
during successive hours in a given class of service are fairly uniform. 
On account of the wide variation of the instantaneous loads, it is im- 
practicable to assign any simple and definite rating to a motor which will 
indicate accurately the absolute capacity of a given motor or the rela- 
tive capacity of different motors under service conditions. It is also 
impracticable to select a motor for a particular service without much 
fuller data with regard both to the motor and to the service than is 
required, for example, in the case of stationary motors which run at 
constant speeds. 

326 Scope of Nominal Rating. It is common usage to give railway 
motors a nominal rating in horse power on the basis of a one-hour test. 
As above explained, a simple rating of this kind is not a proper measure 
of service capacity. The nominal rating, however, indicates approxi- 
mately the maximum output which the motor should ordinarily be 
called upon to develop during acceleration. Methods of determining 
the continuous capacity of a railway motor for service requirements 
are given under a subsequent heading. 

327 The Nominal Rating of a railway motor is the horse-power output 
at the car-axle,^ that is, including gear and other transmission losses, 
which gives a rise of temperature above the surrounding air (referred 
to a room temperature of 25° Cent.) not exceeding 90° Cent, at the 
commutator and 75° Cent, at any other part after one hour's con- 
tinuous run at its rated voltage (and frequency, in the case of an alter- 
nating-current motor) on a stand, with the motor-covers removed, and 
with natural ventilation. The rise in temperature is to be determined 
by thermometer, but the resistance of no electrical circuit in the motor 
shall increase more than 40% during the test. 



1472 lO.—ELECTRlC POWER AND LIGHTING. 

(II) — Selection Op Motor for Specified Service. 

328 General Requirements. The suitability of a railway motor for a 
specified service depends upon the following considerations: 

329 a. Mechanical ability to develop the requisite torque and speeds as 
given by its speed-torque curve. 

330 b. Ability to commutate successfully the ciirrent demanded. 

331 c. Ability to operate in service without occasioning a temperature 
rise in any part which will endanger the life of the insulation. 

332 Operating Conditions, Typical Run. The operating conditions which 
are important in the selection of a motor include the weight of load, the 
schedule speed, the distance between stops, the duration of stops, the 
rate of acceleration and of breaking retardation, the grades and the 
curves. With these data at hand, the outputs which are required of the 
motor may be determined, provided the service requirements are with- 
in the limits of the speed-torque curve of the motor. These outputs may 
be expressed in the form of curves giving the instantaneous values of 
current and of voltage which must be applied to the motor. Such curves 
may be laid out for the entire line, but they are usually constructed 
only for a certain average or typical run, which is fairly representative 
of the conditions of service. To determine whether the motor has 
sufficient capacity to perform the service safely, further tests or in- 
vestigations must be made. 

333 Capacity Test of Railway Motor in Service. The capacity of a 
railway motor to deliver the necessary output may be determined by 
measurement of its temperature after it has reached a maximum in 
service. If a running test cannot be made under the actual conditions 
of service, an equivalent test may be made in a typical run back and 
forth, under such conditions of schedule speed, length of run, rate of 
acceleration, etc., that the test cycle of motor losses and conditions of 
ventilation are essentially the same as would be obtained in the speci- 
fied service. 

334 Methods of Comparing Motor Capacity with Service Requirements. 
Where it is not convenient to test motors under actual service condi- 
tions or in an equivalent typical run, recourse may be had to one of the 
two following methods of determining temperature rise now in general 
use: 

335 1. Method by Losses and Thermal Capacity Curves. The heat de- 
veloped in a railway motor is carried partly by conduction through the 
several parts and partly by convection through the air to the motor- 
frame whence it is distributed to the outside air. As the temperattire of 
the several parts is thus dependent not only upon their own internal 
losses but also upon the temperature of neighboring parts, it becomes 
necessary to determine accurately the actual value and distribution of 
losses in a railway motor for a given service and reproduce them in an 
equivalent test -run. The results of a series of typical runs expressed in 
the form of thermal capacity curves will give the relation between degrees 
rise per watt loss in the armature and in the field for all ratios of losses 
between them met with in the commercial application of a given motor. 

336 This method consists, therefore, in calculating the several internal 
motor losses in a specified service and determining the temperature rise 
with these losses from thermal capacity curves giving the degrees rise 
per watt loss as obtained in experimental track tests made under the 
same conditions of ventilation. 

337 The following motor losses cause its heating and should be carefully 
determined for a given service: PR in the field; PR in the armature; 
PR in the brush contacts, core loss and brush friction. 

338 The loss in the bearings (in the case of geared motors) also adds 
somewhat to the motor-heating, but owing to the variable nature of 
such losses they are generally neglected in making calculations. 

339 2. Method by Continuous Capacity of Motor. The essential losses 
in the motor, as found in the typical run, are in most cases those in the 
motor windings and in the core. The mean service conditions may be 
expressed in terms of the current which would produce the same losses 
in the motor windings and in the voltage which, with the current, 
would produce the same core losses as the average in service. The 
continuous capacity of the motor is given in terms of the amperes 
which it will carry when run on the testing stand — with covers on or off, 
as specified — at different voltages, say, 40, 60, 80 and 100 per cent of 



RAILWAY MOTORS. PHOTOMETRY AND LAMPS, 1473 

the rated voltage — with a temperature rise not exceeding 90° at the 
commutator and 75° at any other part, provided the resistance of no 
electric circuit in the motor increase more than 40 per cent. A com- 
parison of the equivalent service conditions with the continuous capacity 
of the motor will determine whether the service requirements are within 
the safe capacity of the motor. 

340 This method affords a ready means of determining whether a speci- 
fied service is within the capacity of a given motor and it is also a con- 
venient approximate method for comparing the service capacities of 
different motors. 

APPENDIX C— PHOTOMETRY AND LAMPS. 

341 Candle -Power. The luminous intensity o^^ sources of light is ex- 
pressed in candle-power. The unit of candle-power should be derived 
from the standards maintained by the National Bureau of Standards at 
Washington, D. C, which standard unit of candle-power equals 100/88 
of the Hefner unit under Reichsanstalt standard conditions for the 
Hefner. In practical measurements seasoned and carefully standard- 
ized incandescent lamps are more reliable and accurate than the primary 
standard. 

342 Candle-Lumen. The total flux of light from a source is equal to its 
mean spherical intensity multiplied by ^k. The unit of flux is called 

the candle-lumen. A candle-lumen is the -p th part of the total flux 

of light emitted by a source having a mean spherical intensity of one 
candle-power. 

343 Candle-Meter. The unit of illumination is the candle-meter. This is 
the normal illumination produced by one unit of candle-power at a dis- 
tance of one metre. 

344 a. Candle-Foot, Illumination is occasionally expressed in candle-feet. 
A candle-foot is the normal illumination produced by one unit of candle- 
power at a distance of one foot. 

345 1 candle-foot = 10.764 candle-metres. 

The use of the candle-metre unit is preferable and is recommended. 

346 The E-fficiency of Electric Lamps is properly stated in terms of mean 
spherical candle-power per watt at lamp terminals. This use of the 
term efficiency is to be considered as special, and not to be confused 
with the generally accepted definition of efficiency in Sec. 85. 

347 a. Efficiency, Auxiliary Devices. In illuminants requiring auxiliary 
power-consuming devices outside of the luminous body, such as steady- 
ing resistances in constant potential arc lamps, a distinction should be 
made between the net efficiency of the luminous source and the gross 
efficiency of the lamp. This distinction should always be stated. The 
gross efficiency should include the power consumed in the auxiliary 
resistance, etc., The net efficiency should, however, include the power 
consumed in the controlling mechanism of the lamp itself. Comparison 
between such sources of light should be made on the basis of gross 
efficiency, since the power consumed in the auxiliary device is essential 
to the operation. 

348 h. A Standard Circuit Voltage of 1 10 volts, or a multiple thereof may 
be assumed, except where expressly stated otherwise. 

349 _ Watts per Candle. The specific consumption of an electric lamp is 
its watt^ consumption per mean spherical candle-power. "Watt per 
candle" is the term used commercially in connection with incandescent 
lamps, and denotes, watts per mean horizontal candle-power. 

350 Photometric Tests in which the results are stated in candle-power 
should always be made at such a distance from the source of light that 
the latter may be regarded as practically a point. Where tests are made 
at shorter distances, as for example in the measurement of lamps with 
reflectors, the result thould always be given as "apparent candle-power" 
at the distance employed, which distance should always be specifically 
stated. 

351 Basis for Comparison. Either the total flux of light in candle- 
lumens, or the mean spherical candle-power, should always be used as 
the basis for comparing various luminous sources with each other, 
unless there is a clear understanding or statement to the contrary. 

352 Incandescent Lamps, Rating. It is customary to rate incandescent 
lamps on the basis of their mean horizontal candle-power; but in com- 



1474 



7Q.— ELECTRIC POWER AND LIGHTING. 



paring incandescent lamps in which the relative distribution of luminous 
intensity differs, the comparison should be based on their total flux of 
light measured in lumens, or on their mean spherical candle-power. 

353 The Spherical Reduction-Factor of a lamp 

_ mean spherical candle-power 
mean horizontal candle-power 

354 The Total Flux of light in candle-lumens emitted by a lamp = 4ff X 
mean horizontal candle-power X spherical reduction-factor. 

355 The Spherical Reduction-Factor should only be used when properly 
determined for the particular type and characteristics of each lamp. 
The spherical reduction-factor permits of substantially accurate com- 
parisons being made between the mean spherical candle-powers of 
different types of incandescent lamps, and may be used in the absence 
of proper facilities for direct measurement of mean spherical intensity. 

356 '* Reading Distance." Where standard photometric measurements 
are impracticable, approximate measurements of illuminants such as 
street lamps may be made by comparing their "reading distances;" 
*. e., by determining alternately the distances at which an ordinary 
size of reading print can just be read, by the same person or persons, 
when all other light is screened. The angle below the horizontal at 
which the measurement is made should be specified when it exceeds 15°. 

357 In Comparing Different Luminous Sources not only should their 
candle-power be compared, but also their relative form, intrinsic 
brilliancy, distribution of illumination and character of light. 

APPENDIX D.— SPARKING DISTANCES. 

358 Table of Sparking Distances in Air between Opposed Sharp Needle- 



359 



Points, for Various Effective Sinusoidal Voltages, 


in inches 


and in 


centimeters. 


The table applies to 


the conditions specified 


in Sees. 


240-246. 


, 










Kilovolts 






Kilovolts 






Sq. Root of 


Distance. 


Sq. Root of 


Distance. 


Mean Square. 


Inches. 


Cms. 


Mean Square. 


Inches. 


Cms. 


5 


.. 0.225 


0.57 


140 


..13.95 


35.4 


10 


.. 0.47 


1.19 


150 


..15.0 


38.1 


15 


.. 0.725 


1.84 


160 


..16.05 


40.7 


20 


.. 1.0 


2.54 


170 


..17.10 


43.4 


25 


.. 1.3 


3.3 


180 


..18.15 


46.1 


30 


.. 1.625 


4.1 


190 


..19.20 


48.8 


35 


.. 2.0 


6.1 


200 


..20.25 


51.4 


40 


.. 2.45 


6.2 


210 


..21.30 


54.1 


45 


.. 2.95 


7.5 


220 


..22.35 


56.8 


50 


.. 3.55 


9.0 


230 


..23.40 


59.4 


60 


.. 4.65 


11.8 


240 


..24.45 


62.1 


70 


.. 5.85 


14.9 


250 


..25.50 


64.7 


80 


.. 7.1 


18.0 


260 


..26.50 


67.3 


90 


.. 8.35 


21.2 


270 


..27.50 


69.8 


100 


.. 9.6 


24.4 


280 


..28.50 


72.4 


110 


..10.75 


27.3 


290 


..29.50 


74.9 


120 


..11.85 
..12.90 


30.1 
32.8 


300 


..30.50 


77.4 


130 









SPARKING DISTANCES. TEMP. COEFFICIENTS, 1475 



APPENDIX E.— TEMPERATURE COEFFICIENTS. 

360 Table of Temperature Coefficients of Resistivity in Copper at 

Different Initial Temperatures Centigrade. 



Initial 


Temp. Coefficient 


Initial 


Temp. Coefficient 


temperature 


in per cent per 


temperature 


in per cent per 


Cent. 
i 


Degree Cent. 


Cent. 
i 


Degree Cent. 





0.4200 


26 


0.3786 


1 


0.4182 


27 


0.3772 


2 


0.4165 


28 


0.3758 


3 


0.4148 


29 


0.3744 


4 


0.4131 


30 


0.3730 


5 


0.4114 


31 


0.3716 


6 


....0.4097 


32 


0.3702 


7 


0.4080 


33 


0.3689 


8 


0.4063 


34 


0.3675 


9 


0.4047 


35 


0.3662 


10 


0.4031 


36 


0.3648 


11 


...0.4015 


37 


0.3635 


12 


. 3999 


38 


0.3622 


13 


0.3983 


39 


0.3609 


14 


0.3967 


40 


0.3596 


15 


0.3951 


41 


0.3583 


16 


0.3936 


42 


0.3570 


17 


0.3920 


43 


0.3557 


18 


0.3905 


44 


0.3545 


19 


0.3890 


45 


0.3532 


20 


0.3875 


46 


0.3520 


21 


0.3860 


47 


.......0.3508 


22 


0.3845 


48 


0.3495 


23.. 


0.3830 


49 


0.3483 


24 


0.3815 


50 


0.3471 


25 


0.3801 







The fundamental relation between the increase of resistance in 
copper and the rise of temperature may be taken as 
i?t = i?o (1 + 0.0042 i) 

where Rq is the resistance of the copper conductor at 0^ C. and Rt is the 
corresponding resistance at ^° C. This is equivalent to taking a tempera- 
ture coefficient of 0.42% per deg. C. temperature rise above 0°C. For 
initial temperatures other than 0°C., a similar formula may be used 
substituting the coefficients in the above table corresponding to the 
actual initial temperature. The formula thus becomes at 25° C. 

where R\ is the initial resistance at 25° C. R\+r the final resistance 
and r the temperature rise above 25° C. 

In order to find the temperature rise in degrees Centigrade from the 
initial resistance R\ at the initial temperature i° C. and the final resist- 
ance i?i+r we may use the formula 



See Sec. 265. 



(238. 1 + i) {^ - \\ degrees C. 



1476 70.--ELECTRIC POWER AND LIGHTING. 



EXCERPTS AND REFERENCES. 

Generators and Transformers for the Bay Counties Power Co., Cal. 

(By E. Heltmann and W. Currie. Eng. News, Nov. 21, 1901).— Illustrated. 

Electric Switches and Fuses for Currents of Very High Voltage ("The 
Jl. of Elec, Power and Gas," June, 1901; Eng. News, Oct. 3, 1901). 

The 50,000=Volt Transmission Plant of the Missouri River Power Co. 
in Mont. (By W. G. McConnon. Eng. News, June 5, 1902). — Details of 
insulators and spacing of poles. 

Success in Long Distance Electric Power Transmission (By F. A. C. 

Perrine. "Technology Quarterly;" Eng. News, Aug. 21, 1902). — See, also, 
Eng. News, Sept. 29, 1904. 

An Analytical Method of Determining Illumination (By Van R. 

Lansingh. Eng. News, Feb. 19, 1903). — Illustrated diagram for determin- 
ing horizontal distribution of light from a given surface. 

Long Spans for Electric Transmission Lines (By F. O. Blackwell. 
Paper, Am. Inst. E. E., June, 1904; Eng. News, July 7. 1904). — Modulus 
of elasticity: Copper hard-drawn wire, 19,500,000; aluminum hard-drawn 
wire, 10,200,000; iron telegraph wire, 24,000,000; copper hard-drawn wire 
cable. 16,300,000. Coefficient of expansion (F.): Copper, 0.0000096; 
aluminum, 0.000013; steel, 0.0000064. 

The Kern River Company's Hydro=Electric Power Enterprise (By 

Burr Bassell. Eng. News, July 21, 1904). — Sixteen illustrations. 

Medium=Span Electric Transmission Line Construction (By C. A. 

Copeland. Pac. Coast Elec. Transmission Assn., June, 1904; Eng. News, 
Aug. 18, 1904). — Illustration of steel pole construction for 300-ft. span; 
also an experimental medium-span construction. 

The Design of Generators for Electric Power Transmission (By D. B. 

Rushmore. Eng. News, Oct 27, 1904). — Illustrated. 

Storage Batteries for Block Signal Work (By E. L. Reynolds. Paper, 
Ry. Signal Assn., Jan., 1905; Eng. News, Jan. 19, 1905).— Tables: (1) 
First cost; (2) Maintenance cost. 

Long Distance Electric Power Transmission Line in Nevada (By 

E. Prince. Eng. News, July 6, 1905). 

Electric Light and Power Plant of Brigham City, Utah (By W. P. 

Hardesty. Eng. News, Sept. 7, 1905). — Illustrations: Power house; iron- 
work for gates for diverting dam; concrete anchorage for steel pressure 
supply pipe on slope; diverting dam and intake; details wood-stave pipe; 
cast-iron hub for connecting steel pipe with wood stave pipe. 

Gas Engine Electric Plant as Auxiliary and Reserve for a Long- 
Distance Transmission System (Eng. News, Sept. 14, 1905). 

A High Head Water Power Electric Plant on the Animas River, 
Colo. (By G. M. Peek. Eng. News, Jan. 4, 1906).— Illustrations: Cross- 
section of power house; reinforced-concrete pole with spread base, for 
single 3-phase power transmission line; four-post reinforced-concrete tower 
with prismatic base, for single 3-phase power transmission line. 

Transmission and Distributing System, Long Island R. R (Eng. 
News, June 14, 1906). — Illustrations: Top of steel transmission line pole; 
strain insulators for transmission line cables; details of third-rail guard and 
supports; standard wooden side approach block construction; standard 
arrangement of third-rail connecting cables at public crossings. 

Cost of Construction and Operating Expenses of the Municipal Elec- 
tric Lighting Plant at Burlington, Vt. (Eng. News, May 30, 1907).— Six 
tables of costs. 

Some New Methods in High Tension Line Construction (By H. W. 

Buck Eng. News, Aug. 8, 1907). — Illustrated. See, also. Paper by E. M. 
Hewlett, in same issue. 

The McCall Ferry Hydro-Electric Power Plant on the Susquehanna 



MISCELLANEOUS DATA. COSTS. 1477 

River (Eng. News, Sept. 12, 1907). — Illustrations: Alternate dam sections, 
showing expansion joints and butt blocks for steel forms; details of steel 
forms; details of steel traveler; concrete mixer plant; section throtigh 
power house. 

Concrete Telegraph Poles (Eng. Rec, Dec. 17, 1910), 17i ft. high above 
the roof have been erected on one of the buildings of the U. S. Aluminum 
Co., at Niagara Falls. They are 16" sq. at base, 10" sq. at top and rein- 
forced with eight i" bars bound with I" hoops 18" c.-c. 

Cost of Constructing Steam°Driven Electric Power Plants (By P. 

Koester. Eng. News, Dec. 19, 1907). — Superstructure. — $15 to $25 for 
plants up to 5000 K W.; $15 where there is a compact arrangement with 
walls of common brick, wooden doors and window frames, steel roof trusses 
supported by the walls and a roof of the cheapest construction, such as 
corrugated iron, tin, etc.; about $20 to $25 per K W. for construction of 
higher grade masonry with fireproof windows and doors, roof trusses carried 
by steel columns which at the same time carry the crane runway, and the 
roof itself consisting of reinforced -concrete covered by tar and gravel. The 
cost of superstructure for large size plants usually runs from $10 to $20 per 
K W. ; these are constructed of self-supporting steel skeleton and self- 
supporting walls. The superstructure at $20 per K W. may embrace 
multiple boiler floors, while those at $10 per K W. cover single boiler floor 
plants only. Chimney. — A radial brick chimney for large size power 
plants may be built for from $1.75 to $2.25 per K W. Reinforced -concrete 
chimneys and plate steel chimneys may cost from $1.50 to $2 per K W. 
Coal and Ash Handling Systems. — Experience shows that the figures for 
equipment for handling coal and ashes range from $1.50 to $3 per K W. 
Boilers. — The cost of water tube boilers ranges from $8 to $10 per K W., 
depending upon the square feet of heating surface in the boiler. These 
figures do not include mechanical stokers, for which from $2 to $3 may be 
assumed. Breeching, of course, is also a separate item and varies con- 
siderably as to cost per K W. The boiler setting is included in the above 
cost. Blowers. — In many of the modem power plants, especially plants for 
railway purposes, forced or induced draft is adopted. The blowers are 
usually steam-driven. The cost of such equipment is about $1 per K W. 
Economizers. — Where economizers are installed of sufficient capacity to 
heat the water to 200° or 220° F. such apparatus costs about $2 per K W., 
provided that there are not too many additional smoke flues necessary for 
by-passing, etc. Boiler Feed Pumps. — ^The cost of such pumps alone is 
some 50 cents per K W. When storage tanks are necessary the cost of the 
combined outfit amounts to 75 cents or $1, depending on the number and 
size of the tanks. Piping. — From $2 to $6 per K W. For plants of 10,000 
to 20,000-K W. capacity, the piping system not being elaborate but suffi- 
cient for continuous operation, $2.50 to $2.75 has covered the cost. This 
includes a high grade of covering for steam piping valued at about 20 cts. 
per K W. Prime Movers. — A 5000-K W. turbo-generator should cost from 
$20 to $22 per KW. Reciprocating engines of this capacity are sold roughly 
at the same price, and about $10 per K W. needs to be added for the gener- 
ator. The total cost for smaller units, 600 to 3000 K W. capacity, is from 
$20 to $25 per K W., whether they consist of turbine or reciprocating- 
engine apparatus. Condensers. — The cost of jet condenser equipment runs 
from $3 to $5 per K W., depending upon the type of pump used. The cost 
of surface condenser aoparatus will vary from $5 to $8, depending partly 
upon the vacuum to oe carried and whether the casing necessary forms 
part of the condenser equipment. Exciters. — A steam-driven exciter unit 
costs from 35 cts. to 40 cts. per K W. If a condenser should be installed 
in connection with it the cost may run as high at 70 cts. per K W., assum- 
ing that the exciter capacity is, approximately, 1% the total capacity of 
the plant. Switchboards. — For a high tension voltage the cost will run from 
$2 to $3.50, which for a low tension voltage (2,300 volts or lower) the 
switchboard equipment may be obtained for $1 to $2 per K W., depending 
largely upon the system of wiring adopted. Miscellaneous. — Traveling 
cranes, 25 or 50 cts. per KW.; smaller items, like house pumps, water 
meters, blow-off tanks, painting, supervision, etc, from $1 to $2 per K W. 
Tabulations. — A summary of the preceding figures is shown in the table 
below, to which should be added the engineering fee. All these figures 
represent costs (per K W.) of plants of large capacity, but some have cost 
as high as $125 or even, in an exceptional case, $150 per K W. 



1478 



n.— ELECTRIC POWER AND LIGHTING. 



Cost of Steam Plants of Large Capacity. 



Items. 


Cost of 

Steam Turbine 

Plants. 

per K W. 


Cost of 

Reciprocating 

Engine Plants. 

per K W. 


Excavations and foundations 


$ 2.00 to $2.50 
10.00 15.00 
1.75 4.00 
2.50 3.50 
8.50 12.00 
2.00 2.50 
2.00 2.25 
1.50 3.00 
1.00 1.50 
1.00 1.25 
2.25. 4.50 
22.00 25.00 

5.00 8.00 

.75 1.00 

.25 .50 
2.00 3.50 
1.00 2.00 


$ 3.001 
10.00 
1.50 
2.50 
8.50 
1.75 
2.00 
1.50 
1.00 
1.00 
2.50 

18.00 

3.00 

.75 

10.00 

.25 

2.00 

1.00 


.o $5.00 


Building 


20.00 


Tunnels (condenser water conduit). 

Flues and stacks 


2.75 
3.50 


Boilers and stokers 


12.00 


Superheaters 


2.25 


Economizers 


2.25 


Coal and ash handling systems 


3.00 


Blowers and ducts 


1.50 


Pumps and tanks 


1.25 


Piping systems 


5.00 


Turbo-generators 




Engines 


22.00 


Condensers (surface) 




Condensers (jet) 


5.00 


Exciters 


1.00 


Generators 


12.00 


Crane 


.50 


Switchboards 


3.50 


Plumbing, painting, labor, etc 


2.00 






Totals 


$65.50 


$92.00 


$70.25 


$104.50 







Smaller plants of about 3000 K W. capacity have been erected in the 
West at from $120 to $130 per K W., which cost may be reduced if a simple 
combination of machines is provided. 



Rates for Electric Current Furnished by the Municipal Plant of Pasadena, 
Cal. (Eng. News, Feb. 10 and Apr. 7, 1910). — Rates are shown in following 
table : 





Incan- 
descent 
Lighting. 


Arc 
Lighting. 


Power. 


Up to 100 KW.-hrs. per mo 


7 cts. 
6 " 
5 " 


6.75 cts. 
5.75 " 
4.75 " 


4 cts. 


100 to 500 " ** 


2.4 •• 


500 to 1000 " " 




1000 to 1500 " " 


2 


1000 to 2000 " " 


4 •• 
3 " 






Over 2000 *' " 






1500 to 5000 '• " 




1.9 '* 


Over 5000* " " 






1.75 " 


Over 5000t ** *' 






1.5 *• 


Over 10000 " " 






1.5 *• 











* If used between 5 p. m. and 10 p. m. 
t If not used from 5 p. m. to 10 p. m. 



MISCELLANEOUS DATA. COSTS. 



147d 



Cost of Overhead Trolley Systems (By A. D. Williams, Jr. Eng. News, 
Dec. 23, 1909) .--Tables: (1) Cost per mile of overhead materials; (2) Bracket- 
arm construction (a — with 37-ft. poles placed in center; b— with 30-ft. poles 
placed in center); (3) Cross-span construction, with 37 and 30-ft. poles; 
(4) Transmission line, and feeder line; (5) Comparison of costs, per mile: — 

Costs per Mile of Double Track. 





Bracket 
arm. 


Cross span 
30 and 37-ft. poles. 


Cross span 
30 and 30-ft. poles. 


Trolley wire and poles 
Transmission line .... 
Feeder line 


$2,136.05 
732.57 
682.69 


$2,420.99 
732.57 
682.69 


$2,264.99 
732.57 
682.69 






Total 


$3,551.31 


$3,836.25 


$3,680.25 



Illustrations of Electrical Works: — 

Description. Eng. News. 

Transmission line for Bay Counties Power Co. Oct. 3, 1901. 

Electric conduit construction at Cincinnati, O. Aug. 20, '03. 

Section of subway for pipes and wires Feb. 16, '05. 

Line construction for high-pressure electric railroads _ April 6, '05. 
High-pressure line construction for alternating-current rail'ys April 6, '05. 

New high-duty electric storage battery plate May 4, '05. 

High voltage electrostatic voltmeter operating in oil Jan. 11, '06. 

Details of transmission line poles, N. Y. C. & H. R. R. R. June 14, '06. 

Carbon regulator for storage battery Sept. 27, '06. 

^Details of steel towers for high tension transmission Mar. 12, '08. 

Rein. -cone, conduit for electric cables, L. I. R. R. July 23, '08. 

New type of switchboard, Salt River Project Aug. 27, '08. 

Divided manhole for high and low-tension conduit lines Sept. 24, '08. 

Insulator for 15,000 volt trolley line, Switzerland Nov. 11, '09. 

Circuit breaker for 110,000-volt lines Dec. 23, '09. 

Underground conduit construction for large transmissions Sept. 29, '10. 

Eng. Rec. 

Hydro-elec. power development; powerhouse, dam, etc. Apr. 3, '09. 
Power house, penstocks, dam, etc., Wis. Hydro-Elec. Power Co.Sept. 4, '09. 



* Failed. See Eng. News, May 13, 1908. 



71.— MISCELLANEOUS DATA AND 
ILLUSTRATIONS. 



DERRICKS AND CRANES. 

Description. 
A 30-ton gantry crane for C. & O. Ry. wharves 
Modem types of cranes for shipyard service 
A 30-ton -locomotive crane; a 5- ton derrick crane 
An electric traveling crane with transfer carriage 
The 120-ton floating derrick for the Norfolk Navy Yard 
Portable pneumatic revolving cranes 
Electric pillar cranes for handling cupola charges 
A gantry crane with double cantilever bridge 
A 10-ton stiff-leg der. with ball and socket foot-block bearing 
Ore-unlocking machines for use at receiving docks 
Recent heavy cranes in English shops 

A reversible hoist for elevators and derricks on construction 
A combined gantry crane and coal elevator 
Traveling cranes equipped with scales 

Steel guyed derrick (mast 70 ft.) for building erection 

Details of steel stiff -leg derrick 

A detachable seat for a boom derrick 

Derrick tower for erecting the Montana capitol 

CHIMNEYS. 

Section and plan of concrete chimney in Switzerland 

Steel chimneys for power station of St. Louis Transit Co. 

The design of self-supporting steel chimneys 

Details of 300-ft. chimney of reinforced-concrete 

A 350-ft. brick chimney for acid chemical gases 

The design of reinforced-concrete chimneys 

Method of building a steel chimney 

Lightning protection for power plant chimneys. U.S. Navy. 

Heat expansion stresses in chimneys. Not illustrated 

Wind stresses in reinforced-concrete chimneys. Diagram 

Largest Chimney in the world: 50' x 506' 

Tapering concrete chimney, 258 ft. high 

Underpinning a leaning chimney, illustrated 

MECHANISM AND GEARING. 

High-speed toothed gearing 

Chains and chain gearing 

Two new power transmission devices 

A mammoth sheave wood block: ht., 53''; wt,. 640 lbs. 

Friction coefficient of wire rope drives 

The Schmidt silent drive chain 

The design of friction clutches (mainly for automobiles) 

MARINE ENGINEERING. 

A comparison of typical marine engines 
Cunningham-Seaton system of coaling war vessels at sea 
Notes on design of steamships Minnesota and Dakota 
Half cross-section of car-ferry steamer "Detroit;" M. C. R. R. 



Eng, News 
Sept. 19, 1901 
May 28, '01 
Nov. 20, '02 
May 7, '03 
June 25, '03 
July 23 " 



^ 



uly 30, '03. 

lov. 26, '03. 
Sept. 22. '04. 
Aug. 3, '05. 
Aug. 29, '07. 
July 2, '08. 
Sept. 17, '08. 
June 17/09. 
Eng. Rec. 
Mar. 27, '09. 
April 23, '10. 
June 18, '10. 
Aug. 13. '10. 

Eng. News. 
Sept. 19, '01. 
Dec. 19, '01. 
July 20, '05. 
Aug. 3, '05. 
Feb. 15. '06. 
Jan. 3, '07. 
Mar. 14, '07. 
22. '07. 
5, '08. 
Sept. 10, '08. 
Nov. 26, '08. 
Jan. 13, '10. 
Eng. Rec. 
July 3. '09. 



Eng. News. 



Yds. Aug 
Mar. 



Feb. 28, 
Sept. 5, 
June 8, 
Mar. 21, 
June 27, 
Nov. 28, '07. 
Oct. 1.'08. 



'01. 
'01. 
'05. 
'07. 
'07. 



Eng. News. 
Aug. 29, '01. 
Jan. 21, '04. 
Sept. 1, '04. 
May 4, '05. 



1480 



DERRICKS, CRANES, CHIMNEYS, ETC, 



1481 



Description. 
Experience in the design of marine screw propellers 
A pneumatic submarine signalling bell 
Ocean steamers with steam turbines 
The Cunard steamship "Mauretania" 

Steel barges for transporting steel products, O. and Miss. B 
Reinforced-concrete barges on the Panama Canal 

CABLEWAYS AND CONVEYORS. 

A cheap cableway: for building concrete piers 
Lubricating the wheels of chain conveyors 
Method of handling ore at Bingham Canon, Utah 
Conveyor for loading iron ore in ships off rocky coast 
Cable haulage for transporting marl to cement mill 
Aerial tramways of the U, S. Mining Co., Bingham, Utah 
The Ridge way "two-belt" conveyor 
Side and end elevations of extensible 16" belt conveyor 
Cableway for cars in filling; cost compared with trestle 
Formulas for the design of cableways 

Traveling bridge suspended from cableway for making fills 
A unique belt conveyor 2000-ft. long, with equalizer 
The Wetterhom cableway incline, illustrated details 

REVETMENTS. 

Bank Revetment on the Lower Mississippi River 
Recent experiments with bank protection works. Ark. 
The David Neale System of bank protection 
Du Muralt system of reinforced-concrete shore protection 
A wave-break added to a concrete sea wall 
Separately-molded sections for a concrete sea-wall 

WELL BORING. 

An oil-well pump rod joint protector 

Diamond drill work on the deep waterways survey; cost 

Electric rock drills; by E. J. Munby. Not illustrated 

Competitive tests of rock drills for air consumption 

The manufacture and use of diamond tools 

Recovery of a diamond-crown from a deep bore-hole 

A rotary drill core from a steel I-beam 

The work of well-drilling machines on the P. R. R. 

A direct-acting gasoline rock drill 

Double wells, and casings, for pumping diff . waters 

MACHINES. 



Evolution of drop hammer for die forging 

Centrifugal machines and their uses 

Requirements of machine tool operation — motor drive 

Concrete mixing and handling machine — -sea wall 

Machines for briquetting flue dust, fine ore and fuel 

A new pneumatic hammer 

Diagrams for estimating hydraulic machinery 

A trussed wagon for hauling heavy machinery 

A rock crusher of 800 tons per hour capacity 

A molding machine for building cement sidewalks 

A machine for handling coke in storage 



BUNKERS AND BINS. 



Safe and proper design of grain storage elevators 
Grain pressures in deep bins. Tables and illustrations 



Eng. News. 
Nov. 2, '05. 
July 12, '06. 
Aug. 23, '06. 
Sept. 27, '06. 
June 9, '10. 
July 28. '10. 



Eng. News. 
June 26, '02. 
July 10, '02. 
July 24, '02. 
Sept. 4, '02. 
Jan. 21, '04. 
Feb. 11, '04. 
June 16, '04. 
June 21, '06. 
Oct. 10, '07. 
April 16, '08. 
April 22, '09. 
May 13, '09. 
July 22, '09. 



Eng. News. 
Oct. 31, '01. 
Jan. 28, '08. 
Oct. 22, '08. 
Dec. 17, '08. 
Aug. 18, '10. 
Sept. 29, '10. 



Eng. News. 
July 31, '02. 
July 23, '03. 
Sept. 3, '03. 
July 16, '04. 
Jan. 19, '05. 
June 29, '05. 
Nov. 30, '05. 
April 12, '06. 
Nov. 26, '08. 
Eng. Rec. 
Feb. 20, '09. 



Eng. News. 
Jan. 1, '02. 
Dec. 11, '02. 
Jan. 8, '03. 
Jan. 15, '03. 
Feb. 12, '03. 
Mar. 14, '03. 
Oct. 22, '03. 
Jan. 21, '04. 
June 4, '08. 
June 18, '08. 
July 16, '08. 

Eng. News, 
Mar. 10, '04. 
Mar. 10. '04. 



1482 n.— MISCELLANEOUS DATA AND ILLUSTRATIONS. 

Description. Eng. News. 

Hydraulic diaphragms and grain pressure tests April 28, '04. 

Design of reinforced-concrete grain elevator bins June 23, '04. 

Grain pressures in deep bins; strength of wooden bins July 14, '04. 

Tests of grain pressures in deep bins, Argentina Dec. 15, '04. 

A problem in detailing hopper work Aug. 24, '05. 

Large reinforced-concrete coal pocket at Charlestown, Mass. Aug. 27, '08. 

Disastrous grain elevator explosion Oct. 22, '08. 

Reinforced-concrete bins for storage of crushed stone Nov. 26, '08. 

Reinforced-concrete storage bins for crushed stone Nov. 21, '09. 

Reinforced-concrete cylindrical storage bins Dec. 2, '09. 

Rein. -cone, locomotive coaling station, 2,000 tons capacity June 23, '10. 

Rein.-conc. grain bins, Gt. Nor. Ry., Superior, Wis. Aug. 4, '10. 

Eng. Rec. 

Sections, coal and ash handling plant and coal bunker May 1, '09. 

Large, 10,000 ton, concrete and timber coal pocket May 8, '09. 

Plans of rein.-conc. coal bunkers, Annapolis, Md. May 15, '09. 



COMPRESSED AIR. 

Use of comp. air for contractor's plant. 4-stage air comp'r 

Apparatus and methods for testing air motors and hammers 

Caisson illness and diver's palsy; experimental study 

An ingenious and effective air-lift pump 

Ignitions and explosions in discharge pipes and receivers 

Specifications for an air compressor 

New method of pumping sand by means of compressed air 

A new positive-pressure blower; high pressure 

Compressed air plant used in boring the E. River tunnels 

Emergency air-lift equipment for deep wells 

Hydraulic compressed air plant, Victoria mines, Mich. 

A high-speed oil engine air compressor 

Effect of moisture in air on compressed air machinery 

Experimental studies of air-lift pumps; tests 

Controlling the output of the air compressor 

The Rateau centrifugal air compressors and blowers 



Eng. 
Nov. 
Dec. 
May 
Nov. 
Mar. 
Mar. 
Dec. 
Feb. 
Aug. 
Dec. 
May 
May 
June 
June 
Nov. 
Jan. 



News. 

26, '03. 
10, '03. 

5, '04. 
24, '04. 

2, '05. 

2, '05. 



05. 
06. 



2, '06. 
13, '06. 

2, '07. 

7, '08. 
18, '08. 
IS, '08. 

5, '08. 
20, '10. 



HEATING AND VENTILATION. 

Some experiments with ventilating fans 

Recent tests of centrifugal mine-ventilating fans 

Problem of ventilating N, Y. subways and similar tunnels 

A fan blower driven by a steam turbine 

A rapid current hot water heating system 

Data for the design of hot water heating systems 

Air washing and himiidifying 



TELEPHONES. 



Long spans in telephone work 
The development of telephony 



MINING. 



Eng. News. 
Nov. 3, '04. 
Nov. 10, '04. 
June 22, '05. 
Nov. 9, '05. 
Nov. 22, '06. 
Jan. 30, '08. 
Aug. 13, '08. 



Eng. News. 
Mar. 2, '05. 
April 8, '09. 



Eng. News. 
Aug. 30, '06. 
Dec. 7, '08. 
July 28. '10. 



Methods of mining, hauling and screening, in Alabama 
New type of jig for separation of metallic ores 
Hammer drills for overhand stoping in gold mines 

METAL SPRINGS. 

New Helical Spring Formulas (following is excerpt) 

New Helical Spring Formulas have been arranged by Mr. Chester B. 
Albree and are explained in his paper "Spring Formulas Simplified," which 



Eng. News. 
Jan. 7, '09. 



COMPPRESSED AIR. ETC. METAL SPRINGS. 1483 

is published in the November issue of the "Proceedings" of the Engineers* 
Society of Western Pennsylvania. These are not strictly new, being derived 
from the well-known Reuleaux' formulas: 

P = S~ 

B2PR^L 
n Gd* 

in which i? = radius of coil to center of wire, 

L = uncoiled length of wire, 
and the other symbols correspond to those used below. The process of 
simplification is explained by Mr. Albree, in part, as follows: 

In comparing the various formulas, it was found that certain quantities 
could be combined giving formulas of much simpler character and yet 
equally exact. This was accomplished by cancellations and reductions, 
eliminating the third and fourth powers and replacing them with areas, 
diameters and constants. This is done with the intention of rendering the 
solution of helical spring problems easy for anyone having at hand standard 
tables of areas and decimal equivalents. The writer is not in the spring 
manufacturing business and is not an authority on the subject. The formu- 
las derived with the terms used are given below: 

p_^dS 

^ 2D 

P_^f_^Pi 
W h F2 

/ = ;j^ for 5 = 100.000 lbs. 
40 a 

= ^ for 5 = 80,000 lbs. 
50 a 

= ^ for S = 60.000 lbs. 

„ fW D^W 

in which P = closing, or maximum permissible, load of spring. 

5 = torsional strain, outer fiber. 
W = any load on spring. 

/ = deflection of one coil under closing load. 

^2 = deflection under any load. 

F = total deflection under closing load. 
jF2 = total deflection under any load. 

J = diam. of bar of wire. 

a = area of bar or wire. 

Z> = diam. of coil, center to center of bar or wire. 

H = free height of coiled spring. 

w== number of free coils. 

G^= modulus of torsion. 
= 12,500,000 lbs. 
The formulas are based on a spring designed so that when it is closed 
under a certain load, the strain 5, selected, will be reached. The deflection 
formulas give the pitch of coils to produce strain S, when closed. With 
what is known ordinarily as "spring steel," it is safe to use 5= 100,000 lbs., 
which is the practice of the spring manufacturers of Pittsburg. 

The value to be used for 5 should depend, of course, upon the nature of 
the work for which the spring is designed and the conditions under which 
it must operate. The value given abov® (5=100,000 lbs. per sq. in.) is 
rather high for general conditions. The German engineer's pocketbook, 
"Hutte," gives 5=4,500 kg. per sq. cm. (which is closely equivalent to 
64,000 lbs. per sq. in.) for spring steel with fairly constant loading. In 
springs designed for continual removal and application of load, such as 
inlet or exhaust-valve springs of gas engines, 5 should not exceed 45,000 
lbs. per sq. in. 



1484 n.— MISCELLANEOUS DATA AND ILLUSTRATIONS. 



Testsof Steel Springs (Proc. A. S. T. M., Vol. VIII., 1908).— The follow- 
ing table shows the effects of different methods of tempering on the elastic 
limit and modulus of elasticity of steel. 



Annealed 
in lead 
at 


Hardened 

in oil 

at 


Hardened 

in water 

at 


Drawn 
to 


Elas.Limit . 
Lbs. per 

sq. in. 


Mod. of Elas. 

Lbs. per 

sq. in. 


1400° F. 








78 500 
137 500 
160 400 
177 600 
187 400 
180 700 
233 900 
240 800 
219 800 
212 000 


27 550 000 




1450° F. 
1450° F. 
1450° F. 
1450° F. 




560° F. 
560° F. 
400° F. 


28 700 000 






27 150 000 






29 000 000 






28 610 000 




1425° F. 
1425° F. 
1425° F. 
1425° F. 
1425° F. 


1050° F. 
900° F. 
750° F. 
600° F. 


28 070 000 






28 860 000 






29 220 000 broke 






30 420 000 broke 






29 960 000 broke 









METAL HOISTING CHAINS. 

Iron, Oval, Open=Link Chains. — The following formulas give the dimen- 
sions, weights and strengths of iron chains with open, oval links: — 

Let d = diam., in ins,, of round iron used; then — 

1.5 d = transverse inside diam. of oval link, in ins., 

2.6 d = longitudinal inside diam. of oval link, in ins., 

= effective length of each link of chain, in ins., 
/ = length of round iron in each link, in ins., 
= 9.475 d; 
Wo= weight of each link, in lbs., 
= 0.2225 / d\ 
= 2.108^3; 

W = weight per lin. ft. of chain, in lbs., 
= 9.73^3. 

5 = ultimate strength of chain, in lbs., 

= 1.625 X strength of single rod of diam. d. 



SOLAR POWER. 



Power from the sun's heat 



EXPERT VALUATIONS AND REPORTS. 

Mine valuation by mining experts 
Report on Chicago street railways and subways 
Depreciation allowances for various public service industries 
Valuation and inspection work of Wis. Tax & R. R. Com. 
Provision for depreciation by public utility corporations 
Table of freight rates on coal, iron and cement 
Necessary elements for water works valuation. — Alvord 
Valuation of track of Detroit St. Ry. System 

Diagram illustrating method of estimating "Going Value" 
CONTRACTS AND SPECIFICATIONS. 

Schedule of 120 clauses as guide in drawing specifications 



Eng. News. 
May 13. '09. 

Eng. News. 



30, '02. 
19, '06. 
23, '08. 
4, '09. 
4, '09. 
Mar. 11. '09. 
Mar. 10, '10. 
Sept. 8, '10. 
Eng. Rec. 
June 19, '09. 



Oct. 

July 
Jan. 
Mar. 
Mar. 



Eng. News. 
April 21. '04, 



GLOSSARY. 

(See, also, Index, page 1539, etc*) 



Abaciscus. — A diminutive of abacus. 

Abacus. — The flat slab (plinth) forming the upper member of the capital of 

a column to support the architrave. 
Abscissa. — ^The horizontal or x distance, measured parallel with the horizontal 

axis X, from the vertical or inclined coHDrdinate axis Y to any point on 

a curve whose ordinate is y; x and y are co-ordinates to any point on a 

curve, measured from the origin or from the axes F and X. See Analytic 

Geometry, page 256. 
Acclivity. — An upward slope or inclination of the ground; opposed to 

declivity, or a slope considered as descending. 
Adit. — A "horizontal" excavation or drift, specially used to drain or operate 

a mine; it is not continuous as a tunnel proper, which strictly has two 

openings. Word tunnel often wrongly used for adit. 
Adze. — A tool with a curved blade placed at right angle to the position of 

an axe blade, and used by ship- and bridge carpenters in dressing the 

top surfaces of timber, ties, etc. 
Alternate. — To pass from one state, as motion or position, to a second, then 

back to the first, and so on in rotation. 
Alternations. — A complete alternation is a change in the direction of a current 

in a circuit from its former direction back again to that direction; 

symbol '^. 
Alternator. — A common term for an alternate current dynamo. 
Ammeter. — An instrument for measuring the amperes of a current. 

Ampere. — ^The unit of electric current; symbol, C. C = -^ in which .E = 

electromotive force in volts, and K — resistance in ohms. Current flowing 
at the rate of one ampere transmits a quantity equal to one coulomb 
per second. One volt-ampere = one watt = yj5 horse-power. 

Angle=bar. — A bar of angle-iron. 

Angie=bead. — A plaster-bead or staff-bead used to protect plaster from 
injury. 

Angle=beam. — A beam with flange set at an angle with the main portion. 

Angle=bevel. — Bevel-square. 

Angle=block. — ;In Howe trusses, a "triangular" block of cast-iron or wood 
set at the junction of the wooden chords and braces, and through which 
pass the vertical iron or steel rods called ties. 

Angle=rafter. — A rafter joining the inclined planes of a hipped roof. 

Angle^splice. — A splice for rails. 

Anneal. — To remove the brittleness of metals, earthenware, glass, etc., by 
heating them and then allowing them to cool gradually. This toughens, 
but lowers the tensile strength. 

Anticlinal.— The incline or dip of stratified rocks in an upward fold ; opposed 
to synclinal. 

Anticlinal Axis. — ^The ridge of an anticlinal. 

Apex. — ^The top junction of two or more lines or members; as the apex of a 
roof. 

A priori. — From that which precedes; from the former. 

Apron. — Anything resembling a common apron in form or use. The bridge 
of a* ferry dock. A platform or flooring at a dock entrance. A covering 
to protect anything, as a dam, from water flowing over it. 

Arbor. — An axle or spindle of a pinion or wheel. A mandrel, in lathe turn- 
ing. 

Architrave. — ^The lower part of an entablature, which rests directly on the 
columns and supports those parts of the building above. The molding 
around the extrados of an arch. Sometimes applied to ornamental 
moldings on faces of jambs and lintels of doors, windows, etc. 

1485 



1486 GLOSSARY. 

Archivolt. — Ornamental molding on extrados of arch. 

Armature. — That part of a magnet, dynamo, or motor designed to act upon, 
or to be acted upon by, the lines of force set up by the poles of the field- 
magnet, in order to produce motion and power. (When placed directly 
at the pole of a permanent magnet it is called a keeper). 
Classification (1): 

Polarized a. — One made of steel or of another electro magnet and hav- 
ing poles which act upon and are acted upon by the field magnet poles. 
Non-polarized a. — One made of soft iron with coils of wire arranged 
in any form. 

Classification (2): 
Ring a. — Round and usually of circular cross-section. 
Drum a. — An armature of cylindrical form. 
Disc a. — 

Pole {-or -radial) a.- — 
Spherical a. — Thomson-Houston type. 

Classification (3): 
Unipolar a. — One whose polarity is never reversed. 
Bipolar a. — One whose polarity is reversed twice in every revolution 

through the field of the machine. 
Multipolar a. — One whose polarity is reversed "multi" times (more than 
twice) in every revolution. 

Armature, Flat=Ring. — A ring armature with a core shaped like a short 
cylinder. 

Armature, Girder. — H-shaped core. 

Armature, Toothed=Ring.— A ring armature with core provided with a 
number of teeth forming spaces between which the armature coils are 
placed. 

Arrester, Lightning. — An apparatus for protecting an electric circuit from 
lightning. 

Arrester, Lightning, Transformer. — A lightning arrester for protecting trans- 
formers. 

Arris. — The edge-line of an exterior angle formed at the junction of two 
surfaces meeting. 

Arris=gutter. — A V-gutter fixed to the eaves. 

Ashlar.— Cut-stone masonry. See Masonry, page 432. 

AstragaL — A small convex molding in the form of a string of beads. 

Axis. — The imaginary line, relatively motionless, about which a rotating 
body turns. 

Axle. — A shaft in the position of the axis of a rotating body. 

Axle=box. — The box containing the bearings for the spindle of the axle. 

Axle=guard. — The parts of a car in which the axle-box moves vertically 
when the springs yield. 

Axle=seat. — The hole in the car wheel to receive the axle. 

Axle=tree. — A fixed axle as for a carriage, the wheels revolving. 

Azimuth. — An arc of the horizon intercepted between the meridian of a 
place and the vertical circle passing through the center of a celestial 
object. A star's azimuth and altitude determine its exact position. 

B. 

Backing. — ^The rough masonry of an arch, abutment or wall, faced with a 
better class of masonry. Of an arch, it is the course of masonry resting 
upon the extrados. 

Balance=bar = balance=beam. — A long bar or beam attached to canal-lock 
gates and drawbridges, and used in opening and closing them (usually 
serving partly as counterbalances) . 

Balk. — A beam or timber of considerable size. In coal mining: the sudden 
contraction of a bed of coal, for a certain distance. 

Ballast. — Broken stone, gravel, slag, sand, or other suitable material placed 
on the sub-grade of a roadbed to support the railroad ties, give them 
lateral stability, and decrease the dust due to passing trains. 

Ball'°cock. — A lever with a hollow metal ball attached to one end and float- 
ing in a tank, and operating (as the ball rises and falls with the water) 
a valve of the cistern at the other end. 

BalUvalve. — A valve formed by a ball resting on a circular seat when valve 
is closed; but which is free to rise with the upward pressure of the 
water. 



ARCHIVOLT. BOASTER. 1487 

Barge=board. — Gable-board of a house. 

Bars, Bus. — Omnibus bars. (See Bars, Omnibus.) 

Bars, Negative=0nini5us. — The bus-bars that are connected with the nega- 
tive terminal of the dynamos. 

Bars, Neutrai^Omnibus, — The bus-bars that are connected with the neutral 
dynamo terminal in a three-wire system of distribution. 

Bars, Omnibus. — Heavy bars of conducting material connected directly to 
the poles of dynamo-electric machines, in electric incandescent light or 
electric railway installation, and therefore receiving the entire current 

produced by the machine Main conductors common to two or 

more dynamos in an electrical generating plant. (The terms "bus" and 
"omnibus" bars refer to the fact that the entire or whole current is 
carried by them..) 

Bars, Positive=Omnibus. — The bus-bars that are connected with the posi- 
tive terminal of the dynamos. 

Bascule=bridge. — A counterpoised drawbridge, dating back to medieval 
times; as the leaf of the span rises, the counterpoise weights descend. 

Batter (not batir) . — The incline (as of a masonry wall) from the perpendicu- 
lar; the ratio of horizontal to vertical distance, as 1 in 12=1 in. hor. 
to 12 ins. vert. 

Bay. — A panel of one span; sometimes, one span of several in a bridge. 
The plain part of anything enclosed or bordered by features in relief. 

Bead. — Any small projecting cylindrical, globular or annular body. 

Bearing. — The direction of an object by the compass. In mining: the run, 
course or strike. In architecture: the clear, unsupported span of a 
beam or timber. In engineering: the actual surface of contact of and 
with something supported, as of a beam, girder, pivot, axle, etc. In ship- 
building: the widest part of a vessel below the plank -sheer; also, the 
line of flotation of a vessel when ready for sea; using in each case the 
plural, bearings. 

Bed=>molding = bedding=niolding. — A molding of the cornice of an entabla- 
ture, above the frieze and beneath the corona. 

Bed'-plate. — A plate (usually of iron or steel) laid on a foundation (say of 
masonry) and used to give direct support to something (as a machine 
or bridge) and distribute the stresses quite uniformly (above or below 
or both) . 

Beetle. — A heavy wooden mallet (maul) to drive wedges, with handle for 
swinging; a rammer, with handle set in middle of one head, used by 
pavers. 

Bell>=crank. — A right-angle lever pivoted at angle, for changing direction 
of motion, force, etc. 

Bench°niark. — A permanent bench of known or determined elevation with 
reference to a datum plane, in a line of levels. 

Berm (old form, berme) =berm=bank. — A terrace; a strip reserved between 
top of cut and the waste bank in excavation, or between the bottom of 
fill and the borrow-pit in embankment; the bank of a canal opposite 
the tow-path. 

Bessemer steel. — Steel made by the pneumatic process, consisting in blow- 
ing air through molten pig-iron in a "converter" lined with a refractory 
material, decarbonizing the iron; later a certain amount of carbon is 
restored by introducing spiegeleisen or ferromanganese. 

Beton. — Hydraulic-cement concrete. 

Bevel. — An instrument with a blade and a handle or stock, movable on an 
adjustable pivot or joint, for including any angle. 

Bevel°angle. — Any angle except a right angle. 

Bevel=gear. — Toothed wheels which gear at an angle, most often at right 
angle. 

Bilge. — The belly or widest part of a cask; the nearly horizontal part of a 
ship's bottom, adjacent to the keel. 

Bilge^keelson. — A fore-and-aft timber placed inside the bilge to strengthen it. 

Bit. — The biting or boring part of a tool. The boring-bit is held or turned 
by the brace or bit-stock. Old form, "bitt." 

Blast°pipe. — The exhaust-pipe of a steam engine; in locomotives, it leads 
into the smoke stack to create a strong draft. 

Board, Switch. — A board provided with a switch or switches, by means of 
which electric circuits connected therewith may be opened, closed, or 
interchanged. 

Boaster = boasting-chisel. — A broad chisel for rough-hewing and dressing 
stone. The use of such a chisel is called "boasting." 



1488 GLOSSARY, 

Body. — Consistency or density, as in paints, oils, etc. 

Bolster. — A "pillow" of timber for various purposes; a short timber or cap- 
piece resting on a post or column to give more extended bearing to the 
string-piece or beam. 

Bond. — The particular arrangement 9f brick, stone, or timber to give certain 
specified joints and courses. Friction-resistance of steel in concrete. 

Bonnet. — A cast-iron plate for covering the opening in a pipe or the opening 
in the valve-chamber of a pump; the cap or lid of an iron pipe; a 
wire netting for the smoke stack of a locomotive, to serve as a spark 
arrester. 

Bore. — The internal diameter or caliber of a hole, as in a pipe or hollow 
cylinder; need not have been "bored." 

Borrow=pit. — The site where excavated material, as earth, is obtained for 
filling elsewhere. 

Boss. — A projecting mass, as of stone, to be cut or carved later. The en- 
larged part in diam. of a shaft for keying a wheel or (if at end) making a 
coupling. 

Box=drain. — A rectangular (or square) drain of masonry, or timber, etc., 
under an embankment (or under ground) . 

Brace. — A strut, or auxiliary compression member in a frame; a bit-stock, 
or curved handle for holding and turning boring-tools or bits. 

Bracket. — A projecting piece of wood or metal, fastened to a wall, or ceiling, 
etc., and used as a support for some object; hence, wall-brackets, hang- 
ing-brackets, etc. 

Bracket, Telegraphic. — A support or cross-piece placed on a telegraph pole 
for the support of the insulators, which are supported on either arms 
or brackets. 

Brake. — Any mechanical device for retarding motion (as of a vehicle) by 
means of friction. 

Brake=hanger. — A bar or link suspending brake-beams and accessories from 
the truck-frame or from the body of car. 

Brake-^head. — The brake-block (usually cast-iron) fastened to the brake- 
beam and bearing on the circumference of the car wheel, forming at the 
same time the brake-shoe. 

Brake^shaft. — ^The shaft on which the chain operating a car-brake (by hand) 
is wound. 

Brass. — A useful alloy of copper and zinc; harder than copper; malleable 
and ductile. 

Braze. — To solder with "hard" solder, as an alloy of "brass and zinc" 
( = copper and zinc in different proportions than that composing brass). 

Break. — A want of continuity in a circuit. 

Breaker, Circuit. — A device for breaking a circuit. 

Break joint. — To arrange parallel members so the joints will not be opposite; 
thus, with the splicing of the leaves composing the chord of a Howe 
truss, or the two rails composing a track, etc. 

Breast=beam. — The transverse forward beam of a locomotive. 

Breast=board. — The weighted board or sled used in rope walks to keep the 
yarns taut while being twisted into a strand. 

Breast=drill. — A drill-stock having a breast piece against which the workman 
bears while operating the drill. 

Breasting. — The (curved) channel in which a breast -wheel turns. 

Breast=!ine. — The rope used to connect the pontoons of a floating bridge. 

Breast=wall. — A (low) retaining wall at the bottom of a slope. 

Breast-wheel. — A water-wheel with radial floats or buckets on its periphery, 
the water being confined to the floats by the "breasting" of timber or 
masonry, nearly touching the wheel. 

Breech. — The hind part of anything; the angle of a knee-timber, opposite 
the "throat." 

Brest='Sunimer. — A lintel; a beam or summer placed to support an upper 
wall, as over a shop window or door. 

Bricknog. — A timber framing, as a partition, filled with brickwork. 

Brick^trimmer. — A brick arch to protect a wooden trimmer in front of a 
fireplace. 

Bridge=bar. — ^The tension bar of a car-coupling. 

Bridge=board = notch-board. — A notched board of a stair to receive the 
^ ends of the wooden steps (treads) and risers. 

Bridge«pit. — The pit for the counterpoise of a bascule-bridge. 

Bridges. — Heavy copper wires suitably shaped for connecting the dynamo- 
electric machines in an incandescent light station to the bus-rods or wires. 



BODY. BY-WASH, 1489 

Bronze. — An alloy of 85% ± of copper and usually 15% T of tin; variable; 
sometimes no tin; sometimes contains zinc. 

Brush=Holders for Dynamo=Electric Machines. — Devices for supporting the 
collecting brushes of dynamo-electric machines. Brushes should be 
adjusted carefully to speed of machine and resistance of external circuit. 
Carbon brushes are plates of carbon for leading current to electric 
motors. Collecting brushes are conducting brushes which bear on the 
commutator cylinder, and takes off the current generated by the 
difference of potentia,! in the armature coils. Copper is almost univer- 
sally employed. 

Bucket=>engine. — An inprovised water-wheel for a high fall with scarcity of 
water, consisting of a series of buckets on an endless chain running over 
a pair of sprocket-wheels. 

Bucket=lift. — In mining, a set of iron pipes attached to a lifting-pump. 

Bucket=pitch. — ^The circular line intersecting the elbows of the buckets 
of an overshot water-wheel. 

Bucket=valve. — The valve at top of the air-pump bucket in a steam engine. 

Bucket»wheel. — A series of buckets arranged on an endless chain passing 
over a wheel, for raising water; or, the buckets may be attached to 
the rim of the wheel. 

Bucking iron. — A tool for pulverizing ore on a plate called a "bucking- 
plate." 

Buckram. — Coarse linen cloth stiffened with glue and used for binding 
books. 

Buffer. — An apparatus for deadening the concussion of railway cars. 

Bulkhead. — A partition; in a ship, to form separate apartments; in a tunnel, 
conduit or mine, to prevent the passage of water, mud and air. An 
improvised wharf, sometimes constructed of sunken cribs, to form a 
basin, or to run parallel with the shore. 

BuU=pump. — A pumping-engine with piston-rod attached directly to the 
pumping-rod, the weight of the rods producing the down stroke. 

Bumper=tiniber. — A timber to which the cow-catcher of a locomotive is 
fastened. 

Bumping=post. — A fender or buffer at the end of a railroad track to stop 
the cars. 

Bunker. — See "coal-bunker." 

Bunker=plate. — An iron plate covering the hole in a ship's deck leading to 
the coal-bunker. 

Buoy. — A floating body anchored in a harbor or stream to indicate the posi- 
tions of objects beneath the surface, as rocks, .shoals, etc., or to locate 
channels. Among the various kinds are spar-buoy§, can-buoys, bell- 
buoys, whistling-buoys, etc. In the U. S., red buoys mark the right- 
hand, and black buoys the left-hand, side of channels coming into 
port. Buoys with black and red transverse stripes mark dangers in 
mid -channel; while buoys with black and white longitudinal stripes 
indicate fairway. Green buoys indicate sunken wrecks. 

Burnish. — ^To polish by friction, as metals. 

Bur=pump = burr-pump = bilge=pump. — Has a cup^shaped cone of leather 
fastened to end of pump-rod, the sides collapsing as the rod descends. 

Bush = bushing. — A lining of harder material fitted into an orifice to reduce 
wear by friction; used in machinery of all kinds. 

Bushel. — U. S. standard = 2150.42 cu. ins. British imperial bushel = 
2218.192 cu. ins. 

Butt. — A door-hinge. A large cask, containing 110 imperial gallons. 

Butte. — A rising ground or mound. 

Butterfly=valve = butterfly-cock. — Employed in lift-buckets of large water- 
pumps and for air-pump buckets of condensing steam-engines. It is 
a double clack-valve or wing- valve, the two wings being hinged to a 
cross-rib cast in the pump bucket. ^ ^ , 

Butt=joint = butting=joint. — Opposed to lap-joint. A joint formed by the 
two pieces of metal or timber abutting endwise, and usually spliced 
together with other pieces. 

Buttress. — A prop or support of masonry. A bearing or thrusting structure 
built against a wall to give it stability. 

Buzz=saw. — A circular saw; it creates a "buzzing" noise. 

By=pass. — An extra gas- or water-pipe passing around a valve or chamber so 
as to give some flow when valve or chamber is closed. 

By-wash = by-lead. — A channel for surplus water from a reservoir or 
aqueduct, to prevent overflow. 



1490 'GLOSSARY, 



C. 



Cableway. — A taut suspended cable for conveying loads suspended in a 
car and moved by a hauling-rope or other device. 

Caisson. — A water-tight casing used in building the foundation of structures 
in water too deep for the coffer-dam. It is sunk by under-mining and 
the masonry is laid on top as it descends into the mud. 

Calcination. — ^The process of expelling volatile matter from a substance 
by heat, and reducing it to a friable state: as carbonate of lime, to lime. 

Caliber = calibre. — The diameter, especially the inner diameter or bore. 
In firearms, a 44 caliber rifle is one whose bore is 0.44 in. dia. 

Caliper = calipers. — An instrument like a pair of dividers, but with curved 
legs, for measureng inside and outside diameters. 

Calking = caulking. — The operation of filling seams of vessels with oakum 
to prevent leaks; or, the joints of cast iron pipes with oakum and lead, 
the oakum and lead being calked with a calking-iron or chisel, hammered 
with a calking-mallet. 

Calorimeter. — An apparatus for measuring the quantity of heat given off 
by a body, etc. 

Cam. — A device, usually upon a shaft, for converting a regular motion into 
an irregular, an alternating, or a reciprocal one. An eccentric. An 
elliptical lever for raising the ends of drawbridges, when closed, to a 
firm bearing. 

Camber. — In a bridge, a slight upward curve in the span, to allow for settling 
when loaded, or for appearance. In laying cast-iron-pipe culverts 
under a railroad embankment, they should be jointed or laid in a vertical 
curve, convex upward, to allow for settlement in the middle. 

Camel. — A water-tight apparatus which is filled with water, sunk, attached 
to vessel's bottom, water pumped out, and which then assists in floating 
the vessel over a shoal, or raising her from wreckage. 

Candle-foot. — A unit of illumination equal to that produced by a standard 
candle at a distance of 1 foot. The power is inversely proportional to 
the square of the distance; thus, the illumination at 3 feet is one-ninth 
that at 1 foot. 

Candle=power. — ^The standard is a spermaceti burning at the rate of 120 
grains of sperm per hour. 

Cantilever. — A block or bracket framed into the wall of the building, pro- 
jecting from it, and used for supporting a molding, balcony, eaves, etc. 

Capital. — The top of a column, pillar or pilaster. 

Capstan. — An apparatus something on the principle of a windlass, but with 
a vertical axis. THe man-power is applied to capstan-bars inserted 
horizontally in holes near top of windlass, a few turns of the rope, 
cable or chain being wound around the barrel of the latter. At the bottom 
of the barrel is a pawl-Head with pawls to engage a ratchet-ring secured 
to the platform. 

Carbonize. — To combine with carbon, as in the manufacture of steel by the 
cementation process, Hence the term carbonization. 

Carburize. — To combine with carbon or a carbon compound. (The vapors 
of volatile hydrocarbons are often mingled with combustible gases in 
order to produce higher illuminating power in the latter). The process 
is called carburization. 

Case=hardening. — A quick process of cementation or converting the outer 
surface of iron into steel by heating it in contact with charcoal or some 
animal matter as bone, hoof parings, or leather. 

Casemate. — ^The masonry vault in the rampart of a fortress, or the armored 
bulkhead in warships, pierced in front with embrasures or port-holes 
through which guns are fired. 

Casting=pit. — ^The part of a foundry where molds are placed and castings 
made. 

Causeway. — A raised path or road over bad ground. A raised sidewalk. 

Cavetto. — A concave molding of at least ^ circle used in cornices, etc. A 
recessed pattern; opposed to relief. 

Cementation. — In metallurgy, a process of effecting a desired important chem- 
ical change in a substance when heated in contact with another 
substance. Bar-iron may be made into steel by heating it above redness 
while embedded in charcoal-powder. Such a process is termed carburi- 
zation by cementation. Decarburization, or the converting of cast- 
iron into malleable-iron, is effected by embedding the casting in red- 
hematite powder and keeping it some time at a red heat. 



CABLEWAY. CLAW-WRENCH. 1491 

Cement-copper. — Copper precipitated by the process of cementation. 
Center = centering — ^The frame supporting an arch during construction. 
Center of oscillation.^ — Coincides with the center of percussion. 
Center of percussion. — ^That point in a revolving or swinging body, or 

pendulum, at which if all the mass were concentrated the effect would 

remain unchanged; the point of greatest impact with another body, 

remaining immovable without rotation at time of contact if the opposing 

body is fixed. 
Center=plate. — A plate which supports a car-body on the center of a truck. 
Center= valve. — A four-way gas-cock. 
Centrifugal. — Radiating or outward (force) from the center. Opposed to 

centripetal. 
Cesspool. — A shallow well of large diameter for receiving sewerage from 

isolated buildings. Best constructed in sand or gravel, with linings, 

say of brick, for the side walls only. Covered top. 
Chamfer. — ^To cut away the edge of a square comer so as to form a bevel 

edge, generally projecting. 
Chamfered. — Anything in which the surface is beveled. See Side-hatchet. 
Chase. — To decorate metal-work by tooling. 
Check=nut. — A nut placed on a screw or bolt to preverrt the main nut from 

turning when in place. 
Check=stop. — In deep dredging, a device to prevent the dredge-line from 

breaking when the dredge fouls. 
Check=valve. — A valve placed in a pipe to prevent the backward flow of 

the water, steam, or other fluid. 
Cheek. — One of two symmetrical pieces enclosing something between them: 

one of the jaws of a vice, one of the walls of a vein of ore, one of the 

sides of a pillow-block, etc. 
Chill. — A metal mold for rnaking certain kinds or parts of iron castings; 

the surface in contact with the mold cools rapidly and hardens. 
Chimney=cap. — A device on top of a chimney which is turned by the wind 

so the exit-aperture is always to leeward, thus helping the smoke 

to escape. A chimney jack. 
Chimney=stack. — Several chimneys carried up together. 
Chisel=draft. — The dressed edge of a stone, either complete or as a guide in 

dressing the stone. 
Chock. — A piece of wood inserted to prevent movement, as the chock nailed 

on the cap of a trestle between the two lines of stringers. 
Chuck. — A block or device in a lathe for holding anything to be turned. 
Churn=drin. — A long stone-drill operated by hand: raised and let fall. 
Cinder. — A mass of ashes, containing more or less unconsumed coal. Pig- 
iron slag from a blast-furnace. 
Circuit, Closed==Magnetic. — A magnetic circuit which lies wholly in iron 

or other substance of high magnetic permeability. All lines of magnetic 

force form closed circuits. An iron ring forms a closed-magnetic circuit. 

Where an air gap is formed, as in the case of a horse-shoe magnet with its 

keeper, it is called an open-magnetic circuit. In other words a closed 

circuit is formed by a ring of high magnetic permeability. 
Circuit, Electric. — ^The path in which electricity circulates or passes from 

a given point, around or through a conducting path, to its starting 

point. 
Circuit, External. — ^That part of a circuit which is external to, or outside 

the electric source. 
Circuit, Magnetic. — The path through which the lines of magnetic force 

pass. (They always form closed circuits.) 
Circuit, Short. — Shunt circuit. 
Clack= valve. — A hinged valve placed in a clack-box, and consisting of a plate 

of leather strengthened above by a plate of iron larger in dia. than the 

main pipe, and below by a plate of iron smaller in dia. than the main 

pipe. The dia. of box is about li times dia. of pipe. Used in pumping. 

A flap-valve or clapper. 
Clamp. — Any instrument used to hold anjrthing or to hold two or more 

pieces together, by pressure. Various forms for different trades and uses. 
Clapboards. — Long thin boards, thicker on one edge than on the other, 

and nailed horizontally on the sides of a ho.use, lapping shingle-fashion. 
Claw. — A part of a tool resembling a claw (hand). 
Claw=hammer. — A hammer cleft for drawing nails. 

Claw=wrench. — A common wrench with one jaw fixed and the other mova- 
ble. 



1492 GLOSSARY. 

Claying-'bar. — In blasting, a rod for driving clay into crevices, to protect 
the charge. 

Cleat. — Any piece, as wood or iron, for fastening a rope; or by nailing to 
other pieces to fasten them together. 

Clevis. — An iron shaped like a horseshoe, or stirrup, or U, with provision for 
inserting a bolt across or between the ends in order to form a link. 

Click = clicker. — A kind of rachet. A small bar pivoted at one end, free to 
move backward on a toothed rack, but in moving forward it engages 
one of the teeth and moves the object forward, leaving it at rest during 
the backward stroke. 

Climbing=irons = climbers = creepers. — Iron frames with spikes, for climbing 
telegraph poles, trees, etc. 

Clincher=bullt = clinker=built.— Composed of pieces overlapping one another, 
as in clincher-built boats where the boards overlap like clapboards. 

Clinker. — ^The fused or melted ash formed by the combustion of coal; par- 
tially-vitrified bricks. 

Clip. — A metal clasp, as for holding a bunch of paper, or a Y-level telescope, 
etc. 

Clip=yoke. — ^The small plate through which the ends of a U-shaped clip pass, 
and serving as washer-plate for the nuts of the clip. 

Cloister. — A covered walk or arched way around the walls of a building, 
the outer edge being supported on a series of arcades or arches resting 
on columns. 

Close=hauled. — Sailing as close to the wind as possible. 

Clutch. — A movable coupling, as for connecting or disconnecting the ends 
of two adjacent shafts to make them revolve together or to allow them 
to revolve separately. If operating by friction it is called a friction- 
clutch; by engaging prongs, a bayonet-clutch. The cross-head of a 
piston rod is a form of clutch. 

Coal'^'breaker. — A person occupied in breaking the large masses of coal as 
they come from the mine; or who operates a machine for this purpose; 
or the machine itself; or the building or structure in which the breaking 
is done. 

Coal=bunker. — A place for storing coal. 

Coal=gas. — An illuminating-gas obtained by heating coal in closed iron 
vessels free from air; contains about 46% hydrogen, 35% marsh gas, 
7% carbonic oxid, 4% olefiant gas, tetrylene, sulphureted hydrogen, 
nitrogen and carbonic acid and traces of other gases. 

Coal=oil = petroleum = rock=oil. — By refining, we get kerosene, naphtha, etc. 

Coaming = combingu — The raised borders or edges of the hatches, to prevent 
water on the deck of a vessel from running into the hold. 

Coal-tar. — The thick, black liquid which condenses in pipes when gas is 
distilled from coal. Contains anthracene, benzol, carbolic acid, creosote, 
naphtha, naphthalin, paraffin, pitch, etc. Used for metal coatings, as for 
cast-iron pipe; in making asphalt tor pavements, etc. 

Cock. — A faucet, turn-valve, or valve, for regulating the flow of fluids; 
as air-cock, gage-cock, feed-cock, etc. 

Cock-brass = cock=metal. — An alloy of two parts copper and one of lead; 
used for large vessels, cocks or taps. 

Cock-water. — Water used to wash away sand from ores. 

Coefficient. — A quantity, number or constant (as 1/10, 4, b, etc.) used as a 
multiplier into some algebraic expression or physical property of a sub- 
stance or condition: The coefficient ot friction is the tangent of the angle 
of repose of a body; the differential coefficient (in the Calculus) is the 
rate of change of a function; the coefficient c in Kutter's formula, 
c=v'\/rs, is the coefficient of velocity of flow for the particular conditions 
imposed, as roughness ot surface n, hydraulic radius r, and hydraulic 
slope 5; the coefficient of safety, as 5, means that the conditions allowed 
are 1/5 the ultimate. "Coefficient" is synonymous with '"modulus" in 
many cases, as "Coef. of Elasticity" = "mod. of elas." 

Coffer-dam. — A water-tight enclosure built in a body of water in order to 
exclude the water and maintain a dry space for the construction of 
foundations, bridge piers, etc. Sheet piling is useful for this purpose. 

Cog. — ^The tooth, catch or projection, as of a cog-wheel. 

Cog^rail. — A rack or rail with cogs, placed between the rails of a track to 
engage the cogged driving-gear of the locomotive in drawing trains up 
inclined railways, too steep for ordinary traction. The rack is composed 
of cogs fastened between two angle-irons. 



CLA YING'BAR. COMB USTION. 1 49 3 

Coil, Choking. — A coil of wire so wotind on a core of iron as to possess high 
self-induction. Such coils are used to obstruct or cut off an alternating 
current with a loss of power less than with the use of mere ohmic 
resistance. 

Coil, Electric. — A convolution of insulated wire through which an electric 
current may be passed. 

Coil, Impedance. — A term sometimes applied to a choking-coil. (Self- 
induction produces impedance.) 

Coil, Induction. — An apparatus consisting of two parallel coils of insulated 
wire employed for the production of currents by mutual induction. 
It consists essentially of a primary coil, a secondary coil, and an iron 
core, usually laminated. The primary coil is wound around the core, 
and over that the secondary coil. The former is composed of thick wire, 
and the latter of thin wire. If a current is passed through the primary coil 
its voltage is raised in the secondary coil. 

Coil, Induction, Inverted. — An induction coil in which the primary coil is 
made of a long, thin wire, and the secondary coil of a short thick wire. 
Hence, a current passing through the primary coil induces a current of 
lower potential in the secondary coil. 

Coil, Magnet. — A coil ot insulated wire surrounding the core of an electro- 
magnet, and through which the magnetizing current is passed. 

Coil, Primary. — That coil or conductor of an induction coil or transformer, 
through which the rapidly interrupted or alternate inducing currents 
are sent. 

Coil, Secondary. — ^That coil or conductor of an induction coil or transformer, 
in which alternating currents are induced by the rapidly interrupted 
or alternating currents in the primary coil. 

Coil, Shunt. — A coil placed in a derived or shunt circuit. 

Coils, Armature, or Dynamo-Electric Machine. — The coils, strips or bars 
that are wound or placed on the armature core. The wire is as thick as 
possible consistent with the desired electromotive force without requiring 
excessive speed of rotation. The armature coils should enclose as many 
lines of force as possible. If rods or bars are used they should be lami- 
nated in planes parallel to the lines of force so as to avoid eddy currents. 
The coils of pole armatures should be wound near the poles rather than 
on the middle of the cores. Open-circuit coils or simply open coils are 
those which are independent of one another, either for a part or the 
whole revolution. Closed-circuit coils or simply closed coils are con- 
nected coils. In alternate current dynamos the separate coils that are 
used on the armature may be coupled either in sertes or in multiple-arc 
(multiple-arc or multiple circuit means a compound circuit in which all 
positive poles are joined to one end of a conductor, and all negative 
poles to the other end.) They are connected in series usually in alter- 
nate current machines of high electromotive force where the converter is 
at a considerable distance; and in parallel where low electromotive force 
is sufficient, as for incandescent lamps in multiple arc. 

Coke. — A useful product of coal, for the manufacture of iron. Coke is the 
"charcoal" of coal. 

Colatitude. — 90° minus the Latitude of the place. 

Cold^chisel. — A tempered steel chisel with a cutting edge for cutting metal. 

Cold=shot = cold=shut. — In foundry work, small particles of iron found in 
chilled parts of a casting. 

Collar. — A ring, or anything resembling a common collar; an enlarged part of 
a shaft, or the enlarged portion of a car-axle, etc. 

Collar-beam = wind=beam. — A timber beam stretching horizontally between 
two rafters (forming a letter A) in order to prevent sagging, etc. 

Collectors of Dynamo-Electric Machines.-^In a restricted sense collectors 
are brushes or points used to carry off the current generated in alter- 
nate-current machines, being distinguished from commutators which 
carry off the current generated in continuous-current machines. In 
other words, commutators change alternate currents, generated on ro- 
tation of the armature, to continuous currents, while collectors do not. 
Nevertheless, the name "collectors" is sometimes used to embrace 
"commutators". 

Column = pillar. — A vertical "shaft" set on a "base" and surmounted by 
a "capital." In classic architecture we have the Doric, Ionic ana 
Corinthian. 

Combustion. — A rapid oxidation of a combustible substance, caused by the 
chemical union with the oxygen of the air. 



1494 GLOSSARY, 

Commutator. — A device for changing the direction of an electric current. 

Commutator, Dynamo=EIectric Machines. — ^That part of a dynamo-electric 
machine which is designed to cause alternating ciurents, produced in 
the armature, to flow in one and the same direction in the external 
circuit; that is, to change alternate to continuous currents. 

Concrete. — The compact mass formed by an aggregate of broken stone or 
other coarse material mixed with a matrix of (hydraulic-cement) 
mortar. There are other kinds, as asphalt-concrete, etc. 

Concrete rubble=masonry. — Rubble masonry in which concrete is used 
instead of the usual cement mortar: for economy of cement. 

Condenser. — A device for accumulating or condensing, as electricity, water 
etc. The surface condenser in a steam-engine isone in which the ex- 
haust-steam passes through a large number of pipes immersed in cold 
water, constantly renewed. 

Conductivity. — The property which a body or substance has of conducting 
electricity, heat or sound. It vares with the temperature and the physi- 
cal stress (as tension) of the substance. In electricity, it varies inversely 
with the resistance, as C = Et-R; and is defined as the reciprocal of 
electric resistance, R. 

Conductor. — A substance which will permit the passage of an electric 
ciurrent. The term "conductor" is used in a relative sense, as we have no 
knowledge of any material that is absolutely a non-conductor. 

Conductor, Anti=Induction. — A conductor so constructed as to avoid exces- 
sive inductive effects from neighboring circuits. 

Conductor, Armored. — One provided with a covering of metal over the 
insulating covering for protection from abrasion. 

Conductor, Potential of. — The relation existing between the quantity of 
electricity in a conductor and its capacity. For a given quantity: the 
smaller the wire, the higher the potential. For a given conductor: 
the greater the' quantity the higher the potential. 

Cone=gear. — ^Two cones transmitting motion by rolling friction. 

Connecting=rod. — The "rod" which connects the crank of a body having 
a circular motion, in order to transmit motion or force to or from 
some other body, as the connecting-rod of a locomotive or of a beam- 
engine. 

Console. — A bracket or corbel with ornamental carvings or shape like an S; 
used for support of cornice, etc. A wall-bracket for supporting machinery. 

Contacts, Lamp. — Metallic plates or rings connected with the terminals of 
an incandescent lamp for ready connection with the line. 

Contour=line. — A line on a topographical map joining points of equal eleva- 
tion. A 10-ft. -contour map shows contour-lines marking siurface eleva- 
tions 10 ft. apart, vertically. 

Controller. — Any device used to regulate the flow of air, water, electricity, etc. 

Controlling=nozzle. — A device for regulating the size of stream issuing 
from a nozzle. 

Convective, Electric. — ^The air particles, or air streams, which are thrown 
off from the pointed ends of a charged, insulated conductor. These 
streams act magnetically, and are themselves acted on by magnets. 

Converter. — A vessel swung on an axis, lined with some refractory material, 
in which molten pig-iron is converted into Bessemer steel. 

Coping. — The top or finishing coiirse of a masonry wall, usually, projecting 
a few inches beyond the line of neat -work of the facing, and beveled 
for appearance and to shed water. 

Corbel. — A horizontal projecting piece, acting as a cantilever in assisting to 
support a beam or piece resting or partly resting upon it. In mill- 
building construction, the columns are often capped with corbels which 
project a short distance under the superimposed beams. Their utility 
is questioned by many engineers. 

Corduroy=road. — A road constructed of logs laid transversely and close 
together, usually across muddy or marshy ground. 

Core. — The inner portion or filling of a wall. The inner mold of a casting to 
make the hollow space, as of a pipe. The iron body of an electromagnet. 
The comparatively thin wall of masonry constructed in the heart of an 
earth dam to prevent leakage. 

Core, Armature, Filamentous. — An armature core, the iron of which con- 
sists of wire. 

Core, Armature, H. — An armattu-e core in the shape of the letter H, generally 
known as the shuttle- or the girder armature. We also have the I arma- 
ture. 



COMMUTATOR. CRANK, 1495 

Core, Armature, Lamination of, — The subdivision of the core of the arma- 
ture of a djmamo-electric machine into separate insulated plates or 
strips to prevent eddy currents. 

Core, Armature, of Dynamo=EIectric Machine. — ^The iron core, on, or around 
which, the armature coils of a dynamo-electric machine are wound or 
placed. It is laminated to prevent eddy currents. In drum, and in 
ring armatures the laminae are in the form of thin insulated discs or 
soft-iron plates; in pole armatures, in bundles of insulated wires. 

Core, Armature, Radially ^Laminated, — An armature core, the iron of which 
consists of thin iron washers. 

Core, Armature, Ribbed. — A cylindrical armature core provided with longi- 
tudinal projections or ribs that serve as spaced channels or grooves for 
the reception of the armature coils. 

Cornice. — An ornamental or molded projection at the top of a building- 
wall, masonry wall, etc. 

Corrugated. — Wrinkled in regular furrows, as corrugated iron. 

Cotter = cotter=bolt = cotter=key. — A wedge inserted to fasten or tighten. A 
split bolt whose ends are spaced apart when inserted as a key. 

Coulomb. — ^The unit of electrical quantity.^ That quantity of electricity 
that will pass in one second in a circuit whose resistance is one ohm, 
and electric-motive force one volt. 

Coulomb=Volt. — Volt-coulomb or Joule = 0.7373 foot-pound. 

Counterbrace. — In a frame, the brace which crosses the main brace and is 
designed to transmit the compressive stress in a panel due to negative 
shear, i. e., when the stress in the main brace would change from com- 
pression to tension. 

Counterfort. — A buttress, or portion projecting from the face of a wall to 
stiffen it. 

Counter=rod = counter. — In a frame, a rod which crosses the main diagonals 
and is designed to transmit the tensile stress in a panel due to negative 
shear, i. e., when the stress in the main diagonals would change from 
tension to compression. 

Counter=shaft. — A secondary shaft running parallel with and driven by a 
main shaft. 

Countersink. — A drill or bit for reaming or making a countersink ( = counter- 
sunk hole). A hole enlarged at the top to receive the countersunk head 
of a bolt or rivet, so as to be flush with face of plate. 

Counterweight. — A weight used to balance another; a counter-poise. 

Coupling. — A device for connecting two shafts so they will act as one in 
running. Generally, anything that connects, as couplings for cars. 

Coursing=joint. — A joint between two courses of masonry. 

Cover. — ^The cap-head of an upright steam cylinder. 

Crab. — A portable windlass for hoisting; used in building operation for 
hoisting bricks and mortar. A horizontal shaft with one or two cranks 
for turning by hand, and geared to a drum on which the hoisting rope 
winds. The whole thing set in a wooden or iron frame. 

Cradle. — A frame placed under the bottom of a ship, on the "ways," to give 
support in launching or on marine railways. Any contrivance having 
a "cradle" form for enclosing or supporting. 

Cramp = cramp=iron. — A piece of iron bent at the ends for holding together 
pieces of stone, timber, etc., in structures. 

Crane.— A machine for moving heavy weights and placing them in any 
desirable position; hence there must be provision for motion in three 
directions, as vertical, longitudinal and lateral. The two latter may be 
combined in a circular motion by a rotary crane, consisting of a jib 
or swinging arm rigidly attached to a vertical post and rotating together 
about the axis of the post. In a derrick-crane ^ the top of the post is 
held in position by guys or guy-ropes. A traveling crane is one in which 
the longitudinal motion is provided for by the whole crane traveling 
longitudinally on a track, as with the /ocomo^w^ era w^ and various forms 
in heavy machine shops; the lateral motion being obtained by the mova- 
ble hoisting carriage operating on the main transverse girder of the crane. 

Crank. — A bent arm attached to a shaft or axle and forming a radial leverage 
for turning. A single crank is used only at the end of an axis, as in 
the common grindstone. A double_crank is used in the middle of a 
shaft and has four bends, thus I I ; or, sometimes used to denote 

two double cranks for reciprocating motion, thus — ' i — . See also 

Bell-crank. 



1496 GLOSSARY. 

Crematory furnace* — A ftimace for burning garbage. 

Crest. — The top, as of a dam. 

Cross=cut. — In mining, a level driven across so as to connect two other 
levels. 

Cross=hair = cross=wire. — A very fine strand of spider's web or metal wire 
stretched across the diameter of a telescope to mark the direction of 
sight on a distant object. Used in transits and levels. Two together, 
crossing each other at an angle of 90°, form cross-hairs. Two or three 
arranged horizontally and parallel in a transit are used as stadia wires 
for measuring distances instead of by chaining. 

Cross=head. — The sliding bar at the end of the piston-rod of a steam engine. 

Cross=cut saw. — A large saw operated by a man at each end, and used for 
sawing logs or large timbers across the grain. Opposed to whip-saw 
which is used to saw (also by hand) with the grain, as planking from 
logs. 

Cross=section. — A section at right angle to the (longest) axis. 

Cross= valve. — A valve placed at the junction of two or more pipes. 

Crow = crowbar. — An iron bar with end pointed or sligthly bent, and used 
for prying, as a lever, etc. 

Crowfoot. — A mark used by surveyors when chaining, consisting of a 
central line, marking the exact distance, and two flaring lines, forming 
a sort of arrow-head. 

Crown=arch. — ^The arched plate which supports the crown-sheet of the 
fire-box of a boiler. 

Crown=bar. — One of the bars on which the crown-sheet rests. 

Crown=gate. — The head gate of a canal lock. 

Crown=gIass. — A good quality of common window-glass. 

Crown=saw. — A cylindrical saw with teeth on edge of cylinder. 

Crown=sheet. — ^The sheet forming the upper part of the fire-box of the 
furnace of a steam-boiler. 

Crown=tile = hip=tile = ridge=tile.— A bent or curved tile used at the crown 
as a finish for pan-tile or flat-tile roofs. 

Crown=wheel = contrate=wheel = face-wheel. — ^A wheel with teeth or cogs 
at right angles with its plane. 

Crucible. — A pot for melting metals, ores, etc. The hollow at the bottom 
of a chemical furnace for collecting the molten metal. 

Crucible steel. — Cast-steel. 

Crypt. — The part of a church or cathedral below the main floor. 

Cupola=furnace. — A furnace for remelting cast-iron. 

Cup=yalve. — A sort of semi-spherical valve, or balance-valve, over an 
opening to which it fits when valve is closed. 

Curb. — Anything used to curb or check. The outer casing of a turbine- 
wheel. The wall-plate at the bottom of a dome. The casing of masonry, 
wood or iron built inside a well that is being sunk. Stones or timber 
at edge of a well or of a street, etc. 

Current, Alternating. — A current which flows alternately in opposite direc- 
tions; that is, its direction is rapidly reversed. 

Current, Assumed Direction of Flow. — The direction the ctirrent is assumed 
to take, i. e., from the positive pole of the source through the circuit 
to the negative pole of the source. 

Current=breaker. — Any device for breaking the circuit of an electrical 
current. 

Current, Constant. — ^A current that continues to flow in the same direction 
for some time without varying in strength. 

Current, Continuous. — A current which flows in one and the same direction. 

Current, Direct. — A continuous current. 

Current, Electric. — ^The quantity of electricity which passes per second 
through any conductor or circuit; that is, the rate of flow. (See Ampere. 
Coulomb.) 

Current, Induced. — The current produced in a conductor by cutting lines 
of force. It results from differences of potential produced by electro- 
dynamic induction. 

Current=meter. — Any device for measuring the flow of water in streams. 

Current-=meter. — A form of galvanometer. 

Current, iVlulti=phase. — A rotating current. 

Current, Periodic. — A simple periodic current. 

Current, Rotating. — A term applied to a current which results by combining 
a number of alternating currents, whose phases are displaced with 
respect to one another. A rotating current is sometimes called a poly- 



CREMATORY FURNACE. DIE, 1497 

phase or multiple -phase current, particularly if there are more currents 
combined. When three currents are combined the displacement between 
each set of phases is 120 degrees. A rotary current, unlike an alternating 
current, possesses, in a certain sense, a definite direction of flow. Its 
effect on a magnetic needle is to cause rotation. 
Current strength. — The product obtained by dividing the electromotive 

force by the resistance. ^="B- (See Ampere.) 

Currents, Eddy. — Useless currents produced in the pole-pieces, armatures, 
field magnet cores of dynamo-electric machines or motors, or other 
metallic masses, either by their rnotion through magnetic fields, or by 
variation in the strength of electric currents fiowing near them. 

Current=wheel. — A wheel driven by the current of a stream. 

Cut-off. — A device for automatically cutting off the steam from the steam- 
chest to the cylinder before the piston has made its full stroke, the bal- 
ance of the stroke being made by the expansive force of the steam in the 
cylinder. A channel cut across a bend thereby shortening the main 
course of the river. 

Cutwater. — The up-stream angle-edge of a bridge pier, designed to more 
effectively lessen the impact of moving water, ice, logs, etc. 

Cyclopean Masonry. — Rubble concrete masonry. Massive concrete in 
which large rubble stone is added or filled in as the mass is built up; 
each piece of rubble stone must be thoroughly embedded in, and sur- 
rounded by, the concrete, 

Cyma = cyme = cima. — A cornice molding having the profile of an ogee, 
letter S, or curve of contra-flexure. 



a 

Dado. — ^The shaft of a pedestal, between cornice and base. 

Danip = fire=danip. — A gas in coal-mines which explodes when mixed with 
air and ignited ; very dangerous. Black-damp or choke-damp is a carbon- 
dioxide gas in collieries and differs from fire-damp but often found mixed 
with it. See Davy. 

Damper. — A metal plate, slide or door, used to regulate the draft of or to a 
stove or furnace, in order to control the rate of combustion. Many are 
regulated automatically, as by heat or by steam. 

Davy. — A safety-lamp for use in mines. 

Dead=head = sinking=head = sprue. — The extra length of metal in a gun- 
casting, not used because of its inferior quality. 

Dead load. — ^The "dead weight" or non-moving weight of a structure. Op- 
posed to live load and wind load, but may include snow load. The dead 
load should always be specified in detail. The empty or non-paying 
rolling-stock of a train. 

Dead=oil = heavy oil. — The oils obtained in the distillation of coal-tar 
above 340° F., and which are heavier than water. 

Dead=point. — ^That position of the crank of an engine when the engine is on 
its dead-center, i. e., when the crank and connecting-rod are in a straight 
line. 

Declination of a heavenly body is its angle north or south of the equator; 
i. e., it is its distance from the celestial equator measured on a great 
circle passing through the body and the pole. 

Declivity. — A downward slope of ground. 

Densimeter. — An apparatus for finding the specific gravity of a substance. 

Dentil = dentel. — One of a series of small blocks, uniformly spaced, in a 
cornice. 

Derrick. — A machine for lifting heavy weights, something similar to a 
crane but having the boom (corresponding to the jib of the crane) 
pivoted or hinged at its lower end (to the post). It is therefore more 
convenient than a crane, for use in general building operations. 
Floating derricks are large derricks erected on barges or vessels specially 
constructed. 

Diaphragm. — A thin plate, serving as a partition, placed across a small 
opening or hollow tube, as the diaphragm of a telephone. 

Die. — ^The cubical part of a pedestal between its cornice and base. An 
engraved steel for stamping a design. Pieces of hardened steel forming 
a female screw for cutting screw threads; they are fitted into a die- 



1498 GLOSSARY, 

stock and are adjustable for use in cutting threads of different dia- 
meters. 

Dike (formerly dyke). — See Dyke. 

Dip. — In geology, the angle which a stratum of rock makes with the hori- 
zontal. The point of dip is the direction of the compass to which the 
stratum inclines. The dip of a compass needle from a horizontal plane. 

Disk = disc = discus. — A flat circular plate. 

Disk=clutch. — A form of friction -clutch. 

Dock. — An inclosed water-space for vessels while handling cargo; a space 
or structure for loading or unloading cargo, for repairs, etc. See 
"Wharves, 'Piers and Docks," page 892. 

Dog = dog=iron.— An iron hook with one or more points at one end, to 
drive into timber for the purpose of moving it. Used largely in saw- 
mills. Has many forms and uses. A cramp (which see). 

Donkey=engine. — A small engine used for performing light work, as pumping 
water into boilers, hoisting anchors, handling building material, etc. 

Donkey=pump. — A feed-pump for boilers. An extra pump for special 
purposes. 

Doriner=window. — A vertical window in the face of a projection built out 
from a sloping roof. 

Doubling=frame. — A machine for winding double silk threads. 

Dovetail. — One of a series of wedge-like projections or tenons and of corres- 
ponding mortises in boards or timbers for fastening them together. 

Dowel. — A wooden or metallic pin inserted part way into two pieces of 
wood or stone to unite them. 

Down=draft. — A downward draft of air in a mine, chimney, etc. 

Draft. — The vertical depth of water which a vessel requires or "draws." 
The dressed edge of a stone. See Chisel-draft. 

Draw=plate. — A drilled plate of hard steel, or a drilled ruby or diamond, 
for drawing wire to reduce its diameter, make it uniform, or shape it. 
The holes are somewhat conical. The wire may be drawn successively 
through holes of decreasing diameter. 

Drift. — A nearly horizontal tunnel in a mine. Loose material, as timber, 
trees, etc., in a current. In geology, loose rocks, boulders, gravel, sand, 
etc., which have been deposited on bed rock; glacial drift if deposited by 
glacier. 

Drift = drift-pin. — ^A long, round, tapering pin of steel, used in enlarging 
the punched holes in metal plates, or smoothing the inner edges. 

Drift=bolt. — A steel bolt used in driving out other bolts. A round, steel pin 
for driving into auger-holes in timbers to fasten them together; bridge- 
stringers are thus drift-bolted to caps. When pointed it is called a 
pointed drift-bolt. 

Drip. — Any small tube or channel to lead water from a structure and let 
it fall to the ground; as a projecting member of a cornice, or a small 
channel cut under the edge of a coping. 

Drop. — One of a series of short cylinders or truncated cones placed in a row 
in cornices, as ornamental. 

Drum. — A revolving cylinder around which ropes are wound in hoisting. 

Dry=rot. — A rot in timber which has not been seasoned sufficiently. Thor- 
oughly seasoned timber will not rot if protected from dampness; or if 
treated with a preservative the decay will be slow, even in damp places. 

Ductility of a metal is that property which renders it capable of being ex- 
tended by drawing, as through a draw-plate, with lessening diameter, 
and without fracture. Gold is the most ductile; then silver, platinum, 
iron, copper, palladium, aluminum, zinc, tin, lead. 

Dyke (more modem spelling is dike). — A long bank of earth thrown up to 
prevent low lands from being overflowed. A levee. In geology, a 
fissure in rocks, filled with lava or other material while in a molten state. 

Dynamics, Electro. — That branch of electric science which treats of the 
action of electric ciurents on one another and on themselves or on 
magnets. 

Dynamo. — Dynamo-electric machine or generator. 

Dyne. — The unit of force in the centimeter-gram-second system. It is 
about 1.02 times the weight of a milligram. 

E. 

Eccentric. — A sort of crank device for converting a regular circle motion 
into an irregular reciprocating straight-line motion. Thus, in the 



DIKE. FELLOE. 1499 

steam engine, it consists of a circular disk rigidly attached to a shaft, 

but not at center of disk, and revolving around with it ; the circular disk 

being surrounded by a loose ring attached to the eccentric rod leading 

to the valve-gear of the cylinder, thereby regulating the cut-off and 

making the engine self-acting. 
Electrolysis. — Chemical decomposition effected by means of an electric 

current. Water- and gas-pipes are affected by electrolysis when forming 

the return circuit of electric distribution as in street railways. The 

electrolytic action occurs when the current jumps from the pipe, carrying 

atoms of metal along with it. 
Electrometer. — An apparatus for measuring differences of potential. 
Entablature. — In architecture, a sort of lintel construction supported on 

columns and extending toward the roof, and comprises the architrave, 

frieze, and cornice. 
Erg. — The unit of work or the work done when unit force is overcome 

through unit distan:e. A dyne-centimeter. 
Escarpment. — The abrupt face of natural rock or soil in a cliff or high ridge. 

In fortifications, ground cut away forming a nearly vertical slope about 

a position to render it inaccessible. 
Escutcheon. — ^The little plate for protecting the keyhole of a door; or the 

plate to which the handle is attached. 
Expansion=drum. — A drum with adjustable diameter, used in connection 

with driving an endless cable. 
Eye. — A circular hole in a plate, or formed by a loop of iron. The center 

hole of a wheel on a shaft. 
Eye=bolt. — A bolt with an eye or ring at one end. 
Eyepiece. — The lense or combination of lenses in an optical instrument to 

which the eye is applied. 



Face. — ^The front of an3rthing. The face of a valve is the part of the sur- 
face which comes in contact with the seat. 

Face=hammer. — One with a flat face. A hammer with a cutting and blunt 
end, used in preparing stone for finer tool-work. 

Face=lathe. — A lathe for turning face-work. 

Face=wheel. — See Crown-wheel. 

Fall = falUrope. — ^The fall of a tackle, or the rope used with pulleys in 
hoisting. "Fall and tackle" means "block and tackle." 

False=work. — A temporary structure to aid in the erection of the permanent 
one. 

Farad. — The practical unit of electric capacity. 

Farad, Micro. — The millionth part of a farad. 

Fascines. — Sticks or brush tied in bundles and used as a protection for 
river-banks; also used in the construction of sea-walls in connection 
with piling. Fascines are weighted with stone. 

Fatigue. — The weakness of metal, as a bar, produced by repeated applica- 
tion of stress well within the breaking load. 

Faucet. — A device in a pipe for regulating the flow of a liquid. The primi- 
tive form is a hollow plug in a cask, with a transverse hole near the 
outer end to be filled with a hollow plug when not in use. 

Feather. — A thin rib cast on iron-framing to give it strength. A rib cast on 
a shaft to fit a corresponding groove in the eye of a wheel. A small 
steel slip inserted in a shaft and projecting so as to fit the groove in 
the eye of a wheel. One of two pieces of metal placed in a hole in a 
stone, which is to be split by driving a "plug" or steel wedge between 
them. The stone is said to be split by "plug and feather." 

Feather=edge. — A very thin edge. 

Feather=joint. — A joint between boards consisting of a small strip, bead or 
feather fitting into the opposite mortises on the edges of the boards. 

Feeders. — In a system of distribution by constant potential, as in incan- 
descent electric lighting, the conducting wires extending between the 
bus-wires or bars, and the junction boxes. 

Feller = felling-machine. — A machine for cutting standing timber. 

Felling=saw. — The saw in a felling-machine (See Feller.) 

Felloe = felly. — The wooden rim of a cart wheel, into which the outer ends 
of the spokes are driven, and around the outer circumference of which 
the iron tire is fitted. 



1500 GLOSSARY, 

Felt. — A coarse fabric of hair, wool, or wool and fur, matted together by- 
moisture, heat or pressure, but not woven like cloth. 

Fender. — A bundle of rope or piece of timber hung over the side of a vessel 
to protect it from injury by rubbing against another vessel or wharf, 
etc. A guard post at the edge of a pier or wharf. 

Fender-pile. — One of a series of piles driven to protect a structure or work 
from injury by concussion resulting from moving bodies. 

Ferro. — Relating to some compound of which iron is a constituent element. 

Ferrule. — A metal ring around anything to prevent it from splitting or 
breaking. A bushing for expanding the end of a flue of a steam-boiler. 
Many things, in the nature of a ring for protection. A sleeve. 

Ferry-bridge. — The landing-stage of a ferry. 

Field, Alternating. — An electrostatic or magnetic field, the positive direction 
of the lines of force in which is alternately reversed or changed in 
direction. 

Filler. — Anything to fill a space or void, as a long narrow plate between the 
web-plate of a girder and its vertical angle-iron stift'ener, sometimes 
called filler-plate. Washers are often used as fillers. A separator, as 
one of the cast-iron spools near the ends of wooden bridge-stringers to 
space them one or two inches apart so the air can circulate and keep 
them seasoned. In painting, the prime coat for filling in between the 
fibers of the bare wood. 

Fillet. — A small flat molding, as in a cornice. 

Fire-damp. — ^The dangerous and explosive gas from coal in a mine. See 
Davy. 

Fish. — A long piece of timber or iron secured alongside of another to 
strengthen it; or at a joint to give stiffness, as one of two fish-plates to 
stiffen a rail- joint. 

Flag. — A broad flat stone (flagstone) used for paving. A flag-pole uged by 
surveyors. 

Flange — A projecting edge or rim, as the flange of a car-wheel, or of cast- 
iron flange-pipe (the flanges being bolted together when laid). 

Flap. — A heavy valve to prevent back tidewater into a sewer, etc. 

Flap-valve. — See Clack-valve. 

Flashing. — Sheets of lead, copper, zinc, tin, etc., used on roofs and other 
places, at the junction of roof and chimney, and at comers, to prevent 
the rain from leaking through. At a chimney, the upper edge of the 
sheet of metal is inserted into the joints of the brickwork, so the rain 
cannot get beneath the flashing, and the rest of the sheet is flattened 
down against the chimney and passes between two courses of shingles. 

Flask = molding-flask. — A wooden or iron mold used in foundries to hold the 
sand and patterns employed in molding and casting. May be in one or 
two parts, a lower and an upper. 

Flatting-coat. — The last of four or five coats of paint prepared so as to dry 
without gloss; it is of pure white lead diluted with spirits of turpentine. 

Flier. — One of several steps, called fliers, in a straight flight ot stairs. 
Opposed to winding stairs. 

Flint-glass. — Glass in which the silica is combined with oxid of lead in various 
proportions, and also containing potash. The lead gives it a higher 
specific gravity and refractive power, and greater brilliancy. 

Flood-gate. — A gate designed to open on the rising tide to allow water to 
fill a basin, and to close at the flood tide to prevent it from flowing out 
at that point. A gate designed to allow water to escape at floods. 
Various uses. 

Floor-hanger. — A bearing-bracket fastened to the floor and used for sup- 
porting shafts and countershafts. 

Flume. — An artificial channel for a stream of water; used in gold-mining, 
logging, irrigation, etc. 

Flush. — To drench plentifully with water, as flushing a sewer, gutter, etc.; 
having the idea of fullness. Even with the surface. 

Flush-box = flush-tank. — A rectangular box in a water-closet for flushing 
out the bowl. The outlet-valve is opened by pulling a cord attached 
to a lever. As the water in the box lowers, the inlet-valve, consisting 
of a ball-valve or ball-and-lever valve, opens allowing the box to fill; 
and when full the valve closes automatically through the rise of the 
ball float. 

Flux. — Any substance or mixture that will assist the welding or adhesion 
of two metals by preventing the formation of rust, which is very rapid 
at such times. 



FELT. FUSE, 1501 

Fly-wheel. — A wheel with a heavy rim placed on a revolving shaft of a 
machine for equalizing the motion of the machinery. 

Follower. — Any cog-wheel, or other part of a machine, which is driven or 
which follows the motion of another part called the leader. 

Foolscap. — A folded writing-paper 12x 15 to 13x 16 inches in size, making 
the sheet about 8 x 12^. • 

Foot=board = foot-plate. — The platform on which the engineer and fireman 
of a locomotive stand. 

Foot-poundal. — The unit of energy, equal to a foot-pound -^-g (= 32.2 ±) = 
421402 ergs. 

Force, Electromotive. — The pressure which tends to move electricity from 
one place to another. The unit of E. M. F. is the volt, V. 

Force, Lines of. — A term applied to the strength of a magnetic or electro- 
magnetic circuit. They reach between the opposite poles which produce 
them, and never intersect. They lie at right-angle with the direction of 
ether waves. 

Forceps. — Tongs, pincers or pliers for seizing and manipulating things 
which it would be impracticable to handle with the fingers. 

Forebay. — That part of a mill-race (channel where the water flows from the 
dam to the mill-wheel) where the water flows upon the wheel. The 
penstock. 

Forge. — An open furnace provided with a bellows for heating metal to be 
hammered or forged into shape. Portable forges are used for heating 
rivets in bridge-erection. A hearth or furnace for making malleable 
iron by the "direct process." A forging-machine is called a drop-press 
and operates with a hammer, by power. 

Forge-roll. — One of a series of rolls for rolling slabs or blooms into puddled 
bars. 

Forging-machine. — A machine for forging metal, usually heated. 

Foundry iron. — Iron containing sufficient carbon for casting. 

Four-way cock. — A cock or valve with four passages: two in the plug and 
four for delivery. 

Foxtail = fox-wedge. — A wedge inserted into the end of a pin or bolt so that 
when the latter is driven to the bottom of the hole the wedge will be 
forced into the pin, spreading the end of the pin and making it secure 
against withdrawal. 

Frame. — Any construction composed of parts fitted together and designed 
to support itself or other things. 

Friability. — The quality ot being friable, i. e., easily broken or crumbled. 

Friction. — The resistance to the relative motion of surfaces of bodies in 
contact: sliding friction if one body tends to slide on another; and 
rolling friction if the body is on wheels or rollers, as the rolling friction 
of a train is say 7 or 8 lbs. per ton.^ The angle of friction, called the 
angle of repose, is the angle of inclination (with the horizontal) of a sur- 
face at which a body will just tend to overcome the frictional resistance 
and begin to slide, by the force of gravity. The coefficient of friction is 
the tangent of the angle of repose. The friction of liquids is more or 
less associated with viscosity. 

Friction-balls. — Balls used to reduce friction of moving parts, as in bicycles 
and some movable bridges; such bearings are called ball-bearings. 

Friction-brake. — A brake acting by friction. 

Friction-gearing. — A gearing of wheels imparting motion one to another 
by the friction of contact alone. They can be thrown in and out of 
contact readily; when in gear or contact they are called friction -tight. 

Friction-rollers. — Cylinders used to reduce friction of moving parts, as the 
rollers of a movable land pile-driver. 

Friction-wheels. — Wheels especially designed to reduce friction of moving 
parts; or to provide for excess of stress in machinery, as dredging, 
by allowing the outer rim ot wheel to give way to all stress in excess of 
the frictional resistance on the inner section of the rim. 

Frieze. — In architecture, the decorative feature ot an entablature between 
the architrave and cornice; also similar decorative features elsewhere. 

Frustum. — That part next to the base when the top is cut off", as of a cone. 

Fulcrum. — The point of rest or support of a lever when lifting a body. The 
support itself. 

Furring. — Strips nailed on to a wall for subsequent lathing and plastering; 
or to the bottoms of joists and rafters to bring them to a level surface. 
The placing of said strips. 

Fuse = fuze. — A slow-burning tube-like or rope-like attachment to an 



1502 GLOSSARY. 

explosive charge. A time-fuse is one which will explode the charge in 
a certain time. Electric fuses (fired by a spark caused by a break in 
the electric circuit) are most frequently used for blasting, in work of 
magnitude. To fuse is to melt and blend together. 

Fusible=plug. — A plug of fusible metal placed in the shell of a boiler of a 
steam-engine, and intended to melt and allow the steam to escape 
when a dangerously high temperature is reached. 

Fusion=point. — ^The temperature at which a substance melts. 

Gable. — ^The vertical end of a triangular- or pitched roof; the triangular 
canopy over a window. The gable-end of a house is the end-wall. 

Gad. — A pointed steel bar or shorter tool for driving into anything and 
loosening it. ^ 

Qadding=machine = gadder. — In quarrying, a movable platform on which a 
steam-drill is mounted. 

Gage = gauge. — An instrument for determining the dimensions, quantity, 
distance, force, capacity, etc., of anything; or the measurement itself. 
The gage of a wire is its diameter; in shingling, it is the exposed length 
of the slate, tile, etc., below the lap; in gunnery, the bore of a gun. 

Qage=saw. — A saw with a gage-bar to determine the depth of kerf. 

Qage=stuf f = gaged=stuf f . — In plastering, a plaster of Paris mixture for 
quick-setting, in making moldings, etc. 

Gallon. — Four quarts. U. S. gallon = 231 cu. ins. = 3.7853 liters = capacity 
of a cylinder T dia. and 6" high; more accurately, such a cylinder has 
a capacity of 230.90706 cu. ins., or 0.9996 gallon, or almost exactly one 
part in 2310 too small. 

Qallows=frame. — The frame for supporting the beam of a beam-engine. 
The structure for supporting the pulleys and cage in a mine shaft. 

Gang=drill. — ^A machine containing a number of vertical drills in the same 
head. 

Gang=plank = gang-board. — A plank with cleats nailed on transversely for 
steps and used as an inclined stair. 

Gap=window. — A long narrow v/indow. 

Gas=conipressor. — A pump for compressing coal-gas into reservoirs for rail- 
road-cars, etc. 

Gasket. — Any fibrous or soft substance used for packing, in machinery. 
The circular collar used when pouring lead around lead-pipe joints. 

Gas=nieter. — An apparatus for measuring the flow of illuminating gas in 
pipes, etc. 

Gate. — A valve. When placed at the headworks of a water supply it is 
called a head-gate. 

Gate=house. — A small house in which the gate of a reservoir is situated and 
operated. 

Gauss. — The unit of intensity of magnetic field. 

Generator, Dynanio=Electric. — An apparatus for producing electricity by 
the mechanical movement of conductors through a magnetic field, 
cutting lines of force. 

Gear. — ^The connecting parts in machinery for transmitting motion. 

Gearing. — A train of toothed wheels, or worms, belts, ropes, etc., in ma- 
chinery. 

Generator, Motor. — A generator driven by electricity instead of by steam-, 
water-, or other power. 

Gib. — A wooden support under the roof of a coal-mine. An iron clasp used 
in connection with a key for clasping pieces together. The arm of a 
crane. A fixed wedge used with the driving wedge to hold together the 
brasses at the end of a connecting-rod of an engine. 

Gin. — One of the two main uprights of a pile-driver, between which the 
hammer operates. A machine for separating the seeds from cotton, 
also called a cotton-gin. A machine with a drum and winding rope, 
for various purposes as moving houses in streets, etc. 

Gin=block. — A tackle-block over which a rope runs, and suspended by a hook 
attached to it. 

Gin^tackle. — A double and a single block used as a system of pulleys for 
hoisting. 

Girder. — A simple or composite beam of larger dimensions than an ordinary 
beam. Thus, 6^am-girder, tort-girder, plate-girder, etc. 



FUSIBLE-PLUG. liALViNG. 1503 

Glacis. — In fortifications, a gentle slope over which the advancing enemy 
is brought into a direct line of fire. 

Gland = gIand=box.— A stuffing box. A joint tightly packed and capable 
of retaining lubricants for a period of time. 

Glaze. — A vitriable substance, as salt, applied to the surface of brick, tile, 
etc., and giving it a transparent coating. An enamel is an opaque 
coating. 

Glue. — A substance having cement properties; the common gelatin boiled 
out of the hides and hoofs of animals. 

Gooseneck. — A flexible coupling, or a pipe shaped like the letter 5. A 
nozzle with a universal joint similar to that used on the stand-pipe of 
a fire-engine. 

Governor. — An automatic regulator for controlling the supply of steam, 
water or gas. The compass-shaped two-ball apparatus on an engine; 
the supply- valve is connected to the levers which are operated by the 
radial movement of the balls, as the latter revolve faster or slower. 

Grapnel = grapple.- — One or more hooks in a cluster for grasping hold of 
things in deep water; a grappling-iron. 

Graving=dock. — A dry-dock for graving or cleaning the bottoms of ships. 

Grillage. — ^Two or more courses of heavy timbers laid parallel and at right 
angle (sometimes notched at their intersections) and usually drift- 
bolted together, to serve as a foundation resting on piles or on the bot- 
tom, and supporting a masonry pier or other structure. 

Grille. — A grating or open work of metal, usually of wrought-iron, for orna- 
mental work. 

Groin. — The intersection of simple vaults or arches crossing each other at 
right angle. A breakwater constructed across a beach to form a protec- 
■fion from the waves and prevent the drifting and washing of sand and 
mud. Sometimes spelled Groyne. 

Groove. — A long narrow channel as if made by a tool, for something to fit 
into. 

Ground=ice = anchor=ice. — Ice formed at the river bottom, prior to surface- 
freezing. 

Ground=swell. — A deep swell of the sea caused by a distant or late storm. 
The surface of a rolling country. 

Grout. — ^Thin mortor poured or forced in joints of masonry. 

Groyne. — See Groin. 

Grubbing=hoe. — A long-handled instrument for digging up or cutting roots; 
used in "grubbing." A mattock. 

Gudgeon. — ^The metal journal of a horizontal shaft, or that part which tui^s 
in the collar. 

Guide=bar. — One of the two parallel sides fitted on the cross-head of a steam- 
engine, on which the cross-head slides. 

Guncotton. — Cotton or other cellulose substances digested in a mixture of 
nitric and sulphuric acids, or in nitric acid alone. Explodes violently 
by percussion. 

Gun=nietal. — A bronze for making cannon; now supplanted by cast iron, 
and more frequently by steel. 

Gun=penduluni. — An apparatus for determining the strength of gunpowder. 

Gunwale. — The upper edge of a ship's side. 

Gusset = gusset=plate. — A triangular or trapezoidal plate riveted to box- 
girders to stiffen them transversely; or used to connect the ends of 
steel floor-beams with the web of plate-girders, the gusset extending 
usually the full height of the girder or nearly so in order to give trans- 
verse stiffness. In general: a large steel connecting-plate. 

Guy. — A rope, rod or chain fastened to anything to keep it from swinging, 
as the guy of a derrick. 

Gyrate. — To whirl or revolve about a point or axis. 

H. 

Hacking. — In masonry, the cutting up of large courses into smaller ones for 

expediency when the stones run small. 
Hack=iron. — A miners' pick or hack. A chisel for cutting nails. 
Half=trap. — A sinking bend in a sewer-pipe. 
Halving. — The notching of two timbers of equal thickness together, either 

crossing each other or at the ends, so the thickness of joint will be equal 

to that of one of the timbers. 



1504 CLOSSARY. 

Haminer<=beam. — A short beam projecting from the foot of a principal 
rafter, outward toward the center of the truss but not reaching a 
similar one from the opposite rafter: used in church roof-trusses. 

Hatnmer=dressed. — In stone-cutting, dressed with a pick or pointed hammer. 

Hand=lever. — In a steam-engine, the lever for starting, stopping or reversing 
the engine. 

Handscrew. — A jack, or machine for raising heavy weights. 

Handspike. — A wooden lever for raising weights or working a windlass or 
capstan. • 

Hand=wheel. — A form of circular crank, as the hand-wheel of a car-brake. 

Hanger. — A bracket from the ceiling or wall, or a stand from the floor, 
with a box and oiling device, for supporting a line of shafting. The 
plates, straps or yokes at the ends of floor beams, for suspending 
them to bridge-trusses. Yoke-hangers are used in connection with wood- 
en floor-beams and consist of square iron bent over the pin of lower 
truss, the ends passing down through the floor beam and an iron plate, 
using a nut and check-nut at each end of hanger. Plate-hangers are 
riveted directly to the ends of steel floor-beams, a hole being drilled in 
upper end for the truss-pin to pass through. A hanger-board is a board 
for supporting electric arc-lamps, making easy connection poles of 
lamp and line-circuit. 

Hasp. — A metal clasp as for a door, with a slot for folding it over a staple, 
and fastened by a pin or padlock. A metal hook for the same pur- 
pose. 

Hatch. — ^The opening in a ship's deck leading to the hold; usually termed 
hatchway. The cover of such an opening. 

Haunch. — The part of one side of an arch between the crown and springing. 

Headbay. — The water space just above a canal lock. 

Head=block. — ^The forward carriage for supporting logs being sawed in a 
mill. Any block for supporting a pillow-block. 

Header. — A stone or brick with its longest dimension at right angle to 
the face of the wall. 

Heading. — A small passage or opening or driftway excavated in advance in 
the line of a tunnel to facilitate the work. 

Head= valve. — The delivery valve in a steam-engine. 

Headway = headroom. — The clear height of space overhead. In railroading, 
the clear height above rail to the lower part of an overhead bridge or 
other structure. Advance or progress in work. 

Heart«=cam = heart=wheel. — A cam-wheel with a heart-shaped channel on 
face of disk in which a guide-wheel travels at the end of an arm, and used 
for converting rotary into reciprocal motion. 

Heart=shake. — Defects in timber, consisting in cracks or shakes extending 
from the center outward. 

Heat=unit. — (B.T.U.) The amount of heat required to raise 1 lb. of water 
through 1° Fahr. See page 1347. 

Hectrogam. — 100 grams, equal to 1543.235 grains. 

Hectoliter. — 100 liters, equal to 26.4 U. S. gallons. 

Hectometer. — 100 meters, equal to 328' V. 

Helve. — The handle of an ax, hatchet or adz. 

Helver. — ^The handle of a mining tool. 

Hematite. — One of the most valuable of iron ores; red oxid of iron, Fe2 O3. 
Found in large quantities in the Lake Superior region. 

Hemp. — ^The fibre of a plant; used for making hemp rope. 

Henry, A. — The practical unit of self-induction. 

Herring=bone bridging = bridging. — The diagonal pieces nailed between the 
floor-joists to give stiffness, by distributing the resistance to the floor 
loads over several joists. 

Highway. — A road or way of common right for all to pass. 

Hinge. — A device for joining two pieces in such a manner that one may be 
turned or swung around or upon the other, as the hinge of a door or of 
a trunk. A common hinge consists of two straps or leaves, joined by 
the pin or pintle passing through the knuckle. A rising hinge is one 
which rises when the door opens, to clear the carpet, and usually 
closing itself. A hutt hinge is a common door hinge where the leaves 
butt against each other. 

Hinge-pin. — The pin or pintle of a hinge. 

Hip. — ^The external angle or corner formed at the junction of two sloping 
roof faces, and supported by a hip-rafter . Opposed to valley^ which is 
the internal angle, and supported by a valley-rafter. 



HAMMER-BEAM. IDLE-WHEEL, 1505 

Hip-roof = hipped=roof. — A roof with four sloping faces, rising immediately 
from the wall-plates and with the same inclinations. 

Hip-tile. — The tile which saddles the hip of a roof. 

Hitch. — A kind of knot used in making one rope fast to another oir to a spar, 
boom, post or timber. See p. 668. 

Hoarding. — An English term for a fence for enclosing a building or mater- 
ials while building work is in progress; a board fence used as an en- 
closure. 

Hold=beam. — One of the transverse beams in the lowest tier of beams in a 
ship's hold. 

Holding=plate = anchor=plate. — ^The plate at the anchorage of a cable or 
guy, through which the cable passes or is secured; the plates being 
backed with masonry, or loose rocks, earth, etc. 

Hook. — A bent iron for holding a link, or suspending anything. A pulley- 
suspension hook is an S-hook which can be hung over a beam, and sup- 
port a pulley from below. 

Hook=block. — A pulley-block fitted with a hook for suspending it, or 
weights to it. The standing pan of the hook is that part attached to 
the block. 

Hook=bolt. — A bolt with one end in the form of a hook; used in fastening 
the wooden floor of a bridge to the iron stringers. 

Hoop. — A circular band or clasp. Hoops around wooden tanks are often 
adjustable. 

Horizon. — The astronomical or celestial horizon is the great circle of the 
celestial sphere whose plane is perpendicular to gravity at any station. 

Horse. — A wooden frame with four legs for supporting staging; many 
similar things. 

Horse=shoveI. — A road scraper. 

Horsing=iron = iron. — A long-handled calking-iron held by one man and 
driven by another. 

Hour=circle. — A circle perpendicular to a north and south-axis, and graduat- 
ed un-clockwise into 24 radial divisions. Any great circle which 
passes through the two poles. See Hour Angle, page 202. 

Hour, KiIo=Watt. — A unit of electrical power equal to a kilo-watt main- 
tained for one hour. 

Hour, Watt. — A unit of electrical work — one watt for one hour. 

Housing. — A niche in a wall for a statue. The jaw of a frame which holds the 
journal-box or housing-box. 

Housing=franie. — The frame which holds the rollers, in a rolling-mill. 

Kub. — The center of a wagon-wheel from which the spokes radiate and 
through which the axle-tree passes; or, in car-wheels, the metal part in 
the center to which the paper web is clamped. The bell-end of a pipe. 

Hydrant = fire=plug. — An apparatus with a valve and with hose-connec- 
tions for drawing water from a main. 

Hydraulic balance. — A water-wheel regulator. 

Hydraulic jack. — A jack operated by a plunger or piston against some 
liquid as oil. 

Hydraulic main. — In gas-works, a large pipe containing water into which 
the raw gas is brought, and serving as a purifier and to convey the 
crude gas to the condenser. 

Hydrocarbon. — A compound of hydrogen and carbon, alone. 

Hydrometer. — An instrument for determining the specific gravity of fluids. 

Hygrometer. — An instrument for determining the humidity of the at- 
mosphere. 

Hygroscope. — An instrument for determining the approximate humidity 
of the atmosphere. 

Hysteresis. — Molecular friction due to magnetic change of stress. 



I. 

Ice-breaker.-^A structure built in the water to protect bridge-piers from 
moving ice. 

Ice°machine. — A machine for producing ice. Anhydrous ammonia is the 
solution most used, and is most efficient. 

Idle-wheel. — In toothed gearing, a wheel placed between two others to 
preserve the same direction of motion in both of them. In rope trans- 
mission of power, a wheel to make the cable sag and preserve its tension. 



1506 GLOSSARY. 

Impedance. — Opposition to current flow. 



Impedance = V(Ohmic resistance)^ + (inductance resistance)^. 

Impost. — The upper part of a wall or column from which an arch springs. 

Inch. — 2.54 centimeters. One meter =39.37 inches. 

Inductance. — The induction of a circuit on itself, or on other circuits. Self- 
induction. The practical unit is the henry. The coefficient of inductance 
is a constant quantity which, multiplied by the current strength passing 
in any coil or circuit will give the induction due to that current. The 
practical unit of inductance is 1,000,000,000 centimeters. (See Ohm.) 

Induction, EIectro=Dynamic. — Electromotive forces set up by induction in 
conductors which are either actually or practically moved so as to cut 
the lines of magnetic force. Flemings' rule: 



^1 

II 



o'^f 



7 '^.o Direction of 






^° Lineof 
^ Motion 



W 



Induction»pipe. — In a steam-engine, the pipe through which the live steam 
passes to the steam-chest. The induction-port is the opening from the 
steam-chest into the cylinder. The induction-valve is the valve controll- 
ing the steam into the cylinder. 

Indurate. — ^To harden, as with indurated clay. 

Infusorial earth. — Fine white earth composed of minute silicioas shells 
and resembling magnesia. Used as an absorbent in making djniamite 
(with nitroglycerin). 

Ingot. — A cast of metal from a mold (ingot-mold), as pig iron. 

Injector. — An apparatus for forcing water into a steam-boiler. 

Inscribe. — In geometry, to draw within, as a square within a circle. Opposed 
to circumscribe. 

Insulator, Oil. — A fluid insulator filled with oil. 

Insulator, Single=Shed. — An insulator with a single inverted cup. 

Insulator, Telegraphic or Telephonic. — A non-conducting support of tele- 
graphic, telephonic, electric light or other wires. Insulators are generally 
made of glass, porcelain or hard rubber, and assume a variety of forms. 

Interlocking system of signals. — In railroading, a system of operating 
switches and signals jointly by means of locking mechanism, operated 
from a central station, so trainmen can tell the position of the switches 
from a distance. 

Interpolate. — To find the missing number of a series. Bi many mathemati- 
cal tables it is desirable to find (usually by simple proportion, but 
not always) intermediate values to those given, and this is done by 
interpolation. 

Intrados. — The inner line of an arch; the outer line is called the exirados. 

Invert. — An inverted arch, as the floor of the lock-chamber of a canal, 
or the lower part of the brick sewer, or the inverted arches used in the 
foundation walls of buildings in order to distribute the pressure more 
uniformly. 

Ion. — One of the elements of an electrolyte; anions are evolved at the 
anode, and cations at the cathode. 

Isobar. — A contour-like line on a map, connecting places at which the 
barometric pressure is the same, 

Isochime = isocheim. — A contour-like line on a map, connecting places hav- 
ing the same mean winter temperature. 

Isoclinal. — In geology, strata having the same inclination or dip. 

Isoclinal lines. — In magnetism, contour-like lines on a map, through points 
at which the dip of the needle is the same. 



IMPEDANCE. JOURNAL-BOX. 1507 

Isometric. — In crystallography, that system which is characterized by 
three equal axes at right angles, and includes the cube. 

Isotherm. — A contour-like line on a niap, connecting points having the same 
mean temperature. 

J. 

Jack. — ^An instrument for raising weights: "jacking them up". A screw- 
jack consists of a screw or worm working in a thread; a lever-jack is 
a kind of a ratched-jack. Various forms. See Hydraulic jack. 

Jack=engine. — A donkey-engine. 

Jack=rafter. — A short rafter, usual in hip-roofs. A sort of sub-rafter or 
secondary rafter, parallel with the main rafters and supported on the 
main purlins; used for supporting directly the sub-purlins or sheathing 
of the roof. 

Jack=rib. — A rib in a framed arch or dome shorter than the others. 

Jack=timber. — A framing timber shorter than the others as in the floor of 
a bay. 

Jad. — In quarrying, a long deep gash or hole made in quarrying soft rock, 
as a sort of heading for wedging or blasting the balance. In coal-mining, 
a "holing" or "benching" so the mass of coal may fall or be loosened by 
wedging or blasting. 

Jadding=pick. — A sort of pick for cutting a jad in a quarry or coal-mine. 

Jag=bolt. — A bolt with a barbed shank. 

Jamb. — ^The vertical side of an opening or recess in a wall, as of a door or 
window, serving to support in part the weight above, as a lintel. A 
door-jamb, or window-jamb, or fireplace-jamb. 

Jamb=post. — The upright post or timber at the side of an opening or jamb. 

Jam=nut. — A sort of lock nut, or a nut screwed down on a bolt hard against 
another nut to prevent the latter from working loose. Used under 
wooden floor-beams of bridges when suspended by iron hangers. 

Jaw. — Anything in the shape or use of a common jaw, as the jaws of a vice, 
or wrench, or stone-crusher. 

Jaw=bit. — A bar under a journal-box for uniting the two pedestals in a car- 
truck. 

Jaw«boIt. — A bolt with a U-shaped head perforated to carry a pin. 

Jetty. — A sort of pier or arm constructed in the water to divert the current 
and protect banks from washing away, or to scour out the channel, 
or to cause slack-water and deposit of mud in any place. A pier in the 
ordinary sense of a wharf. 

Jews'=harp. — A shackle, or partly-closed link in the shape of a horse-shoe 
and with eyes and bolt, for connecting the ring of an anchor with the 
chain or cable. 

Jib. — ^The projecting arm of a crane. 

Jig^pin. — In mining, a pin to prevent the turn-beams from turning. 

Jig=saw. — A vertical reciprocating saw with a narrow blade for sawing 
scroll-work in boards. 

Jimmy. — A short crow-bar. 

Joggle. — A sub-tenon at the end of a framed timber to prevent it from 
moving laterally. A notch or mortise in a piece of stone or timber, or a 
key engaging such a notch, to prevent a corresponding piece or counter- 
part from movement. 

Joggle=piece. — A piece like the king-post of a truss. 

Jogglework. — In masonry, stones internotched or keyed together, as in 
light-house construction. 

Joggling=tab|e. — A table or machine for dressing or concentrating ore. 

Joist. — One of the spaced beams supporting the boards of a floor, as floor- 
joist; or a ceiling, as ceiling- joist; or the floor of a bridge, as bridge- 
joists; etc. Where beams act singly or independently they are called 
girders, especially if larger than common joists. 

Joule. — The unit of electric energy or work. A volt-coulomb. One joule 
= 0.7373 foot-pound. One joule per second = 1 watt. 

Journal. — The part of an axle or shaft which rests in the bearings. 

JournaUbearing. — The bearing-support of an axle or shaft. In general, 
it consists of the brasses, resting in the pillow-block and inclosed in 
the journal-box. 

Journal-box = housing-box. — The box (cast-iron) which contains the journal 
(of the car-axle or shaft), the journal-bearing and key, and the oil- 
packing for lubricating the journal. 



1508 GLOSSARY. 

JournaI=brass. — The bearing of the journal. (Metals of different kinds 
while in rubbing contact will not wear as rapidly as those of the same 
kind.) 

Jump. — A step in a masonry course to accommodate a rise or fall in ground 
level, or slope. Used in buildings. 

Jump=coupling = thinible=coupIing= ring=coupling. — A coupling with a coup- 
ling-box consisting of a ring or thimble over the two connected ends of 
the shaft, the connection being made by pins through thimble and shaft, 
or by parallel keys feather bedded. 

Jumper. — A drill used for drilling holes in stone: a short drill worked by 
a hammer; a long drill, weighted, raised by hand and let fall, and not 
^ worked by a hammer. 

Junction-box. — A chamber connecting lines of pipes or wires. 

Jute. — A plant producing jute-fiber; swells with moisture and is inferior 
for rope. 

K. 

Kaolin. — A fine variety of clay or decomposed feldspar. 

Kedge. — A small anchor with an iron stock. 

Keeper. — A key which may be inserted in a stationary or sliding bolt or 

other piece, to keep it in place. An armature of a magnet, or a piece of 

soft iron across the poles of a magnet when not in use to maintain (or 

increase) the power. 
Kerf. — A channel or cut made as by a saw. A saw-kerf. 
Key. — Anything that locks or holds fast. 

Key-bed = key=seat. — A groove for a key for locking, as a wheel to a shaft. 
Key=boIt = cotter=bolt. — A bolt with a key or cotter, instead of a nut. 
Keystone. — The center stone of an arch ring, at the apex or crown. 
Kibble. — ^The bucket of a shaft or mine for hoisting material; the hoisting 

chain is called the kibble-chain. 
Kiln. — A ftimace or large oven for baking, burning or drying, as a brick kiln 

for burning or baking brick. 
Kiln»dried. — Anything, as timber, deprived of moistiure by treatment in a 

kiln or furnace. 
Kilogram. — 1,000 grams in weight, equal to 2.20462 lbs. 
Kilogrammeter. — A unit of work, equal to 7.233 ft. -lbs. 
Kiloliter. — A unit of capacity, equal to 1,000 liters. 
Kilometer. — 1000 meters. 
Kilowatt. — 1000 watts. 

Kinetic energy. — Energy in some form of motion. 
King=post = king=piece. — The vertical post in a truss with two sloping 

chords meeting the top of post and framed into it. The two chords may 

be rafters. The middle vertical member of a king-post truss. 
King=rod. — A rod used in place of a king-post, in a king-truss. 
King=truss = king=post truss. — A truss framed with a king-post. 
Knee. — A piece of wood or metal having an angle and used to join two 

pieces together, giving support and stiffness; as the beam of a ship to a 

side timber. 
Knee-strap. — An iron strap used in connection with a knee-timber. 
Knot. — A fastening with a rope. See page 668. 
Knuckle-joint. — A flexible joint, as with two adjoining links. 
K. W. — A contraction for kilo-watt. 

L. 

Laboratory, — A place with suitable apparatus for conducting investiga- 
tions or experiments. 

Labyrinth. — A maze, or combination of passages making exit difficult. 

Lacquer = lacker. — An opaque varnish containing lac. 

Ladder-dredge. — A dredge with buckets carried on a ladder-like chain. 

Lagging. — Falling behind. Narrow strips of wood or planking placed out- 
side of and between the ribs of an arch or tunnel to give support to ex- 
traneous material. Used in tunnel construction. The outer wooden cas- 
ing of boilers, or the wooden strips placed on the periphery of a winding 
drum. Also used in the sense of sheeting. 

Lag-screw. — An iron bolt with a flat square or hexagonal head and with 
the other end sharp and threaded like a wood-screw; used in fastening 
small wooden guard-rails to the ties. 



JOURNAL-BRASS. LINK, 1609 

Lamp, Arc— An electric lamp, the source of whose light is the voltaic arc, 
formed between two or more carbon electrodes. 

Lamp, Incandescent. — An electric lamp in which the light is produced by 
the electric incandescence of a strip or filament of some refractory 
substance, generally carbon. 

Lancet=window. — A long narrow window crowned with an acutely-pointed 
arch. 

Landing. — A resting-place or platform at the end of a flight of stairs, or in- 
terrupting a series of steps. A place on shore for discharging passengers 
or freight from water-craft, usually called 'landing-place. 

Landlocked. — Protected from the wind and waves, as a small body of water 
nearly shut in by land. 

Landmark. — A prominent object locating a line or comer of the boundary 
of a tract of land. 

Lantern=wheel = lantern-pinion == trundle-wheel. — A sort of drum -like wheel 
with two parallel heads joined near their peripheries by parallel rods 
or spindles and so spaced as to engage the cogs of a spur-wheel. 

Lap. — ^The length or width of exposed surface partly covered by another, 
as the lap of shingle or slate in roofing. 

Lap-joint. — Opposed to butt-joint. A joint formed by the leaves overlap- 
ping as in the chord of a Howe-truss bridge. If each piece is composed 
of only one leaf, the ends are halved to form the lap-joint. 

Lap-weld. — Opposed to butt-weld. A weld made by lapping two metals 
before hammering. 

Larboard. — In navigation, the left-hand or port side. Opposed to starboard 
or right-hand side. 

Latch. — A sort of self-locking device which may be disengaged usually 
without the use of a key. 

Lattice=girder. — A girder with web consisting of diagonal pieces crossing 
like latticework. 

Lead (pronounced leed) . — Opposed to lag. In earthwork, the distance from 
c. of g. of cut to c. of g. of fill. The pay-lead may be less than this dis- 
tance. In steam-engines, the advance of a valve or valves so that the 
steam is admitted in front of the piston, or allowed to escape behind 
it, before the end of the stroke. 

Leader. — In mining, the leading vein. A pipe leading from the roof to con- 
duct rain-water. The principal wheel in a mechanism, and giving 
motion to the follower. 

Leading-block. — A block for simply keeping a rope in a certain lateral 
position, without transmitting any of its power. 

Leading-wheel. — In a locomotive, one of the smaller wheels ahead of the 
driving-wheels. 

Leaf. — A tooth of a small pinion. 

Leaf-bridge. — A small draw with leaves swinging vertically. 

Leaf=valve = flap-valve = clack-valve. — A hinged or pivoted valve in a 
pumping engine. 

Lean=to. — A roof or building whose rafters pitch against another structure. 

Ledge. — A shelf or something projecting, as a small horizontal molding, or 
the side of a rebate against which a window or door is stopped. 

Lewis. — A contrivance for securing a hold in a vertical wedge-shaped hole 
in a block of stone, for hoisting it; consists of two side pieces of metal 
wedging a center piece (called lewis-bolt) firmly and to which the hoist- 
ing tackle is attached. 

Lewis-bolt. — A wedge-shaped bolt fastened in a hole drilled in a stone, 
by pouring lead around it or by inserting metal wedges. If used for 
lifting the stones it is provided with an eye and is an eye-bolt. 

Lewis=hole. — The hole drilled in a stone for a lewis. 

Lift. — ^The rise in a canal-lock. The vertical distance from one level to 
another in a mine, or a set of pumps. A machine for lifting. 

Lift-bridge. — A bridge which is raised to accommodate cross-traffic, as in 
a canal. 

Lift-pump. — Opposed to force-pump. 

Lift-wall. — ^The cross-wall of a canal-lock chamber. 

Lighter. — A water-craft or barge used for unloading or loading cargoes of 
vessels while anchored in the harbor. 

Light-ship. — A vessel at anchor used as a sort of light-house. 

Linch'^hoop. — A ring on a carriage-axle fastened by a linch-pin. 

Linch-pin. — A pin on the end of a carriage-axle to hold the wheel on. 

Link. — One of the rings or separate pieces of a chain. In surveying, the 



1510 GLOSSARY. 

hundredth part of a chain; equal to 12 inches in an engineer's chain, 
or 7.92 inches in a surveyor's or Gunter's chain. In a steam-engine, 
the Hnks or parts forming the Hnk-motion. 

Link=lever. — In a steam-engine, the reversing lever controlling the link of 
the link-motion valve-gear. 

Link<°motion. — In a steafti-engine, a system of levers for controlling the 
valves in starting and reversing the engine, and for cut-off. 

Lintel. — A horizontal beam for supporting a wall over a door or window or 
other similar opening of moderate span; termed a breast-summer when 
the opening is large. 

List. — In architecture, a square molding, fillet or listel. A narrow strip 
from the edge of a board. The first or thin coat of tin on iron plates, 
to be followed by a heavier coat. The tipping of a vessel due to unequal 
loading. 

Liter. — A unit of capacity, equal to 1.056 U. S. quarts. 

Litharge. — A protoxid of lead (Pb O) , used in the composition of flint-glass, 
varnishes and drying-oils. 

Load. — In mechanics, the pressure upon any part of a structure. 

Load=line. — The line around a vessel to show the allowable load; when she 
sinks in the water to that mark. Also, see page 313. 

Loam. — In foundry-work, a mixture of clay, sand, sawdust, etc., used in 
making the molds for castings of iron and brass. 

Lock. — A device for fastening doors, gates, etc.; composed of a bolt, wards 
(for guarding against the entrance of a key not of the right pattern), 
tumbler (to hold the bolt in position and render the operation of a 
wrong key difficult), and a spring. A mortise-lock is one concealed, as 
in a house-door. An enclosure or chamber in a canal with gates at each 
end, for allowing boats to pass from one level to another. 

Lock=nut = check=nut=jam=nut = pinch=nut. — A nut screwed down on 
another to keep it in place. 

Log=beam. — A traveling frame for supporting and feeding logs to the saw, 
in a saw-mill. 

Log=scale» — A table showing the quantity of lumber, in B. M., procurable 
from a log when sawed; the length of log and its diam. beneath the 
bark being given. 

Louver = louvre. — A long window, usually at tops of roofs of shops, depots, 
etc., with the opening traversed with broad slats sloping downward and 
outward, like the slats of a window-blind, to provide for ventilation and 
exclude rain. 

Lozenge. — A plane figure shaped like a rhomb or diamond, having foiu- 
equal sides with two acute and two obtuse angles. In art, the lozenge 
pattern is a pattern with diamond-shaped figures, or lines meeting or 
crossing each other at regular intervals but not at right angle. 

Lug. — A short flange or projecting piece on anything, as in a casting, by or 
to which something is fastened or supported or kept rigid, etc. 

Lug=bolt = strap=boIt. — A bolt terminating in a long flat extension or bar 
which takes the place of a head ; made by welding a flat bar to a common 
bolt. The bar often contains holes for bolts or screws to fasten to timber. 
There are various forms. 

Lumber=car. — A railroad car for carrying lumber; usually 34 ft. long. 

Luniber=kiln. — An artificially warmed chamber in which lumber is placed 
to deprive it of its moisture. The heat is often furnished by coils of 
steam-pipes, and the moisture in the air, from the wood, is condensed, 
as on cold-water pipes hung in the room, and the drippings conducted 
out of the chamber. The green lumber may be run into a kiln on cars 
and remain on them until dried. 

Machine=bolt. — A threaded bolt with a square or hexagonal head. 

Machine=too5 = engine=tool. — A machine operated by power (water or steam, 
etc.) for performing operations which may be, or which formerly were, 
accomplished by the use of hand-tools, as drilling, planing, etc. 

Magnet, Electro. — A magnet produced by a pass- 
age of an electric current through a coil of 
insulated wire surrounding a core of magne- 
tizable material. The directions of the currents 
required to produce N and S poles, respective- 
ly, are shown in the accompanying illustra- 
tions. A magnetizing coil is called a helix or 
solenoid. 

Main. — ^The chief pipe-line in a system of waterworks; any pipe-line in a 




UNK'LEVER. MEANDER. 1511 

distributing system tapped for domestic supply. Water-main. Simi- 
larly with gas, as gas-main. 

Main«link. — ^The bar that connects the piston-rod with the beam of the 
engine. 

Malleability. — ^The property of being malleable, i. e., of being shaped, by 
hammering or rolling, without fracture, as malleable brass or iron. 

Mallet. — ^A wooden hammer or small beetle used by stonecutters, carpenters, 
etc. 

Mandrel. — A bar or spindle inserted in any work to hold it or shape it, as 
in a lathe. The spindle of a turning lathe; the arbor or axis of any 
tool, as a circular saw or cutter; a rod for shaping the inside or hollow 
of anything, as the plug-core of a metal casting, 

Mandrel=collar. — A collar formed on the mandrel of a lathe, against which 
the chucks abut. 

Mandrel=lathe. — A lathe for turning long hollow work; the material is 
clasped by a chuck on the end of the mandrel. 

Manhole. — An opening in a sewer, culvert, drain, cesspool, steam-boiler, 
tank, etc., through which a man may enter for the purpose of inspecting, 
repairing, cleaning, etc. 

Marble=saw. — One or more thin iron blades, set in a frame and reciprocated 
on a block of marble to be sawed, the kerfs being fed with sand and 
water. 

Mark. — A German coin of the value of $0,238. 

Marking=gage. — A small graduated rod with a steel point on one end for 
scratching a line on the wood, and with an adjustable block for gaging 
the line from the edge of the board. 

Marl=brick = marl=stock, — A superior brick for the fronts of buildings and 
for arches. 

Marlinespike. — A pointed iron tool used by riggers to separate the strands 
of rope in splicing. 

Marsh=>gas. — Carburetted hydrogen; a constituent of fire-damp. 

Masonry. — Generally defined as anything constructed of the materials used 
by masons, as stonework, brickwork, tilework, etc.; but now it is con- 
sidered as including concrete, cyclopean masonry (which see) or rubble- 
concrete masonry, and reinforced-concrete masonry. Dry masonry 
comprises masonry built without mortar. See subject of Masonry, 
Section 25. 

Mass. — ^The weight of a body divided by the gravity acceleration at the 
place where the weight is measured. 

Mass=center = center of mass. — A point through which if a plane is passed 
in any direction, the sum of the products of all the minute masses or 
particles on one side of the plane each multiplied by its respective dis- 
tance from that plane, will be equal to the sum of similar products of 
particles and distances on the other side of the plane. 

Mass, Magnetic. — A quantity of magnetism which at unit distance produces 
an action equal to Unit force. 

Master=wheel. — The chief wheel or the driving-wheel of a mechanism or 
machine. 

Mat=boat = matting=boat. — A sort of framework on scows for making and 
launching mats to protect river-banks from scour. 

Match-board. — A board with a tongue on one edge and a groove on the other 
for constructing partitions, floors, etc., of match-boarding or matched- 
boarding. 

Match=plane. — One or two planes for preparing the edges of matched- 
boards. 

Mathook. — A long pole with an iron hook used in the construction of mats 
for river-bank protection. 

Mattock. — A kind of pick for digging, but having the edges broad instead 
of pointed. 

Maul. — A heavy wooden hammer. 

Maximum. — ^The greatest value or upper limit. The greatest of ^ several 
maxima is termed the absolute maximum, or maximum maximorum. 
Opposed to minimum. 

Mean. — An intermediate value in a series, from which it is derived. Arith- 
metical mean is the sum of n quantities divided by n. Geometrical mean 
is the square root of the product of two numbers. Mean error is the 
quadratic mean of the errors of observation. 

Meander. — A series of transit or compass lines in a survey, with distances, 
angles or bearings, and perhaps levels. 



1512 GLOSSARY, 

Meander«line. — A part or the whole of a meander. 

Mean proportional. — Geometrical-mean. See Mean. 

Megohm. — A large measure of electrical resistance: one million ohmss. 

Melt, — The charge of metal in a cupola or pot, for melting or after being 
melted. 

Melting=pot. — A pot for melting. A crucible. 

Member. — A subordinate part of a structure, as a post or a diagonal of a 
truss. 

Mercury =furnace. — A furnace for roasting cinnabar so the mercurial fumes 
will arise, to be condensed in a series of vessels. 

Meridian. — Noon. A north and south line, on the earth, or on the celestial 
sphere. 

Meter. — A unit of length; 39.37 ins. by U. S. law. 

Meter, Watt. — An instrument for measuring current flow in watts (volt- 
amperes). 

Mica. — A mineral substance employed as an insulator, as for instance of 
commutator bars. 

Micro. — ^The one-millionth. 

Micrometer. — An instrument for measuring exceedingly small lengths and 
angles. 

Mil. — The unit of length equal to the t^^otj of an inch, or .001 inch, used in 
measuring the diameter of wires. 

Mil, Circular. — A unit of area employed in measuring the areas of cross- 
sections of wires; equal to 0.7854 square mil. One circular mil = 
.000000785 square inch. 

Mill=f urnace. — A furnace for re-heating metals to be rolled into shapes or 
welded under the hammer. 

Milligram. — One thousandth of a gram, equal to about 1/65 of a grain. 

Milliliter. — One thousandth of a liter, equal to about 0.061 cu. in. 

Millimeter. — One thousandth of a meter, equal to 0.03937 inch. 

Mil, Square. — A unit of area employed in measuring the areas of cross- 
sections of wires; equal to . 001 X. 001 = .000001 square inch. One 
square mil = 1.2732 circular mil. 

Miner's Inch. — See page 1313. 

Minimum. — ^The smallest or least quantity. PI., minima. 

Miter = mitre = miter=joint. — A joint whose line makes an angle of 45° with 
each of the two pieces joined together at right angle. A bevel-joint is 
one whose line makes any angle greater or less than 45° with two pieces 
joined together at any angle. 

Miter=sill. — A raised sill against which the bottom of the canal-lock gates 
shut on the floor of the lock -bay. 

Miter=wheel. — One of two wheels of a mechanism whose teeth engage, the 
planes of the wheels being at right-angle with each other. 

Modulus = coefficient. — A constant or positive number used as a measure of 
some function, as the modulus- or coefficient of elasticity of a material, 
the modulus depending upon the material and the method of testing, 
etc. 

Moment. — The product of a force into its shortest leverage distance. See 
page 305. 

Monkey = ram. — ^The hammer of a pile driver. 

Monkey=engine. — A pile-driver. 

Monkey=wrench. — A common wrench, with one jaw adjustable by a screw, 
for screwing on nuts, etc. 

Monument. — An artificial landmark used by surveyors to fix a point or a 
corner on an instrument- or a property line. Granite monuments, 
W square at top and 3 to 5 ft. long are serviceable. Small gas-pipes are 
frequently used in surburban districts for semi-permanent work. 

Mooring.— That to which a ship or anything is secured. 

Mooring=swivel = mooring=shackle. — A swivel used to connect two anchor- 
chains together just above the water, say at the forward end of the ship, 
when both anchors are out. 

Mortise. — A part of a mortise-and-tenon joint ; the hole in timber to receive 
the tenon, and made by a carpenter's mortise-chisel. 

Motor, Electric. — A device for transforming electric power into mechanical 
power. Electric motors are specially designed for continuous current 
or for alternating current or for rotating current. 

M-roof. — A double pitch-roof forming an inverted W. 

Muck=bar. — An iron bar which has been passed through the muck-rolls 
only, of a rolling-mill. 



MEANDER-LINE. OGEE. 1513 

Muck=roIls. — ^The first pair of rolls for rolling iron; the bars are finished by 
passing them through the merchant train or puddle-bar train of rolls. 

Muliion. — A vertical division between the lights of windows, screens, etc. 

Munnion. — In ship-building, a vertical division or piece between the panels 
of framed bulkheads. 

Muntin = niunting. — The central vertical piece dividing the panels of a door. 

Mutule. — A flat block projecting under the carona of the Doric cornice; a 
sort of modillion. 

N. 

Nadir. — The point in the heavens vertically below any station on the earth. 
Opposed to zenith, which is a point vertically above. 

Nail=plate. — A metal plate of the proper thickness for cutting up into nails. 

Naphtha. — A colorless liquid distilled from petroleum. Largely used in the 
manufacture of illuminating gas, and for light and power in general. 

Nave. — ^The long, main, central interior part of a church, including the 
central aisle. The hub of a wheel. 

Nave=box. — ^The metallic ring inserted in the nave or hub of a wheel, to 
reduce the wear. 

Needle^'beam. — ^The floor-beam of a bridge. A transverse bolster placed 
beneath the sills of a car and between the bolsters. 

Nest. — A group of parallel steel rollers in a frame and used at the expansion 
end of a truss. A connected series of pulleys or cog-wheels. 

Net-masonry. — Masonry with joints like the meshes of a net. 

Neutral feeder. — The feeder that is connected with the neutral or interme- 
diate terminal of the dynamos in a three-wire system of distribution. 

Newel. — An upright pillar from which the steps of a winding stair radiate; 
or the large ornamental post supporting the hand-rail at the head or 
foot of a flight of stairs; or a round pillar at the end of the wing-wall 
of a bridge. Sometimes called newel-post. 

Niche. — A nook or recess in a wall for a statue. 

Nippers. — A tool like pincers or tongs for grasping hold of small objects; 
sometimes with cutting edges for cutting wire and small pieces of 
metal. In engineering, two toothed jaws attached to gearing, for cut- 
ting off piles under water. 

Non=Conductors. — Substances that offer so great resistance to the passage 
of an electric current through their mass as to practically exclude a 
discharge passing through them. Insulators. 

Normal. — Perpendicular; at right-angle to the tangent to a curve at the 
point of tangency. According to the rule or right principle. 

Nose=piece. — The nozzle of a hose. 

Nose=pipe. — ^The nozzle of a blast-pipe inside the twyer of a blast-furnace. 

Nosing. — ^The projecting edge of a molding, or of a tread or step of a stair. 

Nulled=work. — In wood-turning, pieces or wood turned to form a series of 
connected beads or knobs, as in the rounds of cheap chairs and bed- 
steads. 

Nut. — A sort of adjustable head to a screw-bolt. A short piece of iron with 
a central female screw to fit the screw of a bolt. 

Nut=coal. — Chestnut -coal. 

Nut=lock = nut=fastening = jam=nut. — A device for fastening the nut on a 
bolt so it will not work loose. 

Nut=niachine. — A machine for making (cutting, stamping and swaging) 
iron nuts from a heated bar fed to it. 



Oakum. — ^The coarse part of hemp or flax, removed in combing or hackling. 

Obelisk. — A large rectangular, monumental shaft, tapering from the base 
upward, and with a pointed top. Many abound in Egypt. One in 
Central Park, New York City, near Metropolitan Museum of Art. 

Octastyle. — An architectural feature of a portico with eight columns in front. 

Odometer. — An instrument to be attached to a wheeled vehicle for measur- 
ing distances traveled ; useful in preliminary surveys in connection with 
the compass, and especially for making maps of country roads. 

Ogee = 0. G. — A reverse curve, as in a sectional outline of some molding, 
or of a cast-iron washer; hence ogee washer or O. G. washer. A cyma. 



1514 GLOSSARY. 

Ohm. — ^The unit of electric resistance. It really expresses a velocity, namely, 
1,000,000,000 centimeters per second. Thus, the formula for resistance 
in electro-magnetic units (see Units, Electro-Magnetic, Dimensions of) 
. ^ E L Length ^^ , .^ 

IS i< = 7T == ^ = -TiT- = Velocity. 

C T Time 

Ohm, British Board of Trade. — ^The resistance of a column of mercury 

106.3 centimeters in length and one square millimeter area of cross- 
section at 0° C. 
Ohm, British Association. — ^The resistance of a column of mercury 104.9 

centimeters in length and one square millimeter area of cross-section 

at 0° C. (Not used.) 
Ohm, Legal. — ^The resistance of a column of mercury 106 centimeters in 

length and one square millimeter area of cross-section at 0° C. or 32° F. 

(Never legalized.) One ohm = 1.00112 B. A. Units. 
Ohm, Standard. — A length of wire having a resistance" of the value of the 

true or legal ohm, employed in standardizing resistance coils. 
Ohmmeter. — A commercial galvanometer for measuring ohmic resistance. 
Oil=ceIlar. — A metal box containing oil for oiling the crank-pin, and attached 

to the under side of the strap of the connecting-rod of the engine. 
Oil=pump. — A pump for discharging oil upon a journal. 
Oil=-tempering. — The tempering of steel with oil (not water). 
Oil=weIl. — A boring made for petroleum. 
O. K.— Correct; all right. (Oil Korrect.) 

Opaque. — Dark; shady; obscure; not transparent; impervious to light. 
Open=hearth furnace. — A furnace used in making steel by the Siemens- 
Martin process, with certain late improvements. 
Openwork = open=cast. — Relates to mining or quarrying done in the open 

air, i. e., not covered. 
Ordinate.-^An offset from a base line to any given point. In analytic 

geometry, one of the coordinates locating a point on a curve from two 

co-ordinate axes. See Abscissa, 
Oscillator. — Anything which has or produces a rapid reciprocating motion 

within a limited range of distance, as a power-hammer, the shuttle of a 

sewing-machine, the piston of a steam-engine, etc' 
Osier. — A specie of willow, much used in river-bank protection. 
Outfall. — ^The discharge end of a river, sewer, culvert, drain, etc. 
Outfall=sewer. — That portion of a large sewer which receives the sewage 

from one or more districts and discharges it, as into a river or ocean. 
Out of wind. — Not winding; straight. Specifications calling for timber out 

of wind, mean that the faces shall not be warped. 
Overshot=wheeI. — A mill-wheel with blades or buckets around the periphery, 

and designed to operate by water shot over the top on the descent. 
Ovolo = quarter=round. — A convex molding forming a quarter of a circle. 

In Greek architecture the moldings are a quarter-ellipse, like an egg, 

instead of a quadrant of a circle. 
Ozone. — ^Modified oxygen; its density is one and one-half times that of 

oxygen. It exists in cold regions and in country districts. Can be 

produced by an electric spark passing throi:igh air or through oxygen. 

It is a great purifier and bleacher. 



Packing. — In machinery, material stuffed around moving parts, as in a 

stuffing-box, to prevent leakage of steam, water, etc. 
Packing=ring. — A metal- or rubber-ring used as seat for a coupling-valve of 

a car, to make an air-tight joint. 
Pack=train.: — Pack-animals with their loads. 
Pallet'xmolding. — In brick-making, a process in which the mold is sanded 

after each using. 
Panel. — Any area slightly sunk, or raised, or more or less distinct, as the 

panel or a door. The panel of a truss is the vertical area embraced 

between the chord-joints or floor-beams; the length of a panel -times 

the number of panels is equal to the span, 
Panel=strip. — A narrow strip on the edge of a panel or between two panels. 
Pantile = pan=tile. — A tile with surfaces curved transversely, and overlapping 

and underlapping adjacent tiles, thus: 

and laid on a roof shingle-fashion. 



OHM. PHOTOMETER. 1515 

Pantograph. — A lever-like instrument with arms for reproducing a sketch 
to the same scale or to another scale. 

Paraffin = paraf f ine. — A substance obtained by the dry distillation of wood, 
wax, peat, bituminous coal, etc. 

Parallax. — In an object glass, an apparent lateral movement of the cross- 
hairs as the eye changes position: occurs when the hairs are not co- 
incident with the focal plane. 

Parapet. — A top wall of a masonry structure, forming a sort of breastwork. 
The parapet of an abutment is the long transverse wall on the back edge 
of the coping, rising nearly to sub-grade, to protect the bridge-seat 
from the earth embankment of the roadbed. 

Parcel. — ^To wind anything, as a rope, with strips of canvas. 

Parget. — To cover, gloss over, or smooth over, the surface of anything 
with parget or plaster. 

Parthenon. — The temple of Athene Parthenos, at Athens. 

Party=wall. — A building-wall centrally located on a property-line or party- 
line, for joint use; it may belong to one or to both property owners. 

Passimeter. — A watch-like pocket-odometer for registering walking- or 
running distance. 

Patent hammer. — A hammer for dressing stone, and having sharp knife- 
like ridges on the face. 

Pattern. — An original, or model, or mold for anything. Patterns for castings, 
as chord-blocks of Howe trusses and combination trusses, are made of 
wood, and due allowance must be made for shrinkage of metal. 

Pawl. — A short iron bar or ratchet to engage the saw-like teeth of a ratchet- 
wheel to prevent a windlass or capstan from turning back. 

Pay. — On a ship, to cover with a coat of tar or pitch, as a seam, or a rope. 

Pay out. — ^To slacken rope, as a cable or main sheet. 

Peak. — The upper comer of a sail. Forepeak is the forward extremity of 
a ship's hold, as opposed to after-peak. Peak load means maximum load. 

Peat=charcoal. — -Charcoal made by carbonizing peat. 

Pediment. — ^The triangular front-end of a building included between the 
portico and the sloping edges of the roof. 

Peen = pean. — The end of the head of a peen hammer (pean -hammer) . 

Peen=hammer. — A , pean -hammer, or hammer with two opposite cutting 
edges for dressing stone. A hammer with a chisel edge, used foi 
straightening iron plates. 

Pendentive. — In architecture, a triangular segment of a hemispherical 
dome rising from four supports formed by two intersecting arches. 

Penstock, — A channel or conduit supplying water from the race to the 
gate, through which the water flows to the wheel of the mill or power- 
plant. 

Percussion°cap. — A small metal cap or cup containing fulminating (deto- 
nating or exploding) powder, for exploding dynamite or gunpowder. 

Perihelion. — A point in the orbit of a planet or comet in which it is nearest 
to the sun. Opposed to aphelion, in which it is farthest from the sun. 

Perimeter. — Circumference or outer boundary. 

Periodicity. — ^The rate of change in the alternations or pulsations of an 
electric current. 

Periphery. — Circumference of a circle; arc. 

Peristyle. — In architecture, columns arranged around an enclosure, or any 
part of same, as a court, or cloister. 

Permeability, Magnetic. — Conductibility for lines of magnetic forces. The 
ratio existing between the magnetization produced, and the magnetizing 

force producing it. Permeability, ^=jf' 

Perron. — In architecture, a flight of steps to a building in which the principal 
floor is raised considerably above the ground level. 

Pestle. — One of the vertical moving bars in a stamp-mill for crushing ore. 

Petroleum-Estill. — A still for separating the hydro-carbons, in the order of 
their volatility, from crude petroleum. 

Phase, Angle of Difference of, between Alternating Currents of Same Period. — 
The angle which measures the shifting of phase of a simple periodic 
current with respect to another due to lag or other cause. 

Phase, Shifting of, of Alternating Current. — A change in phase of current 
due to magnetic lag or other causes. 

Photometer. — An instrument for measuring the intensity of light, or com- 
paring one with another. 



1516 GLOSSARY. 

Pick. — A pointed instrument with a handle used in loosening any material 
by vertical swinging. The common pick for loosening earth. The 
stone-^ick. 

Pickax = pickaxe. — ^A sort of combination of pick and mattock: with one 
end pointed and the other flat. 

Pier. — One of the supports of a bridge; generally, one of the central supports 
of two or more spans, the end supports being termed abutments. The 
solid support from which two or more arches spring. The support of a 
wall, between openings. A structure built out into the water, to sup- 
port something, as freight, traffic, warehouses, etc. Usually conveys 
the ideas of length and support. 

Pierre perdue. — Masses of stone thrown into the water to serve as a sub- 
foundation, as for a breakwater. 

Pig. — A cast of metal in compact form, as iron from a blast -fimiace. 

Pigment. — Any substance used by painters to give the desired color. 

Pilaster. — A sort of quarter- or half-pillar projecting from the face of a 
wall, and having the proportional parts of capital and base. 

Pile. — Any long piece driven or planted in the ground to serve as a sub- 
foundation. May be of wood, iron, concrete, reinforced-concrete, etc. 

Pile«hoop. — An iron ring driven over the head of a pile to prevent it from 
splitting, in driving. 

Pile=plank. — Planks driven into the ground like piles, as the sheet-piling 
of a coffer-dam. 

Pile=shoe. — An iron point fitted to the lower end of a pile so it will penetrate 
more easily and not burr at the end. 

Pillar. — A column. 

Pillow-block = pi umber-block. — ^The metal case or support for the end of a 
revolving shaft or joiunal. 

Pin, Insulator. — A bolt by means of which an insulator is attached to the 
telegraphic support or arm. 

Pinch-=bar = pinching=bar. — An iron bar with a small lever-like snout at 
the foot, for working heavy masses sideways. 

Pinion.-; — A small cog-wheel or gear-wheel geared to a larger one, and usually 
giving it motion. 

Pinnacle. — A relatively small structure rising from tha roof or walls of a 
larger one. 

Pin, Switch. — A metallic pin or plug provided for insertion in a telegraphic 
switchboard. 

Pin=switch. — An electrical switchboard by which connections are made by 
means of pins inserted in holes between plates insulated from each other, 

Pintle. — A pin or dowel or long bolt upon which anything turns or revolves, 
as the cylindrical pins on which a blind or a rudder swings, the bolt on 
which the forward axle of a carriage swings under the body, etc. 

Pipeocoupling. — A sort of sleeve which screws on the ends of two abutting 
pipes. 

Pipe-cutter. — A sort of chisel-tool for cutting iron pipes by forcing it down 
on the pipe and around it. 

Pipe-line. — A pipe-conduit. 

Pipe="tongs. — A pair of tongs, one projection being sharp for pushing and 
the other one hooked for pulling, used in screwing pipe together, or 
into their couplings. 

Piston. — A movable piece operated reciprocally by the steam in the cylinder 
of an engine; consists of the piston-head which fits the inside diameter 
of the cylinder, and the piston-rod which connects with the mechanism 
outside. 

Pitch. — Inclination. In mechanics, the distance center to center (c. to c, 
or c — c.) of two adjacent teeth of a cog-wheel, or of rivets when the 
measurement is along some base line, or of the threads of a screw, etc. 
The residuum of tar. The sap from the bark of pines. 

Pitch=board. — A guide or pattern for carpenters in framing the strings of 
stairs, the right-angled notches for the treads and risers. 

Pitch=circle = pitch=line. — In toothed-wheels, a circle intersecting all the 
teeth near the middle of their length, and which is tangent to a similar 
circle of a wheel geared to it; outlines the theoretical size of a toothed- 
wheel. 

Pitch-wheel. — One of two toothed -wheels geared together. 

Pitman. — A connecting-rod between a rotary and a reciprocating part. 

Pit-saw. — A large saw operated by two men one of whom is below in the 
"pit." 



PICK. POTENTIAL, DIFF, 1517 

Pivot — ^That upon which something turns, as the center-pin or pivot-pin 
of a center-bearing drawbridge or turntable. 

Place=brick = sandel = samel=brick. — In brickmaking, a soft brick, insuffi- 
ciently burned. 

Planimeter. — An instrument for measuring the plane area of any object 
or drawing of any irregular outline; it can be adjusted to the scale of 
the drawing. It is an integrator and is mathematically correct. 

Planish. — ^To make smooth or polish. A planisher or flat-headed tool is 
used by tinners; a planishing-hammer is used by metal workers, also a 
planishing-r oiler. 

Plant. — Machinery, tools, and general outfit used in any mechanical opera- 
tion or construction work, etc. 

Platband. — A wide fillet. A flat molding. A lintel formed with voussoirs 
but with intrados horizontal, as anarch with infinite radius. 

Plate, Arrester, of Lightning Protector. — ^That plate of a lightning protector 
which is directly connected with the circuit to be protected, as dis- 
tinguished from the plate that is connected with the ground. 

Plate=girder. — A girder with a web composed of steel plate. 

Plate, Ground, of Lightning Arrester. — That plate of a comb lightning 
arrester which is connected withthe earth or ground. 

Pliers. — Small pinchers with long jaws for handling and bending small 
pieces of metal. 

Plinth. — ^The fiat square member at the base of a column. 

Plug. — A piece to stop a hole. A cast-iron cap leaded in the end of a cast- 
iron water main. A cap screwed into the end of a pipe. 

Plug and feathers. — An iron wedge or plug inserted in one of a series of 
holes in a stone, and between two semi-cylindrical pieces of iron called 
feathers, all the holes being similarly treated, in order to split the stone 
on the line of the holes, by striking the plugs with a sledge-hammer. 

Plumb. — Vertical, as in the direction of gravity. 

Plumb=bob = plumb. — A top-shaped metallic instrument, with the lower 
end pointed, suspended by a cord or plumb-line and used in surveying, 
carpentry, mason-work, etc., to obtain vertical lines. There are various 
kinds and shapes, suited to different classes of work. 

Plumber=block. — See pillow-block. 

Plumb= joint. — A soldered lap-joint, the edges of the metal not being bent 
or seamed. 

Plumb=level = pendulum=level. — A board with a line perpendicular to its 
edge, used in connection with a plumb for obtaining levels. 

Plumb=rule. — A narrow board with parallel edges and with a straight line 
drawn through the middle, used in connection with a plumb, for obtain- 
ing verticals, in bricklaying, carpentry, etc. 

Plummer=block = plumber=block = pillow=block. — See Pillow-block. 

Plummet. — A plumb or plumb-bob used by carpenters, masons, etc. 

PIummet=level = masons*=level. — Similar to plumb-level, preceding. 

Plunger. — A solid piston, as that of a Cornish pump; one without a valve. 

Pockets, Armature. — Spaces provided in an armature for the reception of 
the armature coils. 

Point. — A pointed chisel for dressing stone. To point is to dress masonry 
with a point; or to finish the outer joints with mortar. 

Pole=plate. — A small wall-plate resting on the ends of the tie-beams of a 
roof, for supporting the lower ends of the common- or jack-rafters. 

Polygon. — A figure with numerous sides and angles. 

Pontoon.-^One of a series of flat-bottomed boats or floating structures, 
used in the construction of a temporary bridge across a stream, or for 
support of a pipe-line in hydraulic dredging, etc. 

Pontoon=bridge. — A bridge or roadway supported on pontoons. 

Port. — One of two passages leading from the steam-chest to the inside of 
the cylinder of an engine, above and below the piston, and controlled 
by valves so that the steam enters and exhausts at the proper time. 

Portal. — A door, gate, opening, entrance, etc., to a passage, as to a tunnel 
or cathedral. 

Post. — A compression member connecting the two chords of a truss. An 
end-post, or an intermediate-post, of a truss. 

Potential, Alternating. — A potential, the sign or direction of which is alter- 
nately changing from positive to negative. 

Potential, Constant. — A potential which remains constant under all condi- 
tions. 

Potential, Difference of» — A term employed to denote that portion of the 



1518 GLOSSARY. 

electromotive force which exists between any two points in a circuit. 
In a battery or dynamo it is the total E. M. F. that is available. It 
may be measured by "method of weighing," by "use of electrometers," 
or by "use of galvanometers." 

Potential, Drop of. — Fall of potential. 

Potential, Electric. — ^The power of doing electric work. Comparing with 
flow of water, it is the "pressure" or "head." 

Power. — ^The rate of work. (Energy is capacity for work.) 

Power, Horse, Electric. — Such a rate of doing electric work as is equal to 
746 watts or 746 volt -coulombs per second. 

Pressure. — A force in equilibrium, i. e., opposed by an equal and opposite 
force. Pressure is usually stated in lbs. per sq. in. or in lbs. per sq. ft. 
The pressure on bridge masonry, under the pedestals, is usually limited 
to about 250 lbs per sq. in., depending upon the quantity of the stone, 
etc. 

Prime. — First of anything, as the prime coat, in painting. To prime means 
to charge, as to pour water into a pump-tube to start suction. 

Principal. — [This word may be used in various branches of mechanics to 
denote the chief or main of anything where there are also similar but 
subordinate parts, as principal axis, principal rafter, etc.] 

Pronaos. — An open vestibule or portico. 

Pro rata.- — In proportion. 

Proscenium. — ^That part of a theater between the drop-scene or curtain and 
the orchestra. Thus we have the proscenium-arch and the proscenium- 
box. 

Pro tem. = pro tempore. — ^Temporary; for the time being. 

Prow. — ^The bow of a vessel. 

Prox. = proximo. — In or of the next or coming month. 

Puddle=bar. — Bar-iron from the puddle-rolls of a mill. 

Puddle=rolls. — Grooved iron-rollers for rolling iron as it comes from the 
puddling-fumace and forge. 

Puddling. — ^The process of ramming plastic clay into a structure to prevent 
leakage, as in making the puddle-core of an earth-dam. Also, the 
converting of pig-iron (cast-iron) into wrought-iron in a puddling- 
fumace (reverberatory furnace). 

Pugging. — Mixing clay for bricks. The deadening of sound through a floor 
or partition by interposing some composition or construction; the 
construction itself. 

Pug=mill. — A machine for mixing and tempering clay for bricks, etc. 

Pug=piling. — Dovetailed piling, the piles being mortised into one another 
with a dovetailed-joint. 

Pulley. — A mechanism consisting of a shell (block) containing one or more 
grooved wheels (sheaves) over which a rope runs for hoisting. One or 
more pulleys together with the hoisting rope comprise what is called 
a tackle. Also, a sort of drum over which a belt or cable or rope runs, 
without winding around it. 

Pulsometer. — A kind of pump without a piston; operates by steam-con- 
densation and partial vacuum. 

Punt. — A small boat, square-ended and with a flat bottom. 

Puppet. — ^The head- or tail-stock of a lathe. 

Puppet= valve. — A valve which lifts bodily from its seat when open. 

Purlin = purline. — One of a series of parallel timbers laid horizontally on 
the main or principal rafters of a roof to support the common or jack- 
rafters. PI.— Horizontal shapes, as tees, for supporting any roofing 
material, as tiling. 

Put=log. — One of several pieces of timber used in forming the floor of a 
scaffold ; one end being inserted into a put-hole in the side of the build- 
ing and the other end supported by a horizontal string secured to poles 
erected from the ground. 

Putty. — A mixture of soft carbonate of lime or whiting, with linseed-oil. 

Pyx. — ^The metallic box in which the nautical compass-card is suspended. 



Quadrangle. — A court, square or rectangular, nearly or quite surrotmded 
by buildings. 

Quadratic. — In algebra, an equation in which the highest power cf the un- 
known quantity is the second power, 



POTENTIAL, DROP. RATLINE. 1519 

Quadrature of the circle. — ^The exact determination in square measure of 
the area of a circle. Never been solved exactly, either arithmetically 
or geometrically, by any limited expression. 

Quantity, Unit of Electric. — A definite amount or quantity of electricity 
called the coulomb. 

Quay. — A landing place for vessels; a wharf. 

Queen^post truss. — A truss with two upright intermediate posts meeting 
the end-posts at their tops. Used as roof-trusses. The posts are called 
queen-posts. Queen-post stays are the long rods running below the 
two queen-posts which support the body of the car between the trucks. 

Quirked=niolding = quick=niolding. — A form of molding having an abrupt 
re-entrant angle at its extreme projection. 

Quoin. — Blocks of stone at the corners of buildings and projecting some- 
what from the face of the wall; the subordinate comers of the stones 
being chamfered off, usually. The recess into which the heel-post of a 
lock-gate is fitted. 

Quoin=post. — The heel-post of a lock-gate, on which the latter turns. 

R. 

Rabbet. — A groove, channel, halving or other cut along the edge of a board 
to fit a corresponding cut on another board to fit it. A joint so formed 
by two boards is called a rabbet-joint. Rabbet-saws and rabbet-planes 
are used in preparing the cuts or grooves. 

Race. — Head-race is a channel, canal or watercourse from a dam to a water- 
wheel; tail-race is such a water-course after it leaves the wheel. ^ 

Rack. — One or more long metal bars fitted end to end, forming either a 
straight or a circular piece, and having teeth on one of its sides or 
edges to engage or work into the teeth of a wheel, pinion or screw. 
If the rack is curved it is a segment-rack; if it is a circle it is called a 
circular-rack and is usually composed of segments, as the cast-iron rack 
in the turntable of a drawbridge. 

Rack=and=Pinion. — A small cog-wheel or pinion geared to a rack. 

Rack=and=pinion=jack. — A lifting-jack operated by a straight rack and 
pinion. 

Rack=and=worm. — A rack, geared to a worm or screw instead of to a pinion. 

Rack=railway. — A railway operated by a gear-wheel of the car or locomo- 
tive engaging the teeth of a rack-rail — a rail laid along the track and 
provided with teeth or cogs. Used on inclined planes, as up the sides 
of mountains. 

Rack^saw. — A saw with wide teeth. 

Raft=dog. — An iron bar with ends pointed and bent at right-angle with 
body of bar; used for securing logs together in a raft. 

Rag=bolt = barb=bolt = sprig=bolt. — A sort of jag-spike, or iron spike or pin 
with its shank barbed so as to make it difficult to withdraw after being 
driven. 

Rag=wheel = chain=wheel = sprocket=wheel. — A wheel with teeth on the rim 
to engage the links of a ihain. 

Rail=bender = rail-^bending machine. — A machine for applying lateral pres- 
sure on a rail supported against a bearer, for the purpose of straight- 
ening or curving it. 

Rail=saw. — A portable saw for sawing steel rails. 

Ram. — ^The hammer of a pile-driver. The steel hammer used in forming 
a bloom, in metal-working. 

Rammer. — An instrument for ramming; thus, pavers' rammers, founders' 
rammers, gunners' rammers, etc. 

Random stone.— Rip-rap; stone dumped but not evenly placed, as for 
slope protection or for the sub-foundation of a breakwater. 

Random stonework = random work. — A masonry construction formed of 
stone laid in irregular courses. 

Rasp. — A coarse file; various forms for the trades. 

Ratchet = ratchet and pawl. — A bar or wheel furnished with teeth which 
engage the end of a pivoted bar or click when it tends to turn backward. 
Used for hoisting, etc. Hence, ratchet-brace, ratchet-coupling, ratchet- 
drill, ratchet-jack, ratchet-lever , ratchet-wheel, ratchet-wrench, etc. 

Ratio, Velocity. — A ratio, in the nature of a velocity, that exists between 
the dimensions of the electro-static and the electro-magnetic units. 

Ratline = ratlin = ratling = rattling. — One of the small horizontal ropes 
forming steps to the shrouds of a vessel, for going aloft, 



1520 GLOSSARY. 

Reamer. — A tool with sharp lateral edges or fluted sides for smoothing 
punched holes in plates of metal, or for enlarging them. They may be 
reamed tapered by using a reamer with tapered flutes. 

Reaumur's scale. — A thermometer with the freezing-point zero, and the 
boiling-point 80. Superseded by the centigrade scale, and 100, 
respectively. 

Rebate = rabbet = rabate. — See Rabbet. 

Reciprocal. — The reciprocal of a number is 1 divided by that number, 
the result being expressed usually in decimal form. Reciprocal motion 
is alternating m.otion.^ 

Reconnaissance = reconnoissance. — A critical examination of a country or 
territory prior to the preliminary survey. 

Reducer. — A short pipe of variable diameter for connecting two pipes of 
different diameters. Also called an increaser. 

Reentering=angle = reentrant angle. — An angle or comer pointing inward. 

Reflux. — Flowing back. A reflux-valve is one designed to prevent back- 
flow; a back-pressure valve. 

Refraction. — Deflection or change of direction of rays of light; due to the 
rays passing through a medium of varying density, as the air, or from 
one medium to another, as air to water or water to air. Witness the 
blade of an oar in the water; it seems to bend at the surface. When 
rays pass into a denser medium they are refracted toward the perpen- 
dicular to the surface, and vice versa. 

Refrigerating=machine. — A machine for absorbing heat or converting it 
into work, and hence producing cold. 

Reluctance, Magnetic. — Magnetic resistance. 

T, 1 ^ The magneto-motive force 

Reluctance = 7^^ -^—5 . 

The magnetic flux 

Replacing=switch. — A device for replacing rolling-stock on the track. 

n . J cix- -o-^ -I- D-S Electromotive force 

Resistance, Electric. — Resistance m ohms = i< = 7T = • 7;; 7 = 

C Current 

Volts 



Amperes * 

Retaining=wall = retain=wall = revetment. — A (steep) wall (of masonry) 
built to prevent a bank of earth from sliding or washing away. A struc- 
ture designed to resist lateral pressure of loose material, as earth. See 
Retaining Walls, Section 48. 

Revetment. — See Retaining-wall above, but applies particularly to fortifi- 
cations, the protection of river-banks, etc. 

Revolution. — ^Turning through 360°, or a complete circle; a cycle. 

Rheostat. — An adjustable resistance. A rheostat enables the current to 
be brought to a standard, i. e., to a fixed value, by adjusting the resist- 
ance. 

Rheostat, Dynamo=6alancing. — An adjustable resistance whose range is 
sufficient to balance the current of one dynamo against another with 
which it is required to run in parallel. 

Rheostat, Water. — A rheostat the resistance of which is obtained by means 
of a mass of water of fixed dimensions. ' 

Rib. — One of the curved pieces of iron or timber, as of a dome, arch, vault, 
vessel, etc., to which the outer shell is secured. 

Ridge. — ^The highest part of a roof; the line of meeting of the upper ends of 
the rafters. 

Ridge=pole = ridge=piece = ridge=plate. — ^The timber or iron piece along the 
ridge of the roof into which the rafters are secured. 

Right and left, — In the frame of structures, certain members are rights 
and their counterparts are lefts, some small details being on opposite 
sides of the members otherwise alike, as the end-posts of a bridge. 
In such a case it is necessary to make only one drawing and call it the 
right, accompanied by explanatory notes describing the left. Similarly, 
we have right-and-left spring-frogs, switches, door-locks, screws, etc. 

Right bank of a river is the bank on the right-hand side in descending a 
stream. 

Right solid. — A solid with axis perpendicular to base. 

Rim-saw. — A saw with a central circular disk over which is fitted a central 
band of teeth lying in the plane of the disk. 

Ring-bolt = eyc=bolt. — See Eye-bolt. 

Ring'chuck. — A chuck to a lathe fitted with a ring over the end. 



REAMER. SACK-HOIST, 1521 

Ring-dog. — Two iron dogs for driving into and hauling timber, and con- 
nected by a ring; called a. sling-dog when connected by ropes or chains. 
Riparian. — Marginal or bordering on, as relating to the shore of an ocean, 

a bay, or a stream. 
Riparian owner = riparian proprietor. — One who has vested control in the 

soil to the thread of a stream or to some line in the water established 

by law. 
Riparian rights. — The rights of a riparian owner, as fishing, ferriage, wharf 

or other construction, filling in, etc. 
Riprap = rip=rap. — Broken stone loosely dumped; used for walls, founda- 
tion-beds, bank protection, etc. 
Rip-saw = ripping=saw. — A saw used in sawing wood in the direction of the 

grain, by hand. 
Rising-main. — The vertical column of pipe through which water is pumped 

from a mine. 
Rising=rod. — The valve-rod of a Cornish pum ping-engine. 
Riveting=machine. — A power machine for driving and heading rivets; may 

be operated by steam, hydraulics, electricity, etc. Many are portable. 
Road-machine. — A large scraper mounted on wheels and used for scraping, 

transporting and dumping earth; used in road- making, shaping and 

repairing. 
Road=plow. — A strong plow for loosening earth, etc. 
Road=roIler, — A heavy roller for compacting the surfaces of roads; operated 

by steam-power usually. 
Road=scraper. — A scraper for handling earth which is fairly loose or has 

previously been loosened. 
Roadstead = road . — An unsheltered place where vessels can anchor; not a 

sheltered harbor. 
Roadway. — ^The width of roadway in excavation or embankment is the 

width at sub-grade between edges of slopes. 
Rock=crusher. — A machine for breaking rock into suitable sizes, as for 

concrete. A stone-breaker. 
Rock=driil. — A machine-drill for quarries, mines, rock-excavation, etc. 
Rocker, Brush. — In a dynam.o-electric machine or electric m.otor, any device 

for shifting the position of the brushes on the commutator cylinder. 
Rock=faced = quarry=faced. — The natural face of the stone, without dressing. 
Rocking=bar. — A bar supporting a furnace-grate so that the grate can be 

tipped when desired. 
Rocking=pie_r. — A bridge-pier hinged at the bottom, to accomm.odate the 

change in length of span, i. e., the expansion and contraction, due to 

temperature changes. Sometimes used in suspension bridges. 
Rock=oii. — Petroleum. 
Rock=shaft = rocker=shaft = rocking=shaft. — A shaft that rocks on its 

journals but does not revolve entirely. 
Rockwork = quarry=f aced masonry = rock=faced masonry. — See Rock-faced. 

Squared masonry with the face of stones left undressed. 
RoU=joint. — A metal joint made by rolling one edge over the other and 

pressing the joint. Used in tinning roofs. 
Roof=plate = wall=plate. — A plate on v/hich the lower ends of the rafters of 

a roof rest. 
Rope-clamp. — The metal attachment on the end of a rope or cable and 

forming a part of it. 
Ropewalk. — A long shed in which rope is made. 

Rosin (and Resin.) — The residuum of distilled turpentine. (See page 481.) 
Rotary pump. — A pump having rotary parts as fans to force the liquid ahead. 

A centrifugal pump is a rotary pump. 
Rubble. — Rough stones used in rubble-masonry or rubble-work; irregular, 

for common rubble-masonry, and squared for ranged rubble-work. 
Rubble=Concrete masonry. — See Cyclopean Masonry. 
Rundle. — The rung or round of a ladder. 
Running=trap. — A depressed U-shaped section of a pipe, to contain water 

at all times and guard against the escape of gases. 
Rustic. — Various classes of facings for masonry including rockwork. 



Sack-hoist. — An endless-chain device for hoisting filled sacks, as grain, 
cement, etc. 



1622 GLOSSARY. 

Saddle. — Anything resembling a common saddle. A block resting on rollers 
on top of the pier of a suspension bridge to give the proper relief for 
expansion and contraction of cables; in such a case the resultant 
pressure is vertical. A chair for rails. The ridge-tile of a roof is often 
saddle-shaped and hence called saddle-tile. 

Saddle=joint. — In tinning, etc., a joint made thus [T| between two 

metal plates. - 'i' 

Saddle=plate = crown=sheet. — In locomotive boilers, the bent plate forming 

the arch of the furnace. 
Safety=cage. — A mining cage or elevator car provided with a parachute or 

a safety clutch in case of breakage and too rapid descent. 
Saf ety=catch = saf ety=stop. — A catch to hold an elevator in case of breakage 

of cable. 
Saf ety=lamp. = A lamp used in coal mining and safe when even the inflam- 
mable coal-gas is present, from igniting the latter. 
Safety=valve. — A relief valve in a steam-boiler. 
Sag. — A downward curved bend, or depression. 
Salient. — Projecting outward. A salient angle is one pointing outward, as 

in a common polygon; opposed to reentrant. 
Saltern. — A salt-works, or place where salt is obtained by boiling or evapora- 
tion. 
Saltpeter = saltpetre. — Potassium nitrate, or nitrate of potash. Common 

name, niter. 
Sand=bag. — -A bag of sand or earth, used for repairing leaks or breaks in 

foundation work under water. 
Sand=blast. — A stream of sand driven through a tube onto iron-work for 

the purpose of removing paint, scale, rust, etc., preparatory to repainting 

or welding. Used in portable form in weldmg rail-joints. The sand is 

forced by a sand-blower. 
Sand=pump. — A sludger used to remove the pulverized rock in rope-drilling 

in the oil regions. A sand-ejector used in caisson-work for bridge 

foundations. 
Sand=trap. — A device consisting of a kind of pocket or chamber for collecting 

sand and the heavy sediment from water in pipes, etc. 
Saw-set = saw=wrest. — A tool for springing the teeth of a saw alternately 

to right and left in order that the kerf will be wide enough not to bind 

the blade. 
Scabble.— To dress off the rough projections of a stone with a broad chisel 

or stone-axe or heavy pointed pick preparatory to finer dressing. 
Scale. — A flake or crust on iron due to oxidation, when hammered or rolled 

(milled) ; hence hammer-scale, mill-scale, etc. A flux is used to prevent 

scale. 
Scaling=hammer. — A hammer used to remove scale. 
Scantling. — A small stick of timber not over say 5x5 ins. in section. 
Scarf = scarf -joint." — A joint for splicing the ends of two timbers together 

so as to make a continuous stick; may be reinforced by iron straps 

and bolts, and also be keyed. A usual form is; 



Scarp. — In fortifications, the inner slope of the ditch. A slope. 

Scoring. — In founding: The cracking of a casting when unequal cooling 

takes place; frequently happens to pipes and cylinders when the core 

does not give way to the contraction of the surrounding metal. 
Scotia. — A receding or concave molding, as at the base of a column. 
Scouter. — In stone-working, one who removes large projections by boring 

slanting or transverse holes and using the necessary tools for splitting, 

as wedges, etc. 
Screed. — One of several strips of plaster 6 or 8 ins. wide and a few feet 

apart dividing a surface to be plastered, into bays; the screeds are 

flushed out to the same plane and serve as guides in bringing the whole 

surface to that plane. 
Screw. — One of the six mechanical powers or simple machines, namely, 

lever, wedge, wheel and axle, pulley, screw, and inclined plane. 
Screw-bolt. — A bolt headed at one end and provided with a screw-thread 

and nut at the other end. 



SADDLE. SHANK. 1523 

ScreW"gear. — An endless screw working in the teeth of a pinion; a worm- 
screw working in a worm-wheel. 
Screw>=pile, — A pile with a screw at the lower end. 
Screw=pin, — A cylindrical pin with a screw (and nut) at the end to hold it 

in position. 
Screw=wheel. — A wheel which gears with an endless screw. 
Scfew=wrench. — A wrench having one or both jaws operated by a screw. 
Scribe. — A pointed instrument for marking lines on wood, brick, metal, etc., 

as guides for cutting or scribing. 
Scroll. — A convolved or spiral ornament resembling a partly unrolled 

sheet of paper or the letter S. The spiral ajutage around a reaction 

water-wheel. 
Scupper = scupper=hole. — One of several openings in the side of a vessel at 

the deck level, for the escape of water. 
Scupper=nail. — A short nail with a broad head, for nailing canvas, etc. 
Scuttler. — A small hatchway in a vessel's deck; a hole in the side of a ship; 

a hole for sinking or scuttling a ship. 
Sea=breeze. — A breeze from the sea toward the land. 
Season. — To dry, as timber. 
Seat=earth = seat=stone = under=clay. — In coal-mining, the bed of clay 

which characteristically underlies coal-seams. 
Sea-wall. — A wall (generally artificial, but sometimes thrown up by the 

waves) to prevent encroachment of the sea. 
Second, Ampere. — One ampere flowing for one second. 
Second, Watt. — A unit of electrical work. A volt -coulomb. 
Secret.— Covered up or hidden; thus, secret block, secret dovetail, secret 

nailing. 
Section=liner. — An instrument for drawing parallel lines at certain distances 

apart, and consisting of triangle, straight-edge, scale and set-screw. 
Sector. — A toothed gear comprising only an arc of a circle, for reciprocating 

motion; sector-gear. An astronomical instrument for measuring dif- 
ferences in declination. Sector of a circle is the area between two radii 

and the included arc. 
Segment. — A distinct part of anything. Segment of a circle is the area in- 
cluded between arc and chord. For rack-segment see " Rack." 
Segment=gear. — A gear extending over an arc only, for reciprocating motion. 
Segment-rock. — A rock or cogged arc oscillating on a center. 
Seismograph = seismometer. — An instrument for recording earthquake 

phenomena. 
Seize. — To fasten together with small rope, cord or twine, by winding around 

it, as the end of a large rope, etc. 
Semi»columnar. — Like a semi-column or half column appearing on the face 

of anything, as on a wall. 
Semi=dome. — A half dome abutting a surface. 
Semi=fused. — Half melted. 
Septangular. — Having seven angles. 
Serrated. — Notched or toothed, as a saw. 
Serve. — ^To bind or wind around with twine, cord or marlin, as a rope, in 

order to protect it from rubbing and wearing. 
Service-pipe. — A pipe for supplying water, or gas, etc., to a building, from 

a main. 
Set. — ^The lateral bend of a saw-tooth. The last coat of plaster on a wall 

preparatory to papering. 
Set-screw. — A screw for binding two or more things together, as in a cramp. 

A screw acting in a collar and with a point, for pressing into the metal 

of a shaft or other member to bind them together. 
Sextant. — An instrument for measuring angular distances between objects. 

Used in navigation for obtaining latitude and longitude. 
Shackle. — An unclosed link at the end of a chain, consisting of a U-shaped 

piece of iron fitted with a bolt (shackle-bolt) across the mouth, the 

shackle-bolt being held in place by a pin called a shackle-pin. 

A sort of clevis. 
Shaft. — In mining and tunneling, a vertical hole, pit or well from the ground 

surface. The main body of a column. The interior of a blast-furnace 

above the hearth. In machinery, a large axle supporting something 

which revolves or oscillates. 
Shank. — The long part of anything as the stem of a key or of an anchor, 

the shaft of a column, the holding part of a drill, the body of a bolt, 

etc. 



1524 GLOSSARY, 

Shear. — ^The transverse stress in a girder at any cross-section, tending to 
make that part of the girder on the left (or right) of the section slide 
along the plane of the cross-section. A shear in any direction has an 
accompanying and equal shear at right angle to it; thus we have 
longitudinal shear in a beam. Rivets may be in single shear or double 
shear, the latter when connecting three bars the middle one of which is 
pulling in a direction opposite to that of the two outside bars. 

Shears (formerly sheers). — Two or more poles fastened together near their 
tops, with their lower ends or legs spread apart as a base, and supporting 
hoisting tackle. Called shear-legs. 

Shear<«steel. — Blister-steel especially prepared and suitable for making 
shears, knives, etc. When re-worked, it is called double shear steel. 

Sheathing. — A thin covering, as with plates, boards, etc. 

Sheave. — A grooved wheel, as the wheel of a pulley. 

Sheeting = sheet •=piling= sheet-timbering. — Timber or metal piles, sheets or 
boarding driven to form a more or less water-tight lining, and used 
in connection with the construction of foundations under water, tunnel 
lining, etc. 

Shellac. — A resinous substance possessing valuable insulating properties, 
^ which is exuded from the roots and branches of certain tropical plants. 

Shim. — A sort of flat wedge to separate the surfaces of two adjacent bodies 
by wedging them or holding them apart. Wooden or iron shims are 
used in spacing rail-joints in track-laying. Heavy machinery is often 
shimmed up from the floor. 

Shingle. — Stones on the sea-shore, a little coarser than gravel. 

Ship=worm = teredo (T. nevalis). — A worm that bores into and honeycombs 
timber and piling under water. 

Shoe. — A metallic piece usually shaped to fit the end of various things, 
as the shoe or metallic point of a pile, or the malleable or cast fitting 
at the connecting ends of the bands or wood-stave pipe, etc. 

Shore. — A prop or temporary support for bracing up anything, as a ship, 
or the sheeting in a sewer-trench, etc. 

Shot. — A blast. 

Shroud. — One of the ropes from the side of a ship to the mast-head, to 
support the mast. 

Shunt. — To establish an additional path for the passage of an electrical 
current or discharge. 

Shuttle. — A gate to allow water to flow on a water-wheel. A section of a 
shuttle-dam. 

Side-beam. — One of the two beams of a side-beam engine. 

Side-hatchet. — A hatchet with only one side of the blade chamfered. 

Siding. — A short piece of railroad-track lying along the main track and 
used for passing of trains, etc. The boarding of, or for, the side of a 
building. 

Silt. — An earthy sediment, or deposit of fine soft mud from standing water 
or a running stream. 

Sink. — To excavate downward, as a shaft. 

Sinusoid. — A curve of sines, the angles being laid off as abscissas, and the 
sines as ordinates. 

Siphon. — A bent tube like an inverted V but with unequal legs, the shorter 
leg inserted in the basin of water. When the tube is thus placed and is 
filled with water, the water in the basin will be discharged through the 
tube unless air collects in the latter and stops the siphonic action. 

Siphon, Electric. — A siphon in which the stoppage of flow, due to the gradual 
accumulation of air, is prevented by electrical means. 

Sister-hooks = clip-hooks = clove-hooks. — A pair of hooks which close 
together like the jams of tongs and fit together side by side. 

Size. — A gelatin wash used by painters. ^^ ^^ 

S-joint. — A metal joint, thus y^^\^ 

Skew. — Opposed to right-angled. Oblique. We have skew gearing, skew 

bridges, skew abutments, skew arches, etc. 
Skewback. — ^The inclined stone or surface which takes the thrust of the 

arch. 
Skid. — Any simple arrangement of one or more (generally two) poles or 

timbers placed on an incline (usually) , for sliding or rolling freight upon 

in unloading from a car or vessel, etc. 
Skirting-board = baseboard = mop-board = wash-board. — A narrow board 

placed around the walls of a room next the floor. 



SHEAR. SPANDREL. 1525 

Sledge. — A large heavy hammer; a sledge-hammer. 

Sleeper. — A piece of wood or metal laid on the ground to support various 
classes of construction as rails of a track, floors of houses, etc. A tie. 

Sleeve. — A hollow pipe, tube, or thimble, slid over the ends of two cylinders 
to be joined together. 

Slide°bar. — A bar which is slid over the draft -opening of a furnace. One of 
the guides for the cross-heads of a piston-rod. 

SIide=box. — The slide-valve chest of a steam engine. 

SIide=rod. — ^The rod which operates the slide-valve of a steam-engine. 

Slide= valve. — A valve which slides two and from its seat. 

Slipe. — A sledge, or skip without wheels; used in coal-mining. 

Sling. — A short rope or chain placed around anything to be hoisted. 

Slip. — A docking place for vessels. 

Slope=wall. — A wall built along a stream to prevent the bank from wash. 

Sludge. — In mining, the finely powdered stone, mixed with water, in a drill- 
hole. Refuse from coal-washing and other operations in mining, and 
in the refining of crude petroleum. The deposited sediment in sewage 
tanks after the sewage is treated with chemicals. 

Slug. — In mining, a loop in a rope through which a man puts his leg when 
being raised or lowered in a shaft. 

Sluice. — A wooden trough which miners use in washing gold from gravel 
and sand. An artificial channel. A body of water held in check or 
flowing through a flood-gate or sluice-gate. The injection- valve to a 
condenser. 

Sluice=gate = f lood>=gate. — The gate of a sluice. 

Smoke=box. — A chamber through which smoke and gases pass from the 
furnace to the chimney. 

Snap. — A tool used in riveting, to form the new head. 

Snap=hook. — A metal hook with a spring-mousing to prevent the object 
hooked from slipping off. 

Snatch=block. — A pulley-block with an opening on the side so that the bite 
of a rope can be passed over the sheave; when in use, the opening is 
covered by a strap. 

Snip. — -To cut off. In carpentry, to cut off nearly or quite at right-angle 
with the length of the timber. But see Snipe. 

Snipe. — To cut off on a long bevel. In carpentry, to make long beveled 
cuts at the end of a timber, in framing the end to smaller section than 
the main timber. 

Snub. — ^To check quite suddenly, as the speed of a boat ; the checking is done 
by a snubbing-line, passed around a post called a snubbing-post. 

Soaking^pit. — A pit in which cast ingots are placed so the mass may cool 
to a uniform temperature suitable for rolling. 

Socket=bolt. — A bolt that passes through a thimble placed between the 
parts connected by the bolt. 

Socket=chisel. — A form of heavy chisel for mortising. 

Socket=drill. — A drill for enlarging or countersinking drilled holes. 

Socket=joint. — An articulated-, flexible-,^ or ball-and-socket joint, as a 
flexible joint in a pipe-line. Flexible-joint pipes are used as water- and 
gas mains across streams; they are jointed together on a scow and sunk 
in a continuous line as fast as connected, the last joint being always 
out of water. 

Soffit. — The lower surface of an arch; extended also to include the lower 
surface of a span over a door or window opening. 

Solder. — A fusible alloy for uniting metal surfaces or joints. There are a 
great variety of solders for dift'erent metals. Hard solders, composed 
of copper and zinc, and called spelter, are used for uniting iron, copper 
and brass. Soft solders, composed of lead and tin, are used for uniting 
tin or lead. 

Sole. — Anything resembling in function the sole of a shoe. The foundation- 
plate of a marine-engine. The lower edge of a turbine. 

Solenoid. — A cylindrical coil of wire the convolutions of which are circular. 

Sounding. — To measure the depth of water, soft mud, etc. In shallow water 
this may be done with a graduated rod or pole. In deep water, various 
kinds of apparatus are used; some of them, in addition to measuring 
the depth of water, are provided with a device for bringing to the surface 
samples of mud from the bottom of the sea. 

Spall. — A piece chipped from a stone, as with a spalling-hammer. 

Spandrel. — ^The triangular-like portion or space of an arch lying above or 
outside the extrados, and below the roadway. 



1526 GLOSSARY. 

Spandrel°filling« — ^The filling of earth, or masonry, etc., in liie spandrel of 

an arch. 

Spandrel=wall. — A wall built on the extrados of an arch in the spandrel; 
used to give rigidity to the arch, or to support the roadway, etc. 

Spanner. — A sort of lever-wrench with a hole or with movable jaws to fit 
over a nut for tightening it ; or any instrument for clasping and tighten- 
ing a nut or screw or wheel, etc. A rod connecting two parts having 
parallel motion. 

Spar. — A round stick of timber used for various purposes, as the jib or boom 
of a derrick, the masts, booms and yards of a ship, the poles or common 
rafters of a roof, etc. 

Spark=arrester. — A netting or device placed over or in the smoke-stack to 
prevent the escape of the sparks. 

Spectroscope. — An instrument for producing a spectrum from rays of light, 
and for studying it. 

Spectrum. — The continuous and successive colors in a band of light seen 
when rays of light are deflected through a prism. 

Speculumc^metal. — An alloy consisting of ten parts copper and one part tin; 
used as a mirror or speculum. 

Speed^puIIey. — A sort of cone-pulley, but stepped; different speeds are 
obtained by placing the belt over different steps or faces of the pulley. 

Spelter. — Zinc. Spelter solder is hard solder composed of copper and zinc. 

Spider, Armature. — A light framework or skeleton consisting of a central 
sleeve or hub keyed to the armature shaft, and provided with a number 
of radial spokes or arms for fixing or holding the armature core to the 
dynamo-electric machine. 

Spiegeleisen = spiegeI=iron — A pig-iron containing from 1^2 to J^ or more of 
manganese; used in the manufacture of Bessemer steel. 

Spigot. — A plug with a hole in it and used as a faucet. The end of a cast- 
iron pipe which fits into the bell-end of an adjoining pipe; such a joint 
is called a spigot-joint. 

Spike. — A large metal nail; may be pointed, chisel-pointed, barbed, grooved, 
or split, etc.; the head also may be variously shaped. 

Spile. — A pile. 

Spillway. — A sort of weir or gap or passage for surplus water from a reser- 
voir; located near one end of the dam or at the dam itself. 

Spindle. — A thin axle or shaft. A solid generated by a curve about its 
chord; may be circular, parabolic, elliptical, etc., depending upon the 
curve. 

Splay. — The flaring or widening at the mouth of anything. The wings of 
a culvert when they spread back from the center line are said to be 
splayed. 

Splice. — ^The joining of two pieces together by overlapping. 

Spoil=bank = waste=bank. — A refuse bank in mining and in general excava- 
tion. 

Springer. — ^The lowest voussoir or arch-stone of an arch. 

Springing=line. — ^The lower face-line of the springers of an arch; the line 
from which the arch springs or rises or begins. 

Spring-pole drilling. — In rock-boring, a simple method of using a spring- 
pole on the outer end of which is suspended the drill-rod. The spring 
of the pole raises the rod, the down motion being effected by hand. 

Sprocket=wheel. — A rag-wheel or wheel with _ projections that engage the 
links of a chain passing over it. The projections may be pins, teeth or 
lugs, etc. 

Spur=gear = spur-gearing. — Gearing in which spur-wheels are used. 

Spur=wheel. — A common cog-wheel, the cogs being on the periphery of the 
wheel, and radial. 

Square. — An instrument for laying off right angles. In roofing and flooring, 
an area equal to ten feet square = 100 sq. ft. 

Square-^headed. — Straight, as the upper edge of the opening of a door or 
window not arched or curved. 

Stack. — One or more main flues, funnels, or chimneys grouped together for 
the passage of smoke. A smoke-stack. 

Staff-angle. — In plastering, a square rod of wood flush with the wall at the 
external angle of a room, to protect the plastering. 

Staging. — A temporary structure, including the flooring and supports, used 
in building operations. 

Stamp-mill = stamping-mill — A mill for crushing ore by the use of vertically- 
acting stamps; may be operated by any kind of power. 



SPANDREL-FILLING. ' STRIKING-PLATE, 1527 

Stanchion. — A vertical support, as a column, post or strut used as a support 

to a vessel's deck, or part of a roof, etc. 
Standing. — Anything quite permanent, as standing-rigging or the shrouds 

and stays of a ship. Anything rigidly fastened to and projecting from, 

as a standing-bolt, or stud-bolt. 
Stand=pipe. — A vertical water-pipe used as a reservoir; or inserted in a 

main to show the hydraulic grade line or to act as an air- valve; and 

for other purposes as in a steam-engine. 
Staple. — A U-shaped loop of metal driven into a door or other object, the 

projecting ends being bent or clinched on the inside, to receive a hook 

or hasp and form a kind of lock. 
Starboard. — ^The right side of a vessel facing her bow. Opposed to larboard 

or port. 
Starling. — A pile structure around, or up-stream or down-stream from, a 

bridge-pier for protection or support. One of such piles. 
Static == statical.— Pertaining to weight, without motion; as static equili- 
brium, static load, static pressure, etc. 
Stay. — A rope, tie, brace, or strut, etc., for keeping anything in place or 

making it "stay" in position. 
Stay<=bolt. — A bolt used to prevent two opposite plates or parts from being 

pressed or pulled apart further, as a stay-bolt in a steam-boiler. 
Stay=rod. — Same as stay-bolt but longer, and used for various purposes as 

in building-construction. 
Steam-box = steam-chest — A reservoir or chamber for steam to be used; 

situated above the boiler, in a locomotive. From this chamber the 

steam passes to the cylinders. 
Steam-chest. — [The common name.] Same as Steam-box, above. 
Steam-pipe. — A pipe which leads from the boiler to the engine or from the 

boiler to the steam-chest. Various pipes conveying steam. 
Stem. — A projecting part, as the stem of a gate-valve. 
Step. — In mechanics, anything resembling a step of a stair, as an offset on a 

cone-pulley. The foot or bearing of a vertical shaft. 
Stile. — One of the main frames of a door to which the secondary or central 

frames are secured. The outer frame or main frame of anything. 
Stilted-arch. — In architecture, an arch apparently raised above its springing 

as if stilted. 
Stirrup. — In carpentry, an iron loop or strap for supporting one beam 

butting another, or a rafter, etc. 
Stock and die. — A die and its holder, for cutting screws. 
Stoker. — A fireman. A mechanical stoker is an automatic device for feeding 

fuel in a furnace and attending to the ashes, etc. 
Stone-breaker. — See Rock-crusher. 
Stone-saw. — -A blade made to reciprocate in its kerf while sand is fed by a 

stream of water. Used for sawing marble, etc. 
Stope. — An (horizontal) excavation in a mine, from a shaft or tunnel or drift, 

to remove the ore laid bare, or to pile the ore, or to receive refuse, etc. 
S-trap. — An S-bend in a waste pipe to prevent gases from rising above the 

point of bend. 
Strap. — A long narrow strap-like piece of metal, either straight or looped, 

and bolted to two pieces to hold them together, as in an iron strap used 

in a timber splice, etc. 
Stratified. — Deposited in layers or strata. One of the layers is called a 

stratum. In geology, stratified rock or earth; bed rock. 
Stress. — A force producing strain; measured in lbs. per sq. in. or per sq. ft. 

on the member to which it is applied. Within the elastic limit of the 

material, stress is proportional to strain — Hook's Law. 
Stress-sheet = strain-sheet. — A diagram of a structure, as a bridge, giving 

stresses and sizes of members and other information which will determine 

the character of the structure to be built. 
Stretcher. — In masonry, a brick or stone laid horizontally in the direction 

of the face of the wall; distinguished from header which heads onto the 

face of the wall. 
Stretcher-bond. — Bricks or stones all laid as stretchers in continuous courses, 

no two joints being opposite transversely. 
Stria. — A fillet between the flutes (base moldings) of columns. 
Strike. — The direction, by the compass, of a stratified formation; at right 

angle to the dip. 
Striking-plate. — In arch-centers, one of a series of compound wedges on 

which the centering rests while the masonry arch is being built; and 



1528 GLOSSARY, 

when the wedges are struck, the centering lowers and the arch becomes 

self-supporting. 

String. — A string-course. A line of pieces used in construction, as timber, 
stone, etc. 

Stringc^course. — In architecture, a projecting molding or other prominent or 
distinct band. 

Stringer. — A string-board or string-piece of a stair, or one of the slanting 
pieces supporting the treads and risers; or an ornamental piece fitted 
outside the supporting stringer. A longitudinal piece for supporting 
anything such as the ties or planking on a bridge. 

Stripping. — In a quarry, mine or gravel-pit, the useless material stripped or 
removed preparatory to opening the quarry, mine or pit. 

Strut. — A member acting in compression. Opposed to tie which acts in 
tension. A prop or brace. 

Stub. — A short end or blunt end. 

Stub<=end. — The enlarged end of a connecting-rod or pitman, to which the 
strap is fastened. 

Stud. — One of the vertical pieces of scantling in a partition and to which 
the laths are nailed. 

Stud-bolt. — See Standing-bolt. 

Studding. — In carpentry, material to be used as studs or joists. 

Stuffing=box. — A sort of cast-iron box or chamber arranged around a 
movable rod to secure a tight joint against water, steam, or air, etc., 
passing along the rod through the wall which the rod pierces. The box is 
packed with greased hemp, or india-rubber, etc., and the ring-cap 
screwed or bolted on. 

Suction. — The removal or the lessening of the atmospheric pressure on any 
part of a liquid so as to disturb its equilibrium and cause it to flow, the 
atmospheric pressure still remaining on some other surface of the liquid. 
The exhaustion of a gas or liquid from a chamber. 

Suction-pipe. — A pipe connected with the bottom of a pump-barrel and 
leading down into a well or body of water to be raised. A pipe leading 
from beneath a water-wheel downward to the level of the tail-race in 
order to make the total head or fall available for power; the pipe must 
be air-tight. 

Suction°punip. — A pump with a barrel or cylinder standing on and fastened 
to a suction-pipe leading to a well; at the junction of barrel and pipe 
is a flap-valve raising upward to allow water to enter from below ,_ but 
which closes down tight against any back pressure; the barrel itself 
being fitted with a piston-rod operating a piston-head containing a 
flap-valve which also raises in the same manner as the other valve. At 
the down-stroke of the piston the lower valve closes and the upper one 
opens; at the up-stroke they reverse. Other kinds of valves may be used. 

Suction-valve. — The lower valve of a suction-pump; the one below the piston. 
See Suction-pump. 

Sump. — A depression in which water collects, as at the head of a land- 
slide; a kind of pool. A reservoir in a mine or other working into which 
drainage water is led and from which it is pumped out of the working; 
a sump-pump is used for this purpose, and the shaft through which it 
is pumped is called a sump-shaft. 

Surbase. — An upper molding above a base, as that above the wainscoting 
or at the top of a pedestal. 

Surd. — In_mathematics, a quantity which cannot be expressed definitely, 
as \/3 . 

Surface-condenser. — A condenser in which the exhaust-steam is condensed 
on metal surfaces which are cooled by flowing cold water in contact 
with the opposite faces. 

Surface-tension. — The adhesion or tension of the surface-skin of a liquid, 
as when anything is made to touch it and is then raised. A kind of 
capillary attraction. 

Swage. — A sort of die for giving shape to a piece of metal being ham- 
mered. 

Swage-block. — A block with various holes, grooves, and other shapes for 
swaging or shaping pieces of metal, as for heading bolts, etc. 

Sway=>brace. — A brace used to cut a four-sided panel of a structure into rigid 
triangles. A sort of diagonal member or brace (strut or rod) between 
^two opposite posts of bridge trusses. 

Swing-saw. — A circular saw suspended from a frame and which can be used 
in sawing large bulky pieces at rest. 



STRING. TERNE-PLATE, 1629 

Switch. — Any device for connecting or disconnecting, as a track, a current 
of electricity, etc. 

Switchback. — A sort of zig-zag location of a railway, for gaining a gradual 
grade over a mountain. 

Swivel. — A fastening comprising an axis which may be turned around freely 
in the other part, the end of the axis being headed like a bolt, to sustain 
tension. 

Synchronize. — To cause to occur or act simultaneously. 

Synclinal. — In geology, the dipping of strata toward each other and forming 
valley shapes. Opposed to anticlinal. A synclinal axis is a line follow- 
ing along the lowest points of the depression. 

System; Three=Wire. — A system of electric distribution for lamps or other 
translating devices connected in multiple, in which three wires are used 
instead of two usually employed. In such a system two dynamos are 
usually emploved, connected in series. 

T. 

T=tee. — Anything resembling the letter T, as a T-vail, T-square, T-iron, etc. 

Table.- — In mechanics, the table-like part of a machine on which the work 
is placed. In architecture, a horizontal molding or a projecting portion 
from a wall. 

Tackle. — A rope operating in one or more pulleys for hoisting or hauling. 

Tail-race. — A channel and stream of water leading from the water-wheel. 

Talus = batter. — Slope, as of a parapet or rampart, in fortifications. 

Tamp. — To force down with strokes or pressures, as tamping a charge of 
powder in a hole, or tamping earth, concrete, etc. 

Tamping-bar = tamping=iron. — In blasting, an iron bar for forcing the tamp- 
ing (material) on the charge of explosive in the hole. 

Tamping«plug. — A form of cast-plug used instead of tamping material in a 
blast-hole. 

Tangent-screw. — A sort of slow-motion screw for revolving circular disks 
slowly. 

Tank=engine. — A locomotive that carries its coal and water without a tender. 

Tank=iron. — Plate iron whose thickness is between that of boiler-plate and 
sheet-iron, the latter being the thinnest and about the same as stove- 
pipe iron. 

Tap. — A tool with an external, tapered, and longitudinally-grooved screw 
for cutting the internal screws of nuts, etc. As a verb, to make a tap 
as in a pipe (pipe-tap) ; to bore or cut into; to pound as with a hammer 
in testing rivets or cast-iron pipe. 

Tap-bolt = tap=screw. — A bolt screwed into a tapped hole, as in a plate, 
the plate acting as a nut. 

Tapping-drill = tapping-machine. — A drill or machine for tapping holes in 
street-mains, or iron pipes. 

Tappet. — A projection or arm on a revolving shaft, which strikes or taps 
something at each revolution. 

Tee = T.— See T. 

Teeth. — Plural of tooth; see Tooth. 

Telford pavement = telford. — A road pavement consisting of a foundation 
of small stones laid by hand; on top of and in the crevices of these 
stones are packed smaller pieces; upon this, is broken stone. The 
whole mass is rolled until the surface becomes compact and smooth. 

Temper. — ^To modify. To bring metal to a proper degree of hardness 
by first heating to a high temperature, and then suddenly cooling in a 
bath of oil, or water, etc. To mix mortar to the right consistency for 
bricklaying, etc. 

Templet == tern plate. — An outline of anything, used as a guide or model in 
shaping it, as the edge of a board cut to shape a molding, or a board 
or plate with holes spaced for punching other plates, etc. 

Tender. — ^That which tenders or waits upon, a:s an engine tender, or]^a vessel 
supplying freight, provisions, etc, to another. 

Tenon. — The framed projection at the end of a timber, to fit into a mortise: 
such a joint is called a mortise and tenon joint. 

Teredo = ship-worm. — A sea-worm which bores into and honeycombs piling 
and timber. 

Terne-plate. — An inferior tin-plate, or a plate of sheet-iron or sheet-steel 
coated with tin which is largely alloyed with lead. 



1630 GLOSSARY. 

Terrace. — A sort of long horizontal step in an embankment; used on the 
sloping up-stream and down-stream faces or slopes of earthen dams. 
A kind of bench or level, on the side of a hill. 

Terra=cotta.-^A fine quality of clay baked very hard; used for brick, 
roofing tile, pipes, etc. 

Test=pump. — A force-pump used for testing the strength or tightness of 
pipes, cylinders, etc. It is provided with ^ pressure-gage. 

Theodolite. — A surveying instrument, something like a transit but whose 
telescope is not reversible. 

Thermometer. — An instrument for measuring temperatures. Centigrade 
thermometer reads zero at freezing and -1-100° at boiling. Deep-sea 
thermometer used to record the temperature of the water at any depth 
in the sea. Differential thermometer is used to record small differences 
in temperature. Fahrenheit thermometer reads + 32° at freezing and 
-f- 212° at boiling. Maximum thermometer registers the maximum tem- 
perature. Minimum thermometer registers the minimum temperature. 
Keamur thermometer reads zero at freezing and -I- 80° at boiling. 

Thimble. — A sleeve or tube used to join the ends of pipes or rods. A circular 
ring, concave outside, to form the inside protection to a loop of rope 
when suspended on a hook or other contrivance. 

Thimble=joint. — A sleeve or thimble slipped over a pipe-joint and packed 
to prevent leakage during expansion and contraction. 

Thread.— The spiral ridge or worm of a screw. 

Three=ply. — ^Three thicknesses, as three-ply roofing felt. 

Throat. — The narrow part of an opening, not always near the end, as the 
throat of a venturi-meter. 

Through=mortise. — A mortise cut entirely through a timber. 

Through-stone = thorough-stone — A header extending entirely through a 
wall. 

Throw. — ^The distance moved, as the throw of a railroad switch is from 
5 to 6 inches. The extreme movement of a slide-valve. The double- 
radius of a crank. 

Thrust. — The crushing of the pillars in a coal-mine. 

Thrust=box. — A box-bearing sustaining a vertical shaft. 

Tide=gate. — A gate through which water passes into a basin when the tide 
flows in, and which is shut to retain the water from flowing back at ebb. 

Tie. — Any beam or rod used in construction to hold certain parts together 
by tension or pull. One of the sleepers or supports to the rails. 

Tie=plate. — A metal plate resting on a tie and supporting the rail; used to 
prevent the flange of the rail from sinking into the wooden tie. 

Tie=rod. — Any rod used as a tie to sustain tension or pull. 

Tile. — A thin shape of baked clay, used for covering roofs, floors, walls, etc.; 
also for pipe, drains and sewers. 

Tire. — An outside ring around the periphery of the wheel of a vehicle. In 
common wagons, the metal tire is heated so as to expand and then 
shrunk over the wooden felloe. 

Tit. — A small projection, as like the end of a bolt on the surface of a casting. 

Toe. — ^The sharp end, or front end, or moving end of many devices. The 
toe of a switch, or of a frog, etc. r 

Toggle=joint. — A joint formed by two bars hinged ' 

together at an obtuse angle so that when 
direct pressure or pull is applied at the joint 
the lateral force is greatly augmented. 

Tongue. — Many things pointed or projecting, as the bead or tongue along 
one edge or a board to fit into the groove of an adjacent board; called 
tongue and groove, and used for flooring, etc. The pointed part of a 
crossing-frog. 

Tooling. — In masonry, dressing with a chisel so the face shows the parallel 
marks of the tool with uniformity. 

Tooth. — One of the teeth or cogs of a wheel. Also applies to other tooth- 
like projections as in a saw. 

Torus.— ;-A large convex molding, used at the base of a column. Opposed to 
scoiia, a concave molding. 

Tower, Electric. — A high tower provided for the support of a number of 
electric arc lamps, employed in systems of general illumination. 

Tower, Electric=Transmission. — A tower on a transmission line, for sup- 
porting long-span wires. 




TERRACE. TUBE-VALVE. 1531 

Traction- wheel. — A wheel which by friction on its circumference draws a 
vehicle, as the driver of a locomotive. 

Traction=engine. — A steam-engine for drawing loads on common roads. 

Trailing=wheel. — One of the wheels situated just behind the driving-wheels 
of a locomotive to support the rear weight. 

Train. — A set of wheels, as cog-wheels, working in series, i. e., connected 
together in a train. 

Trammel. — An instrument for drawing an ellipse; consists of a horizontal 
arm with a vertical pencil at one end, two points of the arm working 
along lines which cross each other at right angle. 

Transformer. — An induction coil used for raising or lowering the electro- 
motive force at any point of a circuit. The E. M. F. is raised by the 
step-up transformer, and lowered by the step-down transformer. 
Other terms are: Commuting-, Constant -current-, Core-, Hedgehog-, 
Lightning-arrester-, Multiple-, Oil-, Open-iron-circuit-, Pilot-, Rotary- 
current-, Rotary-phase-, Series-, etc. 

Transom. — One of the horizontal framing timbers across the stem of a ship. 
A horizontal beam across the opening for a door. The opening above a 
door. A horizontal beam of timber or stone across a window. 

Trap. — A pipe with a depressed bend to hold water and thus form a water- 
seal against the passage of gases; usually U- or S-shaped; used in 
closets in connection with soil-pipes, and elsewhere. 

Trap= valve = clack=valve. — See Clack-valve. 

Traverse. — In surveying, a polygonal base-line around a piece of land to be 
surveyed, showing distances and angles. 

Tread. — ^The horizontal part of a step, the vertical part being the riser. 

Treadle. — A foot-lever, for operating a sewing machine, grindstone, or 
lathe, etc. 

Treenail. — A long wooden pin (hard wood) for fastening planks and timbers; 
used in ship-building. Auger-holes are first bored, and the pins driven in. 

Trestlework. — A stilted framework for supporting the floor of a bridge or 
other structure. 

Trimmer. — In carpentry, a short cross-joint inserted at right angle between 
two lines of joists and butting against them; and in turn supporting 
the ends of other joists parallel with the main ones. The ends may be 
supported by iron stirrups, or mortised into the other joists. Used at 
floor- and roof-openings, at chimneys and stairways. 

Trip=gear. — A gear that trips the valve-closing mechanism in a steam- 
engine when the piston reaches a certain definite position. 

Trip=hammer = tilt=hammer = tilting=liammer. — A hammer operated by a 
cam which trips a lever and allows the hammer to fall; used for heavy 
work. 

Tripod. — A three-legged support, as for surveying-instruments, screw-jacks, 
drills, etc. 

Tripper. — A part of a machine which causes another part to be released, 
as the tripper of a pile-driver hammer. 

Trippet. — Any projecting part of a machine that trips. 

Truck. — A framework with two or more pairs of wheels for supporting one 
end of a car or locomotive. A bogie-truck is used forward of the drivers 
and is carried by the bogie-wheels; this name also applies to street-car 
trucks. Also, a wagon with a solid platform for hauling heavy material. 

Trundle=wheel = lantern-pinion = lantern=wheel= wallower. — See Lantern- 
wheel. 

Trunk. — A long trough or box for conveying water, as from a race to a water- 
wheel; a penstock or flume. 

Trunnion. — One of the cylindrical projections on the side of a cannon, 
which support it in its carriage and on which it revolves in a vertical 
plane. A hollow gudgeon on either side of an oscillating cylinder, for 
supporting the cylinder and through which steam enters and is ex- 
hausted. 

Trunnion=valve. — A valve of an oscillating-cylinder, and operated by its 
oscillating motion. 

Truss. — A framework acting as a beam or girder. 

Truss-beam. — A simple or compound (wooden) beam reinforced by one or 
more tie-rods (or in some cases, A-shaped struts). 

Tube, Crookes'. — A tube containing a high vacuum and adapted for showing 
any of the phenomena of the ultra-gaseous state of matter. 

Tube- valve. — A tube pressed against a seat at outlet, the other end pro- 
jecting above the surface of the water; operated in various ways,. 



1532 GLOSSARY, 

Tudor style. — A style of English architecture of the 16th century. 

Tumbler. — A kind of lever which drops into a notch at a certain definite 
position in the movement of a mechanism, and locks. 

Tumbler-tank. — A tank which automatically discharges its contents when 
filled; if two tanks, they alternate in filling and emptying. 

Tumbling=bay = waste=weir — That part of the outlet or waste from a canal 
or other body of water at which the water falls rapidly. 

Tumbling=box. — A box pivoted at opposite ends or comers and made to 
revolve. A cubical concrete-mixer is a tumbling-box. 

Tumbling=shaft. — A shaft used in stamping mills, in the link-motion of a 
locomotive, in thrashing-machines, etc.; a sort of a cam-shaft. 

Turbine. — In the broadest sense: A motor with vanes (usually curved) and 
acted upon by the pressure or velocity (or both) of fluids, as air, gas, 
water, etc. (as ventilating fans, turbine water wheels, etc.); or acting 
upon fluids (as in the case of the centrifugal pump). In a restricted 
sense: A motor with curved vanes acted upon by fluids (water or steam), 
as the hydraulic turbine and steam turbine (see below). 

Turbine, Hydraulic=Turbine Water Wheel = Turbine, — A turbine deriving its 
motive power from the pressure and velocity of water. See page 1342. 

Turbine, Steam. — A turbine deriving its motive power from the velocity 
and expansive force of steam. 

Turn, Ampere. — A single turn or winding in a coil of wire through which 
one ampere passes. 

Tumbuckle. — A double-nut with right and left threads, used for connecting 
the ends of two rods or bars, making one adjustable member. 

Turning=point. — In leveling, a temporary bench-mark. 

Turn=table = turntable. — A sort of deck-drawbridge swinging in a horizontal 
circle, for turning locomotives; used at terminal points and at shops, 
round-houses, etc. There is a central pin or pivot-pin supporting the 
table, and the outer ends are steadied by wheels on a circular track. 

Turpentine. — An oleoresinous substance obtained from the bark and wood 
of coniferous trees. The oil of turpentine is obtained by distillation. 

Tuscan Order. — In architecture, one of the five Orders; similar to the Roman 
Doric. 

Tusk=tenon. — A tenon stepped or shouldered into a mortise, to give addi- 
tional strength to the connected beams. 

Two-ply. — Having two thicknesses, or double thickness. 

Twyer. — One of the tubes through which air enters a blast-fumace. 

U. 

U=bolt. — A U-shaped rod with nut and thread at each end. 

Unctuous. — Oily; greasy, soapy. Having the nature of an tinguent, 

Underdrain. — A sub-drain, or drain under ground. 

Undermine. — To render unstable by weakening the foundation. To exca« 
vate beneath, as by digging or washing. 

Underpinning. — A support or new foundation placed under a wall not 
properly supported; it may be either temporary or permanent, and 
may consist in merely projecting the wall downward. The act of intro- 
ducing such a support. The term undersetting may be used, especially 
with reference to machinery, pedestals, etc. 

Undershot wheel. — A water-wheel operated by the force of the stream 
acting upon the blades or paddles as they fall below the level of the 
center or axis of the wheel. 

Undertow. — A sub-surface current moving in a direction different from the 
surface-current . 

Unguent. — Any soft substance or composition used for lubrication. 

Uniclinal = monoclinal. — In geology, dipping in one direction: a monoclinal 
fold is half an anticlinal fold. 

Unit. — A standard of quantity in any system, as unit of measure, capacity, 
weight, force, etc. 

Upset. — The end of a rod, or bar, etc., which has been thickened by shorten- 
ing its length, for the purpose of connecting it with some other member 
so that the joint will not be weaker than the body of the rod or bar; 
as for forming the head of an eye-bar, or the upset-end on a rod for 
cutting a sere w- thread . The machine for making upsets, or upsetting, 
is called an upsetting-machine . The shaping of eye-bar heads is called 
forging; welding a separate head on the body of the bar is not allow- 
able in a good class of structural work. 



TUDOR STYLE, VALVE, 1633 



Vacuum-brake. — A device for stopping a train by the operation of brakes 
through partial vacuum created in a pipe connected with the locomo- 
tive; the vacuum being created by a steam-jet escaping through an 
ejector, and the brakes being operated by the drawing of the brake- 
rods which are joined to collapsing bellows connected with the pipes. 
Vacuum=gage. — A gage for indicating the pressure (or the amount of 
vacuum) in any chamber, as in the receiver of an air-pump, a steam- 
condenser, etc. 
Vacuum=valve. — A safety-valve opening inward in a steam-boiler to give 
relief from collapse when the partial vacuum in the boiler reduces the 
internal pressure below the point of safety in ressting the external 
atmospheric pressure. 
Valve. — Any device for controlling the flow of liquid, vapor or gas through 
a pipe or passage. Valve-chest, in a steam-engine, is the steam-chest; 
it is the chamber in which the valve works. Valve face, that part of the 
surface of the valve which comes in contact with the valve seat. Valve 
gear = valve motion, the system which gives motion to a valve, as the 
link motion of a locomotive for regulating the supply and exhaust of 
steam to and from the cylinder. Valve seat, the fixed surface or piece 
on which a valve presses and rests. Valve stem, a rod attached to a 
valve to operate it. Valve yoke, a strap to hold a valve. 
Air valve; a valve to regulate flow of air, as in a steam-boiler. 
Automatic valve; a valve that works automatically, as a clack-valve. 
Back-pressure valve; a valve to prevent back-flow when the direction 

or pressure of fluid is reversed. 
Balance -valve; a valve admitting fluid to both sides under nearly equal 

pressure. 
Ball-cock valve; a sort of ball float valve, as used in water-closet tanks. 
Ball valve; a valve formed by a ball resting upon a seat. 
Blow-through valve; a valve situated in the opening through which 

steam enters a condensing steam-engine, for blowing through. 
Brake-shoe valve; in an air- or vacuum-brake, a valve to relieve the 

excessive pressure upon the wheel. 
Butterfly valve; a double clack-valve, used in pumps. 
Check-valve; Si volve placed in a pipe-line, or boiler, to prevent back-flow. 
Clack valve; a trap-flap, or clapper, hinged to allow flow in one direc- 
tion only. 
Cone-valve; a valve with a conical face and seat. 
Conical valve; a T-valve or puppet valve — a circular metal plate with 

beveled edge and seat. 
Cup-valve; a valve cup-shaped, and becoming a balance-valve if con- 
sisting of two cups connected by a stem through the opening. 
Double-heat valve; a double-seat valve. 
D-valve; a valve resembling the letter D, used in the induction and 

eduction passages of a steam-engine cylinder. 
Equilibrium valve; a balance valve (see above) . 
Flap-valve; a clack-valve (see above). 
Globe-valve; a valve with a globular casing. 
Hinged valve; a butterfly-valve or clack-valve (see above). 
Key-valve; an air valve-plug. 

Lifting valve; a ball-, cone-, poppet-, or safety-valve. 
Long-slide valve; a long- valve, governing parts of both ends of a 

steam-engine cylinder, especially of the Cornish type of engine. 
Long-valve; a long-slide valve (see above). 
Low-water valve; a valve which allows the steam to escape when the 

water in the boiler is too low. 
Oscillating valve; a valve that oscillates. 
Piston-valve; a reciprocating valve, alternately opening and closing the 

port of a steam-engine cylinder. 
Pockete d-valve; a valve fitted into the depression of a pocket. 
Poppet-valve; a valve which lifts bodily from the seat. 
Pot-lid valve; a cap valve used at the end of a pipe, or the cover of 

an air-pump of a steam-engine. 
Puppet-valve; a conical valve or poppet valve (see above). 
Regulator-valve; a throttle valve. 

Relief -valve ; a valve through which fluids escape at a certain deter- 
mined high pressure. 



1534 GLOSSARY, 

Reverse valve or reversing valve; the valve of a reversing cylinder, often 

a plain slide-valve. 

Rotary valve; a rock valve which acts by partial rotation. 

Safety-valve; a valve to relieve excessive pressure, as in a steam-boiler. 

Screw-valve; a screw with a point forming a small valve, for regulating 
flow. 

Slide-valve; a valve which slides over a seat. 

Snifting-valve; the tail valve or blow valve in the cylinder of a steam- 
engine, for the escape or admission of air. 

Spherical valve; a ball valve. 

Throttle-valve; a valve in the steam pipe of a boiler for controlling the 
flow of steam, as to a cylinder. 

Trap-valve; a clack-valve or flap-valve (see above). 

Twin-valve; a double-connecton valve or gate. 

Under shut valve; a valve beneath the sole-plate of a pump, etc., and 
which closes by upward pressure from below. 
Vapor. — A gaseous form of a substance which ordinarily exists in solid or 

liquid form, and while it is in this gaseous form it is physically a real 

gas, which may be defined as a substance which at ordinary tempera- 
tures and pressures exists in the gaseous state; hence, all vapors are 

gases but all gases are not vapors. A saturated vapor is a vapor which 

is on the point of condensation. A non-saturated vapor is one which 

obeys the laws of gases, as superheated steam. 
Variable gear.— Geared wheels or sectors which impart alternating changes 

in speed. 
Vault. — A long arch (not in span but in the direction of the axis), or one 

whose length is great in proportion to its span. The space enclosed by 

or beneath such a vault. 

Conical vault; a vault formed as upon part of the surface of a cone. 

Compound vault; a vault composed of two or more simple vaults. 

Cross vault; a vault which crosses another. 

Cylindrical vault; a vault of cylindrical form. 

Double vault; a vault placed above or enclosing another vault. 

Elliptical vault; a vault of elliptical form. 

Groined vault; a vault formed by two intersecting vaults. 

Pointed vault; a vault pointed. 

Rampant vault; a vault which springs from planes not horizontal. 

Simple vault; an ordinary vault with one axis. 

Single vault; one vault. 

Spherical vault; a vault of spherical form, 

Surbased vault; a circular vault whose height is less than half the span. 

Surmounted vault; a circular vault whose height is greater than half 
the span. 
Vaulting=shaft. — A shaft to receive the spring of a roof- vault rib; may 

' extend downward to the floor or to the capital of a pier. 
Vaulting=tile. — A tile used in vaulting; hollow and of various forms. 
Vault=light. — A vault cover set with glass for the admission of light. 
Vault=shell. — ^The masonry, skin, plate or thin filling between the ribs of a 

vault. 
Veneer. — A thin layer of costly or ornamental wood glued over the surface 

of a cheaper variety composing the frame. 
Venetian blind. — A hanging blind operated with cords, the slats being held 

together by some flexible material. 
Vermicular work. — The surface of architectural stone so dressed or worked 

as to appear thickly covered or indented with worm tracks or shapes. 
Vernier. — A small movable scale sliding parallel with a fixed scale, the 

num.ber of subdivisions varying by 1, in order to obtain more exact 

readings; as the vernier of a transit instrument. 
Viaduct. — A series of (masonry) arches supporting a roadway. 
Viscosity. — Internal friction in the movement of liquids and gases. Viscous 

fluids are those in which viscosity is strongly present. 
Vise = vice. — A tool with two gripping jaws that may be opened or closed 

by a screw worked with a lever; used by carpenters and machinists for 

gripping pieces to be worked. 
Vitreous. — Resembling glass; glassy. 
Vitrified. — The whole body, or the surface only, converted into glass. 

glazed, as vitrified- or glazed-brick, tile, terra cotta, pipes, etc. The 

vitrification is performed by the action of heat. 
Volt. — ^The practical unit of electromotive force. 



VAPOR. WATER-MAIN. 1635 

Volt-Ammeter. — A watt-meter. 

Volt-Ampere. — A watt. 

Volt-Coulomb. — ^The unit of electrical work. The joule. 

Voltage. — Electromotive force or differential of potential. 

Voltmeter. — An instrument for measuring the difference of potential. 

W. 

Wainscot. — A wooden lining or paneling of the walls of a room, and reaching 

upward three feet or more above the floor. 
Wale = wale-piece. — A longitudinal timber fastened to a row of piles to keep 

them more rigid and in position; or fastened along a ship, cofferdam, 

caisson, wharf, quay, or jetty, etc. 
Wall. — In mining, one of the rock surfaces enclosing a vein or lode. 
Wallow. — ^To wabble, as a water-wheel revolving imevenly on its journals. 
Wallower = lantern=wheel. — See Lantern-wheel. 
Wall-plate. — In building, a horizontal timber placed on top of the wall to 

bind it together and stiffen it; and to receive the ends of girders, joists, 

rafters, roof-trusses, etc., and distribute their pressures over the wall. 

In mining, the two long pieces of timbers of the four comprising a set 

of timbering in a shaft. 
Warp. — A twist or bend as in a piece of timber which in drying has twisted. 

A rope smaller than a cable and used in towing or warping a vessel. 

Sediment. 
Warped surface. — A surface which looks as though it had been twisted 

from a true plane; applies to the surface cf boards, stones, soffits of 

arches, etc. 
Warping=bank. — A ridge of earth raised around a.:i area of land for holding 

water let in to enrich the land with warp or sediment. 
Wash-board = mopboard = skirting board. — A board around the walls of a 

room next to the floor. A base-board. 
Washer. — An annular piece of metal, leather, or rubber, etc., placed at a 

joint to prevent leakage, or under a nut to distribute pressure, etc. 

Flat washers are stamped out of plate (metal) ; cast washers are thicker 

and mostly of the O. G. pattern, a section of the edge describing a re- 
versed curve. 
Washout. — Excavation of a bank or hillside by the erosive action of water; 

the cutting or washing away of a road-bed by rains or floods; etc. 
Waste-gate. — A gate placed at the waste-outlet of a reservoir, pond, or 

lake, etc. See Tide gate. 
Waste-pipe. — A pipe for discharging waste water; an overflow-pipe. 
Waste-trap. — A device in a pipe to allow the stirplus water to escape and 

yet prevent gases from returning. 
Waterway. — An opening or passage for waste water or overflow. 
Waste-weir. — A cut in the side of a reservoir, pond, or canal, etc., for the 

discharge of surplus water. 
Waste-well. — A well into which surplus water is discharged; should have a 

permeable bed, as gravel. 
Water-butt. — A large cask used as a reservoir or tank for water. 
Water-craft. — Vessels and boats in general. 
Water-float. — A float placed in a tank, boiler, or cistern, etc., to control a 

valve. 
Water-gage. — A device for indicating the height of water in tank, boiler, or 

reservoir, etc. A connecting glass tube may be used, or a float connected 

with an indicator, or a board with elevations marked upon it; various 

contrivances. 
Water-gate. — A gateway, gate or valve for controlling the passage of water 

through an opening or pipe or channel. 
Water-hammer. — The impact of water when its volume of flow is checked 

suddenly as by a gate in a water-pipe; large gates in long pipes are 

regulated by gearing to close slowly in cases where the combined volume 

and velocity would tend to produce considerable impact if checked 

suddenly, pipes often bursting from this cause. 
Water-inch. — ^The quantity of water which will discharge through a circular 

hole one inch in diameter in 24 hours, the surface of the water being at 

top of hole; about 500 cubic feet. 
Water-main. — ;One of the main pipes in a system of water-works; the 

main pipe in each street, or the main pipe supplying several streets, or 

the main pipe leading from the reservoir or- headworks. 



1536 . GLOSSARY. 

Water-meter. — An instrument, device or apparatus for measuring the 
velocity or rate of discharge of water. 

Water-motor. — A water-wheel or turbine. Any motor deriving its power 
from the pressure and flow of water, either for direct pumping or for 
the transmission of power to any kind of machinery. 

Water=pillar = water=crane. — A vertical pipe with a swinging arm or goose- 
neck, acting as a sort of hydrant for supplying water to locomotives. 

Water=plane. — A plane passing through the water line of a ship, i. e., the 
water-surface plane; hence the terms light water-plane and load water- 
plane. 

Water=ram, — A hydraulic ram, for raising water. 

Watershed. — ^The boiindary of a catchment area or basin, i. e., the high 
ridge or divide which surrounds the area drained by a stream. 

Water=table. — In architecture, a string-course aroimd the base of a building, 
and projecting outward for ornament and as if to throw off water from 
the wall. 

Water=tower. — A large standpipe. 

Water=tube boiler. — A form of boiler containing pipes or tubes in which the 
water circulates, being heated by the surrounding flames 

Watt. — ^The unit of electric power. The volt-ampere. One watt is equiva- 
lent to the work of 0.7373 foot-pound per second = ij^g horse power. 

Watts = volt -amperes = CE = C2i?=-^ , in which (7= current in amperes, 

£ = electromotive force in volts, i? = resistance in ohms. 

Watt-Meter. — A galvanometer, for measuring simultaneously, the current 
and the difference of potential at any point of a circuit. 

Watt-second. — A unit of electrical work. 

Way-gate. — ^The tail-race of a mill. 

Ways. = The inclined timbers on which a ship moves in launching. 

Weather-boarding. — A facing of boards on a building; the boards being 
either (1) clapboards, laid horizontally with feather-edge upward and 
overlapping each other like shingles, or (2) boards nailed vertically 
and having either tongued and grooved joints or else narrower boards 
nailed over the joints, or (3) ordinary shingling. 

Weather-tile. — ^Tile used as a substitute for weather-boards. 

Wedge-valve. — A wedged shaped valve operated by a screw. 

Wedging. — ^The process of driving a wedge into a saw-kerf in the end of a 
tenon which just passes through a through-mortise, in order to expand 
the end of the tenon and make it bind firmly against the sides of the 
mortise, as in securing the helve or handle or an ax into the steel head. 

Weir. — A dam across a stream, over which the water flows. ^ A measiuing 
weir, simply called a weir and more properly a standard weir^ for measur- 
ing the flow of water. 

Weld. — ^To unite metallic substances by hammering or compression, with 
or without previous heating; if heated to fusion, a flux is used to pre- 
vent oxidation or rapid rusting. Electric welding is accomplished by 
bringing the proposed joint into a circuit, the greater resistance at the 
joint causing the abutting surfaces to become intensely hot, and then 
applying great mechanical pressure. 

Weld-iron. — Wrought-iron is weld-iron. 

Weld-steel. — Puddled steel. 

Well-boring. — The process of sinking or driving wells by drilling or boring 
through rock, often to great depths; percussion drills are most fre- 
quently employed. A form of well-sinking. 

Well-trap = sink-trap. — A trap which allows water to pass down, but pre- 
vents air or gases from passing up; such as an S-trap 

Welt. — A strip of metal riveted to two abutting plates, forming a butt- 
joint. Similarly in carpentry, a strip placed over a seam or joint to 
strengthen it. 

Wharf. — A platform or depot for vessels. Plural is wharves or wharfs. 

Wheel. — A circular body or frame revolving on 'an axis. The wheel and 
axle, lever, wedge, pulley, screw and inclined plane comprise the six 
simple machines or mechanical powers. 

Wheel-base. — The distance between centers of the extreme front and rear 
wheels of a locomotive, or car, etc. 

Wheel-window. — In architecture, a special circular window. 

Whitewash. — (1) common, quicklime and water; (2) good, whiting, size 
and water. 

Whiting. — Chalk specially prepared by drying, grinding, etc. 



WATER-METER, YARD. 1537 

Wicket. — A small gate, door or opening in a larger one; a small gate in the 
lock-gate of a canal, by which the chamber can be emptied; a small 
gate in a water-wheel chute, etc. 

Winch. — An axle with one or two bent arms or cranks for turning, as a 
common windlass or a grindstone. The axis may be geared to a separate 
drum, thus giving, more power for hoisting. 

Wind [Pronounced with long i]. — A turn, twist or bend.^ Out of wind means 
free from turns, twists, bends, etc.; used in specifications for timber, 
stone, etc. 

Windage, of Dynamo. — A term proposed for the air gap between the arma- 
ture and the pole- pieces of a dynamo. 

Wind=beam = collar=beam. — A beam joining together the rafters of a 
pitched roof. 

Wind-bore. — The end of the suction-pipe of a pump, and covered with a 
strainer to exclude foreign material. 

Wind-bracing. — Any system of braces to stiffen a frame against the pres- 
sure of the wind. 

Winder. — One of the steps in a stair where the staircase winds or turns. 

Wind-gage = anemometer. — An instrument for determining the velocity and 
pressure of the wind. 

Wind-hatch. — The opening where ore is taken out of a mine. 

Winding-engine = drawing-engine = hoisting-engine. — An engine used in 
turning a drum around which is wound a hoisting- or winding-rope. 

Windlass. — A large axle bulging into a drum at the center, or a modified 
wheel and axle, used with a winding rope in hoisting or hauling weight 
or loads, raising anchors, etc.; the ends of the axle are pierced with 
radial holes in which handspikes are inserted as levers or cranks in 
winding, and the drum is fitted with ratchet and pawls. Also operated 
by steam, as the steam-windlass. 

Wing. — A prefix used with certain names of structures which flare out like 
a wing. 

Wing-dam. — A dam projecting out from the shore so as to divert the cur- 
rent; used to deepen channels, protect river-banks from wash, etc. 

Wing-gudgeon. — A short, winged metal-shaft used as a journal for wheels 
having wooden axles. 

Wing-wall. — One of the lateral, flaring walls of an abutment, and acting as 
a sort of retaining-wall. 

Winze. — In mining, a vertical or inclined shaft which does not reach the 
surface but usually connects one level with another, for ventilation, 
passages, etc. 

Wiper. — In machinery, a sort of lever-cam attached to a (horizontal) shaft 
for the purpose of pressing against the toe or projection from another 
(vertical) shaft and raising it so it may fall again by its own weight; 
used in connection with marine-engines, stamp-mills, etc. 

Wood-screw. — A common metal screw for fastening metal or wood to wood. 
But see Lag-screw. 

Work, Unit of Electrical. — ^The erg. 

Work, Units of. — (See pages 90, 91.) 

Working-beam = walking=beam = beam. — ^The large beam of a steam-engine, 
usually the marine or pumping type, and used as an oscillating lever in 
connecting the piston-rod and crank-shaft or pump-rod. 

Working-point. — ^The part of a tool or machine producing the desired effect 
or work. 

Worm. — A shaft with a screw-thread which in revolving engages the teeth 
of a wheel and turns it; the worm is called an endless screw, and the 
wheel a worm-wheel. 

Wrecking-pump. — A steam-pump of great capacity used in pumping water 
from wrecked or damaged vessels. 

Wrench, — A tool with a lever arm and with jaws for holding or turning 
pipe, rods, bolts, heads, nuts, etc. 

Wrought-iron. — Iron that has been forged or rolled, and may be forged, 
rolled or welded. 



Yard. — A round spar with tapering end or ends. In railroading, the space 
set aside for handling and making up trains, and general switching; 
and by extension it includes the space, tracks, buildings, etc., at railway 
stations. 



1538 GLOSSARY, 

Yard-limit. — ^The extreme end of a yard, at which a sign usually cautions 
extreme care or slower speed in running trains. 

Y=level.— Common engineers' spirit-level. 

Yoke. — Any sort of a U-shaped strap or coupling. 

Yoke, Multiple-Pair Brush. — A device for holding a number of pairs of 
brushes of a dynamo electric machine in such a manner that they can 
readily be moved or rotated on the commutator cylinder. Also: 
Multiple-pair-brush-, Single-brush-, Single-pair-, Single -pair-brush- , etc. 

Y"track. — A Y-shaped arrangement of tracks leading from another track, 
and often used instead of a turntable in turning engines, cars, or 
whole trains. 



INDEX. 

(See, also, Glossary, page 1485; and Contents, page V.) 



Abbreviation of a decimal by sub- 
script, 95. 
Abrasion test of bricks, 1116. 
Abscissa and ordinate defined, 256. 
Absorption process, for ties, cost, 

375. 
Abutments (R. R. ), masonry, quan- 
tities in, table, 436, 437. 
Accelerated motion, equations of, 

279, 280, 281. 
Acceleration, 
gravity, 

. equation of, 287. 
formula, 459. 
table, 283. 
in mech., defined, 278. 
metric and English equivalents, 

table, 89. 
problem, 289. 
Acetic acid from wood, 346. 
Acetylene flame for cutting steel- 
work, 833. 
Acid 
bessemer process, 394. 
combinations, 321. 
in chem., defined, 321. 
open-hearth process, 394, 395. 
-proof compositions, 418 
Acre, acres, 
and hectars, 
equivalents, 88. 

square, equiv. (1-10), table, 80, 
equivalent in varas, 81. 
metric equivalents, 68, 81. 
per station (100-ft.) and per mile, 
(R. R.), tables, 1015. 
Acreage, square, dimensions of, 131 3. 
Acre-feet, 
and cubic feet, equivalents, 88. 
per day and cu. ft. per second, 

equivalents, 90. 
per day, irrigation equivalents, 

table, 1314. 
per month, irrigation equivalents, 
table, 1315. 
Addition and subtraction, in alge- 
bra, 100. 
Adulterants, cement, 407. 
Aerating fountain in reservoir, 1206. 
Aggregate, concrete, 416. 
Air, 
compressed-, reference data, 1482. 
dry, weight of, 1145. 
friction of, in small pipes, formu- 
la, 1189. 
necessary for combustion, calcula- 
tion, 1371, 



Air, — Cont'd, 
physical properties of, table, 514, 
relief valves, 1270. 
valves, 1270. 

weight of, at various tempera- 
tures, table, 463. 
Alabaster, defined, 339. 
Alcohol, 
boiling point of, 514. 
capacity and weight equivalents, 

table, 1376. 
fuel, 
properties of , 1370, 1375.^ 
tests of internal-combustion en- 
gines on, 1368-1370, 1374. 
melting point of, 515. 
physical properties of, table, 514. 
vapor pressure of saturation for, 

1372; table, 1373. 
wood-, a tree product, 346. 
Algebra, 100-103. 
Algebraic functions, differentiation 

of, 267. 
Alinement of tunnels, 935. 
Alkali, in chem., defined, 321. 
Alligation and center of gravity, 57. 
Alloy, alloys, 
alzene, 398. 
antimony-tin, tensile strength of, 

499. 
bismuth, 330. 
copper, 329. 
copper-gold, tensile strength of, 

497. 
lead, 329. 
lead-base, 398. 
manganese, 329. 
metal, 396. 
nickel, 329. 
platinum-iridium, composition of, 

516. 
tin, 330. 
tin-base, 398. 
vanadium steel, 399. 
Alternate-current dynamos, 
classification of, 1383. 
principle of, 1382. 
Alternating current and continuous 

current, compared, 1386. 
Alternating stresses, defined, 487. 
Alternations, in elec, defined, 1485. 
Altitude, 
in astron., defined, 947. 
of cone, defined, 134. 
of pyramid (geom.), defined, 133. 
of star, defined, 201. 
Aluminum (Al.), 318. 
bronze, 397. 
composition of, 496. 



1539 



1540 



INDEX. 



Aluminum-— Cont * d . 
bronze, — Cont'd, 
physical properties of, table, 496. 
weight of, 478. 
expansion coefficient of, 516. 
in metal castings, 330. 
melting point of, 515. 
minerals, 330. 
nickel-, 
composition of, 496. 
physical properties of, table, 496. 
paint, how made, 356. 
physical properties of, table, 496. 
wire as conductor, compared with 

copper, 496. 
wire compared with copper wire, 

in transmission, 1386. 
wire, use of, in transmission, 1386. 
Alzene (alloy), 398. 
Amalgamation, 357. 
American equivalents of Foreign 
weights and measures, table, 
92-94. 
Ammeter, defined, 1485. 
Ampere, 
as a current unit, 1379. 
defined, 1485. 
Analysis of fuels, 1350. 
Analytic Geometry, 256. 
Anchorage for suspension bridge, 

759. 
Anchorages of suspension bridges. 

Anchor-bolts, 618. 

holding power of, (ref.), 890. 
Aneroid barometer, 998. 
Angle, angles, 

and lines , geometric definitions, 128. 

arcs and chords, relation of, 130. 

between two planes, to find, 265. 

circular and time measure, equiva- 
lents, 99. 

dihedral, 261. 
defined, 132. 

flange-, of plate girders, proper- 
ties of, table, 572. 

geometric, planes and lines, 132. 

inscribed in a semicircle, 130. 

methods of plotting, 959. 

minutes and seconds to decimals 
of a degree, table, 1010. 

natural functions of, in the four 
quadrants, 138. 

of friction for various substances, 
517-521. 

of polygon, sum of interior and 
exterior, 129. 

of quadrilateral, sum of interior 
and exterior, 128. 

of repose for various substances, 
517-521. 

of triangle, sum of interior and 
exterior, 128. 

rolled, properties of, 538. ^ 

skeleton section, properties of, 
530, 531. 

steel, 
properties of, tables, 548-553. 
rivet gages for, 614. 



Angle, angles — Cont'd, 
supplement and complement of. 

139 
to lay off, (90°, 60°, 45°, 30°), 130. 
(ic = 0° - 3 60°) , trigonometric val- 
ues, 139. 
{x+y), trig, values of, 139. 
{x—y), trig, values of, 139. 
ihx), trig, values of, 140. 
(2r^), trig, values of, 140. 
(Zx), trig, values of, 140. 
(4%), trig, values of, 140. 
Animals, classification of, 347. 
Anions (in electrolysis), 357. 
Annealing, steel, 395. 
Annuities, various kinds of, 63. 
Annuity, 
and sinking fund tables, 64-65. 
final value of, formula, 63. 
present or initial value of, formula, 
63. 
Anode pole, 357. 
Anthracene, from creosote, 367. 
Anti-logarithm, 
defined, 104. 
of numbers, to find, 105. 
Antimony (56), 318. 
cast, tensile strength of, 496. 
minerals, 330. 

-tin alloy, tensile strength of, 499. 
uses of, 330. 
Antiseptic, borax as, 330. 
Apex, 
of cone, 134. 
of pyramid, 133. 
Apothecaries' 
measure (fluid) , metric equivalents, 

fable, 83. 
weight,. metric equivalents, table, 
86. 
Apothem of polygon, 129, 204. 
Apparatus, electrical-, 
classification of, 1452. 
definitions, 1451. 
Apparent solar day, defined, 202. 
Approach, velocity of, in weirs, how 

measured, 1177. 
Aqueduct, aqueducts, 
masonry, 1208. 
New Croton, size of, 1208. 
reinforced concrete, 1208. 
tunnel, Los Angeles, cost data, 939. 
Arabic 
numbers, abstract, table, 95. 
system of numbers, 1. 
Arbitration bar, mold for, 498. 
Arbor vitass, classification of, 342. 
Arc, arcs, 
angles and chords, relation of, 1 30. 
circular, 
cen. of grav. of, 207. 
defined, 129. 
mensuration of, 207. 
skeleton properties of, 532. 
table of lengths to chord 1, 210. 
tables of lengths to radius 1, 208, 
209. 
flat, circular, formulas and tables, 
211, 212, 213. 



INDEX. 



1541 



Arc, arcs — Cont'd, 
-lamps, 

elec. code rules, 1442. 
in series, elec. code rules, 1405. 
on constant - potential circuits, 
elec. code rules, 1414, 
of a great circle, defined, 135. 
parabolic, to chord 1, table, 238. 
semicircular, skeleton section, 

properties of, 531, 532, 
semi-elliptic, lengths of, table, 241. 
Arch, arches, 761. 
brick-, 764. 
bridges, reinforced concrete, cost 

of, 784. 
catenarian-, 761. 
centers for, 770. 
camber of, 773. 
loads on, 771. 
nomenclature of, 770. 
striking, 773. 
types of, 772. 
classification of, 763, 764. 
curve of intrados, 764. 
dimensions, etc., of, tables, 774- 

781. 
groined, in filter and reservoir con- 
struction, (ref.), 1292. 
ideal, 761. 
kinds of, 763. 
masonry-, 763. 
forces acting on, 767. 
lines of resistance of, 768. 
specifications, 435. 
thickness of rings, tables, 766, 
767. 
no-hinged, 782. 
nomenclature of, 763. 
parabolic-, 761. 
parts of an, 763. 
reference data, 784. 
ring, thickness of, 765. 

tables, 766, 767. 
steel and combination, 782. 
stone, stonecutter's plan, 457, 458. 
three-hinged, 783. 

stress diagram, 315. 
transformed-catenarian-, 761. 
two-hinged-, steel, 782, 783. 
Archimedes, spiral of, equation, 260. 
Area, areas, 
artesian-, defined, 1190. 
equivalents (1-10), English and 

metric, table, 80. 
metric and English equivalents. 

table, 88. 
of circle, 129. 

of curved surfaces, by calculus, 276. 
of curves, by calculus, 275. 
of pipes for given diameters, 1167. 
of plane surfaces, tables, 524. 
of regular polygon, 129. 
of triangle, 128. 

by calculus, 273. 
stress per, metric and English 

equivalents, table, 89. 
to cu. yds. per station, earthwork, 

tables, 1021-1027. 
units of, equivalents, 66. 



Argon {chem.), 318. 
Arithmetic, 
elementary, 1-13. 
practical, 55-65. 
Arithmetical 
mean, 57. 

series of progression, 57. 
Arizona land measure, English 

equivalents, 81. 
Armatures, 1486. 

of magnet, 1382. 
Armored cable, 
table, 1427. 
elec. code rules, 1411. 
Arrester, lightning, defined, 1486. 
Arroba (Philippine weight), English 

equivalent, 81. 
Arsenates, in min., classification of. 

327. 
Arsenic, 
chem., 318. 
minerals, 330. 
white, 330. 
Artesian 
area, defined, 1190. 
basin, defined, 1190. 
definitions, 1190. 
nomenclature, 1190. 
pressure, defined, 1190. 
principle, defined, 1190. 
slope, defined, 1190. 
system, defined, 1190. 
well, defined, 1190. 
Artificial stone, described, 415, 417. 
Asbestos, 
for paint, 355. 
uses of; 331. 
Ascension (right), of a star, defined, 

202. 
Ashes, 
coal, weight of, 478. 
(trees), classification of, 346. 
Ashlar masonry, defined, 432. 
Asphalt, asphalts, 
and bituminous rock deposits, of 

U. S., 1141. 
as protection for iron and steel, 

358. 
block pavement, specifications, 

1122. 
cement, defined, 405. 
coatings for waterproofing, 418. 
concrete, defined, 405. 
described, 404. 

for pavements, kinds of, 1128. 
gravel roofing, 802. 
mastic, defined, 405. 
paints for iron and steel, 372. 
pavement, 
g construction of, 1100. 

specifications, 1104, 1110, 1126. 
1128. 
paving, 
weight of, 478. 
blocks, 1100. 
properties of, 405. 
rock-, experiments, on roads, costs, 

1138. 1139. 
specifications, 1116. 



1542 



INDEX. 



Asphaltic 
cement, specifications, 1125. 
flux, 
specifications, 1125. 
use of, for coating macadam 
roads, cost, 1142. 
Asphalting iron and steel, 358. 
Asphaltum, 
liquid, specifications, 1113. 
mineral, substances related to, 3 28. 
natural, weight of, 478. 
Astronomical time, elements of, 202. 
Atmospheric pressure, 1145. 
Atom, in matter, 317. 
Atomic 
symbols, table, 318. 
theory, 316. 
weights, table, 318. 
Attachments, pressure pipe, 1269. 
Atwood's machine, problem, 288. 
Austro -Hungarian money, U. S. val- 
ues, 95. 
Automatic 
circuit-breakers, elec. code rules, 

1406. 
cut-off engines, performance of, 

1366. 
cut-outs, elec. code rules, 1406. 
fuses, elec. code rules, 1406. 
high-speed engines, performance 
of, 1366. 
Autumnal equinox, defined, 202. 
Avagadro's law of gases, 1372. 
Avenarius carbolineum, for timber, 

361. 
Avoirdupois weight, 

(long ton), metric equivalents, 

table, 86. 
(short tons), metric equivalents, 
table, 86. 
Axe, stone-, described, 429. 
Axes, 
coordinate, 266. 
of ellipse, 258. 
Axis, 
inclined, moment of inertia about, 

535-538. 
of celestial sphere, defined, 201. 
Axles, steel, specifications, 504. 
Azimuth, azimuths, 
and offsets for parallels of latitude, 

tables, 973-975. 
observation of polaris for, 949. 
of polaris, at elongation, table, 950. 
of star, defined, 201. 



B 



Babbitt-metal, 398. 

Backfilling and trenching, for sewlr, 

cost, table, 917, 918. 
Backing, masonry, defined, 431. 
Backstays and towers of suspension 

bridges, 754. 
Bacteria, removed by slow sand fil- 
tration, 1204. 
Bag of cement, weight of, 474. 
Bale, paper measure, 95. 



Ballast, 1073. 
amount required per mile of track, 

1073. 
brick and gravel, weight of, 478. 
Ballasting blasting, 923. 
Ball-mill, for cement making, 405. 
Bands, steel, for wood stave pipe, 

1209-1214. 
Bar, bars, 
moment of inertia of, 302. 
omnibus-, defined, 1487. 
steel, 
areas and weights, table, 544. 
weights and areas, table, 544. 
weight of, from specific gravity, 
table, 484. 
Barium, chem., 318. 
Barometer, aneroid, 998. 
Barometric 
correction, table, 1000. 
elevations, table, 999. 
Barrels, 
liquid, and liters, equivalents, 88. 
liquid, equivalents, 83. 
of cement, weight of, 474. 
Barshall process, for timber, 361. 
Basalt, 
composition of, table, 338. 
defined, 340. 
rock, properties of, 400. 
Bascule bridges, 
highway (ref.), 739. 
weight of, 749. 
Base, 
in chem., defined, 321. 
lines and meridians of U. S. sur- 
vey, table, 972. 
Basic 
bessemer process, 394, 396. 
open hearth process, 394, 396. 
Basin, artesian-, defined, 1190. 
Basket-handle sewers and conduits, 

properties of, table, 1302. 
Batter, masonry, defined, 431. 
Batteries, storage or primary, elec. 

code rules, 1398. 
Bauxite ore, 330. 
Bazin's 
hydraulic formula, 1189. 
weir formula, 1178. 
Beam, beams, 
and girders, properties of, tables, 

562. 
as girders, tor buildings, require- 
ments of, 820. 
box girders, steel, 
problem, 569. 
properties of, table, 568. 
calculation of, examples, 564. 
cast separatois for, table, 623. 
circular, 
moment of inertia of, 300. 
radius of gyration of, 300 
and rectangular, resistance com 
pared 301. 
concrete, slip of rods in, (ref ), 455. 
Cooper's loading, table, 708. 
deflection 
and slope of, formulas, 562. 



INDEX, 



1543 



Beam, beams, — Cont'd, 
deflection — Cont'd, 
and span for plastered ceiling, 
564. 
electric-car loadings for, tables, 

717-719. 
fiber stress in, 299. 
formulas, 562. 
girder (single I), properties of, 

table, 583. 
kinds of loading, formulas, 562. 
loads on, formulas, 562. 
longitudinal shear in, formulas, 

565. 
moments and shears, various load- 
ings, 688. 
moment of inertia of, formula, 299. 
moments of resistance of, formu- 
las, 562. 
rectangular, loads on, table, 566. 
reinforced concrete-, 
bending moment, 823, 825, 827, 

829, 832. 
formulas, 444, 447. 
table, 446. 

tests, formulas, (ref.), 585. 
Thacher's computation, 585. 
time element effect in loading, 

(ref.), 586. 
working stresses, 585. 
resisting and bending moments of, 

298. 
resisting moments of, formulas, 

562. 
shear (longitudinal) in, formulas, 

565. 
slope and deflection of, formulas, 

562. 
special I-, properties of, table, 584. 
standard connection angles for, 

615. 
steel, 
properties of, 554. 
rivet gages for, 614. 
stresses in, formulas, 562. 
stringers, wooden-, bending mo- 
ments, table, 791. 
submerged, formulas for pressure 

and moments in, (ref.), 1189. 
wooden, 
for buildings, 819. 
loads on, table, 566. 
problems in, 567. 
working stresses, table, 495. 
working loads, formulas, 562. 
working stresses, formulas, 562. 
Bearing value of concrete in beams. 

585. 
Beaume's hydrometer, 461. 
Becquerel ray, 316. 
Bed plates, for bridges, pressure on 

masonry, 705. 
Beds, masonry, defined, 432. 
Beeches, classification of, 344. 
Belgian block pavement, described, 

1100. 
Bell 
and spigot joint pipe, 1215. 
holes for pipe in rock trenches, 923. 



Bell— Cont'd, 
of cast iron pipe and special cast- 
ings, dimensions, etc., of, ta- 
bles, 1220, 1221, 1223, 1243. 
1245. 
wires, elec. code rules, 1448. 
Belt conveyor, in screening gravel, 

419. 
Belting, 
cotton, strength of, 512. 
flax, strength of, 512. 
leather, friction of, 517, 518, 519. 
Bending 
and resting moment of beams, 298. 
extreme fiber, values of concrete 

in beams, 585. 
in building materials, safe fiber 

stress, 822. 
modulus of elasticity, of timber, 

table, 493. 
moments 
and chord stresses, 307. 
and shears for engine loading, 

table, 692. 
of pins, table, 630. 
problem in, 637. 
strength of metals, table, 496. 
tests of timber, table, 492. 493. 
Bends, 
in cordage, 669. 

in pipe lines, loss of head in, 1160. 
Bents, 
grass-hopper, 789. 
trestle-, 788-792. 
Bemouilli, lemniscate of, equation 

of, 260. 
Beryllium, chem., 318. 
Bessemer 
processes, 394, 395. 
steel, defined, 1487. 
Beton-Coignet, manufacture of, 417. 
Bevel siding lumber (fir), classified, 

389. 
Binder, asphalt pavement, specifi- 
cations, 1110, 
Binomial formula, 101. 
cube root by, 102. 
square root by, 102. 
Bins and bunkers, reference data, 

1481. 
Birches, classification of, 344. 
Bismuth (Bi.), 318. 
alloys, 330. 
minerals, 330. 
tensile strength of, 496. 
uses of, 330. 
Bitulithic pavement 
patents, 1127. 
specifications, 1105, ll26. 
Bitumastic enamel, coating, 359. 
Bitumen, 
described, 404. 
weight of, 478. 
Bituminized-brick, 
specifications, 1116. 
gutters, specifications, 1115. 
Bituminous 
and asphalt rock deposits of U. S.. 
1141. 



1544 



INDEX, 



Bituminous — Cont ' d . 
compounds, grouping of, 1141. 
rock pavement, 1100. 
Black or galvanized pipe, table, 

1284. 
Blast-furnace, 392. 
Blasting, 
and drilling in tunneling, 934. 
ballasting-, 923. 
gelatin, 352. 
-holes, drilled with well-driller, 

cost, 926. 
in rock, drill holes for, 922. 
mat, woven, 923. 
powder, composition of, 350. 
stumps, cost data, 916. 
submarine, cost, 925. 
Block, blocks, 
paving, size of, 1129. 
shapes, properties of, 533. 
stone, kinds of, 417. 
Blow-off, blow-offs, 
described, 1280. 
branches, 
cast iron pipe, table, 1230, 1257. 
with manhole, cast iron pipe, 
tables, 1231, 1258. 
Blowpipe characteristics, 328. 
Bluestone, 
composition of, 331. 
defined, 402. 

physical properties of, table, 
507. 
Board, boards, 
classification of, 388. 
measure, 379. 
table, 380 
Booster, defined, 1380. 
Boat, 
drill-, for submarine work, 925. 
spikes, table, 628. 
Bodies, 
falling, table, 283. 
impact effect, 304. 
Boiler, boilers, 
cement, 402. 

staybolts, etc., in, (ref.), 1378. 
plate steel (open hearth), specifi- 
cations, 501. 
power reqmred for channeling ma- 
chines, 421. 
steam, 1361-1363. 
efficiency and rating, 1361. 
horse -power of, defined, 1361. 
' settings, notes, 1362. 
tests of coal as fuel, table, 1353. 
Boiling-point, 
absolute j defined, 513. 
defined, 513. 

of chemical elements, table, 318. 
of liquids, table, 514. 
of substances, tables, 514. 
Bolsters, for bridges, specifications, 

705. 
Bolts 
and nuts, tables. 618-621. 
rail-, 1060. 

standard, for fastenings, 618. 
Bomb colorimeter, described, 1352. 



Bond, 
between concrete and steel, tests, 

(ref.), 585-586. 
English, defined, 437. 
Flemish, defined, 437. 
masonry, defined, 432. 
value of concrete in beams, 585. 
in brickwork, 764. 
of concrete to steel, 823. 
Boneblack for paint, 355. 
Borates, 330. 

in min., classification of, 327. 
Borax, uses of, 330. 
Boron 
chem., 318. 
minerals, 330. 
Borings, 
diamond drill, cost data, 917 
in soil, 866. 

wash drill, cost data, 916. 
Bosses on cast iron pipe, 1280. 
Botanical materials, 340. 
Boulder 
foundation, 865. 
pavement, specifications, 1107. 
Bow's notation for trusses, 309 
Box-girder, 
steel beam- . 
problem, 569. 
properties of, table, 568. 
Boxes, 
gate, table, 1288. 
outlet', elec. code rules 1429. 
resistance-, elec. code rules, 1449. 
switch-, elec. code niles, 1429. 
Braces, rail-, 1071. 
Bracing, 
lateral-, i.f bridges, problem, 697. 
portal-, of bridges, 698. 
vertical-, of bridges, 698 
Brake horsepower (B. H. P.), formu- 
la, 1374. 
Branches, 
blow-off, 
cast iron pipe, table, 1230, 1257. 
with manhole, cast iron pipe, 
tables, 1231, 1258. 
hydrant, cast iron pipe, table, 1229. 
pipe, cast iron, 
L's, T's, crosses, table, 1225. 

1250-1254. 
Y's, table, 1227. 1228, 1255, 
1256. 
Brass, 
cast, 
physical properties of, table, 496. 
sheet, wire, etc., weight of, table, 
478. 
expansion coefficient of, 516. 
friction of, 518, 519. 
melting point of, 515. 
table 397. 

wire, physical properties of, 496. 
Brazing-metal, 397. 
Breakers, circuit-, elec. code rules, 

1434. 
Breakwaters, 901 
cost data, 902-901 
materials for concrete of, 904. 



INDEX, 



1545 



Breakwaters — Cont'd, 
notable, table of, 903. 
reaction-, 905. 

reinforced concrete caissons for, 
cost data, 904. 
Breast wheel, described, 1336. 
Brick, bricks, 
abrasion test, 1116. 
arches, 764. 
bituminized-, 

gutters, specifications, 1115. 

specifications, 1116. 
bonding of, 764. 
block 

paving specifications, 1105. 

pavement specifications, 1105. 
clinker, 415. 

common, manufacture of, 415. 
crushing tests, 522. 
expansion coefficient of, 516. 
face-, 415. 
friction of, 518. 
glazed, 415. 
hard, 415. 

kinds of, described, 415. 
manufacture of, 415. 
mdisonry, 437. 

compressive strength of, 511. 

quantities of brick and mortar 
in, table, 438. 
pavement, 

described, 1100. 

cost, (ref.), 1142. 

specifications, 1109, 1129. 
paving, 415. 

grout filling, 1109. 

handling and piling, 1109. 

manner of laying, 1109. 

of country road, cost, 1141. 

rolling and tamping, 1109. 

size of, 1129. 

specifications, 1124. 

tar filling, 1109. 
piers, 

compressive strength of, 511. 

crushing tests, table, 522. 
■ pressed, size of, 438. 
rattler test, 507, 1116. 
sand-, manufacture of, 417. 
sewer-, 415. 

walls, thickness of, formula, 
1306. 

66-in., cost, 1310. 
sidewalks, specifications, 1103. 
size of, 415. 
soft, 415. 

specific gravities of, table, 474. 
specifications, 1116. 
street pavements, proper construc- 
tion of, 1106. 
temperature stress for 1 60° F.,523. 
terra cotta, manufacture of, 415. 
tests of, 507. 
various kinds, physical properties 

of, 507. 
vitrified, 415. 

pavement, specifications, 1121, 
1123. 
weights of, table, 474. 



Brickwork, 437. 
bonds in, 764. 
friction of, 521. 
in buildings, 

safe loads for, 821, 826. 
^ weight of, 821. 
lime mortar for, 403. 
mortar, kinds used, 438. 
Bridge, bridges, 683. 
arch-, 

(see also Arches), 774-784. 

dimensions, etc., of, tables, 774- 
781. 

reference data, 784. 

reinforced concrete-, cost of, 784. 
bascule-, weight of, 749. 
cantilever-, 740. 

references, 741. 
clearance, 699. 

combination highway-, and de- 
tails, 729. 
concrete, surface finish, 454. 
economic length of spans, 683. 
electric-car loadings for, 716. 
electric railway, 716. 
estimating weights of, 685. 
ferry-, and details, 898. 
floor, 

specifications, 700. 

trestle, 788-790. 
highway-, 720. 

live load data for, table, 728. 

nickel and carbon steel, specifi- 
cations, 737; table, 738. 

references, 739. 

typical loading for, 727. 

unit stress sheets, 720. 
impact for, table, 709. 
masonry, specifications, 434. 
movable,-, 

references, 749. 

weights of steel in, 748. 
nickel-steel and carbon-steel, 737. 

738. 
piers, 

contents of, 889. 

masonry, 889. 
pins, 629. 

portals, types of, 698. 
references, 687. 
railroad, 688. 

proportion of parts, specifica- 
tions, 702. 

references, 713. 

stresses allowable in, specifica- 
tions, 702. 

types of, 699. 
railway, electric, 716.^ 
reinforced concrete, highway, cost 

data, 738. 
steel of high grades used in, 499. 
steel, railroad, 

compression formulas, table, 710. 

compressive stresses, specifica- 
tions, 703. 

specifications for, 699. 

tensile stresses, specifications, 
702. 

weight of, 710. 



1546 



INDEX. 



Bridge, bridges — Cont'd, 
steel, 
' specifications, 500. 

weight of, formulas, 686. 
suspension-, 750. 
details and specifications, 756- 

760. 
miscellaneous data, 760. 
weights of materials in, table, 
758. 
swing-, 742. 
timber framing, 730. 
wind pressure for, 697. 
Briggs logarithms, defined, 104, 
Briquette, briquettes, 
cement, 
form of, 410. 
tests of, 411. 

mortar, amount of water to use, 
408. 
molds, form of, 410. 
storage tank (ref.), 418. 
British thermal unit (B. T. U.), 
defined, 1347. 
equivalents of, 90. 
table, 91. 
British-French thermal unit (Lb.- 

Col.), defined, 1347. 
Broach ^ , 

channeling, described, 419. 
machine, in quarrying, 419. 
Broken -stone, 
pavement, construction of, 1099. 
size for concrete, 417. 
voids in concrete, 416. 
Bromides, min., classification of, 325. 
Bromine, chem., 318. 
Bronze, 
aluminum, 397. 
composition of, 496. 
physical properties of, table, 496. 
etc., weights of, table, 478. 
expansion coefficient of, 516. 
gun (metal), physical properties 

of, table, 497. 
manganese-, 397. 
physical properties of, table, 497, 
(ref.), 399. 
melting point of, 515. 
paint, how made, 357. 
phosphor-, 397. 
physical properties of, table, 
497. 
silicon-, 397. 

tensile strength of, 497. 
table, 397. 
tobin-, 397. 
physical properties of, table, 
497. 
Brownstone, 
composition of, table, 334. 
formation of, table, 334. 
Brush-holaers, defined, 1489. 
Building, buildings, 812. 
bearing power of soils for, 867. 
codes, 819-829. 

construction, references, 830-834. 
fireproof-, requirements for, 819. 
foundation loads for, 866. 



Building, buildings,— Cont'd, 
materials, 
fire tests on, 523. 
heat effect on, 523. 
temperature stresses in, 523. 
regulations for reinforced concrete 
construction, by Nat'l Assn. 
Cem. Users, 831. 
stone, 
and cement, 400. 
artificial, 415. 

physical properties of, table, 507. 
safe loads for, 821. 
specific gravities of, table, 474. 
thickness of joints. 457. 
weights of, table, 474. 
wall, stonecutter's plan, 457. 
Bucket dredges, 928. 
Bulkhead and pierhead lines, 892. 
Bumping posts (ref.), 1092. 
Bunkers and bins,ref erence data, 1481. 
Buoyancy, 1152. 
center of, 1153. 
Burden, in tunneling, defined, 933. 
Bumetizing 
process, for ties, cost, 375. 
timber, 360. 
cost, 375. 
Bundle, paper measure, 95. 
Bushel, bushels, 
and cubic feet, equivalents (1-9), 

table, 485. 
and gallons, equivalents (1-9), 

table, 485. 
and hectoliters, equivalents(l-lO), 

table, 84. 
and yards-inch, equivalents (1-9), 

table, 485. 
(dollars per) 
and francs per hectoliter, equiv- 
alents (1-10), table. 98. 
and marks per hectoliter, equiv- 
alents (1-10), table, 98. 
(dollars per U. S.) and shillings 
per British bushel, equivalents 
(1-10), table, 98. 
equivalents, 67. 
heaped, measure of, 84. 
metric equivalents, 68. 
of produce, 
weight of, table, 478. 
weights of, 482. 
per acre and hectoliters per hectar, 

equivalents (1-10), table, 84. 
(shillings per Br.) and dollars per 
U. S. bushel, eqmvalents (1- 
10), table, 98. 
struck, metric equivalent, 84. 
Bush hammer, described, 430. 
Bushings and tubes, elec. code rules, 

table, 1430. 
Butt (liquid) or pipe, equiv., 83. 
Butternut tree, 343. 
By-pass, defi ed, 1489. 



C and N in Kutter's formula, experi- 
mental determination of, 11 88 



INDEX. 



1547 



Caban (Philippine measure), Eng- 
lish equivalent, 81. 
Cabinet, cabinets, 
cut-out-, elec. code rules, 1439. 
projection, 261. 
Cable, cables, 
armored-, 
elec. code rules, 1411. 
table, 1427. 
catenarian, lengths of, table, 752. 
length, equivalents, 68. 
suspension bridge-, curves of, 750. 
telephone, 676. 
wrappings, 755. 
Cableways and conveyors, reference 

data, 1481. 
Cadmium 
chem., 318. 
minerals, 329. 
Caesium, cJtem., 318. 
Caisson 
disease, prevention of, (ref.) 890. 
method of tunneling, 936. 
open-, 876. 
pneumatic, 880. 

reinforced concrete-, for break- 
waters, cost data, 904. 
Calcim.ine, 355. 

Calcination, in cement making, 405. 
Calcined magnesite, 330. 
Calcium, 
cement, properties of, 403. 
chem., 318. 
chloride, 
cost, 1138. 

experiments on roads, costs, 11 38 
minerals, 329. 
oxide, 403. 
Calculating machine, Thatcher, 127. 
Calculus, 266. 
Differential, 266. 
Integral, 272. 
California land measure, English 

equivalents, 81. 
Calomel, 329. 

Calorie (Cal.), defined, 1347. 
Calorimeter test, described, 1352. 
Camber 
in bridges, 705. 
in cantilever bridges, 741. 
of arch centers, 773. 
Canadian canal systems, table, 1323. 
Canal, canals, 
Chicago, material, work, costs, 923 
commercial, of the U. S., table, 

1329, 1330. 
earth-, experimental values of N 
in Kutter's formula for flow in, 
1188. 
evaporation and seepage in, 1199. 
flow, surface and mean velocity of, 

1183. 
for water supply, 1207. 
irrigation, 
locating, (ref.), 1318. 
miscellaneous data, (ref.), 1318. 
velocity in, 1317. 
large irrigation, dimensions and 
grades of, table, 1317. 



Canal, canals, — Cont'd, 
maintenance and operation, cost 

data, 1329. 
miscellaneous data, (ref.), 1331. 
navigable, 1320. 
Panama, 
distance between Atlantic and 

Pacific ports, table, 1328. 
excavation, cost data. 916. 
steam shovel work, cost data, 
919. 
proportioned for maximum dis- 
charge, 1161. 
seepage and evaporation in, 1200. 
Cannon ball, energy of, 294. 
Cantilever bridges, 740. 
camber in, 741. 
references, 741. 
Canvas, strength of, 512. 
Caoutchouc, weight of, 480. 
Capacities, 
dry, 
equivalents, 67. 
equivalents (1-10), English and 

metric, table, 84. 
metric, English equivalents, 
table, 84. 
in gallons, of pipes, table, 246- 

247. 
liquid, 
equivalents, 67. 
equivalents (1-10), English and 

metric, table, 83. 
metric, English equivalents, 
table, 82. 
liquid and dry, metric and English 

equivalents, tables, 88. 
of wires, elec. code rules, table, 

1448. 
overload-, in elec, 1469. 
volumes and weights, equivalents 
(1-9), tables, 485. 
Capitalization of annuity, table, 65. 
Caps, 
castiron pipe, tables, 1234, 1266. 
wooden trestle, 788. 
Car 
axles, specifications, 504. 
gondola-, large capacity, (ref.), 

1091. 
houses, elec. code rules, 1421. 
wiring and equipment, elec. code 
rules, 1417. 
Carbolic acid, from creosote, 367. 
Carbolineum avenarius, for timber, 

361. 
Carbon 
chem., 318. 

Lb. of, oxidized with perfect effi- 
ciency, equivalents of, table, 
91. 
minerals, 330. 
Carbonates 
in min., classification of, 327. 
of lime, 403. 
Carbonic acid, boiling point of, 514. 
Carburetters, for vaporizing liquid 

fuels, 1371. 
Card process, for ties, cost, 375. 



1548 



INDEX. 



Cardinal points of celestial sphere, 

defined, 201. 
Carrying capacities of copper wires, 

table, 1404. 
Cartridge enclosed fuses, elec. code 

rules, table, 1438. 
Castings, 
aluminum in, 330. 
iron (gray), specifications, 498. 
malleable, 393. 
steel, 
physical properties of , 504; 

table, 499. 
specifications, 393, 503. 
test pieces, 504. 
testing, 504. 
Cast-iron, 
columns, loads on, tables, 606, 607. 
details for combination bridge, 

732. 
drilling,' 392. 

expansion coefficient of, 516. 
flange pipe, table, 1236. 
for buildings, 819. 
friction of, 518, 519. 
hard, 392. 

in buildings, safe stresses, 826. 
melting point of, 515. 
physical properties of, table, 497. 
properties of, 392. 
puddling, 392. 
refining, 392. 
separators, table, 623. 

use of, 623. 
temperature stress for 160° F., 523. 
washers, weights and dimensions, 

table, 624. 
weight of, 480. 
Cast-iron pipe, 1214. 

and specials, weights and dimen- 
sions of, tables, 1219, 1267. 
bells of, dimensions, etc., tables, 

1220, 1221, 1223, 1243, 1245. 
curves, table, 1244, 1248, 1249. 
for various pressures, table, 1216. 
formulas for designing, 1215. 
friction heads in, table, 1217. 
lead required per joint, 1216; 
^ table, 1222. 
jute required per joint, table, 

1222. 
specials, described, 1280. 
specifications, 1239. 
standard length of, 1215. 
variation allowed, 1222. 
weights and dimensions, tables, 

1220, 1222, 1243-1246. 
weight of, table, 1216. 
Cast separators for wooden string- 
ers, 623. 
Cast-steel, 
expansion coefficient of, 516. 
open hearth, 396. 
specifications for, 393. 
Cast-tin, 
physical properties of, 499. 
weight of, 679. 
Cast-zinc, physical properties of, 499. 
Catch basins, sewer, 1308. 



Catenarian 

arch, 761. 

arcs, lengths of, table, 752. 

cable of suspension bridge, 751. 
Catenary, 

graphical solution of, 753. 

length of, by calculus, 276. 

parameter of, values of, table, 752. 

sewers and conduits, properties of, 
table, 1301. 

transformed, 753. 
Cathode pole, 357. 
Cations (in electrolysis), 357. 
Catty (Philippine weight), English 

equivalent, 81. 
Caulking pipe joints, 1215. 
Cavil, described, 427. 
Cedar, cedars, 

classification of, 342, 343. 

grading rules, 388, 390. 
Cedar block pavement, specifica- 
tions, 1110, 1128. 
Ceiling 

construction, 818. 

lumber (fir), classified, 389. 

plastered, beam calculations, 564. 
Celestial sphere, elements of, 201- 

202. 
Cellulose nitrate, 351. 
Cement, cements, 402. 

adulterants, 407. 

and lutes, useful to engineers, 418. 

as protection for iron and steel, 
358. 

asphalt, 405. 
described, 404. 

asphaltic, specifications, 1125. 

barrel of, weight, 474. 

bitumen, 404. 

briquette, 
form of, 410. 
molds, form of, 410. 
storage of, 410. 

builder's, 403. 

chemical composition, 406. 

clinker, grinding, 405. 

crucible, (ref.), 418.- 

elastic, (ref.), 418. 

endurance of, 406. 

expansion coefficient of, 516. 

filler in brick pavement, 1107. 

final set of, defined, 406. 

fineness of, discussed, 406. 

fineness test, 407. 

foreign, weights per barrel, 474. 

gravel, 
defined, 339. 
roofing, 802. 
swellage when loosened, 911. 

grout, injected in sub-foundatioi 
442. 

hardening or set, 406. 

hydraulic, described, 404. 

in concrete, economy in, 416. 

initial set of, defined, 406. 

iron, (ref.), 418. 
• leather, (ref.), 418. 

miscellaneous, 402. 

mixing, for testing, 410. 



INDEX, 



1549 



Cement, cements, — Cont'd, 
molding, for testing, 410. 
mortar 
briquettes, amount of water to 

use, 409. 
defined, 408. 
for buildings, 819. 
mix for concrete, 417. 
strength of, 507, 508. 
weight of, table, 475. 
natural-, 
constancy of volume, specifica- 
tions, 412. 
defined, 412. 

fineness specifications, 412, 414. 
hydraulic, manufacture of, 404. 
spec. grav. specifications, 412. 
specifications, 

(A. S. T. M.), 411. 
(Engrs. U. S. A.), 414. 
strength of, 507, 508. 
tensile strength specifications, 

412, 414. 
time of setting, specifications, 

412, 414. 

weight per barrel, 414. 
normal consistency test, 408. 
paste, defined, 408. 
pats, 412, 413, 414. 

for testing, 411. 

tests of, 411. 
pavement, described, 1099. 
paving, specifications, 1121. 
physical properties of, 507. 
plaster of paris, 404. 
Portland-, 

constancy of volume, specifica- 
tions, 413. 

cost, 418. 

defined, 412, 413. 

fineness specifications, 412, 413. 

impurities in, 413. 

manufacture of, 404, 405. 

spec. grav. specifications, 412, 
413. 

soundness specifications, 413. 

specifications, 

(A. S. T. M.), 412. 
(Engrs. U. S. A.), 413. 

strength of, 507, 508. 

temperature stress for 160** F., 
523. 

tensile strength, specifications, 

413, 414. 

time of setting, specifications, 
412, 414. 
preservative qualities of, 444. 
Puzzolan-, 
fineness specifications, 414. 
soundness specifications, 414. 
spec, grav. specifications, 414. 
specifications, (Engrs. U. S. A.)» 

414. 
tensile strength, specifications, 

415. 
time of setting, specifications, 
415. 
quick-setting, 406. 
requirements of good, 406. 



Cement, cements, — Cont'd. 
Rosendale-, manufacture of, 404, 
-sand mix for concrete, 417. 
sea-water-proof, 418. 
set or hardening, 406. 
setting, rate of, 406. 
sidewalk, 

described, 1099. 

specifications, 1117. 
sieve, for fineness test, 407. 
slag, manufacture of, 404. 
slow-setting, 406. 
solvents, 402. 
soundness of, 406, 407. 
specific gravity 

of, apparatus for finding, 40 7o 

of, table, 474. 

test, 407. 
specifications, 

(A. S. T. M.). 411. 

(Engrs. U. S. A.), 413. 
stone, (ref.), 418. 
strength ratios of compression and 

tension, 508. 
testing, 406. 

specific gravity, 407. 

standard methods, 407. 

standard sand, 409. 

standard sieve, 407. 

time of setting, 409. 
tests 

for constancy of volume, 411. 

on specimen cylinders, 508. 

tensile strength, 411. 

Vicat needle test, 408. 

weights of, table, 474. 

Cementation process, 395. 

Cementing materials, 402. 

in rocks, table, 334. 
Cent, U. S. money, 95. 
Center, centers, 
for arches, 770. 

camber of, 773. 

loads on, 771. 

nomenclature of, 770. 

striking, 773. 

types of, 772. 
of celestial sphere, defined, 201. 
of gravity 

by Alligation, 57. 

of distributed force, 296. 

of parallel forces, 295. 

of plane figure, formulas, 301. 

of plane surfaces, table, 524. 

of solids, 303. 

of trapezoid, 847. 
of gyration, 303. 
of oscillation, 303. 
of pendulum, 287. 
of percussion, 303. 
of pressure, 846, 847. 

formulas, 1150. 

on vertical orifices and weirs, 
table, 1151. 
Centigrade and Fahrenheit scales, 

equivalents, table, 465. 
Centigram, English equivalents, 85. 
Centiliter, English equivalents, 8U 
82, 84. 



1550 



INDEX. 



Centimeter, centimeters, 
and inches, 
cubic, 
equivalents (1-10), table, 82. 
equivalent, 88. 
equivalents (1-10), table, 70. 
equivalents, 88. 
square, 
eqmvalents, (1-10), table, 80. 
equivalents, 88. 
cubic, 
and liquid ounces, equivalents 

(1-10), table, 83. 
English equivalents, 68, 81. 
English equivalents, 68, 70. 
-grams and inch-pounds, equiva- 
lents, 89. 
square, English equivalents, 68,79. 
Centrifugal 
force, 
formulas, 297. 
of train on curve, specifications, 

702. 
of train, problem, 297. 
pumps, 1367. 
steam pumps, 1367. 
Century, time measure, 99. 
Cerium, chem., 318. 
Cesspools, 1296. 
Chains, 957. 
and meters, equivalents, 88. 
steel hoisting-, formulas, 1484. 
surveyors', equivalent, 68. 
Chalk, 
-rock, 401. 
weight of, 478. 
Channel, channels, 
block, properties of, 533, 534. 
columns, 
properties and safe loads, tables, 

601-603. 
standard dimensions of, 596. 
rolled, properties of, 537. 
skeleton section, properties of, 530. 
standard connection angles for, 

615. 
steel, 
properties of, table, 556. 
rivet gages for, 614. 
Channeling machines, 
air used, 420. 
boiler power for, 421. 
dimensions, etc., table, 421, 
in canal excavation, 924. 
in quarrying, 420. 
in tunneling, 934. 
weights, etc., table, 421, 
Chapman valve with wedge-shaped 

gate, nomenclature, 1285. 
Characteristic and mantissa, of log- 
arithms, 104-105. 
Charcoal, birch, oak, etc., table, 47-8. 
Check valves, 
described, 1288. 
horizontal, table, 1278. 
vertical, table, 1279. 
Waring's, 1296. 
Chemical 
analysis of fuels, 1350. 



Chemical — Cont'd, 
compounds, 321. 
elements, table, 318. 
energy, examples of, 1346. 
equivalents, table, 318. 
substances, common names of, 
324. 
Chemistry of materials, 316. 
Chestnuts, classification of, trees, 

344. 
Chezy's hydraulic formula, 1167. 
Chicago drainage canal, 
data, 1324. 

material, work, costs, 923. 
Chimneys, reference data, 1480. 
Chisel, 
splitting, described, 427. 
stone, described, 427. 
tooth, described, 427. 
Chlorate explosives, 351. 
Chloride, chlorides, 
in min. classification of, 325. 
of zinc, 
cost, 375. 
for timber, 361. 
Chlorine, chem., 318. 
Chord, chords, 
angles and arcs, relation of, 1 30. 
lengths of curved rails, tables, 

1066, 1067. 
of circle, defined, 129. 
(or circular arc), mensuration of, 

207. 
(of flat circular arc) , formulas and 

tables, 211, 212, 213. 
of truss, stress in, 305. 
platting angles by, 959. 
stresses 
and bending moments, 307. 
in Pratt truss, concentrated 

loads, 695. 
in Warren truss, concentrated 
loads, 696. 
to radius 1, table of, 959. 
Chrome 
steel, 396. 

-vanadium steel, 399. 
Chromium, chem., 318. 
Cinder concrete 
cubes, tests of, 510. 
for buildings, 831. 
physical properties of, 510. 
proportions, tests, 510. 
weight of, 475. 
Circle, circles, 
and octagon, inscribed and circum- 
scribed, 131. 
and square, 
inscribed and circumscribed ,131. 
relations, 220, 221, 222. 
and triangle, inscribed and circum- 
scribed, 130. 
areas in sq. ft. for diameters in ft. 

and ins., table, 1157 
area of, 129. 
center of, to find, 130. 
diameter 

(in fractions) to circum (in deci- 
mals), table, 226-229. 



INDEX. 



1551 



Circle, circles, — Cont'd, 
diameter — Cont'd. 
to area, 
in decimals, table, 232, 233. 
in inches, table, 230, 231. 
(ft and ins.) to area (sq, ft.), 

table, 234, 235. 
to circum. in decimals, table, 
224, 225.- 
equation of, 257. 
normal to, 257. 
tangent to, 257. 
hollow, properties of, 528. 
intersection 
with parabola, ^58. 
with straight line, solution, 257. 
mensuration of, 204. 
properties and parts of, 129. 
properties of, 528. 
(semi-), properties of, 528. 
skeleton section, properties of, 
531. 
Circular 
arc, skeleton section, properties of, 

532. 
beam, 
moment of inertia of, 300. 
radius of gyration of, 300. 
cell, skeleton, properties of, 531. 
conduits 
and sewers, properties of, table. 

1300. 
best for maximum discharge, 
1161. 
cylinder, moment of inertia of, 

302. 
measure (7r),142; table, 99. 
motion, 286. 
orifices, center of pressure on, 

formulas, 1151. 
pendulum, 287. 

plate, moment of inertia of, 302, 
ring, moment of inertia of, 302. 
sector, properties of, 528. 
segment (half-), properties of , 528. 
sewers and conduits, properties of, 
table, 1297. 
Circumference, 
circular measure, 99. 
of circle, ratio to diameter, 129. 
(semi-), circular measure, 99. 
Circuit-breakers, 
automatic-, elec. code rules, 1406. 
elec. code rules, 1404, 1434, 
Circuits, grounding, low-potential, 

elec. code rules, 1401. 
Cities, population of, in U. S., 1202, 
City-lot surveying, 966. 
Clamp fastenings for wire rope, 675. 
Clam-shell bucket dredge, 928. 
Classification of yellow pine lumber, 

387, 388, 
Clay, 
composition of, 331. 
dry, friction of, 521. 
foundation, 864, 865. 
moist, friction of, 521. 
tiles, for roofs, 800. 
weight of, table, 478, 



Clearing 
and grubbing, 906, 
cost data, 916. 
reference data, 920. 
road specifications, 1101. 
Cleats, elec. code rules, 1430. 
Clevices, dimensions of, table, 632. 
Clinker, cement, grinder, 405. 
Clip fastening for wire rope, 675. 
Coagulants used in settling basins, 

1204. 
Coal, 
anthracite, etc., weight of, 478. 
boiler test of, table, 1353. 
classified for heating value, table, 

1350. 
consumption 
per boiler horsepower, 1362. 
per h.-p. hour, 1363. 
heating value of, tables, 1350-51. 
mineral, classification of, 328. 
mines, explosives permissible in, 

354. 
storage of, in salt water, (ref.), 
1378. 
Coal-tar, 
coatings, 361, 

composition and source, 365. 
for roads, specifications, 1135. 
manufacture of, 1131. 
paints, (ref.), 374. 
production of, table, 366, 
properties of , 1131. 
Coating, 
bitumastic-enamel, 359, 
protective, for dry-dock, 359. 
Cobalt, 
blue, 329, 
chem, 318, 
minerals, 329. 
Cobblestone 
gutters, 1099, 

pavement, construction of, 1099. 
Codes, 
building-, 819-829. 
electric, 1393. 
Coefficient, coefficients, 
c, in Kutter's formula, 1168-1172, 
of contraction of jet, 1175. 
differential-, defined, 266, 
of discharge 
of jet, 1175. 

through circular orifices, (ref.), 
1189. 
of elasticity, defined, 486. 
of expansion, 
formulas, 516. 
of gases, table, 464. 
of liquids, table, 468. 
of substances, table, 516. • 
of heat resistance, 1377. 
of impact, for bridges, table, 709. 
of restitution, 304. 
of roughness N, values of, 1168. 
of velocity of jet, 1175. 
temperature-, in elec, table, 1475. 
Coexsecants, 
natural, table, 167-175. 
trigonometric, defined, 136. 



1552 



INDEX, 



Coffer-dams, 
kinds of, described, 868. 
leakage in, 870. 
pneumatic foundation-, 882. 
Coils, 
in eleCy kinds of, defined, 1493. 
economy-, elec. code rules, 1414. 
Coins (Foreign) 
and paper notes, equivalents, 
(1-10,-50-100) in U. S. money, 
table, 97. 
value of, in U. S. money, table, 
96. 
Coke 
concrete, strength of, (ref.), 455. 
weight of, table, 478. 
Cold rolling, for mill scale, 358. 
Collectors, in elec.^ defined, 1383, 

1493. 
Collector rings, defined, 1383. 
Collision (or impact), formulas, 303. 
Colors, 
by mixing, 356. 
conventional, for maps, 966. 
Columbium, chem., 318. 
Columns, 
cast iron, loads on, table, 606, 607. 
channel-, 
properties and safe loads, tables 

601-603. 
standard dimensions of, 596. 
concrete, plain and remforced, 

tests, 610. 
eccentric loading on, 588. 
for buildings, requirements of, 820. 
formulas, 587-609. 
Gordon's formula for, 592. 
H-, properties of, tables, 608. 
ideal-, formulas, 588. 
in buildings, formulas, 821, 824, 

826. 
nickel and carbon steel, tests, 609. 
Phoenix-, properties and safe loads, 

tables, 604-605. 
properties and cables of, 587-610. 
reinforced -concrete , 
formulas, 449. 
in buildings, 823, 825, 827, 829, 

832. 
working stresses for, 609. 
Ritter's formula for, 589. 
shearing effect on, 587. 
steel-, 
discussion, 596. 
moment of inertia of, problem 

in, 637. 
standard sections, 596. 
ultimate strength of, table, 597. 
straight-line formulas for, 593. 
wooden-, 
safe loads on, table, 594. 
Smith's formula for, 593. 
working stresses, talDle, 495. 
Z-bar, dimensions and safe loads, 
tables, 598-600. 
Combination 
and permutation, 56. 
highway bridge and details, 729. 
roof truss, design of, 806-810. 



Combined stresses, tests, (ref.) 522. 
Combustion, 
air, necessary for, calculation, 1371. 
of fuels, calculations, 1352. 
Commutators, defined, 1383, 1494. 
Complement and supplement of an 

angle, 139. 
Complementary angles, defined, 128. 
Composition and gravel filling for 
wood block pavement, specifi- 
cations, 1128. 
Compound 
and simple units, equiv., table, 88. 
chemical, 321. 
curves, (R. R.), 1011. 
engines, performance of, 1366. 
interest, 
methods, 59-62. 
table, 62. 
Compressed-air 
painting, with cost, 374. 
process, 879. 
quarrying by, 423. 
reference data, 1482. 
Compression 
in steel bridge members, table, 

710. 
of earth, how estimated, 910-913. 
tests of timber, table, 490. 
strength of metals, table, 496. 
Compressive strength value of con- 
crete in beams, 585. 
Concrete 
aggregate, 416. 
asphalt-, defined, 405. 
beams, slip of rods in, (ref.) 455. 
block, blocks, 450. 
hollow, building, 
specifications, 450. 
testing, 451. 
hollow, safe loads on, 829. 
masonry, 450. 
bonding new to old, (ref.) 455. 
bridges, highway, cost data, 738. 
broken -stone, voids in, 416. 
cement -sand mix, 417. 
cinder- 
and stone-, weight of, 455. 
corrosion of steel in, 374. 
for buildings, 831. 
physical properties of, 510. 
proportions, tests, 510. 
tests on cubes, table, 510. 
weight of, 475. 
coke-, strength of, 455. 
columns, 
tests, 508. 
formulas, 508. 
corrosion of iron in, (ref.) 456. 
cubes, 
compression tests of, 509. 
trap rock, compression tests of. 
509. 
curbing, specifications, 1108. 
curbs, specifications, 1111. 
defined, 416. 

depositing in water, methods, 441 
dry, medium and wet, 440. 
economy of cement in, 416. 



INDEX. 



1553 



Concrete— Cont'd, 
elastic limit under compression, 

509. 
expansion coefficient of, 510, 516; 

table. 516. 
expansion joints in, 454. 
fire-resisting qualities of, 444. 
for buildings, 819. 
forms, costs, (ref.), 1293. 
frost-resisting, economical, 416, 
German specifications, 442. 
gravel voids in, 416. 
gutters, specifications, 1111. 
heat effect on, 510. 
in buildings, safe loads for, 821. 
in sea-water, tests of, (ref.) 891. 
in sub-foundations, 442. 
kinds of, 416. 

laying, underwater, (ref.) 891. 
masonry, 439. 

material for, of Buffalo break- 
water, 904. 
matrix, 416. 

mix, to determine proportions, 4 16. 
mixers, 439. 
efficiency of, 453. 
traveling, (ref.) 455. 
mixing, 416, 440. 
modulus of elasticity of, 510. 
natural, physical properties of , 510. 
new layer on old, 440. 
oil-mixed, for waterproofing, 455. 
paints for, (ref.) 375. 
pavement, specifications, 1128. 
paving for streets, cost, etc., (ref.) 

1142. 
permeability of, under water pres- 
sure, 453. 
physical properties of, 508-510. 
piles, 875. 

metal-shell, 875. 
piles, 
reinforced-, 675. 
water-jet, 875. 
pile piers for steamship terminal, 

(ref.) 900. 
placing and ramming, 440. 
proportions, 416, 440. 

of mix for bridge, 454. 
rammers, 440. 
reinforced-, 443. 
aqueduct, 1208. 
arch bridges, cost of, 784. 
beams, 
formula, 444, 447. 
table, 446. 

tests, formula, (ref.) 585. 
Thacher's computation, 586. 
workmg stresses, 585. 
bridges, railroad, 712. 
columns, 
formulas, 449. 
working stresses for, 609. 
construction for buildings, 822- 

834. 
design, office methods, (ref .^ 456. 
formulas of A. S. C. E., 446. 
French Gov't rules, (ref). 454. 
proportions, 444, 



Concrete — Cont'd, 
reinforced-, — Cont'd, 
references, 456. 
strength of, 445. 
trestles, 792. 
use of, (ref.) 455. 
sand voids in, 416. 
sidewalks, specifications^ 1111, 

1129. 
size of broken stone for, 417. 
slabs, flat, calculation of, (ref.) 455. 
spreading and ramming, 440. 
-steel 

adhesion tests, (ref.) 454. 
construction, for buildings, 822- 

834. 
ties, 1072. 
stone-, weight of, 475. 
subaqueous-, placing, 440. 
surface finish, 454. 
surfaces, 
scrubbed, specifications, (ref.) 

455. 
treatment of, (ref.) 455. 
telegraph poles, 1477. 
tension test of Portland, 509. 
various mixtures, Portland, tests, 

508. 
voids 
determined for, 440. 
in, formula, 1118. 
waterproofing 

data for. 453, 455. 
work, Chicago rules for measuring, 
891. 
Condensers and reactive coils, elec 

code rules, 1443. 
Condensing engines, performance of, 

1366. 
Conductivity 
in elec, 1494. 
of copper, standard, 1466. 
Conductors, 
aluminum and copper wire com- 
pared, 1386. 
elec. code rules, 1394. 
portable, elec. code rules, 1448. 
size of wire, in transmission, 1386. 
underground-, elec. code rules, 
1404. 
Conduits, 
and flumes, irrigation, 1317. 
and sewers, hydraulic properties 

of, tables, 1296-1306. 
experimental values of A^ in Kut- 
ter's formula for flow in, 1188. 
for water supply, 1207. 
ideal sections for maximum dis- 
charge, 1161. 
interior-, elec. code rules, 1412, 

1450; table, 1428, 
miscellaneous data, (ref.) 1291- 
1294. 
Conduit wire, elec. code rules, 1427. 
Cone, cones, 
geom., 134, 

altitude of (geom.), defined, 134. 
and spheres, relations, 250. 
frustum of {geom.), defined, 134. 



1554 



INDEX, 



Cone, cones, — Cont'd, 
mensuration of, 248. 
of sphere, defined, 135. 
volume of (geom.), 134. 
Conglomerate, 
composition of, table, 334. 
formation of, table, 334. 
Conic frustum, mensiiration of, 248. 
Conical nozzle, 1177. 
sections, 256. 
wedge and frustum, 249. 
Conjugate angles, defined, 128. 
Connections, 
of wood stave with cast iron pipe, 

1280. 
standard, for I-beams and chan- 
nels, 615. 
Conoid, parabolic-, mensuration of, 

254. 
Construction, 
of geometric figures, 1 30. 
railroad, 1016. 
Consumption of water, 1202. 

in cities, table, 1203. 
Continuous-current 
and alternating-current, com- 
pared, 1386. 
dynamos, 
classification of, 1384. 
principle of, 1384. 
Contraction, 
coefficient of, 1175. 
of earth, how estimated, 910-913. 
Contracts and specifications, refer- 
ence data, 1484. 
Convective, electric-, defined, 1494. 
Converse pipe, patent lock joint, 

table, 1282. 
Converter, rotary, defined, 1380. 
Conveyors 
and cableways, reference data, 1481. 
systems, for earth. 907. 
Coordinate, coordinates, 
axes, 256. 

planes, 132, 261-265. 
rectangular, defined, 256. 
Coping, masonry, defined, 432. 
Copper (Cu), 318. 
alloys, 329, 

cast, wire, etc., weight table, 479. 
conductivity of, standard, 1466. 
expansion coefficient of, 516. 
friction of, 519. 

-gold alloy, tensile strength of, 497. 
melting point of, 515. 
minerals, ores, 329. 
physical properties of, table, 497. 
sheeting, for timber, 361. 
uses of, 329. 
wire, 
as conductor, compared with 

aluminum, 496. 
as conductor, compared with 

silicon -bronze, 497. 
carrying capacity of, table, 1404. 
compared with aluminum wire, 

in transmission, 1386. 
table, electric, 1388-1391. 
weight of, table, 479. 



Coppersmith's cement, 402. 
Cord, 

in cordage, 668. 

flexible-, elec. code rules, 1413, 
1426. 

foot, of wood, metric equiv., 82. 

of wood, metric equivalent, 82. 
Cordage, 668. 

terms, technical, 668. 
Corduroy roads, described, 1098. 
Core 

drills, rock, 922. 

for dams 
and reservoir embankment, ma- 
cadam as — , 859. 
concrete as — , 860. 
Cores, in elec, defined, 1495. 
Corinth canal, data, 1320. 
Cork, weight of, 478. 
Corliss engine, cylinder of, 1364. 
Corpuscles in atoms, 316. 
Corpuscular theory, 316. 
Corrosion 

of iron in concrete, (ref.) 455. 

of steel, in cinder concrete, 374. 
Corrosive sublimate, for timber, 361. 
Corrugated 

metal, properties of, 532. 

sheet, properties of, 532. 

sheeting, strength of, (ref.) 561. 

steel, 
roofing, 801. 
strength of, 801. 
Corrugations, cycloidal, properties 

of, 532. 
Cosecant, cosecants, 

defined, 136. 

logarithmic, table, 176-198. 

natural, table, 167-175. 
Cosine, cosines, 

defined, 136. 

logarithmic, table, 176-198. 

natural, table, 144-166. 
Costs (see items in question). 
Cotangent, cotangents, 

defined, 136. 

logarithmic, table, 176-198. 

natural, table, 144-166. 
Cotter pins, 629. 
Cotton , 

belting, strength of, 512. 

tensile strength of, 512. 
Cottonwood tree, 343. 
Coulomb, 1495. 

-volt, 1495. 
Counter rods, 634. 
Countersunk rivets, 616. 
Course, masonry, defined, 431 
Conversed sine, -sines, 

defined, 136. 

natural, table, 144-166. 
Cramps, masonry, defined, 432. 
Crandall described, 428. 
Cranes and derricks, reference data, 

1480. 
Creosote, 

commercial, 367. 

composition and manufacture, 365. 

cost of, 366. 



INDEX, 



1555 



Creosote, — Cont'd, 
extracted from timber, 

analysis of, 367; table, 368. 
for timber, best oils to use, 372. 
from coal tar, 366. 
injected in timber, inspection of 

treatment, (ref.) 374. 
in ties and timber, analysis of, 370. 
in timber well preserved, 365. 
oil, 
cost, 375. 

from timber, analysis of, 371. 
in ties, 361. 
weight of, 479. 
production and importation, table, 

366. •^ 

treatment of wood paving block, 

1126. 
well or tank, 367. 
Creosoted 
poles, effect on linemen, 374. 
wood block pavement, specifica- 
tions, 1126. 
Creosoting 
plant, (ref.) 360, 374. 

for poles, 374. 
timber, 360. 
cost, 375. 
wooden poles, 373. 
works in France, (ref.) 374. 
Crib 
coffer-dams, 869. 
ferry-, and details, 894. 
piers, 877. 

pneumatic foundation-, 880. 
Critical 
point of a gas, defined, 512. 
pressure, 
defined, 513. 
of gases, table, 514. 
of liquids, table, 514. 
temperature 
of a gas, defined, 512. 
of gases, table, 514. 
of liquids, table, 514. 
volume, defined, 513. 
Cronstadt and St. Petersburg Canal, 

data, 1320. 
Cross, 
block, properties of, 533. 
skeleton section, properties of, 530. 
Crosses, 
cast iron pipe, table, 1225, 1250- 

1254. 
Matheson pipe, table, 1281. 
Crossover tracks, frog spacing, table, 

1089. - 
Cross-sections of tunnels, 934, 935, 939. 
Cross ties, railroad, 1069. 
Crosswalks, flagging, 1103. 
Croton (New) aqueduct, size of, 1208. 
Crown (Austrian), equiv. (1-10,-50- 
100) in U. S. money, table, 97. 
Crowning streets, 
formula and table, 1123. 
in Chicago, formula, 1143. 
Crucible 
cast steel, manufacture of, 395. 
cements, (ref.) 418. 



Crude 
oils, 
and residuums, compared, 1134. 
products from, in refining, 1133. 
petroleums, test properties, 1134. 
Crushed-stone sidewalk, specifica- 
tions, 1105. 
Cube, cubes, 
and sphere, relations, 250. 
and squares, tables of, uses, 636. 
and square roots, by slide rule, 126. 
defined, 133. 
roots 
and square roots, common tables, 

31-50. 
by binomial formula, 102. 
engineers' tables, 21-24. 
to find, 20. 
squares and roots,''common tables, 

31-43. 
tables of, for structural detailing, 
639-642. 
Cubic 
centimeters (m 1,), 
and liquid ounces, equiv. (1-10), 

table, 83. 
English equivalents, 81. 
equations, 143. 
feet 
and bushels, equivalents (1-9), 

table, 485. 
and gallons, equivalents (1-9), 

table, 485. 
and meters, equivalents, 88, 
and tons, equiv. (1-9), table, 485. 
and yards-inch, equiv. (1-9), 

table, 485. 
per minute, discharge, equiva- 
lent, 90. 
per second, discharge, equiva- 
lent, 90. 
per second, irrigation equiva- 
lents, table, 1314. 
per time, discharge, equiv., 90. 
foot, metric equivalent, 82. 
inch, metric equivalent, 82. 
inches and centimeters, equiv., 88. 
measure, 
English, metric equiv., table, 82. 
metric, English equiv., table, 81. 
parabola, 1013. 
yards 
and meters, equivalents, 88. 
in pipes, table, 246, 247. 
metric equivalent, 82. 
per station, for areas, earthwork, 

tables, 1021-1027. 
weight of, from specific gravity, 
table, 484. '^ 

Culminations of polaris, 949. 
Culvert, culverts, 
concrete-, 782. 
masonry, specifications, 435. 
pipe, tables, 1307. 
Curb, curbs, 
cement mortar, (ref.) 1142. 
concrete, specifications, 1111. 
on concrete foundation, 1122. 
trench, specifications, 1108. 



1666 



INDEX. 



Curbing, 
concrete, specifications, 1108. 
stone, specifications, 1108. 
vitrified clay, for roads and streets, 
1142. 
Current, ciurents, 
electrical-, definitions, 1451. 
breaker, in elec, defined, 1496. 
meters (water), 1186. 
use of, 1186. 

use and care of, by U. S. G. S., 
(ref.) 1189. 
wheel (water), described, 1336. 
Curvature 
and refraction 
corrections in leveling, 987. 
table, 988. 
earthwork correction for, 1069. 
head, in pipe lines, 1160, 
radius of, of ellipse, 765. 
railroad, economic considerations, 
997. 
Curved 
pipe, cast iron, tables, 1224, 1248, 

1249. 
rails, 
chord lengths -of, tables, 1066, 

1067. 
middle ordinates of, tables, 1064- 
1067. 
surfaces, areas of, by calculus, 276. 
track, 
to find degree of curve of, 1066. 
turnouts from, 1084. 
Curves, 
analysis of, (plane-), 256. 
areas of, by calculus, 276. 
centrifugal force of train on, 297; 

specifications, 702. 
cycloidal, motion on, 286. 
easement-, (R. R.), 1013. 
elevation of outer rail on, 298. 
finding intersection of, 103. 
in pipe lines, loss of head in, 1160. 
lengths of, by calculus, 276. 
of projectile, 285. 
parabolic oval-, (ref.) 766. 
railroad, 1006. 
problems, 1011. 
radii of, table, 1007. 
reversed-, 1011, 1012. 
tangents and externals to 1°, 
table, 1009. 
spiral-, (R. R.) 1013. 
to lay out, 130. 
track gage on, 1073. 
turnout-, formula, 1083. 
vertical (R. R.), 1006. 
Cut-out, cut-outs, 
cabinets, elec. code rules, 1439. 
automatic-, elec. code rules, 1406. 
elec. code rules, 1404. 1434, 1449. 
Cutters for hydraulic dredges, (ref.) 

932. 
Cycloid, 
equation of, 260. 
normal to, 260. 
properties of, 236. 
radius of curvature of, 260. 



Cycloidal 
corrugations, properties of, 532. 
curve, motion on, 286. 
spindle, 254. 
pendulum, 267. 
Cyclopean masonry, 
dam, 860. 
defined, 1497. 
Cylinder, cylinders, geom., 134. 
and sphere, relations, 250. 
area, volume, 244 
hollow, dia. to area, capacity, mean 
radius, volume, weight (water) , 
table, 246-247. 
maximum, inscribed in sphere, 269. 
moment of inertia of, 302. 
of Corliss engine, 1364. 
piers, 878. 
frictional resistance of, 878. 
platform-, 879 
pneumatic, 879. 
volume ot, geom., 134. 
wind pressure on, 797. 
Cypresses, classification of, 342, 343. 



Dalton's atomic theory, 316. 
Dam, dams, 844. 
arched masonry, (ref.) 861. 
backwater of, height, (ref.) 860. 
buttressed-, design of, (ref.) 860. 
Cyclopean masonry, 860. 
earth-, 

quantities in, table, 868. 

shrinkage data, 914. 
fixed, types of, 844, 
foundation of, pressure on, 848- 

851. 
gravity-, 

design of, 862. 

stability of, 846. 
high, masonry, dimensions of 

eight, table, 859. 
hydraulic fill, cost data, 919. 
hydrostatic pressure on, 845. 
masonry, 

design of, 862. 

quantities in, tables, 855, 866. 
movable, (ref.) 861. 
multiple-arch, described, 860. 
profile effect on, 864. 
reference data, 869-862. 
rock fill, quantities in, table, 867. 
rubble concrete, 860. 
safety factor against overturning, 

860. 
shear in, 850. 
steel-, (ref.) 869. 
surcharged, (ref.) 869. 
triangular-, 847. 
Day. days, 
number of, between two dates, 

table, 61. 
sidereal-, defined, 202. 
solar, 

defined, 202. 

and degrees (longitude), equiva- 
lents, 99. 



INDEX, 



1567 



Dead-men piles, 874. 
Dead oil of coal tar, 367. 
Decagon, 
inscribed in circle, 131. 
mensuration of, 204. 
Decay of timber, 359. 
Decigram, English equivalents, 85. 
Deciliter, English equivalents, 81, 

82, 84. 
Decimal, decimals, 
abbreviation of, by subscript, 95. 
and fractions, short methods of 
multiplication and division, 
11-13. 
to find root of, by logarithms, 
105. 
Decimeter, 
English equivalents, 70. 
square, English equivalents, 79. 
Deck cantilever bridges, 741. 
Declination, 
in astron., defined, 947. 
of a star, defined, 202. 
Decorative lighting systems, elec. 

code rules, 1414. 
Deflection 
and slope of beams, formulas, 

562. 
angle, of railway curve, 130. 
Degree, degrees, 
circular and time measure, equiva- 
lents, 99. 
(longitude) and time, equiv., 99. 
mariners', equivalents, 68. 
of curve of laid track, to find, 1066. 
or hour, decimals of, for minutes 
and seconds, table, 1010. 
Dekagram, English eqmvalents, 85. 
Dekaliters, 
and pecks (U. S.), equiv. (1-10), 

table, 84. 
English equivalents, 81, 82, 84. 
Dekameter, 
English equivalents, 70. 
square, English equivalents, 79. 
Delta-metal, 397. 
composition of, 479, 497. 

and weight, 479. 
tensile strength of, table, 497. 
Density, 
defined, 460. 
of steam, defined, 1356. 
of water, metric, 67. 
relative, of gases to air and water, 
table, 464. 
Dependent variables, defined, 256. 
Depreciation diagrams and tables, 

(ref.) 1293. 
Depth 
of plate girder, economic, 684. 
of trusses, economic, 684. 
Derrick, derricks, 
and cranes, reference data, 1480. 
used in sewer excavation, cost 
data, 916. 
Descriptive geometry, 261. 

problems of construction, 2'63-5. 
Design of sewers, modem procedure 
in, (ref.) 1311. 



Details, . 

combination bridge, 730. 

structural, 611. 
Detonation of explosives, 352, 
Dew-point, defined, 1190. 
Diagrams, 

load line, in struc, 311. 

stress, general rules, 310. 
Diameter 

and radius, circular measure, 99. 

of circle, 
defined, 129. 

ratio to circumference, 129. 
Diamond drill borings, cost data, 

table, 917. 
Dicken's run-off formula, 1198. 
Dielectric strength, 1463. 
Differential 

calculus, 266. 

coefficient, defined, 266, 

defined, 266. 
Differentiation, 

defined, 266. 

of algebraic functions, 267. 

of expotential functions, 270. 

of inverse trigonometric functions, 
271. 

of logarithmic functions, 270. 

of trigonometric functions, 270. 

rules for, 267, 270, 271. 

successive, 271. 
Dihedral angles, 261. 

defined, 132. 
Dime, U. S. money, 95. 
Dimension 

stones defined, 4 33. 

stuff, (lumber) , classification of, 388 
Dipper dredge, 928, 931. 
Dipping tank, pipe, 1282. 
Directrix, of parabola, 258. 
Dirt roads, described, 1098. 
Discharge 

and velocity of sewers and con- 
duits, tables, 1296-1306. 

coefficient of, 1175. 
through circular orifices, (ref.) 
1189. 

(cu. ft., galls,, liters, etc.) per time, 
equivalents, table, 90. 

from nozzles, 1175. 

from orifices, 1175; table, 1176. 

from tubes, 1175. 

in circular brick sewers, table, 
1299. 

of water through a pipe, formula, 
1156. 

pipe of dredge, 931. 

through small pipes, table, 1284. 

through wood stave pipe, table, 
1210-1214. 
Discount and interest, 59. 
Disk piles, 874. 
Distance 

between points on Earth's surface, 
to find, 201. 

polar, of a star, defined, 202. 
Distributing 

reservoirs, 1205. 

system, 1280. 



1558 



INDEX. 



Division and roots, algebraic, 102. 
Docks, 

kinds of, 892. 

wharves and piers, 892. 
Dodecagon, 

inscribed in circle, 131. 

mensuration of, 204. 
Dodecahedron, defined, 132. 
Dollar, U. S. money, 95. 
Dolomite, 

compression tests of, 511. 

properties of, 401. 
Dolphin, ferry-, and details, 896. 
Dote, in lumber, defined, 387. 
Double integration for polar moment 

of inertia, 535, 536. 
Dowels, masonry, defined, 432. 
Dozen, equivalent of, 95. 
Drachm (see also dram). 

apoth., fluid, metric equiv., 83. 
Drafted stone, defined, 427. 
Drain pipes, 1295. 
Drains, storm water, design of, (ref .) 

1311. 
Drainage 

of irrigated lands, costs, 1319. 

road specifications, 1101. 
Drams 

(apoth.) and milliliters (c c), 
equivalents (1-10), table, 83. 

(apoth., fluid), metric equiv., 83. 

(apoth.), metric equivalents, 86. 

(avoir.), metric equivalents, 86. 
Drawbridges, 742. 

calculation of, 745. 748. 

center-bearing, three supports,746. 

jack-knife, 748. 
. moments and reactions, table, 745, 
747. 

reactions, tables, 744, 745, 747. 

rim -bearing, four supports, 743. 

stress-diagrams for, 745. 
Drawings, shop-, for structural steel, 

cost of, 665. 
Dredge, dredges, 

discharge pipe of, 931. 

hydraulic-, cutters for, (ref.) 932. 

types of, 927, 931. 
Dredged material, methods of meas- 
uring, 927, 931. 
Dredging, 927. 

Detroit river, cost data, 929-930. 

gold, 930, 932. 

method of tunneling, 936. 

Panama canal, cost data, 919. 
Drift 

and heading methods of tunneling, 
933. 

bolts, 618. 

in tunneling, defined, 933. 
Drill, drills, 

boat, for submarine work, 925. 

compressed-air, for rock excava- 
tion, 923. 

diamond-, borings, cost data, 917. 

holes in rock, spacing, 922. 

in quarrying, 422. 

percussion, described, 422, 

rock-, 922. 



Drill, drills,— Cont'd, 
used in tunneling, 934. 
wash-, borings, cost data, 916. 
Drilling 
and blasting in tunneling, 934. 
blasting holes with well-driller. 

cost, 926 
cast iron, 392. 

holes for rock excavation, 922. 
in tunnel, cost data, 939. 
machine-, in rock cuts, economy 

of, 923. 
submarine, cost, 925, 930. 
Drop 
(apoth.) or minim, metric equiva- 
lent, 83. 
siding lumber (fir), classified, 389. 
test for steel 
castings, 50.4. 
rails, 503. 
Dry 
and liquid capacities, metric and 
English equivalents, tables, 88. 
capacities, 
equivalents, 67. 
equivalents (1-10), English and 

metric, table, 84. 
metric and English equivalents, 
table, 84. 
quarts (U. S.) and liters, equiva- 
lents (1-10), table, 84. 
Dry-dock, -docks, 
coating for, 359. 

concrete expansion joints in, 454. 
steel, floating, (ref.) 900. 
Dry- 
masonry, specifications, 435. 
measure, English (U. S.) metric 

equivalents, table, 84. 
process, in cement making, 405. 
rubble work, road specifications, 
1101. 
Dulong's formula for combustion, 

1352. 
Duodecimo numbers, table, 95. 
Dust 
preventives 
for road surfaces, 1131. 
road experiments, costs, 1136. 
suppression on N. J. roads, cost, 
1143. 
Duty 
6f pumps, formula, 1367. 
of water in irrigation, tables, 1315- 
1317. 
Dynamics, 288. 
Dynamite, 
cartridges, 353. 
charge, how prepared, 353. 
commercial, list of, 354. 
defined, 351. 
for submarine blasting, 930; 

cost 925. 
grades of, 353. 
handling and use of, 353. 
in quarrying, 419. 
in tunnels, cost data, 939* 
kinds of, 352. 
properties of, 353. 



INDEX. 



1559 



D y namit e , — Cont * d . 
thawing, 353. 
used in tunneling, 934. 
Dynamos, 
alternate-current, 

classification of, 1383. 

principle of, 1382. 
continuous-current , 

classification of, 1384. 

defined, 1384. 
defined, 1379 . 
foundations for, 867. 



Eagle and double eagle, U. S. money, 

95. 
Earth, earths, 
canals, experimental values of N 
in Kutter's formula for flow 
in, 1188. 
compression of, how estimated, 

910-913. 
contraction of, how estimated, 

910-913. 
dams, 

quantities in, table, 858. 

saturization of, 860. 

shrinkage data, 91 
defined, 909. 
embankment, 

rolling, 909. 

shrinkage vertical in, 915. 

usual slopes of, 909. 
excavation, 

labor item in, 908. 

Panama canal, cost data, 919. 
fill, 

effect of water on, 910. 

jarring effect on, 910. 

puddling effect on, 910. 

shrinkage recom-mendations,914. 

temperature effect on, 910. 
friction of, 521. 
frozen-, excavation by machine, 

cost data, 921. 
Fuller's, uses of, 331. 
infusorial, uses of, 331. 
loading and conveying, 907. 
loosening, 907. . 
methods of handling, 906. 
pressure, 

Rankine's theory, 839-840. 
. theories, 835-840. 
roads, application of oils to, 1135. 
shrinkage of, 909. 

how estimated, 910-913. 
surface of, distance between points 

on, to find, 201. 
swellage of, how estimated, 910-913. 
voids in, 910. 

table, 911. 
weight of, 475. 
Earthwork, 906. 
calculations, 1016. 

for ground slopes, 1030. 
classification 

in Chicago drainage canal, 923. 

(R. R.). 919. 



Earthwork, — Cont'd, 
computation, 1055. 
correction for curvature, 1059. 
cost data, 915. 
"haul," 1059. 
reference data, 920. 
shrinkage, 
experiments, 913. 
railroad specifications for, 913. 
tables, 
formulas for extending, 104^. 
list of, 1017. 

methods of calculating, 1028. 
Easement curves, (R. R.), 1013. 
East point, of celestial sphere, de 

fined, 201. 
Economy coils, elec. code rules, 

1414. 
Economic 
depth 
of plate girders, 684. 
of trusses, 684. 
length of spans, 684. 
problems in calculus, 269. 
Edge grain (in lumber), defined, 

389. 
Edgestones, specifications, 1102. 
Efficiencies of turbines, 1343. 
Egg-shaped sewers 
and conduits, properties of, table, 

1304. 
velocities in, table, 1305. 
Elastic 
bodies, impact effect, 304. 
limit 
affected by stresses, 487. 
defined, 487. 
of metals, table, 496. 
Elasticity, 
coefficient of, defined, 486. 
modulus of, defined, 486. 
Elbe and Trave canal, data, 1322. 
Electric 
apparatus, cost data, 1477. 
car loadings, for bridges, 716. 
code, 1393. 

heaters, elec. code rules, 1407. 
horse -power, equivalents of, 90. 
hydraulic problem, 1379. 
lighting, cost data, table, 1478. 
line poles, 373. 

creosoting, 373. 
motors, 
railway, 1471. 
speed classification, 1452. 
power 
and lighting, 1379. 
cost data, table, 1478. 
plants, costs, 1477. 
sources and uses of, 1 385. 
units, 1379. 
railway bridges, 716. 

steel, weight of, formulas, 686. 
resistance, formula, 1520. 
steam-, problem, 1379. 
transmission of power, 1385. 
waves, 1380. 

wires, attraction and repulsion 
between, 1381. 



1660 



INDEX, 



Electrical 
apparatus, classification of, 1462. 
conductivity 
of aluminum and copper com- 
pared, 496. 
of copper and silicon - bronze, 
compared, 497. 
currents, definitions, 1451. 
definitions and technical data, 

1451. 
efficiency, 1455. 
energy, examples of, 1346. 
insulation, 1462. 
machines, 
classification of, 1462. 
cost data, 1477. 
defined, 1379. 
mechanical and heat units, equiva- 
lents, table, 91. 
notation signs, 1471. 
rating, 1454. 
regulation, 1461. 
rotating machines, definitions, 

1451. 
standardization rules, 1451. 
stationary apparatus, definitions, 
1451. 
Electricity 
and magnetism, principles of,13€0. 
as a form of energy, 1379. 
composition "of , 317. 
source of, 1380. 
Electro-chemistry, 357. 
Electrolysis, defined, 367, 1499. 
Electrolytes, 367. 
Electro- 
magnet, 1381. 
metallurgy, defined, 367. 
Electromotive force, defined, 1382. 
Electron, defined, 317. 
Electro-plating, 367, 358. 
Elements, 
metallic, table, 318. 
(the), of matter, 317. 
Elevating-grader used in railroad ex- 
cavation, cost data, 916. 
Elevation of outer rail on curves, 

formula, 298. 
Elevator bucket dredge, 928. 
Elimination, in algebra, 103. 
Ellipse, 
axes of, 268. 

circumference, length of, 239. 
equation of, 268. 
(calculus), 268. 
normal to, 269. 
tangent to, 259, 268. 
false, or oval, 259. 
foci {sing, focus) of, 268. 
hollow, properties of, 529. 
how to draw, 238. 
properties of, 238, 529. 
radius of curvature of, 259, 766. 
Ellipsoid, mensuration of, 255. 
Elliptic 
arcs, formulas for lengths of, 239, 

240. 
cone, defined, 134. 
cylinder, geont., 134. 



Elliptic— Cont'd, 
segment, 
area of, 242. 
chord of, length, 242. 
Elm, elms, 
friction of, 519. 
classification of, 345. 
Elongations of polaris, 949. 
Emanation of matter, 317. 
Emery, weight of, 479. 
Enamel (bitumastic) coating, 359. 
Enclosed fuses, 
cartridge type, elec. code rules 

table, 1438. 
elec. code rules, 1436. 
cut-outs, elec. code rules, 1435. 
Energy, 
and matter, phenomena of, 1436. 
electrical, dennitions, 1379. 
equation of, 293. 
forms of, examples, 1346. 
heat a form of, 1346. 
kinds of, 1346. 

law of the conservation of, 1346. 
lost during impact, 304. 
of cannon ball, 294. 
of flowing water, (ref.) 1188. 
of moving mass, formula, 1346. 
transformation of, 1346. 
Engine, engines, 
axles, specifications, 504. 
Corliss, cylinder of , 1364. 
load diagrams, 690, 691, 701. 

Cooper's, 707. 
internal-combustion, tests of, on 
alcohol fuel, 1368-1370, 1374. 
moment diagram, 691. 
steam-, 1363-1366. 
diagrams, described, 1364. 
horsepower, problem, 1363. 
principle of, 1363. 
English 
and metric 
approximate equiv., table, 68. 
areas, equiv. (1-10), table, 80. 
dry capacities, equiv. (1-10), 

table, 84. 
fundamental unit equivalents, 66. 
lengths, equiv. (1-10), table, 70. 
liquid capacities, equiv. (1-10), 

table, 83. 
system of weights and measures, 

66-91. 
volumes, equiv. (1-10), table, 82. 
weights, equiv. (1-10), table, 86. 
bond, defined, 437. 
cubic measure, metric equivalents, 

table, 82. 
dry measure, metric equivalents, 

table, 84. 
land measure, square, metric 

equivalents, table, 81. 
liquid measure, metric equiva- 
lents, table, 83. 
money, U. S. Value, 96. 
Entropy 
diagrams, defined, 1356. 
of the liquid, formula, 1356. 
Entry head, defined, 1160. 



INDEX. 



1561 



Ephemeris (solar) tables, reference 

to, 202. 
Eqtiilateral triangle, inscribed in 

circle, 131. 
Equalizers, elec. code rules, 1395. 
Equation, equations, 

algebraic, defined, 100. 

of Payments, 61. 

of time (astronomical) , 202. 

quadratic, squaring, 102. 

simultaneous, 
examples in, 103. 
graphical, 256. 
Equator, of celestial sphere, defined, 

201. 
Equilateral hyperbola, 260. 
Equilibrium, 

kinds of, 1152. 

of floating bodies, 1153. 

of forces, in mech., 294. 

three laws of, 1153. 

unstable, 303. 
Equinox, 

autumnal, defined, 202. 

vernal, defined, 202. 
Equivalents (see units in question) . 
Erbium, chvm., 318. 
Estimating weights of bridges, 685. 
Ether 

and ether waves, 1 380. 

boiling point of, 514. 
Evaporation, 1199. 

effect on gases, 513. 

from ice, 1199. 

from land surface, 1199. 

from running water, 1199. 

from snow, 1199. 

from water surface, 1199. 

in the U. S., table, 1199. 

of 1 lb. water from and at 212° P., 
equivalents of, table, 91. 
Excavating 

by machine, cost data, 915. 

granite in open cuts (R. R.), cost, 
925. 

Chicago rules for measuring, 891. 

earthwork-, cost data, 915. 

for buildings, 819. 

of Chicago drainage canal, mate- 
rial, work, costs, 923. 

Panama canal, cost data, 916, 919. 

railroad-, cost data, 916. 

road specifications, 1101. 

rock, 922. 
by channeling machines, 924. 

sewer-, cost data, 916. 

subway-, cost data, 916. 
Exciter in elec, defined, 1383. 
Expanded metal, 814. 
Expansion 

bolts, 618. 

by heat, 516. 

coefficient of, 
formulas, 516. 
of gases, table, 464. 
of liquids, table, 468. 
of substances, table, 516. 

joints in concrete, 454, 

linear, surface and volumetric, 5 16. 



Expansion — Cont'd, 
of functions, 271. 
of gases, 513. 
of metals in cooling, 330. 
rollers, 
for bridges, pressure on, 705. 
segmental-, table, 635. 
Expert valuations and reports, refer- 
ence data, 1484. 
Explosion pump, direct-acting, 1378. 
Explosives, 360. 
detonation of, 352. 
in quarrying, 422. 
in tunnels, cost data, 939. 
permissible in coal mines, list, 354. 
unmixed, 363. 
Exponential functions, differentia- 
tion of, 270. 
Exponents, algebraic, examples in, 

100. 
Exsecant, exsecants, 
defined, 136. 
natural, table, 167-175. 
Externals and tangents to a 1° curve, 

table, 1009. 
Eye-bars, 

• bending stresses in, formula, 686. 
in bridges, specifications, 706. 
length to form heads of, formula, 

686. 
properties of, 631. 
Eye-bolt fastenings for wire rope, 
675, 



Face brick, strength of, 507. 
Factor, factors, 
algebraic-, of equation, defined, 

100. 
greatest common, to find, 6. 
of safety, 
defined, 487. 
for timber, 496. 
in building, 821. 
prime, of numbers, table, 3-5. 
Fahrenheit and Centigrade scales, 

equivalents, table, 465. 
Fahrenheit's hydrometer, 462. 
Fallacy (apparent) in algebraic solu- 
tions, 103. 
Falling bodies, tables, 283. 
False ellipse or oval, 269. 
Farm surveying, 964. 
Farraday's ring, 1382. 
Fascines, 905. 
Fastenings, wire-rope, 675. 
Fat, beef, etc., weight of, table, 479. 
Fathom, equivalents, 68. 
Feathers, plug and, described, 426. 
Feet, 
and chains, equivalents, table, 958 
and meters, 
cubic, equivalents, 88. 
cubic, equiv. (1-10), table, 82. 
equivalents, 88. 
equivalents (1-10), table, 70. 
square, equivalents, 88. 
square, equiv. (1-10), table, 80. 



1562 



INDEX. 



I 



Feet, — Cont'd, 
cubic and acre-, equivalents, 88. 
per minute and miles per hour, 

equivalents, 89. 
per second 

and meters per sec, equiv., 89. 
. and miles per hour, equiv., 89. 
and miles per minute, equiv., 
89. 
per sec. per sec. and meters per sec. 

per sec, equivalents, 89. 
to meters (1-1000), equivalents, 
table, 71-74. 
Feldspar, uses of, 331. 
Felsite, 
composition of, table, 338. 
formation of, table, 338. 
Fencing, road specifications, 1101. 
Fender piles, 893. 
Fermentation of timber, 359. 
Ferric structures, protection from 

corrosion, 372. 
Ferry 
bridge and details, 898. 
crib and details, 894. 
dolphin and details, 896. 
house, D. L. & W., Hoboken, - 

(ref.) 900. 
slips and bridge aprons, 893. 
Fertilizer, 
greensand as a, 340. 
marl as a, 340. 
Fifth powers 
and fifth roots, engineers' table, 

26-27. 
square roots of, engineers' table, 
25. 
Filling, fillings, 
for woods, block pavement, specifi- 
cation, 1128. 
masonry, defined, 431. 
Filters, mechanical, 1204. 
Filtration, 
mechanical, 1204. 
of water, cost, 1291. 
rapid sand, 1204. 
slow sand, 1204. 
Fire 
hydrants, table, 1290. 
plugr, described, 1288. 
streams, effect of long lengths of 

hose on, (ref.) 1187. 
tests on building materials, 523. 
tube boilers, described, 1362. 
Fireproof 
buildings, requirements for, 819. 
cement, 402. 
floors, 818. 
requirements for, 820. 
Fir, grading rules, 388, 389. 
Firs (trees), classification of, 342. 
Fittings and materials of construc- 
tion, elec code rules, 1423. 
Fixtures, elec code rules, 1413, 1449. 
Fixture wire. elec. code rules, 1427. 
Flagging crosswalks, 1103. 
Flagstone, 
defined, 402. 
quayrying, 419. 



Flange 
angles of plate girders, properties 

of, table, 572. 
block, properties of, 533. 
pipe, cast iron, table, 1236. 
plates of plate girders, properties 
of, table, 580. 
Flashing, defined, 1500. 
Flat plates, Grashof's analysis of, 

(ref.) 586. 
Flax 
belting, strength of, 512. 
tensile strength of, 512. 
yam fiber, strength of, 512. 
Flemish bond, defined, 437. 
Flexible 
cord, elec. code rules, 1413, 1426. 
joint pipe, cast iron, 1238. 
tubing, elec. code rules, 1431. 
Flitch (lumber), classification of, 388. 
Floating dry-dock, steel, (ref.) 900. 
Floats, 
hydraulic, described, 1183. 
rod-, 1183. 
sub-surface-, 1183. 
surface-, 1183. 
Floor, floors, 
and ceiling construction, 817, 818, 
bridge-, specifications, 700. 
building-, construction, types of, 

817, 818. 
fireproof, 818. 

requirements for 820. 
live loads on, table, 815. 
loads, 

for buildings, 820. 
of buildings, live loads for, 824. 
plates, reinforced concrete, bend- 
ing moment, 823, 825, 827, 
829, 832. 
reinforced concrete, instruction 

sheet for placing, (ref.) 586. 
slabs, reinforced concrete, plank 

flooring for, 831. 
trestle bridge, 788-790. 
wooden-, framing, 817. 
Floorbeam, floorbeams, 
effect of, on bending moments, 69 5o 
end connections of, 700. 
reactions. 
Cooper s loading, table, 708. 
for concentrated loads, table, 

694. 
from electric cars, tables, 717. 

718. 
for highway bridges, table, 728 
Flooring, 
classification of, 388. 
glass, physical properties of, 512. 
lumber (fir), classified 389. 
Florin (Dutch), equiv. (1-10,-50- 
100) in U. S. money, table, 97. 
Flotation, 
depth of, formula, 1152. 
of bodies, 1152. 
Flour cement, 402. 
Flow 
in cu. ft. per sec. reduced to horse- 
power, table, 1333. 



INDEX, 



1563 



iPlow— Cont'd, 
in open channels, surface and mean 

velocity of, 1183. 
of air in small pipes, friction form- 
ula, 1189. 
of liquids, theory of, 1154. 
of steam 
through pipes formula and table, 

1361. 
through orifices, (ref.) 1377. 
of water 
in open channels, diagram, (ref.) 

1189. 
in pipes, measurement of, 1183. 
in wood pipes, (ref.) 1187. 
through submerged tubes, (ref.) 
1189. 
Flowing water, energy of, (ref.) 1188. 
Flue boilers, described, 1362. 
Fluid 
drachm (apoth.), metric equiv.,83. 
measure (apoth.), metric equiva- 
lents, table, 83. 
ounce (apoth.), metric equiv., 83. 
Flume, flumes, 
and conduits, irrigation, 1317. 
for water supply, 1207. 
measuring-, instructions for in- 
stalling, (ref.) 1187. 
proportioned for maximum dis- 
charge, 1161.^ 
semicircular, merit of, 1317. 
steel, for water supply, 1207. 
valves, table, 1279. 
Fluorine, chem., 318. 
Flush hydrants, table, 1290. 
Flux, 
asphaltic, specifications, 1125. 
borax as, 330. 
Flywheel, tension in rim, problem, 

297. 
Focus (pi. foci) 
of ellipse, 258. 
of parabola, 258. 
Foot, 
cord (wood), metric equiv., 82. 
cubic, 
equivalents, 67. 
metric equivalents, 68, 82. 
decimals of, to inches and frac- 
tions, table, 223. 
equivalents, 68. 
metric equivalent of, 66, 68. 
-pounds 

and meter-kilograms, equiv., 89. 
equivalents of , 90. 
table, 91. 
per hour, equivalents of, 90. 
per minute, equivalents of, 90. 
per second, equivalents of, 90. 
square, metric equivalents, 68, 81. 
valves, vertical, table, 1279. 
Footings, foundation-, 867, 868. 
Force and motion, equations o f, 

mech., 278. 
Force, forces, , 

centrifugal, 297. 
component, 305. 
• defined, 278. 



Force, forces, — Cont'd, 
distributed-, 

center of gravity of, 296. 

resultant of, 296. 
electromotive, defined, 1382. 
equations of, in mechanics, 288, 289. 
equilibrium of, in mech., 294. 
lines of, electric, defined, 1382. 
outer and inner, in structures, 305. 
parallel, center of gravity of, 295, 
parallelogram of, 294. 
polygons, 295. 

at joints of truss, 310. 
resolution of, 294. 
tractive problem, 288. 
triangle of, 294 
unbalanced, formulas, 288. 
Foreign 
coins and paper notes, equiv. (1- 
10,-50-100) in U. S. money, 
tables, 96 97. 
money, tables, 95-97. 
weights and measures, American 
equivalents, table, 92-94. 
Forest stumpage in the U. S. 376. 
Forgings, 
carbon and nickel steel, chemical 

properties of, 505. 
nickel steel, physical properties of. 

table, 499. 
steel. 

physical properties of, table, 499. 

specifications, 505. 

testing, 506. 
Forms for concrete, costs, (ref .) 1293, 
Foundation, foundations, 863. 
beds, 

bearing pressures, 863-865. 

classification of, 863. 
coffer-dams for, 869. 
concrete; 867. 
footings, 867, 868. 
for buildings, 819. 
gravel and sand, 864, 865. 
hard-pan, 864. 
I-beam, 867. 

indurated clay, 864, 865. 
loads, for buildings, 866. 
loads on, estimating, 866. 
loam, 865. 

of dams, pressure on, 848-851. 
pile, 871. 
pneumatic, 880. 
references, 890. 
road specifications, 1101. 
sand-, 864, 865. 
sewer, 1306. 
sheet piling for, 869. 
soils, 

bearing power of, for buildings, 
867. 

tests of, 865. 
solid rock, 863, 864. 
spread-, reinforced concrete, (ref.) 

890. 
sub-, 

concrete for, 442. 

cement grout injected in, 442. 
walls, waterproofing for, 418. 



1664 



INDEX. 



Foundry work, 392. 

Fountain, aerating-, in reservoir 

1206. 
Fractions 
and decimals, short methods of 
multiplication and division, 
11-13. 
kinds of, 7. 

reduced to decimals, tables, 9-10. 
reduction of, 7-10 
(12ths) reduced to decimals table, 

9. 
(64ths) reduced to decimals, table, 
10. 
Fractional distillation of gasoline, 

table, 1376. 
Franc (French), equiv. (1-10,-50- 
100) in U. S. money, table, 97. 
Francis' weir formulas, 1178. 
Freezing 
mixtures, 513. 
point, defined, 513. 
point of liquids, table, 514. 
process, in foundation work, 882. 
test for sandstone, 402. 
weather, laying masonry in, 433. 
French 
money, U. S. values, 95. 
thermal unit (Cal.) defined, 1347. 
Frequencies and voltages, in elec, 

1470. 
Friction, 517. 
angle of, for various substances, 

tables, 517-521. 
head, in pipe lines, 1160. 
heads in cast iron pipe, table, 

1217. 
in machines, table, 521. 
laws of, 517. 

losses in pipes, defined, 1161. 
Morin's experiments, 517. 
of air in small pipes, formulas, 

.1189. 
of journals on their pillows, table, 

520. 
of plane surfaces, table, 517. 
of various substances, table, 521. 
rolling, 521. 
sliding, 517-521. 
sliding-, of train, 702. 
Frictional resistance of cylinder 

piers, 878. 
Frictionless orifices, experiments on, 

(ref.) 1187. 
Frog, frogs, 1075. 
and switches, tables, 1079-1082. 
angles, properties of, table, 1076. 
crossing-, 1078. 
kinds of, 1075. 
manganese steel, 1075. 
movable-point, 1076. 
number, formula, 1075. 
spacing 
on crossover tracks, table, 1089. 
on ladder tracks, table, 1089. 
spring rail, 1075^ 
Frost boxes for gate valves, table, 

1288. 
Fruhling suction dredge, (ref.) 932. 



Frustum 
of circular spindle, 254. 
of cone. 248. 

defined, 134. 
of conic wedge, 249. 
of cylinder or prism, area, volume, 

244. 
of parabolic spindle, 254. 
of pyramid, 248. 
defined, 134. 
Fteley and Steams* weir formulas, 

1180, 1181. 
Fuel, fuels, 
chemical analysis of, 1350. 
heat of combustion of, calcula- 
tions, 1352. 
heating power of, 1359. 
liquid, properties of, 1370. 
solid, chemical composition of, 

table, 1352. 
vaporization of, 1371. 
wood as, value of, 1363. 
Fuller's earth, uses of, 331. 
Fulminate of mercury for percus- 
sion, 352. 
Fungus in timber, 359. 
Furlong, equivalents, 68. 
Fuses, 
automatic-, elec. code rules, 1406. 
cartridge enclosed, elec. code rules, 

table, 1438. 
elec. code rules, 1436. 
Fusible 
metal 
• (Rose's), melting point of, 515. 

(Wood's), melting point of, 515. 
plug, 398. 
Fusion, 
latent heat of, defined, 513. 
temperature of, defined, 513. 
vitreous, of glass and iron, 515. 



Q 



Gadolinium, chem., 318. 
Gage, gages, 
hook-, 1182, 1183. 
metal-, tables, 666, 667. 
of track and wheels, railroad, 107-3 
rain-, standard, 1196. 
sheet metal, tables, 667. 
standard wheel-, 1073. 
wire, tables, 666, 671. 
Gallium, chem., 318. 
Gallon, gallons, 
apoth., metric equivalents, 83. 
and bushels, equivalents (1-9), 

table, 485. 
and cubic feet, equivalents (1-9), 

table, 485. 
and liters, 
equivalents (1-10), table, 83. 
equivalents, 88. 
and yards-inch, equivalents (1-9), 

table, 485. 
(dollars per U. S.) 
and francs per liter, equiv. (1- 
10), table, 98. 



INDEX, 



1565 



Gallon, gallons, — -Cont'd, 
(dollars per U. S.)— Cont'd, 
and marks per liter, equiv. (1- 

10), table, 98. 
and shillings per Br. Imp. gallon, 
equivalents (1-10), table, 98. 
equivalents, 67. 
metric equivalents, 68, 83. 
of liquid, weight of, 479. 
per minute (discharge) and liters 

per minute, equiv., 90. 
(shillings per Br. Imp.) and dollars 
per U. S. gallon, equiv. (1-10), 
table, 98. 
Galvanized 
iron covering for bridges, 361. 
or black pipe, table, 1284. 
Galvanizing, 357. 

Gantah (Philippine measure), Eng- 
lish equivalents, 81. 
Gatbage disposal, 1295. 
Gas, gases, 
and steam power, 1346. 
Avagadro's law of, 1372. 
coefficient of expansion of, table, 

464. 
critical point of a, defined, 512. 
critical temperature of a, defined, 

512. 
defined. 512. 
engine principles and management, 

(ref.) 1377. 
engineering, problems in, (ref.) 

1378. 
evaporation effect on, 513. 
expansion of, 513. 
heat effect on a, 512. 
lighting, electric-, elec. code rules, 

1446. 
liquefaction of, how accomplished, 

513. 
physical properties of, table, 514. 
-producer and water-gas process, 

1377. 
proof compositions, 418. 
specific gravities of, 
table, 464. 
to determine, 462. 
standards for specific gravity, 462. 
standard 
pressure of, defined, 462. 
temperature of, 462. 
weights and specific gravities of, 
table, 464. 
Gasfitters' cement, 402. 
Gasket, gaskets, 
compositions, (ref.) 418. 
pipe, 1215. 
Gasoline, 
capacity and weight equiv., 1376. 
fractional distillation of, table, 

1376. 
fuel, properties of, 1370, 1375. 
vapor pressure of saturation for, 
table, 1373. 
Gate, gates, 1271-1279, 1285-1287. 
and valves, Ludlow, tables, 1274- 

1279, 1286, 1287. 
boxes, table, 1288. 



Gate, gates, — Cont'd, 
sluice, 
stand and wheel, 1270. 
table, 1279. 
valves, 1271-1279, 1285-1287. 
Chapman, nomenclature, 1285. 
dimensions and weights of , 1273. 
vertical, geared and ungeared, 
, 1273. 
Gearing and mechanism, reference 

data, 1480. 
Generators (see also Dynamos), 
defined, 1379. 

elec. code rules, 1394, 1447. 
Geometrical 
figures, 
areas of, table, 524. 
center of gravity of, table, 524. 
construction of, 130. 
moments of inertia of, table, 524. 
neutral axis of, table, 524. 
properties of, table, 524. 
radius of gyration of, table, 524. 
section modulus of, table, 524. 
mean, 57. 

series or progression, 57. 
Geometry, 
Analytic, 256. 
Descriptive, 261. 

problems of construction, 263. 
Plane, 128-131. 
Solid, 132-135. 
German 
money, U. S. value, 95. 
-silver, 397. 
Germanium, chem., 318. 
Gill, liquid, metric equivalent, 83. 
Girder, girders (see also Beams), 
and beams, properties of, tables, 

562. 
beam box, steel, 
problem, 569. 
properties of, table, 568. 
Cooper's loading, table, 708. 
deck, plate-, spacing of, 700. 
electric-car loadings for, tables, 

717-719. 
for buildings, requirements of, 820. 
moments and shears, various load- 
ings, 688. 
plate and lattice, section-modulus 

diagrams, (ref.) 586. 
plate-, 
economic depth of, 684. 
specifications, 704. 
steel, properties of, table, 570- 
582. 
railroad, weight of, 710. 
(single I) beams, properties of^ 
table, 583. 
Glass, 
expansion coefficient of, 516. 
flooring, 
physical properties of, 512. 
strength, of, 512. 
melting point indefinite, 515. 
physical properties of, 512. 
sizes and weights, table, 479, 
strength of, 512. 



1566 



INDEX, 



Glass, — Cont'd, 
tiles, 800. 

vitreous fusion of, 515. 
window, cost prices, 479. 
Glazed 
brick, 415. 

pipe, for water supply, 1207. 
Glucinum, cheni., 318. 
Glue, 
cement, 402. 
marine, (ref.) 418. 
Gneiss, 
and granite, weights of, 475. 
composition of, table, 336. 
compression tests of, 511. 
compressive strength of, 511. 
defined, 340. 

transverse strength of, 511. 
Gold, 
cast, 
physical properties of, 497. 
etc., weight of, 480. 
chem., 319. 
-copper alloy, tensile strength of, 

497. 
dredging, 930, 932. 
melting point of, 515. 
minerals, ores, 328. 
plating, 358. 

wire, tensile strength of, 497. 
Gondola cars, large capacity, (ref.) 

1091. 
Gothic sewers and conduits, proper- 
ties of, table, 1303. 
Government land surveying, 967. 
Grade, grades, 
angles 
and % of, equivalents, table, 

1002. 
and rates per mile, equivalents, 
table, 1003. 
cost of haul on, table, 996. 
economic considerations, 992. 
feet per 100 ft. and per mile, 

tables, 1001, 1003. 
limiting-, 992. 

locomotive traction on, 992. 
problem, 993. 
table, 994. 
of large irrigating canals, table, 

1317. 
of sewers, 1296. 
of tunnels, 935. 
on profiles, railroad, 1004. 
reduction (R. R.), allowable ex- 
pense for, 995. 
ruling-, 992. 

determination of, 996. 
traction on, 1097. 
Grader, elevating-, used in railroad 

excavation, cost data, 916. 
Gradients, railroad-, 991. 
Grading, 
for street pavement, specifica- 
tions, 1127. 
lumber, 387. 

railroad, economic problem, 269. 
with wheeled scrapers, cost data, 
917. 



Grain, grains, 
and grams, equiv. (1-10), table, 85. 
(apoth.) metric equivalent, 86, 
(avoir.), metric equivalents, 86. 
metric equivalents, 68. 
troy, ^ 

equivalents, 67. 

metric eqiiivalents, 86. 
Gram, grams, 

and grains, equiv.(l-lO), table, 85. 
and ounces, 

equivalents, 89. 

equivalents (1-10), table, 85. 
English eqtiivalents, 68, 85. 
per cu. centimeter and pounds per 

cu. in., equivalents, 89. 
standard, equivalents, 67. 
Granite, 
block pavement, 

described, 1100. 

specifications, 1102, 1105, lrt9. 
block paving, specifications, 1105. 
building, 400. 
composition of, 331. 

table, 337. 
compression tests of, 511. 
compressive strength of, 511. 
excavation, in open cuts (R. R.). 

cost, 925. 
expansion coefficient of, 516. 
heat effect on, 400. 
high compression tests, 511. 
kinds of, 400. 
paving blocks, 1102. 
properties of, 400. 
temperature stress for 160° F., 523. 
tension tests, (ref.) 511. 
transverse strength of, 511. 
weight of, table, 475. 
Graphical 
methods, in struc., 306, 310. 
solution 

of catenary, 753. 

of Pratt truss, 312. 

of truss, 310. 
Graphite, 
for paint, 355. 
paint, 359. 
weight of, 480. 
Grashof's analysis of flat plates, 

(ref.) 586. 
Grasshopper bents, 789. 
Gravel 
and composition filling for wood 
block pavement, specifications, 
1128. 
foundation, 864, 865. 
friction of, 521. 
quarrying, 419. 
roads, 

application of oils to, 1134, 

construction of, 1098. 
screening, 419. 

sidewalks, construction of, 1098. 
streets, oiled, specifications, 1114. 
swellage when loosened, 911. 
voids in, 911. 

concrete, 416. 
weight of, 475- 



INDEX. 



1567 



Graving dock, 892. 
Gravity- 
acceleration, 
formula, 459. 
equation of, 287. 
table, 283. 

center- of, of trapezoid, 847. 

force of, 278. 

specific, (see specific gravity). 

yards, (ref.) 1091. 
Gray iron castings, specifications, 

498 
Great-circle, 

arc of, defined, 135. 

of sphere, defined, 134 
Greensand, 337. 

use of, 340. 
Greenstone, 

composition of, 338. 

properties of, 400. 

trap, weight of, 480. 
Greatest common factor, to find, 6. 
Groined arch in filter and reservoir 

construction, (ref.) 1292. 
Gross, 

equivalent of, 95. 

great-, equivalent of, 95. 
Grounding low-potential circuits, 

elec. code rules, 1401. 
Ground-slope quantities, 

formulas for, 1030. 

correction table, 1039, 1041. 
Grout, 

cement, in sub-foundation, 442. 

filling, 
for wood block pavement, speci- 
fications, 1128. 
in brick paving, 1109. 
Grubbing and clearing, 906. 

cost data, 916. 

refernece data, 920. 
Guard rails, 

specifications, 700. 

timber. 789. 
Gum 

trees, classification of, 345. 

arabic, weight of, 480. 
Guncotton 

explosives, 351, 352. 

manufacture of, 351. 
Gunmetal, bronze, 

weight of, 478, 480. 

physical properties of, table, 497. 
Gunpowder, 350. 

weight of, 480. 
Gusset plates, 705. 
Guttapercha, 

expansion coefficient of, 516. 

weight of, 480. 
Gutters, 

bit uminized -brick, specifications, 
1115. 

cobblestone, 1099. ^ 

concrete, specifications, 1111. 

plank, described, 1098. 
Gypsum, 329, 404. 

defined, 339. 

variety of, 336. 

weight of, 480. 



Gyration, 

center of, 303. 

radius of, problem in, 637. 
Gyroscope, theory of, (ref.) 1095. 



H 



H 



columns, properties of, tables, 608. 
skeleton section, properties of, 530. 
Halite, composition of, 336. 
Hammer-drill, 422. 
Hammers, stone, described, 426-7. 
Hanger boards for series arc lamps, 

elec. code rules, 1442. 
Hard pan 
excavation, by use of dynamite, 

923. 
foundation, 864. 
Harlem River ship canal, cost data, 

1329. 
Harveyized steel 396. 
Haskell current-meter, 1185. 
Haul, 
earthwork-, 1059. 
on various grades (R. R.), cost of, 
table, 996. 
Hawser, in cordage, 668. 
Head, heads, 
curvature-, in pipe lines, 1160. 
entry-, defined, 1160. 
friction-, in pipe lines, 1160. 
of eye-bars, added length formula, 

686. 
of water, 
forgiven pressures, table, 1147. 
reduced to equivalent velocities, 

table, 1155. 
reduced to equivalent pressures, 
tables, 1148, 1149. 
pressure-, in pipe lines, 1160. 
various hydraulic, defined, 1160. 
velocity-, in pipe lines, 1160. 
Header, masonry, defined, 432. 
Heading 
and drift methods of tunneling, 

933. 
in tunneling, defined, 933. 
Heaped bushel, measure of, 84. 
Heat, 
and power, problems, 1350. 
as energy, 1346. 

conductivity and resistance of ma- 
terials, table, 1377. 
effect 
on building materials, 523. 
on concrete, 510. 
on various substances, 512. 
engines (internal combustion) , 
tests of, on alcohol fuel, 1368- 
1370, 1374. 
equivalent 
of external work, formula, 1356. 
of internal work, formula, 1355. 
expansion by, 516. 
mechanical 
and electrical units, equivalents, 
table, 91. 



1568 



INDEX. 



Heat, — Cont'd, 
mechanical — Cont'd, 
equivalent of, 
defined, 1347. 
equiv. (1-10), table, 1349. 
of combustion 
of fuel, calculations, 1352. 
of liquid fuels, 1370, 1375. 
of the liquid, formula, 1355. 
of vaporization, formulas, 1355, 

1356. 
resistance, 
and conductivity of materials, 

table, 1377. 
coefficients of, 1377. 
units, 
defined, 1347. 
equivalents of, table, 91. 
equivalents (1-10), table, 1348. 
per sq. ft. per minute, eqmva- 
lents, table, 91. 
waves, 1380. 
Heaters, electric-, elec. code rules. 

1407. 
Heating 
and ventilation, reference data. 

1482. 
power of fuels, 1350. 
values of coals, tables, 1350, 1351, 
Heavy oil (creosote), 367. 
Hectar, hectars, 
and quarter section, equiv., 88. 
and township, equivalents, 88. 
English eqtiivalents, 68. 
Hectogram, English equivalents, 85. 
Hectoliter, hectoliters, 
and bushels (U. S.), equiv. (1-10), 

table, 84. 
English equivalents, 81, 82, 84. 
(francs per) and dollars per bushel, 

equivalents (1-10), table, 98. 
(Germ, marks per) and dollars per 
bushel, equiv. (1-10), table, 98. 
per hectar and bushels per acre, 
equivalents (1-10), table, 84. 
Hectometer, 
English equivalents, 70. 
square, English equivalents, 79. 
Height, slant-, defined, 133. 
Helical springs, formulas, 1482. 
Helium, 
chem., 319. 

gas never been liquified, 513. 
Helix or screw, 260. 
Hemlock, hemlocks, 
classification of, 342. 
grading rules, 388, 390. 
Hemp, 
friction of, 518, 519. 
required per joint of cast iron pipe, 
1216. 
Heptagon, 
defined, 129. 
mensuration of, 204. 
Herschel's weir formula, 1181. 
Hertzian ray, 316. 
Hesselmann process for timber, 361, 
Hexagon, 
defined. 129. 



Hexagon, — Cont'd, 
hollow-, properties of, 527. 
inscribed in circle, 131. 
mensuration of, 204. 
regular-, properties of, 526. 
Hexahedron, defined, 132. 
Hickories, classification of, 343. 
Highway, highways, 1097. 
bridges, 720. 
combination-, with details, 729. 
live load data for, 728. 
nickel and carbon steel, specifi- 
cations, 737; table, 738. 
references, 739. 
typical loading for, 727. 
unit stress sheets, 720. 
with solid floors, weight of, 

formulas, 686. 
with wooden floors, weight of, 
formulas, 686. 
History (Natural) of materials, 316. 
Hitches and knots, in cordage, 668-9. 
Hogshead (liquid), equivalents, 83. 
Hoisting, 
rope, tension in, 290. 
work formula, 290. 
Hollow 
circle, properties of, 528. 
concrete blocks, safe loads on, 829. 
ellipse, properties of, 529. 
hexagon, properties of, 527. 
octagon, properties of, 527. 
rectangle, properties of, 525. 
square, 
diagonal axis, properties of, 526. 
properties of, 526. 
tile partitions, 813. 
Hook 
bolts, 618. 

fastenings for wire rope, 675. 
gage, 1182, 1183. 
Horizon, of celestial sphere, defined, 

201. 
Horizontal check valves, table, 1278. 
Horse, power of a, 1097. 
Horsepower, 
brake-, (B. H. P.). formula. 1374. 
electric, 

energy value of, 90, 91. 
equivalents of, 90. 

table, 91. 
from flow of one cu. ft. per sec, 

table, 1333. 
hours, 

equivalents of, table, 91. 
from storage of one acre-ft., 

table, 1335. 
from storage of 1,000,000 cu. ft., 
table, 1334. 
indicated, (I. H. P.), formula, 1375. 
mechanical, equivalents of, 90. 
metric, equivalents of, 90. 
of locomotives, problem, 291. 
of steam boiler, defined, 1361. . 
. of turbines, theoretic, formula, 

1344. 
unit, 291. 
Hose, effect of long lengths of, on 
fire streams, (ref.) 1187. 



INDEX. 



156P 



Hour 

angle, of a star, denned, 202. 
circle, of celestial sphere, defined, 

202. 
or degree, decimals of, for minutes 

and seconds, 1010. 
(time) and degrees (longitude), 
equivalents, 99. 
House, houses, 
car-, elec. code rules, 1421. 
drainage, 1295. 
paints, 356. 
Howe truss 
and details, 711. 
brace problem, 635. 
Human body, weight of, 480. 
Hundred weight (avoir.), metric 

equivalent, 86. 
Hydrant, hydrants, 
branches, cast iron pipe, table, 

1229 
described, 1288. 
Ludlow, weight of, table, 1290. 
nomenclature, 1289, 
Hydraulic, hydraulics, 1154. 
cements, described, 404. 
dredges, 928, 931. 
fill dams, cost data, 919. 
formulas, 1161. 
Basin's, 1189. 
Chezy's, 1167. 
Kutter's, 1167. 
grade line, 1159. 
lime, manufacture and use of, 

404. 
limestone, properties of, 401. 
mean radii of pipes, table, 246-7. 
measurements, 1182. 

of streams, (ref.) 1187. 
method of earth excavation, 908. 
notation, 1154. 
problems, 1167, 1172, 1219, 1298, 

1306. 
properties of conduits and sewers, 

tables, 1296-1306. 
shield used in sewer tunnel, cost 
data, 916. 
Hydrocarbons, 
in min., classificaton of, 328. 
natural, classification of, 1141. 
Hydrodynamics, defined, 1154. 
Hydro-electric problem, 1379. 
Hydrogen, 
boiling point of, 514. 

corpuscles in, 316 
melting point of, 515. 
minerals, 330. 
-oxygen flame, 317. 
physical properties of, table, 514. 
Hydrokinetics, 1154. 
Hydrometers, 461, 462. 
Hydrostatics, 1145. 
head and pressure, 
equiv. (1-10), 1146. 
tables, 1146-1149. 
pressure, 
for lead pipe, 679. 
formulas, 1146. 



Hydrostatics, — Cont'd, 
pressure ,—Cont ' d . 
in pipes and tanks, 1152. 
on dams, 845. 

on submerged surfaces, 846, 847. 
units, 1146. 
Hydroxide, in chem., defined, 321. 
Hyperbola, 
equation of, 259. 
normal to, 259. 
tangent to, 259. 
equilateral, 260. 
Hyperbolic 
logarithms, defined, 104. 
example in, 106. 
table, 108. 
spiral, equation of, 260. 



I 



block, properties of, 533, 534. 
skeleton section, properties of, 530, 
531. 
I-beam, -beams, 
block, properties of, 533, 534. 
cast separators for, table, 623. 
rivet gages for, 614. 
skeleton, properties of, 530, 531. 
special, properties of, table, 584. 
standard connection angles for, 

615. 
steel, properties of, 554. 
Ice, 
evaporation from, 1199. 
expansion coefficient of, 516. 
hard J compressive strength, 512. 
melting point of, 515. 
weight of, 480. 
Icosahedron, defined, 132. 
Impact, 
coefficient of, for bridges, table, 

709. 
effect of, 489. 
formulas, 303. 

railroad bridges, effect of, 701. 
Impedance and inductance in trans- 
mission line, 1387, 1392. 
Impulse 
and momentum, 293. 
water wheels, 1336. 
Ion, positive and negative, 317. 
Incandescent lamps, in series cir* 

cuits, elec. code rules, 1406. 
Inch, inches, 
and centimeters, 
cubic, 
equivalents (1-10), table, 82. 
equivalents, 88. 
equivalents (1-10), table, 70. 
equivalents, 88. 
square, 
equivalents (I-IO), table, 80. 
equivalents, 88 
and millimeters, 
cubic, equiv (1-10), table 82. 
equivalents (1-10), table, 70. 
square, equiv. (1-10), table, 80. 



1570 



INDEX. 



Inch, inches, — Cont'd, 
cubic, 
eqmvalents, 67. 
metric eqmvalents, 68, 82. 
decimals of, to millimeters, table, 

69. 
equivalents, 68. 
fractions of, 
and millimeters, eqmvalents, 88. 
to decimals and millimeters, 
table, 69. 
metric equivalent of, 66, 68. 
miner's, table, 1313. 
-pounds and centimeter-grams, 

equivalents, 89. 
square, metric equivalents, 68, 81. 
to decimals of a foot, table, 223. 
(1-12) to meters, equiv., table, 71. 
Inclined 
plane, 
in mech., formula, 292. 
motion on, 286. 
velocities on, 286. 
railway, (ref.) 1091. 
Increasers, 
cast iron pipe, tables, 1233, 1259- 

1264. 
Matheson pipe, table, 1281. 
India rubber, weight of, 480. 
Indicated horsepower (I. H. P.), 

formula, 1375. 
Indium, chem., 319. 
Inductance and impedance, in trans- 
mission line, 1387, 1392. 
Induction, electric, defined, 1381. 
Inductive load, in elec, 1453. 
Inelastic bodies, impact effect, 304. 
Inertia, 
moment of, 
about inclined axis, formulas, 

535-538. 
about parallel axis, 524. 
of rectangles, table, 539-540. 
of rolled shapes, 537. 
polar moment of, 535-538. 
maximum and minimum values, 
537. 
Infusorial earth, uses of, 331. 
Insect larvae in timber, 359, 
Inspection of yellow pine lumber, 

387, 388. 
Insulated wires, table, 1423. 
Insulating joints, elec. code rules, 

1442. 
Insulation, 
electrical, 1462. 
resistance, 
elec. code rules, 1396. 
tables, 1447, 1450. 
Intake, mouthpiece (bell-shaped) at, 

1177. 
Integral calculus, 272. 
Integration, 
definite, 273. 

double, for polar moment of iner- 
tia, 535, 536. 
formulas for, 274, 275 
method of using current meters, 
1186. 



Interest, 
compound, * 
methods, 59-62. 
table, 62. 
simple, 
common and exactjmethods, 58- 

59 
table, 60. 
Interior conduits, elec. code rules, 

1412, 1450; table, 1428. 
Interlocking plant, railroad, (ref.) 

1093. 
Internal combustion engines, tests 
of, on alcohol fuel, 1368-1370, 
1374. 
Intersection of curves, to find, 103. 
Intrados of arch, curve of, 764. 
Inverse trigonometric functions, 140. 
Iodides, in min., classification of, 325. 
Iodine, chem., 319. 
Ions, in electrolysis, 357. 
Iridium, 
chem.^Zl^. 
minerals, 328. 
-platinum, expansion coefficient of, 

516. 
Iron, 
chem., 319. 
cast, 
expansion coefficient of, 516. 
for buildings, 819. 
melting point of, 515. 
pipe, 1214. 

physical properties of, table, 497. 
properties of, 392. 
temperature stress for 160° F., 

523 
weight of, 480. 
cement, 402. 
(ref.) 418. 
corrosion of, in concrete, (ref.) 455. 
friction of, 518. 519. 
gray, castings, specifications, 498. 
in buildings, stresses for, 824. 
malleable cast, 
high tensile strength of, 398. 
tensile strength of, 497. 
melting point of, 394. 
minerals, ores, 329. 
■ molten, weight of, 480. 
ore, 
treatment of, 392. 
uses of, 329. 
oxide (s) 
for paint, 355. 

paints for iron and steel, 372. 
physical properties of, table, 497. 
pig, 
manufacture of, 392. 
uses of, 392. 
pipe, 
cast, specifications, 1239. 
wrought, 1268. 
preservation of, 358. 
slag block pavement, specifica- 
tions, 1121. 
vitreous fusion of 515. 
weight of, 480. 
wire, physical properties of, 498. 



INDEX, 



1571 



Iron, — Cont'd, 
wrought, 
expansion coefficient of, 516. 
for buildings, 819 
in buildings, safe stresses, 826. 
manufacture of, 393. 
melting point of, 515. 
physical properties of taWe, 

498. 
weight of, 480. 
Irrigated lands, drainage of, costs, 

(ref.) 1319. 
Irrigation, 1313. 
canals, 
large, dimensions and grades of, 

table, 1317. 
.locating, (ref.) 1318. 
'miscellaneous data, (ref.) 1318. 
velocity in, 1317. 
duty of water in, tables, 1315-17. 
flumes and conduits, 1317. 
units, 1313. 
of flow, 1313. 
of volume, 1314. 
works of Southern California, (ref.) 
1318. 
Isometric projection, 261. 
of stonework, 457, 458. 
Isosceles triangle, defined, 128. 
Isthmian canal, proposed American, 

data, 1324. 
Italian money, U. S. values, 95. 
Ivory, 
weight of, 480. 
-black, for paint, 355. 



Jack-knife drawbridge, 748. 
Japan varnish, 357. 
Japanning, 357. 
Jarring effect on earth fill, 910. 
Jets and nozzles, 1 1 75. 
Jetties, 905. 

kinds of, 905. 
Jetty-head, reinforced concrete, 

(ref.) 905. 
Joint, joints, 

cast iron pipe, kinds of, 1215. 

Converse pipe, 1282. 

flexible-, pipe, 1238. 

insulating-, elec. code rules, 1442. 

masonry, defined. 432. 

Matheson pipe, 1281. 

pipe, locking-bar, 1269. 

rail-, kinds of, 1068. 
Joist lumber (fir), classified, 389. 
Joule (J.), joules, 

defined, 1347. 

equivalents of, 90. 
table, 91. 

per second, equivalents of, 90. 
Journals, friction of, on their pillows, 

table, 520. 
Jumper-drill, 422.^ 

Jute required per joint of pipe, table, 
1222, 



K 

Kaiser Wilhelm canal, data, 1322. 
Kalamein pipe, tables, 1281, 1282. 
Kaolin, weight of, 480. 
Kaolinite, uses of, 331. 
Keene's marble cement, 403. 
Kerosene, 
capacity and weight equivalents, 

1376. 
fuel, properties of, 1370, 1376. 
obtained from crude petroleum, 
1133. 
Kieselguhr. 353. 
Kiln 
drying, of timber, 363. 
rotary, for cement making, 405. 
Kilograms 
and pounds (avoir.), 
equivalents, 89. 
equivalents (1-10), table, 85. 
and tons (U. S.), equivalents, 89, 
English equivalents, 68, 85. 
(francs per) and dollars per pound, 

equivalents (1-10), table, 98. 
(German marks per) and dollars 
per pound, equiv. (l-10),table, 

per cu. meter and pounds per cu. 

ft., equivalents, 89. 
per sq. centimeter and pounds per 

sq. in., equivalents, 89. 
per sq. meter and pounds per sq. 

ft., equivalents, 89. 
pounds and tons, equivalents, (1- 

10), table, 87. 
standard, equivalents, 67. 
-degree (Cent.), equivalents, 90. 
-meter, -meters (see Meter-kilo- 
grams), 
and foot-pounds, equiv., 89. 
equivalents of, table, 91. 
Kiloliter, English equiv., 81, 82, 84. 
iCilometers, 
and miles, 
equivalents (1-10), table, 70. 
square, equiv. (1-10), table, 80c 
equivalents, 88. 
English equivalents, 68, 70. 
per hour and miles per hour, 

equivalents, 89. 
per minute and miles per minute, 

equivalents, 89, 
square, English equivalents, 79. 
Kinetic energy, defined, 1346. 
Kilowatt, 
energy value of, 1379. 
equivalents of, table, 91. 
. per second, equivalents, 90. 
-hour, 

energy value of, 1379. 
equivalents of, table, 91. 
Knife switches, elec. code rules, 

1431. 
Knot, British Admiralty, 68. 
Knots 
and hitches, in cordage, 668-669. 
in lumber, defined, 387. 
Krypton, chem.^ 319, 



1572 



INDEX. 



Kutter's 
formula, 
experimental determination 
oi N, 1187. 
of A^ and C, 1188. 
hydraulic formula, 1167. 
Kyanizing, for timber, 361. 



L*s, cast iron pipe, tables, 1225, 

1250-1254. 
Labor 

item in earth excavation, 908. 

(square), Texas land measure, in 
acres, 81. 
Lacing, 

for bridge members, table, 706. 

for steel compression members, 
weight of, (ref .) 609. 
Lacquers, 357. 
Lacquering, 357. 
Ladder 

dredge, (ref.) 932. 

tracks, frog spacing, table, 1089. 
Lag-screws, 

table, 622. 

use of, 622. 
Lake Borgue (La.) canal, data, 1323. 
Lamps, 

and photometry, in elec, 1473. 

arc-, 
elec. code rules, 1442. 
on constant-potential circuits, 
elec. code rules, 1414. 

incandescent, in series circuits, 
elec. code rules, 1406. 

in series, elec. code rules, 1423. 

series arc, elec. code rules, 1405. 
Lampblack for paint, 355. 
Land, 

clearing and grubbing, cost data. 
916. 

Government-, surveying, 967. 

measure, 
of Texas, Mexico, etc., table, 81. 
square, English, metric equiva- 
lents, table, 81. 
Lanthanum, cJtem., 319. 
Lap-welded pipe, 1269. 
Larches, classification of, 341. 
Lard, weight of, 480. 
Latent heat 

of fusion, defined, 513. 

of vaporization, defined, 513. 
Lateral 

bracing, of bridges, problem, 697. 

pins, 629. 
Laths, 

diamond, 814. 

expanded metal, 814. 

metal, 812. 

wooden, 812. 
Lathing, 

and plastering, 812. 

building, 812. 
Latitude 
\ \nd longitude (spher. trig.), 201. 
\ Nngths of a degree of, table, 979. 



Latitude — Cont'd. 

to determine, with solar, 946. 
Lattice and plate girders, section- 
modulus diagrams, (ref.) 586. 
Latus rectum of parabola, 257. 
Lava, 
defined, 340. 
weight of, 480. 
Law of the conservation of energy, 

1346. 
Laying brick pavement, 1107. 
Lb.-Cal. (pound calorie,) defined, 

1347. 
Lead, chem., 319. 
alloys of, 329. 
-base alloys, 398. 
expansion coefficient of, 516. 
-melting furnace, portable, 1280. 
melting point of, 515. 
minerals, ores, 329. 
physical properties of, 498. 
pipe, tables, 679. 
red-, 
for paint, 355. 
weight of, 481. 
required per joint 
of cast iron pipe, 1216. 
of pipe, table, 1222. 
sheet, 679. 

effect of, on masonry, 587. 
tubing, 679. 
uses of, 329. 
weight of, 679; 

table, 480. 
white-, 
for paint, 355. 
paint, 329. 
wire, tensile strength of, 498. 
wool, pneumatic calking with, 
(ref.) 1293. 
League, 
equivalents, 68. 

square, Texas land measure, in 
acres, 81. 
Leakage in coffer-dam, 870. 
Lean-to roof trusses, unit stresses in, 

805, 806. 
Leap year, time measure, 99. 
Least common multiple, to find, 6. 
Leather, 
cements, (ref.) 418. 
friction of, 517, 518, 519. 
ox, 
modulus of elasticity, 512. 
strength of, 512. 
Le Chatelier's apparatus for find- 
ing spec. grav. of cements, 
407. 
Lemniscate of Bemouilli, equation 

of, 260. 
Length, lengths, 
equiv. (1-10), English and metric, 

table, 70. 
metric and English, equivalents, 

table, 88. 
of curves, by calculus, 275. 
of spans, economic, 683. 
units of, equivalents, 66. 
Lentinus lepidens, in timber, 362. 



INDEX. 



1573 



Level, 
adjustment of, 941. 
sections, earthwork, list of tables, 
1017. 
Leveling, 987. 
allowable errors in, table, 989. 
correction for earth's curvature 

and refraction, 987. 
sources of errors in, 987. 
Levelman, duties of, in preliminary 

survey (R. R.), 1004. 
Lever, 
compound, formulas, 292. 
simple, formulas, 291. 
Leverage, in mech., formulas, 291. 
Libra (Philippine weight), English 

equivalents, 81. 
Light-waves, 1380. 
Lights, signal, elec. code rules, 1450. 
Lighting, 
and electric power, 1379. 
electric-, cost data, table, 1478. 
systems, decorative-, elec. code 
rules, 1414. 
Lightning arresters, elec. code rules, 

1396, 1444. 
Lignite, weight of, 480. 
Lignum vitse journals, friction of, 520. 
Lime, 
(cement), properties of, 403. 
(common,) properties of, 403. 
fat, 403. 
hydraulic, manufacture and use, 

404. 
mortar, 403. 
for brickwork, 403. 
for buildings, 819. 
weight of, 475. 
plaster, 403. 
quick-, 403. 
slacked, 403. 
specific gravity of, 403 
weight of, table, 475. 
Limestone, 
building, 400. 
composition of, table, 335. 
compressive 
strength of, 511. 
tests of , 511. 
defined, 339. 
formation of, table, 335. 
hydraulic properties of, 401. 
in min.t 329. 
kinds of, 401. 
properties of, 400. 
quarrying, 419. 
tensile strength of, 511. 
tension tests, (ref.) 611. 
transverse strength of, 511. 
travertine 401. 
weights, of, table, 475. 
Limnoria, in timber, 360. 
Line, lines, 
and angles, geometric, definitions, 

128. 
of force, electric, defined, 1382. 
of resistance of masonry arches, 

768. 
pole-, elec. code rules, 1400. 



Line, lines, — Cont'd, 
right-, projection of, 262, 263. 
skeleton-, properties of, 529, 531. 
straight-, 
defined, 132. 
equation of, 256. 
intersection with circle, solution, 
257. 
transmission-, 1386. 
Linea (Philippine measure), English 

equivalent, 81. 
Lining, 
reservoir, 1206. 
tunnels, 934, 939. 
Link 
-fuse cut-outs, elec. code rules, 

1434. 
fuses, elec. code rules, 1436. 
surveyor's, equivalents, 68. 
Linseed oil, 
boiling point of, 514. 
for iron and steel, 372. 
for steel, 358. 
manufacture and use, 356. 
weight of, 480. 
Lintels, for buildings, requirements 

of, 820. 
Live load data for highway bridges, 

table, 728. 
Liquefaction of gases, how accom- 
plished, 513. 
Liquid, liquids, 
and dry capacities, metric and 
English equivalents, table, 88. 
boiling point of, table, 514. 
capacities, 
equivalents, 67. 

equiv. (1-10), English and met- 
ric, table, 83. 
metric, English equiv., table, 82. 
defined, 512. 
freezing point of, 514. 
gallons (U. S.) and liters, equiva- 
lents (1-10), table, 83. 
measure, English (U. S.), metric 

equivalents, table, 83. 
ounces and milliliters (c.c.) 

equivalents (1-10), table, 83. 
physical properties of, table, 514. 
quarts (U. S.) and liters, equiva- 
lents (1-10), table. 83. 
specific gravities of, 
table, 468, 469. 
to find, 461. 
weights of, table, 468, 469. 
Lira (Italian), equiv. (1-10,-50-100) 

in U. S. money, table, 97. 
Liter, liters, 
ana barrels (liquid), equiv., 88. 
and gallons 

(U. S. ), equivalents, 88. 
(U. S. Hquid), equiv. (1-10), 
table, 83. 
and pecks (U. S.), equiv. (1-10). 

table, 84. 
and quarts 
(U. S.), equivalents, 88. 
(U. S. liquid), equiv. (1-10), 
table, 83. 



1574 



INDEX. 



Liter, liters, — Cont'd. 

English equivalents, 68, 81, 82, 84. 

(francs per) and dollars per gallon, 
equivalents (1-10), table, 98. 

(German marks per) and dollars 
per gal.,equiv. (1-10), table, 98. 

of water, weight of, 67. 

per minute (discharge), equiv., 90. 

standard, equivalents, 66. 
Lithium, 

chem., 319. 

minerals, 328. 
Lithology, 331. 
Live loads on floors and roofs, table, 

815. 
Load, loads, 

data for highway bridges, table, 
728. 

factor, in elec, 1453. 

floor-, for buildings, 820. 

from safes, 816. 
-line diagrams, in struc ,311. 

of crowd of people, 815. 

on floors, 814. 
and roofs, table, 815. 

on railroad bridges, specifications, 
700. 

on structures, 305. 

snow-, on roofs, 797, 798. 
Loading, 

rock on cars by steam shovel, 924. 

sudden, effect of, 489. 
Loam (earth), 

weight of, 475. 

foundation, 865. 

voids in, 911, 
Lobnitz rock breaker for rock exca- 
vation, (ref.) 926. 
Locating 

engineer, duties of, in preliminary 
survey (R. R.), 1000. 

irrigation canals, (ref.) 1318. 
Location, 

railroad-, 998. 
filing with State, 1013. 

survey (R. R.), 1004. 
Locking-bar joint pipe, 1269. 
Locomotive, locomotives, 

horsepower of, 291. 

oil fuels in, use of, (ref.) 1378. 

traction force of, 992. 
Log, logs, 

(mill), lengths of, 378. 
■ rules, (ref.) 391. 

sawing, 379. 

scales, (ref.) 391. 

scaling, 379. 

transportation of, 378. 
Logarithm, logarithms, 

(anti-), defined, 104. 

of numbers, to find, 105. 

(common), table, 108-125. 

(Hyperbolic), table, 108-125. 

mathematical operations by, 105- 
106. 

(Naperian), table, 108-125. 
V of numbers, 104-127. 
\ table, 108-125. 

Systems of, 104. 



Logarithmic 
bases, defined, 104. 
(common) tables, explanation of, 

104-105. 
cosecants, table, 176-198. 
cosines, table, 176-198. 
cotangents, table, 176-198. 
equivalents, 104. 
functions, defferentiation of, 270. 
secants, table, 176-198. 
sines, table, 176-198. 
spiral, equation, 260. 
tangents, table, 176-198. 
Logging, 378. 
Long 
-distance transmission, 1386. 
measure, 
English, metric equiv., table, 68. 
surveyors', metric eqxiiv., table, 

68. 
metric, English equiv., table, 70. 
Longitude, 
and latitude (spher. trig.), 201. 
and time measure, table, 99. 
lengths of a degree of, 981. 
to determine, with solar, 946. 
Longitudinal shear in beams, for- 
mulas, 565. 
Loomis water-gas and producer-gas 

process, (ref.) 1377. 
Loop heading, in tunneling, defined, 

933. 
Loose rock classification (R. R.),919. 
Ix)ss, losses, 
in friction in cast iron pipe, table, 

1217. 
of energy in turbines, 1343. 
of head 
due to friction in pipe lines, 1 1 60. 
during flow in pipe lines, 1159. 
Lowry process, for ties, cost, 375. 
Ludlow 
double-gate valves, 
dimensions, table, 1274, 1275, 

1286. 1287. 
nomenclature, 1272. 
gates and valves, tables, 1274- 

1279, 1286, 1287. 
hydrants, weight of, table, 1290. 
Lugs, cast iron pipe, 1280; 

table, 1247. 
Lumber, 
and lumbering, 376. 
board measure, 379; 

table, 380. 
defects in, 387, 389. 
dimensions, 379. 
dote in, defined, 387. 
edge grain in, defined, 389. 
fiat grain in, defined, 389. 
grading, 387. 
rules, 388. 
kinds of, 379. 
knots in, defined, 387. 
pitch pockets in, defined, 387. 
planing, 379. 

cost of, 379. 
prices in U. S., 377. 
red heart in, defined, 387. 



INDEX. 



1576 



Lumber, — Cont*d . 

rot in, defined, 387. 

rough, 379. 

sawing, 379. 

saws, kinds of, 379. 

seasoning, 379. 

shakes in, defined, 387. 

sizing, 379. 

steam seasoning, 379. 

stumpage of Pacific coast, 377. 

supply in U. S., 376. 

trade weights, 391. 

trees, best, 346. 

wane in, defined, 387. 

yellow pine, 
classification of, 387, 388. 
inspection of, 387, 388. 
Lune, 

circular-, mensuration of, 220. 

of sphere, defined, 135. 
Lutes and cements, useful to engi- 
neers, 418. 



M 

Macadam 
and telford roads, specifications, 

nil. 

roads, 
specifications, 1116. 
construction, 
application of oil in, 1134. 
inverted, 1142. 
roadway, specifications, 1105,1129. 
surfaces, application of oils to, 
1134. 
Machine, machines, 
drilling in rock cuts, economy of, 

923. 
electrical-, 
classification of, 1452. 
defined, 1379. 
definitions, 1451. 
excavation of trenches, cost data, 

921. 
excavator, 915. 
foundations for, 867. 
friction in, table, 521. 
quarrying, 419. 
reference data, 1481. 
work, in bridges, specifications, 
706. 
Maclauren's theorem, 271. 
Magnesia, weight of, 480. 
Magnesite, calcined, 330. 
Magnesium, chem., 319. 

minerals, 330. 
Magnet, 
electro-, 1381. 
horse-shoe, 1382. 
permanent, 1382. 
Magnetic 
field, induced, 1380. 
reluctance, formula, 1520. 
Magnetism and electricity, princi- 
ples of , 1380. 
Mahler's formula for combustion, 
1352. 



Maintenance and operation of can- 
als, cost data, 1329. 
Malleable 
castings, 393. 
cast-iron, 
high tensile strength of, 398. 
specifications, 497. 
iron, physical properties of, 497. 
Mallet, stone-, described, 428. 
Manchester ship canal, data, 1320. 
Manganese, chem., 319. 
alloys, 329. 
bronze, 397. 
physical properties of, table, 497. 
(ref.) 399. 
in cast iron, effect of, 393. 
minerals, ores, 329. 
steel, 396. 

weight of, table, 480. 
Maintenance-of-way, cost data, (ref.) 

1092. 
Manhattan suspension bridge, de- 
tails and specifications, 756- 
760. 
Manhole, manholes, 
sewer, 1308. 

pipes, cast iron, tables, 1232, 1259. 
Manila rope, 669. 
weight and strength of, tables, 669, 
670. 
Manometer, 1174. 

Mantissa and characteristic, of loga- 
rithms, 104-105. 
Mantle of Welsbach lamps, 330. 
Maples (trees), classification of, 346. 
Mapping, 966. 
in preliminary survey (R. R.), 
1004. 
Marble, 
cement, Keene's, 403. 
compressive 
strength of, 511. 
tests of, 511. 
defined, 339. 

expansion coefficient of, 516. 
foreign, 401. 
in min., 329. 
properties of, 401. 
quarried where, 401. 
quanying, 419. 

temperature stress for lOO** P., 523. 
tensile strength of, 511. 
transverse strength of, 511. 
weight of, table, 476. 
Marine engineering, reference data, 

1480. 
Mariners' measure, metric equiva- 
lents, table, 68. 
Mark (German), equiv. (1-10,-50- 
100) in U. S. money, table, 97. 
Marks and Davis equation for total 
heat of steam, dry and satu- 
rated, 1378. 
Marl, 
for cement, 405. 
properties of, 401. 
use of, 340. 
weight of, 480. 
Marline, in cordage, 668. 



1576 



INDEX, 



Masonry, 431. 
abutments, quantities in, table, 

436, 437. 
aqueducts, 1208. 
arch, -arches, 763. 

forces acting on, 767. 

line of resistance of, 768. 

specifications, 435. 

thickness of rings, tables, 766-7. 
ashlar, defined, 432. 
brick, 437. 

compressive strength of, .511. 

quantities in, table, 438. 

weight of, table, 476. 
bridge, specifications, 434. 
classification, 434. 
compressive strength of, 511. 
concrete, (see Concrete), 
concrete, 439. 
-block, 450. 

cinder, weight of, table, 476. 

stone, weight of, table, 476. 
culvert, specifications, 435. 
Cyclopean-, defined, 1497. 
dams, quantities in, tables, 855, 856. 
dressed, weight of, table, 476. 
dry, specifications, 435. 
expansion coefficient of, 516. 
friction of, 521. 
granite, weight of, table, 476. 
in buildings, safe loads on, 826. 
kinds of, 431, 432. 
laying in freezing weather, 433. 
limestone, weight of, table, 476. 
marble, weight of, 476. 
mixed, 449. 
piers, 888. 

pitch-faced, defined, 432. 
pointing, specifications, 434. 
pressure on, allowable, 825. 
quarry-faced, defined, 432. 
railroad, classification of, 431. 
random-work, defined, 432. 
range-work, defined, 432. 
retaining-wall, specifications, 434. 
rubble, 

defined, 432. 

weight of, table, 476. 
sandstone, weight of, 476. 
squared -stone, defined, 432. 
stone-, 

compressive strength of, 511. 

described, 431. 

in buildings, weight of, 821. 

laying, specifications, 433. 

specifications, 433. 
wall, parts of, defined, 431. 
work, in buildings, safe loads for, 
821. 
Mass, 
and weight, of water, metric, 67. 
defined, 459. 
in mech., defined, 278. 
moving, energy of, formula, 1346. 
unit of, defined, 459. 
Masses (weights), 
metric, English equiv,, table, 85. 
metric and English, equiv. (1-10). 
table. 85. 



Mastic, 
asphalt, defined, 406. 
(resin), weight of, 480. 
Materials, 
chemistry of, 316. 
(general) , weight and specific grav- 
ities of, table, 478. 
miscellaneous, physical properties 

of, 512. 
Natural History of, 316. 
quality of, for buildings, 819. 
resistance of, 486. 
roofing-, weight of, 802. 
specific gravities of, 459. 
strength of, 486. 
weight of, 459. 
Matheson pipe, patent lock- joint, 

tables, 1281. 
Matter, 
and energy, phenomena of, 1346. 
composition of, 316. 
defined, 459, 1346. 
in mech., defined, 278. 
radio-activity of, 316. 
states of existance of, 1346. 
the elements of, 317. 
Matrix, concrete, 416. 
Maxima and minima (calctilus), to 

find, 268. 
Maximum and minimum polar mo- 
ments of inertia, 537. 
McMurtrie stone, manufacture of , 417 
Mean, 
arithmetical, 57. 
effective pressure (m. e. p.) of 

steam engines, 1 365, 1 366. 
geometrical-, 57. • 

proportional, 56. 

in semicircle, 131. 
solar 
and siderial time, equivalents, 

table, 202. " 
day, defined, 202. 
Measure, measures, 
and weights 

(Foreign) , American equivalents 

table, 92-94. 
of Philippines, English equiv., 81. 
English long, metric eqmvalents, 
; table, 68. 

lineal, surveyors', metric equiva- 
lents, table, 68. 
mariners', metric equiv., table, 68. 
weights and money, 66-99. 
Measurements, hydraulic, 1182. 
Measuring 
-flumes, instructions for installing, 

(ref.) 1187. 
velocity of approach in weirs, 1 1 77. 
Mechanical, 
electrical and heat units, equiva- 
lents, table, 91. 
energy, example of, 1346. 
equivalents of heat (J.), 
defined, 1347. 
equiv. (1-10), table, 1349. 
filters, 1204. 
filtration, 1204. 
horse-power, equivalents of, 90. 



INDEX. 



1577 



Mechanics, 278. 

Mechanism and gearing, reference 

data, 1480. 
Melaphyr, composition of, table, 

338. 
Melting point, 

defined, 513. 

of chemical elements, table, 318. 

of iron and steel, 394. 

of substances, table, 515. 
Members 

in algeb., of equation, defined, 100. 

in struc, active, cutting of, 305. 
Mensuration, 

of lines, 203. 

of solids, 243. 

of surfaces, 203. 
Mercury, 

boiling point of, 514. 

chem., 319. 

melting point of, 515. 

minerals, 329. 

uses of, 329. 

weight of, table, 480. 
Mergel (marl), for cement, 405. 
Meridian, meridians, 

and base lines of U. S. surveys, 
table, 972. 

from north star, to determine, 948. 

in astron., defined, 947. 

in surv., convergency of, table, 
977. 

of celestial sphere, defined, 201. 

to determine with solar, 946. 
Metacenter, defined, 1153. 
Metal, metals, 

alloys, 396. 

bending strength of, table, 496. 

compressive strength of, table, 496, 

elastic limit of, table, 496. 

expanded-, 814. 

expansion of, table, 516. 

friction of, 521. 

gages, tables, 666, 667. 

in machines, friction of, 521. 

journals, friction, table, 520. 

melting point of, 515. 

modulus of elasticity of, table, 496. 

moldings, elec. code rules, 1412, 

physical properties of, table, 496. 

resistance (strength) of, table, 496. 

shearing strength of, table, 496. 

springs, formulas, 1482. 

strength of, table, 496. 

surfaces, varnishing, 357. 

tensile strength of, table, 496. 
Metallic 

elements, table, 318. 

tiles, 800. 
Metallurgy, 392. 

of steel, (ref .) 399. 
Meter, meters, 

and chains, equivalents, 88. 

and feet, 
cubic, 
equivalent, 88. 
equiv. (1-10), table, 82. 
equivalents, 88, 
equiv. (1-10), table, 70. 



Meter, meters, — Cont'd, 
and feet,— Cont'd, 
square, 
equiv. (1-10), table, 80. 
equivalents, 88. 
and rods, 
equivalents, 88. 
square, equivalents, 88. 
and stations (100 ft.), equiv., 88. 
and yards, 
cubic, 
equivalents, 88. 
equiv. (1-10), table, 82. 
equivalents, 88. 
equiv. (1-10), table, 70. 
square, 
equivalents, 88. 
equiv. (1-10), table, 80. 
cubic, English equivalent, 68. 
current (water), 1185. 
English equivalents, 68, 70. 
(francs per) and dollars per yard, 

equivalents (1-10), table, 98. 
(Germ, marks per) and dollars per 
yard, equiv. (1-10), table, 98. 
per second 
and feet per second, equiv., 89. 
and miles per minute, equiv., 89. 
per sec. and feet per sec. per sec. 
equivalents, 89. 
Pitot tube, 1183, 1184. 
register, 1174, 1186. 
square, English equiv., 68, 79. 
standard, equivalents, 66. 
(1-1,000) to feet, equiv., table, 75- 

78. 
Venturi, 1173. 
-kilograms 

and foot-pounds, equiv., 89. 
equivalents of, 90; 

table, 91. 
per hour, equivalents of, 90. 
per minute, equivalents of, 90. 
per second, equivalents of, 90 
Method 
of moments, in struc. ^ stresses by, 

306. 
of shears, in struc. » stresses by, 306. 
Metric 
and English 
approximate equiv., table, 68. 
areas, equiv. (1-10), table, 80. 
curves, tables, 1007. 
fundamental unit equiv,, 66. 
lengths, equiv. (1-10), table, 70. 
systems of weights and measures 

66-91. 
volumes, equiv. (1-10), table, 82. 
weights, equiv (1-10), table, 85. 
and United States 
dry capacities, equiv. (1-10), 

table, 84. 
liquid capacities, equiv. (1-10), 
table, 83. 
capacities 

(dry), English equiv., table, 84. 
(liquid), English equiv , table, 82 
cubic measure, English equiva- 
lents, table, 81. 



1578 



INDEX, 



Metric — Cont'd, 
horsepower, equivalents of, 90. 
long measure, English equivalents, 

table, 70. 
square measure, English equiva- 
lents, table, 79. 
volumes, English equiv., table, 

81. 
weights (masses), English equiva- 
lents, table, 85. 
Mexico land measure, English equiv- 
alents, table, 81. 
Mica 
schist, composition of, 336. 
uses of, 331. 
weight of, 480. 
Middle ordinate of curved rails, 
formulas, 1063. 
tables, 1064-1067. 
Mils, 
and millimeters, 
equivalents, 88. 
square, equivalents 88. 
areas of wire in, tables, 671^ 
Mile, miles, 
and kilometers, 
equivalents, 88. 

square, equiv. (1-10), table, 80. 
equivalents (1-10), table, 70. 
and stations (100-ft.), equivalents, 

table, 1001. 
equivalent in varas, 81. 
land, 68. 

metric equivalents, 68. 
nautical, equivalents, 68. 
per hour 
and feet per minute, equiv., 89. 
and feet per second, equiv., 89. 
and kilometers per hour, equiva- 
lents, 89 
per minute 
and feet per second, equiv., 89. 
and kilometers per minute, 

equivalents, 89. 
and meters per second, equiv., 89. 
square, 
and hectars, equivalents, 88. 
metric equivalent, 81. 
statute, equivalents, 68. 
-stones, road specifications, 1102. 
U S. C S. nautical, 68. 
Milk, weight of, 480. 
Mill 
buildings, 
cost data, (ref.) 833. 
wind loads on, 833. 
scale, removing, 358. 
U. S. money, 95. 
Miller or tonneau (metric), English 

equivalents, 85. 
Milligram, English equivalents, 85. 
Milliliter, milliliters (c. c), 
and drams (U. S. apoth.), equiva- 
lents (1-10), table, 83. 
and liquid ounces, equiv. (1-10), 

table, 83. 
and scruples (U. S. apoth.), equiv- 
alents (1-10), table, 83. 
English equivalents, 81, 82, 84. 



Millimeters 
and inches, 

cubic, equiv. (1-10), table, 82. 

equivalents, 88. 

equiv. (1-10), table, 70. 

square, equiv. (1-10), table, 80. 
and mils, 

equivalents, 88. 

square, equivalents, 88. 
English equivaletns, 68, 70. 
of water, weight of, 67. 
square, English equivalents, 79. 
to decimals of an inch, table, 70. 
Mine, mines, steel timbering in, (ref.) 

9oy. 

Miner's inch, table, 1313. 
Mineral, minerals, 
chemical composition, table, 332. 
classification of, 325. 
color, table, 332. 
defined, 324. 
hardness of, 324; 

table, 332. 
oils, for roads, specifications, 1136. 
physical characteristics of, 324. 
rock-forming, table, 332. 
species, table, 332. 
specific gravity, table, 332. 
Mineralogy, 324. 
Minim or drop (apoth.), metric 

equivalents, 83. 
Minima and maxima (calculus), to 

find, 268. 
Mining, reference data, 1482. 
Minium, for paint, 355. 
Minute, minutes, 
and seconds to decimals of a de- 
gree or hour, table, 1010. 
circular and time measure, equiva- 
lents, 99. 
time and longitude, equiv., 99. 
Mixed masonry, 449. 
Mixing process, in cement making, 

405. 
Mixtures, explosive, 350. 
Modulus _ 
of elasticity, 
bending, of timber, table, 493. 
concrete and steel, 445. 
defined, 486. 
of metals, table, 496. 
of steel and concrete, ratio, 823, 
825, 832. 
of resilience, defined, 488. 
section-, of plane surfaces, tables, 
524. 
Moisture in timber, effect on 

strength, 490-494. 
Moldings, 
elec code rules, 1429. 
metal-, elec. code rules, 1412. 
wooden-, elec. code rules, 1450, 
Molybdenum, chem., 319. 
in steel, 330. 
minerals, 330. 
Moment, moments, 
and reactions, 
drawbridge-, table, 745, 747. 
of forces, 295. 



INDEX. 



1579 



Moment, moments, — Cont'd, 
and shears for engine loadings, 

table, 692. 
arm, in struc, to find, 305. 
bending, 
and chord stresses, 307, 
problem in, 637. 
diagram for engine loads, 691. 
in beams and girders, various load- 
ings, 688. 
in structures, method of, 305. 
in trusses, various loadings, 693. 
maximum-. 
Cooper's loading, table, 708. 
for highway bridges, table, 7i28. 
from electric cars, table, 717, 718. 
position of load on truss for, 695. 
metric and English, equivalents, 

table, 89. 
of forces in structures, to find, 305. 
of inertia 
about inclined axis, 300. 

formulas, 535-538. 
about parallel axis, 300. 
of beams, formula, 299. 
of circular beam, 300. 
of figure about parallel axis, 524. 
of plane surfaces, 298, 

tables, 524. 
of rectangles, 
(ref.) 586. 
table, 539-540. 
of rolled shapes, 537. 
of solids, table, 302. 
of steel column, problem in, 637. 
polar, 535-538. 
maximum and minimum val- 
ues, 537. 
origin of, 296, 305. 
parabola, to draw, 688. 
resisting-, problem in, 637. 
Momentum, 
and impulse, 293. 
train-, coefficient of sliding fric- 
tion, 702. 
Money, 
Austro-Hungarian, U. S. values, 

95 
English, U. S. values, 95. 
Foreign, tables, 95-97. 
French, U. S. values, 95. 
German, U. S. values, 95. 
Italian, U. S. values, 95. 
Russian, U. S. values, 95. 
U. S., table, 95. 
Monomials, examples in, 101, 
Mortar, 
brickwork, kinds used, 438, 
cement, 
for concrete, 417. 
strength of, 507, 508. 
strength ratios "of compressoin 

and tension, 608. 
weight of, 480. 
table, 475. 
for buildings, 819. 
lime, 403. 
for brickwork, 403. 
weight of, 476, 480. 



Mortar, — Cont'd, 
quantities in brick masonry, table, 

438. 
stone masonry, specifications, 433. 
Motion, 
accelerated, 
equations of, 279, 280, 281, 282. 
table, 283. 
and force, equations of, mech,, 

278 
circular, 286. 

formulas, summary of, 284. 
in mech., defined, 278. 
of projectile, 285. 
on cycloidal curve, 286. 
on inclined plane, 286. 
uniform, equations of, 279. 
Motor, motors, 
elec. code rules, 1397, 1450, 
electric, 
defined, 1380. 
railway, 1471. 
speed classification, 1452. 
water, described, 1336. 
Mouthpiece, bell-shaped, at intake, 

1177. 
Movable bridges, 742. 
references, 749. 
weight of steel in, 748. 
Moving picture machines, elec. code 

rules, 1446. 
Muck, in tunneling, defined, 933. 
Mud, weight of, 480. 
Mueller tapping machine, 1283. 
Multiple, least common, to find, 6. 
Multiplication 
and powers, algebraic, examples 

in, 101. 
tables, 1018-1020, 
Muntz-metal, 397. 
Mushroom floor, analysis of, (ref.) 

586. 
Myriagram, English equivalents, 85. 
Myrialiter, English equivalents, 81, 

^ 82, 84. 
Myriameter, 
English equivalents, 70. 
square, English equivalents, 79. 



N 



N, 



coefficients of roughness, values of, 
1168. 

experimental determination of, in 
Kutter's formula, 1187. 

and C, in Kutter's formula, experi- 
mental determination of, 1188. 
Nadir, of celestial sphere, defined, 

201. 
Nails, 

slating, 402. 

steel, weights and dimensions, 
tables, 625-628. 
Naperian logarithms, 

defined, 104. 

examples in, 106. 

table, 108. 



1580 



INDEX. 



Naphthalin, 
boiling point of, 514. 
from creosote, 367. 
National electric code, 1 39 3. 
Natural 
cement, hydraulic, manufacture of 

404. 
coexsecants, tables, 167-175. 
cosecants, table, 167-175. 
cosines, table, 144-166, 
cotangents, table, 144-166. 
coversed sines, table, 144-166. 
exsecants, table, 167-175. 
secants, table, 167-175. 
sines, table, 144-166. 
tangents, table, 144-166. 
versed sines, table, 144-166. 
Nautical almanac, reference to, 202, 
Naviga'ble canals, 1320. 
Neodymium, chem., 319. 
Neon, chem., 319. 
Neutral axis of plane surfaces, table, 

524. 
New Mexico land measure, English 

equivalents, table, 81. 
Nicaragua and Panama routes com- 
pared, 1324-1328. 
Nicholson's hydrometer, 462. 
Nickel 
alloys, 329. 
-aluminum, 

composition of, 496. 
physical properties of , table, 496. 
expansion coefficient of, 516. 
in chem., 319. 
minerals, 329. 
steel, 396. 
annealed, physical properties of, 

table, 499. 
forged, oil-tempered, physical 

properties of, table, 499. 
forgings, 
chemical properties of, 505, 
physical properties of, 505. 
manufacture of, 398. 
properties of, 398. 
spans, compared with carbon 

steel, 737, 738. 
specifications for Manhattan 
bridge, 758. 
-vanadium steel, 399. 
weight of, 480. 
Niobium, chem., 318. 
Nitrate explosives, 350. 
Nitric acid 
compounds, 351. 
treatment of cellulose, 351. 
Nitro-explosives, 351. 
Nitroglycerin, manufacture of ,351,352 
Nitrogen, 
boiling point of, 514. 
in chem., 319. 
melting point of, 515. 
physical properties of, table, 514. 
Nonagon, mensuration of, 204. 
Non- 
condensing engines, performance 

of, 1365. 
inductive load, in elec, 1453. 



Normal 
and tangent (calculus) , equations 

of, 267. 
to circle, equation of, 257. 
to cycloid, 260. 
to ellipse, equation of, 259. 
to hyperbola, equation of, 259. 
to parabola, equation of, 258. 
North 
point, of celestial sphere, defined, 

201. 
star (see also Polaris), 
to determine meridian from, 948. 
to find, 948. 
Notation, Bow's, for trusses, 309, 
Nozzle, nozzles, 
conical, 1177. 
discharge from, 1175. 
Numbers, 
abstract, Arabic notation, table, 

95. 
Arabic system of, 1. 
duodecimo, table, 95. 
logarithms of, 104-127. 
primes, multiples and factors, 2. 
Roman system of, 1. 
short methods of multiplication 
and division, 11-13. 
Nuts 
and bolts, tables, 618-621. 
pilot-, 629. 
pin-, table, 629. 
sleeve-, 634. 
weights and dimensions, table, 
633. 



Oak, oaks, 
expansion coefficient of, 516. 
classification of, 344. 
friction of, 518-520. 
Obsidian, 
composition of, table, 338. 
defined. 340. 
Ochre for paint, 355. 
Octagon, 
and circle, inscribed and circum- 
scribed, 131. 
hollow, properties of, 527. 
mensuration of, 204. 
regular, properties of, 527. 
Octahedron, defined, 132. 
Offsets, cast iron pipe, tables, 1235, 

1249. 
Ohm, defined, 1514. 
Oil, oils, 
as road dust preventives, 1131. 
crude-, products from, in refining, 

1133. 
experiments on roads, costs, 1139, 

1140. 
fields of the U. S., 1133. 
for graveled streets, specifications, 

1115. 
for roads, 
best kinds, 1133. 



INDEX, 



1581 



Oil, oils,— Cont'd, 
for roads,— Cont'd, 
classification of, 1133. 
properties of, 1133. 
fuels in locomotives, use of, (ref.) 

1378. 
heavy, 
application to roads, 1134. 
application to road surfaces, 
1134. 
linseed, manufacture and use, 356. 
mineral, for roads, specifications 

1136. 
-proof compositions, 418. 
refining crude-, 1133. 
semi-asphaltic, 1133. 
Oiling 
graveled streets, specifications, 

1114. 
iron and steel, 358. 
roads, cost of, (ref.) 1142. 
electronic theory, 317. 
Omnibus bars, defined, 1487. 
Onzo (Philippine weight), English 

equivalent, 81. 
Open 
-cut tunneling, 933. 
hearth 
cast steel, 396. 
process, 394, 395. 
steel (boiler plate), specifica- 
tions, 501. 
Operation and maintenance of ca- 
nals, cost data, 1329. 
Orange-peel bucket dredge, 928. 
Ordinate and abscissa, defined, 256. 
Ore, ores, 
amalgamation, 357. 
extraction, 357. 
iron, treatment of, 392. 
of minerals, 328. 
smelting, 357. 
weights of, 480. • 

Orifice, orifices, 
and tubes, compared, 1176. 
center of pressure on, table, 1151. 
coefficient of discharge through 

circular, (ref.) 1189. 
discharge from, 1175. 

table, 1176. 
flow of steam through, (ref.) 1377. 
frictionless-, experiments on, (ref.) 

1187. 
standard, 1176. 
Origin, 
of curves, defined, 256. 
of moments, 296, 305. 
Orthographic projection, 261. 
Osbom rivet code, 611. 
Oscillation, 
center of, 303. 
of pendulum, 287. 
Osmium, chem., 319. 
Ounce, ounces, 
and grams, equivalents (1-10), 

table, 85. 
apothecary, 
fluid, metric equivalents, 83. 
metric equivalents, 86. 



Ounce, ounces, — Cont'd. 
avoirdupois, 
and grams, equivalents, 89. 
and troy, equivalents, 67. 
metric equivalent, 86. 
liquid, and milliliters (c. c), equiv. 

(1-10), table, 83. 
metric equivalents, 68. 
troy, metric eqtdvalents, 86. 
Outlet 
boxes, elec. code rules, 1429. 
pipe from reservoir, how arranged, 
1205. 
Oval, 
or false ellipse, 259. 
parabolic-, curve of, (ref.) 765. 
Overload capacities, in elec, 1469. 
Overshot wheel, described, 1336. 
Oxidation of 1 lb. carbon with per- 
fect efficiency, equivalents of, 
table, 91. 
Oxide, in chem., defined, 321. 
Oxy -acetylene flame for cutting steel- 
work, 833. 
Oxygen, 
boiling point of, 514. 
chem. ,319. ' 
compounds, in min., classification 

of, 326. 
physical properties of, table, 514. 
Oxy -hydrogen flame, 317. 
Ozone, chem., 319. 



Paint, paints, 355. 

adulterants, 355. 

aluminum, how made, 356. 

bronze, how made, 357. 

coal tar, (ref.) 374. 

driers, 356. 

for concrete, (ref.) 375. 

house, 356. 

iron ground for, 329, 

solvents, 356. 

vehicles, 355. 

white lead, 329. 

zinc white, 329. 
Painting 

and sand-blast cleaning, 373. 

by compressed air, with cost, 374. 

iron and steel, 358. 

metal work, for buildings, 820. 

steel at the mill, 373. 
Palladium, chem., 319. 
Panama _ 

and Nicaragua routes compared, 
1324-1328. 

canal, 
distance via., between Atlantic 
and Pacific ports, table, 1328. 
excavation, cost data, 916. 
steam shovel work, cost data, 
919. 
Panclastite explosives, 353. 
Panel boards, elec. code rules, 1439. 
Paper 

measure, table, 95. 

weight of, 480. 



1582 



INDEX. 



Pappus's theorem, 243. 
Parabola, 
any base and altitude, to draw, 

258. 
area of, by calculus, 275. 
cubic-, 1013. 
equation of, 257. 
normal to, 258. 
tangent to, 258. 
intersection with circle, 258. 
moment-, to draw, 688. 
properties of, 237. 
radius of curvature of, 258. 
to draw, 238. 
Paraboloid, 
mensuration of, 254. 
volume of, by calculus, 277. 
Parabolic 
arch, 761. 

arcs, lengths of, table, 238. 
cable of suspension bridge, 750, 
conoid, mensuration of, 254. 
motion (of projectile), 285. 
oval, curve of, (ref.) 765. 
segment, 237. 

(half-), properties of, 529, 
spandrel, 237. 

properties of, 529. 
vertical curves, (R. R.), 105. 
Paraffin, 
compounds, obtained from petro- 
leum, 1133. 
expansion coefficient of, 516. 
weight of, 480. 
Parallelogram, 
area and definition, 203. 
defined, 128. 
properties of, 525. 
Parallelopipeds, defined, 133. 
Parameter of catenary, values of, 

table, 752. 
Paris green, 330. 
Parmley's weir formula, 1180. 
Partial payments, 63. 
Partitions, 
building, 812. 
expanded metal, 814. 
hollow-tile, 813. 
lumber (fir), classified, 389. 
plaster board, 813. 
wire lath, 813. 
wooden, 813. 
Paste, flour, 402. 
Patent hammer, described, 429. 
Pavement, 
asphalt, 
construction of, 1100. 
specifications, 1104, 1110, 1125, 

1128, 
block, specifications, 1122. 
rock, specifications, ll25. 
sheet, refined, specifications, 1125. 
Belgian block, described, 1100. 
bitulithic, specificat ions, 1 105, 

1126. 
bituminous-rock, 1100. 
boulder, specifications, 1107. 
brick, 
-block, specificatiors, 1105. 



Pavement, — Cont'd, 
brick, — Cont'd, 
cost, (ref.) 1142. 
described, 1100. 
specifications, 1109, 1129. 
street, proper construction of, 
1106. 
broken -St one, construction of, 

1099. 
cedar block, specifications, 1110, 

1128. 
cement, described, 1099. 
cobblestone, construction of, 1099. 
concrete, specifications, 1128. 
grading for street, specifications, 

1127. 
granite-block, specifications, 1102, 

1103, 1119. 
petrolithic, specifications, 1112. 
reinforced concrete base for, (ref.) 

1112. 
sandstone block, specifications, 

1124. 
slag (iron) block, specifications, 

1121. 
specifications, 1101. 
traction on, 1097. 
vitrified brick, specifications, 1121. 

1123. 
wood -block, 
construction of, 1099. 
creosoted, specifications, 1126. 
specifications, 1120. 
Paving 
a country road with brick, cost, 

1141. 
blocks, 
asphalt, 1100. 

specifications, 1122. 
cedar, specifications, 1128. 
granite, 1102. 

iron slag, specifications, 1121. 
sandstone, specifications, 1124. 
size of, 1129. 
vitrified brick, specifications, 

1121. 
wood, 
creosoting treatment, 1126. 
grooved, 1120. 
specifications, 1120. 
brick, 415. 
-block, specifications, 1105. 
specifications, 1124. 
size of, 1129. 
strength of, 507. 
cedar block, specifications, 1128. 
cement, specifications, 1121. 
concrete, for streets, cost, etc., 

(ref.) 1142. 
granite-block, specifications. 1105. 
practice m Chicago, crowning, 1143. 
Payments, 
equation of, 61. 
partial, 63. 
Pean hammer, described, 429, 
Peat, pressed, weight of, 480. 
Peck, pecks, 

(U S.), and dekaliters, equiv. (l- 
10), table, 84. 



INDEX. 



1583 



Peck, pecks, — Cont'd. 
(U. S.), and liters, equiv. (1-10), 

table, 84. 
equivalents, 67. 
metric equivalents, 68, 84. 
Pecul (Philippine weight), English 

equivalent, 81. 
Pelton water wheels, 
quintex nozzle, table, 1342. 
single nozzle, table, 1338-1341. 
Pendulum, 
circular, 287. 
compound circular, 287. 
cycloidal, 287. 
simple circular, 287. 
Pennyweight (troy), metric eqtiiva- 

lent, 86. 
Penstock design, economic, 1332. 
Pentagon, 
defined, 129. 
inscribed in circle, 131. 
mensuration of, 204. 
to construct, 131. 
People, load of crowd of, 815. 
Percentage, interest and discount, 

58. 
Percussion 
caps, 353. 

fulminate of mercury in, 352. 
center of, 303. 

of pendulum, 287. 
drill, described, 422. 
rock-drills, 
dimensions, etc., table, 424. 
weights, etc., table. 424. 
Perimeter 
of ellipse, 239. 
of polygon, 129. 204. 
of triangle, defined, 128. 
Periodic law, in chem., 322, 323. 
Permutation and combination, 56. 
Perspective, 261. 

Peso (Mexican), equiv. (1-10,-50- 
100-) in U. S. money, table, 97. 
Petroleum, 
crude-, 
products from, in refining, 1133. 
test properties, 1134. 
heat of combustion of, 1370. 
residuums, test properties, 1134. 
Petrolithic pavement, specifications, 

1112. 
Pewter, 398. 
expansion coefficient of, 516. 
weight of, 480. 
Phase, single-, alternator, defined, 

1384. 
Philippines, weights and measures, 

English equivalents, 81. 
Philorier's mixture or freezing, 513. 
Phoenix columns, properties and 

safe loads, tables, G04, 605. 
Phosphates, in min., classification 

of. 327. 
Phosphor bronze, 397. 

physical properties of, .table, 497. 
Phosphorus, chem., 319. 

weight of, 480. 
Photometry and lamps, in e/^<r..l473. 



circular measure, 99. 
table of combinations of, with logs, 
205, 206. 

times any number, tables, 224- 
229. 
Pick, stone-, described, 428. 
Pickling, for mill scale, 358. 
Pie (Philippine measure), English 

equivalents, 81. 
Pier, piers, 
concrete pile, for steamship termi- 
nal, (ref.) 900. 
construction, 893. 
crib, 877. 
cylinder, 878. 

frictional resistance of, 878. 
docks and wharves, 892. 
masonry, 888. 

contents of, 889. 
pile, 877. 

platform cylinder, 879. 
pneumatic cylinder, 879. 
reinforced concrete, (ref.) 890. 
river, 889. 

steel, in Africa, (ref.) 900. 
tubular, 877. 
Pierhead and bulkhead lines, 892. 
Piezometer tubes, 1174. 
Pig iron, 
manufacture of, 392. 
use of, 392. 
Pigments for paints, 355. 
Pile, piles, 
-and-timber trestles, 790. 
bearing power of. 819. 

in various materials, 890. 
concrete, 875. 

creosote in, analysis of, 370. 
cutting off, 874. 
dead-men-, 874. 
disk, 874. 
drivers, 871. 
derrick, 873. 
drop-hammer, 872. 
portable, 873. 
power for, 873. 
steam-hammer, 872. 
tilting-, (ref.) 890. 
driving, 
formulas, 871. 
water jet in, 873. 
false, 788. 
fender-, 893. 
foundations, 871. 
iron, 874. 

maximum load on, 787. 
metal-shell, concrete, 875. 
piers, 877. 
concrete, for steamship terminal . 
(ref.) 900. 
planted, 874. 

-pulling machines, (ref.) 871. 
reinforced concrete. 875. 
safe bearing power of, table, 872. 
sand, 875. 
screw, 874. 
shoes, 874. 
supporting power of, 871. 



1584 



INDEX, 



Pile, piles, — Cont'd, 
spacing and driving, 874. 
spliced, 874. 
trestles, 787. 
water-jet concrete, 875. 
Piling, 
cost of creosoting, 360. 
preservation of, 360. 
sheet-, 869.. 
Pillars, (see Columns). 
Pilot-nuts, 629. 
Pin, pins, 
bending moments of, table, 630. 
bending stresses on, specifications, 

704. 
bridge-, 629. 
cotter-, 629. 
lateral-, 629. 
-nuts, table, 629. 
plates, for bridges, 706. 
steel-, properties of, table, 630. 
Pine, pines, 
expansion coefficient of, 516. 
classification of, 341. 
Pint, 

(apoth), metric equivalents, 83. 
(dry), metric equivalent, 84. 
(liquid), metric equivalent, 83. 
metric equivalents, 68. 
Pipe, pipes, 
or butt (liquid), equivalents, 83. 
and tubes, 677. 

references, 682. 
areas of, forgiven diameters, table, 

1157. 
bell and spigot, 1215. 
black or galvanized, table, 1284. 
block tin, 679. 
branches, cast iron, 
L's, T's, crosses, tables, 1225, 

1250-1254. 
Y's, tables, 1227, 1228, 1255, 
1256. 
cast iron, 1214. 
and specials, weights and dimen- 
sions of, tables, 1219-1267. 
flange, table, 1236. 
for various pressures, table, 1216. 
formulas for designing, 1215. 
friction heads in, table, 1217. 
hemp required per joint, 1216. 
jute required per joint, table, 

1222. 
lead required per joint, 1216; 

table, 1222. 
specifications, 1239. 
variation allowed, 1222. 
weights and dimensions, tables, 

1220, 1222, 1243-1246. 
weight of, table, 1216. 
caulking with lead wool, (ref.) 

1293. 
coating, Sabin process, 358. 
Converse patent lock joint, table, 

1282. 
crosses, Matheson, table, 1281. 
culvert, tables, 1307. 
curves, cast iron, tables, 1224, 
1248, 1249. 



Pipe, pipes, — Cont'd, 
diameters and equivalent areas, 

table, 1157. 
dia. to area, capacity, mean radius, 
volume, weight (water), table 
246-247. 
dipping tank, 1282. 
discharge through small, table, 

1284. 
drain-, 1295. 
flexible joint, cost, 1238. 
flow of steam throiigh, formula 

and table, 1361. 
flow of water in, measurement of, 

1183. 
for water works, cost, 1292. 
friction losses in, defined, 1161. 
friction of air in small, formula, 

1189. 
glazed (salt-), for water supply, 

1207. 
increasers, Metheson, table, 1281. 
iron-, cement, 402. 
joints, 
caulking, 1215. 

mortar required for, table, 1 309. 
sewer, 
cement and sand required for, 

table, 1309. 
sulphur and sand required for, 
table, 1310. 
kalamein, tables, 1281, 1282. 
lap-welded, 1269. 
-laying, 1215. 
notes, 1219. 
lead, 
hydrostatic pressure for, 679. 
tables, 679. 
line, 
economic size of, for power in- 
stallation, 1189. 
gains and losses in, 1154. 
hydraulic notation used in, 1154. 
hydraulic terms used in, 1154. 
losses in during flow, 1158. 
practice of increasing the dia- 
meter, 1156. 
locking-bar joint, 1269. 
Matheson patent lock -joint, tables. 

1281. 
plugs, Matheson, table, 1281. 
pressure-, 
attachments, 1269. 
in, hydrostatic, 1152. 
reducers, Matheson, table, 1281. 
riveted steel, design of, 1268. 
sewer, tables, 1307. 
soil-, 1295. 

specials, cast, described, 1280. 
spiral riveted, 1269. 

steel, 680-682. 
steel, 1268. 
experimental values of N in Kut- 
ter's formula for flow in, 1188. 
tees, Matheson, table, 1281.^ 
theoretic velocities of flow in, 

table, 1155. 
velocity ratios to area and diam- 
eter, 1158. 



INDEX, 



1585 



Pipe, pipes, — Cont'd, 
waste-, 1295. 

wood-, bored and banded, 1208. 
wooden (bored), for water supply, 

1207. 
wood stave, 
and cast iron connection, 1280. 
and details, table, 1210. 
details of, 1208. 
durability of, 1187. 
flow of water in, (ref.) 1187. 
wrought iron, 1268. 
welded, standard, tables, 677, 678. 
Pitch 
and tar for waterproofing, 418. 
of screw thread, 618. 
weight of, 480. 
-faced masonry, defined, 432. 
Pitot-tube 
meter, 1183, 1184. 
rating, (ref.) 1189. 
Plane, planes, 
coordinate, 132, 261-265. 
geometric, 
angles and lines, 132. 
determination of, 132. 
geometry, 128-131. 
inclined, 
in mech., formulas, 292. 
motion on, 286. 
velocities on, 286. 
revolved, 261, 262, 265. 

volumes of, by calculus, 277. 
-table, (ref.) 990. 
the, projection of, 263, 264. 
trigonometry, 136-198. 
two, to find angle between, 265. 
Planing, 
in bridge work, 706. 
lumber, 379. 
cost of, 379. 
Plank, 
classification of, 388. 
gutters, described, 1098. 
roads, described, 1098. 
sidewalks, described, 1098. 
Planking, classification of, 388. 
Plaster, 
expansion coefficient of, 516. 
gypsum, weight of, table, 481. 
lime, 403. 
of paris, 404. 
how made, 339. 
manufacture and use, 404. 
physical properties of, 612. 
weight of, 480. 
ordinary, 481. 
Plastered ceiling, beam calculation, 

564. 
Plastering, building, 812. 
Plate, plates, 
bearing value of pins, table, 630. 
circular, moment of inertia of, 302. 
flange, of plate girders, properties 

of, table, 580. 
flat, 
Grashof's analysis of, (ref.) 586. 
washers, weights and dimen- 
sions, table, 624. 



Plate, plates, — Cont*d, 
girders, 
economic depth of, 684. 
steel, properties of, table, 570- 

582. 
specifications, 704. 
steel, 
areas and weights, table, 544. 
gage and weight, table, 667. 
weight and areas, table, 544. 
web, of plate girders, properties of, 
table, 575. 
Platform cylinder piers, 879. 
Plating, 357, 358. 
Platinum, chem., 319. 
cast, etc., table, 481. 
expansion coefficient of, 516. 
-iridium, 

alloy, composition of, 516, 
expansion coefficient of, 516. 
melting point of, 515. 
minerals, 328. 

wire, tensile strength of, 498. 
Platting angles, methods of, 959. 
Plenum process, 879. 
Plug, plugs, 
and feathers, described, 426. 
cast iron pipe, tables, 1234, 1267. 
Metheson pipe, table, 1281. 
Plumbago, weight of, 481. 
Pneumatic 
caissons, 880. 
caulking of mains with lead wool, 

costs, (ref.) 1293. 
cylinder piers, 879. 
foundations, 880. 
process, 879, 

stone-dressing machine, (ref.) 430. 
Point, 
stone-, described, 428. 
(the), projection of, 262. 
Pointing, masonry, 
defined. 432. 
specifications, 434. 
Polar 
distance, 
of a circle, defined, 135. 
of a star, defined, 202. 
of Polaris for latitude 0°, 949. 
moment of inertia, 535-538. 
maximum and minimum values, 
537. 
Polaris, 
azimuth of, 
at elongation, 950. 
tables, 954-955f. 
how to find, 948. 
observations of, 951. 

for azimuth, 949. 
polar distances of, for latitude 0°, 

table, 949. 
time of upper culmination of, table 

953, 
to determine meridian from, 948. 
Pole, poles, 
concrete telegraph, 1477. 
cost of creosoting, 374. 
(geom.) of a circle, defined, 134. 
iron, 373. 



1586 



INDEX, 



Pole, poles, — Cont'd. 

lines, elec. code rules, 1400. 
of celestial sphere, defined, 201. 
preservation of, 373. 
reinforced concrete, (ref.) 1142. 
wooden, 
for electric line, 373. 
life of, 373. 
Polygon, polygons, 
definitions of, 129. 
(general), properties of, 129. 
of forces, 295. 
regular, 
area of, 129. 
formulas and table, 204. 
properties of, 129. 
Polyhedrons, 
(geom.), 132. 
regular, table, 243. 
Polyphase alternator, defined, 1384. 
Poplars, classification of, 343. 
Population of cities in U. S., 1202. 
Porcelain, expansion coefficient of, 

516. 
Porch decking lumber (fir), classi- 
fied, 389. 
Portable conductors, elec. code rules, 

1448. 
Portalj portals, 
bracing, of bridges, 698. 
bridge-, types of, 698. 
skew-, detailing, (ref.) 665. 
Portland Cement (see, also," Cement, 

Portland). 
Portland 
cement, 
cost, 418. 

manufacture of, 404, 405 
concrete (see Concrete, Portland), 
stone, maniifacture of, 417. 
Posts, wooden, 
preserving, 361. 
trestle, 788. 
Position line, of an arch, 782, 783. 
Potash, weight of, 481. 
Potassium, chem., 319. 
minerals, 328. 
weight of, 481. 
Potential energy, defined, 1346. 
Pound, pounds, 
and kilograms, equiv. (1-10), 

table, 85. 
(apoth.), metric equivalents, 86. 
(avoir.), and kilograms, equiv., 

89. 
avoirdupois and troy, equiv., 67. 
(avoir.), metric equivalent, 86. 
-calorie (Lb.-Cal.), defined, 1347. 
-degree (Fahr.), eqmvalents of, 90. 
(dollars per) 
and francs per kilogram, equiva- 
lents (1-10), table, 98. 
and marks per kilogram, equiva- 
lents (1-10). table, 98. 
kilograms and tons, equiv. (1-10), 

table, 87. 
metric equivalents, 68. 
per cu. ft. and kilograms per cu. 
meter, equivalents, 89. 



Pound, pounds, — Cont'd, 
per cu. in. and grams per cu. 
centimeter, equivalents, 89. 
per sq. ft. and kilograms per sq. 

meter, equivalents, 89. 
per sq. in. and kilograms per sq. 

centimeter, equivalents, 89. 
(troy), metric eqmvalent, 86. 
(£) sterling (British), equivalents 
(1-10,-50-100) in U. S. money, 
table, 97. 
Powder, 
Aetna, 354. 
Atlas, 352, 354. 
black, 350. 
blasting, 
composition of, 350. 
specific gravity of, 350. 
weight of, in hole, 350. 
brown, 350. 
Calonia, 354. 
Giant, 352, 354. 
gun-, weight of, 480. 
Hecla, 354. 
Hercules, 352, 354. 
Horsley, 354. 
Judson, 352, 354. 
safety nitro, 352. 
smokeless, 352. 
sporting, 350. 
Vulcan, 353. 
Power 
and heat problems, 1350. 
and work equivalents, metric and 

English, table, 90. 
comparative cost of electricity, 
gas, gasoline and steam for, 
(ref.) 1378. 
electric-, 
and lighting, 1379. 
cost data, table, 1478. 
sources and uses of, 1385. 
transmission of, 1385. 
-factor, in elec, 1453. 
horse-, (see, also. Horsepower), 
mechanical, equivalents of, 90. 
electric, equivalents of, 90. 
metric, equivalents of, 90. 
equivalents of, 91. 
in mech., equations of, 291. 
of a horse, 1097. 
plants, 
electric and steam, costs, 1477. 
railway, elec. code rules, 1398. 
solar-, reference data, 1484. 
steam and gas, 1346. 
steam- and water-, compared, 1385. 
units of, 291. 

electrical, 1379. 
water- installation, economic size 
of pipe line for, 1189. 
Powers, 
and multiplication, algebraic, ex- 
amples in, 101. 
fifth, engineers' table, 26-27. 
of numbers, by logarithms, 106. 
roots and reciprocals of numbers, 
14-54. 
Praseodymium, chem., 319, 



INDEX. 



1587 



Pratt truss, 
calculation of, 306. 
chord stresses in. concentrated 

loads, 695. 
graphical solution of, 312. 
Precipitation, 
average monthly, in United States, 

1191. 
defined, 1190. 
high intensities of, formulas, 1195- 

1197. 
in U. S., for driest years, 1194, 
1195. 
Preliminary survey (R. R.), 1000. 
Preservation 
of iron and steel, 358. 
of timber, 359. 
Preservatives, 355. 
Pressure, 
artesian-, defined, 1190. 
atmospheric, 1145. 
center of, 846, 847. 
formulas, 1150. 

on vertical orifices and weirs, 
table, 1151. 
critical, 
defined, 513. 
of gases, table, 514. 
of liquids, table, 514. 
earth-, 
Rankine's theory, 839-840. 
theories, 835-840. 
head in pipe lines, 1160. 
hydrostatic-, 
for lead pipe, 679. 
formulas, 1146. 
on dams, 845. 
in pipes, hydrostatic, 1152. 
in tanks, hydrostatic, 1152. 
of train on curve, problem, 297. 
of water, 
for given heads, tables, 1148, 

1149. 
reduced to equivalent heads, 
table, 1147. 
on masonry, 
allowable, 825. 
for bridges, 705. 
on submerged planes, 846. 847. 
-pipe attachments, 1269. 
relief valves, described, 1288. 
units, hydrostatic, 1146. 
wind-, 794. 
Price current-meter, 1185. 
Prices per unit weights and meas- 
ures (metric and English), com- 
parison of, equivalents (1-10), 
table, 98. 
Primary or storage batteries, elec. 

code rules, 1398. 
Prime, primes, 
vertical, of celestial sphere, de- 
fined, 201. 
multiples and factors, table, 3-5. 
Prism, prisms, 
area, volume, 224. 
geom., 133. 

truncated, defined, 133. 
volume of, 133. 



Prismoidal 

correction 
formula, earthwork, 1056. 
table, earthwork, 1057. 

formula, 
earthwork, 1056. 
for contents of piers, 889. 
general, 243. 
Principle, artesian-, defined, 1190. 
Produce, weight of, 478, 482. 
Profiles and grades, railroad, 1004. 
Progression, 

Arithmetical, 57. 

Geometrical, 57. 
Projectiles, 

motion of, 285. 

velocity and height of, table, 283. 
Projection, 

Cabinet, 261. 

Isometric, 261. 

of right lines, 262, 263. 

of the plane, 263, 264. 

of the point, 262. 

Orthographic, 261. 
Properties of plane surfaces, tables, 

524. 
Proportion 

and ratio, 55-56. 

by segments of chords of circle, 
131. 
Proportional, mean-, 56. 

in semicircle, 131. 
Proximate analysis of fuels, 1350. 
Puddling, 

cast iron, 392. 

effect on earth fill, 910. 
Pulley, in mech., formulas, 292. 
Pulsation and variation, in elec, 

1453. 
Pulverizer, for cement making, 405. 
Pulverizing process, in cement mak- 
ing, 405. 
Pumice stone, weight of, 481. 
Pump, pumps, 

duty of, formula, 1367. 

explosion-, direct-acting, 1378. 

steam-, 1366, 1367, 
Purification of water, 1204. 
Purlins, roof-, weight of steel in, 810, 

811. 
Pyramid, pyramids, 

altitude of, defined, 133. 

frustum of, defined, 134. 

geom., 133. 

mensuration of, 248. 

of sphere, defined, 135. 

spherical, solution of, 199. 

volume of, 133. 
Pyramidic frustum, mensuration of, 

248. 
Pyroxylin, manufacture of, 351. 

Q 

Quadrant, quadrants, 

geom., defined, 135. 

of circle, defined, 129. 

trig., the four, 137. 
Quadratic equations, squaring, 102. 



1588 



INDEX, 



Quadrilaterals, 

defined, 128. 

mensuration of, 203. 
Quarry-faced 

masonry, defined, 432. 

stone, defined, 427. 
Quarrying, 419. 

block stone, 419, 

by compressed air, 423. 

channeling machines in, 420. 

cost of, 423. 

drills used in, 422. 

explosives in, 422. 

flagstones, 419. 

gravel and sand, 419. 

machines used in, 419. 

riprap, 419. 

sand and gravel, 419. 

stone, building, 419. 

tools used in, 419. 
Quart, quarts, 

and liters, equivalents, 88. 

dry, 
and liters, equiv. (1-10), table, 

84. 
metric equivalents, 84. 

eqmvalents, 67. 

liquid, 
and liters, equiv. (1-10), table, 

83. 
metric equivalent, 83. 

metric eqmvalents, 68. 
Quarter 

(avoir, wt.), metric equiv,, 86. 

section (160a,) and hectars, equiv- 
alents, 88. 
Quartz, 

crystal, weight of, 481. 

uses of, 331. 
Quicklime, 

made how, 403. 

specific gravity of, 403. 
Quintal (metric), English equiv., 85» 
Quire, paper measure, 96. 
Quoin, masonry, defined, 432. 



Rack railways, (ref.) 1095. 
Radian (tt), circular measure, 142. 
Radii of curves (R. R,), tables, 1007. 
Radio-activity of matter, 316. 
Radium, chem., 319. 
corpuscles in, 316. 
energy of, 316. 
from uranites, 330. 
Radius 
and diameter, circular measure, 99. 
of circle, defined, 129. 
of curvature 
of cycloid, 260. 
of ellipse, 259, 765. 
of parabola, 258. 
of gyration 
of circular beam, 300. 
of plane section, 299, 300, 
of plane surfaces, table, 524. 
of solids, 302. 
problem in, 637. 



Radius — Cont'd. 

of polygon, 129. 
Rail, rails, 
and fastenings, (R. R.), 1060. 
braces, 1071 

elevation on curves, formula, 298. 
joints, kinds of, 1068. 
manganese-steel, on curves, (ref.), 

1094. 
sections, standard, J060-1062. 
steel, 
chemical properties of, 503. 
dimensions and weights, table, 

560. 
properties of, table, 560. 
specifications, 503. 
testing, 503. 
traction on, 1097. 
weights and dimensions, table, 
560. 
T-, for street railway tracks, (ref.) 

1142. 
weight of, per mile, table, 1063. 
Railroad, railroads, 991. 
bridges, 688. 
references, 713. 
reinforced concrete, 712. 
steel-, 
specifications for, 699. 
weight of, 710. 
construction, 1016. 
curves, 1005. 

problems, 1011. 
excavation, cost data, 916. 
grading, 
economic problem, 269. 
with wheeled scrapers, cost data, 
917. 
location, 998. 

filing with State, 1013. 
masonry, classification of, 431. 
mileage in U. S„ 991. 
projection of, 991. 
reconnoissance, 998. 
right-of-way, 1013. 
spikes, 
table, 627. 

weight per mile of track, table, 
1068. 
ties, 1069. 
trestles, 787. 
cost of, 793. 
Railway 
bridges, steel-, weight of, formu- 
las, 686. 
curve, to lay out, 130. 
electric motors, 1471. 
embankments, shrinkage vertical 

in, 915. 
inclined-, 1091. 
location, tables, (ref.) 1092, 
power plants, elec. code rules, 1398. 
rack-, (ref,) 1095, 
trestles, steel-, weight of, formula, 
686. 
Rainfall, 
and runoff in storm sewers, formu- 
las, diagrams and tables, (ref.) 
1310. 



INDEX. 



1589 



Ramfall,— Cont'd 
average monthly, in U. S., table, 

1191. 
data most useful to engineers, 1190. 
distribution of, 1190. 
high intensities of , formulas, 1195- 

1197. 
in U. S., for driest years, 1194, 

1195. 
relation of, to runoff, in California, 
(ref.) 1201. 
Rain-gage, standard, 1196. 
Rain-water, weight of, 482. 
Randoms, in surv., correction of, 

976. 
Random-work masonry, defined, 4 32. 
Range-work masonry, defined, 432. 
Rankine's theory of earth pressure, 

839-840. 
Ransome stone, manufacture of, 4 17. 
Rapid sand filtration, 1204. 
Rating, 
electrical-, 1454. 
pitot-tube, (ref.) 1189. 
Ratio and proportion, 55-56, 
Rattler test for bricks, 507, 1116. 
Reaction, reactions, 
and moments, 295. 
breakwater, 905. 

drawbridge-, tables, 744, 745, 747. 
floorbeam-, 
for concentrated loads, table, 

694. 
for highway bridges, table, 728. 
from electric cars, tables, 717, 
718. 
in struc, in any direction, 314. 
of forces, 295. 
to find, 305. 
Reactive 
coils and condensers, elec. code 

rules, 1443. 
-factor, in ^/^c.,1453. 
Ream, paper measure, 95. 
Reaming, in bridge work, 706. 
Reciprocals, 
by slide rule, 126, 
common tables, 51-54. 
engineers' tables, 28-29. 
powers and roots of numbers, 14- 

54. 
to find, 30. 
Reconnoissance survey, (R. R,), 998. 
Rectangle, rectangles, 
angular axis, properties of, 525. 
area and cen. of grav., 203. 
axis at base, properties of, 525. 
defined, 128. 

diagonal axis, properties of, 525. 
hollow-, properties of, 525. 
moments of inertia of, table, 539- 

540. 
properties of, 525. 
skeleton section, properties of , 530. 
Rectangular ' 
beams, 
formulas, 563. 
loads on, table, 566. 
cell, skeleton, properties of, 530. 



Rectangular — Cont'd, 
conduits, proportioned for maxi- 
mum discharge, 1161. 
orifices, center of pressure on, 
formulas, 1151. 
Red 
heart, in lumber, defined, 387. 
lead 
for paint, 355. 
paint, 359. 
weight of, 481. 
oxide for paint, 355. 
rays, length of, 1380. 
Reducers, 
cast iron pipe, tables, 1233, 1259- 

1264. 
Matheson pipe, table, 1281. 
Redwoods, classification of, 342. 
Refraction, 
defined, 1520. 
and curvature, table, 988. 
Refuse disposal, 1295. 
miscellaneous data, (ref.) 1309- 
1312. 
Regenerative method of liquefying 

gases, 513. 
Register current-meter, 1 1 86. 
Regulation, electrical-, 1461. 
Reinforced concrete, 443. 
aqueduct, 1208. 
arch bridges, cost of, 784, 
arches, reference data, 784. 
beam, beams, 
diagram, (ref.) 586. 
formulas, 444, 447. 
table, 446. 

tests, formulas, (ref.) 585. 
Thacher's computation, 585. 
time element effect in loading, 

(ref.) 586. 
working stresses, 585. 
bridges, 
highway, cost data, 738. 
railroad, 712. 
piers, (ref.) 890. 
buildings, unit stresses used, 831. 
caissons, for breakwaters, 904. 
columns, 
formulas, 449. 
working stresses for, 609. 
construction, 
building regulations for, by Nat'l 

Assn. of Cem, Users, 831. 
for buildings, 822, 834. 
design, office methods, (ref.) 455. 
economic design of, (ref.) 585. 
flat plates, methods of computing, 

(ref.) 586, ^ 
floors, instruction sheet for plac- 
ing, (ref.) 586. 
formulas of A. S, C. E., 446. 
French Gov't rules, (ref.) 454, 
in buildings, references, 830-834. 
jetty-head, (ref.) 905. 
members separately-molded, cost 

data, (ref.) 831. 
piles, 875. 
poles, (ref.) 1142. 
proportions of mix, 444. 



1590 



INDEX. 



Reinforced concrete, — Cont'd, 
references, 456. 

retaining walls, (ref.) 841-843. 
roadway base, (ref.) 1112. 
slabs, slide rule for, (ref.) 586. 
T-beam and column tests, (ref.) 

586. 
trestles, 792. 
use of, (ref.) 455. 
wharf, (ref.) 900. 
Reinforcement, steel, in beams, ten- 
sile stress value, 585. 
Relief valves, pressure-, described, 

1288. 
Reluctance, magnetic-, formula, 1520. 
Repeated stresses, defined, 487. 
Reports and valuations, expert, 

reference data, 1484. 
Repose, angle of, for various sub- 
stances, 517-521. ' 
Reservoir, reservoirs, 1205. 
aerating fountain in, 1206. 
concrete expansion joints in, 454. 
distributing, 1206. 
linings, 1206. 
miscellaneous data, (ref.) 1291- 

1294. 
outlet pipe from, how arranged, 

1205. 
storage-, 1205. 

walls, waterproofing for, 418. 
waterproofing, 1293. 
Residuums from refining crude oil, 

1133. 
Resilience, 488. 
elastic, defined, 488. 
formulas, 488. 
modulus of, defined, 488. 
Resin, 
a tree product, 346. 
defined, 481. 
mastic, weight of, 480. 
weight of, 481. 
Resistance 
boxes, elec. code rules, 1395, 1449. 
electric-, formula, 1520. 
insulation-, elec. code rules, tables, 

1447. 1450. 
line of, of masonry arches, 768. 
of materials, 486. 
Resisting 
and bending moment of beams, 298. 
moments, problem in, 637. 
forces, in mech., 294. 
Resultant 
of a distributed force, 296. 
of two velocities, 284. 
velocities, 284. 
Restitution, coefficient of, 304. 
Retaining 
plank, street, specifications, 1111. 
stone, street, specifications, 1109. 
walls, 835. 
dimensions of, table, 842. 
dry, specifications, 435. 
masonry, specifications, 434. 
quantities in, table, 841. 
reference data, 841-843. 
standard type, 841. 



Return wires, elec. code rules, 1400. 
Reversed curves, (R. R.), 1012. 
Revetments, reference data, 1481. 
Revolved planes, 261, 262, 265. 
Rheostats, elec. code rules, 1398. 

1442. 
Rhodium, chem., 319. 
Rhomboid, 
defined. 129. 

area and cen. of grav., 203. 
properties of, 525. 
Rhombus, 
defined, 129. 
and area, 203. 
Rhyolite, 
composition of, table, 338. 
defined, 340. 
formation of, table, 338. 
Right 
angle, circular and time measure, 

equivalents, 99. 
ascension, of a star, defined, 202. 
-of -way, 

purchase and condemnation, 

1014. 
railroad-, 1013. 

required widths of, table, 1014. 
tables, 1014, 1015. 
Ring, rings, 
circular, 
mensuration of, 219. 
moment of inertia of, 302. 
segmental, mensuration of, 253. 
collector-, defined, 1383. 
regular circular, mensuration of, 

253. 
revolving, tension in, 297. 
Riprap, quarrying, 419. 
Rise of temperature, in elec, 1466. 
River, rivers, 
mean velocity depth of thread in, 

1187. 
spans, economic layout of, 683. 
Rivet, rivets, 611. 
button head, 616. 
countersunk, 616. 
gages for standard steel shapes, 

table, 614. 
shearing and bearing values, table, 

612. 
signs, Osbom code, 611. 
spacing and clearance of, table, 

613. 
weights and dimensions of, table, 
616. 
Riveted 
joints, 
application of polar moment of 

inertia to, (ref) 538. 
net areas of, table, 617. 
problem, 612. 
spiral-, steel pipe, 680-682, 1269. 
steel pipe, design of, 1268. 
work, in bridges, specifications, 
706. 
Rivet-steel, 
extra soft, specifications, 502. 
Tire-box, specifications, 502. 
flange or boiler, specifications, 502, 



INDEX. 



1591 



Rivet-steel, — Cont'd, 
open hearth, specifications, 501. 
test pieces, 502. 
testing, 502. 
Road, roads, 
and streets, 1098. 
application of oils to, 1 1 34. 
construction, use of tar in, 1132. 
corduroy, described, 1098. 
dirt-, described, 1098. 
dust preventives, experiments, 

costs, 1136. 
gravel-, construction of, 1098. 
graveled, oiled, specifications, 1114. 
macadam ^ 

and telford, specifications, 1111. 
specifications, 1116. 
oils for, 
best kinds, 1133. 
classification of, 1133. 
properties of, 1133. 
oiling, cost, (ref.) 1142. 
plank, described, 1098. 
sampittic surfacing of, (ref.) 1142. 
specifications, 1101. 
surfaces, 

application of oils to, 1134. 
• application of tars to, 1132. 
care of, 1131. 
dust preventives, 1131. 
tars for, 1131. 
traction on, 1097. 
surfacing materials, tests of, table, 

1144. 
tar macadam, cost, 1142. 
Roadbed, standard-, (R. R.), 1059. 
Roadway, 
board bed, specifications, 1128. 
concrete base, specifications, 1102, 

1109. 
crushed-stone base, specifications, 

1105. 
gravel base, specifications, 1102. 
macadam, specifications, 1105, 

1129. 
reinforced concrete base, (ref.) 1112. 
Rock, rocks, 
-asphalt, experiments on roads, 

costs, 1138, 1139. 
-breaking machines, for rock ex- 
cavation, (ref.) 926. 
calcareous, 400. 

cementing material in, table, 334. 
common,, 
notes on, 339. 
table, 334. 
crushed, voids in, 911. 
crushers, capacity, cost, etc., 439. 
crystal (halite), weight of, 481. 
crystalline, siliceous, 400. 
cuts, 
open-, excavation of, 922. 
side slopes for, 922. 
drills, 922. 
in quarrying, 422. 
Little Giant, 425. 
percussion, 
dimensions, etc., table, 424. 
weights, etc., table, 424. 



Rock, rocks, — C ont'd. 
excavation, 922. 
by channeling machines, 924. 
by Lobnitz rock breaker and 

other machines, (ref.) 926. 
Chicago canal, 924. 
costs, 925. 
methods, 925. 

Panama canal, cost data, 919. 
-fill dams, quantities in, table, 857. 
formations, 331. 
fragmentary, 401. 
granite, excavation, in open cuts, 

(R. R.), cost, 925. 
loading on cars, by steam shovel, 

924. 
loose, classification (R. R.), 919. 
loosened, voids in, 911. 
salt, composition of, 336. 
solid, 
classification (R. R.), 919. 
foundation, 863, 864. 
swellage when broken, table, 911. 
trench excavation, estimating, 92 3. 
trenching in, 923. 
Rod, rods, 
and meters, 
equivalents, 88. 
square, equivalents, 88. 
counter-, 634. 
equivalents, 68. 
floats, for hydraulic measurements, 

1183. 
or bar, moment of inertia of, 302. 
square, metric eqtdvalent, 81. 
steel, 
areas and weights, table, 542. 
weights and areas, table, 542. 
Rolled 
angle, properties of, 538. 
channel, properties of, 537. 
T-beam, properties of, 537. 
tee, properties of, 538. 
shapes, properties of, 535-538. 
Z-bar, properties of, 538. 
Rollers, segmental-, table, 635. 
Rolling 
brick pavement, 1107. 
friction, 521. 
Roman 
numerals, table 1. 
system of numbers, 1. 
Rontgen ray, 316. 
Rood, metric equivalent, 81. 
Roof, roofs, 794. 
angles and pitches of, table, 796, 

797. 
coverings, 798. 
live loads 
for, 824. 
on, table, 815. 
pitches and angles of, tables, 796, 

797. 
reference data, 811. 
snow loads on, 797, 798. 
tile-, specifications, 800. 
trusses, 
combination-, design of, 806- 
810. 



1592 



INDEX. 



Roof, roofs, — Cont'd. 
trusses, — Cont'd, 
four cases, 314. 
stress diagrams 
for, 803-804. 
of, 314, 315. 
timber for, 819. 
types of, 803. 

unit stresses in, tables, 804, 805. 
weight of steel in, 810, 811. 
wind pressure on, tables, 796, 797. 
Roofing, 
asphalt-gravel, 802. 
cement-gravel, 802. 
corrugated steel, 801. 
materials, 798. 

weight of, table, 802. 
patented, 802. 
sheet-steel-, 801. 
shingle, 799. 
slag-, 802. 
slate, 
laying of, 402. 
tables, 799, 800. 
tar-gravel, 801. 
tile 800. 
tin-, 800. 
Root, roots, 
and division, algebraic, 102. 
by slide rule, 126. 
cube, 
engineers' tables, 21-24. 
to find, 20. 
of decimals, by logarithms, 106. 
of numbers, by logarithms, 106. 
powers and reciprocals of num- 
bers, 14-54. 
square 
and cube, common tables, 31- 

50. 
engineers* tables, 16-19. 
of fifth powers, engineers' table, 

25. 
to find, 14. 
Rope, 
hoisting, tension in, 290. 
in cordage^ 668. 
manila, 669. 
weight and strength of, tables, 
669-670. 
tension in traction, 289. 
wire-, 673. 
fastenings, 675. 
tables, 674. 
Rosendale cement (see, also, Ce- 
ment, natural), 
manufacture of, 404. 
Rose's fusible metal, melting point 

of, 515. 
Rosettes, elec. code rules, 1439. 
Rosin, 
a tree product, 346. 
defined, 481. 
weight of, 481. 
Rosseau's hydrometer, 461. 
Rot, in lumber, defined, 387. 
Rotary 
furnace, for cement making, 405. 
pumps, 1367. 



Rotating machines, electrical-, defi- 
nitions, 1451. 
Rotting of timber, 359. 
Rough 
edge (lumber), classification of, 

388. 
lumber (fir), classified, 389. 
Roughness A^, values of, 1168, 
Rubber 
-covered wire, tables, 1424. 
India, weight of, 480. 
vulcanizer, 330. 
Rubble 
concrete dam, 860. 
masonry, defined, 432. 
stonework, in buildings, safe loads 
for, 821. 
Rubidium, chem., 319. 
Ruble (Russian, gold), equivalents 
(1-10,-50-100) in U. S. money, 
table, 97. 
Rueping process, for ties, cost, 375. 
Rimoff, 
distribution of, 1190. 
formulas for, 1197, 1198. 
in storm sewers, formulas, dia- 
grams and tables, (ref.) 1310. 
relation of rainfall to, (ref.) 1201. 
relation of, to rainfall, in Califor- 
nia, (ref.) 1201. 
Rupture (bending) tests of timber, 

table, 492. 
Russian money, U. S. values, 95. 
Rustic lumber (fir), classified, 389. 
Rutgen's process, for timbef, 361. 
Ruthenium, chem., 31^. 



S-trap, 1295. 

Sabin process for pipe coating, 358. 
Safes, 
loads from, 816, 817. 
weight of, table, 816. 
Safety factor, 
defined, 487. 
for timber, 496. 
in building, 821. 
Salt, 
a, in chem., defined, 321. 
and ice for freezing, 513. 
solutions, as road dust prevent- 
ives, 1131. 
(rock), composition of, 336. 
weights of, table, 481. 
Saltpetre, weight of, 481. 
Samarium, chem., 319. 
Sampittic surfacing of roads, (ref.) 

1142. 
Sand 
-blast 

cleaning 
and painting, 373. 
(ref.) 374. 
with cost, 374. 
for mill scale, 358. 
bricks, manufacture of, 417.. 
filtration of water, methods, 1204. 
foimdation, 864, 865. 



INDEX. 



1593 



Sand, — Cont'd. 

impurities in, for concrete, (ref.) 
455. 

Ottawa, in cement testing, 409. 

piles, 875. 

quarrying, 419. 

sieve in cement testing, 409. 

standard, in cement testing, 409. 

voids in, 911. 
concrete, 416. 

weight of, table, 476. 
Sandstone, 

best kind of, 401. 

building, 401. 

block pavement, specifications, 
1124. 

blocks, paving, specifications, 11 24. 

composition of, 331. 
table, 334. 

compression tests of, 512. 

compressive strength of, 512. 

defined, 339. 

expansion coefficient of, 516. 

formation of, table, 334. 

freezing test, 402. 

frost action on, 401. 

kinds of, 401. 

physical properties of, 512. 

quarrying, 419. 

temperature stress for 160° F.,523. 

tensile strength of, 512. 

test for frost action, 402, 

transverse strength of, 512. 

water absorbed by, 401. 

weight of, 476. 
Sanitary 

disposal, miscellaneous data, (ref.) 
1309-1312 

works, designs, (ref.) 1311, 1312. 
Sanitation, 1295. 
Saturated steam, 

formulas, 1355. 

tables, 1355-1360. 
Saturation-factor, in elec, 1453. 
Saturization of earth dams, 860. 
Sault Ste. Marie canals, data, 1322. 
Saws, lumber, kinds of, 379. 
Sawing 

logs, 379. 

lumber, 379. 
Scalene triangle, defined, 128. 
Scaling logs, 379. 

(ref.) 391. 
Scandium., ckem.y 319. 
Scantling, 

classification of, 388. 

lumber (fir), classified, 389. 
Schist, schists, 

composition of, table, 337. 

defined, 340. 

mica-, composition of, 336. 
Score, equivalent of, 95. 
Scrapers, wheeled-, grading with, 

cost data, 917. 
Screening, 

gravel, 419. 

water for domestic use, 1204. 
Screw, screws, 

^olts, 618. 



Screw, screws, — Cont'd, 
in mech., formulas, 292. 
lag-, 
table, 622. 
use of, 622. 
or helix, 260. 
piles, 874. 

threads, standard, 618. 
wood-, table, 622. 
Scruple, scruples, 

(apoth.), metric equivalents, 86. 
(U. S. apoth.) and milliliters, 
equiv. (1-10), table, 83. 
Seasoning, 
lumber, 379. 
steam-, of lumber, 379. 
of timber, 361. 
described, 362. 
Sea-wall, at Havana, (ref.) 904. 
Sea-water at great depths, density 

of, 1145. 
Sea-worms, in timber, 360. 
Secant, secants, 
logarithmic, table, 176-198. 
natural, table, 167-175. 
to circle, defined, 129. 
(trig.), defined, 136. 
Second, seconds, 
and minutes to decimals of a de- 
gree or hour, table, 1010. 
circular and time measure, equiva- 
lents, 99. 
time and longitude equivalents, 99. 
Section, sections, 
conic, 256. 
Government land, subdivision of, 

970. 
-modulus of plane surfaces, table, 

524. 
quarter-, and hectars, equiv., 88. 
Sector, 
circular, 
center of gravity of, 220. 
mensuration of, 220. 
properties of, 528. 
of circle, defined, 129. 
Sedimentation, in settling basins, 

1204. 
Seepage, 1200. 

distribution of, 1190. 
Segment, segments, 
circular, 
area, formulas for, 215. 
areas, etc., tables, 216, 217, 218 
(half-), 
circular, properties of, 528. 
parabolic, properties of, '529. 
of circle, 
cen. of grav. of, 214. 
defined, 129. 

mensuration of, 214, 215. 
of circular spindle, 254. 
of ellipse area of, 242. 
of parabola, 237. 
of sphere, 
defined, 135. 
mensuration of, 252. 
Segmental rollers, table, 635. 
Selenium, chem., 319. 



1594 



INDEX, 



Semicircle, 
axis at base, properties of, 528. 
axis through cen. of grav., proper- 
ties of, 528. 
defined, 129. 
Semicircular 
arc, skeleton section, properties of, 

531, 532. 
cell, skeleton, properties of, 531, 

532. 
orifices, center of pressure on, 
formulas, 1151. 
Semi-tangents and externals to a 1° 

curve, table, 1009. 
Separators, 
cast iron, 
table, 623. 
use of, 623. 
steel diaphragm-, 623. 
Series 
arc lamps, elec. code rules, 1406., 
lamps, elec. code rules, 1423. 
Serpentine, 337. 
defined, 331. 
weight of, 481. 
Settling basins, sedimentation in, 

1204. 
Sewage disposal, 1295. 
miscellaneous data, (ref.) 1309- 
1312. 
Sewer, sewers, 1296. 
and conduits, 
circular, properties of, table, 

1300. 
hydraulic properties of, tables, 
1296-1306. 
basket -handle, properties of, table, 

1302. 
brick, 415. 

66-in., cost of, 1310. 
catch basins, 1 308. 
catenary, properties of, table, 1301. 
circular 
and other sections compared, 

1296, 1300. 
brick, velocities in, table, 1299. 
construction, illustrated, (ref.) 

1306. 
design, modem procedure in, (ref.) 

1311. 
egg-shaped, 
properties of, table, 1304. 
velocities in, table, 1306. 
excavation, cost data, 916. 
foundations, 1306. 
Gothic, properties of, table, 1303. 
grade of, 1296. 
inverts, wear of, 1310. 
kinds of, 1307. 
location of, 1308. 
manholes, 1308. 
pipe, 
tables, 1307. 
joints, 
cement and sand required for, 

table, 1309. 
mortar required for, table, 1309. 
sulphur and sand required for, 
table, 1310. 



Sewer, sewers, — Cont'd, 
runoff in, formulas, diagrams and 

tables, (ref.) 1310. 
size of, 1296. 
trench, cost data, 915. 
trenching and backfilling, cost 

data, 917, 918. 
tunnel work, cost data, 916. 
walls, brick, thickness of, formula, 
1306. 
Shackle fastenings for wire rope, 875. 
Shaft, in tunneling, defined, 933. 
Shakes, in lumber, defined, 387. 
Shale 
rock, quarrying, 419. 
weight of, 481. 
Shapes, 
block, properties of, 533. 
rolled, properties of, 535-538. 
skeleton, properties of, 529. 
steel, 
list of tables, 541. 
properties and tables of, 541. 
Shear, shears, 
and web stresses, 307. 
and moments for engine loading, 

table, 692. 
end-. 
Cooper's loading, table, 708. 
for highway bridges, table, 728. 
from electric cars, tables, 717, 719. 
in' beams and girders, various load- 
ings, 688. 
in building materials, safe"stresses, 

822. 
in struc., method of, 306. 
in trusses, various loadings, 693. 
longitudinal, in beams, formulas, 

565. 
-steel, 396. 
Shearing 
effect on columns, 587. 
strengths of metals, table, 496. 
tests of timber, with grain, table, 

494. 
values of concrete in beams, 585. 
Sheet, sheets, 
asphalt pavement, specifications, 

1110. 
lead, 679. 
metal, 
weight of, from specific gravity, 

table, 484. 
gages, tables, 667. 
paper measure, 95. 
piling, 869. 
steel, 870. 
Sheeting, corrugated, strength of, 

(ref.) 561. 
Shield, 
hydraulic-, used in sewer tunnel, 

cost data, 916. 
method of tunneling, 936. 
Shims, track-, thickness of, 1069. 
Shingle, shingles, 
dimensions of, 390. 
grading of, 390. 
roofing, 799. 
wooden, table, 799. 



INDEX, 



1595 



Shipping weights of lumber, 391. 
Shoes, 
cast-, for wood stave pipe, 1209. 
pile-, 874. 
Shop drawings for structural steel, 

cost of, 665. 
Shrinkage, 
earthwork, reference data, 920, 
in earth dams, 914. 
of earth, 909. 
how estimated, 910-913. 
fills, recommendations, 914. 
of earthwork, 
experiments, 913. 
railroad specifications for, 913. 
vertical in earth embankments, 915. 
Shroud laid, in cordage, 668. 
Sidereal 
and mean solar time, equivalents, 

table, 202. 
day, defined, 202. 
Sidewalk, sidewalks, 
areas, surfacing, 1115. 
brick, specifications, 1103. 
cement, 
described, 1099. 
specifications, 1117. 
concrete, 
specifications, 1111, 1129. 
base for, specifications, 1129. 
crushed -stone, specifications, 1105. 
gravel, construction of, 1098. 
part cement, part gravel, 1115. 
plank-, described, 1098. 
practice in Chicago, costs, 1143. 
Sienite, composition of, 337. 
Sienna for paint, 355. 
Sieve, sand-, in cement testing, 409. 
Sign, signs, 
circular and time measure, equiva- 
lents, 99. 
electrical notation, 1471. 
values (trig.), 137. 
Signal lights, elec. code rules, 1450. 
Signaling systems, elec. code rules, 

1444. 
Silica 
minerals, 331. 
weight of, 481. 
Silicates, 
in min., classification of, 326. 
most important, 331. 
Silicic acid, weight of, 481. 
Silicon, chem., 319. 
-bronze, 397. 

tensile strength of, 497. 
-wire as conductor, compared 
with copper, 497. 
Sills, wooden trestle, 788. 
Silver, chem., 320. 
cast, tensile strength of, 498. 
expansion coefficient of, 516. 
melting point of, 515. 
minerals, ores, 328. 
weight of, 481. 
Simple 
and compound units, equivalents, 

table, 88. 
interest tables, 60. 



Simultaneous equations, 
examples in, 103. 
graphical, 256. 
Sine, sines, 
logarithmic, table, 176-198. 
natural, table, 144-166. 
(trig.), defined, 136. 
Single-phase alternator, defined, 

1384. 
Sinking fund, 63, 

and annuity tables, 64-65. 
formula, 65. 

diagrams and tables, (ref.) 1293. 
Sizing lumber, 379. 
Skeleton figures, properties of, 529. 
Skew portals, detailing, (ref.), 665, 
Slabs, 

concrete, calculation of, (ref.) 455. 
floor-, reinforced concrete, bending 
moment, 82,3, 825, 827, 829, 
832. 
reinforced concrete, 
slide rule for, (ref.), 586. 
Thacher's computation, 585. 
Slag 
block pavement specifications, 

1121. 
cement, manufacture of, 404. 
roofing, 802. 
weight of, 481. 
Slant height, defined, 133. 
Slate, 
composition of, 331. 

table, 334. 
defined, 339. 

expansion coefficient of, 516. 
formation of, table, 334. 
formed how, 402. 
physical properties of, 512. 
quarried where, 402, 
roofing, 402, 799. 
temperature stress for 160° F., 

523 
weight of, 477, 
Sleeve, sleeves, 

cast iron pipe, tables, 1232, 1265. 
nuts, 634. 
weights and dimensions, table, 
633, 
Slide 
rules, 
described, 126-127. 
problems, 126-127. 
for reinforced concrete slabs, (ref.) 

586. 
-valve engines, performance of, 
1365. 
Sliding friction, 517-521. 
Slip, 
in cement making, 405. 
of rods in concrete beams, (ref.) 
455. 
Slope 
and deflection of beams, formulas, 

562. 
artesian-, defined, 1190. 
side-, for rock cuts, 922. 
-staking, 1055. 
walls, dry, specifications, 436, 



1596 



INDEX. 



Slow 

sand filtration, 1204. 

-burning wire insulation, table, 
1425. 
Sludge, defined. 1525. 
Sluice gate, -gates, 

stand and wheel, 1270. 

table, 1279. 
Slurry, in cement making, 405. 
Smelting, 357. 

Smith's durable metal coating, 358. 
Smokeless powders, 352. 
Snap switches, elec. code rules, 1433. 
Snow, 

evaporation from, 1199. 

loads on roofs, 797, 798. 

weight of, 481. 
Soapstone, 340. 

weight of, 481. 
Sockets, 

elec. code rules, 1413, 1440, 1450. 

fastenings for wire rope, 675. 
Sodium, chem., 320. 

chloride, use of, 340. 

minerals, 328. 
Soil, soils, 

bearing capacity of, 819. 

bearing power of, for buildings, 867. 

borings in, 866. 

defined, 909. 

density of, 909. 

for foundation, tests of, 865. 

pipes, 1295. 

seepage in, various, 1200. 
Solar 

and sidereal time, eqmvalents, 
table, 202. 

attachment, 944. 
adjustment of, 947. 

day, defined, 202. 

ephemeris tables, reference to, 202. 

instrument, uses of, 946. 

observations with transit alone, 
947. 

power, reference data, 1484. 
Solder, 398. 

kinds of, 1525. 
Soldering fluid, elec. code formula, 

1447. 
Solid, solids, 

center of gravity of, 302. 

coefficient of expansion of, table, 
516. 

defined, 512. 

determining specific gravity of, 
460. 

Geometry, 132-136. 

melting points of, 515. 

mensuration of, 243. 

moments of inertia of, table, 302. 

radius of gyration of, 302. 

rock classification (R. R.), 919. 
Solvents, cement, 402. 
Sorel stone, manufacture of, 417. 
South point, of celestial sphere, de- 
fined, 201. 
Space, 

Analytic Geometry of, 256. 

in mech., defined, 278. 



Spans, bridge-, economic length of, 

683. 
Spandrel, parabolic, properties of, 

237, 529. 
Spar, weights of, 481. 
Spark arresters, elec. code rules, 

1442. 
Sparking distances, in elec, 1474. 
Specials, 
pipe, described, 1280. 
pipe-castings, bells of, dimensions, 
etc., tables, 1221, 1223, 1243, 
1245. 
Specific 
gravity, gravities, 
by displacement method, 460. 
defined, 460. 
equivalents for any weight, table 

474. 
methods for determining, 460. 
of brick, table, 474. 
of building stones, table, 474. 
of cement by LaChatelier's ap- 
paratus, 407. 
of granular substances, to find, 

461. 
of liquids, table, 468, 469. 
of materials, 459. 

general table, 478. 
of porous substances, to find, 460. 
of tars, 1132. 
of woods, table, 470-473. 
reduced to weight, table, 483, 

484. 
standards for determining, 460. 
heat of the liquid, formula, 1356. 
volume 
of saturated steam,defined,1356. 
of the water, formula, 1356. 
Specifications (see specific items). 
Specifications and contracts, refer- 
ence data, 1484. 
Spelter, weight of, 481. 
Sphere, spheres, 
area great circle of, equivalents, 

250. 
area of surface of, by calculus, 276. 
Celestial, elements of, 201-202. 
circumference of, equivalents, 251. 
cylinder (max.) inscribed in, 269. 
diameter of, equivalents, 251. 
diameter 
(ft. and ins.) 
to surface (sq. ft.), table, 234, 

235. 
to volume( in ft.), table, 234, 
235. 
(in fractions) to surface (in deci- 
mals), table, 226-229. 
to surface, 
in decimals, table, 224, 225, 

OQO 233 
in inches, table, 230, 231. 
to volume, 
in decimals, table, 232, 233. 
in inches, table, 230, 231. 
geometry of, 134-135. 
hollow, mensuration of, 253. 
moment of inertia of, 302. 



INDEX, 



1597 



Sphere, spheres, — Cont'd. 
properties of, 249. 
radius of, equivalents, 251. 
relations of area, diameter, surface, 

volume, etc, 250. 
relations to cone, cube and cylin- 
der, 250. 
surface measure; in radii, 135. 
surface of, 
equivalents, 250. 
table, 251. 
tables listed, 251. 
volume of, 
equivalents, 250. 
measure, in radii, 136. 
table, 252. 
tables listed, 252. 
wind pressure on, 797, 
Spherical 
cone, defined, 135. 
pyramid, 
defined, 135. 
solution of, 199. 
segment, 
defined, 135. 
mensuration of, 252, 
triangles, solution of, 199-201. 
trigonometry, 199-202. 
zone, 
area of, by calculus, 276. 
mensuration of, 253. 
Spindle, 
circular, mensuration of, 253. 
cycloidal, mensuration of, 254. 
oil, weight of, 481. 
parabolic, mensuration of, 254. 
Spikes, 
railway, 
effect of creosote oil on, 361. 
weight per mile of track, table, 
1068. 
steel, weights and dimensions, 

tables, 626-628. 
street railway, table, 628. 
Spiral, 
common, 260. 
curves (R. R.), 1013. 
hyperbolic, equation of, 260. 
logarithmic, equation of, 260. 
of Archimedes, equation, 260. 
riveted steel pipe, 680-682, 
1269. 
Spirit, rectified, weight of, 481. 
Splices, 
in cordage, 669. 
rail-, 1060. 
Split switches, 1084, 1088. 

turnouts for, tables, 1085-1088. 
Splitting chisel, described, 427. 
Spot method of using current meters, 

1186. 
Spread-foundation, reinforced con- 
crete, (ref.) 890. 
Spring, springs, 
metal-, formulas, 1482. 
steel, 
physical properties of, 499. 

1884. 
tests of, table. 1484. 



Spruce, spruces, 
classification of, 341. 
grading rules, 388, 390. 
Square, squares, 
acreage, dimensions of, 1313. 
and circle, inscribed and circum- 
scribed, 131. 
and cubes, tables of, uses, 636. 
and cube roots, by slide rule. 126. 
axis at base, properties of, 526. 
cell, skeleton, properties of. 530. 
centimeters, English equiv., 79. 
cubes and roots, common tables, 

31-43. 
decimeter, English equiv., 79. 
dekameter, English equiv., 79. 
diagonal axis, properties of, 526. 
feet and meters, equivalents, 88. 
foot, metric equivalent, 81. 
yards and meters, equivalents, 88. 
geometric, defined, 128. 
hectometer, English equiv., 79. 
hollow-, 
properties of, 526. 
diagonal axis, properties of, 626. 
inch, inches, 
metric equivalent, 81. 
and centimeters, equivalents, 88. 
kilometers, English equiv., 79. 
land measure, English, metric 

equivalents, table, 81. 
measure, metric, English equiva- 
lent, table, 79. 
mensuration of, 203. 
meter, English equivalents, 79. 
miles and hectars, equivalents, 88. 
mile, metric equivalent, 81. 
millimeter, English equiv., 79. 
mils and millimeters, equiv., 88. 
myriameter, English equiv., 79. 
properties of, 525. 
rod, metric equivalent, 81. 
rods and meters, equivalents, 88. 
root, roots, 
and cube roots, common tables, 

31-50. 
by binomial formula. 102. 
engineers' tables, 16-19. 
of fifth powers, engineers' tables, 

25. 
to find, 14. 
skeleton section, properties of, 530. 
tables of, for structural detailing, 

643-664. 
yard, metric equivalent, 81. 
Squared-stone masonry, defined, 432. 
Squaring quadratic equation, exam- 
ples, 102. 
Stadia 
reduction table, 984. 
surveying, 983. 
surveys, cost data, 990. 
Standard 
connection angles, for I-beams and 

channels. 615. 
orifice, 1176. 
rain gage, 1196. 
tube, 1176. 
weir, 1177, 



1598 



INDEX, 



Standardization, electrical-, 1451. 
Standpipes, 1271. 
miscellaneous data, (ref.) 1292- 

1294. 
steel, design of, 1206. 
Star, 
altitude of, defined, 201. 
azimuth of, defined, 201. 
hour angle of, defined, 202. 
polar distance of, defined, 202. 
right ascension of, defined, 202, 
zenith distance of, defined, 202. 
Static 
equilibrium, in struc, principles of, 

306. 
stress, defined, 487. 
Stations (100 ft.) 
and meters, equivalents, 88. 
and miles, equivalents, table, 1001. 
Stave, wood-, pipe, 
and details, table, 1210. 
details of, 1208. 
Staybolts (iron), tensile strength of, 

497. 
Steam 
and gas power, 1346. 
apparatus, cost data, 1477. 
boilers, 1361-1363. 
efficiency and rating, 1361. 
horsepower of, defined, 1361. 
kinds of, 1362. 
-electric problems, 1379. 
engines, 1363-1366. 
consumption of coal per h.-p. 

hour, 1363. 
economic performance of, 1365, 

1366. 
effect of load upon economy of, 

1366. 
efficiencies of, 1363. 
horsepower, problem, 1363. 
mean effective pressure of. 1365, 

1366. 
principle of, 1363. 
flow through orifices, (ref.) 1377. 
flow through pipes, formula and 

table, 1361. 
ideal, weight and specific gravity 

of, 464. 
joints, waterproofing for, 418. 
kinds of, defined, 1354. 
pipe cement, 402. 
pipes, connections of, etc., (ref.) 

1377. 
plants, cost data, table, 1478. 
power 
and water power compared, 

1385. 
plants, costs, 1477. 
problems solved by the use of dia- 
grams, (ref.) 1378. 
pumps, 1366, 1367. 
seasoning of lumber, 379. 
saturated, 
formulas, 1355. 
tables, 1355-1360. 
shovel work 
at Panama, cost data, 919. 
cost data, 916. 



Steam — Cont'd , 
shovels used in loading rock on 

cars, 924. 
superheated-, formulas, 1355. 
total heat of, new and old formu- 
las, 1378. 
Steams and Fteley's weir formulas, 

1180, 1181. 
Steel, 
acid open hearth process, 394, 395. 
angles, properties of, tables, 548- 

553. 
annealing, 395. 
arches, 782. 

axles, specifications, 504. 
bars, 
areas and weights, table, 544. 
for drills, 922. 

weights and areas, table, 544. 
basic open hearth process, 395. 
beam box girders, properties of, 

table, 568. 
beams, properties of, table, 554. 
carbon-, 
annealed, physical properties of, 

table, 500. 
oil-tempered, physical properties 

of, table, 500. 
ast, 

expansion coefficient of, 516. 
open hearth, 396. 
specifications for, 393. 
castings, 
physical properties of, 504. 

table, 499. 
specifications, 503. 
test pieces, 504. 
testing, 504. 
cementation process, 395. 
channels, porperties of, table, 556. 
chemical properties of, 500. 
chisel-, 396. 
chrome, 396. 

-vanadium, 399. 
corrugated-, roofing, 801. 
cost of cleaning, 373. 
crucible, 395. 

details of combination bridge, 730. 
die-, 396. 

expansion coefficients of, 516. 
flumes, for water supply, 1207. 
for bridges, specifications, 706. 
for buildings, 819. 
for mine timbering, use of, (ref.) 

939. 
forgings, 
physical properties of, 506; 

table, 499. 
specifications, 505. 
testing, 506. 
girder beams (single I), properties 

of, table, 583. 
grades of, used in structures, 499. 
harveyized, 396. 
I-beams, 
properties of, table, 554. 
special, properties of, table, 584. 
in cinder concrete, corrosion of, 
374. 



INDEX, 



1599 



Steel.— Cont'd, 
in concrete, adhesion tests, (ref.) 

454. 
in buildings, 

safe stresses, 826. 

stresses for, 824, 
kinds of, 396. 
manganese-, 396. 
manufacturer's standard. 499. 
melting point of, 394, 515. 
metallurgy of, (ref.) 399. 
molybdenum in. 330. 
nickel-. 396. 

manufacture of, 398. 

properties of. 398. 

specifications for Manhattan 
bridge, 758 

annealed, physical properties of, 
499. 

forged, oil-tempered, physical 

properties of, table 499. 
-vanadium, 399. 
open hearth, 

boiler plate, specifications, 501. 

rivet, specifications, 501. 
physical properties of, 500. 

table, 499. 
pipe, 1268. 

experimental values of iV in Kut- 
ter's formula for flow in, 1188. 

for water works, costs, 1292, 

riveted, design of, 1268, 

spiral riveted, 680-682 
plate, -plates, 

areas and weights, table, 544. 
-girders, properties of, table, 
570-582 

weights and areas, table, 544. 
preservation of, 358. 
rails, 

chemical properties of, 503. 

dimensions and weights, table, 
560. 

properties of, table, 560. 

specifications, 503. 

testing, 503. 

weights and dimensions, table, 
560. 
railroad bridges, specifications, 699. 
razor-, 396. 
reinforcement, 

for buildings, specifications, (ref.) 
825. 

in beams, tensile stress value, 
585. 
rivet-, 

open hearth, specifications, 501. 

test pieces, 502. 

testing, 502. 
rods, 

areas and weights, table, 542. 

weights and areas, table, 542, 
saw-file-. 396, 
set, 396. 
shapes, 

list of tables, 541. 

properties and tables of, 541, 
shear-, 395. 
sheet-, roofing, 801. 



Steel,— Cont'd, 
specifications for Manhattan 

bridge, table, 758. 
spindle-, 396. 
spring-, 

physical properties of, 1484 
springs, 

physical properties of, 499. 

tests of, table, 1484. 
structural, 

analysis of 394. 

manufacture of, 394. 

(bridge), specifications, 500. 
tees, properties of, tablet 558. 
temper of, 396. 

temperature stress for 160° F.,523 
tempering, 396. 
test specimens, 501. 
testing, specifications, 501. 
ties, 1072. 
tool-, 396. 
trestles. 791. 

elevated railroad, (ref.) 792. 
tungsten-, 396. 
uranium in. 330. 
vanadium-, 396, 399. 

alloys, 399. 
weights of, 481. 
welding, 396. 
wire, 

physical properties of, table, 499. 

Roebling, properties of, 672. 

weight of, 481. 
Z-bars, properties of, table, 557. 
Steelwork, 
cleaning by sand blast, with cost, 

374. 
cutting with oxy-acetylene flame, 
833. 
Stepping lumber, 
.classification of, 388. 
(fir), classified, 389. 
Stereotomy, 457« 
Stone, 
-arch, stonecutter's plan, 457, 458, 
artificial, described, 415, 417. 
-axed, defined, 429. 
block-, kinds of, 417. 
bolts, 618. 
building-, 400. 

physical properties of, table, 507. 

quarrying, 419. 

safe loads for, 821. 

specific gravities of, table, 474. 

thickness of joints, 457, 

weights of, table, 474. 
bush -hammered, defined, 430. 
-cements, (ref.) 418. 
chisels, described, 427. 
classified finish, 426. 
crandalled, defined, 428. 
curbing, specifications, 1108. 
cut, defined, 428. 
cutting, 426. 

diamond-paneled, defined, 430. 
dimension-, defined, 433. 
drafted, defined, 427. 
dressing, 

specifications, 433. 



1600 



INDEX. 



Stone, — Cont'd, 
dressing, — Cont'd. . 
tools employed i56. 
machines pneumatic, (ref.) 430 
expansion coefficient of, 516. 
fine-poined defined. 428. 
friction of 518-521. 
hammers, described, 426, 427. 
masonry, 
compressive strength of, 511. 
described. 431. 
in buildings weight of, 821. 
specifications, 433. 
McMurtrie, manufacture of, 417, 
natural building-, 400* 
patent-hammered, defined, 429. 
pean-hammered, defined, 429. 
Portland, manufacture of, 417 
quarry-faced, defined, 427, 
Ransome, manufacture of, 4.17. 
rough-pointed, defined, 428. 
rubbed-, defined, 430. 
sorel-, manufacture of, 417. 
squared, defined, 427, 
tooth-axed, defined. 429. 
unsquared, defined » 426. 
Stonework, in buildings, safe loads 

on. 8.26. 
Storm-water drains, design of, (ref.) 

1311. ' 

Stop valves. 1271 1279. 1285-1287. 
Storage 
in acre-ft. reduced to horsepower 

hours, table, 1335. 
in million cu. ft. reduced to horse- 
power hotirs, table, 1334. 
or primary batteries, elec. code 

rules, 1398. 
reservoirs, 1205. 
Straight 
angle, defined. 128. 
line, defined, 132. 
lines (skeleton), properties of , 529, 
531. 
Strain, defined, 486. 
Strand, in cordage, 668. 
Stream, streams, 
flow, surface and mean velocity of, 

1183. 
method of measuring flow in, (ref.) 
1187. 
Street, streets, 
and roads, 1098. 

crowning, formula and table, 1123. 
grading for pavement, specifica- 
tions, 1127. 
graveled, oiled, specifications, 1 1 1 4. 
pavements, 1099. 
railway tracks, 
and paving, 1094, 1095; 

(ref.) 1142. 
T-railsfor, (ref.) 1142. 
trackway (steel), experimental, 
cost, 1142. 
Strength 
of materials,' 486. 
ultimate, defined, 487. 
Stress, stresses, 
alternating, defined, 487. 



Stress, stresses, — Cont'd, 
chord- and bending moments, 307. 
combined-, tests, (ref.) 522. 
defined, 486. 
-diagrams 

for drawbridge, 746. 
general rules, 310. 
of Pratt truss. 312, 313. 
of roof trusses, 803-804. 
effect of, 
on elastic limit. 487. 
on ultimate strength, 487. 
in beams, formula, 299. 
in structures, theory of , 305. 
per area, metric and English equiv 

table, 89. 
repeated, defined, 487. 
static, defined, 487. 
ultimate, defined, 487. 
unit-, in roof trusses, tables, 804, 

805. 
web-, and shears, 307. 
working-, 
defined, 487. 

for reinforced concrete beams. 
585. 
Stretcher, masonry, defined, 432. 
Striking the arch center, 773. 
String, in cordage, 668. 
Stringers, 
bridge-, 
moments and shears, various 

loadings, 688. 
spacing of, 700. 
wooden-, 
bending moments, table, 791. 
cast separators for, 623. 
floor-, 789. 
Strontium, chem., 320. 
Struck bushel, metric equiv., 84. 
Structural 

details 611. . 

references, 665. 
steel, 
shop drawings for, cost of, 666. 
specifications, 500. 
Structures, theory of stresses in, 

305. 
Struts, see Columns. 
Stub switches, 1078. 
Stumpage, forest. 
Pacific Coast, 377. 
U. S., 376. 
^tumps, blasting, cost data, 916. 
Subaqueous concrete, placing, 440. 
Sub-grade shaping, road specifica- 
tions, 1101. 
Submarine drilling and blasting, 

cost, 925. 
Submerged 
beams, formulas for pressure and 

moments in, (ref.) 1189. 
planes, pressure on, 846, 847. 
tubes, flow of water through, (ref.) 

1189. 
weirs, 1181. 
formulas, 1181. 
Subscript abbrevation of a decimal, 
95. 



INDEX, 



160i 



Sub-surface floats, for hydraulic 

measurements, 1183. 
Subtraction and addition, in algebra, 

100. 
Subway excavation, cost data, 916. 
Successive differentiation, 271. 
Sudden loading, effect of, 489. 
Suez canal, 
dimensions and cost, 1320. 
traffic data, table, 1321. 
Sulphates, in min., classification of, 

327. 
Sulphides, in min., classification of, 

325. 
Sulphur, chem., 320. 
melting point of, 515. 
minerals, 330. 
uses of, 330. 
weight of, 481. 
Sulphuric acid, weight of, 481. 
Superheated steam, 
formulas, 1355. 

in locomotive boilers, use of, (ref .) 
1378. 
Superintendence of work, value of 

good, 908. 
Supplement and complement of an 

angle, 139. 
Supplementary angles, defined, 128. 
Surface, surfaces, 
curved-, areas of, by calculus, 276. 
floats, for hydraulic measure- 
ments, 1183. 
of sphere, measure of, in radii, 
135. 
Surfacing, road specifications, 1101. 
Survey, surveys, 
location, (R. R.), 1004. 
preliminary, (R. R.), 1000. 
reconnoissance, (R. R.), 998. 
stadia-, cost data, 990. 
traverse, 964. 
Surveying, 
city-lot, 966. 
farm-, 964. 

Government land, 967. 
instruments, care of, 941. 
mapping and leveling, 941. 
stadia-, 983. 
Surveyors' measure, lineal, metric 

equivalents, table, 68. 
Suspension 
bridges, 750. 
anchorages of, 755, 759. 
cables vs. chains, 754. 
details and specifications, 756- 

760. 
miscellaneous data, 760.- 
towers and backstays, 754. 
weights of miaterials in, tables, 
758. 
cables, curves of, 750. 
Swedge bolts, 618. 
Sweet gums (trees), classification of, 

345. 
Swellage 
of earth, how estimated, 910-913. 
of rock when broken, table, 911. 
Swing bridges, 742. 



Switch, switches, 
and frogs, tables, 1079-1082. 
and turnouts, 1075. 
boxes, elec. code rules, 1429. 
elec. code rules, 1404, 1407, 1431, 

1449. 
split-, 1084, 1088. 

turnouts for. tables, 1085-1088. 
stub-, 1078. 

table for laying out, 1078. 
ties, bills for, tables, 1070. 
Wharton, 1088 
Switchboards, elec. code rules, 1395, 

1449. 
Swivel hook fastenings for wire rope, 

675. 
Syenite, composition of, 337. 
Symbols, atomic, table, 318. 
System, artesian-, defined, 1190. 



T, T's, 
-bar, -bars, 

block, properties of, 533, 534. 

rivet gages for, 614. 

steel, properties of, table, 558. 

-beam, rolled, properties of, 537. 

block, properties of , 533, 534. 

cast iron pipe, table, 1225, 1250- 

1254. 
-rails for street railway tracks, 
(ref.) 1142. 
Tables 
of cubes 
and squares, uses of, 636. 
for structural detailing, 639-642. 
of squares, for structural detailing, 
643-664. 
Tablet and panel boards, elec. code 

rules, 1439. 
Tacks, table, 627. 
Tael, 

(Chinese), equiv. (1-10,-50-100) 

in U. S. money, table, 97. 
(Philippine weight), English 
equivalent, 81. 
Talc 
schist, 337. 
uses of, 331. 
weight of, 481. 
Tallow, 
melting point of, 515. 
weight of, 481. 
Tangent, tangents, 
and externals to a 1° curve, table, 

1009. 
and normal (calculus), equations 

of, 267. 
logarithmic, table, 176-198. 
natural, table, 144-166. 
to circle, 
defined, 129. 
equation of, 257. 
to ellipse, equation of, 259, 268. 
to hyperbola, equation of, 259. 
to parabola, equation of, 258. 
(trig.), defined, 136. 



1602 



INDEX. 



Tank, tanks, 
dia. to area, capacity, volume, 
weight (water), table, 246-7. 
-measurement of water, 1182. 
pipe-dipping-, 1282. 
pressure in, hydrostatic, 1152. 
water-, 
elevated, 1207. 

miscellaneous data, (ref.) 1292- 
1294. 
Tantalum, chem., 320. 
Tapes, 956. 
sag and stretch of, 956. 
temperature corrections for, table, 
956. 
Tapping machine, Mueller, 1283. 
Tar, tars, 
a tree product, 346. 
acids, 367. 

amount and cost, in road construc- 
tion, 1133. 
and pitch for waterproofing, 418. 
application to road surfaces, 1132. 
as road dust preventives, 1131. 
coal-, for roads, specifications, 11 35. 
composition of, 1132. 
dehydrated, 1131. 
experiments on roads, costs, 1136, 

1137. 
filling, 
for wood block pavement, speci- 
fications, 1128. 
in brick paving, 1109. 
for road surfaces, 1131. 
from coke ovens, 1131. 
from gas plants, 1131. 
-gravel roofing, 801. 
macadam roads, cost, 1142. 
manufacture of, 1131. 
properties of , 1131, 1132. 
refined coal-, 1131, 
specific gravity of, 1132. 
use of, in road construction, 1132. 
water-gas, 1132. 

cost, 1136. 
weight of, 481. 
Taylor's theorem, 272. 
Tee, Tees, 

Matheson pipe, table, 1281. 
rolled, properties of, 538. 
skeleton section, properties of, 

530. 
steel-, 
properties of, table, 558. 
rivet gages for, 614. 
Telegraph poles, 
preserving, 361. 
concrete, 1477. 
Telephone, telephones, 
cable, 676. 
reference data, 1482. 
Telford and macadam roads, specifi- 
cations, 1111, 
Tellurium, chem., 320. 
minerals, 330. 
weight of, 481. 
Temperature, 

absolute zero of, 462, 513. 
coefficients, in elec, table, 1475. 



Temperature, — Cont'd. 

(C. and F.) scales, equivalents, 

table, 465. 
critical, 
defined, 513. 
of gases, table, 514. 
of liquids, table, 514. 
effect on earth fill, 910. 
low, from freezing mixtures, 613. 
lowest attained, 513. 
of fusion, defined, 513. 
provision for bridges, 706. 
reduced 
by exaporation, 513. 
by expansion, 513. 
by freezing mixtures, 513. 
regenerative method, 513. 
rise of, in elec, 1466. 
stresses in building materials, 623. 
Tempering, steel, 396. 
Tension (direct) 
in building materials, safe loads, 

822. 
in hoisting ropes, 290. 
of rope in traction, 289. 
strength of metals, table, 496. 
Terbium, chem., 320. 
Teredo nevalis, in timber, 360. 
Term (algebraic) of equation, de- 
fined, 100. 
Teme plate, 357, 800. 
Terra cotta, 
brick, 415. 

compressive strength of, 512. 
piers, crushing tests, 522. 
tiles, for roofs, 800. 
Tests, 
bending, of timber, table, 492, 493. 
compression, of timber, 
across grain, table, 494. 
table, 490. 
of cement, cylinders, 508. 
of reinforced concrete beams, 

formula, (ref.) 585. 
shearing, of timber, with grain, 
table, 494. 
Testing cement, 406. 
Tetrahedron, defined, 132. 
Texas land measure, English equiva- 
lents, table, 81. 
Texture of rocks, table, 334. 
Thallium, chem., 320. 
Theater wiring, elec. code rules, 1414. 
Theorem, 
Maclauren's, 271. 
Taylor's, 272. 
Thermal 
energy, 
defined, 1347. 
examples of, 1346. 
unit (British), 
defined, 1347. 
equivalents of, 90, 
equiv (1-10), table, 1348. 
Thermodynamics, 
first law of, 1347. 
references, 515. 
Thimble fastenings for wire rope, 676 
Thorium, chem., 320. 



INDEX. 



1603 



Thread, threads, 
in cordage, 668. 
of mean velocity in rivers, depth 

of, 1187. 
screw-, standard, 618. 
Three -hinged arch, stress diagram, 

315. 
Thulium, chem., 320. 
Thurston-metal, 397. 
Tie, ties, 
and timber preserving plant, (ref .) 

374. 
best time for cutting, 1071. 
concrete-steel, 1072. 
cost of various treatments, 375. 
creosote extracted from, 
analyses of, table, 368. 
table, 368. 
life of creosoted, 370. 
plates, 1071. 

for bridges, specifications, 705. 
railroad, 1069. 
steel-, 1072. 

switch-, bills for, tables, 1070. 
wooden-, 
buried in concrete, (ref.) 1093. 
cubic feet in, table, 1069. 
feet B. M. in, table, 1070. 
life of, 1071. 
Tile, tiles, 
glass, 800. 
metallic, 800. 
roof, specifications, 800. 
roofing, 800. 
weight of, 481. 
Timber, 
across grain, compression tests of, 

table, 494. 
air-drying, 362. 

and tie preserving plant, (ref.) 374. 
bending modulus of elasticity of, 

table 493 
bending tests of, table, 492, 493. 
best time for cutting, 378. 
best to use, 360. 
bumettizing, cost, 375. 
checking and splitting, to prevent, 

363. 
compression tests of, table, 490. 
cost of bumettizing, 360. 
cost of creosoting, 360. 
creosote extracted from, table, 368. 
creosoting, cost, 375. 
cut in U. S., 377. 
decay due to presence of water, 

362. 
decay of. 359. 

details or combination bridge, 730. 
evaporation of water from, 362. 
framing for bridge, 731. 
friction of, 521. 

galvanized iron covering for, 361. 
green, compression tests of, table, 

491. 
in buildings, 
safe loads for, 821. 
safe stresses, 825. 
stresses for, 824. 
insect larvse in, 359. 



Timber,— Cont'd, 
kiln drying, 363. 
life of creosoted, 370. 
moisture in, effect on strength, 

490-494. 
oil seasoning, 364. 
old vs. new, crushing tests, 523. 
preservation, 359. 
preserving methods, 360. 
relation between strength and 

weight, 494. 
rotting of, 359. 
safety factors for, 496. 
seasoning, 361, 364. 

advantages of, 363. 

described, 362. 

effect of, 363. 

methods, 363. 

recommendations, 365. 
shipping weights, 391. 
standing-, voliime of, estimated, 

378. 
steam seasoning, 364. 
steaming, effect of, 363. 
structural,- safe unit stress in, 713. 
stumpage of Pacific Coast, 377. 
supply of the U. S., 376. 
treatments, various processes, 

costs, 375. 
trees, best, 346. 
trestles, 787. 

railroad, cost of, 793. 
water in, 361. 
water seasoning, 364. 
water-soaked-, crushing tests, 522. 
well-preserved-, creosote in, 365. 
with grain, shearing tests, table, 

494. 
working stresses, table, 495. 
Timbering, 
steel-, in mines, use of, (ref.) 939. 
tunnels, 934. 
Time, 
and longitude measure, table, 99. 
astronomical, 

and civil, compared, 952. 

elements of, 202. 

equation of, 202. 
between two dates, table, 61. 
mean local, 950. 

mean solar and sidereal, equiva- 
lents, tables, 202. 
minutes and seconds to decimals 

of an hour, table, 1010. 
railway standards, 952. 
to determine, with solar, 946. 
Tin chem., 320. 
alloys, 330. 
-antimony alloy, tensile strength of, 

499. 
-base alloys, 398. 
cast-, 

physical properties of, 499. 

weight of, 679. 

etc., weight of, 481. 
expansion coefficient of, 516. 
lined lead pipe, table, 679. 
melting point of, 515. 
minerals, 330. 



1604 



INDEX. 



Tin,— Cont'd, 
molten, weight of, 481. 
plate, 357. 

rolled, etc., weight of, 481. 
roofing, 800. 
tubing, 679. 
uses of, 330. 
Tinning, 357. 
Titanium, chem., 320. 

minerals, 330. 
Tobin bronze, 397. 

physical properties of, table, 497. 
Toggle, 
in cordage, 668. 
in mech., formulas, 293. 
-joint, principle of, 1530. 
Ton, tons, 
and cubic feet, equiv. (1-9), table, 

485. 
(avoir., short), metric equiv., 86. 
kilograms and pounds, eqmv. (1- 

10), table, 87. 
long, 
and short, equiv. (1-9), table, 

485. 
metric equivalents, 68. 
metric, 
and U. S., equivalents, 89. 
English equivalents, 68. 
per cu. meter and U. S. tons 

per cu. yd., equivalents, 89. 
per sq. meter and U. S. tons 
per sq. ft., equivalents, 89. 
short-, metric equivalents, 68. 
U.S., 
and kilograms, equivalents, 89. 
and metric, equivalents, 89. 
per cu. yd. and metric tons per 

cu. meter, equivalents, 89. 
per sq. ft. and metric tons per sq. 
meter, equivalents, 89. 
Tonneau or miller (metric), English 

equivalents, 85. 
Tools, quarrying, 419. 
Tool-steel, 396. 
Tooth 
ax, described, 429. 
chisel, described, 427. 
Topographer, duties of, in prelimi- 
nary survey (R. R.), 1004. 
Total 
head (hydrostatic), defined, 1160. 
heat, 
formula, 1355. 

of steam, new and old formulas, 
1378. 
Tower, towers, 
and backstays of suspension 

bridges, 754. 
water-, 1207. 
Township, 
and hectars, equivalents, 88. 
metric equivalent, 81. 
Trachylite, 

composition of, table, 338. 
Trachyte, 
composition of, table, 338. 
defined, 340. 
formation of, table, 338. 



Track, tracks, 
and paving for street railways, 

1094, 1095. 
and wheel gage, 1073. 
construction 
for tunnels and subways, (ref.) 

1094. 
on 59 railroads, (ref.) 1093. 
crossover-, frog spacing, table, 1069. 
gage, -gages, 
best standard, 1074. 
increased for curves, 1073. 
table, 1074. 
ladder-, frog spacing, table, 1089. 
shims, thickness of, 1069. 
spikes, 
table, 627. 

weight per mile of track, table, 
1068. 
to find degree of curve of, 1066. 
Trackway, steel-, for street, cost, 1142. 
Traction 
force of locomotives, 992. 
on grades, 1097. 
on pavements, 1097. 
on rails (steel), 1097. 
on railroad grades, 
problem, 994. 
table, 994. 
on roads, 1097. 
on steel rails, 1097. 
problem, 288. 

road-, on pavements, 1142, 
tension on rope, 289. 
Train 
momentum, coefficient of sliding 

friction, 702. 
pressiire on curve, 297. 
resistance formulas, (ref.) 1091. 
Transformed 
catenary, 753. 
-catenarian arch, 761. 
Transformer, transformers, 
defined, 1380. 
elec. code rules, 1398, 1401, 1422. 

1443 
kinds of, 1531. 
Transit, 
adjustment of, 942. 
observation of polaris for azimuth, 
949. 
Transitman, duties of, in prelimi- 
nary survey, (R. R.), 1000. 
Transmission, 
electric, of power, 1385. 
line, 1386. 
cost data, 1479. 
problems, 1387, 1392. 
wire in, kinds and properties of, 
1476. 
long-distance, 1386. 
Transmutation of matter, 317. 
Trap 
rock, 
composition of, table, 337. 
greenstone, weight of, 480. 
properties of, 400. 
weight of, 477. 
S-, 1295. 



INDEX. 



1605 



Trapezium, defined, 129. 

and area, 203. 
Trapezoid, 
center of gravity of, 847. 
defined, 129. 

and area, 203. 
properties of, 526. 
Trapezoidal conduits, proportioned 
for maximum discharge, 1161. 
Traverse, 
adjustment of, 964. 
survey, 964. 
Travertine limestone, 401. 
Trees, 
best time for cutting, 378. 
classification of, 341. 
how they grow, 378. 
interesting facts about, 346. 
life of, 346. 
lumber, best, 346. 
products of, 346. 
rapid growth of, 346. 
tallest, 346. 
timber, best, 346. 
Trench, trenches, 
bracing of, formulas, (ref.) 843. 
excavation, 
by machine, cost data, 921. 
in rock, estimating, 923. 
sewer-, cost data, 915. 
Trenching 
and backfilling for sewer, cost 

data, table, 917. 
in rock, 923. 
Trestle, trestles, 787. 
bents, 788-792. 
pile-, 787. 

and timber, 790. 
railroad-, cost of, 793. 
reinforced concrete, 792, 793. 
steel-, 791. 
elevated railroad, (ref.) 792. 
weight of, formulas, 686. 
timber, 787. 

railroad, cost of, 793. 
wooden-, on curves, 790. 
Triangle, triangles, 
and circle, circumscribed and in- 
scribed, 130. 
area of, 128. 

by calculus, 273. 
center of gravity of, 203. 
equilateral, inscribed in circle, 131. 
geometric, definitions, 128. 
mensuration of, 203. 
properties of, 524, 525. 
skeleton section, properties of , 531. 
solution of, trigonometric, 141- 

142. 
solving, by table of squares, 638. 
spherical, solution of, 199-201. 
Triangular 
cell, skeleton properties of, 531. 
dam, 847. 

orifices, center of pressure on, 
formulas, 1151. 
Trigonometric 
functions, 136-141. 
differentiation of, 270 



Trigonometri c — Cont'd, 
functions , — Cont'd, 
in the four quadrants, 137. 
natural and logarithmic, expla* 

nation of tables, 141. 
(primary), equivalents, tables, 
136-137. 
inverse-, functions, 140. 
differentiation of, 271. 
operations by slide rule, 127. 
Trigonometry, 
plane, 136-198. 
spherical, 199-202. 
Trinidad asphalt, 404. 
Trolley 
systems, cost of, table, 1479. 
wires, elec. code rules, 1399. 
Tropical year, defined, 202. 
Troy weight, table, 86. 
Truncated 
cone, defined, 134. 
prism, defined, 133. 
pyramid, defined, 134. 
Truss, trusses, 
bridge-, 
electric-car loadings for, tables, 

717-719. 
moments and shears, various 
loadings, 693. 
combination, roof-, design of, 806- 

810. 
Cooper's loading, table, 708. 
diagonals, economic angle, 269. 
diagram, 729. 
economic depth of, 684, 
loading on, for maximum moment, 

695. 
Pratt-, 
calculation of, 306. 
chord stress in, concentrated 

loads, 695. 
graphical solution of, 312. 
railroad, weight of, 710. 
roof-, 
four cases, 314. 
stress diagrams of, 314, 315, 

803, 804. 
timber for, 819. 
types of, 803. 

unit stresses in, tables, 804, 805. 
weight of steel in, 810, 811. 
spacing of, 700. 

Warren-, chord stresses in, concen- 
trated loads, 696. 
Tube, tubes, 
and bushings, elec. code rules, 

table, 1430. 
and orifices compared, 1176. 
and pipes, 677. 

dia. to area, capacity, mean radius, 
volume, weight (water), table, 
246-7. 
discharge from, 1175. 
meter, Pitot, 1183, 1184. 
standard, 1176. 
Tubing, 
flexible-, elec. code rules, 1431. 
lead, 679. 
tin, 679. 



1606 



INDEX, 



Tubular piers, 877. 

Tun (liquid), equivalents, 83. 

Tungstates, in min., classification of, 

327. 
Tungsten, chem., 320. 

-steel, 396. 

weight of, 481. 
Tunnel, tunnels, 

alinement and grade of, 935. 

aqueduct-, Los Angeles, cost data, 
939. 

bracing of, formulas, (ref.) 843. 

cross-sections of, 934, 935, 939. 

kinds of, 933. 

lining, 934, 939. 

list of, with costs, 937. 

sewer-, work, cost data, 916. 

timbering, 934. 

ventilation of, 935. 
Tunneling, 933, 

caisson method of, 936. 

dredging method of, 936. 

methods, 
described, 933. 
and cost data, 937. 

shield method of, 936. 
Turbine, turbines, 

efficiencies of, 1343. ' 

horsepower of, theoretic, 1344. 

losses of energy in, 1343. 

nomenclature of, 1342. 

water wheels, described, 1342. 
Tumbuckle, tumbuckles, 

fastenings for wire rope, 675. 

weights and dimensions, table, 633. 
Turnout, turnouts, 

and switches, 1075. 

curves, formulas, 1083. 

for split switches, tables, 1085- 
1088. 

from curved track, 1084. 
Turntable pit, (ref.) 1093. 
Tiirpentine, 

a tree product, 346. 

boiling point of, 514. 

for paint, 356. 

oil of, melting point, 515. 

weight of, 481. 
Tweddell's hydrometer, 461. 
Type metal, cast, weight of, 481, 

u 

Ultimate 

analysis of fuels, 1350. 

strength, defined, 487. 

stress, defined, 487. 
Ultra-violet rays, length of, 1380. 
Umber for paint, 355. 
Undecagon, mensuration of, 204. 
Underground conductors, elec. code 

rules, 1404. 
Undershot wheel, described, 1336. 
Ungula (circular cylindric-), 244. 
Uniform motion, equations of, 278, 

279. 
Unit, units, 

equivalents, simple and compound, 
table, 88-91, 



Unit, units, — Cont'd, 
heat (B. T. U.), equivalents of, 

table, 91. 
heat, mechanical and electrical, 

equivalents, table, 91. 
heat, per sq. ft. per minute, 

equiv. table, 91. 
of electric power, 1379. 
of power, defined, 291. 
stresses in roof trusses, tables, 804, 

805. 
of U. S. money, 95. 
United States 
equivalents of foreign weights and 

measures, tables, 92-94. 
money, 95. 
Upset screw ends, table, 634. 
Urine, weight of, 481. 
Uranite, radium from, 330. 
Uranium, chem., 320. 
in steel, 330. 
minerals, ore, 330. 
weight of, 481. 



Vacuum process, 879. 
Valency, 318. 

Valuations and reports, expert, ref- 
erence data, 1484. 
Valve, valves, 
air, 1270. 
and gates, Ludlow, tables, 1274- 

1279, 1286, 1287. 
boxes, table, 1288. 
Chapman, with wedge-shaped gate, 

nomenclature, 1285. 
flume-, table, 1279. 
gate-, 1271-1279, 1285-1287. 
described, 1288. 

dimensions and weights of, 1273. 
vertical, geared and ungeared, 
1273. 
horizontal check, table, 1278. 
kinds of, 1533. 
Ludlow, nomenclature, 1272. 
-metal, 397. 

pressure relief, described, 1288. 
vertical 
check, table, 1279. 
foot, table, 1279. 
Waring's check, 1296. 
water, 1271-1279. 1285-1287. 
Vanadium, chem., 320. 
-steel, 396, 399. 
alloys, 399. 
Vapor, 
defined, 1534. 

pressure of saturation for liqtiids, 
1372; table, 1373. 
Vaporization, 
latent heat of, defined, 513. 
of fuel, 1371. 
Vaporizers for liquid fuels, 1371. 
Vara, 

(Philippine measure), English 

equivalent, 81. 
(Texas land measiu-e), English 
equivalent, 81, 



INDEX. 



1607 



Vara,— Cont'd, 
square (Texas land measure), Eng- 
lish equivalents, 81. 
Variables, dependent-, defined, 256. 
Variation and pulsation, in elec, 

1453. 
Varnishes, 357. 
Vaults, 
fixed-, loads from, 817. 
kinds of, 1534.^ 
Velocity, velocities, 
and discharge 
in pipes, theoretic, table, 1155. 
of sewers and condtiits, tables, 
1296-1306. 
coefficient of, 1175. 
during impact, 304. 
head in pipe lines, 1160. 
in circular brick sewers, table, 

1299. 
in egg-shaped sewers, table, 1305. 
in irrigation canals, 1317. 
in mech., defined, 278. 
mean-, depth of thread of, in riv- 
ers, 1187. 
metric and English equivalents, 

table, 89. 
of approach, 1159. 

in weirs, how measured, 1177. 
of falling bodies, 
from various heights, table, 1155. 
table, 283. , 
on inclined plane, 286. 
resultant of, 284. 

surface and mean, in open chan- 
nels, 1183. 
Ventilation 
and heating, reference data, 1482. 
of tunnels, 935. 
Venturi meter, 1178. 
measurement of water, 1182. 
standard proportions, 1175. 
(625 cu. ft. per sec.) in India, (ref .) 
1189. 
Verdigris for paint, 355. 
Vermillion, 329. 
Vernal equinox, defined, 202. 
Versed sines, 
natural, table, 144-166. 
(trig.), defined, 136. 
Vertical 
angles, defined, 128. 
bracing, of bridges, 698. 
check valves, table, 1279. 
circle, of celestial sphere, defined, 

201. 
curves, parabolic, 1005. 
foot valves, table, 1279. 
line, of celestial sphere, defined, 
201. 
Viaduct, reinforced concrete, 793. 
Vibration of pendulum, 287. 
Vicat needle test of cement, 408. 
Violet rays, length of, 1380. 
Vitreous fusion of glass and iron, 

515. 
Vitrified brick, 415. 
pavement, specifications, 1121, 
1123. 



Voids, 

determined for concrete, 440. 

in concrete, formula, 1118. 

in earth, 910. 

in gravel, 911. 

in loam, 911. 

in loosened earth and rock, 911. 

in sand, 911. 
Volt, as a pressure unit, 1379. 
Voltages and frequencies, in elec, 

1470. 
Volume, volumes, 

capacities and weights, equiv. (1- 
9), table, 485. 

critical, defined, 513. 

cu. yds., of pipes, table, 246-247. 

defined, 460. 

equivalents (1-10), English and 
metric, tables, 82. 

metric, English equiv., table, 81. 

metric equivalents, table, 88. 

of cone, 134. 

of cylinder, 134. 

of prism, 133. 

of pyramid, 133. 

of solids, by calculus, 277. 

of sphere, measure of, in radii, 135. 

units of, eqmvalents, 66, 67. 

weight per, metric and English 
equivalents, table, 89. 
Vulcanizer, rubber, 330. 

w 

Wall, walls, 
building-, weights of, 821. 
masonry, parts defined, 431, 
of buildings, 
reinforcement, 831. 
stonecutter's plan, 457. 
retaining-, 835. 
Walnuts (trees), classification of, 

343. 
Wane, in lumber, defined, 387. 
Waring 's check valve, 1296. 
Warren truss, chord stresses in, con- 
centrated loads, 696. 
Wash 
drill borings, cost data, 916. 
mill, in cement making, 405. 
Washers, 
cast iron, weights and dimensions, 

table, 624. 
flat plate, weights and dimensions. 

table, 624. 
use of, 624. 
Waste pipes, 1295. 
lead, table, 679. 
Wasteway, length of, formula, (ref.) 

859. 
Water 
as road dust preventive, 1131. 
at maximum density, weight of, 85. 
boiling point of, 514. 
capacity and weight equivalents, 

table, 1376. 
consumption of, 1202. 
in cities, table, 1203. 
cranes, table, 1290. 



1608 



INDEX. 



Water— Cont'd, 
-current meters, 1185. 
decomposition of, 316. 
density of, 67. 
duty of, in irrigation, tables, 1315- 

1317. 
effect of, on earth fill, 910. 
filtration, 1204. 

cost, 1291. 
gallons per capita required in vari- 
ous cities, table, 1203. 
-gas and gas-producer process, 

(ref.) 1377. 
-gas tar, 1132. 
heads 
for given pressures, table, 1147. 
reduced to equivalent pressures, 
tables, 1148, 1149. 
hydrants, table, 1290. 
-jet 

concrete piles, 875. 
for driving piles, (ref.) 890. 
in pile driving, 873. 
(Lb. of) evaporated from and at 

212° F., equiv. of, table, 91. 
measurement of, 
by tank, 1182. 
by venturi meter, 1182. 
by weirs, 1182. 
meter, Pitot tube, 1183, 1184. 
metric weight of, 67. 
motors, described, 1336. 
physical properties of, table, 514. 
power 
and steam power compared, 

1385. 
development, miscellaneous data 

(ref.) 1345. 
formulas, 1332. 

installation, economic size of 
pipe line for, 1189. 
pressures, 
for given heads, tables, 1148, 

1149. 
reduced to equivalent heads, 
table, 1147. 
purification, 1204, 1292. 
rain-, weight of, 482. 
register, 1187. 
instructions for installing, (ref.) 
1187. 
screening, for domestic use, 1204. 
sea-, 
at great depths, density of, 

1145. 
-proof cement, 418. 
weight of, 481, 482. 
storage and irrigation works of 

Southern California, (ref.) 1318. 
supply, 1190. 

miscellaneous data, (ref.) 1201. 
surface, evaporation from, 1199. 
tank, elevated, 1207. 
tower, 1207. 

-tube boilers, described, 1362. 
under pressure, density of, 1145. 
valves, 1271-1279, 1285-1287. 
vapor pressure of saturation for, 
table. 1373. 



Water— Cont'd, 
weight of, 1145. 
table, 481. 
at various temperatures, table, 

465. 
in pipes, table, 246-247. 
wheels, 
described, 1336. 
Pelton, 
quintex nozzle, table, 1342. 
single nozzle, table, 1338-1341. 
works, 1202. 
miscellaneous data, (ref.) 1291- 
1294. 
Waterways, 1320. 
Waterproof compositions, 418. 
Waterproofing 
a reservoir, 1293. 
asphalt for, 418. 
concrete, 455. 
data for concrete, 453. 
oil-mixed concrete for, 455. 
Watt, watts, 
equivalent of, table, 91. 
-meter, defined, 1536. 
per second, equivalents, 90. 
per sq. inch, equiv. of, table, 91. 
units, single and compound, 1379. 
Wave, waves, 
shape, in elec, 1455. 
ether, lengths of, 1380. 
Wax, 
bees-, weight of, 482. 
melting point of, 515. 
Weatherproof wire, table, 1425. 
Web 
plates 
of plate girders, properties of, 

table, 575. 
(vert, line), properties of, 529. 
stresses and shears, 307. 
Wedge, wedges, 
circular cylindric-, properties of , 

245. 
common-, volume of, 249. 
in mech., formulas, 292. 
of cone, 249. 
Week (time measure), 99. 
Weight, weights, 
and dimensions of cast iron pipe 
and specials, tables, 1219- 
1267. 
and measures 

(Foreign) , American equivalents, 

table, 92-94. 
of Philippines, English equiva- 
lents, 81. 
and mass of water, metric, 67. 
apothecaries', metric equivalents, 

table, 86. 
avoirdupois, 
long ton, metric equiv., table, 86, 
short tons, metric equivalents, 
table, 86. 
capacities and volumes, equiva- 
lents (1-9), table, 485. 
defined, 459. 

equivalent for any specific gravity, 
table, 483, 484. 






INDEX, 



1609 



Weight, weights, — Cont d. 
in mech., defined, 278. 
measures and money, 66-99. 
metric and English 
equivalents, table, 89. 
equivalents (1-10), table, 85, 
metric, English equiv., table, 85. 
of brick, table, 474. 
of building stones, table, 474. 
of cast iron pipe, table, 1216. 
of cements, table, 474. 
of concrete, 475. 

cinder- and stone-, 465. 
of lime, 475. 

mortar, 475. 
of limestone, table, 475. 
of liquids, table, 468, 469. 
of marble, table, 476. 
of masonry, table, 476. 
of materials, 459. 

general table, 478. 
of sand, table, 476. 
of water at maximum density, 85, 
of woods, table, 470-473. 
per cubic foot, equiv. (1-9) in ca- 
pacities and volumes, table, 
485. 
per volume, metric and English 

equivalents, table, 89. 
reduced to specific gravities, table, 

474. 
trade-, of lumber, 391. 
troy, table, 86. 
Weir, weirs, 1177. 
center of pressure on, table, 1151. 
contracted, defined, 1177. 
formulas, 1178. 
Bazin's, 1178. 
Francis', 1178. 

Fteley and Stearns', 1180, 1181. 
Herschel's, 1181. 
Parmley's, 1180. 
instructions for installing, (ref.) 

1187. 
measurement of water, 1182. 
sharp-crested, surface, formulas, 

1178. 
standard, 1177. 
submerged, 1181. 
formulas, 1181. 
suppressed-, defined, 1177. 
triangular and trapezoidal, 1181. 
Weisenkalk (marl) for cement, 405. 
Weld, defined, 1536. 
Well, wells, 
artesian-, defined, 1190. 
boring, reference data, 1481. 
dia. to area, capacity, volume, 

weight (water), table, 246-7. 
-driller for drilling blasting holes, 
cost, 926. 
Welded (lap-) pipe, 1269. 
Welding, borax for, 330. 
Welland canal, data, 1322. 
Wellhouse process, forties, cost, 375. 
West point, of celestial sphere, de- 
fined, 201. 
Wet process, in cement making, 405. 
Wharton switch, 1088. 



Wharf, wharves, 
construction, 892. 
piers and docks, 892. 
reinforced concrete, (ref.) 900. 
Wheel gage, standard M. C. B., 1073. 
Wheels, 
Pelton, 
quintex nozzle, table, 1342. 
single nozzle, table, 1338-1341. 
water, described, 1336. 
Wheeled scrapers, grading with, cost 

data, 917. 
White 
lead for paint, 355. 
-metal, 398. 
zinc for paint, 355. 
Willow, willows, 
classification of, 344. 
weight of, 482. 
Wind 
loads on mill buildings, 833. 
pressure, 794. 
for bridges, 697. 
for buildings, 822. 
formulas, 794, 795. 
on cylinders, 797. 
on inclined surfaces, 795-797. 
on roofs, tables, 796, 797. 
on spheres, 797. 
railroad bridges, specifications, 

702. 
tables, 795, 796, 797. 
suction, 795. 
tension, 795. 

velocities attained, 794, 795. 
Window glass, cost prices, 479. 
Wire, wires, 
aluminum and copper compared, 

as electrical conductors, 496. 
bell-, elec. code rules, 1448. 
brass, 
physical properties of, table, 496. 
weight of, 478. 
brush, for mill scale, 359. 
capacity of, elec. code rules, table, 

1448. 
conduit-, elec. code rules, 1427. 
copper, 
and silicon -bronze compared, as 

electric conductors, 497. 
carrying capacity of, table, 1404. 
physical properties of, table, 

497. 
table, 1388-1391. 
weight of, 478. 
delta-metal, tensile strength of, 

table, 497. 
elec. code rules, 1399, 1403, 1405, 

1408, 1422, 1447. 
fixture-, elec. code rules, 1427. 
gages, tables, 666, 671. 
gold, tensile strength of, 497. 
in transmission lines, 
kinds and properties of, 1476. 
kind and size, 1386. 
insulated, table, 1423. 
iron, physical properties of, table, 

498. 
lead, tensile strength of, 498. 



1610 



INDEX. 



Wire, wires, — Cont'd. 
phosphor-bronze, tensile strength 

of, 497. 
platinum, tensile strength of, 498. 
Roebfing steel, properties of, 672. 
rope, 673. 
fastenings, 675. 
tables, 674. 
rubber-covered, tables, 1424. 
silicon-bronze, tensile strength of, 

497. 
slow-burning, table, 1425. 
steel, 
physical properties of , table, 499, 
weight of, 481. 
weatherproof, table, 1425. 
weight of, from specific gravity, 

table, 484. 
car-, and eqmpment, elec. code 

rules, 1417. 
theater-, elec. code rules, 1414. 
Wood (see, also. Timber), 
-alcohol, a tree product, 346. 
as fuel, value of, 1363. 
-block pavement, 

construction of, 1099. 
creosoted, specifications, 1126. 
specifications, 1103, 1120, 1128. 
compression tests of, table, 490. 
expansion coefficient of, 516. 
friction of, 517-521. 
(in machines), friction of, 521. 
oil, weight of, 482. 
paving blocks, 
grooved, 1120. 
treatment, analysis of, 1120. 
pipe, 
bored and banded, 1208. 
flow of water in, (ref.) 1187. 
screws, table, 622. 
specific gravities of, table, 470- 

473. 
stave connection with cast iron 

pipe, 1280. 
stave pipe, 
and details, table, 1210. 
details of, 1208. 
discharge through, table, 1210- 

1214. 
durability of , 1187. 
notes on, 1214. 
vinegar, 346. 

weights of, table, 470-473. 
Wood's fusible metal, melting point 

of. 515. 
Wolfram, weight of, 482. 
Wooden 
beams, 
for buildings, 819. 
loads on, table, 566. 
problems in, 567. 
working stresses, table, 495. 
columns, working stresses, table, 

495. 
moldings, elec. code rules, 1450. 
pipe, bored-, for water supply, 

1207. 
stringers, bending moments, table, 
791. 



Wooden— Cont'd, 
structures, methods of preserving. 

361. 
ties, 
cubic feet in, table, 1069. 
feet B. M. in, table, 1070. 
Work 
and power equivalents, metric and 

English, table, 90. 
formula for, in hoisting, 290. 
in mech., equations of, 290. 
Working stress, -stresses, 
defined, 487. 

for reinforced concrete beams, 585. 
for wooden beams and columns, 
table, 495. 
Worthington steam pump, 1367. 
Wrought iron, 
expansion coefficient of, 516. 
for buildings, 819. 
in buildings, safe stresses, 826. 
manufacture of, 393. 
melting point of, 515. 
physical properties of, table, 498. 
pipe, 1268. 
standard, welded, tables, 677, 
678. 
weight of, 480. 



X-ray, 316. 
Xenon, chem.. 



X 



320. 



Y's, cast iron pipe, tables, 1227,1228, 
1255, 1256. 

Yard, yards, 
and meters, 
cubic, 
equivalents, 88. 
equiv. (1-10), table, 82. 
equivalents, 88. 
equiv. (1-10), table, 70. 
square, 
equivalents, 88. 
equiv. (1-10), table, 80. 
cubic, 
equivalents, 66-67. 
metric equivalent, 68. 
metric equivalent, 68, 82. 
(dollars per) 
and francs per meter, eqtiiva- 

lents (1-10), table, 98. 
and marks per meter, 98. 
equivalents, 68. 
gravity-, (ref.) 1091. 
-inch and bushels, equiv. (1-9), 

table, 485. 
-inch and cubic feet, equiv. (1-9), 

table, 485, 
-inch and gallons, equiv. (1-9), 

table, 485. 
metric equivalents, 66, 68. 
square, metric equivalents, 68, 81. 
Yam 
fiber, flax, strength of, 512. 
in cordage, 668. 



INDEX. 



1611 



Year, 

common and leap-, time measure, 
99. 

tropical, defined, 202. 
Yellow pine lumber, 

classification of, 387, 388. 

inspection of, 387, 388. 
Yen (Japanese), equiv. (1-10,-50- 
100) in U. S. money, table, 97. 
Yield point, defined, 487. 
Ytterbium, chem., 320. 
Yttrium, chem., 320. 



Z-bar, -bars, 

block, properties of, 533, 534. 

columns, dimensions and safe 
loads, tables, 598-600. 

rivet gages for, 614. 

rolled, properties of, 538. 

steel, properties of, table, 557. 
Z, block, properties of, 533, 534. 
Zenith 

distance, of a star, defined, 201. 

in astron., defined, 947. 

of celestial sphere, defined, 201. 



Zmc, chem., 320. 
boiling point of, 514. 
cast-, physical properties of, 

499. 
cast, etc, weight of, 482. 
chloride, 
cost, 375. 
for timber, 361. 
expansion coefficients of, 516. 
melting point of, 515. 
minerals, ores, 329. 
ores, uses of, 329. 
oxide for paint, 355. 
rolled-, tensile strength of, 499. 
uses of, 329. 

-white for calcimining, 355. 
white paint, 329. 
Zirconium, chem., 320. 
Zone, 
of circle, mensuration of, 219. 
of circular spindle, 254. 
of parabolic sprindle, 254. 
of sphere, 
area of, by calculus, 276. 
defined, 135. 
menstu-ation of, 253. 
Zoological materials, 847. 



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